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**Simulated Annealing General Idea: Start with an initial solution**

Generate a neighboring solution. If the neighbor is better, move to it. If the neighbor is not better move to it with some probability, else stay put. Repeat step 2 for n iterations, however continually reduce the probability of moving to a poorer neighbor over time. The continuous reduction in the probability of moving to a poorer neighbor over time effectively reduces the annealing process to a local improvement in the end.

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**Simulated Annealing Issue 1: What to use as the initial solution x?**

Options: a) Heuristic b) Randomly generated Issue 2: What defines a neighbor or neighborhood? Same as local improvement. Issue 3: Search strategy. Randomly generate a neighbor or simple programmatic approach.

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**Simulated Annealing Issue 4: Evaluation Function – How do you know the**

adjacent (or neighboring) solution is better or not? Speed – need efficient mechanism to evaluate solutions.

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Simulated Annealing Issue 5: Probability Function - What probability function to use for determining whether to move to a poorer solution or not? Let V be the value of the current solution and V’ the value of the neighboring solution. Then Johnson et al proposed using the following: for minimization problem, let D = V’ – V if D <= 0, then downhill move, so take it. if D > 0, then uphill move, move with probability e-D/T, where T is some value referred to as the temperature.

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**Simulated Annealing "e(-d/T) Temp delta 100.0 75.0 56.3 42.2 31.6 23.7**

17.8 13.3 10.0 1.00 5 0.95 0.94 0.91 0.89 0.85 0.81 0.76 0.69 0.61 10 0.90 0.88 0.84 0.79 0.73 0.66 0.57 0.47 0.37 15 0.86 0.82 0.77 0.70 0.62 0.53 0.43 0.33 0.22 20 0.14 25 0.78 0.72 0.64 0.55 0.45 0.35 0.25 0.15 0.08 30 0.74 0.67 0.59 0.49 0.39 0.28 0.19 0.11 0.05 35 0.63 0.54 0.44 0.23 0.07 0.03 40 0.02 45 0.34 0.24 0.01 50 0.51 0.41 0.31 0.21 0.12 0.06 55 0.58 0.48 0.38 0.27 0.18 0.10 0.00 60 65 0.52 0.42 0.13 70 0.50 0.29

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Simulated Annealing Issue 6: Cooling Function – How to reduce the temperature as the annealing process runs. After n iterations, set T =rT, where r < 1. r is typically set somewhere between .95 and .8

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Simulated Annealing The classic Johnson, et al. simulated annealing algorithm: 1. Get an initial solution S. 2. Get an initial temperature T > 0. 3. While not yet frozen do the following: 3.1 Perform the following loop l time. 3.1.1 Pick a random neighbor S’ of S. 3.1.2 Let D = cost(S’) – cost(S) 3.1.3 If D <= 0 (downhill move), Set S = S’. 3.1.4 If D > 0 (uphill move), Set S = S’ with probability e-D/T. 3.2 Set T = rT (reduce temperature). 4. Return S.

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**Simulated Annealing Issue 7: What should the initial temperature T be?**

Large enough to make some uphill moves. This becomes problem instance specific. Could be found programmatically by finding comparing the value of S to the value of several neighboring solutions and then ensuring that: where X is some value say 0.8 or 0.9

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Simulated Annealing Issue 8: How do you perform the probability test to see if an uphill move is to be made? Generate a random number R between 0 and 1. If, then set S = S, move uphill. else, do not move. Sample C++ code: x = (rand( )%1000)/1000.0; if(x < exp(-float(result - finalreport->BestLmax)/Temp)) { Accepted++; newjoblist = true; } else { rev_seq_switch(Mach_1,Seq_1,CenterList,offset,n); NotAccepted++;

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**Simulated Annealing Issue 9: How do you know if the process if frozen?**

Rules of thumb – - If loop 3 completes 3 times without any move being accepted (uphill or downhill). - No move being accepted after M iterations. Minimum temperature reached. Issue 10: How many iterations per temperature, or what is the value of l? Use experimentation to determine good value. Possible numbers are 50,000 or 100,000, or 200,000, etc…

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**Simulated Annealing Extensions to Johnson et al. algorithm –**

Alternately heat up and cool down the process. Use a re-centering approach. For example when the process has become frozen, replace the current solution with the best solution found so far, S = S*, then restart the annealing process by setting the T back to the original temperature. Use a different probability function. Vary the number of iterations l as a function of temperature T. Others?

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