32 An Inventory Model for Maximum Life Time Products under the Price and Stock Dependent Demand Rate Vikram Vashisth Department of Computer Science B.K. Birla Institute of Engineering &Technology Pilani, Rajasthan (India) Ajay Tomar Department of MCA B.K. Birla Institute of Engineering &Technology Pilani, Rajasthan (India) Ramakant Soni Department of Computer Science B.K. Birla Institute of Engineering &Technology Pilani, Rajasthan (India) A.K. Malik Department of Mathematics B.K. Birla Institute of Engineering &Technology Pilani, Rajasthan (India) ABSTRACT Today’s due to competitive business scenarios, the suppliers provide his/her retailers a discount in price of a product. To attract the customers to buy more products at one time, the retailer managed the demand rate depends on price and stock dependent which is very realistic in day to day life. The holding cost is assumed to be variable. For this we proposed a model to solve such types of problems to determine the optimal replenishment policy for non-instantaneous deteriorating products. Numerical example is provided to demonstrate the optimal total profit for discussed inventory model. Keywords Inventory, Non-instantaneous deterioration, Variable holding cost, price and Stock-dependent demand. 1. INTRODUCTION In supermarkets, stores, malls, all of the customers not only focuses on the amount of stock but also observes the price of the items. The product in a large quantity and price discount in the supermarkets attracts many customers and generates the high demand. Therefore the effect of price and stock dependent demand rate cannot be ignored for determining the optimal order policy. Levin et al. (1972) discussed the big amount of consumer goods displayed in a supermarket would attract the customer to buy more products. Gupta and Vrat (1986) first proposed an inventory model for utilization environment to optimize the total inventory cost and considering the stock- dependent demand. Baker and Urban (1988) discussed an inventory model for a power-form inventory-level-dependent demand pattern. Datta and Pal (1990) gave a note on an inventory model with inventory level dependent demand rate. Pal et al. (1993) extended the inventory model of Baker and Urban (1988) for perishable products that deteriorate at a constant rate. Bar-Lev et al. (1994) developed an extension of the inventory-level-dependent demand-type EOQ model. Padmanabhan and Vrat (1995) proposed an inventory model of deteriorating items with stock-dependent demand. Urban and Baker (1997) discussed the EOQ model in which the demand is a multivariate function of price, time, and level of inventory. Datta and Paul (2001) analyzed a multi-period EOQ model with stock-dependent and price-sensitive demand rate. Other researchers related to this area such as Mandal and Phaujdar (1989), Giri et al. (1996 and 1998), Ray et al. (1998), etc. Chang (2004) developed an Inventory model with stock- dependent demand and nonlinear holding costs for deteriorating items. Soni and Shah (2008) developed the optimal ordering policy inventory model and stock-dependent demand under the progressive payment scheme environment. Goyal and Chang (2009) developed an inventory model with optimal ordering policy under the stock dependent demand rate. Chang et al. (2010) discussed an optimal replenishment policy inventory model for non-instantaneous deteriorating items with stock-dependent demand rate. Lots of research work has been done in the area of inventory system with stock- dependent/multivariate demand rate. One may refer to the research work of Dye and Ouyang (2005), Alfares (2007), Singh et al. (2011a, b), Malik and Kumar (2011), Singh and Malik (2011), Sana (2011), Malik and Sharma (11), Sarkar (2012), Sarkar and Sarkar (2013).Sana, S.S., (2010) proposed an inventory model for perishable items under stock-dependent demand rate. Sharma et al. (2013) developed a non- instantaneous deterioration inventory model with inflation and stock-dependent demand. Gupta et al. (2013) proposed an optimal ordering policy for stock-dependent demand inventory model with non-instantaneous deteriorating items. This paper deals with an inventory model with price and stock dependent demand rate under the variable holding cost. The optimal replenishment policy for non-instantaneous deteriorating items with maximum life time is discussed in this proposed model. The optimizing conditions of the existence and uniqueness of the optimal solutions are given. Here investigate the possible effects of a temporary price discount offered by a supplier on a retailer’s replenishment policy for deteriorating items with linear and stock dependent demand rate. 2. NOTATION AND ASSUMPTIONS Proposed inventory model used the following notations and assumptions:  The demand rate is dp t cI bt a t D     ) ( ) ( , Where , a , , c b p d , are positive constants and I(t) is the inventory level at time t.  Shortages are not acceptable and lead time is zero.  t R t    1 1 ) (  is the deterioration rate  R is the maximum life time of an items  A is the ordering cost per order  t h h t h 2 1 ) (   is the inventory holding cost per unit time  C d is the deteriorating cost per unit  C p is the purchasing cost per unit