Journal of Intelligent Manufacturing, 15, 439±448, 2004 # 2004 Kluwer Academic Publishers. Manufactured in The Netherlands.
The strategies and parameters of tabu search for job-shop scheduling FA R U K G E Y I K Industrial Engineering Department, University of Gaziantep, 27310 Gaziantep, Turkey E-mail:
[email protected]
ISMAIL HAKKI CEDIMOGLU Industrial Engineering Department, Sakarya University, 54040 EsentepeÐAdapazari, Turkey E-mail:
[email protected] Received February 2003 and accepted December 2003
This paper presents a tabu search approach for the job-shop scheduling problem. Although the problem is NP-hard, satisfactory solutions have been obtained recently by tabu search. However, tabu search has a problem-speci®c and parametric structure. Therefore, in the paper, we focussed on the tabu search strategies and parameters such as initial solution, neighborhood structure, tabu list, aspiration criterion, elite solutions list, intensi®cation, diversi®cation and the number of iteration. In order to compare some neighborhood strategies and tabu list length methods, a computational study is done on the benchmark problems. Keywords: Tabu search, neighborhood, tabu list, job-shop scheduling
1. Introduction Scheduling concerns the allocation of limited resources to tasks over time (Pinedo, 1995) and the ef®ciency of this allocation. The resources and tasks may be in many forms. In a manufacturing environment, e.g., the resources may be machines and the tasks may be operations. Production scheduling together with production planning is an important function which determines the ef®ciency and productivity of a manufacturing system. However, production scheduling is not an independent function; there are many elements which affect it. For instance, precedence constraints, due dates, production levels, lot-size restrictions, priority rules etc. In addition, the basic element which affects the scheduling is essentially process planning (Geyik and Cedimoglu, 1999). There are commonly two kinds of feasibility constraints in production scheduling problem: machine capacity and precedence constraints. A
schedule is any feasible solution of these constraints (Baker, 1994). In principle the number of feasible schedules for any job-shop problem is in®nite because it can be put an arbitrary amount of idle time between successive operations. Even if no idle time is left, the number of possible semi-active schedules will be m excessive, that is about
n! Ðwhere n is the number of jobs and m is the number of machines. In this case, to ®nd an optimum schedule must be consumed excessive computational timeÐeven it sometimes will be impossible. According to Pinedo (1995), if a scheduling problem does not have an ef®cientÐsocalled polynomial timeÐalgorithm, it is called nondeterministic polynomial hard (NP-hard) problem. So, the job-shop scheduling problem (JSP) is NP-hard. Although the solution of scheduling problems optimally is dif®cult, the most ef®cient ways of exact solution methods are the branch and bound algorithm and dynamic programming. Some of works in these ®elds are Carlier and Pinson (1989), Applegate and Cook (1991), Brucker et al. (1994),
440 and Sonmez and Baykasoglu (1998). NP-hard scheduling problems are generally solved by heuristic methods. Some of these are the priority dispatching rules (Panwalker and Iskander, 1977; Saad et al., 2002), expert systems (Fox and Smith, 1984; Smith, 1995), arti®cial neural networks (Foo and Takefuji, 1988; Cedimoglu, 1993). There has recently been a great progress in developing heuristics that ®nd effective schedules. Some of these are the shifting bottleneck (Adams et al., 1988; Dauzere-Peres and Lasserre, 1993), the local search (Aarts et al., 1994; Vaessens et al., 1996), and the intelligent search procedures or meta-heuristics such as genetic algorithms (Dorndorf and Pesch, 1995), simulated annealing (Matsuo et al., 1988; Laarhoven et al., 1992; Alfano et al., 1994; Baykasoglu, 2002) and tabu search (Tailard, 1993, 1994; Dell'Amico and Trubian, 1993; Barnes and Laguna, 1993; Barnes and Chambers, 1995; Nowicki and Smutnicki, 1996). These papers reported that tabu search (TS) is a powerful technique for the job-shop scheduling. However, there are many parameters and strategies to be determined in tabu search. Furthermore, these parameters and strategies affect the solution quality. Therefore, we essentially focus on determining values of these parameters and strategies in this study. Our aim is to form a TS template for the job-shop scheduling applications. The remainder of this paper is organized as follows. In Section 2, the JSP and the disjunctive graph representation is described. In Section 3, the TS methodology, its elements and an example implementation are presented. Section 4 presents neighborhood and memory strategies as well as computational tests. Conclusions are presented in Section 5.
Geyik and Cedimoglu Table 1. A four job, three machine JSP data. (a) Process routes; (b) Processing times Jobs
Processes 1
2
3
(a) Process routes 1 2 3 4
m1 m2 m3 m2
m2 m1 m2 m3
m3 m3 m1 m1
(b) Processing times 1 2 3 4
4 1 3 3
3 4 2 3
2 4 3 1
Let us consider an example JSP consisting of four jobs and three machines, for which data are seen in Table 1. This problem can be represented by a disjunctive graph (Balas, 1969) (see Fig. 1). Where conjunctive arcs satisfy precedence constraints and disjunctive arcs satisfy capacity constraints of machines. Disjunctive arcs are un-directional. We create a partial schedule by ®xing direction of each disjunctive arc. We will de®ne a feasible schedule if every disjunctive arc is ®xed and the ®nal disjunctive graph is acyclic. Figure 2 represents a feasible schedule. According to this, the sequence of operations on machines 1±3 are fO11 ?O22 ?O43 ?O33 g, fO21 ?O41 ?O32 ?O12 g, and fO31 ?O42 ?O23 ? O13 g, respectively. Gantt chat of the schedule is seen in Fig. 3. This Gantt chart is also obtained by Gif¯er and Thomson algorithm (1960) by using the shortest process time (SPT) priority rule as a tie-
2. The job-shop scheduling problem The JSP consists of a ®nite jobs set, Ji
i 1; 2; . . . ; n, to be processed on a ®nite machines set, Mk
k 1; 2; . . . ; m. Each job must be processed on every machine. Each operation has to be scheduled by satisfying the precedence and capacity constraints, i.e., the job sequences on each machine must be found. The objective function in scheduling is almost a function of the completion times of the jobs such as makespan
Cmax , mean ¯ow time, mean tardiness, etc.
Fig. 1. Disjunctive graph representation of the problem in Table 1.
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The strategies and parameters of tabu search
Fig. 2. A feasible solution for the disjunctive graph in Fig. 1.
Fig. 3. Gantt diagram of the feasible solution in Fig. 2.
break. In this case, the value of objective function, Cmax , is 14. The obtained solution is only one of the possible ones and its optimality for Cmax is not guaranteed. Improvement of this solution is possible. For this purpose, we use a TS algorithm.
3. Tabu search for job-shop scheduling Local search techniques are capable of solving problems of combinatorial optimization by searching neighborhood of a given solution. The basic problem to arise here is how to avoid being trapped at a local optimum. Since the selection of a neighbor is stochastic in simulated annealing, it may be avoided from local minimum. However, due to the possibility of returning to the old solutions, the searching may become oscillation in local optimum surrounding. The oscillation causes the searching to exhaust much more time. If some solutions to have been dealt with before are stored on a tabu list and forbidden to return to them for a while then the possibility of the oscillation could be reduced. A
search procedure that was equipped with such a tabu list is called TS. Tabu Search was developed by Glover (1986, 1989, 1990). The philosophy of this approach arises from the intelligent problem solving tendency. TS is a search procedure that limits the searching and negotiates a local minimum, while keeping the history of searching in its memory. According to Brucker (1995), TS is an intelligent search technique that uses a memory function in order to avoid being trapped at a local minimum and hierarchically canalizes one or more local search procedures in order to search quickly the global optimality. According to Laguna and Glover (1996), it is a progressive approach that opens a door in order to solve a variety of business problems. Perhaps, this progressive approach has a transparent and natural memory; its goal is to emulate intelligent uses of memory. That is, TS tries to create memory itself similar to use of some memory functions of people in order to ®nd its way out. General TS framework permits a meta-level ¯exibility for designing of solution procedures. Researchers use this ¯exibility for exploring new strategies to attain stronger solution procedures. The TS procedure is generally simple. The procedure begins with an initial solution and stored it as the current seed and the best solution. The neighbors of the current seed are then produced by a neighborhood structure. These are candidate solutions. They are evaluated for an objective function and a candidate of which is not tabu or satis®es the aspiration criterion is selected as new seed solution. This selection is called a move and added to tabu list in order to create memory. The new seed solution is compared with the current best solution: if better, it is stored as new best solution. Iterations are repeated until a stop criterion is satis®ed. A general TS algorithm is as follows: *
*
*
Step 1. Start with an initial solution, store it as the current seed and the best solution. Step 2. Generate neighbors of the current seed solution by a neighborhood structure. * Select a neighbor which is not tabu or satis®es a given aspiration criterion and move it as new seed solution. * Update the tabu list. * Store it as the new best solution, if the new seed solution is better for an objective function. Step 3. Repeat Step 2 until a termination criterion is satis®ed.
442 3.1. Elements of the TS algorithm The elements of the TS algorithm in connection with JSP are de®ned as follows: Initial solution: The initial solution can be obtained by various methods such as the priority dispatching rules, the diverse insertion and random methods, etc.Ðeven arti®cial intelligence. Jain et al. (2000) show that the initial solution method affects the scheduling solution quality; such that the better initial solution is the better TS solution is. In this paper, the SPT rule used as an initial solution method. Neighborhood structure: A neighborhood structure is a mechanism which can obtain a new set of neighbor solutions by applying a simple modi®cation to a given solution. Each neighbor solution is reached from a given solution by a move (Glover and Laguna, 1997). Neighborhood structure is directly effective on the ef®ciency of TS because TS proceeds iteratively from one neighbor to another in problem solution space. Therefore, a neighborhood structure must eliminate unnecessary and infeasible moves if it is possible. In the following section, a comprehensive computational study has been done in connection to some neighbor generation mechanisms. Move: The best neighbor which is not tabu or satis®es a given aspiration criterion is selected as new seed solution. ``The best'' neighbor is one whose objective function, Cmax , is minimum. If all neighbor is tabu or no neighbor satis®es the aspiration criterion then the oldest neighbor, entering the tabu list at ®rst, is selected as new seed solution. Tabu list and updating: There are two common ways of the use of memory in TS: short- and long-term memory. Glover and Laguna (1997) states that the effect of both types of memories may be viewed as modifying the neighborhood of a given solution. The short-term memory keeps track of the solutions attributes that have changed during the recent history. It is exploited as a tabu list. In this work, we use only a single tabu list. The elements added on the list are attributive. The main aim of using an attributive representation is to save computer memory. The attribute that represents an element in sequencing is two operations that have interchanged in the recent move. It is the arc
j; i which is obtained by swapping a pairs of operations
Geyik and Cedimoglu
i; j. To identify tabu status of the neighbor
j; i it is looked whether or not it is on the tabu list; if so, it is labelled as ``tabu''. Tabu list is updated after each move in so far as the strategic forgetting occurs. The move selected replaces to the top of the list and the elements on the list go down one apiece. The element on the bottom of the list is removed from the list, i.e., the temporary memory is updated. The length of tabu list determines the time limit of remaining on memory for elements. Therefore, it can change the course of the search. In the following section, some methods of the length of tabu list were tested. Aspiration criterion: The aim of the aspiration criterion, when it is necessary, is to override the tabu status of a neighbor. The aspiration criterion used in this work is as follows: If the move yields a solution better than the best obtained so far then the move is performed even as it is tabu. Termination criterion: When the number of the disimproving moves reaches to a maximum value, set to 2000, or no neighbor is generated or an infeasible solution is encountered, the TS algorithm terminates.
3.2. An example implementation of the TS algorithm Let us consider the example problem in Table 1. The initial solution obtained by SPT, the sequences of operations on machines, are as follows: M1 fO11 ?O22 ?O43 ?O33 g; M2 fO21 ?O41 ?O32 ?O12 g; M3 fO31 ?O42 ?O23 ?O13 g: Cmax 14 This solution is stored as the current seed and the best solution. The critical path for this solution is fO11 ?O22 ?O23 ?O13 g. Neighbors are
O22 ?O11 and
O13 ?O23 . That is, neighbor 1 is M1 fO22 ?O11 ?O43 ?O33 g, M2 fO21 ?O41 ?O32 ? O12 g, M3 fO31 ?O42 ?O23 ?O13 g and Cmax 14, and neighbor 2 is M1 fO11 ?O22 ?O43 ?O33 g, M2 fO21 ?O41 ?O32 ?O12 g, M3 fO31 ?O42 ? O13 ?O23 g and its Cmax 15. The move selected is the neighbor 1 because its objective function is minimum. The neighbor 1 is
443
The strategies and parameters of tabu search Table 2. An example implementation Iter no.
Neighbors
O13 ?O23
The values of Cmax
Move
Tabu list
14*, 15 14* 14{, 15* 13*{, 14 13*, 16 16*, 14{ 13{*
O22 ?O11
O11 ?O22
O13 ?O23
O12 ?O32
O41 ?O21
O12 ?O21
O21 ?O12
O11 ?O22
O22 ?O11 ,
O23 ?O13 ,
O32 ?O12 ,
O21 ?O41 ,
O21 ?O12 ,
O12 ?O21 ,
O11 ?O22 ,
O13 ?O23 ,
O22 ?O11 ,
O41 ?O21 ,
O23 ?O13 ,
O11 ?O22 ,
O32 ?O12 ,
O22 ?O11 ,
O21 ?O41 , ...
1 2 3 4 5 6 7
O22 ?O11 ,
O11 ?O22
O22 ?O11 ,
O12 ?O32 ,
O41 ?O21 ,
O12 ?O21 ,
O21 ?O12
8
O12 ?O21 ,
O23 ?O13
16{, 14{*
O23 ?O13
9
O21 ?O41 ,
O13 ?O23
14{*, 13{
O21 ?O41
10
O22 ?O11 ,
O13 ?O23
17{*, 13{
O22 ?O11
11
O11 ?O22 ,
O32 ?O12
14{*, 14{
O11 ?O22
...
...
...
...
O13 ?O23
O23 ?O13
O13 ?O42
O23 ?O13
O11 ?O22 ,
O22 ?O11 ,
O23 ?O13 ,
O32 ?O12 ,
O21 ?O41 ,
O21 ?O12 ,
O12 ?O21 ,
O11 ?O22 ,
O13 ?O23 ,
O22 ?O11 ,
O41 ?O21 ,
O23 ?O13 ,
O11 ?O22 ,
O32 ?O12 ,
O11 ?O22 ,
O22 ?O11 ,
O23 ?O13 ,
O32 ?O12 ,
O21 ?O41 ,
O11 ?O22 ,
O22 ?O11 ,
O11 ?O22 ,
O23 ?O13 ,
O22 ?O11 ,
O11 ?O22 ,
O32 ?O12 ,
O23 ?O13 ,
O22 ?O11 ,
O21 ?O12 ,
O21 ?O41 ,
O32 ?O12 ,
O23 ?O13 ,
O12 ?O21 ,
O11 ?O22 ,
O13 ?O23 ,
O22 ?O11 ,
O41 ?O21 ,
O23 ?O13 ,
O21 ?O12 ,
O21 ?O41 ,
O32 ?O12 ,
O12 ?O21 ,
O21 ?O12 ,
O21 ?O41 ,
O11 ?O22
O13 ?O23 ,
O12 ?O21 ,
O21 ?O12 ,
O22 ?O11
* Refers the move selected. { Refers the move which is tabu. { Refers the best solution found until now.
added to tabu list. It is new seed solution now. An 11iteration implementation is seen in Table 2. 4. Computational study The proposed TS algorithm is implemented in Pascal language on a Pentium PC. A set of benchmark problems from literature have been formed, in so far as the tests are performed on the same instances. The benchmark set contains 98-piece JSP that are known the optimal or upper bound solutions for Cmax . These test problems have been taken from various authors (Adams et al., 1988ÐABZ; Fisher and Thompson, 1963ÐFT; Lawrence, 1984ÐLA; Applegate and Cook, 1991ÐORB; Storer et al., 1992ÐSWV; Tailard, 1993ÐTA). Some information relating to these problems is seen in Table 3 or is available from the OR Library site http://www.ms.ic.ac.uk/jeb/pub/ jobshop1.txt and jobshop2.txt in detail. Initial solutions are obtained by using Gif¯er and Thomson (1960) algorithm. In the program, it is possible to use diverse priority rules, SPT, as a tiebreak. The iterative phase was designed so as to use the various neighborhood and tabu list length (TLL) methods. The various neighborhood and TLL methods were
compared statistically below, as based on the Cmax obtained for each problem, by the paired t-test. Meanwhile, several measures, which gain some statistics relating to implementation of these methods, are created. They are the mean relative improvement of the initial solution (MRI%), the mean relative error (MRE%), i.e., the mean relative deviation from the optimum, the mean number of evaluated neighbors (MEN), the mean number of moves (Iter no.), the number of the problems found equal to the known best solution (LBE) and the mean tabu list length (MTL).
initial Cmax
best Cmax
initial Cmax
1
best Cmax
optimum Cmax
optimum Cmax
2
MRI% MRE%
4.1. Neighborhood strategies Six neighborhood structures are tested below. Five of these are well known neighborhoods in literature and they are denoted by N1 (Laarhoven et al., 1992), N2 (Matsuo et al., 1988), N3 (Brucker et al., 1994), N4 (Nowicki and Smutnicki, 1996), and N5 (Balas and Vazacopoulos, 1998). For further information, see Geyik (2000). The other is a new neighborhood that is
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Geyik and Cedimoglu
Table 3. The best Cmax solutions known and the TS solutionsa found for the benchmark problems in literature Prob. name
Best sol. known
TS sol. found
Prob. name
Best sol. known
TS sol. found
Prob. name
Best sol. known
TS sol. found
abz5 abz6 abz7 abz8 abz9 ft06 ft10 ft20 la01 la02 la03 la04 la05 la06 la07 la08 la09 la10 la11 la12 la13 la14 la15 la16 la17 la18 la19 la20 la21 la22 la23 la24 la25
1234 943 656 645 661 55 930 1165 666 655 597 590 593 926 890 863 951 958 1222 1039 1150 1292 1207 945 784 848 842 902 1046 927 1032 935 977
1238 947 696 697 741 55b 971 1165b 666b 655b 604 598 593b 936 910 863b 951b 1034 1222b 1039b 1159 1374 1207b 959 784b 861 860 909 1099 962 1032b 989 995
la26 la27 la28 la29 la30 la31 la32 la33 la34 la35 la36 la37 la38 la39 la40 orb01 orb02 orb03 orb04 orb05 orb06 orb07 orb08 orb09 orb10 swv01 swv02 swv03 swv04 swv05 swv06 swv07 swv08
1218 1235 1216 1142 1355 1784 1850 1719 1721 1888 1268 1397 1196 1233 1222 1059 888 1005 1005 887 1010 397 899 934 944 1392 1475 1369 1450 1421 1591 1446 1640
1240 1258 1221 1206 1355b 1784b 1850b 1719b 1721b 1888b 1302 1453 1254 1269 1261 1114 915 1065 1037 917 1055 405 935 958 983 1471 1542 1506 1559 1549 1834 1712 1959
swv09 swv10 ta01 ta02 ta03 ta04 ta05 ta06 ta07 ta08 ta09 ta10 ta11 ta12 ta13 ta14 ta15 ta16 ta17 ta18 ta19 ta20 ta31 ta32 ta33 ta34 ta35 ta36 ta37 ta38 ta39 ta40
1604 1631 1231 1244 1206 1170 1210 1210 1223 1187 1247 1241 1321 1321 1271 1345 1293 1300 1458 1369 1276 1316 1764 1774 1778 1828 2007 1819 1771 1673 1795 1631
1833 1900 1260 1259 1282 1214 1247 1290 1253 1268 1339 1310 1440 1429 1453 1375 1403 1424 1515 1526 1402 1424 1950 2013 1889 1998 2048 1910 1900 1896 1959 1847
a
The solutions are achieved by the N6 neighborhood, the mean 4628 moves and T12 tabu list method. The problems found equal to the best solution.
b
denoted by N6. It takes some advantages of the neighborhoods in literature. The N6 neighborhood: The only one critic-path is essential for the N6 neighborhood. If alternative criticpaths exist, the one that has job-predecessor is selected. For example, if we have two operations that can just begin after the critic operation h, let i and j, then the one that is job-successor of operation h is preferred. Let the adjacent operations to be performed on the same machine be a block. N6 generates the neighbors by swapping the ®rst two and the last two
operations on interval blocksÐif the block contains more than two operations. Besides, it will swap the last two operations on the ®rst block if the starting time of the ®rst operation of the block is the same as the head of the schedule (let generally zero) and the block has more than two operations. It will also swap the ®rst two operations on the last block, if the ®nishing time of the last operation of the block is the same as the makespan and the block has more than two operations. One feature of the N6 is that the feasibility (cycle) check for candidate neighbors is done when they are generated; if no cycle exists, the candidate will be
445
The strategies and parameters of tabu search
Fig. 4. The N6 neighborhood strategy.
Table 4. The values of the various measures for six neighborhoods
MCmax MRI% MRE% MEN Iter no. LBE
N6
N5
N3
N2
N4
N1
1268.6 16.6 4.9 32140 4628 18
1271.2 16.5 4.6 36171 4724 19
1278.8 16.0 5.1 48542 4758 16
1278.9 16.1 5.1 37630 4343 16
1292.8 15.3 5.9 27979 4155 13
1355.4 12.3 8.8 65270 3266 9
allowed to become a neighbor. For example, let i and j be two operations to be interchanged, based upon information above. Here operation i precedes operation j. If the operation j is the last operation on a block and there is no path from the job-successor of the operations i to the operation j or if the operation i is the ®rst operation on a block and there is no path from the i to the job-predecessor of the operation j then the interchange of a pair of operations
i, j will generate a neighbor, j?i. That is, operation j will precede operation i on the neighbor schedule with only a difference from a given schedule. Figure 4 illustrates this neighborhood on a Gantt chart. For example, it will generate three neighbors: The ®rst one is that the process sequence of machine 2 is f7; 19; 31; 3; 26; 17g, the second one is that the process sequence of machine 6 is f15; 33; 24; 10; 5; 28g, and the last one is that the process sequence of machine 6 is f15; 33; 10; 24; 28; 5g. Table 4 summarizes computational results relating to these neighborhoods. Table 5 shows results of
Table 5. A comparison of the neighborhoods: paired t-test statistics
N6 N5 N3 N2 N4
N5
N3
N2
N4
N1
0.1451
0.0009 0.0107
0.0008 0.0016 0.4885
0.0000 0.0000 0.0048 0.0022
0.0000 0.0000 0.0000 0.0000 0.0000
paired t-test concerned in comparison of the neighborhoods for Cmax criterion. N6 has minimum MRE value, 2.85%. Also the rate of initial solutions improving is 16.4%. It yields signi®cantly different and better results than that of each neighborhood except for N5 at an alpha level of 0.004. No difference is there between N6 and N5. N5 is better than N3, N2, N4, and N1 at alpha level of 0.0171. N3 is better than N4 and N1 at alpha level of 0.0035, but it is similar to N2. N2 is better than N4 at alpha level of 0.0019 and
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Geyik and Cedimoglu
N4 is better than N1 at alpha level of 0.0000. As a result, we can conclude that N6 is an ef®cient neighborhood for the JSP.
4.2. Length of tabu list In this work, we tested 14 methods of TLL. They are shown in Table 6. In the ®rst ®ve methods (T8±T20), TLLs are constant and they are 8, 10, 12, 15, and 20. In the method TD, TLL is randomly selected within the range of 7±14 from a uniform distribution. The other methods are formed by using the number of blocks
b, jobs
n, machines
m, and neighbors
k. The list lengths of the methods except for the constant ones are updated list length iteration later from the iteration that it is determined. The values of the various measures which were obtained for 14 methods of TLL are shown in Table 7. According to this table, the method with the lowest mean Cmax is the method T12. Table 8 shows the statistical comparison of the method T12 with the others. It is seen that T12 is not different from TM2, TBN3, TBN, T15, TD, TM, and TBN7 at an alpha level of 0.03, but it is better than the others. The list lengths of these methods except for TBN are generally moderate and in the vicinity of 12, see MTL values on Table 7.
5. Conclusions In this paper, a TS model for solving JSP is presented. For representation of JSP, the disjunctive graph method is used. The main elements of TS such as, initial solution, neighborhood structure, move, tabu list, aspiration and termination criteria, are explained as based on this representation and also a numerical example is given. With the goal of determining an ef®cient neighborhood strategy and TLL, a comprehensive computational study is presented. Six strategies of neighborhoods which ®ve of those are from literature and 14 methods of TLLs are tested with a benchmark set containing 98-piece JSPs. The proposed algorithm for the computational study is coded in Pascal. An important conclusion is that the proposed new neighborhood structure, N6, can be used ef®ciently for JSP. Because N6 neighborhood together with N5 is produced the better results from the others. Another conclusion is that the TLL for JSPs is in the vicinity of 12. For TLL, a constant length or a formulation containing the number of jobs, machines and blocks can be used on condition that it is near to 12. It was not mentioned here on intensi®cation and diversi®cation strategies of TS. Therefore, they may be a subject of future works. The stronger hybrid frameworks using ef®ciently both intensi®cation and
Table 6. The TLL methods Methods
T8
T10
T12
T15
T20
TD
TM
TM2
TBM
TBN
TBN3
TBN7
TK
TK2
TLL
8
10
12
15
20
7±14
m
m2
bm
bn
b n/3
b7
k
k2
Table 7. The values of the various measures for 14 TLL methods T12
TM2
TBN3
TBN
T15
TD
TM
TBN7
TBM
T10
TK2
T20
T8
TK
1270.9 1271.1 1272.7 1272.7 1272.4 1273.1 1273.1 1273.4 1274.5 1276.4 1276.9 1278.9 1284.6 MCmax 1268.6 MRI% 16.63 16.42 16.56 16.35 16.38 16.5 16.34 16.4 16.34 16.36 16.29 16.11 16.12 15.59 MRE% 4.50 4.76 4.57 4.81 4.79 4.65 4.83 4.74 4.82 4.80 4.88 5.10 5.05 5.65 MEN 33140 31271 32365 30874 33000 31822 33146 32495 31593 32860 32803 31282 31591 31107 Iter no. 4728 4429 4828 4352 4651 4597 4680 4592 4495 4669 4850 4375 4623 4619 LBE 18 14 19 15 17 17 15 16 17 16 15 15 17 7 MTL 12 14.84 13.73 22.18 15 10.5 13.27 12.01 17.75 10 9.27 20 8 7.23
Table 8. The T12 method vs. the other TLL methods: paired t-test statistics
T12
TM2
TBN3
TBN
T15
TD
TM
TBN7
TBM
T10
TK2
T20
T8
TK
0.1314
0.1584
0.0666
0.0475
0.0927
0.0452
0.0397
0.0179
0.0161
0.0070
0.0010
0.0035
0.0000
The strategies and parameters of tabu search
diversi®cation can be built up by the techniques such as scatter search, path relinking, and genetic algorithms. In addition, the move evaluation phase of TS for JSP takes a very long computational time. Another future work may be about shortening this time.
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