Cloud Computing Considering Both Energy and Time Solved by Two-Objective Simplified Swarm Optimization

Abstract

Cloud computing is an operation carried out via networks to provide resources and information to end users according to their demands. The job scheduling in cloud computing, which is distributed across numerous resources for large-scale calculation and resolves the value, accessibility, reliability, and capability of cloud computing, is important because of the high development of technology and the many layers of application. An extended and revised study was developed in our last work, titled “Multi Objective Scheduling in Cloud Computing Using Multi-Objective Simplified Swarm Optimization MOSSO” in IEEE CEC 2018. More new algorithms, testing, and comparisons have been implemented to solve the bi-objective time-constrained task scheduling problem in a more efficient manner. The job scheduling in cloud computing, with objectives including energy consumption and computing time, is solved by the newer algorithm developed in this study. The developed algorithm, named two-objective simplified swarm optimization (tSSO), revises and improves the errors in the previous MOSSO algorithm, which ignores the fact that the number of temporary nondominated solutions is not always only one in the multi-objective problem, and some temporary nondominated solutions may not be temporary nondominated solutions in the next generation based on simplified swarm optimization (SSO). The experimental results implemented show that the developed tSSO performs better than the best-known algorithms, including nondominated sorting genetic algorithm II (NSGA-II), multi-objective particle swarm optimization (MOPSO), and MOSSO in the convergence, diversity, number of obtained temporary nondominated solutions, and the number of obtained real nondominated solutions. The developed tSSO accomplishes the objective of this study, as proven by the experiments.

1. Introduction

The computing tasks of cloud computing, which play a very important role nowadays due to the high development of technology and the many layers of application, resulting in the increasing application and demand for cloud computing, involve the delivery of on-demand computing resources ranging from applications to remote data centers over the internet on a pay-for-use basis. Therefore, many scholars and practitioners have devoted their efforts to strengthening or innovating this related research. Section 2 summarizes the existing literature and demonstrates how this work differs in its approach.

In essence, the computing tasks of cloud computing, which are distributed across numerous resources for large-scale calculation and resolve the value, accessibility, reliability, and capability of cloud computing, comprise a computing style in which dynamically scalable and often virtualized resources are provided as an Internet service [1]. The service model of computing tasks includes cloud platforms, users, applications, virtual machines, etc. Within each cloud platform, there are multiple platform users, each with the ability to run multiple applications on the platform. Each application corresponds to the jobs requested by a user and uses a certain quota of virtual machines, through which the application finishes and returns the jobs, thus completing the procedure.

Countless discussions and research have been conducted on the two prevailing issues in both grid and cloud computing: resource allocation and job scheduling [2,3,4]. The job scheduling problem revolves around exploring how the provider of cloud computing services assigns the jobs of each client to each processor according to certain rules and regulations to ensure the cost-effectiveness of the job scheduling process. The efficiency and performance of cloud computing services are usually associated with the efficiency of job scheduling, which affects not only the performance efficiency of users’ jobs but also the utilization efficiency of system resources. Hence, the interdependent relationship between the efficiency and performance of cloud computing services and the efficiency of job scheduling necessitates research on the job scheduling problem of the computing tasks of cloud computing.

The job scheduling problem of the computing tasks of cloud computing services assign jobs to individual processors. It is an NP-hard combinatorial problem, which renders it difficult to obtain the global optima within polynomial time. The greater the problem size, e.g., the number of resources or jobs, the more difficult it is for traditional job scheduling algorithms to solve. Hence, alongside improving traditional algorithms, many scholars have introduced machine learning algorithms, e.g., the Pareto-set cluster genetic algorithm [2] and particle swarm optimization [2], the binary-code genetic algorithm [3] and integer-code particle swarm optimization [3], the simulated annealing algorithm [4], particle swarm optimization (PSO) [5], the genetic algorithm [6], the machine learning algorithm [7], Multi-Objective PSO [8], the artificial bee colony algorithm [9], the hybrid optimization algorithm [10], etc., to solve the job scheduling problem of the computing tasks of cloud computing.

Numerous different objectives have been discussed to measure service performance, e.g., cost, reliability, makspan, and power consumption [1,2,3,4,5,6,7,8,9,10]. However, when conducting research on the job assignment problem, most scholars emphasize the single-objective job scheduling problem and overlook job scheduling problems with more than one objective [11,12,13,14,15,16,17]. By doing so, they fail to acknowledge other goals that may influence the quality of the cloud-computing service. It is thus necessary to balance various aspects when evaluating job scheduling problems [15,16,17]; for example, the objective of minimizing total energy consumption and the objective of minimizing makspan are conflicting objectives in real-life applications [17].

With the ever-advancing development of cloud computing, more and more data centers have successively been established to run a large number of applications that require considerable computations and storage capacity, but this wastes vast amounts of energy [12,15,16,17]. Reducing power consumption and cutting down energy costs has become a primary concern for today’s data center operators. Thus, striking a balance between reducing energy consumption and maintaining high computation capacity has become a timely and important challenge.

Furthermore, with more media and public attention shifting onto progressively severe environmental issues, governments worldwide have now adopted a stronger environmental protection stance [12,15,16,17]. This puts increasing pressure on enterprises to pursue higher output and also to focus on minimizing power consumption [12,15,16,17].

Stemming from real-life concerns, as mentioned above, our previous work considered a two-objective time-constrained job scheduling problem to measure the performance of a certain job schedule plan with two objectives: the quality of the cloud computing service in terms of the makspan and the environmental problem based on the energy consumption [12,15,16,17].

There are two different types of algorithms used to solve multi-objective problems. While one converts multi-objective problems to ones that are single objective in nature through methods such as ε-constraint, LP-metrics, goal programming, etc., the other solves multi-objective problems based on the concept of Pareto optimality. The latter is the one we adapted, both here and in our previous study [10,11,12,13,14,15,16,17].

Moreover, a new algorithm called the multi-objective simplified swarm optimization (MOSSO) was proposed to solve the above problem [17]. However, there are some errors in the MOSSO source code, which ignores the multi-objective problem; the number of temporary nondominated solutions is not always only one, and some temporary nondominated solutions may not be temporary nondominated solutions in the next generation. Therefore, the performance comparison is limited to MOSSO and MOPSO, despite both having an iterative local search [17].

In order to rectify our previous source code with new concepts based on the Pareto optimality to solve the two-objective time-constrained job scheduling, we proposed a new algorithm called two-objective simplified swarm optimization (tSSO). It draws from the SSO update mechanism to generate offspring [18], the crowding distance to rank nondominated solutions [19], and new hybrid elite selection to select parents [20], and the limited number of nondominated solutions adapted from multi-objective particle swarm optimization (MOPSO) serves to guide the update [13].

The motivation and contribution of this work are highlighted as follows:

• An improved algorithm named two-objective simplified swarm optimization (tSSO) is developed in this work to revise and improve errors in the previous MOSSO algorithm [17], which ignores the fact that the number of temporary nondominated solutions is not always only one in the multi-objective problem, and some temporary nondominated solutions may not be temporary nondominated solutions in the next generation. The algorithm is based on SSO to deliver the job scheduling in cloud computing.
• More new algorithms, testing, and comparisons have been implemented to solve the bi-objective time-constrained task scheduling problem in a more efficient manner.
• In the experiments conducted, the proposed tSSO outperforms existing established algorithms, e.g., NSGA-II, MOPSO, and MOSSO, in the convergence, diversity, number of obtained temporary nondominated solutions, and the number of obtained real nondominated solutions.

The remainder of this paper is organized as follows. Related work in the existing literature is discussed in Section 2, which also demonstrates how this work is different in its approach. Section 3 presents notations, assumptions, and the mathematical modeling of the proposed two-objective time-constrained job scheduling problem to address energy consumption and service quality in terms of the makspan. Section 4 introduces the simplified swarm optimization (SSO) [18], the concept of crowd distance [19], and traditional elite selection [20]. The proposed tSSO is presented in Section 5, together with the discussion of its novelties and pseudo code. The section also explains the errors in the previous MOSSO algorithm and how the proposed new algorithm overcomes them. Section 6 compares the proposed tSSO in terms of nine different parameter settings with the MOSSO proposed in [17], the MOPSO proposed in [13], and the famous NSGA-II [19,20]. Three benchmark problems, one small-size, one medium-size, and one large-size, are utilized and analyzed from the viewpoint of convergence, diversity, and number of obtained nondominated solutions in order to demonstrate the performance of the tSSO. Our conclusions are given in Section 7.

2. Literature Review

The related work of the existing literature is provided in this section, which also demonstrates how this work is different in its approach. The job scheduling problem of cloud-computing services is very important, meaning that there is a significant amount of related research on it, which can be classified into the following types.

2.1. A Review of Job Scheduling in Cloud-Computing

Houssein et al. presented a review of numerous meta-heuristics for solving job scheduling in cloud-computing [21], and a literature review of the related methods used for solving job scheduling in cloud-computing has been conducted by Arunarani et al. [22] and Kumar et al. [23].

2.2. Algorithm Research for Solving the Single-Objective Problem for Job Scheduling in Cloud-Computing

Chen et al. proposed an improved meta-heuristic whale optimization algorithm to solve resource utilization for job scheduling in cloud-computing [24]. Attiya et al. studied improved simulated annealing (SA) to optimize makspan for job scheduling in cloud-computing [25]. Gąsior and Seredyński worked on a distributed security problem for job scheduling in cloud computing by a method based on the game theoretic approach [26]. Mansouri and Javidi resolved response time using a cost-based job scheduling method [27]. A deep reinforcement learning (DRL) algorithm was proposed by Cheng et al. [28] to reduce the cost of job scheduling in cloud computing.

2.3. Algorithm Research for Solving the Multi-Objective Problem for Job Scheduling in Cloud-Computing

Processing time and cost of job scheduling in cloud computing were optimized by Shukri et al. [29] using an improved multi-verse optimizer. Jacob and Pradeep minimized the multi-objective problem, including makspan, cost, and deadline violation rate, using a hybrid algorithm of cuckoo search (CS) and particle swarm optimization (PSO) [30]. Abualigah and Diabat optimized the makspan and resource utilization using an improved algorithm based on the elite-based differential evolution method [31]. Sanaj and Prathap proposed a chaotic squirrel search algorithm (CSSA) to solve the makspan and cost issues [32], and Abualigah and Alkhrabsheh optimized the cost and service availability in the multi-objective problem [33].

This work aims to optimize a two-objective problem that includes both energy and makspan, but the objectives of this work are different from the multi-objective contents of the existing literatures, as shown in the third part of the literature review [29,30,31,32,33]. Furthermore, this work proposes an improved algorithm to solve the two-objective problem for job scheduling in cloud computing, which is obviously different from the first and second parts of the existing literatures above [21,22,23,24,25,26,27,28].

3. Assumptions, and Mathematical Problem Description

The formal mathematical model of the two-objective constrained job scheduling problem of cloud-computing services [12,15,16,17] is described as follows, together with the assumptions used in the problem, as well as the algorithms.

3.1. Assumptions

In cloud-computing services, all requests from cloud-computing users are collected as jobs and subsequently divided into multiple jobs, which are then executed by data centers with multiple processors [2,3,4,10,11,12,15,16,17]. To simplify our model without foregoing generality, the following assumptions are utilized to prompt the two-objective constrained job scheduling problem to focus on the energy consumption and makspan of the cloud-computing service [2,3,4,10,11,12,15,16,17].

• All jobs are available, independent, and equal in importance with two attributes, size_i and time_i, for i = 1, 2, …, N_var for processing simultaneously at time zero.
• Each processor is available at any time with two attributes, denoted as speed_j and power_j, j = 1, 2, …, N_cpu, and cannot process two or more jobs simultaneously.
• All processing times include set-up time.
• Job pre-emption and the splitting of jobs are not permitted.
• There is an infinite buffer between the processors.

3.2. The Mathematical Model

A feasible job scheduling solution is a plan that assigns jobs to processors in a sequence such that the makspan, which is the time at which the final job is complete before a predefined limited time. Let solution X = (x₁, x₂, …, x_Nvar) be a feasible job scheduling plan and x_i be the type of processor assigned to the job i for i = 1, 2, …, N_var.

To maintain efficiency and ensure environmental protection, the two-objective constrained job scheduling problem considered here is to assign N_var jobs to N_cpu processors to minimize both the energy consumption and makspan so that the makspan is not overdue [17]:

$$Min F_{\mathrm{e}}(X)={\displaystyle {\sum }_{i=1}^n(t_{i,x_i}}·e_{x_i})$$

(1)

$$Min F_{\mathrm{m}}(X)=\underset{j}{Max}{\displaystyle {\sum }_{i=1}^n{t}_{i,j}}$$

(2)

s.t. F_m(X) ≤ T_ub.

(3)

Equation (1) is the total energy consumption of a job scheduling plan, and it is the sum of the usage of energy of all jobs assigned to processors based on X. Note that the energy consumption of each job is the product of the power cost per unit time and the running time is equal to the size of the job divided by the execution speed, as shown in formula (4).

$$t_i{,}_j=\left\{\begin{array}{cc}siz{e}_i/spee{d}_j\hfill & \mathrm{if} \mathrm{task} i \mathrm{is} \mathrm{processed} \mathrm{on} \mathrm{processor} j\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right..$$

(4)

Equation (2) is the makspan, which is the time the last job is finished. Equation (3) is the time constraint for each job, such that each job must be finished before the deadline T_ub.

4. SSO, Crowding Distance, and Elite Selection

The proposed tSSO, based on SSO [18], is used to update solutions from generation to generation intelligently; the crowding distance [19] is used to rank temporary nondominated solutions systematically, and the elite selection [34] is used to select solutions to act as parents. Hence, SSO, crowding distance, and elite selection are introduced briefly in this section.

4.1. Simplified Swarm Optimization

SSO, developed by Yeh in 2009 [18], is a simple but powerful machine leaning algorithm that is a hybrid of leader-solution swarm intelligence and population-based evolutionary computation.

In traditional SSO, all variables need to be updated (called the all-variable update in SSO) such that the jth variable of the ith solution (i.e., x_i,j) is obtained from either the jth variable of the P_gBest (i.e., p_gBest,j) with probability c_g of the P_i with probability c_p of its current value (i.e., x_i,j) with probability c_w of a randomly generated feasible new value (say x) with probability c_r, where the P_gBest is the best solution among all existing solutions; P_i is the best ith solution in its evolutionary history, and c_g + c_p + c_w + c_r = 1.

The above update mechanism of SSO is very simple, efficient, and flexible [17,18,35,36,37,38,39,40], and can be presented as a stepwise-function update:

$$x_{i,j}=\left\{\begin{array}{cc}{p}_{gBest,j}\hfill & {\mathrm{if} \mathsf{\rho }}_{[0,1]}\in [0,C_g)\hfill \\ p_{i,j}\hfill & {\mathrm{if} \mathsf{\rho }}_{[0,1]}\in [C_g,C_p)\hfill \\ x_{i,j}\hfill & {\mathrm{if} \mathsf{\rho }}_{[0,1]}\in [C_p,C_w)\hfill \\ x\hfill & {\mathrm{if} \mathsf{\rho }}_{[0,1]}\in [C_w, 1]\hfill \end{array}\right.,$$

(5)

where C_g = c_g, C_p = C_g + c_p, and C_w = C_p + c_w.

The SSO pseudo code is provided below [17,18,35,36,37,38,39]:

STEP S0.

Generate X_i randomly, find gBest, and let t = 1, k = 1, and P_i = X_i for i = 1, 2, …, N_sol.

STEP S1.

Update X_k.

STEP S2.

If F(X_k) is better than F(P_k), let P_k = X_k. Otherwise, go to STEP S5.

STEP S3.

If F(P_k) is better than F(P_gBest), let gBest = k.

STEP S4.

If k < N_sol, let k = k + 1 and go to STEP S1.

STEP S5.

If t < N_gen, let t = t + 1, k = 1, and go to STEP S1. Otherwise, halt.

The stepwise-function update mechanism is very simple and efficient, but it is also very powerful and has various successful applications, e.g., the redundancy allocation problem [35,36], flexible grid trading [37], the disassembly sequencing problem [38], artificial neural networks [39], power systems [40], energy problems [41,42], and various other problems [17,18,43,44,45,46,47,48,49,50,51,52]. Moreover, the stepwise-function update mechanism is easier to customize by replacing any item of its stepwise function either with other algorithms [36,39] or hybrid algorithms [41], in sequence or in parallel [42].

4.2. Crowding Distance

Let

$${\stackrel{⌢}{d}}_{l,i}=Min\{\frac{F_l(X_i)-F_l(X_j)}{Max(F_l)-Min(F_l)}\}$$

(6)

be the shortest normalized Euclidean distance between the ith temporary nondominated solution and any other temporary nondominated solutions based on the lth objective function, where Min(F_l) is the minimal value of the lth objective function and Max(F_l) is the maximal value of the lth objective function.

The crowding distance is the sum of the individual distance and can be calculated as follows [19]:

$${\displaystyle \sum _i\sqrt{{\stackrel{⌢}{d}}_{1,i}^2+{\stackrel{⌢}{d}}_{2,i}^2}},$$

(7)

for all temporary nondominated solutions, X_i. The crowding distance is only used to rank temporary nondominated solutions for selecting as parents if the total number of determine are over a limited value, which is defined as N_non and N_non = N_sol in this study.

4.3. The Elite Selection

Among numerous different selection policies, the elite selection is the simplest and is thus adopted in the proposed tSSO. The elite selection chooses the best solutions in the current generation to generate and update solutions for the next generation [34]. For example, let N_sol = 50, X₁, X₂, …, X₁₀₀ be the solutions required for selection. Elite selection ranks and selects the best 500 solutions among X₁, X₂, …, X₁₀₀ and renumbers these selected solutions as X₁, X₂, …, X₅₀.

In the tSSO, these best solutions are temporary nondominated solutions. Due to the fact that the number of temporary nondominated solutions may be less than or larger than the number of parents, a new elite selection called the hybrid elite selection is developed and used in the proposed tSSO, and the details are discussed in Section 5.3.

5. Proposed Algorithm

The proposed tSSO is a population-based, all-variable update, and it is a stepwise function-based method, i.e., there are a fixed number of solutions in each generation and all variables must be updated based on the stepwise function for each solution. Details of the proposed tSSO are discussed in this section.

5.1. Purpose of tSSO

The developed tSSO algorithm revises and improves the errors in the previous MOSSO algorithm, which ignores the fact that the number of temporary nondominated solutions in the multi-objective problem is not always only one, and some temporary nondominated solutions may not be temporary nondominated solutions in the next generation.

The developed tSSO algorithm overcomes the errors in the previous MOSSO algorithm by the following methods:

• The novel update mechanism of the role of gBest and pBest in the proposed tSSO, which are introduced in Section 5.3.
• The hybrid elite selection in the proposed tSSO, which is introduced in Section 5.4.

5.2. Solution Structure

The first step of most machine learning algorithms is to define the solution structure [2,3,4,35,36,37,38,39,40,41,42,43,44,45]. The solution in the tSSO for the proposed problem is defined as a vector, and the value of the ith coordinate is the processor utilized to process the ith job. For example, let X₅ = (3, 4, 6, 4) be the 5th solution in the 10th generation of the problem with 4 jobs. In X₅, jobs 1, 2, 3, and 4 are processed by processors 3, 4, 6, and 4.

5.3. Novel Update Mechanism

The second step in developing machine learning algorithms is to create an update mechanism to update solutions [2,3,4,35,36,37,38,39,40,41,42]. The stepwise update function is a unique update mechanism of SSO [17,18,35,36,37,38,39,40,41,42]. In the single-objective problem, there is only one gBest for the traditional SSO. However, in the multi-objective problem, the number of temporary nondominated solutions is not always only one, and some temporary nondominated solutions may not be temporary nondominated solutions in the next generation [17]. Note that a nondominated solution is not dominated by any other solution, while a temporary nondominated solution is a solution nondominated by other found solutions that are temporary in the current generation, and it may be dominated by other updated solutions later [13,17,19].

Hence, in the proposed tSSO, the role of gBest is removed completely and the pBest used in the tSSO is not the original definition in the SSO. The pBest for each solution in the proposed tSSO is selected from temporary nondominated solutions, i.e., there is no need to follow its previous best predecessor. The stepwise update function used in the proposed tSSO is listed below for multi-objective problems for each solution X_i, with i = 1, 2, …, N_sol:

$$x_{i,j}=\left\{\begin{array}{cc}{x}_i^{*}\hfill & {\mathrm{if} \mathsf{\rho }}_{[0,1]}\in [0,C_p)\hfill \\ x_{i,j}\hfill & {\mathrm{if} \mathsf{\rho }}_{[0,1]}\in [C_p,C_w) \hfill \\ x\hfill & \mathrm{otherwise}\hfill \end{array}\right.,$$

(8)

where $X^{*}=(x_1^{*},x_2^{*},\dots ,x_k^{*})$ is one of the temporary nondominated solutions selected randomly, ρ_[0,1] is a random number generated uniformly in [0, 1], and x is a random integer generated from 0, 1, 2, …, N_var.

For example, let X₆ = (1, 2, 3, 2, 4) and X* = (2, 1, 4, 3, 3) be a temporary nondominated solution selected randomly. Let C_p = 0.50, C_w = 0.95, and ρ = (ρ₁, ρ₂, ρ₃, ρ₄, ρ₅) = (0.32, 0.75, 0.47, 0.99, 0.23). The procedure to update X₆ based on the stepwise function provided in Equation (8) is demonstrated in

Table 1. Example of the update process in the proposed tSSO.

Variable	1	2	3	4	5
X5	1	2	3	2	4
X*	2	1	4	3	3
ρ	0.32	0.75	0.47	0.99	0.23
New X5	2	2	4	4 #	3

“^#” indicates that the corresponding feasible value is generated randomly.

.

From the above example, the simplicity, convenience, and efficiency of the SSO can also be found in the update mechanism of the proposed tSSO.

5.4. Hybrid Elite Selection

The last step is to determine the selection policy to decide which solutions, i.e., parents, are selected to generate solutions in the next generation. In the proposed tSSO, hybrid selection is harnessed.

Let π_t be a set to store the temporary nondominated solutions found in the tth generation. It is impossible to have all temporary nondominated solutions because their number is infinite, and also because temporary nondominated solutions may not be real nondominated solutions. The value of |π_t| is limited to N_sol, and parts of temporary nondominated solutions are abandoned to keep |π_t| = N_sol.

If N_sol ≤ |π_t|, the crowding distances need to be calculated for each temporary nondominated solution, and only the best N_sol solutions will be selected from π_t to be parents. However, all temporary nondominated solutions in π_t are used to serve as parents and (N_sol − |π_t|) solutions are selected randomly from offspring if N_sol > |π_t|. As discussed above, there are always N_sol parents for each generation, and these temporary nondominated solutions are usually chosen first to serve their role as parents.

Note that these temporary nondominated solutions are abandoned if they are not selected as parents.

5.5. Group Comparison

Referring to Section 4.3, the temporary nondominated solutions play a paramount role in most multi-objective algorithms. Up to now, the most popular method to achieve the above goal is pairwise comparison, which takes O(N²) [13,17,19] for each solution in each generation, where N is the number of solutions from which we determine temporary nondominated solutions. Hence, the corresponding related time complexity of the tSSO and NSGA-II are both O(4N_sol²). Therefore, the most time-consuming aspect of solving multi-objective problems using machine learning algorithms is the search for all temporary nondominated solutions from all offspring.

In the proposed tSSO, the parents are selected from temporary nondominated solutions found in the previous generation and offspring generated in the current generation. To reduce the computation burden, a new method called group comparison is proposed in tSSO. All temporary nondominated solutions in offspring are found first using the pairwise comparison, which takes O(N_sol²) as the number of offspring in N_sol. The temporary nondominated solutions obtained in the previous generations are then compared with the new temporary nondominated solutions found in the current generation. The number of both sets of temporary nondominated solutions are at most N_sol, i.e., the time complexity is O(N_sol log(N_sol)) based on the merge sort.

Hence, the time complexity is O(N_sol log(N_sol)) + O(N_sol²) = O(N_sol²), which is only one quarter of the pairwise comparison.

5.6. Proposed tSSO

The procedure of the proposed tSSO, which has nine different parameter settings, denoted as tSSO₀, tSSO₁, tSSO₂, tSSO₃, tSSO₄, tSSO₅, tSSO₆, tSSO₇, and tSSO₈, has a detailed introduction in Section 6, based on the solution structure, update mechanism, hybrid elite selection, and group comparison discussed in this section, which are presented in pseudo code as follows.

PROCEDURE tSSO

STEP 0.

Create initial population X_i randomly for i = 1, 2, …, N_sol and let t = 2.

STEP 1.

Let π_t be all temporary nondominated solutions in S* = {X_k | for k = 1, 2, …, 2N_sol} and X_Nsol+k is updated from X_k based on Equation (8) for k = 1, 2, …, N_sol.

STEP 2.

If N_sol ≤ |π_t|, let S = {top N_sol solutions in π_t based on crowding distances} and go to STEP 4.

STEP 3.

Let S = π_t ∪ {N_sol − |π_t| solutions selected randomly from S* − π_t}.

STEP 4.

Re-index these solutions in S such that S = {X_k|for k = 1, 2, …, N_sol}.

STEP 5.

If t < N_gen, then let t = t + 1 and go back to STEP 1. Otherwise, halt.

6. Simulation Results and Discussion

For fair comparison with MOSSO [17], which was the previous work of the extended topic, the numerous experiments of the parameter-setting procedure sof the three different-sized benchmark problems used in [17] were carried out using tSSO. The experimental results obtained by the proposed tSSO were compared with those obtained using MOSSO [17], which were acquired using MOPSO [13] and NSGA-II [19,20].

6.1. Parameter-Settings

All machine learning algorithms have parameters in their update procedures and/or the selection procedure. Thus, there is a need to tune parameters for better results. To determine the best parameters of the proposed tSSO, three different levels of the two factors of c_p and c_w, the low value of 0.1, the middle value of 0.3, and the high value of 0.5, have been combined, e.g., nine different parameter settings denoted as tSSO₀, tSSO₁, tSSO₂, tSSO₃, tSSO₄, tSSO₅, tSSO₆, tSSO₇, and tSSO₈, as shown in

Table 2. Nine different parameter setting of the proposed tSSO.

ID	tSSO0	tSSO1	tSSO2	tSSO3	tSSO4	tSSO5	tSSO6	tSSO7	tSSO8
Cp	0.1	0.1	0.1	0.3	0.3	0.3	0.5	0.5	0.5
Cw	0.2	0.4	0.6	0.4	0.6	0.8	0.6	0.8	1.0

.

Note that C_p = c_p and C_w = c_p + c_w. The following provided the parameter settings for the other algorithms:

NSGA-II: c_crossover = 0.7, c_mutation = 0.3

MOPSO: w = 0.871111, c₁ = 1.496180, c₂ = 1.496180 [17]

MOSSO: C_g = 0.1 + 0.3t/N_gen, C_p = 0.3 + 0.4t/N_gen, C_w = 0.4 + 0.5t/N_gen, where t is the current generation number [17].

6.2. Experimental Environments

To demonstrate the performance of the proposed tSSO and select the best parameter settings, tSSOs with nine parameter settings were utilized for three job scheduling benchmarks [17], namely (N_job, N_cpu) = (20, 5), (50, 10), and (100, 20), and the deadlines were all set as 30 for each test [17].

The following lists the special characteristics of the processor speeds (in MIPs), the energy consumptions (in KW/per unit time), and the job sizes (in MI) in these three benchmark problems [17]:

• Each processor speed is generated between 1000 and 10,000 (MIPs) randomly, and the largest speed is ten times the smallest one.
• The power consumptions grow polynomial as the speed of processors grow, and the value range is between 0.3 and 30 (KW) per unit time.
• The values of job sizes are between 5000 and 15,000 (MIs).

Alongside the proposed tSSO with nine parameter settings, there are three other multi-objective algorithms; MOPSO [13,17], MOSSO [17], and NSGA-II [19,20] were tested and compared to further validate the superiority of the proposed tSSO. NSGA-II [19,20] is the current best multi-objective algorithm and is based on genetic algorithms, while MOPSO is based on particle swarm optimization [13], and MOSSO on SSO [17].

The proposed tSSO with its nine parameter settings and NSGA-II have no iterative local search to improve the solution quality. To ensure a fair comparison, the iterative local search is removed from both MOPSO and MOSSO. All algorithms were coded in DEV C++ on a 64-bit Windows 10 PC, implemented on an Intel Core i7-6650U CPU @ 2.20 GHz notebook with 16 GB of memory.

In addition, for a fair comparison between all algorithms, N_sol = N_non = 50, N_gen = 1000, and N_run = 500, i.e., the same solution number, generation number, size of external repository, and run number for each benchmark problem were used. Furthermore, the calculation number of the fitness function of all algorithms was limited to N_sol × N_gen = 50,000, 100,000, and 150,000 in each run for the small-size, medium-size, and large-size benchmarks, respectively.

Note that the reason for the large value of N_run is to simulate the Pareto front, and the details are discussed in Section 6.3.

6.3. Performance Metrics

The convergence metrics and diversity metrics are always both used to evaluate the performances of the multi-objective algorithms in the solution quality. Among these metrics, the general distance (GD), introduced by Van Veldhuizen et al. [53], and spacing (SP), introduced by Schott [54], are two general indexes for the convergence metrics and diversity metrics, respectively. Let d_i be the shortest Euclidean distance between the ith temporary nondominated solution and the Pareto front, and d be the average sum of all d_i for all i. The GD and SP are defined below:

$$\mathrm{GD}=\frac{\sqrt{{\displaystyle \sum _{i=1}^{{\mathrm{N}}_{\mathrm{sol}}}{d}_i{}^2}}}{{\mathrm{N}}_{\mathrm{sol}}},$$

(9)

$$\mathrm{SP}=.\sqrt{\frac{{\displaystyle \sum _{i=1}^{{\mathrm{N}}_{\mathrm{sol}}}{(\underset{¯}{d}-d_i)}^2}}{{\mathrm{N}}_{\mathrm{sol}}-1}}.$$

(10)

The GD is the average of the sum of the squares of d_i, and the SP is very similar to the standard deviation in probability theory and statistics. If all temporary nondominated solutions are real nondominated solutions, we have GD = 0. The solutions are equality far from d, we have SP = 0. Hence, in general, the smaller the SP is, the higher the diversity of solutions along the Pareto front and the better the solution quality becomes [13,17,19,46,47].

The Pareto front is needed for both the GD and SP in calculating from their formulas [43,44]. Unfortunately, it requires infinite nondominated solutions to form the Pareto front, and this is impossible to accomplish, even with exhaustive methods for the job scheduling problem of cloud computing, which is an NP-hard problem that has no guarantee of having the ability to obtain the global optimal solution under the complexity of polynomial time, and the traditional algorithms are intractable, resulting to a special optimization algorithm having to be used to reduce the system search complexity and improve the overall search quality [12,15,16,17]. Rather than a real Pareto front, a simulated Pareto front is implemented by collecting all temporary nondominated solutions in the final generation from all different algorithm with different values of N_sol for the same problem.

There are 12 algorithms with three different N_sol = 50, 100, and 150 for N_run = 500, i.e., 12 × 500 × (50 + 100 + 150) = 1,800,000 solutions obtained at the end for each test problem. All temporary nondominated solutions are found from these 1,800,000 final solutions to create a simulated Pareto front to calculate the GD and SP.

6.4. Numerical Results

All numerical results attained in the experiments are listed in this section.

Table 3. Result for small-size problem.

Nsol	Alg	Avg (Nn)	Std (Nn)	Avg (Np)	Std (Np)	Avg (GD)	Std (GD)	Avg (SP)	Std (SP)	Avg (T)	Std (T)	Avg (F1)	Std (F1)	Avg (F2)	Std (F2)
50	tSSO0	48.34	2.22	0.012	0.109	0.565	0.611	3.527	4.343	2.8954	0.5839	11,937.28	261.3206	2539.957	4798.1386
	tSSO1	49.634	0.904	0.018	0.133	0.26	0.25	1.475	1.781	3.4103	0.6708	11,689.93	272.5046	1015.790	1926.8740
	tSSO2	49.966	0.202	0.058	0.234	0.138	0.059	0.722	0.445	4.1789	0.6901	11,185.89	273.4064	838.2936	632.7848
	tSSO3	49.494	1.165	0.012	0.109	0.311	0.349	1.819	2.495	3.2951	0.6579	11,766.60	269.7630	1208.2600	2270.2913
	tSSO4	49.936	0.276	0.034	0.202	0.17	0.073	0.915	0.547	3.9837	0.6919	11,317.03	289.0288	779.7255	448.9989
	tSSO5	50	0	0.284	0.544	0.118	0.065	0.71	0.478	4.7604	0.7535	10,603.56	229.7030	938.6334	22.5991
	tSSO6	49.968	0.208	0.054	0.235	0.153	0.101	0.821	0.734	4.2185	0.6928	11,230.14	290.7808	817.7619	631.6859
	tSSO7	50	0	0.218	0.464	0.121	0.064	0.722	0.475	4.86	0.7531	10,561.73	236.0619	936.6982	24.2198
	tSSO8	49.7	0.766	0.188	0.457	0.153	0.103	0.776	0.598	4.4077	0.8044	10,088.10	550.3918	1010.433	888.7098
	MOPSO	9.274	2.026	0	0	2.947	0.908	17.366	6.314	3.1238	0.5445	12,176.37	230.151	29,378.78	16,620.2163
	MOSSO	1.23	0.508	0	0	9.914	0.122	8.417	4.978	1.2282	0.187	9833.03	24.8444	488,648.8	10,786.0609
	NSGA-II	17.22	3.216	0	0	2.213	1.27	13.107	8.676	0.0102	0.0202	11,963.00	487.9212	19,913.23	20,625.1095
100	tSSO0	61.856	5.503	0.024	0.153	1.996	0.502	18.548	4.721	10.3052	1.4032	24,162.67	365.2535	48,016.98	20,979.7233
	tSSO1	71.016	5.808	0.05	0.227	1.284	0.506	12.06	4.968	10.2902	1.3693	23,633.28	438.6975	24,077.66	15,460.6944
	tSSO2	95.07	4.779	0.184	0.463	0.289	0.306	2.712	3.064	11.282	1.6767	22,010.09	475.5156	4596.549	5938.6783
	tSSO3	71.316	5.619	0.064	0.253	1.347	0.508	12.678	4.946	10.5688	1.3536	23,621.20	426.826	26,436.17	16,447.3743
	tSSO4	88.894	6.061	0.116	0.362	0.537	0.387	5.102	3.839	10.8808	1.4016	22,401.50	480.8059	8531.313	8913.8709
	tSSO5	99.98	0.165	0.668	0.824	0.061	0.037	0.514	0.382	21.7762	2.6418	19,982.09	420.3826	1891.765	452.4866
	tSSO6	98.638	2.135	0.278	0.549	0.137	0.174	1.212	1.755	13.335	2.399	21,646.20	473.8394	2588.289	3570.9857
	tSSO7	99.996	0.088	0.714	0.849	0.062	0.035	0.521	0.369	23.0932	2.6965	19,860.81	428.8591	1879.873	57.8411
	tSSO8	99.89	0.546	1.354	1.173	0.059	0.04	0.497	0.401	26.0226	4.3313	19,086.17	614.0985	2056.674	772.1143
	MOPSO	12.042	2.554	0	0	2.113	0.454	17.758	4.29	12.0856	1.6538	24,347.86	307.4949	57,505.36	23,415.3032
	MOSSO	1.58	0.699	0	0	7.01	0.055	9.271	3.117	4.7402	0.4797	19,666.68	33.48	977,197.9	13,878.8233
	NSGA-II	21.096	3.25	0	0	2.25	0.777	19.435	7.076	0.0392	0.0489	24,289.14	713.0953	65,231.95	40,247.3188
150	tSSO0	67.96	5.77	0.036	0.197	1.97	0.343	21.901	3.817	23.0904	3.509	36,497.8	446.2135	99,073.14	31,223.9014
	tSSO1	76.334	6.347	0.078	0.276	1.552	0.349	17.458	4.014	22.8375	3.3043	35,809.46	542.9496	69,700.06	27,524.2792
	tSSO2	102.624	7.73	0.22	0.473	0.929	0.312	10.699	3.723	23.8908	3.3867	33,666.71	657.8649	36,715.2	20,668.0023
	tSSO3	79.2	5.726	0.098	0.304	1.465	0.327	16.518	3.814	23.6139	3.4082	35,679.86	536.1647	64,560.6	24,571.6138
	tSSO4	98.296	6.9	0.166	0.413	0.932	0.276	10.655	3.313	24.0771	3.3979	33,969.23	669.9933	36,254.89	18,845.1508
	tSSO5	149.92	0.427	1.362	1.177	0.042	0.028	0.441	0.361	43.0185	7.8536	29,176.09	593.4306	2833.142	627.5616
	tSSO6	122.092	7.632	0.408	0.631	0.591	0.254	6.907	3.087	25.971	3.4265	32,432.3	657.7752	20,747.74	14,131.2079
	tSSO7	149.954	0.277	1.47	1.151	0.04	0.028	0.414	0.356	51.0654	7.7838	28,979.39	598.0727	2870.726	638.1370
	tSSO8	149.904	0.602	3.548	2.013	0.036	0.028	0.39	0.347	74.6991	13.5849	27,451.57	612.008	3233.779	995.0084
	MOPSO	13.692	2.496	0	0	1.744	0.289	18.011	3.32	26.7054	4.1966	36,486.46	388.5253	87,266.1	26,859.1298
	MOSSO	1.986	0.921	0	0	5.724	0.038	9.331	2.551	10.4472	1.2398	29,494.15	43.4111	1,466,864	18,034.0917
	NSGA-II	23.45	3.732	0	0	2.109	0.579	22.251	6.213	0.0855	0.0743	36,643.47	992.1199	120,913.5	60,590.6631

,

Table 4. Result for medium-size problem.

Nsol	Alg	Avg (Nn)	Std (Nn)	Avg (Np)	Std (Np)	Avg (GD)	Std (GD)	Avg (SP)	Std (SP)	Avg (T)	Std (T)	Avg (F1)	Std (F1)	Avg (F2)	Std (F2)
50	tSSO0	24.472	4.177	0	0	20.001	2.216	109.497	7.657	4.5751	0.7871	32,537.48	456.8619	196,678.1	37,821.6436
	tSSO1	25.954	4.105	0	0	18.074	2.375	103.29	9.431	4.403	0.7533	32,405.09	566.3836	167,400.6	37,900.9380
	tSSO2	29.672	4.648	0	0	14.629	2.778	89.001	13.521	4.3079	0.7238	32,177.28	716.8466	120,611.9	39,137.8242
	tSSO3	26.74	4.27	0	0	17.701	2.308	102.104	9.346	4.5363	0.765	32,413.09	581.7217	161,630.7	37,035.2625
	tSSO4	29.87	4.493	0	0	14.258	2.518	87.075	12.468	4.397	0.7395	32,032.87	750.0289	118,041.1	36,646.8146
	tSSO5	35.19	5.142	0.006	0.077	9.059	2.676	58.733	16.013	4.0844	0.6768	31,237.8	919.8058	65,527.05	32,220.9459
	tSSO6	32.81	4.649	0	0	12.138	2.64	76.712	14.884	4.417	0.748	32,117.32	801.4451	90,607.97	32,073.7435
	tSSO7	36.364	4.946	0.002	0.045	8.258	2.656	54.108	16.584	4.2589	0.7155	31,191.78	946.228	56,523.23	28,417.2914
	tSSO8	44.082	4.463	0.064	0.303	0.976	1.184	6.054	8.46	4.0538	0.6846	28,331.11	1073.477	5418.824	8430.7514
	MOPSO	5.278	1.58	0	0	18.821	1.434	85.456	4.305	6.7045	1.1244	31,253.08	456.2674	286,050.1	35,533.4621
	MOSSO	1	0	0	0	20.393	0.106	5.148	3.944	2.6425	0.3924	28,024.24	44.3314	499,880.3	1086.7815
	NSGA-II	8.528	2.42	0	0	23.357	3.072	109.155	10.135	0.013	0.022	32,648.83	926.4653	272,009.9	62,082.4709
100	tSSO0	27.056	4.507	0	0	16.977	0.907	113.305	3.756	17.8248	2.9748	65,051.87	667.284	558,643.9	50,252.7832
	tSSO1	28.476	4.388	0	0	15.963	1.018	111.674	4.176	17.1294	2.739	64,873.31	824.3303	509,758.2	53,979.6751
	tSSO2	31.5	5.173	0	0	14.406	1.139	107.424	5.392	16.6702	2.6083	64,582.33	1120.178	438,233.4	60,357.1999
	tSSO3	30.116	4.689	0	0	15.331	0.948	110.571	4.615	17.569	2.8203	64,877.63	892.6177	478,286.6	49,968.1420
	tSSO4	32.442	4.55	0	0	13.563	1.04	104.043	5.577	16.9932	2.6475	64,347.25	1159.757	402,642.2	55,190.6790
	tSSO5	37.524	5.4	0	0	10.642	1.067	87.882	6.868	15.7426	2.3916	62,982.56	1581.812	302,505.7	58,114.0986
	tSSO6	37.188	4.95	0.002	0.045	11.909	1.066	96.473	6.802	17.1704	2.702	64,295.33	1245.893	334,714.5	50,535.6466
	tSSO7	40.262	5.529	0.002	0.045	9.49	1.126	80.679	8.005	16.5428	2.4717	62,883.59	1635.266	260,267.7	55,235.1007
	tSSO8	57.302	6.849	0.152	0.435	2.329	0.992	21.916	9.291	15.8048	2.3262	55,830.02	2048.334	57,761.36	42,030.1210
	MOPSO	6.848	1.843	0	0	13.321	0.697	85.525	3.172	25.8886	3.993	62,458.84	631.2151	573,135.1	48,380.8306
	MOSSO	1	0	0	0	14.418	0.054	5.869	2.966	10.1184	1.161	56,046.99	62.3825	999,720.8	1646.6140
	NSGA-II	10.444	2.663	0	0	17.906	1.378	109.495	7.892	0.0476	0.05	65,316.82	1320.111	621,792.5	85,797.1254
150	tSSO0	29.234	4.409	0	0	14.51	0.548	111.228	3.186	39.0909	5.8462	97,500.52	813.8773	924,285.2	56,397.7725
	tSSO1	30.044	4.752	0.004	0.063	13.987	0.624	111.209	3.446	37.5183	5.4441	97,367.75	1050.166	873,603.5	62,586.7269
	tSSO2	32.602	4.976	0	0	12.975	0.721	109.937	3.887	36.3396	5.1288	97,031.77	1314.433	781,236.9	72,066.7149
	tSSO3	32.566	4.636	0	0	13.238	0.588	110.849	3.482	38.4948	5.5112	97,373.33	1186.611	801,509.8	59,333.8052
	tSSO4	34.552	4.915	0	0	12.02	0.672	106.822	4.391	37.3443	5.448	96,646.12	1497.259	702,948.2	67,389.4702
	tSSO5	38.842	5.225	0	0	9.865	0.657	94.662	5.112	34.557	4.8437	94,766.85	2117.23	561,339.4	76,923.5292
	tSSO6	39.38	4.876	0.002	0.045	10.63	0.684	100.761	5.231	37.7004	5.5096	96,480.24	1760.161	593,359.7	66,298.1134
	tSSO7	43.326	5.156	0.002	0.045	8.652	0.66	86.662	5.569	36.3051	5.1832	94,508.79	2363.881	476,971	77,652.8961
	tSSO8	62.466	7.329	0.336	0.713	2.584	0.702	29.27	7.644	34.4403	4.8116	83,208.2	2666.568	142,370.8	69,677.5283
	MOPSO	7.816	1.896	0	0	10.879	0.497	85.598	2.492	56.5194	8.1362	93,759.12	788.4589	857,348.1	63,773.8475
	MOSSO	1	0	0	0	11.777	0.036	5.749	2.406	22.2228	2.7068	84,077.84	77.1704	1,499,801	1397.3776
	NSGA-II	11.27	2.609	0	0	15.074	0.927	107.625	7.159	0.1059	0.0684	97,772.24	1688.187	994,780.5	101,431.9720

and

Table 5. Result for large-size problem.

Nsol	Alg	Avg (Nn)	Std (Nn)	Avg (Np)	Std (Np)	Avg (GD)	Std (GD)	Avg (SP)	Std (SP)	Avg (T)	Std (T)	Avg (F1)	Std (F1)	Avg (F2)	Std (F2)
50	tSSO0	27.174	4.155	0	0	748.62	96.656	4480.39	365.757	8.4167	1.5329	98,628.83	771.7182	143,665.5	35,722.9236
	tSSO1	29.798	4.686	0	0	639.303	109.316	4019.28	526.113	8.0533	1.343	98,279.76	960.7157	106,329.4	34,186.9693
	tSSO2	34.79	5.074	0	0	478.211	136.159	3161.886	809.014	7.7758	1.2132	97,705.9	1259.995	62,884.07	30,649.8855
	tSSO3	30.616	4.385	0	0	633.698	108.007	3994.477	524.318	8.218	1.3142	98,354.38	893.9128	104,469.4	33,485.2153
	tSSO4	34.058	4.867	0	0	484.791	118.408	3210.835	695.26	7.9161	1.2229	97,382.84	1207.66	63,353.57	27,636.0545
	tSSO5	42.104	5.073	0	0	237.875	158.78	1641.271	1075.713	7.293	1.092	95,492.67	1530.155	21,440.25	19,011.2797
	tSSO6	37.198	4.822	0	0	408.566	128.757	2754.903	808.218	7.904	1.2089	97,480.85	1267.524	46,923.7	24,637.2246
	tSSO7	42.934	4.724	0.002	0.045	205.495	154.737	1424.205	1060.277	7.6309	1.108	95,610.53	1573.731	17,520.32	16,311.2192
	tSSO8	45.918	2.998	0.006	0.1	6.73	36.201	45.981	255.548	6.8534	1.0066	89,370.73	2219.432	1315.029	2029.9054
	MOPSO	6.328	1.726	0	0	904.797	76.097	4909.066	144.96	12.4131	1.93	97,663.75	709.6203	207,628.7	34,209.3182
	MOSSO	4.948	1.556	0	0	1152.116	58.678	4687.853	261.903	4.7327	0.6193	98,649.82	634.5005	334,246.6	33,687.3944
	NSGA-II	10.368	2.799	0	0	856.652	145.453	4711.009	392.009	0.0201	0.0245	98,891.73	1471.68	190,118.4	61,317.1989
100	tSSO0	30.448	4.494	0	0	657.704	35.537	4947.358	69.806	33.2646	5.1487	197,248.5	1061.073	436,547.4	46,564.5095
	tSSO1	32.06	4.894	0	0	613.194	42.945	4832.571	135.141	31.8998	4.8814	196,881.6	1430.933	380,499.4	52,806.8613
	tSSO2	36.274	5.131	0	0	539.157	47.963	4531.276	241.08	30.8796	4.6327	196,021.5	1827.626	295,533.3	51,535.3657
	tSSO3	34.158	4.699	0	0	586.502	42.738	4742.246	167.804	32.6676	4.963	196,844.5	1456.553	348,421.8	49,963.8885
	tSSO4	37.156	5.092	0	0	511.663	48.763	4388.164	277.232	31.3592	4.5905	195,426.1	2058.278	266,699.6	49,262.8425
	tSSO5	44.196	5.801	0	0	400.691	56.641	3663.083	419.515	28.8562	4.0773	192,100.9	2664.295	166,154.7	45,483.4355
	tSSO6	41.918	5.683	0	0	453.726	49.978	4037.566	329.065	31.453	4.5322	195,306.7	2181.89	210,798.2	45,526.3109
	tSSO7	47.33	5.836	0	0	363.759	57.152	3381.729	449.962	30.127	4.1393	191,714.3	2919.089	137,930.9	41,874.8504
	tSSO8	72.528	7.324	0.116	0.468	38.469	58.147	382.143	578.595	27.616	3.7534	176,917.2	4417.363	6889.327	8665.4426
	MOPSO	7.948	2.065	0	0	641.551	38.106	4910.514	90.919	49.2814	7.0987	195,334.2	995.1512	416,075.8	48,864.6986
	MOSSO	6.508	1.72	0	0	813.33	28.137	4705.793	166.873	19.1044	2.2699	197,323	843.5352	665,380.8	45,753.2918
	NSGA-II	12.482	2.856	0	0	686.503	64.443	4926.057	119.871	0.0794	0.041	197,624.3	2143.287	478,296.4	87,822.5684
150	tSSO0	31.892	4.499	0	0	571.334	23.553	4986.075	23.806	74.9205	11.0373	295,934.7	1364.945	739,913.7	60,689.2869
	tSSO1	33.948	4.799	0	0	541.511	25.246	4952.636	54.778	71.7156	10.4578	295,532.6	1663.517	665,347.3	61,362.5540
	tSSO2	37.058	5.306	0	0	497.394	31.575	4816.909	128.206	69.2553	9.7601	294,401.4	2369.671	562,929.2	70,555.4639
	tSSO3	36.176	5.035	0	0	515.033	26.975	4884.23	88.475	73.4922	10.7752	295,225.5	1901.874	602,559.3	62,546.5565
	tSSO4	38.654	5.274	0	0	461.168	33.508	4648.141	179.439	70.7016	10.1622	293,575.6	2662.045	485,020.5	69,694.8734
	tSSO5	46.074	6.107	0.002	0.045	375.99	38.045	4077.306	301.451	65.0655	9.094	289,048.3	3537.873	325,141.9	64,553.0441
	tSSO6	44.548	5.498	0	0	407.532	32.447	4317.863	231.212	70.9386	10.2171	292,870.2	2835.374	379,900.7	59,375.2787
	tSSO7	49.592	5.643	0.004	0.063	339.382	35.921	3774.411	317.912	68.0226	9.4273	288,517.2	3575.387	265,771.2	54,642.6111
	tSSO8	77.03	7.705	0.362	1.141	75.415	53.721	915.721	647.916	62.04	8.258	263,672.3	6382.976	22,508.07	20,508.0012
	MOPSO	8.758	2.175	0	0	522.934	26.468	4906.145	78.318	110.3736	14.8236	293,040.9	1222.526	621,413.8	62,184.5910
	MOSSO	7.398	2.044	0	0	666.568	18.525	4689.758	137.576	43.2861	4.9709	295,948.8	1105.613	1,005,113	55,590.2798
	NSGA-II	13.36	3.007	0	0	590.492	41.685	4939.701	88.364	0.18	0.0623	296,189.8	2758.174	792,818.2	110,125.8133

list the averages and standard deviations of the obtained number of temporary nondominated solutions (denoted by N_n), the obtained number of nondominated solutions in the Pareto front (denoted by N_p), the converge metric GD, and the diversity metric SP, the required run time, the energy consumption values, and the makspan values for three different-sized benchmark problems with N_sol = 50, 100, and 150, respectively. In these tables, the best of all the algorithms is indicated in bold, and the proposed BSSO with its nine parameter settings are denoted as BSSO_0–8, respectively, in Table 3, Table 4 and Table 5.

From Table 3, Table 4 and Table 5, we can make the following general observations:

• The lower the c_r value, the better the performance. In the small-size problem, the proposed tSSO₇ with c_p = 0.5 and c_w = 0.3 is the best among all 12 algorithms. However, the proposed tSSO₈ with c_p = 0.5 and c_w = 0.5 is the best one for the medium- and large-size problems. The reason for this is that the number of real nondominated solutions is infinite. Even though in the update mechanism of the proposed tSSO when c_r = 0 there is only an exchange of information between the current solution itself and one of selected temporary nondominated solutions, it is already able to update the current solution to a better solution without needing any random movement.
• The larger the size of the problem, e.g., N_job, the fewer the number of obtained nondominated solutions, e.g., N_n and N_p. There are two reasons why this is the case: (1) due to the characteristic of the NP-hard problems, i.e., the larger the size of the NP-hard problem, the more difficult it is to solve; (2) it is more difficult to find nondominated solutions for larger problems with the same deadline of 30 for all problems.
• The larger the N_sol, the more likely it is to find more nondominated solutions, i.e., the larger N_n and N_p for the best algorithm among these algorithms no matter the size of the problem. Hence, it is an effective method to have a larger value of N_sol if we intend to find more nondominated solutions.
• The smaller the N_job, the fewer the number of obtained nondominated solutions, e.g., There are two reasons for the above: (1) due to the characteristic of the NP-hard problems, i.e., the larger the size of the NP-hard problem, the more difficult it is to solve; (2) it is more difficult to find nondominated solutions for larger problems with the same deadline, which is set 30 for all problems.
• The smaller the value of N_p, the shorter the run time. The most time-consuming aspect of finding nondominated solutions is filtering out these nondominated solutions from current solutions. Hence, a new method, called the group comparison, is proposed in this study to find these nondominated solutions from current solutions. However, even the proposed group comparison is more efficient than the traditional pairwise comparison on average, but it still needs O(N_p²) to achieve the goal.
• There is no special pattern in solution qualities, e.g., the value of GD, SP, N_n, and N_p, from the final average values of the energy consumption and the makspan.
• The one with the better number of obtained nondominated solutions also has better DP and SP values.
• The MOPSO [13] and the original MOSSO [17] share one common factor: each solution must inherit and update based on its predecessor (parent) and its pBest, and this is the main reason that it is less likely to find new nondominated solutions. The above observation is coincident to that observed in item 1. Hence, the proposed tSSO and the NSGA-II [19,20] are much better than MOSSO [17] and MOPSO [13] in solution quality.

In general, the proposed tSSO without c_r has a more satisfying performance in all aspect of measures.

In addition, the histogram graphs for the average and standard deviations of the results of two-objective values, i.e., the energy consumption values and the makspan values, for three different-sized benchmark problems by the proposed tSSO and the compared MOPSO, MOSSO, and NSGA-II algorithms are further drawn, as shown in the following Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, in order to obtain more results and discussion to validate the performance of the proposed model and to have a comparison at a glance. The best results among tSSO₀ to tSSO₈ are taken as the solution of tSSO and plotted in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

On average, the average of the energy consumption for the small-size benchmark obtained by the proposed tSSO is better than those obtained by the compared algorithms, including MOPSO, MOSSO, and NSGA-II, though it is slightly worse than the performance of MOSSO when Nsol equals 50, as shown in Figure 1.

The standard deviation of the energy consumption for the small-size benchmark obtained by the MOSSO is the best, as shown in Figure 2.

The average of the makspan for the small-size benchmark obtained by the proposed tSSO is the best and is superior to MOSSO, as shown in Figure 3.

The standard deviation of the makspan for the small-size benchmark obtained by the proposed tSSO is the best and is superior to the compared algorithms, including MOPSO, MOSSO, and NSGA-II, as shown in Figure 4.

On average, the average of the energy consumption for the medium-size benchmark obtained by the proposed tSSO is better than those obtained by the compared algorithms, including MOPSO, MOSSO, and NSGA-II, though it is slightly worse than the performance of MOSSO when Nsol equals 50, as shown in Figure 5.

The standard deviation of the energy consumption for the medium-size benchmark obtained by the MOSSO is the best, as shown in Figure 6.

The average of the makspan for the medium-size benchmark obtained by the proposed tSSO is the best and is superior to the compared algorithms, including MOPSO, MOSSO, and NSGA-II, as shown in Figure 7.

The standard deviation of the makspan for the medium-size benchmark obtained by the MOSSO is the best and is superior to the compared algorithms, including the proposed tSSO, MOPSO, and NSGA-II, as shown in Figure 8.

The average of the energy consumption for the large-size benchmark obtained by the proposed tSSO is better than those obtained by the compared algorithms, including MOPSO, MOSSO, and NSGA-II, as shown in Figure 9.

The standard deviation of the energy consumption for the large-size benchmark obtained by the MOSSO is the best, as shown in Figure 10.

The average of the makspan for the large-size benchmark obtained by the proposed tSSO is the best and is superior to the compared algorithms, including MOPSO, MOSSO, and NSGA-II, as shown in Figure 11.

The standard deviation of the makspan for the large-size benchmark obtained by the proposed tSSO is the best and is superior to the compared algorithms, including MOPSO, MOSSO, and NSGA-II, as shown in Figure 12.

Overall, the performance of the proposed tSSO outperforms the compared algorithms, including MOPSO, MOSSO, and NSGA-II in all aspects of measures.

7. Conclusions

This study sheds light on a nascent two-objective time-constrained job scheduling problem focusing on energy consumption and service quality in terms of the makspan to find non-dominated solutions for the purpose of ameliorating the service quality and addressing the environmental issues of cloud computing services. In response to this two-objective problem, we proposed a new two-objective simplified swarm optimization (tSSO) algorithm to revise and improve the errors in the previous MOSSO algorithm [17], which, in the multi-objective problem, ignores the fact that the number of temporary nondominated solutions is not always only one, and some temporary nondominated solutions may not be temporary nondominated solutions in the next generation, based on SSO to deliver the job scheduling in cloud computing.

To ensure better solution quality, the tSSO algorithm integrates the crowding distance, a hybrid elite selection, and a new stepwise update mechanism, i.e., the proposed tSSO is a population-based, all-variable update, and stepwise function-based method. From the experiments conducted on three different-sized problems [17], regardless of the parameter setting, each of the proposed tSSOs outperformed the MOPSO [13], MOSSO [17], and NSGA-II [19,20], in convergency, diversity, the number of obtained temporary nondominated solutions, and the number of obtained real nondominated solutions. Among nine different parameter settings, we concluded that the tSSO algorithm with c_p = c_w = 0.5 is the best one. The results prove that the proposed tSSO can successfully achieve the aim of this work.

In order for the readers to understand the real-life cloud problems, the assumptions in Section 3.2 are simplified. Hence, future work will relax the assumptions to further meet and solve real-life cloud problems. In addition, a sequel work that considers real-life assumptions with changes in the algorithms is needed in the near future.

Moreover, in future works, the proposed model will be tested with more data from different domains, and the results will be compared with recently published studies from top journals and conferences.