Using Latin Hypercube Sampling to Improve Solution for a Supply Chain Network Design Problem

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1 Uing Latin Hypercube Sampling to Improve Solution for a Supply Chain Network Deign Problem Kantapit Kaewuwan Department of Indutrial Engineering, Faculty of Engineering King Mongkut Intitute of Technology Ladkrabang, Bangkok, Thailand Tel: (+66) , kantapit.k@gmail.com Udom Janjarauk Department of Indutrial Engineering, Faculty of Engineering King Mongkut Intitute of Technology Ladkrabang, Bangkok, Thailand Tel: (+66) ext. 8, kjudom@gmail.com Kanokporn Rienkhemaniyom Graduate School of Management and Innovation King Mongkut Univerity of Technology Thonburi, Bangkok, Thailand Tel: (+66) , kanokporn.rie@gmail.com Chumpol Yuangyai Department of Indutrial Engineering, Faculty of Engineering King Mongkut Intitute of Technology Ladkrabang, Bangkok, Thailand Tel: (+66) ext. 4, kychumpo@kmitl.ac.th Abtract. In thi paper, we focu on a upply chain network deign problem in determining the number and location of facilitie, a well a the flow between them. Due to globalization and complexity in upply chain environment, recent upply chain network deign tudie have incorporated uncertainty and diruption ri into the deign deciion in order to make upply chain network more robut and reilience. To deal with the unexpected diruption, a Latin Hypercube Sampling (LH method i ued to generate diruptive cenario for model evaluation. We compare the olution with thoe from the Monte Carlo Sampling (MC technique. The reult how that LHS provide maller tandard deviation than the MCS technique a it yield a good approximation of the ample pace. Thi further indicate that LHS ue le ample ize leading to more efficient computational time in obtaining olution. Keyword: Supply chain network deign, Facility diruption, Two-tage tochatic program, Simulated annealing, Latin hypercube ampling. INTRODUCTION Supply chain network deign i a crucial trategic deciion a it effect will lat for many year under dynamic buine environment (Klibi et al., 00). In addition, uncertaintie and ri are intenified in the upply chain network where demand and cot vary and diruption are unpredictable. Therefore, the deign of upply chain network hould be reilience to uncertainty and poible diruption in the network. In many upply chain network deign tudie, Monte Carlo Sampling technique ha been applied in order to generate cenario for model evaluation. In thi paper, we focu on the ue of a ampling technique called Latin Hypercube Sampling for upply chain network deign under facility diruption. In thi tudy we ue a two-tage tochatic programming model from Rienkhemaniyom et al. (06) that formulated for a upply chain network deign problem conidering poibility of facility diruption. The model i ued for a ingle-product upply chain network which include upplier, manufacturing plant, warehoue, and retailer where the facilitie are located in different place around the world. The objective

2 function i to maximize the expected profit under facility diruption. In the firt tage, upplier and warehoue are elected. In the econd tage, the quantity of product purchaed, produced and tranported between facilitie are determined under diruptive cenario o that the expected profit i maximized. A imulated annealing meta-heuritic algorithm (SA) i ued to find near optimal olution. In term of ampling technique, the Monte Carlo Sampling (MC technique i ued in contructing the ample average problem in order to get the etimate of the objective function value. Beide MCS, there are everal ampling technique available to reduce the variance of the etimate and to improve the quality of olution, for example Latin Hypercube Sampling (LH, and Antithetic Variate Sampling (AV. In thi paper, we apply Latin Hypercube Sampling (LH technique to generate diruptive cenario. Several tudie have dicued the advantage of LHS, for intance, Fattahi et al. (04) and Shi et al. (03) tated that LHS cover more of domain of the random variable. Hence, the objective of thi tudy i to improve olution of a upply chain network deign problem by uing LHS technique to ampling diruptive cenario.. LITERATURE REVIEW For a tochatic linear program, it can be olved approximately by generating a ubet of all poible random cenario. The value of the optimal olution to the ampled problem provide an etimate of the true objective function value. However, the etimator i known to be optimitically biaed (Freimer et al., 0). The bia and variance of the olution are alo related to which ampling method ha been ued. Freimer et al. (0) tudied the impact of ampling method: antithetic variate and Latin Hypercube ampling, on the bia and variance reduction in a two-tage tochatic linear program for a newvendor problem. The firt-tage olution i to chooe the order quantity, while the econd-tage olution decide on how much of the available tock after demand ha been realized. The reult how that both ampling method are effective at reducing variance, however LHS ampling approach outperform the antithetic variate method in both bia and variance reduction. K.R.M do Santo et al. (05) preented a benchmark tudie that comparing MCS with four modern ampling technique, including Importance Sampling Monte Carlo, Aymptotic Sampling, Enhanced Sampling, and Subet Simulation. The ampling technique are combined with three cheme for generating the ample, which conit of Simple Sampling, LHS, and Antithetic Variate Sampling. The reult from thi paper how that Importance Sampling i extremely efficient for evaluating mall failure probabilitie but it can be an iue for ome problem. Subet Simulation yield very good performance for all problem. Enhanced Sampling ha performance better than Aymptotic Sampling. LHS outperform Simple Sampling and Antithetic Variate Sampling. The Latin Hypercube Sampling (LH i a tratifiedrandom procedure where ample are obtained from their ditribution. The cumulative ditribution for each variable i divided into N equiprobable interval. Auming that the variable are independent, a value i elected randomly from each interval. (Yuhumito et al., 0). LHS ha been ued in variou application, including upply chain network deign problem a it become a very large and complex problem when diruption i conidered (Garcia-Herrero and Gromann, 03). Keyvanhokooh (05) ha applied LHS to generate cenario for tranportation cot. Govindan and Fattahi (05) ued LHS to generate a et of demand cenario. Yuhumito et al., (0). Deleri et al. (005) propoed a Monte Carlo baed technique to evaluate upply chain network under ri. From the review, we oberve that Monte Carlo Sampling (MC i a common and widely ued technique for cenario generation, including diruption, in upply chain network deign problem. For LHS, it ha alo been ued to generate demand and cot cenario. However, very few tudie have ued Latin Hypercube Sampling for generating diruptive cenario. In thi paper, we ue LHS technique to generate diruptive cenario for a upply chain network deign propoed by Rienkhemaniyom et al. (06). Then, we compare the olution quality, in term of variance, and computational time with the original work. 3. SUPPLY CHAIN NETWORK DESIGN PROBLEM 3. Problem decription The original tudy of a upply chain network deign problem propoed in Rienkhemaniyom et al. (06) involve a two-tage tochatic program for a upply chain network deign under facility diruption for a four-tage upply chain which conit of upplier, manufacturing plant, warehoue, and retailer (a how in Figure ) under ingle-product upply chain network environment. The facilitie are located around the world. Location of manufacturing plant and retailer are known, while upplier and location of warehoue are deciion to be made. The objective function i to maximize the expected profit under facility diruption. Simulated annealing meta-heuritic algorithm i ued to find a good olution, and MCS technique i ued for generating diruptive cenario. In thi tudy, a diruption may occur at any region, which lead to diruption of upplier, manufacturing plant, or warehoue that are located in the dirupted region. The deciion variable are to elect the upplier and the location

3 of warehoue o that the expected profit of the upply chain i maximized. S M Supplier Plant Warehoue Retailer 3. Mathematical model W Figure. Supply chain network The two-tage tochatic programming model of a upply chain network deign problem that propoed in Rienkhemaniyom et al. (06) can be ummarized a follow. Notation: Set S M W C L K Set of upplier Set of manufacturing plant Set of warehoue Set of retailer Set of warehoue capacitie Set of diruptive cenario Indice Index of upplier S m Index of manufacturing plant m M w Index of warehoue w W c Index of retailer c C l Index of warehoue capacitie l L k Index of diruptive cenario k K Parameter cap m Production capacity at manufacturing plant m cap Capacity at upplier cap l w Capacity at warehoue w of ize l d c Demand for product at retailer c mm Minimum tranportation quantity form upplier to manufacturer m f l w Fixed cot of opening a warehoue w of capacity l pm m Purchaing cot of material from upplier by plant m tr m Tranportation cot per unit from plant m to warehoue w Tranportation cot per unit from warehoue w to tr wc C pc m np l c retailer c Production cot for a product at plant m Price of a product Lot ale cot at retailer c Random Parameter 0 p k wk mk 0 0 if upplier i operatedin cenario k if a diruptionoccur at upplier in cenario k if warehoue w i operatedin cenariok if a diruption occur at warehoue w in cenariok if plant m i operatedin cenariok if a diruption occurat plant m in cenariok The probability of diruption occur in cenario k Firt-tage Deciion Variable x l w y if warehoue w i operatedwith ize l 0 otherwie if upplier i elected 0 otherwie Second-tage Deciion Variable QSM mk Quantity of raw material purchaed from upplier by plant m in cenario k QMW mwk Quantity of product hipped from plant m to warehoue w in cenario k QWC wck Quantity of product hipped from warehoue w to retailer c in cenario k Quantity of ale lot at retailer c in cenario k LD ck 3.. Objective Function The objective i to maximize the expected upply chain profit (Z) which i the difference between the total cot and the expected revenue. The firt-tage cot i the fixed cot of opening warehoue and the econd-tage cot i the difference of the expected revenue and operation cot. Maximize l l Z f wxw pk np QWCwck ww ll kk ww cc pm mqsm mk S mm tr mwqmw mwk trwcqwcwck mm ww ww cc pc m QMW mwk lcldck () mm ww cc

4 Information Update 3.. Contraint Supplier capacity QSM mm mk cap y, S, k K () Inter-tage flow mm y QSM cap y, S, k K (3) Production capacity QMW w W mwk mk cap m mk Material flow between upplier and plant QSM S mk Warehoue capacity QMW ww mwk, m M, k K (4), m M, k K (5) l QMW mwk cap wwk xw, ww, k K (6) mm ll l xw, ww (7) ll Product flow between warehoue and retailer QMW mm mwk Demand requirement QWC ww wck QWC CC LD ck wck, ww, k K (8) d, c C, k K (9) Non-negativity QSM mk, QMW mwk, QWCwck, LDck 0 (0) For diruptive cenario, the author ue the number of diater from the Centre for Reearch on the Epidemiology of Diater CRED ( The number of diater occurred in each region i ued to etimate the occurrence or probability of diruption (a hown in Table ). Hence, thi probability will be ued for generating diruptive cenario by LHS. Table. Number of diater reported during Region c Number of diater Probability of occurrence Africa 3, Aia 8, Autralia Europe, North America, South America, Solution Methodology A imulated annealing meta-heuritic algorithm (SA) i applied in Rienkhemaniyom et al. (06) for finding the olution. The firt-tage olution i obtained outide the model, and the econd-tage olution i obtained by re-optimizing the aociated econd-tage problem from a given firt-tage olution. In thi tudy LHS technique i ued to contruct the ample average problem intead of olving problem with all poible cenario. Thi technique i known a the ample average approximation (SAA) technique in A. Shapiro (003). The olver CPLEX i ued to olve the econd-tage ample average problem in order to etimate the objective function value. The general framework of the algorithm i howed in Figure Initialization Generate t tage olution (Simulated Annealing) Contruct nd tage problem with diruptive cenario (Latin Hypercube Sampling) Evaluate the ample problem (CPLEX) Termination Figure. The general framework of the propoed methodology 3.3. Simulated Annealing Simulated annealing (SA) i a probabilitic technique for olving global optimization problem in Aart et al. (005). The

5 algorithm ha inpiration come from annealing in metallurgy, a thermal proce involving heating and controlled cooling of a olid for the particle to arrange themelve in the ground tate. Parameter r Index of iteration c r Control parameter at iteration c 0 Initial value of the control parameter c min Minimum value of the control parameter a, b Firt-tage olution vector T Number of olution evaluated in each iteration N Sample ize (the number of ampled cenario) Ẑ N Etimate of the objective function value by uing ample ize N Ƥ The probability vector of the firt-tage variable (x and y) U Random number in [0, ) ρ Coefficient of the control parameter update λ Coefficient of the probabilitie update ^ ^ Z N ( b) Z N ( a) exp cr () The parameter ρ, λ, c 0, T, N and the probability vector Ƥ are initialize parameter. The algorithm generate a feaible firt-tage olution vector from the probability vector Ƥ. In each iteration, the algorithm will earch for a better olution. The objective value are compared between current olution a and new olution b. The new olution b will be accepted if it objective value i better than the current objective value or if the probability of acceptance in Equation () i greater than a randomly generated number U. The control parameter c r play the role of temperature control in the cooling progre, which control the probability of accepting olution in the algorithm. The value of c r i decreaed by a factor of ρ in each iteration, which reducing the chance of accepting wore olution. At the end of each iteration, the probability vector i updated from the current olution vector. The algorithm terminate when c r i le than a given value c min. 4. LATIN HYPERCUBE SAMPLING LHS technique wa introduced by McKat et al. (979). It i one of the ampling technique that ha been ued in computer experiment for finding imulation olution. In thi tudy, we ue LHS to generate diruptive cenario to enure that all diruptive cenario will be occurred in each input variable. Due to the advantage of LHS technique, we expect to obtain olution that have a maller tandard deviation of the ample mean compared to thoe from the MCS technique in the original paper. 4. Latin Hypercube Deign A Latin hypercube deign (LHD) with N ample and S input variable, denoted by LHD(, i an N S matrix (i.e. a matrix with N row and S column). Hence, a LHS can be generated by the following tep. Step. The matrix P( wa generated, which each column of matrix P conit of a random permutation of the integer to N. p, p, S P( p p S Step. The random number in Monte Carlo method (Simply random ampling) u k, j ~ U(0, ), k =,, j =,, S which are mutually independent are generated to form the matrix U(. u, u, S U( u u S Step 3. The matrix L( wa generated from Equation (), and thi matrix i ued for generating the diruptive cenario. L( P( U( () N Figure 3 how a catter plot of MCS for ample ize N = 40 with two input variable. There i no pattern of point in each row and each column. The portion of it range are not repreented in each of the input variable. Figure 4 how a catter plot of LHS for ample ize N = 40. We can ee that each row and each column contain eight point, and the point are uniformly ditributed in the correponding row and column. It how that each input variable ha all portion of it range covered. For ample ize N=40, when dividing the ample pace of LHS into ten region, reach region will contain exactly four point. In thi tudy, LHS technique i ued to draw N ample from the et of original cenario K. The objective function value Z in Equation () i etimated by ubtituting the firttage olution obtained from SA, and then the ample average econd-tage problem i contructed by uing LHS technique. The random vector in the model i replaced by the ampled cenario, and the probability p k i changed to /N in the objective function. The reulting ample average problem i then olved by uing CPLEX.

6 Input variable Input variable MCS Input variable Figure 3. Scatter plot of ample generated by MCS LHS 5. RESULTS AND DISCUSSIONS Table preent a ummary of comparion of the upply chain profit value under diruptive cenario between the MCS and our propoed LHS ampling from running 0 experiment trial in different cae. In mot cae, the LHS provide better objective value and maller tandard deviation compared to the MCS. For example, at N = 50, the LHS ampling method yield a tandard deviation of profit of $8,3.69, while it require N = 00 for the MCS method to achieve the imilar reult (the tandard deviation of profit under MCS ampling method i $93,.6). When conidering diruption with ample ize 00, the MCS technique ha an average profit value $9,833,097.05, and the tandard deviation of profit value i $93,.6. On the other hand, the upply chain network olution from the LHS ha an average profit value $0,03,344.56, and the tandard deviation of profit value i $39, The reult how that the tandard deviation of olution when uing LHS with ample ize 00 can be reduced by 8.4 percent, and the average profit value i alo improved, but it i not ignificantly different CONCLUSION Input variable Figure 4. Scatter plot of ample generated by LHS Thi paper preent an application of LHS technique in a upply chain network deign problem. LHS i ued to generate diruptive cenario. The reult how that the tandard deviation of olution are reduced a compared to the reult from the MCS technique when uing the ame ample ize a in Rienkhemaniyom et al. (06). In olving large tochatic programming model for a upply chain network problem, LHS i an effective ampling technique that could provide good olution while uing mall ample ize. In future tudy, we will conider other new ampling technique to reduce ample ize and tandard uch a Optimal Latin Hypercube Sampling technique to generate diruptive cenario. Table : Comparion of objective value between two ampling technique with different ample ize Sampling technique Objective value N = 0 N = 50 N = 00 Monte Carlo Average,73,98.4 0,370,.87 9,833, Standard deviation 47, , ,.6 Latin Hypercube Average,897,9.75 0,366, ,03, Standard deviation 400, , , Average Difference.48% -0.04%.93% Standard deviation Difference -5.6% -6.44% -8.4%

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