Optimal replenishment decision in an EPQ model with defective items under supply chain trade credit policy

The traditional inventory of the economic order quantity model assumes perfect items in an ordered lot and an infinite replenishment rate. However, such conditions are rare in actual production environments. Additionally, most studies of this problem have only considered suppliers offering the wholesaler a grace period. In practice, wholesalers often extend a fixed credit period to downstream customers as well. This study therefore proposes a production model for a lot-size inventory system with finite production rate and defective quality under the condition of two-level trade credit policy and the condition that defective items involve both imperfect quality and scrap items. Thus, optimal wholesaler replenishment decisions can be determined for defective items under two-level trade credit policy in the EPQ framework. Four theorems for determining the optimal cycle time and the results in this study can be deduced as a special case of earlier models. Finally, illustrative examples are provided to verify the theoretical 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.     

Optimal inventory replenishment policy for the EPQ model under trade credit derived without derivatives

Within the economic production quantity (EPQ) framework, the main purpose of this article is to deal with Chung and Huang's model (K.J. Chung, and Y.F. Huang,“The optimal cycle time for EPQ inventory model under permissible delay in payments,” Int. J. Prod. Econ., 84, pp. 307–318, 2003) and extend Chung and Huang's model (2003) by considering the unit selling price higher than the unit purchasing cost using the algebraic method to determine the optimal inventory replenishment policy for the retailer under trade credit. This article provides this algebraic approach which could be used easily to introduce the basic inventory theories to younger students who lack the knowledge of calculus. In addition, we develop an easy-to-use procedure to find the optimal inventory replenishment policy for the retailer in the extended model developed in this article. Finally, numerical examples are given to illustrate the result obtained in our extended model.     

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.     

An accurate and reliable solution procedure for the EOQ model with deteriorating items under supplier credits linked to the ordering quantity

The inventory problem consists of two parts: (1) the modelling and (2) the solution procedure. The modelling can provide insight to solve the inventory problem and the solution procedure involves the implementation of the inventory model. Basically, the modelling and the solution procedure to the inventory problem are equally important. Chang et al.’s inventory model [Chang, C. T., Ouyang, L. Y., & Teng, J. T (2003). An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Applied Mathematical Modelling, 27, 983–996] is correct and interesting. However, they ignore the explorations of the functional behaviors of the annual total relevant cost to locate the optimal solutions such that proofs of their solution procedures are not perfect from the viewpoint of logic. The main purposes of this paper are to provide accurate and reliable solution procedures to improve [Chang, C. T., Ouyang, L. Y., & Teng, J. T (2003). An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Applied Mathematical Modelling, 27, 983–996].     

A particle swarm optimization for solving joint pricing and lot-sizing problem with fluctuating demand and trade credit financing

Pricing is a major strategy for a retailer to obtain its maximum profit. Furthermore, under most market behaviors, one can easily find that a vendor provides a credit period (for example 30 days) for buyers to stimulate the demand, boost market share or decrease inventories of certain items. Therefore, in this paper, we establish a deterministic economic order quantity model for a retailer to determine its optimal selling price, replenishment number and replenishment schedule with fluctuating demand under two levels of trade credit policy. A particle swarm optimization is coded and used to solve the mixed-integer nonlinear programming problem by employing the properties derived in this paper. Some numerical examples are used to illustrate the features of the proposed model.     

A study on an inventory model for non-instantaneous deteriorating items with permissible delay in payments

In this study, an appropriate inventory model for non-instantaneous deteriorating items with permissible delay in payments is considered. The purpose of this study is to find an optimal replenishment policy for minimizing the total relevant inventory cost. This mathematical model is a general framework that comprises numerous previous models such as in Ghare and Schrader [Ghare, P. M., & Schrader, G. H. (1963). A model for exponentially decaying inventory system. International Journal of Production Research, 21, 449–460], Goyal [Goyal, S. K. (1985). Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 36, 335–338], and Teng [Teng, J. T. (2002). On the economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 53, 915–918] as special cases. We have developed some useful theorems to characterize the optimal solutions and provide an easy-to-use method to find the optimal replenishment cycle time and order quantity under various circumstances. Several numerical examples are given to test and verify the theoretical results. Finally, a sensitivity analysis of the optimal solution with respect to major parameters is also included. According to the results of numerical analysis, we provided several ways for the retailer to effectively reduce total annual relevant inventory cost.     

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.     

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.     

New mixed integer approach to solve a multi-level capacitated disassembly lot-sizing problem with defective items and backlogging

The disassembly process has attracted mounting interest due to growing green concerns. This paper addresses the capacitated dynamic lot-sizing problem with external procurement, defective and backordered items, setup times, and extra capacity. The problem is to determine how many end-of-life products to disassemble during each period. We propose a new mixed-integer programming (MIP) approach to formulate the problem under study in order to maximize the disassembly-process gain, which is obtained as the difference between the revenue achieved by resale of the items recovered after disassembly and the costs tied to operating the disassembly tasks. Several numerical tests using the well-known CPLEX solver proved that this new model can find the optimal disassembly schedule for most test instances within an acceptable computational time. Furthermore, we led sensitivity studies on disassembly capacity, setup time and procurement cost. Test results validate the power of the suggested model and provide helpful insights for industry practitioners.     

A decision rule for coordination of inventory and transportation in a two-stage supply chain with alternative supply sources

This paper deals with a two-stage supply chain that consists of two distribution centers and two retailers. Each member of the supply chain uses a (Q,R) inventory policy, and incurs standard inventory holding and backlog costs, as well as ordering and transportation costs. The distribution centers replenish their inventory from an outside supplier, and the retailers replenish inventory from one of the two distribution centers. When a retailer is ready to replenish its inventory that retailer must decide whether it should replenish from the first or second distribution center. We develop a decision rule that minimizes the total expected cost associated with all outstanding orders at the time of order placement; the retailers then repeatedly use this decision rule as a heuristic. A simulation study which compares the proposed policy to three traditional ordering policies illustrates how the proposed policy performs under different conditions. The numerical analysis shows that, over a large set of scenarios, the proposed policy outperforms the other three policies on average.     

Optimal inventory control policy and supply chain coordination problem with carbon footprint constraints

Carbon footprint constraints exert pressure on supply chains to reexamine decisions. In this paper, we consider carbon transfer cost and carbon holding cost in a supply chain. A multiperiod dynamic programming model with carbon footprint constraints is presented to investigate the impact of carbon transfer cost and carbon holding cost on inventory control policy as well as the supply chain coordination problem. A two‐control limit inventory control policy is proved to be optimal and a contract with wholesale price, subsidy, and fixed setup cost is verified analytically to coordinate the supply chain. Finally, a numerical study is conducted to reveal managerial insights. We find that when the supply chain is coordinated, the chain's profit is more sensitive to carbon transfer cost while inventory level is more sensitive to carbon holding cost. Additionally, because of the complexity of the coordinated contract, when it is not easy to coordinate the supply chain, it is better to keep the values of wholesale price, subsidy, and fixed setup cost below the corresponding values for the coordinated supply chain.     

Coordination-based inventory management for deteriorating items in a two-echelon supply chain with profit sharing

This study develops a model for inventory management consisting of a two-echelon supply chain (SC) with profit sharing and deteriorating items. The retailer and the supplier act as the leader and follower, in which the supplier faces a huge setup cost and economic order quantity ordering strategy. The market demand is affected by the sale price of the product, and the inventory has a deterioration rate following a Weibull distribution. The retailer executes three profit-sharing mechanisms to motivate the supplier to participate in SC optimisation and to extend the life cycle of the product. A search algorithm is developed to determine the solutions as using the profit-sharing mechanisms. The outcomes from numerical experiments demonstrate the profitability of the proposed 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.     

An optimization approach for joint pricing and ordering problem in an integrated inventory system with order-size dependent trade credit

Under a business trading environment, it is common for the trade credit to depend on the order size. Therefore, it is important to discuss the single-supplier and single-buyer supply chain problem which includes order-size dependent trade credit. In this study, an integrated inventory model with a price sensitive demand rate, determining jointly economic lot size of the buyer’s ordering and the supplier’s production batch, are developed to maximize the total profit per unit time. An efficient algorithm is provided to obtain the optimal solution, and then numerical examples are presented to illustrate the theoretical results. Finally, the comparison between whether an optimal solution is jointly or independently determined is also provided.     

Optimal economic order quantity for buyer–distributor–vendor supply chain with backlogging derived without derivatives

In this article, we first complement an inappropriate mathematical error on the total cost in the previously published paper by Chung and Wee [2007, ‘Optimal the Economic Lot Size of a Three-stage Supply Chain With Backlogging Derived Without Derivatives’, European Journal of Operational Research, 183, 933–943] related to buyer–distributor–vendor three-stage supply chain with backlogging derived without derivatives. Then, an arithmetic–geometric inequality method is proposed not only to simplify the algebraic method of completing prefect squares, but also to complement their shortcomings. In addition, we provide a closed-form solution to integral number of deliveries for the distributor and the vendor without using complex derivatives. Furthermore, our method can solve many cases in which their method cannot, because they did not consider that a squared root of a negative number does not exist. Finally, we use some numerical examples to show that our proposed optimal solution is cheaper to operate than theirs.     

Inventory management in a two-echelon closed-loop supply chain with correlated demands and returns

Reverse logistics or closed-loop supply chains where product returns are integrated with traditional forward supply chains have been one of the major topics of research since about the last one and a half decades. In this paper, we address the inventory management issue in closed-loop supply chains, and develop deterministic and stochastic models for a two-echelon system with correlated demands and returns under generalized cost structures. In particular, we address the following questions – Do closed-loop supply chains cost more than traditional forward supply chains? Does a higher rate of return always translate into lower demand variability and hence lower expected costs? What is the relationship between expected costs and correlations between demands and returns? Models developed and numerical examples shown in the paper reveal that although a higher rate of return and a higher correlation between demand and return reduce the variability of net demand, it may not necessarily lead to cost savings; rather the movement of costs will depend on the values of system parameters. We also quantify the cost savings in case the actual demand and return information is available at the time of decision-making. We conclude the paper by providing managerial implications and directions for future research.     

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.     

Optimal decisions for a two-echelon supply chain with capacity and demand information

Due to the applications of Internet of Things and big data in the Industry 4.0 context, more information in and out of a smart factory can be collected and shared between manufacturers and retailers. In this study, we consider two types of information that can be available in a supply chain consisting of a manufacturer and a retailer in Industry 4.0: the capacity information for the later rush production and the demand information shared between the retailer and manufacturer. In the supply chain, the manufacturer provides two orders with maximum limits by using a capacitated normal production and two capacitated rush production modes. To study the effects of the information, we investigate the optimal decisions and profits for the supply chain with and without the capacity information and demand information sharing. In addition, we propose a coordination mechanism for the supply chain with both the capacity information and demand information sharing. The coordination mechanism does not only rely on cost parameters, but also on the capacity and demand information. The numerical examples show that the supply chain profit can be improved by as large as 16.76% in the coordinated system, compared with the original system without the capacity information and demand information sharing.