Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy

The main purpose of this paper is to investigate the optimal retailer’s replenishment decisions under two levels of trade credit policy within the economic production quantity (EPQ) framework. We assume that the supplier would offer the retailer a delay period and the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer’s replenishment model under the replenishment rate is finite. Furthermore, we assume that the retailer’s trade credit period offered by supplier M is not shorter than the customer’s trade credit period offered by retailer N (M ? N). Since the retailer cannot earn any interest in this situation, M < N.     

Determination of the optimal ordering policy for the retailer with limited capitals when a supplier offers 2 levels of trade credit

In this article, we consider and investigate the cases when the retailer's capitals are restricted and when the supplier offers another kind of 2‐level trade credit. This means that the supplier offers 2‐level trade credit for the retailer to settle the account and the retailer's capitals are restricted, so the retailer decides to pay off the unpaid balance as follows: Firstly, the retailer decides to pay off the unpaid balance at the end of the first credit period if the retailer can pay off all accounts and, in addition, the retailer can use the sales revenue to earn interest throughout the replenishment cycle time. Secondly, the retailer decides to pay off all accounts either after the end of the first credit period, but before the second credit period, or after the second credit period if the retailer cannot pay off the unpaid balance at the end of the first credit period. Additionally, the delay will incur interest charges on the unpaid and overdue balance due to the difference between the interest earned and the interest charged. Consequently, the main purpose of this article is to characterize the optimal solution processes and (in accordance with the functional behavior of the cost function) to search for the optimal replenishment cycle time. Finally, numerical examples are given to illustrate the theoretical results which are proven in this article by means of mathematical solution procedures.     

Two-phase pricing and inventory management for deteriorating and fashion goods under trade credit

This study models a finite horizon inventory problem for deteriorating and fashion goods under trade credit and partial backlogging conditions. Demand may vary with price or time. The supplier can extend credit to the retailer. As a result, the retailer does not have to pay for goods immediately upon acquiring them, and can instead earn interest on the retail price of the goods between the time they are sold and the end of the credit period. The proposed model considers two-phase pricing and inventory decisions. In other words, it determines both the optimal prices and the lengths of the in-stock and stock-out period. This paper is the first to consider different price decisions for in-stock and stock-out periods under trade credit. We develop an algorithm to determine the optimal pricing and replenishment strategy while still maximizing the total profit. Further, this study shows that the proposed two-phase pricing strategy is superior to a one-phase pricing strategy in terms of profit maximization. Computational analysis illustrates the solution procedures and the impacts of the related parameters on decisions and profits. The results of this study can serve as references for business managers or administrators.     

Two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under trade credit

In real life situation, it is observed that demand of an item depends on the length of the credit period offered by the retailer to his customers which has a positive impact on demand of an item. But the impact of credit period on demand has received a very little attention by researchers. Furthermore, by allowing shortages as backlogging, the impact on the cost from the decay of the products can be balanced out. A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the permissible delay in payments. Recently, Jaggi et al. (Eur J Oper Res 190:130–135, 2008) have investigated the impact of credit linked demand on the retailer’s optimal replenishment policy. The objective of this study is to extend Jaggi et al. (Eur J Oper Res 190:130–135, 2008) model by incorporating deterioration and backlogging. That is, we formulate a two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under two-level trade credit financing and determine the retailer’s optimal replenishment policy when both the supplier as well as the retailer offers the credit period to stimulate customer demand. Furthermore, we establish some useful theorems to characterize the optimal solution and provide an easy and useful computational algorithm with the help of computer code using the software Matlab 7.0 to determine the optimal shortage point, cycle length, ordering quantity and credit period. A numerical example is included to illustrate the solution procedure for the mathematical model developed. Finally, we implement sensitivity analysis of the optimal solution with respect to the major parameters of the system and obtain some important managerial insights.     

Determination of the optimal trade credit policy: a supplier-Stackelberg model

This paper considers a two-echelon supply chain, where one supplier sells through a retailer a product with a stable market demand. We focus on how the supplier induces the retailer through trade credit to order more to reduce his/her own inventory-related cost. Under a ‘supplier-Stackelberg’ setting, we provide the supplier with the method of determining two trade credit scenarios: unconditional and conditional trade credit. We show that the unconditional trade credit scenario is always beneficial to the retailer but harmful to the supplier in most situations, while the conditional trade credit scenario is always beneficial to both parties. In addition, we specify the conditions under which the provision of unconditional trade credit is beneficial to the supplier. The three insights obtained in this paper are the following: (i) When the retailer’s per-unit opportunity cost is less than his/her per-unit opportunity gain, unconditional trade credit can induce the retailer to order less instead of more. (ii) If the supplier offers the retailer unconditional trade credit, the length of trade credit offered will have an upper bound. (iii) A well-designed conditional trade credit policy can realize a win-win outcome but also enables the supplier to occupy all the savings in the channel's cost incurred by trade credit, but any unconditional trade credit policy does not.     

Retailer’s optimal replenishment and payment policies in the EPQ model under cash discount and two-level trade credit policy

The main purpose of this paper is to investigate the retailer’s optimal cycle time and optimal payment time under the supplier’s cash discount and trade credit policy within the economic production quantity (EPQ) framework. In this paper, we assume that the retailer will provide a full trade credit to his/her good credit customers and request his/her bad credit customers pay for the items as soon as receiving them. Under this assumption, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal payment time under the replenishment rate is finite. Then, an algorithm is established to obtain the optimal strategy. Finally, numerical examples are given to illustrate the theoretical results and obtain some managerial phenomena.     

An incentive-compatible solution for trade credit term incorporating default risk

This paper studies the optimal trade credit term decision in an extended economic ordering quantity (EOQ) framework that incorporates a default risk component. A principal-agent bilevel programming model with costs minimization objectives is set up to derive the incentive-compatible credit term. The supplier determines the credit term as the leader in the first level programming, by balancing her/his financing capacity with the retailer’s default risk, order behavior and cost shifting. At the second level, the retailer makes decisions on ordering and payment time by reacting on the term offered by the supplier. A first order condition solution procedure is derived for the bilevel programming when credit term is confined within the practically feasible interval. Two key results are obtained – the condition to derive incentive-compatible credit term, and an equation system to derive threshold default risk criterion filtering retailers suitable for credit granting. Numerical experiments show that the capital cost of the supplier is the most important factor determining the credit term. Default risk acts like a filtering criterion for selecting retailers suitable for credit granting. Empirical evidence supporting our theoretical considerations is obtained by estimating three panel econometric models, using a dataset from China’s listed companies.     

An uncooperative order model for items with trade credit,inventory-dependent demand and limited displayed-shelf space

This paper considers a two-echelon supply chain where a supplier sells a single product through a retailer, who faces an inventory-dependent demand. The supplier hopes to incentive the retailer to order more items by offering trade credit. The retailer places the ordered items on the display shelf (DS) with limited space and stocks the remaining items (if any) that exceed the shelf capacity in his/her backroom/warehouse (BW). From the supplier’s perspective, we focus mainly on under which conditions the supplier should offer trade credit and how he/she should design such trade credit policy and corresponding ordering policy to obtain much more benefits. From the retailer’s perspective, we discuss whether the retailer needs BW and exactly how many items need to be stocked in BW when the supplier offers trade credit. We formulate a “supplier-Stackelberg” game model, from which we obtain the conditions under which the presented simple trade credit policy not only increases the overall chain profit but also each member’s profit. We also show that the trade credit policy is always more beneficial to the retailer than to the supplier if it is offered.     

Channel coordination under two-level trade credits and demand uncertainty

This study considers a decentralized supply chain where a retailer has an opportunity to order a product from a supplier prior to the sales season to satisfy uncertain demand. The retailer provides trade credit to end customers and makes credit period and order quantity decisions to maximize profits. The end demand is both random and credit period-dependent. On the basis of the newsvendor model, this paper focuses on channel coordination when a retailer provides trade credit to end customers. When the supplier also provides trade credit to the retailer, we show that the traditional trade credit contract cannot coordinate the channel. Four composite contracts based on trade credit (trade credit cost sharing with buy back or quantity flexibility; modified trade credit with buy back or quantity flexibility) are provided to induce the retailer to make decisions while optimizing the channel profit. This paper shows that the retailer provides a longer credit period to its customers and orders a larger quantity from the supplier under the composite contracts. With these contracts, the profit sharing between both parties depends on the wholesale price (Pareto improvement) for the fixed retail price and the purchasing cost.     

The optimal pricing and ordering policy for an integrated inventory model when trade credit linked to order quantity

In traditional inventory models, it is implicitly assumed that the buyer must pay for the purchased items as soon as they have been received. However, in many practical situations, the vendor is willing to provide the buyer with a permissible delay period when the buyer’s order quantity exceeds a given threshold. Therefore, to incorporate the concept of vendor–buyer integration and order-size-dependent trade credit, we present a stylized model to determine the optimal strategy for an integrated vendor–buyer inventory system under the condition of trade credit linked to the order quantity, where the demand rate is considered to be a decreasing function of the retail price. By analyzing the total channel profit function, we developed some useful results to characterize the optimal solution and provide an iterative algorithm to find the retail price, buyer’s order quantity, and the numbers of shipment per production run from the vendor to the buyer. Numerical examples and sensitivity analysis are given to illustrate the theoretical results, and some managerial insights are also obtained.     

Optimal retailer's ordering policies in the EOQ model under trade credit financing

The main purpose of this note is to modify the assumption of the trade credit policy in previously published results to reflect the real-life situations. All previously published models implicitly assumed that the supplier would offer the retailer a delay period, but the retailer would not offer the trade credit period to his/her customer. In most business transactions, this assumption is debatable. In this note, we assume that the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer's replenishment model. Furthermore, we assume that the retailer's trade credit period offered by supplier M is not shorter than the customer's trade credit period offered by retailer N(M?N). Under these conditions, we model the retailer's inventory system as a cost minimization problem to determine the retailer's optimal ordering policies. Then a theorem is developed to determine efficiently the optimal ordering policies for the retailer. We deduce some previously published results of other researchers as special cases. Finally, numerical examples are given to illustrate the theorem obtained in this note.     

Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing

In a supplier-retailer-buyer supply chain, the supplier frequently offers the retailer a trade credit of S periods, and the retailer in turn provides a trade credit of R periods to her/his buyer to stimulate sales and reduce inventory. From the seller’s perspective, granting trade credit increases sales and revenue but also increases opportunity cost (i.e., the capital opportunity loss during credit period) and default risk (i.e., the percentage that the buyer will not be able to pay off her/his debt obligations). Hence, how to determine credit period is increasingly recognized as an important strategy to increase seller’s profitability. Also, many products such as fruits, vegetables, high-tech products, pharmaceuticals, and volatile liquids not only deteriorate continuously due to evaporation, obsolescence and spoilage but also have their expiration dates. However, only a few researchers take the expiration date of a deteriorating item into consideration. This paper proposes an economic order quantity model for the retailer where: (a) the supplier provides an up-stream trade credit and the retailer also offers a down-stream trade credit, (b) the retailer’s down-stream trade credit to the buyer not only increases sales and revenue but also opportunity cost and default risk, and (c) deteriorating items not only deteriorate continuously but also have their expiration dates. We then show that the retailer’s optimal credit period and cycle time not only exist but also are unique. Furthermore, we discuss several special cases including for non-deteriorating items. Finally, we run some numerical examples to illustrate the problem and provide managerial insights.     

A particle swarm optimization for solving lot-sizing problem with fluctuating demand and preservation technology cost under trade credit

In this paper, we consider the effect of preservation technology cost investing on preservation equipment for reducing deterioration rate under two-level trade credit. The preservation technology cost is allowed for periodical upward or downward adjustments due to the time varying demand and the strategy of trade credit within the planning horizon. We establish a deterministic economic order quantity model for a retailer to determine his/her optimal preservation technology cost per replenishment cycle, the trade credit policies, the replenishment number and replenishment schedule that will maximize the present value of total profit. A particle swarm optimization with constriction factor is coded and used to solve the mixed-integer nonlinear programming problem by employing the properties derived from this paper. Some numerical examples are used to illustrate the features of the proposed model.     

Optimal trade credit and order quantity when trade credit impacts on both demand rate and default risk

In practice, to attract new buyers and to avoid lasting price competition, a seller frequently offers its buyers a permissible delay in payment (ie, trade credit). However, the policy of granting a permissible delay in payment adds an additional dimension of default risk to the seller. In contrast to previous researchers for finding optimal solutions to buyers, we first propose an economic order quantity model from the seller's prospective to determine its optimal trade credit and order quantity simultaneously. In addition, we incorporate the important and relevant fact that trade credit has a positive impact on demand rate but a negative impact on receiving the buyer's debt obligations. Then the necessary and sufficient conditions to obtain the seller's optimal trade credit and order quantity are derived. An algorithm to determine the seller's optimal trade credit is also proposed. Finally, we use some numerical examples to illustrate the theoretical results and to provide some managerial insights.     

Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy

In 2007, Huang proposed the optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy, in which the supplier offers the retailer a permissible delay period M, and the retailer in turn provides its customer a permissible delay period N (with N < M). In this paper, we extend his EPQ model to complement the shortcoming of his model. In addition, we relax the dispensable assumptions of N < M and others. We then establish an appropriate EPQ model to the problem, and develop the proper theoretical results to obtain the optimal solution. Finally, a numerical example is used to illustrate the proposed model and its optimal solution.     

Analysis of a fuzzy economic order quantity model for deteriorating items under retailer partial trade credit financing in a supply chain

This paper investigates the economic order quantity (EOQ) — based inventory model for a retailer under two levels of trade credit to reflect the supply chain management situation in the fuzzy sense. It is assumed that the retailer maintains a powerful position and can obtain the full trade credit offered by the supplier yet the retailer just offers a partial trade credit to customers. The demand rate, holding cost, ordering cost, purchasing cost and selling price are taken as fuzzy numbers. Under these conditions, the retailer can obtain the most benefits. Study also investigates the retailer’s inventory policy for deteriorating items in a supply chain management situation as a cost minimization problem in the fuzzy sense. The annual total variable cost for the retailer in fuzzy sense is defuzzified using Graded Mean Integration Representation method. Then the present study shows that the defuzzified annual total variable cost for the retailer is convex, that is, a unique solution exists. Mathematical theorems and algorithms are developed to efficiently determine the optimal inventory policy for the retailer. Numerical examples are given to illustrate the theorems and the algorithms. Finally, the results in this paper generalize some already published results in the crisp sense.     

An EOQ model for deteriorating items under trade credits

In the classical inventory economic order quantity (or EOQ) model, it was assumed that the supplier is paid for the items immediately after the items are received. However, in practices, the supplier may simultaneously offer the customer: (1) a permissible delay in payments to attract new customers and increase sales, and (2) a cash discount to motivate faster payment and reduce credit expenses. In this paper, we provide the optimal policy for the customer to obtain its minimum cost when the supplier offers not only a permissible delay but also a cash discount. We first establish a proper model, and then characterize the optimal solution and provide an easy-to-use algorithm to find the optimal order quantity and replenishment time. Furthermore, we also compare the optimal order quantity under supplier credits to the classical economic order quantity. Finally, several numerical examples are given to illustrate the theoretical results.     

An inventory model for increasing demand under two levels of trade credit linked to order quantity

During the growth stage of a product life cycle especially for high-tech products, the demand function increases with time. In this paper, we extend the constant demand to a linear non-decreasing demand function of time and incorporate a permissible delay in payment under two levels of trade credit into the model. The supplier offers a permissible delay linked to order quantity, and the retailer also provides a downstream trade credit period to its customers. The objective is to find the optimal replenishment cycle that minimizes the retailer’s annual total relevant cost per unit time. The condition for an optimal solution to the generalized model is presented and some fundamental theoretical results are established. Finally, numerical examples to illustrate the proposed model are provided. Sensitivity analysis is performed and some relevant managerial insights are obtained.     

A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity

This paper explores the inventory replenishment policy 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. As a matter of fact, the inventory system discussed by this paper is the same as that of Chang et al. [C.T. Chang, L.Y. Ouyang, J.T. Teng, An EOQ model for deteriorating items under supplier credit credits linked to ordering quantity, Appl. Math. Model. 27 (2003) 983–996]. However, their approach in solving the problems needs further analysis. This article deals with an alternative approach to present a simple procedure in order to determine the optimal ordering policy when the supplier provides a permissible delay in payments linked to order quantity. Numerical examples reveal that the solution algorithm described in this paper is accurate and rapid.     

Economic order quantity under conditionally permissible delay in payments

Within the economic order quantity (EOQ) framework, the main purpose of this paper is to investigate the retailer’s optimal replenishment policy under permissible delay in payments. All previously published articles dealing with optimal order quantity with permissible delay in payments assumed that the supplier only offers the retailer fully permissible delay in payments if the retailer ordered a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. However, in this paper, we want to extend this extreme case by assuming that the supplier would offer the retailer partially permissible delay in payments when the order quantity is smaller than a predetermined quantity. Under this condition, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal order quantity. Three theorems are established to describe the optimal replenishment policy for the retailer. Some previously published results of other researchers can be deduced as special cases. Finally, numerical examples are given to illustrate all these theorems and to draw managerial insights.