Distributionally Robust Optimization Under a Decision-Dependent Ambiguity Set with Applications to Machine Scheduling and Humanitarian Logistics

We introduce a new class of distributionally robust optimization problems under decision-dependent ambiguity sets. In particular, as our ambiguity sets, we consider balls centered on a decision-dependent probability distribution. The balls are based on a class of earth mover’s distances that includes both the total variation distance and the Wasserstein metrics. We discuss the main computational challenges in solving the problems of interest and provide an overview of various settings leading to tractable formulations. Some of the arising side results, such as the mathematical programming expressions for robustified risk measures in a discrete space, are also of independent interest. Finally, we rely on state-of-the-art modeling techniques from machine scheduling and humanitarian logistics to arrive at potentially practical applications, and present a numerical study for a novel risk-averse scheduling problem with controllable processing times. Summary of Contribution: In this study, we introduce a new class of optimization problems that simultaneously address distributional and decision-dependent uncertainty. We present a unified modeling framework along with a discussion on possible ways to specify the key model components, and discuss the main computational challenges in solving the complex problems of interest. Special care has been devoted to identifying the settings and problem classes where these challenges can be mitigated. In particular, we provide model reformulation results, including mathematical programming expressions for robustified risk measures, and describe how these results can be utilized to obtain tractable formulations for specific applied problems from the fields of humanitarian logistics and machine scheduling. Toward demonstrating the value of the modeling approach and investigating the performance of the proposed mixed-integer linear programming formulations, we conduct a computational study on a novel risk-averse machine scheduling problem with controllable processing times. We derive insights regarding the decision-making impact of our modeling approach and key parameter choices.

A note on parallel-machine scheduling with controllable processing times and job-dependent learning effects

This note studies a unrelated parallel-machine scheduling problem with controllable processing times and job-dependent learning effects, where the objective function is to minimize the weighted sum of total completion time, total load, and total compression cost. We show that the problem can be solved in O(nm+2) time, where m and n are the numbers of machines and jobs. We also show how to apply the technique to several single-machine scheduling problems with total criteria.

Fast and meta heuristics for part selection in flexible manufacturing systems with controllable processing times

This study addresses a part selection problem for flexible manufacturing systems in which part processing times are controllable to optimize the total cost associated with energy consumption, operational performance, and so on. The problem is to determine the set of parts to be produced, part processing times and the number of tools for each tool type in each period of a planning horizon while satisfying processing time capacity, tool magazine capacity and tool life restrictions. The objective is to minimize the sum of part processing, earliness/tardiness, tool and subcontracting costs. Tool sharing among part types is also considered. After an integer programming model is developed, two types of solution algorithms are proposed, that is, fast heuristics useful when decision time is critical and variable neighborhood search algorithms when solution quality is important. Computational experiments were conducted on a number of test instances and the best fast heuristics are specified, together with reporting how much the variable neighborhood search algorithms improve the fast heuristics.