Decomposition Based Quantum Inspired Salp Swarm Algorithm for Multiobjective Optimization

Various real-world problems are essentially multiobjective optimization problems (MOPs), which involve several conflicting objectives. We propose a Quantum-inspired multiobjective Salp Swarm Algorithm based on the Decomposition technique to locate the multiple Pareto-optimal solutions (POS). The main objective while designing the algorithms for the multiobjective optimization problems is to attain a good convergence and uniform dissemination of the solutions, which remains a significant challenge for the algorithms. The proposed Decomposition-based Quantum-inspired Salp Swarm Algorithm for Multiobjective Optimization (DMQSSA) extends the primary form of SSA by using the quantum-inspired framework and a basic decomposition strategy to improve the balance between exploration and exploitation for MOPs. The Delta potential-well model (DPWM) from quantum mechanics is known for enhancing the convergence and diversity in the population, and the decomposition strategy is proved to be effective to generate evenly distributed solutions set on the Pareto front for simultaneous optimization of the subproblems. In this paper, the existing DPWM model is analysed and redesigned for MOPs with a modification in the contraction equation, and decomposition strategy is used along with an intelligent selection technique to ensure non-dominated solutions. The proposed hybrid approach is evaluated and compared with other techniques on a set of well-known benchmark problems. The results show that DMQSSA can handle the multiobjective optimization problems to find better and well-distributed Pareto optimal set. Also, success of the proposed algorithm is further illustrated on a real-world application.


I. INTRODUCTION
Many real-world optimization problems in engineering and other research areas are multiobjective optimization problems (MOPs). These are required to find a set of well-distributed optimal solutions across the Pareto-optimal front in the search space. Generally, in many areas of engineering and science, including mathematical optimization and the controller placement problem, the optimal decision has to be taken in the existence of trade-offs between two or more conflicting objectives. In general, a multiobjective optimization problem can be a minimization or maximization problem h (z) subject The associate editor coordinating the review of this manuscript and approving it for publication was Yu-Da Lin .
to z Q, where h (z) is a real-valued scalar function, often known as objective or cost function, and Q is the feasible set or subset of the search space. Without the loss of generality, it can be mathematically presented as below: where z Q n ≥ 2 set Q includes all the feasible solutions (1) where z = (z 1 , z 1 , . . . z d ) is a d-dimensional vector in the search space R d , h (z) = (h 1 (z) , h 1 (z) , . . . h n (z)) is an n-dimensional vector in the objective space R n . As the objectives in MOP are usually confute to one and all, the Pareto dominance association is often applied VOLUME  Pareto-optimal Set (PS) is the set of non-dominated solutions in the decision space [1] whereas Pareto-optimal Front (PF) is introduced as a vector set corresponding to PS in the objective space [2], [3]. The essence of decoding the multiobjective optimization problems (MOPs) is to identifying a set of well-distributed optimal solutions in the search space.
Salp swarm algorithm (SSA) is a novel metaheuristic approach to imitate the Salp swarming behaviour in the ocean by making a chain. The SSA is similar to other evolutionary computation in several aspects and has been used intelligently to solve many real-world optimization problems. The flocking behaviour can keep away from convergence of one and all solution into a local best solution up to some extent, mainly because of the Salp chain. Still, there are many optimization problems where SSA is unable to obtain the expected optima and gets trapped into a deceptive optimum. The complex unconstrained and constrained multiobjective optimization problems are one of such kind. The difficulty lies for the SSA, mainly due to the search strategy, which is designed originally for single-objective optimization. It must have superior stability in terms of exploration and exploitation to achieve the expected optima for the multiobjective optimization problems. Thus, there is a need to modify the standard SSA algorithm's original design to perform the overall searching process efficiently for MOPs.

A. QUANTUM INSPIRED ALGORITHM
The quantum-inspired algorithm is a new branch of study in the area of evolutionary computation. It is characterized by certain principles of quantum physics such as uncertainty, superposition, interference etc. The approach to merge and design quantum-inspired algorithms for classical computers generally used solution's to quantum probabilistic model representation. The probabilistic model representation from quantum computing offers better diversity during the execution of algorithm. The quantum search strategy intelligently guides the individuals towards the global optima by significantly improving convergence speed and solution efficiency. The diversity can be derived from the representation model of the population. A probabilistic model of linear superposition of states introduces better characteristics of generating diversity in the population. Maintaining a good diversity in the population increases the searchability of the algorithm and resolves search stagnation. Sun et al. proposed Quantumbehaved Particle Swarm Optimization (QPSO) to improve the performance of Particle Swarm Optimization (PSO), including good convergence and global search ability [6]. The experimental results show that the QPSO works better than standard PSO and is an efficient algorithm. Therefore, we propose to employ quantum's inspiration in the current work to extend the standard SSA for the same reason as QPSO for multiobjective optimization problems and compare it to MSSA [7], MOPSO [8] and MOEA/D [9].

B. SOLVING MULTIOBJECTIVE OPTIMIZATION
Metaheuristic algorithm is currently a popular approach in the domain of optimization to solve complex optimization problems. An intuitive approach to explore for the optimum solution among all obtainable ones of a particular occurrence is known as Optimization [4], [5]. The conventional approach for the multiobjective optimization problem is to scalarize the vector into a single objective by averaging the value of the objective function with a weight vector. This process of converting MOP into a single objective optimization problem allows a single-objective algorithm to be used in a straight forward manner. Still, the obtained optimal solution is dependent primarily on the weight vector considered during the scalarization process. Moreover, a decision-maker has to know the problem in advance and provide a weight against each objective.
There are many multiobjective optimization algorithms suggested in the literature. The primary reason for this is to find multiple Pareto-optimal set in one execution of algorithm. Furthermore, the decision-maker would be more interested in learning alternate solutions if obtainable. There are existing algorithms associated with such a predefined assumption where prior knowledge is required to define the weight and optimal solutions obtained after several runs against each weight. Such a predefined assumption, making them ineffective in solving today's composite multiobjective optimization problems. Solving a multiobjective optimization problem is one such kind where a set of well-distributed optimal solutions are required across the Pareto-optimal front in the search space.
In SSA, there are some constraints which are specific to multiobjective optimization problems.
• A single best solution saved by SSA. It is not designed to keep many optimal solutions as best solution, which is necessary for the multiobjective optimization problem.
• The target position updates with the best solution obtained so far in single objective optimization problems. However, there is no single best solution for multiobjective optimization problems.
• No information is shared among the population other than leaders, which is accessing the global solution. Information sharing is critical in initiating the required selection constraints to move the population towards the true Pareto-optimal front during multiobjective optimization process. The DMQSSA enables information sharing across the individuals in population, especially regarding the population to progress towards the true Pareto-optimal front.

C. THE MULTIOBJECTIVE CONTROLLER PLACEMENT PROBLEM
Controller Placement Problem (CPP) is introduced in [10]. It is a complex and tedious task, implemented to encounter an explicated objective and select an expected number of controllers and their location in Software Defined Network (SDN). There are many explicated network objectives presented to solve the CPP in the literature. Heller et al. [10] debated to reduce the latency between each switch and its linked controller in a Wide Area Network (WAN). The primary use case of improving the latency between controllers to the associated switches is considered as facility location problem, which is a well-known NP-hard problem. In this article, the purpose of CPP is to minimize the controller-to-switch latency, inter-controller latency and load balancing network objectives. Designing a method to produce an optimum solution for the placement of controllers is an exhaustive and a time-taking task. Alternatively, meta-heuristics can provide a promising solution. The search space of meta-heuristics comprises of number of search proxy with presumed limitations. In case of CPP for given controller r in the network of n switches, each and every feasible combination could be n C r where r < n. For example, to discover the position of 7 SDN controllers in a network of 60 switches or nodes, the solution could be obtained after carrying out an exploration from a total 38, 620, 6920 feasible placements. An evolutionary algorithm is a substitute for such a scheme, exploring a subset of the continuous search space to produce a good solution.
Many functional parameters are required to be examined during the controllers placement in SDN. In this paper, the factors are switch-to-controller (S2C) latency, inter-controller (C2C) latency and imbalance. The goal is to achieve minimum values of these objectives and provide an exact illustration of these competing objective trade-offs. There are several research papers that introduce propagation delay as one of the key parameters for the controller placement as this parameter influences the accessibility and convergence time in the large-scale networks; this paper also acknowledges it as the most critical parameter for the controller placement. The next metric is C2C latency, which is defined as average latency between any two controllers. The final metric which is examined in this work is the imbalance of identified controllers. This metric is associated with the quality of services offered by controllers. Imbalance is the difference between the lowest and highest number of nodes assigned to the two controllers. In Pareto-based optima controllers placement, each candidate solution is compared according to the Pareto dominance principles to obtain the non-dominated solution set across the Pareto-front. In general, an obtained solution belongs to the Pareto group if there is no other solution that completely dominates the objectives.
This paper presents Multiobjective Quantum-inspired Salp Swarm Algorithm based on Delta potential-well model (DPWM) representation and a decomposition technique. DMQSSA has the following features: • A hybrid approach of two novel paradigms: SSA and Quantum Computing. Besides many other essential properties, this model can find a suitable solution faster using fewer individuals. This approach reduces the required number of evaluations significantly, which is an important performance factor for solving multiobjective optimization problems.
• The DMQSSA's Delta potential well model (DPWM) enhances the convergence speed of traditional SSA. This novel approach helps in maintaining the population's diversity, preventing the population from stagnating in the deceptive optima and increasing the algorithm's searchability. According to DPWM, if individual Salp in the SSA algorithm has quantum behaviour, the algorithm is bound to work differently due to Heisenberg's uncertainty principle of quantum physics.
• Decomposition-based approach while performing crossover operation continually searches in the subpopulation of Salp to discover solutions in the search space. An intelligent archive maintenance scheme and a selection technique are employed to ensure maintaining the non-dominated solutions list in each iteration. To the best of our knowledge, a Decomposition-based Quantum-inspired approach to boost the performance of SSA for complex multiobjective optimization problems is introduced for the first time. The proposed algorithm's performance is evaluated on the multiobjective domain's complex benchmark problems included in CEC09. A comparative study is performed to assess the performance with wellregarded algorithms like MSSA, MOPSO and MOEA/D. Moreover, a real-world multiobjective optimization application is evaluated by DMQSSA to demonstrate its success.
Furthermore, Friedman's test is conducted to decide the statistical significance of the advantage of DMQSSA. Friedman's test is a non-parametric test that is used to evaluate the statistical sense of superiority of the proposed approach. The result of the test shows that the proposed system performance is at the highest ranking together with other algorithms. Hence, the statistical test result also confirms the superiority of the proposed method. This paper is organized as follows: Section II includes a discussion on related work. Section III presents the proposed algorithm. Experimental design work described in Section IV. Results and analysis are discussed in Section V, and then conclusions are drawn in Section VI.

II. RELATED WORK A. STANDARD SALP SWARM ALGORITHM
Mirjalili et al. introduced Salp Swarm Algorithm (SSA) in 2017, which produces the swarming behaviour of Salp in the ocean by building a chain of Salp [7]. There are several research work conducted in the literature to show the improvement and applications of SSA in engineering and technology such as feature selection and the SDN multicontroller placement problem. A random search radius based simplified version of standard SSA is presented to enhance the competence of algorithm [11]. In [12], a particle-best SSA is proposed considering the global and local searchability to improve the convergence speed and reliability of the algorithm. In [13], a gravitational search algorithm was merged with SSA to enhance its searchability. In [14], self-adaptive VOLUME 10, 2022 SSA for the dynamic optimization problems is proposed. Besides, SSA is extensively used in many engineering fields including power system optimization [15], [54] and multilevel colour image segmentation [16], recently a multiple scenario multiobjective salp swarm algorithm (MS-MOSS) is proposed for sizing the photovoltaic system in which number of PV modules and storage battery is crucial and affects the reliability and cost of the system [17]. SSA demonstrates to be an efficient optimization technique for single-objective optimization and, more recently, has also reveal favourable results in solving multiobjective optimization problems [18], [19].
In SSA, each Salp constitute a candidate solution, Like any other EAs, SSA first initialize an initial set of random and trial solutions for a specified optimization problem in the search space and then iteratively start the searching process to obtain the best solution. In each iteration, the individuals update its position with regard to global optima or in-line with other Salp position. The standard SSA framework is presented in Algorithm 1, and the searching process of SSA can be described in five steps below: 1) Initialization: Initialize the population with initial position and maximum population size N , the total iterations (T). The spatial position of Salp is presented in d dimensional search space using a d dimensional vector.  (2) and (3) respectively. Where c 1 is the most important parameter to control the exploration and exploitation of the algorithm. It can be represented mathematically as (4), where t is the current iteration and T maximum number of iterations. The position of leaders and followers presented as x i j for i th Salp in j th dimension. Parameters c 2 and c 3 are uniformly distributed random numbers between 0 and 1. The upper bound and lower bound of the search space are represented as ub j and lb j independently in the j th dimension. Few researchers have proposed mechanism for maintaining the distribution of solutions in the search space. An idea of crowding distance proposed by Deb and Tiwari [20]. They presented an Omni-optimizer to maintain the diverseness of the solution in the search space. The diverseness in search space through hypervolume metric is introduced by Ulrich et al. to escalate the diversity in both decision space and objective space [21]. Another approach to improve the diverseness of the population in the decision and objective space through Lebesgue's contributions and neighbourhood count strategy, adopted by Chan and Ray [22]. A restart mechanism to obtain well distributed solutions across the Pareto front in the search space is employed by Rudolphet et al. [23]. Zhou et al. [24] introduced a probabilistic model build on MOEA to estimate the PS and the PF. In Xia et al. [25] proposed crowding estimation approach into MOEA, where crowding distances put in to guarantee diverseness in the decision space.
Li et al. [48] proposed mechanisms based on Gaussian Sampling to maintain the solution's distribution in the decision space, that has an effect on improving the overall performance of optimization technique while solving the multiobjective optimization problems. Zhang et al. [49] introduces a new approach to address multi-objective optimization based on the decomposition strategy called the decompositionbased archiving approach (DAA), dividing the whole objective space into sub-spaces, which is based on a set of weights and a single non-dominant solution selected to update the external archive to maintain good diversity in each iteration. Li et al. [50] have taken the approach of combining the decomposition strategy with Pareto's dominant principles to introduce MOEA/D with Domination Archive (MOEA/DD) to distinguish and select candidates for multiobjctive optimization, where each weight vector specifies a sub-region to estimate local population density and sub-problems for assessing ability by fitness evaluation.

C. THE CONTROLLER PLACEMENT PROBLEM
The literature of the controller placement problem primarily considers three issues in a given network topology: (1) the number of controllers, (2) the location of the controller, (3) allocation of switches to the controller. The goal is to optimize network objectives such as shortening the network latency between controller to forwarding elements and controller-to-controller, and network imbalance. For example, Heller et al. primarily focuses on propagation latency [10], [52]. The problem is formulated as a facility location problem, and K-center method is adopted to tackle this problem. Wang et al. [26] introduced the concept of network partitioning to decrease the whole latency of the network including queuing Latency of SDN controllers. Also, the load on controllers is considered in their work. Yao et al. [27] investigates one step ahead with the capacity of the controller along with propagation latency. Thus, it is regarded as capacitated K-center problem [31]. Bari et al. [30] proposed an approach by considering the large-scale WAN deployment where centralized network architecture reported many limitations related to network performance and scalability. The trade-off between propagation latency and load on controllers are investigated by Hu et al. [28]. The results depicted that a bit of increment of delay can balance the load on the control plane. Han et al. [29] investigated different approach to maximize the number of controlled switches within a bounded latency instead of minimizing the propagation latency.
The significance of network latency, including queue latency and propagation latency, has been considered in the literature. There are methods proposed to reduce the latencies for an efficient network operation [53]. However, latency is only one aspect of the overall network performance. Other aspects are load balance and resilience, which are equally critical for an optimum performance of network. This paper investigates the latency with load balance and seeks solutions to shorten the obtained values in a large-sized network. We propose to evaluate the allocated controllers based on formulated multiobjective optimization for a given number of controllers. A novel Quantum-inspired SSA is employed instead of brute force for assessing the controller's placement in the entire solution space, which is computationally efficient. The network metrics are considered for this placement is propagation latency between switches-to-controller, inter-controller latency and load balancing. These objectives are competing with each other. For instance, ordering controllers by considering to lower the inter-controller latency increases the propagation latency between node-to-controller.
To analyze this kind of relation between the objectives, it motivates to study the Pareto-based optimal approach for the placement of controllers, which helps with the set of possible arrangements of controllers to inspect and decide best possible solution.

III. PROPOSED ALGORITHM
This section presents main framework of the Decomposition based Quantum-inspired Salp Swarm Algorithm for Multiobjective optimization (DMQSSA). The DMQSSA is composed of Delta potential well model, crossover operation with decomposition technique, an intelligent technique for the archive maintenance, and a constrained handling method. The details of various components are expressed in the subsections.

A. THE FRAMEWORK OF DECOMPOSITION-BASED MULTIOBJECTIVE QUANTUM INSPIRED SALP SWARM ALGORITHM
The proposed DMQSSA employs a well-known method to enhance convergence and diversity in the population, i.e., Delta potential well model representation from quantum mechanics instead of binary representation, which is better suited to the multiobjective optimization. The basic decomposition technique applied during the crossover operation to search in the subpopulation of Salp, and get new possible solutions in the search space. This decomposition approach is based on population in a group for multiple problems unlike the technique adopted by MOEA/D. In DMQSSA, the dominion-based approach has been employed to maintain the archive which is similar to MSSA, which sometimes lead to zero non-dominated solutions and an inferior Pareto optimal set. The existing archive maintenance scheme is modified and employed to upgrade the quality of obtained solutions in the search space. We also adopted a mechanism to select the globally best solutions after crossover operation to get a well-distributed Pareto optimal solution.
In quantum computing, when considering the dynamics of particles and avoiding explosion with guaranteed convergence, particles must be moving in a potential attraction field towards the center point [32]. There are potential field model such as Delta potential-well and Quantum oscillator to confirm the particles movements in a bound state. This paper employs the Delta potential-well model for the convergence of each particle with quantum behaviour because convergence is much faster in the quantum oscillator, lead to prematurity [6].
In the proposed method, the algorithm's representation is changed based on the principle of DPWM and integrated into the standard SSA. The inspiration of the proposed DMQSSA is based on the similar approach to QPSO; the aim is to improve the algorithm's overall performance for the multiobjective optimization problems. Further, this modification aims to adapt SSA and enhance the exploration ability and increase the algorithm's overall performance by an appropriate balance between exploration and exploitation. The VOLUME 10, 2022 embedding approach of SSA with quantum computing is a promising scheme to improve the performance of standard SSA. DPWM representation is known for increasing the speed of convergence and diversity in the population, an integrated approach designed for better search strategy in SSA, which increases the population's global searchability.

B. DELTA-POTENTIAL WELL MODEL
In quantum physics, the Delta potential well model represented by Dirac delta function as a generalized function and qualitatively corresponds to the potential, which is zero everywhere other than points where it takes an infinite value. Each Salp (X i ) has a quantum state, and it can be formulated using the wavefunction (x, t). From the analysis of the movements of Salp towards a centre point A, we assume that each Salp moves in a Delta potential well of search space for which the centre point is A as described in equation (8). Hence, the Salp position update equation is designed to move the chain towards the A.
For the simplicity of the presentation, one-dimensional space with centre point A, the Delta potential well can be represented as follows: where γ is a constant positive value and γ δ(y) denotes the Dirac delta function for y = (x -A).

C. DECOMPOSITION-BASED TECHNIQUE FOR CROSSOVER OPERATION
The basic decomposition strategy is designed to be used during crossover to find more promising solutions in the search space while searching in the subpopulation of Salp. According to decomposition rules, the N size population is divided into an equal group of subpopulations. It calculates the number of subpopulations and the decomposition cost for each group. Newly born individual generated during the execution of DMQSSA for the crossover operation. In crossover operation, two nearby random searches are performed in the subpopulation of Salp and its corresponding index from the Salp population. When considering the Euclidean distance in the middle of two individuals within the subpopulation, it used to be somewhere around in middle. Therefore, the crossover operation in the middle is investigating the suitable individual in that region. But, performing the crossover operation only in the middle of individual of the same region is not very much useful for investigating an open region. The algorithm only by subpopulation search to explore the new region will not be very useful. Hence, we execute the crossover operator between the members and with another individual, randomly chosen from its corresponding Salp population.

D. SALP POSITION MEASUREMENT
Multiobjective Quantum-inspired SSA algorithm employs the Delta potential-well model representation from the quantum mechanics and considers Salp movements towards a centre point A as per equation (8), which is updated based on the lead position (F) stated in section II as the best location. The potential field model and quantum oscillator in quantum physics ascertain the particles movements in a bound state due to the characteristics that each particle oscillate in a potential attraction field. Salp in the delta potential-well model must move in bound state according to this strategy. The position of the Salp needs to be calculated to evaluate the fitness of each Salp. However, only the probability of position for each Salp (X i ) can be learned from the probability density function | (x, t) | 2 , which depends on the potential field the particle lies into. The probability of position indicates that a Salp appears at position x relative to a point A. Hence, it is required to measure Salp position using the collapsing technique, i.e. transformation from a quantum state to the classical form. The Monte Carlo method can reproduce this process of measurement [33], [34].
In this paper, the following iterative equations were used to measure each Salp according to the Monte Carlo method. As stated, Salp moved round and swayed towards A with its kinetic energy declining to zero. With this characteristic in the modified algorithm, the Salp are in inbound states and avoid explosion with guaranteed convergence.
we assumed here first iteration l = 0 and maximum iteration size is L. The c 4 , r j and u j are random numbers between [0, 1] in j − dimension. B l describes in equation (7), a contractionexpansion coefficient that is adjusted dynamically during the optimization process. A j is represented in equation (8), a centre point for Salp chain movements. It is also known as a learning point inclination for Salp to oscillate around. BestMean l presented mathematically in equation (9), mean of the best position. X k j is the k th Salp in j − dimension and X k j+1 is the new position of Salp.
where the current iteration is l, and maximum allowed iteration is L.
A j = r 1j * X k j + r 2j * Leader j r 1j + r 1j (8) where r 1j and r 2j are random numbers in the range [0, 1]. Leader j is the leader position and represented as the best location.
where N is the maximum number of populations. Control parameters calculation 8.
end while 20. Best Solution * Retums the Obtained Pareto-set

E. CONSTRAINT HANDLING TECHNIQUE
The above mentioned DMQSSA algorithm is designed for unconstrained multiobjective optimization problems. However, there is a technique, i.e., ON-OFF, considered for the constrained multiobjective optimization problems with minimal change in the selection and archive maintenance strategy.
In the ON-OFF technique, ON indicates that the constraint is satisfied and OFF indicates that constraint is not satisfied. Each constraint has been considered as another objective function returning values 0 (OFF) and 1 (ON). So, in this case, if the check is matched, the returned value from function is 0; else, the returned value from function is 1. Hence, DMQSSA can also be employed to solve the applications for the constrained multiobjective optimization. A little change would be needed in the method UpdateArchive() and BestPosition(). In UpdateArchive(), the value returned by objective functions with no modification is considered a solution of the estimated set stored. The solutions in the estimated set are satisfying all the constraints. Infeasible solutions are taken out from the estimated set. After that, everything proceeds in the same way as the original UpdateArchive() function does. In the BestPosition(), the best solutions are chosen only if they are nondominated and satisfy all the constraints.

F. THE DECOMPOSITION-BASED MULTIOBJECTIVE QUANTUM INSPIRED SALP SWARM ALGORITHM BEHAVIOR
This hybrid approach with quantum computing and a decomposition strategy is employed to extend the original SSA for the complex multiobjective optimization problems, and boosts the algorithm's overall performance. DMQSSA maintains diversity during the multiobjective optimization and produces a set of well-distributed Pareto-optimal solutions. Preserving diversity is extremely important to avoid entrapment into non-distributed solutions while solving such optimization problems. The exploratory ability depends on the population's diversity, which means the exploration power is lower in the identical elements of the population.
Algorithm 2 presents the pseudo-code of DMQSSA. Equation (6) updates the Salp position and produces new solutions. After the initialization phase of the population and first approximation calculation in step 2 and 4, the algorithm enters into the optimization process's main loop. This loop executes for MAX generations and is formed to perform several tasks. First, the control parameters for the standard SSA and DMQSSA calculated according to the equation (4) and (7), respectively, followed by calculating the objective function values for each Salp and identify the non-dominated ones at step 4.
Step 9 handle the archive to update the obtained non-dominated solutions. The nondominated solutions added if there is a place in the repository. At this stage, non-dominated solutions are compared with available solutions in the archive to keep only non-dominated solutions in the archive. After that, it also deletes the solutions from the crowded region to maintain the maximum number of solutions. For this, it first ranked all the archived solutions and then selects using an intelligent roulette wheel technique.
Further, it checks to maintain minimal optimal solutions in the repository to avoid producing inferior solutions in consecutive runs of the algorithm.
Step 15 executes only once in the first iteration and develop a Salp chain where a division of population happen into leaders and followers. The algorithm at step 16, evaluating the solutions using the crossover operator followed with calculation of decomposed cost and updating the search boundary. The process repeats for MAX generation, and in each generation, it produces a set of well-distributed Pareto-optimal solutions across the Pareto-optimal front. At stage 12, the interference process and proposed quantum-based equations for DMQSSA are executed to generate new solutions. This process consists of updating the individual position using equation (6), calculating the contraction-expansion coefficient, evaluating the converging points and best mean along with the fitness of Salp. At step 20, the algorithm returns the obtained Paretooptimal set. The flowchart is drawn as shown in Fig.1 of the proposed algorithm DMQSSA.
The above explanations demonstrate that the DMQSSA algorithm is capable of efficiently finding accurate Paretooptimal solutions with good distribution for all the objectives.  sub-population size respectively. The result and discussion section also demonstrates the execution time of DMQSSA along with the state-of-the-art algorithm's execution time for each benchmark problem as shown in Appendix F's Table 3,  Table 5, and Table 7 of the supplementary material, which shows clearly the superiority of the algorithm.

G. THE MULTIOBJECTIVE CONTROLLER PLACEMENT PROBLEM
The control plane and data plane are disassociated in the Software-defined networking (SDN) and the problem of controller placement emerged in network. While minimizing propagation latency is a crucial aspect of the controller placement, many other competing objectives requires a thought, such as inter-controller latency, fault tolerance, load balancing, controller capacity, and the relationship between these objectives [35]. The multiobjective controller placement problem introduced in this section is solved by the proposed DMQSSA algorithm. The purpose is to find the controller's location in SDN, considering the multiple competing objectives such as minimizing propagation latency between controller and linked switches, inter-controller latency, and load balancing. The multiobjective approach allows a more precise illustration of the trade-offs between competing objectives. Moreover, available alternatives are always preferable concerning the different purposes for the decision [36]. Motivated by the numerous competing objectives while evaluating the controller placement, the controller placement's first and most crucial step is problem formulation. That means, the problem is to be suitable for an optimizer to execute the optimization process.
• Latency indicates the connectivity from the controller to assigned switches and inter-controller communication.
• Load balancing deal with the imbalanced state of network due to the assignment of switches to its nearest controllers. The result of placements using this mechanism tends to the imbalanced position of switches for the controllers. In this paper, the above-defined network metrics are considered as main objectives for the placement of controllers. These are competing objectives. As stated in section I, such relations between the objectives lead to selecting the Pareto-optimal set from the obtained solutions. The problem representation is an essential step in the placement of controllers. We present a network G = (V , E), where V is set of nodes, and E is a set of edges connecting the nodes. The node set V comprises of a set of switches S and the controller's set H . The controller's set indicated as For a defined k as total controllers to be positioned for the specific deployment can be presented as |H i | = k. Let's examine all possible locations of the controllers H = {H 1 , H 2 , H 3 , H 4 . . . .H m }. This is to discover the location of controller H i ∈ H , which is an optimization problem where an objective function is defined as a network metric to be optimized. The node to controller latency is used by the majority of networking literature as the network metrics for the placement of controller in SDN. It is indicated as an average distance linking the forwarding elements and its connected controller as shown in equation (10).
where d (s, h) is the direct path between switch s to its associated controller h ∈ H i . In the similar analogy of the node to controller latency, inter-controller latency for a given placement H ∈ 2 |V | and any pair of controllers h 1

and h 2 in this placement can be defined as inter-controller Latency
AverageLatency H

(H ) between h 1 and h 2 :
AverageLatency H The defined metrics in equation (10) and (11) strived for the shortest communication paths during the placement of controllers.
For a desired reliable controller placement, controller load balance is another critical metrics. Network imbalance metrics is a minimization problem to solve and to comply with the definition of the problem in this article. In a given placement H and controller h, the total number of switches assigned to h when each switch connects to its nearest controller is defined as r h . The imbalance metric imbalance (H ) indicates the difference in r h for the two controllers with the lowest and highest number of assigned nodes, respectively. Further, in the presence of failures, the imbalance can be quantified by analyzing the imbalance in the assignment for corrective measures.
The goal is to solve the placement of controller problem using DMQSSA that is to achieve minimum values of average latency and imbalance metrics defined as objective equations (10), (11) and (12), i.e., to find a controller placement featuring an adequate trade-off between the goals that are relevant for a particular use case. The strategy to use DMQSSA for the multiobjective controller placement is to efficiently provides practical solutions after exploring the search space and returning the Pareto-optimal set as alternatives to the decision-makers for an efficient network operation.

IV. EXPERIMENTAL DESIGN
In the following, the performance metric and multiobjective test suits used in our experiments are introduced firstly. The algorithms parameters settings and evaluation mechanism are defined, followed with the state-of-the-art algorithms used for the comparison of results on computer running environment.

A. PERFORMANCE ASSESSMENT METRICS
To evaluate the performance of DMQSSA algorithm, four metrics are used in this paper: Pareto Sets Proximity (PSP) [37], Inverted Generational Distance (IGD) [38], [39], Hypervolume (HV) [41] and Spread ( ) [51]. (1) PSP assesses the analogy between the exact Pareto-optimal set and the obtained solution set, which is reflecting the parallelism of the true PSs with the achieved Pareto-optimal set. The more significant PSP value means the obtained solutions are equally distributed in the search space. Algorithm obtaining sizeable PSP value is reviewed as a superior design for the multiobjective optimization. Let U * denotes a set of uniformly distributed exact Paretooptimal set and H represents a set of obtained Pareto-optimal solutions of the final population. The IGD can be calculated as the average distance from U * to H in the decision space using the below equation: where u represents an exact point in U * , d(u, H ) is the minimum Euclidean distance between u and the points in H . Similarly, the PSP value can be calculated using the below equation, which indicates the similarity between the exact PSs and obtains PSs.
CR shows the overlapping ratio between the exact optimal Pareto-set and the obtained optimal Pareto-set, and IGD is the inverse generational distance in the decision space. A larger PSP value means the obtained solutions are well distributed in. the decision space and is satisfactory.
The HV value can be calculated using the below equation.
where Z (s) refers to the hypercube bounded by a solution s in obtained PF. The larger HV value is a better approximation to PF. The Spread ( ) equation is presented as follows.
where dist i is Euclidean distance between two continuous solutions anddist is average of these distances.
Further, a non-parametric test, i.e., Friedman's test employed for the statistical analysis, confirms the results' significance.

B. MULTIOBJECTIVE TEST SUITES
The research on the multiobjective optimization problem is still in the emerging phase. In this study, several multiobjective optimization benchmark functions used to evaluate the proposed techniques in the DMQSSA.

1) Zitzler et.al. proposed four complex and challeng-
ing multiobjective benchmark problems called ZDT [40], [41], is applied to benchmark the performance of DMQSSA. The benchmark problems are ZDT1, ZDT2, ZDT3, and ZDT4. The nature of these functions is both convex and non-convex. These multiobjective benchmark problems are complex due to many decision variables, the disjointedness of Pareto-optimal and multiple local fronts. 2) Most of the functions in the ZDT series are not multimodal. This study employs the CEC09 test functions to benchmark the achievement of the presented algorithm on more difficult testbeds. The above-mentioned benchmark suites are the most complex benchmark functions in the literature of multiobjective optimization and able to confirm whether the superiority of DMQSSA is significant or not.

C. DMQSSA COMPARED WITH COMPETING ALGORITHMS
The efficacy of the presented algorithm DMQSSA can be demonstrated using the metrics PSP, IGD, HV and Spread, and comparing with the well-known multiobjective optimization algorithms in the literature. In this paper, we made a comparative study with MSSA, MOPSO and DMOEA/D. Note that the source code implementation of the MOPSO and MOEA/D algorithm in MATLAB is available and can be found https://yarpiz.com/category/multiobjective-optimization. For impartial comparison with the state-of-the-art algorithms, the maximal number of evaluations is set to 500, and the population size is set to 100. Each algorithm is run 32 times on each benchmark problem.

D. PARAMETER SETTINGS
The maximum repository size for MOPSO is 100. The other parameters c 1 and c 2 are set to 2.05 with inertia weight 0.1 and mutation rate 0.1 [8]. The parameters value of MSSA, and MOEA/D are used with no change and that is same value as used in the standard algorithm. An interface has been created from main program to initialize the parameters of DMQSSA and other applicable algorithms based on the execution flow for the optimum performance of algorithms.

E. DEVELOPMENT FRAMEWORK
The algorithm's development environment is as follows: 1) The computer is based on 16  To realize the implementation of DMQSSA for the placement of controllers in SDN, OS3E topology from Internet2 community has been selected that is being used in most of the literature for the controller placement problem. The most popular use case also being considered in this paper as 34 forwarding elements with k = 4 controllers to be positioned in SDN. The purpose of the controller placement in this paper is to discover all the possible placements of controller k within the constraints of minimizing the network latencies and network node imbalance as described in section III (G). There is an empirical conclusion when the software is being designed considering the controllers to access the real-time events or pushed the expected events in advance to the network nodes [10].
To assess all possible placement of controller for a given number of controllers that has to be positioned in the SDN network considering the objectives to discover an optimal position set in the network, is a time-taking and full-scale task. Sometimes it runs out of resources during the evaluation including RAM capacity in many cases [35]. In that case, the whole search space is combination of controllers r and switches u, where the optimum position of controllers r to find place out of all applicable controller placements using the below equation.
Consequently, for the fairly small-scale network switches (u), the number increases exceedingly when the number of controllers r increases. For an example, OS3E topology network with 34 forwarding elements and 2 controllers results in possible placements as 34 2 = 561 and moving to 34 5 = 278, 256 possible placements. This change in the possible placements is extremely high and to evaluate each and every placement requires a high computational effort and time.

V. RESULTS AND DISCUSSION
We conduct qualitative and quantitative analysis for demonstrating the conceptual assertion made in the preceding sections. The qualitative outcome is obtained from the visualization tools to ascertain if DMQSSA is a better algorithm. And quantitative results are collected in the form of an average of PSP, IGD, HV and Spread to find how much better the DMQSSA is when compared to the other well-known multiobjective optimization algorithms. A group of benchmark problems is considered from the class of unconstrained and constrained functions introduced in the Special Session & Competition on Performance Assessment of Constrained/Bound Constrained Multiobjective Optimization Algorithms CEC09 [43]. The benchmark problems also includes the complex convex and non-convex multiobjective problems in the ZDT series [40]. Further, the proposed procedures are applied to a complex multiobjective controller placement problem involving three conflicting objectives, i.e., node to controller latency, controller to controller latency and imbalance of the controllers.
The DMQSSA is executed and evaluated for the multiobjective optimization benchmark problems, and quantitative results are calculated using the PSP, IGD, HV and Spread of each trial. They are recorded in the form of average and standard deviation (SD), presented in Table 2 to Table 13 in Appendix F of the supplementary material as results for the analysis. The obtained solutions as best Pareto-optimal set is illustrated using visualization tools shown in Fig.2-7 of the supplementary material. Also convergence curve Fig.10-11 shows the IGD, PSP, HV and Spread of DMQSSA for benchmark problems ZDT1, ZDT3, UF1, UF3, UF6 and UF9 in Appendix H of the supplementary material. The largest archive size for DMQSSA is set to 100. The three well regarded algorithms in the literature of multiobjective optimizations are adopted along with Algorithm's Parameters shown in Table 1, for the results endorsement and the comparative study in this paper: MSSA [7], MOPSO [8] and MOEA\D [9].

A. EXPERIMENTAL VERIFICATION OF THE PROPOSED ALGORITHM
The success of the Delta potential well model from quantum computing and an archive maintenance scheme with a crossover operator on decomposed problems illustrated using qualitative and quantitative analysis for the obtained results. The proposed DMQSSA is compared with the standard MSSA [7], MOPSO [8], MOEA\D [9].
The mean PSP, IGD, HV and Spread values on twentyfour benchmark functions are presented in Table 2 to 13 in Appendix F of the supplementary material. Highlighted blue cells in the tables denote the best performing algorithm. The performance of DMQSSA is better than that of the other algorithm on almost all of the benchmark problems. Furthermore, the Pareto optimal set obtained against the Pareto-front by these four algorithms on twenty-four benchmark functions is shown in Fig. 2 to 7 in the Appendix of supplementary material.
It is noted that the PSP values acquired by MOPSO, MOEA\D, and MSSA on most of the benchmark functions are marginally larger or lesser than each other. In contrast, the PSP value attained by DMQSSA is superior than MSSA, MOEA\D and MOPSO on almost all the benchmark functions other than a few, where MOPSO, MOEA\D is slightly greater. Moreover, DMQSSA discovers new Pareto solutions in search space as compared to the other algorithms, as exhibited in Fig. 2 to 7 and explained rationale in the following sections; Delta potential well model allow the algorithm to find sufficient optimal solutions. This model integration into SSA can guide the Salp to explore the promising area with optimal information. The introduction of crossover operation between the decomposed problems and a modified archive maintenance scheme helps in obtaining a good dissemination of obtained solutions in search space. Similarly, solutions to be out and keep minimal non-dominated solutions in the repository using the preferential rules of non-dominated and intelligent route wheel selection method. Thus, if solutions are not alike in the search space, they can remain and maintained in repository.
Although the DMQSSA is designed for unconstrained multiobjective optimization problems, the constraint handling technique in the objective functions with a slight change in the selection mechanism enables DMQSSA to solve the constrained multiobjective optimization problems also efficiently. In conclusion, introductions of Delta potential well model and basic decomposition strategy along with an intelligent archive maintenance technique, makes DMQSSA more effective in solving the multiobjective optimization problems. Table 2, Table 3, Table 8 and Table 11 in Appendix F of the supplementary material show that the DMQSSA algorithm significantly performs better than MSSA, MOPSO and MOEA\D on most of the ZDT series multiobjective optimization problems. Further, when examined, the obtained PF in Fig.2 of Appendix A in the supplementary material indicates that DMQSSA shows a better convergence than MSSA, MOPSO and MOEA\D on all the ZDT functions. The solutions distribution is mostly uniform for the DMQSSA algorithms, that means the high coverage of the algorithm. There is a space observed in between of the obtained Paretooptimal front by MSSA, MOPSO and MOEA\D on some of the ZDT problems, which indicates how the coverage of the algorithm is negatively impacted. The Pareto-optimal front of the ZDT1 function is rounded-shaped. Accordingly, the convergence and coverage of the algorithms on a distinct Pareto optimal front can be bench-marked. When observing the obtained Pareto optimal solutions in Fig.2 and HV, PSP and Spread values in Table 2, Table 8 and Table 11 for DMQSSA, it performs outstandingly better than MSSA, MOPSO and MOEA\D algorithms for ZDT1, ZDT3 and ZDT4. The coverage of the MOPSO on ZDT2 is deficient when noticed in Fig.2 and Table 3. It appears that the convergence of DMQSSA is superior than MSSA with good coverage. When investigating the evolution curve of ZDT1, ZDT3 and UF1 shown in Fig.3 of Appendix G of supplementary material, the consistency of convergence can be bench-marked for DMQSSA which is superior than MSSA, MOPSO, and MOEAD. With a particular analysis of IGD value for the different size of population and converging curve indicates that DMQSSA can find suitable solution faster using fewer individuals. MSSA algorithm observed a space in Pareto optimal front, but the solutions distribution is really consistent for the rest of front. The Pareto optimal front of ZDT2 and ZDT4 is incurved-shaped and every time it VOLUME 10, 2022 demands for accumulation-based approaches. Still, the above mentioned outcomes shown that DMQSSA can efficiently approximate these test problems with good convergence and coverage. Between DMQSSA and MOPSO; PSP value, IGD value, HV value, Spread value and Fig. 2 show that the coverage and convergence of DMQSSA are much better and keen. Table 2-3 and Fig. 2 analysis of ZDT3 function has Pareto-optimal front with isolated regions. Such kind of Pareto-optimal fronts is standard in real-world optimization problems. It is demanding for the optimization algorithms to estimate the Pareto-optimal set of such problem. An algorithm might be deceived in one of the segments and fail to get the Pareto-optimal set in all the segregate areas. When comparing the results on ZDT3 for MSSA, MOEA\D, and MOPSO; the coverage of MOEA\D and MOPSO is good on some of the ZDT functions, but the convergence is inferior. The obtained Pareto-optimal solutions for some of the segregated areas are afar from the actual Pareto-optimal front. When comparing the results in consideration of the obtained Pareto-optimal front by DMQSSA, it is evident that DMQSSA performs outstandingly superior to other algorithms as regards of convergence and coverage. These results show that DMQSSA can successfully discover all the segregated areas of a Pareto optimal front with high dissemination in each area.

B. COMPARATIVE STUDY ON ZDT TEST SUITE
Experimental results and the above discussions prove that DMQSSA can approach in-curved and out-curved shaped Pareto optimal front with very good convergence and coverage on ZDT1, ZDT2, ZDT3 and ZDT4. Table 4-5, Table 9, Table 12 in Appendix F and Fig. 3-5 in the Appendix B of the supplementary material displayed the outcome of DMQSSA on CEC09 benchmark problems while comparing with MSSA, MOPSO and MOEA/D. Table 5, Table 9 and Table 11 indicates DMQSSA performance is superior on IGD, HV and Spread respectively for all the CEC09 unconstrained problems except UF8, where the MOEAD result is slightly better. The Hypervolume and IGD quantifies both convergence and diverseness of the obtained solutions in the search space. The small value of IGD means; convergence and diversity of achieved solutions in the search space are adequate for the stated multiobjective optimization problems. High convergence may result in an inadequate coverage.

C. COMPARATIVE STUDY ON CEC09 BENCHMARK PROBLEMS
When reviewing the results of Table 4-5, Table 9, Table 11 and obtained qualitative metrics in Fig. 3-5 of the Appendix B in the supplementary material, it can be observed that the coverage of DMQSSA is superior than MOPSO, MSSA and MOEA/D on most of CEC09 unconstrained problems. The results show that the presented algorithm benefits from high coverage as well. In Table 4, Table 9 and Table 12, the DMQSSA algorithm provides better results of PSP, HV and Spread on most of the CEC09 unconstrained problems except PSP and HV of UF8, where the MOEA/D result is slightly better. The PSP quantifies the likeness of the exact Pareto-optimal set and the obtained Pareto-optimal set. The considerable PSP value means, the obtained solutions in the search space are well disseminated.
The result of constraint optimization functions CF1 to CF10 in Table 6-7, Table 10, Table 13 in Appendix F of the supplementary material and the obtained Pareto optimal set in Fig. 6-7 of Appendix C in the supplementary material, demonstrates that DMQSSA expeditiously handles Salp towards different regions of true Pareto optimal front and avoid getting solutions in the infeasible areas. Inspecting Table 6 and Table 13, the DMQSSA algorithm provides better Pareto set proximity than MSSA, MOPSO and MOEA/D for almost all of the constraints benchmark problems except the value of PSP for CF1, where MOPSO is better. The convergence and diversity of the algorithm resulted better as IGD value on CF6, where MOEA/D is slightly better.
The experimental results and above discussion prove that DMQSSA can approximate almost all of the CEC09 benchmark problems with very good convergence and unvaried dissemination of solutions on the Pareto optimal front.

D. ALGORITHMS PARAMETERS AND EXPERIMENTAL RESULTS
A total of seven algorithmic parameters are used in the DMQSSA, including the maximum generation and objective function instance. There is an initial computational study performed, which resulted in the parameter setting of the algorithm. The mean and standard deviation of the PSP, IGD HV and Spread values of 32 independent eventual estimation obtained for each benchmark problem, as shown in Table 2-13 in Appendix F of the supplementary material. In the IGD tables of all the problems includes the CPU time also. The CPU time of DMQSSA and MSSA are much shorter than MOPSO and MOEA/D. For some of the problem's CPU time of DMQSSA is larger than MSSA because, for those problems, we obtained a large set of estimation while solving those optimization problems. We need to find the nearby neighbour of each solution in the estimation set when we use UpdateArchieve to lower the size of the defined approximation to the maximum size allowed. Observing Table 3,  Table 5 and Table 7, it is observed that the average IGD values are suitable for all instances except UF8, CF1 and CF6, where MOEA/D, MOPSO and MOEAD have slightly better IGD value. The reason is still not known why the IGD value is a bit higher for DMQSSA.

E. STATISTICAL RESULTS AND ANALYSIS
The outcome of algorithms from the former tables discussed were collected and suggested based on the results achieved in 32 independent runs of the algorithms. The statistical metrics average and standard deviation are employed to reveal how effectively the presented algorithm performs on an average. A non-parametric Friedman's test is carried out in this subsection to comprehend how significant the primacy of the proposed algorithm is in view of each run and prove that the obtained results are not by chance. Friedman's test is a non-parametric test which is used to determine if there is a significant difference between the algorithms compared. The p-values that are less than 0.05 could be considered as solid evidence in opposition to the null hypothesis. The null hypothesis H 0 is set and result R is tested for the provided p-value (X 2 R−1 > P), where P is test statistics [44]. The test statistics result at confidence level α=0.05 shows that the differences between the different approaches are significant, with the result that this null hypothesis is rejected, as shown in Table 2.
When the defined null hypothesis is rejected, that is, there is a significant difference between comparative algorithms, post-hoc analysis can be applied to compare different conditions and treatments [44]. The post-hoc study is suggested to perform once the significant difference is confirmed, where pairwise comparison is made for detecting significantly different sets of algorithms [45], [46]. The leading algorithm on each benchmark problem is selected and collated with other algorithms separately while conducting this statistical test. The Bonferroni-Holm process has been used for this analysis which works on the sequential rejective incremental approach for the calculation of the adjusted p−value [47]. For an instance, if the leading algorithm is DMQSSA, the doublet differentiation is made between DMQSSA and MOPSO, DMQSSA and MSSA, DMQSSA and MOEA/D. The test and results of post-hoc analysis on the obtained value of IGD are presented in Table 1 of Appendix D in the supplementary material, including p-value and adjusted p-value. It is noticeable in the above-mentioned tables that the superiority of the presented algorithm is statistically significant in most of the test cases. The DMQSSA tends to produce p-values less than 0.05, which indicate that this algorithm is really consistent on the benchmark functions.

F. THE MULTIOBJECTIVE CONTROLLER PLACEMENT PROBLEM
DMQSSA algorithm is implemented to assess all the possible placements of controller for the OS3E network topology from Internet2 community. OS3E network topology consists of number of forwarding elements (n = 34) and total controllers to be positioned k = 4. The obtained results of 32 independent runs, are presented in Table 3 for the metrics average latency between switches and controller, inter-controller latency and controller imbalance. The best placements are found as follows: in 18th iterations while achieving minimum average latency between switches to controller indicated controller's position as Portland, El-Paso, Jackson and Raleigh; at 16th iterations, while reaching minimum imbalance demonstrated controller's position as Portland, Kansas-City, Atlanta and Washington; and in 10th iterations, while achieving minimum inter-controller latency with a suggested controller's position as Memphis, Atlanta, Indianapolis and Louisville, as feature in table marked in bold and shown position of controllers in network topology, in Fig.8 of Appendix E in the supplementary material.

G. ROLE OF DELTA POTENTIAL WELL MODEL, DECOMPOSITION AND ARCHIVE MAINTENANCE STRATEGY
The outcome shows that DMQSSA could be perform significantly superior than state-of the art algorithms while solving the multiobjective optimization problems. The DMQSSA algorithm shows high convergence and good coverage. The superior convergence and coverage of DMQSSA is because of upgrading the solutions throughout the best non-dominated solutions acquired until now in the Delta potential well model representation, where attractors are defined around the solutions moving with regard to the best solutions and by applying the basic decomposition strategy described. Also high convergence derived during the optimization, quicken the activity of the Salp chain with regard to the secured best non-dominated solutions until now in the archive.
Decomposition method, archive maintenance strategy and selection technique resulted in the high coverage achieved by DMQSSA. Further to demonstrate the dissemination of obtained solutions across front, non-dominated solutions in the most occupied regions are abandoned by DMQSSA when the archive becomes full. The selection method also highlighting coverage as it chooses solutions from the slightest populated regions to be explored and exploited by the artificial chain.
The role of the Delta potential well model with a basic decomposition strategy is to swiftly recognize the optimal global solution following the leading knowledge of Salp in the initial phase of iteration and to find further potential solutions by making use of the neighborhood of optimal solution in the later stage of iteration. The crossover operator applied on the decomposed problems further enhances the convergence and essentially play a role in exploring the suitable individuals in the obtained solutions area. The purpose is to exploit the new region by performing the crossover operation between the individuals with more considerable distance. The existing archive maintenance strategy is modified for DMQSSA to keep preferential non-dominated solutions according to the dominion rules and avoid zero nondominated solutions in the repository. The preferential solutions are non-dominated solutions obtained out of crossover operation and roulette wheel technique. This approach helps maintain the archive with Pareto optimal set and ensure it is available for the next generation.
To verify its effectiveness, the DMQSSA is differentiated with the basic MSSA. The results of the IGD value are provided in Appendix F of the supplementary material Table 3, 5 and 7. This clearly indicates that the execution of DMQSSA is superior than that of MSSA on all the benchmark problems, which indicates that the proposed strategy enhance the execution of algorithm.

H. FUTURE WORK AND LIMITATIONS OF THIS STUDY
To summarize, the outcome indicate that DMQSSA exceptionally performed better in contrast to the well-regarded algorithms in literature. The rationale is derived by using the two attributes, derived from the result of metrics assessment: high level of the convergence and coverage. Better convergence is because of the Delta potential-well model representation from quantum computing and target selection mechanism. The most leading non-dominated solutions every time improves the position of others. High coverage is one more superiority of DMQSSA algorithm due to Decomposition, archive maintenance technique, and target selection system. As the crossover operator used for the solutions chosen from different subproblems distributed solutions, the algorithm every time discarded solutions from most crowded region and superiors are selected from the least crowded region of the repository. DMQSSA ameliorate the diverseness and coverage of solutions across the objectives of optimization problems. Although, when considering these benefits, an attempt is made to solve the problems with five objectives, DMQSSA is expected to be applied on the problems with three objectives for more promising results. Since DMQSSA is a Pareto dominance-based algorithm, it is less efficient in proportionate to number of objectives. This is mainly because of the problems with more than four objectives produces huge number of solutions as non-dominated, so the archive becomes full swiftly due to the current design. Therefore, DMQSSA is acceptable for solving problems with below four objectives.

VI. CONCLUSION
In this paper, a quantum-inspired multiobjective Salp Swarm Algorithm based on decomposition technique (DMQSSA) is introduced to locate the multiple Pareto-optimal solutions for multiobjective optimization problems. The Delta potentialwell model (DPWM) representation is more suited for multiobjective optimization than the binary representation. DPWM is well known and being used in literature for enhancing the convergence and diversity in the population. Decomposition based approach, while performing crossover continually searches in the subpopulation of Salp to discover better promising solutions in the search space. Furthermore, the traditional repository continuation method is modified and employed to ameliorate the standard of the attained solutions in the search space along with adopting a system to select the globally best solutions.
The proposed DMQSSA algorithm is compared with three well known multiobjective algorithms in the literature, i.e., MSSA, MOPSO and MOEA/D. All the algorithms are evaluated on twenty-four unconstrained and constrained multiobjective benchmark functions, including the ZDT series. The experimental results shows that DMQSSA has superior performance than the well-regarded algorithms in multiobjective optimization domain and can find more Pareto-optimal solutions with better distributions. Ultimately, the presented DMQSSA is applied to a real-world problem, a multiobjective controller placement problem. Friedman's nonparametric test was carried out to determine the statistical significance of the obtained results as compared with other techniques. Post-hoc analysis on the result of Friedman's test indicates that the proposed framework for the multiobjective optimization problems performs better than other methods.
The future study will consider to analyze different constraint handling technique along with DMQSSA for constraint multiobjective optimization problems and exploit the other decomposition approaches, including weighted sum and Tchebycheff when coupling it with other metaheuristics to design a more robust strategy that can effectively deal with more complicated multiobjective real-world problems.
Appendixes included in the supplemental material.