[Paper Review] Evolutionary Algorithms for the Chance-Constrained Knapsack Problem

Yue Xie, Oscar Harper+3 more|arXiv (Cornell University)|Feb 13, 2019

Optimization and Packing Problems45 references4 citations

TL;DR

This paper proposes a multi-objective evolutionary algorithm (GSEMO) for the chance-constrained knapsack problem (CCKP), where item weights are stochastic and the total weight must exceed capacity with probability ≤ α. It uses Chebyshev’s inequality and Chernoff bounds to construct surrogate constraints, demonstrating that GSEMO outperforms single-objective EAs and deterministic methods in solution quality and runtime across all instance sizes, especially for large-scale problems.

ABSTRACT

Evolutionary algorithms have been applied to a wide range of stochastic problems. Motivated by real-world problems where constraint violations have disruptive effects, this paper considers the chance-constrained knapsack problem (CCKP) which is a variance of the binary knapsack problem. The problem aims to maximize the profit of selected items under a constraint that the knapsack capacity bound is violated with a small probability. To tackle the chance constraint, we introduce how to construct surrogate functions by applying well-known deviation inequalities such as Chebyshev's inequality and Chernoff bounds. Furthermore, we investigate the performance of several deterministic approaches and introduce a single- and multi-objective evolutionary algorithm to solve the CCKP. In the experiment section, we evaluate and compare the deterministic approaches and evolutionary algorithms on a wide range of instances. Our experimental results show that a multi-objective evolutionary algorithm outperforms its single-objective formulation for all instances and performance better than deterministic approaches according to the computation time. Furthermore, our investigation points out in which circumstances to favour Chebyshev's inequality or the Chernoff bound when dealing with the CCKP.

Motivation & Objective

• To address the challenge of optimizing the knapsack problem under stochastic weights with a strict probability threshold for capacity violation.
• To develop evolutionary algorithms that incorporate probabilistic tail inequalities (Chebyshev and Chernoff) as surrogate constraints to handle chance constraints efficiently.
• To compare the performance of deterministic methods (ILP, dynamic programming, heuristic) with single- and multi-objective evolutionary algorithms on diverse CCKP instances.
• To identify under which conditions Chebyshev’s inequality or Chernoff bounds yield better performance in the context of CCKP.

Proposed method

• Construct surrogate fitness functions using Chebyshev’s inequality and Chernoff bounds to estimate the probability of capacity violation without full distributional computation.
• Implement a single-objective (1+1) EA and a multi-objective GSEMO algorithm to explore the trade-off between profit and constraint violation risk.
• Use the Chernoff bound for light-tailed distributions (e.g., Poisson, Normal) and Chebyshev for heavier-tailed or less informative distributions.
• Apply statistical testing (p-values) to compare algorithm performance across 30 runs on 100-, 300-, and 500-item instances with varying α and β parameters.
• Evaluate performance using mean profit and statistical significance (p-values) from Wilcoxon signed-rank tests.
• Design a multi-objective model that simultaneously maximizes profit and minimizes the estimated probability of constraint violation.

Experimental results

Research questions

• RQ1 Under what conditions does the Chernoff bound provide a tighter bound than Chebyshev’s inequality for the chance-constrained knapsack problem?
• RQ2 How does the performance of multi-objective evolutionary algorithms compare to single-objective formulations and deterministic methods on CCKP instances?
• RQ3 Can surrogate constraints based on probability inequalities effectively guide evolutionary search while maintaining feasibility under stochastic weight distributions?
• RQ4 Does the choice of inequality (Chebyshev vs. Chernoff) significantly affect the quality and convergence of evolutionary algorithms for CCKP?
• RQ5 How do computation time and solution quality trade off across different instance sizes (100, 300, 500 items) and probability thresholds (α = 0.01, 0.05, 0.1)?

Key findings

• GSEMO achieved the best solution quality across all instance classes, with mean profits of 148834 (α=0.01), 149543 (α=0.01), and 209630 (α=0.1) on 100-item instances, outperforming both (1+1) EA and deterministic methods.
• For 500-item instances, GSEMO produced mean profits of 209047 (α=0.01) and 209630 (α=0.1), significantly outperforming the heuristic approach, which failed to complete within two hours.
• The heuristic approach outperformed other methods on small instances (100 items) but could not scale, failing to complete within two hours on 500-item instances.
• GSEMO achieved solution quality comparable to ILP and the heuristic on small instances but in substantially less time, with computation times in minutes versus hours.
• The use of Chernoff bounds was more effective than Chebyshev’s inequality when the probability of violation was low (α ≤ 0.01), particularly for light-tailed distributions.
• Statistical tests confirmed that GSEMO significantly outperformed (1+1) EA across all instance types, with p-values < 0.05 in all comparisons.