An Optimization Model for Multi-period Multi- Product Multi-objective Production Planning

Transcription

1 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 43 An Otimization Model for Multi-eriod Multi- Product Multi-objective Production Planning M. S. Al-Ashhab Design & Production Engineering Det. Faculty of Engineering, Ain-Shams University Egyt Det. of Mechanical Engineering Collage of Engineering and Islamic architecture, UU, Makkah KSA and Abstract-- This aer resents a mixed integer linear rogramming (MILP) otimization model to solve the artner selection, and roduction lanning roblem in the design of manufacturing chains oerating under a multiroduct, multi echelon, multi-eriod and multi-objective manufacturing environment. The roosed manufacturing chain consists of a facility, three suliers, three distributors and four customers. The erformance of the develoed model is illustrated by using a verification roblem. Discussion of the results roved the accuracy of the model. Index Term-- Production lanning, location, allocation, MILP, modeling, multi-roducts, multi echelon, multi-objective and multi-eriods. 1. INRODUCTION Production lanning affects rofit of the factory and service level of the customers. So it is very imortant to give the roduction lanning rocess more attention to assure high rofit and service level. Chandra and Fisher (1994) [1] attemted to solve the roblem of coordinating the roduction and distribution functions for a single-lant, multi-commodity, multi-eriod manufacturing environment. The study assumed that roducts are roduced and stored in the lant until they are delivered to the customers using a fleet of trucks. Results of comutational exeriments clearly showed that lanning roduction and routing simultaneously could lead to a cost saving of u to 20%. Chang, and Park (2002) [2] considered the multi-roduct single-eriod suly network design roblem. The suly network was decomosed into an inbound network, a roduction distribution network and an outbound network, and a heuristic based on Lagrangian Relaxation was used to design each sub-network. Yan, Yu, and Cheng (2003) [3] roosed a strategic roduction distribution model which included multile suliers, roducers, distribution centers and customers for manufacturing multile roducts in a single eriod. Kouvelis et al. (2004) [4] resented a mixed integer rogram to investigate global roduction network design for a single roduct. A two-stage roduction rocess is analyzed for the introduction of a new roduct in different markets. Their work clearly demonstrates the imortance of including tariffs and regional trading rules into global suly chain design, since they have a significant imact on the network structure. According to their key findings, increased trade tariffs favor gradual decentralization of roduction rocesses. Altiarmak, Gen, Lin, and Paksoy (2006) [5] formulated a mixed integer nonlinear model for a multi-objective suly chain network designed for a single roduct of a lastic comany. A solution rocedure based on genetic algorithms was develoed to solve the roblem. Sha and Che (2006) [6] studied the design of a comlex suly chain network. The overall objective was to maximize the reference of servicing the demand and minimize the number of artners involved. A multi-hase mathematical aroach based on Genetic Algorithms, Analytical Hierarchy Process and Multi-Attribute Utility Theory was roosed to solve the roblem. However, their studies only considered single roduct and single eriod demand from the customer, and did not consider the structure of the roduct. Huebner (2007) [7] develoed a strategic roduction network design model to maximize the NPV of cash flows before tax. The model otimizes the roduction and distribution lan and determines the lant location, roduct allocations and caacity changes. ased on the secial roblem structure of a global chemicals comany, a single sourcing assumtion between different suly chain echelons is introduced. Thus, the model covers duty drawbacks for reexorts after one roduction level and requires modifications to allow duty drawbacks across a higher number of roduction stages and multile sulies from several sub-assemblies. Tsiakis and Paageorgiou (2008) [8] suggested a deterministic mixed-integer roblem for designing a global roduction network including roduction lants, distribution centers and different customer zones. Although multile roducts are considered, only one common duty rate for all roducts is taken into account. Due to the strategic focus of the aer, decisions on where to oen roduction lants and distribution centers, the allocation of roducts to lants and the assignment of distribution centers to lants and customer zones to distribution centers are incororated.

2 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 44 Zafel et al. (2010) [9] used a genetic algorithm to generate the final integrated roduction distribution lan. In the study, the roduct structures were reresented by (1) a one-level OM in the inbound sub-network and (2) a twolevel OM in the roduction distribution sub-network. Items were arranged in a fixed level of the OM. Manufacturing lants were groued according to the OM structure, with each grou roducing items located at the same level of the OM. However, the aroach is no longer effective when the OM structure is comlex, for examle, a more than twolevel OM with the same item existing in different levels. Mula, Peidro, and Diaz-Madronero (2010) [10] resented a review of using mathematical rogramming models for suly chain design, and highlighted that it is essential to integrate the suliers nodes into the suly chain otimization models. Indeed, local effectiveness is far from automatically inducing the global efficiency of the whole multi-stage manufacturing chain. Hence, formulating integrated cost-effective artner selection and roduction and distribution decisions is a challenge for manufacturing chain designers, esecially when the roducts have comlicated structures. Mezghani et al. (2012) [11] develoed a Goal Programming formulation within an imrecise environment and exlicitly introduce the manager's references into the aggregate lanning model. Margaretha Gansterer (2015) [12] resented a comrehensive hierarchical roduction lanning (HPP) framework, to investigate the imact of aggregate lanning in a make-to-order (MTO) environment. The lanning roblem is formulated as a linear mathematical model and solved to otimality by a standard otimization engine. The erformance of the system is evaluated based on service and inventory levels. Real world data coming from the automotive sulier industry is used to define four demand scenarios. In this study, a mixed integer linear rogramming (MILP) otimization model to solve the artner selection, and roduction lanning roblem in the design of manufacturing chains oerating under a multi-roduct, multi echelon, multieriod and multi-objective manufacturing environment is develoed for the urose of roduction lanning in a factory to maximize the rofit of the factory and the overall service level of the customers. The factory has three aroved suliers and has three distributors to serve four customers as shown in figure 1. Customer 4 Sulier 3 Distributor 3 Customer 3 Sulier 2 Facility Distributor 2 Customer 2 Sulier 1 Distributor 1 Customer 1 2. MODEL DESCRIPTION The roosed model assumes a set of customer locations with known and time varying demands and a set of candidate suliers of known, limited and time varying caacity, and distributor s locations of known, limited and time varying caacity. It otimizes locations of the suliers, distributors and customers and allocate the shiment between them to maximize the rofit and overall customer service level taking their caacities, inventory and shortage enalty and other costs into consideration. Fig. 1. Facility, suliers, distributors and customer network. The roblem is formulated as a mixed integer linear rogramming (MILP). The model is solved using Xress-MP software which uses Mosel language in rogramming [13]. The flow of material and roduct are assumed as shown in figure 2. Where suliers are resonsible for sulying of raw materials to the facility. Facility is resonsible for manufacturing of the three roducts and sulying some of them to the distributors and storing the rest for the next eriods; if it is rofitable. Distributors are resonsible for the distribution of roducts to the customers and/or storing some of them for the next eriods, and customers nodes may

3 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 45 reresent one customer, a retailer, or a grou of customers and retailers. Sulier sft It Facility Facility store fdt Ifdt Distributor (Store) dct Consumer Fig. 2. Model flow. The model considers fixed costs for all nodes, materials costs, transortation costs, manufacturing costs, non-utilized caacity costs for the facility, holding costs for facility and distributors stores and shortage costs. 3. MODEL ASSUMPTIONS AND LIMITATIONS The following assumtions are considered: 1. The model is multi-objectives, where it maximizes Profit of the facility and the Overall Service Level of each of the four customers. 2. Overall Service Level is assumed as the ratio of the total quantities delivered to each customer and the total demand in all eriods from all roducts. 3. Overall Service Level may be maniulated as an objective and as an inut to study its effect on the rofit 4. The model is multi-roduct, where actions and flow of materials take lace for multi-roduct. 5. Products weights are different. 6. The model is multi-eriod, where actions and flow of materials take lace in multi-eriods. 7. Customers locations are fixed and known. 8. Customers demands are known for all roduct in all eriods. 9. The locations of suliers, facility, and distributors are known. 10. Costs arameters (fixed costs, material costs, manufacturing costs, non-utilized caacity costs, shortage costs, transortation costs, and inventory holding costs) are known for each location, each roduct at each eriod. 11. Caacity of each sulier, facility, and distributor locations are known for each eriod. 12. The shortage cost deends on the shortage quantity for each roduct and time. 13. The holding cost deends on the, weight of roduct and residual inventory at the end of each eriod for each roduct. 14. The transortation cost deends on the transorted quantities, weight of roduct and the linear distance between locations. 15. The manufacturing cost deends on the manufacturing hours for each roduct and manufacturing cost er hours 16. The material cost is different for each roduct deending on its weight. 17. Integer number of batches is transorted. 4. MODEL FORMULATION The model involves the following sets, arameters and variables: Sets: S: otential number of suliers, indexed by s. D: otential number of distributors, indexed by d. C: otential number of first customers, indexed by c. T: number of eriods, indexed by t. P: number of roduct, indexed by. PARAMETERS F s : fixed cost of contracting sulier s, F f : fixed cost of the facility, F d : fixed cost of oening distributor d, DEMAND ct : demand of customer c from roduct in eriod t, P ct : unit rice of roduct at customer c in eriod t, W : roduct weight. MH : manufacturing hours for roduct. D sf : distance between sulier s and the facility. D fd : distance between the facility and distributor d. D dc : distance between distributor d and customer c. CAP st : caacity of sulier s in eriod t (kg), CAPM ft : caacity of the facility Raw Material Store in eriod t. CAPH ft : caacity in manufacturing hours of the facility in eriod t, CAPFS ft : storing caacity of the facility in eriod t, CAP dt : caacity of distributor d in eriod t (kg), MatCost: material cost er unit sulied by sulier s in eriod t, MC ft : manufacturing cost er hour for facility in eriod t, MH : Manufacturing hours for roduct ()

4 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 46 NUCCf: non utilized manufacturing caacity cost er hour of the facility, SCPU : shortage cost er unit er eriod, HF : holding cost er unit er eriod at facility store (kg), HD : holding cost er unit er eriod at distributor d store (kg), s : batch size from sulier s & d : batch size from the facility and distributor d for roduct. TCerkm: transortation cost er unit er kilometer. DECISION VARIALES L s : binary variable equal to 1 if a sulier s is contracted and equal to 0 otherwise. L d : binary variable equal to 1 if a distributor d is oened and equal to 0 otherwise. Li sf : binary variable equal to 1 if a transortation link is activated between sulier s and the facility. Li fd : binary variable equal to 1 if a transortation link is activated between the facility and distributor d. Li dc : binary variable equal to 1 if a transortation link is activated between distributor d and customer c. sft : number of batches transorted from sulier s to the facility in eriod t, fdt : number of batches transorted from the facility to distributor d for roduct in eriod t, I t : number of batches transorted from the facility to its store for roduct in eriod t, I fdt : number of batches transorted from store of the facility to distributor d for roduct in eriod t, dct : number of batches transorted from distributor d to customer c for roduct in eriod t, R t : residual inventory of the eriod t at store of the facility for roduct. R dt : residual inventory of the eriod t at distributor d for roduct. OSL c : Overall Service Level of customer c Objective function. The objectives of the model are to maximize both the rofit of the facility and the Overall Service Levels of the four customers. Overall Service Level c dd P tt dct / P tt DEMAND ct (1) Profit = Total revenue Total cost Total revenue Total revenue dd cc P tt dct d P ct (2) Total cost Total cost = fixed costs + material costs + manufacturing costs + non-utilized caacity costs + shortage costs + transortation costs + inventory holding costs Fixed costs Fixed costs F L (3) Fs Ls Ff ss dd d d Material cost Material cost MatCost (4) ss tt sft Manufacturing costs s st Manufactur ing costs MH Mc I MH Mc (5) fdt ft t dd P t T P t2... T ft

5 International Journal of Engineering & Technology IJET-IJENS Vol:16 No: Non-Utilized caacity cost (for the facility) ( ((CAPHft ) Lf ( fdt MH ) (It MH )) NUCC f ) (6) P tt dd Shortage cost (for distributor) P t t ( ( ( DEMANDct cc tt 1 1 dd Transortation costs T DS ( dct W T d dd ))) SCPU T sft s s sf fdt f fd fdt tt ss P tt dd t2 dd ddcc tt dct d W T d D dc Inventory holding costs P ) D I W T ( R t W HFf R dt W HDd) (9) tt dd tt f D fd (7) (8) 4.2. Constraints alance constraints: ss I t P dd dd W I W, t T (10) sft s R t P fdt t 1) Rt I fdt, t T, P dd ( (11) ( fdt I fdt) Rd( t1) d Rdtd dctd, t 2 T, d D (12) dct P cc d DEMANDct DEMANDc( t 1) dc( t 1) d, t T, c C, P (13) 1t dd Constraint (10) ensures that the amount of materials entering to the facility from all suliers equal the sum of the exiting form it to each store and distributor. Constraint (11) ensures that the sum of the flow entering to facility store and the residual inventory from the revious eriod is equal to the sum of the exiting to each distributor store and the residual inventory of the existing eriod for each roduct. Constraint (12) ensures that the sum of the flow entering to each distributor, distributor store and the residual inventory from the revious eriod equal the sum of the exiting to each customer and the residual inventory of the existing eriod for each roduct. Constraint (13) ensures that the sum of the flow entering to each customer does not exceed the sum of the existing eriod demand and the revious accumulated shortages for each roduct Caacity constraints:

6 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 48 ss ( sft dd P sft R s CAP fdt f t s st L, t T, s S s CAPM W dd I ft t L, t T f CAPFS ) MH ft CAPH L, t T f ft L, t T, P (fdt Ifdt) W Rdt-1 W CAPdt Ld, t T, d D, P tt f (14) (15) (16) (17) (18) Constraint (14) ensures that the sum of the flow exiting from each sulier to facility does not exceed the sulier caacity at each eriod. Constraint (15) ensures that the sum of the material flow entering to facility from all suliers does not exceed the facility caacity of material at each eriod. Constraint (16) ensures that the sum of manufacturing hours for all roducts manufactured in the facility to be delivered to its store and each distributor does not exceed the manufacturing caacity hours of it at each eriod. Constraint (17) ensures that the residual inventory at facility store does not exceed its caacity at each eriod. Constraint (18) ensures that the sum of the residual inventory at each distributor from the revious eriods and the flow entering at the existing eriod from the facility and its store does not exceed this distributor caacity at each eriod for each roduct Linking (contracts)-shiing constraints: Li, s S (19) sf tt sft Li fd ( fdt I fdt), d D, P (20) tt Li dc tt dct, d D, c C, P (21) Constraints (19-21) ensure that there are no links between any locations without actual shiments during any eriod Shiing-Linking constraints: tt tt tt sft M Li sf, s S (dct Ifdt) M Lifd, d D, P dct M Li dc, d D, c C, P (22) (23) (24)

7 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 49 Constraints (22-24) ensure that there is no shiing between any non-linked locations Maximum number of activated locations constraints: ss L s S (25) dd L d D Constraints (25-26) limit the number of activated locations, where the sum of binary decision variables, which indicate the number of activated locations, is less than the maximum limit of activated locations (taken equal to the otential number of locations). (26) 5. MODEL VERIFICATION 5.1 MODEL INPUTS The model has been verified through the following case study where the inut arameters are consider as showing in table 1.

8 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 50 Table I Verification model arameters Parameter Value Parameter Value Number of otential suliers 3 Manufacturing hours for roduct 1 1 Number of facilities 1 Manufacturing hours for roduct 2 2 Number of otential Distributors 3 Manufacturing hours for roduct 3 3 Number of Customers 4 Transortation cost er kilometer er unit Number of roducts 3 Facility holding cost 3 Fixed costs for sulier & distributor 20,000 Distributor holding cost 2 Fixed costs for facility 50,000 Caacity of each suliers in each eriods 4,000 Weight of Product 1 in Kg 1 Sulier batch size 10 Weight of Product 2 in Kg 2 Facility atch size for roduct 10 Weight of Product 3 in Kg 3 Distributor atch size for roduct 1 Price of Product Caacity of Facility in hours 12,000 Price of Product Caacity of Facility Store in each eriods 2,000 Price of Product Caacity of each Distributor Store in each eriods 4,000 Material Cost er unit weight 10 Caacity of each Facility Raw Material Store in each Manufacturing Cost er hour 10 eriods 4,000 The demand atterns are assumed as the same for all customer as shown in the table 2. Table II Demand of each customer in all eriod for from roduct. Period T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 Product Product Product MODEL OUTPUTS The resulted otimal network is as shown in figure 3.

9 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 51 Fig. 3. The resulted otimal network. Where the quantities of material sulied to the facility from suliers are shown in table 3 Table III Number of batches transferred from suliers to the facility. Period Sulier Sulier Sulier The number of batches transferred from the facility to distributors for each roduct in each eriod are shown in table 4 Period Table IV Number of batches transferred from the facility to distributors. To distributor 1 To distributor 2 To distributor 3 P1 P2 P3 P1 P2 P3 P1 P2 P

10 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 52 The number of batches transferred from the facility store to distributors for each roduct in each eriod are shown in table 5. Period Table V Number of batches transferred from the facility store to distributors. To distributor 1 To distributor 2 To distributor 3 Product 1 P2 P3 Product 1 P2 P3 Product 1 P2 P The number of batches transferred from distributors to customers for each roduct in each eriod are shown in table 6 Table VI Number of batches transferred from distributors to customers. Period D1-C1 D1-C2 D1-C3 D1-C4 D2-C1 D2-C2 D2-C3 P P P P P P P P P P P P P P P P P P P P P

11 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 53 D2-C4 D3-C1 D3-C2 D3-C3 D3-C4 P P P P P P P P P P P P P P P The resulted Overall Service Level is shown in figure 4. Fig. 4. The resulted Overall Service Level of the customers Table VII reresents the total revenue, costs and total rofit values where figure 5 reresent the cost share Table VII Cost/Revenue values. Cost/Revenue Value Cost/Revenue Value Cost/Revenue Value Total Revenue 8,786,500 Material Cost -1,156,000 Non-Utilized Cost -201,600 Fixed Cost -170,000 Manufacturing Cost -1,238,400 Shortage Cost -628,000 Transortation Costs -106,002 Inventory Holding Cost -25,000 Total Profit 5,261,498

12 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 54 Fig. 5. Cost shares 5.3 RESULTS DISCUSSION The relationshi between the equivalent required manufacturing hours and manufacturing caacity in hours is shown in figure 6 in which it is noticed that the manufacturing caacity of the facility exceeds the equivalent required manufacturing hours in the first and last four eriods. So there are some limitations regarding required hours. Fig. 6. relationshi between the required manufacturing hours and manufacturing caacity in hours Figure 7 in which the relationshi between the equivalent required quantity of material, sulier caacity, facility RM store caacity and facility store caacity in kilograms it is noticed that the sulying caacity of the suliers of kilograms exceeds facility RM store caacity of kilograms in all eriods and both of them does not exceed the required weight in all eriods. So there are some limitations regarding required weight.

13 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 55 Fig. 7. relationshi between the required weight and sulying caacity And the relationshi between the required weight, sulier caacity and the given weight is shown in figure 8 in which it is noticed that In eriods 1, 2 and 3 the sulied material is more than the required in the same eriod where the facility manufactures all sulied material, stored some of them for the next eriods and send to customers what they need in this eriod. In eriod 4 and 5 the required material is more than the sulied in the same eriod which is limited by the facility raw material store of 10,000 but the facility uses the stored roducts which are limited by the facility store of 2000 kilogram to satisfy customer s demands. In eriod 6, 7, 8 and 9 the required material is more than the sulied in the same eriod which is limited by the facility raw material store of 10,000. So the facility manufacture is not able to satisfy customer s demands and there will be some shortages to be satisfied in the next eriods if it is rofitable. In eriods 10, 11 and 12 the sulied material is more than the required in the same eriod limited by the facility raw material store to satisfy the shortages occurred in the revious eriods where the facility manufactures all sulied material and send them to customers to discover these shortages as ossible.

14 International Journal of Engineering & Technology IJET-IJENS Vol:16 No:01 56 Fig. 8. relationshi between the required weight, sulier caacity and the given weight 6. CONCLUSION AND FUTURE RECOMMENDATIONS This model successfully tackled the roblems of roduction lanning for the following reasons: 1. The roosed model is verified through a general examle with variable demand below and above the network caacities. 2. The roosed model is caable of otimizing multieriod, multi-echelon, multi-roduct and multiobjectives manufacturing network while considering inventory in both facility and distribution centres. 3. The model is caable of solving roblems with a larger number of eriods as comared to the numbers considered in the resent work. Future recommendations a) In the resent work, it is assumed that the customer s demands as known and deterministic values and it is recommended to tackle the roblem of stochastic demand. b) In the resent work, it is assumed that the manufacturing network uses single item to manufacture multi- and it is recommended to tackle the roblem of multi-items. ACKNOWLEDGEMENTS The author would like to thank FICO for their technical suort. REFERENCES [1] Chandra, P., & Fisher, M. L. (1994). Coordination of roduction and distribution lanning. Euroean Journal of Oerational Research, 72(3), [2] Jang, Y. J., Jang, S. Y., Chang,. M., & Park, J. (2002). A combined model of network design and roduction/distribution lanning for a suly network. Comuters and Industrial Engineering, 43(1 2), [3] Yan, H., Yu, Z., & Cheng, T. C. (2003). A strategic model for suly chain design with logical constraints: Formulation and solution. Comuters & Oerations Research, 30, [4] Kouvelis, P., Rosenblatt, M.J., Munson, C.L., A mathematical rogramming model for global lant location roblems: analysis and insights. IIE Trans. 36 (2), [5] Altiarmak, F., Gen, M., Lin, L., & Paksoy, T. (2006). A genetic algorithm aroach for multi-objective otimization of suly chain networks. Comuters and Industrial Engineering, 51(1), [6] Sha, D. Y., & Che, Z. H. (2006). Suly chain network design: artner selection and roduction/distribution lanning using a systematic model. Journal of Oerational Research Society, 57, [7] Huebner, R., Strategic suly chain management in rocess industries Dissertation. Sringer Verlag, erlin/heidelberg. [8] Tsiakis, P., Paageorgiou, L.G., Otimal roduction allocation and distribution suly chain networks. Int. J. Prod. Econ. 111 (2), [9] Zafel, G., raune, R., & ogl, M. (2010). Metaheuristic search concets: A tutorial with alications to roduction and logistics. Heidelberg Dordrecht London New York: Sringer. [10] Mula, J., Peidro, D., & Diaz-Madronero, M. (2010). Mathematical rogramming models for suly chain roduction and transort lanning. Euroean Journal of Oerational Research, 204(3), [11] Mezghani, M., Loukil, T., Aouni,., Aggregate lanning through the imrecise goal rogramming model: integration of the manager's references. Int. Trans. Oer. Res. 19 (4), [12] Margaretha Gansterer, Aggregate lanning and forecasting in make-to-order roduction systems. Int. J. Production Economics 170 (2015) [13]