An inventory system with service facility and 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.

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]

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.

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 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.

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.

We describe a very elementary direct numerical method to find the average number of backorders, costs and related quantities in a continuous review (s, S) inventory system with exponentially distributed lead time. This method can also be used in the study of E/M/C queues with state-dependent service and arrival rates.

We study the effects of different pricing strategies available to a continuous review inventory system with capacitated supply, which operates in a fluctuating environment. The system has a single server with exponential processing time. The inventory holding cost is nondecreasing and convex in the inventory level, the production cost is linear with no set-up cost. The potential customer demand is generated by a Markov-Modulated (environment-dependent) Poisson process, while the actual demand rate depends on the offerred price. For such systems, there are three possible pricing strategies: Static pricing, where only one price is used at all times, environment-dependent pricing, where the price changes with the environment, and dynamic pricing, where price depends on both the current environment and the stock level. The objective is to find an optimal replenishment policy under each of these strategies. This paper presents some structural properties of optimal replenishment policies, and a numerical study which compares the performances of these three pricing strategies.

We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability δ and is lost forever with probability (1−δ). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability δ and is lost forever with probability (1−δ). In all these models, the interretrial times are exponentially distributed with linear rate. Using matrix-analytic method, we study these inventory models. Some measures of the system performance in the steady state are derived. A suitable cost function is defined for all three cases and analyzed using graphical illustrations.

A perishable inventory system with service facility and finite source

A two-commodity inventory system with a single server is considered in this paper. We assume that the capacity of the buffers (to store the two types of commodities) to be finite. Customers (or demands) arrive according to a Poisson Process and the requirement for either type or both type of commodities are modelled using certain probabilities. Customers are lost when their demands are not met due to shortage only at the time of service offerings as opposed to getting lost when the inventory level is zero at the time of arrival. This is to allow the possibility of inventory being replenished prior to offering services to those who arrive when the inventory level is zero. A customer’s demand for both items may be met with only one item should a situation in which there is only one type of inventory is positive and the other is zero at the time of initiating a service occurs. The processing time for meeting the demands are random and modelled using exponential distribution with parameters depending on the type of demands being processed. We adopt (s, S)-type replenishment policy which depends on the type of commodity. Assuming the lead times to be exponentially distributed with parameters depending on the type of commodity, we employ matrix-analytic methods to study the queueing inventory system and report interesting results including an optimization dealing with various costs.

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.

The material inventory control system in the company shows that the addition of material inventory is carried out repeatedly and the demand (needs) of the material becomes rigid because it is highly dependent on the needs of the production process. The implementation of Make to Order system within the production process creates a demand that is probabilistic in nature as such that it requires proper raw material inventory planning. At present, inventory control policies in the company are not accurate in anticipating probabilistic demand, leading to a high number of lost sale products. The problem with the company is that the company is not aware of the level and timing of orders that are appropriate in controlling raw material inventory. Based on this condition, this research aims to help the company in determining inventory control policies using Q and P lost sale through Monte Carlo simulation approach system testing analysis. After obtaining the results through Q and P models, a simulation will be performed to see the effectiveness of the policy. Based on the simulation results of inventory policy, Model Q has the advantage of saving total inventory costs by Rp13,778,373.58 and decreasing the number of lost sale products by 352 units or with a percentage increase in succession of 76% and 100% compared to the company’s current system, while Model P can also provide a total inventory cost savings of Rp10,457,037.44 and a reduction in the number of lost sale products by 352 units or with a percentage increase in succession of 59% and 84%. Based on this, we can draw a conclusion that the best alternative inventory policy model is obtained in the Q model.

We consider a continuous-review inventory system in which the setup cost of each order is a general function of the order quantity and the demand process is modeled as a Brownian motion with a positive drift. Assuming the holding and shortage cost to be a convex function of the inventory level, we obtain the optimal ordering policy that minimizes the long-run average cost by a lower bound approach. To tackle some technical issues in the lower bound approach under the quantity-dependent setup cost assumption, we establish a comparison theorem that enables one to prove the global optimality of a policy by examining a tractable subset of admissible policies. Since the smooth pasting technique does not apply to our Brownian inventory model, we also propose a selection procedure for computing optimal policy parameters when the setup cost is a step function.

The object of research in this paper is an open Markov queueing network with revenues, limited number of customers, random waiting time for them in the queue, and unreliable queueing systems. The service times of customers in systems and the recovery times of lines in them have an exponential distribution with parameters depending on the number of customers in the systems and the number of service lines in service, and the waiting times in queues and service hours are also distributed according to an exponential law distribution, the parameters of which depend on the number of customers in systems. Using an asymptotic analysis with a large number of customers, the total expected revenue of such network was found. To do this, first derived and solved the differential equation in partial derivatives of the density distribution of revenue, and then received and decided to ordinary differential equation for the expected revenue of the network. A model example is calculated.