Logistics Optimization Using Hybrid Metaheuristic Approach under Very Realistic Conditions

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1 17 th Euroean Symosium on Comuter Aided Process Engineering ESCAPE17 V. Plesu and P.S. Agachi (Editors) 2007 Elsevier B.V. All rights reserved. 1 Logistics Otimization Using Hybrid Metaheuristic Aroach under Very Realistic Conditions Yoshiaki Shimizu, Takeshi Wada, and Yoshihiro Yamazaki Det. o Production Systems Engineering, Toyohashi University o Technology, shimizu@se.tut.ac. Abstract In this study, we have concerned with a hierarchical logistic network design roblem rom two asects that have been scarcely considered reviously though they are very common in real-world. That is, to coe with a multi-commodity roblem and a roblem with stair-wised discount cost o transortation eectively as well as ractically, we have develoed the extended hybrid tabu search method or each roblem. Validity o the methods is veriied through comarison with the commercial sotware or the multi-commodity roblem while the algorithm is imlemented as sotware amenable or suorting a daily logistic lanning or the cost discount roblem. Keywords: Logistics, Large-scale Combinatorial Otimization, Multicommodity Product, Stair-wised Discount Cost, Hybrid Tabu Search 1. Introduction Logistic otimization has been weighed increasingly as a key issue to imrove the eiciency o business rocess under global cometition [1]. In the ield o OR, many studies since the earlier acility location roblems [2] have been taken lace or more than three decades. However, most o these studies concern with the roblems ormulated simly and maor emhasizes have been aid to develoing a new algorithm, and to evaluating its validity through

2 2 Y. Shimizu et al. benchmarking. To coe with the comlex and comlicated real-world situations, we still need more earnest eorts. This is because it causes such a dramatic increase in roblem size that makes it imossible to solve the resulting roblem by any currently available sotware. With this oint o view, this study concerns with a multi-commodity roduct roblem and a roblem with stair-wised discount regarding transortation cost and resents the extended hybrid meta-heuristic methods develoed or each roblem. 2. Problem Statement Taking a hierarchical logistic network comosed o members such like lants, distribution centers (DC), and customers as shown in Fig.1, we ormulate a tyical hierarchical logistic roblem as a mixed-integer rogramming roblem. Its obective unction is a total cost comosed o roduction cost, every transortation cost, holding cost and the ixed-charge cost or oening DC. tably, comared with the conventional ormulation, in this study, we have imosed realistic conditions. Since thus ormulated roblem (Reerence model) belongs to a NP-hard class, its solution becomes extremely diicult according to the increase o roblem size. To coe with various asects under such circumstance, we have roosed a method termed hybrid tabu search (HTS) [3-5]. It is a two-level method whose uer level roblem decides the location o DC in terms o the sohisticated tabu search [6] and the lower derives the routes among those members. At the lower level, the egged DC location roblem reers to a linear rogram (LP) that is ossible to transorm into the minimum cost low roblem (MCF). Hence, we can aly the grah algorithm known as CS2 [7] to solve the resulting roblem extremely ast. These rocedures will be reeated until a certain convergence criterion has been satisied. Figure 2 sketch the rocedure o HTS. In what ollows, the extended rocedures will be described or two cases that should be taken into account or real-world alications. Plants Close Oen Close Oen Distribution Centers Customers Start Generate init. DC location Uer Level Lower Level End Tabu search Relocate DC Converged? Evaluation Transorm to MCF CS2 (Grah algorithm) Decide route Fig.1 Logistic system concerned Fig.2 Scheme o hybrid tabu search (HTS)

3 Logistics Otimization Using Hybrid Meta-heuristic Aroach under Very Realistic Conditions 3 3. Formulation under very realistic conditions 3.1. Multi-Commodity Problem We imose the same mild assumtions as the reerence model excet or the caacitated condition on DC whose uer bound is limited regarding the total sum amount o each kind. Eventually, the maor extensions reer to exansion o the decision variables (rom i to i, P,) and the modiication o the caacitated condition on DC rom Eq.(1) to Eq.(2). i I 1 i + ' J + 2 ' 1 i P i I P ' J U x (or single-commodity) (1) (or multi-commodity) (2) # where i and # i denote resectively the shiing amounts rom i to or single-commodity and multi-commodity case regarding the item reerred by suerscrit #. Moreover, index sets I, J and P corresond to lants, ossible DCs and kinds o roduct, resectively. On the other hand, the right hand side o both equations gives the uer bound o the holding caacity at DC. It could be U i -th DC would be oen (x =1), otherwise 0 (x =0). Ater all, the roblem requires us to decide the location o DC, and delivery routes rom lants to customers via DCs to satisy demand o every roduct so as to minimize the total cost mentioned already. Due to the existence o binding condition on the holding caacity o DC i.e., Eq.(2), getting simly every solution o singlecommodity roblem together cannot be the resent solution. To coe with the multile commodities or the lower level roblem in HTS, thereore, we roose to give the ollowing ingenious rocedure as deicted in Fig.3, and outlined below. Ste 1: Set the initial values. Ste 2: Solve the lower level roblem er each kind and collect every solution. Ste 3: Examine every caacity constraint o DC by summing u the above results. I no 2 ' U x From Uer Level Ste1 For all, set U :=U Ste2 Solve or every kinds, and sum u caacity Ste3 Ste5 Satisy Eq.(2)? Solve by LP, i necessary Ste6 E(S) < E(Sbest)? To Uer Level Ste4 Revise U by Eq.(3) Ste4 E(S) < αe(sbest)? Ste6 Set Sbest := S Fig.3 Lower level rocedures or multicommodity model

4 4 Y. Shimizu et al. violations are observed, go to Ste 5 & 6. Otherwise, go to the next ste. Ste 4: Imose/revise the orcing conditions Eq.(3) on the routes where the violations occur, and go back to Ste 2 i the resent search is romising. Otherwise, return to the uer level. Ste 5 & 6: Based on sohistication by LP i necessary (Ste 5), the result o evaluation will be udated only i it is imroved (Ste 6). Then return to the uer level. U e : = U P, k Σ e k P J vio where e denotes the holding amount o kind at DC obtained in Ste 2 and J vio an index set o the caacity overilling DC. In real imlementation, the algorithm contains more elaborate ideas not described here A Stair-wised discount cost roblem By considering the stair-wised volume discount transortation cost as shown in Fig.4, we need to distinguish the shiing amount deending on the discount level. This induces the introduction o additional 0-1 variables. For examle, relation (3) N l 1l yi 1l i 1l i My l L, i I, (4) means that shiing amount rom lant i to DC must be greater or equal to N l l to aly the l-th level discount or this transortation. Here, yi denotes the 0-1 variable that takes 1 i this is true, and otherwise 0. Moreover, M is a very large number and L index set o discount level. Aarently, necessary shiing amount will be calculated by 1 i = l L 1l i i I, From so ar discussion, even i the location is egged at the uer level roblem, there still remain 0-1 variables in the lower level roblem. In other word, the roblem reers to the mixed 0-1 rogramming roblem. To rerain rom alying the comutationally intensive solution method like branch and bound method, and to kee alying a high-seed grah algorithm to solve the transormed MCF roblem, we invent a sequential substitution method to handle the discount ractically and eectively as ollows. (5) Unit Transort Cost (cost/unit) Volume (unit) Fig.4 Stair-wised discount cost

5 Logistics Otimization Using Hybrid Meta-heuristic Aroach under Very Realistic Conditions 5 CPU Time (sec) Proosed method with LP Proosed method without LP CPLEX Aroximate rate (-) with LP without LP (1) (2) (3) (4) (5) Fig.5 Screen shots o GUI o the Fig.6 Comutation load vs. roblem size ( P ) Ste 1: Set the initial discount at level 1 ( discount) or all transortations. Ste 2: Solve the roblem under the set-u discount level. Ste 3: Examine the consistency between the set-u level and the level ostdetermined rom the result o Ste 2. I there are no conlicts at all, return to the uer level. Otherwise, go to the next ste. Ste 4: Relace the set-u level with the ost-determined one or the inconsistent routes. Then go back to Ste 2. Also, in real imlementation, the algorithm contains more elaborate ideas not described here. Moreover, we develoed sotware amenable or suorting a daily decision making on logistic lanning, and conirm its eectiveness through a real-lie alication or a maor chemical comany in Jaan. Figure 5 shows a ew screen shots o GUI o the sotware Numerical exeriments 0 (1) (2) (3) (4) (5) Number o the kinds o roduct P ( - ) To validate the erormance o the roosed methods, we comared the results with those obtained rom commercial sotware (CPLEX9.0). Figure 6 shows a trend o comutational load along with the number o kinds o roduct. Three line grahs reresent the CPU times to ind the otimal solution using CPLEX, time to attain at the converged solution using roosed method with LP, and that without LP (Reer to Ste 5 o the algorithm in Sec.3.1). Moreover, numbers in the igure are the aroximated rates o the solutions rom the roosed method with LP and without LP to that o CPLEX in terms o the obective value. We can see very high aroximated rates are attained by the roosed method with very short time or the smaller roblems, say u to 50 (limit o the resent comarison). Moreover, the roosed method can solve much larger roblems with aroximately linear increase in comutation load. Relying on such observations, we can assert the advantage o the roosed method.

6 6 Y. Shimizu et al. Table 1 Results o benchmark roblems Prob.ID. o 0-1 vars. CPU time [s] The introduction o stair-wised discount cost makes increase both numbers o 0-1 variables and constraints simultaneously many times o the number o stairs. For the roblems in the chemical comany, the roosed method can converge at the desirable solution within the ractical comutation time as shown in Table 1. It should be noticed this is achieved through a slow seed rimal-dual algorithm o ree sotware to solve the MCF instead o CS2 due to the license. By virtue o GUI shown already, the develoed sotware is being used helully to suort a decision making on the logistic design tasks at the comany. 4. Conclusions In this study, concerning with the two asects that are very common in realworld logistics, i.e., a multi-commodity roblem and a roblem with stair-wised discount transortation cost, we have develoed the extended methods o HTS or each roblem. vel iterative rocedures are invented resectively while alying the grah algorithm to kee the high seed solution or the lower level roblem. Validity o the methods is veriied through comarison with the commercial sotware and evolution or the daily logistic lanning in a real-lie alication. Acknowledgement This study was artly suorted in the 21 st Century COE Program Intelligent Human Sensing, rom the Jaanese Ministry o Education, Culture, Sorts, Science and Technology. Reerences 1. I.A. Karim, R. Srinivasan and P.L. Han, Chemical Engineering Progress, 98 (2002) J.F. Cambell, Studies in Locational Analysis, 6 (1994) Y. Shimizu and T. Wada, Transaction o the Institute o systems, control and inormation engineers, 17, (2004) 241. (in Jaanese) 4. T. Wada and Y. Shimizu, Transaction o the Institute o systems, control and inormation engineers, 19, (2006) 69. (in Jaanese) 5. Y. Shimizu, S. Matsuda and T. Wada, Transaction o the Institute o systems, control and inormation engineers, 19, (2006) (in Jaanese) 6. C. Reeves (ed.), Tabu search, Modern Heuristic Techniques or Combinational Problems, Blackwell Scientiic Publishing, A.V. Goldberg, J. Algorithms, 22 (1997) 1.