Table 1 – Optimal (s, S) values and minimum cost
γ0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
δ
0.1 (3,11) (1,26) (1,12) (1,9) (1,8) (1,7) (1,7) (1,7) (1,6) (1,6)
605.4 958.33 1189.3 1309.1 1381.7 1430.3 1465.1 1491.2 1511.6 1527.9
0.2 (1,10) (2,13) (6,20) (1,27) (1,15) (1,13) (1,13) (1,10) (1,9) (1,9)
515.24 649.96 793.76 983.33 1120 1214.3 1282.5 1334.1 1374.4 1406.7
0.3 (1,10) (1,12) (2,14) (4,18) (7,25) (1,23) (1,16) (1,19) (1,13) ((1,12)
490.34 610.1 689.76 765.15 804.83 1008.3 1105.2 1180.1 1239.3 1287
0.5 (1,10) (1,13) (1,15) (1,15) (1,16) 2,18) (4,21) (6,26) (1,29) (1,24)
472.89 584.32 664.66 722.9 763.58 795.24 838.8 908.47 987.12 1058.3
0.6 (1,10) (1,13) (1,16) (1,16) (1,16) (1,17) (2,18) (3,20) (5,24) (4,29)
468.89 578.74 660.13 721.23 766.65 797.98 821.01 849.05 896.26 959.93
0.7 (1,11) (1,14) (1,16) (1,17) (1,17) (1,17) (1,18) (2,18) (2,20) (4,23)
466.11 574.69 656.36 720.28 769.82 806.51 831.65 849.35 867.81 899.16
0.9 (1,11) (1,14) (1,16) (1,18) (1,18) (1,19) (1,19) (1,19) (1,19) (1,19)
462.32 569.49 651.76 732.47 773.71 818.36 853.53 879.47 896.85 907.9
1(1,11) (1,14) (1,16) (1,18) (1,19) (1,20) (1,20) (1,20) (1,20) (1,20)
461.02 567.74 650.26 717.95 774.64 822.1 860.79 891.35 913.86 928.76
Table 2 – Effect of γand δon various performance measures
Performance Measures γ=1and δ=0.5γ=0.5and δ=1 γ=δ=1
Ls0.00085731 0.10005 0.038268
Ws0.75643 0.0013604 0.07402
Ws1.2436 1.9986 1.926
Einv 1.5852 7.8376 5.9064
Erp 1.2436 0.99932 1.926
Ecycle 580.22 3.9955 10.066
Closs 0.75643 0.0013604 0.07402
(
s, S
)pair. First emptiness time distribution of the inventory is computed for the case when the waiting
room capacity is restricted to one. We propose to study the transient behaviour of such a system. In
addition, we analyze the case in which the transfer time from the production plant to the retail shop is
a positive valued random variable. Also, the case of arbitrarily distributed lead time and/or service
time is being investigated.
ACKNOWLEDGMENT
The research work of Manikandan, R., is supported by DST-RSF research project no. 64800 (DST) and
research project no. 22-49-02023 (RSF).
REFERENCES
[1]
M.P. Anilkumar, K. P. Jose, (2021). Stochastic Decomposition of Geo/Geo/1 Production Inven-
tory System, J. Phys.: Conf. Ser. 1850 012027, https://doi.org/10.1088/1742-6596/1850/1/012027.
[2]
Ann M. Noblesse, Nikki Sonenberg, Robert N. Boute, Marc R. Lambrecht & Benny
Van Houdt. (2022). A joint replenishment production-inventory model as an MMAP[K]/PH[K]/1
queue, Stochastic Models, 38:3, 390-415, https://doi.org/10.1080/15326349.2022.2049822.
[3]
P., Beena., & Jose, K.P. (2020). A MAP/PH(1), PH(2)/2 Production Inventory Model with
Inventory Dependent Production Rate and Multiple Servers. AIP Conference Proceedings 2261,
030052; https://doi.org/10.1063/5.0017008.
[4]
T. G. Deepak, A. Krishnamoorthy, Viswanath C. Narayan and K. Vineetha (2008).
Inventory with service time and transfer of customers and/inventory. Ann. Oper. Res 160: 191 - 213.
[5]
K. P. Jose, P. Beena.,(2020). Investigation of a production inventory model with
two servers having multiple vacations, J. Math. Comput. Sci. 10, No. 4, 1214-1227
https://doi.org/10.28919/jmcs/4361.
https://doi.org/10.17993/3cemp.2022.110250.139-151
[6]
Jose, K.P., Reshmi, P.S.(2021). A production inventory model with deteriorating items and
retrial demands. OPSEARCH 58, 71-82. https://doi.org/10.1007/s12597-020-00471-8.
[7]
A. Krishnamoorthy and Viswanath C. Narayanan (2010). Production inventory with service
time and vacation to the server. IMA Journal of Management Mathematics https://doi.org/10.
1093/imaman/dpp025.
[8]
A. Krishnamoorthy, B. Lakshmy and R. Manikandan (2011). A survey on inventory models
with positive service time. OPSEARCH 48(2):153-169.
[9]
Krishnamoorthy A., Varghese R., Lakshmy B. (2019) Production Inventory System with
Positive Service Time Under Local Purchase. In: Dudin A., Nazarov A., Moiseev A. (eds) Information
Technologies and Mathematical Modelling. Queueing Theory and Applications. ITMM 2019. Communi-
cations in Computer and Information Science, vol 1109. Springer, Cham.https://doi.org/10.1007/978-
3-030-33388-1_20.
[10]
A. Krishnamoorthy, R. Manikandan and B. Lakshmy (2015). A revisit to queueing-inventory
system with positive service time. Ann Oper Res. 233, 221–236 (2015). https://doi.org/10.1007/s10479-
013-1437-x
[11]
A. Krishnamoorthy and Viswanath C Narayanan (2013). Stochastic Decomposition in
Production Inventory with Service Time. European Journal of Operational Research 228: 358-366.
[12]
R. Krenzler and H. Daduna (2014). Loss systems in a random environment - steady state
analysis.Queueing Syst https://doi.org/10.1007/s11134-014-9426-6.
[13]
R. Krenzler and H. Daduna (2013). Loss systems in a random environment - embedded Markov
chains analysis. http://preprint.math.uni-hamburg.de/public/papers/prst/prst2013-02.pdf.
[14]
M. Saffari, S. Asmussen and R. Haji (2013). The
M/M/
1queue with inventory, lost sale,
and general lead times, Queueing Syst https://doi.org/10. 1007/s11134-012-9337-3.
[15]
Otten, S., Krenzler, R.K., & Daduna, H. (2019). Separable models
for interconnected production-inventory systems. Stochastic Models, 36, 48-93.
https://doi.org/10.1080/15326349.2019.1692667.
[16]
Otten, S. (2022). Load balancing in a network of queueing-inventory systems. Ann Oper Res.
https://doi.org/10.1007/s10479-022-05017-3.
[17]
Otten, S., & Daduna, H. (2022). Stability of queueing-inventory systems with different priorities.
https://doi.org/10.48550/arXiv.2209.08957.
[18]
Ning Zhao and Zhanotong Lian. (2011). A queueing-inventory system with two classes of
customers. Int. J. Production Economics 129: 225-231.
[19]
M. F. Neuts (1994). Matrix-Geometric Solutions in Stochastic Models - An Algorithmic Approach,
2nd ed., Dover Publications, Inc., New York.
[20]
Yonit Barron. (2019). Critical level policy for a production-inventory model with lost sales,
International Journal of Production Research.https://doi.org/10.1080/00207543.2018.1504243.
[21]
Yue, D., Qin, Y. (2019a) A Production Inventory System with Service Time and Production
Vacations. J. Syst. Sci. Syst. Eng. 28, 168–180. https://doi.org/10.1007/s11518-018-5402-8.
[22]
Yue, D., Wang, S., Zhang, Y. (2019b). A Production-Inventory System with a Service Facility
and Production Interruptions for Perishable Items. In: Li, QL., Wang, J., Yu, HB. (eds) Stochastic
Models in Reliability, Network Security and System Safety. JHC80 2019. Communications in Computer
and Information Science, vol 1102. Springer, Singapore. https://doi.org/10.1007/978-981-15-0864-6-
21.
https://doi.org/10.17993/3cemp.2022.110250.139-151
3C Empresa. Investigación y pensamiento crítico. ISSN: 2254-3376
Ed. 50 Vol. 11 N.º 2 August - December 2022
150