Heuristic Methods for the Single- Machine Problem Chapter 4 Elements of Sequencing and Scheduling by Kenneth R. Baker Byung-Hyun Ha R2

1 Outline  Introduction  Dispatching and construction procedures  Neighborhood search techniques  Tabu search  Simulated annealing  Summary

2 Introduction  Heuristic procedures  Obtaining good solutions with limited computational effort  Performance measures of heuristic procedure  Frequency with which optimal solutions are produced  Average deviation from optimality  Computational time

3 Dispatching and Construction Procedures  Sorting  Use of a ranking scheme where the relative ranking of two jobs does not change with time  SPT for F-problem  Dispatching  A procedure that uses a decision rule to select the next job when the machine becomes free  Combining SPT and MST for T-problem At time t, select the job with smallest value of u j, where u j = p j (d j – p j – t) Exercise: dispatch the jobs below. Job j123 pjdjpjdj

4 Dispatching and Construction Procedures  Construction  A procedure where a schedule is built from scratch, normally adding jobs to the schedule one at a time Not necessarily adding jobs in order from earliest to latest  It is sometimes called “greedy” procedure Making the next selection in the most favorable way, without regard to the possibilities that might arise later in the algorithm  T-problem by last-to-first structure Select the last job which occurs the smallest tardiness Exercise: construct sequence of the jobs below  T-problem by insertion  choose the best among 3-1-2, 1-3-2, and Job j12345 pjdjpjdj

5 Neighborhood Search Techniques  Neighborhoods of a solution  Schedules formed by a mechanism for generating neighborhoods from a seed  Adjacent pairwise interchange mechanism seed: neighborhood: , ,  Last-insertion mechanism seed: neighborhood: , ,  Pairwise interchange mechanism, insertion mechanism

6 Neighborhood Search Techniques  Algorithm 1: Neighborhood Search 1.Obtain a sequence to be an initial seed and evaluate it with respect to the performance measure. 2.Generate and evaluate all the sequences in the neighborhood of the seed. If none of the sequence is better than the seed with respect to the performance measure, then stop. Otherwise proceed. 3.Select one of the sequences in the neighborhood that improved the performance measure. Let this sequence be the new seed. Return to Step 2.  Tactical options  A method of obtaining the initial seed  A generating mechanism  A method of selecting a particular sequence to be the new seed  Gradient-descent approach  Does not guarantee global optimum

7 p = [5, 9, 4, 3] d = [6, 9, 13, 15] T-problem neighborhood by adjacent pairwise interchange

8 Tabu Search  Overcoming local optimum trap of neighborhood search  Allowing a new seed that is worse than the old seed  Preventing infinite cycle  Termination rule  Concepts  Move Changing from one seed to the next seed from neighborhood  Tabu list List of forbidden moves Possible ways of organizing tabu list keeping sequences, or keeping moves Impact of tabu list size Cycling vs. performance Usually 5 to 10

9 Tabu Search  Example: a tabu search (adopted from Algorithm of Pinedo, 2008) 1.Set k = 1. Select an initial seed S 1 using some heuristic. Set S 0 = S 1. 2.Select a next seed S k+1 that is not prohibited by tabu list from the neighborhood of S k. Update tabu list using the current move. If objective function value by S k+1 is less than that of S 0, set S 0 = S k+1. 3.Increment k by 1. If termination condition holds then stop. Otherwise, go to step 2.

10 Tabu Search  Example: T w -problem by the tabu search  Jobs  Generating neighborhoods mechanism: adjacent pairwise interchanges  Keeping moves in tabu list  Size of tabu list: 2  Initial seed: S 1 = S 0 = (2,1,4,3)  objective value: 500 Job j1234 pjdjwjpjdjwj

11 Tabu Search  Example: T w -problem by the tabu search (cont’d)  Iteration 1: S 1 = (2,1,4,3), S 0 = (2,1,4,3)  500 Tabu list: Neighborhood: (1,2,4,3)  480, (2,4,1,3)  436, (2,1,3,4)  652 Next seed: S 2 = (2,4,1,3)  436  S 0  Iteration 2: S 2 = (2,4,1,3), S 0 = (2,4,1,3)  436 Tabu list: (1,4) Neighborhood: (4,2,1,3)  460, (2,1,4,3)  500, (2,4,3,1)  652 Next seed: S 3 = (4,2,1,3)  460  Iteration 3: S 3 = (4,2,1,3), S 0 = (2,4,1,3)  436 Tabu list: (2,4), (1,4) Neighborhood: (2,4,1,3)  436, (4,1,2,3)  440, (4,2,3,1)  632 Next seed: S 4 = (4,1,2,3)  440  Iteration 4: S 4 = (4,1,2,3), S 0 = (2,4,1,3)  436 Tabu list: (2,1), (2,4) Neighborhood: (1,4,2,3)  408, (4,2,1,3)  460, (4,1,3,2)  586 Next seed: S 5 = (1,4,2,3)  408  S 0  Iteration 5: S 5 = (1,4,2,3), S 0 = (1,4,2,3) 

12 Simulated Annealing  Annealing  Physical process of cooling material slowly, until the material reaches a stable (frozen) state  Tabu search vs. simulated annealing  Bringing objective function value down steeply or slowly iteration objective function value tabu search simulated annealing ?

13 Simulated Annealing  Example of simulated annealing iterations  At stage i, select randomly the next seed S i+1 from the current seed S i.  If Z i+1  Z i, then accept S i+1, where Z i is objective value of S i.  Otherwise, accept S i+1 with the probability q ij = min{1, e –  Z/T(i) }, where  Z = Z i+1 – Z i and T(i) denotes the temperature at stage i.  Features of the probability function, q ij  q ij decreases as temperature decreases, other things being equal As search proceeds, there is a decreasing probability of moving to worse solution  If  Z  0, q ij varies inversely with  Z  Temperature schedule  T(i + 1) =  T(i), with  as 0.9 and T(1) as the mean processing time  Termination rule

14 Random Sampling  Sampling procedure  Construct and evaluate N sequences (sample), using some random device  Identify the best sequence in the sample  Heuristics (1 sequence) random sampling (N sequences) optimizing procedures (all sequences, at least implicitly)  Tactical questions  Random device? (e.g., biased random sampling)  Conclusions about the quality of the best in the sample?  Biased random sampling  Using selection device that are not equally likely  Example Sequencing jobs by consecutively drawing a job from a list of jobs, ordered by a specific rule (e.g., SWPT), by favoring the first job to the second job, the second job to the third job, and so on

15 Summary  Challenge to combinatorial optimization problems  Relatively few cases of optimal approaches known  Usually, burden of computational effort for even medium-sized problems Heuristic methods as alternatives  Some heuristic approaches  Dispatching and construction procedures Building relatively good schedules quickly  Search techniques Neighborhood search, tabu search, simulated annealing  Random sampling  Further topics  Metaheuristics c.f., Luke, Essentials of Metaheuristics ... (too many)