Optimal lot-sizing policy for a manufacturer with defective items in a supply chain with up-stream and down-stream trade credits

In this paper, we establish an economic production quantity model for a manufacturer (or wholesaler) with defective items when its supplier offers an up-stream trade credit M while it in turn provides its buyers (or retailers) a down-stream trade credit N. The proposed model is in a general framework that includes numerous previous models as special cases. In contrast to the traditional differential calculus approach, we use a simple-to-understand and easy-to-apply arithmetic–geometric inequality method to find the optimal solution. Furthermore, we provide some theoretical results to characterize the optimal solution. Finally, several numerical examples are presented to illustrate the proposed model and the optimal solution.     

Two-period supply chain with flexible trade credit contract

This paper studies a two-period supply chain that consists of a retailer and a supplier. A newsvendor-like retailer is capital constrained and orders products using a supplier's trade credits to satisfy uncertain market demand. Most existing studies show that the retailer always postpones payment until the due date. To recall the loans earlier, we present a case in which the supplier, as a Stackelberg leader, offers an incentive of a discounted wholesale price in the second order to entice the retailer to choose flexible early payment. The proposed incentive is related to the retailer's early payment time in the first period. In the presence of bankruptcy risks for both the retailer and supplier, we propose a continuous newsvendor model of a two-period supply chain to analyze the decisions involved in the flexible trade credit contract. The analytic forms confirm that such an incentive can improve the decentralized supply chain efficiency and decreases the supplier's trade credit risk. The retailer always prefers early payment to payment around the due date to increase revenues. Furthermore, the action of paying early might help the retailer adjust cash flow between the two periods. We also find that a revenue sharing contract significantly affects the retailer's payment behavior and supplier's wholesale price. The numerical simulations support our results.     

Optimal dynamic pricing and replenishment policy for perishable items with inventory-level-dependent demand

An inventory system for perishable items with limited replenishment capacity is introduced in this paper. The demand rate depends on the stock quantity displayed in the store as well as the sales price. With the goal to realise profit maximisation, an optimisation problem is addressed to seek for the optimal joint dynamic pricing and replenishment policy which is obtained by solving the optimisation problem with Pontryagin’s maximum principle. A joint mixed policy, in which the sales price is a static decision variable and the replenishment rate remains to be a dynamic decision variable, is presented to compare with the joint dynamic policy. Numerical results demonstrate the advantages of the joint dynamic one, and further show the effects of different system parameters on the optimal joint dynamic policy and the maximal total profit.     

A two-level replenishment frequency model for TOC supply chain replenishment systems under capacity constraint

Good production planning and replenishment management are important for a firm to keep competitive in the market. The theory of constraints-supply chain replenishment system (TOC-SCRS) is a replenishment method under the TOC philosophy. In the application of the TOC-SCRS in a node of a supply chain, the replenishment frequency (RF) and the reliable replenishment time (RRT) are required parameters. Generally, the RF of a node depends on the public transportation schedule such as ship schedules or its private conveyor schedule. If this node is a plant, however, the RF depends on the setup frequency in this plant, and a higher RF (i.e., once a day) is preferred by the TOC because of lower inventory and quick response to different market requirement. Basically, the RF in a plant is determined by its sales or production quantity. When sales increase significantly, the RF in a plant requires to be elongated from higher frequency (i.e., once a day) to lower frequency (i.e., once every two or more days) due to the limited capacity. Therefore, a two-level replenishment frequency model for the TOC-SCRS under capacity constraint is proposed. This model is especially suitable to a plant in which different products have a large sales volume variation. Numerical examples are utilized to evaluate the application of the proposed method. Employing this proposed methodology will facilitate a plant or a central warehouse to implement an effective TOC-SCRS successfully.     

Optimal replenishment policy for perishable items with stock-dependent selling rate and capacity constraint

In this paper, a deterministic inventory model is developed for deteriorating items with stock-dependent demand and finite shelf/display space. Furthermore, we allow for shortages and the unsatisfied demand is partially backlogged at the exponential rate with respect to the waiting time. We provide solution procedures for finding the maximum total profit per unit time. In a specific circumstance, the model will reduce to the case with no shortage. Further, we use numerical examples to illustrate the model.     

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.     

Deterministic inventory model for deteriorating items with trade credit financing and capacity constraints

This work presents an inventory model for optimizing the replenishment cycle time for a single deteriorating item under a permissible delay in payments and constraints on warehouse capacity (owned warehouse capacity, with excess inventory stored in rental warehouses). Rented warehouses are assumed to charge higher unit holding costs than owned warehouses. Furthermore, item deterioration rates are assumed to differ between warehouses. This study has two main purposes: First, the mathematical models of the inventory system are established under the above conditions. Second, this study demonstrates that the optimal solution not only exists but is unique, and two theorems are devised for determining the optimal replenishment cycle time. Finally, numerical examples are presented to illustrate the resulting theorems.     

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.     

Joint pricing and replenishment decisions for deteriorating items with lot-size and time-dependent purchasing cost under credit period

In the real world, the purchasing cost would normally decrease as the replenishment lot-size becomes larger. In other words, the quantity discount effect applies. The purchasing cost may also decrease with the passage of time, for example if the supplier has made effective improvements in their production efficiency, in other words due to the effect of the learning curve. In this article we discuss a purchasing cost pattern which considers these phenomena: i.e., lot-size and time-dependence. The objective of the model is to make decisions related to the pricing and replenishment of deteriorating items over a finite time horizon, given variable purchasing cost and credit period. We provide the properties and develop algorithms for solving the problems described. Also, we discuss the influence of the variable purchasing cost, the length of the credit period, the rate of deterioration, etc., on the retailer behavior.     

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.     

Optimizing a bi-objective multi-product EPQ model with defective items,rework and limited orders: NSGA-II and MOPSO algorithms

In this paper, a bi-objective multi-products economic production quantity (EPQ) model is developed, in which the number of orders is limited and imperfect items that are re-workable are produced. The objectives of the problem are minimization of the total inventory costs as well as minimizing the required warehouse space. The model is shown to be of a bi-objective nonlinear programming type, and in order to solve it two meta-heuristic algorithms namely, the non-dominated sorting genetic algorithm (NSGA-II) and multi-objective particle swarm optimization (MOPSO) algorithm, are proposed. To verify the solution obtained and to evaluate the performance of proposed algorithms, two-sample t-tests are employed to compare the means of the first objective value, the means of the second objective values, and the mean required CPU time of solving the problem using two algorithms. The results show while both algorithms are efficient to solve the model and the solution qualities of the two algorithms do not differ significantly, the computational CPU time of MOPSO is considerably lower than that of NSGA-II.     

持续补货模式下的改良品库存模型研究

改良品的订货问题一直没有引起足够的重视。分别建立持续补货模式下不允许缺货与允许缺货的改良品库存模型,并分析改良率与补货周期、各种费用参数的相关关系,研究表明：改良率与补货周期、单位时间存储费用呈负相关关系,与单位时间订货费用、单位时间改良费用、单位时间缺货费用、单位时间总费用呈正相关关系。同时,对不允许缺货与允许缺货的改良品库存模型单位时间总费用进行比较,并分析库存模型各参数对两种库存模型单位时间总费用差值的影响方式,研究表明：单位时间总费用差值与改良率、补充速率、单次订货费用、单位存储费用、单位改良费用呈正相关关系,与单位缺货费用成呈负相关关系,与需求速率的相关关系取决于补充速率与需求速率的大小关系。     

Supply chain coordination with defective items and quantity discount

This study develops an integrated inventory system involving defective items and quantity discount for optimal pricing and ordering strategies. The model analysed in this study is one in which the buyer orders a quantity, the vendor produces more than buyer's order quantity in order to reduce set-up cost, and then he/she offers an all-units quantity discount to the buyer. Our objective is to determine the optimal order quantity, retail price, mark-up rate, and the number of shipments per production run from the vendor to the buyer, so that the entire supply chain joint total profit incurred has a maximum value. Furthermore, an algorithm of finding the optimal solution is developed. Numerical examples are provided to illustrate the theoretical results.     

Investigating imperfect process and demand effects on inspection scheduling and supply chain replenishment policy

The challenge of supply chain management (SCM) is to have a supervision in managing operations that is critical to coordinate with partners and have high quality product. Global companies are now striving to engage in production improvement and product design efficiency to provide a good support for competition in any parts of the world. Recently, the development and improved capability of production and information technology have changed the ways in which companies and manufacturers operate. With the assist of advanced production philosophy and collaborative design and manufacturing (CDM), the quality of product and manufacturing has been notably enhanced. However, a traditional single-stage inventory model considers a case in which depletion of inventory is merely caused by a constant rare. In practice, there is an inventory loss due to deterioration and imperfect production. The purpose of this paper is to investigate the effects of stock- and warranty-dependent demand, and take the relationship among imperfect, warranty policy and inspection scheduling into consideration. An integrated two-stage production inventory deteriorating model for replenishment policy and inspection plan is developed utilizing time-weighted inventory approach. A numerical example is presented to demonstrate the theory. The results show that fixed selling rate, the holding cost and the unit inspection cost are the critical factors in applying the deteriorating inventory model.     

An inventory replenishment policy for deteriorating items with shortages and partial backlogging

It has long been assumed that the shortages in inventory systems are either completely backlogged or totally lost. However, it is more reasonable to characterize that the longer the waiting time for the next replenishment, the smaller the backlogging rate would be. Moreover, the opportunity cost due to lost sales should be considered since some customers would not like to wait for backlogging during the shortage periods. Without considering these two realistic conditions, study on the inventory modeling for deteriorating items with shortages and partial backlogging cannot be complete and general. In the present article we define an appropriate time-dependent partial backlogging rate and introduce the opportunity cost due to lost sales. Numerical examples are also presented to illustrate the effects of changes in backlogging parameter and unit opportunity cost on total cost and the optimal number of replenishments.Scope and purposeIn a recent article published in this Journal, Giri et al. (Comput. Oper. Res. 27 (2000) 495–505) implemented an existing procedure to the inventory problem of Hariga and Al-Alyan (Comput. Oper. Res. 24 (1997) 1075–83) which concerns with lot-sizing heuristic for deteriorating items with shortages allowed in all cycles except the last one. Giri et al. deviated from the traditional practice and suggested a new policy allowing shortages in all cycles over a finite planning horizon. Their numerical results indicated the proposed policy is cheaper to operate with a cost reduction up to 15%. However, they did not consider the opportunity cost due to lost sales that happen because customers would not like to wait for backlogging. Moreover, for many products with growing sales, the length of the waiting time for the next replenishment is the main factor for determining whether the backlogging will be accepted or not, and the backlogging rate is expected to be time-dependent. Thus the assumption made in Giri et al. that the backlogging rate is a fixed fraction of the total amount of shortages is not reasonable.The purpose of this paper is to present a more realistic discussion of the inventory problem for deteriorating items with time-varying demands and shortages over a finite planning horizon. In contrast to the model by Giri et al., we define an appropriate partial backlogging rate and introduce the opportunity cost due to lost sales. We attempt to complement their model as a practical and general solution for inventory replenishment problems. With these extensions, the scope of applications of the present results is expanded.     

Two-echelon competitive integrated supply chain model with price and credit period dependent demand

This study considers a two-echelon competitive supply chain consisting of two rivaling retailers and one common supplier with trade credit policy. The retailers hope that they can enhance their market demand by offering a credit period to the customers and the supplier also offers a credit period to the retailers. We assume that the market demand of the products of one retailer depends not only on their own market price and offering a credit period to the customers, but also on the market price and offering a credit period of the other retailer. The supplier supplies the product with a common wholesale price and offers the same credit period to the retailers. We study the model under a centralised (integrated) case and a decentralised (Vertical Nash) case and compare them numerically. Finally, we investigate the model by the collected numerical data.     

A supplier-selection model with classification and joint replenishment of inventory items

Since inventory costs are closely related to suppliers, many models in the literature have selected the suppliers and also allocated orders, simultaneously. Such models usually consider either a single inventory item or multiple inventory items which have independent holding and ordering costs. However, in practice, ordering multiple items from the same supplier leads to a reduction in ordering costs. This paper presents a model in capacity-constrained supplier-selection and order-allocation problem, which considers the joint replenishment of inventory items with a direct grouping approach. In such supplier-selection problems, the following items are considered: a fixed major ordering cost to each supplier, which is independent from the items in the order; a minor ordering cost for each item ordered to each supplier; and the inventory holding and purchasing costs. To solve the developed NP-hard problem, a simulated annealing algorithm was proposed and then compared to a modified genetic algorithm of the literature. The numerical example represented that the number of groups and selected suppliers were reduced when the major ordering cost increased in comparison to other costs. There were also more savings when the number of groups was determined by the model in comparison to predetermined number of groups or no grouping scenarios.     

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.     

考虑产品替代的低碳供应链定价和政府财税

基于低碳经济背景,考虑由政府、制造商和零售商三方博弈构成的供应链,使用Stackelberg博弈和纳什均衡的分析方法求出模型的均衡解,并对比了零售商是否联合两种模式下的社会福利、制造商和零售商利润。结果表明：零售商联合情况下的零售商和制造商利润大于零售商非联合情形,同时产品需求量和社会福利与零售商非联合情形相同。高碳产品市场规模越大,社会福利、零售商和制造商利润会先下降后上升,低碳产品的生产成本增加,社会福利、制造商利润会下降,若零售商不联合,销售高碳产品零售商利润上升,销售低碳产品零售商利润下降,若零售商联合则零售商利润下降。