بهینه سازی ازدحام ذرات برای حل مشکل تعیین اندازه دسته تولید و قیمت گذاری مشترک با تقاضای در حال نوسان و تامین مالی اعتباری تجارت

Pricing is a major strategy for a retailer to obtain its maximum profit. Furthermore, under most market behaviors, one can easily find that a vendor provides a credit period (for example 30 days) for buyers to stimulate the demand, boost market share or decrease inventories of certain items. Therefore, in this paper, we establish a deterministic economic order quantity model for a retailer to determine its optimal selling price, replenishment number and replenishment schedule with fluctuating demand under two levels of trade credit policy. A particle swarm optimization is coded and used to solve the mixed-integer nonlinear programming problem by employing the properties derived in this paper. Some numerical examples are used to illustrate the features of the proposed model.

In the inventory models developed, it is often assumed that payment will be made to the vendor for the goods immediately after receiving the consignment. Because the permissible delay in payments can provide economic sense for vendors, it is possible for a vendor to allow a certain credit period for buyers to stimulate the demand to maximize the vendors-owned benefits and advantage. Recently, several researchers have developed analytical inventory models with consideration of permissible delay in payments. Goyal (1985) first studied an EOQ model under the conditions of permissible delay in payments. Chung (1989) presented the discounted cash flows (DCF) approach for the analysis of the optimal inventory policy in the presence of the trade credit. Later, Shinn, Hwang, and Sung (1996) extended Goyal’s (1985) model and considered quantity discounts for freight cost. Chung (1997) presented a simple procedure to determine the optimal replenishment cycle to simplify the solution procedure described in Goyal (1985). Teng (2002) provided an alternative conclusion from Goyal (1985), and mathematically proved that it makes economic sense for a well-established buyer to order less quantity and take the benefits of the permissible delay more frequently. Huang (2003) developed an EOQ model in which a supplier offers a retailer the permissible delay period M, and the retailer in turn provides the trade credit period N (with N ⩽ M) to his/her customers. He then obtained the closed-form optimal solution for the problem. Jaber and Osman (2006) proposed a two-level supply chain model with delay in payments to coordinate the players’ orders and minimize the supply chain costs. Jaber (2007) then incorporated the concept of entropy cost into the EOQ problem with permissible delay in payments. In real situations, “time” is a significant key concept and plays an important role in inventory models. Certain types of commodities deteriorate in the course of time and hence are unstable. As a result, while determining the optimal inventory policy for product of that type, the loss due to deterioration cannot be ignored. To accommodate more practical features of the real inventory systems, Aggarwal and Jaggi, 1995 and Hwang and Shinn, 1997 extended Goyal’s (1985) model to consider the deterministic inventory model with a constant deterioration rate. Since the occurrence of shortages in inventory is a very nature phenomenon in real situations, Jamal et al., 1997, Sarker et al., 2000, Chang and Dye, 2000 and Chang et al., 2002 extended Aggarwal and Jaggi’s (1995) model to allow for shortages and makes it more applicable in real world. Chang, Ouyang, and Teng (2003) then extended Teng’s (2002) model, and established an EOQ model for deteriorating items in which the supplier provides a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. By considering the difference between unit selling price and unit purchasing cost, Ouyang, Chuang, and Chuang (2004) developed an EOQ model with noninstantaneous receipt under conditions of permissible delay in payments. Recently, Taso and Sheen (2007) developed a finite time horizon inventory model for deteriorating items to determine the most suitable retail price and appropriate replenishment cycle time with fluctuating unit purchasing cost and trade credit. Chang, Wu, and Chen (2009) established an inventory model to determine the optimal payment time, replenishment cycle and order quantity under inflation. However, all the above models make an implicit assumption that the demand rate is constant over an infinite planning horizon. This assumption is only valid during the maturity phase of a product life cycle. During the introduction and growth phase of a product life cycle, the firms face increasing demand with little competition. Some researchers Resh et al., 1976, Donaldson, 1977, Dave and Patel, 1981, Sachan, 1984, Goswami and Chaudhuri, 1991, Goyal et al., 1992 and Chakrabarty et al., 1998 suggest that the demand rate can be well approximated by a linear form. A linear trend demand implies an uniform change in the demand rate of the product per unit time. This is a fairly unrealistic phenomenon and it seldom occurs in the real market. One can usually observe in the electronic market that the sales of items increase rapidly during the introduction and growth phase of the life cycle because there are few competitors in market. Recently, Yang, Teng, and Chern (2002) established an optimal replenishment policy for power-form demand rate and proposed a simple and computationally efficient method in a forward recursive manner to find the optimal replenishment strategy. Khanra and Chaudhuri (2003) advise that the demand rate should be represented by a continuous quadratic function of time in the growth stage of a product life cycle. They also provide a heuristic algorithm to solve the problem when the planning horizon is finite. To achieve maximum profit, Chen and Chen (2004) presented an inventory model for a deteriorating item with a multivariate demand function of price and time. Their model is solved with dynamic programming techniques by adjusting the selling price upward or downward periodically. Chen et al., 2007a and Chen et al., 2007b dealt with the inventory model under the demand function following the product-life-cycle shape over a fixed time horizon. Skouri and Konstantaras (2009) studied an order level inventory model when the demand is described by a three successive time periods that classified time dependent ramp-type function. In this paper, to obtain robust and general results, we will extend the constant demand to a generalized time varying demand, which is suitable not only for the growth stage but also for the maturity stage of a product life cycle. In addition, we assume that supplier offers retailer a trade credit period M, and retailer in turn provides a trade credit period N (with N ⩽ M) to his/her customers. The lot sizing problem is then to find the optimal pricing and replenishment strategy that will maximize the present value of total profit. A traditional particle swarm optimization is coded and used to solve the mixed-integer nonlinear programming problem by employing the properties derived in this paper. Finally, numerical examples will be used to illustrate the results.

In this paper, we consider a retailer’s optimal pricing and lot-sizing problem for deteriorating items with fluctuating demand under trade credit financing. We have successfully formulated the problem as a mixed-integer nonlinear programming model and proposed a solution algorithm associated with it. In contrast to the classical fixed selling price policy under trade credit, the pricing policy in this model provides more flexibility by changing price upward or downward. We can also use similar derivations as in Appendix C to prove that View the MathML source∂2TP(p|n,t)/∂pi2>0 where p = {p1, p2, … , pn} and pi denotes the selling price per unit in the ith replenishment cycle. Hence, the model in this paper not only can be easily extended the single price policy to change selling prices upward or downward periodically, but is ideal for managers to design marketing strategies to stay ahead of the challenges their products are likely to face. Furthermore, the PSO algorithm is selected in this paper because of its robustness, simplicity and ease of implementation. The computational results indicated that the PSO algorithm offers acceptable efficiency and accurate search capability. The proposed model can be extended in several ways. For instance, we may generalize the model to allow for shortages, quantity discounts and capacity constraint of owned warehouse. Also, we could extend the deterministic demand function to stochastic demand patterns. Finally, we could extend the sales environment to an advance booking system.