International Journal of Supply and Operation Management IJSOM November 2015, Volume 2, Iue 3, pp. 925-946 ISSN-Print: 2383-1359 ISSN-Online: 2383-2525 www.ijom.com A tochatic programming approach for a multi-ite upply chain planning in textile and apparel indutry under demand uncertainty Houem Felfel *a, Omar Ayadi a and Faouzi Mamoudi a a National Engineering School of Sfax (ENIS), Univerity of Sfax, Tuniia, Road Soukra, Sfax, Tuniia Abtract In thi tudy, a new tochatic model i propoed to deal with a multi-product, multi-period, multi-tage, multi-ite production and tranportation upply chain planning problem under demand uncertainty. A two-tage tochatic linear programming approach i ued to maximize the expected profit. Deciion uch a the production amount, the inventory level of finihed and emi-finihed product, the amount of backorder and the quantity of product to be tranported between uptream and downtream plant in each period are conidered. The robutne of production upply chain plan i then evaluated uing tatitical and rik meaure. A cae tudy from a real textile and apparel indutry i hown in order to compare the performance of the propoed tochatic programming model and the determinitic model. Keyword: multi-ite; upply chain planning; tochatic programming; textile; robutne. 1. Introduction Modern proce indutrie operate no more a traditional ingle-plant but a multi-ite upply chain tructure where different production facilitie are erving a global market. In the lat decade, upply chain management ha received a remarkable interet in order to cope with highly * Correponding author email addre: houem.felfel@gmail.com 925
Felfel, Ayadi and Dadgar continuou competition. Supply chain planning i an important proce within the upply chain management involving deciion undertaken by a company from the procurement of raw material to the hipping of end product to the cutomer. The upply chain planning problem can be claified following the time horizon into three categorie: trategic, tactical, and operational (Chopra and Meindl 2010). The trategic level concern the deign and the tructure of the upply chain over a long time horizon between five and ten year. The operational level i related to hort term deciion lating from few day to few week uch a cheduling, lot izing and equencing. The tactical planning model i between thee two extreme and include procurement, production, and ditribution deciion. Thi tudy i particularly motivated by a tactical upply chain planning problem faced by multi-ite upply network from textile and apparel indutry. Textile manufacturing proce conit of knitting and dyeing, cutting, embroidery, cloth making, and packaging tage. Each production tage may include more than one plant, forming a multi-ite, multi-tage manufacturing environment. The fluctuation of product demand i among the mot important ource of uncertainty in the textile and apparel indutry. In fact, the cutomer demand could be determined only at the end of the planning horizon. The under-etimation of overall demand lead either to lo ale or unatified cutomer. However, the over-etimation of the product demand reult in high production and inventory cot. In thi paper, we deal with a multi-product, multi-period, multi-tage, multi-ite upply chain planning problem under cutomer demand uncertainty. A two-tage tochatic programming model i developed in order to incorporate the effect of the uncertainty in the conidered problem. Deciion variable uch a amount of production and quantity to be tranported between different manufacturing facilitie are conidered a firt-tage variable and are aumed to be made before the realization of the uncertainty. Otherwie, deciion variable related to the inventory level, backorder amount, and tranportation amount of end product to be hipped to the cutomer are conidered a econd-tage variable made after the realization of uncertain demand. Subequently, the robutne of production planning olution i evaluated. Statitical metric and financial rik metric, uch a value at rik (VaR) and conditional VaR (CVaR), are calculated in order to evaluate the robutne of planning olution generated by the tochatic programming model compared to the determinitic model. Beide, a real example from a textile and apparel manufacturer cae in Tuniia i illutrated to compare the propoed tochatic model with the traditional determinitic upply chain planning model. The main cientific contribution of thi work i to develop a new tochatic model for a multi-product, multi-period, multi-tage, multi-ite upply chain production and tranportation planning problem under cutomer demand uncertainty. Beide, the propoed model and the evaluation approach are applied to a real cae tudy from textile and apparel indutry. The ret of the paper i organized a follow. In the next ection, we preent the literature review of related topic. Section 3 decribe the textile and apparel upply network under conideration. In ection 4, a two-tage tochatic formulation i propoed in order to incorporate demand 926
Int J Supply Oper Manage (IJSOM) uncertainty in the upply chain planning problem. Section 5 decribe the tochatic programming algorithm. By conducting a real cae tudy, Section 6 verifie the effectivene and the robutne of the propoed tochatic model compared to the determinitic model. Finally, concluion, limitation of the developed model, and future reearch direction are drawn in Section 7. 2. Literature review To cope with highly competitive and global market, the tructure of manufacturing companie ha changed from traditional ingle-ite to multi-ite tructure. Multi-ite production planning problem have received a lot of attention in the literature. Mot of the paper dealing with multi-ite production planning problem focu on determinitic approache. Toni and Meneghetti (2000) addreed the production planning problem of a textile-apparel indutry upply chain. The author invetigate the influence of production planning period length a well a color aortment in the ytem' time performance. A real cae tudy from an Italian network of firm wa treated uing a imulation model. Lin and Chen (2007) developed a monolithic model of a multi-tage multi-ite multi-item production planning problem. The propoed model combined imultaneouly two different time cale, i.e., monthly and daily time bucket. A practical example from the thin film tranitor-liquid crytal diplay (TFT-LCD) indutry i illutrated to explain the planning model. Leung et al (2003) tudied a multi-ite aggregate production planning problem of a multinational lingerie company located in Hong Kong uing a goal programming approach. Three major objective function were conidered, which are minimization of the cot of worker hiring and laying-off, the maximization of profit and the minimization of the over-or under-utilization of import quota of different product. Shah and Ierapetritou (2012) treated the integrated planning and cheduling problem for multi-ite, multi-product batch plant uing the augmented Lagrangian decompoition method. Given the fixed demand forecat, the model aim to minimize production, torage, hipping, and backorder cot. Felfel et al (2014) propoed a multi-objective, multi-tage, multi-product, and multi-period model for production and tranportation planning in a multi-ite manufacturing network. The developed model aim imultaneouly to minimize the total cot and to maximize product quality level. It hould be noted that mot of the paper dealing with multi-ite production planning problem focu on determinitic olution. However, real production planning problem are characterized by everal ource of uncertainty. Hence, the aumption that all model parameter are known with certainty will lead to non-optimal and even unrealitic reult. Many approache have been propoed in the literature to cope with uncertainty. According to Sahinidi (2004), thee approache can be claified into four major categorie: fuzzy programming approach, robut optimization approach, tochatic programming approach, and tochatic dynamic programming approach. Stochatic programming approach (Birge and Louveaux, 1997; Dantzig, 1955) i one of the mot widely pread technique in the literature ued in upply chain planning problem under uncertainty. In thi approach, the deciion variable of the optimization problem are divided into two et. The deciion variable of the firt tage called here and now deciion have to be made before the realization of uncertainty. Subequently, the econd-tage deciion variable are choen after the preence of uncertain parameter in order to 927
Felfel, Ayadi and Dadgar correct the infeaibilitie caued by uncertainty realization ( wait and ee deciion). Therefore, the value of the objective function i the um of firt-tage deciion variable and the econd-tage expected recoure variable. Several work in the literature have been intereted in tochatic programming model for upply chain planning problem. Gupta and Marana (2000) propoed a multi-ite midterm upply-chain planning problem under demand uncertainty uing two-tage tochatic programming approach. The upply chain deciion are devied into two categorie: manufacturing deciion and logitic deciion. The manufacturing deciion are taken here and now before the realization of uncertainty while the logitic deciion are potponed in a wait and ee mode. A ingle period i conidered in the developed model. Leung et al (2006) developed a two-tage tochatic programming model in order to optimize a multi-ite aggregate medium-term production planning problem under an uncertain environment. The firt-tage deciion include the amount of manufactured product in regular-time and overtime, volume of ubcontracted product and number of required worker, hired worker and laid-off worker. Deciion uch a inventory level of product, and the amount of under-fulfilment product are conidered a econd-tage deciion. The effectivene of the propoed model wa highlighted through a real-world cae tudy from a multinational lingerie company ituated in Hong Kong. Karabuk (2008) conidered a yarn production planning problem under demand uncertainty in a textile manufacturing upply chain. The author developed a tochatic programming model where the rover configuration, frame configuration, and production quantity are the firt-tage deciion. Inventory level i conidered a recoure deciion. A two-tep preproceing algorithm i developed to olve the optimization problem and to reduce computational complexitie of the large-cale reulting model. Neverthele, thee work didn t conider tranportation in the mathematical optimization model. Nagar and Jain (2008) tudied a multi-period upply chain planning problem for new product launche under demand uncertainty. A two-tage tochatic programming approach i developed in order to incorporate uncertainty. Production quantity, raw material procurement, and capacity utilization are preented a here and now deciion. Outourcing, inventory and hipping of end product to cutomer are propoed a wait and ee deciion until the realization of uncertain demand. Subequently, thi model i extended uing a multi-tage tochatic programming formulation. Mirzapour Al-e-hahem et al (2011a) propoed a mid-term multi-product, multi-period, multi-ite production-ditribution planning problem under cot and demand uncertaintie. A two-tage tochatic programming model wa developed to incorporate the uncertain parameter. Mirzapour Al-e-Hahem et al (2011b) developed a multi-ite, multi-product, multi-period aggregate production planning problem. To olve thi problem, a new robut multi-objective mixed integer nonlinear programming model wa propoed. The common critic of thee work i the conideration of a ingle production tage in the planning problem. Awudu and Zhang (2013) developed a two-tage tochatic programming model for a production planning problem in a biofuel upply chain under uncertainty in order to maximize the expected 928
Int J Supply Oper Manage (IJSOM) profit. Amount of product to be produced, and amount of raw material to be purchaed and conumed are conidered a the firt-tage deciion. Deciion uch a backlog, lot ale, and old product quantity are conidered a econd-tage deciion. A cae tudy from a biofuel upply chain i illutrated to demontrate the effectivene of the propoed model. A ingle period and a ingle production tage are taken into account in thi work. In the context of upply chain planning, robutne can be defined a a meaure of reilience of the objective function, uually cot or profit, to change under random event and uncertain parameter. Therefore, the evaluation of robutne repreent an important iue in order to ae the performance of the upply chain planning in the face of parameter uncertainty. Vin and Ierapetritou (2001) developed a trategy to quantify cheduling robutne in the face of uncertainty under uncertainty. To do o, everal robutne metric were ued, uch a the corrected tandard deviation, the determinitic tandard deviation and the extent of violation. Lin et al (2011) propoed a tochatic programming model for trategic capacity planning in thin film tranitor-liquid crytal diplay indutry. The robutne of the capacity plan i evaluated uing financial rik meaure, uch a the value at rik and the conditional value at rik. Although the evaluation of robutne wa performed in cheduling and trategic planning, thi concept wa not extended to tactical multi-ite upply chain production and tranportation planning problem. 3. Problem tatement In thi paper, we conidered a upply network from the textile and apparel indutry wherein the finihed product i proceed by mean of different production tage. The textile and apparel manufacturing proce conit of five main tage: knitting and dyeing, cutting, embroidery, cloth making, and packaging. Each production tage may include more than one plant etablihing a multi-ite upply network manufacturing environment a illutrated in Figure 1. The conidered upply network i compoed of an internal plant (Textile-International TE-INTER ) and four ubcontractor: a dyer and a knitter, an embroiderer, and three cloth maker. The TE-INTER company i formed of three manufacturing department which are cutting, packaging, and cloth making. Other activitie uch a knitting, dyeing, and embroidery are ubcontracted becaue of the lack of technical competence and reource. Cloth making operation can be alo ubcontracted in order to extend the production capacity and to fulfill all the cutomer demand. 929
Felfel, Ayadi and Dadgar Figure 1. The multi-ite upply network environment of textile and apparel indutry In textile and apparel indutry, product are uually characterized by volatile demand and hort life cycle. The objective of the production upply network planning i to maximize the expected profit, computed by ubtracting the expected total cot from the total revenue. The expected total cot comprie production cot, torage cot, hortage cot, and tranportation cot. Deciion to be decided include the production amount at each plant, the amount of inventory of finihed or emi-finihed product, and the flow of material between different plant taking into account product demand uncertainty. The multi-ite upply chain planning model i built on the following aumption: - It i aumed that there i no initial amount of inventory and backorder. - Since the demand i uncertain, hortage of product may occur in each period, which i aumed to be backordered. - The uncertain demand i defined under different cenario and it i aumed to follow a dicrete ditribution aociated with known probability. - A ditribution lead time i taken into account in hipping the finihed product to the cutomer and emi-finihed product between different plant of the network. - There i no wate of product during the tranportation of finihed and emi-finihed product. To formulate the mathematical model, we introduce the following indice parameter and deciion variable: 930
Int J Supply Oper Manage (IJSOM) Indice L i Set of direct ucceor plant of i. ST j i, i k t Set of tage (j= 1,2,..., N). Production plant index (i,i = 1, 2,...,I) where i belong to tage n and i belong to tage n+1. Product index (k = 1,2,..., K). Period index (t = 1,2,..., T). Scenario index ( = 1,2,...,S). Deciion variable P ikt S ikt JS ikt Production amount of product k at plant i in period t in regular-time. Amount of end of period inventory of product k for cenario at plant i in period t. Amount of end of period inventory of emi-finihed product k for cenario at plant i in period t. BD kt Backorder amount of finihed product k for cenario in period t. TR Amount of product k tranported from plant i to i in period t. i i', kt TRi CUS, kt Amount of product k tranported from the lat plant i to cutomer for cenario in period t. Q ik, Amount of product k received by plant i for cenario in period t. Parameter cp ik Unit cot of production for product k in regular-time at plant i. cti i ', k Unit cot of tranportation between plant i and i of production for product k. ct Unit cot of tranportation between the lat plant i and the cutomer. i CUS, k c ik Unit cot of inventory of finihed or emi-finihed product k at plant i. cbk Unit cot of backorder of product k. prk Unit ale price of finihed product k. 931
Felfel, Ayadi and Dadgar capp it Production capacity at plant i in normal working hour in period t. cap it Storage capacity at plant i in period t. captr Tranportation capacity at plant i in period t. i i', t D kt Demand of finihed product k for cenario in period t. b k DL Time needed for the production of a product entity k [min]. Delivery time of the tranported quantity. The occurrence probability of cenario where S 1 1 4. Propoed two-tage tochatic programming model Due to the uncertainty of finihed product demand, the determinitic model i inappropriate to optimize the expected net profit. Therefore, a two-tage tochatic programming model i propoed in order to incorporate uncertainty in the deciion-making. It hould be noted that the tage of the tochatic programing model correpond to different tep of deciion-making and it i not related to time period. Due to the coniderable lead time required in the production proce, the production amount in each plant and the product amount to be tranported between uptream and downtream plant are taken here and now before the realization of the uncertainty. Other deciion variable uch a inventory, backorder ize and flow of finihed product to be hipped to the cutomer can be achieved in a wait and ee mode. Conequently, the two-tage tochatic programming model can be formulated a follow. The objective function (1) aim to maximize the expected profit obtained by ubtracting the total expected cot from the expected revenue. The occurrence probability of each cenario i conidered in order to calculate the expected revenue and the expected cot. The total cot include production cot, inventory cot, backorder cot, tranportation cot of emi-product between uptream and downtream plant, and tranportation cot of finihed product to cutomer. S T K I k i CUS, kt ik ikt ikt 1 t 1 k 1 i 1 Max E[ Profit] pr TR c ( S JS ) T K I i CUS, k i CUS, kt k k, t ik ikt i i', k i i', kt t 1 k 1 i 1 ct TR cb BD cp P ct TR (1) Contraint (2) i the balance for the inventory level of product in each production tage excluding the lat tage. 932
Int J Supply Oper Manage (IJSOM) (2) S S P TR, i ST, k, t, ik, t ik, t 1 ikt i i', kt j N i' L i Contraint (3) provide the balance for end of period inventory in the lat production tage. I I ik, t ik, t 1 ikt i CUS, kt i 1 i 1 S S P TR, i ST, k, t, j N (3) Equation (4) repreent the inventory balance for the emi-finihed product. JSik, t JSik, t 1 Q Pikt, i, k, t, (4) ikt Contraint (5) repreent the balance equation for hortage in end product demand. BDkt BDk, t 1 Dkt TRi CUS, kt, k, t, (5) Contraint (6) provide the balance for tranportation between different production plant. Q TR, i, k, t, i' k, t DL i i', kt i' L i (6) Contraint (7) enure that the production capacity i repected. K bk Pikt cappit, i, t (7) k 1 Equation (8) i the torage capacity contraint. 933
Felfel, Ayadi and Dadgar K Sikt JSikt capit, i, t, (8) k 1 Contraint (9) guarantee that the tranportation capacity i repected. K TRi i', kt captrit, i, t, (9) k 1 Contraint (10) i the non-negativity retriction on the deciion variable. P,,, ',,,,,, ikt Sikt JSikt TRi i kt TRi CUS kt Qi k BDkt 0, i, k, t, (10) 5. Stochatic programming algorithm The major tep of the two-tage tochatic programming algorithm to olve the propoed model are given below: Step 1: make the firt-tage deciion including the production amount in each plant and the product amount to be tranported between uptream and downtream plant. Step 2: Compute the firt tage cot, Cot1 a follow: T K I Cot1 cp P ct TR t 1 k 1 i 1 ik ikt i i', k i i', kt (11) Step 3: at the beginning of tage 2, the realization of all uncertain demand occur. Step 4: at the end of tage 2, having een the realization of the uncertainty, and the firt-tage deciion, make the econd-tage deciion including inventory and backorder ize a well a the amount of product to be hipped to the cutomer. Step 5: compute the econd-tage cenario cot, Cot2 a well the cenario revenue, Revenue. The econd-tage cenario cot i equal to: T K I Cot2 c ( S JS ) ct TR cb BD (12) ik ikt ikt i CUS, k i CUS, kt k k, t t 1 k 1 i 1 The revenue of each cenario i given by: 934
Int J Supply Oper Manage (IJSOM) Revenue T K I (13) t 1 k 1 i 1 pr TR k i CUS, kt Step 6: Calculate the expected total profit E[Profit] a follow: S (14) E[ Profit] ( Revenue Cot2 ) Cot1 1 The propoed two-tage programming model i olved uing the tochatic programming olver for multitage tochatic program with recoure of Lingo 14.0 oftware. 6. Computational experiment The main purpoe of thi ection i to evaluate the effectivene and the robutne of tochatic model in comparion with the determinitic model uing real cae indutrial data from textile and apparel indutry. In Section 6.1, the related input data are decribed. Then, the determinitic and tochatic model are olved and the quality of the obtained olution i compared uing tochatic programming parameter in Section 6.2. It i worthwhile mentioning that the determinitic model i widely ued in the literature (Kall and Wallace, 1994; Birge and Louveau, 1997; Awudu and Zhang, 2013) to evaluate the performance of the tochatic programming model. To olve the determinitic model, the random parameter are aumed to be known with certainty and thu only one cenario with mean random value i conidered. Giving the imulation reult, we evaluate the robutne of the propoed model through many tatitical and rik metric a detailed in Section 6.3. Section 6.4 give other cae tudie under randomly generated cutomer demand to validate the obtained reult. The experiment are conducted uing LINGO 14.0 package program and MS-Excel 2010 with an INTEL(R) Core (TM) and 2 GB RAM. 6.1. Indutrial cae decription In thi ection, real data i provided from a medium and mall enterprie located in Tuniia in textile and apparel indutry. The planning horizon of the planning problem cover two month and the length of a period i one week. On the bai of pat ale record and future long-term and hort-term contract, the future economy can be aumed to be one of four cenario: poor, fair, good, or boom. The market demand of the finihed product P1 and P2 under each cenario i reported in Table1. Different plant indice are lited in Table 2. Table 3 decribed the production capacitie of different plant. It hould be noted that the production capacity varie from one period to another becaue of the abenteeim. Table 4 provide information about production and inventory unit cot. The tranportation unit cot and capacity are hown in Table 5. The proceing time of different manufacturing proce are reported in Table 6. 935
Felfel, Ayadi and Dadgar Table 1. Finihed product demand. Period Scenario T1 T5 T6 T7 T8 P1 P2 P1 P2 P1 P2 P1 P2 1 2350 2230 0.25 2300 2310 0.25 2 430 2210 0.2 2 2650 2520 0.35 2730 2770 0.4 2720 2530 0.3 0 0 3 1860 1550 0.2 1930 2040 0.15 1950 1860 0.25 4 1240 1020 0.2 1180 1060 0.2 1010 930 0.25 P1: product 1, P2: product 2. Table 2. Plant indice and deignation. Plant A1 A2 A3 A4 A5 A6 A7 A8 Deignation Knitting and dyeing proce Subcontractor1 Cutting- TE-INTER Embroidery Subcontractor2 Cloth making- TE-INTER Cloth making Subcontractor3 Cloth making Subcontractor4 Cloth making Subcontractor5 Packaging- TE-INTER 936
Int J Supply Oper Manage (IJSOM) Table 3. Production capacity per week [min]. Plant Period T1 T2 T3 T4 T5 T6 T7 T8 A1 57600 54720 57600 60480 54720 54720 51840 54720 A2 28800 31680 34560 25920 31680 23040 28800 23040 A3 43200 40320 46080 37440 40320 40320 43200 40320 A4 86400 77760 74880 83520 77760 83520 89280 83520 A5 31680 34560 25920 28800 34560 37440 31680 37440 A6 54720 48960 46080 60480 48960 63360 51840 63360 A7 17280 20160 20160 14400 20160 17280 23040 17280 A8 17280 20160 14400 17280 20160 23040 20160 23040 Table 4. Unit production cot and inventory unit cot. Unit cot Product Plant A1 A2 A3 A4 A5 A6 A7 A8 cp P1 1.72 0.72 0.9 1.75 1.9 1.65 1.5 0.38 P2 2.5 0.57 1.42 2.6 2.3 2.83 2.1 0.29 c P1, P2 0.3 0.1 0.15 0.12 0.1 0.11 0.1 0.2 937
Felfel, Ayadi and Dadgar Table 5. Unit cot and capacity of tranportation per week. Plant i Plant j Capacity (captr) Unit Cot (ct) (P1,P2) A1 A2 9100 0.6 A2 A3 8700 0.45 A3 A4 7500 0.37 A3 A5 7500 0.52 A3 A6 7500 0.65 A3 A7 7500 0.34 A4 A8 ----- 0 A5 A8 2500 0.49 A6 A8 5000 0.35 A7 A8 1500 0.27 A8 Cutomer 10000 0.5 Table 6. Proceing time [min]. Product Plant A1 A2 A3 A4 A5 A6 A7 A8 P1 8 4 4.5 11 10.5 12 13 3 P2 10 2.5 6.5 16.5 15.5 14 16 2.5 938
Int J Supply Oper Manage (IJSOM) 6.2. Computational reult In order to evaluate the impact of uncertainty parameter on the planning deciion, two tochatic well-known meaure were ued: the expected value of perfect information (EVPI) and the value of tochatic olution (VSS) (Birge and Louveau 1997). The EVPI parameter help to determine the expected profit lo under uncertainty. It can be calculated a: EVPI= WS TSP (15) Where TSP i the objective value of two-tage tochatic programming model and WS repreent the objective value of the wait and ee model. The WS model involve a family of linear programming model. Each model i aociated with an individual cenario. The olution of the WS model i obtained by weighting each individual cenario with it correponding probability. Such a model would allow to alway make the bet deciion regardle of the uncertain parameter which i not poible in practice. The VSS parameter calculate the poible profit from olving the two-tage tochatic programing model over the determinitic model. If the VSS i poitive, it implie that the olution of tochatic programming model are better than thoe of the determinitic model. It i defined a: VSS= TSP- EEV (16) Where EEV repreent the expected olution of determinitic model. The EVPI i then computed: EVPI=WS-TSP= 127716.8-109267.2 = 18449.6. According to the reult mentioned in Table 7, the EVPI/WS ratio i equal to 14.45%, which how the big influence of product demand uncertainty on the obtained olution. Therefore, it i worthwhile to have better forecat about the demand cenario. Then, the VSS i calculated: VSS= TSP- EEV=109267.2-101688.6=7578.6. Therefore, the two-tage tochatic model can lead to 7.45% more gain than the determinitic model a hown in Table7. Table 7. Stochatic programming parameter WS TSP EEV EVPI= WS-TSP VSS= TSP-EEV EVPI/WS (%) VSS/EEV (%) 127716.8 109267.2 101688.6 18449.6 7578.6 14.45% 7.45% 939
Felfel, Ayadi and Dadgar 6.3. Robutne evaluation 6.3.1. Robutne evaluation metric In order to evaluate the robutne of the production planning, different tatitical and rik metric were ued. Thee metric are baically: 1. Mean value ( ). It i defined a follow : (17) Where are the value of the ample item. 2. Standard deviation of profit ditribution (SD): It meaure the diperion or variation of a et of data from it mean. A high tandard deviation indicate a larger diperion or variability. A low tandard deviation implie that the data point are cloe to the average value. It can be formulated a: (18) 3. Value at rik (VaR): it i a percentile-baed metric widely ued in the literature for rik meaurement purpoe. It i defined a the minimal return or the maximal lo of a production planning over a pecific time horizon at a pecified confidence level.the VaR can be defined a the minimal portfolio return or the minimal profit at a pre-pecified confidence level a follow (Topaloglou and al 2002): VaR( x, ) min{ u : F( x, u) 1 } min{ u : P{ R( x, r) u} 1 }. (19) ~ Where ~ r : the return vector, ~ ~ ~ ~ ( 1, 2... ) T r r r r n R: Uncertain return of the portfolio at the end of the holding period. F: Ditribution function and F( x, u) P{ R( x, r) u} ~ 940
Int J Supply Oper Manage (IJSOM) 4. Conditional value at rik (CVaR): It i alo called the mean hortfall, the mean exce lo, and tail VaR. It i a more conitent and coherent meaure of rik than the VaR ince it give information about the average lo which exceed the VaR. Topaloglou et al (2002) have introduced a general definition of the CVaR for continuou and dicrete ditribution a follow: p { R( x, r ) } 1 z VaR( x, ) (1 ) z pr( x, r) 1 1 { R( x, r ) z} (20) Where z VaR( x, ) ; p Aociated probability to the return value r ~ ~ ~ ~ ( r1, r 2... rn ) T under a cenario. Both VaR and CVaR are calculated for different rik level ( ): 0.85, 0.9, and 0.95. 6.3.2. Meaurement of robutne The cae tudy i olved with a ample ize of 64 cenario. The profit ditribution reult for the two-tage tochatic and determinitic model are illutrated in Figure 2. According to thee ditribution, we make a rik aement analyi by comparing different ditribution and we evaluate the robutne of the production planning of the tochatic and determinitic model. Two-tage tochatic model Figure 2. Profit ditribution for tochatic and determinitic model 941
Felfel, Ayadi and Dadgar Baed on thee ditribution, we calculate the mean, the tandard deviation, and the VaR and CvaR value for different rik level ( =0.85, =0.95, =0.9) a hown in Table 8. The tandard deviation calculated from thee reult are 31623.55 and 23371.93, repectively, for the two-tage tochatic and determinitic model. Thi reult implie that a larger pread of value i obtained when we take uncertainty into account. From the Table 8, we alo oberve that the mean i greater for tochatic programming model. However, the VaR and CVaR for different coefficient ( =0.9, =0.95, =0.99) are greater for determinitic model. The computational reult how that the determinitic model provide a more robut olution than the tochatic model. In order to meaure the improvement performance gap of the tochatic model over the determinitic model, a metric i defined a follow: Expected tochaticvalue Expected determiniticvalue Improvment gap (%) 100 (21) Expected determinitic value We can ee from Table 9 an improvement of 3% in mean value. Beide, we ee a negative improvement in VaR by -12.21%, -13.55% and -18.41%for ( =0.85), ( =0.9) and ( =0.95), repectively. Moreover, we can note a negative improvement in CVaRby -13.79%, -16.53% and -22.66% for ( =0.85), ( =0.9) and ( =0.95), repectively, which mean that the robutne of the obtained olution i reduced after applying the tochatic programming model. Table 8. Statitical and rik metric of the tochatic and determinitic model Mod el SD Mean VAR (85%) CVAR (85%) VAR (90%) CVAR (90%) VAR (95%) CVAR (95%) SPM 31623.55 102315.5 2 68179.36 53691.66 57026.46 45186.98 42019.46 32356.37 DTM 23371.93 99335.22 77661.97 62281.20 65963.17 54134.27 51502.07 41838.98 Table 9. Improvement gap (%) of the tochatic model over the determinitic model Improvement gap (%) Mean VaR CVaR VaR CVaR VaR CVaR (85%) (85%) (90%) (90%) (95%) (95%) 3.00% -12.21% -13.79% -13.55% -16.53% -18.41% -22.66% 942
Int J Supply Oper Manage (IJSOM) 6.4. Validation by other cae tudie In order to validate the obtained reult in Section 6.2 and 6.3, we imulate five et of indutrial data for randomly generated cutomer demand. Then, we calculate the tochatic programming parameter and the tatitical and rik metric a hown in Table 10 and Table 11, repectively. A it i een in Table 10, the VSS in all cenario are poitive with an average of 8.21%. Therefore, the ue of the tochatic model can gain an average of 8.21% over the determinitic model in different cenario. Table 10 alo report that the EPVI/WS average ratio i 16.99 %, which reflect the big impact of uncertain parameter on the olution model. Table 10. Stochatic programming parameter under other indutrial cae tudie. Cae WS TSP EEV EVPI=WS-TSP VSS=TSP-EEV EVPI/WS (%) VSS/EEV (%) 1 114125.8 94567.47 87115.42 19558.33 7452.05 17.14% 8.55% 2 111782.2 92836.5 85592.06 18945.7 7244.44 16.95% 8.46% 3 117844.5 100436.2 94720.9 17408.3 5715.3 14.77% 6.03% 4 118476 97928.61 90186.84 20547.39 7741.77 17.34% 8.58% 5 108886.2 88460.2 80821.45 20426 7638.75 18.76% 9.45% Average 16.99% 8.21% From Table 11, we oberve that the SD of the tochatic ditribution i higher than thoe of the determinitic ditribution. It i clear that a larger diperion and variability i obtained when we conider uncertainty. We can alo ee from Table 11 that the mean value are greater for the tochatic model. Beide, the VaR value are greater for the determinitic model in all cenario at different coefficient expect for ( =0.85). Moreover, CVaR value are higher for determinitic model in all cenario. The performance improvement of the tochatic model over the determinitic model i ummarized in Table 12. The average performance improvement in mean value i 12.14%. However, the average performance improvement in VaR i 3.03%, -2.90% and -12.49% for ( =0.85), ( =0.9), and ( =0.95), repectively. In addition, the average performance improvement in CVaR i -15.37%,-17.60% and -21.04% for ( =0.85), ( =0.9) and ( =0.95), repectively. Thi finding ugget that the tochatic programing model improve profitability. However, it reduce the robutne of upply chain planning reult in comparion with the determinitic model under demand uncertaintie. 943
Felfel, Ayadi and Dadgar Table 11. Statitical and rik metric of the tochatic and determinitic model under other indutrial cae tudie. Cae Mode SD Mean VAR CVAR VAR CVAR VAR CVAR l (85%) (85%) (90%) (90%) (95%) (95%) 1 SPM 23407.33 97051.29 72644.10 47229.49 62797.20 41862.03 46947.60 33922.50 DTM 17740.87 88384.74 70125.75 55110.32 66031.02 49676.22 55164.85 42139.75 2 SPM 22271.98 96518.20 73067.70 46286.98 66306.10 39318.89 54116.20 29463.07 DTM 16738.18 86888.04 71306.34 53558.06 64760.34 48896.60 59279.34 38357.51 3 SPM 20158.92 103115.20 84228.70 59491.84 69924.80 57760.26 62631.80 50594.38 DTM 15840.12 93215.74 76408.80 67316.43 70747.80 63257.36 67106.76 57079.91 4 SPM 25292.45 102639.02 72848.10 46808.37 66464.80 39823.24 49599.90 32448.35 DTM 18402.98 89782.71 69436.64 56574.10 64655.21 51366.09 56944.64 42960.88 5 SPM 26016.38 91962.82 60471.20 38849.96 48820.50 34223.94 39533.70 25892.22 DTM 17236.30 80066.20 64374.31 48623.87 56516.21 43605.30 48958.41 35316.93 Table 12. Improvement gap (%) of the tochatic model over the determinitic model under other indutrial cae tudie. Improvement gap (%) Cae Mean VaR CVaR VaR CVaR VaR CVaR (85%) (85%) (90%) (90%) (95%) (95%) 1 9.81% 3.59% -14.30% -4.90% -15.73% -14.90% -19.50% 2 11.08% 2.47% -13.58% 2.39% -19.59% -8.71% -23.19% 3 10.62% 10.23% -11.62% -1.16% -8.69% -6.67% -11.36% 4 14.32% 4.91% -17.26% 2.80% -22.47% -12.90% -24.47% 5 14.86% -6.06% -20.10% -13.62% -21.51% -19.25% -26.69% Average 12.14% 3.03% -15.37% -2.90% -17.60% -12.49% -21.04% 944
Int J Supply Oper Manage (IJSOM) 7. Concluion In thi paper, we propoe a multi-product, multi-period, multi-tage, multi-ite production and tranportation upply chain planning problem faced by textile and apparel indutry under demand uncertainty. In order to incorporate the effect of the uncertainty in the upply chain planning problem, a two-tage tochatic model i developed. A real cae tudy from a textile and apparel upply network i illutrated to verify the effectivene and the robutne of the developed model. According to the computational reult, the propoed tochatic programming model provide higher expected profit and profit mean value than the determinitic model under demand uncertainty. However, the propoed tochatic model lead to le robut olution in comparion with the determinitic model. Improving the robutne of the planning olution in the face of uncertainty repreent an intereting future work. Thi perpective can be addreed by mean of rik management model that incorporate rik meaure into the tochatic programming model. Acknowledgement Thi reearch and innovation i carried out in the context of a thei MOBIDOC funded by the European Union under the program PASRI. The author are grateful to LINDO Sytem Inc for providing u with a free educational reearch licene of the extended verion of LINGO 14.0 oftware package. Reference Awudu, I., & Zhang, J. (2013).Stochatic production planning for a biofuel upply chain under demand and price uncertaintie. Applied Energy, 103, 189-19. Birge, J. R., & Louveaux, F. (1997). Introduction to Stochatic Programming. New York: Springer. Chopra, S., & Meindl, P. (2010). Supply chain management: Strategy, planning, and operation(4th ed.). Pearon Education, Inc : Upper Saddle River, New Jerey. Dantzig, G. B. (1955). Linear programming under uncertainty. Management Science 1, 197. Felfel, H., Ayadi, O., & Mamoudi, F. (2014). Multi-objective Optimization of a Multi-ite Manufacturing Network. Lecture Note in Mechanical Engineering, 69-76. Gupta, A., & Marana, C. D. (2000). A Two-Stage Modeling and Solution Framework for Multiite Midterm Planning under Demand Uncertainty. Indutrial & Engineering Chemitry Reearch, 39, 3799-3813. Lin J.T., & Chen Y.Y. (2007). A Multi-ite Supply Network Planning Problem Conidering Variable Time Bucket - A TFT-LCD Indutry Cae. International Journal of Advanced Manufacturing Technology,33(9-10), 1031-1044. 945
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