Plane Thermoelastic Waves in Infinite Half-Space Caused Decision Making: Applications in Management and Engineering Vol. 2, Issue 2, 2019, pp. 100-111. ISSN: 2560-6018 eISSN: 2620-0104 DOI: https://doi.org/10.31181/dmame1902076r *Corresponding author Email: royarindamroy@yahoo.com (A. Roy), pkcollege.contai@gmail.com (A. Manna), sumanta@iimcal.ac.in (S. Maity). A NOVEL MEMETIC GENETIC ALGORITHM FOR SOLVING TRAVELING SALESMAN PROBLEM BASED ON MULTI- PARENT CROSSOVER TECHNIQUE Arindam Roy 1*, Apurba Manna 1 and Samir Maity 2 1 Deptartment of Computer Science, Prabhat Kumar College, Purba Medinipur, West Bengal, India 2 Operations Management Group, Indian Institute of Management, Kolkata, India Received: 5 January 2019; Accepted: 4 May 2019; Available online: 5 May 2019. Original scientific paper Abstract: In the present study, a Novel Memetic Genetic Algorithm (NMGA) is developed to solve the Traveling Salesman Problem (TSP). The proposed NMGA is the combination of Boltzmann probability selection and a multi-parent crossover technique with known random mutation. In the proposed multi-parent crossover parents and common crossing point are selected randomly. After comparing the cost/distance with the adjacent nodes (genes) of participated parents, two offspring’s are produced. To establish the efficiency of the developed algorithm standard benchmarks are solved from TSPLIB against classical genetic algorithm (GA) and the fruitfulness of the proposed algorithm is recognized. Some statistical test has been done and the parameters are studied. Key words: TSP, Memetic GA, multi-parent crossover. 1. Introduction One best example of a well known intensively studied the combinatorial optimization problem is TSP. TSP is also too much related to different type of transportation problem (Kundu, 2017; Kar, 2018) with vehicle convenience. It is also an example of NP-hard problem (Lawler & Lenstra, 1985; Das et al., 2010; Das et al., 2011). Many researchers are trying to solve TSP with reasonable time and space. But still, there have lacunae to solve such kind of NP-hard problems. Presently two ways are implemented such as direct method (Lin-Kernighan Helsgaun, Scant-method, Sant-cycle method) and indirect such as heuristic or metaheuristics. The classical TSP involves finding the shortest path/minimum cost with the visit of all cities exactly ones except the starting one. The probe on the efficient algorithm for TSP is an open problem. mailto:royarindamroy@yahoo.com mailto:pkcollege.contai@gmail.com mailto:sumanta@iimcal.ac.in Optimal decisions on pricing and greening policies of multiple manufacturers under… 101 Moscato (1989) introduced the word Memetic Algorithm (MA) as a combination of population-depended global search and the heuristic local search based on every of the individuals. From the different context of view, MAs are recently used with different names like hybrid evolutionary algorithms (Martinez-Estudillo, 2005), many kinds of literature found in (Jampani, 2010; Li & Feng, 2013; Silberholz, 2013; Hiremath & Hill, 2013; Nesmachnow, 2014; Skinner, 2015), a different Lamarckian Evolutionary Algorithms are studied in (Omran, 2016), very recently a Cultural Algorithms developed found in (Reynolds & Peng, 2004). In the case of colonial optimization, the various number of instances of MA have been notifiable across a broad scope of application realm, generally merging to better-tone solutions much expeditious than the established affected counterparts. Real-world complex problems have been successfully satisfied with memetic algorithms. Although various researcher avail procedures nearly bound up to MA, through other names like hybrid genetic algorithms are also exploited. Nowadays MAs are applied in different research areas included pattern recognition (Aguilar & Colmenares, 1998), artificial neural networks (Ichimura & Kuriyama, 1998), circuit design (Harris & Ifeachor, 1998), robotic motion planning, beam orientation (Haas et al., 1998), electric service restoration (Kumar et al., 2006), medical expert systems (Wehrens, 1993), single machine scheduling (Chyu & Chang, 2010), etc. A study on multi-parent MA is found in (Wang et al., 2010) and they concluded that different combination got better results from others. Ye et al., (2014) developed a multi-parent recombination operator for solving Linear Ordering Problem but they do not restrict to chosen the parent because it increases the computational complexity. At the present investigation, only four parents are selected from the mating pool and randomly a common crossing point is chosen. Genetic Algorithms (GAs) were first proposed by Holland (1992) whose ideas were applied and expanded on by Goldberg (1998). The classical GA has three operators, such as selection, crossover and mutation. Different kinds of selection operators (RW, Ranking, Tournament, etc.) and cyclic, partial-map, ordered based, etc. crossovers are available with a random mutation to solve the discrete optimization problem by GA. In our proposed method (NMGA) are the combinations of probabilistic selection and adaptive four- parents crossover with the classical ergodic mutation. Now the crossover is taken from the realistic social observations. We see that some child born with legal parents but they adapted by other parents and grown up under them. In third world countries, it is very common. In the proposed crossover methods four parents are used and modified them finally comparing the costs/distance, the offspring is created. The proposed algorithm has the following key features:  Boltzmann probabilistic selection,  Multi-parent adaptive crossover,  Four parents,  Random crossing point,  Comparing the cost/distance genes are selected,  Test on standard TSPLIB problems. The remaining part of this paper is presented as follows: section 1, a short introduction is presented. In section 2, We describe mathematical pre-requisite. In section 3, the proposed modified memetic algorithm is presented. In section 4, some numerical experiments are done. Again in section 5, a brief discussion is given. Finally, in section 6, a conclusion with future scope is studied. Roy et a., /Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 100-111 102 2. Classical Definition of TSP The goal of a Traveling Salesman Problem (TSP) is that a salesperson would create a path. This path should be an ideal path. Ideal path means path would be the shortest while salesman completes his visit across a finite number of cities, visiting each city only once and finished at the starting city. Let G=(V, A) is a graph. G has n vertices and V is a set of this n vertices. A is also a set of arcs or edges of this G. Then G=(V, A). Let C = c(i,j) be a distance ( or cost) matrix associated with A. The intent of TSP is determining a minimum distance or cost circuit passing through each and every vertex only once except starting node. This type of circuit is familiar as a tour or Hamiltonian circuit or cycle. In case of symmetric TSP c(i,j) = c(j,i) for all i,j∈ V. (n- 1)! path will be generated for symmetric TSP and (n-1)!/2 path will be generated for asymmetric TSP. Now mathematically TSP defined as below. Minimize ( , ) ij i j Z c i j x   subject to 1 1 N ij i x   for 1, 2,...,j N ; 1 1 N ij j x   for , 1, 2,...,i N ; | | 1, N N ij i S j S x S S Q       ; Where {0,1}, , 1, 2..,ijx i j N  . Now, Q  {1, 2, 3,.., N } set of nodes ij x = the decision variable, 1 ij x  if thesalesman visits from city-i to city-j, 0 ij x  otherwise. Then the above TSP reduces todetermine a complete tour 1 2 1 ( , ,..., , ) N x x x x ; 1 1 1 1 ( , ) ( , ) N i i N i Z c x x c x x      Where, , , 1, 2..., . i j x x i j N  along with sub tour elimination criteria Optimal decisions on pricing and greening policies of multiple manufacturers under… 103 | | 1, N N ij i S j S x S S Q       , where, {0,1}, , 1, 2..,ijx i j N  . 3. Proposed Memetic Genetic Algorithm Here we propose a probabilistic selection, multi-parent crossover with the simple random mutation for solving the TSP. 3.1. Representation Considering N cities available to make a complete tour which stands for a solution. Say an integer vector Xi of N-dimensional. Where Xi =(xi1,xi2,··· ,xiN) is used as cities, and xi1, xi2, ···, xNstand for N successive cities in a tour. At the beginning need a group of paths (tours) for the salesman. These paths are randomly generated for GA. These initial paths is a group of possible solutions for the GA part of this algorithm. 3.2 Selection 3.2.1. Probabilistic Selection The main objective of TSP that minimizing the path cost/distance. So here minimum fitness value (say fmin) of the choromosome play a vital role. Matting pool is formed using the Boltzmann-Probability (Roy et al., 2018) of all chromosome from the initial population. Now (( ( ))/ ) ;min j f f X T B p e   T=T0(1-a)k, k=(1+100*(g/G)), g=ongoing generation number, G= maximum generation, T0= rand[10,150], f(Xi) corresponding fitness/objectives of chromosome corresponding to Xi, a=rand[0,1], i=chromosome number. 3.3. Multi-Parent Crossover Nowadays in our society adaptation is very common matter from the different practical situation. Here except original parents (father, mother), there are one more parents (father, mother) taken as a part. Inspiring this realistic happening here a new approach with four parents (first two are original parents and the other two are adoptive parents) are used to produce offspring. This urged crossover method choose four individuals or parents in an ergodic manner to create offspring. To collect optimum results of a TSP, we make a journey from one node to next node maintaining minimum traveling cost based on TSP condition. Following the above conception, we make the crossover procedure in the following condition. At first select (randomly) four individuals ( parents ) from the mating pool. Give an example here. PR1, PR2, PR3, PR4 are the parents and CH1, CH2 are the offspring. PR1: PR2: PR3: PR4: 1 2 0 3 4 0 2 4 3 1 4 0 3 1 2 3 4 1 2 0 Roy et a., /Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 100-111 104 Generate random number between 0 and 4. Suppose it is 2. Then according to our proposed algorithm it would be the starting node of a new offspring (CH1). CH1: Now we have to comparing minimum traveling cost between 2 node··· (1) (1st node of Parent1 [PR1] ) 2 node··· (0) (1st node of Parent2 [PR2] ) 2 node··· (4) (1st node of Parent3 [PR3] ) 2 node··· (3) (1st node of Parent4 [PR4] ) and if say the traveling cost node (2) to node (4) is minimum from rest three paths, then next node of the new offspring (CH1) would be 4. So it should be like as- CH1: The above process will continue until the new offspring (CH1) gets its all nodes maintaining the condition of TSP. Similarly, generate the second offspring but in reverse order than another. Here R1 and R2 are two randomly generated variable two nodes from the given set of nodes. 3.4. Random Mutation An ergodic number r is created for every solution of P(t). Here r is generated between a range [0,1] and r < pm is a condition, if the condition is true, then a solution is selected for mutation. Two nodes are selected ergodic manner from each chromosome and they interchange their positions and replace it in the offspring set. 3.5. Proposed Novel Memetic Algorithm ( NMGA ) NMGA Algorithm: 1. Input: for Crossover procedure (pc), Maximumgen(S0), (pop−size) and for Mutation procedure (pm). 2. Output: The best solution. 3. Begin 4. Approve generation t = 0. 5. (Initialize) ergodic manner and generate approve population p(t), here f(xi),i= 1,2,··· ,(pop − size) state the all chromosomes. 6. All solution will be judged it’s efficiency one by one from the approve population p(t) 7. Repeat up to (18) till (t < S0) 8. Modify the current generation such as t = t + 1 9. Decide (pB) for all chromosome over p(t) to subsection (3.2.1) 10. Construct the mating pool on the basis psand pB 11. For crossover parents will be chosen based on pc over mating pool 12. According to subsection (3.3) the crossover operation will be conducted based on exclusive chromosomes/solutions 13. Produce offspring and the parents will be replaced 14. Repeat (9) to (11) based on pc 15. Followed by the subsection (3.4) mutation process will be executed 16. Offspring will be selected for mutation depend on pm 17. Interchange the position between selected nodes 18. Replace offspring 19. Determine the effectiveness and save the local best and near best solutions 2 4 2 Optimal decisions on pricing and greening policies of multiple manufacturers under… 105 20. Repeat (5) to (18) 21. (for best result) Store the global best and near best results 22. Stop Proposed NMGA pseudo code: Begin generation t = 0; for (i=1 to pop-size) Produce chromosome ergodic manner; end for; for (i=1 to pop-size) Judge fitness; end for; for (gen=1 to max-gen) { for (j=1 to pop - size) r=rand[0,1]; T0= rand[5,100]; a=rand[0,1]; k=(1+100*(i/G)); T=T0(1-a)k; (( ( ))/ ) ;min j f f X T B p e   if ( s r p ) select the current chromosome; j++; else if (r B p ) select f(X j ) ; j++; else Select the corresponding chromosome of f mi n ; j++; end for; end for; for (s=1 to (noc ∗ pc)) R1=rand[0, N-1]; R2=rand[0, N-1]; PR1=rand[0, pv-1]; PR2=rand[0, pv-1]; PR3=rand[0, pv-1]; PR4=rand[0, pv-1]; CH1[0]=R1; i=1; do{ CH1[i] = min {c(R1,PR1[0]),c(R1,PR2[0]),c(R1,PR3[0]),c(R1,PR4[0])}; i=i+1;} while(CH1[ N-1]); CH2[n-1]=R2; i = n − 2 ; do{ CH2[i] = min {(R2,PR1[N−1]),c(R2,PR2[N−1]),c(R2,PR3[N−1]),c(R2,PR4[N− 1])}; i = i − 1 ; } while(CH2[0]) ; End for for(a=0 to noc) { Roy et a., /Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 100-111 106 If (rand[0,1] < pm) mutate; } for (i=1 to noc) Evaluate fitness; end for } Stop 3.6. Termination Criteria The proposed algorithm is terminated if it finishes the user-defined maximum number of generations or iterations, or the difference between consecutive iterations less than some predefined values which are earlier. 4. Numerical Experiments 4.1. Test Results of NMGA We have taken benchmarks from TSPLIB (Reinelt, 1991) and select 53 standard instances form 7 city to 318 cities to test the performance of our proposed algorithm NMGA. Table 1 shows the comparison of performance between proposed NMGA and standard classical GA through the results presented in the form of percentage error. The total comparison held the basis of total cost. We have taken the best, average and the worst outcome of both NMGA and classical GA under 100 independent runs and the best solution is presented with relative percentage error. The parameters of the NMGA given in Table 2 for the same nodes of the benchmarks instance kora100 with 100 cities problem. We have increases pop-size, Maxgen, pc and pm of an instance as a parameter. 4.1.1. Sensitivity of CPU-time w.r.t. pc and pm Sensitivity analysis is performed for CPU-time on the basis of concerned values of the key parameters pm and pc and outcomes are projected in Figure1 (three dimensions linear graph using STATISTICA). It is observed that for fixed value of pc, as pm increases, CPU-time increases. Also, it is observed that for a fixed value of pm, as pc increases, CPU-time also increases. Figure 1. Sensitivity Analysis Optimal decisions on pricing and greening policies of multiple manufacturers under… 107 5. Discussion The superiority of the developed algorithm is established by solving standard benchmarks from TSPLIB which given in Table 1. This proposed algorithm NMGA is coded in C++ based on few keys like the maximum number of chromosomes (100) and a maximum number of iterations (5000). Table 1 is used to show the differences between NMGA and GA for few benchmark TSP references in TSPLIB. It is observed that the percentage of error is lesser in NMGA than the classical standard GA. Here, 53 standard instances from 7 to 318 cities are studied and most of the cases NMGA produced better results. A parameter analysis is done which is given in Table 2 for the standard benchmark kora100 with 100 cities problem. Table 1. Performance (relative error) of benchmarks from Instances NMGA Classical GA Best Worst Average Best Worst Average sh-07 0 0 0 0 0 0 sp11 0 0 0 0 0.15 0.07 uk12 0 0 0 0.04 0.22 0.15 lau15 0 0 0 0.29 0.55 0.44 gr17 0 0 0 0.22 0.52 0.39 wg22 0 0.02 0.01 0.65 1.07 0.94 fri26 0 0.02 0.01 0.76 1.09 0.98 bay29 0 0.06 0.01 1.01 1.25 1.12 bayg29 -0.02 0.04 0.02 0.92 1.22 1.10 wi29 -0.00 0.06 0.02 1.29 1.77 1.59 ha30 0 0.08 0.02 1.14 1.53 1.37 dj38 0.00 0.04 0.01 1.90 2.24 2.12 dantzig42 0.00 0.11 0.03 1.98 2.45 2.28 swiss42 0.00 0.11 0.03 1.75 2.04 1.91 att48 2.16 2.40 2.25 9.07 10.60 10.11 eil51 0.00 0.06 0.03 1.94 2.22 2.11 berlin52 0.00 0.14 0.05 1.90 2.24 2.14 wg59 0.00 0.18 0.07 2.73 3.17 2.99 st70 0.00 0.14 0.04 3.18 3.60 3.48 eil76 0.01 0.09 0.06 3.04 3.59 3.39 pr76 0.01 0.26 0.11 3.20 3.59 3.45 rat99 0.01 0.12 0.06 14.68 16.28 15.80 kroA100 0.00 0.32 0.12 5.25 6.00 5.76 kroB100 0.02 0.27 0.10 5.05 5.64 5.44 kroC100 0.02 0.22 0.10 5.19 6.14 5.90 kroD100 0.03 0.21 0.10 5.08 5.70 5.50 kroE100 0.02 0.25 0.09 5.24 5.87 5.63 rd100 0.01 0.20 0.11 4.65 5.25 5.07 eil101 0.04 0.15 0.09 3.51 3.85 3.71 lin105 0.01 0.26 0.13 5.95 6.50 6.28 pr107 0.01 0.20 0.09 9.28 10.36 9.95 pr124 0.01 0.41 0.14 8.74 9.59 9.26 bier127 0.07 0.39 0.20 3.60 3.83 3.74 ch130 0.03 0.20 0.12 5.37 5.85 5.70 pr136 0.09 0.36 0.20 6.18 6.70 6.49 pr144 0.01 0.39 0.15 10.71 11.60 11.24 Roy et a., /Decis. Mak. Appl. Manag. Eng. 2 (2) (2019) 100-111 108 In Table 2, it is used to calculate the goodness of parameter of selection (ps) in NMGA. It shows that to get the optimal solution of the standard TSP kroA100, psindicates the given space better for ps= 0.34. Table 2. Parameters for NMGA of kroA100 instance Instance pc pm popsize Gen result cpu-time(sec) 0.34 0.01 4673 21417 5526 0.02 4065 21344 5733 0.001 3407 21322 5405 0.003 4957 21298 5373 0.005 2427 21294 5380 0.007 2173 21316 4810 0.008 3376 21285 4470 0.009 3068 21384 4214 kroA100 0.2 0.01 70 2384 21322 3331 0.25 2787 21386 3232 0.30 4958 21333 3916 0.35 2612 21285 5574 0.40 4868 21294 6164 0.45 3883 21412 6256 0.50 3708 21349 6149 0.55 3505 21365 4407 0.60 4406 21316 4438 0.70 4467 21535 6640 0.75 4320 21334 6874 0.80 4975 21831 9982 0.85 3905 21335 9840 0.34 0.01 50 3438 21390 3175 60 4638 21474 8382 kroA150 0.06 0.37 0.18 6.91 7.80 7.52 kroB150 0.08 0.34 0.20 6.92 7.88 7.58 pr152 0.14 21.88 13.26 11.31 11.94 11.67 u159 0.03 0.27 0.16 8.13 8.80 8.45 qa194 0.09 0.27 0.17 2.28 3.20 2.23 rat195 0.06 0.25 0.16 32.79 35.82 34.81 d198 0.10 0.29 0.20 9.20 10.05 9.66 kroA200 0.13 0.45 0.26 8.95 9.73 9.37 kroB200 0.16 0.42 0.28 8.73 9.43 9.14 ts225 0.08 0.35 0.19 10.13 10.75 10.49 tsp225 0.07 0.27 0.17 8.30 8.87 8.61 pr226 0.11 0.84 0.23 16.82 18.62 18.12 gil262 0.12 0.37 0.24 9.03 9.58 9.33 pr264 0.20 0.41 0.30 17.90 20.19 19.37 a280 0.15 0.37 0.27 10.61 11.33 11.00 pr299 0.12 0.37 0.24 12.85 13.74 13.35 lin318 0.17 0.40 0.29 11.57 12.20 11.94 Optimal decisions on pricing and greening policies of multiple manufacturers under… 109 Instance pc pm popsize Gen result cpu-time(sec) 72 4453 21322 4470 85 3460 21322 4952 110 2706 21285 7664 150 4700 21294 8974 It is evident from Table 2 that, for standard three parameters pc, pm and pop − size, our proposed algorithm NMGA give us optimum or near optimum result easily. Thus the importance of the parameters is discussed in Table 2. 6. Conclusion In this paper, a novelty introduced in GA regarding selection and crossover, called Novel Memetic Genetic Algorithm (NMGA). NMGA is tested with few test references from TSPLIB and examined with classical GA. In NMGA, Boltzmann probabilistic selection and a new four parents crossover are worked with ergodic mutation. The concept of MA is not new in the area of TSPs, but the idea of multi-parent(four) crossover on the basis of the memetic concept is new, establishes our proposed algorithm as highly NP-hard combinatorial optimization problem. Realistically, it is true that multi-parent crossover especially four parent crossover may not be so much truthful than two parent crossover for a specific problem or the complexity may be high than other. But the numerical analysis proves the efficiency of our proposed algorithm. 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