A Multi-objective Genetic Algorithm for Reliability Optimization Problem

Transcription

1 International Journal of Performability Engineering, Vol. 5, No. 3, April 2009, pp RAMS Conultant Printed in India A Multi-objective Genetic Algorithm for Reliability Optimization Problem AMAR KISHOR 1, SHIV PRASAD YADAV 1 KUMAR 2* and SURENDRA 1 Department of Mathematic, I.I.T Roorkee, Roorkee, India: Department of Electrical Engineering, I.I.T Roorkee, Roorkee, India: (Received on November 1, 2007; Final Reviion on October 24, 2008) Abtract: Thi paper conider the allocation of maximum reliability to a complex ytem, while minimizing the cot of the ytem, a type of multi-objective optimization problem (MOOP). Multi-objective Evolutionary Algorithm (MOEA) have been hown in the lat few year a powerful technique to olve MOOP.Thi paper uccefully applie a Nondominated orting genetic algorithm (NSGA-II) technique to obtain the Pareto optimal olution of a complex ytem reliability optimization problem under fuzzy environment in which the tatement might be vague or imprecie. Deciionmaker (DM) could chooe, in a poteriori deciion environment, the mot convenient optimal olution according to hi/her level of atifaction. The efficiency of NSGA-II in olving thi problem i demontrated by comparing it reult with thoe of imulated annealing (SA) and nonequilibrium imulated annealing (NESA). Keyword: Reliability optimization, fuzzy optimization, multi-objective optimization problem (MOOP), genetic algorithm (GA), nondominated orting genetic algorithm (NSGA-II) 1. Introduction In the broadet ene, reliability can be defined a meaure of performance of ytem. A ytem have grown more complex, the conequence of their unreliable behavior have become evere in term of cot, effort, live etc. and the interet in aeing ytem reliability along with it improvement ha become very important. Unlike conventional optimization method where it i aumed that all deign data are preciely known and objective are well defined and eay to formulate, many practical optimization problem e.g., reliability optimization, there are incompletene and unreliability of input information. The reaon for unreliability can be many uch a; uncertainty in judgment, lack of evidence, etc. Further more, a DM often ha vague deire uch a, thi objective function hould be le than or greater than or equal to thi certain value. Fuzzy et theory [1] i effectual in handling thee cae. In reliability optimization problem, it i often required to minimize or maximize everal objective ubject to everal contraint. Such problem i formulated a multi-objective optimization problem (MOOP). A MOOP can be olved in two way; firt one i to olve it by tranforming the MOOP into ingle objective problem uing poitive weight (for objective) and penaltie (for contraint), and the other one which i alo better i to obtain a Pareto-optimal olution which give a DM uitable range of choice to adjut trade off between different * Correponding author urendra_iitr@yahoo.com 227

2 228 Amar Kihor, Shiv Praad Yadav and Surendra Kumar objective. Sakawa [2] ued the urrogate worth trade off method to a multi-objective formulation of a reliability allocation problem to maximize the ytem reliability while minimizing the ytem cot. Huang [3] tackled fuzzy multi-objective optimization deciion-making problem on the erie reliability ytem with two objective. Ravi et al. [4] and [5] implemented imulated annealing (SA) algorithm for everal reliability optimization problem. Mahapatra et al. [6] propoed a new fuzzy multi-objective optimization method to olve reliability optimization problem having everal conflicting objective. Genetic algorithm (GA) are well-known tochatic method of global optimization baed on the evolution theory of Darwin and have uccefully been applied in different real-world application including reliability optimization. Since GA work with a population of point, a number of Pareto-optimal olution may be captured uing GA, making it a very powerful tool alo for MOOP. The Non-dominated Sorting Genetic Algorithm (NSGA-II) [7] i a well known and extenively ued algorithm baed on it predeceor NSGA [8]. It i a fat and very efficient Multi-objective evolutionary algorithm (MOEA), which incorporate the feature an elitit archive and a rule for adaptation aignment that take into account both the rank and the ditance of each olution regarding other. Daniel et al. [9] ha applied and compared the efficiency of NSGA-II with exiting method for different reliability optimization problem without including fuzzy environment. Thi paper ue a fuzzy atificing method projected and applied by Huang [3], Ravi et al. [4-5] to tranform the multi-objective reliability optimization problem into fuzzy optimization problem uing linear memberhip function. Then NSGA-II i employed to the reulting fuzzified problem to obtain the Pareto olution becaue NSGA-II i one of the mot popular multi-objective optimization algorithm applied uccefully for variou real world problem. Thi paper ugget ue of NSGA-II to obtain the Pareto-optimal olution unlike Ravi et al. [5] who olved the problem uing NESA. The DM can chooe appropriate olution from the group of Pareto-optimal olution obtained. If the DM i not atified with any of them, then he/he can eek a final olution by modifying parameter interactively according to hi/her preference. 2. The Mathematical Model of Problem Let R j and C j be the reliability and cot of j th component of the ytem and R and C denote the total reliability and cot of the ytem. It i often required to conider, in addition to maximization of ytem reliability, the minimization of the cot. Mathematically, thi problem can be expreed a n R for erie ytem j = 1 j n Maximize R ( R, R, R,..., R ) 1 (1 R ) for parallel ytem n = j 1 j = or combination of erie and parallel ytem n Minimize C ( R, R, R,..., R ) C ( R ) n = j = 1 j ubject to: R j,min R j 1, R,min R 1 for j=1,2,,n

3 A Multi-objective Genetic Algorithm for Reliability Optimization Problem 229 where n repreent total number of component in the ytem while R j,min and R,min are minimum value for the j th component and ytem repectively. Here, we conider a complex ytem which repreent a block diagram of reliability of a life-upport ytem in a pace capule. Fig. 1 how the ytem The mathematical model for the life upport ytem in a pace capule can be formulated uing the block diagram (ee Fig. 1) a follow: 2 2 Maximize R (R) = 1 R ((1 R )(1 R )) (1 R )[1 R (1 (1 R )(1 R ))] (1) α 4 j Minimize C (R) = 2. K R j j j = 1 ubject to : 0.5 R R R R 1 for j = 1,2,3,4. j,min j,min j where different parameter value are K j a 100, 100, 200, 150 repectively and all equal to Genetic Algorithm Fig. 1: Life-upport ytem in a pace capule (2) α j Genetic algorithm have been uccefully applied a an optimization technique. GA introduced by Holland [10] and further decribed by Goldberg [11] and Deb [12], mimic natural election or Darwinian Theory of urvival of the fittet. The baic GA methodology can be preented in the following form 1. Set population ize, tournament ize, croover rate, mutation rate, mutation exponent and elitim ize. Set the parameter of the topping criterion. 2. Initialize the population with random number. 3. Compute the fitne function value. Perform election, croover, mutation and elitim in order to create a new population. 4. If the topping criterion i not atified, return to tep 3. Otherwie, chooe the bet individual found a the final olution. 4. Nondominated Sorting Genetic Algorithm (NSGA-II) There were ome major drawback in NSGA uch a

4 230 Amar Kihor, Shiv Praad Yadav and Surendra Kumar High computational complexity of non-dominated orting. Lack of elitim. Lack of pecification of haring parameter. Deb et al. [7] propoed an improved verion of NSGA [8], called NSGA-II which dealt all the drawback of original NSGA. NSGA-II incorporate an archive and a rule for adaptation aignment that take into account both the rank and the ditance of each olution. Let P t repreent the current population during any generation t, and P A t the population which conit of non-dominated olution archive. The Peudo code for NSGA-II can be tated a follow Input: N (Population ize) M (Archive ize) t max (maximum number of generation) Begin: Randomly initialize P A t, et P 0 = Ø, t=0 while t < t max P t =P t + P A t Aign adaptation to P t P A t+1 ={ M bet individual from P t } Mating Pool={ N individual randomly elected from P A t+1 uing a binary tournament} P t+1 = { N new individual generated by applying recombination (croover and mutation) on Mating Pool t= t +1 Output: Non-dominated olution from P A t 5. Methodology Preent methodology to olve a multi-objective reliability optimization problem conit of following tep Step 1) Fuzzification: Fuzzification of the problem with the help of a linear memberhip function. Step 2) Problem Reformulation: Converion of multi-objective crip reliability optimization problem into fuzzified MOOP of memberhip function. Step 3) Solution: Finding the olution of the reulting MOOP uing NSGA-II decribed in ection 4. Now preent methodology i applied to olve the multi-objective reliability optimization problem of life-upport ytem in a pace capule. 5.1 Fuzzification Let f % and f % be the fuzzy region of atifaction of ytem reliability (R 1 2 ) and ytem cot (C ) repectively and µ % ( R ) and f1 µ % ( C ) be their correponding memberhip function where R and C are defined in (1) and (2). Then µ % ( R ) i defined a f2 f1

5 A Multi-objective Genetic Algorithm for Reliability Optimization Problem if R 0.9 µ ( R ) f % 1 = R (3) if 0.9 R if R if C 641 µ ( C ) f % 2 = 700 C (4) if 641 C if C 700 Parameter for linear memberhip function ued in equation (3 and 4) are taken from Ravi et al. [4-5]. The above formulation repreent fuzzy repreentation for a complex ytem (fig. 1) in term of reliability and cot. DM want to optimize the ytem reliability within 0.9 and 0.99 while maintaining the cot between 641 to 700 unit thu making it a multiobjective problem having two objective function namely reliability (R ) and cot (C ). 5.2 Problem Reformulation Optimal deciion i made by electing the bet alternative from the fuzzy deciion pace D characterized by the memberhip function µ D. In other word, the problem i to find the optimum R* which maximize µ D where µ D [0,1] where R* i the deciion variable, R*=( R 1,R 2,, R 4 ). Within the frame work of Bellman and Zadeh [13] model, and following Zimmermann [14] the optimal olution i obtained by maximizing µ D where µ = µ % ( R ) * µ % ( C ) and * repreent the operator ued in the deciion. In the preent D f1 f2 paper thi problem i olved in purely multiobjective manner. So the reformulated fuzzy problem can be mathematically expreed a Maximize ( µ % ( R ), µ % ( C ) ) (5) f1 Subject to: 0.5 R j 1 j = 1,2,3,4 f2 5.3 Solution of MOOP A NSGA-II i employed to olve the reulting MOOP maximization problem of memberhip function. Varying the different parameter (croover probability, mutation probability, population ize, croover and mutation index, etc.) of NSGA-II different Pareto-optimal front can be obtained. Baed on rigorou experimentation and tuning of the parameter ome better optimal front have been reported here. 6. Reult and Dicuion To olve the tranformed problem (5) NSGA-II ha been applied. Parameter of NSGA-II uch a population ize N, maximum number of generation t max, croover probability p c and mutation probability p m. Different Pareto-optimal front can be obtained varying thee parameter. Thi paper dicue few better one out of them after a number of experimental run. Cae-1 If population ize N i being fixed at 100 and number of generation t max equal to 300 while croover probability (p c ) equal to 0.9 and mutation probability (p m ) = 0.02 then reulting Pareto olution i given in Fig. 2.

6 232 Amar Kihor, Shiv Praad Yadav and Surendra Kumar Cae-2 For Pareto-optimal olution obtained in Fig. 3, NSGA-II parameter uch a p c i taken 0.9 again while p m i 0.03 and population ize N of 200 for 500 (t max ) generation. Similarly many Pareto optimal olution of thi problem can be achieved. More point on Pareto curve require bigger population ize. Table 1 how the top 15 coordinate of the Fig 2. Earlier Ravi et al. [4-5] olved the ame problem uing SA [4] and NESA [5] repectively. Ravi et al. in [4] tackled the problem a ingle objective formulation uing SA while they olved the ame in [5] uing multi-objective formulation and NESA under fuzzy environment. Preent reult are better than thoe obtained by Ravi et al. [4], a it provide the DM more choice. Alo preent methodology provide Pareto optimal front in a ingle run of algorithm (NSGA-II) unlike [5] which need different value of memberhip function of the fuzzy et deciion for different run. It alo improve the accuracy of reult obtained compared to [5] a it provide the reult up to 6 decimal place (ee Table 1) unlike [5] which gave reult up to 5 decimal place only. Hence thi approach i much more efficient and flexible, and thu provide a better range of choice out of which DM can chooe the preferred olution interactively Cot Reliability 7. Concluion Fig. 2: Pareto-optimal front by NSGA-II for Cae-1 A real life reliability optimization problem of a life-upport ytem in a pace capule under fuzzy environment ha been dicued here. Ravi et al.[5] olved the ame problem uing NESA. Thi paper olve the problem uing NSGA-II which i highly efficient for continuou multi-objective optimization problem a it provide a deciion maker more flexibility in obtaining Pareto-optimal front. Alo preent methodology provide Pareto optimal front in a ingle run of algorithm (NSGA-II) unlike [5] which need different value of memberhip function of the fuzzy et deciion for different run. Alo accuracy of the reult ha been increaed. Therefore, the paper prove the effectivene of NSGA-II for continuou multi-objective optimization problem.

7 A Multi-objective Genetic Algorithm for Reliability Optimization Problem Cot Reliability Fig. 3: Pareto-optimal front by NSGA-II for Cae-2 Table 1: Coordinate of lat 15 point i R (i) C (i) R 1 R 2 R 3 R Reference [1] Zadeh L. A., Fuzzy et, Inf. Control, Vol. 8, no. 3, pp , [2] Sakawa M., Multi-objective optimization by the urrogate worth trade-off method, IEEE Tranaction on Reliability, R-27, pp ,1978. [3] Huang H.Z., Fuzzy multi-objective optimization deciion-making of reliability of erie ytem, Microelectronic Reliability, Vol. 37 (3), pp , [4] Ravi V., Murty B. S. N., and Reddy P. J., Nonequilibrium imulated annealingalgorithm applied to reliability optimization of complex ytem, IEEE Tran. Rel., Vol. 46, pp , 1997.

8 234 Amar Kihor, Shiv Praad Yadav and Surendra Kumar [5] Ravi V., Reddy P. J., and Zimmermann Han-Jurgen., Fuzzy global optimization of complex ytem reliability, IEEE Tran. Rel., Vol. 8, pp , [6] Mahapatra G.S. and Roy T.K., Fuzzy multi-objective mathematical programming on reliability optimization model, Applied Mathematic and Computation, Vol. 174, pp , [7] Deb K, Pratap A, Agarwal S and Meyarivan T. A fat and elitit multiobjective genetic algorithm: NSGA-II, Kanpur Genetic Algorithm Laboratory (KanGAL), Indian Intitute of Technology, KanGAL Report No , [8] Sriniva N. and Deb K., Multiobjective function optimization uing nondominated orting in genetic algorithm, Evol. Comput., Vol. 2, no. 3, pp , [9] Salazar Daniel, Rocco C. M. and Galvan B. J., Optimization of contrained multiple-objective reliability problem uing evolutionary algorithm, Reliability Engineering and Sytem Safety, Vol. 91, pp , [10] Holland J. H., Adaptation in natural and artificial ytem, The Univerity of Michigan Pre, [11] Goldberg D., Genetic Algorithm in Search, Optimization and Machine Learning, Addion Weley Pre, [12] Deb K., Optimization for engineering deign: algorithm and example, Prentice- Hall, [13] Bellman R. E. and Zadeh L. Z., Deciion making in a fuzzy environment, Management Sci., Vol. 17, pp. B , [14] Zimmermann H.-J., Fuzzy Set Theory and Application, 2 nd ed. Boton, MA: Kluwer Amar Kihor i a Reearch Scholar in the Department of Mathematic at Indian Intitute of Technology Roorkee, Roorkee, India. He hold M.Sc. in Mathematic from Banara Hindu Univerity, Varanai, India. Hi current reearch area include Reliability Optimization, Genetic Algorithm and Fuzzy Optimization. Shiv Praad Yadav i an Aociate Profeor of Mathematic at Indian Intitute of Technology Roorkee, Roorkee, India. He hold Ph.D. in Mathematic with concentration on Optimal Control Theory from Intitute of Technology, Banara Hindu Univerity, India. Hi current reearch area include Optimal Control, Operation Reearch, Data Envelopment Analyi, Fuzzy Mathematic and Reliability Optimization. Surendra Kumar i an Aitant Profeor of Electrical Engineering at Indian Intitute of Technology Roorkee, Roorkee, India. He hold a Ph.D. in Electrical Engineering with concentration on Sytem Modeling and Optimization from Indian Intitute of Technology Roorkee, Roorkee, India. Hi current reearch area include PID Control, Fuzzy and Neural Control, Fuzzy Reliability, Reliability Optimization and Robotic.