Single-Vendor and Multi Buyers of Supply Chain Management under Fuzzy Environment

A supply chain may be defined as an integrated process wherein a number of various business entities work together in an effort to: (1) acquire raw materials, (2) convert these raw materials into specified final products, and (3) deliver these final products to retailers. Several authors have studied the integrated policies of the manufacturers and the retailers. Most of the research in this area is based on the classic work of Clark and Scarf (1960). The idea of joint total cost of the supplier and the customer was first introduced by Goyal (1976). Later, Cohen and Lee (1988), determined material requirement for all materials at every stage in a supply chain. Pake and Cohen (1993), extended the above study to consider for stochastic sub systems. Gyana and Bhabha (1999), explored a single manufacturing system for procurement of raw materials with a multi-ordering policy that minimized the total inventory costs of both the raw materials and the finished goods. Sarkar et al. (2000) explored a supply chain model for determining an optimal ordering policy under inflation and allowable shortages. Goyal and Nebebe (2000), considered a problem of determining economic production and shipment policy of a product from a vendor to a buyer. Woo et al. (2001), considered an integrated inventory system where a vendor purchases and processes raw materials and delivered the finished items to multiple buyers. Rau et al. (2003) developed a multi-echelon inventory model for a deteriorating item and derived an optimal joint total cost from an integrated perspective among the supplier, the producer, and the buyer. Yang and Wee (2003), developed an integrated inventory model with constant rate of deterioration and multiple deliveries. Chien and Lin (2004), investigated the optimal order interval and discount price such that the joint total cost was minimized during a finite planning horizon. Banerjee (2005), developed a model where a supplier makes an agreement with a buyer, and determines the inventory policies of the supplier. Hans et al. (2006), presented a methodology to obtain the joint economic lot size in distribution system. Ahmed et al. (2008). have coordinated a two level supply chain in which they considered production interruptions for restoring of the quality of the production process. Jha and Shanker (2009) considered a two-echelon supply chain inventory problem consisting of a single-vendor and a single-buyer. In the system under study, a vendor produced a product in a batch production environment and supplied it to a buyer. Also, buyer‘s lead time was controllable which can be shortened at an added cost and all shortages were backordered. Uthayakumar and Geetha (2009). presented an inventory model for non-

 GRADED MEAN INTEGARTION REPRESENTATION METHOD

Chen and Hsieh (1998, 1999, 2000b) introduced Graded Mean Integration Representation Method based on the integral value of graded mean h-level of generalized fuzzy number for defuzzifing generalized fuzzy number. They also found this method is better than the methods of Adamo (1980), Campos and Verdegay (1989), Yager (1981), Kaufmann and Gupta (1991), Lee and Yao (1998), Liou et al (1992), Heilpern 1997). Here, we describe generalized fuzzy number as follows. Suppose à is a fuzzy number it is described as a fuzzy subset of the real line R, whose membership function μA satisfies the following conditions.  μA is a continuance mapping from R to the closed interval [0, 1],  μA = 0, 1xa,  μA = L(x) is strictly increasing on [a1, a2],  231,Aaxa,  μA = R(x) is strictly decreasing on [a3, a4],  40,Aax, Where a1, a2, a3and a4 are real numbers s.t. Also this type of fuzzy number is denoted as By Graded Mean Integration Representation method, 11LandRare the inverse functions of L and R respectively and the graded mean h-level value of fuzzy number 1234(,,,)LRAaaaa is 11(()())/2hLhRh as Figure1. Then the graded mean integration representation of à is P(Ã) where With 0 < h ≤ 1. be a trapezoidal fuzzy number, and be denoted as . Then we can get the graded mean integration representation of B by formula (I) as

Function principle is introduced by Chen (1985) to treat the fuzzy arithmetical operations by trapezoidal fuzzy numbers. We will use this principle as the operation of addition, multiplication, subtract, division of trapezoidal fuzzy numbers, because (1) Function principle is easier to calculate than Extension Principle, (2) Function Principle will not change the shape do trapezoidal fuzzy number after the multiplication of two trapezoidal fuzzy numbers, but the multiplication of two trapezoidal fuzzy numbers will become drum like shape fuzzy number by using Extension Principle, (3) If we have to multiple more than four trapezoidal fuzzy numbers then the Extension Principle cannot solve the operation, but Function Principle can easy to find the result by point wise computation.

The mathematical model in this study is developed on the basis of the following assumptions:  A single item with constant deterioration rate of the on –hand inventory is considered.  Single-vendor multi-buyers with one item is assumed.  Shortage is not allowed.  There is no replacement or repair of deteriorated units.  Holding cost is applied to good units only.

Here we suppose that are non-negative trapezoidal fuzzy number.

The inventory system depicted in Fig.1 is represented by the following differential equations:

On using the boundary condition()0biIT, solution of the above differential equation is Maximum inventory of each buyer is The yearly holding cost for all buyers is The annual deteriorated costs for all buyers is The setup cost per year for all buyers is The buyer‘s total cost is the sum of the Holding cost, deteriorated cost and the setup cost

The inventory system depicted in Fig.2 is represented by the following differential equations

Using the boundary conditions Iv1(0) = 0 and Iv2(T2) = 0 the solutions of the above differential equations are Maximum inventory level of the vendor is By the boundary condition Iv1(T1) = Iv2(0), we have We know that T = T1 + T2, thus Holding cost for the vendor is The annual deteriorated cost for the vendor is The setup cost per year for the vendor is The vendor‘s total cost is the sum of the holding cost, deteriorated cost and the setup cost as The integrated total cost of the vendor and the buyers, TC, is the sum of (7) and (18). By (13) TC is a function of T2 for a fixed value of ni, thus Where ,are the fuzzy arithmetic operations under Function principle. Firstly, we get the fuzzy total integrated total cost in the form of trapezoidal fuzzy number as below Now we want to determine the value of ni that minimize TC where i=1,2,3,………..,N. since the number of deliveries per period, since ni is a discrete variable, so the value of ni can be determined by the following procedure: (I) For a range of ni values, determine the derivative of TC w.r.t. T2 and set it to zero. For each ni denote the value of T2 for minimum of TC by T2(ni) where i = 1, 2, 3… N. (II) The optimal value of ni is derived by satisfying the following condition Secondly, we defuzzify the fuzzy integrated total cost using graded mean integration representation method. i.e. Thirdly we can get the optimal production quantity Q* when F(T,C) is minimization. In order to minimize F(T,C) the necessary conditions is , And solving for we will get the optimal time value of and then from equations (13) and (14), we will find the optimal values of and T.

The preceding theory can be illustrated by considering two buyers, i.e. N=2. The capacity of production is 200000 units per year: the annual demand rate of the first and the second buyers are 4000 and 8000 units, respectively; the yearly percentage of holding cost per dollar for the vendor and the buyers are $(0.12, 0.15, 0.18, 0.20) and (0.15, 0.17, 0.19, 0.21) respectively. The other related factors are as follows the ordering cost is $(160, 180, 210, 240) for the buyers, the production setup cost is $(4000, 4600, 5000, 5500) the unit production cost is $(8, 10, 12, 15) the unit price for buyer is $(10, 11, 12, 14) and the deterioration rate is 0.1 per year. By applying the above solution procedure, obtained results are shown in Tables 1. From the above tables it is clear that the total cost is reduced in both cases when compared to the costs obtained in buyers and vendor‘s separate policies. We see that the total cost is minimum when the number of deliveries are n1*= 3 and n2*= 3. The optimal cost in integrated policies is $15401.94 and the total cost in their separate policies is obtained $15594.39. Thus it is concluded that integrated policy results in an impressive cost reduction.

Figure 3. Variation in

Figure 4. Variation in

Figure 5. Variation in VC Figure 6. Variation in BC

A single vendor multi-buyers inventory model with fuzzy costs has been developed. We used integrated inventory policy instead of independent decisions made by the vendor and buyers. From the above observations it is concluded that the integrated inventory policy results in the impressive cost reduction as compared to the vendor‘s and buyers independent decisions. Fuzzy model provides more realistic situations with market uncertainties and to be prepared to best deal with them.

1. Ahmed, M.A., El Saadany and Jaber, M.Y. (2008). ―Coordinating a two-level supply chain with production interruptions to 2. Banerjee, A. (2005). ―Concurrent pricing and lot sizing for make-to-order contract production‖, International Journal of Production Economics; 93–94: pp. 189–195. 3. Chien, S.T. and Lin, S.D. (2004). ―Optimal buyer seller inventory models in supply chain‘‘, International Journal of Operations Research, Vol. 1(1), pp. 47-58.

5. Cohen, M.A. and Lee, H.L. (1988). ―Strategic analysis of integrated production-distribution system: Model and methods‘‘, Operations Research, Vol.36, 216-228. 6. Goyal, S. K. (1976). ―An integrated inventory model for a single supplier-single customer problem‘‘, International Journal of Production Research, pp. 107-111. 7. Gyana, R.P. and Bhabha, R.S. (1999). ―Operations planning in a supply chain system with fixed interval deliveries of finished goods to multiple customers‘‘, IIE Transactions, Vol.31, pp. 1075-1082. 8. Goyal, S.K. and Nebebe, F. (2000). ―Determination of economic production-shipment policy for a single-vendor-single buyer system‖, European Journal of Operational Research; Vol.121, pp. 175–178. 9. Hans, S., Raafat, N.I. and Paul, B.L. (2006). ―Joint economic lot size in distribution system with multiple shipment policy‖, International Journal of Production Economics, Vol.102, pp. 302-316. 10. Jha, J.K. and Shanker, K. (2009), ―Two-echelon supply chain inventory model with controllable lead time and service level constraint‘‘, Computers & Industrial Engineering, Vol.57(3), pp. 1096-1104. 11. Pake, D.F. and Cohen, M.A. (1993). ―Performance characteristics of stochastic integrated production-distribution system‘‘, European Journal of Operational Research, Vol. 68, pp. 23-48.

13. Sarkar, B.R., Jamal, A.M.M. and Wang, S. (1993). ―Supply chain models for perishable products under inflation and permissible delay in payments‘‘, Computers and Operations Research, Vol.27, pp. 59-75. 14. Uthayakumar, R. and Geetha, K. V. (2009). ―Replenishment policy for single item inventory model with money inflation‘‘ Opsearch, Vol. 46(3), pp. 345-357. 15. Woo, Y.Y., Hsu, S.L. and Wu, S. (2001). ―An integrated inventory model for a single vendor and multiple buyers with ordering cost reduction‖, International Journal of Production Economics; Vol. 73, pp. 203–215. 16. Yang, P.C. and Wee, H.M. (2003). ―An integrated multi-lot-size production inventory model for deteriorating item‖, Computers and Operations Research, Vol. 30, pp. 671–682.

Department of Mathematics, Deshbandhu College, Delhi