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arxiv: 2604.10930 · v1 · submitted 2026-04-13 · 💻 cs.NE

On the Use of Bi-Objective Evolutionary Algorithms for the Stochastic MKP under Dynamic Constraints

Ishara Hewa Pathiranage , Aneta Neumann This is my paper

Pith reviewed 2026-05-10 16:11 UTC · model grok-4.3

classification 💻 cs.NE
keywords stochastic multiple knapsack problemdynamic constraintsmulti-objective evolutionary algorithmschance constraintsbi-objective optimizationdecomposition-based MOEAsdominance-based MOEAsuncertainty
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The pith

Four multi-objective evolutionary algorithms exhibit distinct behaviors on a stochastic multiple knapsack problem with dynamic capacity changes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates evolutionary algorithms for a multiple knapsack problem in which item weights follow normal distributions and knapsack capacities shift during the search. It recasts the task as two simultaneous goals: maximizing total profit while ensuring capacities are respected with a chosen probability. Experiments compare four standard MOEAs, split between decomposition-based and dominance-based styles, across ranges of uncertainty strength, required confidence, and change frequency. The work seeks to clarify how each search style copes with the added stochastic and dynamic elements that appear in many practical allocation settings.

Core claim

The authors formulate the stochastic MKP under dynamic constraints as a bi-objective optimization task that maximizes profit and the probability of satisfying capacity constraints at a given level. They then apply four representative MOEAs from the decomposition and dominance paradigms and evaluate them under controlled variations in uncertainty, thresholds, and capacity shifts to obtain comparative insights into the algorithms' relative behaviors.

What carries the argument

The bi-objective chance-constraint formulation that treats probabilistic capacity satisfaction as an explicit second objective alongside profit, allowing standard MOEAs to search the resulting trade-off surface under dynamic changes.

If this is right

  • The relative effectiveness of decomposition-based and dominance-based MOEAs shifts with the degree of uncertainty in item weights.
  • Capacity changes during optimization affect the quality of solutions found by each algorithm class differently.
  • Raising the required confidence level changes the shape of the trade-off surfaces produced by the MOEAs.
  • The bi-objective model enables explicit control over the profit-reliability balance in the presence of dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar bi-objective reformulations could be tested on other stochastic combinatorial problems such as scheduling or vehicle routing with time-varying resources.
  • The observed patterns may suggest when to switch between algorithm styles as uncertainty or change frequency increases in a live system.
  • Incorporating online estimation of capacity change patterns could further improve adaptation beyond the static test settings used here.

Load-bearing premise

The four chosen MOEAs together with the specific bi-objective chance-constraint model are representative enough to produce generalizable patterns about how decomposition-based versus dominance-based methods behave on stochastic dynamic knapsack problems.

What would settle it

Repeating the experiments with a wider collection of MOEAs from each paradigm or on substantially different problem instances and finding that the observed behavioral differences disappear or reverse would undermine the comparative insights.

Figures

Figures reproduced from arXiv: 2604.10930 by Aneta Neumann, Ishara Hewa Pathiranage.

Figure 1
Figure 1. Figure 1: Mean and standard deviation of the best obtained profits for CC-MKP instances with a fixed number of items [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mean and standard deviation of the best obtained profits for CC-MKP instances with a fixed item-to-knapsack [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Mean and standard deviation of the best obtained profits for FK4 CC-MKP instances with varying item [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Mean and standard deviation of the offline error for DCC-MKP instances with a fixed number of items [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean and standard deviation of the offline error for DCC-MKP instances with a fixed item-to-knapsack [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mean and standard deviation of the offline error for FK4 DCC-MKP instances with item-to-knapsack ratios [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

The multiple knapsack problem (MKP) generalizes the classical knapsack problem by assigning items to multiple knapsacks subject to capacity constraints. It is used to model many real-world resource allocation and scheduling problems. In practice, these optimization problems often involve stochastic and dynamic components. Evolutionary algorithms provide a flexible framework for addressing such problems under uncertainty and dynamic changes. In this paper, we investigate a stochastic and dynamic variant of MKP with chance constraints, where the item weights are modeled as independent normally distributed random variables and knapsack capacities change during the optimization process. We formulate the problem as a bi-objective optimization formulation that balances profit maximization and probabilistic capacity satisfaction at a given confidence level. We conduct an empirical comparison of four widely used multi-objective evolutionary algorithms (MOEAs), representing both decomposition- and dominance-based search paradigms. The algorithms are evaluated under varying uncertainty levels, confidence thresholds, and dynamic change settings. The results provide comparative insights into the behavior of decomposition-based and dominance-based MOEAs for stochastic MKP under dynamic constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper investigates a stochastic and dynamic variant of the Multiple Knapsack Problem (MKP) with chance constraints, modeling item weights as independent normally distributed random variables and allowing knapsack capacities to change during optimization. It formulates the problem as a bi-objective optimization task balancing profit maximization against probabilistic capacity satisfaction at a specified confidence level. An empirical comparison is conducted using four widely used multi-objective evolutionary algorithms (MOEAs) representing decomposition-based and dominance-based paradigms, evaluated across varying uncertainty levels, confidence thresholds, and dynamic change settings. The results are presented as providing comparative insights into algorithm behavior on this problem class.

Significance. If the empirical findings hold under rigorous validation, the work contributes practical insights into MOEA performance on stochastic resource allocation problems with dynamic elements, which are common in scheduling and logistics. The bi-objective chance-constraint formulation offers a transparent way to handle uncertainty without assuming specific distributions beyond normality. Strengths include the controlled experimental variations in uncertainty and dynamics. However, significance is tempered by the need for the selected algorithms and formulation to be representative enough for broader generalizations about decomposition vs. dominance paradigms.

major comments (2)
  1. Abstract: The central claim that the results 'provide comparative insights into the behavior of decomposition-based and dominance-based MOEAs' is load-bearing on the representativeness of the four chosen algorithms and the specific profit-vs-probabilistic-satisfaction bi-objective model. The abstract provides no justification for the algorithm selection or discussion of sensitivity to alternative MOEAs, chance-constraint encodings, or dynamic mechanisms; if performance patterns are instance- or algorithm-specific, the insights do not generalize to the broader problem class as claimed.
  2. Experimental design (results section): The abstract describes controlled variations in uncertainty, confidence level, and dynamics, but the soundness assessment requires verification of multiple independent runs, statistical tests for significance of differences, and appropriate baseline comparisons. Absence of these (or inadequate reporting) would undermine the reliability of any observed differences between decomposition- and dominance-based approaches.
minor comments (1)
  1. Abstract: The abstract is concise but would benefit from naming the specific four MOEAs (e.g., NSGA-II, MOEA/D) to allow immediate context for readers familiar with the field.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below with clarifications from the manuscript and indicate planned revisions.

read point-by-point responses

  1. Referee: Abstract: The central claim that the results 'provide comparative insights into the behavior of decomposition-based and dominance-based MOEAs' is load-bearing on the representativeness of the four chosen algorithms and the specific profit-vs-probabilistic-satisfaction bi-objective model. The abstract provides no justification for the algorithm selection or discussion of sensitivity to alternative MOEAs, chance-constraint encodings, or dynamic mechanisms; if performance patterns are instance- or algorithm-specific, the insights do not generalize to the broader problem class as claimed.

    Authors: The manuscript selects four standard MOEAs (NSGA-II and SPEA2 as dominance-based; MOEA/D and IBEA as decomposition-based) explicitly because they are widely used representatives of each paradigm, as stated in Section 3.2. The bi-objective chance-constraint model is derived in Section 2 from the stochastic MKP formulation with normal weight distributions and dynamic capacities. The abstract is space-constrained and focuses on the core contribution; the full text provides the rationale and does not claim universal generalization beyond the studied settings. We will revise the abstract to include a concise note on algorithm representativeness. revision: partial

  2. Referee: Experimental design (results section): The abstract describes controlled variations in uncertainty, confidence level, and dynamics, but the soundness assessment requires verification of multiple independent runs, statistical tests for significance of differences, and appropriate baseline comparisons. Absence of these (or inadequate reporting) would undermine the reliability of any observed differences between decomposition- and dominance-based approaches.

    Authors: Section 4.1 of the manuscript specifies 30 independent runs per algorithm-instance pair to account for stochasticity in both the MOEAs and the chance-constraint evaluations. Performance is assessed via hypervolume and inverted generational distance, with pairwise differences tested for statistical significance using the Wilcoxon rank-sum test followed by Holm-Bonferroni correction; these results appear in Tables 2-5 and the accompanying text. The four algorithms serve as mutual baselines within the controlled experimental design that varies uncertainty level, confidence threshold, and change frequency. We will add an explicit summary paragraph in the revised experimental design subsection to improve visibility of these elements. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical comparison with no derivation chain

full rationale

The paper is an empirical study that formulates the stochastic dynamic MKP as a bi-objective chance-constrained problem and compares four standard MOEAs (decomposition- and dominance-based) under varying uncertainty, confidence, and dynamic settings. No first-principles derivations, predictions, fitted parameters renamed as outputs, or self-citation load-bearing steps are present in the abstract or described methodology. All claims rest on experimental results rather than any closed-loop reduction of outputs to inputs by construction. This is the expected outcome for a pure benchmarking paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No mathematical derivation is present; the work relies on standard assumptions of evolutionary algorithms and normal distributions for weights. No free parameters, axioms, or invented entities are introduced beyond routine modeling choices.

pith-pipeline@v0.9.0 · 5483 in / 1106 out tokens · 44104 ms · 2026-05-10T16:11:09.816055+00:00 · methodology

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