A particle swarm optimization for solving joint pricing and lot-sizing problem with fluctuating demand and trade credit financing

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Abstract

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.

Research highlights

► Studied a deteriorating inventory system with fluctuating demand and trade credit financing. ► PSO is coded and used to solve the problem by employing the properties derived from this paper. ► The computational results indicated that the PSO algorithm offers acceptable efficiency and accurate search capability.

Introduction

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, 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, 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, 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, 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.

Section snippets

Assumptions and notations

The mathematical model in this paper is developed on the basis of the following assumptions and notations.

Model formulation

As shown in Fig. 1, the depletion of the inventory occurs due to the combined effects of the demand and deterioration in the interval [ti−1, ti]. Hence, the variation of inventory level, I(t), with respect to time can be described by the following differential equation:dI(t)dt=-θI(t)-α(p)f(t),ti-1t<ti,with boundary condition I(ti) = 0, i = 1, 2,  , n. The solution of (1) can be represented byI(t)=e-θtttiα(p)f(t)eθudu,ti-1t<ti.Then, applying (2), the present value of the holding cost in the ith

The background of particle swarm optimization

The PSO is an algorithm for finding optimal regions of complex search spaces through the interaction of individuals in a population of particles. It was proposed by Eberhart and Kennedy, 1995, Kennedy and Eberhart, 1995 and has been widely used in finding solutions for optimization problems. The PSO algorithm is inspired by social behavior of bird flocking or fish schooling. In PSO, the potential solutions, called particles, fly through the problem space by following the current optimum

Numerical examples

To illustrate the results, let us apply the proposed algorithms to solve the following numerical examples. In Example 1, the demand function follows the shape of a product life cycle. In Example 2, we have a quadratic increasing demand and in Example 3 we have a exponential decreasing demand. Algorithm 1, Algorithm 2 are implemented on a personal computer with Intel Core 2 Duo under Mac OS X 10.5.6 operating system using Mathematica version 7.

Example 1

In this example, we consider the demand function for

Concluding remarks

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

Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their valuable and constructive comments, which have led to a significant improvement in the manuscript. This research was partially supported by the National Science Council of the Republic of China under NSC-97-2221-E-366-006-MY2.

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