Abstract: Scheduling is a problem that is often encountered in everyday life, including in manufacturing companies. One type of scheduling problem that is commonly found in the industrial world especially the textile industry is the Flow Shop Scheduling Problem (FSSP). Given a number of jobs that must be processed in a series of stages where at each stage there exists only one machine for processing the jobs, FSSP aims to find a sequence of jobs that meets certain optimal criteria. Flexible FSSP is a variant of FSSP that differs from the classical FSSP in the number of machines at each stage.
In a Flexible FSSP, the number of machines at each stage can be more than one. Thus, in solving Flexible FSSP, not only the job ordering but also the machine selection should be considered. This paper presents an algorithm for solving Flexible FSSP. The proposed algorithm combines a hyper-heuristic and tabu search algorithm. Two kinds of heuristics are used in this work, namely heuristics for job selection and heuristics for machine selection. Tabu search algorithm is used twice:
for selecting the job selection heuristics and for selecting the machine selection heuristics. The proposed algorithm is then compared with the other algorithm using a fixed combination of two kinds of heuristics. The experiments show that in general, the proposed algorithm performs better than the other algorithm.
Index Terms: scheduling problem, flexible flow shop scheduling, tabu search, hyper-heuristic
I. INTRODUCTION
Because of their dynamic, complex, and hard to solve nature, real-life scheduling problems are considered to be NP-complete. When the problem size gets bigger, the deterministic search method will no longer work effectively. Scheduling problems are among the most favorite research topics that have been studied. Until now, a lot of research has been developed to solve scheduling problems. Flow shop scheduling problem (FSSP) is a type of scheduling problem that usually found in the industrial world, in particular in manufacturing. Given a set of jobs, each job consists of a number of operations and each operation will be processed by a specific machine in the same order, the main problem of FSSP is to find a job ordering that meets some optimality constraints. The common optimality constraint is makespan, which is the total time required to complete all the jobs. Another type of scheduling problem is Job Shop Scheduling Problem (JSSP). In JSSP the order of operation processing by the corresponding machine is not unique, which means the operation processing order may vary from job to job.FSSP can be regarded as sorting or sequencing jobs with certain criteria. To do this sorting, generally, there are two commonly used approaches, namely constructively and improvement [1,2].
The principle of the constructive approach is to build job sequences by starting with one job and then iteratively inserting the second job, the third job, and so on until a sequence containing all jobs is generated. Some examples of constructive algorithms are dispatching rules such as First In First Out,
A Tabu Search Based Hyper-heuristic for Flexible Flowshop Scheduling Problems
[1] Cecilia E. Nugraheni, [2] L. Abednego, [3] M. Widyarini
[1][2] Dept. Of Informatics Parahyangan Catholic University, Bandung, Indonesia
[3] Dept. of Business Administration, Parahyangan Catholic University, Bandung, Indonesia
Last In First Out, Shortest Processing Time, etc.The improvement approach begins with an arbitrary sequence consisting of all the jobs to be scheduled. This sequence is then rearranged in a certain way to find a sequence that meets the optimization criteria. This approach is usually called metaheuristic. Some examples of meta-heuristic are genetic algorithm, simulated annealing, bee colony algorithm, etc.Unfortunately, most of those techniques are domain-specific, which means that their application is really only for one specific problem and does not generally apply to other problems. To overcome this problem, a new approach called hyper-heuristic has been proposed. The hyper-heuristic technique provides a more general, not domain-specific, search framework. The hyper-heuristics methodology is more flexible in the search process and can be easily applied to a larger scope of issues [3]. Instead of searches directly on the solution space, hyper-heuristic searches on the heuristic space. In this manner, hyper-heuristics can do the searching process more flexibly, free from the dependency on the problem domain. We have interested in using hyper-heuristic in solving industrial scheduling problems, especially in textile industries. Many scheduling problems in textile industries belong to FSSP. In our previous works, we used genetic programming hyper-heuristic and a combination of meta-heuristic and hyper-heuristic [4, 5, 6]. In our last two works, we proposed a tabu search based hyper-heuristic for FSSP [7] and JSSP [8]. In this work, we focus on a variant of FSSP called Flexible FSSP. It differs from the original FSSP in the number of machines for processing each corresponding operation. Again, we proposed a tabu-search hyper-heuristic for solving Flexible FSSP. We have implemented this algorithm and conducted some experiments for measuring the performance of this algorithm. There is much work conducted in the area of scheduling production problems, in particular, the ones that related to FSSP and Flexible FSSP as well. Some can be found in [9, 10, 11, 12, 13, 14, 15, 16] with various techniques proposed, with heuristic, meta-heuristic, and hyper-heuristic approach. Our approach which combines hyper-heuristic and tabu search uses two kinds of heuristics (job ordering and machine selection) and applies tabu search in two stages can be considered as a new approach. The rest of the paper is organized as follows. Section II briefly explains the definition of Flexible Flow Shop Scheduling Problem, tabu search algorithm, the proposed algorithm and the computational experiments conducted for measuring the performance of the proposed algorithm. Section III is about the experiment designs and section IV presents and discusses the experimental results.
II. METHODOLOGY
A. Flexible Flowshop Scheduling Problems Flexible FSSP can be defined as follows.
Definition 1. Let
• J = {j1, …, jn} be a set consisting of n jobs, where each job, ji , consists of m operations, ji = {oi1, …, oim};
• M = {cm1, …, cmm} be a set consisting of m stages where each stage, cmi, consists of a number of parallel machines;
• f : J → M be a one to one and onto function mapping each job in J to a stage in M such that f(ji) = cmi
holds for every job ji in J and cmi in M;
• p : J → m be a function mapping each job to a tuple consists of m non negative real numbers, representing the time needed to process operation by a machine from its corresponding stage.
The Flexible FSSP is to find a sequence of job such that the time between the beginning of the execution of the first job on the first machine and the completion of the execution of the last job on the last machine (makespan) is minimal while satisfying the following constraints:
• A machine can process only one operation at a time.
• An operation cannot be interrupted.
• The operation processing orders for all jobs are the same.
B. Tabu Search Algorithm
The flow chart in Fig. 1 depicts the general framework of tabu search algorithm. Initially, an arbitrary solution is constructed. As long as the termination condition is not met yet, iteratively, this solution is updated according to chosen non-tabu heuristic.
This algorithm uses two lists: open list and tabu list. The first list is used for memorizing the non-tabu candidate solutions that can be selected to produce new solutions. The second list is used to store the candidate solutions that no longer can be selected.
C. Proposed Algorithm
The main difference between the classical FSSP and Flexible FSSP is the number of machines at each stage. In classical FSSP there is only one machine at each stage, whereas in Flexible FSSP there can be more than one machine at each stage. Therefore, in constructing candidate solutions, besides the job sequence, we have to consider the machine selection algorithm that will be used in each stage as well.
The main idea of the proposed algorithm is to apply the tabu search algorithm twice: first, at the selection of the algorithm for job selection, and second, at the selection of the algorithm for machine selection in every stage.
Fig.1. Flow chart of the general framework of tabu search algorithm.
For Flexible FSSP with n jobs and m stages, a candidate solution can be modeled as a tuple (jt, mt).
The first tuple has (n-1) elements jt = h1, ..., hn-1 where each hi represents the heuristic for job selection (or dispatching rule) for choosing the i-th job to be processed. We need only n-1 heuristics since the n-th job needs not to be chosen, the one last job that not yet selected will be automatically taken. The second tuple has m elements mt = g1, ..., gm where each element gi represents the heuristic for machine selection applied at stage i.
The proposed algorithm is comprised of three main steps:
1. Solution generation, S = (jt, mt).
2. Schedule generation, sched(S).
3. Makespan computation, makespan (sched(S)).
In the first step, an arbitrary solution S = (jt, mt) is generated. The construction consists of two parts:
the jt construction and the mt construction.
For the job selection heuristic, jt, we use the tabu search principle as follows:
• Maintain two lists namely open list and tabu list.
• Use a list T to record how many times a is heuristic already chosen.
• Define the tabu tenure tj.
• Initially, all heuristics are in the open list.
• For i = 1 to n-1, iteratively chose a heuristic h from the open list. Every time a heuristic is chosen, increment its corresponding element in Lh. When Lh > tj, h is removed from the open list and inserted to the tabu list.
The heuristic for machine selection, mt, is conducted similarly.
Based on the constructed solution, S, a schedule is generated by passing S to a function sched(S). This function returns information about the processing of each operation. For every operation sched(S) will define the machine that processes the operation as well as the beginning and the end of time processing of the operation.
The last step is the makespan computation. The output of sched(S) is then passed to the function makespan to produce the makespan of the schedule.
III. EXPERIMENTDESIGN A. Dataset Generation
To measure the performance of the proposed algorithm, we need to conduct a number of experiments.
Unfortunately, we found difficulty in finding benchmarks that could be used as performance references.
For this reason, we have to provide our own dataset for these experiments.
We consider the problems of Flexible FSSP which consists of two groups: problems with 7 jobs and problems with 8 jobs. For each of these groups, we generated 33 problems with the composition of the number of operations and the number of machines in each stage as shown in Table 1.
B. Job and Machine Selection Heuristics
In this study, we consider four heuristics (dispatching rules) for job selection, namely:
1. FIFO (First In First Out) 2. LIFO (Last In First Out)
3. STPT (Shortest Total Processing Time) 4. LTPT (Longest Total Processing Time)
Whereas for machine selection we have used the following algorithms:
1. Random
2. Longest idle time 3. Least workload
4. First fit
Table 1. Dataset Composition.
Job# Operation# Machine#
7 2 1-1, 1-2, 1-3, 1-4, 1-5
2-1, 2-2, 2-3, 2-4, 2-5
3-1, 3-2, 3-3, 3-4, 3-5
4-1, 4-2, 4-3, 4-4, 4-5
5-1, 5-2, 5-3, 5-4, 5-5
3 1-3-4, 4-2-3, 4-3-2
5 1-2-3-3-4,
3-2-2-4-4, 3-4-2-2-4, 4-3-4-2-2
10 2-2-3-4-4-3-3-2-1-1 8 2 1-1, 1-2, 1-3, 1-4,
1-5
2-1, 2-2, 2-3, 2-4, 2-5
3-1, 3-2, 3-3, 3-4, 3-5
4-1, 4-2, 4-3, 4-4, 4-5
5-1, 5-2, 5-3, 5-4, 5-5
3 1-3-4, 4-2-3, 4-3-2
5 1-2-3-3-4,
3-2-2-4-4, 3-4-2-2-4, 4-3-4-2-2
10 2-2-3-4-4-3-3-2-1-1
C. Experiment scenarios
For each composition in Table 1, we have generated 20 problem instances. For each problem instance, we have generated the processing time for each operation. Randomly, we set each processing time is from 1 to 25 time units. For each problem instance, we have run 100 times and recorded the resulted makespans. Besides, we have also run each problem instance with all possible variations of job and machine selection heuristics. As explained before, there are four job selection heuristics and four machine selection algorithms, which mean there exist 16 combination job and machine selection heuristics. For each problem instance and each heuristic combination we run once and recorded the
resulted makespan. The performance of the proposed algorithm will be measured by comparing the results of two types of experiments.
IV. RESULTS
From the resulted makespans, for each type of experiment, we calculated the minimum, the maximum, and the average makespan. We have done this for all problem instances. We give examples of resulted makespans for some problem instances in Table 2, Table 3, Table 4, and Table 5.We then compared the makespans resulted by the proposed algorithm with the ones resulted by the combination of job and machine selection heuristics. Three comparisons are made, namely the minimum, maximum, and average comparisons. The objective of minimum/maximum/average comparison is to know how many instances such that the minimum/maximum/average makespan resulted by the proposed algorithm is smaller than the one resulted by the heuristic combination. For the problem with 7 and 8 jobs, the summary is given in Table 6 and Table 7, respectively.
Table 2. Resulted makespans for 7 jobs, 2 operations, 1-2 number of machines.
No
Job and Machine Selection
Combination
Proposed Algorithm Min Max Ave Min Max Ave 1 123 441 280.56 110 141 121.45 2 89 326 206.31 81 119 93.95 3 116 469 288.75 94 120 104.09 4 128 437 280.06 116 144 123.39 5 109 395 252.19 87 114 94.46 6 101 400 253.25 100 118 108.34 7 128 519 325.00 92 107 96.42 8 91 320 202.31 98 117 103.04 9 120 468 294.50 83 94 88.05 10 126 445 282.75 75 101 86.51 11 104 373 237.38 87 116 93.55 12 118 478 301.81 79 123 99.02 13 93 346 219.75 116 146 132.21 14 100 408 253.75 113 137 123.39 15 93 368 231.88 99 119 109.93 16 105 393 248.75 101 137 111.02 17 85 327 206.25 128 150 140.7 18 78 303 191.19 83 113 92.68 19 91 335 213.94 120 161 132.75 20 85 311 197.75 107 146 121.94
Table 3. Resulted makespans for 8 jobs, 5 operations, 3-4-2-2-4 number of machines.
No
Job and Machine Selection
Combination
Proposed Algorithm Min Max Ave Min Max Ave 1 140 378 244.13 103 153 124.44 2 135 333 235.00 118 190 142.45 3 125 292 202.56 125 178 144.58 4 120 325 216.44 113 199 136.62 5 147 358 247.19 113 173 138.21 6 141 384 247.81 108 163 128.78 7 146 391 257.19 113 200 141.42 8 154 414 256.19 117 193 149.7 9 137 359 233.25 128 194 148.61 10 147 325 226.81 106 172 132.57 11 168 424 279.25 122 193 143.52 12 134 385 245.75 111 169 135.14 13 151 405 257.50 92 146 110.66 14 128 371 235.81 99 170 121.83 15 122 376 233.06 123 179 144.34 16 136 365 243.19 114 184 138.58 17 166 380 265.69 119 195 148.23 18 133 334 228.56 118 172 145.04 19 135 395 263.44 98 176 125 20 141 379 253.31 101 166 125.39
Table 4. Resulted makespans for 7 jobs, 10 operations.
No
Job and Machine Selection
Combination
Proposed Algorithm Min Max Ave Min Max Ave 1 258 636 450.25 250 300 274.74 2 243 565 404.13 233 258 238.36 3 258 623 440.00 233 268 237.96 4 247 524 390.44 226 293 254.86 5 239 523 386.75 226 307 255.81 6 212 568 394.25 235 279 254.82 7 240 506 374.31 235 278 252.52 8 263 618 442.06 199 288 245.59 9 211 469 336.94 202 295 239.22 10 233 550 395.25 230 288 255.35 11 265 536 400.50 229 279 253.46 12 252 610 434.13 217 271 240.45 13 245 564 400.75 215 269 239.73
14 247 649 452.13 217 278 246.37 15 221 557 381.06 214 286 247.42 16 255 566 412.19 196 252 215.17 17 216 469 341.50 200 249 216.51 18 239 596 416.75 219 277 241.58 19 235 600 421.13 219 283 244.22 20 232 565 395.50 219 271 235.03
Table 5. Resulted makespans for 8 jobs, 10 operations.
No
Job and Machine Selection
Combination
Proposed Algorithm Min Max Ave Min Max Ave 1 246 599 427.56 226 278 264.49 2 278 753 509.75 249 314 277.46 3 249 637 444.94 247 318 274.61 4 261 648 446.56 213 270 237.61 5 278 663 467.50 217 264 236.59 6 273 666 470.00 238 295 257.81 7 264 662 460.19 238 312 256.79 8 242 574 408.00 238 295 258.10 9 266 656 462.06 234 291 257.42 10 245 632 434.75 251 290 270.00 11 257 596 420.00 251 292 269.97 12 261 589 423.50 235 297 262.72 13 271 585 426.75 235 297 261.73 14 245 624 434.75 226 283 249.23 15 247 595 420.81 226 273 247.23 16 250 587 413.00 226 277 242.25 17 239 505 372.13 228 278 243.10 18 264 664 460.13 250 309 270.47 19 241 542 394.00 248 299 269.59 20 247 660 451.25 227 304 247.60
Table 6. Summary of 7 jobs problems
Instance Minimum Maximum Average
1 60% 100% 100%
2 60% 100% 100%
3 65% 100% 100%
4 50% 100% 100%
5 50% 100% 100%
6 45% 100% 100%
7 85% 100% 100%
8 90% 100% 100%
9 90% 100% 100%
10 75% 100% 100%
11 50% 100% 100%
12 80% 100% 100%
13 100% 100% 100%
14 95% 100% 100%
15 95% 100% 100%
16 40% 100% 100%
17 85% 100% 100%
18 90% 100% 100%
19 100% 100% 100%
20 95% 100% 100%
21 60% 100% 100%
22 75% 100% 100%
23 95% 100% 100%
24 95% 100% 100%
25 85% 100% 100%
26 65% 100% 100%
27 100% 100% 100%
28 95% 100% 100%
29 85% 100% 100%
30 95% 100% 100%
31 95% 100% 100%
32 95% 100% 100%
33 95% 100% 100%
Average 80% 100% 100%
From Table 6 and Table 7 we can conclude that in general, the proposed algorithm outperforms the other algorithm (the job and machine heuristic combination). For each problem instance group, although the minimum makespan generated by the proposed algorithm is not always smaller than the makespan produced by the other algorithm, almost all the maximum and the average makespans are below the ones generated by the other algorithm.
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