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.     

An EOQ model for deteriorating items with price- and stock-dependent selling rates under inflation and time value of money

This study applies the discounted cash flow (DCF) approach for the analysis of a replenishment problem over a finite planning horizon. Thus, a deterministic economic order quantity (EOQ) inventory model taking into account inflation and time value of money is developed for deteriorating items with price- and stock-dependent selling rates. An efficient solution procedure is presented to determine the optimal number of replenishment, the cycle time and selling price. Then the optimal order quantity and the total present value of profits are obtained. Numerical examples are presented to illustrate the proposed model and particular cases of the model are also discussed.     

A note on the inventory model for deteriorating items with trapezoidal type demand rate

In this paper, we study the inventory model for deteriorating items with trapezoidal type demand rate, that is, the demand rate is a piecewise linearly function. We proposed an inventory replenishment policy for this type of inventory model. The numerical solution of the model is obtained and also examined.     

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.     

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 EPQ model for deteriorating items with inventory-level-dependent demand and permissible delay in payments

This article develops an inventory model for exponentially deteriorating items under conditions of permissible delay in payments. Unlike the existing related models, we assume that the items are replenished at a finite rate and the demand rate of the items is dependent on the current inventory level. The objective is to determine the optimal replenishment policies in order to maximise the system's average profit per unit of time. A simple method is shown for finding the optimal solution of the model based on the derived properties of the objective function. In addition, we deduce some previously published results as the special cases of the model. Finally, numerical examples are used to illustrate the proposed model. Some managerial insights are also inferred from the sensitive analysis of model parameters.     

Inventory model with two rates of production for deteriorating items with permissible delay in payments

Some decentralized smoothing problems are solved by applying a forward-pass fixed-interval smoother formula in discrete-time systems. It is assumed that a simple estimation structure consists of a global processor and of two local processors. Two cases are considered for the problems of decentralized smoothing and smoothing update: when the local backward-pass information filtered estimates are available, and when the local-smoothed estimates are available. Some features of present algorithms are discussed from the point of view of data transmissions and numerical computations, etc.     

An EOQ model for deteriorating items with time-varying demand and shortages

The study explores the inventory replenishment policies for the cases with time-varying demand, linearly increasing deterioration rate, partial back-ordering, constant service level and equal replenishment intervals over a fixed planning horizon. Since it is difficult to solve the problem directly, we derive the upper bound of replenishment number for a specific planning horizon and find the solution of service level under a given number of replenishment. The optimal solutions of replenishment number and service level are then determined. Numerical examples and sensitivity analyses are also provided to illustrate the solution procedure.     

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.     

An economic production quantity model for deteriorating items with preventive maintenance policy and random machine breakdown

In recent years, many researches on economic production quantity (EPQ) models with machine breakdown and preventive maintenance have been developed, but few of them have developed integrated models for deteriorating items. In this study, we develop EPQ models for deteriorating items with preventive maintenance, random machine breakdown and immediate corrective action. Corrective and preventive maintenance times are assumed to be stochastic and the unfulfilled demands are lost sales. Two EPQ models of uniform distribution and exponential distribution of corrective and maintenance times are developed. An example and sensitivity analysis is given to illustrate the models. For the exponential distribution model, it is shown that the corrective time parameter is one of the most sensitive parameters to the optimal total cost.     

A lot sizing model with partial downstream delayed payment,partial upstream advance payment,and partial backordering for deteriorating items

In practice, installment of cost of lots is very common between manufacturers and distributors, and many distributors pay the purchasing cost in equal installments (i.e., an upstream partial prepayment). In a similar fashion, distributors allow retailers to pay the cost of purchased goods after such goods are received (i.e., a downstream partial delay payment). In this research we apply a model of economic order quantity (EOQ) in supply chains with partial downstream delayed payment and partial upstream advance payment for a deteriorating item under three conditions: 1) shortage is not allowed, 2) full back ordering is allowed, and 3) partial back ordering is allowed. We prove the convexity of the cost functions and derive closed form global optimal solutions for the decision variables of all models. Eventually, to demonstrate the theoretical results and managerial options, we perform some computational examples.     

EOQ inventory model for items with Weibull distribution deterioration,time-varying demand and partial backlogging

The paper presents an EOQ inventory model that is depleted not only by time-varying demand but also by Weibull distribution deterioration, in which shortages are allowed and partially backordered. The backlogging rate is variable and dependent on the waiting time for the next replenishment. Further, the optimal procedure was independent of the form of the demand rate. It is then illustrated with the help of four numerical examples. The sensitivity analysis is also studied.     

An inventory model for deteriorating items under stock-dependent demand and controllable deterioration rate

In this paper, we formulate a deteriorating inventory model with stock-dependent demand by allowing preservation technology cost as a decision variable in conjunction with replacement policy. Moreover, it is assumed that the shortages are allowed and partially backlogged, depending on the length of the waiting time for the next replenishment. The objective is to find the optimal replenishment and preservation technology investment strategies while maximizing the total profit per unit time. For any given preservation technology cost, we first prove that the optimal replenishment schedule not only exists but is unique. Next, we show that the total profit per unit time is a concave function of preservation technology cost when the replenishment schedule is given. We then provide a simple algorithm to find the optimal preservation technology cost and replenishment schedule for the proposed model. Finally, we use some numerical examples to illustrate the model.     

Optimal ordering decision for deteriorating items with expiration date and uncertain lead time

The ordering policy for the retailers and the suppliers is a function of deterioration, product expiration date, the supplier’s uncertain lead time, available capital constraint and the retailer’s seasonal pattern demand. We develop a deteriorating inventory replenishment model of the system and present an algorithm to derive the retailer’s optimal replenishment cycle, shortage period, order quantity and the supplier’s managing cost. Numerical example and sensitivity analysis are given to illustrate the model.     

Economic production quantity models for deteriorating/imperfect products and service with rework

In this paper, two economic production quantity (EPQ) models are proposed for deteriorating/imperfect items with rework process. The production process is imperfect, and imperfect quality items are reworked to become serviceable. At the same time, the remaining good quality items may deteriorate. The inspection of deteriorated items is also imperfect, so that deteriorated items may be sold to customers, which will create negative impact on corporate image. In the first model, a single production-rework plant system is considered. The optimal production times and the economic production quantities are obtained analytically. In the second model, a system that consists of n production plants and one rework plant is considered. A solution procedure is developed in order to obtain the optimal operating cost. Numerical examples are provided to compare the performance of the two systems.     

Single supplier multiple cooperative retailers inventory model with quantity discount and permissible delay in payments

We consider in this paper an Economic Order Quantity (EOQ) problem involving a single supplier that offers quantity discounts and allows retailers to delay payments. The retailers are tempted to form coalitions in order to minimize their costs. We propose a solution approach that generates stable coalition structures for the retailers taking into account the delay in payments and the discount quantity offered by the supplier. The proposed approach includes a decision rule that generates preferred coalitions for each retailer. Our decision rule reduces considerably the number of explored coalition structures in order to determine solutions in the core.     

Optimal order policy in response to announced price increase for deteriorating items with limited special order quantity

When a supplier announces an impending price increase due to take effect at a certain time in the future, it is important for each retailer to decide whether to purchase additional stock to take advantage of the present lower price. This study explores the possible effects of price increases on a retailer's replenishment policy when the special order quantity is limited and the rate of deterioration of the goods is assumed to be constant. The two situations discussed in this study are as follows: (1) when the special order time coincides with the retailer's replenishment time and (2) when the special order time occurs during the retailer's sales period. By analysing the total cost savings between special and regular orders during the depletion time of the special order quantity, the optimal order policy for each situation can be determined. We provide several numerical examples to illustrate the theories in practice. Additionally, we conduct a sensitivity analysis on the optimal solution with respect to the main parameters.     

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.     

Finite horizon EOQ model for non-instantaneous deteriorating items with price and advertisement dependent demand and partial backlogging under inflation

This paper deals with an economic order quantity (EOQ) model for non-instantaneous deteriorating items with price and advertisement dependent demand pattern under the effect of inflation and time value of money over a finite planning horizon. In this model, shortages are allowed and partially backlogged. The backlogging rate is dependent on the waiting time for the next replenishment. This paper aids the retailer in minimising the total inventory cost by finding the optimal interval and the optimal order quantity. An algorithm is designed to find the optimum solution of the proposed model. Numerical examples are given to demonstrate the results. Also, the effect of changes in the different parameters on the optimal total cost is graphically presented and the implications are discussed in detail.     

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.