PREEMPTIVE SCHEDULING IN A TWO-STAGE

MULTIPROCESSOR FLOWSHOP WITH RESOURCE

CONSTRAINTS

Ewa Figielska

Warsaw School of Computer Science

Lewartowskiego 17, 00-169 Warsaw, Poland

Keywords:

Two-stage ﬂowshop, Parallel machines, Resource constraints, Column generation algorithm, Genetic algo-

rithm.

Abstract:

A heuristic combining the column generation technique and a genetic algorithm is proposed for solving the

problem of preemptive scheduling in a two-stage ﬂowshop with parallel unrelated machines and renewable re-

sources at the ﬁrst stage and a single machine at the second stage. The objective is to minimize the makespan.

The lower bound on the optimal makespan is derived to be used in the performance analysis of the heuris-

tic. The performance of the heuristic is analyzed by a computational experiment. The results show that the

heuristic is able to ﬁnd near-optimal solutions in reasonable computation time.

1 INTRODUCTION

This paper proposes a heuristic combining the col-

umn generation (CG) technique with a genetic algo-

rithm (GA) for solving the multiprocessor ﬂowshop

scheduling problem which can be brieﬂy described as

follows: there are a number of preemptive jobs to be

processed at two stages, each job being processed ﬁrst

at stage 1 then at stage 2. Stage 1 consists of a num-

ber of parallel unrelated machines, at stage 2 there is

a single machine. Upon completion at stage 1 a job is

ready to be processed at stage 2: it may be processed

at stage 2 when the machine is available there, or it

may reside in a buffer space of unlimited capacity fol-

lowing stage 1 until the machine at stage 2 becomes

available. At stage 1, a job can be processed on any of

the parallel machines, and its processing times may be

different on different machines. The processing of a

job on a machine of stage 1 may be interrupted at any

moment and resumed later on the same or another ma-

chine. A job during its processing at stage 1 requires

some amounts of additional renewable resources. The

total amounts of these resources available at any mo-

ment are limited.

The objective is to ﬁnd a feasible schedule which

minimizes the maximum job completion time in the

two-stage ﬂowshop, C

max

, referred to as makespan.

This problem is NP-hard in the strong sense since

the problem of preemptive scheduling in the two-

stage ﬂowshop with two identical parallel machines

at one stage and one machine at another is NP-hard in

the strong sense (Hoogeveen et al., 1996).

During the last decade the ﬂowshops with multi-

ple machines (FSMP), also called hybrid ﬂowshops,

received considerable attention from researchers.

Most literature in this area addresses the minimum

makespan problems under the assumption that pre-

emptions of jobs are not allowed and the parallel ma-

chines at each stage are identical, e.g. (Gupta, 1988;

Chen, 1995; Haouari and M’Hallah, 1997; Brah and

Loo, 1999; Linn and Zhang, 1999; Oguz et al., 2003).

Only few papers concern the ﬂowshop with parallel

machines that are not identical (Suresh, 1997; Ruiz

and Maroto, 2006).

To the best of our knowledge the multiproces-

sor ﬂowshop scheduling problem with additional re-

source constraints has not been considered in the lit-

erature so far.

In this study, two-stage multiprocessor ﬂowshop

scheduling research is extended by considering the

preemptive scheduling of unrelated parallel machines

with additional resource constraints. Such a problem

may arise in real-life systems that are encountered in

a variety of industries, e.g. in chemical, food and cos-

metics industries. These systems are often subjected

to some additional resource constraints for example

on the availability of the additional resources such as

skilled labor, tools, power. Preemption of jobs usu-

130

Figielska E. (2006).

PREEMPTIVE SCHEDULING IN A TWO-STAGE MULTIPROCESSOR FLOWSHOP WITH RESOURCE CONSTRAINTS.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 130-135

DOI: 10.5220/0001220101300135

 SciTePress

Figure 1: An illustrative example. The resulting schedules: a) the feasible schedule with a random sequence of the partial

schedules, b) the ﬁnal schedule with the sequence of the partial schedules minimizing the makespan.

Figure 2: The data for an illustrative example.

ally results in shortening a schedule. The problem

with parallel unrelated machines at the ﬁrst stage and

a single machine at the second stage may arise in a

manufacturing environment in which products are ini-

tially processed on any of parallel machines and then

each product must go through a ﬁnal testing opera-

tion, which is to be carried out on a common testing

machine.

2 THE FRAMEWORK OF THE

HEURISTIC

The proposed heuristic proceeds in two steps.

Step 1. A column generation algorithm solves the

minimum makespan problem of unrelated par-

allel machines scheduling with additional re-

source constraints which occurs at stage 1.

The solution is composed of a number of par-

tial schedules and its makespan does not de-

pend on the ordering of the partial schedules.

A partial schedule assigns some jobs (or parts

of jobs) to machines for simultaneous process-

ing during a certain period of time, so that

resource constraints are fulﬁlled at every mo-

ment.

Step 2. A genetic algorithm ﬁnds the sequence of the

partial schedules that minimizes the makespan

in the two-stage ﬂowshop which is equal to the

maximum job completion time at stage 2. For

each sequence of the partial schedules gen-

erated in the search process the completion

times of jobs at stage 1 are calculated. Then,

a schedule for the two-stage ﬂowshop is con-

structed and its makespan is calculated taking

into account the job ready times (equal to the

job completion times at stage 1) and process-

ing times of jobs at stage 2.

To illustrate the performance of the heuristic we

present the following example. Consider an instance

of 10 jobs with the job processing times, resource re-

quirements and resource availability as shown in Fig-

ure 2. There are two stages: stage 1 contains two

parallel unrelated machines, stage 2 consists of one

PREEMPTIVE SCHEDULING IN A TWO-STAGE MULTIPROCESSOR FLOWSHOP WITH RESOURCE

CONSTRAINTS

131

machine. Figure 1 presents two schedules for this in-

stance. Each schedule in the two-stage ﬂowshop may

be treated as composed of a schedule of the ﬁrst stage

and a schedule of the second stage. The schedules of

the ﬁrst stage parallel machines are composed of 10

partial schedules. Each partial schedule satisﬁes re-

source constraints at a time. The availability of the

resource at any moment is 10 units. For example, in

the partial schedule of index 1, S

, job 1 and job 9 re-

quire, respectively, 6 and 3 units (see Figure 2) of the

resource at a time, so the total usage of the resource at

any moment in this partial schedule is equal to 9 and

is no greater than 10. Similarly, all remaining partial

schedules satisfy resource constraints at any moment.

In Figure 1a the sequence of the partial schedules

is S

, S

. We can

see that for this sequence of the partial schedules, the

ﬁrst job completed at stage 1 is job 1 (it is ﬁnished

in S

). After completing at stage 1, job 1 starts on

the machine at stage 2 because this machine is free.

The next job completed at stage 1 (in S

) is the job

of index 2. After its completion job 2 is stored in the

buffer space between the stages until the machine of

stage 2 is freed up. The indices of successive jobs

completed at stage 1 are: 1, 2, 9, 7, 10, 3, 4, 5, 8,

6. In Figure 1a, we observe that the machine at stage

2 remains idle after ﬁnishing processing jobs 10 and

3 when it waits for the completion of jobs 3 and 4,

respectively, at stage 1.

In Figure 1b the sequence of the partial schedules is

, S

. This sequence

has been found by the GA so as to minimize the max-

imum job completion time at stage 2 (the makespan

of the whole schedule). For this sequence of the par-

tial schedules, jobs are completed at stage 1 in the

following order: 1, 2, 7, 3, 10, 8, 6, 5, 4, 9, and their

completion times are different than those in Figure 1a,

which results in a much smaller idle time of the sec-

ond stage machine and a much shorter schedule then

those in Figure 1a.

3 NOTATION

In this paper jobs are indexed by j, parallel machines

at stage 1 by i, resource types by r. The parameters

of the problem considered are as follows:

n the number of jobs,

m the number of machines at stage 1,

l the number of types of renewable resources,

the processing time of job j on machine i at

stage 1,

the processing time of job j on the machine at

stage 2,

the number of units of resource r available at a

time,

ijr

the number of units of resource r required at

every moment during processing job j on ma-

chine i at stage 1.

4 HEURISTIC DESCRIPTION

4.1 Solving the Problem of Stage 1

As stated in Section 2, ﬁrst, the problem of resource

constrained preemptive scheduling of parallel unre-

lated machines so as to minimize the makespan is

solved. The solution to this problem is represented

by a set S of partial schedules S

, β ∈ B, where B

is the set of indices of all feasible partial schedules.

Partial schedule is determined by its duration ∆

and

the values of v

(i = 1 . . . m,j = 1 . . . n) represent-

ing an assignment of jobs to machines, where v

= 1

if job j is processed on machine i in partial schedule

and v

= 0, otherwise. The problem is formally

deﬁned as follows:

min

β∈B

∆

(1)

subject to:

β∈B

∆

i=1

= 1, j = 1, . . . , n (2)

j=1

≤ 1, i = 1, . . . , m, β ∈ B (3)

i=1

≤ 1, j = 1, . . . , n, β ∈ B (4)

i=1

j=1

ijr

≤ W

, r = 1, . . . , l, β ∈ B (5)

∈ {0, 1}, i = 1, . . . , m, j = 1, . . . , n, β ∈ B

(6)

∆

≥ 0, β ∈ B (7)

where ∆

(β ∈ B) and v

(i = 1, . . . , m, j =

1, . . . , n, β ∈ B) are decision variables. Constraints

(2) ensure that all jobs are completed at stage 1 of the

two-stage ﬂowshop. Constraints (3) and (4) ensure

that, respectively, each machine works on at most one

job at a time and each job is processed on no more

than one machine at a time. Due to constraints (5)

the usage of each resource at every moment does not

exceed its availability.

In the general case, the above problem is known

to be NP-complete (Slowinski, 1980). It can be opti-

mally solved by means of a CG algorithm. The theo-

retical basis of the CG technique has been provided by

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

132

Dantzig and Wolfe in (Dantzig and Wolfe, 1960) (for

applications of the CG technique see e.g. (Gilmore

and Gomory, 1961; Barnhart et al., 1998; Figielska,

1999; Chen and Lee, 2002)).

A CG algorithm does not generate explicitly all

columns of the problem, which correspond to all par-

tial schedules. It works only with a subset of columns

and adds a new column which improves the solution.

At each iteration of the CG algorithm, the schedule

length is minimized by solving the LP problem of the

form:

min

β∈

∆

(8)

subject to:

β∈

∆

i=1

= 1, j = 1, . . . , n (9)

∆

≥ 0, β ∈

B (10)

where β ⊂

B denotes a subset of indices of columns,

∆

(β ∈

B) are decision variables, and the values of

(i = 1, . . . , m, j = 1, . . . , n, β ∈

B) are ﬁxed

(determined in the previous iterations) or (before the

ﬁrst iteration) given in advance. Let π

∗

(j = 1, . . . , n)

be the optimal solution of the dual problem to the

problem (8)-(10) (π

∗

are dual variables corresponding

to constraints (9)). If there exists a column β ∈ B \

such that

j=1

i=1

∗

pij

− 1 > 0, then the current

set

B can be extended by this new index β and a new

iteration of the CG algorithm is started. Otherwise,

the optimal solution is found and the algorithm stops.

4.2 Finding the Schedule of Minimal

Makespan

The aim is the minimization of the makespan in the

two-stage ﬂowshop. A GA ﬁnds the ordering of the

partial schedules which provides the schedule for the

two-stage ﬂowshop with minimum makespan.

A GA (Holland, 1975) is a search technique that

imitates the natural selection and biological evolu-

tionary process. GAs have been used in a wide vari-

ety of applications, particularly in combinatorial opti-

mization problems and they were proved to be able to

provide near optimal solutions in reasonable time.

A GA starts with a population of randomly gen-

erated candidate solutions (called chromosomes). A

chromosome is represented by a string of numbers

called genes. Each chromosome in the population is

evaluated according to some ﬁtness measure. Certain

pairs of chromosomes (parents) are selected on the

basis of their ﬁtness. Each of these pairs combines

to produce new chromosomes (offspring) and some

of the offspring are randomly modiﬁed. A new pop-

ulation is then formed replacing some of the original

population by an identical number of offspring. The

process is repeated until a stopping criterion is met.

Let P (t) denotes the population at iteration t and

pop

size is the population size. The GA applied in

this paper can be outlined as follows.

1. Generate and evaluate the initial population P (t),

t = 0.

2. Repeat the following steps until stopping condition

is satisﬁed.

2.1. Repeat the following loop

pop size

times

(pop

size is an even number).

2.1.1. Select two parents from P (t).

2.1.2. Apply the crossover operator over the

parent chromosomes and produce 2

offspring chromosomes.

2.1.3. Apply the mutation operator over the

offspring.

2.1.4. Copy the offspring to population P (t+

1).

2.2. Evaluate P (t + 1).

2.3. Replace the worst chromosome of P (t+1) by

the best chromosome found so far.

2.4. Set t = t + 1.

3. Return the best chromosome found.

The factors which characterize the GA applied to

the problem considered in this paper are determined

as follows.

Solution representation. A solution to the sequenc-

ing problem solved by the GA is coded as a single

chromosome whose genes represent the indices of

partial schedules.

Initial population. An initial population of chro-

mosomes is randomly generated.

Evaluation. The value of an objective function,

which is equal to the makespan in the two-stage ﬂow-

shop, is used to measure the ﬁtness of a chromo-

some. For each partial schedule sequence (chro-

mosome) generated in the search process, a sched-

ule for the two-stage ﬂowshop is constructed and

the makespan is calculated taking into account ready

times and processing times of jobs at stage 2.

Parent selection. The binary tournament selection

method is used. In a binary tournament selection,

two chromosomes are randomly chosen. The more ﬁt

(with a smaller objective function value) is then taken

as a parent chromosome. Two binary tournaments are

held to produce two parents.

Crossover. The two-point crossover operator PMX

(Goldberg, 1989) is applied to each pair of par-

ent chromosomes with a probability P

crs

(crossover

probability).

Mutation. The genes of each chromosome in the

population are considered one by one, and the gene

being considered swaps its value with another ran-

PREEMPTIVE SCHEDULING IN A TWO-STAGE MULTIPROCESSOR FLOWSHOP WITH RESOURCE

CONSTRAINTS

133

domly generated gene of the same chromosome with

a probability P

mut

(mutation probability).

Stopping condition. The search process terminates

when the best objective function value (makespan)

found so far is not updated for a predetermined num-

ber of iterations.

On the basis of the preliminary computational ex-

periment the following values of the genetic parame-

ters which ensure a good performance of the algo-

rithm were selected: pop

size = 30, P

crs

= 0.8,

mut

= 0.01, the number of iterations without any

improvement of the best solution found so far is set at

250.

5 LOWER BOUND

Since determination of the optimal solution to the

considered problem is practically impossible (for

larger size problems it is impossible even in the case

of ﬂowshop with identical parallel machines and no

resource constraints (Santos et al., 1995)), a simple

and easily computable lower bound on the optimal

makespan is derived to evaluate the quality of the pro-

posed heuristic.

Let C

∗

denotes the minimal makespan (i.e. the

minimum time needed to complete all jobs at stage 1)

for the problem occurring at stage 1. The ﬁrst lower

bound on the optimal makespan in the two-stage ﬂow-

shop is obtained from the following equation:

= C

∗

+ min

j=1,...,n

} (11)

The second term of the above equation represents the

minimum unavoidable idleness at stage 1 which is

equal to the smallest job processing time at stage 2.

The second lower bound is given by:

= min

i=1,...,m,j=1,...,n

} +

j=1

(12)

since the machine at stage 2 remains idle for at least

the time needed to complete at stage 1 a job with the

smallest processing time at stage 1.

and LB

will be effective for problem in-

stances which are dominated by jobs with large

processing times at stage 1 and stage 2, respectively.

Hence, a lower bound on the optimal makespan in

the considered two-stage ﬂowshop will be

LB = max{LB

, LB

} (13)

6 COMPUTATIONAL

EXPERIMENT

In this section the results of a computational exper-

iment conducted to evaluate the performance of the

Table 1: The computational results.

n m δ (%) CPU time (s)

CG alg GA

20 2 1.16 0.67 0.14

3 2.48 1.46 0.13

4 1.73 1.83 0.16

40 2 0.50 2.01 0.59

3 0.45 4.38 0.52

4 0.61 3.84 0.48

60 2 0.29 4.56 1.00

3 0.37 8.36 1.23

4 0.69 7.21 1.33

80 2 0.19 7.66 1.95

3 0.45 12.96 2.71

4 0.24 12.66 2.59

100 2 0.07 12.29 3.82

3 0.15 21.20 4.45

4 0.22 17.48 5.15

120 2 0.08 18.71 6.20

3 0.18 31.05 9.14

4 0.19 27.54 7.05

average: 0.56 10.88 2.70

proposed heuristic are presented. 360 randomly gen-

erated instances were created and examined. In-

stances were generated for the number of jobs n =

20, 40, 60, 80, 100 and 120, the number of machines

at stage 1, m = 2, 3 and 4, and one resource type.

Resource requirements, α

ijr

, were generated from

U[1, 10] (U [a, b] denotes the discrete uniform distri-

bution in the range of [a, b]), whereas the resource

availability, W

, was set at 10. Processing times at

stage 1, p

, were generated from U[1, 200], U [1, 250]

and U[1, 300] for instances with 2, 3 and 4 machines,

respectively, whereas processing times at stage 2, s

were generated from U[1, 100] for all instances.

To evaluate the performance of the proposed

heuristic two performance measures are used. The

ﬁrst is the relative deviation of a heuristic solution

from the lower bound on the optimal makespan (in

other words the maximum relative deviation from the

optimal makespan) deﬁned as

δ =

C − LB

× 100%,

where C is the best makespan found by the heuristic.

The second performance measure is the CPU time (re-

ported in seconds) consumed by the heuristic to ﬁnd

the best solution.

The results of the experiment are presented in Ta-

ble 1. The ﬁrst two columns show the size of the in-

stances. The relative deviation is presented in column

3. Columns 4 and 5 contain the CPU time (in seconds)

consumed by the CG algorithm and the GA, respec-

ICINCO 2006 - INTELLIGENT CONTROL SYSTEMS AND OPTIMIZATION

134

tively. The entries in Table 1 are average values over

20 instances.

The results show that the presented heuristic per-

forms very well for the entire collection of instances.

We observe that δ decreases with the increase in the

number of jobs. On average δ = 1.79%, 0.52%,

0.45%, 0.29%, 0.15% and 0.15% for problems with

20, 40, 60, 80, 100, and 120 jobs, respectively. The

CPU time of the heuristic grows with the number of

jobs. This increase is more signiﬁcant for the CG al-

gorithm than for the GA. The number of machines

does not seem to affect signiﬁcantly the computation

time of the heuristic, however, we observe that in the

case of 2 machines the CG algorithm consumes less

CPU seconds than in the cases of 3 and 4 machines.

7 CONCLUSIONS

In this paper a heuristic combining the column gener-

ation algorithm with the genetic algorithm for solving

the two-stage ﬂowshop preemptive scheduling prob-

lem with parallel unrelated machines and resource

constraints at the ﬁrst stage, and a single machine at

the second stage has been developed. The heuristic

has been tested as to its effectiveness in ﬁnding a min-

imum makespan schedule and as to its computation

time. The obtained results indicate that for problems

with a large number of jobs the heuristic usually ﬁnds

schedules with makespan close to the optimal value

in reasonable computation time.

REFERENCES

Barnhart, C., Johnson, E., Nemhauser, G., Savelsbergh, M.,

and Vance, P. (1998). Branch and price: column gen-

eration for solving huge integer problems. Oper. Res.,

46:316–329.

Brah, S. A. and Loo, L. L. (1999). Heuristics for scheduling

in a ﬂow shop with multiple processors. Europ. J. of

Opernl Res., 113:113–112.

Chen, B. (1995). Analysis of classes of heuristics for

scheduling a two-stage ﬂow shop with parallel ma-

chines at one stage. J. of Opernl Res. Soc., 46:234–

244.

Chen, Z.-L. and Lee, C.-Y. (2002). Parallel machine

scheduling with a common due window. Europ. J. of

Opernl Res., 136:512–527.

Dantzig, G. B. and Wolfe, P. (1960). Decomposition prin-

ciple for linear programs. Oper. Res., 8:101–111.

Figielska, E. (1999). Preemptive scheduling with

changeovers: using column generation technique and

genetic algorithm. Comp. and Ind. Engin., 37:63–66.

Gilmore, P. C. and Gomory, R. E. (1961). A linear program-

ming approach to the cutting-stock problem. Oper.

Res., 9:849–859.

Goldberg, D. E. (1989). Genetic Algorithms in Search, Op-

timization and Machine Learning. Addison-Wesley.

Gupta, J. N. D. (1988). Two stage hybrid ﬂowshop schedul-

ing problem. J. of Opernl Res. Soc., 39:359–364.

Haouari, M. and M’Hallah, R. (1997). Heuristic algorithms

for the two-stage hybrid ﬂowshop problem. Oper. Res.

Let., 21:43–53.

Holland, J. H. (1975). Adaptation in Natural and Artiﬁcial

Systems. University of Michigan Press, Ann Arbor,

MI.

Hoogeveen, J. A., Lenstra, J. K., and Veltman, B. (1996).

Preemptive scheduling in a two-stage multiproces-

sor ﬂow shop is np-hard. Europ. J. of Opernl Res.,

89:172–175.

Linn, R. and Zhang, W. (1999). Hybrid ﬂow shop schedul-

ing: a survey. Comp. and Ind. Engin., 37:57–61.

Oguz, C., Ercan, M. F., Cheng, T. C. E., and Fung, Y. F.

(2003). Heuristic algorithms for multiprocessor task

scheduling in a two-stage hybrid ﬂow shop. Europ. J.

of Opernl Res., 149:390–403.

Ruiz, R. and Maroto, C. (2006). A genetic algorithm for hy-

brid ﬂowshops with sequence dependent setup times

and machine eligibility. Europ. J. of Opernl Res.,

169:781–800.

Santos, D. L., Hunsucker, J. L., and Deal, D. E. (1995).

Global lower bounds for ﬂow shop with multiple

processors. Europ. J. of Opernl Res., 80:112–120.

Slowinski, R. (1980). Two aproaches to problems of re-

source allocation among project activities - a compar-

ative study. J. Opl Res. Soc., 31:711–723.

Suresh, V. (1997). A note on scheduling of two-stage ﬂow

shop with multiple processors. Int. J. of Prod. Econ.,

49:77–82.

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