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.     

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.     

A two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation

In the business transactions, the supplier usually offers a permissible delay in payment to his retailer to attract more sales. In addition, a permissible delay in payment may be applied as an alternative to price discount. Based on the above phenomena, we incorporate a permissible delay in payment into the model of Yang [1] and develop a two-warehouse partial backlogging inventory model for deteriorating items with permissible delay in payment under inflation. The objective of this study is to derive the retailer’s optimal replenishment policy that maximizes the net present value of the profit per unit time. The necessary and sufficient conditions for an optimal solution are characterized. An algorithm is developed to find the optimal solution. Finally, numerical examples are provided to illustrate the proposed model. Sensitivity analysis is made and some managerial implications are presented.     

Coordinating ordering and pricing decisions in a two-stage distribution system with price-sensitive demand through short-term discounting

We consider a short-term discounting model in which the distributor offers a discounted price for the retailers’ orders placed at the beginning of its replenishment cycle, in a non-cooperative distribution system with one distributor and multiple retailers, each facing price-sensitive demand. We examine the value of the price discount strategy as a mechanism for the distributor to coordinate the retailers’ ordering and pricing decisions under two common types of demand, linear demand in price and constant elasticity demand in price. Our numerical study reveals that, in the presence of homogeneous retailers (namely, retailers with identical demand rates), the distributor’s profit improvement due to coordination generally decreases as the number of retailers or the inventory holding cost rate increases, but increases as price elasticity increases. Although an increase in the inventory holding cost rate has a negative effect on the distributor’s profit, it may have a positive effect on the retailers’ profits. We further find that with heterogeneous retailers (namely, retailers with different demand rates), offering a discounted price under linear demand benefits the distributor when both the inventory holding cost rate and the variation in demand are either small or large. This cross effect, however, is absent under constant elasticity demand.     

Integrating inventory control and a price change in the presence of reference price effects: a two-period model

Demand and procurement planning for consumer electronics products must cope with short life cycles, limited replenishment opportunities and a willingness to pay that is influenced by past prices and decreases over time. We therefore propose the use of an integrated pricing and inventory control model with a two-period linear demand model, in which demand also depends on the difference between a price-history-based reference price and the current price. For this model we prove that the optimal joint pricing/inventory policy for the replenishment opportunity after the first period is a base-stock list-price policy. That is, stock is either replenished up to a base-stock level and a list-price is charged, or it is not replenished and a discount is given that increases with the stock-level. Furthermore, we use real-world cell phone data to study the differences between an integrated policy and traditional sequential optimization, where prices are initially optimized based on the expected demand and ordering cost, and the resulting demand distribution is used to determine an optimal inventory policy. Finally, we discuss possible extensions of the model.     

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.     

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.     

An inventory control problem for deteriorating items with back-ordering and financial considerations

This paper investigates the effects of time value of money and inflation on the optimal ordering policy in an inventory control system. We proposed an economic order quantity model to manage a perishable item over the finite horizon planning under which back-ordering and delayed payment are assumed. The demand and deterioration rates are constant. The present value of total cost during the planning horizon in this inventory system is modeled first, then a three phases solution procedure is proposed to derive the optimal order and shortage quantities, and the number of replenishment during the planning horizon. Finally, the proposed model is illustrated through numerical examples and the sensitivity analysis is reported to find some managerial insights.     

A multiple replenishment contract with ARIMA demand processes

This paper is concerned with a multiple replenishment contract with a purchase price discount in a supply chain. The chain is composed of one supplier, one buyer and consumers for a product. The replenishment contract is based upon the well-known (s, Q) policy, but allows us to contract replenishments at a future time with a price discount. Owing to the larger forecast error of future demand, the buyer should keep a higher level of safety stock to provide the same level of service as the usual (s, Q) policy. However, the buyer can reduce his purchase cost by ordering a larger quantity at a discounted price. Hence, there exists a trade-off between the price discount and the inventory holding cost. For the ARIMA demand processes, we present a model for the contract and an algorithm to find the number of the future replenishments. Computational experiments show that the algorithm finds the global optimum solution very quickly.     

Optimal ordering policies for periodic-review systems with a refined intra-cycle time scale

Chiang [C. Chiang, Optimal ordering policies for periodic-review systems with replenishment cycles, European Journal of Operational Research 170 (2006) 44–56] recently proposed a dynamic programming model for periodic-review systems in which a replenishment cycle consists of a number of small periods (each of identical but arbitrary length) and holding and shortage costs are charged based on the ending inventory of small periods. The current paper presents an alternative (and concise) dynamic programming model. Moreover, we allow the possibility of a positive fixed cost of ordering. The optimal policy is of the familiar (s, S) type because of the convexity of the one-cycle cost function. As in the periodic-review inventory literature, we extend this result to the lost-sales periodic problem with zero lead-time. Computation shows that the long-run average cost is rather insensitive to the choice of the period length. In addition, we show how the proposed model is modified to handle the backorder problem where shortage is charged on a per-unit basis irrespective of its duration. Finally, we also investigate the lost-sales problem with positive lead-time, and provide some computational 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.     

Single-supplier/multiple-buyer supply chain coordination: Incorporating buyers’ expectations under vertical information sharing

We address the coordination problem in a single-supplier/multiple-buyer supply chain. The supplier wishes to coordinate the supply chain by offering quantity discounts. To obtain their complete cost information, the supplier exchanges his own cost parameters with buyers leading to vertical information sharing. The supplier thinks that the buyers, as they have access to supplier’s setup and holding cost information, may demand a portion of the anticipated coordination savings based on the partial information they hold about the cost structure of the entire supply chain. We model each buyer’s expectations based on her limited view of the entire supply chain which consists of herself and the supplier only. These expectations are then incorporated into the modeling of the supply chain, which results in a generalization of the traditional Stackelberg type models. We discuss alternative efficiency sharing mechanisms, and propose methods to design the associated discount schemes that take buyers’ expectations into account. In designing the discount schemes, we consider both price discriminatory and non-price discriminatory approaches. The study adds to the existing body of work by incorporating buyers’ expectations into a constrained Stackelberg structure, and by achieving coordination without forcing buyers to explicitly comply with the supplier’s replenishment period in choosing their order quantities. The numerical analysis of the coordination efficiency and allocation of the net savings of the proposed discount schemes shows that the supplier is still able to coordinate the supply chain with high efficiency levels, and retain a significant portion of the net savings.     

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.     

带短期价格折扣和允许两次特殊补货的库存决策模型

针对供应商提供短期价格折扣且允许零售商两次特殊补货的库存系统, 建立了以零售商库存效益最大化为目标的库存决策模型, 分析了模型的性质, 根据经济订单批量补货决策下补货时间点与折扣时段的关系, 确定了零售商在不同补货策略下的库存效益增值函数. 据此给出零售商相应的最优补货策略函数表达式, 提出了该模型的一个全局优化算法, 并通过数值算例验证了模型和算法的有效性与可行性.     

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.     

考虑需求率变化与延期支付的临时订货模型

供应商给予临时价格折扣、并针对采购商的临时订货给予优惠的延期支付条件;同时考虑到采购商的自身需求即为终端需求并对价格敏感,由于考虑需求量变化的同时考虑到时间因素,因此需求率实际上发生了改变。在这样的背景下,采购商在价格变化时刻面临两种选择:以低价购进大量货物或者放弃这个机会。本文以EOQ模型单位产品成本为参考,从成本节约最大的角度出发,分析了采购商临时订货量的确定过程,并用数例分析了参数变化对最优订货以及成本节约情况的影响。     

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.     

基于M-CVaR和生产资金约束的供应链协调研究

本文针对供应商面临生产资金约束的情况,在需求随机条件下研究了由一个风险中性零售商和一个具有风险偏好的供应商组成的二级供应链的协调问题。文章在零售商享有批发价折扣的提前支付和供应商银行信贷两种融资方式下,分别建立了零售商的最优决策模型以及基于M-CVaR测度工具的供应链整体订购决策模型,并给出了采用两种不同融资模式时零售商和供应链整体的最优订购量以及供应链的协调条件,分析了供应商的风险偏好对供应链整体最优决策及协调条件的影响。最后通过算例验证了文章的主要结论。研究表明,当提前支付价格折扣大于临界值时,零售商会选择提前支付货款;提前支付模式下供应链整体的最优订购量大于银行信贷模式;随着供应商的风险偏好由风险规避向风险喜好转变,供应链整体的最优订购量逐渐增大。     

Optimal pricing,shipment and payment policy for an integrated supplier–buyer inventory model with two-part trade credit

In this article, we develop an integrated supplier–buyer inventory model with the assumption that the market demand is sensitive to the retail price and the supplier adopts a trade credit policy. The trade credit policy discussed in this paper is a “two-part” strategy: cash discount and delayed payment. That is, if the buyer pays within M₁, the buyer receives a cash discount; otherwise, the full purchasing price must be paid before M₂, where _M2>_M1?0$M_2>M_1?0$. The objective of this research is to determine the optimal pricing, ordering, shipping, and payment policy to maximize the joint expected total profit per unit time. An iterative algorithm is established to obtain the optimal strategy. Furthermore, numerical examples and sensitivity analysis are presented to illustrate the results of the proposed model and to draw managerial insights.     

Optimal promotion planning—depth and frequency—for a two-stage supply chain under Markov switching demand

In considering the retailer–supplier supply chain, this paper analyzes how a retailer reasonably decides both the depth and frequency of the price discount promotion including or excluding a supplier’s inventory decision. Assuming that the promotion frequency used by the retailer is probabilistic, we model a promotion-inventory decision under an AR(1) demand with a Markov switching promotion regime. After obtaining the optimal promotion plan, our analysis also considers the behavior of the optimal promotion decision; the retailer’s price format selection, either an Every-Day-Low-Price policy (EDLP) or a Promotion policy (HiLo); and the impact of information sharing of promotion status on the system’s performance. Our results suggest that a retailer tends to overpromote if inventory cost is excluded in its promotion decision, that increasing the market share is a preferable action for both the retailer and the supplier, that total margin and price-elasticity play an important role in selecting the price format, and that the profitability for a supplier of sharing promotion information depends on the transition probabilities of the Markov switching regime.