Integrated inventory models considering the two-level trade credit policy and a price-negotiation scheme

This paper develops the integrated inventory models with permissible delay in payment, in which customers’ demand is sensitive to the buyer’s price. The models consider the two-level trade credit policy in the vendor–buyer and buyer–customer relationships in supply chain management. A simple recursive solution procedure is proposed for the integrated models to determine the buyer’s optimal pricing and production/order strategy. Although the total profit from the buyer and vendor increases together, the buyer’s share lessens. To compensate the buyer’s loss due to the cooperative relationship, a negotiation system is presented in order to allocate the profit increase to the vendor and buyer to determine the pricing and production/order strategy. A numerical example and sensitivity analysis are provided to illustrate the proposed model. The results indicate that the total profit from the buyer and vendor together can increase, although a price discount is given to the buyer in the proposed models.     

A comprehensive extension of an integrated inventory model with ordering cost reduction and permissible delay in payments

Huang (2010) [1] proposed an integrated inventory model with trade credit financing in which the vendor decides its production lot size while the buyer determines its expenditure to minimize the annual integrated total cost for both the vendor and the buyer. In this paper, we extend his integrated supply chain model to reflect the following four facts: (1) generated sales revenue is deposited in an interest-bearing account for the buyer, (2) the buyer’s interest earned is not always less than or equal to its interest charged, (3) the total number of shipments in one lot size is the vendor’s decision variable to minimize the cost, and (4) it is vital to have a discrimination term which can determine whether the buyer’s replenishment cycle time is less than the permissible delay period or not. We then derive the necessary and sufficient conditions to obtain the optimal solution, and establish some theoretical results to characterize the optimal solution. Finally, numerical examples are presented to illustrate the proposed model and its optimal solution.     

Integrated inventory model with quantity discount and price-sensitive demand

Quantity discount has been a subject of study for a long time; however, little is known about its effect on integrated inventory models when price-sensitive demand is placed. The objective of this study is to find the optimal pricing and ordering strategies for an integrated inventory system when a quantity discount policy is applied. The pricing strategy discussed here is one in which the vendor offers a quantity discount to the buyer. Then, the buyer will adjust his retail price based on the purchasing cost, which will influence the customer demand as a result. Consequently, an integrated inventory model is established to find the optimal solutions for order quantity, retail price, and the number of shipments from vendor to buyer in one production run, so that the joint total profit incurred has the maximum value. Also, numerical examples and a sensitivity analysis are given to illustrate the results of the model.     

A genetic algorithm for optimal operating parameters of VMI system in a two-echelon supply chain

This paper deals with the operational issues of a two-echelon single vendor–multiple buyers supply chain (TSVMBSC) model under vendor managed inventory (VMI) mode of operation. The operational parameters to the above model are: sales quantity and sales price that determine the channel profit of the supply chain, and contract price between the vendor and the buyer, which depends upon the understanding between the partners on their revenue sharing. In order to find out the optimal sales quantity for each buyer in TSVMBSC problem, a mathematical model is formulated. Optimal sales price and acceptable contract price at different revenue share are subsequently derived with the optimal sales quantity. A genetic algorithm (GA) based heuristic is proposed to solve this TSVMBSC problem, which belongs to nonlinear integer programming problem (NIP). The proposed methodology is evaluated for its solution quality. Furthermore, the robustness of the model with its parameters, which fluctuate frequently and are sensitive to operational features, is analysed.     

Optimal production and shipment models for a single-vendor–single-buyer integrated system

This paper develops a more general production-inventory model for a single-vendor–single-buyer integrated system. Unlike the hitherto existing production-inventory models for the vendor–buyer system, the present model neither requires the buyer’s unit holding cost greater than the vendor’s nor assumes the structure of shipment policy. Secondly, the model is extended to the situation with shortages permitted, based on shortages being allowed to occur only for the buyer. Thirdly, the paper also presents a corresponding production-inventory model for a deteriorating item for the integrated system. The solution procedures are provided for finding the optimal production and shipment policies and illustrated with numerical examples. Three significant insights are shown: (1) no matter whether the buyer’s unit holding cost is greater than the vendor’s or not, the present model always performs best in reducing the average total cost as compared to the hitherto existing models; (2) if the buyer’s unit holding cost is less than the vendor’s, the optimal shipment policy for the integrated system will only comprise shipments with successive shipment sizes increasing by a fixed factor. It is different from that obtained by Hill [Hill, R.M., 1999. The optimal production and shipment policy for the single-vendor single-buyer integrated production-inventory problem. International Journal of Production Research 37, 2463–2475] for the opposite case; (3) when designing a single-vendor–single-buyer integrated system, making the buyer’s unit holding cost lower than the vendor’s is more beneficial to the system if shortages are not permitted to occur; otherwise it just reverses.     

Integrated vendor–buyer cooperative inventory models with variant permissible delay in payments

The integrated inventory models usually have the advantage of reducing total cost. However, the way to allocate the cost savings from the integration to the buyer and vendor is critical to the success of the joint relationship between both sides. To deal with this problem, this paper develops the integrated models with the permissible delay in payments for determining the optimal replenishment time interval and replenishment frequency. Applying the models and considering the coefficient of negotiation and the maximum delay payment period, a simple solution algorithm is presented to resolve the allocation of cost savings in the integration model. The coefficient of negotiation is adopted to determine the compromise between the buyer’s and vendor’s cost savings. A numerical example is used to demonstrate the feasibility of the proposed integration models. A sensitivity analysis is also conducted to demonstrate some properties. Using the information from the proposed models, the buyer and vendor can achieve an acceptable compromise solution for both sides in the supply chain management.     

The impact of consumer returns policies on consignment contracts with inventory control

We consider a consignment contract with consumer non-defective returns behavior. In our model, an upstream vendor contracts with a downstream retailer. The vendor decides his consignment price charged to the retailer for each unit sold and his refund price for each returned item, and then the retailer sets her retail price for selling the product. The vendor gets paid based on net sold units and salvages unsold units as well as returned items in a secondary market. Under the framework, we study and compare two different consignment arrangements: the retailer/vendor manages consignment inventory (RMCI/VMCI) programs. To study the impact of return policy, we discuss a consignment contract without return policy as a benchmark. We show that whether or not the vendor offers a return policy, it is always beneficial for the channel to delegate the inventory decision to the vendor. We find that the vendor’s return policy depends crucially on the salvage value of returns. If the product has no salvage value, the vendor’s optimal decision is not to offer a return policy; otherwise, the vendor can gain more profit by offering a return policy when the salvage value turns out to be positive.     

Supply chain coordination with controllable lead time and asymmetric information

This paper considers coordinated decisions in a decentralized supply chain consisting of a vendor and a buyer with controllable lead time. We analyze two supply chain inventory models. In the first model we assume the vendor has complete information about the buyer’s cost structure. By taking both the vendor and the buyer’s individual rationalities into consideration, a side payment coordination mechanism is designed to realize supply chain Pareto dominance. In the second model we consider a setting where the buyer possesses private cost information. We design the coordination mechanism by using principal-agent model to induce the buyer to report his true cost structure. The solution procedures are also developed to get the optimal solutions of these two models. The results of numerical examples show that shortening lead time to certain extent can reduce inventory cost and the coordination mechanisms designed for both symmetric and asymmetric information situations are effective.     

Economic selection of process mean for single-vendor single-buyer supply chain

Process mean selection for a container-filling process is an important decision in a single-vendor single-buyer supply chain. Since the process mean determines the vendor’s conforming and yield rates, it influences the vendor–buyer decisions regarding the production lot size and number of shipments delivered from the vendor to buyer. It follows, therefore, that these decisions should be determined simultaneously in order to control the supply chain total cost. In this paper, we develop a model that integrates the single-vendor single-buyer problem with the process mean selection problem. This integrated model allows the vendor to deliver the produced lot to buyer in number of unequal-sized shipments. Moreover, every outgoing item is inspected, and each item failing to meet a lower specification limit is reprocessed. Further, in order to study the benefits of using this integrated model, two baseline cases are developed. The first of which considers a hierarchical model where the vendor determines the process mean and schedules of production and shipment separately. This hierarchical model is used to show the impact of integrating the process mean selection with production/inventory decisions. The other baseline case is studied in the sensitivity analysis where the optimal solution for a given process is compared to the optimal solution when the variation in the process output is negligible. The integrated model is expected to lead to reduction in reprocessing cost, minimal loss to customer due to the deviation from the optimum target value, and consequently, providing better products at reduced cost for customers. Also, a solution procedure is devised to find the optimal solution for the proposed model and sensitivity analysis is conducted to investigate the effect of the model key parameters on the optimal solution.     

An integrated economic lot-size model for vendor–buyer inventory system when input is random

An integrated strategy is discussed for both vendor and buyer when the input is random. It is shown numerically that the cooperative approach is beneficial to reduce the cost when compared with an independent decision by both the parties. Though the integrated total cost decreases, the buyer’s cost increases due to random input in his inventory. To encourage the buyer to order a large quantity, a trade credit is offered by the vendor to the buyer to settle the account. A conciliation factor is suggested to share the benefits.     

An inventory model under two levels of trade credit and limited storage space derived without derivatives

This paper tries to incorporate both Huang’s model [Y.F. Huang, Optimal retailer’s ordering policies in the EOQ model under trade credit financing, J. Oper. Res. Soc. 54 (2003) 1011–1015] and Teng’s model [J.T. Teng, On the economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Soc. 53 (2002) 915–918] by considering the retailer’s storage space limited to reflect the real-life situations. That is, we want to investigate the retailer’s inventory policy under two levels of trade credit and limited storage space. Furthermore, we adopt Teng’s viewpoint [J.T. Teng, On the economic order quantity under conditions of permissible delay in payments, J. Oper. Res. Soc. 53 (2002) 915–918] that the retailer’s unit selling price and the purchasing price per unit are not necessarily equal. Then, an algebraic approach is provided and three easy-to-use theorems are developed to efficiently determine the optimal cycle time. Some previously published results of other researchers can be deduced as special cases. Finally, a numerical example is given to illustrate these theorems and managerial insights are drawn.     

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.     

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.     

Economic order quantity under advance payment

Though advance payment is widely used in practice, its influences on buyer’s inventory policy are rarely discussed. This paper investigates the buyer’s inventory policy under advance payment, including all payment in advance and partial-advanced–partial-delayed payment. The buyer’s ordering policy is derived by minimizing his total inventory costs including inventory holding cost, ordering cost, and interest cost caused by advance payment or delayed payment. The conclusions show that when all the payment is paid in advance, the buyer’s optimal replenishment cycle is influenced only by the price discount associated with advance payment, and the length of advance payment has no effect. For the partial-advanced–partial-delayed payment case, the buyer’s replenishment cycle is also not influenced by the length of advance period. However, in this situation, the delayed period and the price discount may have impacts on the inventory policy. We also use discounted cash flow (DCF) model to derive the buyer’s replenishment cycle and show that the replenishment cycle is negatively related to the length of advance period. Numerical examples are presented to illustrate the results.     

Optimal policy for a dynamic,non-stationary,stochastic inventory problem with capacity commitment

This paper studies a single-product, dynamic, non-stationary, stochastic inventory problem with capacity commitment, in which a buyer purchases a fixed capacity from a supplier at the beginning of a planning horizon and the buyer’s total cumulative order quantity over the planning horizon is constrained with the capacity. The objective of the buyer is to choose the capacity at the beginning of the planning horizon and the order quantity in each period to minimize the expected total cost over the planning horizon. We characterize the structure of the minimum sum of the expected ordering, storage and shortage costs in a period and thereafter and the optimal ordering policy for a given capacity. Based on the structure, we identify conditions under which a myopic ordering policy is optimal and derive an equation for the optimal capacity commitment. We then use the optimal capacity and the myopic ordering policy to evaluate the effect of the various parameters on the minimum expected total cost over the planning horizon.     

A note on supply chain coordination for joint determination of order quantity and reorder point using a credit option

Credit options and side payments are two methods suggested for achieving coordination in a two-echelon supply chain. We examine the credit option coordination mechanism introduced by Chaharsooghi and Heydari [Chaharsooghi, S., & Heydari, J. (2010). Supply chain coordination for the joint determination of order quantity and reorder point using credit option. European Journal of Operational Research, 204(1), 86–95]. This method assumes that the supplier’s opportunity costs are equal to the reduction in the buyer’s financial holding costs during the credit period. In this note, we show that Chaharsooghi and Heydari’s method is not applicable when buyer and supplier opportunity costs are not equal. We introduce an alternate per order rebate method that reduces supply chain costs to centralized management levels.     

Vendor-buyer inventory problem revisited: modeling random yield and trade credit

This paper revisits the vendor-buyer inventory problem considering random yield and trade credit. The problem deals with a single vendor, single buyer setting with deterministic demand and stochastic supply due to the random nature of yield associated with the vendor’s production process. Since a trade credit agreement is in place, the buyer is not required to make a payment to the vendor at the time of delivery of the order. The full payment must be delivered at the end of the trade credit period dictated by the agreement. The problem of interest was originally studied by Chen and Kang (Int J Prod Econ 123(1):52–61, 2010) who treat the delivery frequency, i.e., buyer’s order cycle length, as a decision variable. The current paper demonstrates that the buyer’s order cycle length is in fact a random variable due to the random nature of yield, and there is no guarantee that the trade credit period will be longer or shorter than the buyer’s order cycle length. As a result, the modeling and solution approach presented by Chen and Kang (2010) is flawed. The approach is remedied here by formal application of renewal theory. The exact cost expressions for the vendor-buyer system are redeveloped rigorously. The newly developed expressions are utilized to investigate the optimal decisions under decentralized control. Furthermore, the results are generalized to consider a general distribution of random yield associated with the vendor’s production process. Operational and financial consequences these new and more general results are demonstrated via numerical examples.     

An integrated inventory model under conditions of order processing cost reduction and permissible delay in payments

This article develops an integrated inventory model to determine the optimal policy under conditions of order processing cost reduction and permissible delay in payments. Both the vendor and the buyer participate in order processing cost reduction by applying information technologies. The order processing cost can be reduced by certain expenditures and will affect lot-size decisions. Simultaneously, the existence of the credit period serves to reduce the cost of holding stock to the buyer, because it reduces the amount of capital invested in stock for the duration of the credit period. The article derives the total cost function and shows that the function possesses some kinds of convexities. A solution procedure is provided to determine the optimal order policy. Finally, numerical examples are presented to illustrate the solution procedure.     

Global optimal policy for vendor–buyer integrated inventory system within just in time environment

Traditionally, inventory problems for the vendor and the buyer are treated separately. In modern enterprises, however, the integration of vendor–buyer inventory system is an important issue. This co-operative approach to inventory management contributes to the success of supply chain management by minimizing the joint inventory cost. The joint inventory cost and the response time can further be reduced when the buyer orders and the vendor replenishes the required items just in time (JIT) for their consumption. The inclusion of the JIT concept in this model contributes significantly to a joint inventory cost reduction. A numerical example and sensitivity analysis are carried out. The derived results show an impressive cost reduction when compared with Goyal’s model.     

Solving the vendor–buyer integrated inventory system with arithmetic–geometric inequality

In the past, economic order quantity (EOQ) and economic production quantity (EPQ) were treated independently from the viewpoints of the buyer or the vendor. In most cases, the optimal solution for one player was non-optimal to the other player. In today’s competitive markets, close cooperation between the vendor and the buyer is necessary to reduce the joint inventory cost and the response time of the vendor–buyer system. The successful experiences of National Semiconductor, Wal-Mart, and Procter and Gamble have demonstrated that integrating the supply chain has significantly influenced the company’s performance and market share (Simchi-Levi et al. (2000) [1]). Recently, Yang et al. (2007) [2] presented an inventory model to determine the economic lot size for both the vendor and buyer, and the number of deliveries in an integrated two stage supply chain. In this paper, we present an alternative approach to determine the global optimal inventory policy for the vendor–buyer integrated system using arithmetic–geometric inequality.