International Journal of Physical, Chemical & Mathematical Sciences, Vol. 3; No. 1: ISSN: 2278-683X (Jan –June 2014) www.gtia.co.in 32 An Inventory Model with Exponentially Decreasing Demand and Linearly Increasing Deterioration Vinod Kumar Verma * , B. B. Verma ** *Research Scholar, Mewar University, Chittorgarh, Rajasthan, India **Department of Mathematics, A K G Engineering College, Ghaziabad, India Email: vinodverma1965@gmail.com Abstract: In this paper, we have developed an inventory model with exponentially decreasing demand and linearly increasing deterioration for fixed and finite planning horizon considering shortages which is backlogged. This work is the extension of earlier works for constant deterioration and variable deterioration. Keywords: Exponentially Decreasing Demand, Shortages, Production Cycle, Inventory Holding Cost, Stochastic Nature of Demand, Constant Deterioration, Variable Deterioration. Accepted On: 25.06.2014 1. Introduction Order level inventory models for items deteriorating with time have engaged attention of researchers in recent years. Covert and Philip [1] formulated an inventory model with a variable rate of deterioration with no shortages. Park, K. S. [2] developed Inventory models with partial backorders. Maitreyee Deb and Chaudhary K.S. [3] developed an EOQ model for items with finite rate of production and variable rate of deterioration. Wee, H. M. [4] discussed Joint pricing and replenishment policy for deteriorating inventory with declining market. Gupta, P. N. and Aggarwal, R. N. [5] discussed an order level inventory model with time dependent deterioration. Optimum inventory management for exponentially increasing demand with deterioration was discussed by Aggarwal S. P. and Jain V. [6]. Production inventory model for deteriorating items with an exponentially decling demand over a fix time horizon is given by Sharma and Kumar [7]. They considered uniform deteriorating rate and production rate as a linear combination of on hand inventory and demand at any instant Demand is an exponential function of time, shortages are allowed and excess demand is backlogged as well. They obtained approximation expressions for production scheduling period maximum inventory level, unfilled order backlogging and total average cost. As the particular cases the model reduces to variable production and constant deterioration rate as well as EOQ model by choosing appropriate values of parameter. Dye, C. Y., Ouyang, L. Y. and Hsieh, T. P. [8] developed Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate. Chung-Yuan Dye, Liang-Yuh Ouyang, Tsu-Pang Hsieh [9] discussed an optimal production-inventory model for deteriorating items with multiple-market demand and optimal control of a stochastic production-inventory model with deteriorating items was developed by Ahmad M. Alshamrani [10]. 2. Assumptions and Notations The Inventory Model is developed on the basis of the following assumptions and notations: 1. T is the fixed length of each production cycle. 2. C 1 is the inventory holding cost per unit per unit time. 3. C 2 is the shortage cost per unit per unit time. 4. C 3 is the constant set up cost for each new cycle or cost of deteriorated units. 5. S ( > 0) is initial inventory after fulfilling back order.