MAP/PH/1 Üretim Envanter Modeli

Abstract

Bu çalışmada, müşterilerin Markovian varış sürecine göre sisteme katıldığı faz-tipi hizmet sürelerine sahip bir üretim envanter modeli tartışılmaktadır. Envanter seviyesi pozitif olduğunda, gelen bir müşteri hizmet biriminin boş olduğunu tespit ederse hemen hizmete girer. Hizmet verilen müşteri sistemden ayrılır ve eldeki stok, hizmet tamamlanma anında bir birim azalır. Aksi takdirde müşteri sonsuz büyüklükte bir bekleme alanına (kuyruğa) girer ve hizmet almayı bekler. Üretim tesisi, ürünleri (𝑠, 𝑆) politikasına göre üretir. Envanter seviyesi 𝑠'ye düştüğünde üretim açılır ve envanter seviyesi maksimum 𝑆 seviyesine ulaşana kadar üretim açık kalır. Envanter seviyesi 𝑆 olduğu anda, üretim süreci kapatılır. Matris-geometrik yöntemi uygulayarak, üretim envanter modelinin kararlı durum analizini gerçekleştiriyoruz ve parametrelerin sistem performans ölçüleri üzerindeki etkisini ve envanter politikası için bir optimizasyon çalışmasını içeren birkaç açıklayıcı sayısal örnek gerçekleştiriyoruz.

Keywords

• Üretim envanteri
• Faz-tipi dağılım
• Markovian varış süreci
• Matris-geometrik yöntem
• Yerel satın alma
• Optimizasyon

A MAP/PH/1 PRODUCTION INVENTORY MODEL

Abstract

In this study, a production inventory model with phase type service times where customers join the system occur according to a Markovian arrival process is discussed. When the inventory level is positive, if an arriving customer finds the server idle gets into service immediately. Served customer leaves the system and the on-hand inventory is decreased one unit of item at service completion epoch. Otherwise, the customer enters into a waiting space (queue) of infinite capacity and waits for get served. The production facility produces items according to an (𝑠,𝑆) policy. The production is switched on when the inventory level depletes to 𝑠 and the production remains on until the inventory level reaches to the maximum level 𝑆. Once the inventory level becomes 𝑆, the production process is switched off. Applying the matrix-geometric method, we carry out the steady-state analysis of the production inventory model and perform a few illustrative numerical examples includes the effect of parameters on the system performance measures and an optimization study for the inventory policy

Keywords

• Production inventory
• Phase-type distribution
• Markovian arrival process
• Matrix-geometric method
• Local purchase
• Optimization

References

1. 1. Artalejo, J.R., Gomez-Corral, A. and He, Q.M. (2010) Markovian arrivals in stochastic modelling: a survey and some new results. SORT: Statistics and Operations Research Transactions, 34(2), 101-144.
2. 2. Baek, J.W. and Moon, S.K. (2014) The 𝑀/𝑀/1 queue with a production-inventory system and lost sales, Applied Mathematics and Computation, 233, 534-544. doi.org/10.1016/j.amc.2014.02.033
3. 3. Baek, J.W. and Moon, S.K. (2016) A production-inventory system with a Markovian service queue and lost sales, Journal of the Korean Statistical Society, 45, 14-24. doi.org/10.1016/j.jkss.2015.05.002
4. 4. Barron, Y. (2022) The continuous (𝑆, 𝑠, 𝑆𝑒 ) inventory model with dual sourcing and emergency orders, European Journal of Operational Research, 301, 18-38. doi.org/10.1016/j.ejor.2021.09.021
5. 5. Chakravarthy, S.R. (2001) The batch Markovian arrival process: a review and future work. In: Krishnamoorthy, A., Raju, N. and Ramaswami, V. (eds) Advances in probability and stochastic processes. Notable, New Jersey, 21-49.
6. 6. Chakravarthy, S.R. (2010) Markovian arrival processes, Wiley Encyclopedia of Operations Research and Management Science. doi.org/10.1002/9780470400531.eorms0499
7. 7. Chakravarthy, S.R. (2020) Queueing-inventory models with batch demands and positive service times, Automation and Remote Control, 81, 713-730. doi.org/10.1134/S0005117920040128
8. 8. Chakravarthy, S.R. and Rumyantsev, A. (2020) Analytical and simulation studies of queueing-inventory models with 𝑀𝐴𝑃 demands in batches and positive phase type services, Simulation Modelling Practice and Theory, 103, 1-15. doi.org/10.1016/j.simpat.2020.102092

9. 9. De la Cruz, N.N. and Daduna, H. (2022) Analysis of second order properties of production– inventory systems with lost sales, Annals of Operations Research. doi.org/10.1007/s10479- 022-05061-z
10. 10. He, Q.M. and Zhang, H. (2013) Performance analysis of an inventory–production system with shipment consolidation in the production facility, Performance Evaluation, 70(9), 623- 638. doi.org/10.1016/j.peva.2013.05.007
11. 11. Karthikeyan, K. and Sudhesh, R. (2016) Recent review article on queueing inventory systems, Research Journal of Pharmacy and Technology, 9(11), 1451-1461. doi:10.5958/0974-360X.2016.00421.2
12. 12. Krishnamoorthy, A. and Raju, N. (1998) (𝑠, 𝑆) inventory with lead time-the N-policy. International Journal of Information and Management Sciences, 9, 45-52.
13. 13. Krishnamoorthy, A., Lakshmy, B. and Manikandan, R. (2011a) A survey on inventory models with positive service time, OPSEARCH, 48, 153-169. doi.org/10.1007/s12597-010- 0032-z
14. 14. Krishnamoorthy, A., Viswanath, C. and Narayanan, V.C. (2011b) Production inventory with service time and vacation to the server, IMA Journal of Management Mathematics, 22, 33-45. doi.org/10.1093/imaman/dpp025
15. 15. Krishnamoorthy, A. and Narayanan, V.C. (2013) Stochastic decomposition in production inventory with service time, European Journal of Operational Research, 228, 358-366. doi.org/10.1016/j.ejor.2013.01.041
16. 16. Krishnamoorthy, A., Nair, S.S. and Narayanan, V.C. (2015) Production inventory with service time and interruptions, International Journal of Systems Science, 46(10), 1800- 1816. doi.org/10.1080/00207721.2013.837538
17. 17. Latouche, G. and Ramaswami, V. (1999) Introduction to Matrix Analytic Methods in Stochastic Modelling, ASASIAM, Philadelphia.
18. 18. Melikov, A., Mirzayev, R. and Sztrik, J. (2023) Double-sources queuing-inventory systems with finite waiting room and destructible stocks, Mathematics, 11, 226. doi.org/10.3390/math11010226
19. 19. Neuts, M.F. (1981) Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach, The Johns Hopkins University Press, Baltimore, MD. [1994 version is Dover Edition].
20. 20. Saffari, M., Asmussen, S. and Haji, R. (2013) The 𝑀/𝑀/1 queue with inventory, lost sale, and general lead times, Queueing Systems, 75, 65-77. doi.org/10.1007/s11134-012-9337-3
21. 21. Schwarz, M., Sauer, C., Daduna, H., Kulik, R. and Szekli, R. (2006) 𝑀/𝑀/1 queueing system with inventory, Queueing Systems: Theory and Applications, 54, 55-78. doi.org/10.1007/s11134-006-8710-5
22. 22. Sigman, K. and Simchi-Levi, D. (1992) Light traffic heuristic for an 𝑀/𝐺/1 queue with limited inventory, Annals of Operations Research, 40, 371-380. doi.org/10.1007/BF02060488
23. 23. Shajin, D., Krishnamoorthy, A., Melikov, A.Z. and Sztrik, J. (2022) Multi-server queuing production inventory system with emergency replenishment, Mathematics, 10(20), 3839. doi.org/10.3390/math10203839