Multi-Sine Cosine Algorithm for Solving Nonlinear Bilevel Programming Problems

Yousria Abo-Elnaga; M.A. El-Shorbagy

doi:10.2991/ijcis.d.200411.001

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Volume 13, Issue 1, 2020, Pages 421 - 432

Multi-Sine Cosine Algorithm for Solving Nonlinear Bilevel Programming Problems

Authors

Yousria Abo-Elnaga¹^{, *}, M.A. El-Shorbagy²^{, 3}

¹Department of Basic Science, Higher Technological Institute, Tenth of Ramadan City, Egypt

²Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia.

³Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom, Egypt

^*Corresponding author. Email: yousria_naga@yahoo.com

Corresponding Author

Yousria Abo-Elnaga

Received 31 October 2019, Accepted 1 April 2020, Available Online 24 April 2020.

DOI: 10.2991/ijcis.d.200411.001 How to use a DOI?
Keywords: Nonlinear bilevel programming problems; Sine cosine algorithm; Optimization
Abstract: In this paper, multi-sine cosine algorithm (MSCA) is presented to solve nonlinear bilevel programming problems (NBLPPs); where three different populations (completely separate from one another) of sine cosine algorithm (SCA) are used. The first population is used to solve the upper level problem, while the second one is used to solve the lower level problem. In addition, the Kuhn–Tucker conditions are used to transform the bilevel programming problem to constrained optimization problem. This constrained optimization problem is solved by the third population of SCA and if the objective function value equal to zero, the obtained solution from solving the upper and lower levels is feasible. The heuristic algorithm didn't used only to get the feasible solution because this requires a lot of time and efforts, so we used Kuhn–Tucker conditions to get the feasible solution quickly. Finally, the computational experiments using 14 benchmark problems, taken from the literature demonstrate the effectiveness of the proposed algorithm to solve NBLPPs.
Copyright: © 2020 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Many practical problems such as engineering design, management, economic policy and traffic problems, can be formulated as nonlinear bilevel programming problems (NBLPP). So, it has been studied and received increasing attention in the literatures. The NBLPP is a nested optimization problem with two levels (namely the upper and lower level) in a hierarchy order. The decision maker at the upper level (the leader) firstly optimizes his/her objective function independently. After the leader picks his/her decision, the decision maker at the lower level (the follower) makes his/her decision. The leader knows the objective and constraints of the follower who may be known or not the objective and (or) constraints of the leader. However, the leaders' decision is directly influenced by the decision of the follower. Throughout the most recent decades, some surveys and bibliographic reviews were given by several authors in [1–3]. In addition, reference books on NBLPPs and related issues have emerged in [4,5].

Figure 1 show that the general structure of NBLPP involving the interlinked optimization and decision-making tasks at both levels. from the figure we can see that for any given upper level decision vector, there is a corresponding lower level optimization problem must be solved which provides the rational optimal (response) of the follower for the leader's decision [6].

The NLBPP is a nonconvex problem, which is extremely difficult to solve. Firstly, Jeroslow [7] pointed out to that, then Ben-Ayed and Blair [8] and Bard [9] proved sequentially that the bilevel programming problem is a NP-Hard problem. Also, Vicente et al. [10] showed that even the search for the local optima to the nonlinear bilevel programming is NP-Hard operation. For more detailed discussion of the complexity issues in bilevel programming problem (see [11]). Therefore, many researchers devote their efforts to developing algorithms for solving NLBPP.

Traditional methods for solving NBLPPs can be classified to the following categories [6]: Branch-and-bound method, decent approach, approaches based on Kuhn–Tucker conditions and penalty function approach [12]. But, the properties such as differentiation and continuity are necessary when using these traditional approaches [13,14]. In addition, the NBLPP is nonconvex, which is extremely difficult to solve by traditional methods. Thus, most of researchers tend to use the metaheuristic algorithms for solving the NBLPP because of their good characteristics such as simplicity, global search capability, implicit parallelism and overcome difficulties and limitations of traditional techniques. These methods are population-based approaches which are able to find optimal solution for large scale problems easily [15,16].

The first use of the metaheuristic algorithms to solve NBLPP was through Mathieu et al. [17]; where they developed a genetic algorithm (GA) to handle this type of problems. For the same reason, other kinds of GA were proposed for solving NBLPP in Refs. [18–24]. In addition, in Refs. [25–28], the neural network approach was used to solve NBLPP; where it can converge to the optimal solution rapidly. Furthermore, other metaheuristic algorithms are used to solve NBLPP such as tabu search [29–32], simulated annealing [33], particle swarm optimization (PSO) [34–36], fruit fly optimization algorithm [37] and evolutionary algorithms [38,39].

Sine cosine algorithm (SCA) [40], is a novel population-based optimization algorithm for solving optimization problems. Due to its efficiency and simplicity, it has gained the interest of researchers from various fields for solving optimization problems. In this paper, multi-sine cosine algorithm (MSCA) is presented for solving NBLPPs; where two populations of SCA are used to solve the upper level problem and the lower level problem. On the other hand, the Kuhn–Tucker conditions is used to convert the NBLPP to a constrained optimization problem which is solved by the third population of SCA. If the objective function value of the constrained optimization problem equal to zero, the obtained solution from solving the upper and lower levels is feasible. Many benchmark problems, taken from the literature, used to demonstrate the effectiveness of the proposed algorithm.

This paper is organized as follows: Section 2 presents the definition and properties of the bilevel programming problem. Section 3 is devoted for a brief introduction to SCA and presentation of the proposed approach. Computational experiments are introduced and discussed in Section 4. Finally, the conclusion and future works are given in Section 5.

2. DEFINITION AND PROPERTIES OF NBLPP

The NBLPPs consist of two levels, namely, the upper and lower levels each having its nonlinear objective function. NBLPPs are formulated as follows:

Minx,y Fx,y, Subject to: gx,y≤0 where y solves the following problem Miny fx,y, hx,y≤0;(1)

where, Fx,y,fx,y are the two objective functions of the upper and lower level problems, respectively. gx,y and hx,y are the constraint functions of the upper and lower level problems, respectively. x∈Rn1,y∈Rn2 are the decision variables under the control of the upper lower level problems, respectively. Next we give the following definitions of the NBLPPs [4]:

The constraint region (CR) of NBLPPs
CR=x,y|gx,y≤0,hx,y≤0.(2)
The projection of CR onto the upper level's decision space
CRX=x|there exist y such that x,y∈CR.(3)
For each fixed x∈CRX, the CR of the lower level problem
CRx= y|hx,y≤0.(4)
For each fixed x∈CRX, the rational reaction set of the lower level problem
Mx= y|y∈argmin fx,y,y∈CRx.(5)
The inducible region of NBLPP
IR=x,y|x,y∈CR,y∈Mx.(6)

Firstly, we suppose that CR≠ϕ is compact and CRX≠ϕ. For each x∈CRX, the lower level problem (LP) is formulated as follows:

Miny fx,y Subject to: hx,y≤0.(7)

To avoid situations where (7) is not well posed, it is natural to assume that CRx≠ϕ,Mx≠ϕ. Even so, NBLP may be not well defined when the rational reaction set, Mx, is not single-valued [4]. In [9], Bard used examples to illustrate the difficulties that often arise when Mx is multi-valued and discontinuous. Here, we consider the situation that there is a unique solution to the lower level problem for each fixed x∈CRX. The reader can refer to [4,6] for how to do when that Mx is multi-valued. Then, we can give the definitions of feasible solution and optimal solution to NBLP as follows [35]:

Definition 1.

A solution x,y is said to be feasible to NBLPP if x,y∈IR.

Definition 2.

A feasible solution x∗,y∗ is called to be optimal to NBLPP if Fx∗,y∗≤Fx,y,∀x,y∈IR.

From the definition of the feasible solution to NBLPP, x¯,y¯ is a feasible solution means that y¯ solves problem (7) for fixed x¯. By applying Kuhn–Tucker conditions for problem (7), there exists β, such that

∇y fx¯,y¯+βT∇yhx¯,y¯=0βThx¯,y¯=0,β≥0;(8)

where β∈Rm is a column variable. Obviously, Eq. (8) can be equivalently transformed into the optimization problem as follows:

Minβ ‖∇yfx¯,y¯+βT∇yhx¯,y¯‖2+‖βThx¯,y¯‖2 subject to:β≥0.(9)

Therefore, if x¯,y¯ is a feasible solution to NBLPP, there exists an optimal solution to problem (9) and the optimal value equals zero. That is to say, we can solve Eq. (8) to judge whether the point x¯,y¯∈CR is feasible to NBLPP.

Now, we can give the following definition:

Definition 3.

Denote wx,y as the feasible weighting value of the point x,y is given by

w=minβ≥0∇yfx,y+βT∇yhx,y2+‖βThx,y‖2.(10)

Obviously, the smaller the feasible weighting value is, the closer x,y is near the feasible region. x,y is a feasible solution if the feasible weighting value equals zero. In Section 3 a brief explanation of the sine cosine algorithm is provided with a detailed description of the proposed method.

3. THE PROPOSED ALGORITHM

3.1. Brief Introduction to Sine Cosine Algorithm

SCA [40] is a novel population-based optimization algorithm for solving optimization problems. The SCA starts with creating a multiple initial random candidate solutions. This random set is evaluated by an objective function and improved by a set of rules as any optimization technique. Then, by using a mathematical model based on the sine and cosine functions, these random solutions is moved toward the best solution. There is no guarantee of finding a solution in a single run. However, with enough number of random solutions and optimization steps (iterations), the probability of finding the global optimum increases. The SCA is used the following equation to update the positions of solutions [40].

xi,k+1 d=xi,k d+r1×sinr2×|r3pk d−xi,k d|,r4<0.5xi,k d+r1×cosr2×|r3pk d−xi,k d|,r4≥0.5∀i=1,2,…,PS;(11)

where PS is the population size, xi,kd is the position of the current solution in d-th dimension at k-th iteration, r₁, r₂ and r₃ are random numbers, pkd is position of the destination point in d-th dimension at k-th iteration, ‖ indicates the absolute value and r₄ is a random number in [0, 1]. The parameter r₁ dictates the movement direction (or the next position regions) which could be either in the space between the solution and destination or outside it as shown in Figure 2. The parameter r₂ defines how far the movement should be toward or outward the destination. The parameter r₃ gives random weights for destination in order to stochastically emphasize r3>1 or deemphasize r3<1 the effect of desalination in defining the distance. Finally, the parameter r₄ equally switches between the sine and cosine components [41].

Mathematically, the steps of SCA are described as below:

Step 1. Initialization:

Initially agents are generated randomly and the parameters of SCA are stetted to form the initial population that satisfying the feasibility of the solved problem (the search domain).

Step 2. Evaluation:

For each agent, the desired optimization fitness function is evaluated.

Step 3. Setting the best position:

In the first generation, set the best position pk and its objective value equal to the position and objective value of the best initial agent.

Step 4. Updating agents positions:

The position of search agents is updated according to Eq. (11).

Step 5. Updating the best position pk:

Determine the best agent of the current population with the best objective value. If the objective value is better than the objective value of pk, then update pk and its objective value with the position and objective value of the current best agent.

Step 6. Termination criteria:

If the maximum number of generations has been produced, or when the agents of the population convergences, the algorithm is terminated and the best solution obtained so far is returned as the SCA global optimum. Otherwise, Update the SCA parameters (r₁, r₂, r₃, and r₄) and go to step 2. Convergence occurs when all agents positions in the population are identical.

The steps previously described are used in the proposed algorithm: multi-SCAs to solve the NBLPPs.

3.2. The Proposed Algorithm

The idea of the proposed algorithm is to used three population of SCA, as that described previously, to solve the NBLPPs. One of them is used to solve the upper level problem, while the second is used to solve the lower level problem and the third one is used to solve the problem (9). The flow chart of the proposed algorithm is shown in Figure 3. The steps details of it are described as follows:

Step 1. Solve the upper level's problem using the first population of SCA (1) to obtain x^*.

Step 2. Solve the lower level's problem using the second population of SCA (2) to obtain y^*.

Step 3. By using x∗ and y∗, problem (9) is solved by using the third population of SCA (3). If there exists a solution to problem (9) and the objective function value equal to zero, then the solution x∗,y∗ is feasible, and added to the feasible list and Compute upper level and lower level's objective function values; otherwise, the solution x∗,y∗ is infeasible.

Step 4. Terminal conditions; where if the number of iterations is larger than the maximum number of iterations, go to Step 5, otherwise go to Step 1.

Step 5. Output the optimal solutions and output the upper level and lower level's objective function values.

The MSCA code is implemented in MATLAB, and the results are compared with previous studies, to ensure the ability of the proposed approach to solve NBLPPs. The simulations have been executed on an Intel(R) core (TM) i5 CPU M430 @ 2.27 GHz processor, installed memory (RAM): 6.00 GB. In Section 4, computational experiments are performed and their results discussed.

4. COMPUTATIONAL EXPERIMENTS

4.1. Benchmark Problems

In this section, for computational experiments, 14 benchmark problems were used to illustrate the feasibility and efficiency of the proposed algorithm for solving NBLPPs. In these problems, the properties of the upper/lower objective functions are differentiable/convex, differentiable/nonconvex, nondifferentiable/convex and nondifferentiable/nonconvex. The descriptions and details of these problems are as follows:

Problem 1:

minx F(x,y)=x1−302+x2−202−20y1+20y2, subject to:x1+2x2≥30, x1+x2≤25, x2≤15, min0≤y≤10 f(x,y)=x1−y12+x2−y22.

Problem 2:

min0≤x≤50 F(x,y)=2x1+2x2−3y1−3y2−60, subject to:x1+x2−y1−2y2≤40, min0≤y≤10 f(x,y)=−x1+y1+202+−x2+y2+202, subject to:x1−2y1≥10, x2−2y2≥10.

Problem 3:

minx≥0 F(x,y)=−x12−3x22−4y1+y 22, subject to:x 12+2x2≤4, miny≥0 f(x,y)=2x 12+y 12−5y2, subject to:x12−2x1+x22−2y1+y 2 ≥−3, x2+3y1−4y2≥4.

Problem 4:

minx≥0 F(x,y)=−8x1 −4x2 +4y1−40y 2 −4y3, miny≥0f(x,y)=x1 +2x2 +y1+y 2 +2y3, subject to:y2+y3−y1≤1, 2x1−y1+2y2−0.5y3≤1, 2x2 +2y1−y 2 −0.5y3≤1 y≥0.

Problem 5:

min0≤x≤15 F(x,y)=x2+y−102, subject to: −x+y≤0, min0≤y≤20 f(x,y)=x+2y−302, subject to: x+y≤20.

Problem 6:

max0≤x≤1 F(x,y)=100x+1000y1, maxy1,y2 f(x,y)=y1+y2, subject to:x+y1−y2≤1, y1+y2≤1.

Problem 7:

minx≥0 F(x,y)=x−12+2y1−2x, miny1,y2≥0 f(x,y)=2y1−42+2y2−12+xy1, subject to:4x+5y1+4y 2 ≤12, 4y2−4x−5y1≤−4, 4x−4y1+5y 2 ≤4, 4y1−4x+5y2≤4.

Problem 8:

minx F(x,y)=(x1+y1)(x2+y2)21+x1y1+x2y2, subject to:x12+x22≤100, x1≥0 x2≥0 min0≤y≤20 f(x,y)=−F(x,y), subject to:0≤y1≤x1, 0≤y2≤x2.

Problem 9:

minx F(x,y)=|(x1−30)2+(x2−20)2−20y1+20y2−225|, subject to:30−x1−2x2≤0, x1+x2−25≤0, x2≤15, min0≤y≤10 f(x,y)=x1−y12+x2−y22.

Problem 10: the same problem 9 except

F(x,y)=|sin((x1−30)2+(x2−20)2−20y1+20y2−225)|

Problem 11: the same problem 9 except

F(x,y)=|tan((x1−30)2+(x2−20)2−20y1+20y2−225)|

Problem 12: the same problem 7 except

F(x,y)=|(x−1)2+2y1−2x+1.2097|

Problem 13: the same problem 7 except

F(x,y)=|sin((x−1)2+2y1−2x+1.2097)|

Problem 14: the same problem 7 except

F(x,y)=|tan((x−1)2+2y1−2x+1.2097)|

4.2. Results

The proposed algorithm was executed 50 independent runs on each of the above 14 benchmark problems. In all runs, we record the best solution found x∗,y∗, the upper's objective values at the best solution Fx∗,y∗ and the lower's objective values at the best solution Fx∗,y∗.

In addition, the 14 benchmark problems have been optimized and solved, using new evolutionary algorithms (NEA), by Wang et al. [19] and the combining particle swarm optimization with chaos searching technique (PSO-CST), by Wan et al. [35]. Hence, their results are comparable with the present optimization results that obtained by MSCA.

Figures 4–8 show the obtained feasible results, Fx,y and fx,y by the proposed approach MSCA for the test problems 1–5 and some of these set of feasible solutions are available in the Appendix (Table A.1 to Table A.5). From the figure, we can see that the proposed approach can provide many feasible solutions for the NBLPP in one run. So, the proposed algorithm MSCA is better than the other methods that solve the NBLPPs, where it gives the leader (the upper level) the ability to choose the appropriate solution from many feasible solutions.

The comparison between the best solution x∗,y∗ and the best results, upper level's objective function Fx∗,y∗ and lower level's objective function fx∗,y∗, that obtained by NEA, PSO-CST and the proposed approach MSCA are listed in Tables 1 and 2.

	x∗,y∗
Problem No.	NEA	PSO-CST	MSCA
1	(20, 5, 10, 5)	NA	(16.713, 8.286, 9.999, 4.02)
2	(0, 30, −10, 0)	NA	(19.980, 23.065, −5.733, 5.5127)
3	(4.4E-7, 2, 1.875, 0.9063)	(0.3844, 1.6124, 1.8690, 0.8041)	(0.3365, 1.0785, 1.73627, 0.56497)
4	(1.25E-13, 0.9, 0, 0.6, 0.4)	(0.1324, 0.1754, 0.6935, 0.7327, 0.2273)	(0.1885, 0.06320, 0.8608, 0.8449, 0.4560)
5	(10.0, 10.0)	(10.0020, 9.9961)	(10.0914, 9.9085)
6	(1.4E-12, 1, 7.07E-13)	(0.1511, 0.6256, 0.369)	(0.1510, 0.6256, 0.369)
7	(1.8888, 0.8889, 0)	(1.8602, 0.9073, 0.005)	(1.8602, 0.9073, 0)
8	(7.0709, 7.0713, 7.0709, 7.0703)	(7.0321, 6.842047, 5.9071, 6.8312)	(7.0321, 6.842044, 6.9071, 6.8312)
9	(20, 5, 10, 5)	(17.2024, 7.4665, 7.2189, 2.4251)	(17.2023, 7.4765, 7.2138, 2.4250)
10	(19.5629, 5.2722, 10, 5.2722)	(0.1946, 14.9870, 6.1019, 7.9628)	(0.1946, 14.9870, 6.1019, 7.9628)
11	(6.2048, 12.8594, 6.2048, 10)	(10.6084, 10.0550, 9.4545, 5.1257)	(10.6231, 10.0435, 9.3548, 5.0235)
12	(1.8888, 0.8889, 0)	(0.8606, 1.4599, 0.3138)	(0.8606, 1.4599, 0.3138)
13	(0.6648, 1.5746, 0.0721)	(0.9099, 1.5294, 0.1762)	(0.9099, 1.5294, 0.1762)
14	(0.6648, 1.5746, 0.0721)	(0.9233, 1.5083, 0.1899)	(0.9233, 1.5083, 0.1899)

PSO, particle swarm optimization; MSCA, multi-sine cosine algorithm.

Table 1

Comparison between the best solution x∗,y∗ obtained by NEA, PSO-CST and the proposed approach MSCA for problems 1–14.

	Fx∗,y∗			fx∗,y∗
Problem No.	NEA	PSO-CST	MSCA	NEA	PSO-CST	MSCA
1	225	NA	194.242	100	NA	63.229
2	0	NA	−25.904	100	NA	38.635
3	−12.68	−12.68	−13.1380	−1.016	−1.016	0.4163
4	−29.2	−29.2	−33.9402	3.2	3.2	2.9329
5	100.001	100.58	101.846	3.5E−11	0.001	0.008
6	1000	640.7139	640.71	1	0.9946	0.9946
7	−1.2098	−1.1660	−1.0745	7.6168	7.4441	7.4637
8	1.9802	1.9816	1.9800	−1.9802	−1.9816	−1.9800
9	0.0000	0.0075	0.0000	100	125.0854	124.987
10	6.86E−15	0.0000	0.0000	91.45	84.2367	84.6547
11	1.47E−14	0.0001	0.0000054	8.18	25.6292	25.6543
12	2.22E−16	0.0082	0.0000	7.62	2.5621	2.5611
13	1.22E−16	0.0374	0.00065	2.50	2.6969	2.5644
14	1.22E−16	0.0337	0.0000	2.50	2.7442	2.4446

PSO, particle swarm optimization; MSCA, multi-sine cosine algorithm.

Table 2

Comparison between the best results, Fx∗,y∗ and fx∗,y∗, found by NEA, PSO-CST and the proposed approach MSCA for problems 1–14.

From the Tables 1 and 2, it is clear that results obtained by MSCA are better than that obtained by NEA and PSO-CST for problems 1, 2, 4, 8, 12 and 14 for both upper level's objective function values and the lower level's objective function values. While the results of problems 11 and 13 that obtained by MSCA are only better than PSO-CST for both upper level's objective function values and the lower level's objective function values. But, the results obtained by MSCA for Problem 5 are worse than both algorithms, NEA and PSO-CST, for both upper level's objective function values and the lower level's objective function values.

For problem 3, only the upper level's objective function values that obtained by MSCA is better than that obtained by NEA and PSO-CST. For problem 6, only the upper level's objective function value that obtained by MSCA is better than that obtained by NEA and PSO-CST. For problem 7, the upper level's objective function value by MSCA is better than that obtained by NEA and PSO-CST, while the lower level's objective function value is better than PSO-CST and worse than NEA.

For problem 9, the results obtained by MSCA are better than PSO-CST for both upper level's objective function values and the lower level's objective function values. But, for comparison with NEA, MSCA succeeded only in obtaining a value for the upper level's objective function same as that obtained by NEA. Finally, for problem 10, the results obtained by MSCA are better than NEA for both upper level's objective function values and the lower level's objective function values. But, for comparison with PSO-CST, MSCA succeeded only in obtaining a value for the upper level's objective function to be same as that obtained by PSO-CST.

In general, we can say that the proposed approach MSCA was able to get better solutions for the upper level's objective function values for most benchmark problems in comparison with NEA and PSO-CST. So, the proposed algorithm MSCA is effective and efficient to solve NBLPPs.

5. CONCLUSION

This paper proposed MSCA to solve NBLPPs. Three population of sine cosine algorithm (SCA) are used. The first one is used to solve the upper level problem, while the second is used to solve the lower level problem. In addition, the bilevel programming problem is converted to a constrained optimization problem by the Kuhn–Tucker conditions. The third population of SCA is used to solve this constrained optimization problem to check the feasibility of the obtained solutions; where if the objective function value equal to zero, the obtained solution from solving the upper and lower levels is feasible. The heuristic algorithm didn't used only to get the feasible solution because this requires a lot of time and efforts, so we used Kuhn–Tucker conditions to get the feasible solution quickly. Finally, the proposed approach is tested by using 14 benchmark problems taken from the literature. The proposed approach has many features which can be concluded in many points as follows:

It has flexible adaptation to solve NBLPPs.
The solution quality is increased by using MSCA with different populations to solve the upper level problem, the lower level problem and the optimization problem that made up of transforming the bilevel programming problem using the Kuhn–Tucker conditions.
It is unlike traditional techniques; where it searches by a population of points, not single point. So, it can provide many feasible solutions for the NBLPP.
The numerical results that obtained by the computational experiments prove the superiority of the proposed algorithm, where it is better than those reported in the literature.

In our future works, we will focus on proposed other heuristic algorithms to solve NBLPP with consider other and more complex bilevel programming problems. In addition, this approach could be treating the multilevel programming problem.

CONFLICT OF INTEREST

The authors declare no conflict of interest.

AUTHORS' CONTRIBUTIONS

All authors are equally contributed in this article.

Funding Statement

The authors received no specific funding for this work.

ACKNOWLEDGMENTS

The authors would like to thank the referees for valuable remarks and suggestions that helped to increase the clarity of arguments and to improve the structure of the paper.

APPENDIX: SOME OF THE FEASIBLE SOLUTIONS FOR THE TEST PROBLEMS 1–5

x1	x2	y1	y2	Fx,y	fx,y
16.4834960466894	8.51650395330898	10.0000000000000	2.63861763169650	167.338913208160	76.5852685972360
19.9999610471097	5.00003895289030	10.0000000000000	4.55420770367797	216.083764547691	100.197986446486
19.9994429763476	5.00055702365228	10.0000000000000	9.03645054727345	305.723441329499	116.277296371236
19.9999999997282	5.00000000027182	9.99999999999974	8.85599101619236	302.119820321134	114.868666709429
15.6851161584213	9.31488384157868	10.0000000000000	8.21240746227845	283.335755962416	33.5359999016581
18.3898918090829	6.61010819084000	10.0000000000000	2.85413135279575	171.166441921726	84.4976465760407
17.2385457971006	7.40927578297419	9.99999999999886	1.66210253238730	154.623100329667	85.4265456290013
19.2502404571777	5.74975954282235	9.99999999999992	4.72648563974966	213.156396110878	86.6140379963173
18.9761803650813	6.02381963491871	9.99999999999994	0.211221836439455	121.082653669293	114.358107113358
18.1758460986601	6.82415389989657	9.99999999999574	5.20000368052242	217.413609546716	69.4823233641385
15.5531318369073	9.43458936624393	9.99999999999958	5.78905780241159	236.121057629907	44.1271735809760
19.7401398125080	5.25986018749199	9.99999999999856	6.95678332015329	261.672119162261	97.7498716853925
19.0936753908462	5.90632460915376	9.99999891688560	9.99999993214684	317.579622808003	99.4531294631476
17.2274674197367	7.77253257994694	9.99999999999996	7.65973063136341	265.843160649615	52.2490095829606
19.0321290913861	5.96787090861395	10.0000000000000	8.17160812485705	280.627001602484	86.4358136417173
15.0000000000003	9.91798247504626	9.99999999999998	3.45043811219105	195.655839617287	66.8291300855036
18.9269605855044	6.07303941449359	9.99999999998792	7.65889278422407	269.750288709944	82.2055562056510
19.5424621286193	5.45753787138068	9.99999999999972	8.14708230858107	283.784949265316	98.2922327558147
16.7136534124213	8.28634658757844	9.99999999999985	4.02531451947929	194.242972301187	63.2295364274879
17.5060426160103	7.47159078123141	9.99999999999480	0.130602422568305	115.672057117347	110.230785835468
15.0002970717591	9.99970292824088	10.0000000000000	4.81163219552818	221.229673369476	51.9190487334729
18.6650489169518	6.33495108304819	10.0000000000000	8.45236251712773	284.261928300336	79.5665039143385
17.0179068399914	7.98209315956506	10.0000000000000	6.05130472184851	233.990922077486	52.9789604060176
15.8204430979949	8.50939077578589	9.99999999999870	2.86878651913862	190.469664663594	65.6939742371186
17.0530962421431	7.94690375775025	9.99999999999986	6.74880199335452	247.875485807238	51.1816144387828
15.0001613434793	8.68862772639577	10.0000000000000	8.60127598935184	324.967822220753	25.0092437867890
17.3409044097306	7.65909559026935	9.99999999999924	5.44569881772352	221.464599168172	58.7880028255303
18.0908751374792	6.90912445615208	9.99999999999254	2.01477280896775	153.493733675130	89.4169385366964
17.8848537837812	7.11514605258855	9.99999999529601	6.43942540281612	241.584737236888	62.6275176623175
16.9528200885189	8.04717991147873	9.99999999999988	5.35774689802051	220.253749671525	55.5747571171931
18.1923817058604	6.80761829413955	10.0000000000000	8.44690329813828	282.396850815986	69.8023733388520

Table A.1

The set of feasible solutions obtained by multi-sine cosine algorithm (MSCA) for test problem 1.

x1	x2	y1	y2	Fx,y	fx,y
15.5387897122692	38.1450436155856	−9.15491945801839	4.53573422523622	−39.6027058169920	207.244207857494
2.60740248477875	41.9509110988793	−4.61852559060798	7.36809024165173	−82.8805293574979	375.835577689184
0.18717823930710	43.8931706442540	−6.61513244790278	8.26437186046296	−83.6734168219315	418.438354621163
0.02680932035748	35.2991104178593	−8.80039621280156	−4.7499391602369	−68.8443031387396	526.795725183233
7.07103583706055	19.4657433022623	−7.31896341018606	0.60300138289248	−43.3667813975830	32.7654643878523
19.9802129722810	23.0655234782115	−5.73346216726769	5.51275477227811	−25.9047110318464	38.6350246069557
11.7710636419019	6.54015612646236	−4.25558792658300	−7.5472083282536	−30.2312650629096	50.7467568495800
12.9704706831220	31.3300477541001	−3.95188862592689	9.30234285492468	−53.5334405100755	13.5834593807383
20.9473118708197	8.63797495412984	−0.44246685074850	−0.8408886748432	−25.4159506602448	112.625795432668
12.3302573925890	11.7445465983031	−2.74689258714901	−0.8786690476717	−38.8433540516780	78.6513997277838
7.98626409971397	33.0974022387324	−4.65476892486140	−0.7219464089927	−63.1605672647203	245.128791992776
4.48113373869799	30.4780537544522	−3.31816650112116	1.40888620607920	−71.5612867736927	231.106874658538
1.32154550037616	32.0495948357767	−5.55505693513630	4.31855628859339	−72.7413330296154	231.992520653631
4.21103920420309	30.8252147842564	−5.34277831212989	8.14214086062079	−66.3748014394606	116.321614161765
9.05459700982721	38.6683374581766	−9.61705014608239	4.17990467127684	−51.7079930002750	211.679205898914
0.07989908374101	11.7959341798244	−6.05548370439718	−3.0575269461838	−53.4696848991508	218.714472812962
2.41826382733149	45.2397947136916	−3.94963764352691	10.8150988155511	−88.5543541284479	393.905962062056
11.3500250502695	16.9028783280638	−5.68380781069236	−6.3639026885098	−37.1514047954477	19.4700057069557
14.6180234494580	39.4112881786770	−8.26699332368971	13.0005953223612	−45.3742613086918	49.4203046793626

Table A.2

The set of feasible solutions obtained by multi-sine cosine algorithm (MSCA) for test problem 2.

x1	x2	y1	y2	Fx,y	fx,y
0.99954526995436	1.16948795841407	9.50508979688807	4.50002437717950	−22.8723367936572	69.8447916543844
0.56918155742448	0.781344934570686	3.33349052270050	1.53342431098634	−13.1380393389281	4.09297280062670
0.84859268806525	0.308810945584173	9.98317824524428	4.28011065713497	−22.6195678942323	79.7035136911194
1.22954933869177	0.587053750521192	8.34396693106365	1.93719315666943	−32.1688382922881	62.9594015158913
0.26553572538002	1.83795256408921	3.89613479114882	0.88518781659479	−25.0056997989268	10.8949456707324
1.11557088809851	0.497492625214286	4.01676551735323	0.96032191814755	−17.1318390257390	13.8217924434060
0.81584142820524	1.20434759564482	9.96006144854210	2.52523092754962	−38.4804111860976	87.9078638929383
0.73651799391718	1.55852841371877	6.42547393133041	2.45240890593463	−27.5170774878842	30.1095882232611
1.04512758235492	1.34049718249537	8.53808873750856	1.36062455464306	−38.7841455235697	68.2804198431533
0.76202919664434	1.44867591530624	9.19990189958513	1.14545956671451	−42.3642041986663	80.0722741214944
1.20145811040292	0.95697878066106	4.45272476459092	1.91156235376235	−18.3477551770018	13.1559492424954
0.66285799880180	0.86310985389979	4.54786053773273	1.59179499005126	−18.3318874468509	13.6028219735614
0.62958003973951	0.06523865112941	9.30407513153499	1.39385731965786	−35.6826015698181	80.3892695078353
0.23228828455389	1.72190351617758	7.24846434514192	3.37000776229657	−26.5857180668436	35.7981122455927
0.39663826579551	0.91513735375060	6.24320858120901	0.01107937291180	−27.6424626149141	39.2369003517093
0.72177004130695	1.04467113399381	4.59129313474070	0.34755653828856	−22.0393423187836	20.3840939427307
0.97147411543027	0.26025064644296	7.30591591805685	2.82913573275856	−22.3666078317291	41.1182526518257
0.67499551190213	0.08836974125421	6.36099444670490	2.48734200098138	−19.7361541315696	28.9367782282797
0.66808821967701	0.11114460423195	5.12978559415573	1.72193028383340	−18.0374997126612	18.5977325613830
0.15866041902200	1.00661557603358	5.09921098818522	2.88481570224924	−15.1396801991016	11.6282204479112
0.77282063698029	0.33724842143606	8.44951283743064	2.47910964376220	−28.5905279541538	60.1932224449795
0.33657304637855	1.07853609436302	1.73627648803347	0.56497801480557	−10.2289075310003	0.416328799967051
1.09896346774441	0.99338615332173	7.39461010372874	0.41435888724943	−33.5749159797433	55.0239055567936

Table A.3

The set of feasible solutions obtained by multi-sine cosine algorithm (MSCA) for test problem 3.

x1	x2	y1	y2	y3	Fx,y	fx,y
0.0219789522662545	0.077011832938644	0.055711815243667	0.308524587820997	0.306776636661773	−13.8291217483969	1.15379229453176
0.148067816195352	0.177472256740090	0.725271660036854	0.440460620052212	1.21053529223336	−21.4539108873977	4.08981519423132
0.168021537291266	0.116175395218021	0.124084378838530	0.390268921459030	0.250747086453702	−17.9262815680241	1.41621980093227
0.0123703137499864	0.431178847175924	0.363257842542576	0.364210321817028	0.505830926517453	−16.9623831072842	2.61385802549634
0.843518302596460	0.113291096249447	0.581349533963859	0.154011416917199	1.38093666823368	−16.5601160195367	4.56733478244378
0.414582368495294	0.100595800835866	0.541880760168823	0.160931470802675	0.321502368887112	−9.27478741828599	1.96159093891275
0.127657430738104	0.283815821231952	0.491349291504876	0.248206081969839	1.04438585415068	−14.2969122602094	3.52361615497808
0.0140761551525193	0.0583750824277646	0.261579199793153	0.366371624544478	0.633499007005176	−16.4886537815584	2.02577515835603
0.252324479525196	0.0386892819302381	0.346189906301264	0.182136907727001	1.15726007108396	−12.7031099321333	3.17254999958185
0.265697761819986	0.461664257911050	0.026670974533030	0.000480820231789	0.500389149535003	−5.88634463548354	2.21695637147691
0.143699999342032	0.0053359195584884	0.161327226651221	0.447399856503252	0.411813336196394	−20.0688823712810	1.58672559400627
0.518377547658868	0.223043952665171	0.559980575472251	0.105803815706327	0.978409584122738	−10.9450648547866	3.58706901241326
0.632872932051398	0.0593690400478564	0.864194044803383	0.051786184588027	1.65985680231169	−10.5545580301569	4.98730484616189
0.0823677222322206	0.0045694785637084	0.219563039521511	0.561756819476326	0.624143499303446	−24.7658143102934	2.12111353696437
0.963324106942369	0.0190578149530657	0.601579089920769	0.010725389313378	1.19913176034682	−10.6020503695906	4.01200773677629
0.143788836089682	0.0190012603820694	0.688239480740719	0.757643255540148	0.559687005304804	−31.0178360501080	2.74704810374430
0.0145966468436354	0.569003182561170	0.235304544063625	0.457047476193431	0.653063889962365	−22.3457223363260	3.15108281214776
0.195948136196853	0.0789312669982073	0.716872768205968	0.809817735748292	0.798362302266633	−34.6019777237420	3.47722577868079
0.527145075605880	0.229044481651444	0.427678982149580	0.340308328500235	0.972132172334458	−20.9234844322017	3.69748569422750
0.0365697252020392	0.0550243411412149	0.591029056011186	0.568475692615912	0.862775968845162	−24.3386705221536	3.03167509380189
0.188572921721599	0.0632043643092972	0.860849632739098	0.844946184511238	0.456092038627976	−33.9402178350150	2.93296154484648
0.400626069464438	0.187323838293649	0.786531748282482	0.451499811431901	1.20041765317899	−23.6698399857522	4.41414061212409
0.236480219662212	0.169131522541224	0.172408285000627	0.266531515364797	0.0235352068949085	−12.6341361496316	1.06075347889990
0.357820606982177	0.483245276098604	0.0894287179525100	0.260795908343957	0.341761206418330	−16.2367122478734	2.35805819831251
0.130841683032999	0.146346235320911	0.770780343077803	0.479208757894081	0.883711805134506	−21.2521945695377	3.44094686491572

Table A.4

The set of feasible solutions obtained by multi-sine cosine algorithm (MSCA) for test problem 4.

x	y	Fx,y	fx,y
4.54434711375526	4.54431459818833	50.4155938938367	267.879464464703
9.93189321542748	9.93189017160819	98.6471417913780	0.04174929463726
4.84160820966831	4.84160789016214	50.0501792147651	239.481072540709
13.5573439330933	6.44265606598530	196.456270385054	12.6546958714265
7.76503492696258	7.76503032586637	65.2908568612458	44.9557432988312
12.8977790812233	7.10221469600974	174.749864896063	8.39719573254597
9.19849039252296	9.19847563899968	85.2546668026141	5.78190075986046
4.95455549576239	4.95455508606452	50.0041345401467	229.108617013369
12.0700032264420	7.92999389563460	149.969903158429	4.28493718675652
5.41720241899285	5.41719191379270	50.3482120033490	189.018880735276
8.05159577927594	8.05158458834688	68.6245172092216	34.1667727199599
11.0902067092904	8.90978707056004	124.181249086307	1.18857779413173
9.06993556756303	9.06992747622094	83.1287660992336	7.78526894225452
5.96728465772724	5.96728285202540	51.8712937819153	146.365224668810
5.11983492087859	5.11980061967912	50.0290556087321	214.346109546181
14.2557531911277	5.74422783887468	221.338095733756	18.1117581517450
12.1273766665743	7.87261159343451	151.599046245361	4.52583138356921
5.49721908525848	5.49721736933950	50.4944690903080	182.475416412343
5.02239346585321	5.02239236113826	50.0010139322998	222.989167265069
12.4456383902228	7.55430870753585	160.875320838222	5.98166466593585
4.17896823484413	4.17896272455500	51.3482504699564	304.960082205521
6.96583456382562	6.96580876405524	57.7291676268717	82.8563784185986
12.7371684550677	7.26282840324340	169.727568602862	7.49212554878433
7.89532648981604	7.89527810624009	66.7660346308632	39.8680772491621
12.8152978766264	7.18468857158428	172.157838105634	7.92605474416644
14.4263399738156	5.57362048994727	227.712120607124	19.5931855733581
6.31551130053454	6.31550686996195	53.4611726124771	122.179308681212
10.5193213529967	9.48066709657031	110.925828392197	0.26971866175790
8.13431855516288	8.13431821981965	69.6479068617641	31.3269127902098
5.27988668172122	5.27984871830056	50.1570314939460	200.517377942069
5.41219412384739	5.41219384803432	50.3398105222224	189.431679999866
10.0914679984401	9.90853033524031	101.846093063112	0.00836700440941

Table A.5

The set of feasible solutions obtained by multi-sine cosine algorithm (MSCA) for test problem 5.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 13 - 1
Pages: 421 - 432
Publication Date: 2020/04/24
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.200411.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Yousria Abo-Elnaga
AU  - M.A. El-Shorbagy
PY  - 2020
DA  - 2020/04/24
TI  - Multi-Sine Cosine Algorithm for Solving Nonlinear Bilevel Programming Problems
JO  - International Journal of Computational Intelligence Systems
SP  - 421
EP  - 432
VL  - 13
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.200411.001
DO  - 10.2991/ijcis.d.200411.001
ID  - Abo-Elnaga2020
ER  -

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