Lingrui Liao

Minimizing conditional-value-at-risk for stochastic scheduling problems

This paper introduces the use of conditional-value-at-risk (CVaR) as a criterion for stochastic scheduling problems. This criterion has the tendency of simultaneously reducing both the expectation and variance of a performance measure, while retaining linearity whenever the expectation can be represented by a linear expression. In this regard, it offers an added advantage over traditional nonlinear expectation-variance-based approaches. We begin by formulating a scenario-based mixed-integer program formulation for minimizing CVaR for general scheduling problems. We then demonstrate its application for the single machine total weighted tardiness problem, for which we present both a specialized l-shaped algorithm and a dynamic programming-based heuristic procedure. Our numerical experimental results reveal the benefits and effectiveness of using the CVaR criterion. Likewise, we also exhibit the use and effectiveness of minimizing CVaR in the context of the parallel machine total weighted tardiness problem. We believe that minimizing CVaR is an effective approach and holds great promise for achieving risk-averse solutions for stochastic scheduling problems that arise in diverse practical applications.

Primary pharmaceutical manufacturing scheduling problem

This article addresses an integrated lot-sizing and scheduling problem that arises in the primary manufacturing phase of a pharmaceutical supply chain. Multiple pharmaceutical ingredients and their intermediate products are to be scheduled on parallel and capacitated bays for production in batches. Sequence-dependent setup times and costs are incurred when cleaning a bay during changeovers between different product families. The problem also contains a high multiplicity asymmetric traveling salesman-type of substructure because of sequence-dependent setups and special restrictions. Mixed-integer programming formulations are proposed for this problem and several valid inequalities are developed to tighten the model. A column generation method along with a decomposition scheme and an advanced-start solution are designed to efficiently derive good solutions to this highly complex problem. A computational investigation is performed, based on instances that closely follow a real-life application, and it demonstrates the efficacy of the proposed solution approach.

Stochastic Scheduling for a Network of Flexible Job Shops

In this chapter, we address the problem of optimally routing and sequencing a set of jobs over a network of flexible machines for the objective of minimizing the sum of completion times and the cost incurred, assuming stochastic job processing times. This problem is of particular interest for the production control in high investment, low volume manufacturing environments, such as pilot-fabrication of microelectromechanical systems (MEMS) devices. We model this problem as a two-stage stochastic program with recourse, where the first-stage decision variables are binary and the second-stage variables are continuous. This basic formulation lacks relatively complete recourse due to infeasibilities that are caused by the presence of re-entrant flows in the processing routes, and also because of potential deadlocks that result from the first-stage routing and sequencing decisions. We use the expected processing times of operations to enhance the formulation of the first-stage problem, resulting in good linear programming bounds and inducing feasibility for the second-stage problem. In addition, we develop valid inequalities for the first-stage problem to further tighten its formulation. Experimental results are presented to demonstrate the effectiveness of using these strategies within a decomposition algorithm (the L-shaped method) to solve the underlying stochastic program. In addition, we present heuristic methods to handle large-sized instances of this problem and provide related computational results.

A Scenario Generation-Based Lower Bounding Approach for Stochastic Scheduling Problems

In this paper, we investigate scenario generation methods to establish lower bounds on the optimal objective value for stochastic scheduling problems that contain random parameters with continuous distributions. In contrast to the Sample Average Approximation (SAA) approach, which yields probabilistic bound values, we use an alternative bounding method that relies on the ideas of discrete bounding and recursive stratified sampling. Theoretical support is provided for deriving exact lower bounds for both expectation and conditional value-at-risk objectives. We illustrate the use of our method on the single machine total weighted tardiness problem. The results of our numerical investigation demonstrate good properties of our bounding method, compared with the SAA method and an earlier discrete bounding method.

Robust Scheduling Approaches to Hedge against Processing Time Uncertainty

Introduction Robust scheduling is one of the approaches adopted to deal with uncertainties in stochastic scheduling. Since these uncertainties, arising as they do from variability in the scheduling parameters, significantly affect the performance of a system, a viable approach has been to determine a robust schedule that is least susceptible to disturbances or variations in the input parameters and that has minimum variability in its output performance measure. For instance, when processing times are uncertain, a robust schedule would have the least variance of the output performance measure, although it might not necessarily have the least expected value. As also mentioned in Chapter 1, of the various parameters involved in scheduling, we intend to focus on the uncertainty factor associated with job processing times. The following sections detail different robust scheduling methodologies that are available to model certain scheduling environments and the underlying principles behind those various approaches. Our primary focus is to elicit and delve in detail on the modeling approaches and not to expostulate the solution methodologies. However, we do briefly discuss the various solution approaches that are used to solve these robust scheduling problems and also state relevant results (as propositions) that are used in developing these solution methodologies. We intend to capture the core techniques that have been used in modeling robustness in scheduling and would advise interested readers to refer to the appropriate literature for a detailed and complete exposition.

Single-Machine Models

Introduction In this chapter we consider the problem of sequencing a given number of jobs on a single machine and devise methodologies and develop closed-form expressions (wherever possible) to compute the expectation and variance of various performance measures. Numerical illustrations indicating the significance and applicability of our work are also presented through example problems. Besides being of significance in their own right, single-machine problems also constitute a good starting point for analyzing more complex scheduling environments of flow shop and job shop. The different performance measures that we consider for the single-machine case can be classified into two categories: Completion-time-based , which includes total completion time (total flow time), total weighted completion time, and total weighted discounted completion time Due-date-based , which includes total tardiness, total weighted tardiness, total number of tardy jobs, total weighted number of tardy jobs, mean lateness, and maximum lateness. The processing time of a job is assumed to be a random variable that follows an arbitrary probability distribution. First, we present the notation that is used in this chapter. Some of this notation has been introduced already in earlier chapters. However, we include it here for the sake of completeness.