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.

Appendix: VaR versus CVaR in risk management and optimization

A substantial amount of literature focuses on studying and comparing the properties of risk measures based on several criteria such as compatibility with axiomatic settings, mathematical properties, stability of statistical estimation (robustness), computational tractability (algorithmic possibilities for the resulting optimization problems), and acceptance by the regulators and practitioners. The findings imply that each risk measure has its own advantages and disadvantages. Therefore, one risk measure may be preferred over the other in some decision making context and vice versa in another setting. In particular, the problem of choice between the two important and popular risk measures VaR and CVaR has attracted a lot of attention in the literature, especially in financial risk management (see, e.g., Yamai and Yoshiba 2002; Sarykalin et al. 2008; Danielsson and Zhou 2015; Davis 2015). In the introduction, we provide arguments supporting the use of VaR in our scheduling context and discuss certain mathematical properties and issues related to computational tractability. Here, our focus is on the axiomatic settings—with a particular emphasis on the subadditivity axiom—and robustness.

The main criticism against VaR is that it fails to be subadditive in general and is therefore not coherent (Artzner et al. 1999). The significance of the subadditivity property stems from financial applications, where the lack of subadditivity implies that the diversification (of portfolio positions into different assets) might actually increase risk. However, the concept of diversification in a scheduling context does not have a natural interpretation because the decisions correspond to constructing an order of jobs. Thus, requiring subadditivity in a scheduling application does not appear to be inherently justified. Moreover, some recent studies (Heyde et al. 2006; Kou et al. 2013) question the necessity of the subadditivity axiom in general and suggest replacing it by the comonotonic subadditivity axiom, which only requires subadditivity to hold for comonotone random variables; that is, for random variables characterized by a perfect dependence structure. This leads to a modified coherence concept and provides an axiomatic justification for VaR, which satisfies the comonotonic subadditivity axiom. Kou et al. (2013) also point to some interesting pieces of research, which conclude that VaR is typically subadditive in practical applications (see, e.g., Danielsson et al. 2005; Ibragimov and Walden 2007). A further concern voiced in the literature about the subadditivity axiom stems from its potential conflict with the robustness of risk measurement procedures (Cont et al. 2010) as we elaborate upon next.

The literature emphasizes that CVaR is statistically less robust than VaR in the sense that it is more sensitive to the tail behavior of a distribution and statistical estimation errors. In particular, the CVaR estimate becomes less reliable in the presence of infrequent and large loss events (Yamai and Yoshiba 2002). Consequently, CVaR requires a larger sample size than VaR to provide the same level of estimation accuracy (Cont et al. 2010; Kou et al. 2013; Danielsson and Zhou 2015). In this context, the indifference of VaR to high tail outcomes exceeding VaR–which are usually difficult to measure—is considered as a good property. This may lead to a superior out-of-sample performance of VaR versus CVaR for some applications (see, e.g., Sarykalin et al. 2008). The conclusion from this discussion is that CVaR captures the risks reflected in the tail and may be preferred to VaR under the availability of a good model for the tail of the distribution. Otherwise, the CVaR value computed based on a small historical scenario set may not be accurate, and we should then refrain from adopting CVaR as the risk measure of choice. In this case, one may instead resort to VaR. In summary, VaR and CVaR focus on different aspects of the distribution, and the appropriate choice of the risk measure to be incorporated in a risk-averse optimization problem is also contingent on the available data and a good model for the tail of the distribution of the random outcome.