Embedded Learning Approaches in the Whale Optimizer to Solve Coverage Combinatorial Problems
Abstract
:1. Introduction
 A new smart binarization scheme selector is proposed.
 The Qlearning technique, proposed in [21], is used to binarize the whale optimization algorithm.
 A selector with a much wider repertoire in its actions is obtained from the literature.
2. Set Covering Problem
3. Reinforcement Learning
3.1. QLearning
3.2. SARSA
4. Whale Optimization Algorithm: Fundamentals
4.1. Identifying the Prey
4.2. Encircling the Prey
4.3. Bubble Netting Technique
Algorithm 1 Whale Optimization Algorithm 
Input: The population $X=\{{X}_{1},{X}_{2},\dots ,{X}_{i}\}$ Output: The updated population ${X}^{\prime}=\{{X}_{1}^{\prime},{X}_{2}^{\prime},\dots ,{X}_{i}^{\prime}\}$ and ${X}^{*}$

5. Whale Optimization Algorithm: QBinary Version
Type  Transfer Function 

S1 [38,39]  $T\left({d}_{w}^{j}\right)=\frac{1}{1+{e}^{2{d}_{w}^{j}}}$ 
S2 [39,40]  $T\left({d}_{w}^{j}\right)=\frac{1}{1+{e}^{{d}_{w}^{j}}}$ 
S3 [38,39]  $T\left({d}_{w}^{j}\right)=\frac{1}{1+{e}^{\frac{{d}_{w}^{j}}{2}}}$ 
S4 [38,39]  $T\left({d}_{w}^{j}\right)=\frac{1}{1+{e}^{\frac{{d}_{w}^{j}}{3}}}$ 
V1 [39,41]  $T\left({d}_{w}^{j}\right)=\left(\right)open=""\; close="">erf\left(\right)open="("\; close=")">\frac{\sqrt{\pi}}{2}{d}_{w}^{j}$ 
V2 [39,41]  $T\left({d}_{w}^{j}\right)=\left(\right)open=""\; close="">tanh\left({d}_{w}^{j}\right)$ 
V3 [38,39]  $T\left({d}_{w}^{j}\right)=\left(\right)open=""\; close="">\frac{{d}_{w}^{j}}{\sqrt{1+{\left({d}_{w}^{j}\right)}^{2}}}$ 
V4 [38,39]  $T\left({d}_{w}^{j}\right)=\left(\right)open=""\; close="">\frac{2}{\pi}arctan\left(\right)open="("\; close=")">\frac{\pi}{2}{d}_{w}^{j}$ 
X1 [42,43]  $T\left({d}_{w}^{j}\right)=\frac{1}{1+{e}^{2{d}_{w}^{j}}}$ 
X2 [42,43]  $T\left({d}_{w}^{j}\right)=\frac{1}{1+{e}^{{d}_{w}^{j}}}$ 
X3 [42,43]  $T\left({d}_{w}^{j}\right)=\frac{1}{1+{e}^{\frac{{d}_{w}^{j}}{2}}}$ 
X4 [42,43]  $T\left({d}_{w}^{j}\right)=\frac{1}{1+{e}^{\frac{{d}_{w}^{j}}{3}}}$ 
Z1 [44,45]  $T\left({d}_{w}^{j}\right)=\sqrt{1{2}^{{d}_{w}^{j}}}$ 
Z2 [44,45]  $T\left({d}_{w}^{j}\right)=\sqrt{1{5}^{{d}_{w}^{j}}}$ 
Z3 [44,45]  $T\left({d}_{w}^{j}\right)=\sqrt{1{8}^{{d}_{w}^{j}}}$ 
Z4 [44,45]  $T\left({d}_{w}^{j}\right)=\sqrt{1{20}^{{d}_{w}^{j}}}$ 
Type  Binarization 

Standard  ${X}_{new}^{j}=\left(\right)open="\{"\; close>\begin{array}{cc}1\hfill & if\phantom{\rule{2.84544pt}{0ex}}rand\le T\left({d}_{w}^{j}\right)\hfill \\ 0\hfill & else.\hfill \end{array}$ 
Complement  ${X}_{new}^{j}=\left(\right)open="\{"\; close>\begin{array}{cc}{X}_{w}^{j}\hfill & if\phantom{\rule{2.84544pt}{0ex}}rand\le T\left({d}_{w}^{j}\right)\hfill \\ 0\hfill & else.\hfill \end{array}$ 
Static Probability  ${X}_{new}^{j}=\left(\right)open="\{"\; close>\begin{array}{cc}0\hfill & if\phantom{\rule{2.84544pt}{0ex}}T\left({d}_{w}^{j}\right)\le \alpha \hfill \\ {X}_{w}^{j}\hfill & if\phantom{\rule{2.84544pt}{0ex}}\alpha T\left({d}_{w}^{j}\right)\le \frac{1}{2}(1+\alpha )\hfill \\ 1\hfill & if\phantom{\rule{2.84544pt}{0ex}}T\left({d}_{w}^{j}\right)\ge \frac{1}{2}(1+\alpha )\hfill \end{array}$ 
Elitist  ${X}_{new}^{j}=\left(\right)open="\{"\; close>\begin{array}{cc}{X}_{Best}^{j}\hfill & if\phantom{\rule{2.84544pt}{0ex}}randT\left({d}_{w}^{j}\right)\hfill \\ 0\hfill & else.\hfill \end{array}$ 
Roulette Elitist  ${X}_{new}^{j}=\left(\right)open="\{"\; close>\begin{array}{cc}P[{X}_{new}^{j}={\zeta}_{j}]=\frac{f\left(\zeta \right)}{{\sum}_{\delta \in {Q}_{g}}f\left(\delta \right)}\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}\mathrm{rand}\phantom{\rule{4.pt}{0ex}}\le T\left({d}_{w}^{j}\right)\hfill \\ P[{X}_{new}^{j}=0]=1\hfill & else.\hfill \end{array}$ 
Algorithm 2 QBinary Whale Optimization Algorithm 
Input: The population $X=\{{X}_{1},{X}_{2},\dots ,{X}_{i}\}$ Output: The updated population ${X}^{\prime}=\{{X}_{1}^{\prime},{X}_{2}^{\prime},\dots ,{X}_{i}^{\prime}\}$ and ${X}^{*}$

5.1. Exploration and Exploitation of Actions
5.2. Feasibility and Repair Heuristic
Algorithm 3 Feasibility Heuristic 
Input: Matrix A of incidence and solution Output: coverage, is number of columns covering a row and feasibility

Algorithm 4 Repair heuristics 
Input: Matrix A of incidence and unfeasible solution Output: feasible solution

5.3. Complexity Analysis
6. Experimentation Results
Analyses of Exploration and Exploitation
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Parameter  Value 

Independent runs  31 
Number of populations  40 
Number of iterations  1000 
parameter a of WOA  decreases linearly from 2 to 0 
parameter b of WOA  1 
parameter $\alpha $ of QLearning and SARSA  0.1 
parameter $\gamma $ of QLearning and SARSA  0.4 
QBWOA  BSWOA  BCL  MIR  40BQWOA  40BSWOA  

Inst.  Opt  Best  Avg  RPD  Opt  Best  Avg  RPD  Opt  Best  Avg  RPD  Opt  Best  Avg  RPD  Opt  Best  Avg  RPD  Opt  Best  Avg  RPD 
41  429  431.0  781.87  0.47  429  430.0  610.0  0.23  429  489.0  607.55  13.99  429  638.0  715.87  48.72  429  435  439.48  1.4  429  430  434.6  0.23 
42  512  522.0  839.19  1.95  512  519.0  872.87  1.37  512  679.0  852.94  32.62  512  1079.0  1181.94  110.74  512  538  546.44  5.08  512  523  535.6  2.15 
43  516  519.0  912.03  0.58  516  520.0  979.13  0.78  516  739.0  884.48  43.22  516  1222.0  1293.61  136.82  516  537  543.78  4.07  516  525  532.7  1.74 
44  494  500.0  789.13  1.21  494  503.0  776.26  1.82  494  596.0  779.06  20.65  494  954.0  1052.03  93.12  494  519  526.33  5.06  494  497  508.0  0.61 
45  512  517.0  913.68  0.98  512  520.0  1032.16  1.56  512  755.0  874.68  47.46  512  1067.0  1204.13  108.4  512  537  541.89  4.88  512  523  528.4  2.15 
46  560  565.0  1007.32  0.89  560  564.0  1028.68  0.71  560  815.0  981.26  45.54  560  1389.0  1483.13  148.04  560  573  580.33  2.32  560  567  570.3  1.25 
47  430  433.0  710.55  0.7  430  434.0  645.0  0.93  430  570.0  682.42  32.56  430  854.0  931.84  98.6  430  440  445.29  2.33  430  435  439.5  1.16 
48  492  496.0  950.55  0.81  492  497.0  836.39  1.02  492  713.0  875.42  44.92  492  1148.0  1236.16  133.33  492  505  507.83  2.64  492  496  499.6  0.81 
49  641  659.0  1297.0  2.81  641  664.0  1422.0  3.59  641  922.0  1141.55  43.84  641  1532.0  1706.1  139.0  641  686  690.8  7.02  641  671  677.6  4.68 
410  514  515.0  825.61  0.19  514  517.0  812.45  0.58  514  696.0  847.0  35.41  514  1046.0  1180.97  103.5  514  530  532.4  3.11  514  521  524.1  1.36 
51  253  259.0  418.9  2.37  253  259.0  473.1  2.37  253  347.0  411.1  37.15  253  543.0  614.74  114.62  253  262  267.71  3.56  253  258  262.7  1.98 
52  302  314.0  617.1  3.97  302  316.0  628.74  4.64  302  468.0  577.35  54.97  302  762.0  913.61  152.32  302  326  332.17  7.95  302  319  325.3  5.63 
53  226  228.0  410.39  0.88  226  229.0  325.48  1.33  226  313.0  377.84  38.5  226  506.0  558.06  123.89  226  232  233.5  2.65  226  230  230.8  1.77 
54  242  246.0  503.58  1.65  242  247.0  405.26  2.07  242  318.0  394.97  31.4  242  540.0  580.19  123.14  242  250  252.5  3.31  242  247  249.4  2.07 
55  211  212.0  356.87  0.47  211  213.0  355.42  0.95  211  294.0  339.65  39.34  211  397.0  427.84  88.15  211  216  218.83  2.37  211  212  214.5  0.47 
56  213  213.0  369.68  0.0  213  214.0  335.81  0.47  213  334.0  389.71  56.81  213  507.0  544.58  138.03  213  227  229.0  6.57  213  218  221.3  2.35 
57  293  297.0  538.74  1.37  293  298.0  548.81  1.71  293  387.0  504.06  32.08  293  642.0  710.52  119.11  293  311  313.2  6.14  293  302  304.6  3.07 
58  288  290.0  534.9  0.69  288  290.0  395.81  0.69  288  399.0  497.35  38.54  288  663.0  745.32  130.21  288  298  299.33  3.47  288  291  293.9  1.04 
59  279  282.0  602.74  1.08  279  281.0  562.42  0.72  279  404.0  497.03  44.8  279  673.0  740.35  141.22  279  284  287.4  1.79  279  282  284.3  1.08 
510  265  267.0  461.35  0.75  265  266.0  402.58  0.38  265  390.0  470.13  47.17  265  594.0  669.58  124.15  265  277  278.33  4.53  265  267  273.1  0.75 
61  138  140.0  480.94  1.45  138  140.0  654.55  1.45  138  283.0  403.26  105.07  138  736.0  836.52  433.33  138  144  146.68  4.35  138  141  144.1  2.17 
62  146  146.0  603.32  0.0  146  146.0  619.52  0.0  146  320.0  561.77  119.18  146  1100.0  1211.81  653.42  146  154  155.83  5.48  146  147  152.3  0.68 
63  145  145.0  479.03  0.0  145  147.0  696.0  1.38  145  337.0  537.19  132.41  145  912.0  1125.97  528.97  145  149  150.4  2.76  145  147  148.4  1.38 
64  131  132.0  308.0  0.76  131  131.0  376.81  0.0  131  246.0  366.19  87.79  131  652.0  722.42  397.71  131  132  134.17  0.76  131  131  133.1  0.0 
65  161  162.0  842.87  0.62  161  162.0  652.77  0.62  161  357.0  531.74  121.74  161  1020.0  1155.81  533.54  161  180  181.5  11.8  161  163  172.2  1.24 
a1  253  260.0  785.84  2.77  253  260.0  800.94  2.77  253  455.0  662.71  79.84  253  1243.0  1352.03  391.3  253  263  266.84  3.95  253  260  263.2  2.77 
a2  252  259.0  886.87  2.78  252  263.0  910.58  4.37  252  452.0  651.16  79.37  252  1150.0  1241.0  356.35  252  266  269.83  5.56  252  261  264.0  3.57 
a3  232  239.0  608.71  3.02  232  239.0  582.29  3.02  232  436.0  601.58  87.93  232  1066.0  1185.84  359.48  232  244  245.6  5.17  232  240  243.4  3.45 
a4  234  237.0  794.77  1.28  234  237.0  672.42  1.28  234  467.0  595.97  99.57  234  1080.0  1161.45  361.54  234  251  251.8  7.26  234  238  242.5  1.71 
a5  236  241.0  800.0  2.12  236  240.0  702.77  1.69  236  447.0  618.65  89.41  236  1139.0  1191.23  382.63  236  242  247.33  2.54  236  241  244.2  2.12 
b1  69  69.0  1085.1  0.0  69  69.0  890.42  0.0  69  309.0  561.94  347.83  69  1344.0  1441.68  1847.83  69  70  71.68  1.45  69  69  70.5  0.0 
b2  76  76.0  630.23  0.0  76  76.0  548.48  0.0  76  337.0  547.39  343.42  76  1265.0  1426.32  1564.47  76  78  79.5  2.63  76  76  77.2  0.0 
b3  80  80.0  912.77  0.0  80  80.0  648.65  0.0  80  378.0  689.84  372.5  80  1737.0  1848.74  2071.25  80  82  82.17  2.5  80  81  81.7  1.25 
b4  79  79.0  978.26  0.0  79  79.0  1268.35  0.0  79  334.0  631.39  322.78  79  1514.0  1644.03  1816.46  79  83  83.83  5.06  79  79  81.3  0.0 
b5  72  72.0  785.74  0.0  72  72.0  959.45  0.0  72  299.0  541.81  315.28  72  1372.0  1467.52  1805.56  72  73  74.33  1.39  72  72  72.9  0.0 
c1  227  234.0  812.35  3.08  227  233.0  794.55  2.64  227  523.0  717.65  130.4  227  1488.0  1610.13  555.51  227  243  247.81  7.05  227  234  238.4  3.08 
c2  219  224.0  1061.39  2.28  219  226.0  1099.06  3.2  219  474.0  737.55  116.44  219  1654.0  1779.23  655.25  219  234  238.83  6.85  219  229  232.3  4.57 
c3  243  252.0  1560.45  3.7  243  248.0  1298.35  2.06  243  629.0  919.1  158.85  243  1970.0  2126.94  710.7  243  258  260.83  6.17  243  249  254.2  2.47 
c4  219  226.0  1105.35  3.2  219  225.0  1148.52  2.74  219  570.0  769.13  160.27  219  1582.0  1734.13  622.37  219  232  233.83  5.94  219  227  229.0  3.65 
c5  215  220.0  894.19  2.33  215  221.0  918.71  2.79  215  496.0  754.65  130.7  215  1541.0  1686.84  616.74  215  229  231.33  6.51  215  221  225.4  2.79 
d1  60  60.0  918.32  0.0  60  61.0  951.23  1.67  60  419.0  801.39  598.33  60  1950.0  2080.39  3150.0  60  63  64.97  5.0  60  61  62.3  1.67 
d2  66  66.0  1271.84  0.0  66  67.0  799.77  1.52  66  480.0  806.1  627.27  66  2261.0  2333.68  3325.76  66  68  69.0  3.03  66  67  67.4  1.52 
d3  72  73.0  647.81  1.39  72  72.0  1528.39  0.0  72  506.0  880.23  602.78  72  2445.0  2581.48  3295.83  72  76  77.33  5.56  72  73  74.8  1.39 
d4  62  62.0  1464.52  0.0  62  62.0  1201.81  0.0  62  312.0  755.48  403.23  62  2025.0  2107.97  3166.13  62  62  63.4  0.0  62  62  62.2  0.0 
d5  61  61.0  750.81  0.0  61  61.0  1044.03  0.0  61  344.0  792.87  463.93  61  1930.0  2093.29  3063.93  61  63  64.33  3.28  61  61  62.6  0.0 
Average    257.33  784.68  1.21    257.73  782.6  1.36    463.07  653.83  152.83    1176.27  1280.82  778.69    264.93  267.99  4.27    258.76  262.44  1.73 
QBWOA  BSWOA  BCL  MIR  40BQWOA  40BSWOA  

QBWOA    ≥0.05  ≥0.05  0.00  ≥0.05  ≥0.05 
BSWOA  ≥0.05    ≥0.05  0.00  ≥0.05  ≥0.05 
BCL  ≥0.05  ≥0.05    0.00  ≥0.05  ≥0.05 
MIR  ≥0.05  ≥0.05  ≥0.05    ≥0.05  ≥0.05 
40BQWOA  ≥0.05  ≥0.05  ≥0.05  0.01    ≥0.05 
40BSWOA  ≥0.05  ≥0.05  ≥0.05  0.00  ≥0.05   
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BecerraRozas, M.; CisternasCaneo, F.; Crawford, B.; Soto, R.; García, J.; Astorga, G.; Palma, W. Embedded Learning Approaches in the Whale Optimizer to Solve Coverage Combinatorial Problems. Mathematics 2022, 10, 4529. https://doi.org/10.3390/math10234529
BecerraRozas M, CisternasCaneo F, Crawford B, Soto R, García J, Astorga G, Palma W. Embedded Learning Approaches in the Whale Optimizer to Solve Coverage Combinatorial Problems. Mathematics. 2022; 10(23):4529. https://doi.org/10.3390/math10234529
Chicago/Turabian StyleBecerraRozas, Marcelo, Felipe CisternasCaneo, Broderick Crawford, Ricardo Soto, José García, Gino Astorga, and Wenceslao Palma. 2022. "Embedded Learning Approaches in the Whale Optimizer to Solve Coverage Combinatorial Problems" Mathematics 10, no. 23: 4529. https://doi.org/10.3390/math10234529