Analysis of Two-Echelon Inventory System with Direct and Retrial Demands

Supply Source ◽

Distribution Centre ◽

Abundant Supply

In this Chapter, the author considers a continuous review inventory system (both perishable and non-perishable) with Markovian demand. The operating policies are (s, S) and (0,M) policy, that is the maximum inventory level at lower echelon is Sand whenever the inventory drops to s, an order for Q(= S - s) units is placed at the same time in the higher echelon, the maximum inventory level is fixed as M(= nQ: n = 1, 2, ….) and has an instantaneous replenishment facility from an abundant supply source. The demands that occur directly to the distribution centre are called direct demands. The arrival process for the direct demand follows Poisson process. The demand process to the retailer node is independent to the direct demand process and follows Poisson process. The demands that occur during stock out period are enter into the orbit of finite size. The joint probability distribution of the inventory level at lower echelon, higher echelon and the number of customer in the orbit is obtained in the steady state case.

Perishable Inventory System with Postponed Demands and Negative Customers

Journal of Applied Mathematics and Decision Sciences ◽

10.1155/2007/94850 ◽

2007 ◽

Vol 2007 ◽

pp. 1-12 ◽

Cited By ~ 11

Author(s):

Paul Manuel ◽

B. Sivakumar ◽

G. Arivarignan

Keyword(s):

Steady State ◽

Joint Probability ◽

Inventory System ◽

Arrival Process ◽

Inventory Level ◽

Time Interval ◽

Negative Customers ◽

This article considers a continuous review perishable (s,S) inventory system in which the demands arrive according to a Markovian arrival process (MAP). The lifetime of items in the stock and the lead time of reorder are assumed to be independently distributed as exponential. Demands that occur during the stock-out periods either enter a pool which has capacity N(<∞) or are lost. Any demand that takes place when the pool is full and the inventory level is zero is assumed to be lost. The demands in the pool are selected one by one, if the replenished stock is above s, with time interval between any two successive selections distributed as exponential with parameter depending on the number of customers in the pool. The waiting demands in the pool independently may renege the system after an exponentially distributed amount of time. In addition to the regular demands, a second flow of negative demands following MAP is also considered which will remove one of the demands waiting in the pool. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. The measures of system performance in the steady state are calculated and the total expected cost per unit time is also considered. The results are illustrated numerically.

Perishable Inventory System with Server Interruptions, Multiple Server Vacations, and N Policy

International Journal of Operations Research and Information Systems ◽

10.4018/ijoris.2015040103 ◽

2015 ◽

Vol 6 (2) ◽

pp. 32-52 ◽

Cited By ~ 7

Author(s):

K. Jeganathan ◽

N. Anbazhagan ◽

B. Vigneshwaran

Keyword(s):

Steady State ◽

Poisson Process ◽

Finite Size ◽

Joint Probability ◽

Inventory System ◽

Perishable Inventory ◽

Waiting Area ◽

Negative Exponential

This article presents a perishable inventory system under continuous review at a service facility in which a waiting area for customers is of finite size . The authors assume that the replenishment of inventory is instantaneous. The items of inventory have exponential life times. It is assumed that demand for the commodity is of unit size. 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. If the server finds the customer level is less than when he returns to the system, he immediately takes another vacation. The individual customer is issued a demanded item after a random service time, which is distributed as negative exponential. 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 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 and the total expected cost is also considered. The results are illustrated with numerical examples.

A Perishable Inventory System with Postponed Demands and Multiple Server Vacations

Modelling and Simulation in Engineering ◽

10.1155/2012/620960 ◽

2012 ◽

Vol 2012 ◽

pp. 1-17 ◽

Cited By ~ 9

Author(s):

R. Jayaraman ◽

B. Sivakumar ◽

G. Arivarignan

Keyword(s):

Steady State ◽

Joint Probability ◽

Life Time ◽

Inventory System ◽

Inventory Level ◽

Single Server ◽

Cost Rate ◽

Stochastic Inventory ◽

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.

Stochastic Processes and Models in Operations Research - Advances in Logistics, Operations, and Management Science ◽

A Stochastic Inventory Model with Multiple Vacations and N - Policy

10.4018/978-1-5225-0044-5.ch007 ◽

2016 ◽

pp. 106-123

Author(s):

Jeganathan Kathirvel

Keyword(s):

Poisson Process ◽

Finite Size ◽

Joint Probability ◽

Stochastic Inventory ◽

Left Behind ◽

Stochastic Inventory Model ◽

Waiting Area ◽

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.

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

Yugoslav journal of operations research ◽

10.2298/yjor150326018j ◽

2016 ◽

Vol 26 (4) ◽

pp. 467-506 ◽

Cited By ~ 3

Author(s):

K. Jeganathan ◽

J. Sumathi ◽

G. Mahalakshmi

Keyword(s):

Steady State ◽

Finite Size ◽

Joint Probability ◽

Life Time ◽

The Other ◽

Inventory Level ◽

Cost Rate ◽

Parallel Queues ◽

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.

Stochastic Inventory Model with Reworks

The Chittagong University Journal of Science ◽

10.3329/cujs.v40i1.47908 ◽

2018 ◽

Vol 40 (1) ◽

pp. 59-74

Author(s):

Mohammad Ekramol Islam ◽

M Sharif Uddin ◽

Mohammad Ataullah ◽

Ganesh Chandra Ray

Keyword(s):

Poisson Process ◽

Current Model ◽

Joint Probability ◽

Fixed Time ◽

Inventory Level ◽

Time Duration ◽

Stochastic Inventory ◽

Stochastic Inventory Model ◽

Volatile Liquids

In most of the inventory models, a single stock is considered from where items are served for the customers. In this paper, two stocks are considered. One is for fresh items and another is for returned items. The model is more appropriate where warranties are provided for a fixed time duration. Moreover, inventory which are kept to the stock may not have finite shelf-life is also considered here such as milk, meat, vegetables, radioactive materials, volatile liquids. In this current model, we considered the inventory is decayed in a constant rate θ. It is assumed that inventory level for both the fresh and returned items are pre-determined. When inventory level reaches at s, a replenishment takes place with parameter ϒ The demands that arrive for fresh items and returned items follow Poisson process with parameter λ & δ respectively. Service will be provided with Poisson process for returned items with parameter μ. The joint probability distribution for inventory level of returned items and for fresh items is obtained in the steady state analysis. Some systems characteristics of the model are derived and the results are illustrated with the help of numerical examples. The Chittagong Univ. J. Sci. 40(1) : 59-74, 2018

A Perishable Inventory System with Postponed Demands and Finite

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v12i8.5070 ◽

2016 ◽

Vol 12 (8) ◽

pp. 6500-6515

Author(s):

R Jayaraman

Keyword(s):

Steady State ◽

Joint Probability ◽

Life Time ◽

Inventory System ◽

Inventory Level ◽

Single Item ◽

Perishable Inventory ◽

In this article, we consider a continuous review perishable inventory system with a finite number of homogeneous sources generating demands. The demand time points form quasi random process and demand is for single item. The maximum storage capacity is assumed to be The life time of each item is assumed to have exponential distribution. The order policy is policy, that is, whenever the inventory level drops to a prefixed level an order for 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 the stock out periods either enter a pool or leave the system which is according to a Bernoulli trial. The demands in the pool are selected one by one, while the stock is above the level with interval time between any two successive selections is distributed as exponential. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady state case. Various system performance measures are derived to compute the total expected cost per unit time in the steady state. The optimal cost function and the optimal are studied numerically.

Correlation structure of teletraffic measurements from randomly-scanned calls arrival data

Journal of Applied Probability ◽

10.1017/s0021900200046933 ◽

1980 ◽

Vol 17 (01) ◽

pp. 203-217

Author(s):

P. A. Lee

Keyword(s):

Probability Distribution ◽

Explicit Expression ◽

Correlation Coefficient ◽

Point Processes ◽

Joint Probability ◽

Correlation Structure ◽

Arrival Process ◽

Interarrival Times ◽

Stochastic Point Processes

In teletraffic measurements, a call arrival process is commonly studied using a method with time-uniform or periodic scanning. The information recorded is the number of calls arrived between the scannings, from which data the number of scans between two successive calls is obtained. These later numbers are used as a measure of the interarrival times. For an exponential call arrival process, except in the case of Poisson scanning, all other scanning schemes yield the number of scans which are not independent in any two interarrival intervals. By treating the problem as an interaction of two stationary stochastic point processes, we determine the exact joint probability distribution of the number of scans in two adjacent and non-adjacent interarrival intervals. An explicit expression for the correlation coefficient is also obtained.

Correlation structure of teletraffic measurements from randomly-scanned calls arrival data

Journal of Applied Probability ◽

10.2307/3212937 ◽

1980 ◽

Vol 17 (1) ◽

pp. 203-217

Author(s):

P. A. Lee

Keyword(s):

Probability Distribution ◽

Explicit Expression ◽

Correlation Coefficient ◽

Point Processes ◽

Joint Probability ◽

Correlation Structure ◽

Arrival Process ◽

Interarrival Times ◽

Stochastic Point Processes

In teletraffic measurements, a call arrival process is commonly studied using a method with time-uniform or periodic scanning. The information recorded is the number of calls arrived between the scannings, from which data the number of scans between two successive calls is obtained. These later numbers are used as a measure of the interarrival times.For an exponential call arrival process, except in the case of Poisson scanning, all other scanning schemes yield the number of scans which are not independent in any two interarrival intervals. By treating the problem as an interaction of two stationary stochastic point processes, we determine the exact joint probability distribution of the number of scans in two adjacent and non-adjacent interarrival intervals. An explicit expression for the correlation coefficient is also obtained.

SCALINGS OF A MODIFIED MANNA MODEL WITH BULK DISSIPATION

International Journal of Modern Physics C ◽

10.1142/s0129183109013601 ◽

2009 ◽

Vol 20 (02) ◽

pp. 273-284 ◽

Cited By ~ 2

Author(s):

CHIEN-FU CHEN ◽

CHAI-YU LIN

Keyword(s):

Probability Distribution ◽

Finite Size ◽

Joint Probability ◽

Square Lattice ◽

Finite Size Scaling ◽

Scaling Relation ◽

Critical Scaling ◽

Crossover Behavior ◽

Boundary Dissipation

This study incorporates bulk dissipation described by a losing probability f into a modified Manna model on an L × L square lattice. The crossover behavior between bulk and boundary dissipation is investigated using the characteristic lengths produced by bulk dissipation. The toppling number Nn and area Na are studied. For a probability distribution of Ns, [Formula: see text] where s = n or a, the scaling form including the finite-size scaling (f = 0) and the critical scaling (L → ∞) are determined. Subsequently, this paper investigates the joint probability distribution [Formula: see text] and provides the scaling relation between the toppling number and area.