Skip to main content
Log in

Analytic evaluation of the expectation and variance of different performance measures of a schedule on a single machine under processing time variability

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

In this paper, we present closed-form expressions, wherever possible, or devise algorithms otherwise, to determine the expectation and variance of a given schedule on a single machine. We consider a variety of completion time and due date-based objectives. The randomness in the scheduling process is due to variable processing times with known means and variances of jobs and, in some cases, a known underlying processing time distribution. The results that we present in this paper can enable evaluation of a schedule in terms of both the expectation and variance of a performance measure considered, and thereby, aid in obtaining a stable schedule. Additionally, the expressions and algorithms that are presented, can be incorporated in existing scheduling algorithms in order to determine expectation-variance efficient schedules.

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

Access this article

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

  • Ayhan H, Olsen TL (2000) Scheduling of multi-class single server queves under nontraditional performance measures. Oper Res 48(3):482–489

    Article  MATH  MathSciNet  Google Scholar 

  • Clark CE (1961) The greatest of a finite set of random variables. Oper Res 9:145–162

    Article  MATH  Google Scholar 

  • Daniels RL, Carrillo JE (1997) β-robust scheduling for single-machine systems with uncertain processing times. IIE Trans 29:977–985

    Google Scholar 

  • Daniels RL, Kouvelis P (1995) Robust scheduling to hedge against processing time uncertainty in single-stage production. Manag Sci 41(2):363–376

    Article  MATH  Google Scholar 

  • De P, Ghosh JB, Wells CE (1992) Expectation-variance analysis of job sequences under processing time uncertainty. Int J Prod Econ 28:289–297

    Article  Google Scholar 

  • Dodin B (1996) Determining the optimal sequences and the distributional properties of the completion times in stochastic flow shops. Comput Oper Res 23:829–843

    Article  MATH  Google Scholar 

  • Jung YS, Nagasawa H, Nishiyama N (1990) Bicriterion single-stage scheduling to minimize both the expected value and the variance of the total flow time. J Jpn Ind Man Assoc 39:76–82 (in Japanese)

    Google Scholar 

  • Kouvelis P, Daniels RL, Vairaktarakis G (2000) Robust scheduling of a two-machine flow shop with uncertain processing times. IIE Trans 32:421–432

    Google Scholar 

  • Krokhmal P, Grundell D, Pardalos PM (2007) Asymptotic behavior of the optimal value of multidimensional assignment problems. Math Program 109(2–3):525–551

    Article  MATH  MathSciNet  Google Scholar 

  • McKay KN, Safayeni FR, Buzacott JA (1988) Job-shop scheduling theory: What is relevant? Interfaces 18:84–90

    Article  Google Scholar 

  • Morizawa K, Ono T, Nagasawa H, Nishiyama N (1993) An interactive approach for searching a preferred schedule. J Jpn Ind Man Assoc 39:76–82

    Google Scholar 

  • Murata T, Ishibuchi H, Tanaka H (1996) Multi-objective genetic algorithm and its applications to flowshop scheduling. Comput Ind Eng 30:957–968

    Article  Google Scholar 

  • Nagasawa H, Shing C (1997) Interactive decision system in parallel-machine stochastic multi-objective scheduling. In: Proceedings of the 1st international conference on engineering design and automation, Bangkok, Thailand, pp 421–424

  • Nagasawa H, Shing C (1998) Interactive decision system in stochastic multi-objective scheduling to minimize the expected value and variance of total flow time. J Oper Res Soc Jpn 41(2):261–278

    Google Scholar 

  • Portougal V, Trietsch D (1998) Makespan related criteria for comparing schedules in stochastic environments. J Oper Res Soc 49(11):1188–1195

    Article  MATH  Google Scholar 

  • Rajendran C (1995) Heuristics for scheduling in flowshop with multiple objectives. Eur J Oper Res 82:540–555

    Article  MATH  Google Scholar 

  • Shing C, Nagasawa H (1996) Interactive decision support system in stochastic multi-objective scheduling. Bull Osaka Prefect Univ Ser A 45(2):133–141

    MATH  Google Scholar 

  • Wilhelm WE (1986) The application of lognormal models of transient operations in the flexible manufacturing environment. J Manuf Syst 5(4):253–266

    Article  Google Scholar 

  • Yang J, Gang Y (2002) On the robust single machine scheduling problem. J Comb Optim 6:17–33

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Subhash C. Sarin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sarin, S.C., Nagarajan, B., Jain, S. et al. Analytic evaluation of the expectation and variance of different performance measures of a schedule on a single machine under processing time variability. J Comb Optim 17, 400–416 (2009). https://doi.org/10.1007/s10878-007-9122-0

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10878-007-9122-0

Keywords

Navigation