Minimizing value-at-risk in single-machine scheduling

Abstract

The vast majority of the machine scheduling literature focuses on deterministic problems in which all data is known with certainty a priori. In practice, this assumption implies that the random parameters in the problem are represented by their point estimates in the scheduling model. The resulting schedules may perform well if the variability in the problem parameters is low. However, as variability increases accounting for this randomness explicitly in the model becomes crucial in order to counteract the ill effects of the variability on the system performance. In this paper, we consider single-machine scheduling problems in the presence of uncertain parameters. We impose a probabilistic constraint on the random performance measure of interest, such as the total weighted completion time or the total weighted tardiness, and introduce a generic risk-averse stochastic programming model. In particular, the objective of the proposed model is to find a non-preemptive static job processing sequence that minimizes the value-at-risk (VaR) of the random performance measure at a specified confidence level. We propose a Lagrangian relaxation-based scenario decomposition method to obtain lower bounds on the optimal VaR and provide a stabilized cut generation algorithm to solve the Lagrangian dual problem. Furthermore, we identify promising schedules for the original problem by a simple primal heuristic. An extensive computational study on two selected performance measures is presented to demonstrate the value of the proposed model and the effectiveness of our solution method.

A branch-and-bound approach for the single machine maximum lateness stochastic scheduling problem to minimize the value-at-risk

“…The authors proposes a Lagrangian relaxation-based scenario decomposition method considering random processing times and deterministic due dates. The approach in Atakan et al (2016) does not take into consideration release dates and adopt a scenario based formulation to deal with stochastic variables, differently from what the present work that explicitly considers the stochastic distributions of processing times and release dates.…”

“…Atakan et al (2016) address a similar problem minimizing the VaR of a tardiness-related objective function, namely 6 the Total Tardiness and the Total Weighted Tardiness for a single machine. The authors proposes a Lagrangian relaxation-based scenario decomposition method considering random processing times and deterministic due dates.…”

Risk-averse single machine scheduling: complexity and approximation

“…A sample problem 1|| C j with 4 jobs and 5 processing time scenarios is shown in Figure 2. Let π = (1,2,3,4). It is easily seen that E[F(π)] = 26, VaR 0.5 [F(π)] = 29, CVaR 0.5 [F(π)] = 34 and Max[F(π)] = 36.…”

Single Machine Scheduling under Uncertainty Processing Time with CVAR Constraints

“…Robust decision-making formulations were presented by [9][10], and [11] focused on the scheduling issues in single-stage production environments, and defined a general E -robust scheduling objective when job processing times are independent random variables. Reference [12] imposed a probabilistic constraint on the random TWT(total weighted tardiness) and introduced a risk-averse stochastic programming model. In particular, the objective of the proposed model is to find a non-preemptive static job processing sequence that minimizes the value-at-risk (VAR) measure on the random TWT at a specified confidence level.…”

“…Both VaR theory and CVAR theory seek to guard against unfavorable realization of uncertain parameters by going beyond expectation evaluation when expressing the uncertainty of system parameters. Reference [12] considered the single machine scheduling problem by minimizing the VaR measure in the presence of uncertain problem parameters, they imposed a probabilistic constraint on the random TWT and introduced a risk-averse stochastic programming model. However, primarily due to the facts that VaR does not properly account for risk diversification and that it says nothing about the magnitude of losses beyond the quantile level in question, VaR, the class of coherent risk measures, was axiomatized and popularized by [15].…”