Continuous Review Inventory System Research Articles

An Approximation Approach for Solving a Continuous Review Inventory System Considering Service Cost

【The modular assembly system can make it possible for the variety of products to be assembled in a short lead time. In this system, necessary components are assembled to optional components tailor to customers' orders. Budget for inventory investments composed of inventory and purchasing costs are practically limited and the purchasing cost is often paid when an order is arrived. Service cost is assumed to be proportional to service level and it is included in budget constraint. We develop a heuristic procedure to find a good solution for a continuous review inventory system of the modular assembly system with a budget constraint. A regression analysis using a quadratic function based on the exponential function is applied to the cumulative density function of a normal distribution. With the regression result, an efficient heuristics is proposed by using an approximation for some complex functions that are composed of exponential functions only. A simple problem is introduced to illustrate the proposed heuristics.】

Economic and environmental considerations in a continuous review inventory control system with integrated transportation decisions

Sustainability throughout supply chains is gaining more importance and re-planning inventory operations can help companies curb emissions. In this study, we present two bi-objective integrated continuous review inventory control and transportation models with less-than-truckload and truckload carriers. Solution methods to approximate the Pareto Frontiers are proposed. Numerical studies illustrate the effects of demand variance and lead time on expected costs and carbon emissions as well as the changes in expected costs and carbon emissions due to sustainability considerations. Sample examples illustrate the use of the methods to compare different carriers in terms of not only economic but also environmental considerations.

A comparative study on replenishment policies of inventory system with postponed demands

We consider a continuous review inventory system in which demands arrive according to a Markovian arrival process. The demands that occur during stock-out periods enter a finite pool. Once the pool is full, subsequent demands will be lost. We use (s, S) ordering policy and variable ordering policy for replenishing the inventory. The lead time is assumed to follow a phase-type distribution. When the on-hand inventory level is above s, the demand in the pool is selected one by one with first come first serve discipline. The time interval between any two successive selections of demand from the pool is assumed to have exponential distribution. The joint probability distribution of on-hand inventory level, number of demands in the pool and number of ordered items are obtained in the steady-state case. The measures of system performance in the steady state and total expected cost are derived for both policies. Finally, these two policies are numerically compared, and we obtain results that show the total expected cost rate and the mean waiting time of pooled demands with variable ordering policy are minimal when comparing with (s, S) policy.

Cost-Effective Optimization on a Two Demand Classes Inventory System

This paper investigates impact of change of priority on a continuous review inventory system with a mixture of backorders and lost sales. The inter-arrival processes are assumed to be exponentially distributed. The objective is to minimize the anticipated total cost rate by simultaneously optimizing the maximum inventory level and backorders. An entropy-based uncertainty metric for evaluating the operating performance is demonstrated. The proposed modification offers the practical value of establishing how the long-run cost rate is reduced. Various system performance measures are derived.

Accurate lead time demand modeling and optimal inventory policies in continuous review systems

To construct an accurate probability density function for lead time demand in inventory management models, a mixture of polynomials (MOPs) distributions is estimated using B-spline functions from empirical data on demand per unit time. This is accomplished by summarizing the empirical observations into separate datasets for each lead time value. A mixture distribution approach is then applied to model lead time demand in a continuous review inventory system. Inventory policies can be determined without knowledge of the underlying demand and/or lead time distributions. An improvement to a mixture distribution approach that models the lead time demand distribution with a mixture of truncated exponentials (MTEs) distribution is also presented, and the MOP and MTE techniques are compared. Both methods provide reasonable accuracy, but the MOP approach requires lower computational time to determine optimal inventory policies. The mixture distribution approach is also compared with solutions calculated using optimization through simulation and by compiling a discrete, empirical lead time demand distribution.

An inventory system with two types of customers and retrial demands

We consider a continuous review (s, S) inventory system in which the arriving customers belong to any one of the two types (type-1 or type-2). When the inventory level is above s, the customers are not distinguished as to their type and their demanded items are delivered immediately to them. Once the inventory level drops to s(≥ 0), an order for Q items is placed and thereafter the demands of type-2 customers alone are satisfied. The type-1 customers are sent to a place called orbit which is of infinite size. These orbiting demands retry for their demand after a random time which is assumed to have exponential distribution. The arrivals of customers are assumed to follow a Markovian arrival process and the lead time is assumed to have phase-type distribution. The joint probability distribution of the number of customers in the orbit and the inventory level is obtained in the steady-state case. Various system performance measures in the steady state are derived and total expected cost rate is calculated.

Continuous review inventory model with reducing lost sales rate under fuzzy stochastic demand and variable lead time

The present study considers a continuous review inventory system for the inventory model involving fuzzy random demand, variable lead-time with lost-sales (backorder) caused by stock-out. Two forms of capital investment cost function viz. logarithmic and power are employed to reduce the lost-sales rate. We first formulate the basic model mathematically by assuming expected demand as triangle fuzzy number along with the capital investment to reduce lost-sales rate. As a result, the demand during lead-time is fuzzy random variable which is the linear sum of the weekly demand during the lead-time period. Consequently, the total weekly cost of this model is also fuzzy random in nature. Hence the expected total weekly cost is fuzzy valued function. To defuzzify the expected total weekly cost, we use the signed distance method. A computer program using the software MATLAB is developed to obtain the optimal solution and provide numerical examples to illustrate the models. Moreover, sensitivity analysis is carried out with respect to the key parameters.

Dynamic Pricing and Production Control of an Inventory System with Remanufacturing

We consider optimal pricing and manufacturing control of a continuous-review inventory system with remanufacturing. Customer demand and product return follow independent Poisson processes. Customer demand is filled by serviceable product, which can be either manufactured or remanufactured from the returned product. The lead times for both manufacturing and remanufacturing are exponentially distributed. The objective is to maximize the expected total discounted profit over an infinite planning horizon. We characterize the structural properties of the optimal policy through the optimality equation. Specifically, the optimal manufacturing policy is a base-stock policy with the base-stock level nonincreasing in the return inventory level. The optimal pricing policy is also a threshold policy, where the threshold level is nonincreasing in the return inventory level.

Optimal Control of Service Parameter for a Perishable Inventory System with Service Facility, Postponed Demands and Finite Waiting Hall

This paper deals with the problem of optimally control service rates for a continuous review (s, S) inventory system with a service facility consisting of finite buffer(waiting hall) and a single server. The customers arrive according to a Poisson process. The customer’s demand is satisfied after an exponential service time. An arriving customer, who finds the buffer is full, enters into the pool of finite size or leaves the system according to a Bernoulli trial. The replenishment time of the order is distributed as exponential. The life time of each item in the inventory is assumed to be exponential distribution. Here we determine the service rates to be employed at each instant of time so that the long-run total expected cost rate is minimized. The problem is modelled as a semi-Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.

Optimal control of production time of perishable inventory system with finite source of customers

We consider a continuous-review inventory system with perishable items and make-to-stock production facility. The customers are assumed to be generated by a finite number of sources. Each customer demands a single item. The life time of each item is assumed to be exponential. Customers demand is met from stock, if available, otherwise the customer enters into an orbit. The orbiting customer sends out signal to get his demand satisfied and the duration between two successive retrials is assumed to be distributed as exponential. In order to replenish stock, a production facility is switched on when the inventory level drops to s and items are produced one by one until the stock level reaches a maximum capacity S(> s). The production facility is dynamic in the sense that the rate, at which the item is produced, can be selected from a pre-fixed set of positive values. In other words, the successive production time is assumed to follow an exponential distribution whose parameter is chosen from a given set of positive values. The purpose of this paper is to investigate the optimal set of production rates that minimizes the total expected cost per unit time. We formulate this problem as a semi Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.

Emergency lateral transshipments in a two-location inventory system with positive transshipment leadtimes

Abstract We consider a single-echelon continuous review inventory system for spare parts with two parallel locations. Each location faces independent Poisson demand and backorders are allowed. In this paper we consider the possibility of lateral transshipments between the locations. The transshipment leadtime is positive and deterministic, and there is an additional cost for making a transshipment. We suggest a transshipment policy which is based on the timing of all outstanding orders, and develop and solve a heuristic model by using theory and concepts from doubly stochastic Poisson processes and also partial differential equations. A simulation study indicates that our heuristic works very well, and that the relative cost increase of disregarding the transshipment leadtime may be quite high. Our results also indicate that it is, in general, worth the effort of reducing the transshipment leadtime, even if it is already relatively short.

An inventory system with service facility and finite orbit size for feedback customers

In this article, we consider a continuous review inventory system with service facility consisting of finite waiting hall and a single server. The customers arrive according to a Poisson process and any arriving customer, who finds the waiting hall is full, is considered to be lost. The individual customer’s unit demand is satisfied after a random time of service, which is assumed to be exponential distribution. After a customer is served, he/she will decide either to join the retrial group, which is of finite size, called orbit, for another service or leave the system according to a Bernoulli trial. The customers in the orbit, called feedback customers, compete for service according to classical retrial policy. The service times for these feedback customers are assumed to be independent and exponential distribution. The inventory is replenished according to an (s, S) inventory policy, and the replenishing times are assumed to be exponential. The joint probability distribution of the inventory level, the number of customers in the orbit, number of customers in the waiting hall and the status of the server is obtained in the steady state. Some important system performance measures in the steady state are derived, and the long-run total expected cost rate is also calculated. We have derived the Laplace-Stieljes transforms of waiting time distribution of both primary as well as the feedback customers. The sensitivity analysis has been carried out to examine the effect of different parameters and cost in the system.

A comparative study of the inventory system with service facility and postponed demands

In this article, we present a continuous review (s, S) inventory system with a service facility consisting of finite buffer (capacity N) and a single server. The customers arrive according to a Poisson process. The individual customer’s unit demand is satisfied after a random time of service, which is assumed to be exponential. When the inventory level drops to s an order for Q(= S − s) items is placed. The lead time of reorder is assumed to be exponential distribution. An arriving customer, who finds the buffer is full, enters into the pool of infinite size or leaves the system according to a Bernolli trial. At the time of service completion, if the buffer size drops to a preassigned level L (1 < L < N) or below and the inventory level is above s, we select the customers from the pool according to two different policy: in first policy, with probability p(0 < p < 1) we select the customer from the head of the pool and we place the customer at the end of the buffer; in the second policy, with p(0 < p < 1) the customer from the pool is transferred to the buffer for immediate service and after completion of his service we provide service to the customer who is in the buffer with probability one. If at a service completion epoch the buffer turns out to be empty, there is at least one customer in the pool and the inventory level is positive, then the one ahead of all waiting in the pool gets transferred to the buffer, and his service starts immediately. The joint probability distribution of the number of customers in the pool, number of customers in the buffer and the inventory level is obtained in the steady-state case. Various stationary system performance measures are computed and total expected cost rate is calculated. A comparative result of two models is illustrate numerically.

Analysis of inventory policies for perishable items with fixed leadtimes and lifetimes

This paper deals with a continuous review inventory system with perishable items and Poisson demand. Lifetimes and leadtimes are assumed to be fixed. First, a quite general base-stock model is developed where a number of combinations between backorder- and lost sales policies are evaluated and optimized. The solution technique for all these combinations is exact. Secondly, we consider the case with non-negligible ordering costs and assume that the inventory policy employed is the commonly used (\(R,Q\)) policy. We develop a new heuristic approach for evaluating and analyzing the proposed \((R,Q)\) model and compare our results with those obtained by related papers. This heuristic approach uses the base-stock model developed as a building block. The results reveal that our approach works reasonably well in all cases considered.

On two-echelon inventory systems with Poisson demand and lost sales

We consider a two-echelon, continuous review inventory system under Poisson demand and a one-for-one replenishment policy. Demand is lost if no items are available at the local warehouse, the central depot, or in the pipeline in between. We give a simple, fast and accurate approach to approximate the service levels in this system. In contrast to other methods, we do not need an iterative analysis scheme. Our method works very well for a broad set of cases, with deviations to simulation below 0.1% on average and below 0.36% for 95% of all test instances.

Lead Time Management through Expediting in a Continuous Review Inventory System

We consider a continuous review inventory system where delivery lead times can be managed by expediting in‐transit orders shipped from the supplier. First, we propose an ordering/expediting policy and derive expressions for evaluating the operating characteristics of such systems. Second, using extensive numerical experiments, we quantify the benefits of such an expediting policy. Third, we investigate a number of managerial issues. Specifically, we analyze the impact of the number of expediting hubs and their locations along the shipment network on the performance of such systems and offer insights into the design of the shipment network. We show (i) a single expediting hub that is optimally located in a shipment network can capture the majority of cost savings achieved by a multi‐hub system, especially when expediting cost is not low or demand variability is not high; (ii) when expediting time is proportional to the time to destination, for small‐enough or large‐enough demand variations, a single expediting hub located in the middle of the shipment network can capture the majority of cost savings of an optimally located hub; and (iii) in general, hubs close to the retailer significantly drive down costs, whereas hubs close to the supplier may not offer much cost savings.

An (s, S) retrial inventory system with impatient and negative customers

In this paper, we consider a continuous review inventory system with an orbit of infinite size. A customer turns out to be an ordinary customer and a negative customer. On arrival, an ordinary customer joins the queue and the negative customer does not join the queue and takes away one waiting customer from the orbit if any. If there is no customer when a negative customer arrives, then the negative customer will be serviced. The customers in the orbit are assumed not only to retry for their demand, but may renege from the orbit after a random time. We consider this situation by assuming Poisson arrivals for both types of customers, exponential lead time for orders placed under (s, S) ordering policy, exponential time between successive attempts of retrial of a customer and exponential times for reneging by customers at the orbit. The condition for ergodicity of the system is obtained. The joint probability distribution of the number of customers in the system and the inventory level is obtained in steady...

Analysis of Two-Commodity Continuous Review Inventory System in Supply Chain

Abstract: This paper considers a two commodity continuous review inventory system. Continuous review inventory control of a single item at a single location had been considered by many researchers in past. We extend this inventory control strategy to two-echelon system, which is a building block for serial supply chain. The inventory control system consists of two warehouse (WHi), Two Distribution Centre’s (DCi) each associated with a retailer and handling two non-identical products. A (s, S) type inventory system with Poisson demand and exponentially distributed lead times is assumed at retailer node. The items are supplied to the retailers in packs of Qi (= Si-si) items from the distribution center (DCi) which has instantaneous replenishment facility from an abundant source (manufacturer). The steady state probability distribution and the operating characteristics are obtained explicitly. The measures of system performance in the steady state are obtained. The results are illustrated with numerical examples.

OPTIMAL CONTROL OF A FINITE-CAPACITY INVENTORY SYSTEM WITH SETUP COST AND LOST SALES

One of the most fundamental results in inventory theoryis the optimality of (s, S) policy for inventory systems withsetup cost. This result is established based on a key assumptionof infinite production/ordering capacity. Several studies haveshown that, when there is a finite production/ordering capacity,the optimal policy for the inventory system is very complicatedand indeed, only partial characterization for the optimal policyis possible. In this paper, we consider a continuous reviewinventory system with finite production/ordering capacity andsetup cost, and show that the optimal control policy for thissystem has a very simple structure. We also develop efficientalgorithms to compute the optimal control parameters.

A Minimax Distribution Free Procedure for a Multi-constraint Inventory Model with Deterministic Variable Lead Time

Nowadays, considering real environment limitations, the effect of some constraints such as storage space and budget in determining optimal policies of inventory/production systems is a momentous concern of inventory managers. Hence, here, a budget constraint on total inventory investment and a maximum permissible storage space constraint are added simultaneously to a stochastic continuous review inventory system. The piece-wise linear crashing cost function which is widely used in project management is considered in this paper. The study relaxes the assumption about the form of distribution function of lead time demand by only assuming the mean and standard deviation of lead time demand are known. A minimax distribution free procedure is developed for the constrained problem. This study also shows that the respective budget and storage space constrained inventory model to be minimized is jointly convex in the decision variables. A numerical example is given to illustrate the discussed model.