Journal of Zhejiang University-SCIENCE A

, Volume 10, Issue 2, pp 311–318

# Optimal operating policy for a controllable queueing model with a fuzzy environment

• Chuen-horng Lin
• Jau-chuan Ke
Article

## Abstract

We construct the membership functions of the fuzzy objective values of a controllable queueing model, in which cost elements, arrival rate and service rate are all fuzzy numbers. Based on Zadeh’s extension principle, a set of parametric nonlinear programs is developed to find the upper and lower bounds of the minimal average total cost per unit time at the possibility level. The membership functions of the minimal average total cost are further constructed using different values of the possibility level. A numerical example is solved successfully to illustrate the validity of the proposed approach. Because the object value is expressed and governed by the membership functions, the optimization problem in a fuzzy environment for the controllable queueing models is represented more accurately and analytical results are more useful for system designers and practitioners.

## Key words

Controllable queue Fuzzy sets Membership function Nonlinear programming

O22

## References

1. Arumuganathan, R., Jeyakumar, S., 2005. Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times. Appl. Math. Model., 29(10):972–986. [doi:10.1016/j.apm.2005.02.013]
2. Bell, C.E., 1971. Characterization and computation of optimal policies for operating an M/G/1 queueing system with removable server. Oper. Res., 19(1):208–218. [doi:10.1287/opre.19.1.208]
3. Buckley, J.J., Feuring, T., Hayashi, Y., 2001. Fuzzy queueing theory revisited. Int. J. Uncert., Fuzz., Knowl.-Based Syst., 9(5):527–537. [doi:10.1142/S0218488501001046]
4. Buzacott, J., Shanthikumar, J., 1993. Stochastic Models of Manufacturing Systems. Prentice-Hall, Englewood Cliffs, NJ.
5. Choudhury, G., Madan, K.C., 2005. A two-stage batch arrival queueing system with a modified Bernoulli schedule vacation under N-policy. Math. Comput. Model., 42(1–2):71–85. [doi:10.1016/j.mcm.2005.04.003]
6. Choudhury, G., Paul, M., 2006. A batch arrival queue with a second optional service channel under N-policy. Stoch. Anal. Appl., 24(1):1–21. [doi:10.1080/07362990500397277]
7. Gal, T., 1979. Postoptimal Analysis, Parametric Programming, and Related Topics. McGraw-Hill, New York.
8. Gross, D., Harris, C.M., 1998. Fundamentals of Queueing Theory (3rd Ed.). John Wiley, New York.
9. Heyman, D.P., 1968. Optimal operating policies for M/G/1 queueing system. Oper. Res., 16(2):362–382. [doi:10.1287/opre.16.2.362]
10. Hillier, F.S., Lieberman, G.J., 2001. Introduction to Operations Research (7th Ed.). McGraw-Hill, Singapore.
11. Kaufmann, A., 1975. Introduction to the Theory of Fuzzy Subsets, Volume 1. Academic Press, New York.
12. Kella, O., 1989. The threshold policy in the M/G/1 queue with server vacations. Nav. Res. Logist., 36(1):111–123. [doi:10.1002/1520-6750(198902)36:1〈111::AID-NAV3220360109〉3.0.CO;2-3]
13. Kleinrock, L., 1975. Queueing Systems, Vol. 1: Theory. Wiley, New York.
14. Lee, H.S., Srinivasan, M.M., 1989. Control policies for the M [x]/G/1 queueing system. Manag. Sci., 35(6):708–721. [doi:10.1287/mnsc.35.6.708]
15. Lee, H.W., Park, J.O., 1997. Optimal strategy in N-policy production system with early set-up. J. Oper. Res. Soc., 48(3):306–313. [doi:10.1057/palgrave.jors.2600354]
16. Lee, H.W., Lee, S.S., Chae, K.C., 1994a. Operating characteristics of M X/G/1 queue with N policy. Queueing Systems, 15(1–4):387–399. [doi:10.1007/BF01189247]
17. Lee, H.W., Lee, S.S., Park, J.O., Chae, K.C., 1994b. Analysis of M [x]/G/1 queue with N policy and multiple vacations. J. Appl. Probab., 31(2):467–496. [doi:10.2307/3215040]
18. Lee, S.S., Lee, H.W., Yoon, S.H., Chae, K.C., 1995. Batch arrival queue with N-policy and single vacation. Comput. Oper. Res., 22(2):173–189. [doi:10.1016/0305-0548(94)E0015-Y]
19. Li, R.J., Lee, E.S., 1989. Analysis of fuzzy queues. Comput. Math. Appl., 17(7):1143–1147. [doi:10.1016/0898-1221(89)90044-8]
20. Pearn, W.L., Ke, J.C., Chang, Y.C., 2004. Sensitivity analysis of the optimal management policy for a queueing system with a removable and non-reliable server. Comput. Ind. Eng., 46(1):87–99. [doi:10.1016/j.cie.2003.11.001]
21. Tadj, L., Choudhury, G., 2005. Optimal design and control of queues. TOP, 13(2):359–414. [doi:10.1007/BF02579061]
22. Tadj, L., Choudhury, G., Tadj, C., 2006a. A quorum queueing system with a random setup time under N-policy with Bernoulli vacation schedule. Qual. Technol. Quantit. Manag., 3(2):145–160.
23. Tadj, L., Choudhury, G., Tadj, C., 2006b. A bulk quorum queueing system with a random setup time under N-policy with Bernoulli vacation schedule. Stoch.: Int. J. Probab. Stoch. Processes, 78(1):1–11. [doi:10.1080/17442500500397574]
24. Taha, H.A., 2003. Operations Research: An Introduction (7th Ed.). Prentice-Hall, New Jersey.
25. Wang, K.H., Kao, H.T., Chen, G., 2004. Optimal management of a removable and non-reliable server in an infinite and a finite M/H k/1 queueing system. Int. J. Qual. Technol. Quantit. Manag., 1(2):325–339.
26. Yadin, M., Naor, P., 1963. Queueing systems with a removable service station. J. Oper. Res. Soc., 14(4):393–405. [doi:10.1057/jors.1963.63]
27. Yager, R.R., 1986. A characterization of the extension principle. Fuzzy Sets Syst., 18(3):205–217. [doi:10.1016/0165-0114(86)90002-3]
28. Zadeh, L.A., 1978. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst., 1(1):3–28. [doi:10.1016/0165-0114(78)90029-5]
29. Zhang, B., 2006. A Fuzzy-logic-based Methodology for Batch Process Scheduling. IEEE Systems and Information Engineering Design Symp., p.101–105. [doi:10.1109/SIEDS.2006.278721]Google Scholar
30. Zhang, R., Phillis, Y.A., Kouikoglou, V.S., 2005. Fuzzy Control of Queueing Systems. Springer, New York.
31. Zimmermann, H.J., 2001. Fuzzy Set Theory and Its Applications (4th Ed.). Kluwer Academic, Boston.