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Optimal control of a dynamic production-inventory system with various cost criteria

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Abstract

In this article, we investigate the dynamic control problem of a production-inventory system. Here, demands arrive at the production unit according to a Poisson process and are processed in an FCFS manner. The processing time of the customer’s demand is exponentially distributed. Production manufacturers produce items on a make-to-order basis to meet customer demands. The production is run until the inventory level becomes sufficiently large. We assume that the production time of an item follows an exponential distribution and that the amount of time for the produced item to reach the retail shop is negligible. In addition, we assume that no new customer joins the queue when there is void inventory. Moreover, when a customer is waiting in an infinite FIFO queue for service, he/she does not leave the queue even if the inventory is exhausted. This yields an explicit product-form solution for the steady-state probability vector of the system. The optimal policy that minimizes the discounted/average/pathwise average total cost per production is derived using a Markov decision process approach. We find an optimal policy using value/policy iteration algorithms. Numerical examples are discussed to verify the proposed algorithms.

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References

  • Arivarignan, G., Elango, C. & Arumugam, N. (2002). A continuous review perishable inventory control system at service facilities. Advances in Stochastic Modelling, 29–40.

  • Berman, O., Kaplan, E. H., & Shimshak, D. G. (1993). Deterministic approximations for inventory management at service facilities. IIE Transactions, 25(5), 98–104.

    Article  Google Scholar 

  • Berman, O., & Kim, E. (1999). Stochastic models for inventory managements at service facilities. Communications in Statistics. Stochastic Models, 15(4), 695–718.

    Article  Google Scholar 

  • Berman, O., & Kim, E. (2004). Dynamic inventory strategies for profit maximization in a service facility with stochastic service, demand and lead time. Mathematical Methods of Operational Research, 60, 497–521.

    Article  Google Scholar 

  • Chakravarthy, S. R. (2022). Introduction to matrix analytic methods in queues 1: Analytical and simulation approach—basics. Wiley.

    Book  Google Scholar 

  • Chakravarthy, S. R. (2022). Introduction to matrix-analytic methods in queues 2: Analytical and simulation approach—queues and simulation. Wiley.

    Book  Google Scholar 

  • Chakravarthy, S., & Alfa, A. S. (1986). Matrix-analytic method in stochastic models. CRC Press, Taylor & Francis Group.

    Google Scholar 

  • Federgruen, A., & Zhang, Y. (1992). An efficient algorithm for computing an optimal \((r, Q)\) policy in continuous review stochastic inventory systems. Operations Research, 40(4), 633–825.

    Article  Google Scholar 

  • Federgruen, A., & Zipkin, P. (1986). An inventory model with limited production capacity and uncertain demands II. The discounted-cost criterion. Mathematics of Operations Research, 11(2), 208–215.

    Article  Google Scholar 

  • Federgruen, A., & Zipkin, P. (1986). An inventory model with limited production capacity and uncertain demands I. The average-cost criterion. Mathematics of Operations Research, 11(2), 193–207.

    Article  Google Scholar 

  • Ghosh, M. K., & Saha, S. (2014). Risk-sensitive control of continuous time Markov chain. Stochastic, 86(4), 655–675.

    Article  Google Scholar 

  • Golui, S., & Pal, C. (2022). Risk-sensitive discounted cost criterion for continuous-time Markov decision processes on a general state space. Mathematical Methods of Operations Research, 95, 219–247. https://doi.org/10.1007/s00186-022-00779-9

    Article  Google Scholar 

  • Guo, X. P., & Hernández-Lerma, O. (2009). Continuous time Markov decision processes. Stochastic modelling and applied probability (p. 62). Springer.

    Book  Google Scholar 

  • He, Q.-M., Buzacott, E. M., & Jewkes, J. (2002). Optimal and near-optimal inventory control policies for a make-to-order inventory-production system. European Journal of Operational Research, 141, 113–132.

    Article  Google Scholar 

  • He, Q.-M., & Jewkes, E. M. (2001). Performance measures of a make-to-order inventory-production system. IIE Transactions, 32, 400–419.

    Google Scholar 

  • Helmes, K. L., Stockbridge, R. H., & Zhu, C. (2015). An inventory model with limited production capacity and uncertain demands II. The discounted-cost criterion. SIAM Journal on Control and Optimization, 53, 2100–2140.

    Article  Google Scholar 

  • Hernández-Lerma, O. & Lasserre, J. B. (1996). Discrete-time Markov control processes. Stochastic Modelling and Applied Probability (1996).

  • Krishnamoorthy, A., Deepak, T. G., Narayanan, V. C., & Vineetha, K. (2006). Control policies for inventory with service time. Stochastic Analysis and Applications, 24(4), 889–899.

    Article  Google Scholar 

  • Krishnamoorthy, A., Manikandan, R., & Lakshmy, B. (2015). A revisit to queueing-inventory system with positive service time. Annals of Operations Research, 233, 221–236.

    Article  Google Scholar 

  • Krishnamoorthy, A., & Narayanan, V. C. (2013). Stochastic decomposition in production inventory with service time. European Journal of Operational Research, 228(2), 358–366.

    Article  Google Scholar 

  • Krishnamoorthy, A., Narayanan, V. C., Deepak, T. G., & Vineetha, K. (2006). Effective utilization of server idle time in an \((s, S)\) inventory with positive service time. Journal of Applied Mathematics and Stochastic Analysis. https://doi.org/10.1155/JAMSA/2006/69068

    Article  Google Scholar 

  • Krishnamoorthy, A., Sha**, D., & Narayanan, V. C. (2021). Inventory with positive service time: A survey. In V. Anisimov & N. Limnios (Eds.), Queueing theory (Vol. 2, pp. 201–237). Wiley.

    Chapter  Google Scholar 

  • Kumar, K. S., & Pal, C. (2013). Risk-sensitive control of jump process on denumerable state space with near monotone cost. Applied Mathematics and Optimization, 68, 311–331.

    Article  Google Scholar 

  • Kumar, K. S., & Pal, C. (2015). Risk-sensitive control of continuous-time Markov processes with denumerable state space. Stochastic Analysis and Applications, 33, 863–881. https://doi.org/10.1080/07362994.2015.1050674

    Article  Google Scholar 

  • Malini, S. & Sha**, D. (2020). An \((s,S)\) production inventory system with state dependent production rate and lost sales. Applied Probability and Stochastic Processes, 215–233.

  • Melikov, A. Z., & Molchanov, A. A. (1992). Stock optimization in transportation/storage systems. Cybernetics and Systems Analysis, 28(3), 484–487.

    Article  Google Scholar 

  • Meyn, S. P., & Tweedie, R. L. (1993). Stability of Markovian processes III: Foster–Lyapunov criteria for continuous time processes. Advanced in Applied Probability, 25, 518–548.

    Article  Google Scholar 

  • Neuts, M. F. (1989). Structured stochastic matrices of M/G/I type and their applications. Marcel Dekker.

    Google Scholar 

  • Neuts, M. F. (1994). Matrix-geometric solutions in stochastic models: An algorithmic approach (2nd ed.). Dover.

    Google Scholar 

  • Pal, C., & Pradhan, S. (2019). Risk sensitive control of pure jump processes on a general state space. Stochastics, 91(2), 155–174.

    Article  Google Scholar 

  • Pal, B., Sena, S. S., & Chaudhuri, K. (2012). Three-layer supply chain: A production-inventory model for reworkable items. Journal of Applied Mathematics and Computation, 219, 510–543.

    Google Scholar 

  • Perera, S. C., & Sethi, S. P. (2022). A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)-type policies-the discrete-time case. Production and Operations Management. https://doi.org/10.1111/poms.13820

    Article  Google Scholar 

  • Perera, S. C., & Sethi, S. P. (2022). A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)-type policies-the continuous-time case. Production and Operations Management. https://doi.org/10.1111/poms.13819

    Article  Google Scholar 

  • Piunovsky, A., & Zhang, Y. (2020). Continuous-time Markov decision processes. Probability theory and stochastic modelling. Springer.

    Book  Google Scholar 

  • Puterman, M. L. (1994). Markov decision processes: Discrete stochastic dynamic programming. Wiley series in probability and statistics. Wiley.

    Book  Google Scholar 

  • Rahul, R. (2022). Low delay file transmissions over power constrained quasi-static fading channels. Ph.D. thesis, IISC.

  • Saffari, M., Haji, R., & Hassanzadeh, F. (2011). A queueing system with inventory and mixed exponentially distributed lead times. The International Journal of Advanced Manufacturing Technology, 53, 1231–1237.

    Article  Google Scholar 

  • Sajeev, S. N. (2012). On \((s, S)\) inventory policy with/without retrial and interruption of service/production. Ph.D. thesis, Cochin University of Science and Technology, India.

  • Sarkar, B. (2012). An inventory model with reliability in an imperfect production process. Journal of Applied Mathematics and Computation, 218, 4081–4091.

    Google Scholar 

  • Schwarz, M., Sauer, C., Daduna, H., Kulik, R., & Szekli, R. (2006). M/M/1 queueing systems with inventory. Queueing Systems, 54, 55–78.

    Article  Google Scholar 

  • Schwarz, M., Wichelhaus, C., & Daduna, H. (2007). Product form models for queueing networks with an inventory. Stochastic Models, 23, 627–663.

    Article  Google Scholar 

  • Sigman, K., & Simchi-Levi, D. (1992). Light traffic heuristic for an M/G/1 queue with limited inventory. Annals of Operations Research, 40, 371–380.

    Article  Google Scholar 

  • Tijms, Henk C. (2003). A first course in stochastic models. Wiley.

    Book  Google Scholar 

  • Veatch, H. M., & Wein, M. L. (1994). Optimal control of a two-station tandem production/inventory system. Operations Research, 42, 337–350.

    Article  Google Scholar 

  • Zhao, N., & Lian, Z. (2011). A queueing-inventory system with two classes of customers. International Journal of Production Economics, 129, 225–231.

    Article  Google Scholar 

Download references

Acknowledgements

We thank the anonymous referees for their careful reading of our manuscript and suggestions that have greatly improved the presentation. The research work of Chandan Pal is partially supported by SERB, India, Grant MTR/2021/000307, and the research work of Manikandan R, is supported by SERB, India, Grant EEQ/2022/000229.

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Author notes

  1. Subrata Golui and Chandan Pal have contributed equally to this work.

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Subrata Golui & Chandan Pal

  2. Department of Mathematics, Central University of Kerala, Kasaragod, Kerala, 671320, India

    R. Manikandan

  3. Department of Industrial and Management Systems Engineering, University of South Florida, Tampa, FL, 33620, USA

    Abhay Sobhanan

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  1. Subrata Golui
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Correspondence to R. Manikandan.

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Golui, S., Pal, C., Manikandan, R. et al. Optimal control of a dynamic production-inventory system with various cost criteria. Ann Oper Res 337, 75–103 (2024). https://doi.org/10.1007/s10479-023-05716-5

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