International Journal of Management and Transformation: Vol.5, No.2

In this study, the effect of a temporary price discount for a larger order than the regular order offered by the supplier on the retailer’s ordering policy is studied. The demand is assumed to be stock-dependent. The units in inventory deteriorate at a constant rate. This study will help the retailer to take the decision whether to adopt a regular or special order policy. The optimum special quantity to be procured is determined by maximizing the total cost saving between the special and regular orders for the cycle time. The algorithm based on analytic results is outlined to take the favorable decision. The numerical examples are given to support the derived results. The sensitivity analysis is carried out to determine the critical inventory parameters. INTRODUCTION To boost the demand, increase market share and cash-flow, the supplier offers a temporary price discount to the retailer. For the retailer, the question is to find out whether or not it is advantageous to buy special order at a discounted price or not?” The impact of price discount on the order policy is well cited in review article by Dixit and Shah (2005). The all-unit quantity discounts ordering policy is discussed by Arcelus et al. (2003), Abad (2007), Shah et al. (2005), Bhaba and Mahmood (2006), Abad (2007), Dye et al. (2007), Shah et al. (2008), Mishra and Shah (2009) and their cited references. These formulations assumed that the price discount rate is independent of the special order quantity. However, in market, it is observed that the supplier offers a quantity discount to entice larger orders. For the larger order, the higher price discount rate is offered by the supplier and thereby reducing his inventory. Here,, the retailer has to settle the trade-off for purchase price savings against higher total holding cost. In addition to holding cost, the retailer incur deterioration cost for products like; dairy products, volatile liquids, blood components, fruits and vegetables, medical accessories etc. The literature surveys by Raafat (1991), Shah and Shah (2000) and Goyal and Giri (2001) advices the retailer to take care of ordering policy while dealing with deteriorating items. One can review Moon et al. (2005), Deng et al. (2007), Liao (2007). Ouyang et al. (2009) developed a decision policy when a supplier offers a temporary price discount to a retailer for larger order. The price discount rate depends on the units ordered. They assumed that the demand is constant and deterministic. In this paper, an impact of a temporary price discount on the retailer’s ordering policy is studied when the demand is stock-dependent and price discount rate is linked to special order quantity. The units in inventory are subject to a constant deterioration. This analysis will help the retailer to adopt or decline the sales promotional scheme. The retailer’s optimal special order quantity is obtained by maximizing the total cost savings between the special and regular orders during a special order cycle time. Two scenarios are discussed : (1) the special order time occurs at the retailer’s replenishment time, and (2) the special order time occurs during the retailer’s regular cycle time. A computational procedure is outlined to decide to the optimal solutions. The numerical examples are given to support the validated the theoretical results. The sensitivity analysis of the optimal solutions is carried out with respect to the model parameters. The managerial issues are derived. International Journal of Management and Transformation 5(2): Autumn 2011 4 NOTATIONS AND ASSUMPTIONS The following notations and assumptions are used to formulate the proposed problem : Notations R(I( t)) (= α + βI(t)), stock-dependent demand rate where α > 0 is scale demand and 0 ≤ β < 1 is stock-dependent parameter C Unit purchasing cost A Ordering cost per regular or special order I Holding cost rate per annum Constant deterioration rate; 0 ≤ θ < 1 Q Order quantity under regular policy Q Optimal order quantity when regular order policy is adopted T Cycle time when regular order policy is adopted T Optimal cycle time for using a regular order policy Qs Special order quantity at discounted price (a decision variable) Ts Cycle time for the special order quantity Qs Q Inventory level before the arrival of the special order quantity; q ≥ 0 tq Cycle time when q – units deplete to zero TW Cycle time for depletion of the inventory level W = Qs + q I(t) Inventory level at time t when the regular order policy is adopted; 0 ≤ t ≤ T Is(t) Inventory level at time t when the special order policy is adopted; 0 ≤ t ≤ Ts IW(t ) Inventory level at time t when the special order policy is adopted; 0 ≤ t ≤ TW where W = Qs + q Assumptions (a) The single item inventory system is considered. (b) The supplier offers the retailer a temporary price discount if the order quantity is larger than the regular order quantity Q. The discount rate depends on the quantity ordered and the discount schedule is as follows : Cl ass Special order quantity Discount rate 1 1 2 s Q Q Q   d1 2 2 3 s Q Q Q   d2 ⋮ ⋮ ⋮ n 1 n s n Q Q Q    dn where Qk is the k-th discount rate breaking point, k = 1, 2, ..., n+1 and * 1 2 1 n Q Q Q Q         ; dk is the price discount rate offered by the supplier when the retailer’s order quantity Qs belongs to the interval [Qk, Qk+1) and 1 2 0 n d d d      . (c) The price discount is not passed on to the customers. (d) Only one time price discount is offered. (e) The replenishment rate is infinite. (f) The lead-time is zero or negligible and shortages are not allowed. (g) The units in inventory deteriorate at a constant rate. The deteriorated units can neither be repaired nor replaced during the period under review. International Journal of Management and Transformation 5(2): Autumn 2011 5 MATHEMATICAL MODEL The goal of the study is to decide the advantage of temporary price discount for larger order than the regular order, under the assumption of the stock-dependent demand and constant rate of deterioration of units in retailer’s inventory system. If the retailer follows a regular order policy without a temporary price discount, then the inventory depletes in the retailer’s inventory system due to the stock-dependent demand and deterioration of units. The rate of change of inventory level can be described by the differential equation ( ) ( ( ) ( )), 0 dI t I t I t t T dt          (1) With boundary condition I(T) = 0, the solution of (1) is   ( ) exp( ( )) 1 I t T t       , 0 t T   (2) where      . Hence, the order quantity is   (0) exp( ) 1 Q I T       (3) In this case, the retailer’s total cost per order cycle is