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Insights into Weighted Sum Sampling Approaches for Multi-Criteria Decision Making Problems

Aled Williams, Yilun Cai

TL;DR

The paper addresses weight selection for weighted sum scalarisation in multi-criteria decision making, focusing on how to sample weight vectors on the simplex to balance coverage and redundancy. It analyzes and compares multiple strategies—uniform increment, random sampling, Latin hypercube sampling (LHS), structured Latin hypercube sampling (SLHS), and a structured adaptive approach—for both $p=2$ and $p\ge 3$ objectives, highlighting practical scalability and distributional properties. A key contribution is the identification that LHS can bias weights toward the center ($0.5$), the formal counting of combinations in grid-based schemes $\binom{d+p-1}{p-1}$, and the introduction of Dirichlet-based random schemes plus adaptive, structure-guided refinement to improve exploration. The work provides a transparent, off-the-shelf framework for weight generation in MCDM with potential applicability to non-convex problems and sets the stage for extensive computational benchmarking against alternative scalarisation methods like the $\varepsilon$-constraint and elastic constraint approaches.

Abstract

In this paper we explore several approaches for sampling weight vectors in the context of weighted sum scalarisation approaches for solving multi-criteria decision making (MCDM) problems. This established method converts a multi-objective problem into a (single) scalar optimisation problem. It does so by assigning weights to each objective. We outline various methods to select these weights, with a focus on ensuring computational efficiency and avoiding redundancy. The challenges and computational complexity of these approaches are explored and numerical examples are provided. The theoretical results demonstrate the trade-offs between systematic and randomised weight generation techniques, highlighting their performance for different problem settings. These sampling approaches will be tested and compared computationally in an upcoming paper.

Insights into Weighted Sum Sampling Approaches for Multi-Criteria Decision Making Problems

TL;DR

The paper addresses weight selection for weighted sum scalarisation in multi-criteria decision making, focusing on how to sample weight vectors on the simplex to balance coverage and redundancy. It analyzes and compares multiple strategies—uniform increment, random sampling, Latin hypercube sampling (LHS), structured Latin hypercube sampling (SLHS), and a structured adaptive approach—for both and objectives, highlighting practical scalability and distributional properties. A key contribution is the identification that LHS can bias weights toward the center (), the formal counting of combinations in grid-based schemes , and the introduction of Dirichlet-based random schemes plus adaptive, structure-guided refinement to improve exploration. The work provides a transparent, off-the-shelf framework for weight generation in MCDM with potential applicability to non-convex problems and sets the stage for extensive computational benchmarking against alternative scalarisation methods like the -constraint and elastic constraint approaches.

Abstract

In this paper we explore several approaches for sampling weight vectors in the context of weighted sum scalarisation approaches for solving multi-criteria decision making (MCDM) problems. This established method converts a multi-objective problem into a (single) scalar optimisation problem. It does so by assigning weights to each objective. We outline various methods to select these weights, with a focus on ensuring computational efficiency and avoiding redundancy. The challenges and computational complexity of these approaches are explored and numerical examples are provided. The theoretical results demonstrate the trade-offs between systematic and randomised weight generation techniques, highlighting their performance for different problem settings. These sampling approaches will be tested and compared computationally in an upcoming paper.
Paper Structure (9 sections, 2 theorems, 40 equations, 5 figures)

This paper contains 9 sections, 2 theorems, 40 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Multi-Criteria Decision Making Problems
  3. Efficiency and Nondominance
  4. An Introduction to Weighted Sum Scalarisation
  5. Literature Review
  6. Weighting the Weighted Sum Method
  7. Weight Selection for p = 2 Objectives
  8. Weight Selection for p ≥ 3 Objectives
  9. Conclusion

Key Result

Theorem 1

Suppose the multi-criteria optimisation problem multi_problem is convex. If $\boldsymbol{x}^*$ is efficient (or Pareto optimal), then there exists a weighting vector $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_p)$ with $\lambda_i \ge 0$ for each $i \in \{1,2,\ldots,p\}$ and $\sum_{i=1}^p \lamb

Figures (5)

  • Figure 1: This figure presents four normal Q-Q plots for different (smaller) values of $d$ (number of intervals) and $s$ (number of shuffles) in the Latin hypercube sampling (LHS) approach.
  • Figure 2: This figure presents four normal Q-Q plots for different (larger) values of $d$ (number of intervals) and $s$ (number of shuffles) in the Latin hypercube sampling (LHS) approach.
  • Figure 3: This figure illustrates the growth of the binomial coefficient $\binom{d + p - 1}{p - 1}$ on a logarithmic scale, as $d$ (depth) and $p$ (number of objectives) vary. The colours denote different values of $p$.
  • Figure 4: This figure illustrates the 2-dimensional unit (or standard) simplex in $\mathbb{R}^3$, whose vertices are the 3 standard unit vectors in $\mathbb{R}^3$.
  • Figure 5: This figure illustrates the Dirichlet distribution for symmetric $\boldsymbol{\alpha} = (\alpha_1, \alpha_2, \alpha_3)$ with $K=3$. Note that each plot features 5000 samples, where each point represents a sample consisting of three nonnegative components that sum to one.

Theorems & Definitions (7)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1
  • Example 1
  • Lemma 2
  • proof
  • Example 2