A Study of the Multi-Objective Neighboring Only Quadratic Minimum Spanning Tree Problem in the Context of Uncertainty

Abstract

The pursuit of studying the quadratic minimum spanning tree (QMST) problem has captivated numerous academics because of its distinctive characteristic of taking into account the cost of interaction between pairs of edges. A QMST refers to the minimum spanning tree, which is a graph that is both acyclic and minimally connected. It also includes the interaction cost between a pair of edges in the minimum spanning tree. These interaction costs can occur between any pair of edges, whether they are adjacent or non-adjacent. In the QMST problem, our objective is to minimize both the cost of the edges and the cost of interactions. This eventually classifies the task as NP-hard. The interaction costs, sometimes referred to as quadratic costs, inherently exhibit a contradictory relationship with linear edge costs when solving a multi-objective problem that aims to minimize both linear and quadratic costs simultaneously. This study addresses the bi-objective adjacent only quadratic minimum spanning tree problem (AQMSTP) by incorporating the uncertain nature of the linear and quadratic costs associated with the problem. The focus is on the interaction costs between adjacent edges. Consequently, we have introduced a multi-objective problem called the uncertain adjacent only quadratic minimum spanning tree problem (mUAQMSTP) and formulated it using the uncertain chance-constrained programming technique. Afterwards, two MOEAs—non-dominated sorting genetic algorithm II (NSGAII) and duplicate elimination non-dominated sorting evolutionary algorithm (DENSEA)—and the traditional multi-objective solution approach, the global criterion method, are employed to solve the deterministic transformation of the model. Finally, we provide a suitable numerical illustration to substantiate our suggested framework.

1. Introduction

With its wide range of technical and scientific applications, the minimum spanning tree (MST) problem is one of the cornerstones of network optimization. The goal of the minimum spanning tree (MST) problem is to determine the least expensive minimally connected subgraph of a connected finite network. Assad and Xu [1] presented a version of the minimum spanning tree (MST) problem called the quadratic minimum spanning tree (QMST). This problem finds the minimum spanning tree of a weighted connected network by taking into account both linear and quadratic weights. Quadratic weights often result from the interaction effect between any two edges, while linear weights are attached to every edge individually. According to Cordone and Passeri [2], the QMST problem is useful in many areas of network design, including transportation networks, networks for the transmission of oil or water, and telecommunications networks. Usually, the linear weights in telecom networks often reflect the costs of setting up radio or cable connections, while the quadratic weights indicate the price of a device that converts between different types of linked cables. The cost of laying different pipes is represented by linear weights in a transportation or utility network, whereas the cost of interacting with things like valves or bending losses in T-joints is represented by quadratic weights in the same context. In a transportation network, the linear weights indicate the cost of developing roads, while the quadratic weights represent turn penalties [3].

QMST has been the subject of extensive investigation by several researchers over the past few decades. The branch-and-bound approach was first employed by Assad and Xu [1] to obtain the results. Subsequently, Zhou and Gen [4] proposed an evolutionary approach for quality management system testing (QMST). The iterative tabu search method, hybrid genetic algorithm, and multi-start simulated annealing were presented by Palubeckis et al. [5] as potential solutions for the QMST problem. In 2010, Sundar and Singh [6] employed the artificial bee colony algorithm that was designed in the form of a hive of bees to identify a solution. Afterwards in the same year, Öncan and Punnen [7] developed a local search method that is exceptionally efficient in solving the QMST problem. Cordone and Passeri [2] tackled the problem by introducing a tabu search and variable neighborhood search method. Pereira et al. [8] introduced an enhanced iteration of QMST capable of computing more robust linear programming bounds. In their study, Lozano et al. [9] proposed a methodology for addressing the QMST problem by integrating a tabu search with the strategic oscillation algorithm. The authors proposed a methodology that utilizes the creation of dynamic columns and rows in order to evaluate the bounds. In order to tackle this concern, Rostami and Malucelli [10] employed a reformulation–linearization methodology to depict a range of novel mixed 0–1 linear constructs. Pereira et al. [8] subsequently tackled this problem by employing two parallel branch-and-bound algorithms, while also introducing an innovative formulation of the QMST. The study conducted by Čustić et al. [3] aimed to ascertain the existence of QMST and its three versions through an examination of NP-hard and polynomially solvable connected network problems. The three problems are: (i) the minimum spanning tree problem with conflict pair constraints, (ii) the bottleneck spanning tree problem with conflict pair constraints, and (iii) the quadratic bottleneck spanning tree problem. Finding a collection of node–disjoint cycles that visit all the nodes in a way that minimizes the total sum of interaction costs between subsequent arcs is the quadratic cycle cover problem that de Meijer and Sotirov [11] tackle in 2020. Guimarães et al. [12] established the bottom bounds for the QMST using Semidefinite Programming (SDP) in the same year. Following this, Hu and Sotirov [13] demonstrated that linearization-based limits are provided by the extended Gilmore–Lawler bounding strategy for binary quadratic problems. The authors of the same paper established that the bound derived via the first level reformulation linearization approach is also a type of linearization-based bound. Recently, using an extended formulation for the minimum spanning tree problem, Sotirov and Verchére [14] derived a series of relaxations for the QMST with increased complexity.

The aforementioned research on QMST is viewed as an optimization problem with one objective function, where the linear and quadratic weights are in conjunction. Maia et al. [15] investigated the contrasting characteristics of linear and quadratic weights. In this case, the authors have broken down the problem into two distinct minimization objectives for linear and quadratic weights. Their research suggests a Pareto local search algorithm for a bi-objective QMST that takes into account the quadratic weights for any two neighboring edges.

The accuracy of parameters associated with decision-making problems is frequently compromised due to several variables, including inadequate data, limited input sources, the dynamic nature of input parameters, and measurement inaccuracy, among others. Therefore, a multitude of academics has proposed several advanced theories, including the probability theory, fuzzy set theory (Zadeh [16]), rough set theory (Pawlak [17]), and uncertainty theory (Liu [18]), in order to analyze and represent useful information that is hidden inside imprecise or ill-defined data. The literature has several instances where these ideas have been implemented, as demonstrated by Feng et al. [19], Majumder et al. [20], Rani and Garg [21], Zhou et al. [22] and Wang et al. [23]. The investigation of stochastic MST has been conducted by researchers in response to the problem of uncertain MST (Swamy and Shmoys [24]; Torkestani and Meybodi [25]). There exist several noteworthy scholarly publications pertaining to fuzzy MST, including works by Janik and Kasperski [26], Zhou et al. [27], and Gao et al. [28]. Conversely, a limited number of studies have concentrated on QMST with uncertain parameters, as opposed to uncertain MST. Up till now, only three research studies have focused on the practical use of the QMST problem in an uncertain setting. The problem was solved by Gao and Lu [29] using a genetic algorithm. This research is regarded as an introductory study on QMST with fuzzy parameters. Afterwards, Zhou et al. [30] developed a QMST that incorporated uncertain variables representing linear and quadratic weights. Using the rough–fuzzy hybrid paradigm, Majumder et al. [20] conducted research on the multi-objective QMST problem. Considering the existing research on QMST within the uncertain paradigm, it is evident that a multi-objective QMST problem, under the framework of uncertainty theory (Liu [18]), is yet to be addressed in the literature. Therefore, the present study emphasizes the following contributions to address the observed lacuna:

(i)

A multi-objective uncertain adjacent quadratic minimum spanning tree (mUAQMSTP), where the linear and quadratic weights are treated as uncertain variables.

(ii)

This study uses the chance-constrained programming (CCP) approach to model the mUAQMSTP.

(iii)

The deterministic equivalent of the CCP for the proposed mUAQMSTP is subsequently solved using the global criterion method (Rao [31]), a classical technique in multi-objective programming. Additionally, two multi-objective evolutionary algorithms (MOEAs) are employed: (a) non-dominated sorting algorithm II (NSGAII) (Deb et al. [32]) and (b) duplicate elimination non-dominated sorting evolutionary algorithm (DENSEA) (Greiner et al. [33]).

The subsequent section within the article is delineated as shown below. Section 2 covers the fundamental concepts that are crucial for this research. We model the mUAQMSTP and develop the associated uncertain CCP in Section 3. Then, in Section 4, we presented the crisp transformation of the CCP of mUAQMSTP. In Section 5, the methods used to solve the deterministic models developed for the mUAQMSTP are discussed. In Section 6, we numerically demonstrate the proposed mUAQMSTP using an example. The proposed problem and the results that are required of it are addressed in Section 7. To conclude, in Section 8, the epilogue of the study’s final results is drawn.

2. Preliminaries

We lay the groundwork for our analysis by presenting the essential ideas and key findings of uncertainty theory.

Envision a non-empty set, $\Gamma$, $\mathcal{L}$ is the $\sigma$-algebra over $\Gamma$. $\mathcal{M}$ is the uncertain measure which maps $\mathcal{L}$ to $\mathbb{R}$ (0,1), i.e., $\mathcal{M}:\mathcal{L}\mapsto \left[\mathrm{0,1}\right]$. In order to define the uncertain measure axiomatically, each event $\Lambda$ of $\mathcal{L}$ is assigned a belief degree $\mathcal{M}\left\{\Lambda \right\}$ such that $\mathcal{M}\left\{\Lambda \right\}\in \left[\mathrm{0,1}\right]$ If $\mathcal{M}$ satisfies the conditions of countable subadditivity, normality, and duality (Liu [18]), we consider $\mathcal{M}$ as an uncertain measure. In this scenario, the triplet $(\Gamma ,\mathcal{L},\mathcal{M})$ represents the uncertainty space. Both the product uncertain measure and the product measure axiom were presented by Liu [34]. If there is an arbitrary event, ${\Lambda }_p$, to each ${\mathcal{L}}_p$, where $p=\mathrm{1,2},...,$ for all uncertainty spaces $({\Gamma }_p, {\mathcal{L}}_p, {\mathcal{M}}_p)$, then $\mathcal{M}$ is said to be a product uncertain measure if

$$\mathcal{M}\left\{\prod _{p=1}^{\mathrm{\infty }}{\Lambda }_p\right\}=\underset{p=1}{\overset{\mathrm{\infty }}{\bigwedge }}{\mathcal{M}}_p\left\{{\Lambda }_p\right\}.$$

(1)

Definition 1.

(Liu [18]). In an uncertainty space $(\Gamma ,\mathcal{L},\mathcal{M})$, an uncertain variable, $\xi$, is defined by a measurable function from $(\Gamma ,\mathcal{L},\mathcal{M})$ to $\mathbb{R}$. In simple terms, for each set, $\mathcal{B}$ ∈ $\mathbb{R}$, where $\mathcal{B}$ is the Borel set, the set $\left\{\xi \in \mathcal{B}\right\}=\left\{\lambda \in \Gamma |\xi \left(\lambda \right)\in \mathcal{B}\right\}$ is an event.

Definition 2.

(Liu [18]). A zigzag uncertain variable,$\xi$, denoted by $\mathcal{Z}(p,q,r)$, where $p<q<r$ and $p,q,r\in \mathbb{R}$, is an uncertain variable with the uncertainty distribution as shown in Equation (2).

$$\mathsf{\Phi }\left(t\right)=\left\{\begin{array}{l}\begin{array}{ll}0& ;if t<p\\ {\displaystyle \frac{t-p}{2\left(q-p\right)}}& ;if p\le t<q\\ {\displaystyle \frac{t+r-2q}{2(r-q)}}& ;if q\le t<r\\ 1& ;if t\ge r\end{array}\end{array}\right.$$

(2)

When dealing with mathematical operations involving independent uncertain variables, the inverse uncertainty distribution becomes quite important. For any, $\alpha$, in the interval $\left[\mathrm{0,1}\right]$, a distribution, $\mathsf{\Phi }$, is said to be regular if and only if its inverse function, ${\mathsf{\Phi }}^{-1}\left(\alpha \right)$, exists. Consequently, the zigzag uncertain variable displays a regular uncertainty distribution.

Definition 3.

(Liu [18]). For any Borel sets ${\mathcal{B}}_p$ $\in \mathbb{R}$

$$\mathcal{M}\left\{\bigcap _{p=1}^n\left({\xi }_p\in {\mathcal{B}}_p\right)\right\}={\bigwedge }_{p=1}^n\mathcal{M}\left\{{\xi }_p\in {\mathcal{B}}_p\right\},$$

(3)

when each ${\xi }_p$ is the independent uncertain variable. Furthermore, a strictly increasing real-valued function, $f(x_1,x_2,\dots ,x_n)$, adheres the following properties:

(i)

$f\left(x_1,x_2,\dots ,x_n\right)\le f(y_1,y_2,\dots ,y_n)$, where $x_i\le y_i,i=\mathrm{1,2},\dots n$

(ii)

$f\left(x_1,x_2,\dots ,x_n\right)<f(y_1,y_2,\dots ,y_n)$, where $x_i<y_i,i=\mathrm{1,2},\dots n$

Theorem 1.

(Liu [35]). Let ${\zeta }_1,{\zeta }_2,\dots ,{\zeta }_r$ be the independent uncertain variables with regular uncertainty distributions ${\psi }_1,{\psi }_2,\dots ,{\psi }_r$, respectively. If the function $\zeta =f\left({\zeta }_1,{\zeta }_2,\dots ,{\zeta }_r\right)$ is continuous and strictly growing, then $\zeta$ can be considered an uncertain variable that follows an inverse uncertainty distribution

$${\mathsf{\psi }}^{-1}\left(\alpha \right)=f\left({\mathsf{\psi }}_1^{-1}\left(\alpha \right),{\mathsf{\psi }}_2^{-1}\left(\alpha \right),\dots ,{\mathsf{\psi }}_r^{-1}\left(\alpha \right)\right).$$

Example 1.

Imagine $\zeta$ as a zigzag uncertain variable with a regular uncertainty distribution $\psi$, and identify it as $\mathcal{Z}(p,q,r)$. Following that, here Equation (4), is the representation of the inverse uncertainty distribution of $\zeta$.

$${\psi }^{-1}\left(\beta \right)=\left\{\begin{array}{ll}\left(1-2\beta \right)p+2\beta q& ;if \beta <0.5\\ 2\left(1-\beta \right)q+\left(2\beta -1\right)r& ;if \beta \ge 0.5.\end{array}\right.$$

(4)

3. Multi-Objective Uncertain Adjacent Only Quadratic Minimum Spanning Tree

In the subsequent section, we focus on how the suggested mUAQMSTP is structured for a network whose parameters are expressed as uncertain variables.

Assume that $G=\left(V,E\right)$ is a weighted connected undirected graph (network) (WCUG) with $n=|V$| and $m=\left|E\right|$, where $V=\{v_1,v_2,\dots ,v_n\}$ and $E=\{e_1,e_2,\dots ,e_k\}$ are the finite sets of vertices and edges of $G$, respectively. Any pair of vertices, $v_i$ and $v_j$, connected by an undirected edge, $e_i\in E$, has a linear weight that stands for a construction or running cost. The quadratic weight, on the other hand, is a measure of the interaction cost that results from selecting two neighboring edges, $e_i$ and $e_j$, from a spanning tree at the same time. The linear weights $\left({\xi }_i\right)$ and the quadratic weights $\left({\zeta }_{ij}\right)$ are regarded as zigzag uncertain variables in this context. The fuel price, the enacted toll tax, the expenses of vehicle overhauls, and the time spent in traffic congestion are just a few of the variables that might cause direct and interactive costs to be uncertain in this case. Consequently, we have presented our problem mUAQMSTP within the framework of uncertainty theory in order to address the indeterminacy, as seen in Equation $\left(4\right)$. Within this framework, we have also included a numerical illustration in Section 7 that pertains to our problem.

$$\left\{\begin{array}{l}Min Z_1={\displaystyle \sum _{i=1}^n}{\xi }_i{x}_i\\ Min Z_2={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{\zeta }_{ij}{x}_i{x}_j\\ subject to\\ {\displaystyle \sum _{i=1}^m}{x}_i=\left|V\right|-1\\ \sum _{i\in E_{\kappa }}{x}_i\le \left|\kappa \right|-1, \left|\kappa \right|\ge 3\\ \hspace{1em}\hspace{1em}{x}_i\in \left\{\mathrm{0,1}\right\}, \forall e_i\in E.\end{array}\right.$$

(5)

In this context, ${\xi }_i$ represents the uncertain direct cost (linear weight) and ${\zeta }_{ij}$ stands for the uncertain interactive cost (quadratic weight) linked to the edge, $e_i$, and the pair of neighbouring edges $(e_i,e_j)$, respectively. In particular, the first objective function seeks to reduce $G$’s uncertain direct cost, whereas the second objective function seeks to decrease $G$’s uncertain interaction costs. The subgraph of $G$ produced by the set of vertices, $\kappa$, is denoted as $H$ with edge set $E_{\kappa }$. Here, $E_{\kappa }$ is the subset of edges in $E$, where both of the end vertices are in $\kappa$. The first constraint in Model (5) is the cardinality constraint, which states that the selected edges must be precisely $\left|V\right|-1$. The second constraint guarantees that the chosen edges do not form a cycle in the network. Every decision variable, $x_i$, in this case can take on a value of $0$ or $1$. When $x_i=1$, the suitable edge, $e_i$, is chosen from $G$ to build an MST. However, $e_i$ is not part of the MST, when $x_i=0$. Following this part, we develop the CCM of mUAQMSTP, which is necessary since Model (5) is a problem of uncertain programming.

Chance-Constrained Model of the Proposed mUAQMSTP

Charnes and Cooper [36] proposed the chance-constrained model (CCM) as an alternative method for optimizing uncertain programming problems. At a predetermined confidence level(s) and subject to chance constraints, the goal of CCM is to minimize or maximize the objective function(s) with respect to a deterministic target value(s) such that the optimized value(s) of the objective function(s) does not surpass the corresponding target value(s). The CCM permits the restrictions to be disregarded. Nevertheless, it guarantees that the limitations are feasible at specific confidence levels. The following is the formula for the related CCM of mUAQMSTP if the decision maker wishes to optimize it under the limitations.

$$\left\{\begin{array}{l}Min {\overline{F}}_1\\ Min {\overline{F}}_2\\ subject to\\ M\left\{\sum _{i=1}^n{\xi }_i{x}_i\le {\overline{F}}_1\right\}\ge {\alpha }_1\\ M\left\{\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{x}_i{x}_j\le {\overline{F}}_2\right\}\ge {\alpha }_2\\ constraints of \left(5\right).\end{array}\right.$$

(6)

The first two constraints in Model (6) have target values denoted as ${\overline{F}}_1$ and ${\overline{F}}_2$, respectively. The predefined confidence levels for the first constraint are denoted as ${\alpha }_1$ and for the second constraint as ${\alpha }_2$. Additionally, when it comes to finding the adjacent only quadratic minimum spanning tree (AQMST) for G, the first constraint defines the total linear cost while heading between the cities at the chance level ${\alpha }_1$, and the second constraint defines the total interactive cost of the adjacent cities going between all the connected cities at the chance level ${\alpha }_2$.

The graphical representation of the proposed mUAQMSTP model from its formulation to the solution methodologies involved to solve it is shown in Figure 1.

4. Deterministic Transformation of mUAQMSTP

A deterministic model of the CCM for the suggested mUAQMSTP is demonstrated here.

Theorem 2.

Assume that ${\xi }_i$ and ${\zeta }_{ij}$ are two uncertain variables with regular distributions, denoted as ${\Psi }_{{\xi }_i}$ and ${\Psi }_{{\zeta }_{ij}}$, respectively. Then Equation (6) reports the deterministic transformation of the CCM, which is subsequently demonstrated in Equation (7) as

$$\left\{\begin{array}{l}Min {\overline{F}}_1={\displaystyle \sum _{i=1}^n}{\mathsf{\Psi }}_{{\xi }_i}^{-1}\left({\mathsf{\alpha }}_1\right)x_i \\ Min {\overline{F}}_2={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{\mathsf{\Psi }}_{{\zeta }_{ij}}^{-1}\left({\mathsf{\alpha }}_2\right)x_i{x}_j\\ subject to\\ \hspace{1em}constraints of \left(5\right).\end{array}\right.$$

(7)

Proof. 

Given that ${\xi }_{c_{ij}}$ and ${\xi }_{t_{ij}}$ are the independent uncertain variables in the Model (6), it follows that the constraints $\mathcal{M}\left\{\sum _{i=1}^n{\xi }_i{x}_i\le {\overline{F}}_1\right\}\ge {\alpha }_1$ and $\mathcal{M}\left\{\sum _{i=1}^n\sum _{j=1}^n{\zeta }_{ij}{x}_i{x}_j\le {\overline{F}}_2\right\}\ge {\alpha }_2$ at the respective chance levels ${\alpha }_1$ and ${\alpha }_2$ can be correspondingly expressed as $\mathcal{M}\left\{F_1\le {\overline{F}}_1\right\}\ge {\alpha }_1$ and $\mathcal{M}\left\{F_2\le {\overline{F}}_2\right\}\ge {\alpha }_2$. In accordance with Theorem 1, the reconstructed values of $\mathcal{M}\left\{F_1\le {\overline{F}}_1\right\}\ge {\alpha }_1$ and $\mathcal{M}\left\{F_2\le {\overline{F}}_2\right\}\ge {\alpha }_2$ are, respectively, denoted as $\sum _{i=1}^n{\mathsf{\Psi }}_{{\xi }_i}^{-1}\left({\mathsf{\alpha }}_1\right)x_i\le {\overline{F}}_1$ and $\sum _{i=1}^n\sum _{j=1}^n{\mathsf{\Psi }}_{{\zeta }_{ij}}^{-1}\left({\mathsf{\alpha }}_2\right)x_{ij}\le {\overline{F}}_2$. Accordingly, Model (7) becomes the deterministic equivalent of the Model (6).

$$\left\{\begin{array}{l}Min {\overline{F}}_1={\displaystyle \sum _{i=1}^n}{\phi }_i{x}_i\\ Min {\overline{F}}_2={\displaystyle \sum _{i=1}^n}{\displaystyle \sum _{j=1}^n}{\varrho }_{ij}{x}_i{x}_j\\ subject to\\ \hspace{1em}constraints of \left(5\right).\end{array}\right.$$

(8)

□

Corollary 1.

Consider the independent zigzag uncertain variables ${\xi }_i=\mathcal{Z}(r_i,s_i,t_i)$ and ${\zeta }_{ij}=\mathcal{Z}(u_{ij},v_{ij},w_{ij})$ where, $r_i<s_i<t_i$ and$u_{ij}<v_{ij}<w_{ij}$, and $r_i,s_i,t_i, u_{ij},v_{ij},w_{ij}\in \mathfrak{R}$. Therefore, Model (7) can be analogously framed as Model (8).

Here, ${\phi }_i$ and ${\varrho }_{ij}$ are verbalized as below.

$${\phi }_i=\left\{\begin{array}{ll}\left(1-2{\alpha }_1\right)r_i+2{\alpha }_1{s}_i& ;if {\alpha }_1<0.5\\ \left(2-2{\alpha }_1\right)s_i+\left(2{\alpha }_1-1\right)t_i& ;if {\alpha }_1\ge 0.5\end{array}\right.$$

and

$${\varrho }_{ij}=\left\{\begin{array}{ll}\left(1-2{\alpha }_2\right)u_{ij}+2{\alpha }_2{v}_{ij}& ;if {\alpha }_2<0.5\\ \left(2-2{\alpha }_2\right)v_{ij}+\left(2{\alpha }_2-1\right)w_{ij}& ;if {\alpha }_2\ge 0.5.\end{array}\right.$$

5. Methodologies for Problem Solving

In this section, the approaches that are utilized in order to solve the deterministic equivalent models of mUAQMSTP are presented. First, there is the classical technique, and second, there are the multi-objective evolutionary algorithms (MOEAs). These are the major categories that these solution techniques fall under. As far as the problem that we presented is concerned, we have utilized the global criterion method as the traditional approach to solving the multi-objective optimization problem (MOOP). Furthermore, in order to solve the equivalent deterministic model of mUAQMSTP, we have taken into consideration the non-dominated sorting genetic algorithm II (NSGAII) and the duplication elimination non-dominated sorting evolutionary algorithm (DENSEA) among the MOEAs. These solution methodologies are discussed elaborately in the subsequent sub sections.

5.1. Global Criterion Method

With the global criterion technique, first proposed by Rao [31], a MOP may be reduced to a compromise single-objective optimization problem (CSOOP). A predefined global criterion, such as the sum of the relative deviations between each objective function and its associated ideal solutions, can be reduced using this method. Finding an acceptable compromise solution between competing goals is the key to solving the CSOOP, which is derived from the analogous MOOP. Model (9) uses the global criterion approach to offer the equivalent formulation of CSOOP for Model (8), the crisp transformation of the mUAQMSTP.

Step 1: Separately evaluate each objective function of the multi-objective problem subject to the same set of constraints as mentioned in Model (8).

Step 2: Based on the outcomes of Step 1, identify the optimal solutions $x^{k^{*}}$ for the Model (8) and find their associated ideal values $f_k^{*}$ = ${\overline{F}}_k \left(x^{k^{*}}\right)$ for all objective functions, where $k$ ranges from $1$ to $m$.

Step 3: Formulate the corresponding compromise model as

$$\left\{\begin{array}{l}Min {\left({\displaystyle \sum _{k=1}^m}{\left({\displaystyle \frac{{\overline{F}}_k\left(x\right)-f_k^{*}}{f_k^{*}}}\right)}^{\omega }\right)}^{{\displaystyle \frac{1}{\omega }}}\\ \hspace{1em}subject to\\ \hspace{1em}\hspace{1em}constraints of \left(5\right).\end{array}\right.$$

(9)

If the value of $\omega$ (where $1\le \omega <\infty$) is taken to be $2$, then the approach is referred to as the global criterion technique in $L_2$ norm.

5.2. Multi-Objective Evolutionary Algorithm

Multiple objectives that are inherently at conflict with one another make up a MOOP. The identification of an ideal solution based on just one objective is not possible, especially when the other objectives are equally important. Both convergence and diversity are essential characteristics of a MOOP, which are implied by the fact that the objective functions are competing. In order to provide a collection of solutions for a MOOP, a multi-objective evolutionary algorithm (MOEA) strives for both convergence and diversity at the same time. Due to the inherent randomness of MOEAs, there is no guarantee that a MOEA will always explore the Pareto optimal solutions of a MOOP. Nevertheless, MOEAs have essential operators that allow them to reliably achieve convergence and an assortment of non-dominated solutions, much like many other evolutionary systems, both natural and artificial, do while improving their solutions. One major advantage of MOEAs over traditional multi-objective solution methods is that they may produce several compromise solutions all in one running of the simulation. Classical methods can only provide a single compromise solution for MOOPs, but MOEAs, due to their population-based approach, can generate several compromise solutions.

In the next sections, we introduce two multi-objective evolutionary algorithms (MOEAs), namely NSGAII and DENSEA. These algorithms are used to generate various solutions that strike a balance between conflicting objectives in the given situation.

5.2.1. NSGAII

Deb et al. [32] introduced the widely used non-dominated sorting genetic algorithm II (NSGAII), a multi-objective evolutionary algorithm (MOEA) with elitism. The preservation of better candidate solutions in subsequent populations is the means by which elitism is accomplished. First, from a population known as $P_0$, the process randomly generates $N$ solutions. In order to ensure that each parent population, $P_q$, produces an equal number of progenies, genetic operators including selection, crossover, and mutation are used during a given generation, $q$. These descendants make up the offspring population,${ C}_q$. Due to this, the parent and offspring populations are combined to form a new population $S_q$ with a size of $2N$. To identify the best $N$ solutions for $S_q$ from a pool of $2N$ options, the crowded tournament selection operator is utilized in the next generation of NSGAII. If the rank of solution $r$ ($r_{rank}$) is lower than the rank of solution $s$ ($s_{rank}$), then solution $r$ will be prioritized by the crowded tournament selection operator. Furthermore, if $r$ and $s$ have equal rank, we compute their crowding distances separately, which are expressed as $r_{distance}$ and $s_{distance}$, respectively. If $r_{distance}$ is more than $s_{distance}$, then $r$ is favored over $s$ in the order of priority. The solutions’ ranking and crowding distance are the main topics of the ensuing discussion.

Solution ranking: A rank is given to each solution, $r$, in the population, $S_q$, denoted as $r_{rank}$. Therefore, the solutions are ranked and then divided into separate non-dominated fronts $N_1, N_2,..., N_w$. All solutions in the front,${ N}_k$, are non-dominated to each other, and are given the same rank, $k$, where $k\in \{\mathrm{1,2},\dots ,w\}$. It is essential to remember that the rank of a solution $r$ in $S_q$ and its associated front number are the same. There is an inherent preference for solutions when $k$ is small since they are better. We start by selecting all the non-dominated solutions that stand for the front $N_1$ in order to create the population $P_{q+1}$ for the ${\left(q+1\right)}^{th}$ generation. The solutions on non-dominated fronts with higher ranks in $S_q$ are disregarded if the number of solutions in $N_1$ is equivalent to the initial population size, $N$. However, when $P_{q+1}$ is generated, the solutions of the subsequent higher-order fronts are taken into account. The aforementioned procedure persists until a certain front, $N_k$, possesses solutions that are not entirely encased inside $P_{q+1}$.

Evaluating crowding distances for the solutions: Deb et al. [32] established the crowding distance as a metric to quantify the density of solutions around a given solution. If the solutions in $N_k$ are not entirely contained in $P_{q+1}$, then a crowding distance value $\left(r_{distance}\right)$ is assigned to every non-dominated solution, $r$, in the front, $N_k$. The crowding distance measures how close the neighboring solutions in the search space are to a certain solution. There is a greater crowding distance between solutions in less densely inhabited regions of the search space compared to those in more densely populated regions. In order to fill the remaining slots of $P_{q+1}$, solutions with a larger crowding distance are always given more priority and chosen from $N_k$. In the subsequent generation, $P_{q+1}$ takes the place of $P_q$ as the parent population following the generation of it. This process iteratively creates a new population for each generation until something like the maximum number of function evaluations or generations is encountered, which are examples of termination conditions.

The time complexity of NSGAII is $O\left(M{N}^2\right)$, as suggested by Deb et al. [32] where $M$ and $N$ are, respectively, the number of objective functions and the population size considered for the problem.

5.2.2. DENSEA

As an adaptation of MOEA, Greiner et al. [33] presented the Duplicate Elimination Non-dominated Sorting Evolutionary Algorithm (DENSEA). By highlighting the need to develop and preserve variety within a population, DENSEA intends to avoid the emergence of a small number of non-dominated solutions. In most cases, the evolutionary process is slowed down when a population of multi-objective evolutionary algorithms (MOEAs) struggles to provide enough non-dominated solutions. This can cause the population to become homogeneous too quickly, which could cause the algorithm to converge too quickly. DENSEA employs the non-domination sorting approach (Deb et al. [32]) to introduce an element of elitism. The capacity to maintain population diversity by eliminating and reintroducing duplicate individuals in subsequent generations is the defining characteristic of DENSEA. Detailed below are the stages that make up the algorithm’s process.

To start the DENSEA execution, a random $N$-size preliminary population ($P_0$) is generated. The chromosomes in $P_0$ are then assessed for their fitness. The next step is to use the non-domination sorting method (Deb et al. [32]) to organize the population’s chromosomes. The parent population, $P_q$, is subjected to genetic operators, such as selection, crossover, and mutation, at each succeeding generation, $q$, resulting in a new population of offspring, $O_q$. So, we calculate $O_q$’s fitness evaluation and use the same non-domination sorting approach to sort the progeny population as we did with the parent population.

Replicated solutions that result from producing fewer non-dominated solutions in the functional space are removed by the deletion operator of DENSEA. Here, we split the population in half lengthwise, creating two groups of size $N/2$: $P_{q_1}$ and $P_{q_2}$. Therefore, when half of the population $\left(P_{q_1}\right)$ has duplicate solutions removed, another half $\left(P_{q_2}\right)$ receives a solution of the same order added to it. The process of replacing a solution that is already there is continued until the population size is half of what it was before $(N⁄2)$. If, for example, $N$ solutions make up the population and the seventh one is a duplicate, the ${\left({\displaystyle \raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}+7\right)}^{th}$ solution from the other half of the population will take their place. The diversity of the population is maintained by replacing duplicates with individuals from different regions in the solution space. Both $P_q$ and $O_q$ undergo this operation to generate the reduced parent and offspring populations, $P_f$ and $O_f$, respectively, with a population size of $N⁄2$. The following generation’s $P_{q+1}$ population is picked using a mix of $P_f$ and $O_f$. Since no genetic information is lost in $P_{q+1}$, its members promote exclusivity within the group while maintaining population variation by replacing half of the duplicate solutions in each generation. The DENSEA maintains this practice until the need to terminate the algorithm based on maximum generation or any other termination criterion is recognized. The algorithmic complexity of DENSEA is $O(MN log N)$ as explained by Greiner et al. [33], for $M$ number of objective functions and the $N$ population size for the problem.

6. Numerical Illustrations

Our proposed mUAQMSTP is demonstrated through an example involving the service network of a recently established bus transportation company. Here is a brief overview of the problem. When planning to launch their service, the bus company focused on seven of the country’s most important towns. To that end, twenty-one potential transit routes connecting the cities are being studied. Additionally, in an effort to entice additional customers, the firm provides a discount on the total cost of the bus ticket to customers who use two of their buses in succession during their trip. In order to maximize efficiency and decrease expenses, the company’s management plans the bus route for the upcoming quarter in a way that minimizes the overall cost of providing bus services between every town, taking into account any potential discounts. Within this framework, the organization also recognizes multiple variables, such as the cost of gasoline, labor, and bus maintenance and overhaul, that impact the cost of providing the transportation service and fundamentally change with time. Because of the complex social and economic factors at play in decision-making problems, such as the multi-objective adjacent only quadratic minimum spanning tree problem, it is not always feasible to precisely establish or obtain the problem parameters in many real-world contexts. Problem parameters that could vary over time include, but are not limited to, the cost of gasoline, the cost of labor, the cost of refurbishment and maintenance, the cost of the toll tax, the length of time spent in traffic, and the cost of vehicle maintenance. Therefore, when adding such characteristics into decision-making problems, it is vital to use the opinion of domain experts to estimate and evaluate the belief degrees of indeterminacy in them. The uncertainty theory paradigm Liu [18,34,35] is a good choice for representing individual beliefs. Therefore, the estimations of the decision makers on the cost of the bus service and the discount amount are considered as uncertain variables.

A suitable numerical example is provided in this part while unfolding the feature of the proposed mUAQMSTP. Here, the project managers consider the possible bus services between the cities as an undirected weighted connected network (UWCN) $G$, $G$ = $(V_G,E_G)$ as shown in Figure 2. In Figure 2, each vertex is represented by a city $v_i,i=\mathrm{1,2},\dots ,\left|V_G\right|$, and an edge $e_j,j=\mathrm{1,2},\dots ,\left|E_G\right|$ represents the possible bus service between the pair of cities to be provided by the company. Each edge of $E_G$ is associated with a linear weight, which determines the possible cost to the company while providing the bus service between a pair of cities. Moreover, the quadratic weight, for any pair of adjacent edges, determines the possible discount on the total bus fare of two consecutive buses of the company when availed by a passenger. These linear and quadratic weights, which are considered as zigzag uncertain variables are reported, respectively, in

Table A1. Linear costs associated with each edge of $G$.

Edge	Linear Cost	Edge	Linear Cost
$e_1$	$\mathcal{Z}\left(539.5 543 546.2\right)$	$e_{12}$	$\mathcal{Z}\left(343.5 345.2 347.1\right)$
$e_2$	$\mathcal{Z}\left(409.4 412.7 414.9\right)$	$e_{13}$	$\mathcal{Z}\left(163.9 167.4 169.4\right)$
$e_3$	$\mathcal{Z}\left(570.1 574.4 576.9\right)$	$e_{14}$	$\mathcal{Z}\left(359.9 362.1 364.8\right)$
$e_4$	$\mathcal{Z}\left(240.6 243.1 247.3\right)$	$e_{15}$	$\mathcal{Z}\left(458.9 462.3 464.7\right)$
$e_5$	$\mathcal{Z}\left(160.2 163.3 167.4\right)$	$e_{16}$	$\mathcal{Z}\left(590.5 592.6 595.4\right)$
$e_6$	$\mathcal{Z}\left(671.2 673.8 678.9\right)$	$e_{17}$	$\mathcal{Z}\left(255.2 257.1 259.9\right)$
$e_7$	$\mathcal{Z}\left(432.5 436.7 439.7\right)$	$e_{18}$	$\mathcal{Z}\left(299.6 302.9 307.1\right)$
$e_8$	$\mathcal{Z}\left(172.2 174.1 175.9\right)$	$e_{19}$	$\mathcal{Z}\left(321.5 324.6 327.6\right)$
$e_9$	$\mathcal{Z}\left(154.5 159.7 161.2\right)$	$e_{20}$	$\mathcal{Z}\left(454.2 459.6 461.7\right)$
$e_{10}$	$\mathcal{Z}\left(322.1 326.3 328.7\right)$	$e_{21}$	$\mathcal{Z}\left(399.2 402.7 405.3\right)$
$e_{11}$	$\mathcal{Z}\left(412.2 415.4 417.7\right)$	$--$	$--$

and

Table A2. Quadratic costs of the adjacent edge pairs of $G$.

Adjacent Edge Pairs	Quadratic Cost	Adjacent Edges Pairs	Quadratic Cost
$e_1{e}_2$	$\mathcal{Z}\left(41.2 43.4 45.6\right)$	$e_9{e}_{20}$	$\mathcal{Z}\left(34.5 36.3 38.6\right)$
$e_1{e}_3$	$\mathcal{Z}\left(33.7 35.7 39.3\right)$	$e_{14}{e}_{18}$	$\mathcal{Z}\left(61.2 63.2 65.2\right)$
$e_1{e}_4$	$\mathcal{Z}\left(21.9 26.3 29.1\right)$	$e_{14}{e}_{19}$	$\mathcal{Z}\left(41.7 43.5 45.2\right)$
$e_1{e}_5$	$\mathcal{Z}\left(10.2 13.7 17.8\right)$	$e_{14}{e}_{20}$	$\mathcal{Z}\left(27.9 29.2 31.6\right)$
$e_1{e}_6$	$\mathcal{Z}\left(49.1 52.1 54.9\right)$	$e_{18}{e}_{19}$	$\mathcal{Z}\left(19.8 21.4 23.6\right)$
$e_2{e}_3$	$\mathcal{Z}\left(51.8 53.7 55.2\right)$	$e_{18}{e}_{20}$	$\mathcal{Z}\left(30.2 32.4 34.8\right)$
$e_2{e}_4$	$\mathcal{Z}\left(61.3 63.7 65.3\right)$	$e_{19}{e}_{20}$	$\mathcal{Z}\left(15.4 17.7 19.7\right)$
$e_2{e}_5$	$\mathcal{Z}\left(11.7 12.9 14.8\right)$	$e_4{e}_{10}$	$\mathcal{Z}\left(63.2 64.4 65.3\right)$
$e_2{e}_6$	$\mathcal{Z}\left(31.7 33.7 36.9\right)$	$e_4{e}_{15}$	$\mathcal{Z}\left(34.6 37.9 39.3\right)$
$e_3{e}_4$	$\mathcal{Z}\left(11.8 14.6 17.9\right)$	$e_4{e}_{16}$	$\mathcal{Z}\left(29.3 31.8 33.4\right)$
$e_3{e}_5$	$\mathcal{Z}\left(41.4 45.3 47.9\right)$	$e_4{e}_{17}$	$\mathcal{Z}\left(25.8 27.8 29.8\right)$
$e_3{e}_6$	$\mathcal{Z}\left(15.9 17.2 19.6\right)$	$e_4{e}_{18}$	$\mathcal{Z}\left(44.3 46.4 48.8\right)$
$e_4{e}_5$	$\mathcal{Z}\left(12.9 15.6 17.9\right)$	$e_{10}{e}_{15}$	$\mathcal{Z}\left(23.5 25.7 27.9\right)$
$e_4{e}_6$	$\mathcal{Z}\left(41.2 43.3 45.2\right)$	$e_{10}{e}_{16}$	$\mathcal{Z}\left(51.2 52.9 54.7\right)$
$e_5{e}_6$	$\mathcal{Z}\left(10.9 11.3 14.7\right)$	$e_{10}{e}_{17}$	$\mathcal{Z}\left(21.3 24.5 25.3\right)$
$e_1{e}_7$	$\mathcal{Z}\left(10.7 12.2 14.1\right)$	$e_{10}{e}_{18}$	$\mathcal{Z}\left(33.2 35.7 37.4\right)$
$e_1{e}_{12}$	$\mathcal{Z}\left(51.2 52.3 54.2\right)$	$e_{15}{e}_{16}$	$\mathcal{Z}\left(30.2 32.9 34.2\right)$
$e_1{e}_{16}$	$\mathcal{Z}\left(60.9 62.7 64.5\right)$	$e_{15}{e}_{17}$	$\mathcal{Z}\left(61.2 63.7 65.3\right)$
$e_1{e}_{19}$	$\mathcal{Z}\left(20.3 23.2 25.2\right)$	$e_{15}{e}_{18}$	$\mathcal{Z}\left(21.2 23.4 25.4\right)$
$e_1{e}_{21}$	$\mathcal{Z}\left(37.3 39.2 41.9\right)$	$e_{16}{e}_{17}$	$\mathcal{Z}\left(44.2 46.9 47.3\right)$
$e_7{e}_{12}$	$\mathcal{Z}\left(27.2 29.3 31.2\right)$	$e_{16}{e}_{18}$	$\mathcal{Z}\left(31.2 34.4 35.7\right)$
$e_7{e}_{16}$	$\mathcal{Z}\left(23.1 25.4 27.5\right)$	$e_{17}{e}_{18}$	$\mathcal{Z}\left(33.4 35.8 39.3\right)$
$e_7{e}_{19}$	$\mathcal{Z}\left(61.3 63.9 64.9\right)$	$e_5{e}_{11}$	$\mathcal{Z}\left(21.3 23.4 25.6\right)$
$e_7{e}_{21}$	$\mathcal{Z}\left(24.2 25.3 27.2\right)$	$e_5{e}_{12}$	$\mathcal{Z}\left(22.3 25.4 26.5\right)$
$e_{12}{e}_{16}$	$\mathcal{Z}\left(51.2 53.4 55.4\right)$	$e_5{e}_{13}$	$\mathcal{Z}\left(33.4 36.7 38.9\right)$
$e_{12}{e}_{19}$	$\mathcal{Z}\left(21.2 23.7 25.2\right)$	$e_5{e}_{14}$	$\mathcal{Z}\left(22.3 24.6 26.3\right)$
$e_{12}{e}_{21}$	$\mathcal{Z}\left(25.8 27.1 29.2\right)$	$e_5{e}_{15}$	$\mathcal{Z}\left(32.1 36.4 37.7\right)$
$e_{16}{e}_{19}$	$\mathcal{Z}\left(32.1 34.4 36.2\right)$	$e_{11}{e}_{12}$	$\mathcal{Z}\left(20.2 22.4 27.5\right)$
$e_{16}{e}_{21}$	$\mathcal{Z}\left(31.2 37.1 39.2\right)$	$e_{11}{e}_{13}$	$\mathcal{Z}\left(63.2 64.9 67.3\right)$
$e_{19}{e}_{21}$	$\mathcal{Z}\left(25.2 27.5 29.3\right)$	$e_{11}{e}_{14}$	$\mathcal{Z}\left(22.3 24.4 27.8\right)$
$e_2{e}_8$	$\mathcal{Z}\left(33.3 35.3 38.9\right)$	$e_{11}{e}_{15}$	$\mathcal{Z}\left(43.3 45.5 48.6\right)$
$e_2{e}_{13}$	$\mathcal{Z}\left(23.2 26.1 28.2\right)$	$e_{12}{e}_{13}$	$\mathcal{Z}\left(43.4 42.3 45.7\right)$
$e_2{e}_{17}$	$\mathcal{Z}\left(53.1 54.5 56.2\right)$	$e_{12}{e}_{14}$	$\mathcal{Z}\left(31.8 32.5 33.5\right)$
$e_2{e}_{20}$	$\mathcal{Z}\left(21.3 25.7 29.2\right)$	$e_{12}{e}_{15}$	$\mathcal{Z}\left(51.2 52.7 56.7\right)$
$e_2{e}_{21}$	$\mathcal{Z}\left(51.2 54.3 56.2\right)$	$e_{13}{e}_{14}$	$\mathcal{Z}\left(53.4 55.6 58.4\right)$
$e_8{e}_{13}$	$\mathcal{Z}\left(23.1 25.9 28.2\right)$	$e_{13}{e}_{15}$	$\mathcal{Z}\left(29.3 29.7 29.9\right)$
$e_8{e}_{17}$	$\mathcal{Z}\left(47.1 49.2 51.2\right)$	$e_{14}{e}_{15}$	$\mathcal{Z}\left(17.6 19.3 21.4\right)$
$e_8{e}_{20}$	$\mathcal{Z}\left(23.2 26.3 27.9\right)$	$e_6{e}_7$	$\mathcal{Z}\left(21.4 23.6 24.7\right)$
$e_8{e}_{21}$	$\mathcal{Z}\left(37.2 38.2 39.2\right)$	$e_6{e}_8$	$\mathcal{Z}\left(23.9 25.7 27.7\right)$
$e_{13}{e}_{17}$	$\mathcal{Z}\left(33.2 36.1 38.5\right)$	$e_6{e}_9$	$\mathcal{Z}\left(52.3 56.6 57.4\right)$
$e_{13}{e}_{20}$	$\mathcal{Z}\left(56.2 57.9 59.4\right)$	$e_6{e}_{10}$	$\mathcal{Z}\left(33.2 34.5 38.4\right)$
$e_{13}{e}_{21}$	$\mathcal{Z}\left(37.4 38.9 41.2\right)$	$e_6{e}_{11}$	$\mathcal{Z}\left(55.2 57.6 59.4\right)$
$e_{17}{e}_{20}$	$\mathcal{Z}\left(51.4 53.7 56.2\right)$	$e_7{e}_8$	$\mathcal{Z}\left( 21.4 24.4 26.4\right)$
$e_{17}{e}_{21}$	$\mathcal{Z}\left(42.1 43.2 45.2\right)$	$e_7{e}_9$	$\mathcal{Z}\left(51.4 56.5 57.4\right)$
$e_{20}{e}_{21}$	$\mathcal{Z}\left(40.2 42.1 44.7\right)$	$e_7{e}_{10}$	$\mathcal{Z}\left(70.2 71.4 73.6\right)$
$e_3{e}_9$	$\mathcal{Z}\left(51.3 52.4 54.4\right)$	$e_7{e}_{11}$	$\mathcal{Z}\left(74.3 78.4 79.8\right)$
$e_3{e}_{14}$	$\mathcal{Z}\left(23.4 25.4 26.3\right)$	$e_8{e}_9$	$\mathcal{Z}\left(44.3 46.4 48.9\right)$
$e_3{e}_{18}$	$\mathcal{Z}\left(26.4 28.3 30.2\right)$	$e_8{e}_{10}$	$\mathcal{Z}\left(34.2 36.3 38.8\right)$
$e_3{e}_{19}$	$\mathcal{Z}\left(45.3 47.4 49.2\right)$	$e_8{e}_{11}$	$\mathcal{Z}\left(61.3 63.8 69.9\right)$
$e_3{e}_{20}$	$\mathcal{Z}\left(34.2 36.3 38.4\right)$	$e_9{e}_{10}$	$\mathcal{Z}\left(61.2 63.7 64.3\right)$
$e_9{e}_{14}$	$\mathcal{Z}\left(41.2 42.3 45.2\right)$	$e_9{e}_{11}$	$\mathcal{Z}\left(33.2 34.5 35.6\right)$
$e_9{e}_{18}$	$\mathcal{Z}\left(25.2 27.4 29.3\right)$	$e_{10}{e}_{11}$	$\mathcal{Z}\left(31.2 33.4 35.3\right)$
$e_9{e}_{19}$	$\mathcal{Z}\left(41.4 42.9 44.3\right)$	$--$	$--$

(cf. Appendix A).

Consequently, the parameters of $G$ are modelled within the context of uncertainty theory in this investigation. In this case, the uncertain parameter, ${\xi }_i$, represents the linear cost (weight) associated to an edge, $e_i$, in $G$. While the indeterminate parameter, ${\zeta }_{ij}$, represents the quadratic cost (weight) that corresponds to two adjacent edges, $e_i$ and $e_j$.

In order to determine an optimal bus route, the project managers determine a multi-objective uncertain adjacent quadratic minimum spanning tree (mUAQMST) of $G$ using the deterministic Model, (8), of the CCM (as demonstrated in Model (6)) of the proposed mUAQMST (cf. Model (5)). Model (8) is the MOOP and, thus, we used the global criterion method, a classical multi-objective solution technique, on its related compromise Model (9). The model’s compromise solution is found using an optimization problem solver, Lingo 11.0. Subsequently, it is important to note that by adjusting the the chance (confidence) level to $0.$4 and $0.9$, we may solve Model (9). Consequently,

Table 1. The mUAQMSTs of $G$ as obtained by solving the compromise deterministic equivalents of CCM at confidence levels of 0.4 and 0.9.

CCM
Confidence Level at 0.4			Confidence Level at 0.9
${\overline{\mathit{F}}}_1$	${\overline{\mathit{F}}}_2$	$\mathbf{Minimum} \mathbf{Spanning} \mathbf{Tree} \mathbf{of} \mathit{G}$	${\overline{\mathit{F}}}_1$	${\overline{\mathit{F}}}_2$	$\mathbf{Minimum} \mathbf{Spanning} \mathbf{Tree} \mathbf{of} \mathit{G}$
$1289.46$	$194.28$	$\left\{e_5,e_8,e_{10},e_{11},e_{19},e_{21}\right\}$	$1323.18$	$196.38$	$\left\{e_4,e_5,e_8,e_{13},e_{18},e_{21}\right\}$

reports the compromise solutions of Model (9), where we can observe that with $0.$4 and $0.9$ confidence levels for both the model’s objective functions, respectively, two distinct quadratic minimum spanning trees are produced. Here, in Table 1, ${\overline{F}}_1$ represents the total possible expense for providing bus services between all the cities and ${\overline{F}}_2$ determines the total possible discount to be offered by the company at two different confidence levels (i.e., 0.4 and 0.9) as set by the decision makers (project managers).

It should be noted that the project managers aim to find the most efficient bus route by taking into account (i) the entire cost of running the service between all cities and (ii) the total discount that may be provided on ticket fares of the customers who will avail two successive buses of the company. The associated deterministic multi-objective Model (8) is solved using the two MOEAs, NSGAII, and DENSEA, to determine multiple non-dominated solutions of the CCM of $G$ at confidence levels 0.4 and 0.9. The population size of all the MOEAs is considered as 100 and there is a 500-generations limit for each MOEAs. In this case, NSGAII and DENSEA both employ bit-flip mutation, single-point crossover, and binary tournament selection as genetic operators. On top of that, the crossover probability is 0.9, and the mutation probability is 0.05 for both MOEAs. Furthermore, the non-dominated solutions of Model (8) for $G$ are generated using a jMetal4.5 framework (Durillo and Nebro [37]).

Table 2. For a confidence level of $0.4$, the non-dominated solutions obtained by solving Model (9) of $G$ by employing the MOEAs. The values which are same for the global criterion method, NSGAII and DENSEA are highlighted as bold.

NSGAII		DENSEA
$\mathbf{L}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{r} \mathbf{C}\mathbf{o}\mathbf{s}\mathbf{t} \left({\overline{\mathit{F}}}_1\right)$	$\mathbf{Quadratic} \mathbf{Cost} \left({\overline{\mathit{F}}}_2\right)$	$\mathbf{L}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{r} \mathbf{C}\mathbf{o}\mathbf{s}\mathbf{t} \left({\overline{\mathit{F}}}_1\right)$	$\mathbf{Quadratic} \mathbf{Cost} \left({\overline{\mathit{F}}}_2\right)$
1960.36	114.60	1206.60	230.62
1206.60	230.62	2094.02	116.72
1373.40	134.42	1373.40	134.42
1366.00	183.94	1832.04	131.24
1220.72	210.04	1366.00	183.94
1289.46	194.28	2093.28	127.44
1304.66	184.12	1220.72	210.04
$--$	$--$	1289.46	194.28
$--$	$--$	1304.66	184.12

and

Table 3. For a confidence level of $0.9$, the non-dominated solutions obtained by solving Model (9) of $G$ by employing the MOEAs. The values which are same for the global criterion method, NSGAII and DENSEA are highlighted as bold.

NSGAII		DENSEA
$\mathbf{L}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{r} \mathbf{C}\mathbf{o}\mathbf{s}\mathbf{t} \left({\overline{\mathit{F}}}_1\right)$	$\mathbf{Quadratic} \mathbf{Cost} \left({\overline{\mathit{F}}}_2\right)$	$\mathbf{L}\mathbf{i}\mathbf{n}\mathbf{e}\mathbf{r} \mathbf{C}\mathbf{o}\mathbf{s}\mathbf{t} \left({\overline{\mathit{F}}}_1\right)$	$\mathbf{Quadratic} \mathbf{Cost} \left({\overline{\mathit{F}}}_2\right)$
1224.74	242.18	1224.74	242.18
1834.3	142.68	2374.52	119.02
1392.06	144.66	1392.06	144.66
1323.18	196.38	1721.92	124.42
1237.62	222.38	1383.06	193.42
1306.5	206.50	2118.18	119.94
$--$	$--$	1237.62	222.38
$--$	$--$	1306.50	206.50
$--$	$--$	1323.18	196.38

detail these solutions that did not dominate each other, meaning they are not inferior. Here, each solution represents a possible bus route, which the decision makers can select. In this situation, we find that for both the 0.4 and 0.9 confidence levels of CCM, one of the non-inferior solutions generated by the MOEAs is the same as the solution generated by the global criterion approach (see Table 1). The bold solutions stand out. It can be shown from Table 2 and Table 3 that DENSEA produces a higher number of non-dominated solutions for the CCM of $G$ rather than NSGAII. Figure 3 and Figure 4 provide a visual depiction of the non-dominated solutions that are created for the CCM. Additionally, Figure 5a, and Figure 5b also provide graphical illustrations of the possible bus routes corresponding to the nondominated solutions which are highlighted in bold as reported, respectively, in Table 2 and Table 3.

7. Results and Discussions

We have purposefully developed four arbitrary instances of mUAQMSTP for the sole purpose of this discussion. Instances $mUAQMST\_Instance10$, $mUAQMST\_Instance20, mUAQMST\_Instance30$, and $mUAQMST\_Instance40$ consist of ten, twenty, thirty and forty vertices, which eventually make up these instances, in that order. With $n$ vertices and $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)$ edges, each of these instances is a UWCN and a complete graph. Here, we have expressed the edge weight of an edge, $e_i$, for each occurrence as an imprecise linear cost, ${\xi }_i$. In addition, for any one of these cases, there is an imprecise variable, ${\zeta }_{ij}$, which represents a quadratic cost for any two neighboring edges, $e_i$ and $e_j$. In this context, ${\xi }_i$ and ${\zeta }_{ij}$ are viewed as zigzag uncertain variables. These variables are represented as $\mathcal{Z}(l_{{\xi }_i},m_{{\xi }_i},n_{{\xi }_i})$ and $\mathcal{Z}(l_{{\zeta }_{ij}},m_{{\zeta }_{ij}},n_{{\zeta }_{ij}})$ which are created at random to represent the related ${\xi }_i$ and ${\zeta }_{ij}$, respectively. In particular, the values of $l_{{\xi }_i},m_{{\xi }_i}$, and $n_{{\xi }_i}$ are generated at random from the interval $\left[700.5,150.5\right]$ for every edge, $e_i$, of an instance, with the constraint that $l_{{\xi }_i}<m_{{\xi }_i}<n_{{\xi }_i}$. Thereafter, $l_{{\zeta }_{ij}},m_{{\zeta }_{ij}}$ and $n_{{\zeta }_{ij}}$ are chosen at random from the interval $\left[\mathrm{10.5,80.5}\right]$ for each ${\zeta }_{ij}$, with the condition that $l_{{\zeta }_{ij}}<m_{{\zeta }_{ij}}<n_{{\zeta }_{ij}}$ is satisfied.

Taking into account all four of the aforementioned mUAQMSTP instances, NSGAII and DENSEA, subsequently, solve the corresponding crisp equivalent Model (8) for the CCM at 0.4 and 0.9 confidence levels. The two most important performance measures that we use to evaluate the MOEAs on these four instances are hypervolume (HV) (Zitzler et al. [38]) and inverted generational distance (IGD) (Van Veldhuizen and Lamont [39]). A greater value is always preferred for HV. Nevertheless, lesser values are more likely to be attained for IGD. Consequently, HV and IGD, two performance measurements, corroborate each other that non-dominated solutions generated by an MOEA tend to ensure diversity and convergence. For each instance, at both confidence levels, a jMetal4.5 framework (Durillo, Nebro [37]) is used to execute NSGAII and DENSEA on the Model (8). Because MOEAs are inherently stochastic, we have chosen to run each algorithm with 500 generations, 100 times, separately, with the population size of all the MOEAs considered as 100. Subsequently, for both the MOEAs, NSGAII, and DENSEA we employ bit-flip mutation, single-point crossover, and binary tournament selection as genetic operators with their respective crossover probability as 0.9 and the mutation probability set to 0.05. Here, as indicated in Section 6, the genetic operators and their associated parameters are identical for both methods.

The majority of real-world multi-objective scenarios typically make it impossible to obtain the collection of Pareto solutions indicated in a Pareto front (PF). To estimate the PF of an instance, a reference front is created that considers the non-inferior solutions from the first front. In this study, this front is constructed after each run of NSGAII and DENSEA on a specific instance. Following the establishment of the reference front, the solutions that constitute it are employed in the assessment of performance indicators.

By calculating the $mean$, standard deviation ($sd$), $median$, and interquartile range ($IQR$), we conduct the statistical analysis of the performance metrics. The $mean$ and $sd$ of all performance metrics are shown in

Table 4. $Mean$ and $sd$ of the $HV$ and $IGD$ for the mUAQMSTP instances considering Model (8). The superior values are highlighted in bold.

Confidence Level	MOEA	Uncertain Instances
		$\mathit{m}\mathit{U}\mathit{A}\mathit{Q}\mathit{M}\mathit{S}\mathit{T}\mathit{P}\_\mathit{I}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}10$				$\mathit{m}\mathit{U}\mathit{A}\mathit{Q}\mathit{M}\mathit{S}\mathit{T}\mathit{P}\_\mathit{I}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}20$				$\mathit{m}\mathit{U}\mathit{A}\mathit{Q}\mathit{M}\mathit{S}\mathit{T}\mathit{P}\_\mathit{I}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}30$				$\mathit{m}\mathit{U}\mathit{A}\mathit{Q}\mathit{M}\mathit{S}\mathit{T}\mathit{P}\_\mathit{I}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}40$
		$\mathit{H}\mathit{V}$		$\mathit{I}\mathit{G}\mathit{D}$		$\mathit{H}\mathit{V}$		$\mathit{I}\mathit{G}\mathit{D}$		$\mathit{H}\mathit{V}$		$\mathit{I}\mathit{G}\mathit{D}$		$\mathit{H}\mathit{V}$		$\mathit{I}\mathit{G}\mathit{D}$
0.4		$mean$	$sd$	$mean$	$sd$	$mean$	$sd$	$mean$	$sd$	$mean$	$sd$	$mean$	$sd$	$mean$	$sd$	$mean$	$sd$
	NSGAII	6.178 × 10−1	8.80 × 10−2	6.245 × 10−4	3.46 × 10−4	6.636 × 10−1	8.93 × 10−2	4.436 × 10−4	2.20 × 10−4	6.420 × 10−1	1.04 × 10−1	5.422 × 10−4	2.62 × 10−4	6.851 × 10−1	1.19 × 10−1	7.240 × 10−4	2.48 × 10−4
	DENSEA	6.565 × 10−1	1.55 × 10−1	4.782 × 10−4	2.62 × 10−4	6.747 × 10−1	1.19 × 10−1	3.979 × 10−4	2.17 × 10−4	6.620 × 10−1	1.26 × 10−1	4.877 × 10−4	2.25 × 10−4	7.012 × 10−1	1.11 × 10−1	6.318 × 10−4	2.41 × 10−4
0.9	NSGAII	6.073 × 10−1	8.32 × 10−2	5.823 × 10−4	3.22 × 10−4	6.531 × 10−1	8.44 × 10−2	3.874 × 10−4	3.75 × 10−4	6.315 × 10−1	9.88 × 10−2	5.000 × 10−4	2.38 × 10−4	6.697 × 10−1	8.82 × 10−2	7.007 × 10−4	2.12 × 10−4
	DENSEA	6.158 × 10−1	1.09 × 10−1	4.360 × 10−4	2.38 × 10−4	6.723 × 10−1	1.04 × 10−1	1.93 × 10−4	1.81 × 10−4	6.610 × 10−1	1.14 × 10−1	4.644 × 10−4	1. 89 × 10−4	7.087 × 10−1	1.06 × 10−1	5.895 × 10−4	2.17 × 10−4

, while the $median$ and $IQR$ of both the performance metrics for all mUAQMSTP instances are shown in

Table 5. $Median$ and $IQR$ corresponding to $HV$ and $IGD$ for the mUAQMSTP instances considering Model (8). The superior values are highlighted in bold.

Confidence Level	MOEA	Uncertain Instances
		$\mathit{m}\mathit{U}\mathit{A}\mathit{Q}\mathit{M}\mathit{S}\mathit{T}\mathit{P}\_\mathit{I}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}10$				$\mathit{m}\mathit{U}\mathit{A}\mathit{Q}\mathit{M}\mathit{S}\mathit{T}\mathit{P}\_\mathit{I}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}20$				$\mathit{m}\mathit{U}\mathit{A}\mathit{Q}\mathit{M}\mathit{S}\mathit{T}\mathit{P}\_\mathit{I}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}30$				$\mathit{m}\mathit{U}\mathit{A}\mathit{Q}\mathit{M}\mathit{S}\mathit{T}\mathit{P}\_\mathit{I}\mathit{n}\mathit{s}\mathit{t}\mathit{a}\mathit{n}\mathit{c}\mathit{e}40$
		$\mathit{H}\mathit{V}$		$\mathit{I}\mathit{G}\mathit{D}$		$\mathit{H}\mathit{V}$		$\mathit{I}\mathit{G}\mathit{D}$		$\mathit{H}\mathit{V}$		$\mathit{I}\mathit{G}\mathit{D}$		$\mathit{H}\mathit{V}$		$\mathit{I}\mathit{G}\mathit{D}$
0.4		$median$	$IQR$	$median$	$IQR$	$median$	$IQR$	$median$	$IQR$	$median$	$IQR$	$median$	$IQR$	$median$	$IQR$	$median$	$IQR$
	NSGAII	6.239 × 10−1	1.56 × 10−1	6.484 × 10−4	6.11 × 10−4	6.698 × 10−1	1.58 × 10−1	$4.653\times {10}^{-4}$	$3.88\times {10}^{-4}$	$6.492\times {10}^{-1}$	$\mathbf{1.83}\times {\mathbf{10}}^{-\mathbf{1}}$	$5.603\times {10}^{-4}$	$4.64\times {10}^{-4}$	$6.956\times {10}^{-1}$	$1.95\times {10}^{-1}$	$7.411\times {10}^{-4}$	$4.38\times {10}^{-4}$
	DENSEA	6.499 × 10−1	1.95 × 10−1	4.963 × 10−4	4.64 × 10−4	6.756 × 10−1	1.79 × 10−1	$\mathbf{4.128}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{3.84}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{6.667}\times {\mathbf{10}}^{-\mathbf{1}}$	$2.04\times {10}^{-1}$	$\mathbf{5.032}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{3.98}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{7.122}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{1.81}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{6.484}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{4.26}\times {\mathbf{10}}^{-\mathbf{4}}$
0.9	NSGAII	6.131 × 10−1	1.47 × 10−1	6.045 × 10−4	5.69 × 10−4	6.590 × 10−1	1.49 × 10−1	$4.006\times {10}^{-4}$	$3.41\times {\mathbf{10}}^{-\mathbf{4}}$	$6.383\times {10}^{-1}$	$\mathbf{1.75}\times {\mathbf{10}}^{-\mathbf{1}}$	$5.164\times {\mathbf{10}}^{-\mathbf{4}}$	$4.77\times {10}^{-4}$	$6.758\times {10}^{-1}$	$\mathbf{1.56}\times {\mathbf{10}}^{-\mathbf{1}}$	$7.153\times {10}^{-4}$	$\mathbf{6.05}\times {\mathbf{10}}^{-\mathbf{4}}$
	DENSEA	6.233 × 10−1	1.93 × 10−1	4.524 × 10−4	4.22 × 10−4	6.796 × 10−1	1.85 × 10−1	$\mathbf{3.870}\times {\mathbf{10}}^{-\mathbf{1}}$	$\mathbf{3.21}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{6.689}\times {\mathbf{10}}^{-\mathbf{1}}$	$2.02\times {10}^{-1}$	$\mathbf{4.220}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{3.34}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{7.161}\times {\mathbf{10}}^{-\mathbf{1}}$	$1.87\times {10}^{-1}$	$\mathbf{3.750}\times {\mathbf{10}}^{-\mathbf{4}}$	$6.84\times {10}^{-4}$

. Values that are more favorable are shown in bold in each of these tables. Looking at the $mean$ and $median$, respectively, in Table 4 and Table 5, we observe that DENSEA outperforms NSGAII for all performance measures, for all mUAQMSTP instances. Here, for all the instances in Table 4, DENSEA achieve better $mean$ for HV and IGD at both the confidence levels of 0.4 and 0.9. However, it can be seen from Table 4 that NSGAII generates a better $sd$ for HV for three mUAQMSTP instances. However, DENSEA produces better a $sd$ for IGD for four mUAQMSTP instances at the confidence level 0.4. Similarly, when the confidence level is set to 0.9, NSGAII generates a superior $sd$ for HV for four instances, and DENSEA produces a better $sd$ for IGD for three instances. Similar, analysis can be performed considering the $IQR$ values in Table 5 at both the confidence levels 0.4 and 0.9.

This suggests that NSGAII and DENSEA are more consistent, respectively, for the HV and IGD, which are generated corresponding to each of the 100 executions of the MOEAs. Figure 6 and Figure 7 support the same hypothesis of Table 5 at the 0.4 and 0.9 confidence levels, respectively. In particular, these figures use letter–value (L-V) plots [40] to graphically explain the HV and IGD values that are related to Table 5. Specifically, the L-V plots in Figure 6 and Figure 7 for the respective confidence levels 0.4 and 0.9, display the medians as the horizontal lines along with many quartiles. Here, it can be observed that DENSEA outperforms NSGAII in terms of HV and IGD, since DENSEA has superior median performance indicators for all the instances compared to NSGAII for both the confidence levels.

Moreover, we have graphically elucidated Table 4 in terms of displaying the difference of the mean values of the 100 observations for HV and IGD generated for both NSGAII and DENSEA on the deterministic models of the four uncertain instances at two different confidence levels, 0.4 and 0.9. For this purpose, we have displayed Gardner–Altman plots (Gardner and Altman [41]) as depicted in Figure 8 and Figure 9. In each of these plots the markers ‘X’ and ‘Y’ in the x-axis represent the vector of data points generated by the NSGAII and DENSEA, respectively. Considering all the HV values in Figure 8, it has been clearly observed that the mean corresponding to ‘Y’ is evidently greater than that of ‘X’ for both the confidence levels. Whereas, in Figure 9, the mean values of ‘Y’ are desirably less than that of ‘X’ for the IGD.

We have also compared the execution time of the algorithms used in this study, i.e., NSGAII and DENSEA, for all instances. Here, it is to be mentioned that we also considered another algorithm multi-objective cross generational elitist selection, heterogeneous recombination and cataclysmic mutation (MOCHC) (Nebro et al. [42]) from the literature. The parameter settings of the algorithms are the population size of 100, the 500-generation limit for each MOEA, and the crossover probability and the mutation probability, which are, respectively, set as 0.9 and 0.05. The test platform used for the simulation of our proposed study is a personal computer with Intel (R) Core ™ [email protected] GHz and 4 MB memory. The execution times of all the instances are shown in

Table 6. Computation time of the MOEAs on the four mUAQMSTP instances. The superior values are highlighted in bold.

Instances	Computation Time (in Seconds)
	DENSEA	NSGAII	MOCHC
$mUAQMST\_Instance10$	1.05	1.94	1.27
$mUAQMST\_Instance20$	1.47	2.04	1.78
$mUAQMST\_Instance30$	3.38	4.87	4.32
$mUAQMST\_Instance40$	4.93	6.17	5.59

. Here, we can observe that for all the instances, DENSEA takes minimum computation time (highlighted in bold) to generate the solutions compared to its counterparts.

Furthermore, considering all the four mUAQMSTP instances, we perform a two-level hypothesis testing on the performance metrics, HV and IGD. Here, at first, the null hypothesis ($h_0$) is tested for the two-sample parametric $t$-test for which the p-value is set as $p^{+}$. Here, if the $h_0$ is rejected, then we accept the value of ${ p}^{+}$. However, if $h_0$ is accepted, then we conduct the two-sided nonparametric Wilcoxon signed-rank test for the same $h_0$ and observe its corresponding p-value, which is represented as $p^{-}$. In this context, it should be mentioned that as the distributions of the generation of the HV and IGD are unknown; therefore, we performed two-level hypothesis testing comprising both parametric and non-parametric tests. Accordingly, the $h_0$ for the test is set as follow:

$h_0:$ Significantly equal performance metrics are engendered by NSGAII and DENSEA.

Subsequently, in

Table 7. p-value of hypothesis testing of HV and IGD for the four mUAQMSTP instances.

Uncertain Instance	$\mathbf{Confidence} \mathbf{Level}$
	$0.4$		$0.9$
	$\mathit{H}\mathit{V}$	$\mathit{I}\mathit{G}\mathit{D}$	$\mathit{H}\mathit{V}$	$\mathit{I}\mathit{G}\mathit{D}$
	$\mathbf{DENSEA}\star$NSGAII	$\mathbf{DENSEA}\star$NSGAII	$\mathbf{DENSEA}\star$NSGAII	$\mathbf{DENSEA}\star$NSGAII
$mUAQMST$ $\_Instance10$	${\prec }_{p^{-}=4.314\times {10}^{-3}}$	${\prec }_{p^{-}=8.904\times {10}^{-4}}$	${\prec }_{p^{-}=5.360\times {10}^{-3}}$	${\prec }_{p^{-}=7.566\times {10}^{-4}}$
$mUAQMST$ $\_Instance20$	${\prec }_{p^{-}=4.560\times {10}^{-3}}$	${\prec }_{p^{+}=3.476\times {10}^{-3}}$	${\prec }_{p^{-}=1.545\times {10}^{-3}}$	${\prec }_{p^{-}=6.273\times {10}^{-4}}$
$mUAQMST$ $\_Instance30$	${\prec }_{p^{+}=2.223\times {10}^{-3}}$	${\prec }_{p^{-}=1.158\times {10}^{-4}}$	${\prec }_{p^{-}=5.217\times {10}^{-4}}$	${\prec }_{p^{-}=2.428\times {10}^{-4}}$
$mUAQMST$ $\_Instance40$	${\prec }_{p^{-}=3.258\times {10}^{-3}}$	${\prec }_{p^{+}=3.753\times {10}^{-4}}$	${\prec }_{p^{-}=5.529\times {10}^{-4}}$	${\prec }_{p^{+}=4.779\times {10}^{-4}}$

we provide the result of the hypothesis testing, which is achieved at 5% level of significance. In this table, the column DENSEA$\star$NSGAII conclude either of the following two conditions.

(i)

DENSEA${\prec }_{p^{+}or p^{-}}$NSGAII: At 5% level of significance, DENSEA is significantly better than NSGAII, and $h_0$ is not accepted at $p^{+}or p^{-}$.

(ii)

DENSEA${\succ }_{p^{+} or p^{-}}$NSGAII: At 5% level of significance, DENSEA is significantly better than NSGAII, and $h_0$ is accepted at $p^{+} or p^{-}$.

Accordingly, it is observed from the reported p-values in Table 7, DENSEA is significantly better than NSGAII for HV and IGD at a 5% level of significance and at the corresponding confidence levels 0.4 and 0.9. Furthermore, here in Table 7, we observe that when the confidence level is set to 0.4, then for the instance,$mUAQMST\_Instance30$, $h_0$ is rejected for HV, and for the instances $mUAQMST\_Instance20$ and $mUAQMST\_Instance40$, $h_0$ is rejected for IGD when two-sample parametric $t$-test is conducted. Similarly, by conducting the same hypothesis testing, for the instance, $mUAQMST\_Instance40$, $h_0$ is rejected for IGD at the 0.9 confidence level. However, at both the confidence levels of 0.4 and 0.9, for all the remaining instances, $h_0$ is accepted for two sample parametric $t$-test and, therefore, by conducting the two-sided nonparametric Wilcoxon signed-rank test, the same $h_0$ is rejected and subsequently, the corresponding $p$-values are calculated.

8. Conclusions

This work is unique since it examines an adjacent only quadratic minimum spanning tree problem with indeterminate parameters that involve multiple objectives. In this article, we provide the proposed mUAQMSTP that minimizes the uncertainty parameters in a quadratic minimum spanning tree (QMST) by minimizing the linear cost of each edge and the quadratic cost of any two neighboring edges. In this case, we optimize both the linear and quadratic costs inside a multi-objective framework to investigate their relative merits. We have developed the CCM of mUAQMSTP based on the uncertainty theory and addressed the deterministic transformation of the problem at two distinct confidence levels. The global criterion approach and two MOEAs, namely NSGAII and DENSEA, are used to solve these deterministic models. After that, an effective numerical example is provided on which the models are employed. Thereafter, we evaluate and examine the MOEAs’ performance on four bigger mUAQMSTP instances which are generated randomly.

The proposed model of mUAQMSTP is essentially not suitable to model multi-objective versions of the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict pair constraints under the uncertain framework. Furthermore, the proposed mUAQMSTP is modeled by considering the zigzag uncertain variables, which capture the real-life uncertainty by following the zigzag uncertainty distribution (Liu [13]); however, if the uncertainty is required to be captured considering the normal uncertainty distribution (Liu [13]), then our proposed model is required to be reformulated using the normal uncertain variable. Here, the zigzag uncertainty distribution can model uncertainty, where there is insufficient historical evidence as well as very few expert opinions. However, to model uncertainty with the normal uncertain variable, there should be much more expert opinions that can be approximated to normal uncertainty distribution to describe an uncertain event. In addition, there is also the linear uncertain variable, which follows the linear uncertainty distribution (Liu [13]). However, since the zigzag uncertain variable is the generalization of the linear uncertain variable, therefore, we have not considered the model formulation of mUAQMSTP with the linear uncertain variable.

In the future, mitigating the limitation of the study by extending our study’s suggested mUAQMSTP to several versions of the multi-objective quadratic minimum spanning tree problem, including the multi-objective capacitated quadratic MST and multi-objective degree-constrained quadratic MST under uncertain and uncertain random hybrid environments are the possible future extension of the study. Moreover, the reformulation of our proposed problem with the normal uncertain variable and the linear uncertain variable is also considered as an area of our interest.