Part III. Ant Colony Optimization combined with Taboo Search for the Job Shop Scheduling Problem
4. Implementations of local search method
In this section, we discuss the local search methods employed in MACOFT, including FT and POP.
4.1. Taboo search method
Taboo Search (TS), proposed by Glover, is one of the successful local search methods applied in solving combinatorial optimization problems [23, 24]. The technique can be described as follows.
TS starts from an initial solution. During iterating, a move is performed to the best neighboring solution whose quality may not be better than the current one. To prevent cycling, a short-term memory, called taboo list, is employed for recording a fixed number of recent moves. The use of taboo list avoids returning to a local minimum that has been visited before. In addition, another memory structure called long-term memory is proposed, expecting to expand the searching ability.
Many researchers such as Taillard [32], Dell’Amico and Trunian [13], and Nowick and Smutnicki [27] apply TS to JSSP and show its superiority. Among them, the Fast Taboo search (FT or TSAB), proposed by Nowick and Smutnicki [27], is the championship for JSSP. Basically, FT employs a specific neighborhood definition which effectively reduces the neighborhood size [27].
4.2. The fast taboo method
FT can be characterized as the following components: initial schedule, neighborhood definition, short-term memory (taboo list), long-term memory, stopping criterion. Details are described in the following.
Consider a given critical path and divide it into several blocks, each of which contains at least two operations processed on the same machine. We swap the first (last) two successive operations in the last (first) block. In others, we swap the first and last two successive operations. Obviously, the neighborhood structure of FT is much smaller than those of other taboo search methods and thus the computational effort can be reduced substantially [32, 35].
Once finding a new schedule with a better value during iterating, FT stores the triple (schedule, neighborhood, and taboo list) in the long-term memory. When the search is terminated according to the stopping criterion of short-term memory, the top element in the long-term memory is retrieved to trace back.
In MACOFT, the length of taboo list MaxTL is set initially to the magical number 7 [23, 24]
and changes dynamically depending on the accumulated number of iterations of ACO. The value of MaxTL is increased by one after every quarter of the total iterations, except for the last quarter, is performed (i.e., MaxTL changes between 7 and 10, inclusively). Let MaxLM be the maximum length of the long-term memory and MaxTabooIter be the stopping criterion of short-term memory.
Increasing the sizes of MaxLM and MaxTabooIter may yield a higher probability for getting a better schedule but require more computational effort. As a tradeoff, they are determined experimentally.
Furthermore, to prevent cycling FT exercises a cyclic test function to judge if there exists a cyclic period δ (1≤ ≤δ Max )δ by detecting repetitions of makespan values in time Ο . If there (1) exists some period that repeats MaxCycle times, then we stop the search and back jump tracking by retrieving the long-term memory list.
4.3. Long-term memory and global updating queue
The proposed global updating queue has a structure similar to the long-term memory. While
FT is working, once a better schedule is found, MACOFT immediately updates not only the long-term memory in FT, but also the global updating queue. Experimentally, the length of global updating queue is set at ⎢⎣J / 3+ M / 2⎥⎦ , implying that the larger the problem instance sizes are, the more schedules the global updating queue has. This makes the use of pheromone information more effective.
4.4. The modified makespan calculation
Despite producing high quality results in a reasonable computation time, the FT in MACOFT however has to be executed in each iteration, causing a heavy computational effort. The proposed MACOFT attempts to accelerate the standard FT by using a modified makespan calculation.
We begin by analyzing the time complexity of FT. In each iteration, two dynamic programming formulae are applied to calculate the longest path from operation σ σ( ∈O) to sink and source as follows:
However, not all the neighbors need to be calculated by using the standard Ο(O) makespan calculation. Taillard [32] develops a lower bound to reject the worse moves, which can be executed in time Ο . Supposing that a move (1) (σmi ,σmj) is applied, the lower bound of the makespan for the new critical path is calculated by the following equation:
max{ ( mj) ( mj), ( mi) ( mi)}
It is easy to see that if LB is greater than the previous makespan, then it must be the new value of the critical path; on the contrary, if we get a less one, we should recalculate the makespan value traditionally.
However, not all the ( ) ( ( ))F σ Bσ of the operations change; only those depending upon ( )
j i
m m
σ σ have to be modified. Hence, to obtain a new makespan value, MACOFT modifies only the set { ( ) |F σ σ∈suc(σmj)} in the new graph by applying (3), which can be performed at most in time Ο(O) but less in mean case. In comparison with the standard FT, our experimental results showed that the computational speed was accelerated about 2 times.
4.5. POP with partial disjunctive graph
In MACOFT, POP local search is on the basis of FT but works without long-term memory. As stated earlier, POP is only executed when every quarter of the total operations have been scheduled, except for the last quarter. The length of POP taboo list (MaxPOPTL) is set at 4 initially and increased by one each time after reaching the stopping criterion (i.e., MaxPOPTL changes between 4 and 6, inclusively). Another parameter MaxPOPIter, which specifies the maximum number of
iterations without improving the best partial schedule obtained so far, is much less than MaxTabooIter in order to reduce the computational load.
Furthermore, to accelerate the procedure, we substitute the general disjunctive graph by a partial disjunctive graph (PDG) in our POP. Let O denote the set of operations already scheduled, S where suffix S is the accumulated number of executed machines. Then PDG Gp ={ , , ( )}V A E Π is
Considering only the scheduled operations in PDG implies that we need not only ignore the unscheduled operations along with their adjacent arcs when applying (3) and (4), but also connect the remaining adjacent scheduled operations belonging to the same job with dummy arcs. Note that there exist no edges in PDG, and the weights of the dummy arcs can be calculated in advance.
We now use a numerical example to describe how PDG works. Consider a JSSP with 3 machines and 5 jobs as depicted in Fig. 2. Given the following partial schedule
)}
a general disjunctive graph can be illustrated in Fig. 3, which has exactly one critical path (source,
3,
σ4 σ53,σ52,σ51,σ41,σ31, sink) with length 337. Now, we apply PDG to modify Fig. 3 and illustrate it in Fig. 4, where the unscheduled operations and their adjacent arcs represented by dotted lines are ignored. In addition, five dummy arcs are added to connect the scheduled operations, where their weights are added accordingly. When applying the modified makespan calculation, the time complexity of PDG is reduced to Ο(OS ).
Note that the notation JP(σ , ( )mj ) JS σim , and PT(σ in the proposed modified makespan mj ) calculation should be revised slightly if their adjacent arcs are replaced by the dummy ones. For instance, since arc (σ σ12, 11), operation σ11, and arc (σ σ11, 15) are ignored, the job successor
2 5
1 1
( )
JS σ =σ and the processing time PT(σ12), depending on its successor, may equal either 21 (weight of the original arc) or 74 (weight of the dummy arc). Suppose now that we want to apply the move (σ σ53, 52). The lower bound can then be calculated as follows:
where the modified numbers are underlined. Since the new lower bound is larger than the previous makespan in Fig. 3 (337), the new longest path must pass through σ53 and σ52 with makespan 389.
Fig. 2. An example of 5 machines, 3 jobs, and 15 operations.
Fig. 3. The general disjunctive graph with three machines being scheduled.
Fig. 4. The proposed partial disjunctive graph of Fig. 3.
5. Computational results
A computational experiment was conducted to evaluate the efficiency and effectiveness of our proposed MACOFT algorithm, which were coded in C++ language and run on a PC with AMD MP-1800+ (1533 MHz) processor.
Choosing appropriate parameters is time-consuming and in general depending on the instance, such as processing time and instance size. Thus, we adjust the parameters dynamically when working as follows: When MACOFT yields the same makespan value during successive iterations, the algorithm is probably falling into a local minimum, and hence we raise q (utilization of 0 exploration) and α (global pheromone evaporation rate) to give the algorithm a higher probability to escape from the local minimum. Table 1 lists all the parameter values used in our experiments.
MACOFT was tested on 101 benchmark problem instances of different sizes which can be downloaded from OR-Library web site (http://mscmga.ms.ic.ac.uk/jeb/ orlib/jobshopinfo.html).
The best known lower and upper bounds of Taillard’s instances are provided on Taillard’s web site (http://ina.eivd.ch/collaborateurs/etd/ problemes.dir/ordonnancement.dir/ordonnancement.html).
All the tested problem instances are classified into the following three classes:
(a) Nine instances of three different sizes ( J × M =15 10,× 20×10,15×15) denoted by (LA24, LA25, LA27, LA29, and LA36-LA40) due to Lawrence [26], where problem LA29 is still open.
(b) Five instances denoted by (ORB1-ORB5) due to Applegate and Cook [2], two instances (FT10, FT20) due to Fisher and Thompson [18], and five instances (ABZ5-ABZ9) due to Adams et al. [1]. Problems ABZ8 and ABZ9 are still open.
(c) Eighty instances of eight different sizes denoted by (TA01-TA80) due to Taillard [31].
These problem instances are solved only for 48 out of 80 instances.
Before evaluating the performance of MACOFT, we experimented with MACOFT without the local search phase (FT) for both general and partial disjunctive graphs for comparing their computational efficiency. All the parameters are set equal, including the random seeds, to make a fair comparison. The experiment was carried out on four different sizes of instances LA36, TA01, TA11, and TA21, each of which was run 5 times. Table 2 indicates that using the partial disjunctive graph can accelerate the computational speed about 2 times.
Table 1 The setting values of MACOFT parameters
b The parameter set when falling into local optimum.
For the performance measure of the main experiment, the relative error RE (%) was calculated for each instance of problem, as a percentage by which the solution obtained (SOL) is above the optimum value (OPT) if it is known or the best lower bound value (LB) [28]:
(SOL OPT) 100 (SOL LB) 100
RE or RE
OPT LB
− × − ×
= =
Table 2 Computational comparison between partial disjunctive graph (PDG)and general disjunctive graphs (GDG)
Problem J × M T(PDG) T G( DG) improvement (times) LA36 15×15 45.372 102.279 2.254 TA01 15×15 44.161 96.875 2.194 TA11 20×15 109.914 228.194 2.095 TA21 20×20 232.375 474.681 2.042 Notes: 1. Average computational timeT (in seconds) from 5 runs.
2. Improvement=T G( DG) T(PDG).
In the experiments, we examined two different heuristic rules on MACOFT, MACOFT-MWR and MACOFT-TR. Tables 3 and 4 show the best solutions from 15 runs found for instances (a) and (b). For each algorithm, the best value of the makespan (C*max), relative error of makespan (RE), and mean relative error (MRE) are listed on the selected problem instances.
Table 3 reports the best computational results obtained by Nowicki and Smutnicki [27] (FT), Balas and Vazacopoulos [5] (SB-RGLS2), Pezzella and Merelli [28] (TSSB), and Schultz et al. [30]
(SHKT-720min). Overall, MACOFT-MWR outperforms all the other novel algorithms in terms of solution quality, including TSAB (traditional FT method) and TSSB (hybrid algorithm combining TS with SB). Both MACOFT-MWR and MACOFT-TR can be completed in 10 minutes for all of these instances.
Table 4 shows the comparison of our algorithms with other novel heuristic algorithms proposed by Aiex et al. [3] (GPPR), Schultz et al. [30] (SHKT-720min), Pezzella and Merelli [28]
(TSSB) on problem instances (b). It is observed from the table that the proposed algorithms produce MREs of 0.626% (MACOFT-MWR) and 0.689% (MACOFT-TR), both of which exhibit the competition with other novel algorithms.
Table 5 lists the computational results for problem instances (c), which contain partially hard instances. All the problems were tested for 10 runs, where the best makespan (Cmax* ), average makespan (Cmax), relative error of the best makespan (RE(Cmax* )), and mean relative error (MRE) are reported. Instances (c) include 80 instances, where TA51~TA80 are relatively large-sized instances but easy to solve (instances with J much more than M are easy), except for TA62 and TA67. Hence we just list the computational results of TA01-TA50, TA62, and TA67 and omit the easier ones. For these instances, it is observed that MACOFT-TR performs better than MACOFT-MWR. This indicates that the searching ability of MACOFT-TR is superior for large-sized instances. Furthermore, the columns of Cmax show that both proposed algorithms are robust. In particular, it is noted that MACOFT-TR found a new upper bound 2015 for instance TA41.
Table 6 gives the comparison of MACOFT-MWR and MACOFT-TR with other famous hybrid algorithms, TSSB and BV-best, on instances (c) in terms of MRE, average computational time T , and numbers of optimal solutions obtained (No. OPT). Note that BV-best indicates the best results from all those SB-GLS series, provided by Balas and Vazacopoulus [5]. Overall, MACOFT-TR obtained MRE of 1.798% and 40 optimal solutions that are superior to all the others.
Table 3 Comparison with other algorithms for instances class (a)
TSAB (FT) SB-RGLS2 TSSB SHKT-720min MACOFT-MWR MACOFT-TR Problem OPT or
(LB,UB) *
Cmax RE Cmax* RE C*max RE Cmax* RE Cmax* RE Cmax* RE LA24 935 939 0.428 935 0.000 938 0.321 938 0.321 935 0.000 938 0.000 LA25 977 977 0.000 977 0.000 979 0.205 977 0.000 977 0.000 979 0.000 LA27 1235 1236 0.081 1235 0.000 1235 0.000 1238 0.243 1235 0.000 1235 0.000 LA29 (1142,1152) 1160 1.576 1164 1.926 1168 2.277 1161 1.664 1158 1.401 1163 1.839 LA36 1268 1268 0.000 1268 0.000 1268 0.000 1268 0.000 1268 0.000 1268 0.000 LA37 1397 1407 0.716 1397 0.000 1411 1.002 1397 0.000 1397 0.000 1397 0.000 LA38 1196 1196 0.000 1196 0.000 1201 0.418 1196 0.000 1196 0.000 1196 0.000 LA39 1233 1233 0.000 1233 0.000 1240 0.568 1233 0.000 1233 0.000 1233 0.000 LA40 1222 1229 0.573 1224 0.164 1233 0.900 1224 0.164 1224 0.164 1224 0.164 MRE 0.375 0.232 0.632 0.266 0.174 0.223
Table 4 Comparison with other algorithms for instances class (b)
GPPR SHKT-720min TSSB MACOFT-MWR MACOFT-TR Problem J ×M OPT or
(LB,UB) *
Cmax RE Cmax* RE Cmax* RE Cmax* RE Cmax* RE ORB1 10×10 1059 1059 0.000 1059 0.000 1064 0.472 1059 0.000 1064 0.000 ORB2 10×10 888 888 0.000 888 0.000 890 0.225 888 0.000 890 0.000 ORB3 10×10 1005 1005 0.000 1005 0.000 1013 0.796 1005 0.000 1013 0.000 ORB4 10×10 1005 1005 0.000 1005 0.000 1013 0.796 1005 0.000 1013 0.000 ORB5 10×10 884 887 0.000 887 0.000 887 0.000 884 0.000 887 0.000 FT10 10×10 930 930 0.000 930 0.000 930 0.000 930 0.000 930 0.000 FT20 20×20 1165 1165 0.000 1165 0.000 1165 0.000 1165 0.000 1165 0.000 ABZ5 10×10 1234 1234 0.000 1234 0.000 1234 0.000 1234 0.000 1234 0.000 ABZ6 10×10 943 943 0.000 943 0.000 943 0.000 943 0.000 943 0.000 ABZ7 20×15 656 692 5.488 661 0.762 666 1.524 658 0.305 660 0.601 ABZ8 20×15 (645,665) 705 9.302 672 4.186 678 5.116 670 3.876 671 4.031 ABZ9 20×15 (661,679) 740 11.95 687 3.933 693 4.841 683 3.327 685 3.631
M R E 2.228 0.740 1.148 0.626 0.689
Table 5 Results by MACOFT for instances class (c)
MACOFT-MWR MACOFT-TR Problem OPT or
(LB,UB) Cmax* Cmax RE(Cmax* ) Cmax* Cmax RE(Cmax* )
TA01 1231 1231 1232.1 0.000 1231 1233.4 0.000 TA02 1244 1244 1244.0 0.000 1244 1244.6 0.000 TA03 1218 1220 1221.3 0.164 1220 1223.3 0.164 TA04 1175 1175 1180.3 0.000 1175 1180.1 0.000 TA05 1224 1229 1237.5 0.408 1228 1236.9 0.327 TA06 1238 1240 1244.7 0.162 1238 1242.6 0.000 TA07 1227 1228 1231.2 0.081 1228 1232.0 0.081 TA08 1217 1217 1221.1 0.000 1217 1220.2 0.000 TA09 1274 1274 1281.3 0.000 1274 1279.5 0.000 TA10 1241 1241 1248.3 0.000 1241 1246.2 0.000 TA11 (1323,1361) 1367 1371.9 3.326 1365 1374.5 3.175 TA12 (1351,1367) 1374 1380.4 1.702 1374 1387.2 1.702 TA13 (1282,1342) 1349 1361.1 5.226 1350 1359.2 5.304 TA14 1345 1345 1345.0 0.000 1345 1345.0 0.000 TA15 (1304,1340) 1342 1359.1 2.914 1350 1362.6 3.528 TA35 2007 2007 2007.3 0.000 2007 2007.0 0.000 TA36 1819 1823 1839.4 0.22 1819 1840.9 0.000 TA37 (1771,1778) 1793 1798.6 1.242 1791 1804.2 1.129 TA38 1673 1677 1689.4 0.239 1677 1691.3 0.239 TA39 1795 1795 1805.2 0.000 1797 1812.5 0.111 TA40 (1631,1674) 1688 1713.0 3.495 1690 1711.4 3.617 TA41 (1859,2018) 2038 2051.8 9.629 2015* 2041.3 8.392 TA42 (1867,1956) 1961 1972.1 5.035 1963 1982.1 5.142 TA67 2825 2825 2825.5 0.000 2825 2825.2 0.000
MRE 1.830 1.798
*indicates the new upper bound.
Table 6 Comparison with other algorithms for instances class (c)
Average running timeTavg, in seconds, and mean relative error MRE;
a TSSB (experimented on Pentium 133MHz PC);
b BV-Best (best results among those provided in Balas and Vazacopoulos [5] and experimented on SUNSPARC-330);
TSSBa BV-Bestb MACOFT-MWR MACOFT-TR Problem J ×M
MRE T MRE T MRE T MRE T
TA01-TA10 15 15× 0.450 2175 0.173 1498 0.082 1109.7 0.057 1404.6 TA11-TA20 20 15× 3.473 2526 3.018 4559 2.698 1553.4 2.745 2891.5 TA12-TA30 20 20× 6.500 34910 6.098 6850 5.750 2319.5 5.738 3673.1 TA31-TA40 30 15× 1.921 14133 0.795 8491 0.975 3016.4 0.948 5202.4 TA41-TA50 30 20× 6.043 11512 5.204 16018 5.114 4703.0 4.844 9989.2 TA51-TA60 50 15× 0.019 421 0.000 196 0.000 173.6 0.000 1393.7 TA61-TA70 50 20× 0.396 6342 0.112 2689 0.021 2637.1 0.049 5633.7 TA71-TA80 100 20× 0.000 231 0.000 851 0.000 267.4 0.000 739.2
MRE 2.350 1.925 1.830 1.798
No. OPT 31 35 38 40
6. Conclusions
In this part we have proposed a hybrid algorithm combined ACO with FT for minimizing the makespan in JSSP. To improve the traditional ACO, we have defined a specific pheromone trails definition inspired from SB and tested a dynamic greedy heuristic.
This new definition can also be applied in other shop scheduling problems. Besides, we have exercised POP local search with a proposed PDG to offer better searching guidance for artificial ants. Moreover, the traditional FT is accelerated by a modified makespan calculation.
To combine ACO with FT completely, we have proposed a global updating queue with the stores of a series of good schedules to update the pheromone trails diversely and thereby the history information can be utilized more effectively. Also, to offer the robustness of MACOFT, the parameters of evaporation rate and relative proportion between the exploitation and exploration are adjusted dynamically, making the escape from a local minimum easily.
The proposed algorithm has been experimented on 101 benchmark problem instances and shown its competitive effectiveness. In particular, our proposed algorithm has improved the upper bound on one open benchmark problem instance (TA41).
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