2-WAY AND DYNAMIC TIME OF TRAVELING SALESMAN PROBLEM WITH SIMULATED ANNEALING. Taufan Mahardhika ST. Analis Bakti Asih Kopertis 4

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1 2-WAY AND DYNAMIC TIME OF TRAVELING SALESMAN PROBLEM WITH SIMULATED ANNEALING Taufan Mahardhika ST. Analis Bakti Asih Kopertis 4 Abstract. In this paper we will discuss about delivering goods from manufacture to many of their customer and after finish it must head back to the manufacture. It just another type of Traveling Salesman Problem (TSP). The problem is we need to find the right time to deliver their goods and the right route to minimizing cost. A 2-way compete digraph is a fit model to describe what happen on the street. Dynamic time of traveling is a fit model to describe an estimated time arrives. In the end we used Simulated Annealing to solve the problem for every time of departure. Keyword. 2-way, Dynamic time, Traveling Salesman Problem. 1 Introduction A salesman will deliver goods from manufacture to his entire customer and when it is done the salesman will be back to the manufacture. The salesman must choose the route and the time to depart. The street that the salesman uses could be a 2- way street. We use 2-way because there are two condition of street at the same time. For example in the morning there is traffic on the street to the city but reverse there is less traffic when we go out of the city at that time. We assume that every place (manufacture or customer) connect each other. Even if in reality there is no path from one point to another point we can connect it in model but it had an infinity time to travel. So compete 2-way digraph is a fit model to describe what happen on the street. We use Dynamic time of traveling because when we traveling from one place to another place we will have faster time to arrive at the noon than at the morning when every people go to work or go to school. We use Hamilton Circuit [1] because we will deliver the goods from manufacture to their customer and come back to manufacture again. With a model that we describe, it just another type of Traveling Salesman Problem (TSP) [2,3] with 2-way digraph and dynamic time travel. Our goal is to solve this problem with Simulated Annealing[4,5]so we can reduce cost by choosing the right time to departure and the right route to deliver. 1

2 2 Simulated Annealing Simulated annealing (SA) is a generic probabilistic metaheuristic for the global optimization problem of applied mathematics, namely locating a good approximation to the global optimum of a given function in a large search space. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For certain problems, simulated annealing may be more effective than exhaustive enumeration provided that the goal is merely to find an acceptably good solution in a fixed amount of time, rather than the best possible solution. The name and inspiration come from annealing in metallurgy, a technique involving heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. The heat causes the atoms to become unstuck from their initial positions (a local minimum of the internal energy) and wander randomly through states of higher energy; the slow cooling gives them more chances of finding configurations with lower internal energy than the initial one. By analogy with this physical process, each step of the SA algorithm replaces the current solution by a random "nearby" solution, chosen with a probability that depends on the difference between the corresponding function values and on a global parameter T (called the temperature), that is gradually decreased during the process. The dependency is such that the current solution changes almost randomly when T is large, but increasingly "downhill" as T goes to zero. The allowance for "uphill" moves saves the method from becoming stuck at local optima which are the bane of greedier methods. The method was independently described by Scott Kirkpatrick, C. Daniel Gelatt and Mario P. Vecchi in 1983,[4] and by VladoČerný in 1985.[2] The method is an adaptation of the Metropolis-Hastings algorithm, a Monte Carlo method to generate sample states of a thermodynamic system, invented by N. Metropolis et al. in 1953.[5] 3 An Example of 2-way & Dynamic Time of TSP A salesman will distribute the goods from a warehouse to 4 customers [6]. The salesman will depart at 09:00. Time to unload the goods in every customer is 60 minutes. Let W is a warehouse and C1, C2, C3, C4 is customer. 2

3 Figure 1. Map of Warehouse and Customers (KS) Let imagine that the map had 2-way Street which can be represented with 2- way digraph. Every edge on map had 2 direction edges. Let in this case had 3 time interval. This table wrote on hours:minutes. Table 1.Time of journey, depart at 06:00-09:59 and 16:00-19:59 A1 W C1 C2 C3 C4 W 0:00 2:00 0:30 2:00 2:00 C1 2:00 0:00 2:30 2:00 1:00 C2 1:00 2:00 0:00 1:30 2:00 C3 2:00 1:30 1:00 0:00 2:00 C4 1:30 1:30 1:00 1:30 0:00 Table 2. Time of journey, depart at 10:00-15:59 A2 W C1 C2 C3 C4 W 0:00 1:30 2:30 1:00 1:30 C1 2:00 0:00 1:00 0:30 0:30 C2 1:30 1:30 0:00 2:00 1:30 C3 0:30 2:00 1:00 0:00 2:00 C4 2:00 0:30 1:30 2:00 0:00 3

4 Table 3. Time of journey, depart at 20:00-05:59 A3 W C1 C2 C3 C4 W 0:00 0:30 0:30 0:30 1:00 C1 0:30 0:00 1:00 0:30 0:30 C2 1:00 0:30 0:00 1:00 1:00 C3 0:30 1:00 0:30 0:00 0:30 C4 1:00 0:30 1:00 1:00 0:00 If we choose W - C1 - C2 - C3 - C4 - W for the path we took. Then let count the time for this path 1. Because the salesman will depart at 09:00 so with data on table 1. The time travel of W C1 need 2 hours. So the salesman will be at C1 at11:00 and unloading goods until 12: The salesman will continue send goods to C2, because the time to depart is 12:00 from C1 by using data on table 2. This journey takes 1 hour. So unloading goods on C2 start at 13:00 until 14: He will go to C3 at 14:00. From table 2. Its need 2 hours to arrive in C4. So the salesman arrives at 16:00 and the goods finish unloads at 17: The last shipment starts at 17:00 to C4. From table 1. The salesman arrives at C4 at 19:00 and unloads until 20: In the end the salesman need go back to the warehouse W goes at 20:00 by table 3. The estimated time to arrive at W is an hours. So the salesman finishes his job at 21:00. Conclusion of this cycle W C1 C2 C3 C4 W is12 hours. In Mahardhikapaper [6] show that we must count all possible route (24 route) and also must count every 30 minutes in 24 hours (48 time of departures). So we must count 24 x 48 = This isn t efficient. We try to solve this 1152 problem with Simulated Annealing. 4

5 4 Simulated Annealing Algorithm Simulated Annealing Algorithm is an algorithm that we purpose to solve this problem. The step of SAA is: The Step of Algorithm: 1. Let n is the total A warehouse + customers. v1 is warehouse/manufacture. 2. Let m is the total of time interval. So there will be m matrix Am like this: A3 W C1 C2 C3 C4 W 0:00 0:30 0:30 0:30 1:00 C1 0:30 0:00 1:00 0:30 0:30 C2 1:00 0:30 0:00 1:00 1:00 C3 0:30 1:00 0:30 0:00 0:30 C4 1:00 0:30 1:00 1:00 0:00 Note: If in realty there is no road between two places so the weight of that road will be choose as infinity or a large number. 3. For i=0 to 47 (there will be 48 times of departure) s s0; e E(s) // Initial state, route. sbest s; ebest e // Initial "best" solution k 0 // Route evaluation count. while k <kmax and e >emax // While time left & not good enough: snew neighbour(s) // Pick some neighbour. enew E(snew) // Compute its Route. ifenew<ebest then // Is this a new best? sbest snew; ebest enew // Save 'new neighbour' to 'best found'. if P(e, enew, temp(k/kmax)) > random() then // Should we move to it? s snew; e enew // Yes, change state. k k + 1 // One more evaluation done -return sbest // Return the best solution found. 5

6 5 Result By converting SAA into MATLAB programming here are the result for those problem. No Time of Depart Journey ROUTE W C1 C3 C4 C2 W W C1 C4 C3 C2 W W C1 C3 C4 C2 W W C1 C3 C4 C2 W W C1 C3 C4 C2 W W C2 C1 C4 C3 W W C1 C4 C2 C3 W W C4 C2 C1 C3 W W C2 C4 C1 C3 W W C4 C2 C1 C3 W W C4 C2 C1 C3 W W C4 C1 C3 C2 W W C2 C4 C1 C3 W W C2 C1 C4 C3 W W C1 C4 C3 C2 W W C2 C3 C1 C4 W W C3 C4 C1 C2 W W C2 C4 C1 C3 W W C2 C4 C1 C3 W W C2 C4 C1 C3 W W C4 C1 C3 C2 W W C1 C3 C4 C2 W W C1 C4 C2 C3 W W C4 C1 C3 C2 W W C3 C2 C4 C1 W W C1 C4 C2 C3 W 6

7 W C1 C2 C3 C4 W W C4 C2 C1 C3 W W C3 C2 C4 C1 W W C1 C4 C2 C3 W W C4 C1 C3 C2 W W C1 C4 C3 C2 W W C2 C4 C1 C3 W W C2 C1 C3 C4 W W C2 C3 C4 C1 W W C3 C4 C2 C1 W W C3 C4 C2 C1 W W C2 C1 C4 C3 W W C2 C1 C4 C3 W W C2 C1 C3 C4 W W C3 C2 C4 C1 W W C2 C1 C3 C4 W W C2 C3 C4 C1 W W C2 C3 C4 C1 W W C2 C1 C4 C3 W W C2 C4 C1 C3 W W C2 C3 C4 C1 W W C3 C2 C1 C4 W For every 30 minutes process it just take no more than 8 iterations to find the best result. It sbetter than Mahardhika [6] algorithm who takes 24 iterations to find the best result. 7

8 Figure 2. Result of SAA for every 30 minutes 6 Conclusion This Algorithm had successful to find the optimal route and the optimal time to depart. 7 Further Research There is another algorithm that can solve this problem such as Genetic Algorithm Swarm Algorithm Bee Colony Algorithm 8

9 References [1]. Bryant, Victor (1992), Aspects of Combinatorics: A Wide-ranging Introduction, Ed. 1., Cambridge University Press. [2]. V. Cerny, (1985), Athermodynamical approach to the travelling salesman problem: an efficient simulation algorithm. Journal of Optimization Theory and Applications. [3]. Mahardhika, Taufan, Teguh N. S. (2006), SistemInformasiDistribusiBarangDenganSirkuit Hamilton Pada Graf BerbobotDinamik, KonferensiNasionalSistemInformasi, Informatika. [4]. Kirkpatrick, S.; C. D. Gelatt, M. P. Vecchi ( ). Optimization by Simulated Annealing.Science. New Series 220 [5]. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller (1953).Equations of State Calculations by Fast Computing Machines.Journal of Chemical Physics. [6]. Mahardhika, Taufan (2008), 2-Wayand Dynamic Timeof Traveling Salesman Problem, Proceeding of 2nd International Conference on Mathematics and Natural Sciences (ICMNS) ooo000ooo - TAUFAN MAHARDHIKA Sekolah Tinggi Analis Bakti Asih Kopertis IV Jawa Barat - Banten Jl. PadasukaAtas No. 233 Bandung, Indonesia taufansensei@yahoo.com 9