Water cycle algorithm for solving multi-objective optimization problems

Abstract

In this paper, the water cycle algorithm (WCA), a recently developed metaheuristic method is proposed for solving multi-objective optimization problems (MOPs). The fundamental concept of the WCA is inspired by the observation of water cycle process, and movement of rivers and streams to the sea in the real world. Several benchmark functions have been used to evaluate the performance of the WCA optimizer for the MOPs. The obtained optimization results based on the considered test functions and comparisons with other well-known methods illustrate and clarify the robustness and efficiency of the WCA and its exploratory capability for solving the MOPs.

Appendix A: Mathematical formulation of studied MOPs

This Appendix represents the MOPs used in this paper to conduct a qualitative assessment for performance and efficiency of the MOWCA.

• (1) Test problem 1–DEB: Deb’s function is a problem with two design variables. This problem is defined as follows (Deb 2002):

  $$\begin{aligned} \mathrm{DEB}:\min \left\{ {\begin{array}{l} f_1 (X)=x_1 \\ f_2 (X)=g(X)\times h(X) \\ \end{array}} \right. \!\!, \end{aligned}$$

  (22)

  where

  $$\begin{aligned} g(X)=11+x_2^2 -10\cos (2\pi x_2 ), \end{aligned}$$

  (23)
  $$\begin{aligned}&h(X)=\left\{ {{\begin{array}{*{20}c} {1-\sqrt{f_1 /g} } &{} \quad {\mathrm{if}\;\;f_1 (X)\le g(X)\;} \\ 0 &{} {\mathrm{Otherwise}} \\ \end{array} }} \right. \nonumber \\&\mathrm{where}\;\;0\le x_1 \le 1,\;-30\le x_2 \le 30. \end{aligned}$$

  (24)

  The Pareto optimal front for this problem is convex and defined as $x_{1} \epsilon [0, 1], x_{2}$ = 0.

• (2) Test problem 2–FON: Fonseca and Fleming’s function (FON) is an eight-design variable function suggested as follows (Fonseca and Fleming 1993):

  $$\begin{aligned}&\mathrm{FON}:\min \left\{ {\begin{array}{l} f_1 (X)=1-\exp \left( {-\sum \limits _{i=1}^8 {\left( {x_i -\frac{1}{\sqrt{8} }}\right) ^2} }\right) \\ f_2 (X)=1-\exp \left( {-\sum \limits _{i=1}^8 {\left( {x_i +\frac{1}{\sqrt{8} }}\right) ^2} }\right) \\ \end{array}} \right. \nonumber \\&\quad \mathrm{where}\;-2<x_i <2,\;i=1,2,3,\ldots ,8, \end{aligned}$$

  (25)

  The optimal Pareto front for this bi-objective problem is $x_i^*=[-1/\sqrt{8} ,1/\sqrt{8} ]$   for $i = 1,2,3,{\ldots },8$.

• (3) Test problem 3–POL: This function was first introduced by Poloni (1997) which has been widely analyzed in the literature (Deb et al. 2002a; Kaveh and Laknejadi 2011). The mathematical formulation proposed by Poloni (1997) is given as follows:

  $$\begin{aligned}&\mathrm{POL}:\min \nonumber \\&\quad \times \left\{ {\begin{array}{l} f_1 (X)=[1+(A_1 -B_1 )^2+(A_2 -B_2 )^2] \\ f_2 (X)=[(x_1 +3)^2+(x_2 +1)^2] \\ A_1 =0.5\sin 1-2\cos 1+\sin 2-1.5\cos 2 \\ A_2 =1.5\sin 1-\cos 1+2\sin 2-0.5\cos 2 \\ B_1 =0.5\sin (x_1 )-2\cos (x_1 )+\sin (x_2 )-1.5\cos (x_2 ) \\ B_2 =1.5\sin (x_1 )-\cos (x_1 )+2\sin (x_2 )-0.5\cos (x_2 ) \\ \end{array}} \right. \nonumber \\&\quad \mathrm{where}\;-\pi <x_1 ,x_2 <\pi . \end{aligned}$$

  (26)

  The Pareto optimal front for the POL function is non-convex and discontinuous.

• (4) Test problem 4–KUR: This problem, presented by Kursawe (1991), has three design variables having non-convex and discontinuous Pareto optimal front. The KUR’s mathematical formulation is as follows (Kursawe 1991):

  $$\begin{aligned}&\mathrm{KUR}:\min \left\{ {\begin{array}{l} f_1 (X)=\sum \limits _{i=1}^{n-1} {\left( {-10\exp \left( -0.2\sqrt{x_i^2 +x_{i+1}^2 } \right) }\right) } \\ f_2 (X)=\sum \limits _{i=1}^n {\left( {\left| {x_i } \right| ^{0.8}+0.5\sin x_i^3 }\right) } \\ \end{array}} \right. \nonumber \\&\quad \mathrm{where}\;-5<x_i <5,\;i=1,2,3.\nonumber \\ \end{aligned}$$

  (27)

• (5) Test problem 5–VNT: This problem is a three-dimensional problem in objective space suggested by Viennet et al. (1995). This problem has previously been investigated by many researchers (Freschi and Repetto 2006; Gao and Wang 2010) and is formulated as follows (Viennet et al. 1995):

  $$\begin{aligned}&\mathrm{VNT}=\min \nonumber \\&\quad \times \left\{ {\begin{array}{lll} {f(X)=0.5(x_1^2 +x_2^2 )+\sin (x_1^2 +x_2^2 )} \\ {f(X)=(3x_1 -2x_2 +4)^2/8+(x_1 -x_2 +1)^2/27\!+\!15} \\ {f(X)=(x_1^2 +x_2^2 +1)^{-1}-1.1\exp (-x_1^2 -x_2^2 )} \\ \end{array} }\right. \nonumber \\&\quad \mathrm{where}\;-3\le x_1 ,x_2 \le 3 . \end{aligned}$$

  (28)

  It is worth mentioning that the discontinuous Pareto optimal set and having several local Pareto fronts are considered to be challenging features of the VNT problem (Gao and Wang 2010).

• (6) Test problem 6–ZDT1: The ZDT1 function, was suggested by Zitzler et al. (2000), and has been extensively investigated (Deb et al. 2002a; Kaveh and Laknejadi 2011). This problem is described as follows:

  $$\begin{aligned}&\mathrm{ZDT1}:\min \left\{ {\begin{array}{l} f_1 (X)=x_1 \\ f_2 (X)=g(X)[1-\sqrt{x_1 /g(X)} \\ g(X)=1+9\left( {\sum \limits _{i=2}^n {x_i }}\right) /(n-1) \\ \end{array}} \right. \nonumber \\&\quad \mathrm{where}\;0<x_i <1,\;i=1,2,3,\ldots ,30. \end{aligned}$$

  (29)

  The ZDT1 problem has 30 design variables and its Pareto optimal front is convex and defined as $x_{1}\epsilon [0, 1], x_{i}$ = 0, for $i = 2,{\ldots }, 30$.

• (7) Test problem 7–ZDT3: The ZDT3 problem, suggested by Zitzler et al. (2000), has 30 design variables with a non-convex and discontinuous Pareto optimal front. The mathematical formulation of the ZDT3 problem is as follows (Zitzler et al. 2000):

  $$\begin{aligned}&\mathrm{ZDT3}:\min \nonumber \\&\quad \left\{ {\begin{array}{l} f_1 (X)=x_1 \\ f_2 (X)=g(X)[1-\sqrt{x_1 /g(X)} -\frac{x_1 }{g(X)}\sin (10\pi x_1 )] \\ g(X)=1+9\left( {\sum \limits _{i=2}^n {x_i } }\right) /(n-1) \\ \end{array}} \right. \nonumber \\&\;\mathrm{where}\;0<x_i <1,\;i=1,2,3,\ldots ,30. \end{aligned}$$

  (30)

  The Pareto optimal front is defined as $x_{1} \epsilon [0, 1], x_{i}$ = 0, for $i = 2, 3,4, {\ldots }, 30$.

• (8) Test problem 8–ZDT4: The ZDT4 problem, proposed by Zitzler et al. (2000), has 10 design variables having several local Pareto fronts. This problem is given as follows:

  $$\begin{aligned}&\mathrm{ZDT4}:\min \nonumber \\&\quad \left\{ {\begin{array}{l} f_1 (X)=x_1 \\ f_2 (X)=g(X)[1-\sqrt{x_1 /g(X)} ] \\ g(X)=1+10(n-1)+\sum \limits _{i=2}^n {[x_i^2 -10\cos (4\pi x_i )]} \\ \end{array}}\right. \!,\quad \end{aligned}$$

  (31)

  where $x_{1} \epsilon [0, 1] \mathrm{and} x_{i}$ [$-$5,5], for $i = 2, 3,\dots , 10$. In addition, the optimal Pareto front is defined as $x_{1}\epsilon [0, 1], x_{i} = 0$, for $i = 2, 3, 4,\dots , 10$.

• (9) Test problem 9–ZDT6: The ZDT6, introduced by Zitzler et al. (2000), has 10 design variables with a non-convex Pareto optimal front. The mathematical formulation of this problem is given as follows:

  $$\begin{aligned} \mathrm{ZDT6}:\min \left\{ {\begin{array}{l} f_1 (X)=1-\exp (-4x_1 )\sin ^6(6\pi x_1 ) \\ f_2 (X)=g(X)[1-( {f_1 (X)/g(X)})^2] \\ g(X)=1+9\left[ {( {\sum \limits _{i=2}^n {x_i } })/(n-1)} \right] ^{0.25} \\ \end{array}} \right. \nonumber \\ \;\mathrm{where}\;0<x_i <1,\;i=1,2,3,\ldots ,10.\nonumber \\ \end{aligned}$$

  (32)

  The Pareto optimal front is characterized as $x_{1 }\epsilon [0, 1], x_{i}$ = 0 for $i$ from 2 to 10 for test problem 9.

• (10) Test problem 10–DTLZ series

The DTLZ series, proposed by Deb et al. (2002b), known as scalable problem, are considered in this paper. The mathematical formulation of DTLZ problems are given in Table 9. The Pareto optimal front for all DTLZ problems in this paper is defined as $x_{i} = 0.5$ in which $i \epsilon x_{M}.$ The design space for considered DTLZ problems is between zero and one.

Table 9 Mathematical formulations of the DTLZ series problems

Two and three objective functions ($M =2$ and 3) are considered for the DTLZ series given in Table 9. The DTLZ series are minimization problems and the number of design variables for these problems is calculated as follows:

$$\begin{aligned} n=M+\left| {x_M } \right| -1, \end{aligned}$$

(33)

where $n$ and $M$ are the number of design variables and number of objective functions, respectively. Also, $\left| {x_M } \right|$ is set to 10 for all considered problems in this paper.