05-Chena TC_ You PS - Immune algorithms-based approach for redundant reliability problems with multiple component choices (1158526), страница 3
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The goal ofthe algorithm is to adapt the unfeasible antibodies tothe feasible antigen(s), so as to reduce the constraintviolations of the search for obtaining the optimal ornear-optimal solutions. Like the majority of geneticalgorithms applications, for handling these constraintviolations the penalty function has been defined. Thepenalty function increases the penalty for infeasiblesolutions based on the distance away from the feasibleregion. According to Eq. (2) in the problem formulation, the function has been defined and described asfollows.Assume the individual j within the memory ofN.
For each individual antibody, the constraint (2)violation value for the jth individual is defined as8kin kin XX>< XXai;j;k xi;k bj ; ifai;j;k xi;k > bjVj ¼i¼1 k¼1i¼1 k¼1>:0;otherwiseNote the objective and solution are deemed as theantigen and antibody, respectively. After defining thepenalty function, the fitness of each antibody to theantigen (objective) can be obtained. In other words,the affinity between each antibody and antigen is ableto be determined. The affinity function (fitness function) of any Ab to Ag is described below:Affinity ¼RðxjqÞP1þ mj¼1 VjThe above affinity value is to be maximized when thepenalty is minimized.3.4. Genetic operation processFig.
3. Binary string represents a solution of a series–parallelreliability problem.The implementation of genetic operations is thesame as in genetic algorithms. It including thecrossover operator and mutation operator requiresthe selection of the crossover point(s) and mutationpoint(s) for each antibody (string) under a predetermined crossover probability and mutation probability.T.-C.
Chen, P.-S. You / Computers in Industry 56 (2005) 195–205The crossover operator provides a thorough search ofthe sample space to produce good solutions. Themutation operator performs random perturbations toselected solutions to avoid the local optimum. Note themutation rate must be small enough to avoid degradingthe performance.4. Numerical results and discussionTo evaluate the performance of our artificialimmune algorithms for the integer nonlinear redundant reliability problems, 33 test problems are solved.The input data for a reliability system are described in201Table 1, which includes the component choices, andthe corresponding reliability of each component.
Theinput parameters have the same values as those ofNakagawa and Miyazaki [4], Coit and Smith [5] andHsieh [7]. These test problems based on theparameters in Table 1 are resolved with varying theavailable weight varied incrementally from 159 to 191while fixing the available cost = 130. Numericalresults obtained by using artificial immune algorithmare shown in Table 2, and compared with those foundby Nakagawa and Miyazaki [4], Coit and Smith [5]and Hsieh [7] in Table 3. Recall that, for each problem,both the component choices and the number of thechosen component are to be decided simultaneously.Table 2Numerical results by artificial immune algorithmNo.WeightReliabilitySolution1234567891011121314151617181920212223242526272829303132331911901891881871861851841831821811801791781771761751741731721711701691681671661651641631621611601590.98681100.98641610.98592170.98532970.98444950.98417550.98343630.98269800.98220620.98151830.98102710.98029020.97950470.97820850.97724290.97669050.97570790.97469010.97375800.97302660.97192950.97076040.96929100.96812510.96633510.96504160.96371180.96242190.96064240.95918840.95803460.95571440.9545648333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,333,Note: The cost limitation is 130 for all 33 cases.11,11,11,11,22,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,444, 3333, 222, 22, 111, 1111, 12, 233, 33, 1111, 11, 34444, 3333, 222, 22, 111, 1111, 11, 233, 33, 1111, 12, 34444, 3333, 222, 22, 111, 1111, 23, 233, 13, 1111, 22, 34444, 3333, 222, 22, 111, 1111, 13, 233, 13, 1111, 12, 34444, 333, 222, 22, 111, 1111, 23, 233, 33, 1112, 22, 344444, 333, 222, 22, 111, 1111, 23, 233, 33, 1111, 22, 34444, 333, 222, 22, 111, 1111, 23, 223, 33, 1111, 22, 344444, 333, 222, 22, 111, 1111, 23, 333, 33, 1111, 22, 33444, 333, 222, 22, 111, 1111, 23, 233, 33, 1111, 22, 3334444, 333, 222, 22, 111, 1111, 33, 333, 33, 1111, 22, 33444, 333, 222, 22, 111, 1111, 33, 233, 33, 1111, 22, 334444, 333, 222, 22, 111, 1111, 33, 223, 33, 1111, 22, 334444, 333, 222, 22, 111, 1111, 33, 223, 13, 1111, 22, 33444, 333, 222, 22, 33, 1111, 33, 233, 33, 1111, 22, 33444, 333, 222, 22, 33, 133, 33, 223, 33, 1111, 22, 33444, 333, 222, 22, 33, 1111, 33, 223, 13, 1111, 22, 33444, 333, 222, 22, 13, 1111, 33, 223, 33, 1111, 22, 33444, 333, 222, 22, 33, 113, 33, 223, 13, 1111, 12, 33444, 333, 222, 22, 13, 113, 33, 233, 13, 1111, 22, 33444, 333, 222, 22, 13, 113, 33, 223, 13, 1111, 22, 33444, 333, 222, 22, 13, 113, 33, 222, 13, 1111, 22, 33444, 333, 222, 22, 13, 113, 33, 222, 11, 1111, 22, 33444, 333, 222, 22, 11, 113, 33, 222, 13, 1111, 22, 33444, 333, 222, 22, 11, 113, 33, 222, 11, 1111, 22, 33444, 333, 22, 22, 13, 113, 33, 222, 11, 1111, 22, 3344, 333, 222, 22, 13, 113, 33, 222, 11, 1111, 22, 33444, 333, 22, 22, 11, 113, 33, 222, 11, 1111, 22, 3344, 333, 222, 22, 11, 113, 33, 222, 11, 1111, 22, 3344, 333, 22, 22, 11, 113, 33, 222, 13, 1111, 22, 3344, 333, 22, 22, 11, 113, 33, 222, 13, 1111, 22, 3344, 333, 22, 22, 11, 113, 33, 222, 11, 1111, 22, 3344, 333, 22, 22, 11, 111, 33, 222, 13, 1111, 22, 3344, 333, 22, 22, 11, 111, 33, 222, 11, 1111, 22, 33CostWeight130130130130130129128129128130129128126127129124125123124123122120121119118116117115114115113112110191190189188187186185184183182181180179178177176175174173172171170169168167166165164163162161160159202T.-C.
Chen, P.-S. You / Computers in Industry 56 (2005) 195–205Table 3Comparison of the proposed approach, Nakagawa and Miyazaki [4], Coit and Smith [5] and Hsieh [7] performanceNo.123456789101112131415161718192021222324252627282930313233W191190189188187186185184183182181180179178177176175174173172171170169168167166165164163162161160159Nakagawa and MiyazakiCoit and SmithReliabilityCostWeightReliabilityCostWeightReliabilityHsiehCostWeightReliabilityChen and YouCostWeightNote0.98640.98540.98500.98470.98400.98310.98290.98220.98150.98150.98000.97960.97920.97720.97720.97640.97440.97440.97230.97200.97000.97000.96750.96660.96560.96460.96210.96090.96020.95890.95650.95460.9546130132*131*129133*1291291261301301281261271231231251211211221231191191211201171161181161141121111101101911891881881861861851841821821811801791771771761741741731721701701691681671661651641631621611591590.986750.986030.985560.985030.984290.983620.983110.982390.981900.981020.980060.979420.979060.978100.977150.976420.975520.974350.973620.972660.971860.970760.969220.968130.966340.965040.963710.962420.960640.959120.958030.955670.954321301291301301291281301281301261281291251271251241221231221201211201201191181161171151141141131141101911901891881871861851841831821811801791781771761751741731721711701691681671661651641631621611601590.9867110.9863160.9857240.9850310.9841530.9838790.9833870.9822040.9814660.9796900.9792800.9783270.9780550.9768780.9754000.9749750.9735000.9723280.9705310.9692320.9678960.9667760.9656120.9641500.9629900.9612100.9599230.9586010.9573170.9555470.9541010.9529530.9508001301301301301291281271251241261251241231211221211221201191171181191171181161151131141121111121101081911901891881871861851841831821811801791781771761751741731721711701691681671661651641631621611601590.9868110.9864160.9859220.9853300.9844490.9841760.9834360.9826980.9822060.9815180.9810270.9802900.9795050.9782080.9772430.9766900.9757080.9746900.9737580.9730270.9719290.9707600.9692910.9681250.9663350.9650420.9637120.9624220.9606420.9591880.9580350.9557140.954565130130130130130129128129128130129128126127129124125123124123122120121119120116117115114115113112110191190189188187186185184183182181180179178177176175174173172171170169168167166165164163162161160159*********************************Note: * represents the best solution found is superior than the solution found in literature; * represents the best solution found is as well as thesolution found in literature.Our artificial immune algorithm is implemented inMATLAB1 on the Pentium 42.0 GHz PC with thefollowing parameters: memory size = 120, mutationrate = 0.01, crossover rate = 0.86 and the maximumclone number = 10.
Then number of generations wasspecified to be 3000. The determination of immunealgorithm’s parameters is a significant problem for theimmune algorithm implementation. However, there isno any formal methodology to solve the problembecause different value-combinations of the parameters result to different characteristics as well asdifferent performance of immune algorithms.
Therefore, one should note that the best values for theartificial immune algorithm parameters are casedependent and based upon the experience frompreliminary runs.The numerical results in Table 2 reports the detailedsolutions obtained by the proposed approach for eachtest problem. Also, they are compared with those ofNakagawa and Miyazaki [4], Coit and Smith [5] andHsieh [7] in Table 3.The results in Table 3 indicate that: compared with those of Nakagawa and Miyazaki[4], 32 solutions (1–32) obtained by immunealgorithms-based approach are superior than thoseT.-C. Chen, P.-S. You / Computers in Industry 56 (2005) 195–205found by Nakagawa and Miyazaki [4]. The solutionfound in the 33rd test problem by both approaches isthe same.
compared with those of Coit and Smith [5], theproposed approach finds better solutions for 24 outof 33 test problems. The solutions of the left nineobtained by proposed approach are as well as thoseobtained by those obtained by genetic algorithms[5]. compared with those of Hsieh [7], it is seen that thesolutions found by our approach in all test problemsare superior than those found by Hsieh [7].The comparison of numerical results for 33 testproblems with those in literatures is depicted in Fig.
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