A Mathematically Reduced Approach to Predictive Control of Perishable Inventory Systems

Transcription

1 A Mathematially Redued Approah to Preditive Control of Perishable Inventory Systems Orzehowska, J. E. Submitted version deposited in CURVE Marh 2016 Original itation: Orzehowska, J. E. (2014) A Mathematially Redued Approah to Preditive Control of Perishable Inventory Systems. Unpublished PhD hesis. Coventry: Coventry University Politehnika Łódzka Copyright and Moral Rights are retained by the author. A opy an be downloaded for personal non-ommerial researh or study, without prior permission or harge. his item annot be reprodued or quoted extensively from without first obtaining permission in writing from the opyright holder(s). he ontent must not be hanged in any way or sold ommerially in any format or medium without the formal permission of the opyright holders. Some materials have been removed from this thesis due to third party opyright. Pages where material has been removed are learly marked in the eletroni version. he unabridged version of the thesis an be viewed at the Lanhester Library, Coventry University. CURVE is the Institutional Repository for Coventry University

2 A MAHEMAICALLY REDUCED APPROACH O PREDICIVE CONROL OF PERISHABLE INVENORY SYSEMS Joanna Ewa Orzehowska A thesis submitted in partial fulfilment of the University s award of the degree of Dotor of Philosophy Otober 2014 Control heory and Appliations Centre Coventry University In ollaboration with, Instytut Automatyki, Politehnika Łódzka

3 ABSRAC he design and optimisation of inventory replenishment systems has already been exhaustively studied by the operational researh ommunity. Many lassial mathematial methods and simulation tehniques have been developed and introdued in the literature. However, what an be observed is the fat that in a real ase senario the lead-time, deterioration of goods and demand for produt are likely to be time-varying and unertain, whih traditionally have not neessarily been refleted in the model formulations. herefore, in response to the dynamial nature of inventory systems, the potential of algorithms based on ontrol theory to redue the undesirable influenes of system unertainties on inventory level stability, have been investigated /proposed. Consequently, the mapping of the inventory problem into the ontrol theory domain, for ost-benefit inventory trade-off ahievement has been realised. Although, the appliation of ontrol theory in inventory optimisation appears to be benefiial, there are ertain reasons why the approah has gained yet little attention among the operational researh ommunity. One reason is that it annot be adopted easily by researhers who are unfamiliar with ontrol theory and another is due to a ommuniation gap whih exists between the ontrol theory and operational researh ommunities. Prompted by these observations, the thesis presents a novel, systemati mathematial approah for finding the optimal order quantities. he proposed approah has been mathematially demonstrated to be equivalent in study-sate to model-based preditive ontrol, whih is one of the more well-established produtive ontrol tehniques with industrial appliation today. he mathematially redued approah attempts to bridge the identified gap to fulfil the laking dual pereptions of both ommunities. It enables the straightforward benefits afforded by preditive ontrol without the neessity to beome familiarised with priniples of ontrol theory. he method is shown to be appliable for both perishable and non-perishable inventory. Although the novel tehnique was inspired by MPC and notiing the MPC patterns in the mathematial desription, the resulting proposal is no longer MPC. It is in fat a minimum variane approah, or dear beat ontroller, with an inorporated Smith preditor. herefore using the adjetive preditive in the title of the thesis refers to both, the inspiration of MPC and the preditive nature of the minimum variane ontroller to aommodate lead time, being inorporated within an inherent Smith preditor. he developed approah is onsidered to be transferable to other appliations, where similar model formulations may be appliable. ii

4 ACKNOWLEDGEMENS First and foremost I would like to express my thanks to my diretor of studies Prof. Keith J. Burnham, for his essential guidane on my researh from the ontrol theory point of view. His very substantial feedbak, enouragement and motivation have been a tremendous help and ontributed a vital investment towards my aademi development. I would also like to express my gratitude towards my Internal Supervisor, Professor Dobrila Petrovi, for ruial reommendations and feedbak from an operational researh perspetive. I would like to also thank Prof. Andrzej Bartoszewiz from the ehnial University of Lodz, who beame interested in the researh and offered to beome an External Supervisor. I appreiate his help and support, his guidelines in joining the ontrol theory and operational researh disiplines and his detailed feedbak during all my studies. Last but not least, I would like to express my gratitude to Oommen homas for proofreading my thesis. iii

5 able of Contents ABSRAC... ii ACKNOWLEDGEMENS... iv ABLE OF ABBREVIAIONS... viii ABLE OF SYMBOLS... ix 1 INRODUCION Introdution to logistis Basi definitions and desriptions Storage, warehousing & materials handling Industrial approahes to replenishment poliy Lean management Flutuating inventory level Researh Problem he need for system modelling Operational researh approah Researh desription Aim, Objetives Researh Contribution and Outline of hesis Aim of the researh Objetives Researh ontributions Outline of thesis Publiations Summary LIERAURE REVIEW Introdution to an aademi approah to supply hains iv

6 2.2 Operational Researh ehniques ransportation supporting models Supply hain management supporting models Piking-paking supporting models Manufaturing supporting models Inventory and replenishment supporting models Appliation of Control heory to Logistis Historial review Model preditive approah Summary and gap identifiation PRELIMINARY MODELLING AND SIMULAION MODELS Introdution Inventory Modelling Assumptions Converting the inventory problem to the ontrol domain Inventory state spae representation Disrete state spae representation Model-based Preditive Control Approah Novel Inventory Controller Novel Method Properties Novel Methods vs. Smith Preditor, Dead Beat Controller and MPC Novel method Smith preditor Dead-beat ontroller Model preditive ontrol Calulation effiieny of the novel method Summary v

7 4 VERIFICAION OF HE PROPOSED IC APPROACH Introdution Simulation settings justifiation Smith Preditor MPC Approah Alternative IC Method - Verifiation Against MPC est 1: o illustrate the manual alulation of IC est 2: o show effet of varying deterioration rate est 3: Sinusoidal referene inventory level est 4: Different demand patterns Summary DEMONSRAION OF MAHEMAICAL EQUIVALENCY Non-perishable Case Perishable Case Summary RESULS OF PROPOSED MEHOD Introdution Simulation Settings est 1: o illustrate how the referene inventory affets the system behaviour est 2: o find the most profitable referene inventory level est 3: o illustrate how demand pattern affets system behaviour est 4: o illustrate how demand pattern affets system behaviour ontinued Summary FUURE MODELLING AND SIMULAION RESULS IMPC Results for Non-perishable ase Perishable ase vi

8 7.2 Attempt to Develop the Simplified Model for Non-perishable ase Perishable ase Summary CONCLUSIONS AND RECOMMENDAIONS FOR FUURE DIRECIONS Conlusions Reommendations Overall Summary REFERENCES: LIS OF ABLES: LIS OF FIGURES: Appendix I Guidelines for demonstration of Proposition 5 and Proposition Appendix II List of publiations and other aademi ativities vii

9 ABLE OF ABBREVIAIONS Abbreviation ARMA ARX C EOQ EPIOBPCS FIR GA IC IMPC IOBPCS JI LS MPC OC OR PUrIC RLS UrIC WIP Full Name Autoregressive Moving Average Autoregressive with Exogenous Input Control heory Eonomi Order Quantity Estimated Pipeline Inventory and Order Based Prodution Control System Finite Impulse Response Geneti Algorithm Inventory Control Inventory Model Preditive Control Inventory and Order Based Prodution Control System Just In ime Least Square Model Preditive Control Operational Control Operational Researh Inventory Control with tuning parameter for perishable goods Reursive Least Square Inventory Control with tuning parameter for non-perishable goods Work in Progress viii

10 ABLE OF SYMBOLS Symbol Definition Unit / domain 1 0, I a Estimated value of Kn ( 1) 1 in UrIC ontroller a Estimated value of Kn ( ) in 2 UrIC ontroller A Open loop system state n matrix A Closed loop system state n matrix Prie disount [ ] max n ( 1) ( n1) b Open loop system input b matrix Closed loop system input matrix Open loop system output matrix Closed loop system output matrix Cs () Closed loop transfer funtion - n n1 n n1 C () SP s Smith preditor transfer funtion Deterioration rate 0,1 d Disturbane / demand [items] d otal demand [items] ot e Error vetor n F MPC predition vetor N p f Dead-beat ontroller gain n F Profit funtion - prof Gs () Warehouse transfer funtion - H Holding unit osts [ ] - ix

11 I Identity matrix n n I Current inventory level [items] I otal number of bakorders [items] L I Maximum inventory level max I Referene inventory [items] R I u otal number of stored goods [items] k ime instane K Positive onstant K MPC and IC gain vetor n1 K Last element of MPC and IC y gain vetor K MPC and IC gain vetor n x apart from last element of L Lead time [days] m Row number n Lead time / system order N Control horizon N Predition horizon p Φ MPC predition matrix N p N N p P Prie [ ] p exponent p exponent n R MPC tuning matrix N p N N p N r Column number Q Demand inremental vetor N q Demand inremental salar m th m estimate of RLS algorithm u Input signal / order size [items] U Control trajetory vetor N u uning parameter r 9 x

12 u otal number of orders [items] ot x Open loop state vetor n x Closed loop state vetor n or n1 x Referene state vetor n r Y Predited inventory signal N p y Current inventory level [items] Y arget inventory level signal N p R Y Empty vetor p R y Referene inventory level [items] R Plane points values matrix 910 N xii

13 1 INRODUCION 1.1 Introdution to logistis In the ontext of this thesis, logistis is understood as a disipline of siene that develops strategies for appropriate motion and alloation of goods. It assures that all non-human resoures are always in the right plae at the right time. he origin of logistis siene an be reognised in the times of World War wo when the need for appropriate alloation of military resoures was ruial to aomplish military operations suessfully (Rushton, Crouher and Baker, 2006). Nowadays the term logistis is usually identified with the distribution of goods within a supply hain. Starting from the raw materials needed for prodution, through to the omponents needed for assembly, up until the finished goods reahing the end ustomer. All the goods need to be properly distributed so that they reah the appropriate supply hain node at the right time. Although, logistis does not inlude prodution, the term logistis is often substituted by supply hain management. Logisti operations of any ompany are designed to ahieve ustomer demand satisfation. Eah node within a supply hain beomes a ustomer for the previous one and the supplier for the next. A delay or failure might affet the whole supply network inluding the end ustomer. Although logistis is a relatively new sope of siene, tehniques have already been established to redue the risks of both not satisfying the ustomer demands and of not making the whole osts exessively expensive. he tehniques an be differentiated between the managerial tehniques (usually industrial approahes) and the engineering tehniques (usually aademi approahes). he urrent hapter fouses on the industrial approah to improve logistis operations. In order to aquaint the reader with the motivation for the researh, Setions presents several basi terms and logistis strategies. Equation Chapter (Next) Setion 1 1

14 1.1.1 Basi definitions and desriptions Supply Chain Management: Several different definitions of supply hain management an be found in the logistis literature. Aording to Basu and Wright (2008) for every business transation there is a supplier and a ustomer and there are ativities, failities and proesses linking the supplier to the ustomer. he management proess of balaning these links to deliver best value to the ustomer at minimum ost and effort for the supplier is supply hain management. Aording to Simhi-Levi et al. (2003, p. 1) Supply hain management is a set of approahes utilized to effiiently integrate suppliers, manufaturers, warehouses and stores, so that merhandise is produed and distributed in the right quantities, to the right loations, and at the right time, in order to minimize system-wide osts while satisfying servie level requirements. herefore, supply hain management onsists of the flow of materials as well as the reverse flow of information. he latter refers to the ustomer requirements management, in other words the demand. he goods and information flows for a typial supply hain are illustrated in Figure 1-1. his item has been removed due to 3rd Party Copyright. he unabridged version of the thesis an be viewed in the Lanhester Library Coventry University. Figure 1-1: Supply hain management (soure: Rushton, Crouher and Baker, 2010, p.5) 2

15 In general, the supply hain an be defined as a distribution network of goods, starting from the supply of raw materials to appropriate manufaturers, up to the finished goods being delivered to the retailer. he key objetive of supply hain management is to provide best value to the ustomer by measuring, planning and managing all the links in the hain (Basu and Wright, 2008). herefore, the waste and ost redution plays a vital role in a supply hain design. Bullwhip effet (Forrester effet): he bullwhip effet also known also as the Forrester effet, is the ommonly used term for a dynamial phenomenon in supply hains. It refers to the tendeny of the variability of order rates to inrease as they pass through the ehelons of a supply hain toward produers and raw materials suppliers (Disney and Lambreht, 2008). he bullwhip effet again desribes the variations in order rates in a given supply hain ehelon (ustomer) required to be supplied by a previous ehelon (its supplier), whih affets the ordering poliies of the latter from the anteedent ehelons (its suppliers) along the supply hain. If the end ustomer demand suddenly inreases it auses the raising of the demands in all anteedent supply hain ehelons by the same ratio (inreased by the number of goods ordered to support a pre-defined safety stok poliy). Figure 1-2 presents the flutuations appearing in the inventory levels in eah ehelon due to ustomer demand hange. his item has been removed due to 3rd Party Copyright. he unabridged version of the thesis an be viewed in the Lanhester Library Coventry University. Figure 1-2: Bullwhip effet (soure: Shniederjan, 1999) 3

16 Figure 1-3 presents the surge in demand (from end-produt demand to raw-material demand) due to the bullwhip effet. his item has been removed due to 3rd Party Copyright. he unabridged version of the thesis an be viewed in the Lanhester Library Coventry University. Figure 1-3: Bullwhip or Forrester effet (soure: Rushton, Oxley, Crouher. 2000:208) Lead time: Lead time is the time between an order being plaed and it being delivered. It involves the following ativities: manufaturing or piking (whihever is relevant), paking, shipping, dispathing and delivering (Rushton, Oxley and Crouher, 2000, p.208). Logistis operation: Logistis operation refers to any ativity whih is inluded in the overall supply hain management of a ompany. Several logistis operations are now well established. ypial logistis ativities inlude: he movement of argo in all transportation modes of air, land and water inluding planes, vehiles, trains, truks, lorries and ships he storage of goods in warehouses or other storage rooms he ordering and purhasing of an appropriate number of raw materials by manufaturers he ordering of goods by eah supply hain node to fulfil the expeted demand for goods from their ustomers 4

17 Planning the prodution in advane to enable an unertain ustomer demands to be fulfilled Logistis operations are ativities involving goods only, so that labour and human resoure management is not onsidered as logistis operation (Rushton, Crouher and Baker, 2010). Value-added-ativities: Value-added-ativities are onsidered as the ativities whih inrease the value of the item from the ustomer point of view. For instane, a finished produt has a greater value for the ustomer than the sum of raw materials it was made from. As it an be notied, although all of the above mentioned logistis operations are essential for a suessful design of a supply hain network, none of them, in ontrast to the proess of prodution or assembling, an be onsidered as a value-added ativity. herefore they only generate osts related to holding goods in terms of time wastage, fuel wastage et. Any possible ost redution, whih will not affet the ustomer demand satisfation, is advantageous for the ompany. he aim of a suessful supply hain is to find a way to balane logistis operations benefits against osts. Eonomis of sale: Eonomis of sale is a management strategy whih refers to ost redution due to large bath prodution (or large bath transportation). For some industrial appliations, the prodution set up ost (or empty transportation osts) outweigh the storage osts. In suh situations the prodution quantity may be arbitrarily inreased by manufaturers to redue the prodution osts. Suh a prodution quantity may signifiantly exeed the ustomer demand and may require additional inventory holding until the goods are atually demanded. he eonomis of sale is an opposite of the Just-In-ime (JI) poliy. JI refers to very small bath prodution or even ontinuous prodution (or transportation) in respet of inventory redution or elimination. Eonomis of sale implies a push design of the supply hain. In the push system the stok is provided for the next stage of supply (e.g. buying items to sell or starting manufature) without having the total prodution path lear (Basu and Wright, 2008). It means that the produts are produed and stored in the quantity whih is eonomi in terms of designing a prodution system but not neessarily in the quantity whih will be demanded by the ustomer (often signifiantly more than demanded). his is ontrary to the pull system. Pull poliy: he orders are plaed by the supply hain nodes in response to the ustomer demand only. he flow of goods in the supply hain is pulled from the end ustomer side. 5

18 Push poliy: he supply hain members tend to order, manufature and store goods in order to be prepared to immediately satisfy the ustomer demand. he flow of goods is pushed by suppliers. Replenishment poliy: Replenishment is the at of refilling or omplete again by supplying what has been used up or is laking. In logistis it refers to plaing new orders when the goods are used or sold. he replenishment poliy an be deided for eah individual node based on the purpose of the partiular node in the supply hain and the overall ompany strategy (e.g. pull or push system). Exemplary replenishment poliies are elaborated in Setion Storage, warehousing & materials handling here are several types of inventory, depending on the role of the supply hain node, the stok being held, and the role the inventory has for the node. For instane in some nodes the goods an be stored for several months, possibly kept in speial environments to ensure they do not deteriorate. At other nodes goods are moved and proessed very quikly from one plae to another. Among other inventory types the most ommon are: raw materials, spare parts, work in progress, pipeline stok, finished produts, and stok in warehouse or retail. Regardless of the inventory type the need for inventory holding is usually similar Need for stok holding 1. Enables the logistis operation to run smoothly by providing a buffer between suppliers and ustomers: Compensates the negative onsequenes of prodution delay (suh as break down, lak of raw materials et.) or delivery delay (traffi issues, transportation mode break down et.). Compensates the unertainties related to ustomer demand whih, in most of the real world environments, is partially or highly unertain. It enables immediate ustomer demand response even if the demand foreast was underestimated. Suh kind of inventory is alled the safety stok. 6

19 2. Enables redution of osts of other ativities: Redues the purhasing osts. It an be ahieved as a result of eonomy of sale poliy. Purhasing a large bath of produts usually enables negotiation of satisfatory disounts with the suppliers. In the ase of periodially flutuating goods pries, the goods an be purhased at the moment of lowest prie. his strategy is used if the stok holding is more ost effiient than regular small bathes prourement at any time. Redues osts of prodution. Often it is ostly to set up mahines, so prodution needs to be run as long as possible to ahieve low unit ost. (Rushton, Oxley, Crouher, 2000:183). Redution in the numbers of setting up of mahines signify running the prodution without any break for longer than required, even if the produed goods are not demanded by the ustomer in suh number. It requires the storage of finished goods after the prodution proesses Disadvantages of stok holding: From the previous setion it ould be onluded that keeping inventory is neessary for the ompany, although keeping inventory generates non-value-added osts, whih has already been reognised as one of the higher logisti osts. Among the inventory related ost the most signifiant are as follows (Rushton, Oxley, Crouher 2000:191) Cost related to stok holding and maintaining - ost of spae needed for storage (ost of building the warehouse and maintaining it in terms of eletriity osts, taxes et.), ost of manpower, ost of equipment suh as appropriate raks, forklifts, ranes, piking devises et. Cost assoiated with management of stok the higher the inventory level, the higher the need for proper inventory management systems and people ontrolling it to avoid errors in delivery. Cost assoiated with insurane of the stok the higher the inventory level, the risk of damage and the greater need for insurane. Risk ost ost of possible theft, damage or deterioration of stok, whih is more likely in the ase of higher inventory level. Also it inreases the risk of human aidents on the warehouse floor. 7

20 Capital ost loss of potential profits assoiated with investing apital, whih is urrently frozen in the inventory and redues the ash flow in the ompany. herefore, high standard replenishment management requires ahieving the appropriate balane between inventory holding benefits and expenses or equivalently the balane between proesses organization and ustomer requirements. As a response to suh a need, several different replenishment approahes have been established (Rushton, Oxley, Crouher, 2000). Some of them are reviewed in setion Industrial approahes to replenishment poliy here are several approahes to inventory replenishment strategy ommonly used in the industry. Some of them are listed below. Periodi review (shown in Figure 1-4): Stok level of produt in the warehouse is reviewed at the end of eah period of time. he order is plaed if the inventory level is below the reorder level. his item has been removed due to 3rd Party Copyright. he unabridged version of the thesis an be viewed in the Lanhester Library Coventry University. Figure 1-4: Periodi Review (soure: Rushton, Oxley, Crouher. 2000:206) Fixed order quantity (illustrated in Figure 1-5): he order is plaed when the quantity of produt dereases below the reorder level, before it reahes the safety stok level. he disadvantage of both methods is the flutuating inventory levels, the risk that during one period (in the ase of the first method) or lead time (in the ase of the 8

21 seond method) the inventory might derease under the safety level, and might ause unneessary osts related to inventory holding. his item has been removed due to 3rd Party Copyright. he unabridged version of the thesis an be viewed in the Lanhester Library Coventry University. Figure 1-5: Order quantity (soure: Rushton, Oxley, Crouher. 2000:207) Eonomi order quantity (EOQ): he EOQ method is an attempt to estimate the best order quantity by balaning the onfliting osts between holding stok and plaing replenishment orders (Rushton, Oxley, Crouher 2000:191). Figure 1-6 ompares two exemplary replenishment poliies. Poliy A refers to less frequently replenishing, whih redues the transportation osts and order setting up osts. On the other hand it inreases the inventory level signifiantly whih inreases the holding ost as well as the stok maintenane osts. he poliy B, in turn, redues the inventory osts and inreases the transportation osts due to its delivery frequeny. his item has been removed due to 3rd Party Copyright. he unabridged version of the thesis an be viewed in the Lanhester Library Coventry University. Figure 1-6: Order sizes (soure: Rushton, Crouher and Baker, 2010) 9

22 o ahieve a balane, the EOQ approah aims at minimising the total osts through appropriately designed objetive funtion (ost funtion). Figure 1-7 presents the exemplary EOQ plan. he total ost is represented by the objetive funtion. Its minimum value represents minimal osts for goods number of eonomi order quantity. he funtions of ordering osts and holding osts (the two omponents of the objetive funtion) are shown in Figure 1-7 for omparison. his item has been removed due to 3rd Party Copyright. he unabridged version of the thesis an be viewed in the Lanhester Library Coventry University. Figure 1-7: Eonomi order quantity (soure: Shah, Gor and Soni, 2008:305) Suh an approah already seems more benefiial for the ompany than the previously presented ones. One has to remember, though, that the presented model does not onsider system unertainties suh as varying ustomer demand and the lead time delay. It is still quite a basi approah. Chapter 2 is devoted to the disussion of more sophistiated replenishment deision support tools available in aademi literature. Just in time the zero inventory philosophy he onept of just in time (JI) refers to a type of replenishment system where the goods appear in the warehouse (or prodution line) exatly when they are needed, neither before nor after that time instane. Ideally it refers to zero inventory holding poliy. he idea itself seems to be simple, nevertheless due to system unertainties suh as unknown demand or unexpeted delays, it is not so straightforward in appliation. Adopting JI requires either the patiene of the ustomer (for instane ar industry) or 10

23 an aurate prior knowledge regarding future demand (for instane due to aurate foreasting) and/or a sophistiated material requirement planning system. Nevertheless in most of the real world ases the demand is unertain, whih justifies the importane of appropriate design replenishment deision systems. Implementation of JI is not just about reduing the inventory, but also reorganisation of proesses in the manner that they beome more effiient and the lead time is redued. For instane, in terms of redution of inventory held in the warehouse, the delivery frequeny needs to be inreased ontrary to the eonomis of sale approah. Stok ontrollers have been septial about the effiieny of frequent deliveries of smaller bathes without investigating all the options and potential. here are many instanes where the aepted delivery quantity is now muh smaller than it was a few years ago, and is destined to ontinue to be redued (Basu and Wright, 2008). As it was mentioned in the definition of eonomis of sale in Setion 1.1.1, the JI poliy is an opposite strategy to eonomis of sale and in this respet it refers to more frequent and smaller sizes of order quantity, ideally the ontinuous prodution system. Defining a sale of degree of pull/push strategy, the extreme JI (zero inventory poliy) would be at the pull end of the sale, while the eonomis of sale, would be at the push end. he smaller prodution (or transportation) bathes and the more frequent orders (deliveries), the more shifted to pull the replenishment strategy is on the pull/push sale. From a mathematial modelling perspetive, the JI poliy appears to have no onstraints on order bath size, therefore the bath an be as small as needed. For some supply hains, the appliation of JI is natural beause of the nature of the produts or the proesses, e.g. in fast-yle manufature, where a stage in the prodution proess takes very little time (Basu and Wright, 2008). Nevertheless, in many industries the implementation of JI is not straightforward and many ompanies do not adopt JI simply beause of the fear of dereasing servie levels and disappointing ustomers by the lak of stok when required. here is a need for applying analytial tehniques in replenishment planning so that the likelihood of lost business opportunity is redued. 11

24 1.1.4 Lean Management he Lean Management (or oyota Prodution System) onept was pioneered in the oyota prodution system and adopted by other Japanese manufaturers. Muh later it was appreiated by Western manufaturers as well and reognised as an inredible soure of ost redution as well as improved ustomer response. Lean Management is a management approah, realised by the implementation of established priniples, theories and tools whih, ontrary to eonomy of sale approahes, pratie eonomis of time, when designing logistis operations. Implementation of even a single one of the existing lean tehniques is onsidered to be benefiial for a ompany (Biheno, Holweg and Biheno, 2009). he redued ost is related to redution or, if possible, the elimination of non-value-added ativities (e.g. inventory holding) as well as the elimination or appropriate prevention of other identified wastes. herefore, the lean approah aims at ost redution via better organization of proesses in the time line while maintaining high ustomer servie and the rapid response to ustomer needs. he several lean tools listed below briefly explain onepts from the design of a lean inventory replenishment system point of view. Lean theory is based on the five priniples presented below. he onduted researh espeially refers to the flow (no 3) and pull (no 4). he five lean priniples (Biheno, Holweg and Biheno, 2009): 1. Speify the value from the ustomer point of view. he main stress should be put on the fat that the ustomer buys the result, not a produt itself and the value should be assessed having this in mind. 2. Value Stream. It aims at identifiation of non-value-adding ativities. In the ontext of effiient replenishment system design, inventory holding is not onsidered valuable in terms of both, priniple no. 1 as well as priniple no. 2. It does not add any value to the produt. 3. Flow. Aording to lean philosophy, all goods should be kept in a flow. Any storage is onsidered a waste of money, any waiting for finishing of other omponents or bath to be delivered is a waste of time beause it does not add any value to produts. 4. Pull. In ontrast to the ommonly used push approah, lean management proposes just in time supply and prodution. It means that the produts should be delivered when needed but no later than required by the ustomer. It again refers to elimination of 12

25 stok. With a pull system the first ation in the hain is that the item is demanded. o satisfy this demand there is an item in stok. As soon as this stok is used up, another item is supplied, either from outside or from a prodution proess (Basu and Wright, 2008). In a pull system the quantity of delivery is signifiantly smaller ompared to the push system, thus the delivery takes a muh higher frequeny and refers to JI poliy. 5. Perfetion. Perfetion does not only mean quality it means produing exatly what the ustomer wants, exatly when (with no delay), at a fair prie and with minimum waste (Biheno and Holweg, 2008:11). his priniple summarises the previously presented lean priniples and shows that the aim of lean management is to reorganise the proesses in a way that overall time will be redued and the quik response to ustomer requirements will be possible without any buffer inventory holding Lean Wastes One of the more important lean onepts is identifiation and elimination of wastes. Aording to lean theory wastes an be differentiated into two ategories: wastes from an organization s perspetive and the ustomer s perspetive. Both of them are presented as follows. Servie Wastes (ustomer point of view) (Biheno, Holweg and Biheno, 2009) Delay. In general it refers to waste of a ustomer s time, espeially in the form of delays in servies, deliveries, waiting in queues et. Dupliation of ativities whih are required to be ompleted by the ustomer. Unneessary ustomer movement whilst being served. Unlear ommuniation. Inorret inventory, suh as lak of required stok in required time. Loss of business opportunity Errors in orders or damage of produts Seven Wastes (ompany point of view) (Biheno, Holweg and Biheno, 2009) Overprodution: Goods should be produed, delivered and stored in an appropriate quantity. Safety stok should be minimised as it ontributes to wastage of time, money and storage spae. It might also inrease the likelihood of the deterioration of perishable goods. 13

26 Waiting: his refers to delays from the ompany s point of view with regard to delivery or until a bath of produts or omponents are finished. Attention should be paid to the fat that waiting also reates some bottleneks in the system and might ause the bullwhip effet in the whole supply hain. In general it affets the goods flow whih was previously presented as one of the lean priniples. Unneessary motion: his refers to the movement of produts, people and mahines. Waste of transportation: Besides the waste of energy it also refers to the inreased likelihood of damage. Overproessing: his refers to the unneessary use of sophistiated and expensive software or mahines. he designing of omplex proesses or the implementation of sophistiated software when not neessary is a waste of money as well as time devoted to hange management. Unneessary inventory: As previously mentioned the redution of unneessary inventory auses a ost redution related to storage spae or the number of employees needed to maintain the inventory et. Implementation of a Just in ime (JI) poliy prevents unneessary inventory holding. It also makes a ompany more flexible, simplifies inventory management and failitates a quik response to ustomer requirements. Defets: Defets generate unneessary osts to the ompany. Defetive goods need to be repeated or ordered again. Defets are ommon in the ase of a push system, where mass prodution ours due to eonomy of sale. Lean management also suggests the redution of suppliers in order to simplify the system design and the development of better relationships with those suppliers that are kept so that they beome more reliable. It should be noted that all wastes are related to eah other, one ausing another. High inventory an be identified as a fator that most affets other wastes. Unneessary inventory might result in the inreased likelihood of defets and deterioration, ausing omplexity while stok managing, whih is onsidered over proessing. Defets might also involve ause the ustomer additional and unneessary involvement (e.g. returns). Defets need to be repaired or the goods need to be ordered again, whih auses overprodution as well as overproessing. Overprodution auses unneessary motion, 14

27 transportation, delays, waiting or waste of transportation. his also auses additional inventory and redues the ability to respond quikly to demand hanges. Unneessary transportation inreases the probability of goods being damaged and inreases the waiting time. his in turn, affets the smooth flow of goods within the supply hain. Both the defets as well as high inventory an ause errors in inventory, whih ould result in lost business opportunities. All of the above wastes bring about unneessary osts to the system. Designing of effiient replenishment systems with speial fous on inventory ontrol would diretly and indiretly redue osts signifiantly. It would enable a fair prie to be set for the sold produts without a derease in quality aused by mass prodution and big-bath transportation. his, in turn, ould result in inreased ustomer demand. he urrent researh reognises the importane of appropriate inventory ontrol Flutuating inventory level As noted in Setion 0 it an be identified that the surplus inventory holding affets other lean priniples, unneessary inventory holding generates hidden osts to the whole supply hain. herefore, zero inventory poliy seems ideal. his is the reason why so many aademi researhers were devoted to this topi (see Setion 2.2.5). Nevertheless, in most real life appliations keeping some safety stok is still neessary (as explained in ). he right quantity identifiation is a subjet for aademi researh as well as industrial trial and error pratie. Both the industrial and aademi strategies tend to derease the inventory as muh as possible under the onstraint of demand satisfation. Due to the lak of aurate predition of future demands in most real life replenishment systems, searhing for inventory level balane is also often inaurate. As a response, in this setion the author proposes an innovative way of thinking of replenishment design. It resembles JI poliy in respet of no onstraint on order bath quantity. he bath an be as small as needed. However the introdued strategy involves a redution of inventory flutuations rather than inventory level itself. It has not neessarily been realised by many researhers and industrial managers but it is highlighted here that unontrolled inventory level flutuations an ause several hidden osts and surplus problems ompared to stable inventory level holding suh as: 15

28 Bullwhip effet: as shown in Setion the sudden hange of inventory level in one supply hain node auses inventory flutuations in the others. Unneessary deterioration of produts: the shelf life management beomes more omplex when the inventory level is varying. he short shelf life replenishment system is appropriate for low inventory levels when the stok is replenished often. Sometimes the supply hain node hooses goods with lose end dates on purpose as they are usually less expensive and enable some purhasing ost redution. In turn, long shelf life is neessary for high inventory levels where ertain stok will be kept signifiantly longer. In the ase of flutuating inventory, keeping a balane between long and short shelf life purhases beomes omplex, some unexpeted produt deterioration is likely to our. Also in some storage areas of real life appliation the surplus inventory is stored on top of the regular stok, whih for some produt types inreases the likelihood of deterioration (for instane due to redued oxygen exposure of those at the bottom). In other ases, holding unexpeted inventory may mean using emergeny storage areas where onditions may possibly inrease the likelihood of deterioration (for instane outdoor storage areas). Damage: the damage of goods is more likely to our when the inventory level suddenly inreases as it involves the additional movement of goods, whih inreases the likelihood of damage. It an involve the inventory being kept in inappropriate storage areas where damage to goods is more likely to our. Storage apaity definition problem: either the storage room is not used, albeit still paid for, or exeeded. Flutuations in storage ost: varying storage osts affet the ash flow of the ompany and finanial planning ability. Diffiulty in required employees planning: either the surplus labour is maintained and paid for, or the labour apaity is exeeded. A sudden inrease of inventory requires more labour to be involved in logistis operations suh as paking/repaking, labelling, moving to storage loations et. he thesis devotes its attention to model based ontrol to redue the inventory flutuations and redue hidden osts. No bath onstraint on order quantity is onsidered in the thesis. he bath onstraint in mathematial terms an refer to an order quantity smoothing parameter. For 16

29 the purpose of this thesis the smoothing parameter is onsidered to be set to zero (see Equation (3.63)). 1.2 Researh Problem he need for system modelling he purpose of modelling is to gain a better understanding of real systems at low-ost. Experimenting with the model itself instead of involving the real system is less time onsuming and does not require reorganisation of proesses for experimentation purposes and assoiated investments. As many as needed of different senarios an be onsidered before implementation. Several mathematial and engineering tehniques enable system modelling for optimisation purposes. hey an be further used for optimal or suboptimal deision making Operational researh approah Operational researh (OR) is a field of studies that deals with optimal or suboptimal deision making in real world, albeit mostly non-engineering, environments suh as military operations, businesses proesses, network design, sheduling, supply hain management, routing, optimal searh, faility loation et. It uses analytial tehniques for deision making involving mathematis for modelling and /or sophistiated software for senarios analysis and simulation (Winston 1991) Researh desription Control theory (C) is an interdisiplinary field of study that involves dynamial systems theory. Being appliable in any system where a feedbak loop an be mathematially modelled, it has been developed for a broad range of tehniques and algorithms to ontrol the system input in order to obtain the desired output and desired system behaviour. As methods used in ontrol theory have a quantitative nature, the natural onsequene was to find a wide 17

30 array of appliations in engineering fields suh as automotive, heating, eletronis, robotis et. Control engineering, then, regards appliation of ontrol theory tehniques to engineering disiplines. Nevertheless, the algorithms whih are used for engineering appliation, being already reognised as powerful and well performing, an be benhmarked for a variety of problems suh as inventory, manufaturing, network optimisation et. Control theory, with its straightforward onsideration of system dynamis, an potentially be reognised as powerful and effiient in the operational researh field. he thesis fouses on the inventory problem studied from a ontrol theory perspetive. he approah brings the advantage of overoming the problem of system unertainties related to flutuating ustomer demand (as illustrated in Figure 1-8), flutuating lead times and deterioration of produts by updating the urrent information in a feedbak loop, and ontrolling system behaviour without having prior knowledge about the system dynamis. Figure 1-8: Exemplary varying demand showing how dynamially and randomly the demand an hange he final outome presented in the thesis, whih is an inventory level ontrol methodology, aims to not only obtain good results ompared to purely mathematial tehniques, but rather at giving the reader the idea of the elegany and power of ontrol theory in suh appliations. It also highlights the potential benefits of ollaboration between the ontrol engineering and the operational researh ommunities. Although suh ollaboration seems promising, there is still a gap in effetive ommuniation due to differenes in perspetive and understanding between these two groups. One of the reasons an be attributed to very little awareness of OR problems and its appropriate mapping into the ontrol theory domain among the C ommunity. Another reason is the limited awareness of Control heory siene within the OR ommunity 18

31 Figure 1-9: Operational Control Reognising the above, this researh aims at bridging the gap between the two by developing a new field of studies whih, in the future, would enable the operational researh ommunity to apply the omplex and effiient algorithms of ontrol theory without any need to familiarise themselves with the deep and strutured ontrol theory. For the purposes of this thesis, the name operational ontrol (OC) has been given to the ontrol theory patterns and proedures being benhmarked in the operational researh field. o start with, one of the well-established optimisation algorithms, extensive in mathematial desription and deeply ingrained in ontrol theory siene, known as model preditive ontrol (MPC), has been benhmarked. Further, the algorithm, for the inventory model, has been mathematially reformulated to the form whih is signifiantly simplified in desription, and does not require the user to possess any initial ontrol theory knowledge. As a result, the novel shortened optimisation proedure gives almost exatly the same results MPC, and both have been demonstrated to be mathematially equivalent. he developed methodology appears to be transferable to other appliations, for instane prodution systems. he redued mathematial formulation was based on notiing patterns in the MPC formulation and followed a series of propositions and their demonstrations leading to the final approah. he final formulation is presented in the form of a proposition in Chapter 3, Setion 3.3. he elaboration of its simpliity in appliation and the straightforward manual alulation of the next optimal order size on a daily basis, whih maintains the inventory in balane, are shown in Chapter 4, Setion 4.5. he mathematial demonstration of the equivalene of the initial and final formulation is shown in Chapter 5 and the final results are shown in Chapter 6. 19

32 1.3 Aim, Objetives Researh Contribution and Outline of hesis Aim of the researh he aim of the researh onsists of two parts: Apply the ontrol theory as an optimisation tool to develop a deision support system for replenishment and satisfatory performane Bridge the gap between the preision of ontrol theory and the understanding of the operational researh ommunity Objetives In order to aomplish the aim of the researh projet, the following objetives have been established: 1. Identify the urrent pratie of OR modelling of warehouse management. 2. Identify the state-of-the-art in appliation of ontrol theory to inventory modelling and optimisation 3. Convert the inventory problem to ontrol theory domain 4. Benhmark ontrol as an optimisation tehnique for replenishment deision support system 5. Develop the simplified algorithm whih an be easily adoptable by operational researh ommunity 6. Run a number of managerial senarios in order to improve the urrent pratie Researh ontributions Aording to their importane the PhD researh ontributions an be listed as follows: 1. Making the model preditive ontrol (MPC) tehnique available to the OR ommunity via mathematial redution Chapter 3, Setion 3.4 Based on literature review of ontrol tehniques applied in OR and presented in Setion 2.3, it seems that many of the authors follow the same path. All of them apply various existing 20

33 ontrol tehniques for a range of logistis models. herefore, the aessibility of those tehniques to OR is very limited, as it involves familiarity with ontrol theory priniples. In this thesis the author presents a totally different approah and shifts the researh towards a different diretion, developing the MPC inspired simplified method, whih makes it instantly available to the OR ommunity, but retaining the engineering preision of dynamial ontrol. It is omputationally signifiantly less onsuming (as elaborated in Setion 3.6) and, although initially developed for inventory ontrol, it an be appliable in a wide range of other management/prodution/inventory problems. 2. Demonstration of the mathematial equivaleny of the redued approah to MPC for the inventory model (step-by-step proess) Chapter 5 It involves the presentation of a sequene of propositions and their demonstrations, omprising a whole hapter. Eah proposition is a onsequene of the previous one and the last one is the final version of the novel mathematially redued tehnique. he demonstration is an illustration of the evolving thought proess, whih was inspired initially by MPC and subsequently applied for the inventory model. 3. Quantifiation of benefits of MPC with appliation to inventory model (and as a onsequene the benefits of the novel redued ontroller, as mathematially equivalent) Chapter 4 (Setion 4.4), Chapter 6 he quantifiation of benefits was an innovation in the OR perspetive, with the redution of inventory flutuations elaborated in Setion he profitability of the MPC was assessed using a developed profit funtion (minor ontribution). 4. Establishment of limitations of new method Chapter 6 and Chapter 7 he method is limited to the appliations where there is no onstraint on order quantity. he limitation was reognised and justified in Chapter 6. In Chapter 7 the possible diretions for overoming the limitation was elaborated. It also showed that for the onsidered appliation the limitation beomes strength of the model (justified in Chapter 7). 5. ranslation of inventory problem to ontrol theory framework Chapter 3, Setion 3.2, the model is represented in a state spae form, where the demand is treated as a disturbane, the system dimension refers to lead time and the deterioration of produts is a time varying 21

34 quantity. In ontrast to typial ontrol theory approahes, the disturbane (demand) is subtrated from the urrent inventory level signal Outline of thesis he thesis is organised in the following manner. Chapter 1 introdues the reader with logistis definitions, whih are neessary for the understanding of further hapters in the thesis. It also presents the fundamental onepts and problems of logistis that the thesis addresses. It enables an understanding of the need for mathematial modelling of supply hain and inventory problems. It also justifies the thesis goal of reduing the inventory flutuations, whih is the referene point for performane assessment later in the thesis. Chapter 2 presents the literature review of a general OR approah and the more speifi C approah in logistis problems. It shows the variety of tehniques and approahes to logistis systems modelling in the OR ommunity. It highlights the gap in the OR approah and justifies how C an effortlessly bridge the gap. Chapter 2 also allows the reader to see that the C approahes found in the literature of inventory problem follows the same path initiated by Simon (1952). herefore, in Chapter 2 the need for shifting towards a different diretion is justified. he main ontribution of the thesis (see point 1 Setion 1.3.3), starts a new route whih will hopefully be hosen by other researhers in the future. Chapter 3 presents the translation of the inventory problem to the C framework. It shows both, the ontinuous time and disrete time preliminary modelling. It also presents the first researh attempts of the appliation of C algorithms (Smith preditor in Setion ). Further, the model preditive ontrol approah is shown in Setion 3.3. Although the novel mathematially redued ontroller is presented immediately after the model preditive ontrol in Setion 3.4, in the time sale of the researh evolution the novel ontroller development (shown in Chapter 5) took plae between these two milestones of Setion 3.3 and Setion 3.4. he novel algorithm, although inspired by model preditive ontrol used for the inventory model, transpires to be a minimum variane approah, or dead beat ontroller with an inorporated Smith preditor. Although the redued form was initially inspired by model preditive ontrol, the final formulation whih is essentially an optimal dead beat ontroller with Smith preditor appeared to be immensely benefiial. It s pereived limitation (of lak of order size onstraints) was in fat found to be the ontroller s advantage, for whih justifiation is shown in Chapter 7. 22

35 Chapter 4 is a natural onsequene of Chapter 3. It shows the simulation results of eah modelling stage shown in Chapter 3. It justifies the deision of the abandonment of the Smith preditor applied independently. It shows how the simulation results inform and rediret the researh with new attempts to appropriate algorithm seletion, to finally omplete the yle and obtain the dead beat ontroller, with inorporated Smith preditor. his is mathematially equivalent to the model preditive ontrol applied for the inventory model when the tuning parameter is set to zero. Chapter 4 justifies the initial inspiration of model preditive ontrol applied to the inventory problem and shows the equivaleny in results between the original model preditive ontrol and the developed/proposed novel ontroller. Chapter 5 presents the proess of reognising model preditive ontrol patterns in mathematial desriptions and in obtaining the mathematially redued equivalent formulation. As a sequene of propositions and their demonstrations the overall proess of reahing the final mathematially redued novel ontroller is shown. Chapter 6 presents the results of the proposed method in respet to different numerial settings. Although the lak of order onstraints has been onsidered an advantage of the model in this partiular appliation, Chapter 7 disusses possible future diretions in order to overome this limitation. Instead of finding the mathematially equivalent ontroller, the hapter proposes a good estimation in the ase when the model preditive ontrol tuning parameter is non-zero. It an be applied straightforwardly by the OR ommunity and the generated results are very similar to the original model preditive ontrol Publiations A list ontaining publiations of the author is presented in Appendix II List of publiations and other aademi ativities. he publiations were gradually refleting every ahieved milestone of the researh. (Orzehowska, Burnham, and Petrovi, 2011a) was presented at an OR onferene and the publiation highlighted the advantage of the appliation of ontrol theory to inventory modelling based on a literature review (whih was partiularly inspired by the Smith Preditor appliation of Ignaiuk and Bartoszewiz 2010b) and some preliminary experiments with ontinuous model preditive ontrol ombined with the Smith preditor. After the presentation, one question provided inspiration for the rest of the researh. Being asked, if ontrol engineering should be taught on OR ourses. It was understood then, even 23

36 better, that the OR ommunity needs a tool whih will maintain ontrol engineering advantages but should be straightforward and understandable in OR terms. As a response (Orzehowska, Burnham, and Petrovi, 2011b) foused on onvining the ontrol ommunity to apply their method to inventory problems in a manner whih would be aeptable by OR experts. he paper presented the appliation of ontinuous and disrete-time model preditive ontrol with an inorporated Smith preditor. As onsequene of this onferene presentation a new international ollaboration with Prof. Bartoszewiz, the author of the paper whih signifiantly influened the researh in the beginning (Ignaiuk and Bartoszewiz, 2010b). he next step was a journal review paper of ontrol tehniques applied to supply hain management. Having already been partiularly interested in model preditive ontrol at that stage, the main part of the review was devoted to that tehnique (Orzehowska, Bartoszewiz, Burnham and Petrovi, 2012a). he first steps of the mathematial redution proess of disrete model preditive ontrol an be found in (Orzehowska, Bartoszewiz, Burnham and Petrovi, 2012b). he aomplished mathematially redued ontroller for non-perishable goods only was presented in Orzehowska, Bartoszewiz, Burnham and Petrovi (2012). he first steps of the perishable goods in mathematial redution form an be found in Orzehowska, Bartoszewiz, Burnham and Petrovi (2013a). he ompleted mathematially redued ontroller, appliable for both, perishable and non-perishable goods was presented at an OR onferene (Orzehowska, Bartoszewiz, Burnham and Petrovi 2013b) and a ontrol theory onferene (Orzehowska, Bartoszewiz, Burnham and Petrovi 2013). he first one presented the researh from an OR point of view, fousing on performane, results and simpliity. he seond one foused on demonstrating the equivaleny of the original model preditive ontrol and the mathematially redued model. 1.4 Summary he urrent hapter presents an introdution to logisti siene. It is aimed at familiarising the reader with relevant terms and management strategies neessary to understand the researh appliation. It also highlights the problems that usually our within logistis. Speial fous is devoted to the inventory issues and orresponding management strategies as it relates to the researh appliation. 24

37 As previously mentioned, there are several reasons for holding inventory within the distribution arms of supply hains. Among others, the inventory provides a buffer between the supply and ustomer demand in response to dynami demands and lead times. It ensures a high servie level, whih is ruial for a ompany to survive in a ompetitive market. Although, the inventory holding is onvenient for a ompany, it generates an expense, whih has already been reognized as one of the highest logistis osts. Surplus inventory holding also auses problems also brings indiret osts as it affets the work flow in other areas. A replenishment system requires an appropriate balane between ustomer demand satisfation and stok keeping expenses to be ahieved with due onsideration to system dynamis. aking into aount, only two suh system unertainties, inluding unpreditable demand and lead time delay, managing replenishment inventory without appliation of an engineering deision support framework beomes a omplex proess. his often results in unontrolled inventory level flutuations, whih in turn lead to several other problems suh as deterioration of produts, variation in storage osts, extension of storage apaity and/or bakorders. As logistis is a relatively new field of siene there is still a huge sope for further improvements. Due to the many of onstraints in the real world senario the developed theoretial and analytial tehniques used in inventory replenishment design usually fous on a few seleted aspets of the atual logistis systems and usually only ahieve a trade-off between the theory and pratie. he onlusion that an be drawn is that the omplexity of the ost-benefits balane ahievement justifies the need for the development of deision support systems. Sine the 1940s the logistis operations have eliited the attention of operational researhers using more sophistiated methods in designing more effiient logistis systems. he siene of mathematis possesses a set of optimisation tehniques whih an be applied to improve the logistis performane deision making or/and ost redution. It is suffiient to represent the existing problem in a mathematial language of variables and equations to enable the appliation of the tehniques. he literature review, Chapter 2, is devoted to elaborating the history and the state-of-the art of logistis system design with speial fous on inventory replenishment. he aim of the researh is to use mathematial tehniques of ontrol theory to solve inventory problem. 25

38 2 LIERAURE REVIEW 2.1 Introdution to an aademi approah to supply hains he expansion and globalization of ompanies an be widely observed in everyday life. Barely anything is produed loally in today s world. he produtions sites are loated in one ountry, management in another and the distribution network expands worldwide. What is more, the ompetition beomes stronger in a global market, as it develops dynamially and grows fast. Suh a dynami environment requires ontinuous improvement from the ompany. Lak of improvement in this ontext leads to reession and will eventually exlude the ompany from the market (Rushton, Crouher and Baker, 2006). his new situation ompels the ompany managers to ontinuous rethinking of the ompany proesses, poliies, strategies, strutures et. he need to innovate ideas has beome a matter of utmost importane not only within the ompany s ore tasks but also in all proesses enabling the ompletion of ore tasks; logistis is one of them. Although logistis enables expansion of ompanies in terms of organisation and design of goods flow, it generates additional, not neessarily value-added expenses with respet to details as disussed in Chapter 1. he awareness of possible ost redution in logistis proesses, partiularly in inventory holding, beomes ruial in times of market globalisation. he new logistis strategies are ontinuously developed by both industrial and aademi ommunities, and find their appliation in global orporations as well as medium and small businesses. he industrial strategies have been introdued in the previous hapter (Chapter 1). he urrent hapter presents the aademi approah to logistis systems improvement and optimisation with speial fous on inventory management and replenishment. It is observed that inventory replenishment optimal poliy design has gained inreased attention in the field of operational researh in reent times. hough the origins of the appliation of regulation theory in the field of logisti systems ontrol an be dated bak to the 1950s there have been a relatively limited number of researhers dediating their attention to this issue until reently. herefore, the disussion of ontrol methods in logistis, without mentioning the well-established tehniques of the operational researh field, would not give the reader a holisti view of the topi, disregarding the strengths and weaknesses of ontrol methods. hus, the hapter has been organised as 26

39 follows. Firstly, an overview of OR tehniques applied to overome problems faed in logistis is presented. Seondly, the appliation of the ontrol oriented tehniques addressed to different logistis operations is disussed. Here, the main fous of attention is given to the inventory replenishment appliation, as this is relevant to the urrent researh topi. Furthermore, the strengths and weaknesses of the disussed tehniques in replenishment systems are listed with respet to model omplexity and adequay of the model to the ase of real-world appliations. he further setions of the hapter fous on ontrol theory appliations related to logistis, mostly inventory systems, and the gaps in knowledge are identified. Equation Chapter (Next) Setion Operational Researh ehniques here are many tehniques applied in the field of operational researh to find optimal or quasi optimal solutions for a partiular problem. he mathematial methods or simulation based methods or the ombination of both an be found in the aademi literature. Mathematial methods onsist of mathematial optimisation (ommonly mathematial programming) and other algorithms, among whih the geneti algorithm plays a major role. Mathematial programming, whih inludes linear programming, nonlinear programming, onvex programming, geometri programming, integer programming, dynami programming, quadrati programming and many others, refers to building a mathematial model with an objetive funtion whih minimises ost or maximises profit, and the equality and inequality onstraints representing the physial onstraints of the system. Sine a real system has many unertainties the mathematial desription of the system usually requires some initial assumptions and simplifiations. A geneti algorithm (GA) is an iterative tehnique of searhing for better solution(s) with lose proximity to the optimal solution. In a similar manner to a natural seletion proess, a GA selets better solutions from populations of possible solutions and then ombines them together with eah other to breed a new generation of solutions. Just as in a natural seletion proess, the new generation of solutions beomes loser to the optimal value. he proedure is repeated until satisfatory results are obtained. Simulation is another approah used for system design or proess reengineering. Although it allows for relatively more omplex model building than the pure mathematial methods, it is not typially meant to find the optimal solution. Rather than that, it enables deep system analysis, performane evaluation and experimenting with different senarios for finding 27

40 satisfatory results. For instane Hahiha et al. (2010) use simulation for performane evaluation in multi-stage, multi-produt, multi-loation and multi-resoures within a prodution system. he objetive is to find better lot sizes of eah of the produts being manufatured. Simulation an also be ombined with different optimisation tehniques suh as a GA, applied for instane by Kohel and Nelander (2005) in a multi-ehelon inventory problem or satter searh applied by Keskin, Melouk, Sharif and Ivan (2010) for an integrated vendor seletion and inventory problem. he reviewed models proposed in the operational researh literature are briefly disussed in the urrent setion under six ategories: inventory modelling, supply hain management, routing/transportation problem, distribution network design, optimal paking, and manufaturing/sheduling problems. herefore the presented papers are disussed with respet to their field of appliation as well as tehniques applied. he onsidered tehniques are: linear programming, other mathematial optimisation tehniques (suh as nonlinear programming, integer programming, onvex programming, stohasti programming, dynami programming and heuristi methods), simulation and geneti algorithm. All of the disussed papers are ontained in able 2-1 with respet to applied method and appliations. Linear Other Geneti Simulation programming mathematial algorithm optimisation tehniques Inventory (Janssens and (Padmanabhan (Zhang, et al. (Kohel Ramaekers, and Vrat, 2012), and 2011), 1990) (Guhhait, Nelander, (Amaya, (Samadi, Kumar, Maiti 2005), Carvajal, and Mirzazadeh and Maiti, (Kohel Castaño, and Pedram, 2013), and 2013) 2013) (Chun-Wei Nelander and Hsian- 2005), Jong, 2011) (Keskin, 28

41 Melouk, Sharif and Ivan, 2010), (Healy and Shruben, 1991) Supply hain (Bilgen, (Paksoy, (Zhang, (Kharazi management 2010), Özeylan and Zhang, and (Peidro et al., Weber, 2010), Caiand Jandaghi, 2010). (Cintron, Huang, 2011), Ravindran and 2011), (Seo, (Siddiqui, Ventura, Jeong, Lee, Khan and 2010), Lee and Park, Akhtar, (Poojari, 2012), 2008), Luas and (Kaijun and (Long, Lin Mitra, 2008) Xiangjun, and Sun, 2012) 2011) Routing (Chien, (Berman and (Jiang, Wang, (Suzuki, Balakrishnan Larson, 2001), and Ding, 2011), (Lau and Wong, (Lee, Moon, 2013), (Lin, et al. 2009) 1989), and Park, and Yeh, (Adelman, 2010), 2013) 2003), (Golden, (Adelman, Assad and 2004) Dahl, 1984) Supply network (Bilgen, (Cintron, (Lin, and (Long, Lin design 2010), (Peidro Ravindran and Yeh, 2013), and Sun, et al. 2010) Ventura, (Che, Chiang 2011), 2010), and Che, (urhan, (Poojari, 2012), Vayvay and Luas and (Zhang, Cai Birgun, Mitra, 2008) and Huang, 2011) 2011) 29

42 Piking/paking (Bidgoli, (Roodbergen, (Khanlarzade, (Serna and 2010), Sharp, and Vis Yegane and Pemberthy, (Roodbergen, 2008), Nakhai, 2010) and Koster, (Smoli- 2012) 2001a), Roak, et al., (Roodbergen, 2010), (Hall, and Koster, 1993), (Gu, 2001b), Goetshalkx, (Roodbergen, and MGinnis, and Koster, 2007) 2001) Manufaturing (Bard and (Yong, Wang, (Svanara, (Kumar and Nananukul, Lai, 2009), Kralova and Sridharan, 2010), (Lee and Blaho, 2012), 2010), (Winston, Yoon, 2010) (Qiao, Ma, Li (Božičkovié 1998) and Yu, et al., 2012) 2013) ransportation (Chien, (Berman and (Jiang, Wang Balakrishnan Larson 2001), and Ding, and Wong, (Golden, 2013), ( Lin 1989) Assad and and Yeh, (Adelman, Dahl, 1984) 2013) 2003), (Jung and (Adelman, Mathur, 2007) 2004), able 2-1: he OR tehniques applied in logistis proesses modelling ransportation supporting models he purpose of transportation in logistis is to link failities within the logistis system and provide a flow of goods starting from the manufaturer to the end ustomer (Ghiani, Laporte and Musmanno, 2004). he movement of goods, regardless of the transportation mode being 30

43 utilised, generates signifiant osts to the ompany. Motor vehiles (lorries, truks, vans, ars) are regarded as the most ommonly used transportation mode in a supply hain due to motion flexibility in the well-developed road network (Rushton, Crouher and Baker, 2006). As the most often used mode, it is also the most ommonly targeted mode for a potential ost redution in supply hain network design. he most ommon approah towards transportation ost redution is the Vehile Routing Problem (VRP), whih tends to optimise the total vehile route length under the onstraints and requirements of the distribution system: struture of the supply hain network, ustomer requirements, vehile apaity et. Several researhers (e.g. Chien, Balakrishnan and Wong (1989) and Adelman (2003, 2004)) have dediated their work to the aforementioned problem, and designed routing models, employing linear programming as an optimisation tehnique. Berman and Larson (2001), in turn, applied dynami and stohasti programming methods to find optimal vehile paths, while Golden, Assad and Dahl (1984) and Jung and Mathur (2007) foused on heuristi analysis of the problem. Jiang, Wang and Ding (2013) and Lin and Yeh (2013) approahed the routing problem by applying a geneti algorithm. he simulation based models optimising the vehile routes, in turn, appeared in papers of Clarke and Wright (1964) and Shwart, Ward and Zhai (2006). A few papers presented the routing problem in ombination with supply hain design (e.g. Lee, Moon and Park (2010)) or inventory management (e.g. Golden, Assad and Dahl (1984) and Jung and Mathur (2007)) Supply hain management supporting models he supply hain design involves both movement as well as storage of goods and therefore almost any single logistis aspet an be onsidered in this ategory (Mentzer, 2001). For this reason the problem an be onsidered as a multi-objetive optimisation. Nevertheless, for the purposes of this setion, the supply hain modelling and optimisation should be understood in terms of performane measures of supply hain networks. Alloation of failities, inventory optimisation and routing of vehiles and integrated multi-objetive models have been presented in several papers. Bilgen (2010) and Peidro et al. (2010) proposed fuzzy linear programming models for optimal supply hain design. Paksoy, Özeylan and Weber (2010), developed a mixed integer programming model to find the minimal inventory holding osts, transportation osts and the vaant apaity of a warehouse. While developing the model, 31

44 several fators were taken as onstraints: appropriate distribution entres assigned to appropriate ustomers, storage apaity, demands and throughput of suppliers. Also other mathematial programming tehniques suh as pure integer programming (as in the ase of Cintron, Ravindran and Ventura 2010) and dynamial programming (for instane Poojari, Luas and Mitra, 2008) an be ommonly found in the literature. he supply network planning and design an be found in papers of Lin and Yeh (2013), Che, Chiang and Che, (2012) and Zhang, Zhang, Cai and Huang (2011) using the geneti algorithm as an optimisation method. In the papers of Bottani and Montana (2010) and Wangphanih, Kara and Kayis, (2010) the simulation based approah to this topi an be found Piking-paking supporting models he piking and paking problem refers to several issues: the optimal or near optimal warehouse design, whih would allow for quik stok piking based on pikers path redution, pikers routing problem, order-piking poliies, alloation of inventory, optimal fitting of the stok in the transportation vehile, inluding pallet design and paking et. he piking osts have been reognized as one of the highest osts among warehousing osts, approximately 50-75% of the total (Coyle et al., 1996). It refers mostly to labour osts, therefore the optimisation of the piking time problem is often presented in the literature. Although the automation of the piking proesses definitely brings signifiant ost redution, the manual order-piking still plays a major role, due to diversity in the size and shape of warehouse stok (Bidgoli, 2010). herefore several models have been developed in order to failitate optimal order piking system design. Roodbergen, Sharp and Vis (2008), for instane, developed a linear programming model using statistial estimation of walking distanes of pikers. Hall (1993) optimised the order-piking path within one warehouse blok only. Smoli-Roak et al. (2010), however, built the shortest path dynami programming model for a given entire warehouse layout. In the work of Gu, Goetshalkx and MGinnis (2007) the optimisation of a piker s travel routes was ahieved by optimal assignment of produts to partiular storage loations. Gagliardi, Renaud and Ruiz (2007) reated a disrete event simulation-based model aimed at improvement of warehouse performane at piking and storage operations. It addresses the 32

45 onerns of design of the storage areas and piking harateristis. he storage alloation influene on the warehouse s performane is disussed in the paper. Different senarios are ompared and evaluated to obtain the best results. he piker performane has signifiantly inreased with optimal design of the storage area. Petersen and Aase (2004) presented a simulation-based model of the existing piking area whih was designed to improve the order-piking proess with respet to both redued ost and inreased performane. he sensitivity analysis involved the experimentation with the shape of the warehouse, size of order, warehouse layout, pikers routing and the loation of produts in the storage area. In view of the former, the researhers presented several methods of piking to identify whih of the examined methods brought the most improvement to the piking proess. It onluded that bath-piking grossly affets the ost redution in a given system Manufaturing supporting models Prodution modelling refers to two issues here, namely the minimisation of total prodution ost based on inventory, delivery and planning as well as prodution sheduling. he first issue was addressed in the paper of Bard and Nananukul (2010). he linear programming model of a single produt manufaturing system was built by onsidering the prodution apaity limits. Another linear programming model was introdued by Winston (1998). It onsidered a oneprodut inventory-prodution planning system. he demand was assumed to be known in advane, whih made the model unrealisti and not appliable for most of real life senarios. Yong, Wang and Lai (2009), in turn developed a onvex programming prodution-inventory model for deteriorating produts in order to obtain optimal sheduling of the prodution of goods being sold in multiple markets with different peak seasons. Lee and Yoon (2010) introdued an integer programming model to identify the optimal prodution shedule in addition to bath sizes to minimise the total work-in-progress inventory ost. he geneti algorithm approah to optimal manufaturing system design and sheduling an be found in Svanara, Kralova and Blaho (2012) and Qiao, Ma, Li and Yu (2013). 33

46 2.2.5 Inventory and replenishment supporting models he inventory or replenishment problem is usually approahed by omputing the optimal order quantity or eonomi order quantity (EOQ). Regardless of the ompany strategy, usually the aim is to minimise the storage ost by the redution of inventory held in the warehouse. his frequently results in more regular orders and deliveries. Different strategies are applied, depending on the type of produt, to ahieve the appropriate balane between maximising the hanes of demand satisfation and minimising the inventory. he linear programming (Janssens and Ramaekers 2011, Amaya, Carvajal, and Castaño, 2013) and other mathematial optimisation models - for instane the non-linear programming (Padmanabhan and Vrat, 1990) and fuzzy geometri programming (Samadi, Mirzazadeh and Pedram, 2013) - an be found in the literature. he reviewed models are disussed aording to a few key riteria as listed under the bullet poits below. he more realisti the assumptions used while developing the model, the more omplexity is added to the mathematial desription. For this reason many authors have foused on less realisti assumptions, whih is reasonable when onsidering the omplexity of the mathematial desription. he referenes presented in the ategories below have been given additional loal referene numbers in square parentheses [ ] for ease of presentation in able 2-2. Nature of demand - whether it is deterministi, stohasti or totally random. he models built under the assumption of having a prior knowledge about future demand are appliable only in a limited number of real world ases. Although, in real ase senarios the demand is usually not fully preditable, there might be some demand predition done based on past data reords, for instane with regard to seasonality of produts. hough, the EOQ modelling for deterministi demand is rarely appliable in industry, it an be observed that there are relatively few aademi papers, whih present the demand as totally unpreditable and unknown. Deterministi demand an be found in models of Avinadav and Arponen (2009) [2], Chung and Liao (2009) [3], Hsu and Wen-Kai (2009) [4], Panda, Saha and Basu (2008) [5], Konstantaras and Skouri (2010) [8], Ghiani, Laporte and Musmanno (2004) [11] and Chungt (1998) [12]. he stohasti demand onsideration an be found in papers of Madadi, Kurz and Ashayeri (2010) [6], Baumol and Vinod (1970) [7], Edwin, Cheng, and Wang (2007) 34

47 [9] Strak and Pohet (2010) [10], Xiong and Helo (2006) [13] and Ehrhardt (1997) [14]. he random demand is less ommon in the literature and an be found for instane in Dutta, Chakraborty and Roy (2005) [1]. Nature of lead time whether it is onsidered to be different than zero, fixed or variable. he importane of onsidering the non-zero, albeit fixed lead time was appreiated in the literature by many researhers (Chakraborty and Roy (2005) [1], Avinadav and Arponen, 2009 [2], Chung and Liao, 2009 [3], Hsu and Wen-Kai, 2009 [4], Panda, Saha and Basu, 2008 [5], Konstantaras and Skouri, 2010 [8], Ghiani, Laporte and Musmanno, 2004 [11], Chungt, 1998 [12], Madadi, Kurz and Ashayeri, 2010 [6], Baumol and Vinod, 1970 [7], Edwin, Cheng, and Wang, 2007 [9], Strak and Pohet, 2010 [10] and Xiong and Helo, 2006 [13]). he papers published under the assumption that lead time is negligible and goods appear in the warehouse / fatory the moment orders are plaed, are still prominent in reent literature (Stadtler and Sahling, 2013). Suh an approah seems suitable for some partiular ompanies. Nevertheless the range of appliations of models based upon suh an assumption is narrower. In the most optimisti senario the lead time is fixed and known in advane. Although having a fixed lead time often simplifies the mathematial desription of the model the onsideration of a varying lead time makes the model more realisti and easier to apply in the industry. Ehrhardt, 1997 [14], for instane, onsidered varying lead time in his model. Deterioration of produts whether not onsidered, fixed or inreasing with time Not every produt is perishable, therefore there is not a need to onsider the deterioration of produts at all times. For instane the following authors developed the inventory models for non-perishable produts: Chakraborty and Roy (2005) [1], Konstantaras and Skouri (2010) [8], Ghiani, Laporte and Musmanno (2004) [11], Chungt (1998) [12], Madadi, Kurz and Ashayeri (2010) [6], Baumol and Vinod (1970) [7], Edwin, Cheng, and Wang (2007) [9], Strak and Pohet (2010) [10], Xiong and Helo (2006) [13], Ehrhardt (1997) [14]. However, if one wants to make the model more universal and appliable in the perishable and non-perishable industry, the deterioration of goods should be taken into aount. A fixed deterioration rate might be realisti for some produts. For instane all yoghurts produed on one partiular day will expire on one the same day. he appropriate models for suh a senario an be found in 35

48 Avinadav and Arponen (2009) [2], Chung and Liao (2009) [3], Hsu and Wen-Kai (2009) [4], Panda, Saha and Basu (2008) [5]. If the model allows for the deterioration rate to vary over time it makes the model more realisti for the same type of produts (for e.g. fruits, whih do not have a shelf-life). Suh models beome more frequently enountered in the literature: Sarkar and Sarkar (2013), Sett, Sarkar, and Goswami, (2012) or Sarkar, (2012). Number of produts onsidered he single-produt models, though relatively popular in the literature, suh as in the paper of Avinadav and Arponen (2009) [2], Chung and Liao (2009) [3], Chakraborty and Roy (2005) [1], Hsu and Wen-Kai (2009) [4], Panda, Saha and Basu (2008) [5], Madadi, Kurz and Ashayeri (2010) [6], Baumol and Vinod (1970) [7], Konstantaras and Skouri (2010) [8], Ghiani, Laporte and Musmanno (2004) [11], Chungt (1998) [12] and Ehrhardt (1997) [14], are sutable only for a limited number of industrial appliations. he multi-produt models, suh as those presented by Li, Edwin, Cheng, and Wang (2007) [9] Strak and Pohet (2010) [10] or Xiong and Helo (2006) [13], bring out the advantage of broader appliations in real life ase studies. Stati or dynami model he disadvantage of many models introdued in the literature is the fat that they ondut the optimisation for one period ahead only (Chakraborty and Roy (2005) [1], Avinadav and Arponen, 2009 [2], Chung and Liao, 2009 [3], Hsu and Wen-Kai, 2009 [4], Panda, Saha and Basu, 2008 [5], Konstantaras and Skouri, 2010 [8], Ghiani, Laporte and Musmanno, 2004 [11], Chungt, 1998 [12], Madadi, Kurz and Ashayeri, 2010 [6], Baumol and Vinod, 1970 [7], Edwin, Cheng, and Wang, 2007 [9], Strak and Pohet, 2010 [10] and Xiong and Helo, 2006 [13]). or find the optimal order quantities for several periods ahead based on initial information (Ehrhardt, 1997 [14]). he dynami models bring the advantage of onduting the on-line optimisation by updating urrent information at every time instane. hey are rarely found in mathematial optimisation literature: Hung, Chew, Lee and Liu (2012). Also, additional aspets to those mentioned an be found in the EOQ based models. Several models are inorporated within the delivery osts (e.g. (Madadi, Kurz and Ashayeri, 2010 [6]) 36

49 and (Baumol and Vinod, 1970 [7])). Konstantaras and Skouri (2010) [8], for instane, onsidered a reovery inventory remanufaturing system within the model. Li, Edwin, Cheng, and Wang (2007) [9], in turn, inorporated the postponement poliy. Strak and Pohet (2010) [10] onsidered a limited storage apaity. Laporte and Musmanno (2004) [11], onsidered not only the purhasing ost, but also the order rate osts. Chungt (1998) [12], in turn, developed the permissible delay in payment model. Considering the above listed riteria, the reality of the model an be assessed. For instane, the EOQ non-linear programming model developed by Avinadav and Arponen (2009) [2] an be assessed as follows. he inventory onsists of one produt type only but the deterioration of produt is onsidered as a fixed shelf-life period, the lead time is fixed, the demand is deterministi, expressed as a polynomial funtion. he order is plaed only when the need ours, whih means that the optimal order quantity might be equal to zero. It an be notied that the range of appliations in real life is dereased by the simplifiations of deterministi demand and single produt onsideration. A summary of the riteria in respet to the paper referenes an be found in able 2-2. he riteria have been split into two ategories: namely basi and advaned assumptions. he term basi assumption refers to those, whih derease the omplexity of mathematial model desription, hene simplify the reality, while the term advaned assumption refers to more realisti modelling, whih inreases the model desription omplexity. Basi assumption: Paper index: Advaned assumption: Paper index: One produt model: [1], [2], [3], [4], [5], Multiple produt [1],[2],[3], [4],[5],[6], [6], [7], [8], [9],[10], model: [7],[8], [9], [10], [11], [12], [13], [14] Deterministi demand: [1], [2], [3], [4], [5], Deterministi lead time: Deterioration of produts not onsidered: [6], [7], [8], [9],[10], [11], [12] [13],[14] [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] [1], [2], [3], [4],[5], [6], [7], [8], [9], [10], [11], [12], [13], [14] [11],[12], [13] [14], [1 Stohasti demand: [1],[2],[3], [4], [5], [6], [7], [8], [9], [10], [11],[12], [13], [14] Stohasti lead time: [1],[2],[3], [4],[5],[6], Deterioration of produts onsidered: [7], [8],[9],[10], [11], [12], [13], [14], [14] [1], [2], [3], [4], [5] [6], [7],[8],[9],[10], [11], [12],[13], [14], [1 37

50 Stati model: [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13] [14] Dynami model: [1],[2],[3], [4],[5],[6], [7], [8],[9],[10], [11], [12], [13], [14], [14] able 2-2: he auray of OR models applied in replenishment system optimisation Based on the presented examples drawn from the literature, it an be onluded that the simpliity of the mathematial desription prevails at the expense of the auray of the atual system representation. 2.3 Appliation of Control heory to Logistis he urrent hapter onentrates on those ontrol theory tehniques, whih have been found in the literature of logistis in general and inventory in partiular. It also justifies the appliability and advantages of ontrol theory to logistis and prodution systems. Finally the model preditive ontrol appliations in replenishment systems only are disussed separately (as shown in Figure 2-1). Although the fundamental onept in ontrol theory of a feedbak loop is a natural property of any real life system, it is not neessarily visible in non-engineering fields. Nevertheless, if attention is given, it an be realised that the feedbak loop exists in almost every environment, starting from biologial evolution, to human behaviour, to deision making to business development. It an be notied that every improvement is based on taking a orretive ation with respet to the urrent situation. he orretive ations (whih an be defined as system inputs for ontrol modelling purposes) are taken aording to the obtained outome (whih an be defined as a system output) and are often affeted by independent reasons (system disturbanes). herefore the urrent information is fed bak to the system for future deision making / adjustment, whih enables straightforward onsideration of system dynamis. Just as in the ase of pure engineering systems, where the error signal in the negative feedbak loop is used to redue the unstable flutuation of the output signal, the same an be done in nonengineering disiplines as well. It is just a matter of finding an appropriate mathematial representation of the system in the ontrol theory domain. herefore the inventory or prodution system an be represented as a feedbak loop too. One this representation is obtained, the appropriate ontrol an be applied to improve the overall system performane or 38

51 profitability. Control theory, in fat, offers a suffiient range of mathematial tehniques that failitate modelling and ontrol of inventory-prodution systems, whih makes it worthwhile fous. Utilisation of ontrol theory an be a major breakthrough for deision making within the dynami nature of the manufaturing supply hain industry. Figure 2-1: Narrowing the sope of the researh Historial review he setion herein provides a historial review of the first ontributions made in the appliation of ontol theory to logistis. he origins of appliation of regulation theory in the field of logisti systems ontrol an be dated bak to 1950 s, when Simon (1952) applied the 39

52 servomehanism algorithm to support the replenishment poliy in a ontinuous single produt inventory ontrol system. he representation of inventory features, suh as inventory level or order quantity, as system signals in the ontrol theory domain brought an immediate substantial advantage in realisti inventory system modelling. Nevertheless the model was ontinuous, whih makes its appliation somewhat limited. As a response, the disrete-time model with the appliation of servomehanism theory was presented by Vassian (1954) a few years later. he state spae representation as well as the blok diagram of an inventory system was introdued and ontrol engineering attributes suh as a transfer funtion, desribing dynamis of the stok levels, referene inventory signal and feedbak loop appeared in the above mentioned paper, to support the replenishment deision making proess. he inventory system stability and transient response were determined. his showed the potential power of appliation of ontrol theory tehniques in inventory system modelling. Both authors, Simon and Vassian had applied the lassi ontrol approah to the inventory problem for the first time. he next milestone in this field was ahieved by Christen and Brogan (1971), who foused on optimisation of the simulation model of a prodution system as an early ontribution to industrial ontrol, based on matrix analysis and the states of a system. he analytial approah to a weighting funtion, defined to obtain satisfatory results in different senarios, gained some attention in the literature. his approah was ontinued by several researhers and many tehniques have been developed. Among others the main ontributions were made by Mak, Bradshaw and Porter (1976) and Bradshaw and Porter (1974). Forrester (1958) paid partiular attention to flutuating behaviour of inventory levels at a supply hain s nodes, later alled the bullwhip effet, aused, among others, lead time lags and instabilities. He and his suessors (Roberts, 1978; Coyle, 1997) tended to ontrol the industrial dynamis based on simulation of equations of motion models. In his extensive study, Forrester (1958) showed adverse influenes of lead time delay to the sale of inventory level flutuation in the following periods and in different supply hain nodes. He disussed the dynamial behaviour of an industrial inventory system and suggested the appliation of feedbak as a ontrol. Sine that time several papers have been published whih have ontributed to the field to allow one to predit, analyse, measure and avoid the bullwhip effet using a regulation or ontrol theory approah. he first blok diagram representation of on Inventory and Order Based Prodution Control System (IOBPCS) model and its dynami 40

53 analysis was proposed in the early 1980s by owill (1982). he orders for the next period were assumed to be equal to the average orders plus a fration of the shortage in inventory. In fat, owill (1982) ombined the transfer funtion approah with a suboptimal tuning parameter for the feedbak gains. he demand was averaged as well, whih was not neessarily pratial. Sine then the blok diagram has been adopted by several researhers, see ((Wikner, owill and Naim, 1991), (Agrell and Wikner, 1996), (Grubbström and Wikner, 1996), (Samanta, and Al-Araimi., 2001), (Braun, et al 2003), (Gaalman, 2006), (Rodrigues and Boukas, 2006), (Hoberg, Bradley and honemann, 2007), (Zafra-Cabeza et al, 2007), (Venkateswaran, 2006), (Aggelogiannaki and Sarimveis, 2008), (Ignaiuk, and Bartoszewiz, 2011)) for further researh and development. In the above works a target inventory level beame to be modelled as an input to the system and treated as a referene signal, while the order quantity was taken as a manipulated variable. he atual inventory level was modelled as an output of the system and treated as a ontrolled variable to use for feedbak. Subsequently suh an approah to inventory system modelling has been applied by other researhers. Several tests were onduted for relevant design of inventory-prodution systems suh as stability, traking ability and/or noise rejetion. he IOBPCS model was subsequently extended and improved by adding more system omponents. Beside the already existing features, the new blok diagram omponents inluded: target inventory level, feedbak loop, the lead time delay, demand foreasting poliy and work in progress (WIP). For the appliation of this model the foreast of demand is required, whih is not always pratial. Among IOBPCS ontributions one an differentiate the ontinuous and periodi inventory level review approahes, see (Grubbström and Wikner, 1996). More details about the IOBPCS family and desription of IOBPCS omponents have been presented in (Lalwani, Disney and owill, 2006). Several different ontrol theory attributes whih have been used to model and analyse features of inventory-prodution systems an be found in the literature over the last deade. Important ontributions to this sope of study have been done at Cardiff University, UK, with respet to ontrol of the bullwhip effet ((Dejonkheere, et al., 2002, 2003, 2004), (Disney and owill, 1996, 2002, 2003a, 2003b, 2006), (Gaalmanand and Disney, 2006), (Lalwani, Disney and owill, 2006), (Potter et al., 2009), (Zhou, Disney and owill, 2010)). In the onsidered papers, the researhers aimed at smoothing the ordering poliies as well as inventory levels and 41

54 presented the suitability of utilisation of ontrol theory tools in terms of preventing the bullwhip system osillations. he examples of appliations of ontrol theory attributed to inventory systems whih an be found in the literature are as follows. Autoregressive moving average (ARMA) system struture has been used by several researhers for different purposes. Gaalman (2006) and Gaalman and Disney (2006) used an ARMA system struture to model unertain omponents of demand. Aggelogiannaki, Doganis and Sarimveis (2008), however, used an ARMA struture to model inventory position and a reursive least square (RLS) estimation in terms of demand foreasting. A state spae representation, in turn, has been used for instane by Gaalman (2006), for demand modelling and Rodrigues and Boukas (2006) for stok aumulation representation. A differential equation model approah has been used by Ignaiuk and Bartoszewiz (2011, 2010b) as a stok balane equation. he transfer funtion model has been ommonly applied by different researhers. For instane Dejonkheereet. al. (2003, 2004) used it for order-up-to poliy establishing with respet to the prevention of the bullwhip. Hoberg, Bradle and honemann (2007) formulated the transfer funtion of an inventory system for evaluation of the ordering signal stability with respet to different lead time delay values. Lin et al. (2004) presented a ombined losed-loop transfer funtion representing material balane and information flow of a whole supply hain network. he ontrollability and observability tests of supply hain systems an be found in a paper of Lalwani, Disney and owill (2006). Dejonkheere et al (2003, 2004) introdued a damping fator for smoothing order poliy and spetral analysis to obtain demand patterns and frequeny response of a sinusoid demand. Gaalman and Disney (2006) applied a proportional ontroller in the inventory feedbak loop and desribed the proess of tuning it to prevent the bullwhip effet, while Grubbström and Wikner (1996) as well as Samanta and Al-Araimi (2001) applied a PID ontroller and ombined it with fuzzy logi to maintain the stok at target level. Also estimation tehniques an be identified in inventory management literature. he RLS method and Kalman filter, have been used by Aggelogiannaki and Sarimveis (2008) and Aggelogiannaki, Doganis and Sarimveis (2008) for lead time identifiation and by Gaalman (2006) and Gaalman and Disney (2006) for demand foreasting, respetively. A ontribution to non-zero lead time modelling and ompensation in periodi review systems has reently been reported by Ignaiuk and Bartoszewiz in (2010a, 2010, and 2010d). In these papers the lead 42

55 time delay has been taken into aount in the n-th order state matrix and the optimal ontrol ation has been found by minimisation of a quadrati performane index Model preditive approah Beside lassial ontrol tehniques, whih have been mentioned in the previous setion the Model Preditive Control (MPC) has been also used as an optimisation tool by several researhers. he approah brings several advantages to inventory ontrol. he MPC, being a moving horizon ontrol theory tehnique, aims at finding the urrent and future ontrol ations in the desired optimisation horizon, by on-line optimisation of the problem. he MPC approah then applies the first ontrol ation only. he system dynamis is updated at eah sampling instant. It means that the feedbak gains are updated aording to the urrent situation and an be ontinually adjusted for non-linear models. Perea-Lopez, Ydstie and Grossmann (2003) developed a dynami deision framework for a multi-produt, multi-ehelon supply hain. he supply hain model inludes plant, warehouses, distribution entres and retail levels. he MPC tehnique was applied to maximise the profit by redution of the negative impat of unknown demand, onsidered as a system disturbane. he demand predition is assumed to be known in advane and used by the model. Nevertheless, the demand error is regularly updated based upon past and urrent information. Upstream orders are inputs of a partiular ehelon, while shipments represent the ehelon outputs. he orders are transferred from upstream to downstream ehelons, while the shipment is moving the opposite diretion. Nodes are assumed to handle as many produts as the whole system is allowed to handle. he model allows for onsideration of many produts by splitting eah node to one produt division. he reeived orders from the downstream nodes are aumulated during the day and shipped the day after, unless the inventory level is too low to satisfy the ustomer s requirements. Any kind of transportation, suh as shipment from downstream to upstream nodes, delivery from external suppliers or shipment of goods to the end ustomers, is ompleted at the end of the day when the whole day s orders are aumulated. In the model it is represented as an additional term whih usually is equal to zero beside the irumstanes when the time is equal to the partiular value representing the end of the day. 43

56 he transportation times and their osts between nodes are known with ertainty. he authors onsider two different raw material supply possibilities: the quik and ostly response and the slow and eonomi response. he availability of raw soures is assumed to be infinite. he model objetive funtion is related to the overall net profit and ontains all ost and gross profits in the supply hain related to the prodution proess, storage, transportation and sales. In the onsidered paper two optimal deision making approahes are examined, the entralised and the deentralised. he results showed that the entralised senario leads the supply hain overall profit to be higher than in the ase of the deentralised senario. Braunetal (2003) developed a deision support system for a single produt, six-node and three-ehelon prodution-inventory system. he disrete MPC aims at finding the optimal order quantities (system input) for redued inventory levels. he demand (disturbane) predition is assumed to be known in advane and it is updated based on a real demand pattern. he referene signal of the model is assumed to be equal to the predited demand pattern inreased by a safety stok level. he estimated order pattern (the predited system input) is used as a predited demand pattern (disturbane) for downstream ehelon. he atual urrent value of disturbane is updated at eah step. Goods posted to ustomers are understood as system outputs. he author proposed semi-deentralised deision making system. he separate model preditive ontrollers are used for eah of the ehelons so that they are shared between nodes inluded in eah partiular ehelon. he foreast information is used by the downstream nodes. he transportation times between nodes are assumed to be known with ertainty. he more time onsuming routes are hosen only when neessary. It is ahieved by setting the target order value to zero and applying different penalties for different ordering routes in ase the orders plaed for a partiular route are greater than zero. he daily shipment apaities are onstrained as well. he bakorders are onsidered in the model. he time unit in the developed model is equal to one day, whih enables modelling of lead time as a system delay. he model has been tested for different knowledge sharing strategies. It was shown that sharing the foreasted demand among all the nodes and suppressing the real demand pattern is benefiial to the ompany in order to ahieve smoother order patterns, lower inventory levels and prevent inventory level flutuations. Wang, Rivera and Kempf (2007) presented the effetiveness of an MPC algorithm in strategi deision making in a semiondutor manufaturing proess with respet to the system making sudden hanges. In this ase the MPC is only an element of a omprehensive deision 44

57 making poliy for a single produt, single line and multi ehelon manufaturing supply hain, where a fluid analogy is applied to illustrate the flow of materials. Here work in progress with respet to a manufaturing proess is understood as flow of fluid in a pipe, while storage areas aumulating goods between manufaturing proesses are treated as tanks. With respet to a ontrol engineering representation of the developed model, the inventory levels are understood as system outputs and ontrolled variables, the orders are treated as first system input (manipulated variable) and demand is treated as a seond input (disturbane signal). Several onstraints, represented as linear equations, are taken into aount for optimal deision making. hese are: prodution and storage apaities, magnitude of starts, inventory levels, manipulated variable onstraints, ontrol variable onstraints and work in progress apaity. he aim of MPC appliation in this ase is to maintain the inventory level at a desired set point and satisfy the ustomers requirements at the same time. Some unertainties are taken into aount in the model suh as random breakdowns or mistakes of mahines, whih affet the lead time and storage levels. herefore, the lead time is never known with ertainty. o optimise the prodution sheduling, different speeds of assembling mahines are onsidered and used in the model. he demand predition is used and assumed to be similar to the average value of a real demand pattern. he atual demand is regularly updated. zafestas, Kapsiotis and Kyriannakis (1997) presented a MPC appliation for prodution planning for multi-produt manufaturing systems. he aim was to minimise the total ost of prodution and advertising so that the sales and inventory are maintained at desired levels. Selling pries are assumed to be fixed. In this ase the prodution rate and advertising effort are manipulated variables (system inputs) while inventory and sales levels are system outputs (ontrolled variables). he demand (system disturbane) is ontrolled by an advertising parameter, whih allows for demand predition. he ontrol variables have been onstrained in the model. he paper presents the general idea of the developed model only. he applied ase study is not explained in detail and the struture of studied system is not presented. he simulation results show that after some time the model tunes to ahieve satisfying outomes. Li and Marlin (2009) presented the MPC deision framework to minimise the total supply hain osts with respet to storage osts, manufaturing osts, transportation osts and penalty bakorders osts and at the same time to satisfy ustomers requirements. he manufaturing rate, of semi-finished produts, plant running time and transportation rates are model manipulated variables. Final produt manufaturing rate, lead time and demand are assumed to 45

58 be unertain, whih require a orrelated unertainty desription. In simulation, two senarios were examined: the ase when demand, lead time and manufaturing rate are predited orretly and the ase when the predition is not exat. he model performs very well in the first ase while in the seond ase bakorders our. he reason for this is that the model tends to redue inventory level to minimise osts and the inventory level is not always enough for the ustomers requirements. Consideration of additional safety stok level inreased total osts but prevented bakorders at the same time. Aggelogiannaki, Doganis and Sarimveis (2008) developed a MPC framework for optimisation of order quantities for prodution systems with onsideration of system dynamis. Besides unknown demand, the model s unexpeted behaviour was related to breakdowns of mahines or running out of materials. he adaptive Finite Impulse Response (FIR) model has been applied to approximate the prodution system s dynamis. he output of the FIR system is prodution volume while the FIR system input is an order quantity. he RLS algorithm, as an on-line estimation tehnique has been used to estimate FIR model oeffiients, whih hange over time. he inventory balane equation is represented by an autoregressive with exogenous input model (ARX), where demand represents system disturbane, inventory level represents system output and ontrolled variable and orders volume represent system input and manipulated variable. he MPC framework employs the objetive funtion, whih aims at maintaining a target inventory level. In the numerial example the authors ompare the adaptive MPC framework with the Estimated Pipeline Inventory and Order Based Prodution Control System (EPIOBPCS) of Disney and owill (2003b) and with non-adaptive MPC. It was notied that the adaptive MPC is able to respond faster than EPIOBPCS and also avoids osillations, whih our in ase of non-adaptive MPC. herefore the appliation of adaptive MPC has been justified as an advantage. 46

59 2.4 Summary and gap identifiation he urrent hapter has familiarised the reader with the aademi approah to the logisti problem modelling and optimisation. It has presented an overview of the methods applied in different logisti proesses but fouses partiularly on the inventory problem. From the literature review it an found that there are two shools of inventory optimisation. he purely mathematial or business simulation based approah of the operational researh ommunity and the approah of the ontrol theory ommunity. he typial OR tehniques often do not enable onsideration of system dynamis due to inreased omplexity of the model. It indiates that the appliation of ontrol theory to inventory is a reasonable path to be followed. It an be notied that the appliation of ontrol theory to inventory optimisation, though not a new idea, has not gained muh attention yet, while the pure OR tehniques have been exhaustively battled to death in literature. he sophistiated tehniques of ontrol theory, though benefiial in appliation, might seem deterrent to be broadly applied by the OR ommunity. As a response, this thesis aims firstly at informing both ommunities of the possible benefits of ollaboration. Seondly it aims at bridging the gap and making the ontrol tool of MPC available for the OR audiene. he novel method an be easily adopted by other than ontrol engineering researhers. herefore, the gap has been identified and the researh aims at exploring its rihness and goes some way towards bridging this gap. 47

60 3 PRELIMINARY MODELLING AND SIMULAION MODELS 3.1 Introdution he urrent researh onerns an inventory replenishment system ost redution, whih was deided to be ahieved by maintaining the inventory at a desired level (the benefits of keeping inventory level at a referene point and resulting ost redution have been disussed in Setion 0). he urrent hapter presents the preliminary modelling of the inventory problem and the simulation models. It starts with elaborating the assumptions whih have been used to build the inventory model (see Setion 3.2.1). hen the proess of translation of a oneptual model to the ontrol theory domain is shown (see Setion 3.2.2). A state spae representation of the system (Setion 3.2.3) failitates onsideration of the system dynamis (lead time delay, unknown demand and deterioration of produts). he feedbak loop enables an updating of the urrent inventory level on a sampled time instane basis and omparing it with the referene point. Further, several ontrol tehniques have been applied for the established model (see Setions ). A dead beat ontroller enables the mathematial verifiation of the state spae model auray with the atual system. A speial fous, however, has been devoted to the model preditive ontrol tehnique (Setion 3.3) as it yield an inspiration to develop a novel method within this researh. Finally, the novel tehnique of the inventory ontroller is presented in Setion 3.4. Enabling the operational researh (OR) speialists to use the method without the neessity of familiarising themselves with ontrol theory priniples is one of the main attribute of this researh. Equation Chapter (Next) Setion Inventory Modelling Assumptions In the onsidered inventory system (or distribution entre) the ustomer demand is assumed to be prior unknown. he goods that are needed to fulfil the ustomer s demands are ordered from the remote supplier with a ertain delay. Suh a model is usually onsidered in 48

61 inventory-prodution systems where the inventory model is a link between two other supply hain nodes suh as a raw material supplier and fatory or fatory and wholesaler or wholesaler and retailer. he model allows for both senarios - of onsideration and for nononsideration of perishable goods. he warehouse is assumed to be initially empty, as it is a ommon pratie in OR literature, and a single supplier ase is onsidered Converting the inventory problem to the ontrol domain he urrent setion presents a way of mapping of the exemplary inventory replenishment oneptual system into the ontrol theory domain. o illustrate the transformation, Figure 3-1 shows an example of how the inventory (operational researh) problem an be transformed into a ontrol theory sheme. From Figure 3-1 it an be seen that the same system elements are labelled and understood in different ways within eah domain. Demand DISURBANCE Deision making unit Lead time warehouse Referene inventory CONROLLER DELAY SYSEM Figure 3-1: ransforming the inventory problem to a ontrol sheme Now, the initial problem of finding the optimal order quantities (u ) to maintain an appropriate inventory level ( y ) with onsideration of lead time ( Ls e ) and varying demand ( d ) turns into the following formulation: designing a ontroller to find an optimal system input (u ) to maintain an appropriate system output ( y ) with onsideration of system time delays ( Ls e ) and varying disturbanes ( d ). Although in ontrol theory literature, the disturbane is usually denoted as a positive signal, here it is negative, as it represents 49

62 ustomer demand whih due to the balane equation (3.1) must be subtrated from the inventory Inventory state spae representation he urrent setion presents the initial model of the distribution entre. he ontinuous state spae representation refers to a balane equation of the inventory level. t xt x t x t u t L d t y t (3.1) where xt () is the urrent inventory level (system state) ut () is the urrent order quantity (system input) dt () is the urrent demand, bakorders are allowed (system disturbane) yt () is the urrent inventory level (system output) L 0 is a lead time, the time needed for goods to be reeived in the warehouse after an order has been plaed (system delay) t0,1 is the urrent goods survival rate and 1 t refers to a time varying deterioration rate. he variation an be for instane dependent on the inventory level, suh that t y() t. In any ase it must satisfy t t 1. If t he ondition of 0 1 and if bakorders our ( yt ( ) 0) then is set to 1, the model does not allow for onsideration of perishable goods. t always being greater than zero assures the survival of some goods in the warehouse, where t denotes a time, and the first order is allowed to be plaed at t 0, x0 0 and y

63 Figure 3-2 shows the open loop system blok diagram with the internal loop orresponding to stok aumulation. - + integrator + Figure 3-2: Open loop inventory system (warehouse) blok diagram inuding stok deteriration he next step refers to building the ontroller whih will aim to maintain the inventory level at the referene level and prevent the system osillations whih are related to the lead time delay. 51

64 Smith Preditor Deision Maker Beause of the lead time delay, if the Smith preditor is not used, there would appear osillations in the system, whih in operational researh terms orresponds to the bullwhip effet. he typial Smith preditor onfiguration is presented in Figure 3-3 Mathematially it does not matter if and are interhaned. + u(t) Figure 3-3: ypial Smith preditor onfiguration In this thesis an alternative onfiguration of the Smith preditor (based on the non-perishable goods ase, and developed in Ignaiuk and Bartoszewiz, 2010b) was applied in the form presented in the blok diagram of Figure 3-4. Note that in the non-perishable good ase 1 t is zero, i.e. t 1. 52

65 Warehouse defiit (with Desired inventory level bakorders) + + Number of ordered goods / ontrol ation u(t) Lead time Demand u(t L) Integrator Limited Current inventory level + Warehouse Estimated warehouse model Figure 3-4: Smith preditor deision maker for inventory systems (blok diagram) he Figure 3-4 shows an alternative formulation of the Smith preditor, onfigured within a unity feedbak losed loop system, where the feedbak loop well is understood by the OR ommunity and the Smith preditor is viewed as deision maker. he warehouse defiit is the Order signal whih feeds into to the deision maker. In the ontrol engineering sense the ontrol ation, goods ordered, would be the output of the ontroller. he transfer funtion C SP has the form as follows where C SP s 53 C s (3.2) 1 C s G s 1 Ls e Cs is a seleted proportional ontroller (i.e. gain onstant) and t 1 (1 ), with t G s s 0 1 C s K, where K is a positive

66 In inventory ontrol the transfer funtion Gs () orresponds to the warehouse, where the future inventory level beomes equal to urrent inventory level dereased by the deterioration value inreases by the number of delivered goods with a delay ( u( t L) ) and dereased by number of demanded goods ( dt). () he Smith preditor deision maker is inluded to show the omplexity of the system, whih should be ideally hidden from the OR ommunity. It is elaborated here and in Chapter 4, Setion 4.3, that the appliation of the Smith preditor as a deision maker is adequate but not ideal due to the perishable nature of the goods. he Smith preditor deision maker in this ase is not tuned for the effiient handling of perishable goods (i.e. 1). Exluding the ase 1 or inluding an integrator in Cs would improve the effetiveness of the method. However it would require a deep understanding of ontrol the by the OR user, whih is in opposition to the assumption driving the urrent researh. As stated in this thesis the OR ommunity requires simpler approahes to be developed, whih does not need a deep knowledge of ontrol to be applied Disrete state spae representation he above model assumes that the order quantity as well as the inventory levels are ontinuous, whih is not a neessarily realisti assumption. he urrent setion details the disrete state-spae representation. In this formulation the lead time is taken into aount and represented as a system delay within a state matrix. he inventory replenishment has been assumed to be arried out at regular intervals k, where is a review period and k 0,1, 2 herefore, the lead time is assumed to be an integer multiple of the review period (sampling interval). Consider the following state-spae representation for the inventory system k 1 k uk d k x Ax b v (3.3) k y k x (3.4) 54

67 Where L n is a system order and lead time delay n x k is the state vetor at time instane k uk is the system input, whih represents the order quantity at time instane k n yk is the system output, whih represents the urrent inventory level at time instane k d k is a system disturbane, whih represents demand (bakorders are allowed) at time instane k n n A is the system state transition matrix onsidering the lead time delay, and the time varying deterioration of goods denoted as 1 k 0,1 vetors, suh that and n b, n, n v are ( k ) A, b , v 0 1 (3.5) Note that the matrix A is always the same struture and depends only on the deterioration rate value and its dimension is diretly related to the lead time. Should lead time flutuate, so the dimension will flutuate. he form of the vetor v in (3.5) assures the subtration of the demand from the urrent inventory level. It is assumed that the inventory level yk an be lower than zero, whih orresponds to the allowane of bakorders. he deterioration rate k represents to the fration of the total number of goods in the warehouse in the urrent period, whih will be left 55

68 (i.e., will survive) in the warehouses, until the next period. he remaining fration1 ( k), then, is the number of goods whih atually deteriorates within suh a period. he deterioration rate is assumed to be time varying and for some appliations it an be reasonable to be dependent on ( ) y k, the urrent inventory level, suh that k yk makes the A matrix inventory level dependent. Defining yk. It as a linear or non-linear monotonially dereasing or non-inreasing funtion, it may be dedued that the goods are more likely to deteriorate in the warehouse if the number of goods inreases. Regardless of the deterioration rate definition, it must be stressed that for any natural number k, for the deterioration rate suh that 0 k 1 if bakorders our ( yk 0) then k 1 he latter ondition prevents the goods to deteriorate when bakorders our.. Essentially, if k is set to 1, the model does not allow for the onsideration of perishable goods. he ondition of k always being greater than zero assures the survival of some goods in the warehouse. herefore x1 ( k ) x1 0 1 x x x x [( k 1) ] ( k ) u( k ) d( k ) x n x n1 0 0 xn xn 1 0 y k x1 x 2 x x n1 xn 3 ( ) ( k ) (3.6) (3.7) 56

69 where xi ( k ), for i1, 2,3 n represents the quantity being ordered at time instane k n i 1 herefore n x1 k k x1 k x2 k d k x2 k x3 k x3 k x4 k 1 n 1 x k x k xn k u k (3.8) his leads to the representation as follows, where the first line refers to the inventory balane equation of the form: the future inventory level is equal to urrent level inreased by the number of goods arriving to the warehouse at the urrent time instane and dereased by the urrent sale/demand. 1 [ 1) x1 k k y k u k n d k x2 k u k n x3 k u k n (3.9) xn 1 k u k xn k u k From equations (3.9) it an be dedued that the orders reah the warehouse with a delay of n 1 periods. 57

70 he aim of this work is to maintain the inventory at the desired level with the assumption that the warehouse is initially empty ( x 0 0). o do so, the referene inventory level signal, denoted x R, is defined suh that x r 0 0 x R (3.10) A stepping stone he urrent setion is shown to verify the orretness of disrete inventory model formulation. It verified by identifying what input signal must be applied to a system in order to eliminate the error e xr x in the smallest number of time steps (whih in fat refers to dead-beat ontroller). In this thesis the presentation of appliation of Dead-beat ontroller and its simulation results do not ontribute to the leading thought. herefore the urrent setion is shown as a stepping stone to further mathematial formulations rather than to show preliminary formulation of one of tested algorithms. Define the ontrol vetor f x x 1 r f1 f2 f n, whih will define the number of ordering items in period k suh that uk f e (3.11) It an be dedued that u k f f f xr x1 x 2 x n (3.12) x x n1 n 58

71 he solution is to apply feedbak suh that all poles of the system are at the origin of the omplex z-plane. Substituting (3.11) to (3.3) the losed loop system is obtained: k 1 k k d k x Ax bf xr x v A bf x k bf xr vd k (3.13) An illustration of the losed loop system is given in Figure integrator Figure 3-5: Dead-beat ontroller blok diagram Defining A Abf and b bf (3.14) it an be dedued that 59

72 ( k ) A (3.15) f1 f2 f3 f4 fn b (3.16) f1 f2 f3 f4 f n So that k 1 k d k x A x b xr v (3.17) o plae all system poles at the origin of the z-plane, the eigenvalues of the matrix be equal to zero. o ahieve this one onsiders the harateristi equation of the matrix: A must all 0 det A I (3.18) herefore from the above and (3.15) it an be dedued that ( k ) det f1 f2 f3 f4 f n (3.19) Expanding results in the expression n 1 i n i 2 n 1 f f k k 1 0 (3.20) i2 60

73 Further expansion of equation (3.20) it is possible to obtain the oeffiients of the vetor f : n 1 1 i n i2 n1 f f k k i2 n 1 1 i n i1 i2 n1 f f k fi k i2 n 1 f f k f f k f f k f fn k fn k n n1 n2 n n 2 1 f k f f k f f k f n n n n fn 1 k fn fn k 0 (3.21) Sine the steady state response is required, the poles must be settled at the origin, therefore the eigenvalues are assumed to be zeros. herefore f n k and f 1 k f 2 f 2 k f 3 (3.22) f k f 3 4 f n1 k f n 61

74 herefore f n k f n1 k 2... f 3 k n2 (3.23) f 2 k n1 f 1 k n Finally n n1 k k k f (3.24) herefore substituting (3.23) into (3.12) it an be dedued that xr x1 x 2 n n1 x 3 u k k k k x n1 xn whih results in n n n1 u k k x k x k k x k r 1 2 n2 k x k k x k 3 n (3.25) From (3.9) 62

75 1 ) 1 x1 k k x1 k u k n d k 1 x2 k u k n 2 x3 k u k n... (3.26) 2 xn 1 k u k 1 xn k u k From (3.7), and assuming that deterioration is dependent on inventory level suh that k yk and substituting into (3.25) leads to n n1 n 1 ) u k y k x y k y k y k u k n r n1 n2 1 2 y k u k n y k u k n (3.27) n... y k u k 1 y k d k 1 Equation (3.27) enables a one by one onsideration of the few initial order quantities to be obtained with the proposed approah mainly for verifiation that the results math the logial expetation. herefore, it may be dedued as it will beome lear in the following, that if the initial state spae representation is formulated orretly for the appliation, the dead beat ontroller ahieves its goal. Initially the warehouse is assumed to be empty ( time instanes 0 k ( reahes the warehouse ( 0 y k 0 0), no orders are plaed for u k 0 0) and no goods are demanded before the first order d k n ). Considering the above assumptions from (3.27) the following may be dedued: Let a and b denote two arbitrary time instanes, then for k 1 it an be notied that 63

76 y k 1 0, and for any system dimension n b n a 1, a n b u k a) 0 and d k 1 0, therefore, realling the assumption that deterioration must be set to 1 for inventory less than or equal to zero n n and 0 1 u k y k x y k (3.28) r As if there are zero inventories, the goods annot deteriorate, therefore from (3.28) eventually xr u k, for k 1 (3.29) whih is logial as the warehouse is assumed to be initially empty and the first order is equal to the inventory target level. Further for k 2, if n 2 y k 1 0, sine the ordered goods in time step k 1 have not yet reahed the warehouse, u k 1 xr, from (3.29) and from the assumptions b n a 1, a n b u k a) 0, therefore n and u k y k xr y k u k 1 y k 1 d k 1 0 and (3.30) u k x r x r herefore in the onsidered time instanes uk 0 (3.31) whih reflets the expetation, sine until the first order reahes the warehouse, and demand is zero, there is no need to plae new orders. Analogously for any k, suh that 1k n 64

77 y k 1 0, for suh b that k b 1, a k 1 u k a) 0 herefore for some d k 1 0 u k b x, and r n b 1 u k y k x y u k b and y k (3.32) r sine the warehouse remains empty until the first order reahes the warehouse at the time instane k n. herefore uk 0 (3.33) whih means that there is no order plaed until the first order reahes the warehouse, whih would appear to be logial again. Further for k n 1 it an be observed that y k 1 0, as the first order reahed the warehouse in the urrent time instane only 1 r, ) 0. Also a demand greater than or equal and u k n x a n u k a to zero appears. herefore, n n n 1 u k y k x y k y k n y k d k (3.34) r If yk 1, 1 u k d k (3.35) whih means that the size of the orders plaed is always equal to the amount of goods whih have been sold one period bakward. Otherwise n 1 u k y k d k (3.36) 65

78 whih means that the number of goods hosen by the ontroller to be ordered is smaller than the number of goods demanded. his means that the demand might not be satisfied, depending on the hosen referene inventory level. From the above reasoning, it an be notied that, although, the dead-beat ontroller is hosen as an optimal order quantity finder, it may not be the most profitable in the ase of system time delay. Nevertheless, through mathematial reformulation, its onstrution has highlighted, that the inventory state spae model is designed orretly, whih was indeed the purpose of the verifiation. 66

79 3.3 Model-based Preditive Control Approah Defining k1 x suh that from (3.3) it follows that where k 1 k 1 k x x x (3.37) x k 1 A x k x k 1 b u k u k 1 vq k (3.38) Defining u k, suh that from (3.37), (3.39) and (3.40) it follows 1 q k d k d k (3.39) 1 u k u k u k (3.40) k 1 k uk k x A x b vq (3.41) o reate the losed loop system, x as a new state vetor is defined suh that From (3.4) and (3.41) it follows that k k yk x x (3.42) 67

80 1 1 y k y k x k A x k b u k vq k (3.43) so that (3.42) and (3.43) leads to new state equation k 1 k uk k x A x b vq (3.44) k y k x (3.45) where A O b v A,, 1 b v A b v 1n 1 ( n1) ( n1) n1 n1 (3.46) and n1 0 I (3.47) 1n where n1 01 ( 1) 0000 (3.48) n he matries in (3.46) represent the augmented model. he predition model uses the objetive funtion of the quadrati form to optimise the future order quantities within the optimisation horizon so that the inventory ahieves the referene level. he objetive funtion has the form as follows J YR Y YR Y U U R min (3.49) where Y R is a referene inventory level vetor. Defining N as the ontrol horizon and R = I and u 0 as a tuning parameter, u r N N p r U is a ontrol trajetory vetor of future orders suh that N p as the predition horizon, suh as N N, U u( k ) u[( k 1) ] u[( k 2) ] u[( k N 1) ] (3.50) p 68

81 whereu k i for i 0,1, N 1, are future order ontrol signal values to be optimised. Y is a predited inventory level signal represented by the vetor p Y y k k y k k y k k y k N k (3.51) where yk i k is the i-step ahead predition of the inventory level vetor. Y R defines the target inventory level signal. Assuming that Q k k k k k N q q q q (3.52) is a zero-mean white noise sequene, the predited inventory level variables an be represented by the matrix form Y Fx k ΦU (3.53) where A 2 A 3 F A N p A and b A b b Φ A b Ab b... 0 N p-1 N p-2 N p-3 N p-n A b A b A b... A b (3.54) From the above desription of F and Φ it an be notied that Y shown in (3.53) is already extensive in desription and time onsuming in alulation. 69

82 he MPC approah uses past and urrent information to predit the future inventory levels. he optimisation of order quantities is arried out over a fixed predition horizon N p. From (3.50) and (3.53) the ontrol order quantities ΔU an be derived suh that Assuming that 1 R k U Φ Φ + R Φ Y Φ Fx (3.55) N y k y k Y Y (3.56) R R R R where y R is a referene inventory level, this gives the optimal order quantity vetor for the replenishment inventory. If not, attention is given to the size of orders during the optimisation proess, and, as a result ur 0, then the optimal U for the ontrol order signal is obtained as 1 RyR k k U Φ Φ Φ Y Φ Fx (3.57) or 1 RyR k k U Φ Φ Φ Y Φ Fx (3.58) if ur 0. (3.59) Beause of the MPC priniple the first element of U is applied only, suh that N R R u k Φ Φ Φ Y y k Φ F k (3.60) N or R R u k Φ Φ + R Φ Y y k Φ F k (3.61) is the atual value reommended to be ordered by the warehouse manager at time instane k where 70

83 Denoting u k K y k K x k (3.62) y R suh that K is a first element of 1 Φ Φ Φ Y or 1 y R Φ Φ + R Φ Y R and if ur 0 NpN K Φ Φ Φ F K and K xk y (3.63) or N p N K Φ Φ + R Φ F K and K xk y (3.64) if ur 0. It an be seen that alulation of the order size whih should be ordered at time instane k desribed in the manner shown in (3.37)-(3.64), requires an understanding of ontrol theory priniples and might not be easily adoptable by non-ontrol-engineering researhers and pratitioners. In the thesis initially the tuning parameter is initially negleted, as the smoothness of order quantities is not in a onern of the researh. 71

84 3.4 Novel Inventory Controller he urrent setion introdues a novel ontribution to inventory ontrol. he developed ontroller is non-model based yet adaptive and simple in implementation with low omputational osts. As previously mentioned it an be applied with only a minimum of ontrol theory knowledge. It is mathematially equivalent to MPC, applied for the defined inventory model as shown in equations (3.3)-(3.10). MPC applied for that partiular inventory state spae model will be termed the Inventory model preditive ontrol (IMPC) in this thesis, while the developed novel tehnique will be termed the inventory ontroller (IC). Only the formulation is presented in this setion, further hapters show the development proess, the mathematial equivaleny and benhmarking of the new method against IMPC. Being straightforward in appliation, the IC attempts to bridge the gap between the preision afforded by methods of ontrol theory and the expetations of the OR ommunity. Finally, the developed model, being appliable with a minimum of ontrol theory knowledge, appears to be appliable not only in inventory optimisation, but also in other non-engineering appliations, suh as prodution planning, plant or animal ulture predition, property insurane profit optimisation, extraurriular ourses/lasses planning, food restaurant management and many others. he IC requires the same assumptions as IMPC, of lead time delay between the moment the order is plaed and delivery. he inventory level is reviewed and orders are plaed periodially, the tehnique is appliable for a single supplier ase where demand is assumed to be unknown and the warehouse to be initially empty. he goods are assumed to deteriorate aording to a time-varying deterioration rate. he developed IC is defined in the form of the proposition, the origins of the IC proposition within IMPC are elaborated gradually in the following setions. 72

85 IC Proposition: Denoting x as vetor of the form x ( k) = I( k ) I[( k 1) ] u[( k n) ] - u[( k n 2) ] u[( k n 2) ] - u[( k n 1) ]... u( k ) - u[( k 1) ] I( k ) (3.65) where k represents the and u k defines the th k time instane, I k represents the th k time instane order size, and denoting IR th k time instane stok level k as the referene inventory level at time instane k, k as the stok deterioration rate in time instane k and n as the lead time delay, K an be defined as the transposed vetor of n 1 dimension, where r denotes the gain vetor K olumn number suh that 1 1 K K K r K n. hen the following an be formulated: n r1 nr1 i i and 1 K r k K k (3.66) i0 i1 For suh a formulation of vetor K, the urrent optimal order quantity an be found as follows: n u k Kx 0 if u k 1 IR k k 0 u k 1 IR k Kx k otherwise (3.67) whih defines the proposed inventory ontroller (IC). he above proposition uses the pre-defined 73 x vetor of past information. It proposes a method of alulating the gain vetor K and indiates how to use the appropriate information in alulating non-negative optimal order quantities. he advantage of the proposed method is the fat that the alulation is done on-line, separately for eah time instane. herefore the time-varying deterioration rate k an be deided separately for eah time instane, also the lead time an be set differently for eah time

86 instane. Moreover, the deterioration rate in suh a ase an be judged to be dependent on the urrent inventory level I k, whih for some appliations might be more realisti (for instane the higher the inventory level, the more hanes of deterioration of produts). 1 I( k ) I k, where model, the I k 1 (0, ) ). Sine bakorders are allowed in the I max approah enables the deterioration rate to be set bak to 1, when the inventory level goes below zero. herefore the non-realisti assumption, that the produt an still deteriorate if their assoiated value is lower than zero, is avoided. 3.5 Novel Method Properties It an be realised that although the novel tehnique was inspired by MPC and notiing the MPC patterns in the mathematial desription, the resulting proposal is no longer MPC. It is in fat a minimum variane approah, or dear beat ontroller, with an inorporated Smith preditor. Note that using the adjetive preditive in the title of the thesis refers to both, the inspiration of MPC and the preditive nature of the minimum variane ontroller to aommodate lead time, being inorporated within an inherent Smith preditor. It an be notied that the vetor x ( k) of the form shown in (3.65) is the MPC augmented model state vetor. In the OR perspetive though, it is a past and urrent information vetor of inventory levels and order quantities. he ontrol gain vetor K of the form elaborated in (3.66) is a mathematially redued form of the MPC gain. he mathematial redution of this ontrol vetor is one of the main ontributions of this thesis, and Chapter 5 is dediated to the mathematial demonstration. Here, from an OR point of the view, vetor K an be found aording to the straightforward proedure desribed in (3.66) and does not need to be related with the MPC gain to be understood or applied. he proedure for obtaining optimal order quantity u k, shown in (3.67) is in fat the MPC system input or the inremental ontrol input shown in (3.40) of the form as in (3.62) bounded by a saturation ondition. he OR user does not need to be onerned by the above fat as long as the user follows the reipe desribed in IC Proposition. 74

87 Chapter 4, Setion 4.5 and Chapter 6 present the model response to different demand patterns, different deterioration rates and different lead time delays. 3.6 Novel Methods vs. Smith Preditor, Dead Beat Controller and MPC Novel method he novel method an be used by anyone familiar with adding, subtrating and multiplying vetors, therefore it an be expeted to be usable by the OR ommunity. he novel algorithm is designed to be used on a disrete time basis. Moreover no parameter tuning is relevant in the method, whih emphasise the simplify of the algorithm and eliminate the ontrol theory knowledge requirement. he proedure of the proposed approah is presented below: 1. Define the lead time delay n, referene inventory level I R and the urrent deterioration date ( k) 2. Substitute the urrent and past information of inventory levels and order quantities aordingly to the form of the vetor x ( k) (3.65). Suh information is usually stored in the warehouse system and is easily aessible 3. Using the defined n and ( k) aordingly to the desription given in (3.66). values, build vetor K K 1K rk n 1 4. Use the vetors K and x ( k) as well as the referene inventory level I R and the order quantity of the time instane one step before (i.e. yesterday, if one per day is the sampling time) and substitute in the equation (3.67) to obtain the urrent order quantity. As it an be notied the method does not require definition of state spae representation. herefore it does not require familiarity with differene equations as well as state spae representations, nor transfer funtions not even ontrol engineering priniples suh as: input and output signals, system delay or disturbane. he novel tehnique is suffiiently mathematially redued to be alulable in using Exel, with no need for sophistiated software suh as MALAB. 75

88 Setion 3.6.2, Setion and Setion present the desription of the knowledge required as well as the list of main proedure for the Smith preditor, dead-beat ontroller and MPC respetively, to support the statement that the novel tehniques of IC is both, the less omputationally ostly and the easier to apply by the non-ontrol familiar OR expert Smith preditor o apply the Smith preditor on its own of the form shown in Setion , the following minimum knowledge is required: familiarity with differential equations, understanding of ontrol theory priniples suh as: input and output signals, state spae representation, system delay, transfer funtion, disturbane, open loop system, losed loop system, blok diagram, ways of onverting blok diagrams to transfer funtion. he steps needed to be ompleted to apply the Smith preditor are listed as follows: 1. Define ontinuous time state spae representation 2. Build an open loop system blok diagram with onsideration of the system delay 3. Close the loops of the system 4. Identify the losed loop system transfer funtion Cs () of Figure Apply the Smith preditor in the blok diagram of Figure Find the equivalent transfer funtion of the Smith preditor 7. Implement the new transfer funtion form in the blok diagram o use the method MALAB and/or Simulink are reommended. o do so the familiarity with both of them as well as basis of oding are needed Dead-beat ontroller o apply the dead-beat ontroller on its own of the form shown in Setion Error! Referene soure not found. the following minimum knowledge is required: familiarity with vetor and matrix operations, differene equations, understanding of ontrol theory priniples suh as: input and output signals, disrete state spae representation, system delay, system error, disturbane, open loop system, losed loop system, z-plane, determinant and 76

89 eigenvalues. he steps needed to be ompleted to apply the dead beat ontroller are listed as follows: 1. Define the lead time delay n, referene inventory level I R and the urrent deterioration rate ( k) 2. Define the matrix form of state spae representation, where the matrix dimension is dependent on the system delay 3. Define the open loop system 4. Define a ontrol vetor 5. Close the loop of the system 6. Identify the losed loop state spae representation 7. Find the harateristi equation of the system matrix (3.18) 8. Plae the system poles in the z-plane origin 9. Find the ontrol vetor o apply the dead beat ontroller MALAB and/or Simulink are reommended. o do so the familiarity with both of them as well as basis of oding are needed Model preditive ontrol o apply MPC of the form shown in Setion 3.3 the following minimum knowledge is required: familiarity with vetor and matrix operations, differene equations, objetive funtion of the vetor form definition, and derivatives of funtion of vetor form alulation. Understanding of ontrol theory priniples suh as: input and output signals, disrete state spae representation, system delay, system error, disturbane, open loop system, losed loop system, augmented model and moving horizon onept. he steps needed to be ompleted to apply the model preditive ontrol are listed as follows: 1. Define the lead time delay n, referene inventory level I R and the urrent deterioration rate ( k) 2. Define the matrix form state spae representation, where matrix dimension is dependent on the system delay 3. Define the open loop system 77

90 4. Find the inremental form of state vetor, ontrol variable system input and disturbane signal. 5. Find the losed loop system of the augmented form 6. Denote augmented model form of state spae representations 7. Find the i 1, N p step ahead preditions of inventory levels 8. Reformulate the predited inventory level vetor to the form of (3.53). From suh desription it an be notied how omputationally intensive the MPC algorithm is. he matries desribed in (3.54) are themselves onstruted from elements whih are already multipliation of matries of the losed loop system augmented model. 9. Define objetive funtion of the vetor form 10. Find the minimum of the funtion 11. Determine the future order quantities for the minimum value of the funtion 12. Use order quantities to determine the MPC gain vetor form 13. Apply one time instane only of the predition 14. Repeat the proess o apply the MPC MALAB and/or Simulink are reommended. o do so the familiarity with both of them as well as basis of oding are needed Calulation effiieny of the novel method he novel method was benhmarked against IMPC method using a ti to funtion in MALAB to measure the elapsed time of both algorithms. he simulation was run 30 times and the lowest time was piked for both algorithms for the same simulation settings (200 time instanes, 5 days lead time delay, 100 items referene inventory level, 0.9 deterioration rate). he elapsed time of novel tehnique appeared to be equal to s whih is 81.36% of the IMPC elapsed time. he IMPC elapsed time was equal to s. herefore the novel inventory ontroller appeared to be omputationally less heavy that initial IMPC. 3.7 Summary In the urrent hapter the preliminary modelling of the inventory system have been shown. he inventory state spae representation has been used for the appliation of a number of ontrol algorithms. he dead beat ontroller appliation enabled the verifiation of auray of 78

91 the initial state spae model through the mathematial reformulation and omparison of the results obtained as expeted. he model has been verified to be orret as an inventory model. Further, the MPC has been formulated to show the design proess for the verified inventory model. he objetive funtion of the predited inventory levels has been onstruted for obtaining the vetor of optimal future order quantities within the defined ontrol horizon. he moving horizon rule was used to make the optimisation dynami and update inventory level at every time instane. Further, the novel inventory ontroller has been proposed in the form of a proposition. he ontroller does not require any ontrol theory understanding to be applied. It an be adopted by the operational researh ommunity to obtain the preision of MPC for the given model. he ontroller is mathematially equivalent to the MPC used for a given model and its mathematial formulation is signifiantly redued when ompared with state spae and MPC formulation. Its omputational ost is lower, whih was shown by measuring elapsed time of eah of both algorithms, it does not require ontrol theory knowledge and its appliability is not limited to inventory ontrol only. 79

92 4 VERIFICAION OF HE PROPOSED IC APPROACH 4.1 Introdution he main goal of this hapter is a verifiation of the proposed IC approah, rather than a detailed disussion of the simulation results. Although, initially, the hapter shows a few results for the Smith preditor, there is not muh attention devoted to this tehnique. he hapter shows that the Smith preditor applied in a presene of the perishable goods on its own does not generate results whih fulfil the thesis aim. It does not meet the requirements of keeping the inventory at the set level, but it soothes them instead. Note that without the lead time aommodation of the Smith preditor there would be osillations. On the other hand the dead-beat ontrol, in isolation, would not generate stable inventory levels in ase of system delay and deterioration of goods. As it was elaborated in the Chapter 3, it was mainly applied to mathematially verify the auray of the inventory state spae model, rather to improve inventory performane. Although signifiantly more simulations were run to test these two methods, only a seletion of results have been hosen to support the disussion. One identified that they are not relevant for this researh, there is no additional analysis and detailed presentation of more experimental results done. he main aim here is to show how the proposed IC method is equivalent to IMPC in terms of its steady-state response. he IMPC itself, as a model based approah, inorporates the Smith preditor, and, therefore, the presented results of IMPC and IC are more relevant to what was planned to ahieve in this researh, namely developed a ontrol sheme whih ould be used/adopted by the OR ommunity. o onfirm the equivaleny of results between the proposed IC and the IMPC, several different tests were onduted. he first numerial example was run to illustrate the manual alulation of the IC approah. It was the shown that the manually obtained results were idential to at least the 4 th deimal plae with the IMPC simulation results. he seond test was run to show that both methods give very similar gains in respet to engineering auray regardless of the deterioration rate value. herefore the inventory levels obtained for both methods are again very similar. he third test was run to onfirm that the IC and IMPC approahes generate very similar gains for arbitrarily sinusoidally varying referene inventory 80

93 levels. his test shows that both tehniques generate very similar inventory levels regardless of the demand patterns, seasonal as well as random or mixed patterns. Simulation settings are listed at the beginning of eah setion, to allow the reader to reprodue the results. he disretisation period 1 day at any test. Equation Chapter (Next) Setion Simulation settings justifiation In eah test of the urrent hapter as well as Chapter 6 and Chapter 7 the initial simulation settings are similar. he urrent setion elaborates the hoie of partiular numerial values. Simulations time: 200 time instanes, whih refers to 200 working days. he hoie of 200 days for small (few days lead time) is both popular in the OR literature and in pratie, as it gives a good time perspetive of system behaviour. Deterioration rates: different values hosen for different simulations and different test purposes. he hoie of 0.9 refers to 90% of the items remaining good from one time instane to another. In respet to the OR literature the deterioration of suh a degree seems reasonable (espeially for produts suh as flowers fresh juies or preservative free food suh as organi vegetables). Deterioration of 1 orresponds to 100% of the items remaining good from one time instane to another. It means that the onsidered produts are not perishable. It an refer to produts suh as ar parts. For some tests the value of the deterioration rate was hosen to be lower than 0.9 (0.5 for instane, whih refers to 50% of goods surviving from one time period to another). It is not a neessarily realisti assumption in real life problems, but used here to show system properties and the behaviour to suh a system variable. Referene inventory level: Referene inventory level refers to safety stok. In real-life ase senarios the safety stok often exeeds the expeted demand. Espeially in the ase of unertain lead time, unreliable supplier or long lead times the safety stok kept an be very high. he hoie of 100 items of safety stok for an expeted demand between 0 and 70 items is not unommon in reality as well as in OR literature. he more reliable the replenishment strategy, the lower the referene inventory level an be. Ideally, the referene inventory of zero items refers to Just-in-ime poliy. Demand pattern: he hoie of seasonal demand pattern is popular in the OR literature. It is also pratial for seasonal produts suh as Christmas trees or Easter sweats, or New Year fireworks but also for produts whih an be affeted by weather (for 81

94 instaned inrease sale of soft drinks in the summer, or summer lothes) and other seasonal fators. he demand of seasonal patterns where eah of all values is kept at onstant level before the new hange ours is a popular approah in the OR literature but not neessarily the most realisti, as in most of real-life ases the demand varies. herefore a random pattern is onsidered as well for some of the simulations. he hoie of random pattern is relevant for produts where there is no notieable seasonality. hey an be everyday produts, suh as osmetis or leaning produts. he demand in that ase is more or less stable and the variations of demand are not that drasti. here is also a mixture of both demand patterns onsidered in the thesis as the most realisti demand pattern where the seasonality is visible but the demand slightly varies on a daily bases. his is not a popular approah in the OR literature due to its omplexity. Lead time delay: It has been hosen to be relatively low for most of tests (omparing to simulation period) to inrease the visibility of results. In pratie suh a lead time is popular in real-life problems within a first tier loal supply network and it is neessary if the perishable produts are onsidered. Suh a hoie of small lead time values is ommon in the OR literature. Warehouse is initially empty: this is a very ommon assumption in the OR literature. 4.3 Smith Preditor he urrent setion presents the simulation results for the Smith preditor desribed in Setion It was used for the ontinuous state spae inventory representation to present initial steps taken in the researh. Also it shows the initial stage results on the way to find a satisfatory inventory level ontrol tehnique. he simulation was run for 200 time instanes, whih here refers to 200 working days. he proportional term was set to 1. he deterioration of produt was set to 0.9 (whih means that 90% of the items remain good from one time instane to another). he referene inventory level was set to 100 items, whih in respet to the OR literature represents the safety stok of a realisti level. he demand was arbitrarily assumed to be seasonal and hanging periodially between four allowed values: 70 (for time instanes 6-55), 20 (for time instanes ), 50 (for time instanes ) and zero (for the remaining time instanes). he demand was set to be zero for the first 5 days to prevent unneessary bakorders for those ases where the lead 82

95 time is set to 5 or less days (before the first order reahes the warehouse) and to show what happens for the ases where the lead time is greater than 5 days. In partiular the lead times of 1, 5 and 10 days were hosen. herefore the ontroller of equation (3.2) has a form as follows C SP s 0.5 (4.1) 1 ns (0.5 / ( s 1 0.9)) 1 e where n is a system lead time delay. he inventory level, the order quantities and demand pattern are presented in Figure 4-1 for t , and Figure Figure 4-5 (zoomed for speifi time instanes when the demand suddenly hanges) with respet to different lead time delay values. he goal is to test the model response in general and in respet to hanges in the lead time values. he simulation was run for several different values of lead time delay, nevertheless only three results, for n 1, n 5 and n 10 are presented to support the onlusions. follows For the speifi simulation values the system model of equation (3.1) has the form as 0.9 xt x t x t u t n d t y t (4.2) for n 1, n 5 or n 10 and dt values shown in Figure 4-1 for t

96 I Figure 4-1: Smith Preditor in respet to different lead time values he following representative observations have been arried out for the results shown in this setion. his results are typial of those tested for other ases, but this are omitted here. he inventory levels flutuate smoothly within the phases of demand pattern at relatively low levels: near 20 (for 70 items in demand), near items (for 20 demanded items), near 30 items (for 50 demanded items) and near 60 items (for 0 demanded items). At any time the inventory levels are not kept anywhere near to the referene inventory of 100 items, and always kept below that value. herefore, the Smith preditor if applied in isolation does not generate satisfatory results in respet of the thesis goal (keeping inventory at a desired level). Instead the number of stored goods is smoothly flutuating in response to hanges in demand. However the higher lead time delay value, the less smooth the inventory levels and orders beome. he flutuations of inventory levels and order quantities an be observed to behave in a related manner. In ontrol terms the order quantity is the ontrol variable in a funtion of the 84

97 feedbak inventory level. he differene in results obtained for different lead times are mostly visible lose to the time instanes where the sudden demand flutuation ours. Figure Figure 4-5 show the zoomed parts of Figure 4-1 for speifi demand values. Figure 4-2 presents the zoomed demand, inventory levels and order quantities for time period of 0-30 days. It an be notied that initially, for first 6 days, no items are demanded. As in general terms the demand inrease means the subtration of items from inventory levels (sale of items), onsequently any inrease of demand should be followed by the inrease of orders (to replenish the sold goods), whih indeed an be observed in Figure 4-2 for n 1 and n 5 at 6 th time instane. For n 10 the inrease of orders is observed in the 11 th time instane only, as due to 10 days delay the previous orders have not reahed the warehouse by 11 th time instane. As the warehouse is initially empty, the first order size (in the 1 st time instane) is equal to referene inventory level, to supply the empty warehouse with the required number of goods. It an be notied that within the time delay, some ations happens in inventory level after the first order is plaed for the ases of n 1 and n 5, the results differ then for n 10, though. For n 1 and n 5 the inventory level reahes the level of approximately 50 items and stays at that level until the demand suddenly hanges on 6 th day. Until that time the order quantities dereased to 50 items, to make sure no more goods appear in the inventory levels and no surplus inventory will be kept in the warehouse (apart from the referene inventory one the order reah as the warehouse with relevant delays). 85

98 I Figure 4-2: Smith preditor, zoomed result for 0-30 time instanes For n 1 and n 5 one 70 items the demand appears at the 6 th day, the orders inrease quikly to approximately items, to make sure the demand will be satisfied. he derease of inventory level to slightly zero is a result of demanding more produts then available in the warehouse. he value never goes below zero as the Smith preditor integrator is limited bounds inventory from bakorders. he n 1 response is faster, due to smaller delay, but eventually both n 1 and n 5inventory level signals inrease to some stable level (in this ase of 20 items). he behaviour of the system for n 10 is different due to the appearane of the demand (in the 6 th time instane) before the first order reahes the empty warehouse. Also the delayed reation of orders annot quikly ompensate the demanded items and the inventory level peaks above zero level in 10 th time instane and goes almost immediately bak to zero. It then inreases around 21th time instane. herefore the simulation produes the results as expeted, with the ase of the long delay ( n 10 ) requiring more time to reover. 86

99 I he onverge to other inventory levels of n 1 and n 5 ase is only notieable in Figure 4-1 after 35 th day of simulation. Figure 4-3 presents the zoomed results for time instanes between days As the effet of the initial delay of delivery time (lead time) to the warehouse was overome already and all inventory levels were stable at around the 20 items level. Now, when the demand (sale) suddenly dereased in the 56 th time instane, the traes orresponding to the orders (for n 1, n 5 and n 10 ) derease very smoothly and almost identially to a level of items. he inventory levels slightly overshoot for the time of lead time and onverges to the inreased level of 40 items with respet to the time delay. Figure 4-3: Smith Preditor, zoomed result for time instanes In Figure 4-4 and Figure 4-5 the simulation is analogous to Figure 4-3. he orders inrease or derease smoothly and almost identially regardless of the delay (orders are not delayed by the lead time, but the inventory levels are). hen the inventory onverges to a ommon level for all 87

100 I three delays, to a level of 40 items, within appropriate time delay. In partiular, Figure 4-4 shows the zoomed results of Figure 4-1 for time instanes between days. he order signals (traes) regardless of the delay respond to the sudden hange of demand (from 20 to 50 items) by inreasing the order size to items. he inventory level dereased immediately due to the sudden sale of items and with appropriate time delays to onverge to a ommon level regardless of the delay to a value of about 30 items. Figure 4-4: Smith Preditor, zoomed result for time instanes Figure 4-5 presents the zoomed results for time instane between 150 and 170 days. he demand suddenly dereased in time 157 th instane whih results in smooth, no lead time affeted, order sizes for all three ases of n 1, n 5 and n 10. For eah ase the inventory level initially slightly overshoots as the dereased order signal did not ompensate the stok level kept in the warehouse for the duration of lead time delay. For all this ases the inventory level onverges to a ommon level within the delay time. 88

101 I Figure 4-5: Smith Preditor, zoomed result for time instanes Although the Smith preditor deision maker produes a response whih keeps stok in the warehouse, it an be onluded that the blind appliation of the Smith preditor is not suffiient for this partiular appliation i.e. in the presene of perishable goods. It turns out that this in fat levels itself well to this appliation, whih is further extended in the following. Analysis of the Smith preditor shows that under the ideal ase of non-perishable goods the onfiguration adopted here will keep the inventory at the desired level. From a ontrol engineering perspetive to maintain the inventory level in the perishable ase would require the ontroller gain K to be replaed with a proportional plus integral term. However, sine the aim is to develop shemes that are aessible by the OR ommunity, attention is now forward to the optimal dead-beat ontroller. 89

102 4.4 MPC Approah he purpose of this setion is to present some initial results of inventory responses orresponding to the appliation of the MPC method. As it was mentioned before, the tuning parameter of equation (3.58) was set to zero (to be able to refer to the equivalent IC method, whih does not require a tuning parameter neither). herefore for all simulation settings here, the tuning parameter is arbitrarily set to zero, so that the result an be ompared to IC in further setions. he simulation was set as follows. he simulation was run for 200 time instanes. he referene inventory level was set to 100 items. he predition horizon was set to 14 and the ontrol horizon was set to 8 time instanes. he representative values of predition and ontrol horizon are produing typial system behaviour for most of tested numerial value of the predition and ontrol horizons. he lead time delay was set to 5 time instanes. he deterioration rate was set to 1 (whih effetively means no deterioration of produts as 11 0 ). herefore the inventory state spae representation of equation (3.3) and (3.4) has a form x k x k 0 u k 0 d k (4.3) k y k x (4.4) and the MPC gain from (3.63) has a form K (4.5) hree different demand patterns were tested: the seasonal pattern of four different values - 70 items (days: 6-55), 20 items (days: ), 50 items (days: ) and 0 items (remaining days); random pattern: with 50 as a mean value and variane of 10 items; seasonal with some randomness allowed: the pattern of 70 items (days: 6-55), 20 items (days: ), 50 items (days: ) and 0 items (remaining days) plus one random pattern of zero items 90

103 I as a mean value with a variane of 5 items. Figure 4-6 shows the demand values of three patterns (the seasonal, random and mixed) and respetive system responses in terms of order quantities and inventory levels. herefore the sensitivity of the MPC inventories is tested here in respet to different demands. Figure 4-6: MPC in respet to 3 different demand patterns he trae of demand 1 shown in Figure 4-6 presents the inventory level kept at the referene level of 100 items for most of the time (apart for the time instane when the demand hanges suddenly). he sudden flutuations themselves onverge quikly (within the time delay) bak to the referene level. Analogously, the order quantities flutuate only when the demand suddenly hange and onverge bak to expeted mean values. Observing the trae of demand 2 91

104 it an be notied that the inventory levels flutuates within +/-10 items from the referene inventory level of 100 items and the orders flutuate in a similar manner near the demand mean value. Note that demand 2 is not representative of a pratial ase, but inluded to the test the algorithm. Finally, seeing trae of demand 3 it an be observed that the inventory remains near the desired value (with some osillations in orrelation to demand osillations) for most of the time while it flutuates when the demand suddenly hanges too. hen, within the lead time delay the inventory level returns and onverges bak and remains with small osillations near the referene level. he order quantities reflet the expeted behaviour with the observation that it returns near to the demand mean trend. Figure Figure 4-10 present the zoomed results shown in Figure 4-6 for seleted time instanes, where the demand suddenly hanges and the orresponding hanges in inventory levels and order quantities an be observed before the signals onverge bak to the referene inventory level. he zoomed figures inrease the visibility of the results. Figure 4-7 shows the system behaviour for the time period of 0 20 time instanes. During these time instanes the sudden inreased from zero to 70 items in demand an be observed (for seasonal and mix demand patterns) at 6 th time instane. As it an be notied, the inventory levels and order quantities of demand 1 and demand 3 behave in similar way, with the differene that demand 3 adds some small flutuations around the ore values of the results of demand 1. For the first 5 time instanes, the demand is zero (demand 1) or near zero (demand 3). In Figure 4-7 it an be notied that the order sizes start at the level of 50 items and then slowly dereased to zero (or around zero) for all ases regardless of the demand pattern. his way, the orders shift the inventory levels to the referene value. For demand 1 and demand 3 in respet to the demand sudden inrease at the 6 th time instane the orders inrease up to the level of 200 items for one time instane to quikly ath up with inventory level and slowly redues bak to the demand level to ompensate the delay in inventory response. In Figure 4-7 it an be observed that the inventory initially drops after the 6 th time instane following the sudden demand hange, as the previous, not dereased orders are still being delivered to the warehouse due to lead time delay of 5 days. hen the inventory starts building up again due to urrent dereased order deliveries (smaller quantities). Eventually the inventory reahes the referene level and order stabilizes as well. 92

105 I Figure 4-7: MPC in respet to 3 different demand patterns, zoomed values for 0-20 time instanes Figure 4-8 shows the system behaviour for the time period of time instanes. As it an be notied, again for the demand 2, the orders lightly flutuates to push the inventory levels builds up or down in respet to small demand flutuations. he inventory levels and order quantities of demand 1 and demand 3 behave in similar way to eah other, with a differene that demand 3 adds some small flutuations around the ore value of results of demand 1. For demand 1 and demand 3 in respet to demand sudden derease at 56 th time instane the orders dereases to about -50 items (whih refers to returns) due to a sudden demand derease. It should be notied that returns are allowed in this hapter only for the purpose of testing the model behaviour rather than to use this in pratie. In further hapters the returns are not onsidered. 93

106 I Figure 4-8: MPC in respet to 3 different demand patterns, zoomed values for time instanes In Figure 4-8 it an be observed that the inventory initially slowly builds up, as the previous, not yet dereased orders are still being delivered to the warehouse due to the lead time delay. hen the inventory starts dereasing bak due to new dereased order size deliveries after the time delay. Eventually the inventory reahes the referene level and order stabilizes as well. Figure 4-9 shows the system behaviour for the time period of time instanes. It an be notied, again for the demand 2, that the orders slightly flutuate to push the inventory levels up or down in respet to small demand flutuations. he inventory levels and order quantities of demand 1 and demand 3 behave in similar way to eah other, with a differene that demand 3 adds some small flutuations around the ore value of results of demand 1. For demand 1 and demand 3 in respet to a sudden demand inrease at the 106 th time instant the orders inrease to about 100 items due to sudden demand inrease. he inventory initially slowly redues, as the previous, not yet inreased the orders are still being delivered to the 94

107 I warehouse due to the lead time delay. hen the inventory starts rising bak due to the new inreased order size deliveries after the time delay. Again, as was the ase for Figure 4-8 the inventory reahes the referene level and the orders stabilize. Figure 4-9: MPC in respet to 3 different demand patterns, zoomed values for time instanes Figure 4-10 again shows the system behaviour for the time period of time instanes. Similar observations may be desribed around 156 th time instane as disussed for 56 th time instane. Indeed the MPC approah is formed to provide a onsistene performane. 95

108 I Figure 4-10: MPC in respet to 3 different demand patterns, zoomed values for time instanes It an be onluded that for all the studied demand patterns the inventory level is maintained near to or at the referene inventory level for most of the simulation time. he inventory flutuates (and so do the orders respetively) at the time instanes of a sudden demand hange. As there is a delay in the system, the system onverges with a delay to or near the desired value. It an be notied, that ompared with the Smith preditor, the obtained results would be satisfatory for industrial purposes in the sense of the previously defined goal, namely to redue inventory level flutuations and keep it near the referene point (Setion 1.1.5). herefore, based on the presented results and other onduted simulations whih are omitted from this setion, the MPC as a tehnique an generate satisfatory results for inventory stability. As stated in the introdution, the MPC approah has the Smith preditor and potentially the dead-beat ontroller within its struture. Sine MPC is onsidered as satisfatory for this researh, the next subsetions fous on a verifiation of the results obtained with the novel IC against the IMPC. 96

109 Sine it is to be verified that the IC and IMPC generate the same results for different simulation settings, therefore it is worth notiing that the IC method presented in the IC Proposition (Setion 3.4) does not inlude N p and N in its definitions and therefore does not depend on them. herefore, the urrent test was onduted for the IMPC method itself, to study the sensitivity of the IMPC to hanges of the N p and N values. Several different simulations were run for the following sets of pairs of the N p and N values: N n 1n 10 and defining No N p N, values of No were studied. he model appeared to be insensitive to hanges of N and N when the tuning parameter is set to zero in p equation (3.58). Figure 4-12 shows the exemplary order quantities of the pair N p 18 and N 8 (where N 10) and Figure 4-13 of the pair N 14 and N 11 (where N 3). he values were o p hosen in the manner that not only they differed from eah other but also the differene between N p and N (the N o value) different too for eah figure. Both simulations were run for 200 time instanes and following settings: arbitrarily hosen three values of deterioration rate 1, 0.8 and 0.5, lead time delay of 5 days and the seasonal demand pattern of Figure o Figure 4-11: Demand pattern for IMPC vs. IC verifiation Regardless of the N p and N values the inventory state spae models and IMPC gains have the same or very similar values (differenes notieable on or above 4 th deimal plae only). 1. For 1 the inventory state spae representation of equations (3.3) and (3.4) has the form 97

110 x k x k 0 u k 0 d k (4.6) k y k x (4.7) and the MPC gain from (3.63) has the form K K for Np 18 and N 8 (4.8) for N 14 and N 11 p It an be notied that the gains are idential up to the 4 th deimal plae. 2. For 0.8 the inventory state spae representation of equations (3.3) and (3.4) has the form x k x k 0 u k 0 d k (4.9) k y k x (4.10) and the MPC gain from (3.63) has the form K K for Np 18 and N 8 (4.11) for N 14 and N 11 p It an be notied that the gains are idential up to the 4 th deimal plae. 3. For 0.5 the inventory state spae representation of equations (3.3) and (3.4) has the form 98

111 x k x k 0 u k 0 d k (4.12) k y k x (4.13) and the MPC gain from (3.63) has the form K for N 18 and N 8 K for N 14 and N 11 p p (4.14) It an be notied that the gains are very similar. All gain values are the same up to the 4 th deimal plaes apart from the first element. he first element is the same up to the 3 rd deimal plae and different only on the 4 th deimal plae. Suh two gains an ontrol the system very idential. Figure 4-12 and Figure 4-13 present the order quantities for all three of the above systems (deterioration values) for IMPC of N 18 and 8 p N 14 N and N 11 respetively. p Figure 4-12: Order quantities for different deterioration rates when N 18 and N 8 p 99

112 Figure 4-13: Order quantities for different deterioration rates when N 14 and N 11 p From the results shown have and other simulations onduted whih are omitted here, it an be onluded that the same values of order sizes were obtained regardless of the predition and ontrol horizon values for eah of the values of deterioration rate. It is a result of the fat that the form of the IMPC gain for a given lead time is mainly dependent on the deterioration rate regardless of predition and ontrol horizon values. his appears to be further reinfored when the IMPC appears to be non-sensitive to hanges in the ontrol and predition horizon vales when the tuning parameter is set to zero. Consequently the IC (whih does not depend on and N by definition) and IMPC an be now ompared. he issue of non-zero tuning parameter approah of an alternative IC method is addressed in Chapter 7. N p 4.5 Alternative IC Method - Verifiation Against MPC For the purpose of verifiation, it is assumed that negative orders are allowed (whih orrespond to returns). his way the pure model behavior an be tested. For further hapters, where the fous is devoted to optimization of inventory, rather than verifiation of the model, a saturation module preventing returns will be inluded in the model. herefore from (3.67) u k u k 1 IR k Kx k (4.15) 100

113 at any time instane. he assumption is far from the reality, as even if negative orders are onsidered as returns, they should be returned with a delay (in the same manner as they are delivered with a delay). Nevertheless, in the urrent setion the purpose is to verify equivaleny of the IC and IMPC results for different onditions and settings only. Saturation here ould redue the larity of the analysis. Chapter 6 addresses the IC approah for the inventory appliation, therefore the saturation is onsidered there. his way the results given by the IC an be diretly ompared with IMPC without any additional assumptions taken or onstraint applied. he urrent setion is devoted to verify that the IC behaves in the same manner as IMPC. Several numerial examples have been shown in the following setions to support the verifiation. he proess of manual alulation of order quantities, whih would result in satisfatory stable inventory levels, is shown in the Setion his serves to demonstrate how straightforward IC is in terms of omputation with respet to the original MPC est 1: o illustrate the manual alulation of IC In terms of illustrating the simpliity of the developed ontrol poliy, the urrent numerial example shows the manual alulation proess of finding the optimal order sizes. From the IC Proposition in Setion 3.4, for n 5, 1 and y 100, gain K was alulated and has exatly the same form, as was obtained from MPC, namely K R. he simpliity of determining the K values in this ase is related to the fat that was set to 1. Although, the proposition was developed with the purpose of using alulation spreadsheets or simple (a few line of ode) program in general, the urrent setion enables the reader to gain a deeper understanding of the proposed method by following the alulation proedure step by step. From the IC Proposition it an be notied that the method enables the on-line optimisation based on above the urrent and past information vetor of the inventory system (the inventory levels and sizes of plaed orders) denoted as a urrent time instane vetor x k. Using the mentioned data, the warehouse manager is able to alulate the next periods optimal order quantity as follows k u k x (4.16) 101

114 As the warehouse is initially empty, dimension 1 x shown in (4.17) is a zero vetor of n 1 x (4.17) hen, u (4.18) From (3.40), knowing that there was no order before u 1, the first order should be equal to u (4.19) While reahing the next time period, the warehouse manager is able to read the urrent state vetor x and alulate the optimal orders of the seond period as follows u (4.20) hen u (4.21) Further, knowing x u (4.22) hen u (4.23) Similarly, based on monitoring the urrent value of x k, any u k an be alulated on-line. Continuing the alulations for the ase when demand is assumed to be zero for any further time instane, the value u k is always obtained to be equal to zero. It is logial, sine the 100 items ordered in the first step never get sold and therefore the inventory level 102

115 remains at the referene level. One the demand hanges, the order size adjusts as well. he manual alulation was ontinued for the following demand pattern for 25 periods shown in Figure Figure 4-14: Demand pattern he order sizes for this ase were alulated and presented in able 4-1 for the given time instanes: ( )

116 able 4-1: Manually obtained order quantities for eah time instane by IC Figure 4-15 illustrates the order sizes for the proposed IC from the able 2-1 in graphial form. Figure 4-16 shows the order sizes obtained from the IMPC MALAB simulation of the model and the gain of the form presented in (4.3)-(4.5), where ontrol and predition horizons were set as follows: N 14 and N p 8. Figure 4-15: Manually obtained order quantities by IC Figure 4-16: Simulated order quantities obtained by IMPC 104

117 As an be notied, the results are the same or very similar for the given initial onditions, whih onluded the main objetive of this setion est 2: o show effet of varying deterioration rate he seond test aims at omparing the behavior of both methods regardless of the values of deterioration rate. herefore, both, the IMPC and IC were used to obtain the vetor K values and make sure that they are the same (or very similar). In the onsidered senario the time varying deterioration rate had an arbitrary form of k 1 sin 5 k 1 (see Figure 4-17). he simulation was run for 33 periods. In this 2 way 33 different values of 0,1 are used for omparison purposes. he onstant referene inventory level was set to IR 100. he lead-time delay was set to n 5. he predition and ontrol horizons of the MPC method where set as follows: N p 12, N 8 able 4-2 and able 4-3 show the values of the K vetor obtained for every time instane k 1, 2, 3 33 regarding the k values. Figure 4-17: Varying deterioration rate able 4-2 shows the simulation results for the IMPC ase and able 4-3 for the IC ase. 105

118 k K (1) K (2) K (3) K (4) K (5) K (6)

119 able 4-2: Values of K vetor obtained with IMPC k K (1) K (2) K (3) K (4) K (5) K (6)

120 able 4-3: Values of K vetor obtained with IC It an be notied that the values of the vetor K obtained in two different methods are idential to 4 th deimal plae for eah time instane. In fat, the auray is identified to a higher number of deimal plaes, but this is not shown here. he inventory levels for IMPC (Figure 4-18) and for IC (Figure 4-19) are presented below for three hosen values of deterioration rate. Indeed, the results presented are very similar or idential results, thus the aim of this setion has been demonstrated. he models used for IMPC for eah deterioration rate are presented in (4.6), (4.7), (4.9), (4.10), (4.12) and (4.13). he vetor K (4.8), (4.11) and (4.14) gains for both methods are given for the onsidered values of deterioration. Figure 4-18: Inventory levels obtained with IMPC 108

121 Figure 4-19: Inventory levels obtained with IC est 3: Sinusoidal referene inventory level k he next test was onduted for the onstant deterioration rate 1 to make the model time invariant. he delay was set n 20 and no demand disturbane was onsidered while onduting the test. he referene signal was set to be an arbitrary time varying value suh that I sin 10k R. herefore the IMPC model of equations (3.3) and (3.4) take the form as follows x k 1 x k u k d k (4.24) k y k x (4.25) 120 he gain K for both methods has a form as follows (from (3.63) or (3.66)) K= (4.26) 109

122 he referene signal is shown in Figure 4-20: Figure 4-20: he sinusoidal referene signal he MALAB simulation for IMPC and Exel alulation for IC was run for 200 time instanes. he aim of the test is to ompare the system response using both methods. Figure 4-21 and Figure 4-22 show the inventory levels obtained due to simulation for IMPC and IC, respetively. Figure 4-21: Inventory levels in response to sinusoidal referene signal by IMPC 110

123 Figure 4-22: Inventory levels in response to sinusoidal referene signal by IC As it an be notied, both sets of results presented are exatly the same. Both trak the referene inventory level with a fixed delay n. Similar tests were run for different values of deterioration rate and eah time the same results were obtained for both approahes. he test aimed at showing the inventory level response, whih is important from the system designer perspetive. Although, for the future uses of the method, the simulation of inventory levels will not be neessary, as it involves the ontrol theory perspetive and therefore it is not inluded in the IC Proposition. he urrent and previous setions present the inventory results so that the reader an beome onvined about the auray of formulation of IC method. In pratie, the alulation of the optimal order quantities would only be of interest for the user of the IC method, whih, as a result, would lead to satisfatory stable inventory levels. Chapter 6 fouses more on ommentary of the results est 4: Different demand patterns he urrent test was onduted in order for omparing the results of both methods for different demand patents (seasonal, random, and mixture of seasonal with some randomness as desribed in setion 4.4) and also to verify that they are very similar for both methods. he ommentary on the results themselves is a onern of Chapter 6. he simulation was run for 200 periods. he deterioration rate was set to 0.9. he lead time delay was set to 5, referene inventory level to 50 items. he IMPC state spae model of (3.3) and (3.4) has a form 111

124 I x k x k 0 u k 0 d k (4.27) k y k x (4.28) and the gain K of (from (3.63) or (3.66)) for both methods has the form K (4.29) he result present the three tested demand patterns and the inventory levels and orders obtained for eah of the demand patterns by the IMPC by in Figure 4-23 the IC approah in Figure Figure 4-23: IMPC response to different demand pattern 112

125 I Figure 4-24: IC response to different demand patterns Comparing of the results it reveals that the same values of the inventory levels were obtained regardless of the approah used for eah demand pattern. Similar results were obtained for different demand patterns, whih presentation (due to its repetitive harater) was omitted here. 4.6 Summary Initially, the simulation results of the Smith preditor method used for ontinuous systems were shown. Due to presene of deterioration the results are not onsidered as satisfatory for industrial purposes in the sense of this thesis (maintaining inventory at or near the referene point for storage apaity panning, manpower planning and bullwhip affet redution purposes and maintaining ontrol simpliity). Further, the results of two methods, the IMPC (for ur 0 ) and IC (for no onstraints on order size) were ompared for a range of different test ase senarios. In eah ase both 113

126 methods gave exatly the same results, whih indiated that both of the methods might be mathematially equivalent. Also it was notied that the IMPC (for ur 0 ) and IC (for no onstraints on order size) are sensitive to hanges in demand, deterioration rate, lead time delay and referene inventory level. No hange in IMPC results was observed for hanges of the N p and N values, as long as N N 1. he reason behind this is based on the partiular definition of the inventory p state spae representation as well as the fat that the u r was set to zero. As a result, the IMPC beomes non-preditive. In the Setion 7.2, the non-zero values of the tuning parameter are onsidered for further investigation of the alternative IC approah. 114

127 5 DEMONSRAION OF MAHEMAICAL EQUIVALENCY he urrent setion presents the mathematial demonstration of the equivaleny of the novel IC method to the MPC tehnique applied for the inventory state spae model. he demonstration proess reflets the step by step development proedure, presenting how the IC was mathematially obtained from the initial MPC. he hapter presents the sequene of propositions and their demonstrations. Eah of them has been developed based on reognition of some patterns of the MPC mathematial desription. o familiarise the reader how the simplifiation proedure has evolved in time, first the proess is shown for the non-perishable ase and then, analogously, for the perishable ase. he non-perishable ase refers to deterioration rate arbitrarily set to one. In suh ase, the metris defined as an augmented model of MPC ontain zeros and ones only. Whether the non-perishable or perishable ase onsidered, the augmented model mathematial desription of the inventory system is relatively simple. he matries defined in augmented model, when appropriately multiplied, build the elements of matries F and Φ. he formulation of eah element in matries F and Φ was notied to be dependent on its row and olumn indies, in respet to power index of the augmented state matrix. herefore, both of matries F and Φ ould be formulated in general terms, depending on upon the row and olumn indies. he matries F andφ, in turn, are building elements of the MPC gain K. Again some patterns were notied there and the mathematial desription was established in relation to row and olumn indies. he initial propositions present the way of reognising the mentioned patterns in the F and Φ matries, the building elements used for onstrution of the gain in MPC. heir simplified, yet more general mathematial desription is the subjet of the propositions. he final proposition has the IC Proposition form and refers to the MPC gain K simplified desription. he whole proedure is based on the fat, that the inventory model being used, has a partiular state spae form, as shown in inventory modelling hapter, equations (3.3) - (3.7). he ontent of the urrent hapter is the main originality and novelty of the researh. 115

128 5.1 Non-perishable Case If the elements of Φ of equation (3.54) are onsidered, it an be notied, that eah of them, in general terms is either equal to zero or it has a form, whih an denoted in general terms as A b, where p denotes the exponent, or power index. p Using the following propositions, the simplified form of elements of Φ an be represented. Equation Chapter (Next) Setion 1 Proposition 1: For and A defined in (3.46), the general desription of an n 1 dimentional vetor A, an be formulated regardless of the values of the exponent p and lead time delay p n in the following manner: n pn2 1mn n pn2 1mn p A p p p 1 p 2... p m 2... p n 2 1 or p A p p p 1 p 2... p m (5.1) where any th m element of the vetor beomes zero only when p m 2 0 Remark 1: Note that for a hosen p there is no zero element in the vetor A, if p n 2 0. In p other words, if the dimension of the vetor will be no zero element in the vetor. A is smaller than p 2 for a hosen p there p Demonstration: From (3.46) it is known that A A A O 1 ( n1) 1 and I. Using the model desribed in (3.5) for ( y) 1 the above an be expanded as follows. 0 1 n 116

129 A and (5.2) herefore, for p 1 it an be shown that n1 A (5.3) Further, assuming that for some 1 p n 2 p A p p p 1 p 2... p m (5.4) is true, then, A A A p+1 p p p p 1 p 2... p m A (5.5) p 1 p 1 ( p 1) 1 ( p 1) 2... ( p 1) m Assumung now that for same pn 2 p A p p p 1 p 2... p m 2... p n 2 1 (5.6) is true, then, A A A p+1 p p p p 1 p 2... p m 2... p n 2 1 A (5.7) p 1 p 1 ( p 1) 1 ( p 1) 2... ( p 1) m 2... ( p 1) n 2 1 whih has demonstrated the proposition for both ases, when pn 2 and pn

130 Proposition 2: For, p A and b defined in (3.46) for the non-perishable ase ( y) 1 p p n p A b p n 2, when p n 2 0 else A b 0, where n is the order of the system desribed in (3.3) and also represents the lead time delay. Demonstration: From (3.46) it is known that A A A O 1 ( n1) 1 b, b I. b and 0 1 n Using the model desribed in (3.5) for ( y) 1, the above an be expanded as follows , A b and (5.8) From the formulation of b it an be notied that p A b is the n th (seond last) element of the A vetor, whih was defined in Porposition 1. From Proposition 1 it an be observed that depending on the dimension n of the original system the seond last element is either pn 2 or 0. herefore p as np, then if p n 2 p n 2 0 p A b p n 2 (5.9) else p A b 0 (5.10) So that (5.9) and (5.10) provide the statement of the proposition. 118

131 Remark 2: In the definition of Φ in (3.54) it an be notied that the exponent of A in any A b p element is dependent on the row and olumn denoted m and r respetively of the original Φ matrix,, and it an be notied that for eah mr, Φ, the exponent p is equal to m r. hen from Proposition 2 it an be dedued that the general formultion of Φ is as follows. Φ n n 1 n 2 n n 2 n 1 n 2... N p 1 n 2 0 n 1 n 2 n n 2... N p 2 n n 1 n 2... N p 3 n 2 (5.11) N p N 1 n N p N n 2 Note that the equation (5.11) presents more generi from than one ould expet, and might seem unneessary (for instane the element given by n1 n 2 ould be simply denoted as 1). Nevertheless, retaining this notation enables this notation a regularity in the desription is to be observed, whih is later utilised to redue the formulation of the whole matrix to a simple form. Using the partiular numerial values here, would redue the omplexity of the desription, but would redue the larity of the reasoning. Further, as it was elaborated in Setion 3.3, the MPC uses past and urrent information to predit the future inventory levels. he optimisation of order quantities is arried out over a fixed predition horizon derived suh that N p. From (3.55) and (3.56) the ontrol order quantities U an be 1 R k U Φ Φ R Φ Y Φ Fx (5.12) For the time being assume R to be the null matrix suh that ur 0. Assuming that N p y k y k Y Y (5.13) R R R R where y R is a referene inventory level, this gives as in (3.57) 119

132 1 y R k k R U Φ Φ Φ Y Φ Fx (5.14) Beause of the MPC priniple the first element of U is applied N K Kx u k y k k y R Φ Φ R Φ YR yr k Φ F k (5.15) where K is the first element of 1 y N p N Φ Φ Φ YR and K Φ Φ Φ F and K K xk y as in (3.63). o find the new representation of the vetor K, the relevant representation of its omponents suh as ΦΦ and ΦF need to be obtained first. From Proposition 1 and the system desription in (3.5), for the non-persishible ase ( y) 1, the representation of simplified and the following proposition an be demonstrated. ΦΦ an be Proposition 3: N g1 n2 p i0 ΦΦm, r m r i 1 i 1 n m r m when m r where g r when m r (5.16) Demonstration: Defining Ln 2 (5.17) from (5.11) Φ has the form as follows 120

133 n n 1 L n L n 1 L... N p 1 L n 1 L n L... N p 2 L n 1 L... N p 3 L Φ (5.18) N p ( N p N) 1 L N p ( N p N) L 121

134 From the above, ΦΦ an be obtained suh that n 1 Ln 1 L... N 1 L N 1 L n Ln 1 L... N 1 L N 2 L n 1 Ln 1 L N N... N L N 1 L p p p p p p n L n 1 L... N 1 L N 2 L n 1 Ln 1 L... N 2 L N 2 L n 1 Ln L N N... N L N 2 L p p p p p p ΦΦ n 1 Ln 1 L... N 3 L N 1 L n 1 L n L... N 3 L N 2 L n 1 Ln 1 L N N... N L N 3 L p p p p p p n 1 L n 1 L N N... N L N 1 L n 1 Ln L N N... N L N 2 L n 1 Ln 1 L... N L N L p p p p herefore, oleting terms leads to N 1 L N 2 L N 3 L N ( N n 1) n 1 p p p p p n 1 L in 1 L i n 1 L in L i n 1 L in 1 L i... p L p L N 1i n 1i p i 0 i 0 i 0 i 0 N 2 L N 2 L N 3 L N ( N n 1) n 1 p p p p p n 1 L in L i n 1 L in 1 L i n 1 L in L i... p L N 2 i p L n 1 i p i 0 i 0 i 0 i 0 N 3 L N 3 L N 3 L N ( N n 1) n 1 p p p p p ΦΦ n L i n L i n L i n L i n L i n L i p L p L N 3i n 1i p i 0 i 0 i 0 i 0 N L N L n 1 L in 1 L N N i n 1 L in L N N i n 1 L i n 1 p N L N L L N N i... n 1 L in 1 L i p p i0 i0 i0 i0 (5.20) (5.19) 122

135 It an now be observed that the formulation of eah matrix element follows a pattern and relates to the row and olumn indies of a given element loation, denoted m and r respetively. From the above, the general formulation of a given element of the element an be derived suh that: ΦΦ matrix ΦΦ N p g1 L i0 m, r n 1 m r L i n 1 L i m when m r where g r when m r (5.21) notiing that Ln 2 and from (5.17) it follows that ΦΦ N p g1 n2 i0 m, r n 1 m r n 2 i n 1 n 2 i N p g1 n2 i0 m r i 1i 1 m when m r where g r when m r (5.22) he above demonstrates the proposition. Remark 3: From Proposition 3 it an be notied that the elements of the matrix ΦΦ depend on their row and olumn index, denoted m and r respetively, the order of the original system n defined in (3.5) and the predition horizon N p. Proposition 4: he matrix ΦF denoted hare as M is defined as follows 123

136 m Mm,2 1 1 a) m1<r<n+1 M m, r i n m 1 r 2 i 1 i1 b) M, 1 (5.23) m N p Φ ) M m, n 1 m, r m r1 N p m n where m and r, represent the rows and olumns, respetively. Demonstration: Realling the representation of the matrix F as shown in (3.54) F A A A N A 2 3 p From (5.4) it an be observed that A p p p p 1 p 2... p n Notiing that the exponent of the matrix an be dedued that p m, so that A depends upon the row index of the matrix F, it p A m m m 1 m 2... m n F ( m) (5.24) whih is a general definition of a row in the matrix F in respet to its row index denoted as m. From the above, the matrix F an be presented in the following form 124

137 F n n n 1 n 2 n N p N p N p 1 N p 2 N p (5.25) From the form of F, whether shown in (5.25) or (5.24), it an be observed that the 1 st and 2 nd olumns of ΦFmust have the same form, whih demonstrates part b) of the proposition. From the form of F (where the last olumn ontains ones only) it an be reognised that the last olumn of the matrix of the elements of row index m of the matrix ΦF must be a vetor, in whih any element denoted m is the sum Φ (or equivalently the sum of elements of olumn number r of the matrix Φ ). his demonstrates part ) of the proposition. Further, from the formulation of be obtained as follows Φ in (5.11), and notiing that pn n 1, the matrix Φ an n ( Np ) n ( Np 1) n ( Np 2) n1 Φ (5.26) ( N p N) n ( N p N 1) n 1 Further, the multipliation of Φ in the formulation of (5.26) and F in the formulation (5.25), using the demonstrated parts b) and ) of (5.23), the following expression for the matrix M is obtained 125

138 (note that due to its size the formulation ontinues over two lines). M = M M M 1,2 n 2n 1... N p n 1 N p n 1 2 n... N p n 1 N p 1 2,2 n 1 2n 2... N p 1 n 1 N p n 2n 1... N p 1 n 1 N p 1 3,2 n 2 2n 3... N p 2 n 1 N p n 1 2n 2... N p 2 n 1 N p 1 M N p N,2 n N p N... N p N 1 n 1 N p n 1 N p N... N p N 1 n 1 N p 1 n ( n 2) 2n 1 ( n 2)... N p n 1 N p n ( n 2) Φ r... 1, n ( n 3) 2n 1 ( n 3)... N p 1 n 1 N p n ( n 2) Φ r... 2,... n ( n 4) 2n 1 ( n 4 )... N p r1 N p r1 N p N p 2 n 1 N p n ( n 2) Φ 3, r r1 (5.27) N p... n N p N 2n 1 N p N... N p N 1 n 1 N p n n 2 Φ N p N, r r1 126

139 From the above, it an be observed that for every 1 m n 1 M i max m, r i n m 1 r 2 i 1 where i1 max p 1 1 i N m n (5.28) he above ends the demonstration of part a) of (5.23) and the whole proposition. Remark 4: From the form of Φ presented in (5.26) it an be observed that in the ase when zeros will always be generated in the n N, Φ matrix, whih will always result in a zero vetor of the predited order quantities. herefore, n should always be assumed to be no greater than N p, and the onverse ase will not be onsidered in the thesis. p he next step is to find the final representation of u k K y k Kx k (5.29) y R n whih will allow the optimal order quantities to be found, but avoiding the extensive alulations of MPC at the same time. In (3.63) NpN K Φ Φ Φ F and (5.30) K K xk y 127

140 From the above it an be dedued that only the first row of the inverse matrix ΦΦ 1 is required to be obtained for further alulations. he following proposition shows the general form of the inverse matrix with respet to the values of n, N p and N values. Proposition 5: ΦΦ a b b (5.31) where bφ Φ( S 1, S) Φ Φ( S 3, S 1) 4 Φ Φ( S 1, S 2) a ΦΦ( S1, S1) 1 (, 1) Φ Φ S S Φ Φ ( S 1, S b 2) ΦΦ( S1, S1) 1 Φ Φ( S 2, S 1) Φ Φ( S 1, S 1) Φ Φ( S, S) Φ Φ( S 1, S 1) Φ Φ( S, S 1) Φ Φ( S, S 1) 128

141 where S N N p Demonstration With respet to (5.16) the demonstration serves to show that 1 I Φ Φ Φ Φ, where I is the identity matrix of appropriate dimension. he demonstration is shown in Appendix I. Remark 5a: From Proposition 5, it an be dedued that apart from four elements in the matrix ΦΦ 1 denoted a, b and, none of the matrix elements depend on the values of that partiular state spae model. N p or N values for Remark 5b: From the formulation in (5.31) it an be notied that NpN nn N N 2 ΦΦ (5.32) p p Proposition 6: n K n r d or r K r n (5.33) r r 2 an f 1 \1 Demonstration: NpN From the form of 1 have the following form: ΦΦ in Proposition 5, it an be notied that K will n b b b... b a 2a a a 2 a a... a 2a a n n 2n 3n (5.34) where a M ( mr, ) denotes an element of the matrix and mr th r olumn. From (5.23) part b) for the first element vetor of the ΦF orresponding to the th m row 129

142 b a 2a a a 2a a (5.35) and for any other vetor element br 1 b a 2a a (5.36) r 1r 2r 3r in respet to part a) and ) of (5.23). herefore, following the mathematial reformulation of suh defined elements it an be dedued that N 11 n1 N 21 n1 p b1 i n i 1 2 i n i 1 i1 i1 p N p 3 1 n 1 i1 i n i 1 N p n1 N p n N p n i n i i n i i n i i1 i1 i1 N n1 N n1 p p in i i N p nn p 1 N p n 1 N p 1 i1 i1 N p n1 N p n1 N p n1 p p 2 i n i 2 N n N i n i i i1 i1 i1 ( N n) N 1 N n 1 N 2 N n N p p p p p p N N nn n N nn N 2N 2nN p p p p p p p p and for any 1 r n 1 aording to part a) of (5.23) n (5.37) 130

143 3 1 n 1 N p 11 n1 N p 21 n1 br i n 11 r 2 i 1 2 i n 2 1 r 2 i 1 i1 i1 Np i n 31 r 2 i 1 i1 Np n1 i1 N p n1 N p n N p n i n i r i n i r i n i r i1 i1 i1 1 p p 1 p 1 p 1 1 i n i r N n n N n r N n n N n r Np N pn1 N pn1 in i r N p nn N p n r i n i r n1 i1 i1 i1 0 p p 1 p 1 p 2 2 p p 2 i N n N r N n N r N n N r N rn N nn rn n N rn 2N nn rn 2n N r p p p p p p p p p r r n rn n 2 2N p 2 N p N p 2 N p 2 4 n r 2 (5.38) and finally in respet to part ) of (5.23) and the form of Φ shown in (5.26) b N n 1 2( N 1 n 1) N 2 n 1 n p p p 1 (5.39) n ( n1) 2 whih demonstrates the proposition. 131

144 Remark 6: It an be notied that the vetor K defined in Proposition 6 is a speial ase of the vetor K shown in the IC Proposition. If the deterioration rate is not onsidered in the IC Proposition, k so that 1, the vetor K defined in Proposition 6 has the form nr1 i K 1 K 1 1 (5.40) n r i r \1 and i0 i1 n From the above it an be notied, that from now onwards the next order quantity an be immediately alulated by substituting the newly defined K, the referene inventory level and urrent inventory levels into the equation (3.60) K Kx u k y k k Sine u k is an inremental ation defined in (3.40) suh that 1 u k u k u k, i.e. herefore y R 1 u k K yyr k Kx k u k (5.41) Sine the system output yk for an inventory model refers to the urrent inventory level it an now be denoted as I k Sine the referene signal y IR R yk I k k, and from (5.41) it an be dedued that (5.42) k refers to the referene inventory level it an be denoted as 1 u k K yir k Kx k u k (5.43) From Proposition 6 and knowing that K y is the last element of the gain vetor 132

145 n n n 1 1 K K (5.44) y herefore, eventually, it an be dedued that 1 u k IR k Kx k u k (5.45) whih is the optimal order quantity (inventory system input) derived in the IC Proposition. Note that x defined in the IC Proposition is equivalent to that defined in (3.42), therefore x k xk y k k k k k k k [ x1 x1 1 x2 1 x2 x2 1 x2 (5.46) k 1 x k y k... xn n ] From (3.9) and (5.42) it an be finally dedued that x 1 1 I k I k u k n u k n ( ) u k n u k n u k u k I k (5.47) As defined in the beginning of the IC Proposition for the non-perishable ase. 133

146 5.2 Perishable Case Proposition 7: he general desription of following way A, an be formulated regardless of the values of p and n in the p n pn2 p p1 p2 p3 pm1 p i i i i i i1 i0 i0 i0 i0 A n pn2 1mn 0 k k k k k... k 1 or (5.48) 1mn p p1 p2 p3 pm1 p i i i i i i1 i0 i0 i0 i0 A k k k k k where any th m element of a vetor beomes zero only when p m 2 0 Demonstration From (3.46), and using (3.5) it an be dedued that k A and (5.49) k hen, for p 1, multiplying and A the last row of the matrix A is obtained as follows n1 A k (5.50) 134

147 whih an also be represented as 1 11 i i k k k i1 i0 (5.51) whih fullfils the poposition for the p 1 ase. Asummung, that the Proposition 7 is true for some 1 p n 2, then for p 1 it an be dedued that A A A p1 p p p1 p2 p3 pm1 i i i i i k k k k k i1 i0 i0 i0 i A p p p1 p2 pm2 i i i i i 1 k k k 1 k k... k i1 i1 i0 i0 i0 p1 p1 1 p2 1 i i i i1 i0 i0 i i k k k k k p3 1 ( pm1) 1 i0 i (5.52) whih satisfies the proposition for a given ase. Assuming now, that the Proposition 7 is true for some pn 2, then for p 1 it an be dedued that 135

148 A A A p1 p p p1 p2 p3 pm1 i i i i i 0 k k k k k... k i1 i0 i0 i0 i0 1A p p p1 p2 i i i i 0 1 k k k 1 k k... k 1 i1 i1 i0 i0 (5.53) p1 i i i k k k p1 1 p2 1 p3 1 i i1 i0 i0 i0 0 k k 1 whih satisfies the proposition for a onsidered ase and ompletes the demonstration the proposition. Proposition 8: For, p A and b defined in (3.46) pn1 p i p k when p n else i0 A b 2 0, A b 0, (5.54) n p where n is the order of the system desribed in (3.5). Demonstration: From (3.46) and using (3.5) it an be dedued as follows. 136

149 0 k , A b and (5.55) k From the formulation of b it an be notied that A b is the n th (seond last) element of the p vetor A. p As np, then if p n 2 p n 2 0 else A b p pn1 i i0 k (5.56) A b p 0 whih provides the statement of the proposition. Remark 7: It an be notied that Proposition 1 and Proposition 2 from setion 5.1 are essentially speial ases of Propoistion 7 and Proposition 8 when k 1. Remark 8: From the form of the matrix Φ shown in (5.11) it an be seen that A b are elements of the p matrix Φ and the exponent p depends on elements position in the matrix determined by the row (defined as m ) and olumn (defined as r ) indies, suh that p m r. herefore, from Proposition 8 it an be onluded that 137

150 mr n1 i Φ if m r n 2 then Φ m, r else mr, 0 (5.57) n where n is the order of the system desribed in (3.5). i0 From Remark 8 and the system desription in (3.5) the following proposition an be demonstrated. Proposition 9: N p n1 g jmr j i i nn 1, p N N n mrφφmr k k j0 i0 i0 m when m r where g r when m r (5.58) where the matrix Φ is defined in (3.54) and m and r denote the matrix row and olumn number respetively. Remark 9: i From now onwards, for simpliity and ease of desription, for any i, the ( k) will be i denoted as, whih should avoid onfusion. Demonstration: From Remark 8 the matrix Φ an be expanded in the following way 138

151 Φ m 1 n 1 Np 1 n1 i i i i i i0 i0 i0 i0 i0 0 1 m2n1 Np 2n1 i i i i i0 i0 i0 i0 0 m3n1 Np 3n1 i i i i0 i0 i0 N p ( N pn1) n1 i i0 N p ( N p N ) n1 i i0 (5.59) From the above, the matrix ΦΦ an be obtained suh that 139

152 ΦΦ i i0 i0 i0 0 N N p N n N n N N N n N N p p N n p p N n1 N n2 p 0 N N N n1 N n2 i i i i i i i i i i i i i i i i i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i N n p N n p N n1 p N n p N n2 p N n p 0 N N 1 p N n1 N n p i i i i i i i i i i i i i i i i i i i i i i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i 0 i N n1 p N n p N n1 p N n1 p N n2 p N n1 p 0 N N 2 p N n1 N n1 p i i i i i i i i i i i i i i i i i i i i i i i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i N n2 p N n p N n2 p N n1 p N n p N n2 p 0 N N 3 p N n1 N n 2 p i i i i i i i i i i i i i i i i i i i i i i i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 i0 (5.60) 140

153 herefore Φ Φ = N n p j j N n 1 p j j 1 N n 2 p j j 2 N n m 1 p j j mr N n2 j j N N 1 p N n1 j j N N 1 p i i i i i i i i i i i i j 0 i0 i0 j 0 i 0 i 0 j 0 i0 i0 j 0 i 0 i 0 j 0 i 0 i 0 j 0 i 0 i N n1 p j j 1 N n1 p j j N n2 p j j 1 N nm1 p j j mr N n2 j j N N 2 p N n1 j j N N 2 p i i i i i i i i i i i i j 0 i 0 i 0 j 0 i 0 i 0 j 0 i 0 i 0 j 0 i 0 i 0 j 0 i 0 i 0 j0 i0 i N n2 p j j 2 N n2 p j j 1 N n2 p j j N nm1 p j j mr N n2 j j N N 2 p i i i i i i i i i i j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 j 0 i 0 i 0 N n1 j j N N 3 p i j 0 i0 i0 i j 0 i 0 i 0 j 0 i 0 i 0 N nr 1 p j j r m N nr 1 p j j r m N nr 1 p j j r m N nm1 p j j N n2 j j N N 1r p N n2 j j N N r p i i i i i i i i i i i j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 i N n j j N N p N n j j N N j N N p N n j p N n j j N N m p N n j j N N N N p p N n j j N N N N p p i i i i i i i i i i i i j 0 i 0 i 0 j 0 i 0 i 0 j 0 i 0 i 0 j 0 i0 i0 j 0 i0 i0 j 0 i 0 i 0 N n1 j N N 1 j N N 2 j p N n 1 j p N n j i i i i j N N p N n j j N N m p N n j j N N N N p p N n j j N N N p p N i i i i i i i i j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 j 0 i 0 i 0 j 0 i 0 i 0 j 0 i 0 i 0 (5.61) 141

154 From the above the general formulation of ΦΦ a matrix element an be redued to: N p n1 g jmr j i i nn 1, p nnn p mrφφmr j0 i0 i0 m whenm r where g r whenm r (5.62) he above demonstrates the proposition. Remark 10a: From Proposition 9 it an be notied that the elements of the matrix ΦΦ depend on their row and olumn indies, m and r, respetively, the order of the original system n defined in (3.5), the predition horizon N p and the deterioration rate. Remark 10b: It an be notied that for N n 1 the matrix definition in (5.58) does not hold and that ase will not be onsidered in the thesis. Proposition 10: From matrix ΦF denoted as M=ΦFthe following hold m n N p n 1 N N p N p nm1 j jnmr i i a) mrnn 1, p nnn M mr p j0 i0 i0 N p nm1 j jn m1 i i b) mnn 1,1 p n N N M m p (5.63) j0 i0 i1 N p r1 m n Φ m r ) M, 1, 142

155 where the matries Φ and F are defined in (3.54) and m and r denote matrix row and olumn index respetively. Demonstration: Realling the representation of the matrix F shown in (3.54) F A 2 A 3 A N p A from (5.48) it an be notied that F m A m where m is a matrix row index, it an be observed that F i ( 1) i i i i i i i1 i i i i i1 i0 i0 n2 n3 n4 n5 0 i i i i i i1 i0 i0 i0 i0 n1 n2 n3 n4 1 i i i i i i1 i0 i0 i0 i0 n n1 n2 n3 2 i i i i i i1 i0 i0 i0 i0 N p N p N p N p N p n i1 i0 i0 i0 i0 (5.64) It an be notied that for row indies mn 2 there are no zero elements in the rows. 143

156 From the form of the matrix F shown in (5.64) (the last olumn ontains ones only) it an be reognised that the last olumn of the matrix sum of the elements of the m-th row of the matrix matrix Φ, equivalently). his demonstrates part ) of the proposition. Further multiplying the matrix matrix F (formulation shown in (5.64)) it may be dedued that ΦF must be a vetor of elements whih are the Φ (elements of the r-th olumn of the Φ (formultion shown in (5.59)) by the first olumn of the M m,1 0 n N n N p p i i i i... i0 i1 i0 i1 0 n1 N pn1 N p i i i i... i0 i1 i0 i1 0 n2 N pn2 N p i i i i... i0 i1 i0 i1 (5.65) 0 nn p N 1 N p N p N 1 N p i i i i... i 0 i1 i0 i1 So that M m,1 Np n j jn i i j0 i0 i1 Np ( n1) j jn1 i i j0 i0 i1 Np ( n2) j jn2 i i j0 i0 i1 N p ( N p N 1) j jnn p N 1 i i j0 i0 i1 (5.66) herefore m,1 his demonstrates part b) of the proposition. 1 j 1 N p n r jn r i i M (5.67) j0 i0 i1 Further, multiplying (5.59) by the remaining olumns of the matrix F shown in (5.64) it an be observed that for r 1 144

157 M mr,... N n p j j ( n1) N n p j j ( n2) N n p j j ( n3) N n p j j ( n1) (1n) i i i i i i i i j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 j 0 i0 i0... N n1 p j j n N n1 p j j ( n1) N n1 p j j ( n2) N n1 p j j ( n1) ( 2n) i i i i i i i i j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 N n2 p j j ( n1) N n2 p j j n N n2 p j j ( n1) N n2 p j j ( n1) (3n) i i i i i i i... j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 1 j ( 1) 1 1 j ( 1) 2 j ( 1) 3 i i i i N 1 N n N N 1 ( 1) i i p p j j n N N n p i... N n N N p p j n N N p N n N N p p j n N N p N n N p p j n N N p j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 j 0 i0 i0 i (5.68) i 145

158 herefore N p nm1 j jnmr i i M mr, (5.69) j0 i0 i0 his demonstrates part a) of the proposition and ends the demonstration of Proposition 10. he last step is to obtain the final representation of u k K y k Kx k y R n using ΦΦ and ΦF in the simplified desriptions shown in Proposition 9 and Proposition 10. his will allow the optimal order quantities to be found, avoiding at the same time the extensive alulations related to the multipliation of extensive matries Φ and F desribed in (3.54) for the system desribed in (3.5). As in (3.63) NpN K Φ Φ Φ F K and K xk y It an be notied that only the first row of the inverse matrix ΦΦ 1 is required to be obtained. he following proposition shows the general form of the inverse matrix in respet to the, n, N p and N values. Proposition 11: he inverse of the matrix N ΦΦ has dimension p N and has the following form 146

159 ΦΦ a b b. (5.70) 147

160 where b Φ Φ( S 1, S) Φ Φ( S 3, S 1) 4 Φ Φ( S 1, S 2) a ΦΦ( S1, S1) 1 Φ Φ( S, S 1) Φ Φ( S 1, S 2) b ΦΦ( S1, S1) 1 Φ Φ( S 2, S 1) Φ Φ( S 1, S 1) Φ Φ( S, S) Φ Φ( S 1, S 1) Φ Φ( S, S 1) Φ Φ( S, S 1) are salars and where S N N p Demonstration: Φ Φ Φ Φ to ensure Demonstration of the proposition makes referene to the produt of 1 that the identity matrix is obtained. he proess is shown in Appendix I. Remark 11: NpN ΦΦ (5.71) n N p n N N p the matrix ΦF is defined in (3.54) and m and r denote matrix row and olumn number respetively, and for a system desribed in (3.5), using Proposition 11, and Remark 11, the NpN general form of the vetor K Φ Φ Φ F was obtained and shown in Proposition 148

161 Proposition 12: n 1r n1 r nr1 n i i (5.72) 1 K and K i0 i1 Demonstration: Knowing the tripartite struture of the matrix urrent proposition, the following separate ases need to be onsidered: ΦF shown in Proposition 10, to demonstrate the i for r 1, overing the seond part of the urrent proposition ( K 1 ) with referene to part b) of Proposition 10 for 1 r n 1 and for r n 1 overing the first part of the urrent proposition ( r 1rn1 nr1 i K ), with referene to part a) and ) of Proposition 10 where n 1 is i0 the olumn index of the matrix ΦF. Based on Remark 11, starting from the seond part of the proposition, K the matrix M ( m,1), representing the first olumn of the matrix to be shown that 1 n i1 n i1 i and realling ΦF shown in (5.63), it needs i K M( m,1) (5.73) n i1 herefore, from (5.66) it needs to be shown that N p n j jn N p n j jn1 N p n j jn2 n i i i i i i i K (5.74) j0 i0 i1 j0 i0 i1 j0 i0 i1 i1 Defining z N p n, the priniple of mathematial indution is now used to show (5.74) above. As a first stage, onsider z 2, then 149

162 z j jn z j jn z j jn i i i i i i K 1 1 j0 i0 i1 j0 i0 i1 j0 i0 i1 n 1 n1 2 n2 n1 1 n2 n2 i i i i i i i i i 1 i1 i0 i1 i0 i1 i1 i0 i1 i1 n n1 n2 n1 n2 n2 i i 2 i i i i i1 i1 i1 i1 i1 i1 (5.75) n n1 n2 n1 n2 n i i i i i i i1 i1 i1 i1 i1 i1 n i1 i Assuming now that for some z 2 z j jn z j jn z j jn n i i i i i i i K 1 1 (5.76) j0 i0 i1 j0 i0 i1 j0 i0 i1 i1 then for z 1 it an be dedued that: K z1 j jn z j jn1 z1 j jn2 i i i i i i j0 i0 i1 j0 i0 i1 j0 i0 i1 1 1 z j jn z j jn z j jn i i i i i i 1 j0 i0 i1 j0 i0 i1 j0 i0 i1 (5.77) z1 zn1 z zn1 z1 zn1 i i i i i i 1 i0 i1 i0 i1 i0 i1 From the assumption in (5.76), ontinuing (5.77), the following is observed 150

163 K n z1 zn1 z zn1 z1 zn1 i i i i i i i i1 i0 i1 i0 i1 i0 i1 1 1 n zn1 z1 z z1 i i i i i 1 i1 i1 i0 i0 i0 n zn1 z1 z z z1 i i i i i i i1 i1 i0 i0 i1 i1 n zn1 i i z1 z1 i1 i1 n i1 i 0 n i1 i (5.78) whih, in terms of proof by indution, demonstrates that for every 2 z, 1 n i K. i1 Further, based on Remark 11, fousing on first part of the proposition and the ase when 1 r n 1 and realling the matrix 1 1 M ( mr, ) in (5.63), it needs to be shown that rn K r nr1 i M( m, r) (5.79) i0 herefore, from (5.68), it needs to be shown that K N p n11 j jn1 r N p n21 j jn2r N p n31 j jn3r i i i i i i j0 i0 i0 j0 i0 i0 j0 i0 i0 r 1 N pn j jn1 r N pn1 j jn2r i i i i p i i j0 i0 i0 1 j0 i0 i0 j0 i0 i0 N n2 j j n 3 r (5.80) nr1 i i0 151

164 Defining z N p n the priniple of mathematial indution is used to show the above. As a first stage onsider z 2, then z j jn1r z j jn r z j jn r i i i i i i K r j0 i0 i0 j0 i0 i0 j0 i0 i0 n1 r 1 n2r 2 n3r n2r 1 n3r n i i i i i i i i 1 i0 i0 i0 i0 i0 i0 i0 i0 n r 1 n r 2 n r 3 i i i i0 i0 i0 nr2 nr3 nr3 i i i 1 1 i0 i0 i0 n r 1 n r 2 n r 3 i i i i0 i0 i1 3r i0 i nr nr nr i 2 i i i0 i0 i0 (5.81) nr1 i i0 Assuming now that for some z 2 K z j jn1 r z 1 j jn2r z2 j jn3r i i i i i i j0 i0 i0 j0 i0 i0 j0 i0 i0 r 1 nr1 i i1 (5.82) then for z 1 it an be obtained: 152

165 z1 j jn1 r z j jn2r z1 j jn3r i i i i i i K r 1 j0 i0 i0 j0 i0 i0 j0 i0 i0 1 z j jn1 r z1 j jn2r z2 j jn3r i i i i i j0 i0 i0 j0 i0 i0 j0 i0 i0 i (5.83) z1 zn2r z zn2r z1 zn2r i i i i i i 1 i0 i1 i0 i0 i0 i1 From the assumption in (5.82) and ontinuing (5.83) the following is observed K nr1 z1 zn2r z zn2r z1 zn2r i i i i i i i i1 i0 i1 i0 i0 i0 i1 r 1 nr1 zn2r z1 z z1 i i i i i 1 i1 i1 i0 i0 i0 nr1 zn2r z1 i i i z z z1 i i i i1 i1 i0 i0 i1 ii nr1 znr2 i i 0 0 i1 i1 nr1 i1 i 0 (5.84) nr1 i i0 whih, in terms of proof by indution, demonstrates that for every 2 z, r nr1 i K. he last ase whih needs to be onsidered is r n 1. It needs to be shown that for suh defined r, nr1 i K r. Realling the matrix M ( mn, 1) defined in (5.63), representing the i0 last olumn of the matrix ΦF (olumn number n 1) it needs to be shown that for every n i0 153

166 K r n n n1 1 0 i i M( m, n 1) 1 (5.85) i0 i0 herefore, from (5.59) N p N N p N N p N n 1 2, r 1 2, r 3, r K Φ Φ Φ r1 r1 r1 N p n j N p n1 j N p n2 j i i i 1 j0 i0 j0 i0 j0 i0 (5.86) Defining z N p n, the priniple mathematial indution is used to show the above. As a first stage onsider z 2, then z j z1 j z2 j i i i n 1 1 K j0 i0 j0 i0 j0 i0 2 j 1 j 0 j i i i 1 j0 i0 j0 i0 j0 i i i i i i i 1 i0 i0 i0 i0 i0 i (5.87) 0 i i0 Assuming now that for some z 2 154

167 z j z1 j z2 j 0 i i i i n 1 1 K (5.88) j0 i0 j0 i0 j0 i0 i0 then for z 1 it an be obtained: z1 j z j z1 j i i i n 1 1 K j0 i0 j0 i0 j0 i0 z j z1 j z2 j z1 z z1 i i i i i i 1 1 j0 i0 j0 i0 j0 i0 i0 i0 i0 (5.89) From the assumption in (5.88) and ontinuing (5.89) the following is obtained 0 z1 z z1 i i i i n 1 1 K i0 i0 i0 i0 0 z1 z z1 z i i i i i i0 i0 i0 i1 i1 0 i0 i z1 z1 (5.90) 0 i0 i whih ends the demonstration of the proposition. Remark 12 From (3.63) K K x K y, where K y is the last element of the vetor K, it an be notied that as the dimension of K is equal to 1 K K. hen realling Proposition 12 n, n 1 y n n1 1 i K n (5.91) i0 herefore 155

168 n K 1 (5.92) y Remark 13: From Proposition 12 it an be notied, that from now onwards the next order quantity an be immediately alulated by substitution of the simplified desription of the vetor K, the referene inventory level and the urrent inventory levels into the equation shown in (3.60) suh that: K Kx y R u k y k k Sine u k is an inremental quantity defined in (3.40) suh that 1 u k u k u k herefore 1 u k K yyr k Kx k u k (5.93) Sine the system output yk for the inventory model refers to the urrent inventory level it an now be denoted as I k, suh that Sine the referene signal y IR R yk I k k, from (5.93) it an be dedued that (5.94) k refers to the referene inventory level it an be denoted as 1 u k K yir k Kx k u k (5.95) From equation (5.92) it an be eventually observed that 1 u k IR k Kx k u k (5.96) whih is the optimal order quantity (inventory system input) shown in the IC Proposition for the perishable ase. 156

169 5.3 Summary he demonstration of the mathematial equivaleny between the IC and IMPC methods has been systematially shown. o illustrate the development of the IC method, the mathematial demonstration has been presented initially for the non-perishable ase where ( k) 1 and then the perishable ase, where ( k) (0, 1], separately. In a logial sequene the simplifiation proedure of the initial MPC method for the inventory model has been dedued in a step by step manner, firstly for non-perishable and then for perishable onditions. Although, eventually, the non-perishable ase was elaborated to be a speial ase of the wider perishable ase, suh an organisation of the material of the hapter has enabled the reader to follow how the proposed IC method has been developed. herefore, all propositions established for demonstration of mathematial equivaleny of the non-perishable ase appeared to be the speial ases of the propositions established for the perishable ase. Eventually the final proposition has demonstrated the equivaleny of the gain vetor K from the IC Proposition with gain vetor K of IMPC. he proedure of obtaining the optimal order size for the urrent time instane k was elaborated. he proposed IC method an be used equivalently to IMPC, and an therefore signifiantly derease the omputational ost and omplexity and enable the wider aessibility of the OR ommunity. he findings presented in this hapter onstitute the main originality and novel work of the thesis. Equation Chapter (Next) Setion 1 157

170 6 RESULS OF PROPOSED MEHOD 6.1 Introdution In this hapter, simulation results of the proposed IC method are presented to justify the effiieny of the proposed method for the inventory appliation. In Setion 4.4 and Setion 4.5 the low-bound saturation of order levels were not onsidered (the orders were allowed to be lower than zero, returns). he order quantities in the urrent hapter are prevented from dropping below zero for the purpose of obtaining realisti simulation results of given inventory ontrol. Defining the profit funtion to be of the form: F P d I P I H I Pu (6.1) prof ot L L u ot where: d ot denotes the total number of goods sold in N periods I u denotes the total number of goods stored in the warehouse in N periods (the inventory level greater or equal than zero) I L denotes the total number of bakorders in N periods (the inventory level lower than zero) u ot denotes the total number of goods ordered in N periods P denotes the selling prie per unit P denotes the disounted selling prie per unit, where is a disount rate, appliable when the demand is not satisfied immediately and the ustomer has to wait for a delivery (bakorders) H denotes the holding ost per kept unit in an unit time P denotes the purhasing ost per unit. Using the profit funtion (6.1), it an be analysed to show how the simulation settings affet the profit. Based on the observations, the suggestions for improvement an be made. 158

171 he urrent hapter presents the simulation settings first, where, among others, the time varying and inventory level dependent deterioration rate are shown. he deterioration is assumed to proportionally inrease when the inventory level inreases. Indeed, this is true for some real life appliations, where the higher the inventory level, the greater hanes the stok will deteriorate. he first simulation test was run to show that the referene inventory level affets the system response harateristis. hen the defined profit funtion in equation (6.1) is used to test the influene of the referene inventory levels to profitability (for some hosen numerial values of osts and pries of the profit funtion). he following test is run for the most profitable inventory level aordingly to the test results. It shows the system behaviour with respet to different demand patterns to see if the IC ontrol performs aording to expetations. A more detailed disussion of the system behaviour is presented in the following test for inventory levels, order sizes and deterioration rate separately. 6.2 Simulation Settings In the urrent numerial example, the simulation parameters were set as follows: he lead time delay was set to n 5. he inventory was assumed to be no greater than 1000 items at any time (to assure the deterioration rate is never negative), the deterioration rate was assumed to be dependent on urrent inventory level in the following manner I( k ) if I( k ) 0 I k 1 otherwise (6.2) where the urrent value is dynamially substituted to obtain the ontrol vetor of IC Proposition of (3.66) suh that i i i i i K k k k k k 1 (6.3) i1 i0 i0 i0 i0 159

172 he unit osts and pries of profit funtion (6.1) were set as follows: P 2, 0.2, H 0.3, P 0.5 and the simulation was run for 200 periods. he different demand patterns used in the following tests have the same, or approximately the same (in the ase of randomness) mean values of 50 items (if the first 5 null values time instanes are not onsidered, while the mean alulation) or in general (onsidering the null values into mean alulation), as the demand in eah ase is set to zero for the first n 5 periods. Sine the mean value of demand is always the same regardless of demand pattern, one the profit funtion in (6.1) is used, the assessment of the profit an be onsidered. his way, the differenes in profits obtained and results in general depends only on different demand trends rather than total number of produts sold within the simulation period (as it is exatly the same for all demand patterns shown). It aounts for the fat that the warehouse does not sell any items until the first order reahes the warehouse. he five different demand patterns are presented in Figure 6-1 (all of them on the same sale) and in Figure 6-2 separately to inrease the visibility, respetively, as follows: onstant demand, seasonal demand pattern with two possible values only, seasonal pattern with more values allowed, random demand osillating around the mean value and finally the seasonal pattern with some randomness allowed. he demand, apart from the initial 5 time instanes, is onstant and equal to 50 items. Later in the hapter it is referred to as Demand 1. Demand 1 is used in the hapter for simulation of the system to show that the response of the inventory level is smooth, stable and the osts related to inventory overshoots are very limited. he seasonal demand of two allowed values is equal to 100 items (for 6-45 and time instanes) and 0 (for the remaining time instanes). It is later referred to as Demand 2. he hosen values of demand differ extremely (omparing to other tested demand patterns) therefore they enable the reader to see that the more sudden the hanges in demand, the more ost generated to the warehouse in terms of inventory overshoots (storage osts) as well as order overshoots (purhasing osts). he seasonal demand pattern with more values allowed is equal to 50 (for 6-60 and time instanes), and 100 items (for ). It is later referred to as Demand 3. he hosen values are less extreme than in the ase of Demand 2. herefore, they enable to see the differene in profit inrease in omparison to Demand 2. It enables to notie that the sudden 160

173 inrease of inventory is smoother than in ase of the Demand 2. herefore, the less the storage osts are generated. Also, omparing with Demand 2, the purhase osts are smaller, due to the less sudden inrease of order quantities, when they our. he random demand pattern is set to zero for the first 5 time instanes and then osillates around a mean value of 50 with a variane of 5 items. It is later referred to as Demand 4. Demand 4 has a relatively smooth pattern as the randomness allowed is relatively small. herefore, it enables to see that the profit is relatively high, as in ase of onstant demand. he inventory levels and orders are relatively smooth, whih does redue storage and purhasing osts. he seasonal pattern with some randomness allowed ontains the seasonal pattern of 50 (for 6-60 and time instanes), 100 items (for ) plus randomness of 0 as a mean value and variane of 5 items. It is later referred to as a Demand 5. It tends to highlight the differene in results obtained with omparison to Demand 3 an idential pattern, but the randomness or lak of randomness onstitutes the differene. Figure 6-1: 5 Demand patterns shown on one sale 161

174 Figure 6-2: 5 Demand patterns shown on separate sales 6.3 est 1: o illustrate how the referene inventory affets the system behaviour From the profit funtion in (6.1) it an be notied that the positive values of the inventory level (the atual number of stored goods) generate the storage osts whih, pereptibly, dereases the profit. herefore, it an be assumed that the lower the referene inventory level, the greater the profit for the organisation. On the other hand, the negative values of inventory 162

175 levels (the bakorders), are more likely to ompensate for the low value of referene inventory level, whih generates additional ost for the organisation. here are three different referene inventory levels tested: yr 0, y 100 and y 200 for the last (the fifth) demand pattern of r r Figure 6-2. Figure 6-3 shows the system response for eah of them on the same graph. Figure 6-4 and Figure 6-6 illustrate the inventory system response for eah of yr 0, y 100 and yr 200 separately to inrease the visibility. First the disussion of eah referene inventory level is onduted separately based on analysis of Figure 6-4 and Figure 6-6, later all of the results are ompared with eah other with referene to Figure 6-3. r Figure 6-3: System response for all three referene inventory levels 163

176 Figure 6-4 shows the system response to the referene inventory level set to zero (empty warehouse). As a result of demand a sudden inrease at 10 th time instane is notied where the order size suddenly overshoots to up to 300 items (due to inreased sale, more items must be replenished) around the 10 th day, to ompensate for the bakorders of 220 items (negative inventory) obtained in the inventory levels (sold/demanded items) at the same time instane. hen the orders onverge quikly (within 5 days) to approximately 50 items, whih orrespond to the demand value (50 items too) and whih drives the inventory levels bak to near referene value (zero) within lead time delay. For the times instanes between approximately 15 to 60 days, the order sizes osillate slightly around value of 50 items. hey do so in notieable orrelation to demand behaviour at the respetive time instanes. When the demand inreases slightly, the orders inrease respetively to ompensate for the inventory level redution (due to goods sale), and when the demand slightly dereases, the orders derease respetively as less goods were sold and the urrent inventory inreases slightly. When the demand suddenly inreases to 100 items at the 60 th time instane, the orders overshoot again to the level of 300 items, to onverge quikly (within 5 days lead time delay) to a level of approximately 100 items (whih orresponds to the demand value of 100 items too). Again it happens due to the inventory levels ompensation, whih, as it an be seen in Figure 6-4 drops to 200 items of bakorders when the demand suddenly inreases. Further, in orrelation to demand, at time instanes between 65 and 110, the orders slightly osillate near the value of 100 items (diretly proportionally to demand) and inventories slightly osillate near the referene value of zero items (inverse proportionally to demand). hen, as the demand suddenly dereases to just above zero, the orders suddenly derease to just above zero too, to redue the surplus inventory after the real time delay. If returns were allowed, the orders would have ompensated for the surplus inventory after the real time period of 5 days ompletely by dropping signifiantly below zero. At the same time instane, the inventory suddenly rises as fewer items were sold and due to the system delay the previous order quantities were still delivered to the warehouse. As returns are not allowed in the model, the orders ould not go below zero to ompensate for the surplus inventory and thus the inventory returns to the desired level of about zero at a slower rate than before. 164

177 Figure 6-4: System response to referene inventory of value zero items 165

178 For the next season of 40 days for time instanes (the orders stay at or slightly above zero) none or very little items are ordered to the warehouse. he inventory levels take about half of that period to ome bak to around the referene zero inventory level. hen, on the 150 th day, the orders overshoot to a value of 200 items due to a sudden inrease in demand to around 50 items. In this way they drive the inventory level (within time delay) bak to around referene level faster, as they dropped to 200 items of bakorders due to the sudden demand hange. hen the orders onverge within 5 days to the level near to 50 items, whih orresponds to the level of demand (50 items too). Subsequently, the orders keep osillating near the value of 50 items. he inventory is maintained around the referene level after the lead time delay. he demand hanges one more for the last 5 time instanes by dropping to just above zero. his auses the redution of orders to zero and inreases the inventory to around 180 items. In general it an be notied that the system responds in an expeted manner: the order quantity ontrols the inventory level through quik responses to demand hanges and drives them bak to the referene value usually with respet to a system delay. Figure 6-4 also presents the deterioration rate to onfirm that it behaves aording to expetations. Indeed it never exeeds one (as designed) and drops below one inversely proportionally to the inventory level. he signifiant drop below one (to about 0.7) ours in the time instanes, when the inventory overshoots to 200 items. Figure 6-5 shows the system response to the referene inventory level set to 100 items. With respet to hanges in demand, the order size suddenly overshoot to approximately 250 items at approximately the 10 th day (due to the inreased sale, more items must be replenished) in order to ompensate for the bakorders of about 150 items obtained in inventory levels (sold/demanded items) at the same time instane. It an be notied that the overshoot is less than in the ase when yr 0. he orders then onverge quikly (within 5 days) to the level around 50 items, whih orrespond to the demand value (50 items too) and whih drives the inventory levels bak to near the referene value (100 items) within the lead time delay. For the time instanes between approximately 15 to 60 days the order sizes osillate around value of 50 items. he osillations are slightly greater than in the ase of yr 0. Orders osillate in notieable orrelation to demand behaviour at the respetive time instanes. When the demand inreases slightly, the orders inrease respetively to ompensate for the inventory level redution (due to goods sale), and when the demand slightly dereases, the orders derease respetively as less goods are sold and the urrent inventory inreases slightly. 166

179 Figure 6-5: System response to referene inventory of value 100 items 167

180 When the demand suddenly inreases to 100 items at the 60 th time instane, the orders overshoot again to the level of 220 items (the order overshoot here is smaller than in the ase of yr 0 ), to onverge quikly (within 5 days lead time delay) to a level of approximately 100 items (whih orresponds to the demand value of 100 items too). Again it happens due to the ompensation in the inventory levels, whih, as it an be seen in Figure 6-5 drops to 200 items of bakorders when the demand suddenly inreases. Further, in orrelation to the demand at time instanes between 65 and 110, the orders osillate near the value of 100 items (diretly proportionally to demand) and inventories slightly osillate near the referene value of zero items (inverse proportionally to demand). he osillations of both, orders and inventories are greater than in the ase of yr 0. As the demand suddenly dereases to just above zero, the orders suddenly derease to just above zero too, to redue the surplus inventory after the real time delay. If the returns were allowed, the orders would have ompensated for the surplus inventory after the real time period of 5 days ompletely by dropping signifiantly below zero. At the same time instane the inventory would suddenly rise up to around 250 items as less items are sold and due to the system delay the previous order quantities are still delivered to the warehouse. As returns are not allowed in the model, the orders ould not go below zero to ompensate for the surplus inventory, and thus the inventory returns to the desired level of about 100 items slower than before. However the period of inventory level reovery is smaller than in the ase of yr 0. For the next season of 40 days for time instanes (the orders osillate just above zero) none or very little items are ordered to the warehouse). he osillations are signifiant here and are greater than in ase of yr 0. hen, on the 150 th day, the orders overshoot to a value of 200 items due to a sudden inrease in demand to around 50 items. his way they drive the inventory level faster (within the time delay) bak to around referene level, as they dropped to 120 items of bakorders due to a sudden demand hange. he orders then onverge within 5 days to the level near to 50 items, whih orrespond to the level of demand (50 items too). Subsequently, the orders keep osillating near the value of 50 items and the osillations are signifiant. he inventory is maintained around the referene level after the lead time delay with some osillations whih are greater than in the ase of yr 0. he demand hanges one more for the last 5 time instanes by dropping to just above zero. his auses the redution of orders to just above zero and inreases the inventory to around 180 items. In general it an be notied that the system responds in an expeted manner where the order quantity ontrols the inventory level through quik responses to demand hanges and drives 168

181 them bak to the referene value usually with respet to system delay. he osillations are always more signifiant here than in the ase of yr 0, whih an be refleted in the deterioration rates. Figure 6-4 also presents the deterioration rate to onfirm that it behaves aording to expetations. Indeed it never exeeds one (as designed) and if it does drop below one it is inversely proportionally to the inventory level. he signifiant drop below one (to about 0.7) happens in the time instanes, when the inventory overshoots to 200 items. he learly visible osillations just below one or near 0.9 are visible all over the simulation period, whih, in omparison to yr 0, shows even more learly that the inventory levels osillate here more. Figure 6-6 shows the system response to the referene inventory level set to 200 items. he relation between the behaviour of order sizes and the demand pattern is not learly notieable for the urrent ase. he order sizes flutuate signifiantly omparing to the previous ases. he inventory level osillations reah almost 400 items, but the bak orders never our. he deterioration rate osillates between one and 0.75 whih reflets the signifiant osillations in inventory levels. As a onlusion it an be notied that different referene inventory levels generate different responses of the system. Combining all the results together in Figure 6-3 it an be notied that the inventory level goes below zero the least (atually zero times) for yr 200 and the most often for yr 0. herefore bakorders do not appear at all for y 200, while they our relatively often for the ase where yr 0. Nevertheless, the inventory level beomes less stable for yr 200, while it is the most stable for yr 0 (whih in terms of the thesis objetive is an advantage). Also, respetively, the stok is the greatest for the ase of yr 200 and the lowest for the ase of yr 0. he aim of this thesis is to keep the inventory at a stable level therefore in this ase the yr 0 referene level an be reommended for the industrial purposes. In any ase the profit funtion desribed in (6.1) enables numerial analysis and assessment of the most ost effiient referene inventory level for a given unit ost for a given set of numerial values. r 169

182 Figure 6-6: System response to referene inventory of value 200 items 170

183 6.4 est 2: o find the most profitable referene inventory level able 6-1 represents the profit values obtained for a given numerial example. he profits were alulated separately for eah of the five given demand patterns shown in Figure 6-2 and for eleven different referene inventory level values as given in able 6-1. For eah referene inventory level, the mean profit value of the five respetive demand patterns is given in able 6-1 in order to simplify the analysis. referene inventory level y r Profit values mean Demand 1 Demand 2 Demand 3 Demand 4 Demand able 6-1: Profit values It an be observed that for eah referene inventory level, the profit differs with respet to the demand trend. It an be notied that for most ases of the referene inventory level, the best profits are obtained for Demand 4, (whih is the onstant demand with some randomness allowed), and Demand 1, (whih is the onstant demand). For the very low referene inventory levels the demand 1 is slightly more profitable than Demand 4, for higher inventory levels the Demand 4 generates slightly better profit then Demand 1 (see able 6-2). It is related to the fat that both of these demand patterns are smooth and do not ause signifiant inventory 171

184 overshoots or bakorders (whih both would generate extra osts for the warehouse). It an be also notied that Demand 3 and Demand 5 generate similar profits, smaller than Demand 4 and Demand 1, but greater than Demand 2. heir sudden hanges of value are smaller than in the ase of Demand 2, but greater than in ase of demand 4 and Demand 1. Also, it an be notied that regardless of the referene inventory level value, the lowest profits were obtained for Demand 2 (see able 6-2), whih is the demand pattern with two extremely different values. Based on the results obtained, it an be onluded that the more sudden and extreme hanges in demand pattern, the more ost is generated due to overshoots in inventory levels (more goods stored) and order quantities (more goods bought). able 6-2 presents the demand patterns in the order of profitability with respet to referene inventory levels. referene inventory level y r Profit 4 Profit 2 Profit 3 Profit 4 Profit Demand 4 Demand 1 Demand 5 Demand 3 Demand 2 90 Demand 4 Demand 1 Demand 5 Demand 3 Demand 2 80 Demand 4 Demand 1 Demand 5 Demand 3 Demand 2 70 Demand 4 Demand 1 Demand 5 Demand 3 Demand 2 60 Demand 4 Demand 1 Demand 5 Demand 3 Demand 2 50 Demand 4 Demand 1 Demand 5 Demand 3 Demand 2 40 Demand 4 Demand 1 Demand 5 Demand 3 Demand 2 30 Demand 4 Demand 1 Demand 5 Demand 3 Demand 2 20 Demand 1 Demand 1 Demand 5 Demand 3 Demand 2 10 Demand 1 Demand 4 Demand 5 Demand 3 Demand 2 0 Demand 1 Demand 4 Demand 5 Demand 3 Demand 2 able 6-2: Profit vs. Demand Figure 6-7 represents the monotonially dereasing funtion of a relationship of the mean profit to referene inventory level (the last olumn of able 6-1). herefore, it an be seen that for the onsidered numerial example, the bakorder assoiated osts, do not affet the profit as muh as the holding assoiated osts and yr 0 generates the highest profit for the organisation. 172

185 Figure 6-7: Mean profit in respet to referene inventory level herefore, in general, the profit funtion (6.1) an be used by the system designer to deide whih referene inventory level is the most profitable for the organization for real unit osts and pries of a ase senario, or an help to deide the unit osts and prie adjustment for an arbitrarily seleted inventory referene level. In the onsidered numerial example, setting the referene value to zero is reommended. herefore further experiments are onduted for that partiular value only. 6.5 est 3: o illustrate how demand pattern affets system behaviour he simulation was run for the reommended inventory referene level of zero items. he urrent setion presents results of optimal order quantities, inventory levels and deterioration rate values, respetively, to eah demand pattern. Eah of the mentioned results are presented twie, one on a ommon sale for all demand patterns and the seond time on separate sales for eah demand pattern. Figure 6-8 represents the optimal order quantities for all onsidered demand patterns on one sale, while Figure 6-9 represents the same results on separate sales for eah demand to inrease the visibility of the results. 173

186 Figure 6-8: Order quantities for 5 demands on one sale Observing Figure 6-8 and Figure 6-9 it an be notied that order quantities are always above zero (aordingly to the model design) regardless of the demand pattern. It an be observed that the orders overshoot when the demand suddenly hanges from one value to another and the more the sudden hanges, the higher the overshoots. For instane, Demand 2 generates higher overshoots (at 5 th and 100 th time instanes) in orders sizes (up to 600 items) than Demand 3 (up to 300 items, at 10 th, 60 th and 150 th time instane). It happens beause the differene in demand level at Demand 2 (see Figure 6-1or Figure 6-2) between time instanes no 5 and no 100 is 100 more (100 items) than at demand 3 for time instanes no 10, 60 and 150 (50 items only). It explains partially the reason of Demand 2 being less profitable than demand 3, as disussed in Setion 6.4. In Figure 6-8 and Figure 6-9 it an also be observed that the overshoots of Demand 5 are slightly smaller (about 250 items) at time instanes: 10, 60 and 150 than Demand 3 (about 300 items) for the same time instanes. Although the seasonal fator of the demand pattern in both of them are idential, the random fator is the fator whih differs the two demands from eah other (see Figure 6-1or Figure 6-2). It an be onluded 174