MAXIMIZING THE ACCEPTED BIDS OVER THE ε-NON-DOMINATED NONCONVEX SET FOR A MULTIOBJECTIVE WINNER DETERMINATION PROBLEM USING AN EVOLUTIONARY APPROACH

ABSTRACT

This paper addresses the Winner Determination Problem (WDP) within a multiobjective framework, extending the traditional approach that focuses solely on revenue maximization. We develop a mathematical model that simultaneously integrates four strategic objectives: maximizing the number of accepted bids, total revenue, the value of items (including unsold ones), and sales diversification. To solve this complex problem, we propose a novel approach that consists of optimizing a function over the set of ε-non-dominated solutions. We implement an Estimation of Distribution Algorithm (EDA), an evolutionary metaheuristic that models the probability distribution of promising solutions to efficiently guide the search. The performance and robustness of our approach are validated through numerical experiments, which demonstrate the algorithm’s ability to converge to high-quality solutions in a stable manner, thereby offering a practical and powerful decision-support tool for auction organizers.

Keywords:
Estimation of Distribution Algorithms (EDA); multiobjective optimization; Winner Determination Problem (WDP)

1 INTRODUCTION

Auctions have been demonstrated to be a reliable methodology for establishing the value of assets that are difficult to evaluate through conventional means. Over time, a multitude of auction mechanisms have been devised and executed, encompassing English auctions, Dutch auctions, First-Price Sealed-Bid auctions, and Second-Price Sealed-Bid auctions (Krishna, 2009).

The analysis of auctions hinges upon the Winner Determination Problem (WDP). The fundamental optimization problem under consideration aims to identify the optimal set of winning bids from those submitted, typically with the objective of maximizing a given metric, such as the total profit of the bidder or the seller. The WDP has been the subject of extensive research and detailed study in the literature (see, for instance, (Sariddichainunta & Sinapiromsaran, 2011; Boughaci et al., 2009; Qinghua & Jin-Kao, 2011)). Conventionally, WDP models have been developed with the primary objective of maximizing revenue generated by the auction.

However, the practical application of these principles often necessitates the consideration of additional criteria or objectives beyond revenue maximization. In real-world scenarios, auction organizers must balance financial gain with strategic and social goals. For instance, in spectrum auctions, regulators often prioritize market competition and geographical coverage alongside revenue (Cramton et al., 2006). In industrial procurement (e.g., logistics or supply chain), companies must simultaneously optimize costs, supplier reliability, and risk management (Sandholm, 2006; An et al., 2005). Furthermore, public procurement auctions increasingly incorporate environmental impact and social fairness as critical evaluation criteria (Bichler, 2001). These multidimensional requirements explicitly link the classic WDP to a multiobjective framework, where a single “optimal” solution is replaced by a set of Pareto compromises.

In order to surmount the challenge posed by the multiplicity of solutions in the context of the multiobjective WDP, it is proposed that an approach be adopted consisting of optimizing a function specified by the decision-maker over the identified set of efficient solutions. This approach enables the selection of the optimal solution according to a secondary or preference criterion, subsequent to the establishment of the set of Pareto compromises.

The approach of research concerning the optimization of function over the efficient set for multiobjective problems is well-established. A plethora of exact and approximate methods has been formulated for this particular class of problems, especially in the context of discrete optimization (see the references (Abbas & Chaabane, 2006; Chaabane et al., 2012; Chaabane & Mebrek, 2014; Chaabane & Pirlot, 2010; Jesus, 2009; Zaidi et al., 2024, 2025)). Correspondingly, numerous approaches have been devised for continuous optimization (see reference (Yamamoto, 2004)). However, the implementation of these precise methods is frequently constrained by the magnitude and intricacy of the problem instances, a limitation that can be particularly evident in the context of large-scale WDPs. This underscores the rationale behind our decision to explore the potential of a metaheuristic approach in addressing the optimization problem over the efficient set of the multiobjective WDP.

Among metaheuristics, Estimation of Distribution Algorithms (EDAs) have gained popularity in recent years within the evolutionary optimization community. In contrast to classical evolutionary algorithms, which depend on crossover and mutation operators, EDAs model and sample the probability distribution of promising solutions to guide the search. This approach provides a macroscopic perspective on the search landscape and has proven particularly effective for complex and high-dimensional problems. EDAs possess advantageous characteristics, such as their global search capability and conciseness, as highlighted by (Larranãga & Lozano, 2002).

The efficacy of EDAs has been demonstrated in a multitude of optimization applications, and they have proven particularly pertinent for solving multiobjective optimization problems. The extant literature documents numerous studies that have employed EDAs in the context of multiobjective problems (see, for example, (Larranãga & Lozano, 2002; Jianyong et al., 2005; Pang et al., 2019)).

The decision to employ an EDA for the optimization of a function within the ε-non-dominated set of the multiobjective WDP is driven by the numerous advantages offered by these algorithms (Larranãga & Bielza, 2025; Hauschild & Pelikan, 2011). These advantages include their capacity to manage the inherent complexity of searching over a set of solutions by modeling the global structure of the search space through probabilistic learning (Pelikan et al., 2002). Furthermore, EDAs have demonstrated high adaptability to different problem types and a superior ability to effectively explore the Pareto solution space to identify the specific compromise that maximizes a decision-maker’s preference function (Zhang et al., 2008; Santana et al., 2009).

The present study focuses fundamentally on the following aspects:

• • We first present several fundamental concepts in multiobjective optimization, including ε-dominance, the multiobjective winner determination problem, and the global approach to optimizing a function over the efficient set of a multiobjective problem. We then introduce the general principles of the Estimation of Distribution Algorithm (EDA).

• • In the following section, a novel multiobjective version of the winner determination problem is defined. Initially, three conflicting objectives are considered: maximizing total revenue, maximizing the total value of items, and Maximization of Sales Diversification. The resolution of this phase yields a set of optimal compromise solutions (the ε-non-dominated set). Subsequently, rather than increasing the dimensionality of the Pareto front, a fourth criterion, the maximization of the number of accepted bids (Ψ), is introduced as a scalar preference function. This overarching objective is directly optimized over the previously identified set of efficient solutions.

• While our theoretical model is formulated to optimize over the exact Pareto efficient set, finding these exact solutions is computationally intractable. Therefore, our proposed algorithm is designed to find an ε-non-dominated set, which provides a practical and computationally efficient approximation of the true Pareto front.

• The solution to this problem is presented and adapted in the form of the Estimation of Distribution Algorithm (EDA), a probabilistic optimization method that constructs a probability distribution from the currently known set of solutions. Subsequently, this distribution is employed to generate new potential solutions, which are then evaluated based on the objective function.

• • Finally, the efficacy and pertinence of this approach are demonstrated through the use of a numerical example, which will validate its effectiveness in solving multiobjective optimization problems.

2 MATHEMATICAL MODELING OF THE MO-WDP

2.1 Basics Concepts

2.1.1 Multiobjective optimization problem

Let a multiobjective optimization problem be defined by the minimization of k objective functions:

$\underset{x\in X}{min}F\left(x\right)=\left(f_1\left(x\right),f_2\left(x\right),...,f_3\left(x\right)\right)$

where X⊆ℝⁿ is the decision space (or design variable space), F:X→ℝ^m is the vector objective function, and k≥2 is the number of objectives.

While classical multiobjective optimization relies on standard Pareto dominance to identify the set of efficient solutions, our work introduces a two-fold extension. First, we adopt a specialized modeling approach that optimizes an overarching function strictly over this set of efficient solutions. Second, to ensure computational tractability, we substitute strict Pareto dominance with an ε-dominance relation.

2.1.2 ε-dominance

ε-dominance is a crucial extension of Pareto dominance in multiobjective optimization, introducing a notion of tolerance. It utilizes a tolerance vector ε=(ε₁, ..., ε_m ) where each ε_i >0.

Definition 1. For a maximization problem, a solution x ε-dominates a solution y if f _i (x)≥f _i (y)+ε _i for all objectives i, and if f _j (x)>f _j (y)+ε _j for at least one objective j. ( Laumanns et al., 2002 ).

This approach allows a solution to be considered ”sufficiently better” than another. Its primary advantage is that it generates a finite and well-distributed set of non-ε-dominated solutions, forming the ε-Pareto Front. This is fundamental for the efficiency of multiobjective optimization algorithms, as it facilitates the management of continuous or very dense fronts and improves convergence while ensuring good diversity among the found solutions.

2.1.3 Modeling Approach

In order to model the problem at hand, an approach was adopted that was based on the optimization of a function over the set of efficient solutions to a multiobjective problem.

Let the mathematical formulation of WDP be as follows:

$\left(P\right)\left\{\begin{array}{l}max{F}_j\left(x\right)=c_j^{\top }x,j=1,...,3,\\ s.t.x\in \mathbb{X}.\end{array}\right.$ (1)

Where 𝕏≡{x∈{0, 1}ⁿ /Ax≤b} is the feasible set of the problem, with A∈ℝ^m×n , b∈ℝ^m and $C=\left(c_j^{\top },j=1,...,3\right)$ is a 3×n matrix defining the three objective functions.

Since objective functions are often in conflict, it is impossible to find a feasible solution that optimizes all criteria simultaneously. This is why resorting to the concept of an efficient solution is more than necessary.

We denote by 𝔼 the set of efficient solutions of (P) and the new mathematical model of our problem (P _𝔼) defined as

$\left(P_{\mathbb{E}}\right)\left\{\begin{array}{l}max\psi \left(x\right)={\lambda }^{\top }x,\\ s.t.x\in \mathbb{E}.\end{array}\right.$ (2)

where λ denotes an n-dimensional objective vector.

2.3 Multiobjective Winner Determination Problem

In combinatorial auctions, bidders can place bids on combinations (or bundles) of items rather than solely on individual items. The classic Winner Determination Problem (WDP) aims to determine the optimal allocation of these items among the submitted bids. The WDP is formulated as a single-objective optimization problem, predominantly focusing on maximizing a single financial metric, such as the total revenue of the auction organizer or the overall profit.

However, as auction mechanisms are increasingly applied to complex, real-world resource allocation problems, relying on a single financial metric is no longer sufficient. Practical market designs necessitate the consideration of multiple competing criteria. This necessity explicitly links the classical WDP to a multiobjective framework, resulting in the multiobjective Winner Determination Problem (Mo-WDP).

In alignment with the goals outlined in the introduction, our study restricts this general multiobjective framework to four specific objectives:

1. Maximization of the Number of Accepted Bids,

2. Maximization of Total Revenue,

3. Maximization of Total Portfolio Value,

4. Maximization of Sales Diversification.

A significant challenge in the Mo-WDP arises from the fact that these four objectives are frequently at odds with one another. To illustrate, the pursuit of maximizing revenue does not guarantee the maximization of social welfare or item coverage, as improving one objective generally degrades another. Therefore, a single absolute ”optimal” solution is not present. Instead, solving the Mo-WDP yields a set of non-dominated or Pareto-optimal solutions, which represent the best possible compromises between these conflicting objectives.

The introduction of multiple objectives significantly exacerbates the inherent NP-hard nature of the classic WDP. Since evaluating and selecting from these trade-offs poses a substantial challenge for decision-makers, the exploration and approximation of the Pareto set require advanced methodologies. In this work, this is achieved through the implementation of precise modeling founded on integer programming and multiobjective metaheuristics, specifically, Estimation of Distribution Algorithms (EDAs) (Buer & Kopfer, 2014), which are highly pertinent for capturing the complexity of these four criteria simultaneously.

3 MATHEMATICAL MODEL

This section presents a mathematical model for the Winner Determination Problem (WDP) in the context of multiobjective optimization with four distinct objectives. The WDP consists of selecting a subset of compatible bids in order to optimize several criteria simultaneously.

3.1 Model Description

3.1.1 Definition of Sets and Parameters

Let:

• M={1,..., m} : Set of items to be sold.

• N={1,..., n} : Set of received bids.

For each bid i∈N:

• I_i ⊆M: Set of items requested in bid i.

• P_i>0 : Price offered for the bundle of items I_i in bid i.

• d_ij: The quantity of item j∈M requested by bid i∈N.

For each item j∈M:

• v_j≥0: Intrinsic or reserve value of item j if it is not sold.

• α: The minimum percentage of the available quantity of an item j∈M that must be sold for it to be considered sold.

3.1.2 Decision Variables

• x_i ∈{0, 1} for each i∈N: Binary variable equal to 1 if bid i is accepted (winning), and 0 otherwise.

• y_j ∈{0, 1} is a binary variable equal to 1 if the minimum percentage of item j∈M is sold,

$y_j=\left\{\begin{array}{l}1,if\sum _{i\in N}{d}_{ij}{x}_i>\alpha q_j,\\ 0,otherwise,\end{array}\right.j\in M.$ (3)

3.1.3 Objective Functions

In our multiobjective formulation of the Winner Determination Problem (WDP), we consider the following four main objectives, which the auction entity seeks to maximize:

• 1. Maximization of the Number of Accepted Bids (Ψ)

$max\psi \left(x\right)=\sum _{i\in N}{x}_i$ (4)

• This criterion, not directly tied to revenue, can be strategically important for enhancing bidder satisfaction, or promoting a broader participant base for future auctions.

• 2. Maximization of Revenue (f ₁ )

$max{f}_1\left(x\right)=\sum _{i\in N}{P}_i{x}_i$ (5)

• This traditional objective aims to maximize the direct financial gain from the auction by accepting bids (x _i =1) that offer the highest prices (P _i ). It constitutes the fundamental economic measure of the auction’s performance.

• 3. Maximization of the Total Value of Items (f ₂ )

$max{f}_2\left(x\right)=\sum _{i\in N}{P}_i{x}_i+\sum _{j\in M}{v}_j\left(q_j-\sum _{i\in N}{d}_{ij}{x}_i\right)$ (6)

• This objective introduces a consideration for items that are not sold. By seeking to maximize the sum of values v _j (defined by the auctioneer for each item j) associated with these items, this objective reflects a retention strategy where the auctioneer wishes to keep their most valuable items (v _j high) if the received bids do not sufficiently exceed their non-sale value.

• 4. Maximization of Sales Diversification (f ₃ )

$max{f}_3\left(y\right)=\sum _{j\in M}{y}_j$ (7)

• This objective aims to maximize the number of distinct items actually sold and maximize the overall quantity sold. This is relevant to inventory management, which aims to maximize inventory turnover by finding buyers for as many different items as possible, thereby potentially reducing the costs of carrying unsold stock.

3.1.4 Constraints

The constraints define the set of feasible solutions:

$\sum _{i\in N}{d}_{ij}{x}_i\le q_j,\forall j\in M$ (8)

$y_j\le \frac{1}{q_j}\sum _{i\in N}{d}_{ij}{x}_i-\alpha +1j\in M$ (9)

$y_j\ge \frac{1}{q_j}\sum _{i\in N}{d}_{ij}{x}_i-\alpha j\in M$ (10)

$x_i\in \left\{0,1\right\}\forall i\in N$ (11)

$y_j\in \left\{0,1\right\}\forall j\in M$ (12)

The constraints define the set of feasible solutions as follows. Constraint (8) ensures that the quantity of an item j that is sold through winning bids must not exceed the total quantity q _j . Constraints (9) and (10) ensure the logical relationship y _j =1 if and only if the minimum percentage α of item j is sold: the term $\frac{1}{q_j}\sum _{i\in N}{d}_{ij}{x}_i$ acts as a normalization factor, converting the absolute quantity of item j requested by winning bids into a ratio (ranging from 0 to 1) relative to the total available quantity q _j , and this normalization enables a direct comparison with the threshold parameter α∈[0, 1]; specifically, Constraint (10), $y_j\ge \frac{1}{q_j}\sum _{i\in N}{d}_{ij}{x}_i-\alpha$, forces y _j to 1 as soon as the sold ratio exceeds α, while, conversely, Constraint (9), $y_j\le \frac{1}{q_j}\sum _{i\in N}{d}_{ij}{x}_i-\alpha +1$, ensures that y _j remains 0 if the ratio does not reach the required threshold. Finally, Constraints (11) and (12) define the domain of the decision variables (binary variables).

3.2 Mo-WDP Mathematical Model

The mathematical model considering the previously defined objective is as follows:

$P_{E\left(Mo-WDP\right)}\left\{\begin{array}{c}max\psi \left(x\right)=\sum _{i\in N}{x}_i\\ st\\ Mo-WDP\left\{\begin{array}{c}max{f}_1\left(x\right)=\sum _{i\in N}{P}_i{x}_i\\ max{f}_2\left(x\right)=\sum _{i\in N}{P}_i{x}_i+\sum _{j\in M}{v}_j\left(q_j-\sum _{i\in N}{d}_{ij}{x}_i\right)\\ max{f}_3\left(y\right)=\sum _{j\in M}{y}_j\\ \sum _{i\in N}{d}_{ij}{x}_i\le q_{j}j\in M\\ y_j\le \frac{1}{q_j}\sum _{i\in N}{d}_{ij}{x}_i-\alpha +1j\in M\\ y_j\ge \frac{1}{q_j}\sum _{i\in N}{d}_{ij}{x}_i-\alpha j\in M\\ x_i\in \left\{0,1\right\}i\in N\\ y_j\in \left\{0,1\right\}j\in M\end{array}\right.\end{array}\right.$ (13)

The mathematical model detailed in this paper presents a pivotal advancement in multiobjective Winner Determination Problems (Mo-WDPs) through its innovative twofold approach. First, it significantly expands the scope of the WDP by rigorously incorporating four distinct, yet practically crucial, strategic objectives: beyond traditional revenue maximization, it includes the number of accepted bids, sales diversification, and the intrinsic value of unsold items. This holistic formulation directly addresses the complex, real-world trade-offs faced by auction organizers. Second, and most crucially, our model pioneers a novel optimization strategy: instead of merely identifying the ε-non-dominated set, it explicitly optimizes a decision-maker’s scalar preference function directly over the ε-non-dominated set of Mo-WDPs. This methodology provides a robust solution to the common challenge of selecting a single, definitive best solution from a multitude of optimal compromises, thereby transforming a complex analytical problem into a practical, actionable decision.

4 PROPOSED SOLUTION APPROACH: EDA ALGORITHM

4.1 General Principles of Estimation of Distribution Algorithms

Estimation of Distribution Algorithms (EDAs) (Hauschild & Pelikan, 2011) are a family of evolutionary metaheuristics that distinguish themselves from classical genetic algorithms by replacing traditional crossover and mutation operators with a probabilistic modeling approach. Rather than directly manipulating individual solutions, EDAs build and sample a probability distribution learned from the most promising solutions identified in the current population. At each iteration, the algorithm selects a subset of elite solutions, estimates a probabilistic model that captures the statistical dependencies between decision variables, and then samples this model to generate a new population of candidate solutions. This mechanism allows the algorithm to progressively focus the search toward the most favorable regions of the solution space, guided by the structural information extracted from high-quality solutions. EDAs exhibit several attractive properties that make them particularly suited for complex optimization problems: they possess a strong global search capability, they are highly adaptable to both continuous and discrete problem settings, they maintain a diverse set of candidate solutions to avoid premature convergence to local optima, and they require no prior knowledge about the objective function or the topology of the search space. These characteristics have led to their successful application across a wide range of combinatorial and multiobjective optimization problems, where classical metaheuristics often struggle to efficiently explore high-dimensional and heavily constrained solution spaces (Larranaga & Bielza, 2025).

In the specific context of the Mo-WDP, we opted for a univariate EDA approach for several reasons. First, the problem belongs to the class of multiobjective binary optimization problems, where all decision variables x _i ∈{0, 1} and y _j ∈{0, 1}, which is known to be NP-hard even in its single-objective form. The multiobjective extension further exacerbates this computational intractability, as exact methods such as branch-and-bound or integer programming solvers are limited to very small instances and become prohibitively expensive as the number of bids n and items m grows. This inherent complexity provides a strong motivation for resorting to a meta-heuristic approach, which trades optimality guarantees for computational efficiency and scalability. Second, the primary challenge of our study is the two-fold extension of the optimization process over the ε-non-dominated set, rather than the internal complexity of the probabilistic model itself. Third, univariate models offer significantly lower computational complexity compared to multivariate models, which is crucial when dealing with large-scale auction instances where the number of decision variables can reach several hundreds.

4.2 Adaptation of EDA for the PE(Mo-WDP) Problem

The $ED{A}_{P_{E\left(Mo-WDP\right)}}$ algorithm consists of two main steps. The first step involves constructing solutions for the population P ^t by generating each component of the solution vectors through a binomial probability distribution $B\left(p_i,N_S^t\right)$, where i=1, ..., N. Here, p _i represents the probability of success, for instance by explicitly stating that x _i assumes value 1 with probability p _i and $N_S^t$ is the population size at iteration t. The second step is more challenging as it involves updating the parameters of $B\left(p_i,N_S^t\right)$ for each solution component.

Since our problem is binary, we begin by randomly generating values for each decision variable for all solutions in the initial population P ¹ with p _i =0.5. At each iteration t=0, 1, ..., T-1, the algorithm performs an update of the p _i parameter for each variable x _i using the following steps:

• 1. Eliminate the infeasible solutions S ⁰ of P ^t .

• 2. Select the non-dominated solutions S of P ^t: We use the concept of ε-dominance of Pareto dominance, so that each of the three objectives has a tolerance ε=(ε₁, ε₂, ε₃) allowing it to be dominated.

• 3. For the dominated solutions S ¹ of P ^t , we rank them in a non-increasing order according to the Ψ function value and we select the first (θ|S ¹|) solutions, called S ², where θ represents a regulation rate. This allows us to ensure an adequate size of population P ^t in order to estimate the parameters of the probability distribution that will generate the next population, denoted P ^t+1 . This rate is not fixed, it is initially set to 1 and decreases progressively with a step $\Delta =\frac{\theta }{T}$ in order to keep only the non-dominated solutions in the last population. |S ¹| is calculated as follows:

$\left|S^1\right|=N_S^t-\left|S^0\right|-\left|S\right|$ (14)

• 4. Before proceeding to update the p _i parameters using the formula (15) below, it is necessary to create a new set that includes the non-dominated solutions as well as the selected part of the dominant solutions S ³=S∪S ².

$p_i=\frac{{\displaystyle \sum _{j=1}^{\tilde{N}}}{x}_i^j}{\left|S^3\right|},\forall i\in \left\{1,...,n\right\}$ (15)

• where $x_i^j$ denotes the i-th component (the state of bid i) of the j-th solution vector in S ³.

5 DIDACTIC EXAMPLE

To facilitate the understanding of the approach and its solution algorithm, we solve a conceptual example quantitatively, step by step, using the EDA algorithm over 4 iterations.

5.1 Problem and Parameters

The problem data is as follows:

Consider 6 bids and 3 items

• Items (q_j, v_j): Item 1 (20, 2), Item 2 (5, 50), Item 3 (10, 15).

• Bids (P _i ,d _ij ):

With an α value set to $\frac{1}{3}$, the example model is as follows:

$P_{E\left(Mo-WDP\right)}\left\{\begin{array}{c}max\psi \left(x\right)=x_1+x_2+x_3+x_4+x_5+x_6\\ St\\ Mo-WDP\left\{\begin{array}{c}max{f}_1\left(x\right)=210x_1+70x_2+55x_3+80x_4+20x_5+6x_6\\ max{f}_2\left(y\right)=y_1+y_2+y_3\\ max{f}_3\left(x\right)=440+10x_1-21x_2+5x_3-4x_4+10x_5-2x_6\\ St\\ 8x_2+2x_4+5x_5+4x_6\le 20\\ 4x_1+x_3+x_4\le 5\\ 5x_2+2x_4\le 10\\ y_1-\frac{1}{20}\left(8x_2+2x_4+5x_5+4x_6\right)\le \frac{2}{3}\\ y_2-\frac{1}{5}\left(4x_1+x_3+x_4\right)\le \frac{2}{3}\\ y_3-\frac{1}{10}\left(5x_2+2x_4\right)\le \frac{2}{3}\\ y_1-\frac{1}{20}\left(8x_2+2x_4+5x_5+4x_6\right)\ge \frac{-1}{3}\\ y_2-\frac{1}{5}\left(4x_1+x_3+x_4\right)\ge \frac{-1}{3}\\ y_3-\frac{1}{10}\left(5x_2+2x_4\right)\ge \frac{-1}{3}\\ x_i\in \left\{0,1\right\}i=1,...,6\\ y_j\in \left\{0,1\right\}j=1,...,3\end{array}\right.\end{array}\right.$ (16)

5.2 Steps of solving

We start by initializing the parameters.

N_S=8, T=3, θ=1, ε=(5, 0.1, 20), t=0, p _i =0.5 for i=1, ..., 6.

Table 1
Initial population of candidate solutions P ¹.

Iteration 0: Candidate Population
Sol	Vector x	Vector y	Feasible?	Ψ	f 1	f 2	f 3
s 1	[1,0,0,0,0,0]	[0,1,0]	Yes	1	210	1	450
s 2	[0,1,0,0,1,0]	[1,0,1]	Yes	2	090	2	339
s 3	[0,0,0,1,1,1]	[0,1,0]	Yes	3	106	1	444
s 4	[1,0,1,1,0,0]	[0,1,0]	No	-	-	-	-
s 5	[0,1,0,1,0,0]	[1,0,1]	Yes	2	150	2	415
s 6	[0,0,1,0,1,1]	[1,0,0]	Yes	3	081	1	453
s 7	[0,0,0,0,0,1]	[0,0,0]	Yes	1	006	0	438
s 8	[1,0,0,0,1,0]	[0,1,0]	Yes	2	230	1	460
p i	[0.5, 0.5, 0.5, 0.5, 0.5, 0.5]

Table 2
Selection of ε-non-dominated solutions.

Iteration 0: Elite Selection
Sol	Vector x	Vector y	Feasible?	Ψ	f 1	f 2	f 3
s 5	[0,1,0,1,0,0]	[1,0,1]	Yes	2	150	2	415
s 8	[1,0,0,0,1,0]	[0,1,0]	Yes	2	230	1	460
p i	[0.5, 0.5, 0.5, 0.5, 0.5, 0.5]

Table 3
New population generated based on the updated probability vector p _i .

Iteration 1: Candidate Population
Sol	Vector x	Vector y	Feasible?	Ψ	f 1	f 2	f 3
s 5	[0,1,0,1,0,0]	[1,0,1]	Yes	2	150	2	415
s 8	[1,0,0,0,1,0]	[0,1,0]	Yes	2	230	1	460
s 9	[0,1,0,1,1,0]	[1,0,1]	Yes	3	170	2	425
s 10	[1,1,0,0,1,0]	[1,1,1]	Yes	3	300	3	439
s 11	[1,1,0,1,1,0]	[1,1,1]	Yes	4	380	3	435
s 12	[1,1,0,1,0,0]	[1,1,1]	Yes	3	360	3	425
s 13	[0,0,0,1,0,0]	[0,0,0]	Yes	1	080	0	348
s 14	[1,0,1,1,0,0]	[0,1,0]	No	3	-	-	-
s 15	[1,1,0,0,0,0]	[1,1,1]	Yes	2	280	3	429
s 16	[0,0,0,1,1,0]	[1,0,0]	Yes	2	100	1	446
p i	[0.5, 0.5, 0, 0.5, 0.5, 0]

Table 4
Identification of the ε-non-dominated set.

Iteration 1: Elite Selection
Sol	Vector x	Vector y	Feasible?	Ψ	f 1	f 2	f 3
s 8	[1,0,0,0,1,0]	[0,1,0]	Yes	2	230	1	460
s 11	[1,1,0,1,1,0]	[1,1,1]	Yes	4	380	3	435
p i	[0.5, 0.5, 0, 0.5, 0.5, 0]

Table 5
Candidate solutions generated in the third generation.

Iteration 2: Candidate Population
Sol	Vector x	Vector y	Feasible?	Ψ	f 1	f 2	f 3
s 8	[1,0,0,0,1,0]	[0,1,0]	Yes	2	230	1	460
s 11	[1,1,0,1,1,0]	[1,1,1]	Yes	4	380	3	435
s 17	[1,0,0,0,1,0]	[0,1,0]	Yes	2	230	1	460
s 18	[1,1,0,0,1,0]	[1,1,1]	Yes	3	300	3	439
s 19	[1,0,0,1,1,0]	[1,1,0]	Yes	3	310	2	456
s 20	[1,1,0,1,1,0]	[1,1,1]	Yes	4	380	3	435
s 21	[1,0,0,1,1,0]	[1,1,0]	Yes	3	310	2	456
s 22	[1,0,0,0,1,0]	[0,1,0]	Yes	2	230	1	460
s 23	[1,1,0,0,1,0]	[1,1,1]	Yes	2	180	3	429
s 24	[1,0,0,1,1,0]	[1,0,0]	Yes	2	100	1	446
p i	[1, 0.5, 0, 0.5, 1, 0]

Table 6
Updated set of non-dominated solutions.

Iteration 2: Elite Selection
Sol	Vector x	Vector y	Feasible?	Ψ	f 1	f 2	f 3
s 11	[1,1,0,1,1,0]	[1,1,1]	Yes	4	380	3	435
s 19	[1,0,0,1,1,0]	[1,1,0]	Yes	3	310	2	456
p i	[1, 0.5, 0, 0.5, 1, 0]

Table 7
Final population generated by the EDA.

Iteration 3: Final Population
Sol	Vector x	Vector y	Feasible?	Ψ	f 1	f 2	f 3
s 11	[1,1,0,1,1,0]	[1,1,1]	Yes	4	380	3	435
s 19	[1,0,0,1,1,0]	[1,1,0]	Yes	3	310	2	456
s 25	[1,0,0,1,1,0]	[0,1,0]	Yes	3	230	1	460
s 26	[1,1,0,1,1,0]	[1,1,1]	Yes	4	300	3	439
s 27	[1,0,0,1,1,0]	[1,1,0]	Yes	3	310	2	456
s 28	[1,1,0,1,1,0]	[1,1,1]	Yes	4	380	3	435
s 29	[1,0,0,1,1,0]	[1,1,0]	Yes	3	310	2	456
s 30	[1,1,0,1,1,0]	[0,1,0]	Yes	2	230	1	460
s 31	[1,1,0,1,1,0]	[1,1,1]	Yes	2	180	3	429
s 32	[1,0,0,1,1,0]	[1,0,0]	Yes	2	100	1	446
p i	[1, 0.5, 0, 1, 1, 0]

Table 8
Final ε-non-dominated set.

Iteration 3: Final Elite Set
Sol	Vector x	Vector y	Feasible?	Ψ	f 1	f 2	f 3
s11	[1,1,0,1,1,0]	[1,1,1]	Yes	4	380	3	435
s19	[1,0,0,1,1,0]	[1,1,0]	Yes	3	310	2	456
p i	[1, 0.5, 0, 1, 1, 0]

After solving the Mo-WDP problem, the solution is as follows: s ₁₁.

The illustrative example considers a combinatorial auction problem consisting of 6 bids and 3 items, with an initial population of size N _S =8 and a number of iterations T=3. Initially (iteration 0), the probabilities are uniformly initialized to p _i =0.5 for each decision variable, and the randomly generated population produces 7 feasible solutions out of 8, among which two ε-non-dominated solutions are identified: s ₅ and s ₈. From iteration 1 onwards, the update of the probabilities based on the retained non-dominated solutions allows the algorithm to generate a significantly richer population, within which the solution s ₁₁=[1, 1, 0, 1, 1, 0] appears for the first time, achieving the values Ψ=4, f ₁=380, and f ₃=3, and which will dominate all subsequent generations. At iteration 2, the probabilities p ₁ and p ₅ converge to 1 while p ₃ and p ₆ vanish, reflecting the progressive learning of the structure of high-quality solutions by the probabilistic model. The algorithm then identifies two stable non-dominated solutions: s ₁₁, which maximizes the number of accepted bids and the total revenue, and s ₁₉=[1, 0, 0, 1, 1, 0], which offers a slightly higher residual value of unsold items. At iteration 3 (the last one), the generated population concentrates heavily around the vectors [1, 1, 0, 1, 1, 0] and [1, 0, 0, 1, 1, 0], confirming the convergence of the algorithm toward these two compromise solutions. At the end of the process, solution s ₁₁ is selected as the final solution, as it maximizes the preference function Ψ over the ε-non-dominated set, accepting 4 out of the 6 submitted bids, covering all 3 available items, and generating a total revenue of 380.

6 NUMERICAL EXPERIMENTATION

6.1 Performance and convergence of the EDA

In order to evaluate the efficiency and robustness of our approach, we conducted extensive numerical experiments. We generated a benchmark consisting of 70 problem instances, structured into 7 distinct categories based on their dimensions and complexity. Each category contains 10 randomly generated instances to ensure statistical significance.

Full instance data are provided in Table 9 and are publicly available at: https://doi.org/10.5281/zenodo.20764109. The problem size scales from small-scale (10×15) to large-scale (800×1000), where m represents the number of items (constraints) and n represents the number of bids (decision variables). This variety of scenarios, ranging from low to high constraint density, allows for a comprehensive assessment of the algorithm’s performance across different levels of NP-hard complexity.

Table 9
Computational results.

Instance (m×n)	CPU Time (seconds)					Hypervolume HV×10 10	Progress Metric (PM)
	Mean	CV (%)	CI	Min	Max		PM f1	PM f2	PM f3
10×15	1.107	8.0	[1.08, 1.16]	1.06	1.20	0.00172	0.035	0.030	0.029
20×30	4.283	14.7	[4.14, 4.52]	4.02	4.65	0.03194	0.050	0.032	0.048
50×70	6.160	9.1	[5.98, 6.25]	5.86	6.42	0.622	0.032	0.027	0.040
70×100	7.810	9.7	[7.60, 7.98]	7.38	8.14	1.852	0.029	0.029	0.026
120×150	12.960	15.7	[12.50, 13.80]	12.07	14.11	10.386	0.031	0.027	0.052
300×500	346.89	2.7	[345.2, 348.9]	342.04	351.25	272.7	0.062	0.055	0.061
800×1000	1805.2	5.8	[1756, 1831]	1729	1834	4542.2	0.057	0.053	0.032
Summary Statistics (Mean±SD)						0.042±0.01	0.036±0.01	0.041±0.01
Wilcoxon signed-rank test (p-value)						0.0078***	0.0078***	0.0078***

Our algorithm was implemented in Python and executed on a machine equipped with an Intel i7 1.8 GHz processor and 8 GB of RAM, running Ubuntu 22.04.

To ensure the reproducibility of our results, the parameter settings used for the EDA in all benchmark experiments are detailed as follows. The population size is set to N _S =1000, and the maximum number of iterations is T=500. The initial regulation rate is set to θ=1, $\alpha =\frac{1}{3}$, and the initial probability for each decision variable is initialized to p _i =0.5 for all i∈{1,..., n}.

To maintain consistency and fairness across instances of varying sizes and price scales, the tolerance thresholds for the ε-dominance relation are dynamically computed relative to the average coefficients of the objective functions:

• ${\epsilon }_1=0.01\times \frac{1}{n}\sum _{i=1}^n{P}_i$, which represents 1% of the average price of the submitted bids.

• ${\epsilon }_2=0.2\times \frac{1}{n}\sum _{i=1}^n\left(P_i-\sum _{j=1}^m{v}_j{d}_{ij}\right)$, representing 20% of the average net value coefficient associated with the bids in f₂(x).

• ε₃=0.2, which is a static threshold appropriate for the integer-valued sales diversification objective f ₃(y).

To rigorously assess the quality, diversity, and convergence of the solutions obtained under these parameter configurations, we utilize two widely recognized multiobjective performance metrics, which are defined below.

• • Hypervolume indicator (HV), also known as the S-metric, is one of the most widely used metrics in multiobjective optimization because it is strictly monotonic with respect to Pareto dominance. Formally, it measures the Lebesgue measure of the portion of the objective space that is dominated by the set of non-dominated solutions S and bounded by a reference point r.

• In this study, since we evaluate the Pareto front based on three objectives (f ₁, f ₂, f ₃), the reference point r=(r ₁, r ₂, r ₃) belongs to ℝ³. The notation r≤s denotes a component-wise inequality (i.e., r _i ≤s _i for all i∈{1, 2, 3}), ensuring that the reference point is dominated by all solutions in the set.

$H{V}_{3D}\left(s,r\right)=\left(r_1-f_1^{\left(1\right)}\right)\left(r_2-f_2^{\left(1\right)}\right)\left(r_3-f_3^{\left(1\right)}\right)+\sum _{i=2}^k\left(f_1^{\left(i-1\right)}-f_1^{\left(i\right)}\right)\times H{V}_{2D}\left(\left\{s^1,...,s^{i-1}\right\}\right)$

• The formula above calculates the 3D hypervolume by decomposing the volume into k slices along the first objective. The parameters are defined as follows:

• - k: represents the total number of non-dominated solutions found in the Pareto set S.

• - s ¹, ..., s ^k: are the non-dominated solutions sorted in non-increasing order according to the first objective (f ₁).

• - HV _2D (·): denotes the 2D hypervolume (the area) of the projection of the solutions onto the plane formed by the remaining objectives (f ₂ and f ₃).

• This recursive decomposition allows for an accurate measurement of the space ”covered” by the solutions relative to the ”worst-case” boundary defined by r.

• • Progress Metric (PM) is the indicator that measures the progress of the solutions of objectives between the iterations 0 and T.

$P{M}_{fi}=\ln \sqrt{\frac{max{f}_i^T}{max{f}_i^0}}.$

The Table 9 is structured to provide a detailed overview of several key metrics:

• CPU Time (seconds): This section includes the average execution time (Mean), the Coefficient of Variation (CV%), the 95% Bootstrap Confidence Interval (CI), and the extreme values (Min and Max) recorded for each category.

• Hypervolume (HV): The magnitude of the objective space dominated by the obtained ε-non-dominated sets is reported as HV×10¹⁰.

• Progress Metric (PM): The convergence progress is detailed for each of the three primary objectives (PM _f1 , PM _f2 , PM _f3 ).

Additionally, the bottom rows of the table provide summary statistics (Mean±SD) for the Progress Metric series. The results of the Wilcoxon signed-rank test are also included, where the *** notation indicates that the observed progress is statistically significant at the p<0.01 level and SD is the standard deviation.

To visualize the stability of CPU time in Figure 1, we used the Z-score normalization method. For each category, the raw data (Min, Mean, Max) were centered by subtracting the mean and scaled by dividing by the standard deviation. This transformation allows us to compare the relative variation of all categories on a single scale, regardless of whether the execution time is 1 second or 1800 seconds. It clearly demonstrates that the algorithm’s dispersion remains very low across all problem sizes.

Figure 1
CPU Time centered and standardized.

The analysis of computation time reveals remarkable stability of the EDA algorithm across all seven tested instance categories. The coefficient of variation (CV) remains below 16% for all categories, and falls below 10% for the four largest instances. The 95% bootstrap confidence intervals, constructed without any distributional assumption (10 000 resamples), are narrow relative to their respective means: the ratio of CI half-width to mean never exceeds 4%, confirming that the reported mean values are reliable estimates of the true expected runtime. A Spearman rank correlation between problem size and mean CPU time yields ρ=1.00 (p<0.0001), confirming a perfectly monotone relationship. A log-log regression reveals an empirical complexity scaling exponent of approximately 1.70 (R ²=0.998), indicating polynomial rather than exponential growth of computation time. This property is particularly favourable for a metaheuristic applied to NP-hard problems. Taken together, these results demonstrate that the runtime behaviour of the EDA is predictable, reproducible and scalable, regardless of the random variability of the generated instances.

The Hypervolume indicator HV assesses the quality of the produced Pareto front by measuring the volume of objective space dominated by the non-dominated solutions obtained. The HV values grow substantially with problem size, ranging from 0.00172×10¹⁰ for small instances (10×15) up to 4542.2×10¹⁰ for large ones (800×1000). A Spearman rank correlation between problem size and HV yields ρ=1.00 (p<0.0001), reflecting a perfectly monotone relationship. A log-log regression confirms a power law with exponent 1.70 (R ²=0.998), showing that front quality grows in a structured and proportional manner with problem scale. This result rules out the possibility that good performance is an artefact specific to particular instances. The homo-geneity of point distribution within the front, inferred from this monotone growth, attests that the ε-dominance mechanism embedded in the EDA effectively prevents premature convergence toward a narrow portion of the Pareto front. The quality of the solutions is therefore robust and systematic across all tested scales.

The Progress Metric PM measures the logarithmic improvement of each objective between the initial iteration and the final iteration T. All 21 PM values (7 instances×3 objectives) are strictly positive, a necessary condition for asserting convergence. To go beyond this qualitative observation, the one-sided non-parametric Wilcoxon signed-rank test was applied to each of the three PM series against the null hypothesis H ₀:PM≤0, yielding p=0.0078 for f ₁, f ₂ and f ₃, thereby enabling rejection of H ₀ at the 1% significance level. The 95% bootstrap confidence intervals for the mean PM are entirely contained in the strictly positive half of the real line: [0.033, 0.052] for f ₁, [0.028, 0.044] for f ₂, and [0.033, 0.050] for f ₃, further reinforcing this conclusion. A Kruskal-Wallis test comparing the PM distributions of the three objectives yields H=1.28 and p=0.53, indicating that no objective is favoured at the expense of the others. The multiobjective search is therefore balanced between f ₁, f ₂ and f ₃. Collectively, these results rigorously demonstrate that the EDA converges reliably and uniformly across all objectives, validating its status as an effective decision-support tool for auction organisers.

6.2 Comparison of our approach with the classical single-objective WDP

We provide in this subsection a direct empirical comparison between two formulations of the Winner Determination Problem on twelve dedicated test instances. The first formulation, hereafter referred to as Method A, is the classical revenue-maximizing WDP. It consists in solving

$P_A\left\{\begin{array}{c}max{f}_1\left(x\right)=\sum _{i\in N}{P}_i{x}_i\\ St\\ \sum _{i\in N}{d}_{ij}{x}_i\le q_j,\forall j\in M\\ x_i\in \left\{0,1\right\}i=1,...,N\end{array}\right.$ (17)

The second formulation, denoted Method B, is the model P _E(Mo-WDP) introduced in this paper, which selects the allocation that maximizes Ψ over the Pareto-efficient set associated with the three criteria (f ₁, f ₂, f ₃).

With α=1/3 and ε=0 (strict dominance), Table 10 report the results for the four criteria (Ψ, f ₁, f ₂, f ₃) obtained by both methods, along with the relative gains and losses of Method B with respect to Method A.

Table 10
Comparison of Method B with Method A.

Inst.	m×n	Method A				Method B				Variation B vs A
		Ψ	f 1	f 2	f 3	Ψ	f 1	f 2	f 3	∆Ψ	∆f 1 %	∆f 2 %	∆f 3
I1	10×15	8	1159	2358	8	11	1083	2504	8	+3	-6.6	+6.2	+0
I2	10×15	9	1083	1844	8	10	918	2219	8	+1	-15.2	+20.3	+0
I3	10×15	8	952	1856	7	9	805	1971	8	+1	-15.4	+6.2	+1
I4	10×15	11	1070	1851	10	11	1070	1851	10	+0	+0.0	+0.0	+0
I5	10×15	6	1096	2314	7	10	1007	2391	8	+4	-8.1	+3.3	+1
I6	10×15	8	773	1565	7	9	692	1830	9	+1	-10.5	+16.9	+2
I7	10×15	8	1076	2154	9	10	1011	2318	9	+2	-6.0	+7.6	+0
I8	10×15	7	1135	2058	8	8	873	2553	8	+1	-23.1	+24.1	+0
I9	10×15	9	1039	1541	9	9	1039	1541	9	+0	+0.0	+0.0	+0
L1	20×30	18	2116	4626	16	21	2013	4446	16	+3	-4.9	-3.9	+0
L2	20×30	19	2050	3397	18	21	1912	3467	18	+2	-6.7	+2.1	+0
L3	20×30	16	2310	4300	16	20	1847	4110	17	+4	-20.0	-4.4	+1
LL1	300×500	288	33518	57057	239	329	29238	57982	254	+41	-12.8	+1.6	+15
LL2	300×500	277	31886	58905	239	328	27661	61188	249	+51	-13.3	+3.9	+10
LL3	300×500	286	33103	62840	233	330	28203	63188	258	+44	-14.8	+0.6	+25
LL4	300×500	291	32846	61407	235	331	28391	62495	248	+40	-13.6	+1.8	+13
LL5	300×500	278	31953	60418	237	328	27510	61234	254	+50	-13.9	+1.3	+17

Table 10 reveals a robust and economically interpretable trade-off pattern between Method A and Method B across the seventeen instances. The two methods coincide on only two small configurations (I4 and I9), confirming that the proposed model agrees with the classical baseline whenever no genuine multi-criteria conflict is present in the data; on all the other fifteen instances, including all five of the large-scale 300×500 problems, Method B produces strictly different and economically superior allocations. The revenue cost paid by Method B is consistently moderate: ∆f ₁ never exceeds -23.1% for the whole benchmark and is bounded between -14.8% and -12.8% on every large instance, a remarkably stable loss of approximately 13% irrespective of the problem size. In exchange, the three other criteria improve substantially and simultaneously. The number of accepted bids Ψ rises by+40 to+51 winning bidders on the 300×500 instances, which corresponds to a relative gain ranging from+13.7% to+18.4% in bidder satisfaction; the total portfolio value f ₂ is preserved or increased on thirteen of the fifteen non-coincident instances, with gains of up to+24.1% on instance I8, and is only marginally reduced (by less than 5%) on the two intermediate-scale instances L1 and L3; and the diversification objective f ₃ improves on all five large-scale instances, with up to+25 additional items meeting the sales threshold. On every 300×500 instance, the relative gain on Ψ is larger than the relative loss on f ₁, so a single percentage point of revenue conceded by Method A unlocks more than one percentage point of additional bidder satisfaction together with positive movements on f ₂ and f ₃. Overall, the proposed model exchanges a controlled revenue concession against a four-fold improvement profile that the classical single-objective formulation cannot detect, and this advantage becomes proportionally stronger as the size of the auction increases.

The complete data for the seventeen test instances used in this comparison are publicly available at https://doi.org/10.5281/zenodo.20764109.

7 CONCLUSION

In this study, we have introduced a novel approach to the Winner Determination Problem (WDP) by formulating it as a multiobjective optimization problem. Breaking from traditional models that focus solely on revenue maximization, our model incorporates a richer and more pragmatic set of four critical objectives: increasing the number of winning bidders, diversifying sales, and considering the intrinsic value of unsold goods, in addition to total revenue. This holistic approach more accurately reflects the complex trade-offs faced by auction organizers in real-world scenarios.

The primary challenge of selecting a single solution from a potentially infinite set of Pareto-optimal trade-offs was addressed through an optimization method over the ε-non-dominated set. We implemented an Estimation of Distribution Algorithm (EDA) to solve this problem, lever-aging its ability to efficiently explore complex search spaces and model the interdependencies between decision variables.

The numerical experiments conducted on a set of 70 instances of varying sizes have validated the efficiency and robustness of our algorithm. The results demonstrate high stability in computational time and consistent convergence towards high-quality solutions, as indicated by the Hypervolume (HV) and Progress Metric (PM) performance indicators. The proposed approach successfully transforms a complex analytical problem into a practical and actionable decision for the decision-maker.

Future work could explore the integration of additional operational constraints, such as limited bidder budgets or logistical constraints. Furthermore, comparing the performance of the EDA with other metaheuristics or exact methods on very large-scale instances could help to better define the limits and advantages of our approach. Finally, the development of interactive decision-support tools based on this model could greatly facilitate the exploration of trade-offs by decision-makers.

Data Availability

The data that support the findings of this study are available from the link: https://doi.org/10.5281/zenodo.20764109.

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• Funding
  The authors declare no funding was received for this work.

Edited by

• Editor responsible for the review
  Editor-in-Chief: Antônio Augusto Chaves.

Publication Dates

• Publication in this collection
  28 Aug 2026
• Date of issue
  2026

History

• Received
  16 Oct 2025
• Accepted
  11 June 2026