Optimal replenishment policies for deteriorating items with stock sensitive demand under two-level trade credit and limited capacity

Recently, Min et al. [18] established an inventory model for deteriorating items under stock-dependent demand and two-level trade credit and obtained the optimal replenishment policy. Their analysis imposed a terminal condition of zero ending-inventory. However, with a stock-dependent demand, it may be desirable to order large quantities, resulting in stock remaining at the end of the cycle, due to the potential profits resulting from the increased demand. As a result, to make the theory more applicable in practice, we extend their model to allow for: (1) an ending-inventory to be nonzero, (2) a maximum inventory ceiling to reflect the facts that too much stock leaves a negative impression on the buyer and the amount of shelf/display space is limited.     

Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime

Due to evaporation, obsolescence, spoilage, etc., some products (e.g., fruits, vegetables, pharmaceuticals, volatile liquids, and others) not only deteriorate continuously but also have their expiration dates. To attract new buyers and increase sales, a seller frequently offers its buyers a trade credit period to settle the purchase amount. There is no interest charge to a buyer if the purchasing amount is paid within the credit period, and vice versa. On the other hand, granting a credit period from a seller to its buyers increases default risk. In this paper, we propose an economic order quantity model for a seller by incorporating the following relevant facts: (1) deteriorating products not only deteriorate continuously but also have their maximum lifetime, and (2) credit period increases not only demand but also default risk. We then characterize the seller’s optimal credit period and cycle time. Furthermore, we discuss a special case for non-deteriorating items. Finally, we run several numerical examples to illustrate the problem and provide some managerial insights.     

The simplified solution procedure for deteriorating items under stock-dependent demand and two-level trade credit in the supply chain management

Min et al. [1] (J. Min, Y.W. Zhou, J. Zhao, An inventory model for deteriorating items under stock-dependent demand and two-level trade credit, Appl. Math. Model. 34 (2010) 3273–3285.) develop an inventory model for deteriorating items under stock-dependent demand and two-level trade credit. They provide the necessary and sufficient conditions of the existence and uniqueness of the optimal solutions that could maximize the retailer’s average profit per unit time. Basically, their paper is correct and interesting. Recently, several researchers have been showing a huge interest in developing simple and easy to implement solution procedures in management science. Therefore this paper indicates that Min et al.’s solution procedure can be further improved and simplified. So, the main purpose of this paper is to present simple and easy to understand solution procedures to locate the optimal solutions of an inventory model that considers deteriorating items under stock-dependent demand and two-level trade credit.     

An optimal production-inventory model for deteriorating items with multiple-market demand

The global markets of today offer more selling opportunities to the deteriorating items’ manufacturers, but also pose new challenges in production and inventory planning. From a production management standpoint, opportunities to exploit the difference in the timing of the selling season between geographically dispersed markets for deteriorating items are important to improving a firm’s profitability. In this paper, we examined the above issue with an insightful production-inventory model of a deteriorating items manufacturer selling goods to multiple-markets with different selling seasons. We also provided a solution procedure to find the optimal replenishment schedule for raw materials and the optimal production plan for finished products. A numerical example was then used to illustrate the model and the solution procedure. Finally, sensitivity analysis of the optimal solution with respect to major parameters was carried out.     

The inventory model for trade credit in economic ordering policies of deteriorating items in a supply chain system

In 1994, professors Jaggi and Aggarwal presented the economic ordering policies of deteriorating items in the presence of trade credit using a discounted cash-flows (DCF) approach. This paper discusses the same problem as that of Jaggi and Aggarwal and indicates that some approximations to the optimal cycle times proposed by Jaggi and Aggarwal are inappropriate sometimes. A theorem is derived out to find the optimal cycle time. With that theorem, a simple algorithm is developed to locate the optimal cycle time.     

A note on “Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime”

In 2014, Wang et al. (2014) extended the model of Lou and Wang (2012) to incorporate the credit period dependent demand and default risk for deteriorating items with maximum lifetime. However, the rates of demand, default risk and deterioration in the model of Wang et al. (2014) are assumed to be specific functions of credit period which limits the contributions. In this note, we first generalize the theoretical results of Wang et al. (2014) under some certain conditions. Furthermore, we also present some structural results instead of a numerical analysis on variation of optimal replenishment and trade credit strategies with respect to key parameters.     

On an EPQ model for deteriorating items under permissible delay in payments

This paper derives a production model for the lot-size inventory system with finite production rate, taking into consideration the effect of decay and the condition of permissible delay in payments, in which the restrictive assumption of a permissible delay is relaxed to that at the end of the credit period, the retailer will make a partial payment on total purchasing cost to the supplier and pay off the remaining balance by loan from the bank. At first, this paper shows that there exists a unique optimal cycle time to minimize the total variable cost per unit time. Then, a theorem is developed to determine the optimal ordering policies and bounds for the optimal cycle time are provided to develop an algorithm. Numerical examples reveal that our optimization procedure is very accurate and rapid. Finally, it is shown that the model developed by Huang [1] can be treated as a special case of this paper.     

An uncooperative order model for items with trade credit,inventory-dependent demand and limited displayed-shelf space

This paper considers a two-echelon supply chain where a supplier sells a single product through a retailer, who faces an inventory-dependent demand. The supplier hopes to incentive the retailer to order more items by offering trade credit. The retailer places the ordered items on the display shelf (DS) with limited space and stocks the remaining items (if any) that exceed the shelf capacity in his/her backroom/warehouse (BW). From the supplier’s perspective, we focus mainly on under which conditions the supplier should offer trade credit and how he/she should design such trade credit policy and corresponding ordering policy to obtain much more benefits. From the retailer’s perspective, we discuss whether the retailer needs BW and exactly how many items need to be stocked in BW when the supplier offers trade credit. We formulate a “supplier-Stackelberg” game model, from which we obtain the conditions under which the presented simple trade credit policy not only increases the overall chain profit but also each member’s profit. We also show that the trade credit policy is always more beneficial to the retailer than to the supplier if it is offered.     

A note on an EOQ model for deteriorating items under supplier credit linked to ordering quantity

This paper explores the inventory replenishment policy for deteriorating items in which the supplier provides a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. As a matter of fact, the inventory system discussed by this paper is the same as that of Chang et al. [C.T. Chang, L.Y. Ouyang, J.T. Teng, An EOQ model for deteriorating items under supplier credit credits linked to ordering quantity, Appl. Math. Model. 27 (2003) 983–996]. However, their approach in solving the problems needs further analysis. This article deals with an alternative approach to present a simple procedure in order to determine the optimal ordering policy when the supplier provides a permissible delay in payments linked to order quantity. Numerical examples reveal that the solution algorithm described in this paper is accurate and rapid.     

On an inventory model for deteriorating items and time-varying demand

In this paper, we present an optimal procedure for finding the replenishment schedule for the inventory system in which items deteriorate over time and demand rates are increasing over a known and finite planning horizon.     

Optimal policy for deteriorating items with trapezoidal type demand and partial backlogging

Inventory model for time-dependent deteriorating items with trapezoidal type demand rate and partial backlogging is considered in this paper. The demand rate is defined as a continuous trapezoidal function of time, and the backlogging rate is a non-increasing exponential function of the waiting time up to the next replenishment. We proposed an optimal replenishment policy for such inventory model, numerical examples to illustrate the solution procedure.     

Retailer’s optimal ordering policy for deteriorating items with maximum lifetime under supplier’s trade credit financing

Many products such as fruits, vegetables, pharmaceuticals, volatile liquids, and others not only deteriorate continuously due to evaporation, obsolescence, spoilage, etc. but also have their expiration dates (i.e., a deteriorating item has its maximum lifetime). Although numerous researchers have studied economic order quantity (EOQ) models for deteriorating items, few of them have taken the maximum lifetime of a deteriorating item into consideration. In addition, a supplier frequently offers her/his retailers a permissible delay in payments in order to stimulate sales and reduce inventory. There is no interest charge to a retailer if the purchasing amount is paid to a supplier within the credit period, and vice versa. In this paper, we propose an EOQ model for a retailer when: (1) her/his product deteriorates continuously, and has a maximum lifetime, and (2) her/his supplier offers a permissible delay in payments. We then characterize the retailer’s optimal replenishment cycle time. Furthermore, we discuss a special case for non-deteriorating items. Finally, we run several numerical examples to illustrate the problem and provide some managerial insights.     

Optimal deteriorating items production inventory models with random machine breakdown and stochastic repair time

This study develops deteriorating items production inventory models with random machine breakdown and stochastic repair time. The model assumes the machine repair time is independent of the machine breakdown rate. The classical optimization technique is used to derive an optimal solution. A numerical example and sensitivity analysis are shown to illustrate the models. The stochastic repair models with uniformly distributed repair time tends to have a larger optimal total cost than the fixed repair time model, however the production up time is less than the fixed repair time model. Production and demand rate are the most sensitive parameters for the optimal production up time, and demand rate is the most sensitive parameter to the optimal total cost for the stochastic model with exponential distribution repair time.     

An EOQ model for deteriorating items under supplier credits linked to ordering quantity

In the classical inventory economic order quantity (or EOQ) model, it was assumed that the purchaser must pay for the items received immediately. However, in practices, the supplier usually is willing to provide the purchaser a permissible delay of payments if the purchaser orders a large quantity. As a result, in this paper, we establish an EOQ model for deteriorating items, in which the supplier provides a permissible delay to the purchaser if the order quantity is greater than or equal to a predetermined quantity. We then characterize the optimal solution and provide an easy-to-use algorithm to find the optimal order quantity and replenishment time. Finally, several numerical examples are given to illustrate the theoretical results.     

Coordination mechanism for a deteriorating item in a two-level supply chain system

The subject of the coordination between the suppliers and the buyers in multi-echelon inventory-distribution systems has been studied by many researchers. This paper considers a supply chain including a manufacturer and several buyers and assumes that the inventory items deteriorate over time and its inventory level decreases. In order to determine the order policies, coordination over the supply chain is achieved by scheduling the buyers’ delivery days and their coordination with the manufacturer’s production cycle. A mathematical model is developed and analyzed. To test the efficiency of the proposed model, two other models with the supposition of lot-sizing policies with common order cycle and independent deciding are also developed. In comparison to the other two models, the numerical results show that the synchronizing model of production and delivery cycles works better and has less total cost. In addition, in order to encourage the buyers in cooperation, a model on profit sharing is proposed that equitably shares the total savings with all the parties.     

Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate

In this paper, a deterministic inventory model for deteriorating items with two warehouses is developed. A rented warehouse is used when the ordering quantity exceeds the limited capacity of the owned warehouse, and it is assumed that deterioration rates of items in the two warehouses may be different. In addition, we allow for shortages in the owned warehouse and assume that the backlogging demand rate is dependent on the duration of the stockout. We obtain the condition when to rent the warehouse and provide simple solution procedures for finding the maximum total profit per unit time. Further, we use a numerical example to illustrate the model and conclude the paper with suggestions for possible future research.     

Two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under trade credit

In real life situation, it is observed that demand of an item depends on the length of the credit period offered by the retailer to his customers which has a positive impact on demand of an item. But the impact of credit period on demand has received a very little attention by researchers. Furthermore, by allowing shortages as backlogging, the impact on the cost from the decay of the products can be balanced out. A profitable decision policy between a supplier and the retailers can be characterized by an agreement on the permissible delay in payments. Recently, Jaggi et al. (Eur J Oper Res 190:130–135, 2008) have investigated the impact of credit linked demand on the retailer’s optimal replenishment policy. The objective of this study is to extend Jaggi et al. (Eur J Oper Res 190:130–135, 2008) model by incorporating deterioration and backlogging. That is, we formulate a two-echelon inventory model for deteriorating items with credit period dependent demand including shortages under two-level trade credit financing and determine the retailer’s optimal replenishment policy when both the supplier as well as the retailer offers the credit period to stimulate customer demand. Furthermore, we establish some useful theorems to characterize the optimal solution and provide an easy and useful computational algorithm with the help of computer code using the software Matlab 7.0 to determine the optimal shortage point, cycle length, ordering quantity and credit period. A numerical example is included to illustrate the solution procedure for the mathematical model developed. Finally, we implement sensitivity analysis of the optimal solution with respect to the major parameters of the system and obtain some important managerial insights.     

The inventory models under conditional trade credit in a supply chain system

In several recent investigations dealing with the economic order quantity with permissible delay in payments, the following assumptions are made:     

Optimal ordering policies for perishable multi-item under stock-dependent demand and two-level trade credit

In this paper, a multi-item inventory model for perishable items is developed, where the demand rates of the items are stock dependent, two-level trade credit is adopted and the restriction of inventory capacity is also considered. The major objective is to determine the optimal cycle time and order quantities such that the total profit is maximized. The existence and uniqueness of the optimal cycle is discussed by Lagrange approach, and line search algorithms are developed to find the optimal solution of the model. Furthermore, numerical examples are given to illustrate the methods. The sensitivity of the solution to changes in the values of different parameters is also discussed.     

An inventory model for non-instantaneous deteriorating items with quadratic demand rate and shortages under trade credit policy

noindent In this paper, we propose an appropriate inventory model for non-instantaneous deteriorating items over quadratic demand rate with permissible delay in payments and time dependent deterioration rate. In this model, the completely backlogged shortages are allowed. In several existing results, the authors discussed that the deterioration rate is constant in each cycle. However, the deterioration rate of items are not constant in real world applications. Motivated by this fact, we consider that the items are deteriorated with respect to time. To minimize the total relevant inventory cost, we prove some useful theorems to illustrate the optimal solutions by finding an optimal cycle time with the necessary and enough conditions for the existence and uniqueness of the optimal solutions. Finally, we discuss the numerical instance and sensitivity of the proposed model.