Optimal Number Of Replenishments Research Articles

An Ordering Policy for Deteriorating Items with Hybrid Demand Under Permissible Delay in Payment and Price Inflation

ABSTRACT: The paper studies an in ventory model for deteriorating items under price inflation and permissible delay in payment. The demand rate is assumed to be hybrid, which is observed for commodities like coffee, cotton, tin, copper, wristwatch, automobile, etc. The model is studied over a finite planning horizon with unequal cycle lengths. A solution procedure is indicated to determine the optimal number of replenishments, cycle times and optimal order quantities.

A two warehouse deterministic inventory model for deteriorating items with a linear trend in time dependent demand over finite time horizon by Elitist Real-Coded Genetic Algorithm

This paper deals with a deterministic inventory model developed for deteriorating items having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linear time dependent demand (increasing) over a fixed finite time horizon. The model is formulated with infinite replenishment and the successive replenishment cycle lengths are in arithmetic progression. Partially backlogged shortages are allowed. The stocks of rented warehouse (RW) are transported to the owned warehouse (OW) in continuous release pattern. For this purpose, the model is formulated as a constrained non-linear mixed integer programming problem. For solving the problem, an advanced genetic algorithm (GA) has been developed. This advanced GA is based on ranking selection, elitism, whole arithmetic crossover and non-uniform mutation dependent on the age of the population. Our objective is to determine the optimal replenishment number, lot-size of two-warehouses (OW and RW) by maximizing the profit function. The model is illustrated with four numerical examples and sensitivity analyses of the optimal solution are performed with respect to different parameters.

Deterministic ordering policy with price- and stock-dependent demand under fluctuating cost and limited capacity

In this paper, we extend the classical economic order quantity model to allow for the demand rate influenced by both selling price and displayed stock level. A finite period system is considered under inflation for deteriorating items. In addition, we impose a ceiling on the number of on-display stocks. We also allow for shortages and general partial backlogging. The objective is to maximize the discounted total profit by finding the optimal replenishment number, replenishment schedule and selling price changed periodically. An effective search procedure is provided to find the optimal solution by employing the properties derived in this paper and particle swarm optimization algorithm. Several numerical examples are used to illustrate the features of the proposed model.

Economic order quantity models for deteriorating items and partial backlogging when demand is quadratic in time

In this paper, we extend the inventory lot-size model introduced in Ghosh and Chaudhuri (2006) to allow for partial backlogging. In addition, we also relax their assumptions of equal replenishment cycles and constant shortage lengths. For any given number of replenishment cycles, we show that the optimal replenishment schedule exists uniquely. We then prove that the total relevant cost is a convex function of the number of replenishments. Consequently, we propose an algorithm to find the optimal replenishment number and schedule. Finally, we run several numerical examples, and obtain some managerial implications by using sensitivity analyses. [Received 18 November 2009; Revised 18 February 2010; Revised 27 April 2010; Accepted 28 April 2010]

A particle swarm optimization for solving joint pricing and lot-sizing problem with fluctuating demand and unit purchasing cost

In this paper, we extend the classical economic order quantity model to allow for not only a function of price-dependent and time-varying demand but also fluctuating unit purchasing cost. The joint replenishment problem is subject to continuous decay and a general partial backlogging rate. The objective is to find the optimal replenishment number, time scheduling and periodic selling price to maximize the discounted total profit. An effective search procedure is provided to find the optimal solution by employing the properties derived in this paper and particle swarm optimization algorithm. Several numerical examples are used to illustrate the features of the proposed model.

Optimal pricing for a perishable item under time-price dependent demand and time-value of money

The present article develops an inventory model for a perishable item assuming that the demand rate depends not only on time, but also on selling price of the item. The deterioration rate of the item is taken to be constant and time-value of money is also considered. Shortages in inventory are allowed in all cycles and are completely backlogged. Planning horizon in the inventory problem is considered to be finite. The solution of the model determines the optimal number of replenishments, order quantity and price over finite planning horizon so that time-value of total profit associated with the inventory system is maximised. A heuristic approach is presented to determine the optimal solution of the proposed problem. The solution procedure is illustrated by four numerical examples. Sensitivity analysis of each example is carried out to study the effect of changes in the input parameters. All the results are graphically illustrated.

An inventory model under inflation for deteriorating items with stock-dependent consumption rate and partial backlogging shortages

In this paper, we extend Teng, J.T., Chang, H.J., Dye, C.Y., Hung, C.H. [2002. An optimal replenishment policy for deteriorating items with time-varying demand and partial backlogging. Operations Research Letters 30(6), 387–393.] and Hou, K.L. [2006. An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting. European Journal of Operational Research 168(2), 463–474.] by considering an inventory lot-size model under inflation for deteriorating items with stock-dependent consumption rate when shortages are partial backlogging. The proposed model allows for (1) partial backlogging, (2) time-varying replenishment cycles, and (3) time-varying shortage intervals. Consequently, the proposed model is in a general framework that includes numerous previous models as special cases. We then prove that the optimal replenishment schedule exists uniquely, and provide a good estimate for finding the optimal replenishment number. Furthermore, we briefly discuss some special cases of the proposed model related to previous models. Finally, numerical examples to illustrate the solution process and some managerial implications are provided.

Inventory control as a discrete system control for the fixed-order quantity system

This paper shows how to model a problem to find optimal number of replenishments in the fixed-order quantity system as a basic problem of optimal control of the discrete system. The decision environment is deterministic and the time horizon is finite. A discrete system consists of the law of dynamics, control domain and performance criterion. It is primarily a simulation model of the inventory dynamics, but the performance criterion enables various order strategies to be compared. The dynamics of state variables depends on the inflow and outflow rates. This paper explicitly defines flow regulators for the four patterns of the inventory: discrete inflow – continuous/discrete outflow and continuous inflow – continuous/discrete outflow. It has been discussed how to use suggested model for variants of the fixed-order quantity system as the scenarios of the model. To find the optimal process, the simulation-based optimization is used.

Cost-Based Filtering Techniques for Stochastic Inventory Control Under Service Level Constraints

This paper1 considers a single product and a single stocking location production/inventory control problem given a non-stationary stochastic demand. Under a widely-used control policy for this type of inventory system, the objective is to find the optimal number of replenishments, their timings and their respective order-up-to-levels that meet customer demands to a required service level. We extend a known CP approach for this problem using three cost-based filtering methods. Our approach can solve to optimality instances of realistic size much more efficiently than previous approaches, often with no search effort at all.

A backward recursive algorithm for inventory lot-size models with power-form demand and shortages

In 1978, Barbosa and Friedman established a ‘general root law’ for power-form demand rate. Shortages are prohibited. However, in reality, shortages may occur and the planning time horizon is usually finite as the product life cycle is short. Therefore, the author here extends their inventory lot-size model to allow for shortages during a finite planning time horizon and propose an alternative algorithm in a backward manner to determine the optimal replenishment timing without using iterative search schemes. The study not only shows that the total relevant cost is a strictly convex function of the replenishment number, but also provides the condition for determining the optimal replenishment number. Finally, a numerical example is given for illustration.

Deterministic inventory lot-size models with shortages for fluctuating demand and unit purchase cost

The classical economic order quantity (EOQ) model assumes not only a constant demand rate but also a fixed unit purchasing cost. In today's time-based competition, the unit cost of a high-tech product declines significantly over its short product life cycle while its demand increases. Therefore, using the classical EOQ formulation for a high-tech product will cause varying magnitudes of error. In addition, the cost of purchases as a percentage of sales is often substantial. Consequently, adding the purchasing strategy into the EOQ model is vital. In this paper, we assume that not only the demand function but also the unit purchase cost is fluctuating with time. We then provide an easy-to-use algorithm to find the optimal replenishment number and schedule. In a numerical example, we show that the total cost obtained by our proposed model is 32.4% less expensive than that obtained by the classical EOQ model.

Deterministic economic order quantity models with partial backlogging when demand and cost are fluctuating with time

In today's time-based competition, the unit cost of a high-tech product declines significantly over its short life cycle while its demand increases. In this paper, we extend the classical economic order quantity model to allow for not only time-varying demand but also fluctuating unit cost. In addition, we also allow for shortages and partial backlogging. We then prove that the optimal replenishment schedule not only exists but also is unique. In addition, we also show that the total cost is a convex function of the number of replenishments, which simplifies the search for the optimal number of replenishments to find a local minimum. Moreover, we further simplify the search process by providing an intuitively good starting search point.

AN INVENTORY MODEL FOR DETERIORATING ITEMS WITH STOCK-DEPENDENT DEMAND UNDER THE CONDITIONS OF INFLATION AND TIME-VALUE OF MONEY

ABSTRACT In this paper, we incorporate the effects of inflation and time-value of money in inventory decision making when demand, at each time moment rather than being constant, is considered to be dependent upon current stock level. In addition, the shortages are neither completely backlogged nor completely lost assuming the backlogging rate to be linearly dependent on the amount of demand backlogged. We shall be concerned with finding the optimal number of replenishments and service rate to minimize the total relevant costs over a finite planning horizon. Numerical examples are presented to illustrate the proposed models.

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 purpose In 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.

A forward recursive algorithm for inventory lot-size models with power-form demand and shortages

Barbosa and Friedman (L.C. Barbosa, M. Friedman, Management Science 24 (8) (1978) 819) establish an optimal replenishment policy for power-form demand rate. In this paper, we extend their inventory lot-size model to allow for shortages. The goal is to find the optimal number and time of replenishments in order to keep the total relevant cost as low as possible during a finite planning horizon. We develop a simple forward recursive algorithm to determine the optimal replenishment timing. Furthermore, we propose an intuitively accurate estimate for the optimal number of replenishments, which significantly reduces computational complexity in finding the optimal replenishment number. A numerical example is provided to illustrate the solution procedure.

Exact solution of inventory replenishment policy for a linear trend in demand – two-equation model

This study presents a model for obtaining the exact optimal solution of inventory replenishment policy problem. The proposed model, called the “two-equation model”, includes two governing equations and a time-frequency algorithm: the first equation determines the optimal replenishment times for a specified number of replenishments; the second equation determines the optimal number of replenishments. The time-frequency algorithm includes two main procedures to solve the first and second equations. In contrast to many approximation approaches, this model exactly solves the first equation, a simultaneous non-linear equations system using a generalized matrix-based solver. This study also examines the classical no-shortage inventory replenishment policy for linear increasing and decreasing trend demand. According to these results, the solution given by the two-equation model is an exact solution.

Inventory lot‐size policies for the bass diffusion demand models of new durable products

The Bass model and its revised forms have been used for forecasting dynamic demand growth in retail service, industrial technology, consumer durable goods, and others. However, the inventory models currently available do not recognize the Bass diffusion demand pattern. In this paper, we first establish the necessary and sufficient conditions to solve the problem in determining the timing and number of replenishments for the Bass model. We then develop several fundamental theoretical results. Furthermore, a one‐dimensional iterative method is proposed to find the optimal replenishment schedule. Finally, we provide an intuitively accurate estimate for the optimal number of replenishments, which significantly reduces computational complexity in finding the optimal replenishment number.

An inventory model for deteriorating items with stock-dependent demand and shortages under inflation and time discounting

This paper develops a finite time-horizon deterministic economic order quantity (EOQ) inventory model for deteriorating items. The items deteriorating at a constant rate and the demand at any instant depends on the on-hand inventory (stock-level) at that instant, shortages are allowed and completely backordered and the effects of inflation and time-value of money are taken into consideration. The proposed model facilitates managers in determining the optimal number of replenishments and optimal reorder time point. The results are discussed with two numerical examples and a sensitivity analysis of the optimal solution with respect to parameters of the system is carried out.

Deterministic inventory lot-size models under inflation with shortages and deterioration for fluctuating demand

In this paper, we extend the inventory lot-size models to allow for inflation and fluctuating demand (which is more general than constant, increasing, decreasing, and log-concave demand patterns). We prove that the optimal replenishment schedule not only exists but is also unique. Furthermore, we show that the total cost associated with the inventory system is a convex function of the number of replenishments. Hence, the search for the optimal number of replenishments is simplified to finding a local minimum. Finally, several numerical examples are provided to illustrate the results. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 144–158, 2001

Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand

We establish various inventory replenishment policies. We then analytically identify the best alternative among them based on the minimum total relevant costs. Finally, we prove that the relevant cost is convex with the number of replenishments. Consequently, the search for the optimal replenishment number is reduced to finding a local minimum.