7.1 INTRODUCTION. In this era of extreme competition, each subsystem in different

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2 7.1 INTRODUCTION In this era of extreme competition, each subsystem in different echelons of integrated model thrives to improve their operations, reduce costs and increase profitability. Currently, the competition is not confined to the subsystems of the same echelon levels, the necessity of long term and reliable business relation has created competition among the supply chains. Hence, the consideration of joint optimization of supply chain cost is of interest. Globalization of market and increased competition force organizations to rely on effective supply chains to improve their overall performance. Supply chain has become a vital topic in management science and industry. The logical progression of the inventory model is to investigate the supply chain that consists of suppliers, manufacturers, distributors and retailers. Each one of them holds inventory in some form to support the requirement of the customer at the end of the chain. In supply chain many problems still need a careful consideration regarding solution procedure to support respective systems. Deteriorating inventory, in general, is defined as decay, damaged, spoilage, evaporation, obsolescence, pilferage, Loss of marginal value of

3 a commodity that results in decreasing usefulness from the original one. Therefore, time dependent deterioration rate is used. Although it is considered as a negative aspect if a businessman runs out of stock and cannot fulfill the demand of the customers, but in certain situations this shortage may actually prove beneficial for the businessman instead. This happens as, even when the unavailability of the stock puts a penalty cost on the businessman, the backlogging of demand allows him to order a larger lot size and hence allows for a larger cycle inventory and reduces cost for him. Many delivery policies have been proposed in literature for this problem. Clark and Scarf (1960) presented the concept of serial multi-echelon structures to determine the optimal policy. Datta and Pal (1990) developed an inventory model with stock dependent demand until the stock level reached a particular point, after which the demand became constant. Giri et al. (1996) extended Datta and Pal by relaxing their restriction of zero inventories at the end of order cycle and including deterioration effects. Yang and Wee (2000) developed an integrated economic ordering policy of deteriorating items for a vendor and a buyer. Datta and Pal (2001) presented an inventory model in which demand was influenced by the stock level displayed as well as the selling price of -181-

4 the item. Chung and Huang (2003) studied the optimal cycle time for EPQ inventory model under permissible delay in payments. Balkhi and Benkherouf (2004) analyzed a deteriorating stock dependent model for a finite horizon. Huang et al. (2005) considered the optimal inventory policies under permissible delay in payments depending on the ordering quantity. Mahapatra and Maiti (2005) set forth a study on multi objective inventory models with stock and quality dependent demand. Other researches related to this area such as Yang and Wee (2003) and Lee and Wu (2006). Roy and Chaudhari (2006) studied a model with stock dependent demand with constant deterioration. Song and Cai (2006) has been taken on optimal payment time for a retailer under permitted delay of payment by the wholesaler. Liao (2007) assumed on an EPQ model for deteriorating items under permissible delay in payments. Singh and Singh (2008) developed a perishable inventory model with quadratic demand, and partial backlogging. Xu and Leung (2009) proposed an analytical model in two-party under managed system where the retailer restricts the maximum space allocated to the vendor. a very difficult task for the management in most of the countries. So it is economical to order the inventory according to available storage space. This problem is developed with the concept of space restriction in which -182-

5 demand is exponential, and deterioration is time dependent. Production is taken as a function of demand. All these practical aspects are incorporated into the model with the purpose of making it more realistic. The numerical illustration of this problem is also been discussed and shown that the integrated approach is better than the independent. 7.2 ASSUMPTIONS AND NOTATIONS The following assumptions and notations have been taken for model development. Assumptions 1. Inventory position is continuously monitored for the retailer and an order of size Q is made whenever the inventory level hits the reorder point R. 2. Units are demanded in small quantities; overshooting of the reorder point is not appreciable. 3. Deterioration rate is taken as time dependent. 4. Demand rate is taken as exponential function. 5. Production is taken as demand rate dependent

6 6. An area with limited space W, is reserved for the storage of the product for the retailer. 7. The over ordered quantity that can not be accommodated in the available space at the delivery time is returned to the supplier. 8. The ordering quantity is smaller than the storage space capacity. 9. The time when the system is out of stock during a cycle is small compared to the cycle length. 10. The supplier charges of the purchasing cost for each unit of the product returned because of over-ordering. Notations P : Production Rate. T : Cycle Time. T 1 : The time for which production occurs. 2 : The non-production time for +ve inventory. 0 : Production cost per unit. p 1 : Selling price per unit for the supplier. p 2 : Selling price per unit for the retailer

7 c ds : Deterioration cost per unit for the supplier. c d : Deterioration cost per unit for the retailer. 1 : Inventory holding cost per unit for the supplier. h 2 : Inventory holding cost per unit for the retailer. : Set up cost for the supplier. R : Reorder point for the retailer. x : Lead time demand for the retailer. W : Storage capacity for the retailer. Q : Order quantity for the retailer. : Rate of backlogging. O : Ordering cost for the retailer. v : the time for ve inventory in the case of shortage. 7.3 MODEL FORMULATION We have discussed the supplier and buyer model: -185-

8 A simple production system constituting of a single unit which produces a single item is considered and the time interval between two successive production start points is one cycle. The production starts at the very beginning of the cycle. As production is continuous, inventory begins to pile up continuously after meeting demand at deterioration. Production stops at time T. The accumulated inventory depletes due to combined effect of demand and deterioration over the interval T 2. The cycle ends with zero inventory. Production restarts at T 2. The differential equations for this system are given by: (7.1) -186-

9 di s dt ( t) kti s ( t) ae bt,....(7.2) With boundary conditions:, s (7.3) The solutions of these equations are given by: 1....(7.4).... (7.5) The total cost function for the supplier is given by: T.C =Production cost+ deterioration cost+ inventory holding cost+ set up cost....(7.6) Production cost= p T1 0 0 bt ae dt a 1 = p 0 ( e bt 1)....(7.7) b Total deteriorated units =Total production Total demand

10 a bt a 1 bt2 Deterioration cost = c ds{ ( e 1) ( e 1)}....(7.8) b b Holding cost = h 1 T 1 2 ( I ( t)) dt I ( t)) dt 0 s T T 1 s = T1 b 2 T1 k 3 T1 k T1 T ( T T ) ( T T ) ( T T ) ( T )}} (7.9) Set up cost =.... (7.10) Put all these values in equation (7.6): a bt a 1 bt a 1 bt2 T.C s = po ( e 1) cds{ ( e 1) ( e 1)} b b b 2 2 T1 b 3 k 4 T2 b 3 k 4 1{ ( 1)( 1 1 ) { 2 2 h a T T a T T T1 b 2 T1 k 3 T1 k T1 T ( T T ) ( T T ) ( T T ) ( T )}} (7.11) T.A.C= T.C s

11 T.A.C s = 1 ( T T ) 1 2 T.C s. (7.12) When an order quantity of size Q is placed, the actual quantity unloaded into the storage facility depends on the inventory level immediately after the receipt of the order. In fact three different cases have to be distinguished depending on the values of lead time, the reorder point, and the inventory level just after the receipt of an order and the storage capacity W. Case 1 If x is the demand during the lead time: R-x 0 and R-x+Q W. This condition status that the quantity demanded during the lead time is smaller than the reader point

12 Fig. 7.1: Inventory level profile for case 1. The differential equation governing the transition of the system for the relation is given by: r bt r T. (7.13) The solution is given by: Ir 2 3 bt Kt ( t ) = {(R-x+Q)-a(t+ 2 6 kt 2 )} e 2 T....( 7.14) The total cost for the retailer: T. C R = Purchasing cost + Inv. Holding cost + Det. Cost + Ordering cost. (7.15) -190-

13 Purchasing cost = Q p (7.16) Inv. Holding Cost = h 2 0 T I ( t).) dt. r Holding Cost = h {(R-x+Q)T- T b 3 k a( 4 k T ak T T T ) ( R x Q ) }....( 7.17) Total Det. Units = Total inventory Total demand. Total Inventory = (R-x+Q). Total Demand = T 0 bt ae. dt. Det. Cost = {(R-x+Q) - ( e bt - 1)} C d. 7.18) Ordering Cost = O R. 7.19) Then, Total cost for the retailer in this case: T. C R1 = Q p 1 + h 2 {(R-x+Q)T T b 3 k a( 4 k T ak T T T ) ( R x Q ) } {(R-x+Q) - ( e bt - 1)} C + O. (7.20) d R -191-

14 T.A.C 1 1 ( T. C. R ).... (7.21) T R = 1 Case 2 When R-x+Q>W. In this case (R-x) is non-negative and after the receipt of an order, inventory level exceeds the storage capacity. In this case, the quantity required that can be accommodated within the available space is {W-(Rx)} and the max. inventory level over the cycle is now W. The retailer has to pay the material return cost for extra inventory as a penalty cost. Fig. 7.2: Inventory level profile for case

15 given by: The differential equation governing the transition of the system is dir ( t) dt kti ( t) ae r bt. (7.22) With boundary condition: I. The solution of this equation is given by: I r 2 t 2 k k 3 2 b ( t) {( W ( R x)) a( t t t )} e 0....( 7.23) 2 6 The total cost for the retailer in this case: T.C R 2 = Purchasing cost+ inventory holding cost+ deterioration cost+ material return cost+ ordering cost....( 7.24) Purchasing cost = (W-(R-x))p.... (7.25) Holding cost = h 2 Ir( t) dt. 0 T T bt KT H R h2 {( W ( R x)) T a ( ) K T akt ( W ( R x )) }. (7.26)

16 Deteriorated units = Total inventory Total demand = ( W ( R x)) - T 0 bt ae dt. Deteriorated Cost = {( W ( R x)) - e 1)} c. 7.27) bt d Ordering Cost = O R.... (7.28) Material Returned = (R-x+Q-W). Material Returned Cost = p 1(R-x+Q-W). 7.29) T.C 2 R = T bt KT (W-(R-x))p 1 +h2 {( W ( R x)) T a ( ) K T akt ( W ( R x )) } + p 1(R-x+Q-W) +{(W-(R-x)) bt e 1)} c + d O R.. (7.30) T.A. C R 2 = (T.C R 2 ).... (7.31) Case 3 When R-x <0: -194-

17 Fig. 7.3: Inventory level profile for case 3. This case corresponds to the shortage situation, since the demand during late time is greater than the reorder point. So, the minimum inventory in this case is zero and after the arrival of stock maximum inventory level is Q-(x-R). It is assumed that the inventory level starts with shortage. At t = 0, inventory level is zero and it results in shortage. At t=v, after the arrival of stock and satisfying backlogging demand, the inventory level becomes Q-(x-R). The differential equations governing the transition of the system are given by: -195-

18 dir ( t) bt ae dt.... (7.32) dir ( t) dt kti ( t) ae r bt.... (7.33) With boundary conditions:.... (7.34) The solutions of these equations are given by:- a bt Ir ( t) (1 e ) b.... (7.35) 2 2 v t 2 2 k k k b Ir ( t) { a{ v t ( v t ) ( v t )} ( Q ( x R)) e } e, 2 6. (7. 36) Total cost for the retailer in this case is given by: T.C R 3 = Purchasing cost + Inv. Holding cost + Ordering cost + Det. Cost + Shortage Cost.... (7.37) Purchasing cost = Qp.... (7.38) T Inv. Holding Cost h2 I ( t) dt v r -196-

19 T b T k T h2 { a( vt ) ( v T ) ( v T )} ( Q ( x R )) k ak vt T k v bv (1 v ) T ( ) T ( Q ( x R)) a {( k 4 k 2 ak 4 k 3 v )} Q ( x R))(1 v ) v v v ( Q ( x R ))} (7.39) Ordering Cost = O R....( 7.40) Deteriorated units = Total inventory Total demand. Deteriorated Cost = {( Q ( R x)) - e e )} c.... (7.41) bt bv d Shortage Cost = c s r bv a (1 e ) = c s ( v )....( 7.42) b b { ( T ) b ( T ) k R 6 ( T TC =[Qp +h a vt v T v T 4 )} k ak vt T k ( Q ( x R))(1 v ) T ( ) T ( Q ( x R )) v bv k k ak k a{( v )} Q ( x R))(1 v ) v v v

20 a bt bv ( Q ( x R))} OR {( Q ( R x)) ( e e )} c b d ] c s bv a (1 e ) ( v )....( 7.43) b b T.A.C R 3 = 1 TC T R3.... (7.44) Total Average Cost for whole the supply chain is given by: T.A.C. (Supply) = 1 (TC s TC R)....( 7.45) T The Objective Function is given by: Mini T.A.C. (T, T 2 ), s.t. T 0, T ( 7.46) 7.4 NUMERICAL EXAMPLE We use the following numerical data to analyze the model: a = 250, b = 0.2, c ds = 7, c d = 11, = 1.2, p 0 = 6, 1 = 0.04, h 2 = 0.05, R = 350, Q = 800, x = 270, T 2=15, -198-

21 k = , p = 8.5, = 0.1, W = 1000, = 500, o = 200. The optimal values are: 1 = 40, TAC= Table 7.1: Variation in 1 and T.A.C. with the variation in T 2 T.C. T.A.C.(10 ) Sensitivity Analysis Table 7.2: Variation in T.A.C. with the variation in a. A T.C. T.A.C.(10 )

22 Table 7.3: Variation in T.A.C. with the variation in b. B T.C. T.A.C.(10 ) Table 7.4: Variation in T.A.C. with the variation in K. K T.C. T.A.C.(10 )

23 Table 7.5: Variation in T.A.C. with the variation in h 1 and h 2. h T.C. T.A.C.(10 ) Table 7.6: Variation in T.A.C. with the variation in x. X T.C. T.A.C.(10 )

24 7.5 OBSERVATIONS Here we observe that the total cost is minimum for T =40 and T 2 =15. With the increment in the T, the value of T 2 decreases and the value of T.A.C. increases. The value of T.A.C. vary with the variation in a and b. With the variation in h and h 2 the T.A.C. also varies from point to point. 7.6 CONCLUSIONS Manufacturing is a wealth-producing sector of an economy. Manufacturing provides important material support for national infrastructure and for national development. Needless to say, this particular field of inventory management needs careful introspection and study on the part of the management before any major decisions are taken. An extensive study has been done on the topic till date, but there is a major requirement of a model which can not only take care of the demand and supply forces working over the economy of that organization, but also the effect that market forces has over it. Here, we have attempted this task only

25 In this chapter, we developed a single item deterministic production inventory model in which time dependent rate of deterioration is taken. Production is demand dependent and demand is considered to be exponential. This model is developed in the presence of space restriction. It is assumed that the over ordered quantity is returned to the supplier and cost is paid for this. In totality this model can present itself as a remedy for the quick changes in the market performance of any product and hence, can act as a savior for the inventory manager by offering him quick remedies. The model has also been exemplified numerically and it has proved itself financially viable also

26 REFERENCES [1] Clark A.J. and Scarf H. (1960). Optimal policies for a multi-echelon inventory problem, Management Science, 6, [2] Datta T.K. and Pal A.K. (1990). Deterministic inventory systems for deteriorating items with inventory level dependent demand and shortages, Journal of the Operational Research Society, 27, [3] Giri B.C. et al. (1996). An inventory model for deteriorating items with stock dependent demand rate, European Journal of Operational Research, 95, [4] Yang P.C. and Wee H.M. (2000). Economic order policy of deteriorated items for vendor and buyer: An integral approach, Production Planning and control, 11(5), [5] Datta T.K. and Pal K. (2001). An inventory system stock-dependent, price sensitive demand rate, Production Planning and Control, 12, [6] Chung K.J. and Huang Y.F. (2003. The optimal cycle time for EPQ inventory model under permissible delay in payments, International Journal of Production Economics, 84, 3, [7] Balkhi Z.T. and Benkherouf L. (2004). On an inventory model for deteriorating items with stock dependent and time varying demand rates, Computers & Operations Research, 31, [8] Huang Y.F., Chung K.J. and Goyal S. K. (2005). The optimal inventory policies under permissible delay in payments depending on the ordering quantity International Journal of Production Economics, 95, 2,

27 [9] Mahapatra N.K. and Maiti M. (2005). Multi objective inventory models of multi items with quality and stock dependent demand and stochastic deterioration, Advanced Modeling and optimization, 7(1), [10] Yang P.C. and Wee H.M. (2003). An integrating multi-lot size production inventory model for deteriorating item, Computers and Operations Research, 30, [11] Lee H.T. and Wu J.C. (2006). A study on inventory replenishment policies in a two-echelon supply chain system, Computers & Industrial Engineering, 51, 2, [12] Roy T. and Chaudhuri K.S. (2006). Deterministic inventory model for deteriorating items with stock-level dependent demand and shortage, Nonlinear Phenomena in Complex Systems, 9(1), [13] Song X. and Cai X. (2006). On optimal payment time for a retailer under permitted delay of payment by the wholesaler, International Journal of Production Economics, 103, 1, [14] Liao J.J. (2007). On an EPQ model for deteriorating items under permissible delay in payments, Applied Mathematical Modeling, 31(3) [15] Singh S.R. and Singh C. (2008). Optimal ordering policy for decaying items with stock-dependent demand under inflation in a supply chain, International Review of Pure and Applied Mathematics, 1, [16] Xu K. and Leung M.T. (2009). Stocking policy in a two-party vendor managed channel with space restrictions. International Journal of Production Economics 117,2[5]

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