The inventory models analyzed in this paper explore the problem in which the lead time and ordering cost reductions are inter-dependent in the continuous review inventory model with backorder price discount. The objective is to minimize total related cost by simultaneously optimizing the order quantity, reorder point, lead time and backorder price discount. Moreover, we assume that the mean and variance of the lead time demand are known, but its probability distribution is unknown. We apply a minimax distribution free procedure to find the optimal solution, and three numerical examples are given to illustrate the results.
In classical economic order quantity (EOQ) model dealing with inventory problems, either using deterministic or probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable. Therefore, lead time is not subject to control (see, e.g., Naddor, 1966; Johnson and Montgomery, 1974; Silver and Peterson, 1985). However, this may not be realistic. In many practical situations, lead time can be shortened at an added crashing cost; in other words, it is controllable. By shortening the lead time, we can lower the safety stock, reduce the stockout loss and improve the service level to the customer so as to increase the competitive edge in business.
Recently, several authors have presented various inventory models with lead time reduction. Initially, Liao and Shyu (1991) presented an inventory model in which the lead time is a unique decision variable and the order quantity is predetermined. Ben-Daya and Raouf (1994) extended Liao and Shyu's (1991) model by allowing both the lead time and the order quantity as decision variables. Ouyang et al. (1996) generalized Ben-Daya and Raouf's (1994) model and considered shortages with partial backorders, while Pan and Hsiao (2001) revised Ouyang et al.'s (1996) model to consider the backorder price discount as one of the decision variables.
It is noticed that the above papers Liao and Shyu (1991), Ben-Daya and Raouf (1994), Ouyang et al. (1996), Pan and Hsiao (2001) are all focused on the continuous review inventory model to derive the benefits from lead time reduction, and the ordering cost is treated as a fixed constant. In a recent paper, Ouyang et al. (1999) proposed two continuous review inventory models to study the effects of lead time and ordering cost reductions. We note that the lead time and ordering cost reductions in Ouyang et al. (1999) are assumed to act independently. However, this is only one of the possible cases. In practice, the lead time and ordering cost reductions may be related closely; the reduction of lead time may accompany the reduction of ordering cost, and vice versa. For example, according to Silver and Peterson (1985, p. 150), the implementation of electronic data interchange (EDI) may reduce the lead time and ordering cost simultaneously. Therefore, it is more reasonable to assume that the ordering cost reduction vary according to different lead times.
In the real market, as unsatisfied demands occur, we can often observe that some customers may prefer their demands to be backordered, and some may refuse the backorder case. There is a potential factor that may motivate the customers’ desire for backorders. The factor is an offering of a backorder price discount from a supplier Pan and Hsiao (2001). In general, provided that a supplier could offer a backorder price discount on the stockout item by negotiation to secure more backorders, it may make the customers more willing to wait for the desired items. In other words, the bigger the backorder price discount, the bigger the advantage to the customers, and hence, a larger number of backorder ratio may result. This phenomenon reveals that, as unsatisfied demands occur during the stockout period, how to find an optimal backorder ratio through controlling a backorder price discount from a supplier to minimize the relevant inventory total cost is a decision-making problem worth discussing.
In this paper, we attempt to modify Pan and Hsiao's (2001) model for a minimax distribution free inventory model that includes a controllable backorder price discount and the reduction of lead time accompanies a decrease of ordering cost. For this case, we solve the problem by using the minimax distribution free approach, which was originally proposed by Scarf (1958). Recently, Gallego and Moon (1993) presented a new and very compact proof of the optimality of Scarf's (1958) ordering rule. Also, Hariga and Ben-Daya (1999), Moon and Choi, 1995 and Moon and Choi, 1997, Moon and Silver (2000), Ouyang and Wu (1998), Ouyang and Chang (2002), Ouyang et al. (2004), Silver and Moon (2001) applied this approach to some production/inventory models. Moreover, note that the previous works on distribution free approach and partial lost sales (or backorders) are well documented in Silver et al. (1998). In this study, the objective is to minimize the total related cost by optimizing the order quantity, reorder point, backorder price discount and lead time, simultaneously. Furthermore, the effects of parameters are also included and three illustrative numerical examples are given.
The purpose of this paper is to extend the Pan and Hsiao's (2001) model by simultaneously optimizing the order quantity, reorder point, lead time and backorder price, and the reduction of lead time accompanies a decrease of ordering cost. In this paper, we only assume that the first and second moments of the lead time demand are known and finite, but its probability distribution is unknown. And apply the minimax distribution free procedure to find the optimal solution. The results of the numerical examples indicating that when the reduction of lead time accompanies a decrease of ordering cost and larger savings of total expected annual cost can be realized.
In future research on this problem, it would be interesting to deal with an arrival order lot including some defective items. Another possible extension of this work may be conducted by considering that the functional relationships of lead time and ordering cost reductions are other functional forms.
