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Using 3-Objective Evolutionary Algorithms for the Dynamic Chance Constrained Knapsack Problem

Ishara Hewa Pathiranage, Frank Neumann, Denis Antipov, Aneta Neumann

TL;DR

This paper addresses the dynamic chance constrained knapsack problem (DCCKP) where item weights are stochastic and the knapsack capacity varies over time. It introduces a 3-objective Pareto optimization approach that optimizes profit, expected weight, and weight variance, enabling solutions feasible for any confidence level $\\alpha$ without fixing it a priori. The method is evaluated against a 2-objective baseline using GSEMO and MOEA/D across diverse benchmarks (uncorr and bsc) and dynamic settings, demonstrating that the 3-objective formulation yields superior performance, especially under frequent capacity changes and higher weight uncertainty. The work provides a practical framework for robust dynamic optimization under uncertainty, delivering a rich set of trade-offs that cover a range of risk levels in a single run.

Abstract

Real-world optimization problems often involve stochastic and dynamic components. Evolutionary algorithms are particularly effective in these scenarios, as they can easily adapt to uncertain and changing environments but often uncertainty and dynamic changes are studied in isolation. In this paper, we explore the use of 3-objective evolutionary algorithms for the chance constrained knapsack problem with dynamic constraints. In our setting, the weights of the items are stochastic and the knapsack's capacity changes over time. We introduce a 3-objective formulation that is able to deal with the stochastic and dynamic components at the same time and is independent of the confidence level required for the constraint. This new approach is then compared to the 2-objective formulation which is limited to a single confidence level. We evaluate the approach using two different multi-objective evolutionary algorithms (MOEAs), namely the global simple evolutionary multi-objective optimizer (GSEMO) and the multi-objective evolutionary algorithm based on decomposition (MOEA/D), across various benchmark scenarios. Our analysis highlights the advantages of the 3-objective formulation over the 2-objective formulation in addressing the dynamic chance constrained knapsack problem.

Using 3-Objective Evolutionary Algorithms for the Dynamic Chance Constrained Knapsack Problem

TL;DR

This paper addresses the dynamic chance constrained knapsack problem (DCCKP) where item weights are stochastic and the knapsack capacity varies over time. It introduces a 3-objective Pareto optimization approach that optimizes profit, expected weight, and weight variance, enabling solutions feasible for any confidence level without fixing it a priori. The method is evaluated against a 2-objective baseline using GSEMO and MOEA/D across diverse benchmarks (uncorr and bsc) and dynamic settings, demonstrating that the 3-objective formulation yields superior performance, especially under frequent capacity changes and higher weight uncertainty. The work provides a practical framework for robust dynamic optimization under uncertainty, delivering a rich set of trade-offs that cover a range of risk levels in a single run.

Abstract

Real-world optimization problems often involve stochastic and dynamic components. Evolutionary algorithms are particularly effective in these scenarios, as they can easily adapt to uncertain and changing environments but often uncertainty and dynamic changes are studied in isolation. In this paper, we explore the use of 3-objective evolutionary algorithms for the chance constrained knapsack problem with dynamic constraints. In our setting, the weights of the items are stochastic and the knapsack's capacity changes over time. We introduce a 3-objective formulation that is able to deal with the stochastic and dynamic components at the same time and is independent of the confidence level required for the constraint. This new approach is then compared to the 2-objective formulation which is limited to a single confidence level. We evaluate the approach using two different multi-objective evolutionary algorithms (MOEAs), namely the global simple evolutionary multi-objective optimizer (GSEMO) and the multi-objective evolutionary algorithm based on decomposition (MOEA/D), across various benchmark scenarios. Our analysis highlights the advantages of the 3-objective formulation over the 2-objective formulation in addressing the dynamic chance constrained knapsack problem.
Paper Structure (13 sections, 10 equations, 3 figures, 24 tables, 1 algorithm)

This paper contains 13 sections, 10 equations, 3 figures, 24 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our contribution
  4. Preliminaries
  5. Problem Definition
  6. Baseline Evolutionary Algorithms
  7. Dynamic 2-objective Formulation
  8. Dynamic 3-objective Formulation
  9. Experimental Investigations
  10. Experimental Setting
  11. Experimental Results
  12. Conclusion
  13. Appendix

Figures (3)

  • Figure 1: Examples of the knapsack capacity changing over time for low ($r = 500$) and high ($r = 2000$) magnitude of changes.
  • Figure 2: The mean value and the standard deviation of the total offline error for different algorithms when the magnitude of dynamic knapsack bound change follows $\mathcal{U}({-500}, 500)$
  • Figure 3: The mean value and the standard deviation of the total offline error for different algorithms when the magnitude of dynamic knapsack bound change follows $\mathcal{U}({-2000}, 2000)$