A perishable inventory system with service facility and finite source

This article presents a perishable stochastic inventory system under continuous review at a service facility in which the waiting hall for customers is of finite size M. The service starts only when the customer level reaches N (< M), once the server has become idle for want of customers. The maximum storage capacity is fixed as S. It is assumed that demand for the commodity is of unit size. The arrivals of customers to the service station form a Poisson process with parameter λ. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. The items of inventory have exponential life times. It is also assumed that lead time for the reorders is distributed as exponential and is independent of the service time distribution. The demands that occur during stock out periods are lost.The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. Some measures of system performance in the steady state are derived. The results are illustrated with numerical examples.

A multi-server perishable inventory system with negative customer

We consider a continuous review perishable inventory system at a service facility with a finite waiting capacity. The maximum inventory level is fixed and the customers arrive according to a Markov arrival process. The life time of each item and the service time are assumed to have independent exponential distributions. Unlike the conventional method of placing an order at a prefixed level, we consider a set of reorder levels with a specified probability for placing an order at a particular reorder level. This allows the modelling of a situation in which the decision maker may advance or postpone the placement of reorder as a result of his/her memory on the past supply behaviour. The reordering quantity depends upon the reorder level at which an order was triggered and the lead time is distributed as negative exponential. The joint probability distribution of the number of customers in the system and the inventory level is obtained in the steady state. We also derive some stationary system performance measures and compute the total expected cost rate under a cost structure. We also present a numerical illustration.

In this paper, we consider a continuous review perishable inventory system with multi-server service facility. In such systems the demanded item is delivered to the customer only after performing some service, such as assembly of parts or installation, etc. Compared to most inventory models in which inventory is depleted at the demand rate, it is depleted, in this model, at the rate that the service occurs. We assume that the arrivals of customers are according to a Markovian arrival process (MAP) and that service time has exponential distribution. The ordering policy is based on (s; S) policy. The lead time is assumed to have exponential distribution. The customer who finds either all servers are busy or no item (excluding those in service) is in the stock, enters into an orbit of infinite size. These orbiting customers send requests at random time points for possible selection of their demands. The interval time between two successive request-time points is assumed to have exponential distribution. In addition to the regular customers, a second flow of negative customers following an independent MAP is also considered which will remove one of the customers in the orbit. The joint probability distribution of the number of busy servers, the inventory level and the number of customers in the orbit is obtained in the steady state. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated.

A mathematical modelling of a continuous review stochastic inventory system with a single server is carried out in this work. We assume that demand time points form a Poisson process. The life time of each item is assumed to have exponential distribution. We assume(s,S)ordering policy to replenish stock with random lead time. The server goes for a vacation of an exponentially distributed duration at the time of stock depletion and may take subsequent vacation depending on the stock position. The customer who arrives during the stock-out period or during the server vacation is offered a choice of joining a pool which is of finite capacity or leaving the system. The demands in the pool are selected one by one by the server only when the inventory level is aboves, with interval time between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The joint probability distribution of the inventory level and the number of customers in the pool 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.

We consider a perishable inventory system under continuous review at a service facility in which a waiting area for customers is of finite size. We assume that the replenishment of inventory is instantaneous. The items of inventory have exponential life times. The service starts only when the customer level reaches a prefixed level, starting from the epoch at which no customer is left behind in the system. The arrivals of customers to the service station form a Poisson process. The server goes for a vacation of an exponentially distributed duration whenever the waiting area is zero. The service process is subject to interruptions, which occurs according to a Poisson process. The interrupted server is repaired at an exponential rate. Also the waiting customer independently reneges the system after an exponentially distributed amount of time. The joint probability distribution of the number of customers in the system and the inventory levels is obtained in steady state case. The results are illustrated with numerical examples.

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.

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]

Stochastic optimization of level-dependent perishable inventory system by Jackson network

In this article, we consider a discrete-time inventory model in which demands arrive according to a discrete Markovian arrival process. The inventory is replenished according to an $$(s,S)$$ policy, and the lead time is assumed to follow a discrete phase-type distribution. The demands that occur during stock-out periods either enter a pool which has an infinite capacity or leave the system with a predefined probability. The demands in the pool are selected one by one, if the on-hand inventory level is above $$s+1,$$ and the interval time between any two successive selections is assumed to have a discrete phase-type distribution. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady-state case. We derive the system performance measures under steady state and using these measures, the total expected cost rate of the system is calculated. The impacts of arrival rate on the performance measures are graphically illustrated. Finally, we study the impact of cost on the optimal values of the total expected cost rate, inventory level and the reorder point.

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.

This paper examines a retrial inventory system with an orbit of infinite size. An arriving customer, finding the stock out in the system, proceeds to the orbit with probability a and is lost forever with probability (1 – a). The customers in the orbit are assumed not only to retry for their demand, may renege from the orbit at random time. We model this situation by assuming Poisson arrivals for 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 state case. The measures of system performance in the steady state are derived and the long run total expected cost rate is derived. The results are illustrated with numerical examples.

Two-commodity inventory system with retrial demand

In this article we analyze a lost sales two-commodity perishable inventory system, under Poisson demands and exponential lifetimes. The maximum inventory level for the i-th commodity is fixed as . If the total net inventory level drops to a prefixed level , an order will be placed for units of -th commodity . The lead time for replenishment is assumed to be exponential. The limiting probability distribution for joint inventory level, mean inventory levels and mean reorder rates and mean shortage rates in the steady state are computed. The total cost function is expressed in terms of system performance measures obtained. The results are illustrated numerically. Key words : Perishable item, Joint replenishment, Stochastic lead time, Markov process.

This paper analyses a production inventory model with varying production rates where demand is according to Marko­vian arrival process (MAP). The time between additions of two successive items by production to the inventory follows Phase type distribution. When the inventory level decreases to a predetermined level production starts and the rate of production is higher until the number of items in the inventory crosses that level. The arriving customers are directed to a buffer of finite capacity equal to the current inventory level and service times are exponentially distributed. An arriving customer who notices the buffer full, proceeds to an orbit of infinite capacity and retry from there. Inter-retrial times follow an exponential distribution. Various system performance measures of the model are defined. A suitable cost function is also constructed.