Heuristic algorithms for the two-stage hybrid flowshop problem

E L S E V I E R Operations Research Letters 21 (1997) 43-53

Heuristic algorithms for the two-stage hybrid flowshop problem

Mohamed Haouari a'*, Rym M'Hallah b' 1

aEcole Polytechnique de Tunisie-BP 51, 2070 La Marsa, Tunisia b lnstitut Rdgional des Sciences Informatique et T~l~communications, BP 212, 1082 Tunis, Tunisia

Received 20 September 1995; revised 1 September 1996

Abstract

A two-stage Hybrid Flowshop Problem (FS . . . . . ) is a two-center shop with several parallel machines per center and n jobs to be processed on at most one machine per center. The objective consists of minimizing the maximum completion time. Two two-phase methods based on Simulated Annealing and Tabu Search are proposed. The results are compared with solutions provided by existing heuristics, and with a new derived lower bound. These comparisons show the superiority of the derived lower bound and the efficiency of the proposed heuristic. The Tabu search based heuristic yields the optimal solution for 35% of the problems. Its average relative error is 0.82%. © 1997 Published by Elsevier Science B.V.

Keywords: Flowshop scheduling; Simulated annealing; Tabu search

I. In trod uc t ion

Textile and apparel production units can hardly fit in any classical scheduling model. Instead, these units have a special structure combining some ele- ments of both the flow shop problem and the parallel machine scheduling problem. This combined model is referred to as the Hybrid FlowShop problem (HFS) [9, 12]. It is to be noted that this problem is also known as the flowshop problem [1, 3] since the operating procedure follows a flowshop, and the duplication of machines in the shop intro- duces additional flexibility to the production process. The HFS is the adequate model for several industrial settings [2, 14, 15, 17]. A special but

*Corresponding author. 1 E-mail: [email protected]

common case of HFS is the two-stage HFS, de- noted hereafter as FS . . . . . .

Formally, the FS . . . . . is defined as follows. A production unit has two centers with ml identical parallel machines in the first center and m2 identical parallel machines in the second center (mi > 1, i = 1, 2.) A set of n jobs, ready at time 0, must be successively processed by the two centers. Let pij be a positive integer number corresponding to the processing time of job (j = 1, . . . , n ) on center (i = 1, 2.) This job can be executed on any of the m~ machines of center i. It must be noted that a machine can process only one job at a time, and that a job can be processed only by a single machine at a time. Moreover, no preemption is al- lowed. The objective is to minimize the maximum completion time (makespan).

Despite the important theoretical and practical interest of the FS . . . . . . relatively few papers treated

0167-6377/97/$17.00 © 1997 Published by Elsevier Science B.V. All rights reserved PH S0 1 67-63 77(97)00004-7

44 M. Haouari, R. M'Hallah / Operations Research Letters 21 (1997) 43-53

this problem. Arthanary and Ramaswamy [2] developed an exact algorithm and tested it only for very small instances (less than 10 jobs). Gupta [9], and Gupta and Tunc [103 developed heuristic methods for the special case where one of the two centers has exactly one machine. Recently, Lee and Vairaktarakis [ 12] presented heuristics for both the two-center and the general case. They also developed lower bounds for the two-center case, and presented worst case and empirical analysis of their heuristics. A detailed literature review is given by Vignier and Proust [18].

The FS . . . . . is a generalization of the classical two-stage flow shop problem (ml=m2 = 1). However, while this latter problem can be solved efficiently (in polynomial time) using Johnson's algorithm [11], the FS . . . . . is NP-complete, since minimizing the makespan in parallel identical machine environment is NP-complete [6]. This observation motivates the interest in heuristic algorithms.

In this paper, a new heuristic approach for solv- ing the FS . . . . . is developed. In Section 2, a new lower bound for the FS . . . . . is derived. In Section 3, a construction method for obtaining a good feasible schedule is proposed. This schedule is later used as the initial solution for the local improvement heuristics based on Simulated Annealing (SA) and Tabu Search (TS), presented respectively, in Section 4 and 5. The approaches are evaluated and compared with the derived lower bound, and with existing methods.

2. A new lower bound

It is usually difficult to evaluate the performance of approximate algorithms. However, a practical approach, that is often used, consists of running the heuristic on large problem sets and computing the average relative gap defined as (CH- LB)/LB x 100%, where CH is the heuristic's solution cost and LB a lower bound. In the following, a new lower bound, to be subsequently used in evaluating the performance of the proposed heuristics, is derived.

Let SPT(m2) be the minimum sum of completion times, on center 1, of the m2 jobs whose processing

times on center 1 are the shortest. Similarly, let SPT(mI) be the minimum sum of completion times, on center 2, of the m~ jobs whose processing times on center 2 are the shortest.

Lemma 1

L B - ( . . ) Max SPT(m2) + ~j=x P2~ SPT(ml) + ~j=x Pig \ m2 ml

is a lower bound of the optimal makespan, C*.

Proof. A lower bound, LB, for the optimal makespan C* is the average total idle time and total processing time on the second center, i.e.,

LB = [(total idle time in center 2)

+ (total processing time on center 2)]/m2.

A lower bound of the total idle time of the m2 machines of center 2 is given by considering the sum of completion times of the m2 jobs having the least processing time on center 1, and finding the minimum completion times of these jobs in center 1. This can be obtained by applying the shortest processing time rule [3, 4]. Thus,

(SPT(m2) + ZT= 1 pzj)/m2 <<. C*. (1)

By using the symmetry of the FS . . . . . . and by interchanging the roles of centres 1 and 2, we obtain

(SPT(ml) + ~=1 pl~)/ml <~ C*. (2)

As the data is integer, the lower bound is rounded to the closest higher integer. []

Lee and Vairaktarakis [12] proposed

LBLv =

Max ~]j~l PI(j) + (m2 -- m1)+Pl(1) + 2j=1 P2j,

m2

nl I n t ~',j=~ Pz(j) + (ml - mz)+Pz(l) + F.j=I Plj ml

M. Haouari, R. M'Hallah / Operations Research Letters 21 (1997) 43-53 45

as a lower bound to C*, where Pi~j) is the jth ordered value of Pij ( j = 1,2 . . . . . n), (i = 1,2.) A comparison of LBLv to the new derived lower bound is provided by Lemma 2.

Lemma 2

LB/> LBLv.

Proof. Since

m2 SPT(m2) ~> ~ Pltj) + (m2 - ml)+Pltl), (3)

j=l

and

mt SPT(ml) >~

j = l P2~j) + ( m l - - m2)+P2(l), (4)

LB/> LBLv. []

Along with the LBLv, Lee and Variaktarakis [12] proposed another lower bound CLB. CLB is the completion time found by applying the Jackson's algorithm to an auxiliary 2-machine flowshop problem. This problem is defined as follows. Each of the two centers is replaced by a single machine and the processing times Pij are replaced by piJml.

One can easily check that the derived bound does not dominate the CLB bound. Similarly, CLR does not dominate LB. It is however noteworthy that in all the problems considered, LB was consistently better than CLB.

3. A construction heuristic

The following simple heuristic yields a feasible schedule.

Initialization Define for each center i, i = 1, 2 a priority list Li

Current Step For i:= 1 to 2 do (Schedulin9 of Jobs in Center i)

Repeat Assign to the first available machine the

waiting job with highest priority until all jobs are scheduled in center i

End for

This heuristic consists mainly in successively scheduling the jobs on centers one and two. For the second center, the completion time of the job at the first center is taken as the earlist ready time of the job. Thus, jobs are ordered in a sequential manner according to the previously defined priority list.

The priority list that has been empirically found most efficient, consists in ranking the jobs in centers one and two according to the Most Work Remain- in9 rule (MWKR). This rule favors those jobs re- quiring the longest remaining processing time (in- cluding the processing time on the current center.) Moreover, the ordering is performed so as to minimize the idle time of the machines.

The solution obtained by this heuristic is later improved using SA and TS. In the next section, the SA based method is exposed. Some tactical choices and implementation issues are addressed.

4. A simulated annealing based method

SA is a metastrategy for optimization by local improvements [5, 16]. It was developed by analogy to the annealing process studied in mechanical stat- istics [13]. It consists of moving from an initial solution to a minimal cost one. To apply this procedure to the FS,.,,,,~, a clear definition of a solution and its neighborhood must be given and a number of tactical implementation issues must be addressed (such as the choice of the parameter T, the stopping criterion, etc.)

Note that a priority list defines a permutation of the jobs. In the following, a solution to the FS,, .... will be completely identified by a permutation list. This list will be used in the construction heuristic, described in the previous section, as the priority list of the first center.

Let X be a job permutation. We define two types of single moves (transformations) a-opt, and na- opt. The a-opt consists in interchanging two adjacent jobs, whereas, the na-opt consists in swapping any two jobs (i.e.,jobs are not necessarily adjacent). It is clear that the a-opt is a special case of the na-opt. The whole purpose, however, of using the a-opt is to avoid destroying the structure of the solution by modifying it only slightly at each step.


On the other hand, the a-opt reduces the size of the neighborhood in comparison with na-opt (i.e., (n - 1) neighbors versus n(n - 1)/2.) In practice, the a-opt is systematically used until all neighbors are explored. The na-opt is subsequently used to ex- plore a wider neighborhood, increasing the likelihood of finding a better solution.

The parameter T (referred to as the temperature) is adjusted all along the process as follows.

Initialization Let To -- 1.1Ao, where Ao is the positive difference between the cost of the first neighbor having a cost greater than that of the current solution and the cost of the current solution. This temperature guarantees a 0.4 probability of acceptance.

Current step Let L = (n - l)(n + 2)/2, where L is the length of the plateau. Set T k + 1 = 0 . 9 5 Tk, every L iterations.

Stopping criterion Stop the algorithm if the solution has not improved by at least 2% after 3 consecutive pla- teaus.

5. A Tabu search based method

The Tabu search (TS) method is another optimization metastrategy for local improvement [7, 8, 16]. To apply TS to the FS . . . . . . a solution is again recognized by a permutation list. It is then proceeded in a way similar to that used in simulated annealing. The initial solution will be given by the construction heuristic of Section 3.

Two types of neighborhoods are combined: a- opt and na-opt. First, a-opt is used. As it converges, the algorithm is restarted with the best found solution using na-opt to perform the first move, and a-opt subsequently. This procedure is reiterated a finite number of times. The advantage of using a-opt is to limit the number of solutions examined (at most (n - 1) neighbors) instead of n ( n - 1)/2. When the neighborhood is exhausted, an enlarged one (starting with the best solution) is explored so as to maximize the likelihood of finding an improv- ing solution. A Tabu list of fixed length L -- 10 has

been kept. This length seemed to offer the best compromise between the need to avoid cycling and the computation time which increases with the length of L. It is important to emphasize that the cost of the solution and not the solution itself is saved in the Tabu list. The algorithm is restarted when k = 3 consecutive iterations yield no improvement. The whole process is reiterated 3 times.

6. Numerical results

Table 1 reports the average relative gap of the solutions yielded by the proposed heuristics for different problem sizes and different processing time ratios. These were selected as suggested by Lee and Vairaktarakis [12]. The problem sizes (PS) ml x m2 are 2 x 4, 4 × 4, and 4 x 2 with 30, 40, and 50 jobs. Additional cases, corresponding to 20 and 100 jobs, were also considered. The processing time ratios (PTR) are, as suggested by Lee and Vairak- tarakis [ 12], 2: 4, 4: 4, and 4: 2. For example, a ratio 2:4 means that Plj (j = 1 . . . . ,n) is generated randomly from a Uniform distribution on [1, 20] while P2j is generated randomly from a Uniform distribution on [1, 40]. Twenty instances were generated for each problem size and processing time ratio. The main results are summarized in Table 1.

Prior to analyzing these results, it is noteworthy that the proposed construction method yields in several cases better solutions than the H heuristic proposed by Lee and Vairaktarakis [12]. This construction solution is further improved using either TS or SA; leading efficient approximate solutions.

As displayed in Fig. 1, it might seem that the proposed heuristics perform better for larger problems. It is however suspected that the heuristics are very efficient for all problem sizes. On the other hand, the approximation of the lower bound is most likely best when the number of jobs is large.

In most problems, the TS procedure yields optimal or very near-optimal solutions. For example, for the n = 40 case, the worst relative gap is 0.4%. Considering all the problems, TS reaches the optimal solution for 35% of the cases (316 out of 900) while SA yields the optimal solutions in 15% of the

M. Haouari, R. M'Hallah / Operations Research Letters 21 (1997) 43-53

Table 1 Relative gap of the proposed heuristics from the lower bound LB

47

No. of jobs PS PTR = 2:4 PS PTR = 4:4 PS PTR = 4:2

2 x 4 4 x 4 4 x 2 2 x 4 4 x 4 4 x 2 2 x 4 4 x 4 4 x 2

Average

20 TS 2.90 1.20 0.35 0.92 5.72 0.13 0.56 3.43 1.22 SA 6.19 2.81 0.35 1.06 8.77 0.33 0.49 5.15 2.79 H 7.04 10.73 3.03 3.29 10.17 1.76 3.09 10.42 5.69

30 TS 1.43 0.85 0.06 0.57 3.10 0.05 0.27 1.45 1.46 SA 3.82 1.63 0.15 1.03 6.75 0.31 0.37 3.72 3.44 H 4.07 6.46 1.85 2.19 8.30 2.54 2.19 5.99 5.27

40 TS 0.96 0.43 0.12 0.50 1.57 0.12 0.34 1.08 0.89 SA 2.89 1.38 0.31 0.80 5.05 0.14 0.41 2.39 2.19 H 3.11 5.77 2.07 1.79 6.19 1.26 1.82 4.98 3.36

50 TS 0.54 0.30 0.02 0.26 1.09 0.04 0.20 0.95 0.42 SA 2.53 1.50 0.18 0.49 3.46 0.20 0.32 2.01 1.00 H 2.12 4.89 1.09 0.91 4.79 1.34 1.56 4.80 1.83

100 TS 0.19 0.15 0.02 0.11 0.39 0.01 0.07 0.41 0.18 SA 1.23 0.78 0.10 0.16 2.33 0.10 0.14 0.87 0.54 H 1.20 2.39 0.80 0.59 2.19 0.67 0.76 2.45 0.98

Average TS 1.20 0.59 0.11 0.47 2.37 0.07 0.29 1.46 0.83 SA 3.33 1.62 0.20 0.71 5.27 0.22 0.35 2.83 1.99 H 3.51 6.05 1.77 1.75 6.33 1.51 1.88 5.73 3.43

1.83 3.10 6.14

1.03 2.36 4.32

0.67 1.73 3.37

0.42 1.30 2.59

0.17 0.69 1.34

0.82 1.84 3.55

Percent Relanve Enor

7

! 0 m w

2O 3O 40 5O 100

Fig. 1. Relative gap of TS, SA, and H for different problem sizes.

TS

- - I - - S A

--dk-- H

Number of Jobs

Frequency

600

500

400

300

200

Percent Relallve Gap

,OO o

l iSA

C)TS

IIH

Percent Reiatlvo 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8

Fig. 2. Frequency distribution of the percent relative error for TS, SA, and H heuristics.

2.5

2

I

oT


1.5

0.5

2:4

2x4

---E---4X4

~ 4 x 2

, P roemslng T line Rallo

4:4 4:2

Fig. 3. Relative gap of the Tabu search heuristic for different processing times.

cases (138 out of 900) and the H heuristic in 2% of the cases (16 out of 900). The distributions of the relative gap of the 900 problems for TS, SA, and H are displayed in Fig. 2. Observing the three

distributions, it is obvious that the distribution of TS is the best as it is clustered in the [0, 1] interval (70% of the cases have a relative gap less than 1%). The H distribution is the most spread


Percent Rekalve Gap

~. 2x4

--ff i - -4x4

.-.I-.-4x2

2:4 4:4 4:2

Fig. 4. Relative gap of the simulated annealing heuristic for different processing times.

Processing T line Ratio

Table 2 Run time of the TS and SA heuristics(s) a

n m k Avarage run time Standard deviation Minimum Maximum Percent of times the run times run time run time of

TS SA TS - SA TS SA TS - SA TS SA TS SA T S < S A S A < T S T S = S A

30 2 6.92 30 4 7.02 40 2 19.13 40 4 19.78 50 2 42.38 50 4 43.93 60 2 82.90 60 4 85.05 70 2 145.66 70 4 154.45 80 2 242.04 80 4 251.36 90 2 376.10 90 4 390.87

6.34 0.58 0.73 0.72 0.08 5 5 9 8 6 60 34 6.44 0.58 0.77 0.76 0.08 6 5 9 8 11 58 31

17.37 1.76 1.78 1.90 0.23 17 15 25 22 13 85 2 18.42 1.36 2.12 2.30 0.25 17 16 26 23 22 75 3 38.81 3.57 4.28 4.37 0.50 39 35 57 50 10 89 1 40.79 3.14 5.10 5.49 0.54 39 35 58 51 14 86 0 75.34 7.56 8.07 8.80 1.11 75 68 113 97 10 89 1 79.80 5.25 9.37 10.77 1.12 76 68 116 101 18 80 2

137.49 8.17 11.51 18.53 1.98 135 122 200 182 17 83 0 145.57 8.88 20.83 22.06 2.09 136 123 209 183 20 80 0 225.27 16.77 25.82 30.20 3.05 219 201 333 295 12 88 0 239.18 12.02 29.20 38.61 3.95 221 201 343 378 22 78 0 356.03 20.07 42.69 53.14 4.87 343 313 524 463 15 85 0 370.57 20.30 60.48 60.39 8.36 347 314 540 476 22 88 0

a On a 486DX with a clock of 66 Mhz.

(wides t r a n g e ) a n d h a s t h e h i g h e s t a v e r a g e re la -

t ive e r r o r ( 3 . 6 % v e r s u s 1 .8% for S A a n d 0.8°,/0

for TS).

O b s e r v i n g Figs . 3 a n d 4, i t is c l e a r t h a t t h e b e s t

r e su l t s a r e a c h i e v e d w h e n t h e c e n t e r w i t h t h e h i g h -

es t p r o c e s s i n g t i m e h a s t h e l ea s t n u m b e r o f m a c h i n -

es a n d vice v e r s a (e.g. 4 x 2, 2 : 4 a n d 2 x 4, 4: 2).

S imi l a r ly , t h e w o r s t r e su l t s a r e o b t a i n e d w h e n t h e r e

a r e m a n y m a c h i n e s a n d t h e t w o c e n t e r s a r e e q u a l l y

l o a d e d (e.g. 4 x 4, 4 : 4).


Extensive experimentation has been undertaken to evaluate the average run time of SA and TS heuristics. Problems of size 30, 40, 50, 60, 70, 80, and 90 with a maximum of two machines per center and a maximum of 4 machines per center have been considered. Processing times were generated randomly from a [1, 100] Uniform distribution. One

hundred instances of each problem type have been tallied. The problems were executed on a 486 66 PC using a Microsoft Power Station FORTRAN com- piler. Table 2 displays the average and standard deviation of the run times for TS and SA, and the average difference between the two heuristics. Sim- ilarly, it displays the minimum and maximum run

Frequency

30,

26

20

15

10

r l TS

l I S A

0 1 . . . . . . . I ,~ ," ,n, II, I I...n..n ,n ,~ ,1 J], Execution Time(see,)

35 39 43 47 61 56

Fig. 5. Frequency distribution of the run time of the Tabu search and the simulated annealing heuristics for mk = 4 and n = 50.

Execution Time (sec.)

400. • TS mk : 2

• - - 4 1 - - SA m k : 2

350. - - ~ , - - TS mk ,, 4

" ' / - - -X-- - SA m k : 4

300. , ' / /

250 " ~

200 o . ~ / /

150

1005O J / 0 Number o f Jobs

30 40 60 60 70 80 90

Fig. 6. Average run time of the Tabu search and the simulated annealing heuristics for mk = 2 and mk= 4.


times over each set of hundred problems as well as the percent of time the run time of TS was less than, greater than, and equal to the run time of SA. Fig. 5 depicts a typical frequency distribution of the run time of SA and TS heuristics. Finally, Fig. 6 shows that the average run time of SA and TS increases

slightly as the number of machines per center increases but grows rapidly as the number of jobs increases.

Observing Table 2 and Fig. 6, it is clear that TS takes generally longer than SA (in 80% of the cases) even though there are cases where SA requires

Execution Time (sec,) 4O0

(a)

- / / / 300

26o /

160 / /

100

60 ~ ~ "

0 30 40 60 SO 70 80 90

Fig. 6a. Average run time of TS and SA as a function of the number of jobs for mk= 2.

e TS

---I-- SA

Number of Jobs

Execution Time (sec.) 400

(b)

350

3OO

260

2OO

150

100

50

0

3O

&

RB f a

S #

"/° %

°.~°

~ s

.r ";'j"

40 60 60 70 80 90

- - ~ - -TS

- - -X-- - SA

N m of ,ka~

Fig. 6b. Average run time of TS and SA as a function of the number of jobs for mk= 4.



(c)

4OO

36O

3OO

260

200

160

100

5 0

oi 40 60 60 70 80 90

i nk - -2

- - ~+~, - - r a k e 4

Number of Jobs

Fig. 6c. Average run t ime of TS for mk = 2 and mk = 4 as a funct ion of the n u m b e r of jobs.


400.

360,

300

260

200

160

100

60

•j• - - 411-- m k m 2

• - - -X - - mk • 4 / /

, / / /

/;, . /

.g J j

/ V

0

(d) 30 40 60 60 70 80 90

Number of Jobs

Fig. 6d. Average run t ime of SA for mk = 2 and m k = 4 as a function of the n u m b e r of jobs.

a longer run time than TS (15% of the cases). A paired t-test based on the 1400 problems cannot refute the hypothesis that TS run time is superior to SA run time. It is however noteworthy that both

run times are of the same order. Further statistical testing supports the argument that TS and SA run times have equal variances despite the seemingly higher variance of SA run time.


Acknowledgements

The authors would like to thank Dr.G.L. Vairak- tarakis for his encouragement. The first author would like to acknowledge support from the Tu- nisian Secr6tariat d'Etat/t la Recherche Scientifique et ~t la Technologie.

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Documents

Heuristic algorithms for the two-stage hybrid flowshop problem