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.
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References
Ayhan H, Olsen TL (2000) Scheduling of multi-class single server queves under nontraditional performance measures. Oper Res 48(3):482–489
Clark CE (1961) The greatest of a finite set of random variables. Oper Res 9:145–162
Daniels RL, Carrillo JE (1997) β-robust scheduling for single-machine systems with uncertain processing times. IIE Trans 29:977–985
Daniels RL, Kouvelis P (1995) Robust scheduling to hedge against processing time uncertainty in single-stage production. Manag Sci 41(2):363–376
De P, Ghosh JB, Wells CE (1992) Expectation-variance analysis of job sequences under processing time uncertainty. Int J Prod Econ 28:289–297
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
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)
Kouvelis P, Daniels RL, Vairaktarakis G (2000) Robust scheduling of a two-machine flow shop with uncertain processing times. IIE Trans 32:421–432
Krokhmal P, Grundell D, Pardalos PM (2007) Asymptotic behavior of the optimal value of multidimensional assignment problems. Math Program 109(2–3):525–551
McKay KN, Safayeni FR, Buzacott JA (1988) Job-shop scheduling theory: What is relevant? Interfaces 18:84–90
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
Murata T, Ishibuchi H, Tanaka H (1996) Multi-objective genetic algorithm and its applications to flowshop scheduling. Comput Ind Eng 30:957–968
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
Portougal V, Trietsch D (1998) Makespan related criteria for comparing schedules in stochastic environments. J Oper Res Soc 49(11):1188–1195
Rajendran C (1995) Heuristics for scheduling in flowshop with multiple objectives. Eur J Oper Res 82:540–555
Shing C, Nagasawa H (1996) Interactive decision support system in stochastic multi-objective scheduling. Bull Osaka Prefect Univ Ser A 45(2):133–141
Wilhelm WE (1986) The application of lognormal models of transient operations in the flexible manufacturing environment. J Manuf Syst 5(4):253–266
Yang J, Gang Y (2002) On the robust single machine scheduling problem. J Comb Optim 6:17–33
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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
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DOI: https://doi.org/10.1007/s10878-007-9122-0