Applying Genetic Algorithms for Inventory Lot-Sizing Problem with Supplier Selection under Storage Space

Transcription

1 Inernaional Journal of Indusrial and Manufacuring Engineering Applying Geneic Algorihms for Invenory Lo-Sizing Problem wih Supplier Selecion under Sorage Space Vichai Rungreunganaun and Chirawa Woarawichai Inernaional Science Index, Indusrial and Manufacuring Engineering wase.org/publicaion/7361 Absrac The objecive of his research is o calculae he opimal invenory lo-sizing for each supplier and minimize he oal invenory cos which includes join purchase cos of he producs, ransacion cos for he suppliers, and holding cos for remaining invenory. Geneic algorihms (GAs) are applied o he muli-produc and muli-period invenory lo-sizing problems wih supplier selecion under sorage space. Also a maximum sorage space for he decision maker in each period is considered. The decision maker needs o deermine wha producs o order in wha quaniies wih which suppliers in which periods. I is assumed ha demand of muliple producs is known over a planning horizon. The problem is formulaed as a mixed ineger programming and is solved wih he GAs. The deailed compuaion resuls are presened. Keywords Geneic Algorihms, Invenory lo-sizing, Supplier selecion, Sorage space. L I. INTRODUCTION OT- sizing problems are producion planning problems wih he objecive of deermining he periods where producion should ake place and he quaniies o be produced in order o saisfy demand while minimizing producion and invenory coss [1]. Since lo-sizing decisions are criical o he efficiency of producion and invenory sysems, i is very imporan o deermine he righ lo-sizes in order o minimize he overall cos. Lo-sizing problems have araced he aenion of researchers. The muli-period invenory lo-sizing scenario wih a single produc was inroduced by Wagner and Whiin [2], where a dynamic programming soluion algorihm was proposed o obain feasible soluions o he problem. Soon aferwards, Basne and Leung [3] developed he muli-period invenory lo-sizing scenario which involves muliple producs and muliple suppliers. The model used in hese former research works is formed by a single-level unconsrained resources indicaing he ype, amoun, suppliers and purchasing ime of he produc. This model is no able o consider he capaciy limiaions. One of he imporan modificaions we consider in his paper is ha of inroducing sorage capaciy. In his paper based on Basne and Leung [3] geneic algorihms (GAs) are applied o he muli-produc and muli-period invenory lo-sizing problem wih supplier selecion under sorage space. Also a maximum sorage space for he decision maker in each period is considered. The decision maker needs o deermine wha producs o order in wha quaniies wih which suppliers in which periods. The objecive of his research is o calculae he opimal invenory lo-sizing for each supplier and minimize he oal invenory cos. II. METHODOLOGY A. Geneic Algorihms Approach The geneic algorihms (GAs) approach is developed o find opimal (or near opimal) soluion. Deailed discussion on GAs can be found in books by Holland [4], Michalewicz [5], Gen and Cheng [6] [7], Davis [8] and, Goldberg [9]. In his secion, we explain GAs procedure is illusraed in Fig. 1. Topics covered include (1) Chromosome srucure (2) Iniial populaion (3) Evaluaion (4) Selecion (5) Crossover (6) Muaion, and (7) Terminaion rule. Iniial populaion Evaluaion Selecion Crossover Muaion Vichai Rungreunganaun is wih he Deparmen of Indusrial Engineering, Faculy of Engineering, KMUTNB, Bangkok, Thailand (phone: ; fax: ; r_vichai@yahoo.com). Chirawa Woarawichai is wih he Deparmen of Indusrial Engineering, Faculy of Engineering, Rajamangala Universiy of Technology Lanna Tak, Thailand (phone: ; fax: ; Chirawaw@homail.com). No Yes Terminaion rule Fig. 1 The geneic algorihm procedure Sop Inernaional Scholarly and Scienific Research & Innovaion 7(2) scholar.wase.org/ /7361

2 Inernaional Journal of Indusrial and Manufacuring Engineering Inernaional Science Index, Indusrial and Manufacuring Engineering wase.org/publicaion/ Chromosome Srucure In his problem, we ake each chromosome as a model soluion, where I, J and T are he number of producs, suppliers and periods, respecively, and each chromosome is a real values vecor (we make i by X) by lengh of (I x J x T) and a binary values vecor are 0 or 1 (we make i by Y) by lengh of (J x T), appropriae by each X ij and Y j (decision variables). For example, he represenaion of a chromosome is illusraed in Fig. 2. Chromosome X X X ij XIJT 1 Y YJT 1 Y 11 Y 12 j Fig. 2 Chromosome srucure XIJT 2. Iniial Populaion The populaion iniializaion echnique used in he GAs approach is a randomly generae soluions for he enire populaion. Populaion size depends only on he naure of problems and i mus balance beween ime complexiy and search space measure. More populaion size may increase he probabiliy of finding opimal soluion, bu may induce a longer compuer ime. In his paper, we use a populaion size is se no less han wice he lengh of he vecor of he chromosomes [10]. 3. Evaluaion or Finess Funcion I is evaluaed by he chromosome srucure which resuls in posiive value in [11]. Finess value defines he relaive srengh of a chromosome compared wih he ohers, and he opimaliy of he soluion o he problem. The finess funcion of his model is an objecive one (o minimize cos). 4. Selecion The selecion of parens o produce successive generaions plays an exremely imporan role in he GAs. The goal is o allow he fies individuals o be seleced more ofen o reproduce. However, all individuals in he populaion have a chance of being seleced o reproduce he nex generaion. In his paper, he roulee wheel selecion echnique is used [12]. 5. Crossover Operaor Crossover operaors combine informaion from wo parens in such a way ha he wo children (soluions for he nex populaion) resemblance o each paren. There are several available mehods o do so [13]. This paper adaps wo poin crossover operaors o solve GAs [12]. 6. Muaion Operaor Muaion operaors aler or muae one chromosome by changing one or more variables in some way or by some random amoun o form one offspring. For muaion, we use a linear muaion by probabiliy (1/I x J x T) for muaing X vecor and bi-wise muaion by probabiliy (1/J x T) for Y vecor [14], [15]. YJT 7. Terminaion Rule The GAs moves from generaion o generaion selecing and reproducing parens unil a erminaion crierion is me. The mos frequenly used sopping crierion is a specified maximum number of generaions. In his paper, here are wo sop crieria. Firs, he process is sopped when he number of ineraions has reached he maximum generaions. Second, he process is sopped when he maximum ime exceeds (se a 120 minues) [3]. B. Formulaion We also make he following assumpions and mahemaical for he model: Assumpions Demand of producs in period is known over a planning horizon. All requiremens mus be fulfilled in he period in which hey occur: shorage or backordering is no allowed. Transacion cos is supplier dependen, bu does no depend on he variey and quaniy of producs involved. Holding cos of produc per period is produc-dependen. Produc needs a sorage space and available oal sorage space is limied. The sorage space is for finished goods. Base on he above assumpion of model, Fig. 3 shows he behavior of he model considering he scenario of muli-period invenory lo-sizing problem wih supplier selecion under sorage space. The characerisics of he model used o deermine wha producs i, wih which suppliers j, and in which periods o order ( X ij). Fig. 3 Behavior of he model in period Mahemaical Modeling This paper is buil upon Basne and Leung [3] model. We formulae he muli-produc and muli-period invenory losizing problem wih supplier selecion under sorage space using he following noaion: Indices: i = 1,., I index of producs j = 1,., J index of suppliers = 1,., T index of ime periods Parameers: Inernaional Scholarly and Scienific Research & Innovaion 7(2) scholar.wase.org/ /7361

3 Inernaional Journal of Indusrial and Manufacuring Engineering Inernaional Science Index, Indusrial and Manufacuring Engineering wase.org/publicaion/7361 D i = demand of produc i in period P ij = purchase price of produc i from supplier j H i = holding cos of produc i per period O j = ransacion cos for supplier j w i = sorage space produc i S = oal sorage space Decision Variables: X ij = number of produc i ordered from supplier j in period Y j = 1 if an order is placed on supplier j in ime period, 0 oherwise Inermediae Variable: R i = Invenory of produc i, carried over from period o period + 1 Regarding he above noaion, he mixed ineger programming is formulaed as follows: + Minimize (TC) P ijxij OjY j + Subjec o: Ri = i j i = k= 1 j T Dik k i wi Hi k= 1 j X Xijk Dik k = 1 Yj Xij Xijk Dik k= 1 j k = 1 Yj Xij = j ijk k = 1 Dik 0 i, (1) (2) 0 i, j, (3) S (4) 0 or 1 j, (5) 0 i, j, (6) The objecive funcion as shown in (1) consiss of hree pars: he oal cos (TC) of 1) purchase cos of he producs, 2) ransacion cos for he suppliers, and 3) holding cos for remaining invenory in each period in + 1. Consrain in (2) all requiremens mus be filled in he period in which hey occur: shorage or backordering is no allowed. Consrain in (3) here is no an order wihou charging an appropriae ransacion cos. Consrain in (4) each producs have limied capaciy. Consrain in (5) is binary variable 0 or 1 and Consrain in (6) is non-negaiviy resricions on he decision variable. According o a large opimal problem, a GAs approach is applied o solve his problem. C. A Numerical Example In his secion we solved a numerical example of he model using real parameer geneic algorihms. We consider a scenario wih hree producs over a planning horizon of five periods whose requiremens are as follows: demands of hree producs over a planning horizon of five periods are given in Table I. There are hree suppliers and heir prices and ransacion cos, holding cos and sorage space are show in Table II and Table III, respecively. TABLE I DEMANDS OF THREE PRODUCTS OVER A PLANNING HORIZON OF FIVE PERIODS ( D i) Planning Horizon (Five Periods) Producs A B C TABLE II PRICE OF THREE PRODUCT BY EACH OF THREE SUPPLIERS X, Y, Z ( Pij) AND TRANSACTION COST OF THEM ( O j). Price Producs A B C Transacion Cos TABLE III HOLDING COST OF THREE PRODUCTS A, B, C ( H i) AND STORAGE SPACE OF THEM ( w i) Producs Producs A B C Holding Cos Sorage Space The oal sorage space (S) is equal o 200. The resuls of applying he proposed mehod are shown in Table IV. The soluion of his problem (I = 3, J = 3, and T = 5) is o place he following orders. All oher X ij = 0: Inernaional Scholarly and Scienific Research & Innovaion 7(2) scholar.wase.org/ /7361

4 Inernaional Journal of Indusrial and Manufacuring Engineering Inernaional Science Index, Indusrial and Manufacuring Engineering wase.org/publicaion/7361 TABLE IV ORDER OF THREE PRODUCTS OVER A PLANNING HORIZON OF FIVE PERIODS ( X ij) Producs Planning Horizon (Five Periods) A X 131 = 12 X 132 = 15 X 113 = 37 - X 135 = 13 B X 231 = 20 X 231 = 21 X 213 = 22 X 234 = 23 X 235 = 24 C X 131 = 12 X 132 = 15 X 113 = 37 - X 135 = 13 Cos calculaion for his soluion: Purchase cos for produc 1 from supplier 1, 3 = (37 30) + ( ) 32 = 2,390. Purchase cos for produc 2 from supplier 1, 3 = (22 32) + ( ) 30 = 3,344. Purchase cos for produc 3 from supplier 1, 3 = (18 45) + ( ) 45 = 4,050. Transacion cos from supplier 1, 3 = (1 110) + (4 102) = 518. Holding cos for produc 1 R 13 = X D13 = = 20. = H 1 R1 = 1 ( ) = 20. Thus, he oal cos for his soluion = 2, , , = 10,322. III. RESULTS Compuaion Resuls In his secion he comparison of he wo mehods solved problem size is using a commercially available opimizaion package like LINGO12 and GAs code is developed in MATLAB7. Experimens are conduced on a personal compuer equipped wih an Inel Core 2 duo 2.00 GHz, CPU speeds, and 1 GB of RAM. The ransacion coss are generaed from in [50; 200], a uniform ineger disribuion including 50 and 200. The prices are from in [20; 50], he holding coss from in [1; 5], he sorage space from in [10; 50], and he demands are from in [10; 200]. The resul in Table V shows he GAs comparing wih LINGO12 for he nine problem sizes. A problem size of I; J; T indicaes number of suppliers = I, number of producs = J, and number of periods = T. Compuaion ime limi is se a 120 minues. For comparison, he percenage error is calculaed by (7) and (8). Percenage error of LINGO12 Upper bound - Lower bound = 100 Upper bound (7) Percenage error of GAs Upper bound LINGO - GAs = 100 (8) Upper bound LINGO The soluion ime of LINGO12 o opimal is a shor ime as he small problem size (wih he problem sizes 3 x 3 x 5; 3 x 3 x 10; 3 x 3 x 15; and 4 x 4 x 10). For large problems sizes LINGO12 canno obain opimal soluions wihin limi ime due o as he larger problem size (wih he problem sizes 4 x 4 x 15; 5 x 5 x 20; 10 x 10 x 50; 10 x 10 x 80; and 15 x 15 x 50). The GAs can opimally solve when he problem size is small (wih he problem sizes 3 x 3 x 5; 3 x 3 x 10; 3 x 3 x 15; 4 x 4 x 10; 4 x 4 x 15; 5 x 5 x 20; and 10 x 10 x 50). There are wo problems which GAs canno obain opimal soluions (wih he problem sizes 10 x 10 x 80; and 15 x 15 x 50). Nex, we sudy differences in he problem sizes beween soluions from he opimizaion wih LINGO12 and he GAs. The resuls are show in Fig. 4, a plo of he problem size versus soluion ime. LINGO12 uses longer compuaion ime more han GAs wih seven problem sizes, bu uses equal ime wih wo problem sizes. As show in Fig. 5 a plo of he problem size versus % error when he problem size is very large, LINGO12 used a maximum % error from he opimal soluions is found o be 4.41% (a he problem size 10 x 10 x TABLE V COMPARATIVE RESULTS OF THE TWO METHOD Opimizaion approach wih LINGO12 Geneic Algorihms (GAs) Problem size Soluion ime Soluion ime Toal cos % Error Toal cos (minue) (minue) % Error 3 x 3 x 5 10, , x 3 x 10 20, , x 3 x 15 30, , x 4 x 10 25, , x 4 x 15 38,154 a, 37,828 b , x 5 x 20 60,218 a, 59,527 b , x 10 x ,344 a, 274,758 b , x 10 x ,494 a, 436,317 b , x 15 x ,800 a, 405,155 b , a LINGO12 = Upper bound, b LINGO12 = Lower bound. Inernaional Scholarly and Scienific Research & Innovaion 7(2) scholar.wase.org/ /7361

5 Inernaional Journal of Indusrial and Manufacuring Engineering Inernaional Science Index, Indusrial and Manufacuring Engineering wase.org/publicaion/ ) which has more % error han GAs. The GAs can solve small % error of wo problem sizes (a he problem size 10 x 10 x 80; and 15 x 15 x 50). Fig. 6 and Fig. 7 show compares resul beween LINGO12 and GAs in problem size 3 x 3 x 5. Thus, i is eviden ha GAs is an effecive means for solving he problem. GAs soluion is opimal when he problem size is small. For larger problems GAs can find feasible soluion wihin ime limi for which LINGO12 fails o find he opimum. However, he GAs provides superior soluions o hose from LINGO12 ha are close o opimum in a very shor ime, and hus appears quie suiable for realisically sized problems. Addiionally, he compuaion ime when using GAs is also shor, making i a very pracical means for solving he muliple producs and muli-period invenory lo-sizing problem wih supplier selecion under sorage space. Soluion ime (minue) % Error x3x5 3x3x10 3x3x15 LINGO12 4x4x10 4x4x15 Problem size GAs Fig. 4 Plo of he problem size vs. soluion ime (minue) x3x5 3x3x10 3x3x15 LINGO12 4x4x10 4x4x15 Problem Size 5x5x20 GAs 5x5x20 10x10x50 10x10x50 Fig. 5 Plo of he problem size vs. % error 10x10x80 10x10x80 15x15x50 15x15x50 Fig. 6 The bes objecive of LINGO 12 Fig. 7 The finess value of GAs IV. DISCUSSION In his paper, we presen geneic algorihms (GAs) applied o he muli-produc and muli-period invenory lo-sizing problem wih supplier selecion under sorage space. Also a maximum sorage space for he decision maker in each period is considered. The decision maker needs o deermine wha producs o order in wha quaniies wih which suppliers in which periods. The mahemaical model is give and he use of he model is illusraed hough a numerical example. The problem is formulaed as a mixed ineger programming and is solved wih LINGO12 and he GAs. As compared o he soluion of opimizaion package like LINGO12, he GAs soluions are superior. REFERENCES [1] G.H. Goren, S. Tunali, and R. Jans, A review of applicaions of geneic algorihms in lo sizing, Journal of Inelligen Manufacuring. Springer Neherlands Inernaional Scholarly and Scienific Research & Innovaion 7(2) scholar.wase.org/ /7361

6 Inernaional Journal of Indusrial and Manufacuring Engineering Inernaional Science Index, Indusrial and Manufacuring Engineering wase.org/publicaion/7361 [2] H. M. Wagner, and T. M. Whiin, Dynamic version of he economic lo-size model, Managemen Science. 5, pp, 89 96, [3] C. Basne and J. M. Y. Leung, Invenory lo-sizing wih supplier selecion, Compuers and Operaions Research. 32, pp, 1 14, [4] J. H. Holland, Adapaion in Naural and Arificial Sysems. The Universiy of Michigan Press, Ann Arbor [5] Z. Michalewicz, Geneic Algorihms + Daa Srucures = Evoluion Programs. AI Series. Springer-Verlag, New York [6] M. Gen, and R. Cheng, Geneic Algorihms and Engineering Design. Wiley, New York [7] M. Gen, and R. Cheng, Geneic Algorihms and Engineering Opimizaion. Wiley, New York [8] L. Davis, The handbook of geneic algorihms, Van Nosrand Reinhold, New York [9] D. E. Goldberg, Geneic Algorihms in Search, Opimizaion and Machine Learning. Addison-Wesley, Reading, MA [10] S. R. M. Mogador, A. Afsar, and B. Sohrabi, Invenory lo-sizing wih supplier selecion using hybrid inelligen algorihm. Applied Sof Compuing. vol, 8, pp, , [11] S. H. Chan, W. Chung, and S. Wadhwa, A hybrid geneic algorihm for producion and disribuion. Omega. 33, pp, , [12] R. Sarker, and C. Newon, A geneic algorihm for solving economic lo size scheduling problem. Compuers and Indusrial Engineering. 42, [13] K. Deb, Muli-Objecive Opimizaion using Evoluionary Algorihms Wiley, Chicheser, [14] J. Rezaei, and M. Davoodi, Geneic algorihm for invenory lo-sizing wih supplier selecion under fuzzy demand and coss. Advances in Applied Arificial Inelligence. Springer-Verlag Berlin Heidelberg. 4031, [15] J. Rezaei, and M. Davoodi, A deerminisic, muli-iem invenory model wih supplier selecion and imperfec qualiy. Applied Mahemaical Modelling. 32, pp, , Inernaional Scholarly and Scienific Research & Innovaion 7(2) scholar.wase.org/ /7361