- Email: [email protected]
Fuzzy job shop scheduling
international journal of
production economics
ELSEVIER
Int. J. Production
Economics
44 (1996) 45-51
Fuzzy job shop scheduling* Mitsuru
Kurodaa**,
Zeng Wangb
aDepartment ofIndustrial and Systems Engineering, Aoyama Gakuin University, Tokyo, Japan bShibuya Optical Co.. Ltd., Tokyo, Japan
Abstract
This paper discusses job shop scheduling from the viewpoint of dealing with fuzziness inherent in the problem. Some static problems with fuzzy information regarding due dates and/or operation times are solved using a branch-and-bound algorithm and the meaning of solving such scheduling problems is described. The method for handling problems with fuzzy information, i.e., fuzzy job shop simulation is also discussed, with the intention of applying theory to real-life job shop scheduling. Keywords: Job shop scheduling; Simulation
Fuzzy
information;
Static problem;
1. Introduction
problem;
handle the job shop fuzzy information.
The purpose of the present paper is to introduce the vagueness (or fuzziness) of scheduling information into the classical job shop scheduling problem and discuss the meaning of solving the fuzzy job shop scheduling problem. In a real-life situation, it can often be observed that some kinds of information such as due dates, processing times, technological constrains and so forth are not necessarily deterministic (or crisp) and the human schedulers make the schedules considering some vague (or fuzzy) scheduling information. However, most models of job shop scheduling have dealt with only crisp information [l-3]. In this paper Zadeh’s fuzzy theory [4,5] is applied to ” Original version presented at the 1 lth International Conference on Production Research, Hefei, China, August 18-23, 1991. * Corresponding author. 0925-5273/96/%15.00 Copyright SSDI 0925-5273(95)00091-7
Dynamic
cl
Branch-and-bound
scheduling
problems
dynamic the fuzzy
method;
using
2. Fuzzy model The job shop scheduling problems are classified into static types and dynamic types. Static problems are such types in which the information about jobs to be scheduled are previously known. On the other hand, dynamic problems are such types in which jobs arrive at randomly over a scheduling period and the scheduler has no information on the jobs prior to scheduling them. For most of this paper, static problems are handled as they are suitable for theoretical analysis. In the final part of this paper, the dynamic problems will be discussed based on the results of analysis on the fuzzy static problems. The elements of a job shop scheduling problem are a set of machines and a collection of jobs to be
1996 Elsevier Science B.V. All rights reserved
M. Kuroda, Z. WanglInt.
46
J. Production
Economics
44 (1996) 45-51
scheduled. Each job is composed of several operations which are processed using different machines in a certain order. In most cases, schedules are made so as to optimize or suboptimize the criteria of the performance with respect to the due dates of the jobs. We denote by di the due date of the ith job and by Ci the completion time of the ith job. Then, we can define two criteria, i.e. the lateness Li and tardiness Ti of the ith job: Li = Ci - di
(1)
0
7
di
mi
Completion time
and
Fig. 1. Membership
ri = maX{Li,O}.
function
of due date.
(2)
Here, we introduce the fuzziness on due dates and processing times which are the amounts of time required to carry out operations. Generally, the firmness of a due date greatly depends on the customer placing the order or the character of the product to be produced and it varies from order to order. Then, we represent due date by a membership function which is as follows: 1 pi(Ci) =
Ci
when
’ - mi - di when di
I 0
when
(3)
Fig. 2. Possibility
i
TV
-
t~j
when (t~j,
to,
TV)
=
mij
tP
ij
mi
t - t~j
=
t
Processing time
where mi is the margin of tardiness for the ith job. That is, we use the membership function corresponding to the tardiness for the ease of its treatment. Fig. 1 is the graphic representation of the membership function. Processing times are generally estimated from standard time data. However, in a real-life situation, it is sometimes difficult to estimate the fixed time for lacking the enough experience regarding an operation. Then, we represent the processing time by the following possibility distribution:
7tij(t)
t'ij
t~j~t
(4)
’ t
-
t;
lt:j
I
when
-
tt
tzdt
distribution
of processing
time.
where tyj, tt and tc stands for the optimistic time, the most likely time and the pessimistic time for carrying out the jth operation of job i, respectively. Fig. 2 is a graphic representation of the possibility distribution. Hereafter, we handle three fuzzy models: (a) due dates are presented by membership functions, (b) processing times are presented by possibility distributions and (c) both due dates and processing times are presented by membership functions and possibility distributions, respectively. For convenience, we call the first “F-C model,” the second “C-F model” and the third “F-F model,” where capital “F” stands for “fuzzy” and capital “c” stands for “crisp.” The “C-C model,” i.e., the classical model, will be used for the comparison with these models later.
47
M. Kuroda, Z. Wang/M. J. Production Economics 44 (1996) 45-51
where Si is the definite
3. Solution approach In this section, we discuss the solution approach of the fuzzy job shop scheduling problems mentioned above. We use a branch-and-bound method based on the Brook and White algorithm [6] to obtain the optimal solutions for the given performance criteria.
3. I. The criteria of performance The performance criteria of jobs are defined cording to the characters of each model.
ac-
Si =
Vfmc = max {Ci - di, 0} .
(5)
as follows:
$(t)dt
(8)
s 0 2, and 7$(t) = (t?, ty, tf)
(9)
is the possibility distribution of the completion time (see Fig. 4). The distribution is defined by three points: the optimistic time t:, the most likely time ty and the pessimistic time tP like the possibility distribution of the processing times. (4) F-F
(1) C-C model
integral
model
VFmF = [pi(t)Xf(t)dt/Si,
(10)
0 f,
(2) F-C model
I/FmC = /ti(Ci)
(6)
)
where pi(Ci) is the grade of the membership function when the fuzzy variable is Ci (see Fig. 3).
3.2. The sum of processing possibility distribution
(3) C-F model t” max(t - di,0)7lf(t)dt/Si
I’CmF =
where pi(t) is the membership function, n;(t) possibility distribution and Si the definite integral mentioned above. Fig. 5 shows that the values pi(t) and r:(t) are determined for any t.
(7)
s
di
Ci Completion
Fig. 3. Performance
criteria
>
mi
model.
by
To obtain the completion time or its lower bound, we need the sum of processing times. In the
toi
t
tm.L Completion
time for F-C
times represented
Fig. 4. Performance
criteria
tPi
time for C-F
model.
*
48
M. Kuroda, Z. Wang/M.
Completion Fig. 5. Performance
criteria
J. Production Economics
44 (1996) 45-51
time for F-F
model
52
J3
case of the C-F model or F-F formula is applicable.
model, the following
71*(t)+ 7cg(f)= (ti + t;, tg + t;;: tS;+ tb)
(11)
where A and B represent two different operations. They are not necessarily the operations for the same jobs or the operations carried out using he same machine. When it is required to obtain the sum of processing times for more than two operations, formula (11) is used repeatedly.
Fig. 6. Critical path for job 1 as an example; partial resolution of conflicts and (b) the case olution of conflicts. The critical path shown consists of a series of processing times. Mk in that the operation is carried out at machine k
eration
= 3.3. The possibility
distribution
of completion
times
When the processing times are deterministic, the completion time or its lower bound is obtained summing up the processing times on a critical path for the ith job. Fig. 6(a) shows the lower bound of completion time for job 1 in a small job shop scheduling problem and Fig. 6(b) shows the completion time for it. However, in the case that processing times are represented by the possibility distributions, the effect of other operations which are not on the critical path should be given consideration, where it is defined as an operation path determining the lower bound of completion time or the exact completion time of a certain job. Now we denote by P, the set of operations influencing the starting time of operation X on any stage of a critical path. The possibility distribution of the starting time of op-
X is obtained
max
( jEPx
by the following
ty , max tj”, max tP jsPx jsPx 1
(a) the case of of complete resby a bold line a bar represents in the job shop.
formula:
(12)
where ty, tj” and tP are the optimistic time, the most likely time and the pessimistic time for the completion of operation j belonging to set P,. Fig. 7 illustrates the meaning of formula (12) using the example in the case of obtaining the starting time of the third operation of job 2 which is the third stage of the critical path shown by Fig. 6(b). This formula guarantees a monotonous increase in the parameters of the possibility distribution in the conflict resolution process, which is indispensable for applying a branch-and-bound algorithm to the problem.
3.4. Penalty function For a more exact calculation, we replace membership function pi by the following one:
M. Kuroda, 2. Wang/Int. J. Production
Economics
44 (1996) 45-51
49
/ 1.0 0 z & l- /.f *I (Ci)
0
Ci
di Completion
Fig. 8. Performance
Fig. 7. Illustration of the meaning 0 (rz,tz”,t!) = (maxj.P~tt9,max,,p,tj",maxj,p,t,P), {Y,.q.
(1
when
of formula
(12); x”(X) = where P, =
(13)
I
when
di < Ci.
AS pr(Ci) will be negative when Ci > mi, we also replace criteria (6) and (10) by the following ones: I/FmC = 1 - /J*‘(Ci)
(14)
and tp I’FmF =
s f,0
{ 1 - p:(t)} n;(t) dt/S,.
(15)
Eq. (14) represents the grade of the penalty caused by the tardiness as shown in Fig. 8 and these criteria need to be minimized.
4. Computational
experience
We carried out the numerical experiments on small job shop scheduling problems (N = 8, M = 4;
>
time for F-C
model.
N is the number of jobs and M is the number of machines) to examine the effect of handling fuzzy information. Eight sets of data were prepared for the model group which consists of C-C, F-C, C-F and F-F model. Using the branch-and-bound method, we obtained computational results which, respectively, minimize one of the four objective functions: max. I I/c-‘, maxi I/F-‘, maxi VC-F and I ITlaXi
Ci
criterion
mi
I/,FmF.
Now we assume that the F-F model represents the real-life scheduling problem much most precisely. In other words, the model ignores the uncertainty inherent in the problem least of all. Based on this assumption, we applied the performance criterion of F-F model to the computational results of other models and reevaluated them (Table 1). As a consequence, F-F model always gives the best solution for all data set as the table shows. It suggests how the schedules made by traditional models are distinct from what human schedulers will want to obtain as a result of ignoring the uncertainty inherent in the job shop. Another important knowledge obtained from the computational results is the sensitivity of fuzzy information to the performance of produced schedules. In this case, we can see that remarkable differences on the results are not found between C-C model and C-F model, or F-C model and F-F model. This means that fuzzy information on processing times are not so sensitive as the one on due dates. Therefore. the job shop scheduling approach presented in this paper will provide a means for foreseeing whether or not a kind of
50 Table 1 Computational information
M. Kuroda, Z. WanglInt.
results
of job scheduling
problems
J. Production
with fuzzy
Economics
44 (1996) 45-51
where Si is the integral (17) and ni is the number remaining operations for job i:
of
22
Data
Model
Si
C-C”
F-Cb
C-F’
F--F”
1.42
0.55 0.92 0.87 1.45 0.51 0.25 0.62 0.71
1.50 3.60 5.80 3.90 1.14 0.18 1.50 0.64
0.53 0.92 0.87 0.68 0.33 0.18 0.54 0.64
3.72 5.80 1.45 1.72 0.25 2.00 0.71
=
s
{pi(t) - n*(t)} dt)
(17)
21
where n*(t) is the possibility distribution of the completion time defined for the remaining operations of the ith job at present time t*. Now, we can denote the set of the remaining operations by Ri, then xi(t) is represented as follows: 7$(t) =
“C-C: model with crisp information on both due dates and processing times. b F-C: model with fuzzy information on due dates and crisp one on processing times. ‘C-F: model with crisp information on due dates and fuzzy information on processing times. dF-F: model with fuzzy information on both due dates and processing times.
c
1 t; + t*, jcR,
1
I
tFj + t* >
jpR,
when
t d 1 tt + t* jpR,
when
2
tt
+
t*
<
[O
fuzzy information is sensitive to any criterion of schedule under the given conditions of the problem.
The analysis mentioned above will be applied to the dynamic problem. Here we briefly discuss the method of handling the fuzzy dynamic job shop scheduling. The job shop simulation has been considered to be the only practical method for solving the dynamic scheduling problem and used widely to make schedules for real-life job shops. The key idea of job shop simulation is to sequence the jobs constructing a queue at each machine by using some priority rules. The fuzzy information treated above can be considered when we obtain the priority level for each job. For example, the priority level can be obtained by executing only a small amount of computation using the following rule: Si 2 0,
when
Si
C
C
tfj
(16)
+
t*
tFj+t*
jsRi
The range of integral for Eq. (17) is obtained the following formula:
5. Remarks on fuzzy simulation
when
when
t Q
jsRi
jsR,
tr = min
0
C t~+t*,di jsR,
‘J tmij +t* , EKI
(19)
di
Completion Fig. 9. Meaning
from
time of priority
rule.
M. Kuroda, Z. WangJlnt. J. Production
Economics
Caluculation prority
44 (1996) 45-51
of
level
Fuzzy priority level
Random numbers
Fig. 10. Data and information
and t2 = max
1 jeR,
tf)+ t*,Wli
.
(20)
>
The priority level shows a kind of slack considering the fuzziness of the due data and processing times. Therefore, the highest priority will be given to the job with the smallest Pi. Fig. 9 shows the meaning of the priority rule stated as the example. In the simulation run, the processing times are determined successively by random sampling from the possibility distribution of processing time. Thus, the accuracy on the completion time of each job becomes higher as the processing of the job proceeds. Fig. 10 shows the concept of fuzzy job shop simulation. According to the simulation experiments which we performed on larger job shop models, extremely similar results to the analytical ones mentioned above could be obtained.
flow of fuzzy job shop simulation.
(2) The sensitivity of fuzzy information to the performance of produced schedulers differs depending on the kind of information. The presented methods will show us which fuzzy information should be considered and which one can be neglected under the given conditions. The discussions on the usefulness of fuzzy job shop scheduling approach through empirical research seem to be needed for wider acceptance.
Acknowledgements The authors are grateful to Mr. H. Kurozu of Aoyama Gakuin University (at present Matsushita Electric Industrial Co., Ltd.) for coding the programs and other contribution to this research.
References 6. Conclusions The present paper shows the potential value of applying fuzzy theory to job shop scheduling through the use of analytical and experimental method for treating fuzzy information inherent in the problem. The primary results of this research are summarized as follows: (1) The schedules made considering the fuzzy information are distinct from ones made neglecting them. The former seems to be closer to the schedules which human schedulers want to obtain.
R.W., Maxwell W.L. and Miller L.W., 1976. Cl1 Conway Theory of Scheduling. Addison-Wesley Reading, MA. 121 Baker, K.R., 1974. Introduction to Sequencing and Scheduling. Wiley, New York. c31 French, S., 1982. Sequencing and Scheduling: An Introduction to the Mathematics of the Job-Shop. Ellis Horwood, Chichester, UK. M Zadeh, L.A., 1965. Fuzzy Sets, Inform. Control, 8: 338-353. c51 Zadeh, L.A., 1978. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, l(1): 3-28. for C61 Brooks, G.H. and White, C.R., 1965. An algorithm finding optimal or near optimal solution to the production scheduling problem. J. Ind. Eng., 16(l): 34440.
production economics
ELSEVIER
Int. J. Production
Economics
44 (1996) 45-51
Fuzzy job shop scheduling* Mitsuru
Kurodaa**,
Zeng Wangb
aDepartment ofIndustrial and Systems Engineering, Aoyama Gakuin University, Tokyo, Japan bShibuya Optical Co.. Ltd., Tokyo, Japan
Abstract
This paper discusses job shop scheduling from the viewpoint of dealing with fuzziness inherent in the problem. Some static problems with fuzzy information regarding due dates and/or operation times are solved using a branch-and-bound algorithm and the meaning of solving such scheduling problems is described. The method for handling problems with fuzzy information, i.e., fuzzy job shop simulation is also discussed, with the intention of applying theory to real-life job shop scheduling. Keywords: Job shop scheduling; Simulation
Fuzzy
information;
Static problem;
1. Introduction
problem;
handle the job shop fuzzy information.
The purpose of the present paper is to introduce the vagueness (or fuzziness) of scheduling information into the classical job shop scheduling problem and discuss the meaning of solving the fuzzy job shop scheduling problem. In a real-life situation, it can often be observed that some kinds of information such as due dates, processing times, technological constrains and so forth are not necessarily deterministic (or crisp) and the human schedulers make the schedules considering some vague (or fuzzy) scheduling information. However, most models of job shop scheduling have dealt with only crisp information [l-3]. In this paper Zadeh’s fuzzy theory [4,5] is applied to ” Original version presented at the 1 lth International Conference on Production Research, Hefei, China, August 18-23, 1991. * Corresponding author. 0925-5273/96/%15.00 Copyright SSDI 0925-5273(95)00091-7
Dynamic
cl
Branch-and-bound
scheduling
problems
dynamic the fuzzy
method;
using
2. Fuzzy model The job shop scheduling problems are classified into static types and dynamic types. Static problems are such types in which the information about jobs to be scheduled are previously known. On the other hand, dynamic problems are such types in which jobs arrive at randomly over a scheduling period and the scheduler has no information on the jobs prior to scheduling them. For most of this paper, static problems are handled as they are suitable for theoretical analysis. In the final part of this paper, the dynamic problems will be discussed based on the results of analysis on the fuzzy static problems. The elements of a job shop scheduling problem are a set of machines and a collection of jobs to be
1996 Elsevier Science B.V. All rights reserved
M. Kuroda, Z. WanglInt.
46
J. Production
Economics
44 (1996) 45-51
scheduled. Each job is composed of several operations which are processed using different machines in a certain order. In most cases, schedules are made so as to optimize or suboptimize the criteria of the performance with respect to the due dates of the jobs. We denote by di the due date of the ith job and by Ci the completion time of the ith job. Then, we can define two criteria, i.e. the lateness Li and tardiness Ti of the ith job: Li = Ci - di
(1)
0
7
di
mi
Completion time
and
Fig. 1. Membership
ri = maX{Li,O}.
function
of due date.
(2)
Here, we introduce the fuzziness on due dates and processing times which are the amounts of time required to carry out operations. Generally, the firmness of a due date greatly depends on the customer placing the order or the character of the product to be produced and it varies from order to order. Then, we represent due date by a membership function which is as follows: 1 pi(Ci) =
Ci
when
’ - mi - di when di
I 0
when
(3)
Fig. 2. Possibility
i
TV
-
t~j
when (t~j,
to,
TV)
=
mij
tP
ij
mi
t - t~j
=
t
Processing time
where mi is the margin of tardiness for the ith job. That is, we use the membership function corresponding to the tardiness for the ease of its treatment. Fig. 1 is the graphic representation of the membership function. Processing times are generally estimated from standard time data. However, in a real-life situation, it is sometimes difficult to estimate the fixed time for lacking the enough experience regarding an operation. Then, we represent the processing time by the following possibility distribution:
7tij(t)
t'ij
t~j~t
(4)
’ t
-
t;
lt:j
I
when
-
tt
tzdt
distribution
of processing
time.
where tyj, tt and tc stands for the optimistic time, the most likely time and the pessimistic time for carrying out the jth operation of job i, respectively. Fig. 2 is a graphic representation of the possibility distribution. Hereafter, we handle three fuzzy models: (a) due dates are presented by membership functions, (b) processing times are presented by possibility distributions and (c) both due dates and processing times are presented by membership functions and possibility distributions, respectively. For convenience, we call the first “F-C model,” the second “C-F model” and the third “F-F model,” where capital “F” stands for “fuzzy” and capital “c” stands for “crisp.” The “C-C model,” i.e., the classical model, will be used for the comparison with these models later.
47
M. Kuroda, Z. Wang/M. J. Production Economics 44 (1996) 45-51
where Si is the definite
3. Solution approach In this section, we discuss the solution approach of the fuzzy job shop scheduling problems mentioned above. We use a branch-and-bound method based on the Brook and White algorithm [6] to obtain the optimal solutions for the given performance criteria.
3. I. The criteria of performance The performance criteria of jobs are defined cording to the characters of each model.
ac-
Si =
Vfmc = max {Ci - di, 0} .
(5)
as follows:
$(t)dt
(8)
s 0 2, and 7$(t) = (t?, ty, tf)
(9)
is the possibility distribution of the completion time (see Fig. 4). The distribution is defined by three points: the optimistic time t:, the most likely time ty and the pessimistic time tP like the possibility distribution of the processing times. (4) F-F
(1) C-C model
integral
model
VFmF = [pi(t)Xf(t)dt/Si,
(10)
0 f,
(2) F-C model
I/FmC = /ti(Ci)
(6)
)
where pi(Ci) is the grade of the membership function when the fuzzy variable is Ci (see Fig. 3).
3.2. The sum of processing possibility distribution
(3) C-F model t” max(t - di,0)7lf(t)dt/Si
I’CmF =
where pi(t) is the membership function, n;(t) possibility distribution and Si the definite integral mentioned above. Fig. 5 shows that the values pi(t) and r:(t) are determined for any t.
(7)
s
di
Ci Completion
Fig. 3. Performance
criteria
>
mi
model.
by
To obtain the completion time or its lower bound, we need the sum of processing times. In the
toi
t
tm.L Completion
time for F-C
times represented
Fig. 4. Performance
criteria
tPi
time for C-F
model.
*
48
M. Kuroda, Z. Wang/M.
Completion Fig. 5. Performance
criteria
J. Production Economics
44 (1996) 45-51
time for F-F
model
52
J3
case of the C-F model or F-F formula is applicable.
model, the following
71*(t)+ 7cg(f)= (ti + t;, tg + t;;: tS;+ tb)
(11)
where A and B represent two different operations. They are not necessarily the operations for the same jobs or the operations carried out using he same machine. When it is required to obtain the sum of processing times for more than two operations, formula (11) is used repeatedly.
Fig. 6. Critical path for job 1 as an example; partial resolution of conflicts and (b) the case olution of conflicts. The critical path shown consists of a series of processing times. Mk in that the operation is carried out at machine k
eration
= 3.3. The possibility
distribution
of completion
times
When the processing times are deterministic, the completion time or its lower bound is obtained summing up the processing times on a critical path for the ith job. Fig. 6(a) shows the lower bound of completion time for job 1 in a small job shop scheduling problem and Fig. 6(b) shows the completion time for it. However, in the case that processing times are represented by the possibility distributions, the effect of other operations which are not on the critical path should be given consideration, where it is defined as an operation path determining the lower bound of completion time or the exact completion time of a certain job. Now we denote by P, the set of operations influencing the starting time of operation X on any stage of a critical path. The possibility distribution of the starting time of op-
X is obtained
max
( jEPx
by the following
ty , max tj”, max tP jsPx jsPx 1
(a) the case of of complete resby a bold line a bar represents in the job shop.
formula:
(12)
where ty, tj” and tP are the optimistic time, the most likely time and the pessimistic time for the completion of operation j belonging to set P,. Fig. 7 illustrates the meaning of formula (12) using the example in the case of obtaining the starting time of the third operation of job 2 which is the third stage of the critical path shown by Fig. 6(b). This formula guarantees a monotonous increase in the parameters of the possibility distribution in the conflict resolution process, which is indispensable for applying a branch-and-bound algorithm to the problem.
3.4. Penalty function For a more exact calculation, we replace membership function pi by the following one:
M. Kuroda, 2. Wang/Int. J. Production
Economics
44 (1996) 45-51
49
/ 1.0 0 z & l- /.f *I (Ci)
0
Ci
di Completion
Fig. 8. Performance
Fig. 7. Illustration of the meaning 0 (rz,tz”,t!) = (maxj.P~tt9,max,,p,tj",maxj,p,t,P), {Y,.q.
(1
when
of formula
(12); x”(X) = where P, =
(13)
I
when
di < Ci.
AS pr(Ci) will be negative when Ci > mi, we also replace criteria (6) and (10) by the following ones: I/FmC = 1 - /J*‘(Ci)
(14)
and tp I’FmF =
s f,0
{ 1 - p:(t)} n;(t) dt/S,.
(15)
Eq. (14) represents the grade of the penalty caused by the tardiness as shown in Fig. 8 and these criteria need to be minimized.
4. Computational
experience
We carried out the numerical experiments on small job shop scheduling problems (N = 8, M = 4;
>
time for F-C
model.
N is the number of jobs and M is the number of machines) to examine the effect of handling fuzzy information. Eight sets of data were prepared for the model group which consists of C-C, F-C, C-F and F-F model. Using the branch-and-bound method, we obtained computational results which, respectively, minimize one of the four objective functions: max. I I/c-‘, maxi I/F-‘, maxi VC-F and I ITlaXi
Ci
criterion
mi
I/,FmF.
Now we assume that the F-F model represents the real-life scheduling problem much most precisely. In other words, the model ignores the uncertainty inherent in the problem least of all. Based on this assumption, we applied the performance criterion of F-F model to the computational results of other models and reevaluated them (Table 1). As a consequence, F-F model always gives the best solution for all data set as the table shows. It suggests how the schedules made by traditional models are distinct from what human schedulers will want to obtain as a result of ignoring the uncertainty inherent in the job shop. Another important knowledge obtained from the computational results is the sensitivity of fuzzy information to the performance of produced schedules. In this case, we can see that remarkable differences on the results are not found between C-C model and C-F model, or F-C model and F-F model. This means that fuzzy information on processing times are not so sensitive as the one on due dates. Therefore. the job shop scheduling approach presented in this paper will provide a means for foreseeing whether or not a kind of
50 Table 1 Computational information
M. Kuroda, Z. WanglInt.
results
of job scheduling
problems
J. Production
with fuzzy
Economics
44 (1996) 45-51
where Si is the integral (17) and ni is the number remaining operations for job i:
of
22
Data
Model
Si
C-C”
F-Cb
C-F’
F--F”
1.42
0.55 0.92 0.87 1.45 0.51 0.25 0.62 0.71
1.50 3.60 5.80 3.90 1.14 0.18 1.50 0.64
0.53 0.92 0.87 0.68 0.33 0.18 0.54 0.64
3.72 5.80 1.45 1.72 0.25 2.00 0.71
=
s
{pi(t) - n*(t)} dt)
(17)
21
where n*(t) is the possibility distribution of the completion time defined for the remaining operations of the ith job at present time t*. Now, we can denote the set of the remaining operations by Ri, then xi(t) is represented as follows: 7$(t) =
“C-C: model with crisp information on both due dates and processing times. b F-C: model with fuzzy information on due dates and crisp one on processing times. ‘C-F: model with crisp information on due dates and fuzzy information on processing times. dF-F: model with fuzzy information on both due dates and processing times.
c
1 t; + t*, jcR,
1
I
tFj + t* >
jpR,
when
t d 1 tt + t* jpR,
when
2
tt
+
t*
<
[O
fuzzy information is sensitive to any criterion of schedule under the given conditions of the problem.
The analysis mentioned above will be applied to the dynamic problem. Here we briefly discuss the method of handling the fuzzy dynamic job shop scheduling. The job shop simulation has been considered to be the only practical method for solving the dynamic scheduling problem and used widely to make schedules for real-life job shops. The key idea of job shop simulation is to sequence the jobs constructing a queue at each machine by using some priority rules. The fuzzy information treated above can be considered when we obtain the priority level for each job. For example, the priority level can be obtained by executing only a small amount of computation using the following rule: Si 2 0,
when
Si
C
C
tfj
(16)
+
t*
tFj+t*
jsRi
The range of integral for Eq. (17) is obtained the following formula:
5. Remarks on fuzzy simulation
when
when
t Q
jsRi
jsR,
tr = min
0
C t~+t*,di jsR,
‘J tmij +t* , EKI
(19)
di
Completion Fig. 9. Meaning
from
time of priority
rule.
M. Kuroda, Z. WangJlnt. J. Production
Economics
Caluculation prority
44 (1996) 45-51
of
level
Fuzzy priority level
Random numbers
Fig. 10. Data and information
and t2 = max
1 jeR,
tf)+ t*,Wli
.
(20)
>
The priority level shows a kind of slack considering the fuzziness of the due data and processing times. Therefore, the highest priority will be given to the job with the smallest Pi. Fig. 9 shows the meaning of the priority rule stated as the example. In the simulation run, the processing times are determined successively by random sampling from the possibility distribution of processing time. Thus, the accuracy on the completion time of each job becomes higher as the processing of the job proceeds. Fig. 10 shows the concept of fuzzy job shop simulation. According to the simulation experiments which we performed on larger job shop models, extremely similar results to the analytical ones mentioned above could be obtained.
flow of fuzzy job shop simulation.
(2) The sensitivity of fuzzy information to the performance of produced schedulers differs depending on the kind of information. The presented methods will show us which fuzzy information should be considered and which one can be neglected under the given conditions. The discussions on the usefulness of fuzzy job shop scheduling approach through empirical research seem to be needed for wider acceptance.
Acknowledgements The authors are grateful to Mr. H. Kurozu of Aoyama Gakuin University (at present Matsushita Electric Industrial Co., Ltd.) for coding the programs and other contribution to this research.
References 6. Conclusions The present paper shows the potential value of applying fuzzy theory to job shop scheduling through the use of analytical and experimental method for treating fuzzy information inherent in the problem. The primary results of this research are summarized as follows: (1) The schedules made considering the fuzzy information are distinct from ones made neglecting them. The former seems to be closer to the schedules which human schedulers want to obtain.
R.W., Maxwell W.L. and Miller L.W., 1976. Cl1 Conway Theory of Scheduling. Addison-Wesley Reading, MA. 121 Baker, K.R., 1974. Introduction to Sequencing and Scheduling. Wiley, New York. c31 French, S., 1982. Sequencing and Scheduling: An Introduction to the Mathematics of the Job-Shop. Ellis Horwood, Chichester, UK. M Zadeh, L.A., 1965. Fuzzy Sets, Inform. Control, 8: 338-353. c51 Zadeh, L.A., 1978. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, l(1): 3-28. for C61 Brooks, G.H. and White, C.R., 1965. An algorithm finding optimal or near optimal solution to the production scheduling problem. J. Ind. Eng., 16(l): 34440.