Multi-Server Queuing Production Inventory System with Emergency Replenishment
We consider a multi-server production inventory system with an unlimited waiting line. Arrivals occur according to a non-homogeneous Poisson process and exponentially distributed service time. At the service completion epoch, one unit of an item in the on-hand inventory decreases with probability &l...
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MDPI AG
2022-10-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/20/3839 |
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| author | Dhanya Shajin Achyutha Krishnamoorthy Agassi Z. Melikov Janos Sztrik |
| author_facet | Dhanya Shajin Achyutha Krishnamoorthy Agassi Z. Melikov Janos Sztrik |
| author_sort | Dhanya Shajin |
| collection | DOAJ |
| description | We consider a multi-server production inventory system with an unlimited waiting line. Arrivals occur according to a non-homogeneous Poisson process and exponentially distributed service time. At the service completion epoch, one unit of an item in the on-hand inventory decreases with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>, and the customer leaves the system without taking the item with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>δ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. The production inventory system adopts an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> policy where the processing of inventory requires a positive random amount of time. The production time for a unit item is phase-type distributed. Furthermore, assume that an emergency replenishment of one item with zero lead time takes place when the on-hand inventory level decreases to zero. The emergency replenishment is incorporated in the system to ensure customer satisfaction. We derive the stationary distribution of the system and some main performance measures, such as the distribution of the production on/off time in a cycle and the mean emergency replenishment cycle time. Numerical experiments are conducted to illustrate the system performance. A cost function is constructed, and we examine the optimal number of servers to be employed. Furthermore, we numerically calculate the optimal values of the production starting level and maximum inventory level. |
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| last_indexed | 2024-03-09T09:46:13Z |
| publishDate | 2022-10-01 |
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| series | Mathematics |
| spelling | doaj.art-da48e9de27044eb3afc5a76029c8440a2023-12-02T00:36:07ZengMDPI AGMathematics2227-73902022-10-011020383910.3390/math10203839Multi-Server Queuing Production Inventory System with Emergency ReplenishmentDhanya Shajin0Achyutha Krishnamoorthy1Agassi Z. Melikov2Janos Sztrik3Department of Mathematics, S. N. College, Chempazhanthy 695587, Kerala, IndiaCentre for Research in Mathematics, CMS College, Kottayam 686001, Kerala, IndiaInstitute of Control Systems, National Academy of Science, AZ 1141 Baku, AzerbaijanDepartment of Informatics and Networks, Faculty of Informatics, University of Debrecen, 4032 Debrecen, HungaryWe consider a multi-server production inventory system with an unlimited waiting line. Arrivals occur according to a non-homogeneous Poisson process and exponentially distributed service time. At the service completion epoch, one unit of an item in the on-hand inventory decreases with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>, and the customer leaves the system without taking the item with probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>δ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. The production inventory system adopts an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> policy where the processing of inventory requires a positive random amount of time. The production time for a unit item is phase-type distributed. Furthermore, assume that an emergency replenishment of one item with zero lead time takes place when the on-hand inventory level decreases to zero. The emergency replenishment is incorporated in the system to ensure customer satisfaction. We derive the stationary distribution of the system and some main performance measures, such as the distribution of the production on/off time in a cycle and the mean emergency replenishment cycle time. Numerical experiments are conducted to illustrate the system performance. A cost function is constructed, and we examine the optimal number of servers to be employed. Furthermore, we numerically calculate the optimal values of the production starting level and maximum inventory level.https://www.mdpi.com/2227-7390/10/20/3839(<i>s</i>, <i>S</i>) production inventory systemnon-homogeneous Poisson processmulti-serveremergency replenishmentcost function |
| spellingShingle | Dhanya Shajin Achyutha Krishnamoorthy Agassi Z. Melikov Janos Sztrik Multi-Server Queuing Production Inventory System with Emergency Replenishment Mathematics (<i>s</i>, <i>S</i>) production inventory system non-homogeneous Poisson process multi-server emergency replenishment cost function |
| title | Multi-Server Queuing Production Inventory System with Emergency Replenishment |
| title_full | Multi-Server Queuing Production Inventory System with Emergency Replenishment |
| title_fullStr | Multi-Server Queuing Production Inventory System with Emergency Replenishment |
| title_full_unstemmed | Multi-Server Queuing Production Inventory System with Emergency Replenishment |
| title_short | Multi-Server Queuing Production Inventory System with Emergency Replenishment |
| title_sort | multi server queuing production inventory system with emergency replenishment |
| topic | (<i>s</i>, <i>S</i>) production inventory system non-homogeneous Poisson process multi-server emergency replenishment cost function |
| url | https://www.mdpi.com/2227-7390/10/20/3839 |
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