Department of Production Engineering, Birla Institute of Technology, Mesra, Ranchi, Jharkhand, India.

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1 International Journal of Baic and Applied Science Vol. 4. No Pp Copyright by CRDEEP. All Right Reerved. Full ength Reearch Paper Supply Chain Optimization Model under Uncertainty Anurag Singh 1*, Roopa Singh 2 and S. C. Srivatava 3 1 Cognizant Telecom Solution, 24- Paragana (S), Wet Bengal, India. 2 Department of Indutrial & Production Engineering, G. B. Pant Univerity of Agriculture & Technology, Pantnagar, India. 3 Department of Production Engineering, Birla Intitute of Technology, Mera, Ranchi, Jharkhand, India. *Correponding author: Anurag Singh Abtract The upply-chain an integrated effort by a number of entitie - from upplier of raw material to producer, to the dtributor - to produce and deliver a product or a ervice to the end uer. Planning and managing a upply chain involve making decion which depend on etimation of future cenario (about demand, upply, price, etc). Not all the data required for thee etimation are available with certainty at the time of making the decion. The extence of th uncertainty greatly affect thee decion. If th uncertainty not taken into account, and nominal value are aumed for the uncertain data, then even mall variation from the nominal in the actual realization of data can make the nominal olution highly uboptimal. The problem of deign, analy and optimization under uncertainty central to decion upport ytem, and extenive reearch ha been carried out in both Probabiltic (Stochatic) Optimization and Robut Optimization (contraint) framework. In th work, overview of previouly publhed work on incorporating demand uncertainty in midterm planning of multite upply chain ha been taken. The propoed model provide an effective tool for evaluating and actively managing the expoure of an enterpre aet (uch a inventory level and profit margin) to market uncertaintie..in th reearch literature problem of one organization which ha 3 ite of production ha been taken. The problem cont of 3 ite. A total of 10 product which are grouped into 5 part familie are manufactured which are characterized by different proceing and cot attribute. The model cont of 5 contraint, 10 variable and 14 parameter. The model olved manually to 3 iteration to optimal but a th proce time conuming o for th we have to develop the oftware. The problem formula olved uing genetic algorithm and programming of c language. The implex method alo ued to make the problem formula to optimal. Keyword: Supply chain, Uncertainty, Optimization, Simplex method, Genetic algorithm Introduction A upply chain cont of all partie involved directly or indirectly in fulfilling a cutomer requet. The upply chain include not only manufacturer and upplier but alo tranporter, warehoue, retailer and even cutomer themelve Fig1 (Hugo, 2011). Within each organation uch a manufacturer the upply chain include all function involved in receiving and filling a cutomer requet. Thee function include but are not limited to new product development, marketing, operation, dtribution, and finance and cutomer ervice. Fig 1: Activitie and firm in a upply chain In addition to defining the upply chain, everal author have further defined the concept of upply chain management. A defined by Ell ram and Cooper [1] (1993), upply chain management an integrating philoophy to manage the total flow of a dtribution channel from upplier to ultimate cutomer. They believe that upply chain, not firm, compete and that thoe who will be the tronget competitor are thoe that can provide management and leaderhip to the fully integrated upply chain including external cutomer a well a prime upplier, their upplier, and their upplier upplier. 118 Online verion available at:

2 Fig 2: Supply chain management flow diagram The upply chain begin with a need for a computer. In th example, a cutomer place an order for a Dell computer through the Internet. Since Dell doe not have dtribution centre or dtributor, th order trigger the production at Dell manufacturing centre, which the next tage in the upply chain. Microproceor ued in the computer may come from AMD and a complementary product like a monitor may come from Sony. Dell receive uch part and component from thee upplier, who belong to the up-tream tage in the upply chain. After completing the order according to the cutomer pecification, Dell then end the computer directly to the uer through UPS, a third party logtic provider. The impact of Collaborative Tranportation Management on upply chain Performance: A imulation approach (Felix T.S. Chan, T. Zhang DATED, 2011) Collaborative Tranportation Management (CTM) baed on the interaction and collaboration between trading partner and carrier participated in the upply chain, appropriate application of CTM can improve the flexibility in the phyical dtribution and minimize the inefficiency of upply chain management. Th paper propoe new concept of CTM and carrier flexibility. A imulation approach ued to (i) evaluate the benefit of the propoed CTM, (ii) explain the concept of carrier flexibility, and (iii) optimize the delivery peed capability. Baed on a imple upply chain including one retailer and one carrier, three different imulation model have been developed with changeable delivery lead time a follow: (1) Uncontrained delivery peed capability without CTM. (2) Contrained delivery peed capability without CTM; and (3) Contrained delivery peed capability with CTM. Simulation reult reveal that CTM can ignificantly reduce the retailer total cot and improve the retailer ervice level. Giri 2011(3) reported a ingle-product ingle-period inventory model in which the retailer can ource from two upplier in which the rk-avere retailer face yield uncertainty from the primary upplier; the econdary upplier being reliable though capacity contrained. aumption of two unreliable upplier intead of one and/or other uncertaintie uch a demand and upplier lead time uncertaintie would be worthful contribution. Qahtani et al (2) 2010 reported two-tage tochatic MIP model for deigning an integration trategy under uncertainty and plan capacity expanion, a required, in a multite refinery network. Robutne analyzed baed on model robutne and olution robutne, where each meaure aigned a caling factor to analyze the enitivity of the refinery plan and integration network due to variation. In th reearch, I have taken a literature problem of one organization which ha 3 ite of production. The problem cont of 3 ite. A total of 10 product which are grouped into 5 part familie are manufactured which are characterized by different proceing and cot attribute. The model cont of 5 contraint, 10 variable and 14 parameter. The model olved manually to 3 iteration to optimality but a th proce time conuming o for th we have to develop the oftware. The extended work decribed in the next ection. Methodology Simplex Method Simplex method conidered one of the baic technique from which many linear programming technique are directly or indirectly derived. The implex method an iterative, tepwe proce which approache an optimum olution in order to reach an objective function of maximization or minimization (Kumar, 2009). Optimization Optimization the term often ued for minimizing or maximizing a function. It ufficient to conider the problem of minimization only; maximization of F(x) achieved by imply minimizing F(x). In engineering, optimization cloely related to deign. The function F(x), called the merit function or objective function, the quantity that we wh to keep a mall a poible, uch a the cot or weight. The component of x, known a the deign variable, are the quantitie that we are free to 119 Online verion available at:

3 adjut. Phyical dimenion (length, area, angle, etc.) are common example of deign variable. The optimization very vat and many type of explanation are provided to it but the bet way, in limited pace to introduce a few baic method that are good enough for problem that are reaonably well behaved and do not involve too many deign variable. The algorithm for minimization are iterative procedure that require tarting value of the deign variable x. If F(x) ha everal local minima, the initial choice of x determine which of thee will be computed. There no guaranteed way of finding the global optimal point. One uggeted procedure to make everal computer run uing different tarting point and pick the bet reult. More often than not, the deign alo ubjected to retriction, or contraint, which may have the form of equalitie or inequalitie (Geune et al, 2005). Genetic algorithm The concept of GA wa developed by Holland and h colleague in the 1960 and GA inpired by the evolutiont theory explaining the origin of pecie. In nature, weak and unfit pecie within their environment are faced with extinction by natural election. The trong one have greater opportunity to pa their gene to future generation via reproduction. In the long run, pecie carrying the correct combination in their gene become dominant in their population. Sometime, during the low proce of evolution, random change may occur in gene. If thee change provide additional advantage in the challenge for urvival, new pecie evolve from the old one. Unucceful change are eliminated by natural election. The procedure of a generic GA given a follow: A. Step.1: The objective function hould be defined which ha to be minimized. It mut be in form of #.M file and hould return a calar value. B. Step.2: Population: generate the population and mention the population ize that pecifie how many individual there are in each generation. C. Step.3: Fitne value: defined for each population. D. Step.4: Scaling: tournament election ued for fitne caling. In tournament fitne caling the each parent elected by chooing individual at random. Number of parent elected can be pecified by the tournament ize. The bet parent elected out of the formed et. E. Step.5: Reproduction: option for genetic algorithm create children at each new generation. It done by elite count and croover fraction. In elite count the number of individual that are generated to urvive the next generation wherea croover determine the fraction of next generation that the croover produce. F. Step.6: Croover: It ued to combine two individual or parent to form a new individual or child. The parent are elected among exting chromoome in the population with preference toward fitne o that offpring expected to inherit good gene which make the parent fitter. By iteratively applying the croover operator, gene of good chromoome are expected to appear more frequently in the population, eventually leading to convergence to an overall good olution. G. Step.7: Migration: the movement of individual between the ubpopulation which the algorithm create. It can be forward, backward or both way in direction. H. Step.8: Mutation: The mutation operator introduce random change into charactertic of chromoome. Mutation generally applied at the gene level. In typical GA implementation, the mutation rate (probability of changing the propertie of a gene) very mall and depend on the length of the chromoome. Therefore, the new chromoome produced by mutation will not be very different from the original one. Mutation play a critical role in GA. A dcued earlier, croover lead the population to converge by making the chromoome in the population alike. Mutation reintroduce genetic diverity back into the population and at the earch ecape from local optima. I. Step.8: Hybrid function: No hybrid function ued. J. Step.9 :(Termination): Output give a the et containing non-dominated olution when the termination criteria met otherwe again the proce to be repeated from evaluation of function (Sivanandam et al, 2007). Multi-Objective Genetic Algorithm The objective function of formulation compoed of two term. The firt term ubjected to the outer optimization problem contraint account for the cot incurred in the production tage. Second term Z quantitie the cot of the logtic decion and obtained by applying the expectation operator to an embedded optimation problem. Being a population-baed approach, GA well uited to olve multi-objective optimization problem. A generic ingle-objective GA can be modified to find a et of multiple non dominated olution in a ingle run. The ability of GA to imultaneouly earch different region of a olution pace make it poible to find a divere et of olution for difficult problem with non convex, dcontinuou, and multi-modal olution pace. The croover operator of GA may exploit tructure of good olution with repect to different objective to create new non dominated olution in unexplored part of the Pareto front. In addition, mot multi-objective GA doe not require the uer to prioritize, cale, or weigh objective. Therefore, GA ha been the mot popular heurtic approach to multi-objective deign and optimization problem (. (Jone et al., 1998) reported that 90% of the approache to multi-objective optimization aimed to approximate the true Pareto front for the underlying problem. A majority of thee ued a meta-heurtic technique and 70% of all met heurtic approache were baed on evolutionary approache. 120 Online verion available at:

4 Extended work For the formulation variou indice are: i= et of product f= et of product familie j= et of proceing equipment = et of production ite Parameter: FC f = etup cot for family f at ite v ij = variable production cot for product i on unit j at ite p = price of raw material i at ite t = inter-ite tranportation cot form ite to ite t = cutomer-ite tranportation cot Z = afety tock violation penalty for product i at ite J i = revenue per unit of product i old to cutomer c MR f = minimum run length for family f on unit at ite H fj = total available proceing time R ij = rate of production of product i on unit j at ite β i = yield adjuted amount of product i conumed to produce product i I o = initial inventory I = afety tock level for product i at ite θ i = uncertain demand Variable: A = availability of product i for upply at ite P ij = production amount of product i on unit j at ite R ij = run length of product i on unit j at ite C = raw material conumption of product i at ite W = interite hipment of product from ite to ite Y fj = etup (binary variable indicating whether product family f manufactured on unit j at ite ). S = upply of product I = inventory of product I = deviation below afety tock of product I - = cutomer hortage Mathematical model The upply chain formulation taken from Gupta and Marana [2003] a reference for the work: Min FC fyfj vijpij pc t W Z f, j, i, j, i, i,, p ij R ij FR fj A C W 0 Y fj {0,1} Z E i min S, I, I, I 0 t S h I Z I J I S i...( a) I A S...( b) i S Ii i...( c) I I I I...( d) i i i 121 Online verion available at:

5 Subject to P R R...(1) ij ij ij C P W ii i j i j A I P W ij j f i: if 1 R ij H MR Y R H Y fj fj fj ij fj fj i: if 1...(2)...(3)...(4)...(5) The objective function of the determintic midterm planning capture the combined cot incurred in the manufacturing and logtic phae. The manufacturing phae cot include fixed and variable production charge, cot of raw material purchae and tranportation charge incurred for the interite hipment of intermediate product. The ubequent logtic phae cot are compred of the tranportation charge incurred for hipping the final product to the cutomer, inventory holding charge, afety tock violation penaltie and penaltie for lot ale. The decion made in the manufacturing phae etablh the location and timing of production run, length of campaign, production amount and conumption of raw material. Specifically, Pij, Rij, FRfj, A, C, W and Yfj contitute the manufacturing variable, and uniquely define the production level and reource utilization in the upply chain. Thee manufacturing variable are contrained by the manufacturing contraint given by Eq. (1)_(5). The production amount of a particular product defined in term of the rate of production and the campaign run length by Eq. (1). The input-output relationhip between raw material and final product, accounting for the bill-of-material and redundancy in the interite hipment of intermediate product eliminated by Eq. (2), which force the product hipped to a particular ite in a particular period to be conumed in the ame period. The allocation of product to product familie achieved. Grouping of product into product familie typically done to account for the relatively mall tranition time and cot between imilar product. Eq. (4) model the capacity retriction while Eq. (5) provide upper and lower bound for the family run length. The amount available for upply in the logtic phae following the manufacturing phae defined through Eq. (3). The decion made in the logtic phae, termed the logtic variable, are S, I, I,I. The correponding logtic contraint are given by Eq. (a)_(d). The linking between the manufacturing and logtic phae captured by Eq. (b). The inventory level, which determined by the amount available for upply and the actual upplie to the variou cutomer, defined by Eq. (b). No overtocking permitted at the cutomer (Eq. (a)) The cutomer hortage are carried over time (Eq. (c)). Eq. (d) model the violation of the afety tock level.. Updated Work Solving the objective function by Simplex method: Min FC fyfj vijpij pc t W Z f, j, i, j, i, i,, p ij R ij FR fj A C W 0 Y fj {0,1} Z E i min S, I, I, I 0 t S h I Z I J I S i I A S i S Ii i I I I I i i i 122 Online verion available at:

6 FC Y v P p C t W Z f fj ij ij f, j, i, j, i, i,, FC Y v R R p W t W Z f fj ij ij ij f, j, i, j, i, i,, i,, FC Y v R R p W t W f fj ij ij ij f, j, i, j, i, i,, i,, i,, Z FC fy fj vijrijrij W p t Z f, j, i, j, i,, i, i,, FC fy fj vij Rij H fj W p t Z f, j, i, f, j, i,, i, i,, FC1 fy fj v1 ij R1 ijh1 fj W1 p1 t1 Z... 1 now again Z E i min S, I, I, I 0 t S h I K I J I S i I A S i S Ii i I I I I i i t S h I K I J I i i i t h I K I J I i i i i t h A S K I J I i i i i t h A S K I I 0 J I i i i i 123 Online verion available at:

7 t h A S K I I I i i i i, i, i, i t h ( A S ) K ( I I ) I i putting value of Z in eq 1 F C Y v R H W p t [ t h ( A S ) 1 f fj 1ij 1ij 1 fj K 1 ( I1 I ) J 1I ]...(2) 1 i Reult and Dcuion The objective wa to optimize the upply chain under uncertainty a developed in mathematical model. The overall function include uncertainty along with parameter like; i. Tranportation factor which include upply due to uncertain demand with cutomer ite tranportation cot (A). ii. Holding cot include inventory and the holding cot aociated with it (B). iii. Deficit afety tock penalty factor include the afety tock violation penalty and deficit in the afety tock (C). iv. Revenue lo factor cont of the lot revenue cot and the cutomer hortage (D). v. Production cot factor include the et up cot with the variable production cot in producing certain amount (E). vi. Raw material cot include the cot along with the conumption of raw material (F). vii. Inter-ite tranportation cot include the hipment cot along the ite (G). The developed model wa olved into two tage. In firt tage uncertainty parameter were optimized then the optimized parameter of uncertainty wa called in main function to get the final optimized value of objective function. The model wa olved uing both hard computing and oft computing tool. For hard computing an algorithm wa developed in C uing implex. Wherea for oft computing Matlab wa ued a a platform for olving problem uing genetic algorithm. The model wa olved uing above methodology for different cae a dcued later in the chapter. Input Data The data ued to validate the model developed have been conidered from the literature and the optimization ha been done uing the ame. Table 1: Input parameter for cae tudie Parameter Notation Value at ite 1 Value at ite 2 Value at ite 3 Fixed cot FC f 4.5,4.8,5.5,6.2, ,3.5,6.5,4.5, ,5.2,5.1,4.7,6.5 Set up Y fj Variable production cot v ij Raw material cot P Inter-ite tranportation cot t Cutomer ite tranportation t cot Inventory holding cot h Production amount P ij Raw material conumption C Inter-ite hipment W Supply under uncertain S demand Inventory I Safety tock violation K Safety tock deficit I Cutomer hortage - I i ot revenue cot J i The value are conidered from the above table and conidering the given 3 ite a dcued in the problem environment. The developed model wa olved uing the aforeaid method and wa validated for 3 ite taken a cae tudie. 124 Online verion available at:

8 CASE 1: SITE 1 Reult from programming C Fig 3: Output (a) for Site 1 Fig 4: Output (b) for ite 1 Fig 5: Output (c) for ite 1 Taking different value for the different iteration of the value for ite 1 the reult Reult from GA at Site 1. Fig 6: Fitne value of objective function for ite Online verion available at:

9 The Fig 6 how the iterative convergence of objective function toward optimal value. A the iteration tart with initial value and become contant from iteration 14 and the value tabilize for later iteration at 191. Table 2: Optimized value for ite 1 Parameter Notation Optimized Value Optimized value of objective function Tranportation cot A 75 Holding cot B 15 Deficit Safety tock C Revenue o factor D 8 Production cot E 25 Raw Material Cot F 43 Inter-ite tranportation G.774 Comparing the both reult a the model wa olved with help of both hard computing and oft computing the reult from GA more optimized then the reult of other. A hard computing provide the crp value taking the crp data & the oft computing the fuzzy value are taken o which provide better reult. The optimum value from the C Program (hard computing) wherea the GA (Soft computing) provide the reult a A the optimal equence wa generated by the GA for each problem, the problem wa repeatedly run for more time to enure the contency of the olution and the mean of optimization. In C the earch pace le wherea in GA the earch pace comparatively more. CASE 2: SITE 2 RESUTS FROM C Fig 7: Output (a) for ite 2 Fig 8: Output (b) for ite 2 Fig 9: Output (c) for ite 2 Taking different value for the different iteration of the value for ite 2 the reult Online verion available at:

10 Reult from GA at Site 2 Fig 10: Fitne Value of objective function at ite 2 Fig 10 how the iterative convergence of objective function toward optimal value. A the iteration tart with initial value and become contant from iteration 12 and the value tabilize for later iteration at 259. Table 3: Optimized value for ite 2 Parameter Notation Optimized Value Optimized value of objective function Tranportation cot A 121 Holding cot B 18 Deficit Safety tock C Revenue o factor D 11 Production cot E 31 Raw Material Cot F 50 Inter-ite tranportation G Comparing the both reult a the model wa olved with help of both hard computing and oft computing the reult from GA more optimized then the reult of other. A hard computing provide the crp value taking the crp data & the oft computing the fuzzy value are taken o which provide better reult. The optimum value from the C Program (hard computing) 273 wherea the GA (Soft computing) provide the reult a A the optimal equence wa generated by the GA for each problem, the problem wa repeatedly run for more time to enure the contency of the olution and the mean of optimization CASE 3: SITE 3 RESUTS FROM C Fig 11: Output (a) for ite 3 Fig 12: Output (b) for ite Online verion available at:

11 Figure 13: Output (c) for ite 3 Taking different value for the different iteration of the value for ite 3 the reult Reult of GA at Site 3 Fig 14: Fitne value for objective function at ite 3 Fig 14 how the iterative convergence of objective function toward optimal value. A the iteration tart with initial value and become contant from iteration 10 and the value tabilize for later iteration at 388. Table 4: Optimized value for ite 3 Parameter Notation Optimized Value Optimized value of objective function Tranportation cot A 110 Holding cot B 18 Deficit Safety tock C Revenue o factor D 22 Production cot E 44 Raw Material Cot F 150 Inter-ite tranportation G 4 Comparing the both reult a the model wa olved with help of both hard computing and oft computing the reult from GA more optimized then the reult of other. A hard computing provide the crp value taking the crp data & the oft computing the fuzzy value are taken o which provide better reult. The optimum value from the C Program (hard computing) wherea the GA (Soft computing) provide the reult a 388. A the optimal equence wa generated by the GA for each problem, the problem wa repeatedly run for more time to enure the contency of the olution and the mean of optimization. Summary and Concluion A upply chain cont of all partie involved directly or indirectly in fulfilling a cutomer requet. The upply chain include not only manufacturer and upplier but alo tranporter, warehoue, retailer and even cutomer themelve. Within each organization uch a manufacturer the upply chain include all function involved in receiving and filling a cutomer requet. Thee function include but are not limited to new product development, marketing, operation, dtribution, and finance and cutomer ervice. The problem aociated with upply chain Product demand variability which a ource of uncertainty in any upply chain. Failure to account for ignificant product demand fluctuation in the midterm by determintic planning model may lead to exceively high production cot (tranlating to high inventory charge) or unatfied cutomer demand and lo of market hare. Incorporation of demand uncertainty in midterm planning of multite upply chain having (emi)continuou proceing attribute dcued a the trade off involved between the inventory depletion and production cot face of uncertainty. 128 Online verion available at:

12 The mathematical model wa developed for the relevant problem cenario. The objective function of formulation wa compoed for two term. The firt term ubjected to the outer optimization problem contraint account for the cot incurred in the production tage. Second term Z quantitie the cot of the logtic decion and obtained by applying the expectation operator to an embedded optimization problem. The developed mathematical model wa firt olved uing C program (Hard computing) but the hard computing find the optimal olution uing the crp data and the earch pace le. Then the model wa olved with the help of Matlab a platform uing Genetic Algorithm (Soft computing). A oft computing provide larger earch pace a compared to hard and ue fuzzy value which provide the better reult. A the optimal equence wa generated by the GA for each problem, the problem wa repeatedly run for more time to enure the contency of the olution and the mean of optimization. Reference Chan F., Zhang T. The impact of Collaborative Tranportation Management on upply chain performance: A imulation approach Expert Sytem with Application 38 (2011) Cooper, M. C., and. M. Ellram. Charactertic of Supply Chain Management and the Implication for Purchaing and ogtic Strategy. The International Journal of ogtic Management, 4,2 (1993) Giri B. Managing inventory with two upplier under yield uncertainty and rk averion. Int. J. Production Economic 133 (2011) Gupta A, Marana C.D. Managing demand uncertainty in upply chain planning. Computer and Chemical Engineering 27 (2003) 1219_/1227 Jone Rachel Maon, Towill Den R. Time compreion in the upply chain: information management the vital ingredient. ogtic Information Management.11,2,(1998) Joeph Geune, Pano M. Pardalo, Supply Chain Optimization, (2005 Springer Science + Buine Media, Inc.). Michael H. Hugo, Eential of Supply chain Management, 2011 Mituo Gen, Runwei Cheng, in in, Network Model and Optimization: Multiobjective Genetic Algorithm Approach, 2008 Springer- Verlag ondon imited. S.N. Sivanandam, S. N. Deepa, Introduction to Genetic Algorithm, 2007 Vivek Kumar, Operation Reearch, (New Delhi: S. K. Kataria & Son, 2009) 129 Online verion available at: