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This paper presents an inventory model for deteriorating items in which shortages are allowed. It is assumed that the production rate is proportional to the demand rate and greater than demand rate. The inventory model is developed by considering four different circumstances. The optimal of the problem is obtained with the help of Mathematica 7 software. Numerical examples are given to illustrate the model for different parameters. Sensitivity analysis of the model has been developed to examine the effect of changes in the values of the different parameters for optimal inventory policy. Truncated Taylor’s series is used for finding closed form optimal solution.

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This paper deals with a deterministic inventory model for deteriorating items under the condition of permissible delay in payments with constant demand rate is a function of time which differs from before and after deterioration for a single item. Shortages are allowed and completely backlogged which is a function of time. Under these assumptions, this paper develops a retailer\'s model for obtaining an optimal cycle length and ordering quantity in deteriorating items of an inventory model. Thus, our objective is retailer\'s cost minimization problem to nd an optimal replenishment policy under various parameters. The convexity of the objective function is derived and the numerical examples are provided to support the proposed model. Sensitivity analysis of the optimal solution with respect to major parameters of the model is included and the implications are discussed.