Total Expected Cost Rate Research Articles

Lost sales obsolescence inventory systems with positive lead time: a system-point level-crossing approach

AbstractIn this article, we provide a comprehensive analyses of two continuous review lost sales inventory system based on different replenishment policies, namely $(s,S)$ and $(s,Q)$. We assume that the arrival times of demands form a Poisson process and that the demand sizes have i.i.d. exponential distribution. We assume that the items in stock may obsolete after an exponential time. The lead time for replenishment is exponential. We also assume that the excess demands and the demands that occurred during stock out periods are lost. Using the system point method of level crossing and integral equation method, we derive the steady-state probability distribution of inventory level explicitly. After deriving some system performance measures, we computed the total expected cost rate. We also provide numerical examples of sensitivity analyses involving different parameters and costs. As a result of our numerical analysis, we provide several insights on the optimal $(s,S)$ and $(s,Q)$ policies for inventory systems of obsolescence items with positive lead times. The better policy for maintaining inventory can be quantified numerically.

Quantity-based emergency shipment policies

Either structurally or in an ad hoc manner, emergency shipments are widely utilized for preventing the potentially negative impacts of stock-outs. In addition to having policies for their regular orders, companies need to decide when and for how many units to place emergency orders. In this article, we consider the periodic-review problem of a retailer who uses a quantity-based emergency shipment policy. Under this policy an emergency order is triggered if the inventory level within the review period falls below a certain level. We consider the base-stock policy for the regular replenishment orders. The goal is to determine the optimal base-stock level, the optimal period length as well as the optimal size and threshold value of emergency orders such that the total expected cost rate is minimized. We use renewal reward theory to derive the expressions of operating characteristics. The expected cost rate’s properties are analyzed to develop an optimization algorithm for finding the optimal policy parameters. We compare our policy with the time-based and hybrid emergency shipment policies and conclude that the quantity-based policy can bring substantial benefits compared to other policies.

(<i>s</i>, <i>S</i>) stochastic inventory system in Jackson network

In this work, we develop and analyse an (s, S) stochastic perishable inventory system at each node into Jackson network with a service facility in which the waiting hall of first queue has infinite capacity and second queue has finite capacity for the customers. Service times are exponentially distributed. We assume that demands arrive in the system according to a Poisson process with rate λi(> 0, ∀i) and demands only single unit item at a time. The maximum storage capacity of the ith warehouse is fixed as Si; i = 1, 2. Whenever the inventory level reaches the reorder level si(0 ≤ si < Si), an order Qi(= Si – si) units is placed to bring the level to Si. The lead-time is exponentially distributed. The items of inventory have exponential life times. The joint probability distribution of the number of customers in the system and the inventory level is obtained in the steady state case. Matrix analytical method is applied to solve for the steady state occupancy probabilities. Various system performance measures in the steady state are derived. A suitable cost function is defined and the long-run total expected cost rate is calculated. Sensitivity analysis has been carried out to study the effect of variation of parameters. Numerical examples and graphical illustrations are provided to illustrate the proposed model.

An inventory system with replacement and refurbishment of failed items

We consider a continuous review inventory system with two types of inventory, namely type-I and type-II. Type-I and type-II inventory, respectively, contain fresh and refurbished items. Two streams of customers, called type-A and type-B customers arrive to the system in which type-A customer demands for an item either from type-I or type-II inventory according to a Bernoulli trial and type-B customer occurs to replace the failed item with new one from type-I inventory. The arrival time points of type-A and type-B customers follow independent Poisson Processes. Failed items are stored in a place, called pool, of infinite size for refurbishment. After refurbishment, these items are kept in type-II inventory. The refurbishment of pooled item is stopped once type-II inventory reaches its maximum level S 2 and is again commenced only when type-II inventory level falls to s 2. The refurbishment time follows exponential distribution. Type-I inventory is replenished according to a (s, S) policy with exponential lead time. The joint pdf of number of items in the pool, type-II inventory level, status of refurbishment and type-I inventory level is obtained in the steady state. Various system performance measures are derived and total expected cost rate is calculated. Results are illustrated numerically.

Perishable Inventory System with N-Policy, MAP Arrivals, and Impatient Customers

In this study, we consider a perishable inventory system that has an (s, Q) ordering policy, along with a finite waiting hall. The single server, which provides an item to the customer after completing the required service performance for that item, only begins serving after N customers have arrived. Impatient demand is assumed in that the customers waiting to be served lose patience and leave the system if the server’s idle time overextends or if the arriving customers find the system to be full and will not enter the system. This article analyzes the impatient demands caused by the N-policy server to an inventory system. In the steadystate, we obtain the joint probability distribution of the level of inventory and the number of customers in the system. We analyze some measures of system performance and get the total expected cost rate in the steadystate. We present a beneficial cost function and confer the numerical illustration that describes the impact of impatient customers caused by N-policy on the inventory system’s total expected cost rate.

A Finite Source Two Commodity Inventory System with Retrial Demands and Multiple Server Vacation

Consider a two commodity continuous review inventory system with a finite number of homogeneous sources generating demands. The maximum storage capacity for the commodity - i is S i (i = 1, 2). We adopt a joint reordering policy for placing orders. The lead time for the delivery of orders is assumed to have an exponential distribution. An arriving demand receives either commodity - 1 or commodity - 2 according to a Bernoulli trial. The commodities are assumed to be substitutable in the sense that during the zero stock of any commodity, the demand for that commodity can be met by the unit of the other commodity. The server goes for a vacation of an exponentially distributed duration whenever the inventory level of both commodities reaches zero. If the server finds empty stock when he returns to the system, he immediately takes another vacation. The demands that occur during stock-out period or during the server vacation period enter into the orbit of finite size. These orbiting demands retry for their demand after a random time, which is assumed to be an exponential distribution. The joint probability distribution of the inventory level and the number of demands in the orbit is obtained in the steady state case. Various system performance measures in the steady state are derived and the long-run total expected cost rate is calculated.

An inventory system with postponed and renewal demands

This paper deals a continuous review inventory system with postponed demands. We generalise the existing models in the literature of postponed inventory system by assuming arbitrary demand distributions for the inter-arrival times. The inventory is replenished according to a policy and the replenishing times are assumed to follow an exponential distribution. The demands that occur during stock-out periods enter a pool of infinite capacity. The demands in the pool are selected one by one, if the replenished stock is above some prefixed level. The inter-selection time is distributed as exponential. Using the matrix geometric method, the stationary joint probability distribution of the number of customers in the pool and the inventory level is calculated. The measures of system performance in the steady state are derived and the total expected cost rate is also calculated. We employ artificial intelligence techniques to find the optimal values. The sensitivity of system costs and parameters on the optimal values are studied numerically.

Optimal service rates of a perishable inventory system with service facility

This paper studies a continuous-review perishable inventory system with service facility. We consider a finite capacity inventory system with Markovian arrivals and exponentially distributed service times. The inventory is replenished according to an (s, S) policy and the lead times follow phase type distribution. The life time of each item is assumed to be exponential. By controlling the service rate at each instance of time, we can minimise the total expected cost rate. We formulate this model as a semi Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.

A stochastic inventory system with replacement of perishable items

This article presents a continuous review perishable inventory system in which the perished items will be replaced by the supplier at a later time. Demands occur according to a Markov arrival process. The items in the inventory have exponential life times and these perished items are stored in a place, called pool for replacement. The (s, S) ordering policy is adopted. At the time of placing an order, the ordering quantity is adjusted with number of items in the pool. The lead time is assumed to have phase type distribution. The joint probability distribution of the inventory level and the number of pooled items is obtained in the steady state case using the matrix-geometric methods. Various system performance measures in the steady state are derived and the total expected cost rate is calculated under a prefixed cost structure. The results derived in this work are numerically illustrated.

A continuous review (&lt;i&gt;s&lt;/i&gt;, &lt;i&gt;S&lt;/i&gt;) inventory system with postponed demands at service facility

In this article, we present a continuous review (s, S) inventory system with a service facility consisting of finite waiting hall (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. An arriving customer, who finds the waiting hall is full, enters into the pool of infinite size or leaves the system which is according to a Bernolli trial. The joint probability distribution of the number of customers in the pool, number of customers in the waiting hall 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. The results are illustrated numerically. [Received: 8 January 2017; Revised: 28 February 2018; Accepted: 30 July 2018]

A continuous review (&lt;i&gt;s&lt;/i&gt;, &lt;i&gt;S&lt;/i&gt;) inventory system with postponed demands at service facility

In this article, we present a continuous review (s, S) inventory system with a service facility consisting of finite waiting hall (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. An arriving customer, who finds the waiting hall is full, enters into the pool of infinite size or leaves the system which is according to a Bernolli trial. The joint probability distribution of the number of customers in the pool, number of customers in the waiting hall 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. The results are illustrated numerically. [Received: 8 January 2017; Revised: 28 February 2018; Accepted: 30 July 2018]

Optimal Service Rates of a perishable Inventory System with Service Facility

This paper studies a continuous-review perishable inventory system with service facility. We consider a finite capacity inventory system with Markovian arrivals and exponentially distributed service times. The inventory is replenished according to an (s, S) policy and the lead times follow phase type distribution. The life time of each item is assumed to be exponential. By controlling the service rate at each instance of time, we can minimise the total expected cost rate. We formulate this model as a semi Markov decision problem. The stationary optimal policy is computed using linear programming algorithm and the results are illustrated numerically.

Integrated production quality and condition-based maintenance optimisation for a stochastically deteriorating manufacturing system

This paper investigates the problem of optimally integrating production quality and condition-based maintenance in a stochastically deteriorating single- product, single-machine production system. Inspections are periodically performed on the system to assess its actual degradation status. The system is considered to be in ‘fail mode’ whenever its degradation level exceeds a predetermined threshold. The proportion of non-conforming items, those that are produced during the time interval where the degradation is beyond the specification threshold, are replaced either via overtime production or spot market purchases. To optimise preventive maintenance costs and at the same time reduce production of non-conforming items, the degradation of the system must be optimally monitored so that preventive maintenance is carried out at appropriate time intervals. In this paper, an integrated optimisation model is developed to determine the optimal inspection cycle and the degradation threshold level, beyond which preventive maintenance should be carried out, while minimising the sum of inspection and maintenance costs, in addition to the production of non-conforming items and inventory costs. An expression for the total expected cost rate over an infinite time horizon is developed and solution method for the resulting model is discussed. Numerical experiments are provided to illustrate the proposed approach.

A retrial queueing-inventory system with J-additional options for service and finite source

A continuous review (s; S) inventory system at a service facility with nite homogeneous sources of demands and retrial is analysed. The lifetime of each item is assumed to be exponential. Before items are delivered to the customers, some basic service on the item must be performed. It is known as a regular or main service. The service may get interrupted according to a Poisson process and it restarts after an exponentially distributed time. If the server is idle at the time of arrival of a customer and the inventory level is positive, then the service begins immediately. After the completion of regular service, a customer may either abandon the system forever or demand for a second service from the same server, which is multi-optional. If any arriving customer nds that the server is busy or inventory level is zero, he/she either enters into the orbit with probability p or balks (does not enter) with probability 1 - p. The stationary distribution of the number of customers in the system, server status and the inventory level is obtained by the matrix method. The Laplace-Stieltjes transform of the waiting time of the tagged customer is derived. Various system performance measures are derived and the total expected cost rate is computed under a suitable cost structure. A numerical illustration is given. Key words : (s; S) policy, service interruption, nite source, retrial, repair, essential and optional service.

Integrated strategy of Vehicle Routing and Maintenance

This paper deals with the Vehicle Routing Problem (VRP) considering specific situations of transcontinental transportation where heavy vehicles travel over long distances between cities in a difficult environment favoring a non-negligible probability of failure of critical components. Hence, these transportation vehicles are subject to random breakdowns and repairs of random duration, which can obviously delay deliveries. A preventive maintenance (PM) policy is proposed which suggests performing PM actions (generally preventive replacement of the critical component) when reaching selected customers (cities). The purpose of this work is to determine simultaneously the optimal delivery sequence and the optimal sequence of PM actions (i.e. the customers cities at which PM should be performed) in order to minimize the expected total cost per time unit, considering penalties on late arrival and maintenance costs. A mathematical model is developed expressing the total expected cost rate taking into account the vehicle's reliability, maintenance actions costs and durations (PM and minimal repair), transportation cost, and penalties corresponding to late arrivals. For cases of numerous customers to be served, a genetic algorithm is proposed to obtain a nearly optimal integrated vehicle routing and maintenance strategy for any given settings of the problem. A numerical example is presented and the obtained results are discussed.

A finite source perishable inventory system with second optional service and server interruptions

In this article, a service facility inventory system with server interruptions and a nite number of sources are considered. The inventory is replenished according to ( s; S ) ordering policy. Using the matrix methods, the stationary distribution of the stock level, server status and waiting area level is obtained in the steady state case. The Laplace-Stieltjes transform of the waiting time of the tagged customer is derived. Many impartment system performance measures are derived and the total expected cost rate is computed under a suitable cost structure. The results are illustrated numerically. Keywords: Essential and optional service, inventory with service time, service interruption, repair, nite source

Optimal Control Policy of an Inventory System with Postponed Demand

This paper deals with the problem of controlling the selection rates of the pooled customer of a single commodity inventory system with postponed demands. The demands arrive according to a Poisson process. The maximum inventory level is fixed at S . The ordering policy is (s,S ) policy that is as and when the inventory level drops to s an order for Q ( = S − s ) items is placed. The ordered items are received after a random time, which is distributed as exponential. We assume that the demands that occur during stock out period either enter a pool of finite size or leave the system according to a Bernoulli distribution. Whenever the on-hand inventory level is positive, customers are selected one-by-one and the selection rate can be chosen from a given set. The problem is to determine a decision rule that specifies the rate of these selections as a function of the on-hand inventory level and the number of customers waiting in the pool at each instant of time to minimise the long-run total expected cost rate. The problem is modelled as a semi-Markov decision problem. The optimal policy is computed using Linear Programming algorithm and the results are illustrated numerically.

An analysis of replenishment policies for perishable inventory system with postponed demand and multiple vacations

In this article, we consider a perishable inventory system with postponed demand and single server. The server is allowed to take multiple vacations during stock out period. The demands that arrive during a stock out period or server vacation period either postponed their demand or considered to be lost according to a Bernoulli distribution. The postponed demands are sent to a place called pool and its capacity is finite. When the server is not in vacation and the inventory level is above s, the demands in the pool are selected one by one with FCFS discipline. The waiting time distribution for demands in the pool is derived for both (s, S) and the variable ordering policy. System performance measures for model with variable ordering policy and (s, S) policy are also computed under steady-state case. And these performance measures are used to calculate the total expected cost rate for both policies. Finally, using numerical results, we compare the optimal total cost rate corresponding to two policies and also we compare the expected waiting time of the pooled demands.

Markovian inventory model with two parallel queues, jockeying and impatient customers

This article presents a perishable stochastic inventory system under continuous review at a service facility consisting of two parallel queues with jockeying. Each server has its own queue, and jockeying among the queues is permitted. The capacity of each queue is of finite size L. The inventory is replenished according to an (s; S) inventory policy and the replenishing times are assumed to be exponentially distributed. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The life time of each item is assumed to be exponential. Customers arrive according to a Poisson process and on arrival; they join the shortest feasible queue. Moreover, if the inventory level is more than one and one queue is empty while in the other queue, more than one customer are waiting, then the customer who has to be received after the customer being served in that queue is transferred to the empty queue. This will prevent one server from being idle while the customers are waiting in the other queue. The waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the inventory level, the number of customers in both queues, and the status of the server are obtained in the steady state. Some important system performance measures in the steady state are derived, so as the long-run total expected cost rate.

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.