FUZZY MULTI OBJECTIVE MIXED INTEGER LINEAR PROGRAMMING TECHNIQUE FOR SUPPLY CHAIN MANAGEMENT

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1 International Journal of Pure and Applied Mathematics Volume 101 No , ISSN: (printed version); ISSN: (on-line version) url: PAijpam.eu FUZZY MULTI OBJECTIVE MIXED INTEGER LINEAR PROGRAMMING TECHNIQUE FOR SUPPLY CHAIN MANAGEMENT C. Kavitha 1, C. Vijayalakshmi 2 1 Department of Mathematics Sathyabama University Chennai, , India 2 SAS, Mathematics Division VIT University Chennai, , INDIA Abstract: This paper describes a mathematical model for supply chain network consists of multiple objectives with uncertain constraint, deterministic constraint solved by fuzzy mixed integer linear program. The model consists of two different objective functions such as minimizing the overall cost and maximizes the profit in supply chain. Imprecise objectives and uncertain constraints are modeled by triangular fuzzy numbers which help the decision makers to achieve the aspiration level and the constraints are solved by possibilistic membership approach. Fuzzy approach is applied to overcome the vagueness, imprecision and uncertainty in real life situations and the model has been applied to supplier selection of a high technology company named Multi-Flex Lami-Print Ltd which manufactures Flexible Packaging materials. AMS Subject Classification: 03E72, 03F55, 90B05 Key Words: supply chain management, uncertainty, imprecise, fuzzy multi objective integer linear programming Received: March 12, 2015 c 2015 Academic Publications, Ltd. url:

2 782 C. Kavitha, C. Vijayalakshmi 1. Introduction Fuzzy mathematical programming model for supply chain planning which considers supply, demand and process uncertainties as a fuzzy mixed-integer linear programming model where data are ill-known and modeled by triangular fuzzy numbers[david Peidro et al 2009]. Possibilistic decision making in an uncertain environment [ H.M.Hsu et al ]. A possibilistic programming method to solve multi-product and multi-time period production/distribution planning decision (PDPD) problems involving imprecise goals forecast demand and cost/time coefficients in uncertain environments [Tien-Fu Liang 2013]. Supply chain uncertainty model by fuzzy sets and develops a fuzzy linear programming model for tactical supply chain planning in a multi-echelon, multi-product, multi-stage with different methods of manufacturing in each stage, multi-distribution centre and multi-period supply chain network [ Hamid Reza Feili et al 2011]. 2. Fuzzy Mathematical Model for Multi Objective Linear Programming with Fuzzy and Deterministic Constraint This model represents multi raw material, multi production resources, multi manufacturing plant, multi manufacturer, multi supplier, multi distributor, whole seller and retailer to prepare multi product. Uncertainties associated with various cost are represented by triangular membership function which provides the decision maker with possible outcomes as worst, moderate and best. It is defined on the basis of values for the most pessimistic, most optimistic and most likely value. MOLP with triangular numbers for minimization objective function involves simultaneously i) minimizing the most likely value of uncertain total cost Z2 m, ii) maximizing the possibility of obtaining lower total cost (Z2 mzo 2 ) and iii) minimizing the risk of obtaining higher total cost (Z p 2 Zm 2 ) MOLP with triangular numbers for maximization objective function involves simultaneously i) maximizing m which is the most possible value of uncertain total cost, ii) minimizing (Z1 mz1p ) which is the possibility of obtaining lower total cost and iii) maximizing (Z1 ozm 1 ) which is the risk of obtaining higher total cost. To Minimize Z p 1,Zm 1,Z0 1 Min Z 1 = Z m 1, Max (Z m 1 Z 0 1), Min (Z p 1 Zm 1 ).

3 FUZZY MULTI OBJECTIVE MIXED INTEGER LINEAR In the objective functions PIS (positive ideal solution lower bound) and the (negative ideal solution-upper bound) is defined as follows: Z11 PIS = Min Z1 m, ZPIS 12 = Max (Z1 m Z0 1 ), ZPIS 13 = Min (Z1 m Zm 1 ) Z11 PIS = Max Z1 m, ZPIS 12 = Min (Z1 m Z0 1 ), ZPIS 13 = Max (Z1 m Zm 1 ) Linear membership functions for three objective functions are 1 if Z 11 < Z11 PIS Z11 f 11 (Z 11 ) = Z 11 Z11 Z PIS if Z PIS Z 11 Z11 0 if Z 11 > Z11 f 12 (Z 12 ) = f 13 (Z 13 ) = 1 if Z 12 < Z12 PIS Z 12 Z12 12 Z 12 if Z12 Z 12 Z12 PIS 0 if Z 12 > Z12 Z PIS 1 if Z 13 < Z13 PIS Z13 Z Z PIS 13 if Z13 PIS Z 13 Z13 0 if Z 13 > Z13 Z Similarly for Maximization Membership function for fuzzy constraint is 0 if A i x b p i p A i x b i b π bi (x) = m i b i p if bp i < A ix b m i b 0 i A ix b 0 i b if b m i m i < A i x b 0 i 0 if b 0 i < A ix In general the multi objective fuzzy linear programming is converted into single objective ordinary linear programming M axl such that L f 1i (Z 1i ), i = 1,2,3 L f 2j (Z 2j ), j = 1,2,3 L π bi (x) n a ij x j b i, x j 0, 0 L 1 j=1

4 784 C. Kavitha, C. Vijayalakshmi where L represent the overall satisfaction of the decision makers. If L = 1 then each goal is fully satisfied. If 0 < L < 1 then all the goals are satisfied at the level L. If L = 0 then none of the goals are satisfied. If the decision maker is not satisfied with the initial solution then the model will be modified to get a satisfactory solution. MinTPC +TPRC +TTC +TRMC +TRPC +TDBC +TIC, (1) Max TSV, (2) TPC = i=1 j=1 k=1 t=1 [ C VP ijkt FP OP + C ijkt + C ijkt ]Ap ijkt, (3) TPRC = j=1 i=1 k=1 t=1 C jikt RTPR RTjikt PR + j=1 i=1 k=1 t=1 C jikt ROPR OTjikt PR, (4) TTC = + R M r=1 i=1 s=1 m=1k=1 t=1 L i=1 n=1w=1 l=1 t=1 TRMC = TIC = R C TD inwta TD inwt + r=1 i=1 s=1 k=1 t=1 R TDBC = TRPC = r=1 i=1 k=1 s=1 t=1 C rismkt TRM ATRM riskt + L i=1 w=1 l=1 t=1 i=1 k=1 n=1 t=1 C R riskt RM ARM riskt + i=1 k=1 t=1 + M i=1 k=1m=1 t=1 C DP ikt + C TW iwlta TW iwlt + [ C TMP iknt + r=1 s=1 i=1 j=1 k=1 t=1 C rikst IRM AIRM rikst + i=1 n=1 t=1 i=1 k=1 t=1 C DU iknt ]ATMP iknt (5) C TU ikdta TU ikdt, C DRM rsijkt ADRM rsijkt, (6) C RP ikmt ARP ikmt, (7) R i=1 r=1 t=1 i=1 k=1 s=1 t=1 C int ID AID int + C BO irt A BO irt (8) i=1 w=1 t=1 C TMP ikst A IMP ikst (9) C IW iwt AIW iwt,

5 FUZZY MULTI OBJECTIVE MIXED INTEGER LINEAR TSV = L i=1 k=1 s=1 l=1 t=1 Subjact to raw material constraint: Capacity of Supplier: R r=1 j=1 [ P iklt + D ikslt ]A SA ikslt. (10) A RM MAX riskt C P srik Y sijkt s S, i I, k K, t T (11) Manufacturing plant storage capacity: i=1 s=1 t=1 Defective item storage capacity: t=1 Reprocessing: Balance : M m=1 A RM MAX riskt C P rk r R, k K (12) A DRM MAX rsijkt D F rsijk r R, k K (13) F k,i + k=1 V i i=1 i=1 P min i (14) A TRM rismkt = ARM riskt r R, i I, s S, k K, t T (15) Manufacturer: A IRM rikst = AIRM riks(t 1) +ARM RM riskt T L skt Productions Constraint: j=1 A rijkt A P ijkt ATRM rismkt m M, r R, s S, k K, t T (16) A p ijkt ÃMAX ijk X ijkt, i I, j J, k K, t T (17) Processing time: P T ijkt A p ijkt ÃMAX ijk X ijkt, i I, j J, k K, t T (18)

6 786 C. Kavitha, C. Vijayalakshmi Regular and over timeprocessing: RTjikt P R+OTPR MAXPR jikt T jikt X ijkt, j J, i I, k K, t T (19) Production Running: A p ijkt P R MIN ijkt X ijkt, i I, j J, k K, t T (20) FIHED PRODUCT CONSTRAINT: Distributor capacity: w=1 t=1 Whole seller capacity: L l=1 t=1 Demand of retailer: w=1k=1 s=1 A TD MAX inwt C P in i I, n N (21) A TW MAX iwlt C P iw i I, w W (22) Manufacturing Plant storage capacity: j=1 t=1 i=1 k=1 s=1 t=1 Manufacturer: A P iknt A TW iwlt D ikslt i I, l L, t T (23) C P MAX ik i I, k K (24) A TMP MAX iknt C P in Z ikt i I, t T (25) A IMP MAX ikst C P ik i I, k K (26) L l=1 D ikslt +A IMP iks(t 1) = AIMP iks(t 1) +AP PM ijkt T L knt N n=1 A TMP iknt,

7 FUZZY MULTI OBJECTIVE MIXED INTEGER LINEAR BALANCE CONSTRAINT: Manufacturing plant = Distributor: k=1 A TMP iknt = i I, k K, t T, s S, n N (27) w=1 Distributor = Whole seller n=1 A TD inwt = L l=1 Whole seller =Retailer w=1 A TW iwlt = K k=1 s=1 A TD inwt i I, n N, t T (28) A TW iwlt i I, w W, t T (29) A SA ikslt i I, l L, t T (30) DEMAND BACKLOG CONSTRAINT: i=1 w=1 t=1 A TW iwlt i=1 k=1 t=1 [ D ikslt +ÃDB ikt ] n N, l L s S (31) A DB ikt = A DB ik(t 1) + S s=1 l=1 L D ikslt n=1 A TMP iknt i I, k K, t T (32) FLOW CONSTRAINT: Distributor: A ID int = AID in(t 1) + K k=1 A TMP PD iknt T L nwt W w=1 A TD inwt l L, i I, n N, t T (33)

8 788 C. Kavitha, C. Vijayalakshmi Whole seller: A IW iwt = AIW iw(t 1) + N n=1 A TD PW inwt T L wlt L l=1 A TW iwlt l L, i I, w W, t T (34) DISPOSAL SITE CONSTRAINT: Storage Capacity: k=1 t=1 SALES CONSTRAINT : A TU MAX ikdt C P id, i I, d D (35) A SA ikslt D ikslt, i I, k K, s S, l L, t T (36) D ikst A SA ikslt L D iklt, i I, k K, s S, l L, t T (37) NON NEGATIVITY: A P ijkt,rtpr jikt,otpr jikt,atrm rismkt,atmp iknt,a TD inwt,atw iwlt,abo irt,atu ikdt,arm riskt, (38) A DRM rsijkt,arp ijmt,adp ikt,airm rikst,aimp ikst,aid int,aiw iwt,asa ikslt,x ijkt,y sijkt,z ikt 0 3. Numerical Calculations for Supply Chain Model has applied to a professionally managed company namely Multi-Flex Lami-Print Ltd., who manufactures Quality Flexible Packaging Materials against specific orders from their customers like Hindustan Unilever Ltd., ITC Ltd., Tata Tea Ltd., Cavinkare Pvt Ltd.. will produce important raw materials like Polyester film, Bi-axially oriented Poly Propylene film, Polyethylene film and

9 FUZZY MULTI OBJECTIVE MIXED INTEGER LINEAR also Printing inks, Lamination adhesives, Diluting Solvents.from the best suitable suppliers for various production processes such as Printing-Lamination- Slitting-Finishing. Network sizes for the data sets are Products -4, Production Resources-3, Manufacturing plant-2, Suppliers-7, Manufacturer-4, Retailer-12, Time periods-5, Distributor-6, wholesaler-10, Disposal site-3. The solution of the model is obtained in Table 1. Objective Function Min Z 11 Max Z 12 Min Z 13 Z Z Z Z 1 = ( , , ), Objective Function Max Z 21 Min Z 22 Max Z 23 Z Z Z Z 2 = ( , , ), L = Conclusion In real life situation, Supply chain management handle imprecise goal, which governs constrained resources where the uncertain datas and environmental coefficients are fuzzy. The objective of this paper was to develop a fuzzy mathematical programming using triangular numbers for supply chain with multi products, multi periods, multi suppliers, managers, customers, retailers etc. To obtain fuzzy goals of the decision maker, having imprecise, uncertain datas in fuzzy constraint, linear membership function is provided. As a future work, fuzzy non linear program using triangular numbers can be developed to obtain optimal solutions for multi objective supply chain model. References [1] DavidPeidro, JosefaMula, RalPoler, Jos-LuisVerdegay,Fuzzy optimization for supply chain planning under supply, demand and Process uncertainties. Fuzzy Sets and Systems, 160, (2009),

10 790 C. Kavitha, C. Vijayalakshmi [2] Hamid Reza Feili, Mojdeh Hassanzadeh Khoshdooni,A Fuzzy Optimization Model for Supply Chain Production Planning with Total Aspect of Decision Making, The Journal of Mathematics and Computer Science Vol.2 No.1 (2011) [3] H.M.Hsu and W.P.Wang,Possibilistic programming in production planning of assemble to order environment, Fuzzy sets and systems, Vol 119, pp , [4] Tien Fu Liang,Interactive Multi-objective Transportation planning decisions using Fuzzy Linear Programming, Asia Pac. J. Oper Res., 25,11, (2008). [5] Tien-Fu Liang,Imprecise multi-objective production / distribution planning deisions using possibilistic programming method, Journal of Intelligent and fuzzy systems, Volume 25, 1, [6] H.J. Zimmermann,Description and optimization of fuzzy system, Int. J. of Gernaral system, vol. 2, pp , 1976.