Revista de Matemática: Teoría y Aplicaciones 2016 23(1) : 111–142 cimpa – ucr issn: 1409-2433 (Print), 2215-3373 (Online)

sc: a novel fuzzy criterion for solving engineering and constrained optimization problems

sc: un nuevo criterio difuso para resolver problemas de ingeniería y de

optimización con restricciones

Sergio G. de-los-Cobos-Silva^{∗}

Miguel A. Gutiérrez-Andrade^{†}  Eric A. Rincón-García^{‡}
Pedro Lara-Velázquez^{§}  Roman A. Mora-Gutiérrez^{¶}

Antonin S. Ponsich^{�}

Received: 2 Oct 2014; Revised: 5 Oct 2015; Accepted: 6 Oct 2015

∗Universidad Autónoma Metropolitana-Iztapalapa, Departamento de Ingeniería Eléctrica, Av. San Rafael Atlixco 186, Col. Vicentina, Del. Iztapalapa, México D.F., C.P. 09340, México. E-Mail: cobos@xanum.uam.mx

†Same address as S. de-los-Cobos-Silva. E-Mail: gamma@xanum.uam.mx

‡Universidad Autónoma Metropolitana-Azcapotzalco, Departamento de Sistemas, Av. San Pablo 180, Colonia Reynosa Tamaulipas, México D.F., C.P. 02200, México. E-Mail: rigaeral@correo.azc.uam.mx

§Same address as S. de-los-Cobos-Silva. E-Mail: plara@xanum.uam.mx

¶Same address as E. Rincón-García. E-Mail: mgra@correo.azc.uam.mx �Same address as E. Rincón-García. E-Mail: aspo@correo@correo.azc.uam.mx

112 s.d.l.cobos–m.gutiérrez–e.rincón–p.lara–r.mora–a.ponsich

Abstract

In this paper a novel fuzzy convergence system (SC) and its fundamentals are presented. The model was implemented on a monoobjetive PSO algorithm with three phases: 1) Stabilization, 2) generation and breadthfirst search, and 3) generation and depth-first. The system SC-PSO-3P was tested with several benchmark engineering problems and with several CEC2006 problems. The computing experience and comparison with previously reported results is presented. In some cases the results reported in the literature are improved.

Keywords: particle swarm optimization (PSO); optimization.

Resumen

En este trabajo se presenta un novedoso sistema de convergencia (SC), sus fundamentos y la experiencia computacional. Se implementó en un algoritmo PSO monoobjetivo de tres fases: Estabilización, generación y búsqueda en amplitud, generación y búsqueda a profundidad, el cual se probó con diversos problemas benchmark tanto de ingeniería como de la serie CEC2006. La experiencia computacional y la comparación con resultados previamente reportados se presenta. En algunos casos, se mejoran los resultados de la literatura.

Palabras clave: optimización por enjambres de partículas; optimización.

Mathematics Subject Classification: 90C26, 90C29, 90C59.

1 Introduction

In general, an optimization problem can be defined as:

min_{x∈X} h(x).

Many real world problems can be expressed as continuous optimization problems, either with or without restrictions. Due to the complexity and intractability of many of these problems different heuristic solution methods have been developed, among others: Tabu Search (TS) [9], Simulated Annealing (SA) [21], Genetic Algorithms (GA) [15], Scatter Search (SS) [10], Particle Swarm Optimization (PSO) [19], just for PSO, there are so many different versions that a treaty on its taxonomy can be found in [31]. For a review of its variants, see [16].

With respect to constrained optimization: Toscano-Pulido and Coello (2004) [33]presented a simple mechanism to handle constraints with PSO based on

Figure 1: Convergence with SC for Griewank function dimension=120,000.

closeness of a particle to feasible region in order to select a leader, this approach also incorporates a mutation operator. The authors perform 340,000 objective function evaluations. Lu & Chen (2008) [26] presented an algorithm called selfadaptive velocity PSO. To handle constrains, they adopted a dynamic-objective constraint-handling method. Liu et al. (2010) [25] proposed an hybrid PSO named PSO-DE, which integrates PSO with differential evolution (DE) in order to force jump out of stagnation. Mazhoud et al. (2013) [27] proposed constrainthandling mechanism consists of closeness evaluation of a particle to the feasible region, at each iteration, the constraint total violation function is introduced as a second objective, in order to solve the bi-objective optimization problem they use a basic lexicographic method, the algorithm is named CVI-PSO and the algorithms have been tested over 24 benchmark functions with 25,000 fitness function evaluations(FFE) for problems with 2 until 24 variables.

In particular, with the SC-PSO-3P model, evidence of convergence for the Grienwank function with 120,000 variables x_{i} ∈ [−600, 600] is presented (see Figure 1) using only 3 particles and reaching global optimum in less than 90 iterations on average and 40 seconds on average. The instances were solved using Matlab and running it on a Notebook with an Intel Atom N280 processor at 1.66 Ghz. Subsequently, in the section of numerical examples, an Intel Core i5-3210M processor computer at 2.5 GHz computer was used.

This paper presents a novel approach that guarantees the solution in a single algorithm. The work is divided as follows: background of fuzzy numbers are presented in the second section; the proposed Convergence System SC, as well as its definitions and fundamentals are presented in the third section. The general guidelines of PSO are presented in the fourth section. Numerical examples are presented in the fifth section. Finally, conclusions and future research are presented.

2 Fuzzy numbers

In this section, we introduce the basic concepts of fuzzy numbers based on [6].

A fuzzy number A = (a, b, c, d; w) is defined as a fuzzy subset of the real line ℜ with membership function h_{A} such that:

1. h_{A} is continuous mapping from ℜ to the closed interval [0, w].

2. h_{A} = 0, ∀x ∈ (−∞, a].

3. h_{A} is a strictly increasing function on [a, b].

4. h_{A} = w, ∀x ∈ [b, c], where w is a constant and 0 ≤ w ≤ 1.

5. h_{A} is strictly decreasing on [c, d].

6. h_{A} = 0, ∀x ∈ [d, +∞),

where 0 ≤ w ≤ 1, and a, b, c, d ∈ ℝ, and a ≤ b ≤ c ≤ d.

If w = 1, the generalized fuzzy number A is called a normal trapezoidal fuzzy number (see Fig. 2) denoted as A = (a, b, c, d). If a = b and c = d, then A is a crisp interval. If b = c, then A is a generalized triangular fuzzy number. If a = b = c = d, then A is a real number.

The membership function h_{A} of A can be expressed as:

h_{A} = \{ \begin{matrix}h_{A}^{L}(x), & a ≤ x ≤ b, \\ w & b ≤ x ≤ c, \\ h_{A}^{R}(x), & c ≤ x ≤ d, \\ 0, & otherwise. \end{matrix}

where: h_{A}^{L}(x) : [a, b] ⟶ [0, w] and h_{A}^{R}(x) : [c, d] ⟶ [0, w] are continuous, h_{A}^{L}(x) is strictly increasing and h_{A}^{R}(x) strictly decreasing. The inverse functions of h_{A}^{L}(x) and h_{A}^{R}(x) are denoted by g_{A}^{L}(x) and g_{A}^{R}(x), respectively. These functions are continuous on [0, w], this means both ∫_{0}^{w} g_{A}^{L}(x) and ∫_{0}^{w} g_{A}^{R}(x) exist [23].

Let be two trapezoidal fuzzy numbers. B̃_{1} = (a_{1}, b_{1}, c_{1}, d_{1}; w) and B̃_{2} = (a_{2}, b_{2}, c_{2}, d_{2}; w) and c ∈ ℝ, then:

1. cB̃_{1} = (ca_{1}, cb_{1}, cc_{1}, cd_{1}; w).

2. B̃_{1} + B̃_{2} = (a_{1} + a_{2}, b_{1} + b_{2}, c_{1} + c_{2}, d_{1} + d_{2}; w).

Figure 2: Trapezoidal fuzzy number.

3 Convergence system, SC

This section is based on [5]. Let:

1. 0 ≤ f_{1}, f_{2}, f_{3} ≤ 1,

2. f_{1} + f_{2} + f_{3} = 1,

3. 0 < w ≤ 1,

4. a_{i} ∈ ℝ, α_{i} ∈ ℝ. Consider the following class of fuzzy numbers:

C (B̃) = {B̃_{i} = (a_{i}, f_{1}, f_{2}, f_{3}, α_{i}; w)| satisfying the preceding four conditions}, where:

1. b_{i} = a_{i} + f_{1}α_{i};

2. c_{i} = a_{i} + (f_{1} + f_{2})α_{i};

3. d_{i} = a_{i} + (f_{1} + f_{2} + f_{3})α_{i} = a_{i} + α_{i}. Property 1 For any two fuzzy numbers in B̃_{1}, B̃_{2} ∈ C (B̃), such that B̃_{1} = (a_{1}, f_{1}, f_{2}, f_{3}, α_{1}; w) and B̃_{2} = (a_{2}, f_{1}, f_{2}, f_{3}, α_{2}; w) and ∀c ∈ ℝ, satisfied:

1. B̃_{1}c = (ca_{1}, f_{1}, f_{2}, f_{3}, cα_{1}; w).

2. B̃_{1} + B̃_{2} = (a_{1} + a_{2}, f_{1}, f_{2}, f_{3}, α_{1} + α_{2}; w).

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

Definition 1 Given a function G : C(B̃) ⟶ ℝ, and B̃_{1}, B̃_{2} ∈ C (B̃) it is said that they are SC-equivalent if and only if:

G(B̃_{1}) = G(B̃_{2}).

Definition 2 Given a function G : C (B̃) ⟶ ℝ, and a g_{G} ∈ ℝ, in the codomain ofG, the following SC-equivalence class is defined B̃_{g_{G}} ⊂ C (B̃) as:

B̃_{g_{G}} = {B̃ ∈ C (B̃) | G(B̃) = g_{G}} .

Comments:

1. B̃_{g_{G}} is an equivalence class.

2. B̃_{g_{1}} ⋂ B̃_{g_{2}} = ∅, ∀g_{1} ≠ g_{2}

3. ⋃_{g∈ℝ} B̃_{g} = C (B̃). Definition 3 Let f_{1}, f_{2}, f_{3}, w such that they satisfy:

1. 0 ≤ f_{1}, f_{2}, f_{3} ≤ 1,

2. f_{1} + f_{2} + f_{3} = 1,

3. 0 < w ≤ 1.

We define:

1. F_{1} = (1 + f_{2}).

2. F_{2} = (2f_{1}^{2} + 6f_{1}f_{2} + 3f_{2}^{2} + 3f_{3} − 2f_{3}^{2}).

3. F_{3} = (3f_{1}^{3} +4f_{2}^{3} +3f_{3}^{3} +12f_{1}f_{2}^{2} +12f_{2}f_{1}^{2} +6f_{3} −8f_{3}^{2}), It can be proven that F_{3} > 0.

4. A = \frac{12}{w^{3}(1+3f_{2})}. Theorem 1 Let:

G : C (B̃) ⟶ ℝ

such that:

G(B̃) = A [α^{2}F_{3} − 4αF_{2} + 6aF_{1}] .

Then G is bijective on each SC-equivalence class. Proof. See [5], Appendix A.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

Proposition 1 Let B̃ ∈ C (B̃), s^{∗}, s, ϵ ∈ ℝ such that s = s^{∗} + ϵ, then:

lim_{ϵ→0} G(B̃s) = G(B̃s^{∗}).

Proof. See [5], Appendix B.

Proposition 2 Let B̃_{i} ∈ C (B̃), and s_{i}, ϵ_{i} ∈ ℝ such that s_{i} = s_{i}^{∗} + ϵ_{i}, i = 1,2,…, n

lim_{{ϵ_{i}}_{i=1}^{n}→0} G (∑_{i=1}^{n} B̃_{i}s_{i}) = G (∑_{i=1}^{n} B̃_{i}s_{i}^{∗}) .

Proof. See [5], Appendix C.

Definition 4 Consider the following multiobjective optimization problem h(x), x ∈ X ⊆ ℝ^{n}, where:

h(x) = (h_{1}(x), h_{2}(x), …, h_{k}(x)); h_{i} : ℜ^{n} → ℝ, i = 1, 2, …, k.

Then:

1. Given y = (y_{1}, y_{2}, …, y_{k}) is said that it dominates z = (z_{1}, z_{2}, …, z_{k}) if and only if ∀i ∈ {1, 2, …, k} y_{i} ≤ z_{i} and ∃i_{0} ∈ {1, 2, …, k} such that y_{i_{0}} < z_{i_{0}}.

2. Pareto Frontier (PF) = { h(x)| are non dominated}.

3. A solution vector x^{∗} is said to be Pareto Optimal if and only if there is no other vector x such that h(x) dominates h(x^{∗}).

Definition 5 The SC-Frontier (SC-F) is defined as:

SC-F = {h(x)|∀ϵ > 0∃y ∈ Codomine(h)−Image(h) such that ||y−h(x)|| < ϵ}.

Proposition 3 PF ⊆ SC-F.

Proof. See [5], Appendix D.

Definition 6 Given a multiobjective optimization problem h(x), x ∈ X ⊆ ℝ^{n}, where:

h(x) = (h_{1}(x), h_{2}(x), …, h_{k}(x)); h_{i} : ℜ^{n} → ℝ, i = 1, 2, …, k,

and

B̃ = (B̃_{1}, B̃_{2}, …, B̃_{k})

where B̃_{i} ∈ C (B̃), i = 1, …, k. Define:

G (B̃h(x)) = G (∑_{i=1}^{k} B̃_{i}h_{i}(x)) .

Definition 7 Given a multiobjective optimization problem and a B̃, define:

x∗ ; B̃ = argmin {G(B̃h(x)) | h(x) ∈ SC-F}.

Definition 8 The optimal set of SC-(SC-O) solutions is defined as:

SC-O = {x∗ ; B̃, B̃ ∈ C (B̃)}.

3.1 Empirical boundary conditions

Given an optimization problem with k objective functions, the following values have been used empirically:

1. f_{1} = f_{3} = 0.25, f_{2} = 0.5, w = 1,

2. α_{i} ∈ [−12K_{1}, 12K_{1}], i = 1, 2, …, k,

3. a_{i} = K_{2} + K_{3}g_{i},

where: K_{1} = \frac{2F_{2}}{F_{3}} , K_{2} = \frac{2F_{2}}{3kF_{1}} , K_{3} = \frac{A^{−1}F_{3}}{6kF_{1}} , g_{i} = \{ \begin{matrix}1 & if & h_{i}(x^{∗}) > 0 \\ 0 & if & h_{i}(x^{∗}) = 0 \\ −1 & if & h_{i}(x^{∗}) < 0. \end{matrix}

3.2 SC algorithm

Step 0:

1. Provide stopping criteria

2. Set f_{1}, f_{2}, f_{3}, w such that they satisfy the conditions of Theorem 1.

3. Given the function to optimize h = (h_{1}, h_{2}, … h_{k}), find the boundary conditions (or use the suggested empirical conditions), take B̃_{i} ∈ C (B̃), i = 1,2,…, k.

4. Set values for a_{i} and α_{i}.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

5. Take any value x in the Domain of h and evaluate:

G (B̃h(x)) = G (∑_{i=1}^{k} B̃_{i}h_{i}(x)) .

Set:

g_{G}(x) ← G (B̃h(x)) .

Step 1: Through m neighborhoods, find x_{j} ∈ P hase^{j}(x) and evaluate G (B̃h(x_{j})) , j = 1, 2, …, m

g_{G}(x_{1}) = min_{j}{G(B̃h(x_{j})), G(B̃h(x))},

x ← x_{1}.

Step 2: Stop in the following conditions:

1. If for a certain number of iterations g_{G}(x_{1}) = g_{G}(x), then an optimum was reached.

2. Stop criteria is met.

In any other case go to Step 1.

Remark: when g_{G} is very close to zero, it can be used, for example: (1 − g_{G})^{2}.

4 Particle swarm optimization (PSO)

The particle swarm optimization is a metaheuristic based on swarm intelligence and has its roots in artificial life, social psychology, engineering and computer science. PSO differs from evolutionary computation (c.f. [20]) because the population members or agents, also called particles, are “flying” through the problem hyperspace.

PSO is an adaptive method that uses agents or particles moving through the search space using the principles of evaluation, comparison and imitation [20].

PSO is based on the use of a set of particles or agents that correspond to states of an optimization problem, where each particle moves across the solution space in search of an optimal position or at least a good solution. In PSO, agents communicate with each other, and the agent with the best position (measured according to an objective function) influences the others by attracting them towards itself.

120 s.d.l.cobos–m.gutiérrez–e.rincón–p.lara–r.mora–a.ponsich

The population is initialized by assigning a random position and speed to each agent. At each iteration, the velocity of each particle is randomly accelerated towards its best position (where the value of the fitness function or objective function improves) and also considering its neighbors’ best position.

To solve a problem, PSO uses a dynamic management of particles; this approach allows breaking cycles and diversifying the search. In this work, a rparticle swarm is represented at time t under the form:

θ_{1t}, θ_{2t}, …, θ_{rt}

with θ_{jt} ∈ D, j = 1, 2, …, r, then a movement of the swarm is defined according to equation 1:

θ_{jt+1} = θ_{jt} + V_{jt+1}  (1)

where the velocity V_{jt+1} is given in equation 2:

V_{j,t+1} = αV_{j,t} + rand(0, φ_{1})[θ′_{j,t} − θ_{j,t}] + rand(0, φ_{2})[θ′_{g,t} − θ_{j,t}], (2)

where:

D: space of feasible solutions,

V_{j,t}: speed at time t of the j-th particle,

V_{j,t+1}: speed at time t + 1 of the j-th particle,

θ_{j,t}: j-th particle at time t,

θ′_{g,t}: the particle with the best value found so far (i.e., before time t), θ′_{j,t}: best position found so far by the j-th particle (before time t), rand(0, φ): random number uniformly distributed over the interval [0,φ], α: inertia weight factor.

The PSO algorithm is described in Table 1.

Table 1: PSO algorithm.

1. Create a population of particles distributed in the feasible space.

2. Evaluate each position of the particles according to the objective function (fitness function).

3. If the current position of a particle is better than the previous one, update it.

4. Determine the best particle (according to the best previous positions).

5. Update the particle velocities j = 1, 2, …, r according to equation 2.

6. Move the particles to new positions according to equation 1.

7. Go to Step 2 until the termination criterion is satisfied.

4.1 SC-PSO-3P

The three-phase PSO algorithm along with the SC criterion is named SC-PSO-3P. The SC-PSO-3P algorithm is described in Table 1.

As can be seen in the algorithm in table 1, the key modification is to consider the SC criterion, and the 3 phases, the rest of the algorithm remains unchanged. It should be mentioned, that any other search algorithm can be used instead of PSO, although the respective tests must be performed.

We describe the main characteristics of the proposed algorithm, called PSO-3P. This algorithm is based in a traditional PSO heuristic. However, the position of the particles can be modified using different strategies, that are applied in different times, or phases, of the searching process.

In phase 1, called stabilization, according to the description presented in section 4, the PSO-3P algorithm generates randomly a set of particles in the solution space. Then, during itF_{1} iterations the position of the particles is modified using equations (1) and (2). Thus, at the end of this phase the particles are concentrated, or stabilized, in a promising region.

When phase 1 is completed, a breadth-first search strategy, called phase 2, is incorporated. In this phase, if the global best solution is not improved after three consecutive iterations, the position of M_{2} particles is modified randomly. However, the particle with the best known position is preserved. Thus, the population is dispersed in the solution space, but it can be attracted to the best region visited so far. This diversification strategy is considered during itF_{2} iterations.

Finally, phase 3 is initialized. During itF_{3} iterations the following depth-first search strategy is applied. If the global best solution is not improved after three consecutive iterations, M_{3} particles are randomly positioned in a neighborhood of the best known solution. Thus, phase 3 includes an intensification process in a promising region.

The algorithm was implemented in Matlab R2008a and was run in an Intel Core i5-3210M processor computer at 2.5 GHz, running on Windows 8.

5 Computational results

The parameters were set as follows, ϕ_{1} = ϕ_{2} = 2.05, and ϕ = ϕ_{1} + ϕ_{2}, then:

α = \frac{2}{ϕ−2+\sqrt{ϕ^{2}−4∗ϕ}}; φ_{1} = α ∗ ϕ_{1}; φ_{2} = α ∗ ϕ_{2}.

For phase 2 and phase 3 the parameters were set as c = 3, itF_{1} = 75, itF_{2} = 150, itF_{3} = MaxIt, prop = 0.25.

The select problems have been well studied before as benchmarks by various approaches. The 24 constrained problems(G1-G24) have been solved before using the following algorithms : self-adaptive multi-strategy differential evolution with local search(Memetic-SAMSDE) [7], adaptive penalty formulation with GA(APF-GA) [32], modified differential evolution(MDE) [29], adapted constrained particle swarm optimization(CVI-PSO) [27], a differential evolution combined variants (DECV) [29] and a heuristic inspired by T-Cell model of the immune system (T-CELL) [1].

The three engineering problems considered are a pressure vessel design problem(eng01) [18], a welded beam design problem(eng02) [30] and a tensioncompression string design problem(eng03) [2]. Eng01 and eng03, have been solved by the following approaches: constraint violation with interval arithmetic PSO(CVI-PSO) [27], a co-evolutionary PSO(CPSO) [11], a hybrid PSO(HPSO) [12], an accelerating technique(AATM) [35], a T-Cell heuristic [1]. Problem eng02 has been solved by these approach: CVI-PSO, a GA with binary representation (GAPF) [4], a GA based co-evolution model (CGA) [3], CPSO and HPSO.

In this section the results obtained for constrained optimization will be presented, in order to show the efficiency of SC. A study in depth for multiobjective optimization, and unconstrained optimization, will be presented in future work.

In this paper, the functions for restricted optimization were taken from [22].

Begin while Termination criterion is not satisfied do

Set variables c, itF_{1}, itF_{2}, itF_{3}, MaxIt and prop. Create a population of nP op random particles. Set cont = 0 and it = 1. Evaluate each position of the particles according to the SC function. If the current position of a particle is better (respect to the fitness function) than the previous update it. Determine the best particle (according to the best previous positions against the optimization criterion). If a better particle cannot be founded, let cont = cont + 1. Update the particle velocities j = 1, 2, …, nP op according to equation (2). (Phase 1: Stabilization) if it ≤ itF 1 then

go to Step 34. end (Phase 2: Breadth-first search) if itF 1 < it ≤ itF 2 then

if cont = c then

Set n=1. while n ≤ nP op ∗ prop do Create a random particle and, with a probability bigger than 0.5 substitute randomly a particle in the swarm. Set n = n + 1.

end

Set cont = 0. end go to Step 34.

end end (Phase 3: Depth-first search). if itF 2 < it ≤ itF 3 then

if cont = c then

Set n = 1. while n ≤ nP op ∗ prop. do Create a random particle in a variable neighborhood of θ′_{g,t} and substitute randomly a particle in the swarm. Set n = n + 1.

end end Set cont = 0. go to Step 34.

end Select the best nP op particles according to SC optimization criterion. Set it = it + 1. Go to Step 3 until the termination criterion is satisfied.

Algorithm 1: SC-PSO-3P algorithm.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

5.1 Constrained optimization

In this section we consider the global optimization problem with restrictions:

Minimize:  h(x)
subject to:  f_{i}(x) ≤ 0 i = 1, 2, …, m
t_{j}(x) = 0 j = 1, 2, …, k
x ∈ D ⊆ S = [L, U]

(3)

where:

• [L, U] = {x = (x_{1}, x_{2}, …, x_{n})|l_{i} ≤ x_{i} ≤ u_{i}} ⊂ ℝ^{n}.

• the feasible region (D) is defined as:

D = {x ∈ S, f_{i}(x) ≤ 0, i = 1, 2, …, m, t_{j}(x) = 0, j = 1, 2, …, k}. The problem given in (3) can be transformed into the problem (4) [24] given

by:

Minimize_{x∈D} : F (x) = (h_{1}(x), h_{2}(x), h_{3}(x))
with:  h_{1}(x) = h(x)
h_{2}(x) = P_{1} ∑_{i=1}^{m} \frac{c_{i}(x)}{c(x)+ϵ}
h_{3}(x) = P_{2} ∑_{j=1}^{k} d_{j}

(4)

where:

• c_{i}(x) = max{0, f_{i}(x)}.i = 1, 2, …, m.

• c(x) = max_{i=1,2,...,m}{c_{i}(x)}.

• d_{j}(x) = max{0, |t_{j}(x)| − δ}.j = 1, 2, …, k.

• ϵ, δ > 0, P_{i} >> 0, i = 1, 2.

In this section, the efficiency is defined as:

efficiency = \{ \begin{matrix}1 − abs(\frac{h(x)−h(x^{∗})}{h(x^{∗})}) & if & h(x^{∗}) ≠ 0 \\ 1 − abs(h(x)) & if & h(x^{∗}) = 0. \end{matrix}

Table 2 shows the ratio(ρ) for problems G1 to G24, as well as the engineering problems eng01-eng03 and a summary of some characteristics of these problems [27]. sc: a novel fuzzy criterion for solving engineering... 125

Function(n) Type of h Ratio* ρ %
G1(13)  Quadratic  0.0001
G2(20)  Non-linear  99.996
G3(12)  Polynomial 0.0000
G4(5)  Quadratic  26.925
G5(4)  Cubic  0.0000
G6(2)  Quadratic  0.0054
G7(10)  Quadratic  0.0002
G8(2)  Non-linear  0.8611
G9(7)  Polynomial 0.5314
G10(8)  Linear  0.0002
G11(2)  Quadratic  0.0000
G12(3)  Quadratic  4.5420
G13(5)  Non-linear  0.0000
G14(10)  Non-linear  0.0000
G15(3)  Quadratic  0.0000
G16(5)  Non-linear  0.0186
G17(6)  Non-linear  0.0000
G18(9)  Quadratic  0.0000
G19(15)  Non-linear  35.548
G20(24)  Linear  0.0000
G21(7)  Linear  0.0000
G22(22)  Linear  0.0000
G23(9)  Linear  0.0000
G24(2)  Linear  74.268
eng01(4)  Non-linear  39.737
eng02(4)  Non-linear  2.6638
eng03(3)  Non-linear  0.7560

Table 2: Summary ([27]), ∗ρ = \frac{number of solution∈D}{number of solution∈S}.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

SC-PSO-3P  Memetic  APF-GA  MDE  CVI-PSO  DECV  T-CELL
FFE  150000  240000  500000  500000  200000  500000  350000
Best  -14.999999  -15.000000  -15.000000  -15.000000  -15.000000  -15.000000  -15.000000
G1 Average  -14.216335  -15.000000  -15.000000  -15.000000  -15.000000  -14.855000  -15.000000
Std. Dev. 1.205257E+00 0.000000E+00 0.000000E+00 0.000000E+00 4.000000E-16 4.590000E-01 0.000000E+00
Best  -0.760896  -0.803619  -0.803601  -0.800362  -0.800098  -0.704009  -0.801367
G2 Average  -0.560853  -0.803619  -0.803518  -0.786160  -0.790876  -0.569458  -0.752975
Std. Dev. 9.614486E-02 0.000000E+00 1.000000E-04 1.260000E-02 1.091200E-02 9.510000E-02 3.209500E-02
Best  -1.000588  -1.000500  -1.001000  -1.000500  -1.000000  -0.461000  -1.000000
G3 Average  -1.000399  -1.000500  -1.001000  -1.000500  -1.000000  -0.134000  -1.000000
Std. Dev. 2.452163E-04 0.000000E+00 0.000000E+00 0.000000E+00 3.700000E-16 1.170000E-02 0.000000E+00
Best  -30665.465015 -30665.539000 -30665.539000 -30665.538600 -30665.821702 -30665.539000 -30665.538500
G4 Average -30665.287211 -30665.539000 -30665.539000 -30665.538600 -30665.820996 -30665.539000 -30665.538400
Std. Dev. 1.155335E-01 0.000000E+00 1.000000E-04 0.000000E+00 3.391000E-03 1.500000E-06 1.000000E-04
Best  5126.490316 5126.497000 5126.497000 5126.497000 5127.277667 5126.497000 5126.625500
G5 Average  5161.796768 5126.497000 5127.542300 5126.497000 5127.277667 5126.497000 5378.267800
Std. Dev. 3.437387E+01 0.000000E+00 1.432400E+00 0.000000E+00 0.000000E+00 0.000000E+00 2.980173E+02
Best  -6961.773245 -6961.813875 -6961.814000 -6961.814000 -6961.813876 -6961.814000 -6961.813870
G6 Average  -6960.340642 -6961.813875 -6961.814000 -6961.814000 -6961.813876 -6961.814000 -6961.813860
Std. Dev. 1.163449E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 3.900000E-05
Best  25.343143  24.306200  24.306200  24.306200  24.473827  24.306000  24.320900
G7 Average  28.080169  24.306200  24.306200  24.306200  26.561295  24.794000  24.653400
Std. Dev. 2.217613E+00 0.000000E+00 0.000000E+00 0.000000E+00 1.642519E+00 1.370000E+00 2.198150E-01
Best  -0.095825041 -0.095825000 -0.095825000 -0.095825000  -0.105495  -0.095825  -0.095825
G8 Average  -0.095824540 -0.095825000 -0.095825000 -0.095825000  -0.105495  -0.095825  -0.095825
Std. Dev. 3.298078E-07 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 4.230000E-17 0.000000E+00
Best  680.698889  680.630000  680.630000  680.630000  680.635400  680.630000  680.630000

Table 3: Comparative results for several algorithms.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

SC-PSO-3P  Memetic  APF-GA  MDE  CVI-PSO  DECV  T-CELL
FFE  150000  240000  500000  500000  200000  500000  350000
G9 Average  681.152138  680.630000  680.630000  680.630000  680.755705  680.630000  680.630000
Std. Dev. 3.630035E-01 0.000000E+00 0.000000E+00 0.000000E+00 7.923200E-02 3.450000E-07 1.670000E-02
Best  7097.106239 7049.248020 7049.248020 7049.248020 7049.276586 7049.248000 7050.834200
G10 Average  7446.397229 7049.248020 7077.682100 7049.248020 7053.214311 7103.548000 8020.755100
Std. Dev. 3.768273E+02 0.000000E+00 5.124000E+01 0.000000E+00 1.061564E+01 1.480000E+02 6.217231E+02
Best  0.749994  0.749900  0.749900  0.749900  0.750000  0.750000  0.749900
G11 Average  0.750000  0.749900  0.749900  0.749900  0.750000  0.750000  0.749900
Std. Dev. 3.588812E-06 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 1.120000E-16 0.000000E+00
Best  -1.000000  -1.000000  -1.000000  -1.000000  -1.000000  -1.000000  -1.000000
G12 Average  -0.999995  -1.000000  -1.000000  -1.000000  -1.000000  -1.000000  -1.000000
Std. Dev. 3.154629E-06 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00
Best  0.053449  0.053942  0.053942  0.053942  0.055556  0.059798  0.054638
G13 Average  0.055757  0.053942  0.053942  0.053942  0.065591  0.382401  0.458857
Std. Dev.  1.93E-03  0.00E+00  0.00E+00  0.00E+00  1.02E-02  2.68E-01  3.45E-01
Best  -48.29818726  -47.764888  -47.764790  -47.764887  -47.453011  -47.764888  -47.517100
G14 Average  -47.9751512  -47.764888  -47.764790  -47.764874  -44.424690  -47.722542  -45.310800
Std. Dev.  3.16E-01  0.00E+00  1.00E-04  1.40E-05  1.41E+00  1.62E-01  1.12E+00
Best  961.7218517  961.715020  961.715020  961.715020  961.715707  961.715022  961.715020
G15 Average  962.2834191  961.715020  961.715020  961.715020  961.718595  961.715022  963.374820
Std. Dev.  9.29E-01  0.00E+00  0.00E+00  0.00E+00  6.87E-04  2.31E-13  2.28E+00
Best  -2.156472  -1.905155  -1.905155  -1.905155  -1.905150  -1.905155  -1.905155
G16 Average  -1.833968  -1.905155  -1.905155  -1.905155  -1.905150  -1.905155  -1.905155
Std. Dev. 1.837774E-01 0.000000E+00 0.000000E+00 0.000000E+00 8.520000E-15 1.100000E-06 0.000000E+00
Best  8684.768462 8853.539700 8853.539800 8853.539700 8853.538913 8853.541289 8861.821000

Table 4: Comparative results for several algorithms.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

SC-PSO-3P  Memetic  APF-GA  MDE  CVI-PSO  DECV  T-CELL
FFE  150000  240000  500000  500000  200000  500000  350000
G17 Average  8897.083482 8853.539700 8888.487600 8853.539700 8853.538913 8919.936362 8990.997000
Std. Dev. 1.372256E+02 0.000000E+00 2.903470E+01 0.000000E+00 3.700000E-12 2.590000E+01 1.061932E+02
Best  -0.857012  -0.866025  -0.866025  -0.866025  -0.864631  -0.866025  -0.865960
G18 Average  -0.765061  -0.866025  -0.865925  -0.866025  -0.809109  -0.859657  -0.784550
Std. Dev. 1.125068E-01 0.000000E+00 1.000000E-04 0.000000E+00 6.271000E-02 3.480000E-02 9.746000E-02
Best  32.180687  32.655593  32.655593  32.648270  32.827027  32.655593  33.390780
G19 Average  34.116005  32.655593  32.655593  33.341250  35.067337  32.660587  38.927610
Std. Dev. 2.411570E+00 0.000000E+00 0.000000E+00 8.475000E-01 2.286728E+00 2.370000E-02 3.191020E+00
Best  5.024074E-02 2.146897E-01  NA  NA  NA  NA  NA
G20 Average 2.181222E-01  NA  NA  NA  NA  NA  NA
Std. Dev. 1.125954E-01  NA  NA  NA  NA  NA  NA
Best  199.6299601  193.724500  196.633010  193.724500  193.786925  193.724510  NA
G21 Average  729.0817718  193.724500  199.515810  193.724500  193.786935  198.090578  NA
Std. Dev. 1.726319E+02 0.000000E+00 2.356500E+00 0.000000E+00 3.300000E-05 2.390000E+01  NA
Best  0  236.370331  NA  NA  NA  NA  NA
G22 Average  0  245.738829  NA  NA  NA  NA  NA
Std. Dev.  0 9.059390E+00  NA  NA  NA  NA  NA
Best  -400.008150 -400.055100 -399.762400 -400.055100 -400.000000 -400.055093  NA
G23 Average  -399.990059 -400.055100 -394.762700 -400.055100 -400.000000 -392.029610  NA
Std. Dev. 2.052941E-01 0.000000E+00 3.865600E+00 0.000000E+00 0.000000E+00 1.240000E+01  NA
Best  -7.000000  -5.508013  -5.508013  -5.508013  -5.508013  -5.508013  -5.508013
G24 Average  -7.000000  -5.508013  -5.508013  -5.508013  -5.508013  -5.508013  -5.508013
Std. Dev. 0.000000E+00 0.000000E+00 0.000000E+00 0.000000E+00 9.460000E-15 2.710000E-15 0.000000E+00

Table 5: Comparative results for several algorithms.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

Tables 3, 4 and 5 show that the best solutions obtained by SC-PSO-3P are competitive with those achieved by other methods. In particular, for the problem G24 a better feasible solution was consistently reached as can be seen in Table 6. It is noteworthy that for this particular problem, in all runs the solution was reached in the first iteration.

best known  SC-PSO-3P
h(x^{∗}) -5.5080132771  -7.000000000
x_{1}^{∗}  2.39520197477613 3
x_{2}^{∗}  3.178493074117714 4
f_{1} ≤ 0 -1.7763568394e-15 -160.00
f_{2} ≤ 0 0.00  -152.00

Table 6: Table comparing the best known solution vs. SC-PSO-3P in G24.

From literature it is known that regarding the function G20 there is not a single feasible solution while for the function G22, a feasible solution is rarely found [22], for this reason is very common that these instances are not considered when they propose some method. However, it may be noticed that SC-PSO-3P is competitive results even consider that improves the results reported in the literature as can be seen from tables 7 and 8.

For instance G20, in [22] is said "This solution is a little unfeasible and no feasible solution is found so far":

[22] G20: x^{∗}=(1.28582343498528086E-18, 4.8346030252613066E-34, 0, 0, 6.30459929660781851E-18, 7.57192526201145068E-34, 5.0335069837240437E-34, 9.282680796166E-34, 0, 1.76723384525547359E-17, 3.55686101822965701E-34, 2.99413850083471346E-34, 0.158143376337580827, 2.29601774161699833E-19, 1.06106938611042947E-18, 1.31968344319506391E-18, 0.530902525044209539, 0, 0.289148310257773535E-18, 3.34892126180666159E-18, 0, 0.310999974151577319, 5.41244666317833561E-5, 4.84993165246959553E-16); h^{∗}=204.979400285636e-003.

With SC several improved solutions. Below are two of them:

SC-PSO-3P (Example 1) G20: x^{∗}=(1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 0.296470402, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 0.703529139); h^{∗}=8.999996E-02.

SC-PSO-3P (Example 2) G20: x^{∗}=(1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.37E-06, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 1.00E-300, 0.502406032, 1.00E-300); h^{∗}=50.24074020e-003.

[22]  SC-PSO-3P (Example 1) SC-PSO-3P (Example 2)
2.049794E-01 5.024074E-02 8.999996E-02

f(x) ≤ 0

1.317752E-01  2.847350E-300  1.666667E-300
1.766032E-19  2.492499E-300  1.538462E-300
7.578526E-19  2.216294E-300  1.428572E-300
2.224047E-19  2.492499E-300  1.538462E-300
2.092945E-18  1.814211E-300  1.250000E-300
0.000000E+00  2.492499E-300  1.538462E-300

t(x) = 0

1.000000E-04  -2.358454E-294  -1.227952E-299
-2.197257E-17  -4.585322E-295  -1.294290E-300
1.553971E-18  -6.610392E-295  -2.515003E-300
8.175374E-19  -8.994252E-296  7.952623E-302
1.000000E-04  -5.889714E-295  -2.843776E-300
-1.038308E-17  -1.362617E-295  -3.215523E-301
-3.346898E-17  -6.685073E-295  -2.663043E-300
-3.208538E-18  -7.027115E-296  5.820753E-301
0.000000E+00  -1.323114E-296  5.604647E-301
1.000000E-04  -1.803505E-295  -5.182852E-302
1.000000E-04  9.787500E-01  1.760682E-300
6.866522E-16  -8.951397E-297  9.840000E-01
1.000000E-04  4.975926E-01  4.586120E-07
1.000000E-04  1.200004E-01  1.433791E-07

Table 7: Table comparing G20 [22] vs. SC-PSO-3P.

In Table 7 a comparison of the values of the function and constraints is presented for the function G20. Two solutions found by SC-PSO-3P are provided, can be observed that both improve over the best reported value, however, example 2 is "less unfeasible" than example 1.

Regarding the problem G22, the best reported solution in the literature and the solution obtained by SC-PSO-3P are provided below.

[22] G22: x^{∗}=(236.430975504001054, 135.82847151732463, 204.818152544824585, 6446.54654059436416, 3007540.83940215595, 4074188.65771341929, 32918270.5028952882, 130.075408394314167, 170.817294970528621, 299.924591605478554, 399.258113423595205, 330.817294971142758, 184.51831230897065, 248.64670239647424, 127.658546694545862, 269.182627528746707, 160.000016724090955, 5.29788288102680571, 5.13529735903945728, 5.59531526444068827, 5.43444479314453499, 5.07517453535834395); h^{∗} = 236.430975504001e+000.

With SC-PSO-3P an improved solution:

SC G22: x^{∗}=(0, 0, 0, 0, 0, 0, 0, 100, 100„100.010, 100, 100, 101.9292514, 0, 13.653013870, 100.00034450, 0.010, -4.700, -2.7379525440, 4.6039351080, -0.7642880070, -2.7215211340); h^{∗}=0.0000E00.

In Table 8 a comparison of the values of the function and constraints is presented for the function G22.

Regarding the engineering problems eng01–eng03, the results obtained by various methods are shown in Tables 9, 11 and 12. We can see that for the pressure vessel design problem, in Table 10 a better feasible solution was found with respect to those reported previously. For the other two problems, the results obtained by SC-PSO-3P are very competitive over other proposed algorithms.

In Table 13, results of the bootstrap test are shown.

132 s.d.l.cobos–m.gutiérrez–e.rincón–p.lara–r.mora–a.ponsich

[22]  SC-PSO-3P
h^{∗}=2.36E+02 h^{∗}=0.00E+00

f_{1}(x) ≤ 0

-2.21E-07  0.00E+00

t_{i}(x) = 0

-2.93E-05  0.00E+00
9.20E-05  0.00E+00
-3.42E+07  -6.00E+07
-6.00E+07  -4.30E+07
-7.99E+07  -5.40E+07
-6.62E+07  -7.60E+07
5.61E-05  0.00E+00
1.67E-05  0.00E+00
5.41E-05  0.00E+00
4.07E+06  3.44E-04
1.67E-05  1.00E-02
5.74E-05  9.48E-02
5.74E-05  8.04E+00
7.48E-05  1.24E-03
7.47E-05  6.47E+00
-6.16E-07  -1.88E+00
6.12E-06  -2.84E-05
-8.78E-05  3.00E+02
9.59E-05  -3.14E-02

∑_{i} abs(t_{i})

2.44E+08  2.33E+08

Table 8: Table comparing G22 [22] vs. SC.

FFE  Best  Average  Worst  Std. Dev.
SC-PSO-3P 50000 5898.549386 6278.578324 7335.299154 2.551714E+02
CPSO  200000 6061.077700 6147.133200 6363.804100 8.645450E+01
HPSO  81000 6059.725500 6099.932300 6288.677000 8.620220E+01
AATM  30000 6059.725500 6061.987800 6090.802200 4.700000E+00
T-Cell  81000 6390.554000 6737.065100 7694.066800 3.570000E+02
CVI-PSO  25000 6059.714300 6292.123100 6820.410100 2.884550E+02

Table 9: Results of different algorithms for pressure vessel design problem.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

sc: a novel fuzzy criterion for solving engineering... 133

SC-PSO-3P  Best Known
h(x^{∗})  5898.549386  6059.7143
x_{1}^{∗}  0.778643603  0.8125
x_{2}^{∗}  0.38712201  0.4575
x_{3}^{∗}  40.33557909  42.0984456
x_{4}^{∗}  200  176.6365958
f_{1} ≤ 0 -1.66926028E-04 -1.05992992E-12
f_{2} ≤ 0 -2.32058574E-03 -3.58808290E-02
f_{3} ≤ 0 -1.13500617E+03 6.23567029E-06
f_{4} ≤ 0 -4.00000000E+01 -6.33634042E+01

Table 10: Results comparative pressure vessel problem best known vs. SC-PSO-3P.

FFE  Best  Average  Worst  Std. Dev.
SC-PSO-3P 50000 1.724958 1.738032 1.843013 2.555789E-02
GAPF NA 2.433116 NA NA NA
CGA  900000 1.748309 1.771973 1.785835 1.122000E-02
CPSO  200000 1.728024 1.748831 1.782145 1.292600E-02
HPSO  81000 1.724852 1.749040 1.814295 4.004900E-02
CVI-PSO  25000 1.724852 1.725124 1.727665 6.120000E-04

Table 11: Results of different algorithms for the welded beam design problem.

FFE  Best  Average  Worst  Std. Dev.
SC-PSO-3P 50000 0.0126687 0.0128715 0.0137703 2.538650E-04
CPSO  200000 0.0126747 0.0127300 0.0129240 5.190000E-05
HPSO  81000 0.0126652 0.0127072 0.0127191 1.580000E-05
AATM  25000 0.0126682 0.0127080 0.0128613 4.500000E-05
T-Cell  36000 0.0126650 0.0127320 0.0133090 9.400000E-05
CVI-PSO  25000 0.0126655 0.0127310 0.0128426 5.500000E-05

Table 12: Results of different algorithms for the tension-compression string problem.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

134 s.d.l.cobos–m.gutiérrez–e.rincón–p.lara–r.mora–a.ponsich

Problem Low limit Upper limit
1  -13.69776169 -14.62498168
2  -0.524584754 -0.599460572
3  -1.000298967 -1.00048471
4  -30665.24483 -30665.34067
5  5174.327258 5148.021136
6  -6959.868837 -6960.770545
7  28.99774975 27.28710176
8  -0.095824409 -0.095824666
9  681.3461332 681.0367763
10  7618.087189 7321.516701
11 0.750001544 0.7499987
12  -0.999994195 -0.999996666
13  0.056508561 0.055016703
14  -47.83817353 -48.08651249
15  962.6392106 961.9545713
16  -1.76784791 -1.903749563
17  8955.82206 8848.421709
18  -0.684804785 -0.777015372
19  38.29876149 34.66589102
21  794.7558171 661.3324645
23  -399.9122071 -400.0644544
24  -7  -7

Table 13: Results of bootstrap test with an α = 0.05.

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

6 Conclusions and further research

In this work, a novel fuzzy criterion called System of Convergence SC is proposed, which was implemented using a PSO based algorithm with three phases:

1) Stabilization, 2) Generation and breadth-first search, and 3) Generation and depth-first. This algorithm SC-PSO-3P was tested in a set of benchmark instances for constrained optimization and engineering problems.

As an important remark, a deeper study concerning the boundary conditions of the parameters a and α of SC, as well as their implementation with an algorithm different from PSO-3P must be performed.

The SC-PSO-3P generates best solutions by problems G20 and G22 and G24 that results reported in the literature. Based on information of Table 14, we can say that our method has a behaviour similar that DECV and T-CELL.

SC-PSO-3P Memetic APF-GA MDE CVI-PSO DECV T-CELL
SC-PSO-3P  false  true  true  true  true  false  false
Memetic  true  false  false  false  false  true  true
APF-GA  true  false  false  false  false  true  true
MDE  true  false  false  false  true  true  true
CVI-PSO  true  false  false  true  false  false  false
DECV  false  true  true  true  false  false  false
T-CELL  false  true  true  true  false  false  false

Table 14: Results of Wilcoxon Rank test for average normalized values.

It is also necessary to conduct a further study about the topology generated by SC. It was observed that in all the cases studied, SC can reach the global optimum or being very close to it with shorter iteration times and less iterations.

A Proof of theorem 1

Proof. G is injective by construction.

G is surjective: Let g ∈ ℝ, to prove that there is B̃_{g} ⊂ C (B̃) such that:

G(B̃) = g, ∀B̃ ∈ B̃_{g}

it will be proven by construction.

136 s.d.l.cobos–m.gutiérrez–e.rincón–p.lara–r.mora–a.ponsich

Let B̃_{g} = {B̃ = (a, f_{1}, f_{2}, f_{3}, α; w)} which satisfy the conditions of theorem 1, and let:

α = \frac{2F_{2}}{F_{3}} ,
a = \frac{4F_{2}^{2} + A^{−1}F_{3}g}{6F_{1}F_{3}} .

By definition it is known that:

G(B̃) = A [α^{2}F_{3} − 4αF_{2} + 6aF_{1}]
= A [(\frac{2F_{2}}{F_{3}} )^{2} F_{3} − 4\frac{2F_{2}}{F_{3}} F_{2} + 6\frac{4F_{2}^{2} + A^{−1}F_{3}g}{6F_{1}F_{3}} F_{1}]
= A [\frac{4F_{2}^{2}}{F_{3}} − \frac{8F_{2}^{2}}{F_{3}} + \frac{4F_{2}^{2}}{F_{3}} + A^{−1}g] = g.

Therefore, function G is bijective by classes.

B Proof of proposition 1

Proof. Let B̃ ∈ C (B̃) such that B̃ = (a, f_{1}, f_{2}, f_{3}, α; w), and s ∈ ℝ then:

B̃s = (as, f_{1}, f_{2}, f_{3}, αs; w).

Using the definition of G we have that:

G(B̃s) = A [(αs)^{2}F_{3} − 4(αs)F_{2} + 6(as)F_{1}]
A [((α(s^{∗} + ϵ))^{2}F_{3} − 4α(s^{∗} + ϵ)F_{2} + 6a(s^{∗} + ϵ)F_{1}]
A [(αs^{∗})^{2}F_{3} + 2α^{2}s^{∗}ϵF_{3} + (αϵ)^{2}F_{3} − 4αs^{∗}F_{2} − 4αϵF_{2}
+6as^{∗}F_{1} + 6aϵF_{1}]
= A [(αs^{∗})^{2}F_{3} − 4αs^{∗}F_{2} + 6as^{∗}F_{1}]
+A [2α^{2}s^{∗}ϵF_{3} + (αϵ)^{2}F_{3} − 4αϵF_{2} + 6aϵF_{1}] .

Therefore:

lim_{ϵ→0} G(B̃s) = lim_{ϵ→0} A [(αs^{∗})^{2}F_{3} − 4αs^{∗}F_{2} + 6as^{∗}F_{1}]

+ lim_{ϵ→0} A [2α^{2}s^{∗}ϵF_{3} + (αϵ)^{2}F_{3} − 4αϵF_{2} + 6aϵF_{1}]

= G(B̃s^{∗}).

Rev.Mate.Teor.Aplic. (ISSN print: 1409-2433; online: 2215-3373) Vol. 23(1): 111–142, January 2016

C Proof of proposition 2

Proof. It will be demonstrated by mathematical induction. First it will be proven for k = 2:

G(B̃_{1}s_{1} + B̃_{2}s_{2})

= A [(s_{1}α_{1} + s_{2}α_{2})^{2}F_{3} − 4(s_{1}α_{1} + s_{2}α_{2})F_{2} + 6(s_{1}a_{1} + s_{2}a_{2})F_{1}]
= A [((s_{1}^{∗}α_{1})^{2} + 2s_{1}^{∗}α_{1}s_{2}^{∗}α_{2} + (s_{2}^{∗}α_{2})^{2})F_{3}
− 4(s_{1}^{∗}α_{1} + s_{2}^{∗}α_{2})F_{2} + 6(s_{1}^{∗}a_{1} + s_{2}^{∗}a_{2})]
= A [(s_{1}^{∗}α_{1} + s_{2}^{∗}α_{2})^{2}F_{3} − 4(s_{1}^{∗}α_{1} + s_{2}^{∗}α_{2})F_{2}
+6(s_{1}^{∗}a_{1} + s_{2}^{∗}a_{2})F_{1}] + A[H(ϵ_{1}, ϵ_{2})]

where:

H(ϵ_{1}, ϵ_{2})
= (2ϵ_{1}s_{1}^{∗}α_{1} + (ϵ_{1}α_{1})^{2} + 2α_{1}α_{2}(s_{1}^{∗}ϵ_{2} + s_{2}^{∗}ϵ_{1} + ϵ_{1}ϵ_{2}) + 2ϵ_{2}s_{2}^{∗}α_{2}
+(ϵ_{2}α_{2})^{2})F_{3} − 4(ϵ_{1}α_{1} + ϵ_{2}α_{2})F_{2} + 6(ϵ_{1}a_{1} + ϵ_{2}a_{2}).

Therefore:

lim_{{ϵ_{1},ϵ_{2}}→0} G(B̃_{1}s_{1} + B̃_{2}s_{2})

= lim_{{ϵ_{1},ϵ_{2}}→0} A [(s_{1}^{∗}α_{1} + s_{2}^{∗}α_{2})^{2}F_{3} − 4(s_{1}^{∗}α_{1} + s_{2}^{∗}α_{2})F_{2}

+6(s_{1}^{∗}a_{1} + s_{2}^{∗}a_{2})F_{1}] + lim_{{ϵ_{1},ϵ_{2}}→0} A[H(ϵ_{1}, ϵ_{2})]

= G(B̃_{1}s_{1}^{∗} + B̃_{2}s_{2}^{∗}).

Using mathematical induction, it is assumed true for k = n − 1 and will be proven for k = n.

Note that ∀i = 1, 2, …, k ≤ n − 1 it is true that:

G (∑_{i=1}^{k}(B̃_{i}s_{i})^{2}) = G (∑_{i=1}^{k}(B̃_{i}s_{i}^{∗})^{2}) + A [H({ϵ_{i}}_{i=1}^{k})]

such that

lim_{{ϵ_{i}}→0} H ({ϵ_{i}}_{i=1}^{k}) = 0.

For k = n, we have:

let B̃ = ∑_{i=1}^{n} B̃_{i}s_{i}, such that B̃_{i} ∈ C (B̃), i = 1, 2, …, n

It can be seen that:

1. B̃ = ∑_{i=1}^{n} B̃_{i}s_{i} = ∑_{i=1}^{n−1} B̃_{i}s_{i} + B̃_{n}s_{n} = B̃n−1^{′} + B̃_{n}s_{n},

2. α = ∑_{i=1}^{n−1}(α_{i}s_{i}) + α_{n}s_{n} = (αn−1^{′}) + α_{n}s_{n},

3. a = ∑_{i=1}^{n−1}(a_{i}s_{i}) + a_{n}s_{n} = (an−1^{′}) + a_{n}s_{n}.

Therefore:

G (∑_{i=1}^{n} B̃_{i}s_{i}) =

A [(αn−1^{′} + s_{n}α_{n})^{2}F_{3} − 4(αn−1^{′} + s_{n}α_{n})F_{2}+

+ 6(an−1^{′} + s_{n}a_{n})F_{1} + H({ϵ_{i}}_{i=1}^{n−1})]

= A [(αn−1^{′} + (s_{n}^{∗} + ϵ_{n})α_{n})^{2}F_{3} − 4(αn−1^{′} + (s_{n}^{∗} + ϵ_{n})α_{n})F_{2}+

+ 6(an−1^{′} + s_{n}a_{n})F_{1} + H({ϵ_{i}}_{i=1}^{n−1})]

= A [((αn−1^{′})^{2} + (s_{n}^{∗}α_{n})^{2} + 2αn−1^{′}s_{n}^{∗}α_{n}) F_{3} − 4 (αn−1^{′} + s_{n}^{∗}α_{n}) F_{2}+

+ 6 ((an−1^{′} + s_{n}^{∗}a_{n})F_{1} + H({ϵ_{i}}_{i=1}^{n−1}) + H(ϵ_{n})]

= A [(∑_{i=1}^{n}(α_{i}s_{i}^{∗})^{2}) F 3 − 4 (∑_{i=1}^{n}(α_{i}s_{i}^{∗})) F_{2}+

+ (∑_{i=1}^{n} a_{i}s_{i}^{∗}) F_{1} + H({ϵ_{i}}_{i=1}^{n−1}) + H(ϵ_{n})]

where:

H(ϵ_{n}) = ϵ_{n}(2s_{n}^{∗}α_{n}^{2} + 2αn−1^{′}α_{n} + α_{n}^{2}ϵ_{n} + α_{n}F_{2} + a_{n}F_{1}).

Hence:

lim_{{ϵ_{i}}_{i=1}^{n}→0} G (∑_{i=1}^{n} B̃_{i}s_{i})

= lim_{{ϵ_{i}}_{i=1}^{n}→0} A [(∑_{i=1}^{n}(α_{i}s_{i}^{∗})^{2}) F 3 − 4 (∑_{i=1}^{n}(α_{i}s_{i}^{∗})) F_{2}+

+ (∑_{i=1}^{n} a_{i}s_{i}^{∗}) F_{1} + H({ϵ_{i}}_{i=1}^{n−1}) + H(ϵ_{n})]

= G (∑_{i=1}^{n} B̃_{i}s_{i}^{∗}) .

D Proof of proposition 3

Proof. The proof will be done by contradiction. Suppose:

1. y^{∗} = (y_{1}^{∗}, y_{2}^{∗}, …, y_{k}^{∗}) ∈ PF

2. y^{∗} ∉ SC-F

Then we have:

1. y^{∗} ∈ Image(h), and y^{∗} is non dominated.

2. ∃ ϵ > 0, such that if ||y^{∗} − y|| ≤ ϵ ⇒ y ∈ Image(h).

consider a point: y_{0} = (y_{1}^{∗} − \frac{ϵ}{\sqrt{k}}, y_{2}^{∗} − \frac{ϵ}{\sqrt{k}}, …, y_{k}^{∗} − \frac{ϵ}{\sqrt{k}}). Note that:

||y^{∗} − y_{0}|| = \sqrt{(\frac{ϵ}{\sqrt{k}})^{2} + … + (\frac{ϵ}{\sqrt{k}})^{2}} = ϵ,

for that reason y_{0} ∈ Image(h) and y_{0} dominates to y^{∗}, which is a contradiction, therefore: PF ⊆ SC-F.

Acknowledgements

The authors wish to thank the three anonymous referees since their comments helped to considerably improve the present work. S.G. de C. would also like to thank D.Sto. and to P.V.Gpe. for their inspiration, and to Ma, Ser, Mon, Chema, and to his Flaquita for all their support

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