A fuzzy framework for coordinating pricing and inventory policies for deteriorating items under retailer partial trade credit financing

In this paper, we proposed a generalized economic order quantity (EOQ) – based inventory model using a trade credit policy in a fuzzy sense. The trade credit policy adopted here is a two-level trade credit policy in which the supplier offers the retailer a permissible delay period M, and the retailer, in turn, partially provides customers a permissible delay period N. This study considers fuzzy EOQ model to allow for: (1) selling price dependent demand rate which is imprecise in nature, (2) a profit maximization objective and (3) an imprecise holding cost, ordering cost, purchasing cost, interest earned and interest charged rate. Besides, the cases N ? M and N ? M are explored thoroughly. The objective function for the retailer in fuzzy sense is defuzzified using Modified Graded Mean Integration Representation Method. For the defuzzified objective function sufficient conditions for the existence and uniqueness of the optimal solution are provided. An efficient algorithm is designed to determine the optimal pricing and inventory policies for the retailer. Finally, numerical examples are presented to illustrate the proposed model and the effect of key parameters on optimal solution is examined.     

Retailer's optimal policy under inflation in fuzzy environment with trade credit

Trade credit plays an important role in financing many industries. In the classical inventory model it is assumed that the buyer must pay for the items as soon as the items are received. In this problem, it is considered that the retailer can pay the supplier either at the end of the credit period or later pay interest on the unpaid amount for the overdue period. Here, the retailer's inventory model for the optimal cycle time and payment time for a retailer is developed. The effects of the inflation rate, deterioration rate and delay in payment have been discussed. The whole study is performed in a fuzzy environment by taking the opportunity cost, interest earned and interest paid rate as a triangular fuzzy number. Fuzzy profit functions, which involve fuzzy arithmetic operation, are defined using the function principle. We use the signed distance method to defuzzify the fuzzy profit function. Moreover, numerical and sensitivity analysis is performed to validate the proposed model.     

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

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.     

The optimal retailer's ordering policies with trade credit financing and limited storage capacity in the supply chain system

The traditional economic order quantity model assumes that the retailer's storage capacity is unlimited. However, as we all know, the capacity of any warehouse is limited. In practice, there usually exist various factors that induce the decision-maker of the inventory system to order more items than can be held in his/her own warehouse. Therefore, for the decision-maker, it is very practical to determine whether or not to rent other warehouses. In this article, we try to incorporate two levels of trade credit and two separate warehouses (own warehouse and rented warehouse) to establish a new inventory model to help the decision-maker to make the decision. Four theorems are provided to determine the optimal cycle time to generalise some existing articles. Finally, the sensitivity analysis is executed to investigate the effects of the various parameters on ordering policies and annual costs of the inventory system.     

贸易信用下考虑零售商违约风险的供应链协调问题研究

本文研究延迟支付和提前支付同时存在下的供应链协调问题。考虑由一个供应商和一个零售商构成的二级供应链,市场需求随机,延迟支付下零售商会增加订购数量,但由于其违约风险的存在,供应商会降低其交付量,最终导致供应商交付量小于零售商订购量,降低供应链收益。要求零售商提前支付部分货款则可以有效避免零售商的违约问题,通过建立延迟支付和提前支付同时存在下的零售商收益模型和供应商的收益模型,给出了零售商的最优订购量决策和供应商的最优交付量决策,通过调整延迟支付期限和提前支付比例,使得二者相等,并等于供应链整体收益最大化下的生产数量,实现供应链协调。最后通过数值算例分析了相关参数的敏感性等问题。     

Retailer's optimal ordering policies with trade credit financing

In this article, we extended Goyal's model to develop an Economic Order Quantity (EOQ) model in which the supplier offers the retailer the permissible delay period M, and the retailer in turn provides the trade credit period N (with N?≤?M) to his/her customers. In addition, we assume that (1) the retailer's selling price per unit is necessarily higher than its unit cost, and (2) the interest rate charged by a supplier or a bank is not necessarily higher than the retailer's investment return rate. We then establish an appropriate EOQ model with trade credit financing, and provide an easy-to-use closed-form solution to the problem. Furthermore, we find it is possible that a well-established buyer may order a lower quantity and take the benefit of the permissible delay more frequently, which contradicts to the result by the previous researchers. Finally, we perform some sensitivity analyses to illustrate the theoretical results and obtain some managerial results.     

考虑信用融资的期权和实物产品最优组合采购策略

分析由零售商和银行组成的供应链系统, 在零售商面临资金约束的条件下, 通过引入期权采购和信用融资的组合方式, 实现企业的资金风险降低和运作效率提升. 通过建立博弈模型分析得出零售商的最优实物产品采购量和最优期权采购量的显性表达式, 证明银行的最优信用借款利率的存在性和唯一性条件. 通过比较存在资金约束情形下的其他运作方式, 发现信用融资是最有效的决策方式, 可以提升零售商的绩效, 最大可能实现全局最优.

     

Two-echelon trade credit financing for perishable items in a supply chain when demand depends on both selling price and credit period

A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the trade credit scenario such as permissible delay in payments. In real life business, we observe that the demand is a function of both the selling price and credit period rather than the constant demand. Incorporating this demand function to the retailer of a supply chain, we develop an EPQ – based model for perishable items under two-echelon trade financing. The purpose of this paper is to maximize the profit by determining the optimal selling price, credit period and replenishment time. It is shown that the model developed by Jaggi et al. [Jaggi, J. K., Goyal, S. K., & Goel, S. K., 2008. Retailer’s optimal replenishment decisions with creditlinked demand under permissible delay in payments. European Journal of Operational Research, 190, 130–135] can be treated as a special case of this paper. Finally, through numerical examples, sensitivity analysis shows the influence of key model parameters.     

The effect of inflation‐induced demand and trade credit on ordering policy of exponentially deteriorating and imperfect quality items

This paper develops an inventory model for exponentially deteriorating and imperfect quality items when a trade credit is offered by the supplier. As it has been seen that the changing rates of inflation do not just have an effect on the ordered quantity but they also stimulate the demand for products, so this paper considers the combined effect of inflation, deterioration, imperfect quality and ‘’ trade credit policy on the economic ordering policy using a discounted cash flow (DCF) approach. In the study, mathematical models are derived under two different circumstances, i.e., Case I: When the discount offered by the supplier with a condition of making the payment by a specified period, M₁ is accepted; and Case II: Discount is not taken and instead the payment is made by the end of the full credit period M (M > M₁). Finally, a numerical example demonstrating the applicability of the proposed model has been included.     

How to make the replenishment and payment strategy under flexible two-part trade credit

In the existing inventory models concerning the two-part trade credit, a common assumption is that the retailer either pays for all the ordered items within a short permissible delay period and receives a cash discount or pays for all the ordered items within a long permissible delay period at the regular price. In this paper, this unrealistic assumption is relaxed. We assume that the retailer may pay any fraction of the purchase cost within the short permissible delay period and receives a cash discount and then the rest is paid within the long permissible delay period. A decision model is proposed for a retailer to determine the optimal ordering policy and payment plan. The closed-form optimal solution to the model is developed and analyzed. Numerical studies show that a retailer can obtain more benefits from the proposed payment plan than from the extreme payment plan in the existing literature.     

Optimal policies for a finite-horizon batching inventory model

This paper is concerned with finding an optimal inventory policy for the integrated replenishment-production batching model of Omar and Smith (2002). Here, a company produces a single finished product which requires a single raw material and the objective is to minimise the total inventory costs over a finite planning horizon. Earlier work in the literature considered models with linear demand rate function of the finished product. This work proposes a general methodology for finding an optimal inventory policy for general demand rate functions. The proposed methodology is adapted from the recent work of Benkherouf and Gilding (2009).     

An optimization approach for joint pricing and ordering problem in an integrated inventory system with order-size dependent trade credit

Under a business trading environment, it is common for the trade credit to depend on the order size. Therefore, it is important to discuss the single-supplier and single-buyer supply chain problem which includes order-size dependent trade credit. In this study, an integrated inventory model with a price sensitive demand rate, determining jointly economic lot size of the buyer’s ordering and the supplier’s production batch, are developed to maximize the total profit per unit time. An efficient algorithm is provided to obtain the optimal solution, and then numerical examples are presented to illustrate the theoretical results. Finally, the comparison between whether an optimal solution is jointly or independently determined is also provided.     

Optimal pricing and lot-sizing for perishable inventory with price and time dependent ramp-type demand

Product perishability is an important aspect of inventory control. To minimise the effect of deterioration, retailers in supermarkets, departmental store managers, etc. always want higher inventory depletion rate. In this article, we propose a dynamic pre- and post-deterioration cumulative discount policy to enhance inventory depletion rate resulting low volume of deterioration cost, holding cost and hence higher profit. It is assumed that demand is a price and time dependent ramp-type function and the product starts to deteriorate after certain amount of time. Unlike the conventional inventory models with pricing strategies, which are restricted to a fixed number of price changes and to a fixed cycle length, we allow the number of price changes before as well as after the start of deterioration and the replenishment cycle length to be the decision variables. Before start of deterioration, discounts on unit selling price are provided cumulatively in successive pricing cycles. After the start of deterioration, discounts on reduced unit selling price are also provided in a cumulative way. A mathematical model is developed and the existence of the optimal solution is verified. A numerical example is presented, which indicates that under the cumulative effect of price discounting, dynamic pricing policy outperforms static pricing strategy. Sensitivity analysis of the model is carried out.     

Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives

Within the economic production quantity (EPQ) framework, the main purpose of this article is to deal with Chung and Huang's model (K.J. Chung, and Y.F. Huang,“The optimal cycle time for EPQ inventory model under permissible delay in payments,” Int. J. Prod. Econ., 84, pp. 307–318, 2003) and extend Chung and Huang's model (2003) by considering the unit selling price higher than the unit purchasing cost using the algebraic method to determine the optimal inventory replenishment policy for the retailer under trade credit. This article provides this algebraic approach which could be used easily to introduce the basic inventory theories to younger students who lack the knowledge of calculus. In addition, we develop an easy-to-use procedure to find the optimal inventory replenishment policy for the retailer in the extended model developed in this article. Finally, numerical examples are given to illustrate the result obtained in our extended model.     

A single-period inventory and payment model with partial trade credit

In this paper, we consider the ordering and payment issues for a retailer facing stochastic demand. We assume that the retailer can enjoy the partial trade credit from his supplier and borrow money from bank as well if needed, and he can also earn return by investing his superfluous on-hand cash (if any). The retailer’s objective is to maximize the expected cash level at the end of the selling period. We formulate the model of this problem by taking initial inventory and capital levels as the two-dimensional state. First, given the exogenous fraction of immediate payment, we show that unlike the critical fractile solution the retailer’s optimal ordering strategy is a two-threshold policy, which is independent of the retailer’s initial inventory level and capital level. Second, we consider an extensive model where the fraction of immediate payment is decided by the retailer. We employ the sequential optimization procedure to solve the extensive problem, and present the structure of the retailer’s optimal policies under different partial-trade-credit penalty rates. Numerical experiments show that if the fraction of immediate payment is exogenous, both partial trade credit and loan opportunity are detrimental to the capital-constrained retailer in many cases, although they can stimulate the retailer to order more.     

The EPQ model under conditions of two levels of trade credit and limited storage capacity in supply chain management

An inventory problem involves a lot of factors influencing inventory decisions. To understand it, the traditional economic production quantity (EPQ) model plays rather important role for inventory analysis. Although the traditional EPQ models are still widely used in industry, practitioners frequently question validities of assumptions of these models such that their use encounters challenges and difficulties. So, this article tries to present a new inventory model by considering two levels of trade credit, finite replenishment rate and limited storage capacity together to relax the basic assumptions of the traditional EPQ model to improve the environment of the use of it. Keeping in mind cost-minimisation strategy, four easy-to-use theorems are developed to characterise the optimal solution. Finally, the sensitivity analyses are executed to investigate the effects of the various parameters on ordering policies and the annual total relevant costs of the inventory system.     

Three level partial trade credit with promotional cost sharing

In this paper, an economic ordered quantity (EOQ) model, specifically for a newly launched product has been developed with selling price, customers’ credit period and customers’ credit amount induced demand under three levels of partial trade credit policy, where a supplier, a wholesaler and a retailer offer some credit periods on some fraction of the total purchased amount to the wholesaler, the retailer and the customer respectively. Also, here it is assumed that the retailer obtained a quantity discount from the wholesaler on purchased units above a certain level. In addition, the wholesaler and the retailer both enjoy freight charge discount according to the ordered quantity. Retailer introduces a promotional cost to increase the base demand of the item. Objective of this investigation is to maximize the profit of the retailer as well as the wholesaler. It is established that if the wholesaler contributes some portion of the promotional cost then individual profits as well as the joint profit increases. Due to the uncertainty and vagueness of different inventory costs, the proposed model is also discussed in fuzzy and rough environments. Combining the features of particle swarm optimization (PSO) and simulated annealing (SA) a hybrid algorithm named Particle Swarm-Simulated Annealing (PSSA) is developed to find the most appropriate strategies for the proposed model. Efficiency of this algorithm is tested and compared with PSO and genetic algorithm (GA) for a set of benchmark test problems. The model is illustrated with numerical examples and some managerial insights are outlined.     

基于行为模式的电子交易过程中信任评估研究*

在开放动态的网络环境中,交易者的行为信任问题已经成为电子商务发展的重要障碍。提出了以交易过程中呈现出的行为模式为证据的信任评估方法。该方法通过构建行为模式匹配级和可信属性为节点的贝叶斯网可信评估模型,计算推导出电子交易过程中消费可信指数和销售可信指数的等级。在一个实际的电子交易平台上验证了该方法的有效性。     

基于行为模式的电子交易过程中信任评估研究*

在开放动态的网络环境中,交易者的行为信任问题已经成为电子商务发展的重要障碍。提出了以交易过程中呈现出的行为模式为证据的信任评估方法,该方法通过构建行为模式匹配级和可信属性为节点的贝叶斯网可信评估模型,计算推导出电子交易过程中消费可信指数和销售可信指数的等级。在一个实际的电子交易平台上验证了该方法的有效性。     

Inventory policy of a deteriorating item with variable demand under trade credit period

In this paper, an inventory model of a deteriorating item with stock and selling price dependent demand under two-level credit period has been developed. Here, the retailer enjoys a price discount if he pays normal purchase cost on or before the first level of credit period, or an interest is charged for the delay of payments. In return, retailer also offers a fixed credit period to his customers to boost the demand. In this regard, the authors develop an EOQ model incorporating the effect of inflation and time value of money over all the costs. Keeping the business of seasonal products in mind, it is assumed that planning horizon of business is random and follows a normal distribution with a known mean and standard deviation. The model is formulated as retailer’s profit maximization problem for both crisp and fuzzy inventory costs and solved using a modified Genetic Algorithm (MGA). This algorithm is developed following fuzzy age based selection process for crossover and gradually reducing mutation parameter. For different values of MGA parameters, optimum results are obtained. Numerical experiments are performed to illustrate the model.