A mathematical model for multi product-multi supplier-multi period inventory lot size problem (MMMILP) with supplier selection in supply chain system

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1 nernaional ournal of Saisics and Applied Mahemaics 2017; 2(6): SSN: Mahs 2017; 2(6): Sas & Mahs Received: Acceped: Binay Kumar Deparmen of Saisics, Pana Universiy, Pana, Bihar, ndia Anchala Kumari Deparmen of Saisics, Pana Universiy, Pana, Bihar, ndia A mahemaical model for muli produc-muli supplier-muli period invenory lo size problem (MMMP) wih supplier selecion in supply chain sysem Binay Kumar and Anchala Kumari Absrac Supplier selecion is one imporan decision facors in he supply chain managemen. Suppliers are necessary eniies o any business, however wrong selecion may affec he whole business processes; herefore he process of selecing suppliers is exremely imporan. he success of a supply chain is dependen on selecion of beer suppliers. Decision makers and managers always face challenges o selec suppliers, for procuremen of raw maerial and componens for heir manufacuring process. n his paper we develop a Mixed neger Programming Model for Muli produc-muli supplier -Muli period invenory lo size problem (MMMP) wih supplier selecion. he soluion of he opimal mahemaical model is obained in erms of deermining (i) which producs should be ordered o which supplier. (ii) he opimal quaniy o be ordered and he ime of placing orders. We inroduce budge consrain, sorage capaciy consrain and quaniy discoun approach. A numerical case sudy is solve wih Geneic algorihms. Keywords: supplier selecion, mixed ineger programming, supply chain, geneic algorihm Correspondence Binay Kumar Deparmen of Saisics, Pana Universiy, Pana, Bihar, ndia 1. nroducion Supplier selecion is defined as he process of finding he righ suppliers, a he righ price, a he righ ime, in he righ quaniies, and wih he righ qualiy Ayhan, (2013b). is recorded ha, 70% of oal producion cos is composed by he purchases of goods and services Ghodsypour & O Brien, (1998). Hence, selecing he righ supplier will resul in reducing operaional coss, increasing profiabiliy and qualiy of producs, improving compeiiveness in he marke and responding o cusomers demands rapidly Abdollahi, Arvan, & Razmi, 2015; Onu, Kara, & sık, (2009). n he comparaive marke, i is necessary having a good producion planning and replenishmen conrol hrough effecive invenory managemen. he single produc, muliperiod invenory lo-size problem is one of he mos Common and basic problems. he presen work considers an environmen wih muliple producs-muliple periods and muliple suppliers. his paper is based on he work of Basne and aung (2005) [1] which developed he muliperiod invenory lo size wih muliple producs and muliple suppliers. Woarawichai, Kuruvi & Vashirawongpinyo (2012) has inroduced sorage capaciy limiaion wih his model. n his paper a supplier selecion wih muli-period invenory lo size wih muliple producs and muliple suppliers under budge and sorage capaciy consrains and all uni quaniy discoun is developed using GA. Among various algorihms, GAs is develop by Wang, Yung, and p (2001), is mos suiable for selecing bes supplier combinaion and Hokey, Gengui, Misuo and Zhenyu (2005), suggesed ha GA is he bes populaion-based heurisic algorihm, capable of generaing a group of bes soluions a once. ee e al. (2013) inroduced a MP model and geneic algorihm (GA) o solve he lo sizing problem wih muliple suppliers. incorporaes he incremenal and all-uni quaniy discouns and is applicable o deermine he replenishmen sraegy for a manufacurer for muli periods. Bu i is only suiable for single iems. ~23~

2 nernaional ournal of Saisics and Applied Mahemaics he paper is organized as follows. Secion 1 describes he inroducion and lieraure review and secion 2 inroduces assumpions and noaions and iniial analysis of mahemaical model has developed on secion 3. n secion 4 we define he problem wih is mahemaical formulaion and numerical case sudy has discussed in secion 5. Secion 6 inroduces solve he numerical case sudy by Geneic algorihm and NGO. Finally, concluding remarks are given in Secion Assumpion & Noaion n his secion, we inroduce following he assumpions and noaions Assumpion 1. niial invenory of he firs period and he invenory a he end of he las period are assumed o be zero. 2. Demand of producs in each period is known over a planning horizon. 3. ransporaion cos is supplier dependen, bu does no depend on he variey and quaniy of producs involved. 4. Produc needs a sorage space and available oal sorage space is limied. 5. Shorage or backordering is no permied. 6. Holding cos of produc per period is produc dependen. 7. Budge is fixed for each period. 8. All uni quaniy discoun is considered. Parameer i = 1,., index of producs j = 1,., index of suppliers = 1,.,.. index of ime periods k = 1, K index for shipmen ime insan l = 0,1, index for uaniy discoun break poin Noaions d ij = demand of produc i in period c ij H i V w i = purchase price of produc i from supplier j = holding cos of produc i per period = ransporaion cos for supplier j = sorage space produc i S = oal sorage capaciy C = budge available for each period K= Se of shipmen ime insans & { K shipping ime insans lower han or equal o. ijl = uaniy hreshold beyond which a price break become valid a period for produc i from supplier j for lh price break. a ijl ijl = Discoun facor ha is valid if more han uni are aijl purchased, 0 < < 1. 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 Z ijl = 1 if ijl ij x < ij( l 1), oherwise 0. his binary variable indicae ha order size a period is larger han hen discouned prices for he ordered producs. nermediae variable i ijl = nvenory of produc i, carried over from period o period + 1 uaniy discoun wih price break As a markeing policy, Suppliers gran discouns o buyers who buy in quaniy larger han of minimum accepable order. We consider all-uni quaniy discouns. An all-unis quaniy discoun is a discoun given on every uni ha is purchased afer he purchasing exceeds a given level (breakpoin). As discussed above, variable specifies he fac ha he order size a period is larger han and herefore resuls in discouned prices for he ordered quaniy and price breaks are defined as: 1 ij ij0 aij1 ij0 ij ij1 D l aij2 ij1 ij ij aij ij( 1) ij i 1,2...; j 1,2...; l 0,1... ; 1, niial analysis of Mahemaical Model All demands mus be filled in he period in which hey occur: shorage or backordering is no allowed. i = ijk d ik j=1 0 i, j here is no an order wihou charging an appropriae ransacion cos. Y j ( d ik ) ij 0 i, j & Each produc has limied capaciy w i ( ijk d ik ) S i=1 j=1 Budge is fixed of each produc ik C ij i=1 j=1 C Z i, j, ij ijl ijl l0 ~24~

3 nernaional ournal of Saisics and Applied Mahemaics l0 Z ijl 1 i, j, he objecive funcion consiss of hree pars: he oal cos (C) of purchase cos of he producs, ransporaion cos for he suppliers, and holding cos for remaining invenory in each period in +1. C Z a V Y H d ij ij ijl ijl j j i ijk ik l1 i1 j1 1 j1 1 i1 1 j1 k1 k1 Decision variables Yj 0and 1 i, j is binary variable 0 or 1 ijk 0 i, j, is non-negaive resricions Zijl 0 and 1 i, j, i, is binary variable 0 or Formulaion of Mahemaical model he objecive funcion of he Mahemaical model C Z a V Y H d ij ij ijl ijl j j i ijk ik l1 i1 j1 1 j1 1 i1 1 j1 k1 k1 Subjec o consrains i = ijk d ik j=1 0 i, Y j ( d ik ) ij 0 i, j & k= w i ( ijk d ik ) S i=1 j=1 ik C ij i=1 j=1 C Z i, j, l0 ij ijl ijl l0 Z ijl 1 i, j, Decision variables Y 0and 1 i, j j ijk 0 i, j, Z 0 and 1 i, j, i, ijl 5. Numerical case sudy 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 able 1. here are hree suppliers and heir prices and ransporaion cos, holding cos and sorage space are show in able 2 and able 3, respecively. Produc Planning Horizons (five periods) A B C Budge able 1: Demands of hree producs over a planning horizon of five periods (di) and budge for each period. Producs Price Y Z A B C ransporaion Cos able 2: Price of hree producs by each of hree suppliers, Y, Z (Cij) and ransporaion cos of hem (Vj). Cos Produc A B C Holding Cos Sorage Space able 3: Holding cos of hree producs A, B, C (Hi) and sorage space of iem (wi). All uni quaniy discoun as follow: D l 1 ij ij ij Soluion of Numerical case sudy We applied ingo and GAs approach o solve his numerical case sudy. We use NGO 11.0 and MAAB R2013 sofware and experimens are conduced on a personal compuer equipped wih an nel Core 2 duo 2.00GHz, CPU speeds, and 2 GB of RAM. ingo is Opimizaion Modeling Sofware for inear, Nonlinear, and neger Programming. We use his sofware for soluion of numerical case sudy. We find ingo resul of his case sudy in he following able. Producs ij able 4: Order of hree producs over a planning horizon of five periods ( ) Planning Horizon (Five Periods) A B C ~25~

4 nernaional ournal of Saisics and Applied Mahemaics Global opimal soluion found. Objecive value : Objecive bound : nfeasibiliies : E-14 Exended solver seps : 12 oal solver ieraions : 1343 Geneic Algorihm approach is based on a naural selecion process ha mimics biological evoluion. belongs o he larger class of evoluionary algorihms (EA). Gas code is developed in MAAB. he ransporaion coss are generaed from in [50; 200], a uniform ineger disribuion including 50 and 200. he prices are from in [20; 50], he holding coss from in [1; 5], he sorage space from in [10; 50], he quaniy discoun for producs are from in [15; 55] and he demands are from in [10; 200]. A problem size of ; ; indicaes number of suppliers =, number of producs =, and number of periods =. Compuaion ime limi is se a 120 minues. We obain same resul as NGO and soluion ime is 0.01 minues. 7. Conclusion n his paper, we presen geneic algorihms (GAs) and NGO applied o he muli-produc, muli supplier and muli-period invenory lo-sizing problem wih supplier selecion under budge consrain and all uni quaniy discoun wih maximum sorage space for he decision maker in each period is considered. he decision maker needs o deermine wha producs o order in wha quaniies wih which suppliers in which periods. he mahemaical model is formulaed as a mixed ineger programming and illusraed hough a numerical case sudy. he numerical case sudy is solved wih NGO and he GAs. GAs provides beer soluion han NGO ha are close o Opimum in a very shor ime. 8. References 1. Basne C, eung MY. nvenory lo-sizing wih supplier selecion. Compuers and Operaions Research. 2005; 32: Holland H. Adapaion in Naural and Arificial Sysems. he Universiy of Michigan Press, Ann Arbor Woarawichai, Kuruvi, Vashirawongpinyo. nernaional ournal of Mechanical, Aerospace, ndusrial, Mecharonic and Manufacuring Engineering, 2013; 7:2. 4. Michalewicz Z. Geneic Algorihms + Daa Srucures = Evoluion Programs. A Series. Springer-Verlag, New York Gen M, Cheng R. Geneic Algorihms and Engineering Design. Wiley, New York Gen M, Cheng R. Geneic Algorihms and Engineering Opimizaion. Wiley, New York Davis, he handbook of geneic algorihms, Van Nosrand Reinhold, New York Goldberg DE. Geneic Algorihms in Search, Opimizaion and Machine earning. Addison-Wesley, Reading, MA Mogador SRM, Afsar A, Sohrabi B. nvenory lo-sizing wih supplier selecion using hybrid inelligen algorihm. Applied Sof Compuing. 2008; 8: Chan SH, Chung W, Wadhwa S. A hybrid geneic algorihm for producion and disribuion. Omega. 2005; 33: ~26~

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