MAGISTERARBEIT. Titel der Magisterarbeit. Verfasserin. Sally Tischer, Bakk. Angestrebter akademischer Grad

Transcription

1 MAGSERARBE iel der Magiserarbei Losizing and Scheduling Problems - A Review Verfasserin Sally ischer, Bakk. Angesreber akademischer Grad Magisra der Sozial- und Wirschafswissenschafen (Mag. rer. soc. oec.) Wien, im Okober 2007 Sudienkennzahl l. Sudienbla: A Sudienrichung l. Sudienbla: Beriebswirschafslehre Bereuer: O.Univ.Prof.Dipl.-ng.Dr. Richard F. Harl

2

3 . ABLE OF CONENS. LS OF FGURES...V. LS OF ABLES...V NRODUCON.... Producion Planning and Conrol Sysem Producion Planning Producion Conrol LOSZNG AND SCHEDULNG Losizing Scheduling he machine environmen (α ) he ob characerisics ( β ) he opimaliy crierion (γ ) Relaionship beween losizing and scheduling OVERVEW OF LOSZNG AND SCHEDULNG PROBLEMS Classificaion of problems SNGLE-LEVEL UNCAPACAED LOSZNG AND SCHEDULNG PROBLEMS Economic Order Quaniy Model (EOQ) Wagner-Wihin Model (WW) Single-level Capaciaed Losizing and Scheduling Problems Capaciaed Losizing Problem (CLSP) Capaciaed Losizing Problem wih Linked Losizes (CLSPL) Discree Losizing and Scheduling Problem (DLSP) Coninuous Seup Losizing Problem (CSLP) Economic Losizing and Scheduling Problem (ELSP) Proporional Losizing and Scheduling Problem (PLSP) General Losizing and Scheduling Problem (GLSP) GLSP CS GLSP LS...45

4 5.7 General Losizing and Scheduling Problem wih Seup imes (GLSPS) Baching and Scheduling Problem (BSP) MUL-LEVEL LOSZNG AND SCHEDULNG PROBLEMS (MLLSP) MLLSP - Srucure MLLSP - Example LOSZNG AND SCHEDULNG PROBLEM WH MULPLE MACHNES (LSPMM) Muliple Machine ypes LSPMM - Example Muli-level, muliple machine example HERARCHCAL NEGRAON OF LOSZNG AND SCHEDULNG COMPARSON OF HE DFFEREN MODELLNG APPROACHES Single-Level Single-em Uncapaciaed Models Single-Level Muli-em Capaciaed Models SOLVNG MEHODS FOR LOSZNG AND SCHEDULNG PROBLEMS SUMMARY APPENDX V. REFERENCES ABSRAC... 8 CURRCULUM VAE V

5 . Lis of figures Figure : he PPS srucure...2 Figure 2: he producion planning srucure...2 Figure 3: he producion process srucure...4 Figure 4: he producion conrol srucure...5 Figure 5: he relaionship beween losize and coss...6 Figure 6: he relaionship beween losizing and scheduling...0 Figure 7: An overview of he losizing and scheduling problem... Figure 8: Rolling Horizon...4 Figure 9: Characerisics of small bucke problem, big bucke problem and CLSPL...25 Figure 0: Gan char for he DLSP example...32 Figure : Gan char for he CSLP example...34 Figure 2: Gan char for he PLSP example...39 Figure 3: A serial produc srucure...50 Figure 4: An assembly produc srucure...50 Figure 5: An aborescen produc srucure...5 Figure 6: A general produc srucure...5 Figure 7: Gozinho-srucure...57 Figure 8: Gan char for he muli-level, muli-machine LSP example...58 Figure 9: Simulaed annealing flowchar...68 Figure 20: abu search flowchar...70 V

6 . Lis of ables able : Soluion for he WW example...20 able 2: Daa for he CLSP example...23 able 3: Soluion for he CLSP example...23 able 4: Soluion for he CLSPL example...27 able 5: Daa for he DLSP example...3 able 6: Soluion for he DLSP example...3 able 7: Soluion for he CSLP example...34 able 8: Soluion for he PLSP example...38 able 9: Daa for he muli-level, muli-machine LSP example...58 able 0: Single-level single-iem uncapaciaed model condiions...60 able : Single-level single-iem uncapaciaed model complexiy...60 able 2: Single-level muli-iem capaciaed model condiions...6 able 3: Single-level muli-iem capaciaed model complexiy...63 able 4: Single-level muli-iem capaciaed model solvabiliy...65 able 5: Single-level muli-iem capaciaed model complexiy (mahemaical noaion)...75 V

7 nroducion oday every cusomer wans o have low-priced producs wih good qualiy. his increases he cos pressure of he producers, and because of ha cossaving producion is very imporan for hem o survive among he compeiors. So producers ry o improve heir producion processes. is no possible o opimize he whole process for reasons of complexiy, only pars can be looked a. his diploma hesis looks a he losizing and scheduling problems. Losizing and scheduling are imporan pars of he producion process sysem. is necessary o know how many iems and in which order hey have o be produced, o minimize he oal coss. n he conex of his work an overview of he losizing and scheduling problems and he solving mehods are given. he problems are described hrough definiions, mahemaical models and examples. n chaper. an ouline is given of he producion planning and conrol sysem (PPS). hen he definiions and he relaionship of losizing and scheduling are specified in chaper 2. n chaper 3 a shor overview of he losizing and scheduling problems is noed. he differen problems follow hen in chaper 4 (single-level uncapaciaed losizing and scheduling problems), chaper 5 (single-level capaciaed losizing and scheduling problems), chaper 6 (mulilevel losizing and scheduling problems) and chaper 7 (losizing and scheduling problems wih muliple machines). n chaper 8 a hierarchical inegraion is represened. Also a comparison of differen models is illusraed in chaper 9. A las he solving mehods are menioned in chaper 0.

8 . Producion Planning and Conrol Sysem he producion planning and conrol sysem, shor PPS, is divided ino wo componens (see figure ), which are producion planning and producion conrol. he producion planning plans procedures in advance (shor and medium erm planning), and he producion conrol is responsible for he order clearance and order conrol on he basis of he producion planning. Producion Planning and Conrol Producion Planning Producion Conrol Figure : he PPS srucure.. Producion Planning he producion planning consiss of hree pars (see figure 2), he producion program planning, he provision planning and he producion process planning. Producion Planning Producion Program Planning Provision Planning Producion Process Planning Figure 2: he producion planning srucure hese pars can be divided ino sraegic, acical and operaional producion planning. he sraegic par conains of he long erm decisions from he producion program planning and provision planning. Domschke, Scholl, Voß, 993, p. 8 2

9 here are decisions o make abou: wha o produce (assormen) where o produce (locaion) how o produce (mehod) wih wha o produce (facors of producion) he acical producion planning is he deailed deerminaion of he producion program concerning amoun of differen produc varians, and medium erm adusmen of capaciy for example machines and personal. he operaional par consiss of he shor erm program planning, he supply of maerials and he producion process planning.... Producion Program Planning he producion program planning deermines he ype, he amoun and he producion dae of he primary requiremens. ells which producs in wha amoun are going o be produced in a cerain ime. can be divided on he basis of mauriy ino poenial and acual producion program. he poenial one consiss of long erm decisions for wha producs should be produced by looking a he business obecive. he acual producion program defines by looking a he poenial producion program and facors of producion he in realiy o produced producs (ype, amoun and imeframe). Here also he business obecives are o be considered....2 Provision Planning he provision planning acquires he amoun of secondary and eriary requiremens for he primary requiremen. hese producion facors are looked a by qualiy, quaniy and dae of supply. he producion facors can be for example: resources (machines, faciliies) personal maerials 3

10 ...3 Producion Process Planning he producion process planning looks a he inpus of acual producion program and he provision planning, wha makes i possible o plan he precise producion orders. mporan poins o look a are: Which orders can be processed ogeher? Wha maerials are used? n which sequence can he orders be planed? ime regulaion of he producion of he orders? he producion process planning is divided ino hree pars (see figure 3):. Losizing (see secion 2.) 2. Lead ime scheduling and capaciy scheduling: he lead ime scheduling plans he earlies and laes ime for processing he orders. he capaciy scheduling looks if he needed capaciies for he producion program are available. 3. Scheduling (see secion 2.2) Producion Process Planning Losizing Lead ime Scheduling and Capaciy Scheduling Scheduling Figure 3: he producion process srucure 4

11 ..2 Producion Conrol he producion conrol consiss of wo componens (see figure 4), he order release and he order monioring. Producion Conrol Order Release Order Monioring Figure 4: he producion conrol srucure..2. Order Release he funcion of he order release is o sar he producion of he orders, which are incoming from he producion planning sage Order Monioring he order monioring checks he producion flow, by looking a he response from he acual daa of producion. he acual daa is compared wih he arge daa. When here are any variaions beween he acual and he arge daa, hen for example he producion amoun or producion deadline has o be examined. n his work we are looking a he producion process planning, in special a losizing and scheduling. 5

12 2 Losizing and Scheduling his chaper gives he definiions of losizing and scheduling and poins ou he relaionship beween each oher. 2. Losizing Losizing is he aciviy o ransform cusomer demands ino los o minimize he oal coss of seup coss, produc change coss and invenory coss. A lo is an amoun of equal iems produced on a machine one afer he oher wihou changing he seup sae. he amoun of iems in a lo is called losize 2. he relaionship beween losize and coss 3 : he number of seups of an iem depends on he size of he lo. he greaer he lo is he smaller is he number of seups and he seup coss per iem. On he oher side he invenory coss are increased. So he seup and invenory coss have o be balanced o minimize he oal coss (see figure 5). Noe: he minimal oal coss are found in he inercep poin of he ordering and he holding coss. his is only valid for he EOQ model which has linear holding coss and fixed ordering coss. Figure 5: he relaionship beween losize and coss 4 2 Haase, 994, p. 3 Haase, 994, p. 2 4 hp://en.wikipedia.org/wiki/eoq,

13 2.2 Scheduling Scheduling is he aciviy o decide when and on which machine differen iems should be produced so ha here are no shorages in he producion plan. gives he sar and finish ime of every ob and he ob sequence of each machine 5. Scheduling problems are specified hrough hree aribues 6. hese are: he machine environmen (α ), he ob characerisics ( β ) and he opimaliy crierion (γ ) he machine environmen (α ) he machine environmen has wo global parameers. One ells on which machine each ob has o be processed, and he oher gives he amoun of machines. he ypes of machine environmen 7 are: Single machine n his environmen here is only one machine available ha can process he obs. Every ob is done on he same machine. Parallel machine Here here are muliple machines, which work parallel, available o process he obs. Every ob can be done on every machine. 5 Quad, 2004, p. 2 6 Brucker, 995,p. 2 7 hp://rio.ieor.berkeley.edu/~vinhun/machine.hml,

14 Flow Shop here are a series of machines. he firs ask of every ob is done on he firs machine, he second ask on he second machine and so on. Every ob goes hrough all machines in a one-way order. ob Shop Here here are a se of machines. Every ob can go hrough he machines in a differen, bu given order. he flow of he asks of a ob does no have o be one-way, so each ob can use a machine more han once. Open Shop Here here is a se of machines, oo. he flow of he asks is given, bu when and on which machine he ask has o be done is no given he ob characerisics ( β ) here are six ob characerisics ( β, β 2, β 3, β 4, β 5, β 6 ), which build he se β 8 :. β is he ob characerisic ha ells if ob spliing is allowed. ha means ha a ob can be inerruped and hen goes on a a laer ime or even on anoher machine. When ob spliing is allowed hen when no hen β does no appear in β. β = pmn, and 2. β 2 describes he precedence relaions beween obs. hese relaions can be represened hrough differen precedence graphs. f β 2 = prec hen he relaion is shown by an acyclic direced graph. When hen he graph is a ree. For β 2 = ree β = sp graph he relaion is given by a 2 series-parallel graph. When here is no precedence hen β 0 and i does no appear in β. 2 = 3. β 3 ells if here are release daes for each ob. When β = r release 3 daes are specified, bu when r = 0 he ob characerisic β 0 and so i does no appear in β. 3 = 8 Brucker, 995,p. 3 8

15 4. β 4 gives he resricions on he processing imes or on he number of operaions. f β 4 is p = hen every ob has processing ime, bu when β = 0 4 i does no appear in β. i 5. β 5 ells if here is a deadline for a ob. When β 5 = d hen a deadline is specified for a ob bu when β 5 = 0 hen i is no in he se β. 6. β 6 indicaes ha here is a baching problem, wha means ha a number of obs have o be summarized o a bach and hen scheduled ogeher. n he bach all obs have o be finished a he same ime as he finish ime of he las bach ob. he number of obs can reach from one o n obs. When β = bach hen a baching problem occurs, oherwise if 6 β 6 = 0 i does no appear in β he opimaliy crierion (γ ) 9 he opimaliy crierion is o find a feasible schedule of he obs ha minimizes he oal coss. here are wo ypes of oal cos funcions, he boleneck obecives and he sum obecives. Boleneck obecive funcions are for example makespan and maximum laeness. A sum obecive is for example he oal flow ime. he makespan is he compleion ime of he las ob. he oal flow ime is he oal sum of he compleion imes of all obs. Laeness is he difference beween he compleion ime and he due dae of a ob. When he compleion ime exceeds he due dae uni penaly coss arise. i 9 Brucker, 995,p. 6 9

16 2.3 Relaionship beween losizing and scheduling Losizing and scheduling decisions depend on each oher. Losizing is concerned wih he produc quaniies and scheduling wih he machines and produc sequences. Losizing looks a he number of iems of a produc which can be produced per period. his depends on he seup imes, which again depend on he assignmen of he machine and he produc sequence which are fixed by scheduling. Reverse scheduling needs he producion volume o decide abou he machine assignmen and he sequence 0. he relaionship is shown in figure 6. Losizing: Wha quaniies in which period? c b a a c c b c a Scheduling: Which machine and sequence? Seup b Seup a Seup b Seup c c Seup c Seup a a a Figure 6: he relaionship beween losizing and scheduling 0 Quad, 2004, p. 3 Quad, 2004, p. 4 0

17 3 Overview of losizing and scheduling problems Firs of all a general problem descripion of he losizing and scheduling problem is given. Losizing and scheduling problems have he obecive o make an opimal decision abou he losizes of he iems and also he sequence of he iems on he machine o make he producion plan feasible. n he losizing par here are iems wih a given demand ha have o be produced on a producion faciliy. he problem consiss in deermining he iem quaniy ha has o be produced in each period of a given ime horizon for every demanded iem. he demand has o be saisfied while remaining wihin he capaciy of he producion faciliy and looking a he obecive which is he minimizaion of coss (producion coss, invenory coss and/or se-up coss). n he scheduling par decisions are made abou when and on which machine he iems (obs) are processed. Each ob has a release dae and a due dae. he aim of he scheduling problem is o find a feasible plan, wha means he sequencing of he iems, which minimizes he coss from he losizing problem. An overview of he losizing and scheduling problem is shown in figure 7: Figure 7: An overview of he losizing and scheduling problem 2 2 hp:// Sizing and Scheduling- Problem,

18 here are many losizing and scheduling problems named in he lieraure. Before he deailed models are going o be explained a shor overview: One of he firs works on losizing was he economic order quaniy model (see chaper 4.) by Harris in 93 which is based on consan demand. n 958 he Wagner-Wihin model (see chaper 4.2) for dynamic demand followed. An ongoing developmen of his model is he Capaciaed Losizing Problem. his is a capaciaed model where only one iem can be produced per period, and he seup can only occur a he beginning of a period or no a all. A seup carryover is no possible. he model is shown in chaper 5.. he Capaciaed Losizing Problem wih Linked Losizes is an exension of he Capaciaed Losizing Problem. Here a seup can be carried-over from one period o he nex one. is explained in chaper 5... he Discree Losizing and Scheduling Problem has a capaciy consrain, he so called all-or-nohing-assumpion. he consrain means ha an iem can only be produced over a full period or no a all. he model is defined in chaper 5.2. A similar model is he Coninuous Seup Losizing Problem, bu here he all-or-nohing-assumpion is given up, wha means ha an iem has no o be produced over a full period anymore. More deails o his model in chaper 5.3. he Economic Losizing and Scheduling Problem is a model ha shall find cyclical producion schedules. Here no wo iems can be produced a he same ime. More deails on his model in chaper 5.4. Anoher developmen followed 994, he Proporional Losizing and Scheduling Problem. n his model he seup can ake place a he beginning or he end of a period and capaciy spliing is possible. he model is shown in chaper 5.5. n 997 he General Losizing and Scheduling Problem by Fleischmann and Meyr was inroduced. n his model he periods are divided ino sub-periods and because of ha more han one seup per period is allowed. he model is defined in chaper 5.7. Based on his problem Meyr hen developed he General Losizing and Scheduling Problem wih Seup imes in is an exension of he General Losizing and Scheduling Problem where seup coss are included ino he model. As shown in chaper 5.8. All of he problems named above are single-level problems, wha means ha only one producion level is considered. here is also lieraure on losizing and scheduling models for muli-level (see chaper 6) and muliple machines (see chaper 7) problems. Furher on here is looked a hierarchical inegraion of losizing and scheduling (see chaper 8) by Dauzere-Peres and Lasserre. 2

19 3. Classificaion of problems depends on he losizing and scheduling model which problem crieria s are imporan. Below some of he crieria s are lised 3 : Demand here are several demand ypes 4 : saic demand dynamic demand (saic: demand is saionary or consan, he value does no change over ime; dynamic: he value changes over ime) deerminisic demand - sochasic demand (deerminisic: value of he demand, which can be saic or dynamic, is known in advance; sochasic: value of he demand is no known in advance, i is random) independen demand dependen demand (independen: one iems demand does no depend on decisions of an oher iems demand, he demand comes from ouside for example cusomer demand; dependen: demand of one iem depends on he demand of he previous iem, here is a relaionship beween he iem demands) Problems wih dynamic demand and dependen demand are more complex han problems wih saic demand or independen demand. Number of iems here are single-iem and muli-iem problems. n single-iem problems here is only one end iem, he final produc. While in muli-iem problems here are several end iems. he complexiy of muli-iem problems is higher han from he single-iem problem. Number of levels here are single-level and muli-level problems. n he single-level problem he end produc is direcly produced from he raw maerial, or for 3 Haase, 994, p. 3 4 Karimi e al., 2003, p

20 simplificaion he inermediae seps from he sar of he producion o he end are disregarded. Only one operaion is needed. he demand herefore is independen. n he muli-level problem he raw maerial is changed in more hen one operaion o a final produc. here are so called paren-componen relaionships beween he iems. he demand a one level depends on he demand of anoher level, he paren level. his demand is dependen. Muli-level problems are more difficul o solve han single-level problems. hese problems have muli-level produc srucures. Generally here are four srucures, he serial produc srucure, he assembly produc srucure, he arborescen produc srucure and he general produc srucure (see chaper 6.). Planning horizon he planning horizon is he lengh of ime which a producion schedule plans ino he fuure. he horizon can be finie (a given number of periods wih he same lengh) wih a dynamic demand or infinie (an endless operaing procedure is assumed) wih a saionary demand. Anoher planning horizon is he rolling horizon. here only he firs period is implemened. hen an updae of he demand akes place. Wih his informaion he producion plan is calculaed again for he same number of periods. ha akes place a every beginning of a new period (see figure 8) 5. 5 Dauzere-Peres, Lasserre, 994, p.94 6 see foonoe 5 Figure 8: Rolling Horizon 6 4

21 ime scale he ime scale can be discree or coninuous. he discree ime scale can be for example small like hours or shifs, or large like weeks or monhs. here are wo caegories for he ime periods, which are big bucke and small bucke problems. A big bucke problem has long ime periods where several iems can be produced in each period, while a small bucke problem has shor ime periods where only one iem can be produced per period. he coninuous ime scale models are he Economic Order Quaniy Model (see chaper 4.) and he Economic Losizing and Scheduling Problem (see chaper 5.4). he periods of he coninuous ime scale models are real, for example 2.7 periods. Relevan coss Relevan coss are for example seup relaed coss, invenory relaed coss and capaciy relaed coss. Seup relaed coss are coss ha arise when a seup akes place for an iem on a machine. hese coss can be iem specific coss, sequence dependen coss, sar up coss or reservaion coss. nvenory relaed coss are coss for keeping an iem in invenory during one or more periods. hese coss can be holding coss or shorage coss, which are coss ha resul from no having invenory, or no enough invenory. nvenory relaed coss are capial coss, hey reduce he capial. Capaciy relaed coss are coss for using he given capaciy or when needed exra capaciy. n he second case an example is overime coss. Capaciy or resource consrains here are wo kinds of problems. One is he uncapaciaed problem where no resricions are and he oher is he capaciaed problem where capaciy consrains are saed. Capaciaed problems are more complex han uncapaciaed ones. Resource consrains are for example machine capaciy, ransporaion capaciy and he number of workers. 5

22 Seup srucure here are wo seup srucures, he simple and he complex srucure. When he seup ime and seup coss of a period are independen of he sequence of he previous period he seup srucure is called simple, bu when hey are dependen he srucure is called complex. he complex seup srucure is more difficul o solve han he simple one. Service policy Here are decisions made in case of arising shorages. hen for example backlogging, backorders or sockous can be used. Problems wih shorages are more complex han wihou shorages. Obecives he mos common obecive is o minimize he oal coss. Oher ones are for example maximizaion of he service level or maximizaion of he capaciy uilisaion. 6

23 4 Single-level Uncapaciaed Losizing and Scheduling Problems his chaper gives a deailed review of he single-level uncapaciaed losizing and scheduling models menioned in chaper Economic Order Quaniy Model (EOQ) 7 he Economic Order Quaniy Model, developed by Harris in 93, defines he opimal producion quaniy ha minimizes he oal coss of invenory. Harris appoined he problem as How many pars o make a once 8. he model assumes no capaciy consrains, infinie planning horizon, coninuous ime scale, consan demand, no invenory shorages, no backlogging and consan seup and invenory holding coss. he Economic Order Quaniy Model is a single-level single-iem uncapaciaed losizing model. he variables are: q * = opimal order quaniy D = demand of he produc s = coss per order h = holding cos per uni and per monh C = oal coss he formula for he opimal order quaniy ( q *) is 9 : D q C ( q) = * s + * h q 2 d( C( q)) dq D * s h = + q 2 q* = 2* D * s h 7 Salomon, 99, p Harris, 990, p Skripum Grundzüge der BWL V, WS 97/98, p. 58 7

24 he graphical presenaion of he opimal order quaniy is shown in figure 5 (page 6). An example for he EOQ model is given by: here is one produc wih demand D, ordering cos s and holding cos h which are given below. D = 00 s = 200 h = he opimal order quaniy q * is: 2 *00 * 200 q * = = 200 8

25 4.2 Wagner-Wihin Model (WW) he Wagner-Wihin Model, developed by Wagner and Wihin in 958, is an exension of he Economic Order Quaniy Model. is a model wih a finie ime horizon, no capaciy consrains and he demand is dynamic. is a dynamic algorihm for a single-level single-iem uncapaciaed losizing model. he Wagner-Wihin Model finds an opimal soluion, by using a dynamic programming algorihm 20. he decision variables are: x = producion quaniy in period = invenory a he end of period he parameers are: d = demand in period = number of periods v (x) = producion coss of x unis in he period h () = invenory coss of unis a he end of he period he mahemaical formulaion of he WW model is 2 : Min ( v ( x ) + h ( )) = he obecive (4.) is o minimize he producion coss and invenory coss. (4.) = + x d,..., he equaion (4.2) shows he invenory balance. = (4.2) = 0 = 0 (4.3) his equaion (4.3) ells ha he saring and he ending invenory is zero. 20 Salomon, 99, p Salomon, 99, p. 24 9

26 x 0 =,..., (4.4) Here he non-negaiviy condiion (4.4) is given. An example for he WW is given by 22 : here is one produc wih demands d ( = 4 periods), ordering cos s and holding cos h which are given below. d = (20, 40, 20, 30) s = 70 h = he soluion is given in able : E able : Soluion for he WW example 23 he opimal order quaniies q and he oal coss RC are: q = (60, 0, 50, 0) RC = Skripum Grundzüge der BWL V, WS 97/98, p see foonoe 22 20

27 5 Single-level Capaciaed Losizing and Scheduling Problems his chaper gives a deailed review of he single-level capaciaed losizing and scheduling models menioned in chaper Capaciaed Losizing Problem (CLSP) he Capaciaed Losizing Problem is a single-level muli-iem capaciaed losizing model. is an exension of he Wagner-Wihin Model wih capaciy consrains. he CLSP is a large bucke problem 24, also called big bucke problem, because several iems requiring differen seup saes can be produced in each period. he planning horizon is finie, demand is dynamic and here are no shorages allowed. he ime scale is weeks or monhs. he CLSP is based on a fundamenal assumpion 25 : here is exacly one seup per period for each iem which is produced in he period. he seup occurs a he beginning of he period. f no hen here is no seup a all. ha means ha a lo never coninues over periods, and ha a he beginning of every period a new seup has o occur. he decision variables for he CLSP are: = invenory for iem a he end of period q = producion quaniy for iem in period x = binary variable which ells if a seup for iem occurs in period or no he parameers for he CLSP are: C = capaciy of machine in period d = demand for iem in period h = holding coss for iem 0 = invenory for iem a he beginning period = number of iems 24 Saggemeier, Clark, hp:// Fleischmann, 990, p

28 p = capaciy needs for he producion of one uni of iem s = seup coss for iem = number of periods he mixed-ineger model for he CLSP can be formulaed as followed 26 : Min ( s x + h ) = = (5.) he obecive (5.) of he Capaciaed Losizing Problem is o minimize he oal coss. = ( ) + q d,..., = =,..., (5.2) he equaion (5.2) shows he invenory consrain. p q C x =,..., =,..., (5.3) his resricion (5.3) shows ha an iem can only be produced when he machine is seup for his iem. p q = C =,..., (5.4) Here he capaciy consrains (5.4) are menioned. x {0,} =,..., =,..., (5.5) he seup variables (5.5) are binary., q 0 =,..., =,..., (5.6) Here he non-negaiviy condiions (5.6) are given. he Capaciaed Losizing Problem does no include scheduling decisions. Usually he CLSP is solved firs and afer ha he scheduling of each period is done Drexl, Kimms, 999 (997), p Fleischmann, 990, p

29 An example for he CLSP is given by 28 : he number of iems = 3, here are = 5 periods and he capaciy of he machine in period is coss C = 00. he demands h and seup coss s are given in he able 2 below. d, capaciy needs p, holding p h s d d d able 2: Daa for he CLSP example 29 he opimal order quaniies q and he obecive funcion value Z * are given in able 3: Z * q q q able 3: Soluion for he CLSP example Commens o he example: he CLSP is a big bucke problem, because several iems requiring differen seup saes can be produced in each period, wha can be seen for example in period 2 where wo differen iems (iem 2 and 3) are produced wih differen seup coss s. Only one differen iem can be produced per period, wha means only wo iems like iem 2 and 3 in period 2 or iem and 2 in period Haase, 994, p see foonoe see foonoe 28 23

30 he CLSP is a NP-hard problem o solve opimal, and because of ha heurisics were developed. here are wo classes of heurisics 3 : he single-resource heurisics and he mahemaical-programming-based heurisics. he single-resource heurisics, also called common sense heurisics, are divided ino wo groups. he firs one is he period-by-period heurisic, where he problem is looked a from he firs period o he las period. he second one is he improvemen heurisic. Here he model sars wih a soluion, which can be feasible or no, and hen ries o improve i by local improvemen seps. So he model ges a feasible soluion. he mahemaical-programming-based heurisics are divided ino relaxaion heurisics, branch and bound procedures and LP-based heurisics. hese models are opimum seeking. 5.. Capaciaed Losizing Problem wih Linked Losizes (CLSPL) he Capaciaed Losizing Problem was criicized because he model does no allow a seup o be carried over from one period o he nex, even if he las produc in one period and he firs produc in he nex period are he same. his has led o he Capaciaed Losizing Problem wih Linked Losizes model, which is an exension of he CLSP, and so also a big bucke problem. n he CLSPL model he seup can be carried-over from one period o he nex period. A mos one seup for one produc can be carried-over, so ha no new seup is necessary in he second period. his preservaion of seup sae is a characerisic of a small bucke problem, so he model combines he small and he big bucke problem 32 (see figure 9). 3 Maes, Van Wassenhove, 988, p Suerie, Sadler, 2003, p

31 Figure 9: Characerisics of small bucke problem, big bucke problem and CLSPL 33 he decision variables and he parameers for he CLSPL are he same as for he CLSP. here are wo new decision variables: v = his variable shows if more han one iem is produced in period or no z = he binary variable ells if quaniies of iem in period and he following period are linked or no x = binary variable which ells if a quaniy for iem is produced in period or no he mixed-ineger model for he CLSPL can be formulaed as followed 34 : Min [ s ( x z ) + h ] = = (5.7) he obecive (5.7) of he Capaciaed Losizing Problem wih Linked Losizes Problem is o minimize he oal coss. he difference o he obecive funcion of he CLSP is ( x z ) s wha means ha he seup of an iem is only calculaed, when ha iem in a period is no linked wih he producion quaniy of he previous period. 33 see foonoe Haase, 994, p. 8 25

32 he following resricions (5.8) (5.) are he same as in he CLSP model: = ( ) + q d =,..., =,..., (5.8) he equaion (5.8) shows he invenory consrain. p q C x =,..., =,..., (5.9) his resricion (5.9) shows ha an iem can only be produced when he machine is seup for his iem. p q = C =,..., (5.0) Here he capaciy consrains (5.0) are menioned., q 0 =,..., =,..., (5.) Here he non-negaiviy condiions (5.) are given. New resricions for he CLSPL are: = z =,..., (5.2) his consrain ells ha a mos one seup can be carried over from one period o he nex one. z 0 =,..., =,..., (5.3) x, his consrain shows ha a seup can be carried over from one period o he nex one when he iem was seup in he period. z x 0 =,..., =,..., (5.4) his hree resricions (5.2) (5.4) show ha only one produc can be produced a he end of a period and so be linked o he following period. 26

33 x + v 0 =,..., (5.5) = z z + v 2 =,..., =,..., (5.6) +, hese consrains (5.5) + (5.6) indicae ha when more han one iem is produced in a period, ha is when v > 0, he seup can be eiher linked wih he previous period or wih he following period + seup. v 0 =,..., (5.7) Here he non-negaiviy condiion (5.7) is given. x, z {0,} =,..., =,..., (5.8) he seup variables (5.8) are binary. An example for he CLSPL is given by 35 : he daa (,, C, d, p, CLSP on page 23 (see able 2). he opimal order quaniies in able 4: h, s ) is he same as in he example for he q and he obecive funcion value Z * are given Z * q q q able 4: Soluion for he CLSPL example Commens o he example: n he CLSPL quaniies of an iem in a period and in he following period can be linked, for example iem 3 is linked in period 4 and 5. So he seup is carriedover and no new seup has o ake place in period 5 for he iem 3. n his 35 Haase, 994, p see foonoe 35 27

34 model more han wo iems can be produced per period, for example in period here is producion of all hree iems. Knu Haase, who gave he name o he CLSPL, provided in 994 a heurisic o solve he Capaciaed Losizing Problem wih Linked Losizes named Backward Add CLSPL (BACLSPL). his heurisic moves backwards from he las period o he firs period and adds seup and losizing decisions o every period 37. n 200 a abu search heurisic was presened. A ha ime he CLSPL was called Capaziaed Losizing Problem wih Seup Carryover (CLSP-SC) Suerie, Sadler, 2003, p Suerie, Sadler, 2003, p

35 5.2 Discree Losizing and Scheduling Problem (DLSP) he Discree Losizing and Scheduling Problem is a single-level muli-iem capaciaed losizing model. is a small bucke problem, wha means ha only one iem can be produced per period. he planning horizon is finie and demand is dynamic. here is a small ime scale, wha means hours, days or shifs. he fundamenal assumpion for he DLSP is he all-or-nohing assumpion 39 : says ha one iem is produced over a full period or no a all. his is a discree producion policy. here is no changeover allowed. o produce one iem per period he periods are divided in several subperiods. ha means ha he macro-period is divided ino micro-periods. he seup occurs only a he beginning of a period. he seup sae is los over idle periods. he DLSP considers he sequencing of he los 40. he decision variables and he parameers for he DLSP are he same as for he CLSP. As he producion of a lo can now las more han one period, here is a new decision variable and a new parameer needed for he seup sae in a cerain period. he decision variable is: y = variable which ells us if he machine is se up for iem in period or no he parameer is: y 0 = ells us if he machine is se up for iem a he beginning of he period or no 39 Salomon e al., 99, p Fleischmann, 990, p

36 he mixed-ineger model for he DLSP can be formulaed as followed 4 : s + = = Min ( x h ) (5.9) he obecive (5.9) of he Discree Losizing and Scheduling Problem is o minimize he oal coss. he funcion is he same as for he CLSP. = ( ) + q d =,..., =,..., (5.20) he equaion (5.20) shows he invenory consrain. p q = C y =,..., =,..., (5.2) his equaion (5.2) is he all-or-nohing assumpion. he difference o he CLSP is ha he wo sides have o equal. = y =,..., (5.22) he resricion (5.22) makes sure ha only one iem can be produced per period. x y y =,..., =,..., (5.23) ( ) he inequaliy (5.23) shows he beginning of a new lo. y {0,} =,..., =,..., (5.24) he seup sae variables (5.24) are binary., q, x 0 =,..., =,..., (5.25) Here he non-negaiviy condiions (5.25) are given. 4 Drexl, Kimms, 999 (997), p

37 An example for he CLSP is given by 42 : he number of iems = 3, here are = 0 periods and he capaciy of he machine in period is coss C = 50. he demands h and seup coss s are given in able 5 below. d, capaciy needs p, holding p h s d d ,5 50 d able 5: Daa for he DLSP example 43 he opimal order quaniies q and he obecive funcion value Z * are given in able 6: Z * q q q able 6: Soluion for he DLSP example Commens on he example: he DLSP has he all-or-nohing assumpion, wha means ha one iem is produced over a full period or no a all. n he example always only one iem is produced per period, for example iem in period 2, iem in period 3, iem in period 4 and so on. When producion akes place hen he whole capaciy of C = 50 is used. he seup sae is los over idle periods and because of ha he obecive funcion value is high. 42 Haase, 994, p see foonoe Haase, 994, p. 29 3

38 he following Gan char (see figure 0) shows he schedules for he example above: C / p producion of iem Figure 0: Gan char for he DLSP example 45 he char shows ha iem is scheduled from period 2 o period 4, hen iem 3 in period 5 and iem 2 in period 6. Aferwards again iem 3 in period 9 and iem 2 in period 0. he scheduled periods are always fully used because of he allor-nohing assumpion. he advanage of he DLSP over he CLSP is ha he seup sae is no los when he same iem is produced over adacen periods. n he CLSP a new seup is needed in every period. o solve he Discree Losizing and Scheduling Problem opimal is NP-hard. can be solved by models based on ineger programming formulaions. Bernhard Fleischmann used in 990 a Lagrangean relaxaion and a dynamic programming algorihm for he DLSP see foonoe Fleischmann, 990. p

39 5.3 Coninuous Seup Losizing Problem (CSLP) he Coninuous Seup Losizing Problem is a sequence independen kind of a small bucke problem, wha sill means ha only one iem can be produced per period, bu he all-or-nohing assumpion is given up 47. ha is he difference o he Discree Losizing and Scheduling Problem. is single-level muli-iem capaciaed model. he planning horizon is finie and demand is dynamic. he decision variables and he parameers for he CSLP are he same as for he DLSP. he mixed-ineger model for he CSLP can be formulaed as followed 48 : s + = = Min ( x h ) (5.26) = ( ) + q d =,..., =,..., (5.27) p q C y =,..., =,..., (5.28) = y =,..., (5.29) x y y =,..., =,..., (5.30) ( ) y {0,} =,..., =,..., (5.3), q, x 0 =,..., =,..., (5.32) he models for he DLSP and he CSLP are nearly equal. he only difference is in he resricion (5.28) p q C y, where now no equaliy has o be anymore. An iem has no o be produced over a full period. he capaciy of a period which is no fully used says unused. 47 Brahimi e al., 2006, p Drexl, Kimms, 999 (997), p

40 An example for he CSLP is given by 49 : he daa (,, C, d, p, DLSP on page 3 (see able 5). he opimal order quaniies in able 7: h, s ) is he same as in he example for he q and he obecive funcion value Z * are given Z * q q q able 7: Soluion for he CSLP example Commens on he example: n he CSLP sill only one iem can be produced per period, bu he all-ornohing assumpion is given up. n he example here is only one iem per period like iem in period 2, iem in period 3 and so on, bu he full capaciy does no have o used anymore. A negaive poin of his model is ha remaining capaciy of a period says unused. n he period only 30 of he machine capaciy is needed and so 20 is lef unused. he following Gan char (see figure ) shows he schedules for he example above: C / p producion of iem Figure : Gan char for he CSLP example 5 49 Haase, 994, p see foonoe 49 34

41 he char shows ha iem is scheduled in period 2, 3 and 4, bu he periods 2 and 3 are no fully used because he all-or-nohing assumpion is given up. hen iem 3 in period 5 and iem 2 in period 6, 7 and 8, where period 6 and 7 are again no fully used. Afer ha iem 3 in period 9 and iem 2 in period 0. he advanage of he CSLP over he DLSP is ha when he same iem is produced over more periods he seup only akes place in he firs period. hen he seup sae says, alhough a period has no o be fully used. n he DLSP here would be wo seups because of he no fully used period. he CSLP is a NP-hard problem o solve opimally. Karmarkar presened an algorihm based on Lagrangean relaxaion in combinaion wih dynamic programming o solve he CSLP see foonoe Salomon, 99, p

42 5.4 Economic Losizing and Scheduling Problem (ELSP) he Economic Losizing and Scheduling Problem is a capaciaed model for a single-level muli-iem machine. he planning horizon for he ELSP is infinie, he ime scale is coninuous, he demand is consan, and backlogging is no allowed. he deerminaion of he losizes and he sequencing of he los (he cycles) are done simulaneously. he obecive of he model is o minimize he sum of seup coss and invenory holding coss, while deermining he cyclical producion schedule 53. n he model no wo iems can be produced a he same ime, oherwise here would be he phenomenon of inerference 54, wha is physically impossible. he Economic Losizing and Scheduling Problem o solve opimal is a NP-hard. here are hree analyical approaches o solve he ELSP 55 :. he Common Cycle (CC) or Roaion Cycle approach Here all produc cycle imes have o be equal. ha means ha each produc can be produced exacly once in every cycle. 2. he Basic Period (BP) approach Here are differen cycle imes for differen producs allowed. 3. he Dobson s approach Here he losizes of a given produc can vary over a cyclic schedule. Furher here are also heurisic approaches 56, for example he Madigan s procedure, a heurisic from Sankard and Gupa, a heurisic from Doll and Whybark and a heurisic from Goyal. he heurisic approaches lead o a beer performance hen he analyical approaches. 53 Salomon, 99, p Hsu, 983, p Carreno, 990, p Elmaghraby, 978, p

43 5.5 Proporional Losizing and Scheduling Problem (PLSP) he Proporional Losizing and Scheduling Problem is a small bucke problem, where a mos wo iems can be produced per period. is single-level muli-iem capaciaed model. he planning horizon is finie and demand is dynamic 57. he fundamenal assumpion for he PLSP is: A mos one changeover of he seup is allowed wihin each period. 58 f he capaciy of a period is no fully used from one iem, he remaining capaciy can be used for a second iem, bu here is no more han one changeover allowed. When here are wo iems in one period he order in which hey are produced is imporan. he firs iem mus be he same as he las iem in he previous period. he seup can occur a he beginning of a period or a he end of a period. here is only one seup per period, bu i can be carried over from he previous period o he nex period. he seup sae is preserved over idle periods. he decision variables and he parameers for he PLSP are he same as for he DLSP. he mixed-ineger model for he PLSP can be formulaed as followed 59 : s + = = Min ( x h ) (5.33) = ( ) + q d,..., = =,..., (5.34) p q C y + y ) =,..., =,..., (5.35) ( ( ) p q = C =,..., (5.36) = y =,..., (5.37) 57 Drexl, Haase, 995, p see foonoe Drexl, Kimms, 999 (997), p

44 x y y =,..., =,..., (5.38) ( ) y {0,} =,..., =,..., (5.39), q, x 0 =,..., =,..., (5.40) he models for he PLSP and he CSLP are nearly equal. he difference is in ( ( ) he resricion (5.35) p q C y + y ). his inequaliy shows ha he producion of an iem in a period only akes place when here is a seup eiher a he beginning or he end of he period. A new consrain (5.36) is p q = C. secures ha when wo iems are produced per period he capaciy requiremens are no harmed. his spliing of capaciy o produce wo iems wihin a period proporional o he quaniies leads o he name of he Proporional Losizing and Scheduling Problem 60. An example for he PLSP is given by 6 : he daa (,, C, d, p, DLSP on page 3 (see able 5). he opimal order quaniies in able 8: h, s ) is he same as in he example for he q and he obecive funcion value Z * are given Z * q q q able 8: Soluion for he PLSP example Drexl, Haase, 995, p Haase, 994, p see foonoe 6 38

45 Commens on he example: n he PLSP wo differen iems can be produced per period. he capaciy of a period ha is no fully used from one iem can be used for a second iem, bu here is only one changeover allowed. n he period 6 for example iem needs 20 of he machine capaciy, and so 30 says unused. his 30 can be used for he iem 2 in he same period. When here are wo iems in one period hen he order in which hey are produced is imporan, for example he iem 3 in period and 2. he firs iem has o be he same as he las iem in he previous period. ha means ha in period iem 3 is produced and in period 2 iem 3 is furher produced and aferwards iem. he seup for iem 3 is carried-over from period o period 2. he following Gan char (see figure 2) shows he schedules for he example above: C / p producion of iem Figure 2: Gan char for he PLSP example 63 he char shows ha iem 3 is scheduled in period and 2. Here he seup is carried-over from one period o he following one. n period 4, 5 and 6 iem is scheduled. he no used capaciy from he iem in period 6 is used for he iem 2. hen iem 2 in period 8 and 9, and iem 3 in period 9 and 0. he advanage of he PLSP over he oher models discussed unil now is ha he seup sae is preserved over idle periods and ha wo differen iems can be produced per period. o solve he PLSP a randomized-regre-based biased sampling mehod 64 can be used. his mehod consiss of hree seps. he firs par is ha he losizing 63 see foonoe 6 64 Drexl, Haase, 995, p

46 and scheduling is done backwards from he las period o he firs. he second sep of he model is ha in every period a mos wo iems can be added, and he hird one is ha he regres are used o decide in which period which iem is going o be produced. hese hree pars describe a Backward Add mehod. Some exensions of he PLSP are: Proporional Losizing and Scheduling Problem wih Seup imes (PLSPS) Proporional Losizing and Scheduling Problem wih Sequence Dependen Seup Coss (PLSPSDSC) 40

47 5.6 General Losizing and Scheduling Problem (GLSP) he General Losizing and Scheduling Problem is a large bucke problem. is single-level muli-iem capaciaed model. he planning horizon is finie, he demand is dynamic and no backlogging is allowed. n he GLSP he periods, so called macro-periods, are divided ino micro-periods wih variable lengh, which allows sequencing of he producs. he macro-periods indicae he demand and he holding coss, and he micro-periods consis of he changes referred o he decision variables. he all-or-nohing -assumpion exiss for he microperiods 65. he fundamenal assumpion 66 for he GLSP is ha he number of los per period is resriced. his is done wih he new parameer N. he decision variables for he GLSP are basically he same as for he DLSP. hey are: = invenory for iem a he end of period q n = producion quaniy for iem a he posiion n x n = variable which ells if a seup for iem occurs a he posiion n or no y n = variable which ells if he machine is ready o produce iem a he posiion n or no he parameers for he GLSP are he same as for he DLSP, and he new one is: N = is he maximum number of los in period (When N = for all =,...,, he GLSP is he same as he CSLP.) 65 Fleischmann, Meyr, 997, p Drexl, Kimms, 999 (997), p

48 Furher in his model: + F = N = τ τ = shows he firs posiion in period L F + N = shows he las posiion in period = N = = N = oal number of posiions he mixed-ineger model for he GLSP can be formulaed as followed 67 : Min N = n= s x n + = = h (5.4) he obecive (5.4) is o minimize he invenory coss and he sequence dependen seup coss. L = ( ) + q n d,..., n= F = =,..., (5.42) he equaion (5.42) shows he invenory balance. An iem now can be produced a several posiions in a period. p q C y =,...,,..., n n = F L n =,..., (5.43) his resricion (5.43) shows ha a lo for an iem on a posiion can only be produced when he machine is se up for i. L = n= F p q n C =,..., (5.44) Here he capaciy consrains (5.44) are menioned. = y n =,..., N (5.45) n he resricion (5.45) makes sure ha here is unique seup sae. x y y =,..., n =,..., N (5.46) n n ( n ) he inequaliy (5.46) shows he posiion where he seup akes place. 67 Drexl, Kimms, 999 (997), p

49 x n, y n {0,} =,..., n =,..., N (5.47) he seup sae variables (5.47) are binary. 0 =,..., =,..., (5.48) q n, x n 0 =,..., n =,..., N (5.49) Here he non-negaiviy condiions (5.48) + (5.49) are given. he GLSP is a NP-hard problem o solve opimally, because of ha exac soluion mehods are oo complex. So heurisics have o be used, which only can provide near-opimal soluions. he GLSP can be solved for example wih local search heurisics based on hreshold acceping GLSP CS he GLSP CS is a varian of he General Losizing and Scheduling Problem. he shorcu CS means Conservaion of Seup Sae. he mixed-ineger model for he GLSP-CS 69 is: Min si zis + i,, s, h (5.50) he obecive (5.50) is o minimize he invenory coss and he changeover coss. n his funcion s i is he seup coss for a changeover from he produc i o he produc, and z is shows if he changeover akes place a he beginning of he micro-period s or no. he difference o he General Losizing and Scheduling Problem are new consrains (he consrains (5.42) (5.45) from he GLSP say he same): x m y y ) =,..., s =,..., S (5.5) s ( s ( s ) z y + y =,..., i =,..., s =,..., S (5.52) is ( s ) s 68 Fleischmann, Meyr, 997, p. 69 Fleischmann, Meyr, 997, p. 2 43