Bi-Objective Optimization for Transportation: Generating Near-Optimal Subsets of Pareto Optimal Solutions

Abstract

Bi-objective optimization seeks to obtain Pareto optimal solutions that balance two trade-off objectives, providing guidance for decision making in various fields, particularly in the field of transportation. The novelty of this study lies in two aspects. On the one hand, considering that Pareto optimal solutions are often numerous, finding the full set of Pareto optimal solutions is often computationally challenging and unnecessary for practical purposes. Therefore, we shift the focus of bi-objective optimization to finding a subset of Pareto optimal solutions whose resulting set of nondominated objective vectors is the same as, or at least a good approximation of, the full set of nondominated objective vectors for the problem. In particular, we elaborate three methods for generating a near-optimal subset of Pareto optimal solutions, including the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, and the augmented $ϵ$-constraint method. More importantly, the near-optimality of the Pareto optimal solution subset obtained by these methods is rigorously analyzed and proved from a mathematical point of view. This study helps to offer theoretical support for future studies to find the subset of Pareto optimal solutions, which reduces the unnecessary workload and improves the efficiency of solving bi-objective optimization problems while guaranteeing a pre-specified tolerance level.

1. Introduction

Many decision problems related to transportation involve optimizing two or even multiple objectives simultaneously to better simulate the real-world scenarios [1,2,3]. For instance, a logistics company aims to maximize customer satisfaction with its delivery services while minimizing logistics costs [4]. Bi-objective optimization is a specialized type of multi-objective optimization, aiming to identify solutions that satisfy two trade-off objective functions to the maximum extent within a feasible set of decision variables. Compared with the optimization problems with a single objective, the bi-objective optimization problems are more challenging [5].

Generally, in bi-objective optimization problems, it may be difficult for a solution to achieve high performance in both objective dimensions. Thus, bi-objective optimization seeks to balance two conflicting objectives as much as possible while meeting all constraints [6,7]. As a result, the optimal solutions for bi-objective optimization problems are typically a set of compromise solutions, that is, Pareto optimal solutions, also referred to as efficient solutions, nondominated solutions, or noninferior solutions [8,9].

In the context of bi-objective optimization, this study investigates whether finding the full set of Pareto optimal solutions is an excessive requirement and how to identify an appropriate subset of Pareto optimal solutions. The main contributions of this study address the two questions. First, this study demonstrates that it is not necessary to find the full set of Pareto optimal solutions for practical purposes; instead, the focus should be on finding a subset of Pareto optimal solutions whose resulting set of nondominated objective vectors is the same as, or at least a good approximation of, the full set of nondominated objective vectors for the problem. As a result, the workload of solving bi-objective problems is significantly reduced. Second, we provide a detailed and rigorous justification of the methods for generating a near-optimal subset of Pareto optimal solutions, including the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, and the augmented $ϵ$-constraint method. We also present strict mathematical proofs for the proposed lemmas, propositions, and corollaries.

The remainder of this study is organized as follows. In Section 3, we introduce some basic concepts of bi-objective optimization and propose the idea of adopting a subset of Pareto optimal solutions to solve bi-objective optimization problems. In Section 4, we analyze the near-optimality of the Pareto optimal solution subset and provide rigorous mathematical proofs. In Section 5, computational results are carried out based on a real-world bi-objective optimization problem to validate the effectiveness and efficiency of our methods. Section 6 summarizes the main findings of this study and discusses potential directions for future studies.

2. Literature Review

Our literature review focuses on two key aspects: the methodologies of bi-objective optimization and the applications of bi-objective optimization, providing readers with a thorough understanding of both theoretical frameworks and practical implementations of bi-objective optimization.

The first research stream is related to the methodologies for solving bi-objective optimization problems, which are summarized in

Table 1. Summary of studies related to methodologies for bi-objective optimization problems.

Literature	Methods	Objectives
[15]	The weighting method	Minimize total operational costs and time penalty costs
[17]	The $ϵ$-constraint method	Effectively identifying high risk ship and retaining the original ship risk profile framework
[18]	The $ϵ$-constraint method	Minimize transportation risks and costs
[19]	The $ϵ$-constraint method	Minimize transportation risks and costs
[20]	The lexicographic method	Minimize transportation risks and costs
[21]	The global criterion method	Minimize transportation costs of retailers and maximize profits of distributors

. A mainstream way of dealing with two objectives in the case that the unit of neither objective can easily be converted into the unit of the other objective is multi-objective optimization methods [10], such as the weighting method and the $ϵ$-constraint method [11,12]. The weighting method, a classic approach for solving multi-objective optimization problems, involves assigning an appropriate weighting coefficient to each objective function and optimizing the linear combination of these weighted objectives [13,14]. Hence, the multiple objective functions are transformed into a single objective function. Zhao et al. [15] addressed a vehicle routing problem of a mixed fleet of conventional and electric vehicles, aiming to minimize total operational costs and time penalty costs. To solve this problem, a linear weighting method was adopted to transform the bi-objective programming model into a single-objective programming model. The weighting method, however, has several pitfalls. A major pitfall is that nonsupported efficient solutions, which are dominated by the convex combination of two or more other Pareto optimal solutions, are unable to be generated, even though these solutions are still Pareto optimal. Another pitfall of the weighting method is that the “quality” of the subset of Pareto optimal solutions generated is hard to control or quantify. In other words, whether the subset of Pareto optimal solutions generated is a “good representation” of the full set of Pareto optimal solutions is unknown. The $ϵ$-constraint method is also a primary approach for solving multi-objective optimization problems. The fundamental principle of the $ϵ$-constraint method is to optimize a single objective while transforming the remaining objectives into constraints by setting an upper bound for them [16]. Yan et al. [17] focused on the optimization of the thresholds in the ship risk profile, which is a method used for quantifying ship risk. A bi-objective mixed-integer nonlinear model was formulated. To facilitate linearization, a big-M method was adopted to obtain a linear model. Additionally, an $ϵ$-constraint method was implemented to obtain all Pareto optimal solutions. Wang et al. [18] proposed an $ϵ$-constrained multi-objective optimization model considering road network data, population density, and building characteristics to optimize transportation routes, thus reducing transportation risks and costs. Hassanpour et al. [19] focused on a location-routing problem for hazardous materials, and formulated a bi-objective optimization model aiming to minimize transportation costs and risks, which was solved by an augmented $ϵ$-constraint method. A real-world case study indicated that the decrease in warehouse capacity led to more costs and higher risks. With the development of multi-objective optimization, several other well-known techniques, such as the lexicographic method [20] and the global criterion method [21], have been introduced and continuously improved. Interested readers can refer to Stewart et al. [22] and Pereira et al. [23] for a comprehensive understanding. Moreover, emerging technologies, such as machine learning, are increasingly applied to the field of multi-objective optimization in recent years [24].

The second research stream is relevant to the applications of bi-objective optimization. Bi-objective optimization has a wide range of applications. Readers can refer to Zajac and Huber [1], Daş et al. [25], and Tadaros and Migdalas [26] for a comprehensive overview. In the field of transportation, Mousavi et al. [27] proposed a mixed-integer linear programming (MILP) model for the bi-objective optimization of public transportation network timetables and vehicle scheduling. The MILP model simultaneously optimized two objectives: the primary objective of reducing the waiting time of passengers, as well as the secondary objective of minimizing the total costs of the public transportation network. The $ϵ$-constraint method was used to solve small-scale problems, in order to validate the effectiveness of the meta-heuristic algorithms designed in their study. Apart from the transportation field, bi-objective optimization is also utilized in other fields, such as the supply chain management field, the marketing field, and the engineering field. In the field of supply chain management, Lei et al. [28] addressed the supply chain network design problem, aiming to minimize both total supply chain costs and environmental effluents. A bi-objective MILP model was proposed, where the objective of reducing environmental effluents was cleverly transformed into reducing fuel consumption, which was easier to quantify. Additionally, they combined a specified fuzzy decision method with the iterative-based $ϵ$-constraint method to identify Pareto optimal solutions that contributed to building an efficient and environmentally friendly supply chain network. In the marketing field, Tan and Guo [29] focused on the bi-objective capacitated assortment planning problem, with the aim of balancing the expected revenue and the expected market share, and designed a multinomial logit choice model. Experimental results showed that the market share could increase significantly without substantially diminishing the total revenue when the trade-off parameter was set at a lower level. In the engineering field, Guerra-Gómez et al. [30] used the nondominated sorting genetic algorithm (NSGA-II) to determine the optimal sizes for analog integrated circuits. Given the high sensitivity of these optimal sizes to process variations, they conducted a multi-parameter sensitivity analysis on all feasible solutions in the Pareto front to identify those feasible solutions with low sensitivities. As mentioned above, bi-objective optimization is widely applied in various fields, indicating the significance and potential value of further exploration into the underlying theories related to bi-objective optimization.

Although previous studies have explored the methodologies for solving bi-objective optimization problems and the applications of bi-objective optimization in different fields, the existing studies have seldom indicated whether a reasonable subset of Pareto optimal solutions is sufficient to solve bi-objective optimization problems. To address this, this study proposes that it is feasible to use a subset of Pareto optimal solutions as an approximation of the full set of Pareto optimal solutions, and explores the theoretical foundation for it. Additionally, three methods for generating a near-optimal subset of Pareto optimal solutions, including the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, and the augmented $ϵ$-constraint method, are elaborated in this study. Furthermore, the near-optimality of the Pareto optimal solution subset obtained by these methods is rigorously proved.

3. Overview and Insights on Bi-Objective Optimization

Bi-objective optimization problems aim to find a set of Pareto optimal solutions that balance two trade-off objectives, providing scientific guidance for decision makers. For ease of elaboration in the subsequent content, a typical bi-objective optimization model (1) with both objectives to be minimized is introduced.

$$\left\{\begin{array}{c}\underset{\mathit{x}\in X}{min}{f}_1(\mathit{x})\hfill \\ \underset{\mathit{x}\in X}{min}{f}_2(\mathit{x}),\hfill \end{array}\right.$$

(1)

where X is a nonempty compact set and both $f_1(\mathit{x})$ and $f_2(\mathit{x})$ are lower semi-continuous over X. $(f_1(\mathit{x}),f_2(\mathit{x}))$ is an objective vector and it is a feasible objective vector if $\mathit{x}\in X$. The set of all feasible objective vectors represented by $\mathbb{Y}=\left\{(f_1(\mathit{x}),f_2(\mathit{x}))\right|\mathit{x}\in X\}$ is the feasible region in the criteria space. An objective vector $\overline{\mathit{y}}\in \mathbb{Y}$ is nondominated if there does not exist another $\mathit{y}\in \mathbb{Y}$ such that $y_i\le {\overline{y}}_i$ for each $i\in \{1,2\}$ and $y_i<{\overline{y}}_i$ for at least one i. A Pareto optimal solution is a feasible solution with a nondominated objective vector. In other words, the Pareto optimal solutions are the solutions that cannot be improved in one objective function without deteriorating their performance in at least one of the rest [31]. The set of Pareto optimal solutions, denoted by $\mathsf{\Phi }=\{\mathit{x}\in X|(f_1(\mathit{x}),f_2(\mathit{x}))\in \mathbb{Y}\}$, is called the Pareto frontier.

A natural approach for multi-objective optimization is to generate the whole efficient frontier and then decide about the preferred solution in a higher-level decision-making process. However, we find that generating a partial efficient frontier is sufficient to deal with the multi-objective optimization problem. There are three reasons that we generate only a subset of Pareto optimal solutions. First, the full set of Pareto optimal solutions is often too large, e.g., it may contain 10,000 Pareto optimal solutions. It will then be hard for the decision makers to choose one solution. Second, the full set of Pareto optimal solutions may contain an infinite number of solutions, or, even if the cardinality of the full set (i.e., the number of elements in the full set) is finite, it may be computationally challenging to generate all solutions in the full set. Third, some Pareto optimal solutions have the same objective vector and we only need one of them.

It is worth noting that the subset of Pareto optimal solutions generated is not arbitrary. In fact, we aim to generate a subset of Pareto optimal solutions whose resulting set of nondominated objective vectors is the same as, or at least a good approximation of, the full set of nondominated objective vectors for the problem. For instance, in a bi-objective optimization problem that aims to minimize transportation time and fuel consumption, suppose that the full set of nondominated objective vectors is composed of $(38,275)$, $(38.1,274.5)$, $(38.2,274)$, $(50,220)$, and $(50.1,215)$, where the first element of each objective vector represents the hours required for transporting cargo, and the second element represents the fuel consumption in liters. Since several nondominated objective vectors are close in value, a subset consisting of Pareto optimal solutions with objective vectors $(38.2,274)$ and $(50.1,215)$ can be used as a good approximation of the full set of Pareto optimal solutions.

4. Methods for Generating a Near-Optimal Subset of Pareto Optimal Solutions

In this section, we first introduce the basic $ϵ$-constraint method. Then, we elaborate three methods for generating a near-optimal subset of Pareto optimal solutions, including the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, and the augmented $ϵ$-constraint method. More importantly, the near-optimality of the subset generated by the latter three methods is rigorously justified and proved.

4.1. Basic $ϵ$-Constraint Method

The basic $ϵ$-constraint method treats one objective function as a constraint and optimizes the other objective function. In addition, the basic $ϵ$-constraint method can generate nonsupported Pareto optimal solutions, which are dominated by the convex combination of two or more other efficient solutions. A detailed introduction to the basic $ϵ$-constraint method is as follows.

Based on the bi-objective optimization model (1), let $f_1^{min}:={min}_{\mathit{x}\in X}{f}_1(\mathit{x})$ and $f_2^{min}:={min}_{\mathit{x}\in X}{f}_2(\mathit{x})$. Let $f_1^{max}:=min{f}_1(\mathit{x})$ subject to $f_2(\mathit{x})\le f_2^{min},\mathit{x}\in X$. Let $ϵ$ be a small positive number representing the tolerance specified by the decision makers. For ease of exposition, suppose $f_1^{max}-f_1^{min}$ is an integer multiple of $ϵ$. We solve the following single-objective optimization model by varying k from 0, 1, …, all the way to $(f_1^{max}-f_1^{min})/ϵ$:

$$[\mathrm{M}1(k)]:min{f}_2(\mathit{x})$$

(2)

subject to

$$\begin{array}{ccc}\hfill f_1(\mathit{x})& \le & f_1^{min}+kϵ\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \mathit{x}& \in & X.\hfill \end{array}$$

(4)

Solutions with the same objective vector should be discarded and only one of them should be retained. Then, we are able to obtain an approximate Pareto frontier.

It is important to note that a deficiency of the basic $ϵ$-constraint method is that a large number of weakly efficient solutions may be generated. A weakly efficient solution is one whose objective functions cannot be improved simultaneously. Specifically, weakly efficient solutions that allow one objective function to be improved while keeping the other objective functions unaffected are non-Pareto optimal solutions. Thus, a weakly efficient solution is not necessarily Pareto optimal. To address this, additional steps are required to remove weakly efficient solutions that are non-Pareto optimal, through pairwise comparison of the obtained solutions. The pseudocode for the basic $ϵ$-constraint method is shown in Algorithm 1.

Algorithm 1 Basic $ϵ$-constraint method

1: $f_1^{min}$←${min}_{\mathit{x}\in X}{f}_1(\mathit{x})$ // $f_1^{min}$ represents the minimum objective value of the first dimension.   2: $f_2^{min}$←${min}_{\mathit{x}\in X}{f}_2(\mathit{x})$ // $f_2^{min}$ represents the minimum objective value of the second dimension.   3: $f_1^{max}$←$min{f}_1(\mathit{x})$ subject to $f_2(\mathit{x})\le f_2^{min},\mathit{x}\in X$ // $f_1^{max}$ represents the minimum objective value of the first dimension when the second dimension achieves its minimum objective value.   4: $ϵ$← a small positive number such that $f_1^{max}-f_1^{min}$ is an integer multiple of $ϵ$ // $ϵ$ represents the tolerance.   5: $\mathit{x}(k)$← null, $k\in \{0,1,\dots ,(f_1^{max}-f_1^{min})/ϵ\}$ // $\mathit{x}(k)$ represents an optimal solution obtained by the model [M1$(k)$].   6: for  $k\in \{0,1,\dots ,(f_1^{max}-f_1^{min})/ϵ\}$ do   7:        Solve the model [M1$(k)$], and obtain an optimal solution $\mathit{x}(k)$.   8: end for   9: Discard solutions $\mathit{x}(k)$ for $k\in \{0,1,\dots ,(f_1^{max}-f_1^{min})/ϵ\}$ with the same objective vector and retain only one of them. 10: Remove weakly efficient solutions that are non-Pareto optimal through pairwise comparison of the obtained solutions.

However, if a weakly efficient solution that is non-Pareto optimal cannot be dominated by other solutions in the obtained solution set, this weakly efficient solution cannot be eliminated. Therefore, the solutions obtained by the basic $ϵ$-constraint method may be non-Pareto optimal, which is another deficiency of the basic $ϵ$-constraint method.

4.2. Revised $ϵ$-Constraint Method

Considering that numerous weakly efficient solutions may be generated by the basic $ϵ$-constraint method, a revised $ϵ$-constraint method is introduced to address the drawback of the basic version.

We solve the following single-objective optimization models by varying k from 0, 1,…, all the way to $K:=(f_1^{max}-f_1^{min})/ϵ$:

$$[\mathrm{M}2(k)]:\mathrm{Objective}(2)$$

subject to

$$\begin{array}{c}\hfill \mathrm{Constraints}(3)–(4),\end{array}$$

and let $\widehat{\mathit{x}}(k)$ be an optimal solution obtained. We then solve the following model:

$$[\mathrm{M}3(k)]:min{f}_1(\mathit{x})$$

(5)

subject to

$$\begin{array}{cc}\hfill \mathrm{Constraint}& (4)\\ \hfill f_2(\mathit{x})& \le & f_2(\widehat{\mathit{x}}(k)),\hfill \end{array}$$

(6)

and let ${\mathit{x}}^{*}(k)$ be an optimal solution obtained.

The following lemmas, propositions, and corollaries are proposed.

Lemma 1.

$f_1({\mathit{x}}^{*}(k))\le f_1(\widehat{\mathit{x}}(k))$ and $f_2({\mathit{x}}^{*}(k))\le f_2(\widehat{\mathit{x}}(k))$.

Proof. 

Since $f_2(\widehat{\mathit{x}}(k))\le f_2(\widehat{\mathit{x}}(k))$, $\widehat{\mathit{x}}(k)$ is feasible to [M3$(k)$], and therefore, $f_1({\mathit{x}}^{*}(k))\le$$f_1(\widehat{\mathit{x}}(k))$ due to the optimality of ${\mathit{x}}^{*}(k)$ to [M3$(k)$].

Since ${\mathit{x}}^{*}(k)$ is optimal, and thus, feasible to [M3$(k)$], it satisfies the constraint $f_2({\mathit{x}}^{*}(k))\le f_2(\widehat{\mathit{x}}(k))$.    □

According to Lemma 1, Corollary 1 is derived.

Corollary 1.

${\mathit{x}}^{*}(k)$ is an optimal solution to [M2$(k)$].

Proposition 1.

${\mathit{x}}^{*}(k)$ is a Pareto optimal solution.

Proof. 

Suppose there is an ${\mathit{x}}^{\#}\in X$ such that $f_1({\mathit{x}}^{\#})<f_1({\mathit{x}}^{*}(k))$ and $f_2({\mathit{x}}^{\#})\le f_2({\mathit{x}}^{*}(k))$. Then, this contradicts the optimality of ${\mathit{x}}^{*}(k)$ to [M3$(k)$].

Suppose there is an ${\mathit{x}}^{\#}\in X$ such that $f_1({\mathit{x}}^{\#})\le f_1({\mathit{x}}^{*}(k))$ and $f_2({\mathit{x}}^{\#})<f_2({\mathit{x}}^{*}(k))$. Combined with Lemma 1, $f_1({\mathit{x}}^{\#})\le f_1(\widehat{\mathit{x}}(k))$ and $f_2({\mathit{x}}^{\#})<f_2(\widehat{\mathit{x}}(k))$. Then, this contradicts the optimality of $\widehat{\mathit{x}}(k)$ to [M2$(k)$].    □

Define a multiset $\mathsf{\Omega }:=\{{\mathit{x}}^{*}(k);k=0,\dots ,K\}$. If there are multiple solutions in $\mathsf{\Omega }$ that have the same objective vector, keep only one of these solutions.

Proposition 2.

Let Φ be the full set of Pareto optimal solutions. Then, Ω is a “good representation” of Φ in the sense that for any $\tilde{\mathit{x}}\in \mathsf{\Phi }$, there exists an ${\mathit{x}}^{*}\in \mathsf{\Omega }$ satisfying $f_1({\mathit{x}}^{*})<f_1(\tilde{\mathit{x}})+ϵ$ and $f_2({\mathit{x}}^{*})\le f_2(\tilde{\mathit{x}})$.

Proof. 

For any $\tilde{\mathit{x}}\in \mathsf{\Phi }$, let k be an integer such that $f_1^{min}+(k-1)ϵ<f_1(\tilde{\mathit{x}})\le f_1^{min}+kϵ$; evidently, $k\in \{0,\dots ,K\}$. We now prove that ${\mathit{x}}^{*}(k)$ satisfies $f_1({\mathit{x}}^{*}(k))<f_1(\tilde{\mathit{x}})+ϵ$ and $f_2({\mathit{x}}^{*}(k))\le f_2(\tilde{\mathit{x}})$.

Lemma 1 states that $f_1({\mathit{x}}^{*}(k))\le f_1(\widehat{\mathit{x}}(k))$. Since $\widehat{\mathit{x}}(k)$ is optimal, and thus, feasible to [M2$(k)$], $f_1(\widehat{\mathit{x}}(k))\le f_1^{min}+kϵ$. Therefore, $f_1({\mathit{x}}^{*}(k))\le f_1^{min}+kϵ$. Combined with $f_1^{min}$$+(k-1)ϵ<f_1(\tilde{\mathit{x}})$, we have $f_1({\mathit{x}}^{*}(k))<f_1(\tilde{\mathit{x}})+ϵ$.

Evidently, $\tilde{\mathit{x}}$ is feasible to [M2$(k)$]. Since $\widehat{\mathit{x}}(k)$ is optimal to [M2$(k)$], $f_2(\widehat{\mathit{x}}(k))\le f_2(\tilde{\mathit{x}})$. Lemma 1 states that $f_2({\mathit{x}}^{*}(k))\le f_2(\widehat{\mathit{x}}(k))$. Therefore, $f_2({\mathit{x}}^{*}(k))\le f_2(\tilde{\mathit{x}})$.    □

Proposition 2 immediately implies Corollary 2 as follows.

Corollary 2.

If $f_1(\mathit{x})$ can only take integer values (e.g., integer coefficients multiplied by integer decision variables), we can set $ϵ=1$ and use the revised ϵ-constraint method to obtain a subset of Pareto optimal solutions whose objective vectors form the full set of nondominated objective vectors for the problem.

The pseudocode for the revised $ϵ$-constraint method is shown in Algorithm 2.

Algorithm 2 Revised $ϵ$-constraint method

1: $f_1^{min}$←${min}_{\mathit{x}\in X}{f}_1(\mathit{x})$ // $f_1^{min}$ represents the minimum objective value of the first dimension.   2: $f_2^{min}$←${min}_{\mathit{x}\in X}{f}_2(\mathit{x})$ // $f_2^{min}$ represents the minimum objective value of the second dimension.   3: $f_1^{max}$←$min{f}_1(\mathit{x})$ subject to $f_2(\mathit{x})\le f_2^{min},\mathit{x}\in X$ // $f_1^{max}$ represents the minimum objective value of the first dimension when the second dimension achieves its minimum objective value.   4: $ϵ$← a small positive number such that $f_1^{max}-f_1^{min}$ is an integer multiple of $ϵ$ // $ϵ$ represents the tolerance.   5: $\widehat{\mathit{x}}(k)$← null, $k\in \{0,1,\dots ,(f_1^{max}-f_1^{min})/ϵ\}$ // $\widehat{\mathit{x}}(k)$ represents an optimal solution obtained by [M2$(k)$].   6: ${\mathit{x}}^{*}(k)$← null, $k\in \{0,1,\dots ,(f_1^{max}-f_1^{min})/ϵ\}$ // ${\mathit{x}}^{*}(k)$ represents an optimal solution obtained by [M3$(k)$].   7: $\mathsf{\Omega }\leftarrow ⌀$ // $\mathsf{\Omega }$ represents a multiset composed of obtained solutions by the revised $ϵ$-constraint method.   8: for  $k\in \{0,1,\dots ,(f_1^{max}-f_1^{min})/ϵ\}$ do   9:        Solve the model [M2$(k)$], and obtain an optimal solution $\widehat{\mathit{x}}(k)$. 10:        Solve the model [M3$(k)$], and obtain an optimal solution ${\mathit{x}}^{*}(k)$. 11:        $\mathsf{\Omega }\leftarrow \mathsf{\Omega }\cup \left\{{\mathit{x}}^{*}(k)\right\}$. 12: end for 13: Discard solutions with the same objective vector in $\mathsf{\Omega }$ and retain only one of them. 14: Return $\mathsf{\Omega }$.

4.3. Improved Revised $ϵ$-Constraint Method

The following improved revised $ϵ$-constraint method, which generates exactly the same subset of Pareto optimal solutions as the above revised method, can improve the computational efficiency in many cases.

Define a multiset $\mathsf{\Omega }:=⌀$. $K:=(f_1^{max}-f_1^{min})/ϵ$. Set $k\leftarrow K$. We solve the model [M2$(k)$] and obtain an optimal solution $\widehat{\mathit{x}}(k)$. We then solve the model [M3$(k)$] and obtain an optimal solution ${\mathit{x}}^{*}(k)$. Let $\mathsf{\Omega }\leftarrow \mathsf{\Omega }\cup \left\{{\mathit{x}}^{*}(k)\right\}$. Let $\widehat{k}$ be the largest integer satisfying $f_1^{min}+\widehat{k}ϵ<f_1({\mathit{x}}^{*}(k))$. If $\widehat{k}<0$, return $\mathsf{\Omega }$ and stop. Otherwise, set $k\leftarrow \widehat{k}$ and repeat the above procedure of solving [M2$(k)$] and [M3$(k)$].

It is easy to see Propositions 3 and 4.

Proposition 3.

All solutions in Ω are Pareto optimal, and no two solutions have the same objective vector.

Proposition 4.

The improved revised ϵ-constraint method generates the same subset of Pareto optimal solutions (more accurately, the same subset of nondominated objective vectors) as the revised ϵ-constraint method.

Proof. 

To differentiate the models and solutions of the revised method and the improved revised method, we add a prime ′ to the models and solutions of the improved revised method.

Let us consider “let $\widehat{k}$ be the largest integer satisfying $f_1^{min}+\widehat{k}ϵ<f_1({\mathit{x}}^{*}{(k)}^{\prime })$” in the improved revised $ϵ$-constraint method. If $\widehat{k}=k-1$ in all cases, then the two methods are equivalent and generate the same subset of Pareto optimal solutions.

Suppose that we have $\widehat{k}<k-1$ in at least one of the loops. The question is, is it possible that there is a $\overline{k}$ satisfying $\widehat{k}+1\le \overline{k}\le k-1$ and the objective vector of solution ${\mathit{x}}^{*}(\overline{k})$ generated by the revised $ϵ$-constraint method is not an objective vector of any solution generated by the improved revised $ϵ$-constraint method? The answer is no, for the following reasons.

[M2${(k)}^{\prime }$] is a relaxation of [M2$(\overline{k})$] (because $\overline{k}\le k-1<k$) and they have the same objective function. Corollary 1 states that ${\mathit{x}}^{*}{(k)}^{\prime }$ is optimal to [M2${(k)}^{\prime }$] and ${\mathit{x}}^{*}(\overline{k})$ is optimal to [M2$(\overline{k})$]. Therefore, the optimal objective value of [M2${(k)}^{\prime }$] is smaller than or equal to that of [M2$(\overline{k})$], i.e., $f_2({\mathit{x}}^{*}{(k)}^{\prime })\le f_2({\mathit{x}}^{*}(\overline{k}))$. Moreover, since we “let $\widehat{k}$ be the largest integer satisfying $f_1^{min}+\widehat{k}ϵ<f_1({\mathit{x}}^{*}{(k)}^{\prime })$”, $f_1^{min}+(\widehat{k}+1)ϵ\ge f_1({\mathit{x}}^{*}{(k)}^{\prime })$; combined with $\widehat{k}+1\le \overline{k}\le k-1$, we have $f_1({\mathit{x}}^{*}{(k)}^{\prime })\le f_1^{min}+\overline{k}ϵ$, meaning that ${\mathit{x}}^{*}{(k)}^{\prime }$ is feasible to [M2$(\overline{k})$]. Thus, we have $f_2({\mathit{x}}^{*}{(k)}^{\prime })\ge f_2({\mathit{x}}^{*}(\overline{k}))$ (${\mathit{x}}^{*}(\overline{k})$ is optimal to [M2$(\overline{k})$]). Therefore, $f_2({\mathit{x}}^{*}{(k)}^{\prime })=f_2({\mathit{x}}^{*}(\overline{k}))$. Hence, model [M3${(k)}^{\prime }$] is identical to model [M3$(\overline{k})$]. Therefore, the objective vector of solution ${\mathit{x}}^{*}(\overline{k})$ is the same as the objective vector of solution ${\mathit{x}}^{*}{(k)}^{\prime }$.    □

The pseudocode for the improved revised $ϵ$-constraint method is shown in Algorithm 3.

Algorithm 3 Improved revised $ϵ$-constraint method

1: $f_1^{min}$←${min}_{\mathit{x}\in X}{f}_1(\mathit{x})$ // $f_1^{min}$ represents the minimum objective value of the first dimension.   2: $f_2^{min}$←${min}_{\mathit{x}\in X}{f}_2(\mathit{x})$ // $f_2^{min}$ represents the minimum objective value of the second dimension.   3: $f_1^{max}$←$min{f}_1(\mathit{x})$ subject to $f_2(\mathit{x})\le f_2^{min},\mathit{x}\in X$ // $f_1^{max}$ represents the minimum objective value of the first dimension when the second dimension achieves its minimum objective value.   4: $ϵ$← a small positive number such that $f_1^{max}-f_1^{min}$ is an integer multiple of $ϵ$ // $ϵ$ represents the tolerance.   5: $k\leftarrow (f_1^{max}-f_1^{min})/ϵ$   6: $\widehat{\mathit{x}}(k)$← null // $\widehat{\mathit{x}}(k)$ represents an optimal solution obtained by [M2$(k)$].   7: ${\mathit{x}}^{*}(k)$← null // ${\mathit{x}}^{*}(k)$ represents an optimal solution obtained by [M3$(k)$].   8: $\mathsf{\Omega }\leftarrow ⌀$ // $\mathsf{\Omega }$ represents a multiset composed of obtained solutions by the improved revised $ϵ$-constraint method.   9: $\widehat{k}\leftarrow 0$ // $\widehat{k}$ represents the largest integer satisfying $f_1^{min}+\widehat{k}ϵ<f_1({\mathit{x}}^{*}(k))$. 10: while  $\widehat{k}\ge 0$  do 11:        Solve the model [M2$(k)$], and obtain an optimal solution $\widehat{\mathit{x}}(k)$. 12:        Solve the model [M3$(k)$], and obtain an optimal solution ${\mathit{x}}^{*}(k)$. 13:        $\mathsf{\Omega }\leftarrow \mathsf{\Omega }\cup \left\{{\mathit{x}}^{*}(k)\right\}$. 14:        Calculate the largest integer $\widehat{k}$ satisfying $f_1^{min}+\widehat{k}ϵ<f_1({\mathit{x}}^{*}(k))$. 15:        $k\leftarrow \widehat{k}$. 16: end while 17: Return $\mathsf{\Omega }$.

4.4. Augmented $ϵ$-Constraint Method

A deficiency of the improved revised $ϵ$-constraint method is that we must solve two models each time: [M2$(k)$] and [M3$(k)$]. Hence, the augmented $ϵ$-constraint method that only needs to solve one model was developed [31]. The augmented $ϵ$-constraint method is elaborated as follows.

Define a multiset $\mathsf{\Omega }:=⌀$. $K:=(f_1^{max}-f_1^{min})/ϵ$. Set $k\leftarrow K$. We solve the following single-objective optimization model:

$$[\mathrm{M}4(k)]:min\left\{f_2(\mathit{x})+ϵ{\displaystyle \frac{f_1(\mathit{x})-f_1^{min}}{f_1^{max}-f_1^{min}}}\right\}$$

(7)

subject to

$$\begin{array}{c}\hfill \mathrm{Constraints}(3)–(4),\end{array}$$

and let $\dot{\mathit{x}}(k)$ be an optimal solution obtained and $\mathsf{\Omega }\leftarrow \mathsf{\Omega }\cup \left\{\dot{\mathit{x}}(k)\right\}$. Let $\widehat{k}$ be the largest integer satisfying $f_1^{min}+\widehat{k}ϵ<f_1(\dot{\mathit{x}}(k))$. If $\widehat{k}<0$, return $\mathsf{\Omega }$ and stop. Otherwise, set $k\leftarrow \widehat{k}$ and repeat the above procedure of solving [M4$(k)$].

Then, Propositions 5 and 6 are shown below.

Proposition 5.

$\dot{\mathit{x}}(k)$ is a Pareto optimal solution.

Proof. 

Suppose there is an ${\mathit{x}}^{\#}\in X$ such that $f_1({\mathit{x}}^{\#})<f_1(\dot{\mathit{x}}(k))$ and $f_2({\mathit{x}}^{\#})\le f_2(\dot{\mathit{x}}(k))$ or such that $f_1({\mathit{x}}^{\#})\le f_1(\dot{\mathit{x}}(k))$ and $f_2({\mathit{x}}^{\#})<f_2(\dot{\mathit{x}}(k))$. Evidently, ${\mathit{x}}^{\#}$ is feasible to [M4$(k)$] and has a smaller objective value than that of $\dot{\mathit{x}}(k)$, which contradicts the optimality of $\dot{\mathit{x}}(k)$.    □

Proposition 6.

Let Φ be the full set of Pareto optimal solutions. Then, Ω is a “good representation” of Φ in the sense that for any $\tilde{\mathit{x}}\in \mathsf{\Phi }$, there exists an $\dot{\mathit{x}}\in \mathsf{\Omega }$ satisfying $f_1(\dot{\mathit{x}})<f_1(\tilde{\mathit{x}})+ϵ$ and $f_2(\dot{\mathit{x}})\le f_2(\tilde{\mathit{x}})+ϵ$.

Proof. 

Similar to the equivalence between the revised $ϵ$-constraint method and the improved revised $ϵ$-constraint method, we can modify the augmented $ϵ$-constraint method by enumerating all values of $k=0,\dots ,K$. Next, we consider this modified version.

For any $\tilde{\mathit{x}}\in \mathsf{\Phi }$, let k be an integer such that $f_1^{min}+(k-1)ϵ<f_1(\tilde{\mathit{x}})\le f_1^{min}+kϵ$; evidently, $k\in \{0,\dots ,K\}$. We now prove that $\dot{\mathit{x}}(k)$ satisfies $f_1(\dot{\mathit{x}}(k))<f_1(\tilde{\mathit{x}})+ϵ$ and $f_2(\dot{\mathit{x}}(k))\le f_2(\tilde{\mathit{x}})+ϵ$.

Since $\dot{\mathit{x}}(k)$ is optimal, and thus, feasible to [M4$(k)$], $f_1(\dot{\mathit{x}}(k))\le f_1^{min}+kϵ$. Combined with $f_1^{min}+(k-1)ϵ<f_1(\tilde{\mathit{x}})$, we have $f_1(\dot{\mathit{x}}(k))<f_1(\tilde{\mathit{x}})+ϵ$. Evidently, $\tilde{\mathit{x}}$ is feasible to [M4$(k)$]. Since $\dot{\mathit{x}}(k)$ is optimal to [M4$(k)$], $f_2(\dot{\mathit{x}}(k))+ϵ(f_1(\dot{\mathit{x}}(k))-f_1^{min})/(f_1^{max}-f_1^{min})\le f_2(\tilde{\mathit{x}})+ϵ(f_1(\tilde{\mathit{x}})-f_1^{min})/(f_1^{max}-f_1^{min})$, which implies that $f_2(\dot{\mathit{x}}(k))\le f_2(\tilde{\mathit{x}})+ϵ$ because $(f_1(\tilde{\mathit{x}})-f_1^{min})/(f_1^{max}-f_1^{min})\in [0,1]$.    □

Proposition 6 immediately implies Corollary 3 as follows.

Corollary 3.

If both $f_1(\mathit{x})$ and $f_2(\mathit{x})$ only take integer values (e.g., integer coefficients multiplied by integer decision variables), we can set ϵ to be a number less than 1 (e.g., 0.9) and use the augmented ϵ-constraint method to obtain a subset of Pareto optimal solutions whose objective vectors form the full set of nondominated objective vectors for the problem.

The pseudocode for the augmented $ϵ$-constraint method is shown in Algorithm 4.

Algorithm 4 Augmented $ϵ$-constraint method

1: $f_1^{min}$←${min}_{\mathit{x}\in X}{f}_1(\mathit{x})$ // $f_1^{min}$ represents the minimum objective value of the first dimension.   2: $f_2^{min}$←${min}_{\mathit{x}\in X}{f}_2(\mathit{x})$ // $f_2^{min}$ represents the minimum objective value of the second dimension.   3: $f_1^{max}$←$min{f}_1(\mathit{x})$ subject to $f_2(\mathit{x})\le f_2^{min},\mathit{x}\in X$ // $f_1^{max}$ represents the minimum objective value of the first dimension when the second dimension achieves its minimum objective value.   4: $ϵ$← a small positive number such that $f_1^{max}-f_1^{min}$ is an integer multiple of $ϵ$ // $ϵ$ represents the tolerance.   5: $k\leftarrow (f_1^{max}-f_1^{min})/ϵ$   6: $\dot{\mathit{x}}(k)$← null // $\dot{\mathit{x}}(k)$ represents an optimal solution obtained by [M4$(k)$].   7: $\mathsf{\Omega }\leftarrow ⌀$ // $\mathsf{\Omega }$ represents a multiset composed of obtained solutions by the augmented $ϵ$-constraint method.   8: $\widehat{k}\leftarrow 0$ // $\widehat{k}$ represents the largest integer satisfying $f_1^{min}+\widehat{k}ϵ<f_1(\dot{\mathit{x}}(k))$.   9: while  $\widehat{k}\ge 0$ do 10:        Solve the model [M4$(k)$], and obtain an optimal solution $\dot{\mathit{x}}(k)$. 11:        $\mathsf{\Omega }\leftarrow \mathsf{\Omega }\cup \left\{\dot{\mathit{x}}(k)\right\}$. 12:        Calculate the largest integer $\widehat{k}$ satisfying $f_1^{min}+\widehat{k}ϵ<f_1(\dot{\mathit{x}}(k))$. 13:        $k\leftarrow \widehat{k}$. 14: end while 15: Return $\mathsf{\Omega }$.

5. Computational Experiments

In order to validate the effectiveness and efficiency of our methods, using the basic $ϵ$-constraint method and the weighting method as benchmark methods, we assess the solution quality and computational efficiency of the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, and the augmented $ϵ$-constraint method. In the computational experiments with the methods related to $ϵ$-constraint, the value of $ϵ$ is set to 0.1. In the computational experiments with the weighting method, the weight of the first objective function varies from 0, 0.01, …, all the way to 1. The computational experiments are conducted on a PC (14 cores of CPUs, 2.5 GHz, Memory 64 GB). All the five methods are implemented in Gurobi 10.0.0 (Anaconda, Python). In this section, we adopt the five methods to address a realistic bi-objective optimization problem in the field of urban traffic, which is introduced in Section 5.1. Then, we show the computational results in Section 5.2.

5.1. Case Setting

We consider a safe transportation route selection problem for trucks, aiming to minimize transportation time of trucks and safety risks of drivers, as investigated by Guo et al. [32].

Following the notation of Guo et al. [32], an urban traffic network consists of a set V of nodes indexed by i or j, and a set E of arcs indexed by $(i,j)$. To be specific, $V=\{0,1,2,\dots ,n\}$, where 0 and n represent the starting point and the ending point, respectively. $E\subseteq \{(i,j)|i,j\in N\}$ denotes arcs connecting nodes in the urban traffic network. Let $t_{ij}$ and T denote the travel time between node i and node j, and the maximum travel time, respectively. Let $w_{ij}$ and C represent the safety risk of the trucks traveling on the arc $(i,j)$, and the maximum acceptable risk, respectively. $x_{ij}$ denotes the binary decision variable, which equals 1 iff arc $(i,j)$ is selected, and 0 otherwise. Based on the notation given above, a bi-objective programming model can be formulated.

$$\begin{array}{cc}\hfill & min\left(\begin{array}{c}\sum _{(i,j)\in E}{t}_{ij}{x}_{ij}\\ \sum _{(i,j)\in E}{w}_{ij}{x}_{ij}\end{array}\right)\hfill \end{array}$$

(8)

$$\begin{array}{cccc}\hfill \mathrm{s}.\mathrm{t}.& \sum _{(i,j)\in E}{t}_{ij}{x}_{ij}\le T\hfill & & \end{array}$$

(9)

$$\begin{array}{cccc}& \sum _{(i,j)\in E}{w}_{ij}{x}_{ij}\le C\hfill & & \end{array}$$

(10)

$$\begin{array}{cccc}& \sum _{(0,j)\in E}{x}_{0j}=1\hfill & & \end{array}$$

(11)

$$\begin{array}{cccc}& \sum _{(i,n)\in E}{x}_{in}=1\hfill & & \end{array}$$

(12)

$$\begin{array}{cccc}& \sum _{(i,j)\in E}{x}_{ij}=\sum _{(j,i)\in E}{x}_{ji}\hfill & & \forall j\in V\setminus \{0,n\}\hfill \end{array}$$

(13)

$$\begin{array}{cccc}& x_{ij}\in \{0,1\}\hfill & & \forall (i,j)\in E\hfill \end{array}$$

(14)

In a randomly generated urban traffic network, the travel time and safety risk of the trucks traveling on arcs are set to random integers uniformly distributed within the range of 1 to 20, and within the range of 1 to 50, respectively. Both the maximum travel time and maximum acceptable risk are set to 200.

5.2. Experimental Results

The basic $ϵ$-constraint method, the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, the augmented $ϵ$-constraint method, and the weighting method are implemented to solve the problem and find a series of solutions. We conduct five sets of small-scale instances, five sets of medium-scale instances, and five sets of large-scale instances, corresponding to 60 nodes, 200 nodes, and 500 nodes in the urban traffic network, respectively. The time limit for each instance is set to half an hour.

Table 2. Experimental results of the basic $ϵ$-constraint method and revised $ϵ$-constraint method.

Scale Type	Number of Nodes	ID	Basic $\mathit{ϵ}$-Constraint Method			Revised $\mathit{ϵ}$-Constraint Method
			NOSO	HR (%)	CPU Time (s)	NOSO	HR (%)	CPU Time (s)
Small	60	1	6	100.00	10.66	6	100.00	20.13
		2	9	100.00	16.59	9	100.00	33.90
		3	5	100.00	8.08	5	100.00	15.53
		4	8	100.00	21.47	8	100.00	51.26
		5	9	100.00	24.23	9	100.00	48.62
		Avg.	7.40	100.00	16.20	7.40	100.00	33.89
Medium	200	1	7	100.00	173.07	7	100.00	372.35
		2	13	100.00	273.82	13	100.00	532.00
		3	6	100.00	123.80	6	100.00	306.41
		4	11	100.00	375.81	11	100.00	661.75
		5	13	100.00	219.51	13	100.00	598.17
		Avg.	10.00	100.00	233.20	10.00	100.00	494.14
Large	500	1	7	100.00	1122.57	7	100.00	1755.86
		2	6	100.00	1120.02	3	94.58	1801.69
		3	5	100.00	1529.64	2	85.56	1805.72
		4	6	100.00	1449.08	4	97.96	1804.32
		5	7	100.00	739.79	7	100.00	1430.97
		Avg.	6.20	100.00	1192.22	4.60	95.62	1719.71

Abbreviations: NOSO, number of solutions obtained; HR, hypervolume-based ratio.

,

Table 3. Experimental results of the improved revised $ϵ$-constraint method and augmented $ϵ$-constraint method.

Scale Type	Number of Nodes	ID	Improved Revised $\mathit{ϵ}$-Constraint Method			Augmented $\mathit{ϵ}$-Constraint Method
			NOSO	HR (%)	CPU Time (s)	NOSO	HR (%)	CPU Time (s)
Small	60	1	6	100.00	0.57	6	100.00	0.32
		2	9	100.00	0.83	9	100.00	0.41
		3	5	100.00	0.48	5	100.00	0.27
		4	8	100.00	1.02	8	100.00	0.55
		5	9	100.00	0.83	9	100.00	0.47
		Avg.	7.40	100.00	0.75	7.40	100.00	0.40
Medium	200	1	7	100.00	7.64	7	100.00	4.00
		2	13	100.00	15.23	13	100.00	7.92
		3	6	100.00	9.66	6	100.00	4.70
		4	11	100.00	12.76	11	100.00	6.52
		5	13	100.00	14.92	13	100.00	7.61
		Avg.	10.00	100.00	12.04	10.00	100.00	6.15
Large	500	1	7	100.00	49.61	7	100.00	25.24
		2	6	100.00	38.83	6	100.00	19.13
		3	5	100.00	32.42	5	100.00	16.50
		4	6	100.00	39.41	6	100.00	18.50
		5	7	100.00	46.53	7	100.00	23.45
		Avg.	6.20	100.00	41.36	6.20	100.00	20.56

Abbreviations: NOSO, number of solutions obtained; HR, hypervolume-based ratio.

and

Table 4. Experimental results of the weighting method.

Scale Type	Number of Nodes	ID	Weighting Method
			NOSO	HR (%)	CPU Time (s)
Small	60	1	5	98.43	3.95
		2	5	99.37	3.39
		3	3	97.47	3.92
		4	5	98.59	4.99
		5	6	99.58	3.67
		Avg.	4.80	98.69	3.98
Medium	200	1	5	98.17	46.59
		2	7	97.44	50.06
		3	5	99.92	67.34
		4	7	99.25	46.19
		5	6	97.77	47.36
		Avg.	6.00	98.51	51.51
Large	500	1	6	99.97	322.49
		2	4	94.58	300.11
		3	4	98.94	297.81
		4	3	92.70	298.98
		5	6	99.33	303.88
		Avg.	4.60	97.10	304.65

Abbreviations: NOSO, number of solutions obtained; HR, hypervolume-based ratio.

show the computing time and value of performance metrics (i.e., number of solutions obtained (NOSO) and hypervolume-based ratio (HR)). The solution sets of the basic $ϵ$-constraint method, the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, the augmented $ϵ$-constraint method, and the weighting method are denoted by $S_1$, $S_2$, $S_3$, $S_4$, and $S_5$, respectively. Let S represent the union of solution sets of all the five methods, i.e., $S=S_1\cup S_2\cup S_3\cup S_4\cup S_5$. The maximum objective values of solutions in S in the first dimension and the second dimension, represented by $O_1^{max}$ and $O_2^{max}$, determine the position of the reference point in the two-dimensional plane. For each obtained solution $\mathit{x}$, the rectangle whose vertices are $\mathit{x}$ and the reference point ($O_1^{max}$, $O_2^{max}$), with each edge perpendicular to the coordinate axes, is denoted by $g_{\mathit{x}}$. The hypervolume of solution set $S_k$, $k\in \{1,2,3,4,5\}$, and S, denoted by $h_k$ and h, are the hypervolume of the union of $g_{\mathit{x}}$ for $\mathit{x}\in S_k$ and the union of $g_{\mathit{x}}$ for $\mathit{x}\in S$, respectively. Then, the value of the metric HR for method k, i.e., $HR(S_k)$, can be calculated as $h_k/h$. Generally, the higher values of metrics NOSO and HR indicate better performance of the obtained solutions. To strengthen readability, the CPU time is rounded to two decimal places, and the value of HR is expressed as a percentage with two decimal places.

As shown in Table 2, Table 3 and Table 4, the difficulty for all five methods to solve the safe transportation route selection problem increases as the number of nodes in the urban traffic network grows. In terms of solution quality, for the same instance, the basic $ϵ$-constraint method, the improved revised $ϵ$-constraint method, and the augmented $ϵ$-constraint method are able to find more solutions with broader coverage than the weighting method. This is because the weighting method cannot generate nonsupported efficient solutions, even if these solutions are Pareto optimal. Additionally, it is worth noting that compared with the improved revised $ϵ$-constraint method and the augmented $ϵ$-constraint method developed in this study, the basic $ϵ$-constraint method requires extra steps to remove weakly efficient solutions that are not Pareto optimal. In terms of the computing time, for the same instance, the revised $ϵ$-constraint method is the most time-consuming. However, the improved revised $ϵ$-constraint method and the augmented $ϵ$-constraint method, developed by enhancing the revised $ϵ$-constraint method, show relatively high computational efficiency. Specifically, for small-, medium-, and large-scale instances, the average solution times of the improved revised $ϵ$-constraint method and the augmented $ϵ$-constraint method are no more than 5.16% and 2.64% of that of the basic $ϵ$-constraint method, and no more than 23.38% and 11.94% of that of the weighting method, respectively. Therefore, the computational experiments demonstrate the effectiveness and efficiency of both the improved revised $ϵ$-constraint method and the augmented $ϵ$-constraint method for instances of different scales.

6. Conclusions

In numerous practical transportation problems, two conflicting objectives are required to be considered simultaneously. In general, an improvement in one objective is accompanied by a compromise in the other. Therefore, the goal of bi-objective optimization is to find a set of Pareto optimal solutions that balance both objectives. Since it is inefficient and unnecessary to obtain the full set of Pareto optimal solutions for practical purposes, this study proposes that a near-optimal subset of Pareto optimal solutions is sufficient to tackle bi-objective optimization problems. In addition, we elaborate the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, and the augmented $ϵ$-constraint method for solving bi-objective optimization problems. A key challenge in bi-objective optimization is evaluating the quality of the obtained solutions. Therefore, we rigorously analyze the near-optimality of the subset of Pareto optimal solutions obtained by the revised $ϵ$-constraint method, the improved revised $ϵ$-constraint method, and the augmented $ϵ$-constraint method from a mathematical perspective. To this end, we present a series of lemmas, propositions, and corollaries, which are supported by rigorous proofs. Moreover, in order to validate the effectiveness and efficiency of our methods, computational experiments are conducted based on a real-world bi-objective optimization problem.

This study establishes a solid foundation for the development of bi-objective optimization by mathematically proving the near-optimality of the Pareto optimal solution subsets and offering feasible methods for finding such near-optimal subsets of Pareto optimal solutions, contributing to balancing trade-off objectives in practical transportation problems. However, this study has some limitations, which can be further investigated from the following aspects. First, future work can extend our methods to multi-objective optimization problems in the field of transportation. Second, the computational efficiency of our methods could be enhanced. Deeper exploration of advanced optimization algorithms [33] and machine learning [34,35] could be considered in future studies.