Multi-objective interior search algorithm for optimization: A new multi-objective meta-heuristic algorithm

Abstract

This paper proposes a new multi-objective interior search algorithm (MOISA) for solving multi-objective optimization problems. Multi-objective complex mathematical models need to be solved by meta-heuristic algorithms in such a way that Pareto-optimal solutions are obtained; therefore, a new algorithm is presented in this paper for solving such models. The process of the interior search algorithm (ISA) is based on principles of interior design and decoration. This algorithm divides all elements, except the most suitable one, into two groups. In the first group, which is called the artistic composition group, algorithm changes the composition of elements to achieve a more desirable view. In the second group, which is called the mirror group, the algorithm places a mirror between the group elements and the most suitable element to find a better view. This paper uses the principles of the ISA in conjunction with the concepts of the non-dominated sorting and crowding distance to present the proposed MOISA, which is capable of obtaining near-optimal non-dominated solutions from solution space and identifying accurate Pareto fronts. To evaluate the performance of the foregoing algorithm, the related results of solving six models and a maximal covering location-allocation model are compared with several standard multi-objective evolutionary algorithms in terms of different metrics. This comparison shows that the results of the proposed MOISA are better than those obtained from other tested algorithms. Based on the solved numerical examples, the algorithm presented in this paper has many advantages over existing algorithms.

Keywords

Multi-objective interior search algorithm multi-objective optimization Pareto fronts meta-heuristics performance analysis

1 Introduction

Multi-objective models typically have a lot of complexity, so having an appropriate algorithm to solve this model is very necessary. The main purpose of the innovation of this paper is to respond to this essential need, because new algorithms provide better solutions to solving multi-objective models.

A large portion of mathematic problems can be described by multi-objective models, and solving these models requires using a set of algorithms specialized for this purpose [1]. Multi-objective optimization methods are generally classified into two categories, namely classical and evolutionary algorithms. Classical algorithms are those search and optimization ones that find a single initial solution and then update it at each iteration while using a specific rule for changes. Goal programming is one of such algorithms. Their rate and quality of convergence to an optimal solution depend on the quality of an initial solution and these algorithms often tend to get stuck in a sub-optimal solution. Furthermore, these algorithms cannot be used for the problems, whose search spaces are discrete. Thus, evolutionary algorithms, which can cover some of the flaws of the classical algorithms, have replaced these classical algorithms as the solution of practical problems. Evolutionary algorithms emulate the principles of natural evolution in order to search for optimal solutions. They use two distinct processes of search and selection for this purpose. These algorithms do not use a specific principle or structure to solve the problem [2]. In the situations, in which the problem complexity prevents us from using exact methods to approximate the Pareto set, the importance of multi-objective evolutionary algorithms becomes even more evident. These algorithms have minimum dependency on the form and formulation of the problem, so objectives can be easily added, removed or modified. In addition, these algorithms work with a set of solutions (instead of a single one), so they can accurately approximate the Pareto set [3]. From the mathematical point of view, a multi-objective problem is defined as follows [4]:

Minimizing or maximizing vector f(x) as (1) and a set of constraints as (2), in which x is an n-dimensional vector of decision variables x = (x₁, x₂,..., x_n) of the set S. In other words, we have: $Min f (x) = (f_{1} (x), \dots, f_{m} (x))$ (1)

s.t. $g_{i} (x) \leq 0; i = 1 \dots q; x \in S$ (2)

Thus, a multi-objective problem consists of n variables, q constraints and m objectives, whose objective functions can be linear or non-linear. The evaluation function F in the multi-objective problem projects the set on the decision variable; that is: →fF. In weighting methods, before solving, the problem is converted to a single objective problem as shown in Equation (3). In other words, this method finds the effective solutions of the multi-criteria problem: $Max f (x) = [f_{1} (x), f_{2} (x), \dots, f_{m} (x)]$ (3)

The optimum solution to the following single-criterion problem should be found as follows: $Max f (x) = \sum_{k = 1}^{m} W_{k} f_{k} (x)$ (4)

where f_m(x) are objective functions, f(x) is the ultimate objective function created by combining the problem objective functions and the goal is to optimize it, and W_k is weights of the objective functions. In this case, the necessary condition for a point to be a global optimum is that the Kuhn Tucker conditions are met and the sufficient condition is that the programming is convex [5, 6].

It is expressed in the Kuhn Tucker conditions that:

the objective function z(x) is concave for the case of maximization,

the set of acceptable solutions is a convex set for the case of maximization, and

the objective functions are differentiable.

As seen, in addition to limitations of executions (i.e., Kuhn Tucker conditions), the method faces problems in weighting so that the output solution is dependent on input weights [7]. In addition, in order to obtain the effective set of solutions of these methods, it is necessary to solve the algorithm repeatedly and new solutions should be obtained in each execution [8]. In weighting methods, after solving, there are not the problems resulting from weighting as stated for the mentioned method. Furthermore, using evolutionary algorithms in these methods, a single solving of the problem will lead to a set of effective solutions [9]. In addition, the algorithms function insisting on the movement toward the optimum solution and it will be possible to define optimality conditions defining its cost function [7]. On the other hand, there is no emphasis on the Kuhn Tucker conditions before solving in these algorithms as what was in weighting methods [10].

The question that this article seeks to answer is the need to provide novel solving methods for solving multi-objective models. This need should be met in a proper way. The main objective of this paper is to provide a new meta-heuristic algorithm for solving multi-objective models, in which this new algorithm is superior to the existing algorithms.

In Section 2 of this article reviews the literature. Section 3 addresses the basics of the topic. Section 4 presents numerical examples to prove the efficiency of the proposed algorithm. In Section 5, the conclusions and suggestions for future research are raised.

2 Literature review

To review the literature on multi-objective algorithms, first the concept of Pareto solutions needs to be defined. A Pareto solution is one of the basic concepts in multi-objective problems. If f₁, f₂,...,f_k are the objective functions of a problem, x is the Pareto or non-dominated solution; however, if the following two conditions are met for all i = 1,..., k [11].

Solution x_i is not worse than x_j in all objectives; it means:

f_{k} (x_{i}) < f_{k} (x_{j}) for all k \in {1 \dots m}

(5)

Solution x_i is better than x_j at least in one of the objectives; it means:

f_{k} (x_{i}) > f_{k} (x_{j}) for at least one k \in {12 \dots m}

(6)

In the search space of S ∈ Rⁿ, objective functions space of F ∈ R^m and the objective function f : S → F = {f (x : x ∈ S}, vector x ∈ S is a Pareto-optimal solution in S and F if there is no y ∈ F such that f_k (y) ≤ f_k (x) for ∀k = 1, …, m and at least for one k, f_k (y) < f_k (x) [12]. In fact, a point is a Pareto solution if there is no other point in the search space such that it is better than the point in all objective functions [13]. The related literature contains various algorithms designed specifically for optimization of multi-objective models, among which genetic algorithm (GA) stands out. The non-dominated sorting genetic algorithm (NSGA) [14] and niching-based genetic algorithm [15] are other effective approaches for solving multi-objective problems. In these algorithms, ranking procedures are based on Pareto’s ranking.

Elitism-based algorithms are another category of solution methods, which include the modified version of the non-dominated sorting genetic algorithm (NSGA-II) [11] and strength Pareto evolutionary algorithm (SPEA) [16] and its modified version (SPEA-II) [17]. Another group of methods uses the concept of swarm intelligence to solve multi-objective optimization problems, which include multi-objective particle swarm algorithm (MOPSO) [18] and multi-objective ant colony optimization (MOACO) [19]. Many multi-objective evolutionary algorithms have been proposed in the last three decade. Among these algorithms, the vector evaluated genetic algorithm (VEGA) [20], Fonseca and Fleming genetic algorithm (FFGA) [1], the GA has features that make it easier to be applied on multi-objective problems than other algorithms [21].

Binary codes are used to code decision variables in genetic algorithm.

Roulette Wheel is used in genetic algorithm to select the parent, which itself causes to get closer to the best solution.

In recent years, many new algorithms with similar logics have been presented; include, multiobjective simulated annealing combined with tabu search (MOSATS) [22], multi-objective cat swarm optimization (MOCSO) [23], multi-objective gravitational search algorithm (MOGSA) [24], Multi-objective teaching learning based optimization (MOTLBO) [25], multi-objective artificial bee colony (MOABC) [26], multi-objective vibration damping optimization (MOVDO) [27, 28], multi-objective grey wolf optimizer [29], and dragonfly algorithm (DA) [30]. In new study, multi-objective cuckoo search approach for fractional order PID control design presented, in this paper they used that algorithm for semi-active control of smart base-isolated structures [31]. A multi-objective meta-heuristic approach for the designing and planning of green supply chains, namely MBSA, is one of the new meta-heuristic algorithm [32]. A novel Bi-Ant colony optimization algorithms is another method that used for multiobjective service selection problem [33]. In another study, a new optimization algorithm is presented that is extremely robust in solving mathematical and engineering problems. The algorithm combines the deterministic nature of classical methods of optimization and global converging characteristics of meta-heuristic algorithms [34]. In another study, proposed propose five multi-objective meta-heuristic algorithms, namely NSGA-II with an elitism solution, NSGA-II without an elitism solution, NRGA with an elitism solution, the NRGA without an elitism solution, and MOPSO, for uncapacitated hub location problem [35]. In other study, presented a new hybrid optimization algorithm based on hybridization of cuckoo search and particle swarm optimization (CSPSO) [36]. In another study, a new multi-objective algorithm inspired from the navigation of grass hopper swarms in nature is proposed and used for multiobjective optimization problems [37].

New multi-objective evolutionary algorithms (MOEAs) is used for practical portfolio optimization, in which a novel repair algorithm is proposed in order to effectively handle equality constraints without any requirement of any constraint handling technique [38]. Another study develops a bi-objective ant colony algorithm to solve a novel multi-objective mixed load school bus routing model [39]. In other study, a new meta-heuristic algorithm based on an improved shuffled frog leaping algorithm (ISFLA) is presented that belongs to memetic evolution [40]. In other study, a new approaches proposed by hybridizing a multi-objective evolutionary algorithm based on decomposition (MOEA/D) with a meta-Lamarckian (ML) learning strategy that learns from the problem’s properties and objective functions [41]. A novel evolutionary multi-objective discretization algorithm for binary imbalanced datasets (EMDID) is presented that takes advantage of evolutionary multi-objective optimization to simultaneously optimize functions [42]. A fast and efficient adaptation of evolutionary algorithms is presented to solve large-scale multi-objective controller placement problems [43]. An EDOSIM model is a simulation-optimization model for surface irrigation systems that was developed by combining the volume balance model and 20 meta-heuristic algorithms [44].

3 Multi-objective interior search algorithm

The multi-objective interior search algorithm (MOISA) proposed in this paper is based on merging the interior search algorithm (ISA) with the principles of the non-dominated sorting and crowding distance. Before further explanation the mentioned method and principles need to be described.

3.1 Interior search algorithm

The interior search algorithm (ISA) was first presented by Gandomi [45]. In this algorithm, the following principles are considered.

3.1.1 Designing artistic combination

The procedure of interior designing follows a coordinated method. This procedure involves analysis, research, and integration of knowledge to creative process. In producing an interior space, the project’s objectives, needs and resources are accomplished that must lead to customer satisfaction. Interior designing usually begins from borders to center.

3.1.2 Mirror-work

Mirror-work is a standout amongst the most engaging developments as far as style, which has been used by Iranian decoration designers. A mirror-laborer uses different mirrors to make a more beautifying condition. A colossal bit of this method is that the mirror is placed in the district of the loveliest parts to push their perfection. This excellent and multifaceted process can be used for upgrade to such an extent that by setting a segment of the mirrors near the comprehensive best or the most fitting parts, some other lovely viewpoints can be come to.

3.2 Non-dominated sorting

Non-dominated sorting can be considered as the act of categorizing the solutions. This concept can be better explained with the help of following example. As Fig. 1 shows, the solution space is divided into four sections and axes (i.e., F1 andF2) show the values of two objective functions of a given multi-objective problem. A point representing a single solution of this problem is marked with a star. This point dominates all points located within space A; however, it is dominated by all points located within space C because the values of its two objective functions are greater than the values of the objective functions in the space C. It is not clear whether it dominates the points of spaces B and D, since it dominates them in one function, but is dominated in the other. So in overall, these points have no clear advantage over each other and can be placed in one Pareto Front.

Fig.1

Principles of the Pareto solutions.

3.3 Crowding distance

The crowding distance is used to obtain a smoother set of a solution from other algorithms and a density estimation of points around the solutions. It is worth mentioning that the crowding distance is a factor that is used to make a better selection of a solution from the perspective of distribution across a front and is defined as follows [8, 46]:

Its values is considered infinity for the beginning and end point of a front.

For other point of the front and from 2 to (k-1), it is defined by: $CD [i] = \frac{f_{m}^{i + 1} - f_{m}^{i - 1}}{f_{m}^{imax} - f_{m}^{\min}}$ (7)

where, CD[i] is the crowding distance of the i-th individual on front F, f_mⁱ is the value of the m-th objective function in the i-th individual of front F and $f_{m}^{\min}$ and $f_{m}^{imax}$ is the minimum and maximum value of objective function m in front F, respectively. A solution is better if it has a more crowding distance [47].

3.4 Multi-objective interior search algorithm

In the multi-objective interior search algorithm (MOISA), concepts of the ISA are used in conjunction with the principles of the non-dominated sorting and crowding distance. In the proposed algorithm, those steps of the process that require sorting out the original population use the two mentioned principles for this purpose. The pseudo code of this algorithm is as follows:

Initialization

while any stopping criteria is not satisfied

use the non-dominated sorting

use the crowding distance

find the Pareto fronts

choose one of the solution in the first Pareto front as x ^j _gb

for i = 1 to n

if x _gb

x^j_gb = x^j - 1_gb +r_n*λ

else if r₁≤α

x^j_m,i = r₃ x^j–1_i +(1-r₃) x^j_gb

x^j_i = 2 x^j_m,I - x^j–1_i

else

x^j_i = LB^j+(UBⁱ -LB^j)*r₂

end if

check the boundaries except for decomposition elements

end for

for i = 1 to n

evaluate the f(x_i^j)

$x_{i}^{j} = {\begin{matrix} x_{i}^{j} if f (x_{i}^{j}) dominate f (x_{i}^{j - 1}) \\ x_{i}^{j - 1} else \end{matrix}}$

end for

end while

The main steps of this algorithm are as follows:

Randomly create element locations between upper borders (UB) and lower borders (LB) and obtain their suitability value.

Use the non-dominated sorting.

Use the crowding distance.

Finding the Pareto fronts. Choose one of the elements in first Pareto front. This element is the best on the overall level, in the most j iteration, $x_{gb}^{j}$ .

Randomly, they divide the rest of the elements into two groups. To do this, a parameter, α, has been defined. For each element, if r₁ < α, it goes to the mirror group; otherwise, it goes to the combination group. r₁ is a random amount between 0 and 1. It is evident that, theoretically, α is a value in this range. However, it must be arranged accurately to create balance between addition and variety, since it is the main parameter of the algorithm.

For the combination group, the combination of each of the elements changes randomly in a limited search space, which can be demonstrated by:

$x_{i}^{j} = {LB}^{j} + ({UB}^{j} - {LB}^{j}) \times r_{2}$ (8)

where, $x_{i}^{j}$ equals the element i in the j iteration, LB^j and UB^j are the lower and upper borders of the combination group in the j iteration, respectively. They are the minimum and maximum values of all the elements in iteration (j-1), respectively. r₂ is a random value between 0 and 1.

For components of the mirror gathering, mirror-work is used. Initial, a mirror is haphazardly put between one of the elements and the most suitable element (the best overall element or gb). The location of a mirror for element i in iteration j is created by:

$x_{mi}^{j} = r_{3} x_{i}^{j - 1} + (1 - r_{3}) x_{gb}^{j}$ (9)

where, r₃ is a random value between 0 and 1. The location of the image and or the virtual location of an element depend on the mirror location, and can be written by: $x_{i}^{j} = 2 x_{m, i}^{j} - x_{i}^{j - 1}$ (10)

For the best overall element, it is better to alter its location a bit using the random step. This element can be written by:

$x_{gb}^{j} = x_{gb}^{j - 1} + r_{n} λ$ (11)

where, r_n is a vector of random numbers and is distributed normally, which contains the same number of x, and λ is the scale factor that depends on the search space size.λ is adjusted to 0.01 × (UB - LB). This random step acts as a local search since conducts its search around the best universal element.

Calculate the appropriate values for new areas of components and pictures (i.e., virtual components). Then, update each location if it contains suitable value for improved redesigning. For an optimization problem, the following equation can be stated by:

$x_{i}^{j} = {\begin{matrix} x_{i}^{j} if f (x_{i}^{j}) dominate f (x_{i}^{j - 1}) \\ x_{i}^{j - 1} else \end{matrix}}$ (12)

If each of the stopping criteria is not met, the above steps must be repeated from Step 2.

4 Result and discussion

After presenting the proposed MOISA, the performance of this algorithm needs to be evaluated through some numerical examples. This section describes the examples used for this purpose and presents the obtained results.

4.1 Standard models

Six standard multi-objective models obtained from [48] are used to evaluate the effectiveness of the proposed algorithm. The models listed in Tables 1 and 2 are bi-objectives and tri-objectives, respectively.

Table 1
Standard bi-objective optimization models

Name Model

F1 $f 1 = x_{1} + \frac{1}{| J_{1} |} \sum_{j \in J_{1}} {[x_{j} - \sin (6 π x_{1} + \frac{j π}{n})]}^{2}$

$f 2 = 1 - \sqrt{x_{1}} + \frac{1}{| J_{2} |} \sum_{j \in J_{2}} {[x_{j} - \sin (6 π x_{1} + \frac{j π}{n})]}^{2}$

where J₁ ={ j|j is odd and 2 ≤ j ≤ n } and J₂ ={ j|j is even and 2 ≤ j ≤ n }

F2 $f 1 = x_{1} + \frac{2}{| J_{1} |} \sum_{j \in J_{1}} y_{j}^{2}$

$f 2 = 1 - \sqrt{x_{1}} + \frac{2}{| J_{2} |} \sum_{j \in J_{2}} y_{j}^{2}$

where J₁ ={ j|j is odd and 2 ≤ j ≤ n } and J₂ ={ j|j is even and 2 ≤ j ≤ n }

$y_{j} = {\begin{matrix} x_{j} - [0.3 x_{1}^{2} cos (24 π x_{1} + \frac{4 j π}{n}) + 0.6 x_{1}] cos (6 π x_{1} + \frac{j π}{n}) j ɛ J_{1} \\ x_{j} - [0.3 x_{1}^{2} cos (24 π x_{1} + \frac{4 j π}{n}) + 0.6 x_{1}] sin (6 π x_{1} + \frac{j π}{n}) j ɛ J_{2} \end{matrix}}$

F3 $f 1 = x_{1} + \frac{2}{| J_{1} |} (4 \sum_{j \in J_{1}} y_{j}^{2} - 2 \prod_{j \in J_{1}} cos (\frac{20 y_{i} π}{\sqrt{j}}) + 2)$

$f 2 = \sqrt{x_{1}} + \frac{2}{| J_{1} |} (4 \sum_{j \in J_{2}} y_{j}^{2} - 2 \prod_{j \in J_{2}} cos (\frac{20 y_{i} π}{\sqrt{j}}) + 2)$

where J₁ and J₂ are the same as those of F2, $y_{j} = x_{j} - x_{1}^{0.5 (1 + \frac{3 (j - 2)}{n - 2})}; j = 2 \dots n$

F4 $f 1 = x_{1} + \frac{2}{| J_{1} |} \sum_{j \in J_{1}} h (y_{j})$

$f 1 = 1 - x_{1}^{2} + \frac{2}{| J_{2} |} \sum_{j \in J_{2}} h (y_{j})$

where J₁ and J₂ are the same as those of F2, $y_{j} = x_{j} - sin (6 π x_{1} + \frac{j π}{n}); j = 2 \dots n$

$h (t) = \frac{| t |}{1 + e^{2 | t |}}$

Name	Model
F1	$f 1 = x_{1} + \frac{1}{\| J_{1} \|} \sum_{j \in J_{1}} {[x_{j} - \sin (6 π x_{1} + \frac{j π}{n})]}^{2}$
	$f 2 = 1 - \sqrt{x_{1}} + \frac{1}{\| J_{2} \|} \sum_{j \in J_{2}} {[x_{j} - \sin (6 π x_{1} + \frac{j π}{n})]}^{2}$
	where J₁ ={ j\|j is odd and 2 ≤ j ≤ n } and J₂ ={ j\|j is even and 2 ≤ j ≤ n }
F2	$f 1 = x_{1} + \frac{2}{\| J_{1} \|} \sum_{j \in J_{1}} y_{j}^{2}$
	$f 2 = 1 - \sqrt{x_{1}} + \frac{2}{\| J_{2} \|} \sum_{j \in J_{2}} y_{j}^{2}$
	where J₁ ={ j\|j is odd and 2 ≤ j ≤ n } and J₂ ={ j\|j is even and 2 ≤ j ≤ n }
	$y_{j} = {\begin{matrix} x_{j} - [0.3 x_{1}^{2} cos (24 π x_{1} + \frac{4 j π}{n}) + 0.6 x_{1}] cos (6 π x_{1} + \frac{j π}{n}) j ɛ J_{1} \\ x_{j} - [0.3 x_{1}^{2} cos (24 π x_{1} + \frac{4 j π}{n}) + 0.6 x_{1}] sin (6 π x_{1} + \frac{j π}{n}) j ɛ J_{2} \end{matrix}}$
F3	$f 1 = x_{1} + \frac{2}{\| J_{1} \|} (4 \sum_{j \in J_{1}} y_{j}^{2} - 2 \prod_{j \in J_{1}} cos (\frac{20 y_{i} π}{\sqrt{j}}) + 2)$
	$f 2 = \sqrt{x_{1}} + \frac{2}{\| J_{1} \|} (4 \sum_{j \in J_{2}} y_{j}^{2} - 2 \prod_{j \in J_{2}} cos (\frac{20 y_{i} π}{\sqrt{j}}) + 2)$
	where J₁ and J₂ are the same as those of F2, $y_{j} = x_{j} - x_{1}^{0.5 (1 + \frac{3 (j - 2)}{n - 2})}; j = 2 \dots n$
F4	$f 1 = x_{1} + \frac{2}{\| J_{1} \|} \sum_{j \in J_{1}} h (y_{j})$
	$f 1 = 1 - x_{1}^{2} + \frac{2}{\| J_{2} \|} \sum_{j \in J_{2}} h (y_{j})$
	where J₁ and J₂ are the same as those of F2, $y_{j} = x_{j} - sin (6 π x_{1} + \frac{j π}{n}); j = 2 \dots n$
	$h (t) = \frac{\| t \|}{1 + e^{2 \| t \|}}$

To demonstrate the good performance of the proposed MOISA, it is compared with four widely-known algorithms of the MOEA-D, NSGA-II, MOPSO and SPEA-II. This comparison is based on three criteria, namely spacing (S), number of Pareto solutions (NOS), and maximum spread (MS), whose definitions are presented below.

4.2 Spacing

The spacing (S) metric uses Equations (13) to (15) to measure the relative distance between consecutive solutions [49]. $S = \sqrt{\frac{1}{n - 1} \sum_{i = 1}^{n} (d_{i} - \bar{d})^{2}}$ (13) $d_{i} = min_{k ∋ n | k \neq i} \sum_{j = 1}^{m} | f_{j}^{i} - f_{j}^{k} |$ (14) $\bar{d} = \sum_{i = 1}^{n} \frac{d_{i}}{n}$ (15)

The measured distance equals the minimum value of the sum of absolute values of the difference in the values of the objective functions between the i-th solution and the solutions of the final non-dominated set. It should be mentioned that this distance is not the same as the minimum Euclidean distance between different solutions. The above metric measures the standard deviation of the values of d_i. When solutions are located uniformly next to each other, s will be small.

4.3 Number of Pareto-optimal solutions

The higher number of Pareto solutions (NOS) points to a better performance of the algorithm.

4.4 Maximum spread

The maximum spread (MS) metric measures the length of the diagonal of the hyper-box formed by the extreme function values in the non-dominated set. The following equation shows how this parameter can be calculated by [50]: $MS = \sqrt{\frac{1}{r} \sum_{i = 1}^{r} [\frac{\min (f_{i}^{\max} F_{i}^{\max}) - \max (f_{i}^{\min} F_{i}^{\min})}{F_{i}^{\max} - F_{i}^{\min}}]^{2}}$ (16)

In the above equation, $f_{i}^{\max}$ and $f_{i}^{\min}$ represent the maximum and minimum values of the i-th object of the obtained Pareto set, and $F_{i}^{\max}$ and $F_{i}^{\min}$ are the highest and lowest values of the real Pareto front. The higher MS values are more desirable. To compare the performance of the proposed algorithm with that of mentioned algorithms, number of function evaluation is considered equal to 200000. Thus, after setting the basic algorithm parameters, the six models listed in Tables 1 and 2 are each executed 20 times. The results of this test, which include the mean, standard deviation, and best and worst solutions, are presented in Tables 3 to 8. It should also be mentioned that the number of model parameters considered for mentioned six models is 30.

Table 2

Standard tri-objective optimization models

Name	Model
F5	$f 1 = cos (0.5 x_{1} π) cos (0.5 x_{2} π) + \frac{2}{\| J_{1} \|} \sum_{j \in J_{1}} [x_{j} - 2 x_{2} \sin (2 π x_{1} + \frac{j π}{n})^{2}]$
	$f 2 = cos (0.5 x_{1} π) cos (0.5 x_{2} π) + \frac{2}{\| J_{2} \|} \sum_{j \in J_{2}} [x_{j} - 2 x_{2} \sin (2 π x_{1} + \frac{j π}{n})^{2}]$
	$f 3 = sin (0.5 x_{1} π) + \frac{2}{\| J_{3} \|} \sum_{j \in J_{3}} [x_{j} - 2 x_{2} \sin (2 π x_{1} + \frac{j π}{n})^{2}]$
	where J₁ ={ j \|3 ≤ j ≤ n and j - 1 is a multipication of 3 }
	J₂ ={ j\|3 ≤ j ≤ n and j - 2 is a multipication of 3 }
	J₃ ={ j\|3 ≤ j ≤ n and j is a multipication of 3 }
F6	$f 1 = cos (0.5 x_{1} π) cos (0.5 x_{2} π) + \frac{2}{\| J_{1} \|} \sum_{j \in J_{1}} [4 y_{j}^{2} - cos (8 π y_{j}) + 1]$
	$f 2 = cos (0.5 x_{1} π) cos (0.5 x_{2} π) + \frac{2}{\| J_{2} \|} \sum_{j \in J_{1}} [4 y_{j}^{2} - cos (8 π y_{j}) + 1]$
	$f 3 = \sin (0.5 x_{1} π) + \frac{2}{\| J_{3} \|} \sum_{j \in J_{1}} [4 y_{j}^{2} - cos (8 π y_{j}) + 1]$
	where J₁ ={ j\|3 ≤ j ≤ n and j - 1 is amultipication of 3 }
	J₂ ={ j\|3 ≤ j ≤ n and j - 2 is a multipication of 3 }
	J₃ ={ j\|3 ≤ j ≤ n and j is a multipication of 3 }
	$y_{j} = x_{j} - 2 x_{2} sin (2 π x_{1} + \frac{j π}{n}); j = 3 \dots n$

Table 3

Results obtained by solving the model F1

	S				MS				NOS
	Average	STD	Worst	Best	Average	STD	Worst	Best	Average	STD	Worst	Best
F1	MOEA-D	0.0119	0.0031	0.0173	0.0082	0.7451	0.1307	0.4300	0.9036	80.28	4.7035	70	87
	NSGAII	0.0092	0.0017	0.0133	0.0062	0.7182	0.1334	0.5197	0.9168	71.9	7.2031	59	90
	MOPSO	0.0099	0.0018	0.0137	0.0072	0.7425	0.1643	0.4731	0.9473	74.95	5.8712	64	84
	SPEAII	0.0136	0.0032	0.0195	0.0084	0.7173	0.1576	0.4152	0.9314	69.4	7.3154	58	85
	MOISA	0.0084	0.0022	0.0129	0.0034	0.7842	0.1553	0.4123	0.9707	78.8	6.0053	70	91

Table 4

Results obtained by solving the model F2

	S				MS				NOS
	Average	STD	Worst	Best	Average	STD	Worst	Best	Average	STD	Worst	Best
F2	MOEA-D	0.0099	0.0021	0.0155	0.0062	0.6840	0.1648	0.3218	0.9550	77.25	6.5925	66	80
	NSGAII	0.0102	0.0048	0.0194	0.0020	0.7485	0.1233	0.5503	0.9434	76.45	8.0621	62	90
	MOPSO	0.0144	0.0054	0.0255	0.0022	0.7736	0.1305	0.4728	0.9392	73.9	7.2830	59	88
	SPEAII	0.0157	0.0049	0.0267	0.0083	0.7536	0.1038	0.5557	0.9298	71.20	9.6878	53	92
	MOISA	0.0096	0.0030	0.0148	0.0021	0.7905	0.1084	0.5325	0.9723	81.8	8.5385	60	95

Table 5

Results obtained by solving the model F3

	S				MS				NOS
	Average	STD	Worst	Best	Average	STD	Worst	Best	Average	STD	Worst	Best
F3	MOEA-D	0.0128	0.0035	0.0215	0.0081	0.8386	0.1106	0.6356	0.9756	73.75	7.7042	53	84
	NSGAII	0.0200	0.0032	0.0268	0.0151	0.8157	0.1068	0.6324	0.9565	70.15	6.4341	61	80
	MOPSO	0.0130	0.0035	0.0200	0.0071	0.8717	0.0750	0.6436	0.9493	68.9	7.4756	59	85
	SPEAII	0.0214	0.0051	0.0283	0.0127	0.7782	0.1279	0.5420	0.9679	66.75	8.9141	51	83
	MOISA	0.0119	0.0024	0.0168	0.0081	0.8477	0.0971	0.6308	0.9622	78.9	9.049	56	92

Table 6

Results obtained by solving the model F4

	S				MS				NOS
	Average	STD	Worst	Best	Average	STD	Worst	Best	Average	STD	Worst	Best
F4	MOEA-D	0.0072	0.0017	0.0097	0.0043	0.8156	0.1287	0.5537	0.9887	72	7.8001	49	84
	NSGAII	0.0078	0.0020	0.0104	0.0033	0.7854	0.1741	0.4747	0.9700	80.5	5.7720	71	92
	MOPSO	0.0082	0.0025	0.0126	0.0037	0.7907	0.1250	0.4538	0.9892	72.95	8.8999	55	89
	SPEAII	0.0099	0.0025	0.0146	0.0043	0.7466	0.1610	0.4333	0.9291	68.7	6.0359	57	78
	MOISA	0.0067	0.0014	0.0096	0.0036	0.8513	0.1190	0.4631	0.9513	78.45	7.4089	65	91

Table 7

Results obtained by solving the model F5

	S				MS				NOS
	Average	STD	Worst	Best	Average	STD	Worst	Best	Average	STD	Worst	Best
F5	MOEA-D	0.0087	0.0021	0.0143	0.0055	0.7318	0.2317	0.3987	0.8413	72	5	49	81
	NSGAII	0.0091	0.0032	0.0133	0.0043	0.6529	0.3100	0.4187	0.7629	78	5	45	88
	MOPSO	0.0071	0.0022	0.0139	0.0050	0.7001	0.1987	0.4028	0.8311	66	4	51	78
	SPEAII	0.0127	0.0035	0.0170	0.0067	0.5981	0.3874	0.3176	0.7132	60	6	41	71
	MOISA	0.0079	0.0019	0.0101	0.0049	0.8319	0.2781	0.5971	0.9012	75	5	48	84

Table 8

Results obtained by solving the model F6

	S				MS				NOS
	Average	STD	Worst	Best	Average	STD	Worst	Best	Average	STD	Worst	Best
F6	MOEA-D	0.0173	0.0035	0.0258	0.0114	0.3156	0.1406	0.0937	0.6430	31.15	6.5957	20	43
	NSGAII	0.0178	0.0036	0.0246	0.0111	0.2896	0.1250	0.1142	0.5275	30	3.8933	23	37
	MOPSO	0.0194	0.0030	0.0270	0.0136	0.2602	0.1226	0.0708	0.5146	29	4.8558	22	37
	SPEAII	0.0212	0.0055	0.0305	0.0094	0.2311	0.0888	0.1106	0.4554	20.75	3.8645	14	29
	MOISA	0.0166	0.0044	0.0235	0.0098	0.3580	0.1052	0.2100	0.6839	35.15	4.2087	29	43

4.5 Computational results for standard models

This section presents the results obtained from running the proposed MOISA in comparison with those obtained from running MOEA-D, NSGA-II, MOPSO and SPEA-II. All algorithms are run in MATLAB R2013 software on a PC with 2.6 GHz processor and 6G RAM. The obtained results are shown in Tables 3 to 8.

For function F1, the performance of the proposed MOISA in the S and MS metrics is better than that of other algorithms. However, in the NOS metric, MOEA-D outperformed other algorithms. For function F2, the proposed MOISA outperforms other algorithms in all three metrics. For function F3, the proposed MOISA shows the best performance in the NOS and S metrics; however, MOPSO is the best algorithm in terms of the MS metric. For function F4, the performance of the proposed MOISA is better than other algorithms in terms of the S and MS metrics; however, NSGA-II had the best performance in terms of the NOS metric. For function F5, MOISA, MOPSO and NSGA-II achieve the best performance in terms of the MS, S, and NOS metrics, respectively. For function F6, MOISA outperformed other algorithms in all three metrics. Figures of the true Pareto front and Pareto solution obtained from solving models F1 to F6 are shown in Figure 2.

Fig.2

True Pareto solutions and obtained Pareto solutions for models F1– F6.

4.6 Maximal covering location-allocation model

In this section, the two-objective model for maximal covering location-allocation is considered. The parameters and variables of this model are as follows.

Parameters:

d_ir Distance i from the location r

α_t Percentage of trip type t

P_tj The number of trips generated by type t from the traffic area j

U Maximum passenger of each station

Variables:

k_i Station capacity created at location i

$C_{ij} = {\begin{matrix} 1 If location i covers j \\ 0 otherwise \end{matrix}}$

The two-objective mathematical model is written as model 17: $min Z_{1} = \sum_{i = 1}^{m} \sum_{j = 1}^{n} K_{i} C_{ij}$ (17) $max Z_{2} = \sum_{i = 1}^{m} \sum_{r = 1}^{s} \sum_{j = 1}^{n} d_{ir} C_{ij} C_{rj}, i \neq r$ s.t. $\sum_{i = 1}^{m} \sum_{j = 1}^{n} K_{i} C_{ij} ⩾ C$ $K_{i} \leq U, i = 1, \dots, m$ $\sum_{i = 1}^{m} \sum_{j = 1}^{n} C_{ij} \leq C$ $K_{i} ⩾ 0, i = 1, \dots, m$ $C_{ij} is 0 or 1, i = 1, \dots, m; j = 1, \dots, n$ $C_{rj} is 0 or 1, r = 1, \dots, s; j = 1, \dots, n$

In this example, i = 84, r = 84, j = 175, t = 7, U = 50 are considered, also α is shown in Table 9.

Table 9

α values

α _t

0.211

0.105

0.237

0.184

0.132

0.026

4.7 Computational results for maximal covering location-allocation model

The number of function evaluations is set to 10000 and the average results are considered at 100 runs. The results are also given in Table 10. According to the obtained results, the proposed algorithm MOISA in terms of the MS and NOS metrics is superior to the other four algorithms.

Table 10
Results obtained by solving the model 17

MS S NOS

MOEA-D 9.0410∧6 1.5710∧6 15

NSGAII 8.8510∧6 1.7710∧6 11

MOPSO 8.810∧6 1.6310∧6 11

SPEAII 8.7410∧6 1.8110∧6 13

MOISA 9.110∧6 1.610∧6 21

	MS	S	NOS
MOEA-D	9.04*10∧6	1.57*10∧6	15
NSGAII	8.85*10∧6	1.77*10∧6	11
MOPSO	8.8*10∧6	1.63*10∧6	11
SPEAII	8.74*10∧6	1.81*10∧6	13
MOISA	9.1*10∧6	1.6*10∧6	21

5 Conclusion

The presentation of a new meth-heuristic algorithm for solving multi-objective models was the main goal of this paper that responds to the need for new algorithms for solving multi-objective models. This paper presented a new multi-objective algorithm, namely multi-objective interior search algorithm (MOISA), for solving multi-objective optimization models. The interior search algorithm (ISA) uses the principles of interior design and decoration to optimize the mathematical models. This algorithm can be used to solve NP-Hard mathematical models and obtain near-optimal solutions. These features of this algorithm was merged with the principles of the non-dominated sorting and crowding distance to create a new algorithm for solving multi-objective models. In this paper, after presenting an introduction to the subject, the literature on multi-objective meta-heuristic algorithms was reviewed. Then, the theoretical basis of the concept related to the work including the method of the ISA and the principles of the non-dominated sorting and crowding distance were discussed. The next section presented the Pseudo code of the new algorithm along with full explanation about its functions. In the next step, six standard multi-objective models and a maximal covering location-allocation model were used to prove the effectiveness of the proposed algorithm; this was done by comparing the performance of the proposed algorithm with the performance of four widely-known meta-heuristic algorithms, namely MOEA-D, NSGA-II, MOPSO and SPEA-II. This comparison was devised based on three metrics, namely spacing (S), number of Pareto solutions (NOS), and maximum spread (MS) in order to ensure a complete evaluation on the performance of all the multi-objective meta-heuristic algorithms.

The results of this comparison showed that in many instances the proposed algorithm (MOISA) showed significant advantages over other algorithms. For future studies, we can use this new algorithm for optimization of real models (e.g., inventory control, supply chain management, reliability, location and transportation) that need to be optimized by meta-heuristics. Also, other meta-heuristic algorithms based on the principles of multi-objective models can be used for solving multi-objective models. By combining single-objective meta-heuristic algorithms, we can also create new methods for solving multi-objective models by combining their properties together. Also, by creating inspiration from natural phenomena and mathematical methods, new multi-objective meta-heuristic algorithms can be created. The MOISA algorithm can also be used for examples in the field of fuzzy, interval and gray numbers.

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Name	Model
F5	$f 1 = cos (0.5 x_{1} π) cos (0.5 x_{2} π) + \frac{2}{\| J_{1} \|} \sum_{j \in J_{1}} [x_{j} - 2 x_{2} \sin (2 π x_{1} + \frac{j π}{n})^{2}]$
	$f 2 = cos (0.5 x_{1} π) cos (0.5 x_{2} π) + \frac{2}{\| J_{2} \|} \sum_{j \in J_{2}} [x_{j} - 2 x_{2} \sin (2 π x_{1} + \frac{j π}{n})^{2}]$
	$f 3 = sin (0.5 x_{1} π) + \frac{2}{\| J_{3} \|} \sum_{j \in J_{3}} [x_{j} - 2 x_{2} \sin (2 π x_{1} + \frac{j π}{n})^{2}]$
	where J₁ ={ j \|3 ≤ j ≤ n and j - 1 is a multipication of 3 }
	J₂ ={ j\|3 ≤ j ≤ n and j - 2 is a multipication of 3 }
	J₃ ={ j\|3 ≤ j ≤ n and j is a multipication of 3 }
F6	$f 1 = cos (0.5 x_{1} π) cos (0.5 x_{2} π) + \frac{2}{\| J_{1} \|} \sum_{j \in J_{1}} [4 y_{j}^{2} - cos (8 π y_{j}) + 1]$
	$f 2 = cos (0.5 x_{1} π) cos (0.5 x_{2} π) + \frac{2}{\| J_{2} \|} \sum_{j \in J_{1}} [4 y_{j}^{2} - cos (8 π y_{j}) + 1]$
	$f 3 = \sin (0.5 x_{1} π) + \frac{2}{\| J_{3} \|} \sum_{j \in J_{1}} [4 y_{j}^{2} - cos (8 π y_{j}) + 1]$
	where J₁ ={ j\|3 ≤ j ≤ n and j - 1 is amultipication of 3 }
	J₂ ={ j\|3 ≤ j ≤ n and j - 2 is a multipication of 3 }
	J₃ ={ j\|3 ≤ j ≤ n and j is a multipication of 3 }
	$y_{j} = x_{j} - 2 x_{2} sin (2 π x_{1} + \frac{j π}{n}); j = 3 \dots n$

Multi-objective interior search algorithm for optimization: A new multi-objective meta-heuristic algorithm

Abstract

Keywords

1 Introduction

3.1 Interior search algorithm

3.1.1 Designing artistic combination

3.1.2 Mirror-work

3.2 Non-dominated sorting

4.1 Standard models

4.4 Maximum spread

Table 10 Results obtained by solving the model 17 MS S NOS MOEA-D 9.04*10∧6 1.57*10∧6 15 NSGAII 8.85*10∧6 1.77*10∧6 11 MOPSO 8.8*10∧6 1.63*10∧6 11 SPEAII 8.74*10∧6 1.81*10∧6 13 MOISA 9.1*10∧6 1.6*10∧6 21

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Table 10
Results obtained by solving the model 17

MS S NOS

MOEA-D 9.0410∧6 1.5710∧6 15

NSGAII 8.8510∧6 1.7710∧6 11

MOPSO 8.810∧6 1.6310∧6 11

SPEAII 8.7410∧6 1.8110∧6 13

MOISA 9.110∧6 1.610∧6 21