An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity

To attract more sales suppliers frequently offer a permissible delay in payments if the retailer orders more than or equal to a predetermined quantity W. In this paper, we generalize [Goyal, S.K., 1985. EOQ under conditions of permissible delay in payments. Journal of the Operational Research Society 36, 335–338] economic order quantity (EOQ) model with permissible delay in payment to reflect the following real-world situations: (1) the retailer’s selling price per unit is significantly higher than unit purchase price, (2) the interest rate charged by a bank is not necessarily higher than the retailer’s investment return rate, (3) many items such as fruits and vegetables deteriorate continuously, and (4) the supplier may offer a partial permissible delay in payments even if the order quantity is less than W. We then establish the proper mathematical model, and derive several theoretical results to determine the optimal solution under various situations and use two approaches to solve this complex inventory problem. Finally, a numerical example is given to illustrate the theoretical results.     

Economic order quantity under conditionally permissible delay in payments

Within the economic order quantity (EOQ) framework, the main purpose of this paper is to investigate the retailer’s optimal replenishment policy under permissible delay in payments. All previously published articles dealing with optimal order quantity with permissible delay in payments assumed that the supplier only offers the retailer fully permissible delay in payments if the retailer ordered a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. However, in this paper, we want to extend this extreme case by assuming that the supplier would offer the retailer partially permissible delay in payments when the order quantity is smaller than a predetermined quantity. Under this condition, we model the retailer’s inventory system as a cost minimization problem to determine the retailer’s optimal inventory cycle time and optimal order quantity. Three theorems are established to describe the optimal replenishment policy for the retailer. Some previously published results of other researchers can be deduced as special cases. Finally, numerical examples are given to illustrate all these theorems and to draw managerial insights.     

An Inventory Model for Deteriorating Items with?Two?Levels of Trade Credit Taking Account of Time Discounting

This paper presents an extended inventory model of Huang (J. Oper. Res. Soc. 54, 1011–1015, 2003), which investigated the retailer’s optimal inventory policy under two levels of trade credit. Herein, we consider the impact of a replenishment policy on the timing of the cash flows associated with payments to suppliers and revenue streams from customers. That is, the same cash amount will possess different money value at different future time. To see this, we adopt the more appropriate net present value (NPV) object instead of the average cost objective. In addition, the deteriorating effects will be incorporated in this inventory model, and the time to deterioration of each item follows an exponential distribution. The discounted cash flow (DCF) approach is used to derive the optimal solution in this study. Furthermore, we first show that the optimal solution not only exists bus also is unique. Then, we provide a theorem to locate the optimal ordering policy. Finally, a numerical example for illustration is provided.     

Some comments on “A simple method to compute economic order quantities”

In this note, we emphasize that the arithmetic–geometric-mean-inequality approach proposed by Teng [Teng, J.T., 2008. A simple method to compute economic order quantities. European Journal of Operational Research. doi:10.1016/j.ejor.2008.05.019] is not a general solution method. Teng’s approach happens to work and give the correct results when the two terms in an objective function are any functions such that their product is a constant. The classical EOQ model works fine since the product of the two terms is indeed a constant! When the product is not a constant, Teng’s approach is of little use. This is exemplified in Comment 1 via solving the EOQ model with complete backorders (where the model is regarded as having two decision variables). Comment 2 is generally valid for an algebraic method when it is used to solve an objective function with two decision variables.     

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.     

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.     

A deterministic EOQ model with delays in payments and price-discount offers

It is the purpose of this paper to model the retailer’s profit-maximizing strategy when confronted with supplier’s trade offer of credit and price-discount on the purchase of merchandise. Generally, retailers have to face many types of demands for different kinds of goods. In real situation, retailers have to correlate between the selling price and supplier’s trade offer, keeping in mind profit-maximization strategy. In the proposed model, all increasing deterministic demands are discussed analytically, numerically and graphically in the environment of permissible delay in payment and discount offer to the retailer.     

Mond–Weir type second order symmetric duality in non-differentiable minimax mixed integer programming problems

A pair of Mond–Weir type non-differentiable second order symmetric minimax mixed integer primal and dual problems in mathematical programming is formulated. Symmetric and self-duality theorems are then established under second order F-pseudo-convexity assumptions. Several known results including that of Gulati and Ahmad [Eur. J. Oper. Res. 101 (1997) 122], Hou and Yang [J. Math. Anal. Appl. 255 (2001) 491] and Mond and Schechter [Bull. Aust. Math. Soc. 53 (1996) 177], as well as others are obtained as special cases.     

Optimal retailer’s replenishment decisions in the EPQ model under two levels of trade credit policy

The main purpose of this paper is to investigate the optimal retailer’s replenishment decisions under two levels of trade credit policy within the economic production quantity (EPQ) framework. We assume that the supplier would offer the retailer a delay period and the retailer also adopts the trade credit policy to stimulate his/her customer demand to develop the retailer’s replenishment model under the replenishment rate is finite. Furthermore, we assume that the retailer’s trade credit period offered by supplier M is not shorter than the customer’s trade credit period offered by retailer N (M ? N). Since the retailer cannot earn any interest in this situation, M < N.     

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 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.     

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.     

A Production Lot-Size Model for Perishable Items Under Two Level Trade Credit Policy for a Retailer with a Powerful Position in a Supply Chain System

This paper investigates a production lot-size inventory model for perishable items under two levels of trade credit for a retailer to reflect the supply chain management situation. We assume that the retailer maintains a powerful position and can obtain full trade credit offered by supplier yet retailer just offers the partial trade credit to customers. Under these conditions, retailer can obtain the most benefits. Then, we investigate the retailer’s inventory policy as a cost minimization problem to determine the retailer’s inventory policy. A rigorous mathematical analysis is used to prove that the annual total variable cost for the retailer is convex, that is, unique and global-optimal solution exists. Mathematical theorems are developed to efficiently determine the optimal ordering policies for the retailer. The results in this paper generalize some already published results. Finally, numerical examples are given to illustrate the theorems and obtain a lot of managerial phenomena.     

A note on fuzzy inventory model with storage space and budget constraints

In 1997, Roy and Maiti developed a fuzzy EOQ model with fuzzy budget and storage capacity constraints where demand is influenced by the unit price and the setup cost varies with the quantity purchased [T.K. Roy, M. Maiti, A fuzzy EOQ model with demand-dependent unit cost under limited storage capacity, Eur. J. Oper. Res. 99 (1997) 425–432]. However, their procedure has some questionable points and their numerical examples contain rather peculiar results. The purpose of this paper is threefold. First, for the same inventory model with fuzzy constraints, based on the max–min operator, we proposed an improved solution procedure. Second, we review the solution procedure by Roy and Maiti that is based on Kuhn–Tucker approach to point out their questionable results. Third, we compare Roy and Maiti’s approach with ours to explain why our approach can solve the problem and theirs cannot. Numerical examples provided by them also support our findings.     

Viscosity approximation methods for a family of finite nonexpansive mappings in Banach spaces

Viscosity approximation methods for a family of finite nonexpansive mappings are established in Banach spaces. The main theorems extend the main results of Moudafi [Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55] and Xu [Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291] to the case of finite mappings. Our results also improve and unify the corresponding results of Bauschke [The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996) 150–159], Browder [Convergence of approximations to fixed points of nonexpansive mappings in Banach spaces, Archiv. Ration. Mech. Anal. 24 (1967) 82–90], Cho et al. [Some control conditions on iterative methods, Commun. Appl. Nonlinear Anal. 12 (2) (2005) 27–34], Ha and Jung [Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990) 330–339], Halpern [Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967) 957–961], Jung [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005) 509–520], Jung et al. [Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2) (2005) 125–135], Jung and Kim [Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1) (1997) 93–102], Lions [Approximation de points fixes de contractions, C.R. Acad. Sci. Ser. A-B, Paris 284 (1977) 1357–1359], O’Hara et al. [Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003) 1417–1426], Reich [Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292], Shioji and Takahashi [Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (12) (1997) 3641–3645], Takahashi and Ueda [On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984) 546–553], Wittmann [Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992) 486–491], Xu [Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2) (2002) 240–256], and Zhou et al. [Strong convergence theorems on an iterative method for a family nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., in press] among others.     

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.     

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.     

Pricing non-storable perishable goods by using a purchase restriction: General optimality results

The aim of this paper is to investigate the validity of the optimality results derived from [Eur. J. Oper. Res. 134 (2001) 631] by relaxing the following two assumptions: (a) the firm offers restricted units first and then unrestricted units later at higher price levels, and (b) only one type of product is available during the whole selling process. In the absence of (a), we establish a general optimality theorem, which shows that any optimal policy in [Eur. J. Oper. Res. 134 (2001) 631] remains to be optimal in the class of general policies that allow the restriction to be attached at any price level. For the simultaneous availability issue associated with (b), we demonstrate that there always exists an optimal policy that is sustainable even when all active prices are made available at the same time. These two results assure the relevance of the theoretical model in [Eur. J. Oper. Res. 134 (2001) 631] to current yield management practices and further improve our understanding on the role and the impact of a well-designed purchase restriction on pricing decisions.     

Weak subdifferential for set-valued mappings and its applications

In this paper, the existence theorems of two kinds of weak subgradients for set-valued mappings, which are the generalizations of Theorem 7 in [G.Y. Chen, J. Jahn, Optimality conditions for set-valued optimization problems, Math. Methods Oper. Res. 48 (2) (1998) 187–200] and Theorem 4.1 in [J.W. Peng, H.W.J. Lee, W.D. Rong, X.M. Yang, Hahn–Banach theorems and subgradients of set-valued maps, Math. Methods Oper. Res. 61 (2005) 281–297], respectively, are proved by virtue of a Hahn–Banach extension theorem. Moreover, some properties of the weak subdifferential for set-valued mappings are obtained by using a so-called Sandwich theorem. Finally, necessary and sufficient optimality conditions are discussed for set-valued optimization problems, whose constraint sets are determined by a fixed set and a set-valued mapping, respectively.     

The role of retailer’s performance in optimal wholesale price discount policies

The main goal of this paper is to model the effects of wholesale price control on manufacturer’s profit, taking explicitly into account the retailer’s sales motivation and performance. We consider a stylized distribution channel where a manufacturer sells a single kind of good to a single retailer. Wholesale price discounts are assumed to increase the retailer’s motivation thus improving sales. We study the manufacturer’s profit maximization problem as an optimal control model where the manufacturer’s control is the discount on wholesale price and retailer’s motivation is one of the state variables. In particular in the paper we prove that an increasing discount policy is optimal for the manufacturer when the retailer is not efficient while efficient retailers may require to decrease the trade discounts at the end of the selling period. Computational experiments point out how the discount on wholesale price passed by the retailer to the market (pass-through) influences the optimal profit of the manufacturer.