Title of the Paper (18Pt Times New Roman, Bold) s7

Extended Simulated Annealing for the TSPs

KYUNGHEE LEE Dept. of Information and Communication, PyongTaek University 111, YongI-Dong, PyongTaek-Si, KyungKi-Do, KOREA

Abstract: - An extended simulated annealing(ESA), based on grand canonical ensemble, is proposed. An ESA is used to solve the augmented traveling salesman problem(ATSP) and the multiple traveling salesman problem(MTSP). Experimental results show that ESA has salient features such as simplicity and ability to find high-quality solutions as simulated annealing has.

Key-Words: - Grand Canonical Ensemble, Optimization, Extended Simulated Annealing, TSP

1 Introduction the set, N1, N2,…, molecules and is in the energy state The traveling salesman problem(TSP) is one of the Ej(N1, N2,…,V) is hardest combinatorial optimization problems and (E j ( N1 ,N2 ,...,V ) k Nk ) / kT remains NP-hard. Up to now, many researchers tried e k P (N , N ,...;V ,T,  ,  ,...)  to solve large-scale multiple traveling salesmen j 1 2 1 2 (V ,T,  ,  ,...) problems for real world applications such as multi- 1 2 (1) depot vehicle routing problems and topological design of computer networks using many heuristic where (E j (N1 ,N 2 ,...,V ) k N k ) / kT methods[1][2]. k (V ,T, 1 ,  2 ,...)  e Most of the heuristic methods use a local search j,N1 ,N2 ,... (2) algorithm which depends on a neighborhood structure or a complex transform scheme[2][3]. Simulated annealing(SA) is a powerful algorithm for solving 2.2 ATSP and MTSP combinatorial problems[4]. But SA can be applied The augmented traveling salesman problem(ATSP) only to the case of standard TSP because SA is based and the multiple traveling salesman problem(MTSP) on the canonical ensemble which has the properties are extensions of the well-known TSP. In ATSP, n of fixed number of molecules with fixed volume. customer cities with their own profits are given. A salesman at a base city is required to visit k customer cities, kn, and return to the base city. A cost is 2 Grand Canonical Ensemble, ATSP, incurred and a profit is gained if a salesman visits and MTSP customers. The ATSP is to determine the tour route to maximize the total benefit acquired from visited cities. The benefit and the cost function for solving 2.1 Grand Canonical Ensemble the ATSP are defined as the following equations: The system in grand canonical ensemble(GCE), one of the representative ensembles, has a fixed volume, BATSP  i Ni CATSP , isolated temperature, predefined chemical potentials, i (3) n1 varying number of molecules with opened energy C ATSP   d(i,i 1)  d(n,1) barrier. We consider an isothermal system with any i1 (4) number of components is characterized by the thermodynamic variables, volume(V), temperature where i, Ni, and  denote the profit of the city i, the (T), chemical potentials(1, 2,…) and molecules(N1, number of appearance of city i in the tour route, and N2,…). The probability that such a system contains the weighting factor of the cost, respectively. d(i,j) exchanging a molecule in the system itself denotes the (Euclidian) distance between city i and j. with a molecule in the other systems. The MTSP can itself be generalized to a wide variety of routing and scheduling problems. In MTSP, there are n customer cities with m salesmen. Each salesman at a base city is required to visit k customer cities, kn-m+1, and return to the base city and each customer city is to be visited by one salesman only. A cost is incurred if a salesman is actively used to visit customers. The MTSP is to determine the tour route to maximize the total benefit (a) (b) acquired by all the salesmen.

m i B   i N i CMTSP i i1 (5) n1 i CMTSP   d( j, j 1)  d(n,1)  Si j1 (6)

| Li | | Li | Si  ( m  Log10 m ) | Li | | Li | (c) (d) i1 i1 (7) Fig. 1 Examples of perturbation schemes of GCE : where m denotes the number of salesmen. Si is the (a)Increasing, (b)Decreasing, (c)Internal swapping, entropy term to make the salesmen’s workload be (d)External swapping scheme distributed equally, and  is the weighting factor. Li denotes the number of cities that salesman i has These perturbation schemes are used in the visited. following ESA algorithm for the TSPs.

Step 1: Set Initial and final temperature(T) values. 3 Extended Simulated Annealing by Step 2: (For the ATSP) Select one of the above GCE perturbation schemes (i,ii,iii,iv) randomly and perturb a current tour route s to the route In GCE the states of the system are to form a Markov s’ by the scheme. chain by changing the number of molecules. GCE is (For the MTSP) Perform one of the an open system and this gives rise to the following following steps(2.1, 2.2) randomly. four perturbation schemes to decide a next state of the (2.1) Select two tour routes randomly, perturb one system with N molecules(see Fig. 1): route by the perturbation scheme(ii) and another by the perturbation scheme(i) to (i) Increasing scheme : make a system to have make the route s’. It should be noted that the N+1 molecules by extracting one molecule city to be deleted in one route must be added from the other systems and putting it into the into another route. system itself. (2.2) Select one tour route randomly, perturb it by (ii) Decreasing scheme : make a system to have the perturbation scheme(iii) to make the N-1 molecules by excluding one molecule route s’. from the system itself. Step 3: Compute the benefit B(s’) and compare it (iii) Internal swapping scheme : make a system to with the B(s) of the current route s, and let have the same number of molecules by the route s’ be the new route with probability interchanging a molecule with another

molecule in the system. ' 1 if B(s )  B(s) (iv) External swapping scheme : make a system to P(s '  s)   (B(s' )B(s)) / kT ' have the same number of molecules by e if B(s )  B(s) (8) Step 4: Repeat step 2, 3 until the equilibrium state 0 1 2 3 4 0 1 2 3 4 occurs and anneal with an annealing schedule. 5 6 7 8 9 5 6 7 8 9

Step 5: Repeat step 2, 3, and 4 until final T is 10 11 12 13 14 10 11 12 13 14 reached.

15 16 17 18 19 15 16 17 18 19

20 21 22 23 24 20 21 22 23 24 4 Experimental Results (a) (b) An ESA algorithm is tested with 400 cities, denoted by circles, having their own profit values(i) on the ATSP and the MTSP. The tests are performed on an Progress of the convergence(TSP)

IBM PC-586(400MHz). They take less than 51 300 minutes to output the result of the tests with the 250 starting temperature=1.0*103, the final temperature =1.0*10-2, =1.0, respectively. The temperature(T) is 200 decreased as T = 0.95T. 150

Fig. 2 shows the result of the SA in solving 100 traditional the TSP with 25 cities and the same parameters. There is, in the beginning, a large degree 50 No of iteration of randomness at high temperature, and as the 0 1 22 43 64 85 106 127 148 169 190 211 temperature is decreased the system converges to a stable state. (c) Fig. 3 shows the effectiveness of the ESA in solving the ATSP with 400 cities. There is a large Fig. 2 The result of the TSP with 1 salesman(by SA) : degree of randomness at high temperature and the (a)Initial tour route, (b)Final tour route, (c)Progress system converges to a stable state same as in Fig. 2. It of the convergence is noticed that cost means the negative value of benefit in ATSP. The same output result also comes 0.31 8.80 0.62 0.12 1.56 1.74 0.43 0.24 0.78 2.50 0.94 0.15 0.66 7.85 0.87 1.89 0.09 3.78 0.77 0.57 out when different initial tours are tried. It should be 0.53 7.18 1.29 5.43 1.44 6.50 7.95 0.28 2.23 0.97 0.99 0.02 1.39 8.02 1.95 6.54 0.94 0.04 0.95 0.88 noted that some cities (dark gray colored cities in Fig. 1.35 0.25 0.01 0.44 7.90 0.89 1.71 1.71 7.50 0.65 0.69 1.59 9.02 1.49 0.11 2.76 2.24 0.01 0.92 0.54

3(a)) with the profit value of greater than 1.0 are not 1.64 1.03 0.33 0.11 0.35 0.57 8.41 7.83 3.00 6.20 0.51 0.21 8.66 1.20 0.96 0.31 0.79 0.54 1.38 0.01 included and some cities(light gray colored cities in 1.50 5.17 1.00 1.65 0.67 1.18 5.00 0.49 0.84 2.26 0.01 8.52 6.03 0.81 0.24 0.17 3.24 1.36 7.51 1.76 Fig. 3(a)) with the profit value of less than 1.0 are 5.17 9.27 0.25 1.94 2.42 2.78 0.08 0.65 8.43 1.43 0.49 5.48 1.07 1.99 0.06 2.85 9.04 0.07 1.13 0.98 included in the final tour. This shows that a city is 1.55 1.87 6.49 0.90 1.19 1.11 4.32 1.99 3.20 4.91 0.79 0.83 1.73 0.54 0.94 0.25 0.93 1.20 8.89 0.62 included in the tour not just because of its high profit 1.30 0.62 1.53 0.96 1.88 4.76 5.93 9.53 0.15 1.00 1.52 1.10 1.23 0.43 0.98 1.24 0.09 2.47 0.13 3.22 1.93 0.04 5.39 0.80 1.06 0.51 0.44 4.97 0.53 0.22 0.11 0.04 0.38 2.06 0.74 1.92 0.74 1.16 1.44 1.83 and low cost but because of the high profits of its 0.23 9.95 1.24 0.86 4.03 1.90 4.29 1.03 1.02 1.07 0.66 9.50 4.45 0.90 1.28 0.96 7.91 8.93 1.25 9.87

neighborhood included in the tour. 8.96 1.69 6.34 7.55 9.77 0.01 0.15 0.98 5.25 0.56 1.59 0.95 1.21 8.14 0.94 1.23 2.35 1.85 5.99 5.53

Fig. 4 shows the results of the test on the MTSP 1.71 0.68 4.38 1.68 5.58 0.23 2.36 3.16 8.37 0.35 1.92 3.96 5.09 0.63 0.36 8.88 0.92 1.59 1.50 0.69 with 3 salesmen. The parameters used in this test are 0.66 1.20 0.46 0.91 1.01 3.56 0.24 1.96 5.81 3.27 6.94 8.10 3.52 0.11 1.39 5.72 7.20 0.50 2.44 1.92

1.73 0.50 0.76 0.12 0.18 3.47 0.32 1.43 2.27 1.73 8.87 1.74 9.68 1.04 1.55 8.08 9.39 7.09 1.08 0.91 the same as the ones used before except i=1.0 and =80.0. The final tour routes, 3 colored routes for 3 1.52 0.34 7.70 3.62 0.77 0.14 6.42 0.67 0.07 2.62 1.94 3.81 1.98 0.19 0.23 0.74 0.62 0.32 0.07 0.15 salesmen, are shown in Fig. 4(a). This shows that the 5.77 0.44 0.89 0.26 0.46 1.04 1.69 4.96 0.52 0.55 0.42 0.24 0.49 0.23 0.07 1.09 7.41 4.49 8.12 4.71 3.83 0.30 3.32 1.71 0.33 0.07 9.44 1.71 7.55 0.12 1.69 1.89 9.54 0.83 0.98 0.47 0.88 1.34 8.83 9.83 entropy plays a role to help the convergence. It is also 0.70 1.20 8.54 0.21 3.52 9.64 1.70 3.13 0.69 0.50 4.19 4.07 1.00 0.83 6.36 1.64 0.74 9.27 4.90 1.35 noted that ESA can solve the MTSP without any 1.56 0.87 0.87 0.27 0.98 0.18 0.32 0.67 3.78 6.76 1.75 1.32 1.14 6.28 0.84 0.93 0.65 3.51 1.31 0.77 transformation into a standard form, unlike most 4.64 1.96 5.27 0.97 1.21 1.44 0.03 0.62 1.61 1.57 1.67 0.07 1.18 0.98 0.66 0.32 4.77 8.52 0.86 0.44 heuristic algorithms. (a) Progress of the convergence (ATSP) 4 Conclusion In this paper we propose ESA, based on GCE, and

4500 Profit show how ESA can be used to solve the ATSPs and 3500 Benefit

2500 Cost the MTSPs which can not be solved by original SA. # of Cities 1500 SA can find the configuration of all particles arranged 500 themselves only, ESA can also find the configuration -500 No. of iterations 1 51 101 151 201 -1500 of all or partial particles arranged themselves -2500 according to the problem conditions. Therefore ESA -3500 for ATSP causes the unselected cities (1,4,7,8,9, etc.) in final route and makes the total cost(see Fig. 3(b)) (b) reduced. For the efficiency of ESA we presented an augmented TSP that can be applied to most Fig. 3 The result of the ATSP(by ESA) : (a)Final tour combinatorial problems in real world, and showed it route, (b)Progress of the convergence. also can be solved by ESA. ESA also has no need to transform MTSP into a standard TSP as most heuristic algorithms have. Experimental results, such as Fig. 3 and Fig. 4, show that ESA has salient features such as simplicity(in Section 3) and ability to find high-quality solutions as

5.1 SA has in general. By adding the entropy constraints 7 to the energy function, MTSPs can be solved by ESA effectively. Another attractive feature of the ESA, when it is applied to MTSP, is that finding every salesman’s route can be performed in parallel and can therefore take maximum advantage of parallel processing capabilities.

References: [1] Johnson, D., and McGeoch, L., Experimental Analysis of Heuristics for the STSP, The Traveling Salesman Problem and its Variations, Kluwer Academic Publishers, (a) Combinatorial Optimization Series, Vol. 12 , 2002 . Progress of the convergence (MTSP) [2] Guoxing,Y., Transformation of multidepot multisalesmen problem to the 1000 standard traveling salesman problem, European 500 0 Journal of Operational Research, 81, 1995, pp. -500 -1000 557-560. -1500 Total Benefit (3 Salesmen) -2000 Benefit (Salesman-1) [3] Gu, J., and Huang, X. , Efficient -2500 Entropy (Salesman-1,x1000) # of Visited Cities (Salesman-1) Local Search With Search Space Smoothing: A -3000

-3500 No. of iterations Case Study of the Traveling Salesman 1 51 101 151 201 Problem(TSP), IEEE Trans. on SMC, 24, (5), 1994, pp. 728-735. (b) [4] Kirkpatrik, S., and Gelatt. C., and Vecch, M., Optimization by simulated annealing, Fig. 4 The result of the MTSP with 3 salesmen(by Science, 220, 1983, pp. 671-680. ESA) : (a)Final tour routes, (b)Progress of the convergence(in case of a light gray colored route).