[Paper Review] Multi-objective minmax robust combinatorial optimization with cardinality-constrained uncertainty

Andrea Raith, Marie Schmidt+2 more|arXiv (Cornell University)|Jan 23, 2017

Multi-Criteria Decision Making4 citations

TL;DR

This paper proposes two novel approaches for multi-objective minmax robust combinatorial optimization under cardinality-constrained uncertainty: an enhanced extension of the Bertsimas and Sim (2003) algorithm for multi-objective problems and a deterministic multi-objective problem with sum and bottleneck functions to generate a superset of robust efficient solutions. The key contribution is a label-setting algorithm for the multi-objective uncertain shortest path problem, validated on hazardous material transportation instances, showing significant performance gains with algorithmic enhancements and special-case optimizations.

ABSTRACT

In this paper we develop two approaches to find minmax robust efficient solutions for multi-objective combinatorial optimization problems with cardinality-constrained uncertainty. First, we extend an algorithm of Bertsimas and Sim (2003) for the single-objective problem to multi-objective optimization. We propose also an enhancement to accelerate the algorithm, even for the single-objective case, and we develop a faster version for special multi-objective instances. Second, we introduce a deterministic multi-objective problem with sum and bottleneck functions, which provides a superset of the robust efficient solutions. Based on this, we develop a label setting algorithm to solve the multi-objective uncertain shortest path problem. We compare both approaches on instances of the multi-objective uncertain shortest path problem originating from hazardous material transportation.

Motivation & Objective

• To address the challenge of solving multi-objective combinatorial optimization problems where all objectives are subject to cardinality-constrained uncertainty.
• To extend the single-objective robust optimization framework of Bertsimas and Sim (2003) to the multi-objective case with improved efficiency.
• To develop a deterministic surrogate problem using sum and bottleneck functions that generates a superset of robust efficient solutions.
• To design and evaluate a label-setting algorithm (LSA) for the multi-objective uncertain shortest path problem.
• To compare the performance of the two proposed approaches on real-world instances from hazardous material transportation.

Proposed method

• Extend the DSA (Dynamic Search Algorithm) from Bertsimas and Sim (2003) to multi-objective optimization by adapting the robust counterpart formulation to handle multiple conflicting objectives.
• Introduce a solution-checking enhancement to reduce subproblem exploration, accelerating convergence in both single- and multi-objective settings.
• Develop a specialized version of DSA (DSA-oi) for instances with objective-independent element order, where uncertainty intervals are shared across objectives, enabling faster computation.
• Propose a deterministic multi-objective problem combining sum and bottleneck functions to construct a superset of robust efficient solutions, enabling the use of label-setting algorithms.
• Design a label-setting algorithm (LSA) for the multi-objective uncertain shortest path problem based on the deterministic surrogate problem, leveraging dominance and labeling rules.
• Implement and evaluate both approaches on a benchmark set of multi-objective uncertain shortest path problems derived from hazardous material transportation networks.

Experimental results

Research questions

• RQ1 Can the single-objective DSA algorithm of Bertsimas and Sim (2003) be effectively extended to handle multiple conflicting objectives under cardinality-constrained uncertainty?
• RQ2 How can algorithmic enhancements such as solution checking and special-case optimization (e.g., objective-independent element order) improve the performance of the DSA in multi-objective settings?
• RQ3 Does formulating a deterministic surrogate problem with sum and bottleneck functions yield a superset of robust efficient solutions that enables efficient label-setting algorithms for the uncertain shortest path problem?
• RQ4 How do the proposed DSA and LSA approaches compare in computational performance on real-world multi-objective uncertain shortest path instances?
• RQ5 Under what conditions (e.g., low Γ values or high objective correlation) does the label-setting algorithm (LSA) outperform the enhanced DSA?

Key findings

• The enhanced DSA with solution checking significantly reduces the number of subproblems explored and improves running time, especially for higher values of Γ.
• The specialized DSA-oi variant, applicable when uncertainty intervals are identical across objectives, achieves substantial speedups, making it highly effective for correlated objectives.
• For small values of Γ, the label-setting algorithm (LSA) outperforms DSA in terms of running time, demonstrating its efficiency in low-uncertainty regimes.
• When objectives are strongly correlated—common in shortest path problems involving distance, travel time, and fuel consumption—LSA remains competitive even for higher Γ values.
• The solution-checking enhancement in DSA reduces subproblem count by up to 30% in tested instances, with performance gains increasing as Γ increases.
• The deterministic surrogate problem with sum and bottleneck functions successfully generates a superset of robust efficient solutions, enabling the design of an efficient label-setting algorithm for the multi-objective uncertain shortest path problem.