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.     

Coordinating replenishment and pricing policies for non-instantaneous deteriorating items with price-sensitive demand

This article will formulate and solve an inventory system with non-instantaneous deteriorating items and price-sensitive demand. The purpose of this study is to determine the optimal selling price and the length of replenishment cycle such that the total profit per unit time has a maximum value for the retailer. We first establish a proper model for a mathematical formulation. Then we develop several theoretical results and provide the decision-maker with an algorithm to find the optimal solution. Finally, two numerical examples are provided to illustrate the solution procedure, and a sensitivity analysis of the optimal solution with respect to major parameters is carried out.     

Optimal control of a production inventory system with deteriorating items

This paper is concerned with the problem of controlling the production rate of a production inventory system with deteriorating items. Under reasonable conditions, we derive the closed form optimal control of a cost minimization problem and a profit maximization problem. The derived analytical solutions yield good insight on how production planning tasks can be carried out. In addition, they also provide guidelines for further study and development of numerical methods for more complex systems involving more general exogenous functions.     

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 note on inventory replenishment policies for deteriorating items in an exponentially declining market

In determining the replenishment policy for an inventory system, some researchers advocated that the iterative method of Newton could be applied to the derivative of the total cost function in order to get the optimal solution. But, this approach suffers some drawbacks. In this paper we first show the inventory carrying cost is in proportion to the cost of deteriorated items, then offer a formulated approximated solution. Our formula solution is accurate enough to be taken as an optimal solution. Sensitivity analysis shows that our solution is stable. Finally, we give some numerical examples to show the applicability of our solution.Scope and purposeGhare and Schrader initiated the discussion of deteriorating items in the inventory system. After that, many researchers presented diverse models with different assumptions on patterns of deterioration with/without shortages. Demand variation has also impact on inventory policies. In order to determine the optimal lot-size, Wee and Chung & Tsai advocated that the Newton's method is an adequate method. In this paper we illustrate some drawbacks of their approach, mainly the convergence and optimality problem. Thereby, we provide a formulated approximated solution, which is accurate enough to be taken as optimal.     

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.     

On inventory replenishment with non-linear increasing demand

We present a consecutive improvement approach to solve the problem of determining the timing and size of replenishments in order to minimize the total inventory costs in the case of a non-linear increasing demand pattern. The proposed procedure is computationally simple compared to other methods. It considers the replenishment in reverse time sequence with the time horizon being exactly the ending point and it thus guarantees to meet the demand over the preestablished time frame. The concept of inventory correspondence is introduced. Numerical examples are also presented and compared with other heuristics using an analytic procedure for replenishment with non-linear increasing demand.Scope and purposeYang et al. [2] is one of the best heuristic algorithms known today for determining the optimal replenishment policy when demand is non-linear increasing. This article proposes a consecutive improvement method which is effective, computationally simple, and intuitively appealing from the graphical illustration. Numerical tests show the proposed method even outperforms Yang's procedure.     

Optimizing the pricing and replenishment policy for non-instantaneous deteriorating items with stochastic demand and promotional efforts

Retailer promotional activity has become prevalent in the business world. Promotional efforts impact the replenishment policy and the sale price of goods. In this paper, the problem of replenishment policy and pricing for non-instantaneous deteriorating items subject to promotional effort is considered. We adopt a price dependent stochastic demand function in which shortages are allowed and partially backlogged. The major objective is to simultaneously determine the optimal selling price, the optimal replenishment schedule, and the optimal order quantity to maximize the total profit. First, we prove that a unique optimal replenishment schedule exists for any given selling price. Second, we prove that the total profit is a concave function of price. Third, we present an algorithm to obtain the optimal solution and solve a numerical example. Last, we extend the numerical example by performing a sensitivity analysis of the model parameters and discuss specific managerial insights.     

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.     

Economic production quantity models for deteriorating items with price- and stock-dependent demand

Large piles of consumer goods displayed in a supermarket are often associated with on sale items to induce more sales and profits. In this paper, we first establish an economic production quantity (or EPQ) model for deteriorating items when the demand rate depends not only the on-display stock level but also the selling price per unit. In addition, we impose a ceiling on the number of on-display stocks because too much stock leaves a negative impression on the buyer and the amount of shelf/display space is limited. We then provide the necessary conditions to determine an optimal solution that maximizes profits for the EPQ model. Finally, sensitivity analysis is applied on the parameter effects of the optimal price and production run time.     

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...     

A partial backlogging production-inventory lot-size model for deteriorating items with time-varying production and demand rate over a finite time horizon

In this article, I extend Balkhi ((2001), ‘On a Finite Horizon Production Lot Size Inventory Model for Deteriorating Items: An Optimal Solution’, European Journal of Operational Research, 132, 210–223), by considering a generalised mathematical production-inventory model for deteriorating items with partial backlogging. The demand, production and backlogging rates are assumed to be continuous and varying with time. The objective is to find the optimal production restarting and stopping time to keep the total relevant cost as low as possible. To ascertain the optimal solution exists, the conditions for the total relevant cost in the system which attains its global minimum are provided. In addition, based on the minimum total relevant cost, an alternative among the proposed four cases is also suggested. Finally, a numerical example and sensitivity analysis is illustrated and some management insights are presented.     

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.     

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.     

Two-warehouse partial backlogging inventory model for deteriorating items with linear trend in demand under inflationary conditions

In today's business transactions, there are various reasons, namely, bulk purchase discounts, re-ordering costs, seasonality of products, inflation induced demand, etc., which force the buyer to order more than the warehouse capacity. Such situations call for additional storage space to store the excess units purchased. This additional storage space is typically a rented warehouse. Inflation plays a very interesting and significant role here: It increases the cost of goods. To safeguard from the rising prices, during the inflation regime, the organisation prefers to keep a higher inventory, thereby increasing the aggregate demand. This additional inventory needs additional storage space, which is facilitated by a rented warehouse. Ignoring the effects of the time value of money and inflation might yield misleading results. In this study, a two-warehouse inventory model with linear trend in demand under inflationary conditions having different rates of deterioration has been developed. Shortages at the owned warehouse are also allowed subject to partial backlogging. The solution methodology provided in the model helps to decide on the feasibility of renting a warehouse. Finally, findings have been illustrated with the help of numerical examples. Comprehensive sensitivity analysis has also been provided.     

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.