A production lot size inventory model for deteriorating items with time-varying demand

This paper presents a production lot size model for deteriorating items with time-varying demand. The replenishment cycle and deterioration rates are allowed to vary over a finite planning horizon. A solution procedure for this model is developed to provide the optimal replenishment cycle number. Also three numerical examples are solved for different deterioration rates. Finally, a sensitivity analysis is conducted to study the effect of changes in the related parameters on the objective function.     

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 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.     

A multi-warehouse inventory model for items with time-varying demand and shortages

This paper develops a deterministic replenishment model with multiple warehouses (one is an owned warehouse and others are rented warehouses) possessing limited storage capacity. In this model, the replenishment rate is infinite. The demand rate is a function of time and increases at a decreasing rate. The stocks of rented warehouses are transported to owned warehouse in continuous release pattern. The model allows shortages in owned warehouse and permits part of the backlogged shortages to turn into lost sales—which is assumed to be a function of the currently backlogged amount. The solution procedure for finding the optimal replenishment policy is shown. As a special case of the model, the corresponding models with completely backlogged shortages and without shortages are also presented. The models are illustrated with the help of numerical examples. Sensitivity analysis of parameters is given in graphical form.Scope and purposeIn practical inventory management, there exist many factors like an attracted price discount for bulk purchase, etc. to make retailers buy goods more than the capacity of their owned warehouse. In this case, retailers will need to rent other warehouses or to rebuild a new warehouse. However, from economical point of views, they usually choose to rent other warehouses. If there are multiple warehouses available, an important problem faced by the retailers is which warehouses to be selected to hold items replenished, when to replenish as well as what size to replenish. For such a problem, the existing two-warehouse models, based on an unrealistic assumption that the rented warehouse has unlimited storage capacity, presented some procedures for determining the optimal replenishment policy. This paper extends the existing two-warehouse models in three directions. Firstly, the traditional two-warehouse models assumed the storage capacity of the rented warehouse unlimited. The present paper relaxes this impractical assumption and considers the situation with multiple rented warehouses having a limited capacity. Secondly, the traditional two-warehouse models considered a constant demand rate or a linearly increasing demand rate. In this model, the demand rate varies over time and increases at a decreasing rate, which implies an increasing market going to saturation. Thirdly, we extend the two-warehouse models to the case with partially backlogged shortages. The purpose of this paper is to build a multi-warehouse replenishment model to help decision-makers solve the problem of which warehouses to be chosen to store items replenished and how to replenish.     

A production lot size inventory model for deteriorating items

In this paper, an attempt has been made to develop a production lot size model which incorporates an unfilled-order backlog for an inventory system with exponential decaying items. Approximate expressions are obtained for the optimum production lot size, the production cycle time and the total cycle time. The theory is illustrated by considering a numerical example of this class. Also it is shown that earlier models developed by Ghare and Schrader [3] and Misra [5] can be obtained as particular cases by choosing appropriate values for the various parameters of the model.     

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.     

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.     

A two-warehouse inventory model for non-instantaneous deteriorating items with interval-valued inventory costs and stock-dependent demand under inflationary conditions

Neural Computing and Applications - This research work develops a two-warehouse inventory model for non-instantaneous deteriorating items with interval-valued inventory costs and stock-dependent...     

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.     

Theory and methodology on the global optimal solution to a General Reverse Logistics Inventory Model for deteriorating items and time-varying rates

In this paper, we present a unified general inventory model for integrated production of new items and remanufacturing of returned items for an infinite planning horizon. Our model considers a production environment that consists of three shops. The first shop is for remanufacturing returned items, the second shop is for manufacturing new items, while the third shop is for collecting returned items to be remanufactured in the first shop. The system is subject to joint production and remanufacturing options, the first one is to produce new items while the second one is to reproduce/recycle the returned items “as-good-as new”. Items deteriorate while they are in storage, and production, remanufacturing, demand, return, and deterioration rates are arbitrary functions of time. A closed form for the total relevant costs as well as a rigorous mathematical proof, which shows the global optimality of the solution to the underlying inventory system are introduced. Illustrative examples, which explain the application of the theoretical results as well as their numerical verifications, are also given.     

On conflict and cooperation in a two-echelon inventory model for deteriorating items

Four scenarios are proposed concerning cooperative behavior for inventory policies between suppliers and retailers: no information is shared; the supplier is dominant during negotiations with retailers; the retailer is dominant during negotiations with suppliers; and the supplier and retailer cooperate. Unlike other studies, we consider deteriorating items and permit completed backorders, with a fixed service rate, in the models for these four scenarios. We explore the optimality of these models and present a procedure to find the optimal solution. Numerical examples are provided to illustrate the procedure, which are also used for sensitivity analysis. The results show that the cooperation scenario with information sharing is the best way to reach a win–win position. However, some compensation programs might be required to persuade suppliers or retailers to cooperate when one of them faces a loss of profits in a cooperative scenario.     

Optimal inventory policies for deteriorating complementary and substitute items

The optimal production for an inventory control system of deteriorating multi-items where items are either complementary and/or substitute is formulated with a resource constraint. Here, the production function is unknown and considered as a control variable. Also, the deterioration rates of the items are either stock dependent or constant. The demand is stock dependent, shortages are not allowed and deteriorated items are salvaged. The total profit, which consists of the sales proceeds, production cost, inventory holding cost, salvage value, is formulated as a Pontryagin's Optimal Control problem for both steady and transient states and evaluated using Taylor's theorem, generalised reduced gradient technique and optimal control theory satisfying the Generalised Legendre conditions. The model is formulated in general form for n-items, and in particular, is illustrated with three items for some numerical data. The optimum results are presented both in tabular form and graphically.     

Combined optimal price and optimal inventory replenishment policies when a sale results in increase in demand

When a supplier reduces the price of a product temporarily a buyer might place a large order and offer a sale on these units to its customers. In most cases a price discount results in an increase in demand. In this paper we relax the constant demand assumption made in most studies of inventory systems with price changes. We analyze the options available to a buyer and develop profit functions for different combinations of sales period and replenishment time and present optimal ordering policies. The paper also presents a procedure to include any relationship between price and demand to determine the combined optimal price and optimal order quantity.     

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.     

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.     

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.     

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.     

Analyzing an inventory system with multiple reorder points and periodic replenishment

This paper investigates multiple reorder point, periodic replenishment systems similar to those utilized on board some U.S. naval vessels for Selected Item Management (SIM). This inventory system is of interest since it involves both regular and priority replenishment, three reorder points, and cyclical restocking of regular orders. A simulation model is developed to study the impact of demand distribution, cycle time, priority order leadtime, and the quantity of units ordered on the performance of the system. It is found that the frequency of service and days without shortages performance measures can be maximized by setting the reorder points at equal intervals between the zero level and the high level. Also, a power approximation model is presented that estimates the high level that will lead to a desired percent days without shortage percentage.     

An inventory model for deteriorating items with partial backlogging and permissible delay in payments

The paper deals with an inventory model with a varying rate of deterioration and partial backlogging rate under the condition of permissible delay in payments. The existing literature on the subject generally deal with situations where the payment of an order is made on the receipt of items by the inventory system and shortages are either completely backlogged or fully lost. In this paper, a varying deterioration rate of time and the condition of permissible delay in payments used in conjunction with the economic order quantity model are the focus of discussion. In addition, the shortages are neither completely backlogged nor completely lost assuming the backlogging rate to be inversely proportional to the waiting time for the next replenishment. Numerical examples are presented to illustrate the model.     

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.