Aled Williams

Department of MathematicsLondon School of Economics and Political ScienceLondon UK

a.e.williams1@lse.ac.uk USA

yiluncai@uchicago.edu

###### 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 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.

Keywords: multi-criteria decision making problems, weighted sum method, weight sampling, Pareto efficiency, Pareto frontier, nondominated points.

## 1 Introduction

### 1.1 Multi-Criteria Decision Making Problems

We consider multi-criteria optimisation problems of form

$\displaystyle\text{\leavevmode\ltxml@oqmark@open\textquotedblleft\penalty 1000%0\hskip-0.0002pt\hskip 0.0002ptminimise\textquotedblright\ltxml@oqmark@close{}%}\quad\big{(}f_{1}(\boldsymbol{x}),f_{2}(\boldsymbol{x}),\ldots,f_{p}(%\boldsymbol{x})\big{)}$ | (1) | |||

$\displaystyle\text{subject to }\quad\boldsymbol{x}$ | $\displaystyle\in\mathcal{X},$ |

where $\mathcal{X}\subset\mathbb{R}^{n}$ denotes the feasible set (or set of alternatives of the decision problem) and $f_{i}:\mathbb{R}^{n}\rightarrow\mathbb{R}$ for each $i\in\{1,2,\ldots,p\}$. Let us assume, for simplicity of presentation, that each objective $f_{i}$ is linear. It should be noted that each objective function $f_{i}(\boldsymbol{x})$ represents a different criterion (or aspect) of the decision making problem. The aim is to minimise the $p$ objective functions simultaneously, which typically involves in a trade-off between the objectives.

### 1.2 Efficiency and Nondominance

To clarify βminimiseβ in (1), we formally define efficient solutions and nondominated points, which are defined by the component-wise order over the $p$ objectives. Let $\mathcal{X}$ denote the feasible set of solutions to the above problem. Further, denote by $\mathcal{Y}:=c(\mathcal{X})$ the objective function mapping of the feasible set $\mathcal{X}$, where $c=(\boldsymbol{c}_{1},\boldsymbol{c}_{2},\ldots,\boldsymbol{c}_{p})$ for $\boldsymbol{c}^{T}_{i}\in\mathbb{R}^{n}$ for each $i$. Note that $\mathcal{X}\subset\mathbb{R}^{n}$ and $\mathcal{Y}\subset\mathbb{R}^{p}$.

###### Definition 1.1.

A feasible solution $\boldsymbol{x}^{*}\in\mathcal{X}$ is called efficient (or Pareto optimal) if there is no other $\boldsymbol{x}\in\mathcal{X}$ such that $c(\boldsymbol{x})\leq c(\boldsymbol{x}^{*})$, i.e. no other feasible $\boldsymbol{x}$ satisfies $\boldsymbol{c}_{i}^{T}\boldsymbol{x}\leq\boldsymbol{c}_{i}^{T}\boldsymbol{x}^{*}$ for all $i\in\{1,2,\ldots,p\}$ and $\boldsymbol{c}_{j}^{T}\boldsymbol{x}<\boldsymbol{c}_{j}^{T}\boldsymbol{x}^{*}$ for at least one $j\in\{1,2,\ldots,p\}$. If $\boldsymbol{x}^{*}$ is efficient, then $c(\boldsymbol{x}^{*})$ is called a nondominated point.

In other words, a solution $\boldsymbol{x}^{*}$ is efficient if there is no $\boldsymbol{x}\in\mathcal{X}$ such that $\boldsymbol{c}_{k}^{T}\boldsymbol{x}\leq\boldsymbol{c}_{k}^{T}\boldsymbol{x}^{*}$ for $k=1,2,\ldots,p$ and $\boldsymbol{c}_{l}^{T}\boldsymbol{x}<\boldsymbol{c}_{l}^{T}\boldsymbol{x}^{*}$ for some $l\in\{1,2,\ldots,p\}$. Informally, an efficient solution is a solution that cannot be improved in any of the objectives without degrading at least one of the other objectives. Thus, the fundamental importance of efficiency lies in the fact that any solution that is not efficient cannot represent the most preferred alternative for a decision maker. Next, let us define weakly efficient solutions and nondominated points.

###### Definition 1.2.

A feasible solution $\boldsymbol{x}^{*}\in\mathcal{X}$ is called weakly efficient (or weakly Pareto optimal) if there is no other $\boldsymbol{x}\in\mathcal{X}$ such that $c(\boldsymbol{x})<c(\boldsymbol{x}^{*})$, i.e. no feasible $\boldsymbol{x}$ satisfies $\boldsymbol{c}_{i}^{T}\boldsymbol{x}<\boldsymbol{c}_{i}^{T}\boldsymbol{x}^{*}$ for all $i\in\{1,2,\ldots,p\}$. If $\boldsymbol{x}^{*}$ is weakly efficient, then $c(\boldsymbol{x}^{*})$ is called weakly nondominated.

It follows from the above definitions that

$\mathcal{Y}_{N}\subset\mathcal{Y}_{wN}\subset\mathcal{Y}\subset\mathbb{R}^{p}%\quad\text{ and }\quad\mathcal{X}_{E}\subset\mathcal{X}_{wE}\subset\mathcal{X}%\subset\mathbb{R}^{n},$ |

$\mathcal{Y}_{N}$, $\mathcal{Y}_{wN}$, $\mathcal{X}_{E}$ and $\mathcal{X}_{wE}$ denote the set of all nondominated points, weakly nondominated points, efficient solutions and weakly efficient solutions, respectively. Informally, a weakly eο¬cient solution is a solution for which there is no way to improve every objective simulaneously while remaining feasible. Note that the images $\mathcal{Y}_{N}$ and $\mathcal{Y}_{wN}$ are often called the Pareto frontier (or the Pareto front or nondominated front) and the weak Pareto frontier, respectively.

### 1.3 An Introduction to Weighted Sum Scalarisation

The traditional approach to solving problems with multi-criteria such as (1) is by scalarisation, which involves formulating a single objective optimisation problem that is related to the multi-criteria problem. We begin by outlining one of the most commonly applied scalarisation techniques, namely the weighted sum scalarisation approach, before discussing more formal details around weight selection later. To introduce the method, let us once more denote by $\mathcal{X}$ the feasible set of solutions to problem (1).

The weighted sum method (WSM) converts the original problem to

$\min_{\boldsymbol{x}\in\mathcal{X}}\,\sum_{i=1}^{p}\lambda_{i}\,\boldsymbol{c}%_{i}^{T}\boldsymbol{x}=\lambda_{1}\boldsymbol{c}_{1}^{T}\boldsymbol{x}+\lambda%_{2}\boldsymbol{c}_{2}^{T}+\cdots+\lambda_{p}\boldsymbol{c}_{p}^{T}\boldsymbol%{x},$ | (2) |

where $\sum_{k=1}^{p}\lambda_{i}=1$ and $\lambda_{i}\geq 0$ for all $i\in\{1,2,\ldots,p\}$. Note that this approach converts the $p$ objectives into an aggregated scalar objective function by assigning each objective function a weighting factor, before summing yields the overall (single) objective function. Each (original) objective is given a weight to denote its relative importance during the overall aggregation. The method enables the computation of weakly efficient solutions by successively varying the weights $\lambda_{i}$ for convex problems. Being a little more precise, the following result connecting convexity with efficient solutions is known (see e.g. [67, Theorem 3.1.4]).

###### Theorem 1.

Suppose the multi-criteria optimisation problem (1) 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 $w_{i}\geq 0$ for each $i\in\{1,2,\ldots,p\}$ and $\sum_{i=1}^{p}\lambda_{i}=1$ such that $\boldsymbol{x}^{*}$ is a solution to the problem (2).

Thus, the above result suggests that any Pareto optimal solution of a convex multi-criteria optimisation problem can be found via the weighted sum method. The method may however work poorly for non-convex problems, such as a multi-criteria set covering or travelling salesman problems. It should be noted that different approaches for varying the weights are outlined in the subsequent section.

It should be noted for completeness that if the underlying problem is non-convex, then other scalarisation techniques are (perhaps) more appropriate for finding weakly efficient solutions. The celebrated $\varepsilon$-constraint method [38], hybrid method [34], Bensonβs Method [8], or the elastic constraint method (see e.g. [24, 88, 40]) are examples of such scalarisation approaches.

## 2 Literature Review

The weighted sum method (WSM) and its variants have established themselves as fundamental techniques in decision-making and optimisation across numerous fields. These methods provide a straightforward and flexible approach for evaluating multiple criteria by assigning relative importance to different factors, making them particularly well-suited for both objective and subjective decision-making processes. The following review presents key applications of WSM and its adaptations in areas ranging from technology and energy systems to urban planning and multi-objective optimisation, showcasing its versatility in tackling complex problems.

Sarika [78] evaluated and ranked servers from IBM, HP, and Sun Microsystems by employing the WSM and Revised Weighted Sum Decision Model. The analysis considered both objective parameters such as processor, memory, and hard disk, as well as subjective parameters like after-sales service, product quality, and brand name, to identify the optimal server for business applications based on their calculated worth.

Baumann et al. [6] provided a comprehensive review of MCDM approaches for evaluating energy storage systems in grid applications, systematically analysing the goals, methodologies, and criteria used in existing studies to provide guidance on the selection and assessment of energy storage technologies based on economic, technical, environmental, and social dimensions.

Yao et al. [99] conducted a detailed review on the application of advanced industrial informatics in road engineering, focusing on smart sensing technologies, AI-based algorithms, and cooperative vehicle infrastructure systems to enhance road construction, maintenance, and safety, and discussing their potential to develop smart, safe, and sustainable road infrastructures.

Lee and Chang [58] conducted a comparative analysis of MCDM methods to rank renewable energy sources for electricity generation in Taiwan, utilising the WSM, VIKOR, TOPSIS, and ELECTRE methods, along with the Shannon entropy weight method to assess the importance of each criterion, ultimately recommending hydropower as the top renewable energy source, followed by solar, wind, biomass, and geothermal energy.

Stanujkic, Djordjevic and Djordjevic [83] performed a comparative analysis of several prominent MCDM methods, including SAW, MOORA, GRA, CP, VIKOR, and TOPSIS, by applying them to the problem of ranking Serbian banks. Their study emphasised that different MCDM methods can yield varying ranking orders of alternatives, highlighting that these differences are not random but rather inherent to the methodsβ characteristics and the specificities of their aggregation and normalisation procedures.

Rahman and SzabΓ³ [73] conducted a systematic review of 55 studies on multi-objective urban land use optimisation using spatial data, identifying common objectives, constraints, and methodologies, and highlighting the limited incorporation of sustainability dimensionsβparticularly social aspectsβin existing research. They emphasised the need for developing standardised methods to evaluate the economic, environmental, and social benefits of land use optimisation and proposed integrating participatory approaches with mathematical optimisation to improve decision-making in urban land use planning.

Mofidi and Akbari [68] provided an extensive review of the state-of-the-art technologies and methodologies for optimising the operation of intelligent buildings with a focus on occupant comfort and energy consumption, covering topics such as occupant comfort conditions, productivity, building control strategies, computational optimisation methods, and occupant behaviour modelling, while also discussing the challenges and future directions for developing advanced intelligent energy management systems that balance energy efficiency and indoor environmental quality.

Kim and de Weck [55] developed an adaptive weighted sum (AWS) method for multi-objective optimisation that extends the traditional bi-objective approach to problems with more than two objective functions. The proposed method begins with the conventional weighted sum technique to approximate the Pareto front, followed by the use of equality constraints to refine the Pareto front patches in the objective space, resulting in a well-distributed mesh for effective visualisation. The authors demonstrated the methodβs ability to identify Pareto optimal solutions, even in non-convex regions, through numerical examples and a structural optimisation problem, highlighting its advantages over traditional methods like NBI.

Ayan, AbacΔ±oΔlu and Basilio [4] conducted a comprehensive review of novel weighting methods in MCDM, including CILOS, IDOCRIW, FUCOM, LBWA, SAPEVO-M, and MEREC. The authors analysed these methods in terms of their characteristics, application areas, and integration with other techniques, using bibliometric and content analyses based on publications from Web of Science and Scopus databases. Their study provides insights into the strengths, limitations, and potential applications of these methods, offering valuable guidance for researchers and practitioners in the MCDM field.

Dolatnezhadsomarin, Khorram and Yousefkhoshbakht [22] proposed new decision-making methods for ranking non-dominated points in multi-objective optimisation problems (MOPs). They introduced a scalarisation approach that assigns relative importance to objective functions for convex MOPs and developed two decision-making methods to assist decision-makers in selecting preferred solutions from sets of Pareto optimal points for both convex and non-convex MOPs. Their methods demonstrated the effectiveness of these approaches through numerical examples and provided a user-friendly decision support tool that does not require prior familiarity with MOPs.

Jakob and Blume [47] compared Pareto optimisation and the cascaded weighted sum (CWS) method for multi-objective optimisation, focusing on scenarios with multiple conflicting objectives. They discussed the limitations of Pareto optimisation in handling problems with more than four objectives and proposed the CWS method as a more effective alternative for real-world applications, where objectives have varying importance and specific regions of interest need to be explored. The authors demonstrated that CWS can better concentrate solutions in these regions while maintaining computational efficiency, making it suitable for tasks such as automated scheduling and robotic path planning.

Kim and de Weck [54] proposed an adaptive weighted-sum (AWS) method for bi-objective optimisation, designed to address the limitations of the traditional weighted-sum approach in generating well-distributed Pareto fronts, especially in non-convex regions. The AWS method adaptively adjusts weights and imposes additional inequality constraints to focus on unexplored regions of the Pareto front, avoiding non-optimal solutions. The authors demonstrated the methodβs effectiveness through numerical examples and a structural optimisation problem, showing that AWS can produce evenly distributed solutions along the Pareto front and successfully identify Pareto optimal solutions in non-convex regions, offering a robust alternative to traditional methods like Normal Boundary Intersection (NBI).

Madsen and Browning [63] introduced a weighted-sum method for groupwise association testing of rare mutations in genetically heterogeneous diseases. This method aggregates mutations based on their functional grouping (e.g., within genes) and assigns weights according to their frequency in unaffected individuals. The authors demonstrated that this approach enhances the power to detect disease-associated genes compared to existing methods, such as CAST and CMC, using both simulated and Encode data. Their findings suggest that the weighted-sum method is particularly effective for identifying associations in scenarios where multiple rare variants contribute to disease risk, and it can be used to identify significant genetic associations with as few as 1,000 affected and unaffected individuals, depending on the genetic model.

Gunantara [35] provided a comprehensive review of multi-objective optimisation (MOO) methods, focusing on the Pareto and scalarisation techniques, and their applications across various fields such as economics, finance, mechanics, and network optimisation. The paper highlights that these two methods simplify MOO problems by not requiring complex mathematical equations, making them accessible for a wide range of practical applications. The Pareto method addresses optimisation by finding non-dominated solutions to construct a Pareto optimal front, while the scalarisation method combines multiple objectives into a single scalar function using pre-determined weights. The review also discusses the strengths and limitations of each method and suggests potential applications for simplifying multi-objective problems in real-world scenarios.

Grodzevich and Romanko [33] explored various normalisation techniques and multi-objective optimisation methods in the context of financial applications. They focused on ensuring the consistency of optimal solutions with decision-maker preferences and analysed different approaches such as the weighted sum and $\varepsilon$-constraint methods. The report presented a practical algorithm combining both methods, designed for convex multi-objective problems with linear and quadratic objectives. Their analysis included a case study on portfolio optimisation, demonstrating the impact of normalisation and method selection on the efficiency of finding desired Pareto optimal solutions, and proposed a conflict indicator metric to aid decision-makers in multi-objective scenarios.

Augusto, Bennis and Caro [3] proposed methodology for decision-making in multi-objective optimisation problems by introducing a control function that guides the optimisation process over the Pareto set without explicitly generating the set. This approach differentiates itself from classical methods by focusing on a single-objective optimisation procedure that minimises the control function, while ensuring that the final solution belongs to the Pareto optimal set of the performance functions group. The authors demonstrated the effectiveness of this method through several application examples, including the optimisation of a cantilever beam and the conceptual design of a bulk carrier, highlighting its potential to reduce computational complexity and assist in selecting a single optimal solution from the set of non-dominated alternatives.

Jin, Olhofer and Sendhoff [48] developed the Evolutionary Dynamic Weighted Aggregation (EDWA) method for multi-objective optimisation, addressing the limitations of Conventional Weighted Aggregation (CWA) methods in handling problems with concave Pareto fronts. They provided a theoretical explanation for why CWA fails in such scenarios and demonstrated through simulations that EDWA effectively navigates both convex and concave Pareto fronts by dynamically adjusting weights during the optimisation process. Their results showed that EDWA can achieve a comprehensive set of Pareto solutions in a single run, making it an efficient approach for solving multi-objective optimisation problems.

Kumar et al. [56] conducted a critical review of the entropy weights method (EWM) applied in multi-objective optimisation for machining operations. The review classified 65 academic articles into 18 categories of conventional and non-conventional machining operations, highlighting the methodβs ability to assign objective weights and its widespread application in machining. The authors presented the implementation of EWM with examples, discussed its benefits and limitations, and suggested future research directions for expanding its use beyond machining to other fields, emphasising the methodβs capability in achieving precise and rational assessments in multi-criteria decision-making scenarios.

Hua [42] provided a detailed survey of evolutionary algorithms designed to solve multi-objective optimisation problems (MOPs) with irregular Pareto fronts. The paper categorises these algorithms into four groups based on their main strategies for handling irregularities, including methods based on fixed reference vectors, reference vector adjustment, reference point adjustment, and clustering or grid-based techniques. The authors reviewed existing algorithms, analysed their strengths and weaknesses, and highlighted the challenges and future research directions, emphasising the importance of developing more effective methods to handle the complexities of irregular Pareto fronts in real-world applications.

Tanabe and Ishibuchi [87] introduced a comprehensive suite of 16 real-world bound-constrained multi-objective optimisation problems, four of which are multi-objective mixed-integer problems. They provided source codes for these problems in various programming languages, enabling easy access for researchers. Furthermore, they conducted a performance analysis of six representative evolutionary multi-objective optimisation algorithms on the problem set, highlighting its utility for benchmarking and more accurate algorithmic assessment.

Li and Yao [60] proposed an adaptive weight adaptation method (AdaW) for decomposition-based evolutionary multi-objective optimisation, addressing the challenge of achieving consistent performance across a variety of Pareto front shapes. They developed a dynamic weight adjustment strategy that progressively adapts the weight distribution during the optimisation process, based on the evolving population. Their approach incorporates mechanisms for weight generation, addition, deletion, and archive maintenance, demonstrating improved performance on problems with complex and irregular Pareto fronts, as validated by extensive experimental results across various test problems.

Murata, Ishibuchi and Tanaka [71] introduced a multi-objective genetic algorithm (MOGA) and applied it to flowshop scheduling problems. Their approach incorporates a variable weighted sum of multiple objective functions and an elite preserve strategy to maintain diversity within the population. They demonstrated the algorithmβs effectiveness by applying it to two-objective and three-objective flowshop scheduling problems, optimising makespan, tardiness, and flowtime. The experimental results highlighted the algorithmβs capability to handle multi-objective problems with concave Pareto fronts, outperforming traditional methods like Schafferβs VEGA and single-objective genetic algorithms.

Cai et al. [12] proposed a clustering-ranking evolutionary algorithm (crEA) for addressing many-objective optimisation problems (MaOPs). The crEA employs a clustering mechanism based on the non-dominated sorting genetic algorithm III (NSGA-III) to promote diversity and a ranking mechanism to enhance convergence towards the true Pareto front. The algorithm was tested on nine benchmark problems from the Walking Fish Group (WFG) and the multi-objective traveling salesman problem (TSP), demonstrating superior performance in both convergence and diversity compared to six state-of-the-art algorithms through extensive experimentation.

Feliot, Bect and Vazquez [26] introduced an expected weighted hypervolume improvement (EWHI) criterion for Bayesian multi-objective optimisation, particularly tailored for expensive-to-evaluate black-box functions. The EWHI criterion extends the expected hypervolume improvement (EHVI) criterion by incorporating user preferences through a continuous weight function. To address the computational challenges of the criterion, the authors proposed an importance sampling approximation method using a sequential Monte Carlo (SMC) approach. The effectiveness of the EWHI criterion was demonstrated through numerical experiments on a bi-objective optimisation problem, highlighting its ability to focus optimisation efforts on user-preferred regions of the Pareto front.

Brockhoff et al. [11] presented a methodology for directed multi-objective optimisation using the weighted hypervolume indicator, which allows incorporating user preferences into the search process. They proposed the W-HypE algorithm, which uses Monte Carlo sampling to efficiently compute the weighted hypervolume, enabling optimisation in high-dimensional objective spaces. The algorithm adjusts the search direction according to user-defined weight functions that represent preferences, making it scalable and adaptable for many-objective problems. Extensive experiments demonstrated the effectiveness of the approach, highlighting its ability to guide the search towards user-preferred regions of the Pareto front.

Hughes [46] explored the performance of evolutionary algorithms in many-objective optimisation by comparing three methods: the Non-dominated Sorting Genetic Algorithm II (NSGA-II), Multiple Single Objective Pareto Sampling (MSOPS), and repeated single objective optimisations (RSO). The study found that while NSGA-II performed well on problems with two objectives, it became less effective as the number of objectives increased, due to the diminished selective pressure in Pareto ranking. In contrast, MSOPS and RSO, which do not rely on Pareto ranking, consistently outperformed NSGA-II in many-objective scenarios. The experiments demonstrated that generating the entire Pareto set in a single run is superior to multiple single objective optimisations, with MSOPS offering better performance than RSO on higher-dimensional problems.

Chiu, Yen and Juan [16] introduced a minimum Manhattan distance (MMD) approach to MCDM for multi-objective optimisation problems (MOPs). Their method identifies the final solution by selecting the point with the minimum Manhattan distance from a normalised ideal vector, effectively incorporating a preference model based on geometric interpretations. The MMD approach offers computational efficiency through matrix calculations and avoids the common issues associated with subjective weighting in conventional methods. The authors demonstrated the approachβs applicability to evolutionary multi-objective algorithms (MOEAs) and validated its effectiveness across multiple decision-making scenarios.

Lalwani et al. [57] provided a comprehensive survey of the applications of the Multi-Objective Particle Swarm Optimisation (MOPSO) algorithm across various fields. They reviewed existing work by organising the applications into areas such as aerospace engineering, biology, chemical engineering, and industrial applications, among others. The paper emphasises the adaptability of MOPSO in solving complex real-world multi-objective problems. They also highlighted the algorithmβs variants and provided insights into its growing popularity and future research directions, establishing a framework for further exploration of MOPSOβs potential.

Zou et al. [105] proposed a new evolutionary algorithm called the Dynamical Multi-Objective Evolutionary Algorithm (DMOEA) for solving many-objective optimisation problems. They conducted a comparative study between DMOEA and other state-of-the-art algorithms, such as IBEA, MSOPS, and NSGA-II, on three scalable test problems (DTLZ1, DTLZ2, and DTLZ6). The study demonstrated that DMOEA achieved better convergence and maintained a wider diversity of solutions. Additionally, the authors introduced a new concept of optimality, termed L-optimality, which considers both the number and values of improved objectives. They developed the MDMOEA algorithm based on L-optimality, and numerical experiments confirmed its ability to generate well-distributed L-optimal solutions, particularly for many-objective problems.

Sun er al. [86] proposed a two-stage Many-Objective Evolutionary Algorithm based on Independent Two-Stage approach (MaOEA-IT) to effectively address many-objective optimisation problems (MaOPs). The algorithm independently handles convergence and diversity in two separate stages. The first stage employs a non-dominated dynamic weight aggregation method using a genetic algorithm to find the Pareto-optimal solutions, while the second stage ensures diversity by solving a set of single-objective optimisation problems within the learned Pareto-optimal subspace. Their experiments demonstrated that MaOEA-IT significantly outperforms six state-of-the-art MaOEAs on benchmark test problems.

Yang [98] introduced a multi-objective bat algorithm (MOBA), an extension of the bat algorithm for solving multi-objective optimisation problems. The algorithm is validated on a subset of test functions and applied to solve engineering design problems, such as the bi-objective welded beam design problem. The MOBA algorithm uses a weighted sum to combine multiple objectives into a single objective and adapts parameters like loudness and pulse emission rates for each bat. Experimental results demonstrate the efficiency of MOBA in handling complex optimisation problems with nonlinear constraints, achieving a good approximation to the Pareto front.do the

Beretta and TΔtek [9] presented improved algorithms for sum estimation in both proportional and hybrid sampling settings, significantly enhancing the performance over previous methods by Motwani, Panigrahy and Xu [69]. They developed new upper and lower bounds that are tight with respect to both the size of the universe n and the error parameter $\epsilon$. Their algorithms provide more efficient sample complexity, reducing it to $\mathcal{O}(\sqrt{n}/\epsilon)$ in the proportional setting and $\mathcal{O}(n^{1/3}/\epsilon^{4/3})$ in the hybrid setting. Additionally, they introduced techniques for estimating sums when n is unknown, extending the applicability of these methods to more general settings.

Gupta et al. [37] proposed WAPS, a weighted and projected sampling technique designed to handle constrained sampling over formulas, with runtime performance that is agnostic to the underlying weight distribution. WAPS employs knowledge compilation techniques, compiling the constraint formula into a deterministic decomposable negation normal form (d-DNNF), which enables linear-time sampling. Their experiments demonstrated that WAPS outperformed the state-of-the-art sampler WeightGen by up to three orders of magnitude, solving significantly more instances across various benchmarks, thus bridging the gap between theoretical efficiency and practical performance in constrained sampling tasks.

Tao and Scott [89] analysed Markov Chain Monte Carlo (MCMC) sampling methods to improve the efficiency of estimating weighted sums in the Winnow multiplicative weight update algorithm. They proposed an optimised version of the Metropolis sampling method introduced by Chawla et al., reducing computational complexity while maintaining classification accuracy. Additionally, they empirically compared three MCMC techniques, namely Gibbs sampling, Metropolized Gibbs sampling, and parallel tempering, showing that the proposed optimisations significantly decreased the number of Markov chain simulations required, without compromising the accuracy of the estimated weighted sums or prediction performance of the Winnow algorithm.

HΓΌbschle-Schneider and Sanders [44] developed a highly efficient and scalable parallel algorithms for weighted random sampling (WRS), providing solutions for both shared-memory and distributed-memory architectures. Their approach focuses on constructing data structures such as alias tables and compressed structures to support fast, output-sensitive sampling of items with replacement and without replacement. By optimising for communication efficiency and linear work, their algorithms achieve near-linear speedups in both shared and distributed-memory systems. They validated their methods through extensive experiments on alias tables and distributed reservoir sampling, demonstrating robust performance across up to 5120 cores.

Naidu, Mokhlis and Bakar [72] implemented a multi-objective optimisation using the Artificial Bee Colony (ABC) algorithm to optimise the gains of PID controllers for Load Frequency Control (LFC) in a two-area interconnected reheat thermal power system. They applied a weighted sum approach to balance between minimising settling time and maximum overshoot in the frequency response. The researchers utilised performance indices such as Integral of Time Multiplied Absolute Error (ITAE) and Integral of Time Weighted Squared Error (ITSE) to characterise the optimisation. The proposed ABC-based PID controller outperformed conventional PI and PID controllers, demonstrating robustness under varying load demands and system parameter fluctuations.

Wansasueb, Bureerat and Kumar [96] developed a hybrid optimisation algorithm called Ensemble of Genetic algorithm, Grey wolf optimiser, Water cycle algorithm, and Population base increment learning using a Weighted sum (E-GGWP-W). This method was applied to the optimisation of aircraft composite wing designs, focusing on minimising the structural weight while adhering to aeroelastic and structural constraints. The proposed approach was tested on standard benchmark functions from the CEC-RW-2020 test suite and an aeroelastic composite wing design problem. The results showed that E-GGWP-W outperformed several well-established metaheuristics in terms of both convergence and robustness, as confirmed through statistical analysis using Friedmanβs rank test.

Stanujkic, Karabasevic, and Zavadskas [84] proposed a new MCDM approach for personnel selection, combining the adapted Weighted Sum (WS) method with the Step-Wise Weight Assessment Ratio Analysis (SWARA) method. This approach allows each decision-maker to use individual weights for criteria and rank alternatives independently. The final decision is reached through negotiation, where consensus is formed on the most appropriate alternative. An empirical illustration of the personnel selection process validated the efficiency of this approach, demonstrating its applicability in solving complex decision-making problems.

Rehman and Khan [74] developed a multi-criteria decision-making model for wind turbine selection using the Weighted Sum approach. The study identified five critical decision criteria: hub height, rotor diameter, cut-in wind speed, rated wind speed, and rated turbine output. The researchers applied the WSM to evaluate 18 commercially available turbines, determining that the Fuhrlander FL 600 turbine was the most suitable based on these criteria. The proposed model demonstrated its efficiency in simplifying the turbine selection process by providing a computationally efficient method for evaluating turbines while addressing the limitations of previous studies.

Zavadskas, Turskis, Antucheviciene and Zakarevicius [103] introduced the Weighted Aggregates Sum Product Assessment (WASPAS) method as an approach for improving the accuracy of MCDM. The authors compared the accuracy of two traditional methods, the WSM and the Weighted Product Model (WPM), and proposed the WASPAS method as a hybrid that enhances ranking precision by aggregating the strengths of both. Through theoretical analysis and empirical evaluation, they demonstrated that WASPAS outperforms WSM and WPM in terms of accuracy, providing a robust tool for decision support systems.

Karabasevic, Stanujkic, Djordjevic and Stanujkic [50] introduced the Weighted Sum Preferred Levels of Performances (WS PLP) method as an approach to solving problems in human resources management, specifically for personnel selection. The method utilises the Step-wise Weight Assessment Ratio Analysis (SWARA) to determine the weights of decision criteria and applies the WS PLP approach to rank alternatives based on decision-makersβ preferred performance levels. Through a case study involving the selection of a human resource manager, the authors demonstrated the approachβs adaptability, simplicity, and effectiveness in a multi-criteria decision-making context, highlighting its potential application in other decision-making scenarios.

Hua and Abdullah [41] developed the Weighted Sum-Dijkstraβs Algorithm (WSDA), which combines the WSM with Dijkstraβs algorithm to address multi-criteria decision-making problems in path selection. Their method allows for identifying optimal paths based on multiple criteria, such as cost, distance, and time, rather than a single criterion. By normalising and weighting these criteria, the WSDA efficiently calculates the best path through a network graph. Through numerical examples, the authors demonstrated that WSDA outperforms traditional Dijkstraβs algorithm in multi-criteria scenarios, offering a more user-friendly approach for non-mathematical users while maintaining computational efficiency.

Sianturi [81] applied the WSM in the development of a decision support system for selecting football athletes. The system evaluates athletes based on five key criteria: personality, age, marital status, ability, and experience. By using the WSM method, which assigns weights to each criterion, the system objectively ranks athletes according to their performance in these categories. The study demonstrated that the WSM-based system improved the selection process by providing clear, data-driven recommendations, simplifying decision-making for coaches when choosing athletes.

Harahap et al. [39] employed the MCDM WSM technique to analyse and rank university housing options based on five key criteria: location, affordability, amenities, room type, and social environment. Using the Analytic Hierarchy Process (AHP), they determined the weights for each criterion based on student preferences. The study applied the WSM to calculate weighted scores for various housing alternatives, providing a ranked list of options. The research highlights the usefulness of the MCDM-WSM methodology in aiding studentsβ housing decisions by transforming complex choices into a structured and transparent process.

Xie and Bo [104] proposed an improved WSM for evaluating weapon systems by introducing optimal weights to replace subjective weights traditionally used in such evaluations. By combining subjective weights derived from expert knowledge with objective weights calculated through grey theory, the authors developed a mathematical programming model to determine the optimal weights. Their approach improves the accuracy and stability of weapon system evaluations by reducing reliance on subjective judgment. The case analysis demonstrated that the improved WSM raised evaluation precision by over 5%, offering a more reliable method for weapon performance assessment.

Christensen et al. [17] proposed a low-complexity algorithm for weighted sum-rate maximisation in MIMO broadcast channels using weighted minimum mean square error (WMMSE) optimisation. The researchers established a relationship between the weighted sum-rate and WMMSE, allowing for an efficient iterative solution based on alternating optimisation of the transmit and receive filters. Their method was shown to converge to a local optimum with few iterations, achieving high performance in terms of sum-rate maximisation when tested on MIMO systems with multiple users.

Yong [100] applied the WSM to analyse breast cancer data and evaluate different carcinoma types based on various clinical parameters. The study utilised ten criteria, including lung, pleura, liver, bone, adrenal glands, gastrointestinal, skin, brain, pancreas, and kidney, to rank six breast carcinoma alternatives. The WSM method was employed to compute preference scores for each carcinoma type, with tubular/invasive cribriform carcinoma achieving the highest ranking. The results demonstrated the efficiency of the WSM approach in providing an objective framework for assessing different carcinoma types based on clinical characteristics.

Fisal, Hamdy and Rashed [28] proposed an adaptive weighted sum bi-objective Bat algorithm (AW-ABBA) to optimise regression testing by addressing the test suite reduction problem (TSR). They formulated the TSR as a bi-objective optimisation problem, with objectives to minimise the test suite execution time and maximise fault detection capability. Using the adaptive WSM, AW-ABBA efficiently approximated Pareto-optimal solutions. The authors evaluated the algorithmβs performance across five Java programs, demonstrating that AW-ABBA outperforms traditional weighted sum approaches and NSGA-II in terms of convergence and diversity.

Mukhametzyanov [70] conducted a comparative analysis of objective methods for determining the weights of criteria in MCDM problems, focusing on the Entropy, CRITIC, and Standard Deviation (SD) methods. The study highlighted limitations in the formal processing of decision matrices, particularly in the Entropy methodβs sensitivity to state probability evaluations. The author proposed two modifications of the Entropy methodβEWM.df and EWM.dspβthat address these contradictions. The analysis demonstrated the potential of integrated methods for more reliable weight determination and provided a framework for redistributing weights among correlated criteria, particularly through the new EWM-Corr method.

Keshavarz-Ghorabaee et al. [52] introduced an objective weighting method called MEREC (MEthod based on the Removal Effects of Criteria) for determining criteria weights in MCDM. Unlike traditional methods that rely on the variation of criteria values, MEREC assigns weights based on the effect of removing each criterion on the overall performance of the alternatives. The authors validated the effectiveness and stability of the method through comparative analyses with existing weighting methods, such as CRITIC, Entropy, and Standard Deviation, and performed simulation-based analyses to ensure the consistency of the results across different decision matrices.

Brauers et al. [10] developed and implemented a methodology for multi-objective optimisation in road construction. They employed the MOORA (Multi-Objective optimisation on the basis of Ratio Analysis) method to analyse various road design alternatives and determine the best option. A case study was used to illustrate the application of the method, where they compared six different highway design variants based on multiple criteria, such as cost, construction time, noise levels, and longevity, ultimately ranking the alternatives to select the optimal design.

Mathew and Sahu [65] compared various MCDM methods to solve material handling equipment selection problems. They applied four MCDM methodsβCODAS, EDAS, WASPAS, and MOORAβto select conveyors and automated guided vehicles based on conflicting criteria. The researchers calculated Spearman rank correlation coefficients to assess the consistency between the methods and found that the ranks produced by these newer methods were largely in agreement, demonstrating their effectiveness in handling material selection problems.

Deb [19] introduced a robust evolutionary framework for multi-objective optimisation, focusing on the popular elitist non-dominated sorting genetic algorithm (NSGA-II). The paper decomposes NSGA-II into three core principles: domination, diversity preservation, and elite preservation, demonstrating how modifying these principles allows NSGA-II to address a wide range of multi-objective optimisation problems. Debβs framework shows that NSGA-IIβs modularity makes it versatile for solving tasks like finding a diverse set of Pareto-optimal solutions or emphasising certain areas of the Pareto front, contributing to its popularity in evolutionary multi-objective optimisation research.

Tofallis [91] introduced the Automatic Democratic Method for determining objective weights in MCDM. This method uses Data Envelopment Analysis (DEA) to generate upper-bound scores for each entity based on its optimal performance. These DEA scores are then regressed against the attributes, using least squares to calculate common weights in the scoring formula. The approach avoids subjective judgments by deriving a data-driven, common set of weights applicable to all entities, thus promoting transparency and comparability in performance assessment.

Kesireddy and Medrano [53] developed the elite Multi-Criteria Decision MakingβPareto Front (eMPF) optimiser, an improved version of the M-PF optimiser. This new method combines multi-objective optimisation and multi-criteria decision-making techniques to explore and exploit the Pareto front efficiently. Using an evolutionary algorithm, the eMPF optimiser generates a Pareto front and ranks solutions through the TOPSIS method. The researchers evaluated its performance against the M-PF, NSGA-II, and NSGA-III methods, demonstrating superior performance in terms of solution diversity and quality across several test functions.

Ebrie and Kim [23] proposed a reinforcement learning-based multi-objective optimisation framework for generation scheduling in power systems. The framework utilises a multi-agent deep reinforcement learning (MADRL) algorithm, which decomposes the generation scheduling problem into sequential Markov decision processes. The authors evaluate the performance of the proposed method on several test systems, demonstrating its superiority over established techniques like TLBO, RCGWO, and NSGA-II, particularly in handling both economic and environmental objectives in power generation.

Meghwani and Thakur [66] proposed an adaptively weighted decomposition-based multi-objective evolutionary algorithm (AWMOEA/D) to address the limitations of the traditional MOEA/D algorithm in terms of convergence and diversity. Their approach modifies the scalarisation weights periodically by using a crowding distance operator to assess the crowdedness of solutions. Through experiments on several benchmark problems, they demonstrated that this adaptive strategy improves the algorithmβs ability to converge to the true Pareto front while maintaining diversity. The results show that AWMOEA/D outperforms other state-of-the-art multi-objective algorithms on most benchmark problems.

Verma et al. [93] conducted a comprehensive review of the NSGA-II algorithm, focusing on its application to multi-objective combinatorial optimisation problems (MOCOPs), such as assignment, allocation, traveling salesman, vehicle routing, scheduling, and knapsack problems. The review categorised the studies into conventional, modified, and hybrid NSGA-II implementations, analysing performance assessment techniques, modifications made to NSGA-II, and benchmarking with other algorithms. The paper provides insights into NSGA-IIβs adaptability to various combinatorial problems and suggests future research directions in this domain.

UlutaΕ et al. [92] developed an integrated MCDM model for logistics centre location selection, combining fuzzy SWARA and CoCoSo methods. The model was applied to determine the optimal logistics centre location for Sivas province, Turkey, utilising both qualitative and quantitative criteria. The study introduced the CoCoSo method in this context for the first time, alongside GIS-based mapping to compare alternatives. Sensitivity analysis and validation using other MCDM methods, such as COPRAS, VIKOR, and ARAS, confirmed the accuracy and robustness of the results.

Guo et al. [36] investigated a reconfigurable intelligent surface (RIS)-aided multiuser multiple-input single-output (MISO) downlink communication system, aiming to maximise the weighted sum-rate (WSR) of all users. The researchers proposed a low-complexity algorithm to jointly optimise the beamforming at the access point (AP) and the phase shift of the RIS elements under both perfect and imperfect channel state information (CSI) conditions. Their approach employed fractional programming and stochastic successive convex approximation techniques to solve the optimisation problem, and they validated the effectiveness of the proposed methods through numerical simulations.

Tirkolaee et al. [90] developed a novel hybrid method using fuzzy decision-making and multi-objective programming for sustainable and reliable supplier selection in a two-echelon supply chain design. The approach integrates Fuzzy ANP for ranking criteria and sub-criteria, Fuzzy DEMATEL for identifying relationships among criteria, and Fuzzy TOPSIS for prioritising suppliers. A tri-objective mixed-integer linear programming model was then proposed to optimise the supply chain based on cost minimisation, supplier prioritisation, and reliability maximisation. The methodology was validated using a case study from the lamp supply chain, and sensitivity analyses were conducted to assess the modelβs performance.

Zarepisheh, Khorram and Pardalos [102] developed a nonlinear weighted sum scalarisation method to generate properly efficient points in multi-objective optimisation problems (MOPs). The study focused on establishing the relationship between the set of scalarisation solutions and properly efficient points. The authors derived conditions under which the optimal solutions of the scalarised problem are guaranteed to be properly efficient and provided a characterisation of properly efficient points in terms of the scalarisation solutions. The method was demonstrated through theoretical analysis and examples.

Liu et al. [62] proposed an approach to multi-locus family-based association analysis by developing an adaptive weighted sum test using the LASSO method. Their approach improves upon traditional WSMs, which tend to perform poorly with rare variants. By employing LASSO regression, the researchers created a data-driven weight selection process that optimises the power to detect both common and rare genetic variants. Simulations demonstrated the superiority of this method over existing multi-locus tests, particularly in terms of statistical power while maintaining a well-controlled type I error rate. The approach was further validated using a real rheumatoid arthritis dataset.

Sheng et al. [80] proposed a Weighted Sum Validity Function (WSVF) for clustering, which aggregates several g cluster validity functions to improve the confidence and robustness of clustering solutions. To optimise the WSVF, the authors developed a Hybrid Niching Genetic Algorithm (HNGA) that evolves both the appropriate number of clusters and the partitioning of the data set. Their niching method preserves population diversity while preventing premature convergence, and the HNGA was further hybridised with the k-means algorithm to enhance computational efficiency. Experimental results demonstrated that the proposed approach consistently converges to the best-known solutions, outperforming other related clustering algorithms.

Goh, Tung and Cheng [31] proposed a revised weighted sum decision model for robot selection that incorporates expert opinions to evaluate both objective and subjective factors. The model eliminates the highest and lowest expert values for each factor to mitigate the impact of outliers and distorted preferences, ensuring a more balanced decision-making process. The authors demonstrated the effectiveness of this approach through a numerical example, showing how the elimination of extreme values can prevent rank reversal compared to traditional models that retain all expert inputs.

Sasao [79] analysed and synthesised weighted-sum (WS) functions, presenting a design method for their efficient implementation using look-up table (LUT) cascades. The paper derived upper bounds on the column multiplicities of decomposition charts for WS functions, providing estimates for the size of LUT cascades necessary to realize these functions. Furthermore, the research explored the arithmetic decomposition of WS functions and demonstrated their application in practical digital systems such as bit counting circuits, radix converters, and digital filters, offering significant reductions in circuit complexity.

Rowley et al. [75] critically examined the use of multi-criteria decision analysis (MCDA) methods for aggregating sustainability indicators, moving beyond the traditional weighted sum approach. They analysed the theoretical implications of selecting different MCDA methods, providing guidance to sustainability analysts on how to choose and apply appropriate techniques based on the specific decision-making context. The paper emphasised the importance of matching MCDA methodologies with the decision-makerβs needs, illustrating the impact of various methodological choices on the aggregation of environmental performance indicators.

Aristoff and Zuckerman [2] developed parameter optimisation techniques for weighted ensemble sampling of Markov chains in the steady-state regime. They proposed strategies for optimising the choice of parameters, such as bins and the number of replicas in each bin, to enhance the performance of weighted ensemble sampling. Their approach was validated through a numerical example, comparing the new strategies with traditional methods and direct Monte Carlo simulations, showing significant improvements in computational efficiency, especially for rare event calculations in molecular dynamics and reaction networks.

Chagas and Wagner [14] developed a weighted-sum method to solve the bi-objective traveling thief problem (BITTP), which integrates the traveling salesperson problem (TSP) and the knapsack problem (KP). Their approach converts the bi-objective problem into multiple single-objective ones using a convex combination of the objectives. By employing randomised versions of existing heuristics, the authors demonstrated that their method outperformed competition participants on 6 out of 9 instances, and found new best solutions for 379 single-objective problem instances, showcasing the effectiveness of their heuristic strategy.

Faia et al. [25] developed a genetic algorithm (GA) metaheuristic for portfolio optimisation in electricity markets, employing a weighted sum approach to handle multi-objective optimisation of profit maximisation and risk minimisation. The GA was adapted to solve the complex and dynamic problem of electricity allocation across different markets, offering a faster execution time compared to traditional exact methods. The results demonstrated that the GA could achieve near-optimal solutions with significantly reduced simulation time, making it suitable for real-world decision support in electricity market negotiations.

Yuan and Lu [101] proposed an efficient methodology for reliability-based optimisation (RBO) that uses a weighted importance sampling approach. The method approximates the failure probability function (FPF) as a weighted sum of sample values obtained from a single reliability analysis, thus decoupling reliability and optimisation procedures. The approach reduces computational costs by avoiding repeated reliability analyses. The authors demonstrated the effectiveness of the method through engineering examples, showing how it can handle optimisation problems with probabilistic constraints while maintaining accuracy and efficiency.

Huang et al. [43] developed an adaptive method for seeking the Pareto front in multi-objective spatial optimisation problems, addressing the limitations of traditional weighted sum approaches. The method iteratively explores the objective space by adjusting search directions based on unexplored regions, ensuring a well-distributed set of Pareto-optimal solutions. The authors applied this methodology to a case study involving multi-objective route selection in Singapore, using GIS-based tools to manage spatial data. The results demonstrated the approachβs effectiveness in generating diverse and representative Pareto solutions, overcoming the shortcomings of existing methods in handling concave regions of the Pareto front.

Wagner, Beume and Naujoks [94] conducted a comprehensive benchmarking study on the performance of various evolutionary multi-objective optimisation algorithms (EMOA) in many-objective optimisation scenarios. They evaluated Pareto-based methods such as NSGA-II and SPEA2, along with newer algorithms like $\varepsilon$-MOEA, IBEA, and SMS-EMOA, across scalable test problems with three to six objectives. Their results demonstrated that while traditional EMOA suffer performance degradation as the number of objectives increases, newer algorithms using aggregation and indicator functions, such as SMS-EMOA, performed significantly better in high-dimensional objective spaces.

Ferreira et al. [27] proposed methodology for selecting solutions in multi-objective optimisation problems, called the Weighted Stress Function Method (WSFM). This approach uses a stress-strain analogy from material science to quantify the decision makerβs preferences in terms of weights and selects solutions that best align with those preferences. The method was tested against benchmark problems and compared with other preference-based methods, such as the reference point evolutionary multi-objective optimisation (EMO) and the weighted Tchebycheff metric, showing improved consistency in aligning decision-maker preferences with the selected solutions.

Li et al. [59] provided a comprehensive survey on many-objective evolutionary algorithms (MaOEAs) to address the scalability issues encountered when applying traditional multi-objective evolutionary algorithms (MOEAs) to problems with more than three objectives. They categorised MaOEAs into seven key classes based on their underlying strategies: relaxed dominance, diversity-based, aggregation-based, indicator-based, reference set-based, preference-based, and dimensionality reduction approaches. The survey analysed and compared these methods across various test problems, identifying strengths and limitations, and proposed future research directions in tackling the challenges associated with many-objective optimisation problems (MaOPs).

Hughes [45] introduced the Multiple Single Objective Pareto Sampling (MSOPS) method, a non-Pareto evolutionary multi-objective optimisation algorithm designed to perform parallel searches using multiple target vector-based optimisations, such as weighted min-max approaches. The method is capable of efficiently generating and analyzing Pareto sets, especially in problems with a large number of objectives, by identifying boundaries and discontinuities in the Pareto surface. The approach was tested with differential evolution and demonstrated its ability to find and explore βinteresting regionsβ in both convex and concave Pareto sets.

Ahrari et al. [1] developed a weighted pointwise prediction method (WPPM) for dynamic multi-objective optimisation (DMO) problems. The method integrates a multi-model prediction framework with a novel information-sharing strategy, where each model utilises information from adjacent solutions to improve robustness against input errors. A new similarity metric and a directional variation strategy were introduced to enhance solution diversity. Through experiments on the CEC2018 test suite, WPPM demonstrated superior performance compared to existing methods, particularly in capturing dynamic Pareto optimal fronts with frequent and moderate changes.

Wilde, Smith and Alonso-Mora [97] proposed an alternative approach to scalarising multi-objective robot planning problems using a weighted maximum (WM) of objectives instead of the traditional weighted sum (WS). The authors demonstrated that the WM approach, also known as Chebyshev scalarisation, can identify a wider range of trade-offs between objectives, including non-convex Pareto fronts that the WS method fails to capture. They presented a path planning algorithm optimized for the WM cost function and validated its effectiveness through extensive simulations, showing that it provides more expressive and diverse Pareto-optimal solutions across various motion planning scenarios.

Deshpande et al. [20] proposed a multi-objective optimisation method using an adaptive weighting scheme designed to handle black-box simulation-based optimisation problems with bound constraints. The method employs two derivative-free direct search techniquesβDIRECT for global exploration and MADS for local refinementβand adapts weighting dynamically to improve the approximation of the Pareto front. By utilising surrogate models to minimise computational costs and a generalised star discrepancy measure to assess solution distribution, the approach demonstrated efficiency and robustness across several test problems, yielding well-distributed nondominated points on the Pareto front.

Chen and Li [15] conducted an extensive empirical study comparing Pareto search and weighted search in the context of multi-objective Search-Based Software Engineering (SBSE). The study involved 38 systems from three representative SBSE problems and explored the performance of both strategies under different search budgets and preferences. Their findings challenge the conventional belief that weighted search is superior when stakeholder preferences are available, showing that Pareto search consistently outperforms weighted search in most cases when sufficient search budget is provided. The study concludes with practical guidance for choosing between the two methods in SBSE problems.

Cai et al. [13] introduced the Grid Weighted Sum Pareto Local Search (GWS-PLS) method for combinatorial multi-objective and many-objective optimisation problems (CMOPs). The method combines the Pareto dominance and weighted sum approaches in a grid system to enhance the efficiency of local search (LS) by reducing computational and space complexity. By maintaining only one representative solution in each grid, GWS-PLS ensures diverse Pareto front approximations. The study demonstrated GWS-PLSβs superior performance compared to classical Pareto local search and decomposition-based approaches, particularly in handling CMOPs with more than two objectives.

Cohen [18] developed a multi-objective weighted sampling framework to address the challenge of estimating segment statistics over large datasets. The method enables efficient estimation of multiple f-statistics, such as count, sum, moments, and threshold statistics, using a smaller sample with statistically guaranteed quality. Cohen introduced a sampling algorithm that provides high-quality estimates for any function that is a positive linear combination of predefined objectives. The method was demonstrated to significantly reduce the sample size required for accurate estimation across diverse objectives, making it suitable for real-world applications involving large-scale data.

Dhiman et al. [21] introduced a new multi-objective optimisation algorithm called the Multi-objective Seagull optimisation Algorithm (MOSOA), an extension of the Seagull Optimisation Algorithm (SOA). MOSOA utilises dynamic archive caching to store non-dominated Pareto solutions and incorporates a roulette wheel selection method to enhance exploration and exploitation during the search process. The algorithm was tested on 24 benchmark test functions and six real-world engineering design problems, demonstrating superior convergence and diversity in comparison to other well-known multi-objective optimisation algorithms such as NSGA-II, MOPSO, and MOEA/D.

Kashfi, Hatami and Pedram [51] presented multi-objective optimisation techniques for VLSI circuits, focusing on power dissipation and delay as conflicting objectives. They proposed convex and non-convex models to represent these objectives and applied three optimisation methods: the Weighted Sum (WS) method, Compromise Programming (CP), and the Satisficing Trade-off Method (STOM). The authors demonstrated the effectiveness of each method in finding Pareto-optimal solutions using analytical gradients, showing that CP performs better in handling non-convex problems compared to WS. Their results highlight the trade-offs between precision, computational efficiency, and optimisation robustness for circuit design problems.

Grandoni et al. [32] introduced new approaches to multi-objective optimisation by developing multi-criteria Polynomial Time Approximation Schemes (PTAS) for classical $\mathcal{NP}$-hard problems, such as spanning tree, matroid basis, and bipartite matching, under multiple budget constraints. Their key innovation was using iterative rounding techniques to handle multi-objective problems with up to $\mathcal{O}$(1) budget constraints, which slightly violate the budget while guaranteeing near-optimality. Additionally, they presented mechanisms for transforming multi-criteria approximation schemes into pure approximation schemes for certain problems, improving upon existing methods in combinatorial optimisation.

Ryu, Seo and Min [77] developed a multi-objective topology optimisation method that incorporates an adaptive weighted-sum approach and a configuration-based clustering scheme. The adaptive weight determination ensures evenly distributed solutions across both convex and non-convex regions of the Pareto front, while the clustering scheme groups similar Pareto-optimal designs to reduce computational costs. This method was validated on benchmark topology optimisation problems, demonstrating its ability to efficiently find diverse and uniformly spaced Pareto solutions while reducing the overall optimisation time.

Giagkiozis and Fleming [30] analysed the effectiveness of decomposition-based methods in multi-objective optimisation, particularly focusing on the Chebyshev scalarising function compared to Pareto-based methods. They explored the theoretical relationship between these approaches, showing that the Chebyshev scalarising function under certain assumptions yields performance almost identical to Pareto-based methods. The authors demonstrated that the performance disparity between the two methods is largely due to the inability of Pareto-based algorithms to follow balanced search trajectories. Their findings provide insights into the selection of optimal scalarising functions for different problem geometries.

Lin, Gao and Bai [61] developed an intelligent sampling approach (ISA) for metamodel-based multi-objective optimisation, guided by the adaptive weighted-sum (AWS) method, to improve computational efficiency in solving multi-objective optimisation problems with expensive black-box simulation models. The ISA iteratively refines the Pareto front by updating the metamodel using radial basis functions (RBF) and selecting new sample points based on the proximity to the current Pareto front. The method was validated through mathematical examples and an engineering case study on the design of a bus body frame, demonstrating its ability to efficiently generate well-distributed Pareto solutions.

Kaddani et al. [49] proposed a WSM incorporating partial preference information to tackle multi-objective optimisation problems, where precise preference elicitation is often challenging. Their approach generates a set of preferred points by defining a set of possible weights rather than requiring precise values, allowing for a more flexible decision-making process. The authors demonstrate the effectiveness of this method by integrating it with standard multi-objective optimisation algorithms and applying it to several benchmark problems. The results show that this approach significantly reduces computation time while still producing high-quality preferred solutions.

Gharehkhani et al. [29] extended the weighted-sum-of-gray-gases (WSGG) model to simulate radiative heat transfer in a utility boiler with a gas-soot mixture. The researchers introduced a new soot absorption coefficient based on temperature, which they coupled with data from Taylor and Fosterβs model. They applied the zone method to predict heat transfer and temperature distribution within the boiler under different load conditions, validating their model through experiments measuring furnace exit gas temperature and steam production rates. The results demonstrated good agreement with measured data, confirming the modelβs accuracy in handling gas-soot radiative heat transfer.

Soor et al. [82] introduced the Weighted Sum of Segmented Correlation (WSSC) method for material identification in hyperspectral images. This approach calculates correlation indices between segments of a test spectrum and a library spectrum, applying a weighted sum to produce a matching index that emphasises positive correlations and penalises negative ones. The method was tested on AVIRIS Earth data and CRISM Martian data, demonstrating superior performance over full-spectrum matching techniques like cosine similarity and correlation coefficients, particularly for detecting nuanced absorption features in minerals.

Ryu, Kim and Sujin [76] proposed the Pareto front Approximation with Adaptive Weighted Sum (PAWS) method for solving multi-objective simulation optimisation problems, particularly when the objective functions are complex or unknown. The PAWS algorithm combines a weighted sum approach with a trust region method, adaptively adjusting the weights based on non-dominated solutions found during the optimisation process. The method was tested on problems with convex and non-convex Pareto fronts, demonstrating superior performance compared to the BIMADS algorithm in terms of convergence, computational efficiency, and uniform distribution of Pareto-optimal points.

Stanujkic et al. [85] proposed the Integrated Simple Weighted Sum Product (WISP) method, which combines the weighted sum (WS) and weighted product (WP) approaches for solving MCDM problems. The method simplifies normalisation procedures and incorporates four utility measures to facilitate ranking of alternatives. The researchers demonstrated the methodβs reliability through illustrative examples, comparing its results with other well-established MCDM methods like TOPSIS and VIKOR. The results showed that WISP offers a simpler and more accessible approach to decision-making, suitable for users without advanced knowledge in MCDM.

Wang et al. [95] proposed the Localized Weighted Sum (LWS) method for many-objective optimisation, combining the computational efficiency of the weighted sum approach with a local search strategy to handle non-convex Pareto fronts. The LWS method operates within a hypercone defined for each weight vector, ensuring that only neighbouring solutions are evaluated to prevent convergence issues in non-convex regions. The authors demonstrated the effectiveness of the LWS method by integrating it into a multi-objective evolutionary algorithm (MOEA/D-LWS), which outperformed several state-of-the-art many-objective optimisation algorithms across various benchmark problems with up to seven objectives.

Marler and Arora [64] provided new insights into the WSM for multi-objective optimisation, addressing its conceptual significance and limitations. They analysed how weights influence the solution of multi-objective problems, focusing on the methodβs ability to articulate preferences a priori. The authors identified deficiencies in the method, particularly in capturing non-convex Pareto fronts and ensuring an even distribution of Pareto-optimal solutions. They proposed guidelines to improve the effectiveness of the WSM, emphasising the need for careful weight selection and highlighting the methodβs limitations in handling complex preference structures.

## 3 Weighting the Weighted Sum Method

Recall that the weighted sum method converts the original problem to (2). We focus initially on the bi-objective setting (i.e. $p=2$), before extending th approaches to the case with $p\geq 3$ objectives.One natural question explored within this section is how to vary these weights in order to find many (distinct) weakly efficient solutions while additionally avoiding redundancy. It should be noted that such redundancy could occur given that multiple choices of weights could lead to the same solution. Several approaches for varying the weights will be outlined below.

### 3.1 Weight Selection for $p=2$ Objectives

Note that the bi-objective setting yields

$\min_{\boldsymbol{x}\in\mathcal{X}}\,\sum_{i=1}^{2}\lambda_{i}\,\boldsymbol{c}%_{i}^{T}\boldsymbol{x}=\lambda_{1}\boldsymbol{c}_{1}^{T}\boldsymbol{x}+\lambda%_{2}\boldsymbol{c}_{2}^{T}\boldsymbol{x},$ |

where $\lambda_{1}+\lambda_{2}=1$, $\lambda_{1},\lambda_{2}\geq 0$ and $\mathcal{X}$ denotes the feasible set of solutions. The method successively varies the weights $\lambda_{1}$ and $\lambda_{2}$ in order to find weakly efficient solutions.

The first approach, which we call the uniform increment approach, divides the range $[0,1]$ into $d$ equal subintervals for each weight. Note that each subinterval clearly has length $1/d$. In the bi-objective case, we simply vary the weight $\lambda_{1}$ from 0 to 1 in increments of $1/d$, while setting $\lambda_{2}=1-\lambda_{1}$. The parameter $d$ can be intuitively thought of as the βdepthβ of search, where a larger $d$ provides a finer resolution, allowing for more precise sampling at the cost of increased computational effort (and likely greater redundancy). Observe that in this approach we solve $d+1$ problems, namely with weights $(\lambda_{1},\lambda_{2})=\big{\{}(0,1),(1/d,1-1/d),\ldots,(1,0)\big{\}}$.

The second approach, which we call the random sampling approach, instead randomly samples weights from the feasible range, while ensuring that they sum to 1. Since $p=2$ by assumption, it is sufficient to sample only $\lambda_{1}$ and set $\lambda_{2}=1-\lambda_{1}$. The sampling for $\lambda_{1}$ could be done from distributions such as the uniform distribution or the beta distribution, allowing for flexibility in controlling the spread and concentration of weights over the feasible range. Note that this method is less systematic, however, it could be useful for higher-dimensional problems. An important observation is that it is no longer clear when the sampling procedure should terminate and stop searching for nondominated points.

The third approach, which we call the Latin hypercube sampling (LHS) approach, informally divides $[0,1]$ into $d$ equal subintervals for each weight. Then if $d$ is even, we randomly sample one value from each interval for each weight, shuffle those sampled values to create multiple combinations, before normalising each combination so that the sum of the (combined) weights equals 1. If instead $d$ is odd, one value is randomly sampled from each interval other than the $\lceil d/2\rceil$-th interval in the ordered sequence, from which we sample two values for technical reasons.

Note that each shuffle creates an ordered sequence of the $d$ (or $d+1$) randomly sampled weights. This process is repeated $s$ times, where $s$ denotes the number of (overall) shuffles, yielding an ordered sequence of $sd$ (or $s(d+1)$) weights. Adjacent weights in the sequence are then paired to create $sd/2$ (or $s(d+1)/2$) combinations. Each combination is subsequently normalised such that the sum of the (combined) weights equals 1.

This approach interestingly leads to the normalised weights being roughly normally distributed, with most values concentrated around the mean of 0.5, rather than near 0 or 1. This is since the process draws from equally spaced intervals across $[0,1]$, with fewer values coming from the extreme ends near 0 and 1. For instance, to generate a combination where $\lambda_{1}$ is near 0, we would need one sample from an interval close to 0 and another from an interval close to 1 in the same combination, a statistically less likely event. Thus, the weights are much more likely to be close to 0.5 rather than the extremes. This weight concentration may therefore not result in a good spread of solutions in an optimisation context.

This claim is further supported by presenting several normal quantile-quantile (Q-Q) plots (namely Figures 1 and 2) of normalised weights drawn from $d$ intervals and shuffled $s$ times, for selected even values of $d$ and selected $s$. Recall that the process yields $sd/2$ combinations. Then, from each combination, we randomly select one value for plotting, resulting in $sd/2$ points on each plot. Note that in each normal Q-Q plot, for each point $(x,y)$, $x$ corresponds to one of the $sd/2$ quantiles from a normal distribution, and $y$ corresponds to one of the $sd/2$ weights. When the points lie close to the line $y=x$, this indicates that the values are approximately normally distributed.

Furthermore, at the tails of the normal Q-Q plots (Figures 1 and 2), we observe deviations from the line. In particular, smaller $x$-values (representing the lower tail) tend to lie above the line, indicating that the sampled values are larger than expected, suggesting a lighter left tail. Conversely, larger $x$-values (representing the upper tail) tend to lie below the line, indicating that the sampled values are smaller than expected, implying a lighter right tail. This suggests that the distribution of weights has thinner tails compared to a normal distribution.

The concentration of weights in the LHS approach is problematic, as we would ideally like our weights to be more evenly distributed in order to explore a broader range of potential nondominated points. This motivates us to introduce additional structure to the LHS approach, leading to the development of a more refined approach.

The fourth approach, which we call the structured Latin hypercube sampling (SLHS) approach, works similarly to the LHS approach, however, before sampling the $d$ intervals are structured. In particular, firstly the approach once more divides $[0,1]$ into $d$ equal subintervals. Then assuming $d$ is even, we pair all intervals $[a_{1},a_{2}]$ and $[b_{1},b_{2}]$ such that $a_{1}+b_{2}=a_{2}+b_{1}=1$. Observe that given $d$ is by assumption even, we will clearly have $d/2$ pairs of intervals satisfying this property. Then we randomly sample one value from each interval for each weight. Next, we form $d/2$ pairs using the sampled values from the matched intervals. Finally, we normalise each combination such that the sum of the weights equals 1.This approach can be thought of as the structured analogue to the LHS approach given that it is guaranteed that the sampled pairs after normalisation must remain within their corresponding initial (paired) intervals, which is formally proven for completeness below (Lemma 2). Note that if instead $d$ is odd, the $\lceil d/2\rceil$-th interval will remain unpaired and instead we simply sample two values from this interval before applying normalisation similarly.

###### Lemma 2.

Suppose $[a_{1},a_{2}]$ and $[b_{1},b_{2}]$ are (ordered) subintervals of $[0,1]$ with $a_{1}<a_{2}<b_{1}<b_{2}$ satisfying $a_{1}+b_{2}=a_{2}+b_{1}=1$. Then any $a\in[a_{1},a_{2}]$ and $b\in[b_{1},b_{2}]$ satisfy

$\frac{a}{a+b}\in[a_{1},a_{2}]\quad\text{ and }\quad\frac{b}{a+b}\in[b_{1},b_{2%}].$ | (3) |

###### Proof.

Suppose for contradiction that (3) does not hold, i.e. $a/a(a+b)\notin[a_{1},a_{2}]$ or $b/(a+b)\notin[b_{1},b_{2}]$. Note that this is the case when $a/(a+b)<a_{1}$, $a/(a+b)>a_{2}$, $b/(a+b)<b_{1}$ or $b/(a+b)>b_{2}$ hold. Each such case will be considered in turn.

Firstly, suppose that $a/(a+b)<a_{1}$ holds. This yields $a(1-a_{1})<a_{1}b$ through algebraic manipulation. Observe that if $a_{1}=0$, then we deduce that $a<0$, which is clearly a contradiction. If instead $a_{1}\neq 0$, then upon dividing by $a_{1}$ we have ${a}/{a_{1}}\,(1-a_{1})<b$. Note that $a\geq a_{1}$ by assumption and hence ${a}/{a_{1}}\geq 1$. In particular, this implies that $b>1-a_{1}$ holds, which is a contradiction given $b\leq b_{2}$ and $b_{2}=1-a_{1}$.

Secondly, suppose that $a/(a+b)>a_{2}$ holds. This yields $a(1-a_{2})>a_{2}b$. Observe that the assumed ordering $0\leq a_{1}<a_{2}<b_{1}<b_{2}\leq 1$ implies that $a_{2}\neq 0$. Upon dividing by $a_{2}$ we have $a/a_{2}\,(1-a_{2})>b$. Note that $a\leq a_{2}$ by assumption and hence $a/a_{2}\leq 1$. It follows that $1-a_{2}>b$ holds, which is a contradiction given that $b\geq b_{1}$ and $b_{1}=1-a_{2}$.

Thirdly, suppose that $b/(a+b)<b_{1}$ holds. This yields $b(1-b_{1})<b_{1}a$ via manipulation. Note that the assumed ordering implies that $b_{1}\neq 0$. Upon dividing by $b_{1}$ we yield $b/b_{1}\,(1-b_{1})<a$. Note that $b\geq b_{1}$ by assumption and hence $b/b_{1}\geq 1$. This implies that $1-b_{1}<a$ holds, which is a contradiction given that $a\leq a_{2}$ and $a_{2}=1-b_{1}$.

Finally, suppose that $b/(a+b)>b_{2}$ holds. This yields $b(1-b_{2})>b_{2}a$. Note that $b_{2}\neq 0$ and then upon dividing by $b_{2}$ we deduce that $b/b_{2}\,(1-b_{2})>a$. Note that $b\leq b_{2}$ and hence $b/b_{2}\leq 1$. This implies that $1-b_{2}>a$ holds, which is a contradiction given that $a\geq a_{1}$ and $a_{1}=1-b_{2}$. In particular, the four cases considered demonstrate that (3) holds, which completes the proof as required.β

The fifth approach, which we call the structured adaptive approach, starts with an initial set of weights before adapting them as required before terminating searches based on the solutions found. The method informally identifies βlarge gapsβ between nondominated points (in terms of some distance metric) and then subdivides those regions to introduce finer resolution. In particular, we initially follow the uniform increment approach by dividing $[0,1]$ into $d$ equal subintervals, before then adapting our weights by subdividing (some of) our $d$ subintervals into $d$ equal subintervals and repeating until termination. Note that the decision regarding if a subinterval will be further subdivided will depend on the previously found nondominated points that correspond to the end points of the subinterval being considered and toleration threshold and redundancy bounding parameters $\tau\geq 0$ and $\rho\in[0,1]$, respectively. Being more precise, suppose that we are considering if we should subdivide the interval $[a_{1},a_{2}]$ as described above and that the corresponding weight pairs, namely $(a_{1},1-a_{1})$ and $(a_{2},1-a_{2})$, yield the nondominated points $\boldsymbol{n}_{1}$ and $\boldsymbol{n}_{2}$, respectively. In particular, one should subdivide only if $\|\boldsymbol{n}_{1}-\boldsymbol{n}_{2}\|_{2}>\tau$, where $\|\cdot\|_{2}$ denotes the $\ell_{2}$-norm. Further, the (overall) approach should terminate if $\mathcal{N}/\mathcal{D}<\rho$ holds, where $\mathcal{N}$ and $\mathcal{D}$ denote the total number of distinct nondominated points found and the total number of intervals searched, respectively. It should be noted that the second condition enforces termination when the overall level of redundancy has grown significantly. Observe that the stopping conditions introduced here mean that the running time of such an implementation will be intrinsically related to $\tau$ and $\rho$ and, as such, it is natural to consider what is a suitable βsizeβ of $\tau$ and $\rho$ in order to avoid redundancy wherever possible.

### 3.2 Weight Selection for $p\geq 3$ Objectives

Notice that the setting with $p\geq 3$ objectives converts the original problem to (2), where $\sum_{i=1}^{p}\lambda_{i}=1$ and $\lambda_{i}\geq 0$ for all $i\in\{1,2,\ldots,p\}$. The method once more successively varies the weights $\lambda_{i}$ in order to find weakly efficient solutions.

The first approach, namely the uniform increment approach, similarly divides the range $[0,1]$ into $d$ equal subintervals for each weight. Here each weight $\lambda_{i}$ can take values from $\{0,1/d,\ldots,1\}$. Then we generate all possible combinations of weights $\{\lambda_{1},\lambda_{2},\ldots,\lambda_{p}\}$ such that $\sum_{k=1}^{p}\lambda_{i}=1$ and $\lambda_{i}\geq 0$ for all $i\in\{1,2,\ldots,p\}$. This can be done systematically to ensure normalisation. The natural approach is to make use of a nested loop to iterate over all combinations of $\lambda_{1},\lambda_{2},\ldots,\lambda_{p-1}$, before calculating the remaining weight $\lambda_{p}$ to ensure that their sum is 1, where any combination with $\lambda_{p}\not\in[0,1]$ is discarded. This results in solving at most $(d+1)^{p-1}$ problems, as exemplified below.

###### Example 1.

Suppose that $p=3$ and $d=2$. Observe that $\lambda_{1},\lambda_{2},\lambda_{3}\in\{0,\frac{1}{2},1\}$ in such case. Then we iterate over $\lambda_{1}$ and $\lambda_{2}$ before calculating $\lambda_{3}=1-(\lambda_{1}+\lambda_{2})$, namely:

- 1)
for $\lambda_{1}=0$:

- β’
$\lambda_{2}=0$ and $\lambda_{3}=1-(0+0)=1$,

- β’
$\lambda_{2}=\frac{1}{2}$ and $\lambda_{3}=1-(0+\frac{1}{2})=\frac{1}{2}$, and

- β’
$\lambda_{2}=1$ and $\lambda_{3}=1-(0+1)=0$.

- β’
- 2)
for $\lambda_{1}=\frac{1}{2}$:

- β’
$\lambda_{2}=0$ and $\lambda_{3}=1-(\frac{1}{2}+0)=\frac{1}{2}$,

- β’
$\lambda_{2}=\frac{1}{2}$ and $\lambda_{3}=1-(\frac{1}{2}+\frac{1}{2})=0$, and

- β’
$\lambda_{2}=1$ and $\lambda_{3}=1-(\frac{1}{2}+1)=-\frac{1}{2}$, which is invalid as $\lambda_{3}<0$.

- β’
- 3)
for $\lambda_{1}=1$:

- β’
$\lambda_{2}=0$ and $\lambda_{3}=1-(1+0)=0$,

- β’
$\lambda_{2}=\frac{1}{2}$ and $\lambda_{3}=1-(1+\frac{1}{2})=-\frac{1}{2}$, which is invalid as $\lambda_{3}<0$, and

- β’
$\lambda_{2}=1$ and $\lambda_{3}=1-(1+1)=-1$, which is invalid as $\lambda_{3}<0$.

- β’

Thus, the valid combinations are

$\left\{\left(0,0,1\right),\left(0,\frac{1}{2},\frac{1}{2}\right),\left(0,1,0%\right),\left(\frac{1}{2},0,\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2},0%\right),\left(1,0,0\right)\right\}.$ |

It should be noted that the aforementioned upper bound of $(d+1)^{p-1}$ can be refined to instead state an explicit closed formula for the number of valid combinations. The number of valid combinations is

$\binom{d+p-1}{p-1}=\frac{(d+p-1)!}{(p-1)!\,d!},$ |

which is the binomial coefficient for integers $d+p-1$ and $p-1$, respectively. This follows immediately in light of the stars and bars theorem (or formula) (see e.g. [7, Chapter 3]), which provides a way to count the number of nonnegative solutions to the equation$\lambda_{1}+\lambda_{2}+\cdots+\lambda_{p}=d$.

The rapid growth of this binomial coefficient as both $d$ and $p$ increase is demonstrated in Figure 3, which illustrates how the number of valid combinations grows exponentially as we vary $d$ (depth) and $p$ (number of objectives). The figure particularly highlights the steep growth rate for larger $p$, reflecting the complexity and computational effort required to generate and evaluate combinations as the dimensionality increases.

The second approach, namely the random sampling approach, can be extended as follows. Since $p\geq 3$ by assumption, it is no longer to sufficient to sample only one $\lambda_{i}$. One natural extension relies on the Dirichlet distribution, which is the multivariate generalisation of the beta distribution. The Dirichlet distribution (of order $K\geq 2$) with parameters $\alpha_{1},\alpha_{2},\ldots,\alpha_{K}>0$ (which are denoted for convenience by $\boldsymbol{\alpha}$) has probability density function with respect to the $(K-1)$-th dimensional volume or Lebesgue measure (see e.g. [5, Chapter 13]) given by

$f(x_{1},x_{2},\ldots,x_{K};\alpha_{1},\alpha_{2},\ldots,\alpha_{K})=\frac{1}{B%(\boldsymbol{\alpha})}\prod_{i=1}^{K}x_{i}^{\alpha_{i}-1},$ |

where $\sum_{i=1}^{K}x_{i}=1$ and $x_{i}\in[0,1]$ for all $i\in\{1,2,\ldots,K\}$, i.e. that $x_{1},x_{2},\ldots,x_{K}$ belong to the unit (or standard) $(K-1)$-dimensional simplex in $\mathbb{R}^{K}$, as illustrated in Figure 4. The normalising constant of the Dirichlet distribution is the multivariate beta function, namely

$B(\boldsymbol{\alpha})=\frac{\displaystyle\prod_{i=1}^{K}\Gamma\left(\alpha_{i%}\right)}{\Gamma\big{(}\displaystyle\sum_{i=1}^{K}\alpha_{i}\big{)}},$ |

where $\Gamma(\cdot)$ denotes the gamma function.

Note for completeness that the parameters $\alpha_{i}$ in the Dirichlet distribution play a crucial role in shaping the distribution of weights. Being more specific, each $\alpha_{i}$ for $i\in\{1,2,\ldots,K\}$ controls the expected concentration of $x_{i}$ within the unit simplex. Suppose, for simplicity, that $\boldsymbol{\alpha}$ is symmetric. If $\alpha_{i}>1$, then the distribution tends to favour larger values of $x_{i}$, geometrically pushing the probability mass toward the centre of the unit simplex. If $\alpha_{i}<1$, then the distribution tends to favour smaller values of $x_{i}$, concentrating the probability mass around the facets and vertices of the simplex. If $\alpha_{i}=1$ for all $i\in\{1,2,\ldots,K\}$, then the distribution becomes uniform over the simplex. This is illustrated in Figure 5. Thus, by carefully adjusting the parameters $\alpha_{i}$, it is possible to control how the weights are distributed while ensuring $\sum_{i=1}^{K}x_{i}=1$ and $x_{i}\in[0,1]$ for each $i$ hold as required.

The third approach, namely the LHS approach, can be extended as follows. The approach begins by similarly dividing the range $[0,1]$ into $d$ equal subintervals for each weight. Then we randomly sample one value from each interval for each weight, shuffle these samples to create combinations before normalising each combination such that the sum of the combined weights is 1.

The fourth approach, namely the SLHS approach, can be extended as follows. The approach begins by similarly dividing the range $[0,1]$ into $d$ equal subintervals for each weight, as in the LHS approach. Thus, we yield the $d$ subintervals

$\left[0,\frac{1}{d}\right],\left[\frac{1}{d},\frac{2}{d}\right],\ldots,\left[%\frac{d-1}{d},1\right]$ | (4) |

with (set theoretic) union is $[0,1]$.

Denote by $m_{i}\in[0,1]$ the midpoint of the subinterval $[a_{i},b_{i}]$, i.e. $m_{i}=(a_{i}+b_{i})/2$. Intuitively, we aim to select $p$ subintervals such that the sum of their midpoints is βcloseβ to 1. We then collect samples from the subintervals, form $p$-element tuples of the sampled values from the selected subintervals, and finally normalise each combination such that the sum of the (normalised) weights is 1. To measure this βclosenessβ, let $\delta\geq 0$ be a parameter representing the allowable deviation of the sum of midpoints from 1.

Thus, we require the selected subintervals to satisfy the inequalities

$1-\delta\leq\sum_{i=1}^{p}m_{i}\leq 1+\delta,$ | (5) |

where, to simplify notation, $m_{1},m_{2},\ldots,m_{p}$ denotes the midpoints of the $p$ selected subintervals from which we randomly sample.

Upon selecting the subintervals whose sum of midpoints is βcloseβ to 1, we sample a value for each $\lambda_{i}\in[a_{i},b_{i}]$ from each selected subinterval. Since the selected values of $\lambda_{i}$ are sampled from intervals centred around the midpoints $m_{i}$, we expect that their sum is βcloseβ to the sum of the midpoints, namely that

$\sum_{k=1}^{p}\lambda_{i}\approx\sum_{k=1}^{p}m_{k}.$ |

However, in contrast to the case $p=2$ (as shown in Lemma 2), the samples will not necessarily remain within their corresponding grouped intervals upon normalisation. Despite this, it is possible to bound $\sum_{k=1}^{p}\lambda_{k}$ from above and below, allowing us to estimate the maximal variation after normalisation.

Observe that

$\displaystyle\sum_{k=1}^{p}\lambda_{k}$ | $\displaystyle=\sum_{k=1}^{p}\big{(}m_{k}+(\lambda_{k}-m_{k})\big{)}$ | ||

$\displaystyle=\sum_{k=1}^{p}m_{k}+\sum_{k=1}^{p}\lambda_{k}-m_{k}$ | |||

$\displaystyle\leq 1+\delta+\sum_{k=1}^{p}\big{|}\lambda_{k}-m_{k}\big{|},$ |

where the last inequality holds since the subintervals were chosen to satisfy (5). Moreover, note that

$\big{|}\lambda_{k}-m_{k}\big{|}\leq\frac{b_{k}-a_{k}}{2}.$ |

Thus, we obtain

$\displaystyle 1+\delta+\sum_{k=1}^{p}\big{|}\lambda_{k}-m_{k}\big{|}$ | $\displaystyle\leq 1+\delta+\sum_{k=1}^{p}\frac{b_{k}-a_{k}}{2}$ | ||

$\displaystyle=1+\delta+p\cdot\frac{b_{i}-a_{i}}{2}\quad\text{ for any %subinterval }[a_{i},b_{i}]$ | |||

$\displaystyle=1+\delta+\frac{p}{2d},$ |

since $\frac{b_{i}-a_{i}}{2}=\frac{1}{2d}$ for any interval $[a_{i},b_{i}]$ from (4).

A similar argument yields the lower bound

$1-\left(\delta+\frac{p}{2d}\right)\leq\sum_{k=1}^{p}\lambda_{k}.$ |

Thus, we conclude that

$1-\left(\delta+\frac{p}{2d}\right)\leq\sum_{k=1}^{p}\lambda_{k}\leq 1+\left(%\delta+\frac{p}{2d}\right).$ |

This inequality tells us that the sum of the sampled values will be close to 1, with small deviations depending on how tightly the midpoints sum to 1 and the variability introduced by random sampling within each selected subinterval. Further, since the sum of samples is close to 1, the normalisation factor $1/\sum_{k=1}^{p}\lambda_{k}$ will be close to 1, ensuring that the normalised values $\lambda_{i}/\sum_{k=1}^{p}\lambda_{k}$ will remain close to original sampled values for each $i$.

The fifth approach, namely the structured approach, can be extended as follows. The central idea of this approach is to iteratively refine the sampling space by dividing it into structured subintervals, adapting the sample distribution to ensure better coverage of the decision space. The approach begins by following the uniform increment approach by dividing $[0,1]$ into $d$ equal subintervals for each weight. Recall that the uniform increment approach involves solving precisely $\binom{d+p-1}{p-1}$subproblems for fixed $d$, which is upper bounded by $(d+1)^{p-1}$. It is natural therefore to select a small value for $d$ (such as $d=2$). Upon following the uniform increment approach, we then subdivide intervals adaptively based on the $\ell_{2}$-distance between the nondominated points in $\mathbb{R}^{p}$. In particular, we similarly use hyperparameters $\tau\geq 0$ and $\rho\in[0,1]$ to define our toleration distance threshold and redundancy bounding parameters, which control the subdivision and termination criteria, respectively.

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 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.

## 4 Conclusion

In this paper, we presented a range of techniques for selecting weights in the weighted sum scalarisation method for solving multi-criteria decision making problems. We explored both systematic and random sampling methods, offering an initial framework for generating weights efficiently. While the uniform increment approach provides a straightforward solution for problems of smaller dimension, its scalability is limited due to redundancy caused by the superlinear growth in both $d$ (depth) and $p$ (number of objectives). In contrast, random sampling and Dirichlet-based methods show promise for higher-dimensional problems, though their ability to ensure comprehensive coverage of the decision space warrants further study. Structured sampling methods, such as structured Latin hypercube sampling (SLHS), offer more control over weight selection and mitigate redundancy, yet a formal comparison to simpler approaches like random sampling is necessary to assess their practical performance.

There should be a focus in future research on extensive computational testing to evaluate the efficiency, scalability, and redundancy of the proposed sampling methods across a variety of multi-criteria decision making problems. In addition, it will be valuable to investigate how these techniques compare to other scalarisation methods (e.g., $\varepsilon$-constraint [38], hybrid [34], Bensonβs Method [8], or the elastic constraint method (see e.g. [24, 88, 40]), particularly in non-convex settings, where weighted sum approaches are known to encounter challenges. Furthermore, the development of hybrid sampling techniques that combine the strengths of different strategies offers a promising direction to enhance the coverage and efficiency of weight selection, especially for complex, high-dimensional decision-making scenarios.

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