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Improved Jaya Algorithm with Inertia Weight Factor

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Women in Soft Computing

Part of the book series: Women in Engineering and Science ((WES))

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Abstract

The newly established Jaya algorithm has a simple structure and just requires a small set of control parameters to be optimized. Though it has scope of improvement in terms of outcome accuracy with a fast convergence speed, intensification (exploitation), and diversification (exploration). This work suggests an enhanced variant of the Jaya algorithm, named “W-Jaya,” by setting up a time-varying inertia weight factor with it. This method offers high-quality solutions in a reasonable timeframe. Further, W-Jaya tested with 10 constrained and 20 unconstrained test problems taken from the literature. Finally, the comparison and results show that the proposed W-Jaya has obtained desire goals successfully in an effective manner.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Chandigarh University, Ludhiana, Punjab, India

    Sonal Deshwal & Sandeep Kumar Mogha

  2. Department of Mathematics, REC Bijnor (AKTU Lucknow), Bijnor, Uttar Pradesh, India

    Pravesh Kumar

Authors
  1. Sonal Deshwal
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  2. Pravesh Kumar
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  3. Sandeep Kumar Mogha
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Editor information

Editors and Affiliations

  1. Mathematics, School of Basic Sciences, Galgotias University, Noida, Uttar Pradesh, India

    Vanita Garg

  2. Department of Mathematics, Joint Faculty MF School of Data Science and Artificial Intelligence, Indian Institute of Technology Roorkee, Roorkee, India

    Kusum Deep

  3. Aurel Vlaicu University of Arad, Arad, Arad, Romania

    Valentina E. Balas

Appendices

Appendices

1.1 Appendix 1: List of Unconstrained Test Problems (Tables 5, 6, and 7)

Table 5 Unimodal benchmark problems
Full size table
Table 6 Multimodal benchmark problems
Full size table
Table 7 Fixed multidimensional benchmark problems
Full size table

1.2 Appendix 2: List of Constrained Test Problems (Table 8)

Table 8 Constrained benchmark problems
Full size table

Problem-1: G01

$$ \operatorname{Min}\;f(x)=5\sum \limits_{i=1}^4{x}_i-5\sum \limits_{i=1}^4{x}_i^2-\sum \limits_{i=5}^{13}{x}_i $$

  • Subject to:

    $$ {\displaystyle \begin{array}{l}{g}_1(x)=2{x}_1+2{x}_2+{x}_{10}+{x}_{11}-10\le 0\\ {}{g}_2(x)=2{x}_1+2{x}_3+{x}_{10}+{x}_{12}-10\le 0\\ {}{g}_3(x)=2{x}_2+2{x}_3+{x}_{11}+{x}_{12}-10\le 0\\ {}{g}_4(x)=-8{x}_1+{x}_{10}\le 0\\ {}{g}_5(x)=-8{x}_2+{x}_{11}\le 0\\ {}{g}_6(x)=-8{x}_3+{x}_{12}\le 0\\ {}{g}_7(x)=-2{x}_4-{x}_5+{x}_{10}\le 0\\ {}{g}_8(x)=-2{x}_6-{x}_7+{x}_{11}\le 0\\ {}{g}_9(x)=-2{x}_8-{x}_9+{x}_{12}\le 0\end{array}} $$

  • Properties:

    $$ {\displaystyle \begin{array}{l}0\le {x}_i\le 1\kern0.24em \left(i=1,2,..\dots, 9\right),0\le {x}_i\le 100\kern0.24em \left(i=10,11,12\right),\kern0.36em 0\le {x}_{13}\le 1\\ {}{x}^{\ast }=\left(1,1,1,1,1,1,1,1,1,3,3,3,1\right)\\ {}\textrm{Constraints}\;{g}_1,{g}_2,{g}_3,{g}_7,{g}_8,{g}_9\kern0.24em \textrm{are}\;\textrm{active}.\end{array}} $$

Problem-2: G02

$$ \operatorname{Max}\;f(x)=\left|\frac{\sum \limits_{i=1}^n{\cos}^4\left({x}_i\right)-2\prod \limits_{i=1}^n{\cos}^2\left({x}_i\right)}{\sum \limits_{i=1}^n{ix}_i^2}\right| $$

  • Subject to:

    $$ {\displaystyle \begin{array}{c}{g}_1(x)=0.75-\prod \limits_{i=1}^D{x}_i\le 0\\ {}{g}_2(x)=\sum \limits_{i=1}^D{x}_i-7.5D\le 0\end{array}} $$

  • Properties:

    $$ 0\le {x}_i\le 10\left(i=1,2,.\dots .,D\right),D=20 $$

The known best value is f(x*) = 0.803619

Constraint g1 is an active constraint.

Problem-3: G03

  • $$ \operatorname{Min}\;f(x)=-{\left(\sqrt{n}\right)}^n\prod \limits_{i=1}^n{x}_i $$

    Subject to:

    $$ {h}_1(x)=\sum \limits_{i=1}^n{x}_i^2-1=0 $$

  • Properties:

    $$ {\displaystyle \begin{array}{l}0\le {x}_i\le 10\left(i=1,2,..\dots, n\right)\\ {}{x}^{\ast }=\left(\frac{1}{\sqrt{n}}\right),n=10\end{array}} $$

Problem-4: G04

  • $$ \operatorname{Min}\;f(x)=5.3578547{x}_3^2+0.8356891{x}_1{x}_5+37.293239{x}_1-40792.141 $$

    Subject to:

    $$ {\displaystyle \begin{array}{c}{g}_1\left(\textrm{x}\right)=85.334407+0.0056858{x}_2{x}_5+0.0006262{x}_1{x}_4-0.0022053{x}_3{x}_5\\ {}{g}_2\left(\textrm{x}\right)=80.51249+0.0071317{x}_2{x}_5+0.0029955{x}_1{x}_2-0.0021813{x}_3^2\\ {}{g}_3\left(\textrm{x}\right)=9.300961+0.0047026{x}_3{x}_5+0.0012547{x}_1{x}_3+0.0019085{x}_3{x}_4\\ {}0\le {g}_1(x)\le 92\\ {}90\le {g}_2(x)\le 110\\ {}20\le {g}_3(x)\le 25\end{array}} $$

  • Properties:

    $$ {\displaystyle \begin{array}{l}78\le {x}_1\le 102,33\le {x}_2\le 45,27\le {x}_i\le 45\left(i=3,4,5\right)\\ {}{x}^{\ast }=\left(78,33,29.995256025682,45,36.775812905788\right)\end{array}} $$

Problem-5: G05

  • $$ \operatorname{Min}\;f(x)=3{x}_1+0.000001{x}_1^3+2{x}_2+\left(\frac{0.000002}{3}\right){x}_2^3 $$

    Subject to:

    $$ {\displaystyle \begin{array}{l}{g}_1(x)=-{x}_4+{x}_3-0.55\le 0\\ {}{g}_2(x)=-{x}_3+{x}_4-0.55\le 0\\ {}{h}_3(x)=1000\sin \left(-{x}_3-0.25\right)+1000\sin \left(-{x}_4-0.25\right)+894.8-{x}_1=0\\ {}{h}_4(x)=1000\sin \left({x}_3-0.25\right)+1000\sin \left({x}_3-{x}_4-0.25\right)+894.8-{x}_2=0\\ {}{h}_5(x)=1000\sin \left({x}_4-0.25\right)+1000\sin \left({x}_4-{x}_3-0.25\right)+1294.8=0\end{array}} $$

  • Properties:

    $$ {\displaystyle \begin{array}{l}0\le {x}_i\le 1200\left(i=1,2\right),-0.55\le {x}_2\le 0.55\left(i=3,4\right).\\ {}{x}^{\ast }=\left(679.9463,1026.067,0.1188764,-0.3962336\right).\end{array}} $$

Problem-6: G06

  • $$ \mathit{\operatorname{Min}}\;f(x)={\left({x}_1-10\right)}^3+{\left({x}_2-20\right)}^3 $$

    Subject to:

    $$ {\displaystyle \begin{array}{c}{g}_1(x)=-{\left({x}_1-5\right)}^2-{\left({x}_2-5\right)}^2+100\le 0\\ {}{g}_2(x)={\left({x}_1-6\right)}^2+{\left({x}_2-5\right)}^2-82.81\le 0\end{array}} $$

  • Properties:

    $$ {\displaystyle \begin{array}{l}13\le {x}_i\le 100,0\le {x}_2\le 100\\ {}{x}^{\ast }=\left(14.095,0.84296\right).\end{array}} $$

Problem-7: G07

$$ \operatorname{Min}\;f(x)={x}_1^2+{x}_2^2{+}{x}_1{x}_2{-}14{x}_1{-}16{x}_2{+}{\left({x}_3-10\right)}^2{+}4{\left({x}_4-5\right)}^2{+}{\left({x}_5-5\right)}^2\\{+}2{\left({x}_6-1\right)}^2{+}5{x}_7^2{+}7{\left({x}_8-11\right)}^2{+}2{\left({x}_9-10\right)}^2{+}{\left({x}_{10}-7\right)}^2+45$$

  • Subject to:

$$ {\displaystyle \begin{array}{l}{g}_1(x)=-105+4{x}_1+5{x}_2-3{x}_7+9{x}_8\le 0\\ {}{g}_2(x)=10{x}_1-8{x}_2-17{x}_7+2{x}_8\le 0\\ {}{g}_3(x)=-8{x}_1+2{x}_2+5{x}_9-2{x}_{10}-12\le 0\\ {}{g}_4(x)=3{\left({x}_1-2\right)}^2+4{\left({x}_2-3\right)}^2+2{x}_3^2-7{x}_4-120\le 0\\ {}{g}_5(x)=5{x}_1^2+8{x}_2+{\left({x}_3-6\right)}^2-2{x}_4-40\le 0\\ {}{g}_6(x)={x}_1^2+2{\left({x}_2-2\right)}^2-2{x}_1{x}_2+14{x}_5-6{x}_6\le 0\\ {}{g}_7(x)=0.5{\left({x}_1-8\right)}^2+2{\left({x}_2-4\right)}^2+3{x}_5^2-{x}_6-30\le 0\\ {}{g}_8(x)=-3{x}_1+6{x}_2+12{\left({x}_9-8\right)}^2-7{x}_{10}\le 0\end{array}} $$

  • Properties:

$$ {\displaystyle \begin{array}{l}-10\le {x}_i\le 10\;\left(i=1,2,\dots \dots, 10\right)\\ {}{x}^{\ast }=\Big(2.171996,2.363683,8.773926,5.095984,0.99065481,1.430574,\\ {}1.321644,9.828726,8.280092,8.375927\Big)\\ {}\textrm{Constraints}\;{g}_1,{g}_2,{g}_3,{g}_4,{g}_5\;\textrm{and}\;{g}_6\;\textrm{are}\;\textrm{active}.\end{array}} $$

Problem-8: G08

$$ \operatorname{Max}\;f(x)=\frac{\sin^3\left(2\pi {x}_1\right)\sin \left(2\pi {x}_2\right)}{x_1^3\left({x}_1+{x}_2\right)} $$

  • Subject to:

$$ {\displaystyle \begin{array}{c}{g}_1(x)={x}_1^2-{x}_2+1\le 0\\ {}{g}_2(x)=1-{x}_1+{\left({x}_2-4\right)}^2\le 0\end{array}} $$

  • Properties:

$$ {\displaystyle \begin{array}{l}0\le {x}_i\le 10\left(i=1,2\right).\\ {}{x}^{\ast }=\left(1.2279713,4.2453733\right).\end{array}} $$

Problem-9: G09

$$ \begin{aligned} \operatorname{Min}\;f(x)&={\left({x}_1-10\right)}^2+5{\left({x}_2-12\right)}^2+{x}_3^4+3{\left({x}_4-11\right)}^2+0{x}_5^6+7{x}_6^2\\ & \quad +{x}_7^4-4{x}_6{x}_7-10{x}_6-8{x}_7 \end{aligned} $$

  • Subject to:

    $$ {\displaystyle \begin{array}{c}{g}_1(x)=-127+2{x}_1^2+3{x}_2^4+{x}_3+4{x}_4^2+5{x}_5\le 0\\ {}{g}_2(x)=-282+7{x}_1++3{x}_2+10{x}_3^2+{x}_4-{x}_5\le 0\\ {}{g}_3(x)=-196+23{x}_1+{x}_2^2+6{x}_6^2-8{x}_7\le 0\\ {}{g}_4(x)=4{x}_1^2+{x}_2^2-3{x}_1{x}_2+2{x}_3^2+5{x}_6-11{x}_7\le 0\end{array}} $$

  • Properties:

    $$ {\displaystyle \begin{array}{l}-10\le {x}_i\le 10\left(i=1,2,.\dots \dots, 7\right)\\ {}{x}^{\ast }= \left(\begin{array}{l} 2.330499,1.951372,-0.4775414,4.365726,\\ -0.624487,1.038131,1.5942270\end{array}\right).\end{array}} $$

Problem-10: G10

$$ \operatorname{Min}\;f(x)={x}_1+{x}_2+{x}_3 $$

  • Subject to:

    $$ {\displaystyle \begin{array}{c}{g}_1(x)=-1+0.0025\left({x}_4+{x}_6\right)\le 0\\ {}{g}_2(x)=-1=0.0025\left({x}_5+{x}_7-{x}_4\right)\le 0\\ {}{g}_3(x)=-1+0.01\left({x}_8-{x}_5\right)\le 0\\ {}{g}_4(x)=-{x}_1{x}_6+833.33252{x}_4+100{x}_1-83333.333\le 0\\ {}{g}_5(x)=-{x}_2{x}_7+1250{x}_5+{x}_2{x}_4-1250{x}_4\le 0\\ {}{g}_6(x)=-{x}_3{x}_8+1250000+{x}_3{x}_5-2500{x}_5\le 0\end{array}} $$

  • Properties:

    $$ {\displaystyle \begin{array}{l}-100\le {x}_1\le 10000,1000\le {x}_i\le 10000\left(i=2,3\right),\\ 10\le {x}_i\le 1000\left(i=4,.\dots, 8\right).\\ {}{x}^{\ast }=\left(\left(\begin{array}{l} 579.19,1360.13,5109.5979,182.0174,295.5985,\\ \quad 217.9799,286.40,395.5979\end{array} \right)\right..\end{array}} \vspace*{-5pt} $$

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Deshwal, S., Kumar, P., Mogha, S.K. (2024). Improved Jaya Algorithm with Inertia Weight Factor. In: Garg, V., Deep, K., Balas, V.E. (eds) Women in Soft Computing. Women in Engineering and Science. Springer, Cham. https://doi.org/10.1007/978-3-031-44706-8_5

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  • DOI: https://doi.org/10.1007/978-3-031-44706-8_5

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