Skip to main content
SpringerLink
Log in

On a Retrial Production Inventory System with Vacation and Multiple Servers

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, the model of a production inventory system with heterogeneous servers, the vacation of one of the servers (vacationing server), and retrial customers is considered. The customers reach the system each demanding exactly one unit of the item, according to a Poisson process. Service times and lead time are independent and follow exponential distributions. A single unit of the item is produced at a time according to (s, S) policy. It is assumed that when the inventory level reaches zero or when the orbit is empty or both, one of the servers(server 2) takes multiple, exponentially distributed vacation time. At the end of the vacation, that server immediately takes another vacation, if he/she finds empty stock or empty orbit or both. During the stock out period or server busy period or vacation period, the demands that arrive enter into the orbit of infinite size. If a customer finds both the servers busy or server 1 busy and server 2 in vacation or inventory level is zero, in accordance with Bernoulli trials may wait in the orbit or leave forever. When either the customer in the orbit cannot make an attempt due to the busy servers or inventory level zero, he/she under Bernoulli trials may come back to the orbit or leave the orbit. The rate of retrial customers from orbit is linear. The production process is switched on when the inventory level reaches to s, and similarly when the level of inventory reaches back to S the production process is switched off. The production process releases items that have inter arrival times that are exponentially distributed. Ergodicity condition is obtained and matrix analytic method is used to calculate the steady-state probabilities of the constructed 4D Markov chain. Minimum expected total cost per unit time and many significant system performance measures are obtained. Sensitivity analysis is relevant to check various performance measures that is highly applicable in the realistic situation and it also provides managerial insights.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ahmed, W., Sarkar, B.: Impact of carbon emissions in a sustainable supply chain management for a second generation biofuel. J. Clean. Prod. 186, 807–820 (2018)

    Google Scholar 

  2. Ahmed, W., Sarkar, B.: Management of next-generation energy using a triple bottom line approach under a supply chain framework. Resour. Conserv. Recycl. 150, 104431 (2019)

    Google Scholar 

  3. Artalejo, J.: Accessible bibliography on retrial queues: progress in 2000–2009. Math. Comput. Model. 51, 1071–1081 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Falin, G., Templeton, J.G.: Retrial queues, vol. 75. CRC Press, Boca Raton (1997)

    MATH  Google Scholar 

  5. Gao, S., Wang, J., Li, W.W.: An m/g/1 retrial queue with general retrial times, working vacations and vacation interruption. Asia-Pac. J. Op. Res. 31(02), 1440006 (2014)

    MathSciNet  MATH  Google Scholar 

  6. Jose, K.P., Nair, S.S.: Analysis of two production inventory systems with buffer, retrials and different production rates. J. Ind. Eng. Int. 13(3), 369–380 (2017)

    Google Scholar 

  7. Kathiresan, S., Anbazhagan, N., Jeganathan, K.: An inventory system with retrial demands and working vacation. Int. J. Sci. Res. Publ. 4(12), 1–25 (2014)

    Google Scholar 

  8. Kim, M.-S., Kim, J.-S., Sarkar, B., Sarkar, M., Iqbal, M.W.: An improved way to calculate imperfect items during long-run production in an integrated inventory model with backorders. J. manuf. syst. 47, 153–167 (2018)

    Google Scholar 

  9. Krishnamoorthy, A., Jose, K.P.: Three production inventory systems with service, loss and retrial of customers. Int. J. Inf. Manag. Sci. 19(3), 367–389 (2008)

    MathSciNet  MATH  Google Scholar 

  10. Krishnamoorthy, A., Nair, S.S., Narayanan, V.C.: An inventory model with server interruptions and retrials. Op. Res. 12(2), 151–171 (2012)

    MATH  Google Scholar 

  11. Kumar, B., Soundararajan, P.: Analysis of an m/m/n queue with bernoulli service schedule. Int. J. Op. Res. 5, 48–72 (2009)

    MATH  Google Scholar 

  12. Kumar, M.S., Arumuganathan, R.: Analysis of single server retrial queue with batch arrivals, two phases of heterogeneous service and multiple vacations with n-policy. Int. J. Op. Res. 5(4), 213–224 (2008)

    MathSciNet  MATH  Google Scholar 

  13. Laxmi, P.V., Soujanya, M.: Retrial inventory system with negative customers and service interruptions. Opsearch 52(2), 212–224 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Levy, Y., Yechiali, U.: An m/m/s queue with servers’ vacations. Inf. Syst. Op. Res. 14(2), 153–163 (1976)

    MathSciNet  MATH  Google Scholar 

  15. Melikov, A.Z., Rustamov, A.M., Ponomarenko, L.A.: Approximate analysis of a queueing-inventory system with early and delayed server vacations. Autom. Remote Control 78(11), 1991–2003 (2017)

    MathSciNet  MATH  Google Scholar 

  16. Melikov, A.Z., Shahmaliyev, M.: Queueing system m/m/1/\(\infty \) with perishable inventory and repeated customers. Autom. Remote Control 80(1), 53–65 (2019)

    MathSciNet  Google Scholar 

  17. Neuts, M.F.: Matrix-geometric solutions in stochastic models: an algorithmic approach. Courier Corporation (1994)

  18. Neuts, M.F., Rao, B.: Numerical investigation of a multiserver retrial model. Queueing syst. 7(2), 169–189 (1990)

    MATH  Google Scholar 

  19. Noh, J.S., Kim, J.S., Sarkar, B.: Stochastic joint replenishment problem with quantity discounts and minimum order constraints. Op. Res. 19(1), 151–178 (2019)

    Google Scholar 

  20. Omair, M., Ullah, M., Ganguly, B., Noor, S., Maqsood, S., Sarkar, B.: The quantitative analysis of workers’ stress due to working environment in the production system of the automobile part manufacturing industry. Mathematics 7, 627 (2019)

    Google Scholar 

  21. Padmavathi, I., Sivakumar, B., Arivarignan, G.: A retrial inventory system with single and modified multiple vacation for server. Ann. Op. Res. 233(1), 335–364 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Sarkar, B.: Mathematical and analytical approach for the management of defective items in a multi-stage production system. J. clean. prod. 218, 896–919 (2019)

    Google Scholar 

  23. Sarkar, B., Ahmed, W., Kim, N.: Joint effects of variable carbon emission cost and multi-delay-in-payments under single-setup-multiple-delivery policy in a global sustainable supply chain. J. Clean. Prod. 185, 421–445 (2018)

    Google Scholar 

  24. Sarkar, B., Guchhait, R., Sarkar, M., Cárdenas-Barrón, L.E.: How does an industry manage the optimum cash flow within a smart production system with the carbon footprint and carbon emission under logistics framework? Int. J. Prod. Econ. 213, 243–257 (2019)

    Google Scholar 

  25. Sarkar, B., Guchhait, R., Sarkar, M., Pareek, S., Kim, N.: Impact of safety factors and setup time reduction in a two-echelon supply chain management. Robot. Comput. Integr. Manuf. 55, 250–258 (2019)

    Google Scholar 

  26. Sarkar, B., Tayyab, M., Kim, N., Habib, M.S.: Optimal production delivery policies for supplier and manufacturer in a constrained closed-loop supply chain for returnable transport packaging through metaheuristic approach. Comput. Ind. Eng. 135, 987–1003 (2019)

    Google Scholar 

  27. Sarkar, B., Ullah, M., Kim, N.: Environmental and economic assessment of closed-loop supply chain with remanufacturing and returnable transport items. Comput. Ind. Eng. 111, 148–163 (2017)

    Google Scholar 

  28. Sarkar, B., Zhang, C., Majumder, A., Sarkar, M., Seo, Y.W.: A distribution free newsvendor model with consignment policy and retailer’s royalty reduction. Int. J. Prod. Res. 56(15), 5025–5044 (2018)

    Google Scholar 

  29. Sarkar, M., Kim, S., Jemai, J., Ganguly, B., Sarkar, B.: An application of time-dependent holding costs and system reliability in a multi-item sustainable economic energy efficient reliable manufacturing system. Energies 12(15), 2857 (2019)

    Google Scholar 

  30. Sivakumar, B.: An inventory system with retrial demands and multiple server vacation. Qual. Technol. Quant. Manag. 8(2), 125–146 (2011)

    Google Scholar 

  31. Sreenivasan, C., Krishnamoorthy, A.: A queueing models with vacations and working vacations. Cochin University of Science and Technology, Dyuthi, Digital Repository (2012)

  32. Suganya, C., Sivakumar, B.: Map/ph (1), ph (2)/2 finite retrial inventory system with service facility, multiple vacations for servers. Int. J. Math. Op. Res. 15(3), 265–295 (2019)

    MathSciNet  Google Scholar 

  33. Tiwari, S., Ahmed, W., Sarkar, B.: Multi-item sustainable green production system under trade-credit and partial backordering. J. Clean. Prod. 204, 82–95 (2018)

    Google Scholar 

  34. Tiwari, S., Ahmed, W., Sarkar, B.: Sustainable ordering policies for non-instantaneous deteriorating items under carbon emission and multi-trade-credit-policies. J. Clean. Prod. 240, 118183 (2019)

    Google Scholar 

  35. Tweedie, R.L.: Sufficient conditions for regularity, recurrence and ergodicity of markov processes. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 78, pp. 125–136. Cambridge University Press, Cambridge (1975)

  36. Ullah, M., Sarkar, B.: Recovery-channel selection in a hybrid manufacturing-remanufacturing production model with rfid and product quality. Int. J. Prod. Econ. 219, 360–374 (2020)

    Google Scholar 

  37. Yadavalli, V., Anbazhaga, N., Jeganathan, K.: A retrial inventory system with impatient customers. Appl. Math. Inf. Sci. 9(2), 637 (2015)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. P. G. and Research Dept. of Mathematics, St.Peter’s College, Kolenchery, Kerala, 682311, India

    K. P. Jose

  2. Sree Neelakanta Govt.Sanskrit College, Pattambi, Palakkad, Kerala, 679303, India

    P. Beena

Authors
  1. K. P. Jose
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Beena
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. P. Jose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jose, K.P., Beena, P. On a Retrial Production Inventory System with Vacation and Multiple Servers. Int. J. Appl. Comput. Math 6, 108 (2020). https://doi.org/10.1007/s40819-020-00862-x

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40819-020-00862-x

Keywords

Mathematics Subject Classification

Search

Navigation