An integrated inventory model with capacity constraint and order-size dependent trade credit

Trade credit has many forms in today’s business practice. The most common form of trade credit policy that is used to encourage retailers to buy larger quantities is order-size dependent. When the number of ordered units exceeds the capacity of the own warehouse, an additional rented warehouse is required to store the excess units. Therefore, to incorporate the concept of order-size dependent trade credit and limited storage capacity, we proposed an integrated inventory model with capacity constraint and a permissible delay payment period that is order-size dependent. In addition, the unit production cost, which is a function of the production rate, is considered. Three theorems and an algorithm are developed to determine the optimal production and replenishment policies for both the supplier and the retailer. Finally, numerical examples are presented to illustrate the solution procedure and the sensitivity analyses of some key parameters are provided to demonstrate the proposed model.     

Optimal transfer,ordering and payment policies for joint supplier–buyer inventory model with price-sensitive trapezoidal demand and net credit

This study aims at formulating an integrated supplier–buyer inventory model when market demand is variable price-sensitive trapezoidal and the supplier offers a choice between discount in unit price and permissible delay period for settling the accounts due against the purchases made. This type of trade credit is termed as ‘net credit’. In this policy, if the buyer pays within offered time M₁, then the buyer is entitled for a cash discount; otherwise the full account must be settled by the time M₂; where M₂ > M₁ ? 0. The goal is to determine the optimal selling price, procurement quantity, number of transfers from the supplier to the buyer and payment time to maximise the joint profit per unit time. An algorithm is worked out to obtain the optimal solution. A numerical example is given to validate the proposed model. The managerial insights based on sensitivity analysis are deduced.     

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 ordering policies for deteriorating items using a discounted cash-flow analysis when a trade credit is linked to order quantity

In today’s competitive market, in order to obtain a competition advantage, the supplier often offers the purchaser a longer permissible delay in payments or a price discount if the order quantity is greater than or equal to a predetermined quantity. As a result, in this paper, we establish an inventory model for the purchaser in which the supplier provides different trade credits. We then solve the inventory problem by using a discounted cash-flow (DCF) approach, characterize the optimal solution, and obtain some theoretical results to find the optimal order quantity and the optimal replenishment time. Finally, we provide several numerical examples to illustrate the results.     

The optimal integrated inventory policy with price-and-credit-linked demand under two-level trade credit

In this paper, we proposed a generalized, integrated, supplier–retailer inventory model using a trade credit policy. 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 provides customers a permissible delay period N. Cases where M > N and M ? N are explored thoroughly. In addition, the demand rate is assumed to be a function of both retail price and the customers’ credit period. Consequently, this paper deals with the problem of determining the optimal retail price, economic order quantity, and the number of shipments from the supplier to the retailer in one production run for an integrated inventory system under both two-level trade credit and price-and-credit-linked demand rate. Algorithms are developed in order to determine the joint optimal policies. Numerical examples are presented to illustrate the proposed models, as well as sensitivity analysis of key parameters.     

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.     

基于订货阈值的供应商延期支付策略设计

针对完全信息下以供应商为核心企业的二级供应链库存系统,将全单位延期支付与部分延期支付两种手段相结合,并在考虑延期支付期限与订货量相关的条件下,设计了基于供应商视角和订货阈值的延期支付策略,从而得出此策略能使系统利润达到帕累托最优.最后,通过数值算例对相关结论进行了验证和灵敏度分析.     

A comprehensive note on “Lot‐sizing decisions for deteriorating items with two warehouses under an order‐size‐dependent trade credit”

To reduce inventory and increase sales, the supplier frequently offers the retailer a permissible delay in payments if the retailer orders more than or equal to a predetermined quantity. In 2012, Liao et al. proposed an economic order quantity model for a retailer with two warehouses when the supplier offers a permissible delay linked to order quantity. In this paper, we attempt to overcome some shortcomings of their mathematical model. Then, we apply some existing theoretical results in fractional convex programs to prove that the annual total variable cost is pseudoconvex. Hence, the optimal solution exists uniquely, which simplifies the search for the global minimum solution to a local minimum solution. Finally, we run a couple of numerical examples to illustrate the problem and compare the optimal solutions between theirs and ours.     

Trade credit financing in the vendor–buyer inventory system with ordering cost reduction,transportation cost and backorder price discount when the received quantity is uncertain

Previous ordering cost reduction vendor–buyer inventory models with backorder price discount usually assumed that the buyer must pay to the vendor for the ordered items as soon as the items are received, the received quantity is same as the ordered quantity and the transportation cost is independent of the shipment lot-size. In practice, however, the vendor is willing to offer the buyer a certain credit period without interest to promote market competition as well as the buyer's quantity received may not match with the ordered quantity due to unavailability of the raw material, worker's strike, human errors in counting, transcribing, etc. Furthermore, the discounts are offered for the transportation cost of large ordered quantities. This paper derives a single-vendor single-buyer supply chain model for the ordering cost reduction inventory system with backorder price discount, taking into consideration the effect of transportation cost discount and the condition of permissible delay in payments include the case where the buyer's received quantity does not necessarily match the quantity requisitioned. We take the transportation cost as a function of the shipment lot-size and it is taken to be in an all-unit-discount cost format. Thus we incorporate transportation cost explicitly into the model and develop optimal solution procedures for solving the proposed inventory problem. Numerical example and sensitivity analyses are given to demonstrate the applications and performance of the proposed methodology.     

基于现金折扣和延期支付的零售商补货和付款策略

在考虑供应商给予零售商延期支付、现金折扣的基础上,加入了非瞬时补货的条件,建立了相应的零售商库存决策模型.通过模型的分析求解,可以得出零售商在上述情况下的最优订货周期及最优付款时间的简单判定方法.最后,通过算例验证了模型的可行性,得出了与实际相符的结论.     

Coordinated ordering decisions for products with short lifecycle and variable selling price

The supplier–buyer coordination is an important policy in the supply chain management. The buyer in the two-echelon inventory system with regular selling season has to face the uncertainty of customer demand, supplier’s delivery time and variable price change. At the same time, the supplier has to consider the inventory holding and delay cost. The objective of this study is to develop an integrated supply chain strategy for products with short lifecycle and variable selling price to entice cooperation. The strategy must provide a win–win situation for both the supplier and the buyer. A numerical case example, sensitivity analysis and compensation mechanism are given to illustrate the model.     

Supply chain financing with advance selling under disruption

We study a financing problem in a supply chain (SC) consisting of one supplier and one buyer under supply disruption. The supplier could face a disruption at its end which could effectively reduce its yield in case of disruption, thereby resulting in supply yield uncertainty. The retailer can finance the supplier using advance selling that can help mitigate the impact of disruption. We model this problem as a Stackelberg game, where the supplier as the leader announces the wholesale price and the retailer responds by deciding its optimal order quantity given stochastic demand and an exogenous fixed retail price. The supplier then commences production and a disruption can happen with a known probability. We assume that under disruption the quantity delivered is a fraction of the initial quantity ordered by the retailer. The retailer loses any unmet demand. We analyze three different scenarios of the Stackelberg game, namely no advance selling with disruption, advance selling without disruption, and advance selling with disruption. Our results indicate that advance selling can be used to mitigate the impact of supply disruption and at the same time could lead to an increase in the overall SC profit.     

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.     

An integrated retail space allocation and lot sizing models under vendor managed inventory and consignment stock arrangements

In this paper, we develop integrated retail shelf space allocation and inventory models for a single item with a stock dependent demand. The integrated models are developed for a supply chain operating under vendor-managed inventory (VMI) and consignment stock (CS) agreement. More precisely, the supplier is responsible for initiating orders on behalf of the retailer and decides about the size of each order, the quantity to be displayed on the shelves, and the reorder point. In addition, the supplier owns the stock at the retailer’s premises until it is sold. We develop mathematical models to assess the benefits accrued by both parties as a result of the adoption of VMI–CS partnership. Results from the numerical experimental study show that such partnership is more attractive to all supply chain members when the retailer provides a flexible display capacity. Moreover, the supplier can use his/her selling price and the maximum allocated shelf space as negotiation means to benefit from the partnership.     

Discounted multi-item inventory model via genetic algorithm with Roulette wheel selection,arithmetic crossover and uniform mutation in constraints bounded domains

Genetic Algorithm (GA) with different logic structures for price breaks has been developed and implemented for a multi-item inventory control system of breakable items like the items made of glass, mud, porcelain, etc. with all unit discount (AUD), incremental quantity discount (IQD) and a combination of these discounts. Here, AUD and IQD on purchasing price with two price breaks are allowed. Also, demand and breakability of the items are stock-dependent. Shortages are not allowed. Replenishment is instantaneous. Selling price is a mark-up of the purchasing cost. For storage, warehouse capacity is limited. For the present model, GA has been developed in real code representation using Roulette wheel selection, arithmetic crossover and uniform mutation. This algorithm has been implemented successfully to find the optimum order quantities for the above inventory control system to achieve the maximum possible profit. The algorithm and the inventory model have been illustrated numerically and some sensitivity analyses with respect to breakability and demand are presented.     

An inventory model for deteriorating items with partial backlogging and permissible delay in payments

The paper deals with an inventory model with a varying rate of deterioration and partial backlogging rate under the condition of permissible delay in payments. The existing literature on the subject generally deal with situations where the payment of an order is made on the receipt of items by the inventory system and shortages are either completely backlogged or fully lost. In this paper, a varying deterioration rate of time and the condition of permissible delay in payments used in conjunction with the economic order quantity model are the focus of discussion. In addition, the shortages are neither completely backlogged nor completely lost assuming the backlogging rate to be inversely proportional to the waiting time for the next replenishment. Numerical examples are presented to illustrate the model.     

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.     

P2P分层流媒体数据分配的粒子群遗传算法

现有P2P分层流媒体中的数据分配算法是基于贪婪思想的确定性启发式算法,不能得到全局最优解。为此,提出一种基于备选数据块编码方式的粒子群遗传算法。定义备选数据块,建立问题的无约束整数规划模型。仿真实验表明,该算法在优化效果上能比现有算法提高5%~25%。     

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。     

Lot-sizing decisions under trade credit depending on the ordering quantity

This paper deals with the problem of determining the economic order quantity for exponentially deteriorating items under the conditions of permissible delay in payments. The delay in payments depends on the quantity ordered. When the order quantity is less than at which the delay in payments is permitted, the payment for the product must be made immediately. Otherwise, the fixed trade credit period is permitted.Based upon the above arguments, this paper incorporates both Hwang and Shinn (Comput. Oper. Res. 24 (1997) 539) and Khouja and Mehrez (J. Manuf. Systems 15 (1996) 334) under above conditions. In addition, the objective function is modeled as a total variable cost-minimization problem. An algorithm to determine the economic order quantity is developed. Examples are solved to illustrate the results given in this paper. Finally, the results in this paper generalize some already published results.