Accepted Manuscript

Group shop scheduling with sequence-dependent setup and transportation times

Fardin Ahmadizar, Parmis Shahmaleki

PII: S0307-904X(14)00138-3DOI: http://dx.doi.org/10.1016/j.apm.2014.03.035Reference: APM 9923

To appear in: Appl. Math. Modelling

Received Date: 13 December 2012Revised Date: 6 January 2014Accepted Date: 16 March 2014

Please cite this article as: F. Ahmadizar, P. Shahmaleki, Group shop scheduling with sequence-dependent setup andtransportation times, Appl. Math. Modelling (2014), doi: http://dx.doi.org/10.1016/j.apm.2014.03.035

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customerswe are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, andreview of the resulting proof before it is published in its final form. Please note that during the production processerrors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

1

Group shop scheduling with sequence-dependent setup

and transportation times

Fardin Ahmadizar*, Parmis Shahmaleki

Department of Industrial Engineering, University of Kurdistan, Pasdaran Boulevard,

Sanandaj, Iran

* Corresponding author. Tel./fax: +98-871-6660073

f.ahmadizar@uok.ac.ir (F. Ahmadizar); parmis.shahmaleki@uok.ac.ir (P. Shahmaleki)

Abstract This paper considers a group shop scheduling problem (GSSP) with sequence-

dependent setup and transportation times. The GSSP provides a general formulation

including the job shop and the open shop scheduling problems. The consideration of setup

and transportation times is among the most realistic assumptions made in the field of

scheduling. In this paper, we study the GSSP with transportation and anticipatory

sequence-dependent setup times, where jobs are released at different times and there are

several transporters to carry jobs. The objective is to find a job schedule that minimizes

the makespan, that is, the time at which all jobs are completed and transported to the

warehouse (or to the customer). The problem is formulated as a disjunctive programming

problem and then prepared in a form of mixed integer linear programming (MILP). Due to

the NP-hardness of the GSSP, large instances cannot be optimally solved in a reasonable

amount of time. Therefore, a genetic algorithm (GA) hybridized with an active schedule

generator is proposed to tackle large-sized instances. Both Baldwinian and Lamarckian

versions of the proposed hybrid algorithm are then implemented and evaluated through

computational experiments.

Keywords: Group shops scheduling; Sequence-dependent setup times; Transportation

2

times; Makespan; Genetic algorithm.

1 Introduction

The GSSP, which frequently occurs in manufacturing environments, includes both

the job shop and the open shop scheduling problems. Hence, it is NP-hard and difficult to

solve optimally. In the GSSP a set of jobs has to be processed on a set of machines, where

the operations of each job are partitioned into a number of groups by taking the

technological constraints into account. That is, the operations that are not subject to

precedence constraints and can be processed in any order are placed in the same group,

while those operations that have to satisfy precedence constraints are placed in distinct

groups.

In the majority of research efforts conducted on scheduling problems, researchers

usually have either ignored setup times or combined them with their corresponding

processing times to simplify the analysis. In recent years, however, it has been of interest

to consider setup times in scheduling decisions. The main reason why scheduling

problems involving setup times (or costs) have attracted a lot of attention is due to the fact

that there are tremendous savings when setup times are explicitly considered [1,2]. The

setup times could be either sequence-independent or sequence-dependent. In the sequence-

independent type, setup depends only on the job to be processed. In general, the sequence-

independent setup times can be simply included in the job processing times. On the other

hand, in the case of sequence-dependent setup times (SDST), setup depends not only on

the job to be processed, but also on the job just completed. For example, Pinedo [3] has

described a paper bag factory where a setup is required whenever a machine switches from

one type of paper bag to another; the setup duration clearly depends on the similarities

between the two consecutive products (e.g., the similarities in size and the number of

3

colours). In such situations, it is not valid to include the setup times in the job processing

times [4].

The SDST can be either non-anticipatory or anticipatory [2]. A setup is non-

anticipatory if it can begin only when both the corresponding job and machine are

available. On the other hand, a setup is anticipatory if it can begin when the machine is

available even when the job is not available to be processed.

In the past, researchers have usually assumed that transportation times between

stages/machines are negligible. Recently, however, a popular assumption made in

scheduling decisions is that it may be impossible to begin the processing of a job on a

machine immediately after the completion of the preceding operation of the job because of

the transportation time; a transporter must first deliver the job between the two stages.

When a transporter reaches a stage, the processing of the delivered job can be started only

if one of the machines available at that stage is ready to receive the job and the setup

required has been completed. If not, the transporter leaves the job in the buffer of the stage

until the processing can begin.

The transportation times could be either job-independent or job-dependent [5]. In

the job-independent type, the magnitude of a transportation time depends only on the

distance between the two consecutive stages/machines, while in the job-dependent type it

is determined by the distance as well as the job to be carried. Moreover, the transportation

system may be multi-transporter or single-transporter [6]. In a multi-transporter system,

there are several (unlimited) transporters to carry jobs; so, a job never has to wait for a

transporter before its transportation. However, in a single-transporter system, all

transportations between stages are carried out by a single transporter; so, a job may wait

for the transporter to return.

When considering simultaneously the SDST and transportation times, there are two

4

cases to be considered. In the first case that corresponds to the anticipatory SDST, the

overlapping of the setups and transportations is allowed, and so, the processing of a job on

a machine can be started if both the transportation and the setup required have been

completed (see, e.g., Fig. 1). However, in the second case that corresponds to the non-

anticipatory SDST, no overlapping of the setups and transportations is allowed, that is, the

setup of a machine can be started after the transportation has been completed. Fig. 2 shows

an example of this case in which, although machine i' is available, it is kept idle before

being set up to process job j.

- Figures 1 & 2 -

Assuming that jobs are released at different times, this paper investigates a GSSP

with two realistic and rarely considered assumptions: (1) the setup times are anticipatory

and sequence-dependent, and (2) the transportation times are job-dependent and the

transportation system is a multi-transporter system. The aim is to minimize the makespan.

A mathematical formulation of the problem is proposed; it is first formulated as a

disjunctive programming problem, and then prepared in a form of MILP. Moreover, to

tackle large size problem instances, a GA hybridized with an active schedule generator is

proposed. Both Baldwinian and Lamarckian versions of the hybrid algorithm are then

implemented and evaluated through a series of computational experiments. To the best of

our knowledge, no similar studies are known in the scheduling context.

The rest of the paper is organized as follows. The next section goes over the

literature on the GSSP as well as the literature on shop scheduling problems with the

SDST and transportation times. In Section 3, the problem is introduced. Sections 4 and 5,

respectively, describe the proposed MILP model and hybrid algorithm, followed by

Section 6 providing the computational results. Finally, the conclusions and future research

directions are stated in Section 7.

5

2 Literature review

The GSSP was first introduced in a mathematical contest in 1997 [7]. Blum and

Samples [8] are among the pioneers considering the GSSP. Assuming that all jobs are

available at time zero, they have developed an ant colony optimization (ACO) algorithm

for the problem with the makespan criterion, in which the setup times are sequence-

independent and the transportation times are ignored. Liu et al. [9] have proposed a tabu

search (TS) algorithm for the same problem. Ahmadizar et al. [10,11] have considered the

GSSP subject to random release dates and processing times. In [10], the total weighted

completion time minimization problem has been formulated as a deterministic MILP

problem, and an ACO algorithm has then been developed for the problem. Moreover,

suggesting a lower bound on the expected makespan, Ahmadizar et al. [11] have proposed

a simulation-based ACO approach for the makespan minimization problem. Recently,

Ahmadizar and Zarei [12] have considered the GSSP subject to fuzzy release dates and

processing times. They have prepared the makespan minimization problem in a form of

deterministic MILP, and then developed a GA for the problem. However, like most other

research efforts conducted on scheduling problems, these studies have neglected the setup

and transportation times.

Although scheduling problems with setup considerations have been widely studied

over the last decade, there are a few research efforts considering simultaneously setup and

transportation times. Rossi and Dini [13] have developed an ACO-based algorithm for

flexible manufacturing systems scheduling in a job shop environment with routing

flexibility, sequence-dependent setup and transportation times. Gröflin and Klinkert [14]

have proposed a TS algorithm to solve a blocking job shop with sequence-dependent

transfer and setup times, where the objective is to minimize the makespan. Moreover,

making the same assumptions, Gröflin et al. [15] have developed an ACO algorithm for a

6

flexible blocking job shop. Naderi et al. [5] have discussed a flexible flow shop with

anticipatory SDST and job-dependent transportation times. They have proposed an

electromagnetism algorithm to minimize the total weighted tardiness. In addition, Naderi

et al. [16] have studied hybrid flow shop problems with non-anticipatory SDST and

transportation times, and developed a simulated annealing algorithm to minimize the total

completion time and total tardiness.

3 Group shop scheduling problem

The GSSP considered in this paper can be stated as follows. There are n jobs to be

processed on a set of m machines. Each job has at most one operation to be processed on

each machine. Each job may be processed on at most one machine at a time, and each

machine can process at most one job at a time. All operations must be processed without

interruption, that is, preemption is not allowed. All machines are available and ready at

time zero, and there exists an unlimited storage capacity between every two machines.

Moreover, each job j has a release date j

r before which its processing cannot be started.

The processing of job j on machine i, referred to as operation (i, j), has a duration ij

p . The

setup times are anticipatory and sequence-dependent; a setup time ij js ′ is incurred between

the processing of jobs j' and j on machine i, where 0i js denotes the setup time to process

job j first on machine i. Similarly as in [15], the setup times are assumed to satisfy the

triangle inequality [17,18], that is, ij j ijj ij js s s′ ′′ ′ ′′+ ≥ , a reasonable assumption according to

practical applications (see, e.g., [19]). In addition, the transportation times are job-

dependent, and it is assumed that there are enough transporters to carry the jobs from a

machine to another, i.e., there is always a transporter available. Let ji it ′ be the time

required to transport job j from machine i' to machine i, where 0j it and

*jit denote the

7

times required to transport job j from the warehouse to machine i and from machine i to

the warehouse (or to the customer), respectively. The time to load and unload a transporter

is included in the transportation time. Like the setup times, the transportation times are

supposed to satisfy the triangle inequality, that is, ji i jii ji i

t t t′ ′′ ′ ′′+ ≥ ; this is, of course, not a

real restriction [20].

It should be noted here that a distinct feature of the GSSP is that, taking the

technological constraints into account, the operations of each job are partitioned into a

number of groups, such that the operations in the same group are unordered while those in

two distinct groups satisfy the predefined precedence constraint between the groups. If

each group consists of just one operation, the GSSP is equivalent to a job shop scheduling

problem. While in the other special case where all operations of each job belong to just

one group, the GSSP is equivalent to an open shop scheduling problem.

In the GSSP under consideration, processing orders for all machines as well as all

groups have to be determined so as to minimize the makespan, that is, the time at which all

jobs are completed and transported to the warehouse.

Let the decision variable ij

x be the earliest possible starting time of operation (i, j),

and let max

C denote the makespan. Given a feasible solution to the problem, the operation

starting times clearly satisfy the following conditions: if (i, j) is the first operation of job j

and the first operation on machine i, ( )0 0max ,ij j j i i j

x r t s= + ; if (i, j) is the first operation

of job j and is processed just after operation (i, j') on machine i,

( )0max ,ij j j i ij ij ij jx r t x p s′ ′ ′= + + + , where ij ijx p′ ′+ is the completion time of operation (i,

j'); if job j is processed on machine i' and then on machine i and (i, j) is the first operation

on machine i, ( )0max ,ij i j i j ji i i jx x p t s′ ′ ′= + + ; and finally, if job j is processed on machine i'

and then on machine i and (i, j) is processed just after operation (i, j') on machine i,

8

( )max ,ij i j i j ji i ij ij ij jx x p t x p s′ ′ ′ ′ ′ ′= + + + + . The makespan of the schedule is then equal to

{ }*( , )max ij ij ji

i jx p t+ + .

4 MILP model

In the following, the problem under consideration is first formulated as a

disjunctive program and then prepared in a form of MILP. The GSSP can be formulated as

the following disjunctive programming model:

Minimize maxC , subject to:

( , )&( , ) with ( , ) preceding ( , )ij i j i j ji ix x p t i j i j i j i j′ ′ ′′ ′− ≥ + ∀ (1)

or ( , )&( , ) belonging to the same groupij i j i j ji i

i j ij ij jii

x x p t

i j i j

x x p t

′ ′ ′

′ ′

− ≥ +

′∀

− ≥ +

(2)

or ( , ) &( , )ij ij ij ij j

ij ij ij ijj

x x p s

i j i j

x x p s

′ ′ ′

′ ′

− ≥ +

′∀

− ≥ +

(3)

0 ( , )ij j j ix r t i j≥ + ∀ (4)

0 ( , )ij i jx s i j≥ ∀ (5)

max * ( , )ij ij ji

C x p t i j− ≥ + ∀ (6)

Constraints (1) ensure that each operation (i, j) cannot start before all operations

belonging to preceding groups of job j are completed and the job is transported to machine

i. Disjunctive constraints (2) ensure that some ordering exists among operations belonging

to every group. Disjunctive constraints (3) ensure that some ordering exists among

operations requiring the same machine for their execution. It is noted that under these

constraints if, for example, job j is scheduled on machine i immediately after job j', then

the processing of job j cannot be started before the processing of job j' is completed and

9

the machine is set up to process job j. Constraints (4) ensure that the processing of each

job cannot start before it is released and transported from the warehouse to the machine

performing its first operation. Constraints (5) ensure that each machine cannot start to

process its first job before it is set up. Finally, constraints (6) ensure that the makespan is

equal to the time at which all jobs are completed and transported to the warehouse.

Let ji i

y ′ and ij j

z ′ be binary variables related to select one ordering in each

disjunction, and L a large number. Disjunctive constraints (2) and (3) can then be

respectively reduced to the following linear constraints, thus preparing the GSSP under

consideration in a form of MILP.

(1 )

( , )&( , ) belonging to the same groupij i j i j ji i ji i

i j ij ij jii ji i

x x p t L y

i j i j

x x p t Ly

′ ′ ′ ′

′ ′ ′

− ≥ + − −

′∀

− ≥ + −

(7)

(1 )

( , ) &( , )ij ij ij ij j ij j

ij ij ij ijj ij j

x x p s L z

i j i j

x x p s Lz

′ ′ ′ ′

′ ′ ′

− ≥ + − −

′∀

− ≥ + −

(8)

5 Genetic algorithm

Due to the NP-hardness of the problem, large instances cannot be optimally solved

in a reasonable amount of time through the proposed MILP model. Thus, an efficient GA

is proposed to tackle large size problem instances.

GAs are effective, practical and robust meta-heuristics inspired from the

evolutionary process occurring in natural systems. In a GA, a population of individuals

(candidate solutions) evolves through generations by the use of selection, crossover and

mutation. Important issues that must be considered in designing a GA are chromosome

representation, population initialization, genetic operators and termination criterion. So

far, many researchers have successfully applied GAs to solve complicated optimization

10

problems in different fields of scientific and engineering disciplines, including shop

scheduling problems; see, e.g., [12,21-29]. To study more details about GAs, the reader is

referred to [30].

The general structure of the algorithm proposed to solve the GSSP under

consideration is then expressed as follows:

Step 1. (Initialization) For the initial population, generate randomly a set of feasible

solutions, and then, improve each of them.

Step 2. (Parent selection) Apply an operation to the population to select parents.

Step 3. (Crossover) Apply a crossover operation to the parents.

Step 4. (Mutation) Apply a mutation operation to the resulting offspring, and then,

improve each of them.

Step 5. (Survivor selection) Apply an operation to the population together with the

resulting offspring to select individuals for the next generation.

Step 6. (Termination) Repeat steps 2 to 5 until the termination criterion is satisfied.

5.1 Chromosome representation

The chromosome representation, which maps solution characteristics into a string

of symbols, is the first step in designing a GA for a particular problem. It sets up a bridge

between the original problem space and the space being searched by the evolutionary

process. Thus, defining an appropriate chromosome representation strategy is an important

issue that affects the algorithm performance.

To describe and manipulate a solution to the GSSP, in the proposed GA a two-part

chromosome representation, similar to that of [12], is utilized. The first part of a

chromosome is associated with disjunctive constraints (2) and indicates processing orders

for all groups by employing one permutation for each group having more than one

11

operation. Taking the first part into account, the other part of the chromosome associated

with disjunctive constraints (3) then indicates processing orders for all machines by

employing just one permutation with repetition of job numbers, allowing each job to be

repeated exactly the number of times equal to its operations. In this manner, each

chromosome presents a feasible solution.

Fig. 3 shows an example of a chromosome representation for a simple problem

instance consisting of three jobs and three machines, where each job has three operations.

Suppose that, taking into consideration the technological constraints, the operations of job

1 are partitioned into three groups, those of job 2 into two groups, and those of job 3 into

one group. The first group of job 1 consists of operation (1, 1), its second group has

operation (2, 1), and its third group has operation (3, 1). The first group of job 2 consists

of operations (1, 2) and (3, 2), and its second group has operation (2, 2). Moreover, the

unique group of job 3 consists of operations (1, 3), (2, 3) and (3, 3). From part 1 of the

chromosome shown in Fig. 3, the processing order for each of the two groups having more

than one operation is first fixed. Then, from part 2, the first ‘2’ denotes the first operation

of job 2, that is, operation (3, 2), the second ‘2’ denotes the second operation of job 2, that

is, operation (1, 2), and so on. Accordingly, from this part of the chromosome, the

processing order for each of the three machines is determined; e.g., the job sequence on

machine 2 is job 1, job 3, and then, job 2.

- Figure 3 -

5.2 Active schedule generation

To improve a feasible solution represented in form of a chromosome, an active

schedule generation algorithm (ASGA) is proposed which transforms the chromosome

into an active schedule, i.e., a schedule characterized by the property that it is not possible

12

to start an operation any earlier without delaying another operation [3]. The proposed

ASGA is applied to a given chromosome in the following way. A list of gaps, that is, idle

time intervals, is determined for each machine as well as each job. Starting from the

operation at the first position in the second part of the chromosome, each operation (i, j) is

scheduled at the earliest time possible, by considering both the technological constraints

and the gaps on machine i and job j; the gaps are then immediately updated. In order to

explain this algorithm in detail, let i

gapm and j

gapj be the sets of gaps for machine i and

job j, respectively. The ASGA can then be described as follows:

Step 1. For each machine i and each job j, initialize i

gapm to [0, )+∞ and j

gapj to

[ , )j

r +∞ .

Step 2. For k=1 to total number of operations, do the following:

Step 2.1. Consider operation (i, j) at position k in part 2 of the chromosome;

Step 2.2. Determine intervals 1 2[ , ]I I so that operation (i, j) can be assigned to time

slot 1 2[ , ]I I ;

Step 2.3. Among the intervals determined in step 2.2, select interval 1 2[ , ]st st with

minimum 1st , and set

1ijx st= ;

Step 2.4. Remove from i

gapm interval [ , ]ij ij ijx x p+ . If operation (i, j) is the last

scheduled operation among all operations belonging to the same group,

remove from jgapj all the intervals before the latest completion time among

all operations belonging to the group. Otherwise remove from jgapj only

interval [ , ]ij ij ijx x p+ .

In order to clarify step 2.2 of the above proposed ASGA, let us assume

1 2[ , ]i

Im Im gapm∈ is the gap between two scheduled operations (i, j') and (i, j''), that is,

13

1Im and 2Im are, respectively, the completion time of job j' and the starting time of job j''

on machine i. Clearly, operation (i, j) can be assigned to time slot 1 2[ , ]Im Im if

2 1 ij j ij ijjIm Im s p s′ ′′− ≥ + + . Now, assume 1 2[ , ]

jIj Ij gapj∈ is the gap between two scheduled

operations (i', j) and (i'', j), that is, under the current partial schedule job j is completed on

machine i' at 1Ij , and then, it is started on machine i'' at 2Ij . It is clear that operation (i, j)

can be assigned to time slot 1 2[ , ]Ij Ij if 2 1 ji i ij jii

Ij Ij t p t′ ′′− ≥ + + (considering step 2.4, since

1 2[ , ]j

Ij Ij gapj∈ , operations (i, j) and (i'', j) certainly belong to the same group). Therefore,

step 2.2 can be stated as follows: determine intervals

1 2 1 2 1 2[ , ] [ , ] [ , ]ij j ijj ji i jii

I I Im s Im s Ij t Ij t′ ′′ ′ ′′= + − ∩ + − so that 2 1 ijI I p− ≥ . It is noteworthy

that there are some special cases, e.g., in the case where 1 0Im = , that is, where machine i

processes job j first, 1 ij jIm s ′+ is replaced by 0i j

s ; or, in the case where 1 jIj r= , that is,

where job j is first processed on machine i, 1 ji iIj t ′+ is replaced by 0j j i

r t+ .

As an illustration, let us consider the previously mentioned problem instance with

the release dates, processing, setup and transportation times given in Tables 1–4,

respectively. We assume, for simplicity, that the transportation times are job-independent.

Applying step 1 of the proposed ASGA then results in [0, )i

gapm = +∞ for i=1,…,3,

1 [2, )gapj = +∞ , 2 [3, )gapj = +∞ , and

3 [2, )gapj = +∞ . In addition, Table 5 summarizes the

application of step 2 to the chromosome shown in Fig. 3, resulting in the (active) schedule

presented in Fig. 4.

- Tables 1–5 -

- Figure 4 -

The proposed GA hybridized with the ASGA may then be implemented in two

different versions, namely Baldwinian (GAB) and Lamarckian (GA

L) versions. According

14

to the Baldwinian evolution, applying the ASGA to a chromosome only affects its fitness

value, but not its genetic code; the chromosome in selection can then benefit from the

improved fitness. On the contrary, according to the Lamarckian evolution, applying the

ASGA not only affects the fitness value of the chromosome, but also causes changes to its

genetic code that can be inherited in following generations. Accordingly, in GAL

developed for the GSSP under consideration, considering the active schedule generated,

the chromosome's genetic code is altered in a way as follows. All operations are first

sequenced in the ascending order of their starting times, where ties are broken randomly.

Then, for each group in the first part of the chromosome, the operation permutation is

changed in accordance with the sequence, and finally, replacing every operation in the

sequence by its job number, the resulting sequence is inserted into the second part of the

chromosome.

Applying the above method to the schedule shown in Fig. 4 then results in a new

genetic code, as presented in Fig. 5.

- Figure 5 -

It should be noted here that the total number of operations could be at most equal

to n × m (as a result, the chromosome’s length could be at most twice n × m), and for

every machine i and every job j, i

gapm and j

gapj could contain at most n and m slots,

respectively. Thus, the time complexity of the ASGA, clearly dictated by steps 2, 2.2 and

2.3, is 2 2( )O n m . Since all the operation starting times are determined by the ASGA, in

GAB as well as in GA

L calculating the makespan takes (log( ))O nm time. Moreover, the

time complexity of the method applied in GAL to alter the chromosome's genetic code is

obviously ( log( ))O nm nm . Therefore, the overall time complexity of the improvement

process together with the objective evaluation is determined by the ASGA, and

consequently, GAB owns the same worst-case time complexity as GAL, that is, Baldwinian

15

version is not asymptotically faster than Lamarckian version. In the other hand, the time

spent doing the improvement process together with the objective evaluation clearly

dominates the time required by all the other phases of the proposed GA hybridized with

the ASGA. Let PopS and Iter be, respectively, the population size and the number of

generations, i.e., the number of times steps 2 to 5 of the GA are executed. The time

complexity of GAB as well as GAL is then

2 2( )O Iter PopS n m .

5.3 Selection operators

In the proposed GA, the classical roulette wheel mechanism [30,31] more likely

selecting the fitter chromosomes is adopted as the parent selection operation. The roulette

wheel method is implemented in the current population to select, with replacement, a

number of parents equal to PopS, where the fitness of each individual q is calculated as

follows:

max max( )Pop q

qfit C C= − (9)

where max

PopC is the maximum makespan among the population and max

qC the makespan for

individual q.

Moreover, in the survivor selection phase individuals are selected for the next

generation among both the current population and their offspring. A combination of the

elitism strategy and the roulette wheel method is adopted as the survivor selection

operation. First, a predefined percentage ElitP of the new population is filled by the

current best chromosomes. The rest is then filled according to the result of the roulette

wheel mechanism, where the fitness of each individual q is calculated as follows:

max max( )Pop Off q

qfit C C+= − (10)

where max

Pop OffC

+ is the maximum makespan among the set of non-elite individuals of the

16

current population and their offspring.

5.4 Variation operators

Crossover as the main variation operator of a GA recombines the genetic codes of

a pair of parents in order to generate two offspring, i.e., two new solutions in as yet

unvisited regions of the search space. In this study, a combination of two different

operators is adopted as the crossover operator, which is then applied to each pair of

parents according to a predefined crossover probability CrosP. For each group in part 1,

the order crossover (OX) [32] is separately applied with a probability OXP, and for part 2

the generalized position crossover (GPX) [33] with a probability GPXP.

Mutation, performed by randomly changing the gene values of a chromosome,

takes place after recombination in order to prevent premature convergence. As the

mutation operator, the well-known swap mutation exchanging the values of two randomly

selected genes is adopted, which is then applied with a probability MutP. With a

probability SwP, the swap mutation is applied to (just) one randomly selected group in part

1, while it is certainly applied to part 2.

6 Computational results

A computational experiment is conducted in order to assess the performance of the

proposed MILP model as well as the performance of the proposed GA hybridized with the

ASGA. In the literature, there is only one GSSP instance, called whizzkids97, consisting of

15 machines, 20 jobs, and 197 operations partitioned into 124 groups [7]. In the original

instance, of course, the jobs are all available at time zero, and the transportation and setup

times are not considered. Therefore, a number of problem instances with different sizes,

from 5 jobs and 5 machines to 25 jobs and 25 machines, are randomly generated as

17

follows. For an instance with n jobs and m machines, assuming that each job consists of m

operations, a permutation of the integers 1 to m is randomly generated for each job; the

first integer in the permutation then specifies the machine required to process its first

operation, the second integer the machine required to process its second operation, and so

on. The operations of the job are then partitioned into groups by taking into consideration

the permutation; two consecutive operations belong to the same group only if they require

machines i and i+1 or i and i-1 for their execution (e.g., let the permutation generated for

job j – in a problem instance with 5 machines – be 4, 3, 2, 5, 1; then the operations of job j

are partitioned into 3 groups: the first group consists of operations (4, j), (3, j) and (2, j),

the second and third groups consist of operations (5, j) and (1, j), respectively). The job

release dates are generated from a uniform distribution between 5 and 30, as in [10,11].

Moreover, similarly as in [5], the processing and transportation times are uniformly

sampled from the ranges [1, 99] and [1, 30], respectively; the duration of SDST is then

defined as 25%, 50%, 100%, or 125% of the maximum processing time. Accordingly, we

have four scenarios for each problem instance; in the first scenario the setup times are

generated from a uniform distribution between 1 and 25, in the second scenario from a

uniform distribution between 1 and 50, and so on. For whizzkids97, of course, only the

release dates, transportation and setup times are randomly generated. It should be noted

that, in the data generation process, SDST as well as transportation times have been

chosen such that they satisfy the triangle inequality.

The proposed GA has been coded in Visual C#.NET 2008 and run on laptop with

Intel Pentium-4 2.0 GHz CPU and 2 GB RAM. In the preliminary experiment, different

values for the numeric parameters have been considered; PopS has been tested between

100 and 700 in increments of 100, ElitP between 0.1 and 0.5 in increments of 0.1, CrosP

between 0.8 and 1 in increments of 0.05, OXP between 0.3 and 0.7 in increments of 0.1,

18

GPXP between 0.8 and 1 in increments of 0.05, MutP between 0.1 and 0.5 in increments

of 0.1, and SwP between 0.5 and 1 in increments of 0.1. To make the performance of the

GA robust, each combination of the parameter values has been tested on some instances

selected randomly. The following values have then been superior: PopS = 500, ElitP = 0.3,

CrosP = 0.95, OXP =0.5, GPXP = 0.95, MutP = 0.3 and SwP = 0.7. The algorithm has

then been run 10 replicates for each problem instance, and the maximum (Max), average

(Avg) and minimum (Min) of the makespans obtained have been provided. In addition, the

CPU time limit of each run at which the GA is terminated has been set by referring to the

problem size.

As mentioned earlier, the proposed GA hybridized with the ASGA is implemented

in its Baldwinian version (GAB) as well as its Lamarckian version (GAL). A question that

may arise is whether the GA does not perform better without the ASGA but with

generating an additional number of generations. Accordingly, in order to evaluate the

impact of the proposed ASGA, each problem instance has been solved using the

Darwinian version of the proposed GA (GAD) in which the ASGA is not applied at all. In

GAD, no gaps are determined and a schedule is deduced from a chromosome by simply

putting the operations in the schedule according to their order in the second part of the

chromosome; when operation (i, j) is scheduled, the next operation of job j as well as the

next operation on machine i can only be started after the completion of operation (i, j). As

the schedule obtained in this way only has the property that no operation can be completed

earlier without changing the order of processing on the machines as well as inside the

groups, it is semi-active (not necessarily active) [3]. Furthermore, for each problem

instance the MILP model has been solved using CPLEX 11 with a time limit of 2 hours.

The results obtained are summarized in Table 6, where the solution quality is measured by

the mean percentage difference from the best found solution (BFS). If the BFS is not

19

bracketed, it means that the optimality of the BFS has been proved. In addition, LB stands

for lower bound. We derive a lower bound for the problem under consideration. Let j

last

be the set of all operations belonging to the last group of job j. For each operation (i, j)

belonging to gth group of job j, let ij

set denote the set of all operations belonging to

groups g and g-1; if operation (i, j) belongs to the first group of job j, ijset consists of all

operations belonging to this group as well as dummy operation (0, j) denoting the

warehouse. It is then easy to establish the following lower bound for the makespan:

( )*( , ) ( , ) 0,..,1,.., 1,..,

1 1

max max min min , max minij j

m n

j ij ji i ji ij ij ji j set i j last j nj n i m

i j

LB r p t t p s′ ′′ ′∈ ∈ == =

= =

= + + + +

∑ ∑ (11)

- Table 6 -

From Table 6, as expected, it becomes difficult to use the proposed MILP model

when the problem size – particularly the number of jobs – increases; large instances

cannot be optimally solved in a reasonable amount of computing time. Moreover, although

GAD performs well for small instances, it is the worst among the versions of the GA,

confirming the impact of the proposed ASGA. Finally, from the computational results it is

evident that the proposed hybrid algorithm performs better when mimicking the

Lamarckian evolution than when mimicking the Baldwinian evolution; indeed, it is

suggested to apply the ASGA to a chromosome so as to not only improve its fitness value,

but also cause changes to its genetic code inherited in subsequent generations.

7 Conclusions and future research

The GSSP provides a general formulation including the other shop scheduling

problems such as the job shop and the open shop scheduling problems. This study is the

first dealing with a GSSP with sequence-dependent setup and transportation times.

Assuming that jobs are released at different times, this paper investigates the GSSP where

20

the SDST are anticipatory, the transportation times are job-dependent and the

transportation system is a multi-transporter system. The objective is to find a job schedule

that minimizes the makespan. A mathematical formulation of the problem is developed; it

is first formulated as a disjunctive program, and then prepared in a form of MILP.

Moreover, due to the NP-hardness of the GSSP, a GA hybridized with an ASGA is

proposed to tackle large-sized instances. Both Baldwinian and Lamarckian versions of the

hybrid algorithm are then implemented and evaluated through computational experiments.

The computational results are given to show the power of the proposed algorithm, in

particular its Lamarckian version, for producing very good solutions at a reasonable CPU

time.

A number of possible, significant extensions to this study could be investigated in

future research. It would be interesting to consider the case in which the SDST are non-

anticipatory; so, the setup of a machine cannot be started before the transportation has

been completed. Another extension would be to consider the system in which all

transportations between machines are carried out by a single transporter (e.g., a single

robot); as a result, a job may wait for the transporter to return.

References

1. A. Allahverdi, H.M. Soroush, The significance of reducing setup times/setup costs, European

Journal of Operational Research 187 (2008) 978–984.

2. A. Allahverdi, C.T. Ng, T.C.E. Cheng, M.K. Kovalyov, A survey of scheduling problems with

setup times or costs, European Journal of Operational Research 187 (2008) 985–1032.

3. M. Pinedo, Scheduling: Theory, Algorithms, and Systems, 3rd ed., Springer, New York, 2008.

4. K.R. Baker, D. Trietsch, Principles of Sequencing and Scheduling, John Wiley & Sons, New

Jersey, 2009.

5. B. Naderi, M. Zandieh, M.A.H.A. Shirazi, Modeling and scheduling a case of flexible

flowshops: Total weighted tardiness minimization, Computers & Industrial Engineering 57 (2009)

1258–1267.

21

6. D.R. Sule, Industrial scheduling, PWS Publishing Company, USA, 1996.

7. http://www.win.tue.nl/whizzkids/1997.

8. C. Blum, M. Sampels, An ant colony optimization algorithm for shop scheduling problems,

Journal of Mathematical Modelling and Algorithms 3 (2004) 285–308.

9. S.Q. Liu, H.L. Ong, K.M. Ng, A fast tabu search algorithm for the group shop scheduling

problem, Advances in Engineering Software 36 (2005) 533–539.

10. F. Ahmadizar, M. Ghazanfari, S.M.T. Fatemi Ghomi, Application of chance-constrained

programming for stochastic group shop scheduling problem, International Journal of Advanced

Manufacturing Technology 42 (2009) 321–334.

11. F. Ahmadizar, M. Ghazanfari, S.M.T. Fatemi Ghomi, Group shops scheduling with makespan

criterion subject to random release dates and processing times, Computers & Operations Research

37 (2010) 152–162.

12. F. Ahmadizar, A. Zarei, Minimizing makespan in a group shop with fuzzy release dates and

processing times, International Journal of Advanced Manufacturing Technology 66 (2013) 2063–

2074.

13. A. Rossi, G. Dini, Flexible job-shop scheduling with routing flexibility and separable setup

times using ant colony optimisation method, Robotics and Computer-Integrated Manufacturing 23

(2007) 503–516.

14. H. Gröflin, A. Klinkert, A new neighborhood and tabu search for the blocking job shop,

Discrete Applied Mathematics 157 (2009) 3643–3655.

15. H. Gröflin, D.N. Pham, R. Bürgy, The flexible blocking job shop with transfer and set-up

times, Journal of Combinatorial Optimization 22 (2011) 121–144.

16. B. Naderi, M. Zandieh, A.K.G. Balagh, V. Roshanaei, An improved simulated annealing for

hybrid flowshops with sequence-dependent setup and transportation times to minimize total

completion time and total tardiness, Expert Systems with Applications 36 (2009) 9625–9633.

17. C. Artigues, P. Lopez, P.D. Ayache, Schedule generation schemes for the job-shop problem

with sequence-dependent setup times: Dominance properties and computational analysis, Annals

of Operations Research 138 (2005) 21–52.

18. C. Artigues, D. Feillet, A branch and bound method for the job-shop problem with sequence-

dependent setup times, Annals of Operations Research 159 (2008) 135–159.

19. P. Brucker, O. Thiele, A branch and bound method for the general-shop problem with

sequence-dependent setup times, OR Spektrum 18 (1996) 145–161.

20. J. Hurink, S. Knust, Tabu search algorithms for job-shop problems with a single transport

robot, European Journal of Operational Research 162 (2005) 99–111.

21. L.D. Giovanni, F. Pezzella, An improved genetic algorithm for the distributed and flexible job-

shop scheduling problem, European Journal of Operational Research 200 (2010) 395–408.

22. J. Behnamian, S.M.T. Fatemi Ghomi, Hybrid flowshop scheduling with machine and resource-

22

dependent processing times, Applied Mathematical Modelling 35 (2011) 1107–1123.

23. S.M. Kamrul Hasan, R. Sarker, D. Essam, Genetic algorithm for job-shop scheduling with

machine unavailability and breakdowns, International Journal of Production Research 49 (2011)

4999–5015.

24. C.C. Wu, W.H. Wu, P.H. Hsu, K. Lai, A two-machine flowshop scheduling problem with a

truncated sum of processing-times-based learning function, Applied Mathematical Modelling 36

(2012) 5001–5014.

25. J. Behnamian, S.M.T. Fatemi Ghomi, F. Jolai, O. Amirtaheri, Minimizing makespan on a

three-machine flowshop batch scheduling problem with transportation using genetic algorithm,

Applied Soft Computing 12 (2012) 768–777.

26. J.C. Chen, C.C. Wu, C.W. Chen, K.H. Chen, Flexible job shop scheduling with parallel

machines using genetic algorithm and grouping genetic algorithm, Expert Systems with

Applications 39 (2012) 10016–10021.

27. Q. Zhang, H. Manier, M.A. Manier, A genetic algorithm with tabu search procedure for

flexible job shop scheduling with transportation constraints and bounded processing times,

Computers & Operations Research 39 (2012) 1713–1723.

28. F. Ahmadizar, M. Hosseinabadi Farahani, A novel hybrid genetic algorithm for the open shop

scheduling problem, International Journal of Advanced Manufacturing Technology 62 (2012)

775–787.

29. S.H. Chen, M.C. Chen, P.C. Chang, V. Mani, Multiple parents crossover operators: A new

approach removes the overlapping solutions for sequencing problems, Applied Mathematical

Modelling 37 (2013) 2737–2746.

30. A.E. Eiben, J.E. Smith, Introduction to Evolutionary Computing, Springer, Berlin, 2003.

31. S.N. Sivanandam, S.N. Deepa, Introduction to genetic algorithms, Springer, New York, 2008.

32. L. Davis, Job shop scheduling with genetic algorithms, in: Proceedings of the First

International Conference on Genetic Algorithms and Their Applications, Hillsdale, NJ, 1985, pp

136–140.

33. D.C. Mattfeld, Evolutionary Search and the Job Shop: Investigations on Genetic Algorithms

and Production Scheduling, Springer, Berlin, 1995.

23

Table 1 Release times for the example

Job j 1 2 3

rj 2 3 2

24

Table 2 Processing times for the example

Job j 1 2 3

p1j 4 3 4

p2j 2 4 5

p3j 2 3 2

25

Table 3 Sequence-dependent setup times for the example

Machine i 1 2 3

Job j 1 2 3 1 2 3 1 2 3

si0j 2 1 1 4 3 3 5 4 5

si1j - 3 2 - 5 4 - 1 1

si2j 1 - 3 2 - 5 2 - 1

si3j 2 2 - 1 2 - 1 1 -

26

Table 4 Transportation times for the example

Machine i 1 2 3

tj0i 3 2 1

tj1i - 4 2

tj2i 4 - 2

tj3i 2 2 -

tji* 3 2 2

27

Table 5 Application of step 2 of the proposed ASGA to the numerical example

k (i, j) [st1, st2] xij gapmi gapjj

1 (1,1) [5, )+∞ 5 [0,5] [9, )∪ +∞ [9, )+∞

2 (3,2) [4, )+∞ 4 [0,4] [7, )∪ +∞ [3, 4] [7, )∪ +∞

3 (2,1) [13, )+∞ 13 [0,13] [15, )∪ +∞ [15, )+∞

4 (3,1) [17, )+∞ 17 [0,4] [7,17] [19, )∪ ∪ +∞ [19, )+∞

5 (1, 2) [12, )+∞ 12 [0,5] [9,12] [15, )∪ ∪ +∞ [15, )+∞

6 (1, 3) [18, )+∞ 18 [0,5] [9,12] [15,18] [22, )∪ ∪ ∪ +∞ [2,18] [22, )∪ +∞

7 (2, 3) [4,12] 4 [0,4] [9,13] [15, )∪ ∪ +∞ [2, 4] [9,18] [22, )∪ ∪ +∞

8 (2,2) [20, )+∞ 20 [0,4] [9,13] [15, 20] [24, )∪ ∪ ∪ +∞ [24, )+∞

9 (3, 3) [11,16] 11 [0,4] [7,11] [13,17] [19, )∪ ∪ ∪ +∞ [2,4] [9,11] [13,18] [22, )∪ ∪ ∪ +∞

28

Table 6 Computational results for problem instances

MILP

GA

GAD GAB GAL

n-m Scenario LB BFS Quality Time Min Avg Max Min Avg Max Min Avg Max Time

5-5 1 370 426 0 1 0 0 0 0 0 0 0 0 0 10

2 370 431 0 1 0 0.27 0.94 0 0 0 0 0 0 10

3 370 465 0 2 0 0.27 0.94 0 0.21 0.64 0 0 0 10

4 523 626 0 2 0 0.47 1.43 0 0 0 0 0 0 10

Average 0 1.5 0 0.25 0.83 0 0.05 0.16 0 0 0 10

5-6 1 411 501 0 1 0 0 0 0 0 0 0 0 0 10

2 411 513 0 1 0 0 0 0 0 0 0 0 0 10

3 437 555 0 2 0 0.11 0.46 0 0 0 0 0 0 10

4 475 589 0 2 0 0.14 0.53 0 0 0 0 0 0 10

Average 0 1.5 0 0.06 0.25 0 0 0 0 0 0 10

6-5 1 432 512 0 1 0 0 0 0 0 0 0 0 0 11

2 447 523 0 2 0 0.26 0.81 0 0.18 0.52 0 0 0 11

3 447 537 0 2 0 0.43 0.91 0 0.24 0.58 0 0 0 11

4 508 610 0 2 0 0.34 0.87 0 0 0 0 0 0 11

Average 0 1.6 0 0.26 0.65 0 0.10 0.27 0 0 0 11

6-6 1 583 625 0 4 0.32 0.32 0.32 0 0.32 0.32 0 0 0 13

2 583 629 0 6 0 2.43 4.29 0 0.95 2.86 0 0.47 2.86 13

3 583 698 0 6 0 1.14 3.15 0 0.57 2.57 0 0 0 13

4 583 791 0 6 0 0.93 2.52 0 0.35 2.27 0 0.27 0.75 13

Average 0 5.5 0.08 1.21 2.57 0 0.55 2.01 0 0.19 0.90 13

6-7 1 534 667 0 7 0 0.81 1.71 0 0 0 0 0 0 14

2 534 678 0 7 0.14 1.28 2.45 0 0.76 1.38 0 0 0 14

3 551 681 0 10 0.11 1.31 2.66 0 0.27 1.63 0 0 0 14

4 596 705 0 10 0.21 0.85 3.53 0.09 0.42 0.92 0 0.14 0.61 14

Average 0 8.5 0.11 1.06 2.59 0.02 0.36 0.98 0 0.03 0.15 14

7-6 1 572 693 0 345 0.18 0.49 0.83 0 0.28 0.66 0 0 0 19

2 572 704 0 452 0 1.16 2.73 0 0.37 0.75 0 0 0 19

3 589 722 0 1210 0.46 1.88 2.59 0.26 1.62 2.10 0 0.18 0.54 19

4 623 761 0 1890 0.25 2.07 3.06 0.20 0.57 1.15 0 0.14 0.42 19

Average 0 974 0.22 1.40 2.30 0.11 0.71 1.16 0 0.08 0.24 19

7-7 1 520 689 0 420 0 0.87 2.17 0 0.50 1.01 0 0 0 25

2 525 730 0 732 0 0.91 1.98 0 0.68 1.02 0 0.41 0.90 25

3 547 790 0 3240 0.50 0.75 3.51 0 0.63 2.65 0 0.25 0.75 25

4 612 (848) 0 7200 0.23 0.54 3.30 0.23 0.37 1.65 0.23 0.23 0.23 25

Average 0 2898 0.18 0.77 2.74 0.05 0.55 1.58 0.05 0.22 0.47 25

7-8 1 563 678 0 1250 0 0.34 1.02 0 0.22 0.76 0 0 0 27

2 563 691 0 4320 0 1.56 2.04 0 0.53 0.88 0 0 0 27

3 588 (702) 0.05 7200 0.09 0.71 2.51 0 0.60 1.21 0 0.11 0.51 27

4 615 (782) 0.11 7200 0.27 0.94 2.82 0.08 0.67 1.43 0 0.17 0.47 27

Average 0.04 4992 0.09 0.89 2.10 0.02 0.50 1.07 0 0.04 0.24 27

29

8-7 1 681 842 0 1140 0.15 0.32 0.79 0 0.09 0.56 0 0.04 0.13 28

2 705 876 0 6750 0.28 0.51 1.94 0.17 0.34 0.75 0 0.10 0.22 28

3 735 (892) 0.09 7200 0.78 1.83 3.04 0.59 1.58 2.90 0 0.87 1.10 28

4 776 (991) 1.71 7200 1.21 2.77 4.09 0.31 1.05 2.11 0 0.66 1.97 28

Average 0.45 5572 0.60 1.36 2.46 0.27 0.76 1.58 0 0.42 0.85 28

8-8 1 623 853 0 1690 0.61 0.94 1.88 0 0.35 0.96 0 0.22 0.51 34

2 673 (871) 0.67 7200 0.74 1.07 1.75 0.18 0.49 1.43 0 0.37 1.26 34

3 710 (931) 0.42 7200 1.02 2.72 4.32 0.64 2.19 3.28 0 0.74 1.48 34

4 742 (1174) 0 7200 1.31 3.65 4.88 0.72 1.81 3.47 0.14 1.15 2.51 34

Average 0.27 5822 0.92 2.09 3.21 0.38 1.21 2.28 0.03 0.62 1.44 34

5-10 1 444 770 0 5 0 1.48 2.98 0 0 0 0 0 0 14

2 444 857 0 17 0 0.80 3.50 0 0.14 1.40 0 0.14 1.4 14

3 451 937 0 24 0 0. 32 0.90 0 0 0 0 0 0 14

4 461 979 0 27 0 0.83 1.83 0 0.10 0.30 0 0 0 14

Average 0 18.3 0 0.86 2.30 0 0.06 0.43 0 0.04 0.35 14

10-5 1 712 760 0 632 0.92 1.46 1.97 0 0.75 1.44 0 0.22 1.31 25

2 735 787 0 2431 1.90 2.90 5.97 1.90 2.54 3.17 1.39 2.19 3.17 25

3 791 (908) 0.33 7200 0.66 1.53 2.20 0.66 1.28 1.87 0 1.14 1.87 25

4 1228 (1385) 0.43 7200 1.73 2.88 5.55 1.08 2.99 4.11 0 1.79 3.53 25

Average 0.19 4366 1.30 2.19 3.92 0.91 1.89 2.65 0.34 1.34 2.47 25

10-10 1 718 915 0 733 1.31 4.19 6.88 0.43 1.74 3.06 0 1.63 2.95 75

2 718 (977) 0.92 7200 0.71 4.68 9.00 3.07 3.99 4.50 0 2.38 3.78 75

3 718 (1086) 0 7200 1.65 4.37 9.20 1.47 3.31 5.70 0.55 1.66 4.78 75

4 1104 (1684) 9.67 7200 0.83 3.04 5.10 0.77 1.76 3.08 0 1.92 4.69 75

Average 2.65 5583 1.13 4.07 7.55 1.43 2.70 4.09 0.13 1.90 4.05 75

10-15 1 701 943 0 1380 0.88 3.61 8.14 0.88 2.78 4.41 0.03 0.72 1.04 95

2 721 966 0 6430 1.63 3.93 7.84 1.10 2.52 5.21 0 0 0 95

3 773 (1044) 0 7200 1.19 3.49 7.29 0.75 2.53 6.01 0.12 0.23 0.71 95

4 843 (1177) 0.29 7200 2.01 2.91 8.71 0.89 2.09 5.42 0 0.26 1.01 95

Average 0.07 5552 1.43 3.48 7.99 0.90 2.48 5.26 0.04 0.30 0.69 95

10-20 1 657 787 0 5100 0.81 1.17 3.02 0.63 0.98 2.43 0 0.17 0.66 130

2 677 (831) 1.11 7200 1.72 2.43 5.65 0.21 1.32 3.11 0 0.45 1.02 130

3 698 (885) 0.14 7200 1.55 3.23 5.51 0.33 1.78 3.03 0 1.21 2.56 130

4 762 (992) 0.68 7200 2.43 4.76 7.11 1.15 1.76 2.76 0 1.12 1.85 130

Average 0.48 6675 1.63 2.90 5.32 0.58 1.46 2.83 0 0.74 1.52 130

10-25 1 923 (1325) 0.51 7200 3.19 5.76 11.23 1.63 4.02 6.82 0 0.37 0.69 150

2 944 (1405) 0 7200 5.01 7.99 11.61 2.22 4.25 7.10 0.31 0.93 1.82 150

3 1094 (1532) 0 7200 4.31 8.06 12.12 3.21 5.32 8.11 0.11 0.41 0.77 150

4 1105 (1672) 6.23 7200 4.22 7.65 12.08 2.87 5.12 6.54 0 0.43 1.12 150

Average 1.68 7200 4.18 7.36 11.76 2.48 4.68 7.14 0.10 0.53 1.10 150

15-10 1 724 (971) 0.22 7200 3.32 6.34 13.21 3.11 6.21 11.17 0 0.19 0.59 140

2 772 (1003) 0.17 7200 4.25 7.37 14.02 3.88 6.10 12.22 0 0.26 0.61 140

3 839 (1127) 0 7200 3.74 7.23 13.70 2.42 5.98 10.43 0.09 0.28 0.76 140

4 936 (1443) 1.23 7200 4.18 7.65 13.76 2.55 6.09 10.54 0 0.64 1.26 140

Average 0.40 7200 3.87 7.15 13.68 2.99 6.09 11.09 0.02 0.34 0.80 140

30

15-15 1 678 (945) 0.05 7200 5.02 8.33 15.41 2.71 5.66 9.15 0 0.29 0.68 170

2 692 (971) 0 7200 3.09 7.14 13.01 1.97 2.71 6.04 0.17 0.17 0.17 170

3 712 (1063) 3.07 7200 2.27 6.03 10.33 2.27 3.11 6.42 0 0.61 1.21 170

4 749 (1231) 2.98 7200 3.91 5.56 9.01 1.96 3.42 7.77 0 0.47 1.16 170

Average 1.52 7200 3.57 6.76 11.94 2.23 3.72 7.34 0.04 0.38 0.80 170

15-20 1 557 (783) 0.16 7200 4.98 7.36 12.77 4.21 6.87 12.21 0 0.17 0.38 210

2 557 (811) 0 7200 8.41 12.23 18.43 5.21 9.55 13.00 0.19 1.54 4.03 210

3 586 (934) 0 7200 7.47 9.11 15.32 5.63 8.25 11.98 0.15 2.42 6.89 210

4 654 (1230) 0 7200 8.04 10.12 17.51 3.31 4.67 7.01 1.00 1.26 2.57 210

Average 0.04 7200 7.22 9.70 16.00 4.59 7.33 11.05 0.33 1.35 3.47 210

15-25 1 781 (910) 0 7200 6.90 8.53 13.61 2.34 4.71 6.43 0 0 0 230

2 802 (967) 0.27 7200 10.01 13.32 18.24 3.01 5.67 8.91 0 1.54 2.97 230

3 834 (1031) 3.11 7200 10.34 14.02 19.77 6.11 8.39 13.44 0 3.88 7.10 230

4 966 (1362) 0 7200 9.71 12.47 17.21 3.16 6.24 12.00 0.08 4.02 6.37 230

Average 0.84 7200 9.24 12.10 17.21 3.65 6.25 10.19 0.02 2.36 4.11 230

20-15 1 430 (643) 0 7200 13.53 18.36 22.86 7.93 9.95 12.90 2.48 5.41 7.15 300

(whizzki

ds97)

2 430 (734) 9.53 7200 10.89 15.81 22.47 5.31 7.42 9.40 0 3.63 7.90 300

3 810 (1483) 0 7200 11.19 15.75 21.24 7.34 11.92 14.96 0.74 3.05 6.54 300

4 1138 (1806) 10.24 7200 13.17 16.88 19.98 1.49 5.16 8.30 0 4.54 7.86 300

Average 4.94 7200 12.19 16.70 21.63 5.51 8.61 11.39 0.80 4.16 7.36 300

20-20 1 983 (1529) 0 7200 8.92 10.44 15.32 4.52 6.41 10.21 1.03 2.41 4.52 350

2 1095 (1633) 7.41 7200 10.27 16.55 20.13 6.21 8.59 13.08 0 2.37 5.89 350

3 1095 (1782) 5.78 7200 11.24 14.32 19.72 3.19 5.93 10.11 0 2.09 4.98 350

4 1176 (1936) 0 7200 13.67 18.83 24.99 9.77 11.49 16.92 0.69 3.35 5.02 350

Average 3.30 7200 10.87 15.03 20.04 5.92 8.10 12.58 0.43 2.25 5.10 350

20-25 1 981 (1542) 0 7200 11.54 18.76 29.32 7.90 9.68 13.29 0.05 2.47 4.86 410

2 1034 (1579) 5.72 7200 12.34 16.32 25.95 6.31 7.81 9.12 0 4.65 8.34 410

3 1089 (1839) 3.63 7200 9.43 12.65 18.71 2.98 4.53 6.85 0 3.41 5.78 410

4 1188 (2046) 6.75 7200 14.24 17.61 22.53 3.91 5.62 8.88 0 4.44 8.67 410

Average 4.02 7200 11.89 16.33 24.13 5.27 6.91 9.53 0.01 3.74 6.91 410

25-20 1 1066 (1632) 1.04 7200 8.72 12.77 18.39 3.88 6.17 9.17 0 1.12 3.58 480

2 1128 (1765) 7.78 7200 10.12 15.20 19.44 4.71 6.56 8.77 0 2.97 7.63 480

3 1201 (2561) 11.65 7200 7.98 14.53 24.76 3.14 5.10 6.84 0 2.68 5.76 480

4 1310 (2947) 9.11 7200 7.41 17.25 30.33 2.16 4.71 6.91 0 3.35 6.64 480

Average 7.39 7200 8.56 14.94 23.23 3.47 5.63 7.92 0 2.53 5.90 480

25-25 1 1141 (1831) 10.54 7200 11.93 17.35 28.74 7.27 11.88 18.56 0 1.84 3.02 550

2 1207 (2074) 12.11 7200 14.21 19.38 26.69 6.59 14.12 19.83 0 6.15 13.2 550

3 1348 (2349) 19.78 7200 17.55 20.41 31.65 3.32 5.14 9.81 0 2.64 8.78 550

4 1754 (2793) 18.99 7200 16.62 21.59 27.70 2.01 3.63 4.59 0 1.86 3.96 550

Average 15.35 7200 15.10 19.68 28.69 4.80 8.69 13.20 0 3.12 7.24 550

Overall average 1.75 4579 3.78 5.94 9.40 1.82 3.18 5.11 0.09 1.07 2.25 141

All CPU times are in seconds.

31

Figure 1 Overlapping of transportation and setup when (a) setup determines the starting time and

(b) transportation determines the starting time

Figure 2 No overlapping of transportation and setup

Figure 3 An example of chromosome representation

Figure 4 Active schedule generated for the numerical example

Figure 5 Chromosome shown in Fig. 3 after applying the ASGA

 

setup Jobj transportation

Jobj' setup Jobj

Machinei'

(a)

Machinei

setup Jobj transportation

Jobj' setup Jobj

Machinei

Machinei'

(b)

 

setup Jobj transportation

Jobj' setup Jobj

Machinei'

Machinei

 

(3,2) (1,2) (1,3) (2,3) (3,3) 1 2 1 1 2 3 3 2 3

Groupofjob3Group1ofjob2

Part2Part1

 

(3,2) (3,3) (3,1)

(2,3) (2,1) (2,2)

(1,1) (1,2) (1,3)

Operations Setuptimes Transportationtimes Releasetimes

Machine2

Machine1

Machine3

262421

 

(3,2) (1,2) (2,3) (3,3) (1,3) 2 3 1 3 2 1 1 3 2

Groupofjob3Group1ofjob2

Part2Part1