An Integrated Markovian Queueing-Inventory Model in a Single Retailer- Single Supplier Problem with Imperfect Quality and Destructive Testing Acceptance Sampling

Document Type : Research Paper


School of Industrial Engineering, K. N. Toosi University of Technology (KNTU), Tehran, Iran.


This paper proposes a retailer-supplier queueing-inventory problem (RSQIP) in which the imperfect lots are investigated using a single sampling inspection plan. We integrate an M/M/m response queueing system for handling and responding to customers’ demands with a classical retailer-supplier inventory model considering defective items and inspection process for the first time. Customers whose demand is met leave the retailer system with exactly one item unless the inventory shortage occurs. The retailer places an order once the inventory level reaches an economic reorder point. The lead time is assumed exponential, and due to the imperfect incoming items, the retailer conducts a destructive acceptance sampling plan. The rate of inspection depends on the sample size. Provided that a lot is rejected, the supplier is required to provide a defect-free shipment. We present the stationary distribution of the number of demands in the response system. Then the joint stationary distribution of the order status and inventory level of the retailer is derived. Several performance measures and the expected total cost are presented steady-state, and a non-linear integer programming model is proposed to minimize the expected total cost. The results are numerically illustrated and reveal the convexity of the expected total cost. The optimal reorder point, order quantity, and the number of servers is computed for some numerical examples. A comprehensive sensitivity analysis is conducted to examine the effect of defective items, response system, and some important parameters on the entire developed model. Finally, useful managerial insights are presented.


       [1]        Saffari, M., Haji, R. and Hassanzadeh, F., 2011. A queueing system with inventory and mixed exponentially distributed lead times. The International Journal of Advanced Manufacturing Technology, 53(9-12), pp.1231-1237.
       [2]        Jeganathan, K., Reiyas, M.A., Padmasekaran, S. and Lakshmanan, K., 2017. An $$ M/E_ {K}/1/N $$ M/EK/1/N Queueing-Inventory System with Two Service Rates Based on Queue Lengths. International Journal of Applied and Computational Mathematics, 3(1), pp.357-386.
       [3]        Schwarz, M., Sauer, C., Daduna, H., Kulik, R. and Szekli, R., 2006. M/M/1 queueing systems with inventory. Queueing Systems, 54(1), pp.55-78.
       [4]        Yue, D. and Qin, Y., 2019. A production inventory system with service time and production vacations. Journal of Systems Science and Systems Engineering, 28(2), pp.168-180.
       [5]        Naimi Sadigh, A., Chaharsooghi, S.K. and Sheikhmohammady, M., 2016. Game-theoretic analysis of coordinating pricing and marketing decisions in a multi-product multi-echelon supply chain. Scientia Iranica, 23(3), pp.1459-1473.
       [6]        Mohtashami, Z., Aghsami, A. and Jolai, F., 2020. A green closed loop supply chain design using queuing system for reducing environmental impact and energy consumption. Journal of cleaner production, 242, p.118452.
       [7]        Zokaee, M., Nazari, A., Aghsami, A. and Jolai, F., 2021. An inventory system with coordination among manufacturers and retailers under buyback contract, vertical integration, retailer’s effort and carbon footprint constraint. International Journal of Sustainable Engineering, pp.1-21.
       [8]        Mokhtari, H., Asadkhani, J. (2019). 'Economic Order Quantity for Imperfect Quality Items Under Inspection Errors, Batch Replacement and Multiple Sales of Returned Items', Scientia Iranica, doi: 10.24200/sci.2019.52075.2520
       [9]        Gavish, B. and Graves, S.C., 1981. Production/inventory systems with a stochastic production rate under a continuous review policy. Computers & Operations Research, 8(3), pp.169-183.
     [10]      Sigman, K. and Simchi-Levi, D., 1992. Light traffic heuristic for an M/G/1 queue with limited inventory. Annals of Operations Research, 40(1), pp.371-380.
     [11]      Berman, O., Kaplan, E.H. and Shevishak, D.G., 1993. Deterministic approximations for inventory management at service facilities. IIE transactions, 25(5), pp.98-104.
     [12]      Berman, O. and Sapna, K.P., 2000. Inventory management at service facilities for systems with arbitrarily distributed service times. Stochastic Models, 16(3-4), pp.343-360.
     [13]      Schwarz, M. and Daduna, H., 2006. Queueing systems with inventory management with random lead times and with backordering. Mathematical Methods of Operations Research, 64(3), pp.383-414.
     [14]      Chang, K.H. and Lu, Y.S., 2010. Queueing analysis on a single-station make-to-stock/make-to-order inventory-production system. Applied Mathematical Modelling, 34(4), pp.978-991.
     [15]      Zhao, N. and Lian, Z., 2011. A queueing-inventory system with two classes of customers. International Journal of Production Economics, 129(1), pp.225-231.
     [16]      Krishnamoorthy, A. and Viswanath, N.C., 2013. Stochastic decomposition in production inventory with service time. European Journal of Operational Research, 228(2), pp.358-366.
     [17]      Saffari, M., Asmussen, S. and Haji, R., 2013. The M/M/1 queue with inventory, lost sale, and general lead times. Queueing Systems, 75(1), pp.65-77.
     [18]      Sivashankari, C.K. and Panayappan, S., 2014. Production inventory model for two levels production with defective items and incorporating multi-delivery policy. International Journal of Operational Research, 19(3), pp.259-279.
     [19]      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, pp.534-544.
     [20]      Krishnamoorthy, A., Manikandan, R. and Lakshmy, B., 2015. A revisit to queueing-inventory system with positive service time. Annals of Operations Research, 233(1), pp.221-236.
     [21]      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(1), pp.14-24.
     [22]      Manikandan, R. and NAIR, S.S., 2017. M/M/1/1 queueing-inventory system with retrial of unsatisfied customers. Communications in Applied Analysis, 21(2), pp.217-236.
     [23]      Baek, J.W., Bae, Y.H., Lee, H.W. and Ahn, S., 2018. Continuous-type (s, Q)-inventory model with an attached M/M/1 queue and lost sales. Performance Evaluation, 125, pp.68-79.
     [24]      Yue, D., Zhao, G. and Qin, Y., 2018. An M/M/1 Queueing-Inventory System with Geometric Batch Demands and Lost Sales. Journal of Systems Science and Complexity, 31(4), pp.1024-1041.
     [25]      Saffari, M., Sajadieh, M.S. and Hassanzadeh, F., 2019. A queuing system with inventory and competing suppliers. European Journal of Industrial Engineering, 13(3), pp.420-433.
     [26]      Shajin, D. and Krishnamoorthy, A., 2020. Stochastic decomposition in retrial queueing-inventory system. RAIRO-Operations Research, 54(1), pp.81-99.
     [27]      Shajin, D., Krishnamoorthy, A. and Manikandan, R., 2020. On a queueing-inventory system with common life time and Markovian lead time process. Operational Research, pp.1-34.
     [28]      Chakravarthy, S. and Hayat, K., 2020. Queueing-Inventory Models for a Two-Vendor System with Positive Service Times. Queueing Models and Service Management, 3(1), p.1.
     [29]      M Manikandan, R. and Nair, S.S., 2020. An M/M/1 Queueing-Inventory System with Working Vacations, Vacation Interruptions and Lost Sales. Automation and Remote Control, 81, pp.746-759.
     [30]      Shajin, D., Jacob, J. and Krishnamoorthy, A., 2021. On a queueing inventory problem with necessary and optional inventories. Annals of Operations Research, pp.1-26.
     [31]      Ozkar, S. and Kocer, U.U., 2021. Two-commodity queueing-inventory system with two classes of customers. OPSEARCH, 58(1), pp.234-256.
     [32]      Jeganathan, K., Selvakumar, S., Anbazhagan, N., Amutha, S. and Hammachukiattikul, P., 2021. Stochastic modeling on M/M/1/N inventory system with queue-dependent service rate and retrial facility. AIMS Mathematics, 6(7), pp.7386-7420.
     [33]      Graves, S.C., 1982. The application of queueing theory to continuous perishable inventory systems. Management Science, 28(4), pp.400-406.
     [34]      Manuel, P., Sivakumar, B. and Arivarignan, G., 2008. A perishable inventory system with service facilities and retrial customers. Computers & Industrial Engineering, 54(3), pp.484-501.
     [35]      Karthick, T., Sivakumar, B. and Arivarignan, G., 2015. An inventory system with two types of customers and retrial demands. International Journal of Systems Science: Operations & Logistics, 2(2), pp.90-112.
     [36]      Albrecher, H., Boxma, O.J., Essifi, R. and Kuijstermans, R., 2017. A queueing model with randomized depletion of inventory. Probability in the Engineering and Informational Sciences, 31(1), pp.43-59.
     [37]      Chakravarthy, S.R., Maity, A. and Gupta, U.C., 2017. An ‘(s, S)’inventory in a queueing system with batch service facility. Annals of Operations Research, 258(2), pp.263-283.
     [38]      Marand, A.J., Li, H. and Thorstenson, A., 2019. Joint inventory control and pricing in a service-inventory system. International Journal of Production Economics, 209, pp.78-91.
     [39]      Melikov, A., Krishnamoorthy, A. and Shahmaliyev, M., 2018. Perishable queuing inventory systems with delayed feedback. In Information Technologies and Mathematical Modelling. Queueing Theory and Applications (pp. 55-70). Springer, Cham.
     [40]      Melikov, A.Z. and Shahmaliyev, M.O., 2019. Queueing System M/M/1/∞ with Perishable Inventory and Repeated Customers.Automation and Remote Control, 80(1), pp.53-65.
     [41]      Gowsalya, V., Selvakumar, C. and Elango, C., 2019. Finite Source Retrial Queue with Inventory Management: Semi MDP.Journal of Computer and Mathematical Sciences, 10(5), pp.1032-1042.
     [42]      Hanukov, G., Avinadav, T., Chernonog, T. and Yechiali, U., 2020. A multi-server system with inventory of preliminary services and stock-dependent demand. International Journal of Production Research, pp.1-19.
     [43]      Anilkumar, M.P. and Jose, K.P., 2020. A Geo/Geo/1 inventory priority queue with self-induced interruption. International Journal of Applied and Computational Mathematics, 6(4), pp.1-14.
     [44]      Keerthana, M., Saranya, N. and Sivakumar, B., 2020. A stochastic queueing-inventory system with renewal demands and positive lead time. European Journal of Industrial Engineering, 14(4), pp.443-484.
     [45]      Krishnamoorthy, A., Joshua, A.N. and Kozyrev, D., 2021. Analysis of a Batch Arrival, Batch Service Queuing-Inventory System with Processing of Inventory While on Vacation. Mathematics, 9(4), p.419.
     [46]      Rasmi, K. and Jacob, M.J., 2021. Analysis of a multiserver queueing inventory model with self-service. International Journal of Mathematical Modelling and Numerical Optimisation, 11(3), pp.275-291.
     [47]      Aghsami, A., Samimi, Y. and Aghaei, A., 2021. A novel Markovian queueing-inventory model with imperfect production and inspection processes: a hospital case study. Computers & Industrial Engineering, p.107772.
     [48]      Salameh, M.K. and Jaber, M.Y., 2000. Economic production quantity model for items with imperfect quality. International journal of production economics, 64(1-3), pp.59-64.
     [49]      Khan, M., Jaber, M.Y. and Wahab, M.I.M., 2010. Economic order quantity model for items with imperfect quality with learning in inspection. International journal of production economics, 124(1), pp.87-96.
     [50]      Al-Salamah, M., 2011. Economic order quantity with imperfect quality, destructive testing acceptance sampling, and inspection errors. Advances in Management and Applied Economics, 1(2), p.59.
     [51]      Moussawi-Haidar, L., Salameh, M. and Nasr, W., 2013. An instantaneous replenishment model under the effect of a sampling policy for defective items. Applied Mathematical Modelling, 37(3), pp.719-727.
     [52]      Hsu, J.T. and Hsu, L.F., 2013. Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns. Computers & Industrial Engineering, 64(1), pp.389-402.
     [53]      Moussawi-Haidar, L., Salameh, M. and Nasr, W., 2014. Effect of deterioration on the instantaneous replenishment model with imperfect quality items. Applied Mathematical Modelling, 38(24), pp.5956-5966.
     [54]      Priyan, S. and Uthayakumar, R., 2015. Mathematical modeling and computational algorithm to solve multi-echelon multi-constraint inventory problem with errors in quality inspection. Journal of Mathematical Modelling and Algorithms in Operations Research, 14(1), pp.67-89.
     [55]      Chang, C.T., Cheng, M.C. and Soong, P.Y., 2016. Impacts of inspection errors and trade credits on the economic order quantity model for items with imperfect quality. International Journal of Systems Science: Operations & Logistics, 3(1), pp.34-48.
     [56]      Hasanpour Rodbaraki, J. and Sharifi, E., 2016. Economic Order Quantity for Deteriorating Items with Imperfect Quality, Destructive Testing Acceptance Sampling, and Inspection Errors. Advances in Industrial Engineering, 50(2), pp.235-246.
     [57]      Cheikhrouhou, N., Sarkar, B., Ganguly, B., Malik, A.I., Batista, R. and Lee, Y.H., 2018. Optimization of sample size and order size in an inventory model with quality inspection and return of defective items. Annals of Operations Research, 271(2), pp.445-467.
     [58]      Maleki Vishkaei, B., Niaki, S.T.A., Farhangi, M. and Mahdavi, I., 2019. A single-retailer multi-supplier multi-product inventory model with destructive testing acceptance sampling and inflation. Journal of Industrial and Production Engineering, 36(6), pp.351-361.
     [59]      Wangsa, I.D. and Wee, H.M., 2019. A vendor-buyer inventory model for defective items with errors in inspection, stochastic lead time and freight cost. INFOR: Information Systems and Operational Research, 57(4), pp.597-622.
     [60]      Safarnezhad, M., Aminnayeri, M. and Ghasemy Yaghin, R., 2020. Joint pricing and lot sizing model with statistical inspection and stochastic lead time. INFOR: Information Systems and Operational Research, pp.1-34.
     [61]      Taleizadeh, A.A., 2021. Imperfect Inventory Systems: Inventory and Production Management. Springer Nature.
     [62]      Asadkhani, J., Mokhtari, H. and Tahmasebpoor, S., 2021. Optimal lot-sizing under learning effect in inspection errors with different types of imperfect quality items. Operational Research, pp.1-35.
     [63]      Wu, K.S., Ouyang, L.Y. and Ho, C.H., 2007. Integrated vendor-buyer inventory system with sublot sampling inspection policy and controllable lead time. International Journal of Systems Science, 38(4), pp.339-350.
     [64]      Jolai, F., Gheisariha, E. and Nojavan, F., 2011. Inventory Control of Perishable Items in a Two-Echelon Supply Chain. Advances in Industrial Engineering, 45(Special Issue), pp.69-77.
     [65]      Datta, T.K., 2013. An inventory model with price and quality dependent demand where some items produced are defective. Advances in Operations Research, 2013.
     [66]      Taleizadeh, A. and Hasani, M., 2015. A multi-product inventory control model with imperfect production process and rework under delayed payment policy. Advances in Industrial Engineering, 49(2), pp.223-235.
     [67]      Jauhari, W.A., Sofiana, A., Kurdhi, N.A. and Laksono, P.W., 2016. An integrated inventory model for supplier-manufacturer-retailer system with imperfect quality and inspection errors. International Journal of Logistics Systems and Management, 24(3), pp.383-407.
     [68]      Pal, S. and Mahapatra, G.S., 2017. A manufacturing-oriented supply chain model for imperfect quality with inspection errors, stochastic demand under rework and shortages. Computers & Industrial Engineering, 106, pp.299-314.
     [69]      Amirhosseini, K., Mohammadi, M. and Pasandideh, S.H., 2017. Modeling and Solving a Multi-product Economic Order Quantity Problem with Imperfect Items and Emergency Buy or Repair Options. Advances in Industrial Engineering, 51(4), pp.405-416.
     [70]      Manna, A.K., Dey, J.K. and Mondal, S.K., 2017. Imperfect production inventory model with production rate dependent defective rate and advertisement dependent demand. Computers & Industrial Engineering, 104, pp.9-22.
     [71]      Taheri-Tolgari, J., Mohammadi, M., Naderi, B., Arshadi-Khamseh, A. and Mirzazadeh, A., 2019. An inventory model with imperfect item, inspection errors, preventive maintenance and partial backlogging in uncertainty environment. Journal of Industrial & Management Optimization, 15(3), pp.1317-1344.
     [72]      Sarkar, S. and Giri, B.C., 2020. Stochastic supply chain model with imperfect production and controllable defective rate. International Journal of Systems Science: Operations & Logistics, 7(2), pp.133-146.
     [73]      Karakatsoulis, G. and Skouri, K., 2021. Optimal reorder level and lot size decisions for an inventory system with defective items. Applied Mathematical Modelling, 92, pp.651-668.
     [74]      Adak, S. and Mahapatra, G.S., 2021. Two-echelon imperfect production supply chain with probabilistic deterioration rework and reliability under fuzziness. Journal of Management Analytics, pp.1-25.
     [75]      Khan, M., Jaber, M.Y. and Bonney, M., 2011. An economic order quantity (EOQ) for items with imperfect quality and inspection errors. International Journal of Production Economics, 133(1), pp.113-118.
     [76]      Ghosh, P.K., Manna, A.K. and Dey, J.K., 2017. Deteriorating manufacturing system with selling price discount under random machine breakdown. Int. J. Comput. Eng. Manage, 20, pp.8-17.
     [77]      World Health Organization, 2005. WHO guidelines for sampling of pharmaceutical products and related materials. WHO Technical Report Series, 929, pp.59-93.
     [78]      Ross, S.M., 2014. Introduction to probability models. Academic press.
     [79]      Montgomery, D.C., 2009. Introduction to statistical quality control. John Wiley & Sons.
     [80]      Shortle, J.F., Thompson, J.M., Gross, D. and Harris, C.M., 2018. Fundamentals of queueing theory.
     [81]      Aghsami, A. and Jolai, F., 2020. Equilibrium threshold strategies and social benefits in the fully observable Markovian queues with partial breakdowns and interruptible setup/closedown policy. Quality Technology & Quantitative Management, 17(6), pp.685-722.