a fuzzy-neural approach for supporting three-objective job scheduling in a wafer fabrication factory

ORIGINAL ARTICLE

A fuzzy-neural approach for supporting three-objective jobscheduling in a wafer fabrication factory

Toly Chen • Yu-Cheng Wang

Received: 6 September 2012 / Accepted: 5 July 2013

� Springer-Verlag London 2013

Abstract This study is dedicated to three-objective

scheduling in a wafer fabrication factory, which has rarely

been discussed in the literature but is a very important task.

Optimizing a single objective in a complex production

system like a wafer fabrication factory is already quite

complicated. Optimizing three objectives at the same time

is obviously even more complicated. To this end, this study

presents a fuzzy-neural approach that fuses three existing

rules in a nonlinear way, and which can be tailored, and

even optimized, for a wafer fabrication factory. To assess

the effectiveness of the proposed methodology, production

simulation is also applied in this study. According to the

experimental results, the proposed methodology is better

than some existing approaches in reducing the average

cycle time, the maximum lateness, and cycle time standard

deviation.

Keywords Wafer fabrication � Scheduling � Fuzzy �Neural

1 Introduction

Real-world decision making usually has multiple aspects.

Multi-objective models are therefore considered useful to

conduct a comprehensive consideration. Similarly, multi-

objective models have been widely applied to many deci-

sion-making problems involving multiple aspects, attri-

butes, or decision makers. In a factory, sequencing and

scheduling are obviously the most important decision-

making problems. To this end, a variety of targets have

been considered, such as the maximum completion time,

the mean cycle time, cycle time standard deviation, the

maximum lateness, the mean lateness or tardiness, number

of tardy jobs. Most of them are directly related to the

completion time (the so-called regular measures). Since the

completion time is equal to the release time (a constant)

plus the cycle time, the question of how to reduce the cycle

time is usually the focus of most studies. On-time delivery

is another area of concern; missing a deadline not only

incurs a fine, but also increases the possibility of losing

customers.

Although there has been a consensus for the importance

of multi-objective scheduling, the research in this area is

still inadequate, especially for large complex production

systems. In the literature, most previous studies addressed

multi-objective scheduling problems by combining the

basic scheduling rules. Each of these rules is optimal for a

single target. Loukil et al. [1] mentioned five ways to deal

with multi-objective scheduling:

1. Simultaneous (or Pareto) approach: Their combination

should be formed in such a way that ensures the

performances along different dimensions are Pareto

optimal. For minimization problems with K objective

functions, if the value of objective function fk can only

be decreased by increasing the value of some other

objective function fj, k = j, j, k e {1, …, M}, then all

feasible solutions that fulfill this property are called

Pareto optimal solutions.

2. Utility (or compromise) approach: To simplify the

finding of the best solution, the linear or nonlinear

combination of the objectives can be optimized

instead.

T. Chen (&) � Y.-C. Wang

Department of Industrial Engineering and Systems Management,

Feng Chia University, No. 100, Wenhwa Road, Seatwen,

Taichung City, Taiwan 407, ROC

e-mail: [email protected]

123

Neural Comput & Applic

DOI 10.1007/s00521-013-1460-5

3. Goal programming (or satisfying) approach: Some

objectives are formulated as constraints, for which the

satisfaction levels are defined.

4. Hierarchical approach: Objectives are not optimized at

the same time but sequentially.

5. Interactive approach: As a number of steps are

required, the decision maker expresses his/her prefer-

ences to the solution proposed at each step of the

process. This revolutionary approach will result in the

best possible outcome.

Grimme and Lepping [2] considered bi-objective single-

machine scheduling problems. They mentioned that there are

two ways to solve multi-objective scheduling problems—

problem-specific approaches [3] and black-box optimizers

like randomized search or evolutionary algorithms. van

Wassenhoven and Gelder [4] proposed an efficient algorithm

for the 1||RCj, Lmax problem, where RCj and Lmax mean the

total completion time and maximum lateness, respectively. In

order to optimize the total completion time and makespan on

m identical machines, for example, the Pm||RCj, Cmax prob-

lem, Stein and Wein [5] proposed a general algorithmic

framework. In these methods, some operations such as trun-

cation and composition are applied to merge schedules that are

optimal for different goals (assuming that after the merger, the

new schedule is still satisfactory for achieving the goals).

Cochran et al. [6] proposed a multi-population genetic algo-

rithm to solve two-objective scheduling problems, makespan

and total weighted tardiness, for parallel machines. Similar to

a compromise solution, the multiplication of the relative

measure of each objective was optimized. Loukil et al. [1]

proposed a multi-objective simulated annealing process to

solve multiple-objective scheduling problems for one

machine, parallel machines, and permutation flow shops.

Most cases contained at most two objectives, and the only case

with three objectives was on a single machine.

On the other hand, a detailed review of the application

of evolutionary algorithms in multi-objective scheduling

can be found in Deb [7] and Coello et al. [8]. Most studies

used genetic algorithms (GAs) to combine various good

schedules that could in turn evolve into the optimal

schedule.

There are several reasons that multi-objective schedul-

ing in a wafer fabrication factory is a complicated process:

1. A wafer fabrication factory is a very complex manu-

facturing system featured by changing demand, a

variety of product types and priorities, unbalanced

capacity, job reprogramming for machines, alternative

machines with unequal capacity, sequence-dependent

setup times, and shifting bottlenecks.

2. In such a complex production system, it is even

difficult to find a heuristic optimizing a single objec-

tive [3, 9, 10].

3. A naive aggregation of single-objective heuristic does

not necessarily yield feasible non-dominated solutions

[2].

4. The weighted sum of the objectives often leads to

unsatisfactory results.

A few studies applied the response surface method (RSM)

and the desirability function to handle multiple-factor and

multiple-objective optimization in scheduling [11]. How-

ever, the commonly used second-order multiple-factor

regression may not be accurate enough. The desirability

function is a very subjective approach. Most dispatching

rules are focused on a single performance measure. In fact,

minimizing any performance measure in such a complex job

shop is a strongly NP-hard problem. However, simulta-

neously optimizing multiple performance measures is an

objective still being pursued. Taking into account two

simultaneous performance measures, average cycle time and

cycle time variation, Chen and Wang [3] proposed a bi-cri-

teria nonlinear fluctuation smoothing rule that also has an

adjustable factor (1f-biNFS). To increase the flexibility of

customization, Chen et al. [12] extended the above rules and

proposed the bi-criteria fluctuation smoothing rule with four

adjustable factors (4f-biNFS). However, the adjustment

factors in these rules are static. In other words, they will not

change over time. Chen [9] therefore designed a mechanism

to dynamically adjust the values of the factors in Chen and

Wang’s bi-criteria nonlinear fluctuation smoothing rule

(dynamic 1f-biNFS). However, the adjustment of the factors is

based on a predefined rule. This process is too subjective and,

as a result, does not take into account the status of the wafer

fabrication factory. These rules have, furthermore, not been

optimized, so there is considerable room for improvement.

A fuzzy-neural approach is proposed in this study for

three-objective job scheduling in a wafer fabrication fac-

tory. The unique features of the proposed methodology

include the following:

1. Three performance measures—the average cycle time,

cycle time standard deviation, and the number of tardy

jobs—are optimized at the same time. As far as we

know, the existing dispatching rules in this field were

not designed for this purpose. Most of them considered

at most two objectives or only evaluated the perfor-

mance of the rules along multiple dimensions for

simple systems. They simply do not apply to a

complex manufacturing system.

2. We estimate the remaining cycle time of a job with the

fuzzy c-means and fuzzy back propagation network

(FCM–FBPN) approach [13, 14]. According to Chen

and Wang [10], with a more accurate remaining

cycletime estimation, the scheduling performance of

a fluctuation smoothing rule can be significantly

improved. In addition, Chen and Wang used a fuzzy


123

gradient search algorithm for training the FBPN, which

is time consuming and may not be accurate. In this

study, a novel training method, including the Leven-

berg–Marquardt algorithm and goal programming

models, is proposed. It is suggested that this model is

more efficient than that in Chen and Wang’s study and

can produce more accurate forecasts.

3. The new rule is formed by fusing three traditional

dispatching rules in a nonlinear way. There are two

advantages [9, 10] to this procedure: first, the effects of

the parameters in the rule can be more accurately

balanced, and second, the new rule is more responsive

to the changes in the parameters.

4. The content of the new dispatching rule can be tailored

for a specific wafer fabrication factory with an

adjustable factor.

5. The new dispatching rule can be localized. It can even

be tailored to each machine in the wafer fabrication

factory. In previous literature (e.g., [11]), most

heuristics can only be tailored to two machines in a

wafer fabrication factory—one for bottlenecks and the

other one for non-bottlenecks.

To assess the effectiveness of the proposed methodology,

production simulation is also applied in this study. The rest of

this paper is organized as follows. Sect. 2 is divided into two

parts. In the first part, the FCM–FBPN approach is applied to

estimate the remaining cycle time of a job. Subsequently, the

fuzzy-neural approach for three-objective job scheduling in a

wafer fabrication factory is detailed. To assess the effec-

tiveness of the proposed methodology, a simulated case

study is described in Sect. 3. According to the results of the

analyses, some discussion points are made. Finally, con-

cluding remarks are made in Sect. 4.

2 Methodology

The variables used in the proposed methodology are

defined as follows:

1. Ri: the release time of job i; i = 1 * n.

2. BQi: the total queue length before bottlenecks at Ri.

3. CRij: the critical ratio of job i at step j.

4. CSi: the current stage of job i.

5. CTj: the cycle time of job j.

6. gCTEj : the estimated cycle time of job j.

7. DðlÞi : the delay of the l-th recently completed job at Ri,

l = 1 * 3.

8. DDi: the due date of job i.

9. FQi: the total queue length in the whole factory at Ri.

10. LSi: the size of job i.

11. Qi: the queue length on the processing route of job

i at Ri.

12. RCTEij: the remaining cycle time forecast of job

i from step j.

13. RPTij: the remaining processing time of job i from

step j.

14. SCTij: the step cycle time of job i until step j.

15. SKij: the slack of job i at step j.

16. TSi: the total stage of job i.

17. TPTi: the total processing time of job i.

18. Ui: the average factory utilization before job i is

released. If the utilization of the factory is reported on

a daily basis, then Ui is the utilization of the day

before job i is released.

19. WIPi: the factory work-in-progress (WIP) at Ri.

20. k: mean release rate.

21. xk: inputs to the FBPN, k = 1 * K.

22. ~hl: the output from hidden-layer node l, l = 1 * L.

23. ~wol : the weight of the connection between hidden-

layer node l and the output node.

24. ~whkl: the weight of the connection between input node

k and hidden-layer node l, k = 1 * K; l = 1 * L.

25. ~hhl : the threshold for screening out weak signals by

hidden-layer node l.

26. ~ho: the threshold for screening out weak signals by

the output node.

The process of the proposed methodology is described

below:

Step 1 Classify jobs with FCM. The inputs of this step are

job attributes. To determine the optimal number of job

categories, the S test is applied. The output from this step is

the category of each job.

Step 2 Predict the remaining cycle time of each job using

the FBPN approach. Jobs of different categories will be

sent to different FBPNs. The inputs to FBPN are the

attributes of a job. The output from FBPN is the estimated

remaining cycle time of the job.

Step 3 Form a new rule by fusing three traditional rules—

the earliest due date (EDD), the fluctuation smoothing rule for

mean cycle time (FSMCT), and the fluctuation smoothing rule

for the variation of cycle time (FSVCT) in a nonlinear way.

Step 4 Incorporate the estimated remaining cycle time

into the new rule.

Step 5 Assess the effectiveness of the new dispatching

rule with a simulation study.

2.1 The FCM approach

Some past studies (e.g., [3]) have shown that the accuracy

of the remaining cycle time forecasting can be improved by


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job classification. Soft computing methods (e.g., [15]) have

received much attention in this field.

Jobs are pre-classified into K categories with FCM.

FCM performs classification by minimizing the following

objective function:

MinX

K

k¼1

X

n

i¼1

lmiðkÞe

2iðkÞ ð1Þ

where K is the required number of categories; n is the

number of jobs; liðkÞ represents the membership of job

i belonging to category k; eiðkÞ measures the distance

from job i to the centroid of category k; m [ [1, ?) is a

parameter to increase or decrease the fuzziness. The

procedure of applying FCM to classify jobs is as

follows:

1. Establish an initial classification result.

2. (Iterations) Obtain the centroid of each category as

�xðkÞ ¼ f�xðkÞjg ð2Þ

�xðkÞj ¼X

n

i¼1

lmiðkÞxij

,

X

n

i¼1

lmiðkÞ ð3Þ

liðkÞ ¼ 1=X

K

l¼1

ðeiðkÞ=eiðlÞÞ2=ðm�1Þ ð4Þ

eiðkÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X

all j

ðxij � �xðkÞjÞ2s

ð5Þ

where �xðkÞ is the centroid of category k. lðtÞiðkÞ is the

membership of job i belonging to category k after the

t-th iteration.

3. Re-measure the distance of each job to the centroid of

every category and then recalculate the corresponding

membership.

4. Stop if the following condition is satisfied. Otherwise,

return to step (2):

maxk

maxijlðtÞ

iðkÞ � lðt�1ÞiðkÞ j\d ð6Þ

where d is a real number representing the threshold of

membership convergence.

Finally, the separate distance test (S test) proposed by

Xie and Beni [16] can be applied to determine the optimal

number of categories K:

Min S ð7Þ

subject to

Jm ¼X

K

k¼1

X

n

i¼1

lmiðkÞe

2iðkÞ

ð8Þ

e2min ¼ min

p 6¼q

X

all j

ð�xðpÞj � �xðqÞjÞ2 !

ð9Þ

S ¼ Jm

n� e2min

ð10Þ

K 2 Zþ ð11Þ

The K value minimizing S determines the optimal

number of categories.

2.2 The FBPN approach

The remaining cycle time of a job that is being fabricated in a

wafer fabrication factory is the time still required to complete

the job. If the job is just released into the wafer fabrication

factory, then the remaining cycle time of the job is its cycle

time. In other words, the remaining cycle time is an impor-

tant attribute (or performance measure) for the work-in-

progress (WIP) in the wafer fabrication factory. There are

various ways to predict the remaining cycle time of a job:

1. Predict the remaining cycle time from each step

according to the attributes of the job when it is released.

2. Predict the remaining cycle time from each step

according to the attributes of the job when it is at the

step.

3. Predict the cycle time according to the attributes of the

job when it is released. Then, estimate the remaining

cycle time by subtracting the step cycle time from the

cycle time forecast:

RCTEij ¼ CTEij � SCTij ð12Þ

4. Predict the cycle time according to the attributes of the

job when it is released. Then, estimate the remaining

cycle time by considering the delay in the step cycle

time:

RCTEij ¼ ðCTEij � SCTijÞ �SCTij

SCTEij

ð13Þ

The second method is accurate, but computationally

intensive. In this study, we use the fourth way. For this

reason, we need to predict both the cycle time and the step

cycle time.


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After clustering, a portion of the jobs in each category is

feedback as the ‘‘training examples’’ into the FBPN to

determine the parameter values for the category. The

configuration of the FBPN is established as follows:

1. Inputs: Eight parameters are associated with the i-th

example/job including Ui, Qi, BQi, FQi, WIPi, and DðlÞi

(l = 1 * 3).

2. Single hidden layer.

3. Number of neurons in the hidden layer is the same as

that in the input layer.

4. Output: the estimated (normalized) cycle time (CTEi)

or estimated step cycle time (SCTEij) of the example.

In other words, there will be two groups of FBPNs.

The first group is for estimating the CTEi’s of all the

jobs to be scheduled, while the other group is for

estimating their SCTEij’s.

5. Network learning rule: Delta rule.

6. Transformation function: Sigmoid function,

f ðxÞ ¼ 1=ð1þ e�xÞ ð14Þ

7. Learning rate (g): 0.01–1.0.

8. Batch learning.

The procedure for determining the parameter values is

now described. After pre-classification, a portion of the

adopted examples in each category is fed as ‘‘training

examples’’ into the FBPN to determine the parameter

values for the category. Two phases are involved at the

training stage. At first, in the forward phase, inputs are

multiplied with weights, summated, and transferred to the

hidden layer. Then, activated signals are outputted from the

hidden layer as:

~hl ¼ ðhl1; hl2; hl3Þ ¼1

1þ e�~nhl

¼ 1

1þ e�nhl1

;1

1þ e�nhl2

;1

1þ e�nhl3

� �

; ð15Þ

where

~nhl ¼ ðnh

l1; nhl2; n

hl3Þ ¼ ~Ih

l ð�Þ~hhl

¼ ðIhl1 � hh

l3; Ihl2 � hh

l2; Ihl3 � hh

l1Þ; ð16Þ~Ihl ¼ ðIh

l1; Ihl2; I

hl3Þ ¼

X

all k

~whkl � xk

¼X

allk

minðwhkl1xk;w

hkl3xkÞ;

X

all k

whxk2xk;

X

all k

maxðwhkl1xk;w

hkl3xkÞ

!

;

ð17Þ

(-) and (9) denote fuzzy subtraction and multiplication,

respectively; ~hl’s are also transferred to the output layer

with the same procedure. Finally, the output of the FFNN is

generated as:

~oi ¼ ðoi1; oi2; oi3Þ ¼1

1þ e�~no

¼ 1

1þ e�no1

;1

1þ e�no2

;1

1þ e�no3

� �

; ð18Þ

where

~no ¼ ðno1; n

o2; n

o3Þ ¼ ~Ioð�Þ~ho

¼ ðIo1 � ho

3; Io2 � ho

2; Io3 � ho

1Þ;ð19Þ

~Io ¼ ðIo1 ; Io

2 ; Io3Þ ¼

X

all l

~wol ð�Þ~hl

ffiX

all l

minðwol1hl1;w

ol3hl3Þ;

X

all l

wol2hl2;

X

all l

maxðwol1hl1;w

ol3hl3Þ

!

;

ð20Þ

Subsequently in the backward phase, the training of the

FBPN is decomposed into three subtasks: determining the

center value, upper, and lower bounds of the parameters.

First, to determine the center value of each fuzzy

parameter (such as whkl2, hh

l2, wol2, and ho

2), the FBPN is

treated as a crisp one. Some algorithms are applicable for

training a crisp BPN, such as the gradient descent algo-

rithms, the conjugate gradient algorithms, the Levenberg–

Marquardt algorithm, and others. In this study, the

Levenberg–Marquardt algorithm is applied. The Leven-

berg–Marquardt algorithm was designed for training with

second-order speed without having to compute the Hessian

matrix. In training a BPN, the Hessian matrix can be

approximated as

H ¼ JT J ð21Þ

and the gradient can be computed as

g ¼ JT e ð22Þ

where J is the Jacobian matrix containing the first

derivatives of network errors with respect to the weights

and biases; e is the vector of network errors. The

Levenberg–Marquardt algorithm uses this approximation

and updates the network parameters in a Newton-like way:

xepþ1 ¼ xep � ½JT J þ lI��1JT e ð23Þ

where ep is the epoch number. Newton’s method is faster

and more accurate near an error minimum, so its purpose is

to move as quickly as possible to Newton’s method. Thus,

l is decreased after each successful step and is increased

only when a tentative step would increase the performance

function. In this way, the performance function is always

reduced after each epoch [17].

Subsequently, the parameters in the FBPN are fuzzified


123

to determine the lower bound of each fuzzy parameter

(e.g., whkl3, hh

l3, wol3, and ho

3) [14]. The satisfaction level of

the (normalized) actual cycle time (i.e., N(CTj)) in the

(normalized) estimated cycle time (N( gCTEj) or ~oj) has to

be greater than or equal to a certain level s:

NðCTjÞ � oj3

oj2 � oj3

� s: ð24Þ

Therefore,

NðCTjÞ � ð1� sÞoj3 � soj2� 0; ð25Þ

where

oi3 ¼1

1þ e�no3

; ð26Þ

no3 ¼ Io

3 � ho3 ¼ Io

3 � ðho2 þ Dho

2Þ; ð27Þ

Io3 ¼

X

all l

wol3hl3 ¼

X

all l

ðwol2 þ Dwo

l2Þhl3; ð28Þ

in which the outputs from the hidden-layer nodes are

equal to

hl3 ¼1

1þ e�nhl3

; ð29Þ

nhl3 ¼ Ih

l3 � hhl3 ¼ Ih

l3 � ðhhl2 þ Dhh

l2Þ; ð30Þ

Ihl3 ¼

X

all k

whkl3xk ¼

X

all k

ðwhkl2 þ Dwh

kl2Þxk: ð31Þ

Therefore,

hl3 ¼1

1þ e�P

all lðwh

kl2þDwh

kl2Þxk�ðhh

l2þDhhl2Þð Þ ; ð32Þ

or

ln1

hl3

� 1

� �

¼ ðhhl2 þ Dhh

l2Þ �X

all l

ðwhkl2 þ Dwh

kl2Þxk: ð33Þ

Substituting these equations into the inequality generates

the following result:X

all l

ððwol2 þ Dwo

l2Þhl3Þ� ðho2 þ Dho

2Þ

� ln1� s

NðCTjÞ � soj2

� 1

� �

; ð34Þ

which is a nonlinear constraint. In addition, the upper

bound has to be greater than the actual cycle time:

oj3�NðCTjÞ; ð35Þ

which is equivalent to

X

all l

ððwol2 þ Dwo


2Þ � ln1

NðCTjÞ� 1

� �

:

ð36Þ

On the other hand, the objective function is to minimize the

sum of the allowances added to all completion time

forecasts:

MinX

all j

ðoj3 � oj2Þ ð37Þ

where oj2 are pre-determined results, and therefore, the

objective function becomes

MinX

all j

oj3 ð38Þ

which after expansion becomes

MinX

allj

1

1þ e�P

all lðwo

l2þDwo

l2Þhl3�ðho

2þDho2Þð Þ ð39Þ

Finally, the following NP model is constructed:

MinX

allj

1

1þ e�P

all lðwo

l2þDwo

l2Þhl3�ðho

2þDho2Þð Þ ð40Þ

s:t:

ln1

hl3

� 1

� �

¼ ðhhl2 þ Dhh

l2Þ �X

all l

ðwhkl2 þ Dwh

kl2Þxk

;

ð41ÞX

all l

ððwol2 þ Dwo


2Þ

� ln1� s

NðCTjÞ � soj2

� 1

� �

; ð42Þ

X

all l

ððwol2 þ Dwo


2Þ � ln1

NðCTjÞ� 1

� �

;

ð43Þ

Dwhkl2;Dhh

l2;Dwol2; Dho

2 2 R; ð44Þ

k ¼ 1 8; ð45Þl ¼ 1m the number of hidden � layer nodesð Þ; ð46Þ

which is intractable and has to be converted into a simpler

form, e.g., a linear or goal programming model. For this

purpose, the concepts of goal programming are applied.

For example, an upper bound can be established for the

objective function value:

1

1þ e�P

all lðwo

l2þDwo

l2Þhl3�ðho

2þDho2Þð Þ �Wj; ð47Þ

which after expansion becomesX

all l

ðwol2 þ Dwo

l2Þhl3 � ðho2 þ Dho

2Þ� � lnð1=Wj � 1Þ;

ð48Þ

which is a new linear constraint. The new objective

function is


123

MinX

all j

Wj ð49Þ

which is linear. Finally, the following goal programming

problem is constructed for solving the original NP model:

MinX

all j

Wj ð50Þ

s:t:X

all l

ðwol2 þ Dwo


2Þ� � lnð1=Wj � 1Þ; ð51Þ

ln1

hl3

� 1

� �

¼ ðhhl2 þ Dhh

l2Þ �X

all l

ðwhkl2 þ Dwh

kl2Þxk; ð52Þ

X

all l

ððwol2 þ Dwo


2Þ

� ln1� s

NðCTjÞ � soj2

� 1

� �

; ð53Þ

X

all l

ððwol2 þ Dwo


2Þ � ln1

NðCTjÞ� 1

� �

;

ð54Þ

Dwhkl2; Dhh

l2; Dwol2; Dho

2 2 R; ð55Þk ¼ 1 8; ð56Þl ¼ 1m the number of hidden� layer nodesð Þ; ð57Þ

where Wj’s are established positive goals. Various values

of them will be fed into the goal programming problem,

and therefore, the goal programming problem is solved

many times. From these optimization results, the best one

giving the minimum upper bound is chosen.

In a similar way, the following GP problem is solved to

determine the lower bound of each fuzzy parameter (e.g.,

whkl1, hh

l1, wol1, and ho

1):

MinX

all j

Wj ð58Þ

subject toX

all l

ðwol2 þ Dwo


2Þ� � lnð�1=Wj � 1Þ; ð59Þ

ln1

hl1

� 1

� �

¼ ðhhl2 þ Dhh

l2Þ �X

all l

ðwhkl2 þ Dwh

kl2Þxk; ð60Þ

X

all l

ððwol2 þ Dwo


2Þ

� ln1� s

NðCTjÞ � soj2

� 1

� �

; ð61Þ

X

all l

ððwol2 þ Dwo


2Þ � ln1

NðCTjÞ� 1

� �

;

ð62Þ

Dwhkl2;Dhh

l2;Dwol2;Dho

2 2 R; ð63Þ

k ¼ 1 8; ð64Þl ¼ 1m the number of hidden� layer nodesð Þ; ð65Þ

A fuzzy cycle time (or step cycle time) forecast is

defuzzified using the center-or-gravity method [18].

Finally, the FBPN can be applied to estimate the cycle

time or the step cycle time of a new job. When a new job

is released into the factory, the eight parameters associ-

ated with the new job are recorded. Then, the FBPN is

applied to estimate the cycle time or step cycle time of

the new job.

2.3 The three-objective rule

Most of the existing methods in production scheduling

are concerned with the optimization of a single criterion.

However, the analysis of the performance of a schedule

often involves more than one aspect and therefore

requires a multi-objective treatment. A naive aggregation

of single-objective heuristic does not necessarily yield

feasible non-dominated solutions. In addition, consider-

ing the weighted sum of the objectives often leads to

unsatisfactory results. In our opinion, a single dispatch-

ing rule to optimize multiple objectives at the same time

is obviously the most convenient way. To this end, we

fuse FSMCT, FSVCT, and EDD that can be generalized

into the same form, which makes their combination more

natural.

In traditional fluctuation smoothing (FS) rules, there are

two different formulation methods, depending on the

scheduling purpose [19]. One method is aimed at mini-

mizing the mean cycle time with the fluctuation smoothing

rule for mean cycle time (FSMCT):

SKijðFSMCTÞ ¼ i=k� RCTEij ð66Þ

The other method is aimed at minimizing the variance of

cycle time with the fluctuation smoothing rule for cycle

time variation (FSVCT):

SKijðFSVCTÞ ¼ Ri � RCTEij ð67Þ

Jobs with the smallest slack values will be given higher

priority. These two rules and their variants have been

proven to be very effective in shortening the cycle time in

wafer fabrication factories [3, 9, 10, 19].

On the other hand, the earliest due date (EDD) rule is

theoretically the optimal rule to minimize the maximum

lateness for simple production systems. It is also the

most commonly used dispatching rule by make-to-order


123

(MTO) wafer fabrication factories to achieve on-time

delivery. EDD gives higher priorities to jobs with the

earliest due dates, so we can define the slack value as:

SKijðEDDÞ ¼ DDi ð68Þ

If the due date is determined in an internal way, then

equation (68) can be re-written as

SKijðEDDÞ ¼ Ri þ CTEi þ j ð69Þ

where j is a constant allowance. Substituting Eq. (12) into

Eq. (69), we get

SKijðEDDÞ ¼ Ri þ RCTEij þ SCTij þ j ð70Þ

If RCTEij is large, then SKijðEDDÞ is high, but

SKijðFSMCTÞ and SKijðFSVCTÞ are low, which means

that the three objectives are contradictory in nature. To

solve this problem, the following steps are followed:

1. Normalize all terms for the three rules. Its purpose is

to balance the effects of the parameters in the

rules [3].

2. Calculate SKi using the multiplication and division

operators instead. According to Chen and Wang [3],

this treatment magnifies the difference in the slack

value, which seems to be a good way of improving

scheduling performance.

3. Derive a general expression of the three types of SKis.

First of all, the components in Eqs. (66), (67), and (70)

are normalized:

Nði=kÞ ¼ i=k� 1=kn=k� 1=k

¼ i� 1

n� 1ð71Þ

NðRCTEijÞ ¼RCTEij �min

iRCTEij

maxi

RCTEij �mini

RCTEij

ð72Þ

NðRiÞ ¼Ri �min

iRi

maxi

Ri �mini

Ri

ð73Þ

NðSCTijÞ ¼SCTij �min

iSCTij

maxi

SCTij �mini

SCTij

ð74Þ

where N() is the normalization function. Subsequently,

multiplication and division are used to replace the original

addition and subtraction operators, respectively. After

replacement, Eqs. (66) and (67) become

SKijðFSMCTÞ ¼ i� 1

N � 1=

RCTEij �mini

RCTEij

maxi

RCTEij �mini

RCTEij

ð75Þ

SKijðFSVCTÞ ¼Ri �min

iRi

maxi

Ri �mini

Ri

=

RCTEij �mini

RCTEij

maxi

RCTEij �mini

RCTEij

ð76Þ

while equation (70) changes to

SKijðEDDÞ ¼Ri �min

iRi

maxi

Ri �mini

Ri

�RCTEij �min

iRCTEij

maxi

RCTEij �mini

RCTEij

�SCTij �min

iSCTij

maxi

SCTij �mini

SCTij

ð77Þ

A general expression of the three equations can be

derived as

SKijðnew ruleÞ ¼ i� 1

n� 1

� �a

�Ri �min

iRi

maxi

Ri �mini

Ri

0

@

1

A

b

�RCTEij �min

iRCTEij

maxi

RCTEij �mini

RCTEij

0

@

1

A

c

�SCTij �min

iSCTij

maxi

SCTij �mini

SCTij

0

@

1

A

g

ð78Þ

where a, b, c, and g and are positive real numbers

satisfying the following constraints:

If a ¼ 0; then b ¼ 1; and vice versa ð79ÞIf a ¼ 1; then b; g ¼ 0; c ¼ �1; and vice versa ð80ÞIf c ¼ 1; then b; g ¼ 1; a ¼ 0; and vice versa ð81ÞIfc ¼ �1; then g ¼ 0; and vice versa ð82Þ

There are many possible models to form the

combinations of a, b, c, and g. For example,

Linear modelð Þc ¼ 1� 2a; b; g ¼ 1� a ð83Þ

Nonlinear modelð Þc ¼ ð1� 2aÞu; b; g ¼ ð1� aÞu;u ¼ 1; 3; 5; . . .

ð84Þ

ðLogarithmic model1Þc ¼ lnð2� aÞ=ln 2� a;b; g ¼ lnð2� aÞ=ln 2

ð85Þ

ðLogarithmic model2Þg ¼ lnð2þ cÞ=ln3 ð86Þ

Equation (83) requires the value of a to be less than 1.

With any model, the proposed methodology tries various

combinations of a, b, c, and g to optimize the scheduling

performance in the target wafer fabrication factory. In this

way, the new rule becomes tailored to the specific factory.


123

In addition, the values of a, b, c, and g can be dynamically

adjusted to reflect the changes in the production conditions

of the factory. Clearly, EDD, FSMCT, and FSVCT are

special cases of the new rule:

EDD : ða; b; c; gÞ ¼ 0; 1; 1; 1ð ÞFSMCT : ða; b; c; gÞ ¼ 1; 0; �1; 0ð ÞFSVCT : ða; b; c; gÞ ¼ 0; 1; �1; 0ð Þ

An example is given in Table 1 to illustrate the

calculation of the slack value in the new rule (k = 1.18;

(a, b, c, g) = (0.4, 0.6, 0.2, 0.6)). The sequencing result

according to the new rule is 1; 5; 11! 17! 3! 15!14! 16! 7! 18! 6! 2! 9! 8! 10! 12! 4

! 19! 13. On the other hand, the sequencing result

according to EDD (j = 72) is5! 17! 15! 18! 1

2! 8! 10! 9! 2! 14! 3! 1! 4! 19! 6!13! 16! 11! 7 while that according to FSMCT and

FSVCT are, respectively,

7! 13! 19! 6! 16! 11! 1! 4! 9! 8! 15

! 18! 2! 12! 17! 14! 10! 3! 5

and

7! 13! 19! 11! 1! 16! 6! 4! 9! 15

! 17! 8! 3! 12! 2! 18! 14! 5! 10

These schedules are compared in Fig. 1. We note that if

the sequencing results of the traditional rules are very

different, then that of the new rule will be a compromise

among them.

3 Simulation study

To evaluate the effectiveness of the fuzzy-neural approach

for three-objective job scheduling in a wafer fabrication

factory, we used simulated data to avoid disturbing the

regular operations of the wafer fabrication factory. The real

time scheduling systems will input information very rap-

idly into the production management information systems

(PROMIS). To this end, a real wafer fabrication factory

located in the Taichung Science Park in Taichung, Taiwan,

with a monthly capacity of about 25,000 wafers was sim-

ulated. The simulation program has been validated and

verified by comparing the actual cycle times with the

simulated values and by analyzing the trace report,

respectively. The wafer fabrication factory is producing

more than 10 types of memory products and has more than

500 workstations for performing single-wafer or batch

operations using 58 nm * 110 nm technologies. Jobs

released into the fabrication factory are assigned three

types of priorities, i.e., ‘‘normal,’’ ‘‘hot,’’ and ‘‘super hot.’’

0

2

4

6

8

10

12

14

16

18

20

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

job #

orde

r

new ruleEDDFSMCTFSVCT

Fig. 1 The schedules by

various rules

Table 1 An example

No Ri i/k RCTEij j SCTij SKij

1 102 39 1,399 159 881 0

2 756 540 1,127 37 227 0.90778

3 826 733 1,223 37 157 0.72537

4 652 463 1,822 86 331 0.57034

5 208 154 530 55 775 ?

6 783 579 2,040 84 200 0.51283

7 800 656 2,366 96 183 0.47195

8 478 270 942 52 505 0.73744

9 469 232 1,116 65 514 0.51705

10 699 502 995 32 284 1.08266

11 836 772 2,151 85 147 0

12 497 309 883 45 486 0.91569

13 596 386 2,047 101 387 0.45543

14 798 618 1,146 34 185 0.87264

15 197 116 743 79 786 0.38795

16 804 695 2,092 85 179 0.53404

17 163 77 647 78 820 0.29211

18 457 193 810 44 526 0.67214

19 523 347 1,851 100 460 0.45944


123

Jobs with the highest priorities will be processed first. Such

a large scale accompanied with reentrant process flows

make job dispatching in the wafer fabrication factory a

very tough task. Currently, the longest average cycle time

exceeds three months with a variation of more than 300 h.

The wafer fabrication factory is therefore seeking better

dispatching rules to replace first-in first-out (FIFO) and

EDD, in order to shorten the average cycle times and

ensure on-time delivery to its customers. One hundred

replications of the simulation are successively run. The

time required for each simulation replication is about

30 min using a PC with an Intel Dual CPU E2200 2.2 GHz

and 1.99G RAM. A horizon of twenty-four months is

simulated.

To assess the effectiveness of the proposed methodology

and to make comparison with some existing approaches—

FIFO, EDD, shortest remaining processing time (SRPT),

critical ratio (CR), FSVCT, FSMCT, and nonlinear fluc-

tuation smoothing rule (NFS) [20] were all applied to

schedule the simulated wafer fabrication factory to collect

the data of 1000 jobs, and then, we separated the collected

data by their product types and priorities. That is about the

amount of work that can be achieved with 100 % of

monthly capacity. In some cases, there was too little data,

so they were not discussed. These rules are traditionally

used for different purposes:

1. The average cycle time: FIFO, FSMCT, SRPT, NFS.

2. Cycle time standard deviation: FSVCT, NFS.

3. The maximum lateness: EDD, CR.

To determine the due date of a job, the FCM-FBPN

approach was applied to estimate the cycle time. Jobs were

first pre-classified into K categories with FCM, in which

the value of K was determined using the S test. Then, the

data of jobs of a category were fed into the same FBPN.

The FBPN has eight inputs and a single hidden layer also

with eight neurons. For training the BPN, the Levenberg–

Marquardt algorithm rather than the gradient descent

algorithm was applied to speed up the network conver-

gence. The number of epochs was set to 1000. In addition,

the initial values of all parameters in the BPN were ran-

domly generated 20 times. Network training stopped when

mean squared error (MSE) was less than 10-6. After

training, the network output was adjusted using the GP

models to determine the upper and lower bounds of the

remaining cycle time.

The forecasting performance of the FCM-FBPN

approach was also compared with those of six existing

methods, including multiple linear regression (MLR), back

propagation network (BPN), FBPN, case-based reasoning

(CBR), evolving fuzzy rules (EFR) [21], the hybrid self-

organization map, and the Wang and Mendal method

(SOM-WM) [22]. The comparison results are shown in

Fig. 2, which confirmed the effectiveness of the FCM-

FBPN approach. After incorporating each method with

FSMCT, it was clear that more accurate remaining cycle

time estimation led to better scheduling performance (see

Fig. 3). Then, we added a constant allowance of three days

to the estimated cycle time, i.e., j = 72, to determine the

internal due date.

Jobs with the highest priorities are usually processed

first. In FIFO, jobs were sequenced on each machine first

by their priorities and then by their arrival times at the

machine. In EDD, jobs were sequenced first by their

priorities and then by their due dates. In CR, jobs were

sequenced first by their priorities and then by their

critical ratios. The critical ratio of a job is calculated as

follows:

CRij ¼ ðt � DDiÞ =RPTij ð87Þ

In the proposed methodology, nine possible sets of the

four adjustable parameters were tried (see Table 2).

However, the new rule with (0, 1, 1, 1), (1, 0, -1, 0),

and (0, 1, -1, 0) reduces to EDD, FSMCT, and FSVCT,

respectively, and were therefore not tried.

A fair way of comparison is to compare multiple per-

formance measures at the same time [23]. Subsequently,

0

20

40

60

80

100

120

140

RM

SE (

hrs)

Fig. 2 The forecasting performances of various approaches (product

A, normal priority, the 80 % step)

1200

1250

1300

1350

1400

1450

1500

1550

avg.

cyc

le t

ime

Fig. 3 More accurate remaining cycle time estimation leads to better

scheduling performance (product A, normal priority)


123

the average cycle time and maximum lateness of all cases

were calculated to assess the scheduling performance. With

respect to the average cycle time, the FSMCT policy was

used as the basis for comparison, while the EDD policy

was used as the basis for comparison with respect to

maximum lateness, and FSVCT was compared in evalu-

ating cycle time standard deviation. The results are sum-

marized in Tables 3, 4, 5.

According to the experimental results, the following

points can be made:

Table 3 The performances of various approaches in the average cycle time

Avg. cycle time (h) A (normal) A (hot) A (super hot) B (normal) B (hot)

FIFO 1,254 400 317 1,278 426

EDD 1,094 345 305 1,433 438

SRPT 948 350 308 1,737 457

CR 1,148 355 300 1,497 440

FSMCT 1,313 347 293 1,851 470

FSVCT 1,014 382 315 1,672 475

NFS 1,456 407 321 1,452 421

The new rule (0.5, 0.5, 0, 0.5) 1,195 339 260 1,369 394

The new rule (0.3, 0.343, 0.064, 0.343) 1,183 325 266 1,444 397

The new rule (0.6, 0.064, -0.008, 0.064) 1,163 323 265 1,229 356

The new rule (0.9, 0.001, -0.512, 0.001) 1,256 348 278 1,266 348

The new rule (0.25, 0.807, 0.557, 0.807) 1,147 316 260 1,478 397

The new rule (0.5, 0.585, 0.085, 0.585) 1,181 335 269 1,390 391

The new rule (0.75, 0.322, -0.428, 0.322) 1,246 347 269 1,314 363

Table 4 The performances of various approaches in the maximum lateness

The maximum lateness (h) A (normal) A (hot) A (super hot) B (normal) B (hot)

FIFO 401 -122 164 221 172

EDD 295 -181 144 336 185

SRPT 584 -142 174 718 194

CR 302 -159 138 423 192

FSMCT 875 -165 125 856 171

FSVCT 706 -112 174 686 260

NFS 627 10 161 331 151

The new rule (0.5, 0.5, 0, 0.5) 556 -96 81 363 139

The new rule (0.3, 0.343, 0.064, 0.343) 557 -165 107 460 124

The new rule (0.6, 0.064, -0.008, 0.064) 411 -181 91 181 103

The new rule (0.9, 0.001, -0.512, 0.001) 511 -159 98 236 97

The new rule (0.25, 0.807, 0.557, 0.807) 594 -94 91 454 101

The new rule (0.5, 0.585, 0.085, 0.585) 537 -129 100 397 123

The new rule (0.75, 0.322, -0.428, 0.322) 510 -150 96 265 93

Table 2 Some possible sets of the four adjustable factors

Rule (a, b, c, g)

Linear (0, 1, 1, 1), (0.5, 0.5, 0, 0.5), (1, 0, -1, 0), etc.

Nonlinear (u = 3) (0.3, 0.343, 0.064, 0.343), (0.6, 0.064, -0.008, 0.064), (0.9, 0.001, -0.512, 0.001), etc.

Logarithmic (0.25, 0.807, 0.557, 0.807), (0.5, 0.585, 0.085, 0.585), (0.75, 0.322, -0.428, 0.322), etc.


123

1. For the average cycle time, the proposed methodology

outperformed the baseline approach, the FSMCT

policy. The most obvious advantage was up to 17 %

when (a, b, c, g) = (0.6, 0.064, -0.008, 0.064).

2. Meanwhile, the proposed methodology also achieved

good performances in reducing maximum lateness.

The new rule with (a, b, c, g) = (0.6, 0.064, -0.008,

0.064) surpassed the baseline EDD policy in most

cases with an average advantage of up to 18 %, which

revealed that the treatments carried out in this study

did indeed improve the performances of the traditional

policies.

3. Among the variants of the new rule, the one with (a, b,

c, g) = (0.6, 0.064, -0.008, 0.064) also achieved the

best performance in regard to the cycle time standard

deviation. The average advantage over the comparison

basis was 39 %.

4. The advantages of the proposed methodology over the

existing methods are shown in Figs. 4, 5, 6.

5. To statistically compare the performance of these

approaches in all cases, we first ranked them for each

Table 5 The performances of various approaches in cycle time standard deviation

Cycle time standard deviation (hrs) A (normal) A (hot) A (super hot) B (normal) B (hot)

FIFO 55 24 25 87 51

EDD 129 25 22 50 63

SRPT 248 31 22 106 53

CR 69 29 18 58 53

FSMCT 419 33 16 129 104

FSVCT 280 37 27 201 77

NFS 87 49 19 44 47

The new rule (0.5, 0.5, 0, 0.5) 324 49 13 125 49

The new rule (0.3, 0.343, 0.064, 0.343) 352 34 21 124 61

The new rule (0.6, 0.064, -0.008, 0.064) 216 29 17 84 34

The new rule (0.9, 0.001, -0.512, 0.001) 216 36 14 97 31

The new rule (0.25, 0.807, 0.557, 0.807) 361 45 15 126 70

The new rule (0.5, 0.585, 0.085, 0.585) 328 49 16 135 52

The new rule (0.75, 0.322, -0.428, 0.322) 277 38 14 91 38

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

FIFO EDD SRPT CR FSMCT FSVCT NFS

adva

ntag

e

(avg. cycle time)

Fig. 4 The advantages of the proposed methodology over the

existing methods in the average cycle time ((a, b, c, g) = (0.6,

0.064, -0.008, 0.064))

0%

50%

100%

150%

200%

250%

300%

350%

400%

450%


adva

ntag

e

(maximum lateness)


existing methods in the maximum lateness ((a, b, c, g) = (0.6,

0.064, -0.008, 0.064))

(cycle time standard deviation)

-10%

0%

10%

20%

30%

40%

50%


adva

ntag

e


existing methods in cycle time standard deviation ((a, b, c,

g) = (0.6, 0.064, -0.008, 0.064))


123

case and then added up the ranks of the same approach

for comparison. The results are summarized in

Table 6. Four variants of the new rule were among

the top five rules in the experiment, which supported

the Pareto optimality of the proposed methodology

because most of its variants dominated the other

approaches.

6. The efficiency of the simulation experiment can be

improved by the technique of reduced simulation [24],

in which only the bottlenecks are considered, and the

other machines are combined into blocks.

7. For a comparison, the fusion of FIFO, SPT, and CR

was also applied to schedule the simulated wafer

fabrication factory. In addition, instead of a general-

ized form, only the product of the slacks of the three

rules was derived:

SKij ¼ Ri RPTij CRij ð88Þ

The scheduling performances of FIFO ? SRPT ? CR

in various cases were summarized in Figs. 7, 8, 9, which

was not comparable to the scheduling performance of

the proposed methodology, and sometimes was even

worse than those of the three original rules. Since most

Table 6 The sum of the ranks of each approach

Sum of ranks Average

cycle time

Rank

no

Maximum

lateness

Rank

no

Cycle time

standard

deviation

Rank

no

Total

rank no

FIFO 49 11 37 7 26 3 8

EDD 36 8 30 4 31 6 5

SRPT 47 9 58 13 41 9 11

CR 47 9 38 9 26 3 8

FSMCT 56 13 49 11 52 13 13

FSVCT 52 12 64 14 59 14 14

NFS 59 14 50 12 31 6 12

The new rule (0.5, 0.5, 0, 0.5) 26 3 35 5 40 8 4

The new rule (0.3, 0.343, 0.064, 0.343) 30 5 37 7 48 10 9

The new rule (0.6, 0.064, -0.008, 0.064) 14 1 14 1 23 1 1

The new rule (0.9, 0.001, -0.512, 0.001) 31 6 22 3 25 2 2

The new rule (0.25, 0.807, 0.557, 0.807) 24 2 40 10 51 11 10

The new rule (0.5, 0.585, 0.085, 0.585) 27 4 35 5 51 11 6

The new rule (0.75, 0.322, -0.428, 0.322) 31 6 21 2 30 5 3

0 200 400 600 800

1000 1200 1400 1600 1800 2000

1 2 3 4 5

avg.

cyc

le ti

me

(hrs

)

case #

FIFO+SRPT+CR

the proposed methodology

FIFO

SRPT

CR

Fig. 7 The performances of FIFO ? SRPT ? CR in various cases

(the average cycle time)

0

50

100

150

200

250

300

1 2 3 4 5

cycl

e tim

e st

d. d

ev. (

hrs)

case #

FIFO+SRPT+CR


FIFO

SRPT

CR


(cycle time standard deviation)

-400

-200

0

200

400

600

800

1 2 3 4 5

max

imum

late

ness

(hr

s)

case #

FIFO+SRPT+CR


FIFO

SRPT

CR


(the maximum lateness)


123

factors considered in FIFO ? SRPT ? CR and the

proposed methodology are the same, the way to form

the fusion may be the decisive issue.

4 Conclusions and directions for future research

Multi-objective scheduling in a wafer fabrication factory

is a challenging but important task. For such a complex

production system, to optimize a single objective has

been tough enough, needless to say taking into account

three objectives at the same time. As an innovative

attempt, this study presents a fuzzy-neural approach for

three-objective job scheduling in a wafer fabrication

factory, to optimize three performance measures at the

same time, which has rarely been discussed in past

studies.

The fuzzy-neural approach is a modification from

three traditional dispatching rules—EDD, FSMCT, and

FSVCT with some innovative treatments. First, the

remaining cycle time of a job is estimated with the

FCM-FBPN approach to improve the estimation accu-

racy. Second, the three traditional dispatching rules are

fused in a nonlinear way to generate the new rule. To

assess the effectiveness of the proposed methodology and

to compare it with some of the existing approaches, a

production simulation was carried out. Then, the pro-

posed methodology and some existing approaches were

all used for scheduling a simulated wafer fabrication

factory. The experimental results were as follows:

1. Through improving the accuracy of estimating the

remaining cycle time, the performance of a scheduling

rule can indeed be strengthened.

2. In particular, the nonlinear way of rule fusion appears

as an appropriate tool to analyze multi-objective

scheduling problems.

3. Some variants of the new rule dominated most of the

traditional rules compared in the experiment. There-

fore, the proposed methodology can be concluded as

an effective means to optimize the average cycle time,

cycle time standard deviation, and maximum lateness

at the same time.

However, to further assess the effectiveness and effi-

ciency of the proposed methodology, the only way is to

apply it to an actual wafer fabrication factory. In addition,

different objectives can be fused in the same way in future

studies.

Acknowledgments This work was supported by the National Sci-

ence Council of Taiwan.

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a fuzzy-neural approach for supporting three-objective job scheduling in a wafer fabrication factory

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