SYMMETRIC TRAVELING SALESMAN PROBLEM

Transcription

1 Chapter 6 SYMMETRIC TRAVELING SALESMAN PROBLEM 1 1 The part of this chapter has been published as Randomized gravitational emulation search algorithm for symmetric traveling salesman problem, Applied Mathematics and Computation,Vol.192, , 2007.

2 CHAPTER 6. SYMMETRIC TRAVELING SALESMAN PROBLEM Introduction Many and varied are the applications of Symmetric traveling salesman problem(stsp). To name a few Scheduling problems, vehicle routing problems [130], mixed Chinese postman problems, integrated circuit board chip insertion problem [37], printed circuit board punching sequence problems [136] and a wing-nozzle design problem in the aircraft design are worth mentioning. Suppose a salesman has to visit n cities. He wishes to start from a particular city, visit each city only once and then return to his starting point. Minimizing the total traveling distance by selecting a sequence of all n cities ( distance between cities i and j is same as j and i) is known as a symmetric traveling salesman probem. Mathematical description of STSP: Consider a symmetric traveling salesman problem with n cities to visit. Let c ij be the distance from city i to j. Moreover, let x ij be 1 if the salesman travels from city i to city j and 0 otherwise. Then, the symmetric traveling salesman problem can be formulated as follows: n n min i=1 j=1 c ij x ij (6.1) such that n x ij =1,j =1,..., n, (6.2) i=1

3 105 n j=1 x ij =1,i=1,..., n, (6.3) i P j P,i j x ij P, P {1, 2,..., n}, 2 P n 2 (6.4) x ij {0, 1} (6.5) Although there exist several heuristic procedures and branch and bound algorithms for it, this problem is still a difficult NP-complete problem. The main purpose of this chapter is to enhance the GELS algorithm to a Global Algorithm by introducing randomization concept in the available parameters of GELS and to show its effectiveness. A justification is also given for the improvement of the GELS through the results, by solving STSP of various sizes from 5 to Improvement of GELS through randomization has also been established by making a comparative study of other available heuristic algorithms too. This chapter is organized as follows. Section 6.2 explains previous work in STSP. In section 6.3, a short description of GELS is presented. It also includes GELS as described by Webster[200]. In section 6.4, the RGES algorithm, based on GELS with concept of randomization is described and a rationale for its anticipated good performance is provided.it also includes probabilistic performance analysis of RGES. In section 6.5, results of computational experiments are given, results are tabulated, efficiency of the algorithm is established through comparative study, parameters used in various algorithms are tabulated and algorithm features are elaborated. Finally section 6.6 comprises merits and concluding remarks.

4 Previous Work The traveling salesman problem(tsp) has been studied intensively during the last 50 years and many exact and heuristic algorithms have been developed. Comprehensive review of the techniques developed for the TSP can be found in [8, 132] Exact Algorithms Direct search algorithms of exponential complexity, which give optimal solutions, but are applicable only to a small number of nodes and Backtracking can search all possible tours. This method has small memory requirements, but largest execution time. One of the earliest exact algorithms is due to Dantzig[48] in which linear programming(lp) relaxation is used to solve the integer formulation by adding suitably chosen linear inequality to the list of constraints continuously. Other famous algorithm available in literature are dynamic programming, branch and bound method, branch and cut. Branch and bound (B&B) algorithms are widely used to solve the TSPs. Several authors have proposed B&B algorithm based on assignment problem (AP) relaxation of the original TSP formulation. These authors include Eastman [61], Held and Karp [96], Smith et al. [186], Carpaneto and Toth [33], Balas and Crhistofides[12]]. Some branch and cut (B&C) based exact algorithms were developed by Crowder and Padberg [47], Padberg and Hong [161], Grotschel and Holland [87]. Not many heuristic algorithms of polynomial complexity, which give solutions near optimal, but are applicable to large number of nodes are found in literature. The symmetric version of the traveling salesman problem, in which bi-directional distances between a pair of cities are identical, has been studied by

5 107 many researchers and the understanding of the polyhedral structure of the problem has made it possible to optimally solve large size problems[132]. Prim or Kruskal algorithm [8]gives us solutions to STSP which is not more than twice of optimal solution length. Its applicability is bounded only to Euclidean TSP. This algorithm works in polynomial time Heuristic Algorithms Nonetheless, efforts to develop good heuristic procedures have continued because of their practical importance in quickly obtaining approximate solutions of large size problems. Heuristic methods such as the K-opt heuristic by Lin and Kernighan [135] and the Or-opt heuristic by Or[158] are well known for the symmetric traveling salesman problem. There also have been a few attempts to solve it by various meta-heuristics such as tabu search [125, 194], genetic algorithms [27, 38, 170, 171, 180], simulated annealing algorithms[23, 82] and ant colony optimization algorithms[55, 56]. Interestingly, Brest and Zerovnik [25] report that a heuristic algorithm, based on arbitrary insertion, performs remarkably well on benchmark test problems in terms of both solution quality and computation time. Karp[115] also presents a heuristic, patching algorithm, and shows that the heuristic solution asymptotically converges to the optimal solution of the asymmetric TSP.

6 6.3 Gravitational emulation local search( GELS) Algorithm Preliminary work done with GELS For the study of GELS, a conceptual frame work was first developed by Voudouris and Tsang [198] and that preliminary designed algorithm was called Gravitational Local search Algorithm (GLSA). As there are a number of places in the literature where abbreviations, GLS are used to refer guided local search [198], this name was later changed to GELS. Two separate versions of GLSA were implemented using C language. They have two key differences. The first version, dubbed GLSA1, used as its heuristic Newton s equation for gravitational force between two objects and the pointer object moved through the search space one position at a time, while the second, dubbed GLSA2, used as its heuristic Newton s method for gravitational field calculation and the pointer object was allowed to move multiple positions at a time. These issues are precisely elaborated in [200]. Both the procedures had operational parameters that the user could set to tune its performance, the parameters used were density(dens), drag(drag)[182], friction(fric), Gravity(GRAV) [182], initial velocity (IVEL), iteration limit(iter), mass(mass), radius(radi), silhouette(silh) and threshold(thre) among which GELS uses only four parameters namely, velocity, iteration limit, radius and direction of movement. Webster [200] has beautifully designed this robust algorithm (GELS) which had overcome the following difficulties (i) increased the number of iterations caused by hypothetical search space (ii) poor solution caused by random determination of objective function values of neighboring solution Webster assumed objective function

7 109 values of neighboring solutions as the functions of earlier solutions being randomly determined. Because of this objective function values of the invalid solutions of the problem were tactically avoided in the beginning itself. He has also avoided the sensitive and redundant parameters of GLSA, because with these parameters finding the points of equilibrium was extremely difficult GELS algorithm Parameters used in GELS algorithm (a) Max velocity: Defines the maximum value that any element within the velocity vector can have to prevent velocities that became too large to use. (b) Radius: Sets the radius value in the gravitational force formula; used to determine how quickly the gravitational force can increase (or) decrease. (c) Iterations: Defines a number of iterations of the algorithm that will be allowed to complete before it is automatically terminated ( used to ensure that the algorithm will terminate) (d) Pointer : It is used to identify the direction of Movement of the elements in the vectors. Gravitational Force (GF) GELS algorithm uses the formula for the gravitational force between the two solutions as GF = G (CU - CA ) / R 2 Where G = ( Universal constant of gravitation)[182] CU = objective

8 function value of the current solution CA = objective function value of the candidate solution. R = value of radius parameter. 110 Webster s algorithm Webster [200] designed GELS algorithms by using two methods and two stepping modes. They are as under : GELS11: (i) to find the GF between single solution and current solution. (ii) Movement within current local search. GELS12 (i) to find the GF between single solution and current solution (ii) movement to outside of the neighbourhood GELS21 (i) to find GF between each solution within neighbourhood (ii) movement within current local search GELS22 (i) to find GF between each solution within neighbourhood (ii) Movement to outside of the neighbourhood. The GELS algorithms have various distinguishing features from other algorithm like GA, SA, HC, in terms of search space, number of iterations, etc. During the development of GELS the setting of aforesaid parameters have been arrived at through trial and error and some settings caused repetitive visits in a neighbourhood, while others caused large numbers, thus causing the algorithms to behave erratically. Webster, after a number of tests had settled the algorithm with 10 for maximum velocity, 4 for radius and for the iterations. After a number

9 111 of tests we have settled the RGES algorithm with 7 for maximum velocity, 7 for radius and 1000 for iterations. The RGES algorithm and its explanation is given in section Distinguishing properties of GELS algorithm: 1. Introduction of the velocity vector and relative gravitational force in each direction of movement emulated the acceleration effect of the process towards its goal. 2. Multiple stepping procedures helped the solution pointer to show the next direction leading to the other solutions. 3. The algorithm is designed in such a way that it terminates on the two conditions either all the elements in the velocity have gone to zero or the maximum allowable number of iterations has been completed. 6.4 Randamized Gravitational Emulation Search Algorithm In addition to the introduction of Gravitational emulation as a search algorithm, in order to strengthen to solve STSP, the following things have been given due considerations: (i) initiation and construction of neighborhood (ii) Evaluation of fitness and updating.

10 Initiation and neighbourhood RGES search technique starts out with a guess of n likely candidate, actually chosen at random in the search space. This candidate is called the current solution (CU). The RGES algorithm generates a neighbourhood.the algorithm is given below The basic mechanisms are similar to the GELS in the construction of neighbourhood. Step 1: groups(l) = size(n) / radius(r) Step 2: set random numbers for each groups ( maximum velocity ). Step 3: for i = 1 to l Swap the members of group based on the velocity end Step 4: form the neighbourhood by combining the members of groups. Step 5: total number of candidate solutions(total) = r1 * r2 * r3 *...*rl. ALGORITHM 6.1 Algorithm for Neighbourhood generation. For n city STSP, the total number of cities is divided into l groups by means of parameter called radius(r) and a random number (r i )toeachofl groups such that r i R, i=1tol. Here ( R - ri ) for each i does not undergo any swapping. The number of cities in which the swapping takes place is the total number of candidate solutions in the neighbourhood. During swapping within a group, a number of solutions are generated and objective values of the tours are also found.

11 Evaluation of fitness and updating The current solution for the next iteration is obtained by updating the objective function value by finding the lowest objective value in the neighbourhood. The algorithm is given below. Update the velocity of each group of the current solution with the GF (between lowest objective value function (LOVF) and the objective value of the current solution (CS) If the LOVF found in the iteration one is less than the CS, the LOVF is considered as the CS for the next iteration. If not, we choose one candidate solution as CS randomly from candidates solutions as discussed earlier. The algorithm is given below. Step 1: best = obj value of the first candidate solution in the neighbourhood Step 2: best candidate solution = first candidate solution in the neighbourhood Step 3: for i = 2 to total obj ith = the obj value of the ith candidate soluition if(objith best) best = obj ith, best candidate solution = ith candidate solution end Step 4: update the velocity of each groups, Step 5: using GF = G (CU - CA ) / R 2 Step 6: obj current = obj value of the current solution Step 7: if ( best < obj current ) current solution = best candidate solution else current solution = random ( neighbourhood) end ALGORITHM 6.2 Algorithm for Evaluation and Updation

12 RGES Algorithm We repeat the process till either velocity vanishes or number of iterations exceed the assigned value The complete algorithm for RGES is given below. Step 1: Set the parameters(radius, velocity, direction of movement) Step 2: Set the current solution Step 3: While( termination condition are not met) do -Construct neighbourhood solutions based on direction of movement -Compute gravitational force between current solution and -best(candidate) solution in the neighbourhood -Update the velocity and direction of movement by using gravitational Force End ALGORITHM 6.3 RGES Algorithm for STSP Probabilistic Performance Analysis In this section we prove our RGES converges with probability one to the global optimal solution. In order to describe the RGES, we need the following definitions: A neighbourhood structure is a mapping N form M into total ( M is a finite set, i.e. for each solution x it defines a neighbourhood N(x) of x and each y N(x) is called a neighbour of x. In the case of the STSP, the neighbourhood of given tour x can be defined as the set of tours which can be generated by the RGES ( ALGORITHM 6.1). A generation mechanism is a rule of selecting a solution y from the A neighbourhood N(x) of given solution x. In the context of RGES such a generation rule is usually called an updation rule. The generation mechanism can be described

13 115 probability matrix R such that R(x, y) =P {X t+1 = y X t = x} (6.6) where X t denotes the state of the system at time( iteration) t. Clearly R(x, y) > 0 if and only if y N(x). By(6.6) a Markov chain is defined over the set M of feasible solutions. However, in order to solve STSP this Markov chain has to be modified by some acceptance criterion so that good solutions are selected more often with higher probability than bad ones. A local optimal solution is an x M such that F (x) F (y) for all y N(x) while a global optimal solution is defined by x M such that F (x) F (y) for all y M. For the RGES not to get stuck in a local optimum(which is not globally optimal) it is necessary to accept also deteriorations of the objective function with probability. A state y is reachable from state x if there exixts z 1,z 2,..., z m M such that z 1 N(x), z 2 N(z 1 ),..., y N(z m ). The algorithm starts from an initial candidate solutions X 0 = {x 1 0,..., xp 0 }. In each iteration t, from the current parent candidate solutions X t agroupof candidate solutions X t of c candidate solutions is generated by updation, where at least one child is generated by an updation. Then the following selection rule is chosen. (a) Select x as best of all the p+c solutions x 1 t,..., xp t, x 1 t,..., x c t. (b) Select x 2 t+1 arbitrarily among all the candidate solutions x 1 c t,..., x t which have not already been selected in step (a). (c) Select x 3 t,..., xp t+1 if p>2 by any selection rule. (c1) Select those candidate solutions( not already selected) with the

14 116 best values of the objective function. (c2) Select those solutions ( candidate solutions not already selected) with the best value of the objective function. already selected) already selected). (c3) Select p-2 solutions arbitrary among the candidate solutions(not (c4) Select p-2 solutions arbitrary among the candidate solutions(not From the classical literature of Markov processes one can also obtain expressions for the time to absorption. We mention the following: Let the transition matrix of the new Markov process {Y t } be given by P = 1 0, whererisak 1 matrix, T is a k k,0 is a 1 k-zero matrix, and k + R T 1 is the dimension of the state space of {Y t }. Then, with P(n)denoting the vector of probability that the time to absorption in state zero starting in the transient states is n, we get is P (n) =T n 1 (I T ) e where is the k-dimensional unity vector. Hence, the average time to absorption π [ n=0 nt n 1 ](I T ) e = π [ n=0 T n ] e where π is the initial distribution of Y. However, since k is extremely large in practical problems and therefore T cannot be computed any more, this formula does not seem to be of much practical use. For the same reason, we refrain from applying some further sophisticated results from the theory of absorbing Morkov chains.

15 117 We can now formulate the following global convergence result: Theorem Let the set of feasible solutions M be finite and assume, that for all x, y M the state y is reachable from x by the updations considered. Then the RGES with selection rule in any of the variants (c1)-(c4) has the following property limp {At least one solution in X t is globally optimal} =1. Proof. Define a new Markovian-chain Y t as follows. The state y M P is given by the combined vector of p elements of the original state space M. However, for technical purpose we make the following exception for state y 0 : the system is in state y 0 if at least one globally optimal solution x is in the current candidate solutions. Now the sequence of populations computed by RGES can be described by a Markov-chain in the above framework. The state Y t in iteration t is the combined vector of all solutions in group of candidate solutions X t. On the other hand Y t = y 0 if one of the elements x 1,..., x p of X t is a globally optimal solution. According to part (a) of the selection rule the state y 0 is absorbing. This is because of the assumption of our theorem each globally optimal solution is reachable from any other solution. According to part(b) of the selection rule this implies that (absorbing)state y 0 of the new Markov chain is reachable from any other state. This implies that every state y y 0 is transient and lim t P {Y t = y 0 } =1.

16 Experimental Analysis RGES solutions of Test Problems Fischetti [67] describe a branch-cut-algorithm to solve the symmetric Generalised TSP, using partitioning method to 46 standard TSP instances from TSPLIB library[[174] since they provide adequate description of the partitioning method to provide optimal objective values for each of the problem. Here we apply our heuristic to the selected eleven problems in order to ensure the robustness of the proposed GELS algorithm and compare the results with other GELS algorithm proposed by Barry Webster. The results of the problems having sizes 5,6,10,11,15,20,21,24,51 and 76 are shown in TABLE 6.1. Solution of problems having other sizes are not shown here for the sake of brevity The randomization procedure introduced in the GELS algorithm ensure the efficiency of the algorithm to handle STSP of large size up to 76, for each problem we ran RGES 5 times to examine algorithms performance and its variations from trail to trail. TABLE 6.1 summarizes the results. For each of the solved problem the columns in the table are finished as follows: Problem: The name of the test problem the digits of the beginning of the name give the clusters (m) these at the end give the number of nodes (n) Opt obj val : The Optimal objectives value for the problems reported here are taken from TSPLIB [174] some of which have been obtained through direct search method Opt: The number of trials out of 5 in which the RGES found the optimum solution Best: The number of trails out of 5 in which the RGES found its best solution (for problems in which the optimal solution was found, this is equal to opt column).

17 119 Table 6.1.RGESresults Problem opt N.O.O N.O.B mean min max value Pct value Pct value Pct 5sample sample sample sample eil eil kroa Gr Gr Eil Eil N.O.O = number of optimum N.O.B = number of best Pct = Percentage of deviation from optimum No. of runs = 5 Max no. of iterations = 1000 Mean, Min, Max: The mean, min and maximum objective values returned in the 5 trials ( the value column) and the respective percentages above the optimal value (in pct column) The RGES found optimal solutions in at least one of the 5 trials for all the 11 tested problems. For 9(81 percentage) of the problems the RGES found the optimal solution in every trials. The RGES solved 11 problems and in only one case Eil 76 it returns a solution more than 2 percentage above the optimal. The heuristic to return consistent solutions for smaller size problems, for large size problems if returns solutions that vary a little bit best higher (but close to each other in objective value) Comparison with other Heuristics In order to bring out the efficiency of the proposed RGES bring algorithm set of 49 bench mark problems have been solved by the other several algorithm (SA, HC, GA, GELS11, GELS 12, GELS 21, GELS 22) reported in the literature,

18 120 these algorithms have been coded in C Language. The results of solutions of these problems under various category are shown in TABLE 6.3 and TABLE 6.4. The values of the parameters into each of the algorithms to solve the above problems use anticipated in TABLE 6.2. Table 6.2. parameters set for various methods Methods Parameters GA Population size =10, crossover(1-point) = 0.9 mutation(1-point) = 0,05 iteration = 1000 SA Temp =2000, reconfig interval = 10 attempt interval =10 annealing rate =0.01 threshold = 0.01 GELS Radius =10 velocity = 4 iterations = 1000 RGES Radius=7 velocity = 7 iteration =1000. To compare the results of RGES algorithm with other algorithms and to test its efficiency the number of iterations is fixed 1000, number of trials 5 and only best solutions out of 5 are taken into account. Based on the data from the experiment and the solutions from TABLE 6.1, TABLE 6.3 and TABLE 6.4, a number of observations can be made: SA, GELS22, GELS21, HC, GA are provided better solutions or near optimal solutions only. However, the RGES algorithms maintains both optimality and the quality of the solutions. Hence the introduction of randomization concept to GELS algorithm is found to be advantageous in two ways. 1.Randomization enables the algorithm to escape from the local search and pave a way leading to global search with the help of established GELS. 2. In all the problems up to the size 2392, RGES provide optimum solutions.

19 121 Table 6.3. RGES versus Other Algorithms pbm SA HC GA GELS12 GELS22 GELS21 GELS11 RGES Eil Eil Gr Kroa Eil Lin Gr Gr Pr Kroa Pr Rat Kroa tsp Gil A Pr Lin Rd Fl Gr Pr Pcb Att Features of Algorithm When examining the relative contribution of the features of RGES algorithm to the heuristic overall performance, we find that swapping in terms of groups by using random numbers is effectively utilized in making local search to be the global search, which the GELS in its two by two swapping gets entangled into the neighbourhood of the current solution and unable to escape from the local domain. We also find that RGES exhibits an excellent performance over GELS in

20 122 Table 6.4. RGES versus Other Algorithms pbm SA HC GA GELS12 GELS22 GELS21 GELS11 RGES Ali Pa Rat P D Gr U Rat Pr Sil U Vm Pcb Rl Rl Fl U Fl D Vm U D U U Pr Average deviation terms of obtaining optimum solutions even for larger size problems at the same computational cost as in GELS. In our algorithm, if R is constant then there are O(n/R) groups. Hence it appears the product r x i is in O (R n/r ) and so grow exponentially. But since R is chosen as a x n, for finite n, O(R n/r ) becomes O(R 1/a ) some finite a hence

21 123 RGES becomes polynomial time algorithm. 6.6 Conclusion Webster[200] used two stepping modes and two methods to enhance Gravitational emulation properties of a local search algorithm to compete with other algorithms available in literature to provide best solutions, though not focusing the optimality. This new proposed algorithm (RGES) is capable of bringing the optimal solutions without affecting the quality of GELS and also paves a way for Global search through randomization. In our attempt we have solved 49 bench mark problems up to the problem of size The performance of RGES (TABLE 6.3 and TABLE 6.4) has been compared with other heuristics: Simulated Annealing, Hill climbing, Genetic Algorithm, GELS11, GELS12, GELS21, and GELS22. The experimental data shows that RGES maintains optimality and quality. In the next chapter we design the binary version of gravitational emulation search based meta heuristic to solve vertex covering problem.