Optimal lot size for EPQ inventory model for items of different qualities.

Author:Kassar, A.N. El-
Position:Economic production quantity

    The classic economic production quantity (EPQ) model assumes that 100% of the items produced by a manufacturing process during a production run are of perfect quality. Due to deterioration or other factors, the production process may shift and produce some items of a lesser quality. In this paper, we extend the classical economic production quantity model to the case where the item produced is of different qualities. During production, items are produced at a certain rate and screened. Each item is classified as type A, for perfect quality, or type B, otherwise. For simplicity, we assume that the percentage of type A items produced is a constant p. Therefore, the production rates for type A and type B items are constant. The demands for both types of items are continuous during the cycle. Throughout the production period the finished inventory of both types of items will accumulate, and at the end of the production period the accumulated inventory will be consumed at the demand rates until the end of the inventory period. It will be assumed that the demands for both types of items stocked by the system are known with certainty and are constant over time. However, in the real world, demands can almost never be predicted with certainty, instead they must be described in probabilistic terms. The differences between the production and demand rates establish the replenishment rates of the model. Furthermore, in order for the accumulation of the goods to occur in inventory, and for the system to work, the production rates must be greater than the demand rates.

    In this paper, we present inventory models that account for the production of such an item. The optimal operating policy that maximizes the total profit per unit time for the EPQ model under the effect of imperfect quality items is derived. The different costs that are incurred in inventory models and the revenues are:

    * The setup cost of each production run which represents the fixed charges of machine preparation, inspection, maintenance, and supervision that are incurred at the beginning of each production run.

    * Production cost that includes the cost of operating supervisors and laborers, plus the cost of inspection and rework of the defected items.

    * The holding cost of the product which covers storage, handling, insurance, taxes, rent (if applicable), maintenance, breakage and pilferage at the storage site plus cost of warehouse operation.

    * The shortage cost which is the cost of running out of stock.

    * Selling price per unit of each type of items.

    * Discounted selling price of unsold items at the end of period.

    Numerous papers tackling the inventory problem have appeared in the literature. Considerable research, with many different assumptions, has already been done on inventory models. The effect of a single price change on the classical instantaneous replenishment economic order quantity (EOQ) models have been reported by Goyal (1980), and Hannan and Smith (1981). Infinite horizon EOQ models with continuous price changes have been studied by Goyal (1975) and the effect of a single price change on these models have been examined by Naddor (1966), and Brown (1967, 1982). Salameh and Abdul-Malak (1994) studied an exact mathematical approach to determine the optimum order quantity under the effect of time inflation. Salameh et al. (1999), investigated the effect of time-discounting on the EOQ model. The effect of time-discounting on the EPQ model has been examined by Salameh and El-Kassar (2003). Salameh and El-Kassar (1999), studied the optimality of the single period inventory model with credit facility. Salameh et al. (2003), considered the continuous review inventory model with delay in payments. Hayek and Salameh (2001) studied the production lot sizing with the reworking of imperfect quality items produced. Chiu (2003) focused on determining the optimal lot size for the finite production model with random defective rate, the rework process, and backlogging. More recently, Chiu et al. (2006) studied numerical method for determination of reworking or scraping the defective items in a finite production rate model. Ozdemir (2007) examined an EOQ model for that each ordered lot contains some defective items and shortages backordered. El-Kassar and Salameh (2007) introduced an EPQ model that accounts for the cost of raw material.

    In the next section, we present a mathematical model that describes the EPQ model where the item produced is of different qualities and the shortage cost is ignored. This model will answer question of how much of produce in order to maximize the...

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