research article a fuzzy rule for improving the performance of multiobjective...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 986172 18 pageshttpdxdoiorg1011552013986172

Research ArticleA Fuzzy Rule for Improving the Performance ofMultiobjective Job Dispatching in a Wafer Fabrication Factory

Toly Chen and Yi-Chi Wang

Department of Industrial Engineering and Systems Management Feng Chia University No 100 Wenhwa Road SeatwenTaichung City 407 Taiwan

Correspondence should be addressed to Toly Chen tcchenfcuedutw

Received 1 January 2013 Revised 21 August 2013 Accepted 22 August 2013

Academic Editor Constantinos Siettos

Copyright copy 2013 T Chen and Y-C Wang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper proposes a fuzzy slack-diversifying fluctuation-smoothing rule to enhance the scheduling performance in a waferfabrication factory The proposed rule considers the uncertainty in the remaining cycle time and is aimed at simultaneousimprovement of the average cycle time cycle time standard deviation the maximum lateness and number of tardy jobs Existingpublications rarely discusse ways to optimize all of these at the same time An important input to the proposed rule is the jobremaining cycle time To this end this paper proposes a self-adjusted fuzzy back propagation network (SA-FBPN) approach toestimate the remaining cycle time of a job In addition a systematic procedure is also established which can solve the problem ofslack overlapping in a nonsubjective way and optimize the overall scheduling performanceThe simulation study provides evidencethat the proposed rule can improve the four performance measures simultaneously

1 Introduction

The task of job dispatching is to determine which jobs mustbe processed next on the available machines However jobdispatching in a wafer fabrication factory is a very difficulttask A single recipe may contain more than 500 steps anda wafer fabrication factory can produce as many as 200products In addition some machines in a wafer fabricationfactory for example steppers are very expensive and thereare only a very limited number of them Therefore wafershave to revisit these machines for the processing of differentlayers This gives rise to the characteristic re-entrant flows

The scheduling of a wafer fabrication factory usuallyneeds to consider various points of view To shorten the cycletime is one and to meet the due date is another However forsuch a complex production system it is difficult to optimizeeven a single measure [1 2] let alone the simultaneousoptimization of multiple measures To solve this problemmany attempts have beenmade in the literature Some resultsare as follows First a naıve aggregation of single-objectiveheuristics does not necessarily yield feasible nondominatedsolutions [3] In addition considering the weighted sum

of multiple objectives often leads to unsatisfactory resultsFurther scaling parameters can be used to weigh the differentobjectives and the determination of appropriate weights is akey research issue

On the other hand job scheduling in a wafer fabricationfactory is subject to many sources of uncertainty or random-ness Such uncertainty or randomness is partly due tomanualoperations including loading and unloading jobs setting upor repairing machines visual inspection and others Othercauses include unexpected releases of emergency orders andmachine breakdowns that are beyond the control of thefactory To consider the uncertainty or randomness fuzzymethods are easier to use than probabilistic (stochastic)methods

A fuzzy slack-diversifying fluctuation-smoothing rule isproposed in this study for multiobjective job dispatching in awafer fabrication factoryThe existing fuzzy dispatching rulesusually take the form of fuzzy inference rules for example ldquoifthe total processing time is long and the due date is tight thenthe job priority is highrdquo [4 5]

To give a precise schedule usually multiple fuzzy dis-patching rules are used simultaneously The results of these

2 Journal of Applied Mathematics

Table 1 Some existing fuzzy dispatching rules

References System type Rulesinputsoutputs Number of objectivesXiong et al [4] TSK 221 2 (average cycle time lateness)Benincasa et al [5] Mamdani 2731 2 (average cycle time WIP level)Lee et al [9] Single-rule-based 241 2 (average cycle time cycle time standard deviation)Tan and Tang [10] Mamdani 841 3 (throughput average cycle time no of vehicles)Dong and Liu [11] ANFIS 1641 1 (average cycle time)

Tsai and Chen [12] Fusion 451 4 (average cycle time cycle time standard deviationmaximum lateness number of tardy jobs)

lowastWIP work-in progress

dispatching rules must be aggregated To this end there areat least three types of fuzzy inference systemsmdashMamdanirsquosfuzzy inference systems [6] Takagi-Sugeno-Kangrsquos (TSKrsquos)[7] fuzzy inference systems and adaptive neurofuzzy infer-ence systems (ANFISs) [8] These systems use differentaggregation and defuzzification methods Xiong et al [4]scheduled a flexible manufacturing system (FMS) using twofuzzy dispatching rules of the TSK type Lee et al [9]established fuzzy inference rules to select a combination ofsome existing dispatching rules for scheduling a FMS Theeffectiveness of a fuzzy inference system depends critically onthe way in which the related variables are partitioned Forthis reason Tan and Tang [10] applied Taguchirsquos design ofexperiment (DOE) techniques to improve the design of somefuzzy dispatching rules for a test facility According to Benin-casa et al [5] up to 27 rules (each with three inputs and oneoutput) were established for scheduling automated guidedvehicles According to Dong and Liu [11] the uncertainty inthe processing time was modeled with a fuzzy number andan ANFIS was established to schedule a job shop Inputs tothe ANFIS were the differences between any two jobs whilethe output from the ANFIS determined the sequence of thetwo jobs If the output was greater than zero then the first jobshould be processed before the second job Tsai andChen [12]fuzzified four traditional dispatching rules and added fiveadjustable parameters to merge the four rules A summaryof some existing fuzzy dispatching rules is shown in Table 1

The existing approaches have the following problems(1) Establishing fuzzy inference rules is a subjective

process that is not easy to optimize(2) The fuzziness of parameters causes the uncertainty in

scheduling and therefore must be controlled(3) The parameters that have the greatest relevance for

the scheduling performance must be estimated moreaccurately (such as the remaining cycle time) in orderto enhance the effectiveness of scheduling

(4) How to solve ties that is jobs with the same slackswhen the parameters are fuzzy valued has not beenfully discussed

To tackle these problems and to enhance the performanceof multiobjective job scheduling in a wafer fabrication fac-tory a fuzzy slack-diversifying fluctuation-smoothing rule isproposed in this study The unique features of the proposedmethodology include the following

(1) Considering the uncertainty in the remaining cycletime Dispatching rules that consider the remainingcycle time are more able to respond to the changesin the production environment [13] To consider theuncertainty in the remaining cycle time it is estimatedwith a fuzzy number and fed into a fuzzy dispatchingrule To this end a self-adjusted fuzzy back propaga-tion network (SA-FBPN) approach is proposed Com-pared with the existing methods this SA-FBPNapproach can generate a very precise estimate of theremaining cycle time in an efficient manner

(2) Establishing a fuzzy dispatching rule directly from theexisting rules A fuzzy slack-diversifying fluctuation-smoothing rule is proposed by fuzzifying the four-objective fluctuation-smoothing dispatching rule [14]and diversifying the job slack This rule accepts thefuzzy remaining cycle time as an input

(3) Diversifying the slackwith a new approach accordingto Wang et al [15] the slack was defuzzified by maxi-mizing the standard deviation However such a treat-ment has several drawbacks This study establishes asystematic procedure to overcome those drawbacksso that the slacks of jobs can be evenly distributedand the possibility of forming ties may be reduced asmuch as possible

(4) Solving the problem of slack overlapping without theneed to defuzzify the slacks subjectively

(5) Proposing the fuzzy intersection (FI) and generalizedaverage (GAV) approach to aggregate the estimationresults from various FBPNs

Some predictive scheduling methods have been pro-posed in recent years The differences between the pro-posed methodology and these methods were summarized inTable 2 The distinct advantages of the proposed methodol-ogy over these predictive scheduling methods include thefollowing

(1) The upper and lower bounds of the remaining cycletime used in the proposed methodology are tighter

(2) The proposed methodology provides an easier wayto optimize the scheduling rule which improves theusability of the proposed methodology

To assess the effectiveness of the proposed methodologyproduction simulation is also applied in this studyThe rest of

Journal of Applied Mathematics 3

Table 2 The differences between the proposed methodology and some predictive methods

Scheduling methodRemaining cycletime estimation

method

Types of inputsto the rule

Types of outputsfrom the rule

Number ofobjectives

Number ofadjustableparameters

Optimization method

Tsai and Chen [12] Effective FBPN Crisp + fuzzy Fuzzy 4 5Minimizing the sum of thestandard deviations ofslacks by FNLP

Chen and Wang [21] FCM-FBPN Crisp Crisp 2 1Minimizing the standarddeviation of slack bypolynomial fitting

Chen [22] FCM-BPN-GP Crisp + fuzzy Fuzzy 2 1 A systematic procedure fordiversifying the fuzzy slack

Chen and Wang [37] PostclassifyingFBPN Crisp Crisp 2 2sim3 Enumeration

Chen andRomanowski [23] FCM-BPN Crisp Crisp 2 2

Minimizing the standarddeviation of slack bypolynomial fitting

The proposedmethodology

SA-FBPN-aggregation Crisp + fuzzy Fuzzy 4 5 A systematic procedure for

diversifying the fuzzy slacklowastFCM fuzzy c-means BPN back propagation network GP goal programming FNLP fuzzy nonlinear programming

Table 3 The steps of fabricating a semiconductor product

Step Step no Machine Machine noOXWS 1 4HTB 6HTB 2DOD 3DOD 1WOD 3WOD 4WOD 5WOD 1 2 3 4 5 6 7 8MKWF 2 1PFC 2PFC 9 10ALZER 3 7B-I200 11ETZER 4 1B-DP5KZ 12STZER 5 1B-DASP 3B-DASP 1B-DULT 2B-DULT 9B-DULT 10B-DULT 13 14 15 16 17 18ALNW 6 10B-I100 9B-I100 5B-I200 7B-I200 11B-I80 19 20 21 11 22

QCOG 121 4HTB 6HTB 2DOD 3DOD 1WOD 3WOD 4WOD 5WOD 1 2 3 4 5 6 7 8

this paper is organized as follows Section 2 briefly describesthe production environment of a wafer fabrication factoryThen the scheduling problem to be solved is defined Theproposed method for this is described Section 3 is dividedinto three parts SA-FBPN the fuzzy slack-diversifyingfluctuation-smoothing rule and systematic slack diversifi-cation The SA-FBPN approach is proposed to estimate theremaining cycle time of a job with a fuzzy number To acceptthe fuzzy remaining cycle time as an input the four-objectivedispatching rule is fuzzified resulting in fuzzy-valued slacksthat may overlap In order to solve the problem of slackoverlapping a systematic procedure is established to diver-sify the slack In Section 4 a wafer fabrication simulationsystem is applied to test the effectiveness of the proposedmethodology so that its advantages and disadvantages canbe discussed Finally some concluding remarks are given inSection 5

2 Problem Description

At a very high level integrated circuit production may bedivided up into five blocks raw substrate wafer fabricationwafer test mark and final test and packaging A wafer

fabrication factory is one of the most complex productionsystemsThe complexity of a wafer fabrication processmay becharacterized by the number of mask layers or major processsteps Inputs to a wafer fabrication factory include 8 12 or18 inch wafers on which various types of integrated circuitscan be made There are four basic steps to fabricate a waferfilm etching diffusion and photo lithography These stepswill be repeated many times finally a wafer may need togo through hundreds of production steps In addition 20 to30 pieces of wafers are grouped into a joblot and may beprocessed together In total theremay bemore than 1000 jobs(or 30000 wafers) that are being processed or to be processedin a wafer fabrication factory To this end typically hundredsof machines are used simultaneously in a wafer fabricationfactory In addition a wafer may visit the same machinemultiple times which constitutes a special type of productionsystemmdasha re-entrant job shopAn example is given inTable 3In theory the problem of scheduling a simple job shop tooptimize a single regular measure for example 119869

119898 119862max is

alreadyNP-hard Scheduling a reentrant job shop to optimizemultiplemeasures at the same time is muchmore difficult Totackle such difficulties the following treatments are taken inthis study


Table 4 The nomenclature table

Variableparameter Meaning119905 The current time119877

119895The release time of job 119895 119895 = 1 sim 119899

CT119895

The cycle time of job 119895CTE

119895The estimated cycle time of job 119895

RCT119895119906

The remaining cycle time of job 119895 from step 119906RCTE

119895119906The estimated remaining cycle time of job 119895 from step 119906

RPT119895119906

The remaining processing time of job 119895 from step 119906SCT

119895119906The step cycle time of job 119895 until step 119906

SK119895119906

The slack of job 119895 at step 119906120582 Mean release rate119909

119895119901The inputs to the three-layer FBPN of job 119895 119901 = 1 sim 119875

ℎ

119897The output from hidden layer node 119897 119897 = 1 sim 119871

119908

119900

119897The connection weight between hidden layer node 119897 and the output node

119908

ℎ

119901119897

The connection weight between input node p and hidden layer node 119897 119901 = 1 sim 119875119897 = 1 sim 119871

120579

ℎ

119897The threshold on hidden layer node 119897

120579

119900 The threshold on the output node

(1) Dispatching rules have often been criticised for beingtoo simple suboptimal and myopic Neverthelessdispatching rules are still prevalent in the semicon-ductor manufacturing industry For this reason thisstudy aims to propose a new dispatching rule

(2) Predictive information such as the remaining cycletime estimate is incorporated into the schedulingrule to improve the responsiveness of the rule to thechanging conditions of a wafer fabrication factory

(3) Some of the existing dispatching rules that are effec-tive for four measures (the average cycle time cycletime standard deviation the maximum lateness andthe number of tardy jobs) are fused to optimize thesemeasures at the same time Therefore the schedulingproblem to be investigated can be indicated with119869

119898|119903

119895 119889

119895 rent|119862 120590

119862 119871max 119873119879

A flow chart of the proposed methodology is shown in

Figure 1

3 Methodology

The variables and parameters that will be used in theproposed methodology are defined in Table 4

Before any job is scheduled the remaining cycle timeof the job needs to be estimated To this end the SA-FBPN approach is proposed and the remaining cycle time isestimated with a fuzzy value

31 Step 1 Estimating the Remaining Cycle Time

311 The SA-FBPN Approach The remaining cycle time of ajob being processed in a wafer fabrication factory is the timestill needed to complete the job The remaining cycle time is

Feed the fuzzy remaining cycle time estimate to the new rule

Estimate the remaining cycletime using the SA-FBPN

approach

Fuzzify the four-objectivefluctuation-smoothing rule

Establish a systematicprocedure to defuzzify the slack

Use the slack-diversifying ruleto sequence jobs

Figure 1 The flowchart of the proposed methodology

an important attribute (or performance measure) for work-in-progress (WIP) in a wafer fabrication factory Past studies(eg [14]) have shown that the accuracy of remaining cycletime estimation can be improved by job classification Softcomputing methods (eg [16]) have received much attentionin this field


In the SA-FBPN approach jobs are classified into 119870

categories using FCM First in order to facilitate the sub-sequent calculation all raw data are preprocessed In theliterature there are two ways of preprocessing One is partialnormalization [17] the other is variable replacement throughprincipal component analysis (PCA) [18] However thesimple combination of PCA and FBPN does not have mucheffectThemain effect of PCA is to improve the correctness ofthe job classification [19] Then we place the (pre-processed)attributes of a job in vector x

119895= [119909

119895119901] 119901 = 1 sim 119875

FCM classifies jobs byminimizing the following objectivefunction

Min119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896) (1)

where119870 is the required number of categories 119899 is the numberof jobs 120583

119895(119896)indicates that job 119895 belongs to category 119896 119890

119895(119896)

measures the distance from job 119895 to the centroid of category119896 119898 isin [1infin) is a parameter to adjust the fuzziness and isusually set to 2 The procedure of FCM is as follows

(1) Set 119870 = 1

(2) Produce a preliminary clustering result

(3) (Iterations) Calculate the centroid of each category as

119909

(119896)= 119909

(119896)119901 119901 = 1 sim 119875

119909

(119896)119901=

sum

119899

119895=1120583

119898

119895(119896)119909

119895119901

sum

119899

119895=1120583

119898

119895(119896)

120583

119895(119896)=

1

sum

119870

119902=1(119890

119895(119896)119890

119895(119902))

2(119898minus1)

119890

119895(119896)=

radic

sum

all119901(119909

119895119901minus 119909

(119896)119901)

2

(2)

where 119909(119896)

is the centroid of category 119896 120583(119905)119895(119896)

is themembership function that indicates that job 119895 belongsto category 119896 after the 119905th iteration

(4) Re-measure the distance from each job to the centroidof each category and then recalculate the correspond-ing membership

(5) If the following condition is met go to step (6)Otherwise return to step (3) as follows

max119896

max119895

1003816

1003816

1003816

1003816

1003816

120583

(119905)

119895(119896)minus 120583

(119905minus1)

119895(119896)

1003816

1003816

1003816

1003816

1003816

lt 119889 (3)

where 119889 is a real number representing the thresholdfor the convergence of membership

1

2

Input layer

1

2

Hidden layer

1

Output layer

RMSE

xj1xj2

Ih1 nh1 120579h1 h1

wo1

wo2

oj

Io o

IhL nhL 120579hL hL

LP

woL

whP1

whP2

xjP

whPL

middotmiddotmiddot

middotmiddotmiddot

wh11

wh12

wh1L

aj

no

Figure 2 The architecture of the three-layer FBPN

(6) Calculate the following indexes

119869

119898=

119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896)

119890

2

min = min1198961

= 1198962

(sum

all119901(119909

(1198961)119901minus 119909

(1198962)119901)

2

)

119878 =

119869

119898

119899 times 119890

2

min

(4)

(7) 119870 = 119870+1 If119870 = 119870max stop the119870 value minimizing119878 determines the optimal number of categories [20]Otherwise return to step (2)

After clustering for each category a three-layer FBPN isestablished to estimate the remaining cycle times of jobs inthis category A portion of the jobs is input as the ldquotrainingexamplesrdquo to the three-layer FBPN to determine the param-eter values The joint use of fuzzy logic and artificial neuralnetworks is becoming more common in recent years Forexample fuzzy classifiers were applied in [21ndash23] to separatejobs in a wafer fabrication factory into different clustersbefore estimating their cycle times using artificial neuralnetworks Taghadomi-Saberi et al [24] established an ANFISto estimate the antioxidant activity and the anthocyanincontent at different ripening stages of sweet cherries Buiet al [25] also used an ANFIS to determine the landslidesusceptibility in theHoa Binh province of VietnamOcampo-Duque et al [26] proposed a fuzzy inference system tocompute ecological risk points (ERPs) thereby estimating thechanges in water quality over time

The configuration of the three-layer FBPN is as follows(see Figure 2) First inputs are the 119875 parameters of job119895 Subsequently there is a single hidden layer with 2119875

neurons Finally the output from the three-layer FBPN is the(normalized) remaining cycle time estimate (119873(

RCTE119895119906)) of

job 119895 where119873() is the normalization functionThe procedure for determining the parameter values is

now described Two phases are involved at the training stage


First in the forward phase inputs aremultipliedwithweightssummed and transferred to the hidden layer Then activatedsignals are output from the hidden layer as

ℎ

119897= (ℎ

1198971 ℎ

1198972 ℎ

1198973) =

1

1 + 119890

minus119899ℎ

119897

= (

1

1 + 119890

minus119899ℎ

1198971

1

1 + 119890

minus119899ℎ

1198972

1

1 + 119890

minus119899ℎ

1198973

)

(5)

where

119899

ℎ

119897= (119899

ℎ

1198971 119899

ℎ

1198972 119899

ℎ

1198973) =

119868

ℎ

119897(minus)

120579

ℎ

119897

= (119868

ℎ

1198971minus 120579

ℎ

1198973 119868

ℎ

1198972minus 120579

ℎ

1198972 119868

ℎ

1198973minus 120579

ℎ

1198971)

119868

ℎ

119897= (119868

ℎ

1198971 119868

ℎ

1198972 119868

ℎ

1198973) = sum

all 119901119908

ℎ

119901119897sdot 119909

119895119901

= (sum

all 119901min (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901)

sum

all 119901119908

ℎ

1199011198972119909

119895119901 sum

all 119901max (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901))

(6)

where (minus) and (times) denote fuzzy subtraction and multiplica-tion respectively

ℎ

119897values are also transferred to the output layer with the

same procedure Finally the output of the FBPN is generatedas follows

119900

119895= (119900

1198951 119900

1198952 119900

1198953) =

1

1 + 119890

minus119899119900

= (

1

1 + 119890

minus119899119900

1

1

1 + 119890

minus119899119900

2

1

1 + 119890

minus119899119900

3

)

(7)

where

119899

119900

= (119899

119900

1 119899

119900

2 119899

119900

3) =

119868

119900

(minus)

120579

119900

= (119868

119900

1minus 120579

119900

3 119868

119900

2minus 120579

119900

2 119868

119900

3minus 120579

119900

1)

(8)

119868

119900

= (119868

119900

1 119868

119900

2 119868

119900

3) = sum

all 119897119908

119900

119897(times)

ℎ

119897

cong (sum

all 119897min (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973) sum

all 119897119908

119900

1198972ℎ

1198972

sum

all 119897max (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973))

(9)

Subsequently in the backward phase the training of theFBPN is decomposed into three subtasks determining thecenter value and upper and lower bounds of the parametersFirst to determine the center of each parameter (such as119908

ℎ

1199011198972 120579ℎ

1198972 119908119900

1198972 and 120579

119900

2) the FBPN is treated as a crisp net-

work Some algorithms are applicable for this purpose suchas the gradient descent algorithms the conjugate gradientalgorithms the Levenberg-Marquardt algorithm and others

In this study the Levenberg-Marquardt algorithm is applied[27]

Subsequently the upper bound of each parameter (eg119908

ℎ

1199011198973 120579ℎ

1198973 119908119900

1198973 and 120579

119900

3) is to be determined so that the actual

value will be less than the upper bound of the networkoutput Chen and Wang [28] and Chen and Lin [29] havedescribedhowanonlinear programming (NLP)model can beconstructed to adjust the connection weights and thresholdsin the FBPNHowever theNP problem is not easy to solve Inthe proposed methodology only the threshold on the outputnode will be adjusted in an iterative way This way is muchsimpler and can also achieve good results

Substituting (8) into (7)

119900

1198952=

1

1 + 119890

minus119899119900

1198952

=

1

1 + 119890

minus(119868119900

1198952minus120579119900

2)

=

1

1 + 119890

120579119900

2minus119868119900

1198952

(10)

Therefore

ln( 1

119900

1198952

minus 1) = 120579

119900

2minus 119868

119900

1198952 (11)

So

119868

119900

1198952= 120579

119900

2minus ln( 1

119900

1198952

minus 1) (12)

Assume that the adjustment made to the threshold on theoutput node is denoted asΔ120579119900 = 120579

119900

3minus120579

119900

2 After adjustment the

output from the new FBPN 1199001198953 determines the upper bound

of the remaining cycle time

119900

1198953=

1

1 + 119890

minus119899119900

1198953

(13)

where

119899

119900

1198953= 119868

119900

1198953minus 120579

119900

3= 119868

119900

1198953minus (120579

119900

2+ Δ120579

119900

) (14)


119900

1198953=

1

1 + 119890

minus(119868119900

1198953minus120579119900

2minusΔ120579119900)

(15)

And substituting (12) into (15)

119900

1198953=

1

1 + 119890

minus(120579119900

2minusln(1119900

1198952minus1)minus120579

119900

2minusΔ120579119900)

=

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900=

1

1 + 119890

Δ120579119900

(1119900

1198952minus 1)

(16)

Obviously the maximum ofΔ120579119900 determines the lowest upperbound

Since 1199001198953is the upper bound of the remaining cycle time

119900

1198953ge 119873(RCT

119895119906)

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900ge 119873(RCT

119895119906) (17)

Δ120579

119900

le ln( 1

119873(RCT119895119906)

minus 1) minus ln( 1

119900

1198952

minus 1) (18)


Upper bound 1

Upper bound 2

Upper bound 3

Upper bound 5Upper bound 4

Upper bound 6

Final result

⊤

⊤⊤

⊤⊤

⊤

⊤

Actualcycle time

Figure 3 Iterative reduction of the upper bound

Equation (18) holds for all jobs so

Δ120579

119900

le min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1)) (19)

According to (19) the optimal value of Δ120579119900 should be set tothe maximum possible value

Δ120579

119900lowast

= min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(20)

The optimization results of the FBPN are dependenton the initial conditions and therefore are different everyiteration Assume that the optimal value of 119900

1198953in the 119905th

replication is denoted by 1199001198953(119905) then after some iterations

119900

1198953(all iterations) = min

119905

119900

1198953(119905) (21)

In this way the upper bound of the remaining cycle timeis decreased gradually (see Figure 3) Another merit of thisapproach is that it does not rely on the parameters of theFBPN

In a similar way the lower bound of each parameter (eg119908

ℎ

1199011198971 120579ℎ

1198973 119908119900

1198973 and 120579119900

3) can be determined so that each actual

value will be greater than the lower boundThe optimal valueof Δ120579119900 can be obtained as

Δ120579

119900lowast

= max119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(22)

Assume that the optimal value of 1199001198951in the 119905th replication

is denoted by 1199001198951(119905) then after some iterations

119900

1198951(all iterations) = max

119905

119900

1198951(119905) (23)

In this way the upper bound of the remaining cycle timeis increased gradually (Figure 4) Δ120579119900lowast does not rely on theparameters of the FBPN either

Actual cycle time

Final result

Lower bound 6Lower bound 1Lower bound 3Lower bound 2Lower bound 4Lower bound 5

perp

perp

perpperp

perpperp

perp

Figure 4 Iterative reduction of the lower bound

From Figure 5 it can be seen that if only the remainingcycle time is considered then the sequence should be 3 rarr

2 rarr 1 By contrast the sequence based on imprecise fuzzyremaining cycle time estimates is 3 rarr 1 rarr 2 This problemcan be solved by increasing the precision of the remainingcycle time estimate resulting in the correct sequence 3 rarr

2 rarr 1

312 Considering theUncertainty in JobClassification In paststudies the remaining cycle time of a job is usually deter-mined by the FBPN of the cluster with the highest member-ship However that makes fuzzy classification meaninglessTo tackle this problem various treatments have been takenin the literature [30] Recently Wu and Chen [31] proposedthe GAV approach to aggregate the estimation results fromvarious BPNs which is modified by incorporating in theconcept of FI in this study Consider

RCTE119895119906= (max (RCTE

1198951199061(119896))

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)sdot RCTE

1198951199062(119896)

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)

min (RCTE1198951199063

(119896)))

(24)

where RCTE119895119906(119896) is the remaining cycle time of job 119895

estimated by the FBPN of cluster 119896

32 The New Rule

321 Two Basic Fluctuation Smoothing Rules Lu et al [13]proposed two fluctuation smoothing rulesmdashthe fluctuationsmoothing policy for mean cycle time (FSMCT) and thefluctuation smoothing policy for variation of cycle time(FSVCT) FSMCT effectively diminishes the burst of arrivalsto all buffers simultaneously thereby reducing themean cycletime On the other hand FSVCT attempts to make every jobequally late or equally early thereby reducing the standard


Job no1 Job no2 Job no3

Actual remaining cycle time

Imprecise fuzzy estimate

Job no1

Job no2

Job no3

Job no1

Job no2 Job no3

Precise fuzzy estimate

Job no1

Job no2

Job no3

Job no1Job no2 Job no3

Defuzzified imprecise fuzzy estimate

Defuzzified precise fuzzy estimate

Figure 5 Precise remaining cycle time estimation eliminates misscheduling

deviation of lateness In fluctuation smoothing rules each jobis assigned a slack value and the processing order of the jobis dependent on the slack value

(FSMCT)

SK119895119906(FSMCT) =

119895

120582

minus RCTE119895119906

(25)

(FSVCT)

SK119895119906(FSVCT) = 119877

119895minus RCTE

119895119906 (26)

Jobs with the smallest slack values will be given higherpriorities

322 Step 2 The Fuzzy Fluctuation-Smoothing Rule Sub-sequently if the remaining cycle time is estimated with atriangular fuzzy number then we have two fuzzy fluctuationsmoothing rules as

(fuzzy FSMCT FFSMCT)

SK119895119906(FSMCT) =

119895

120582

minus

RCTE119895119906

(27)

(fuzzy FSVCT FFSVCT)

SK119895119906(FSVCT) = 119877

119895minus

RCTE119895119906 (28)

To determine the sequence of jobs the fuzzy slacks needto be compared To this end various methods have beenproposed in the literature such as the method based on theprobability measure [32] the coefficient of variance (CV)index [8] the method considering the area between thecentroid and original points [33] and the method basedon the fuzzy mean and standard deviation [34] For acomparison of these methods refer to Zhu and Xu [34] Inthis study the method based on the fuzzy mean and standarddeviation is applied because it is relatively simple and canyield reasonable comparison results To put this in contextthe following theorems are introduced

Theorem 1 The fuzzy mean of a triangular fuzzy number 119860 =(119909

0minus 119886 119909

0 119909

0+ 119887) is

120583119860= 119909

0+

119887 minus 119886

3

(29)

while the fuzzy standard deviation of 119860 is

120590119860=

radic

119886

2

+ 119886119887 + 119887

2

18

(30)

Proof (see Zhu and Xu [34]) It is in fact the center-of-gra-vity (COG) method

The following definition details the comparison methodbased on the fuzzy mean and standard deviation

Definition 2 For any two fuzzy numbers 119860 and

119861 isin 119865(119877) thesequence of 119860 and

119861 can be determined according to theirfuzzy means and standard deviations as follows

(1) 120583119860gt 120583

119861if and only if 119860 ≻

119861

(2) 120583119860lt 120583

119861if and only if 119860 ≺

119861

(3) if 120583119860= 120583

119861 then

(i) 120590119860gt 120590

119861if and only if 119860 ≺

119861(ii) 120590

119860lt 120590

119861if and only if 119860 ≻

119861(iii) 120590

119860= 120590

119861if and only if 119860 =

119861

323 The Slack-Diversifying Rule The idea behind the fluc-tuation smoothing rules is to disperse the arrivals of jobs toa machine and the tool to control this is the slack value Forthis reason to diversify the slack values of jobs seems to be a


possible way to disperse their arrivals To this endWang et al[15] maximized the standard deviation of the slack value

120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)

However to achieve this the slack formula should contain atleast one parameter that is adjustable and differentiable Inaddition Wu and Chen [31] showed that such a treatmentmay lead to the situation that most slacks concentrate on thetwo extremes

324 Step 3 The Four-Factor Fuzzy Fluctuation-SmoothingRule Chen [14] combined four traditional dispatching rules-EDD critical ratio (CR) FSMCT and FSVCT and proposedthe four-objective dispatching rule In the four-objectivedispatching rule the slack of job 119895 at processing step 119906 isdefined as

SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (

RPT119895119906minusmin

119895RPT

119895119906

max119895RPT

119895119906minusmin

119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE119895119906minusmin

119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (

SCT119895119906minusmin

119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)

where 120572 120573 120574 and 120578 and are positive real numbers that satisfythe following constraints

If 120572 = 1 then 120573 120574 120599 = 0 120578 = minus1 and vice versa

If 120573 = 1 then 120572 = 0 120574 120578 120599 = minus1 and vice versa

If 120578 = 1 then 120572 120573 = 0 120574 120599 = 1 and vice versa(33)

Jobs with the smallest slack values will be given higherpriorities There are many possible models that can form thecombinations of 120572 120573 120574 120578 and 120599 For example

(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)

(Logarithmic model 1)

120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)

The values of 120572 and 120573 are within [0 1]

Theorem 3 The four-objective nonlinear fluctuation smooth-ing rule is more responsive than the four original rules if119877119862119879119864

119895119906is large which is a common phenomenon in a wafer

fabrication factory

Proof See the Appendix

If the remaining cycle time is estimated with a triangularfuzzy number then (34) becomes

SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)

which is equivalent to

SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)


Table 5 An example (120582 = 118)

119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)

1 102 159 881 560 1399 minus1264 minus12972 756 37 227 451 1127 minus1096 minus3713 826 37 157 489 1223 minus1192 minus3974 652 86 331 729 1822 minus1749 minus11705 208 55 775 212 530 minus483 minus3226 783 84 200 816 2040 minus1969 minus12577 800 96 183 946 2366 minus2285 minus15668 478 52 505 377 942 minus898 minus4649 469 65 514 446 1116 minus1061 minus64710 699 32 284 398 995 minus968 minus29611 836 85 147 860 2151 minus2079 minus131512 497 45 486 353 883 minus845 minus38613 596 101 387 819 2047 minus1961 minus145114 798 34 185 458 1146 minus1117 minus34815 197 79 786 297 743 minus676 minus54616 804 85 179 837 2092 minus2020 minus128817 163 78 820 259 647 minus581 minus48418 457 44 526 324 810 minus773 minus35319 523 100 460 740 1851 minus1766 minus1328

SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)

Job 119895 is processed before job 119896 if SK119895119906lt

SK119896119906

325 Step 4 The Four-Factor Fuzzy Slack-DiversifyingFluctuation-Smoothing Rule Wang et al [15] diversified theslack by maximizing the standard deviation of the slackHowever such a practice causes slacks to concentrate onthe two extremes rather than being evenly dispersed Tosolve this problem the following procedure is established todiversify the slack instead

(1) Set 120595max to 0

(2) Vary the values of the five parameters

(3) Sequence the jobs by Definition 2 in ascending order

(4) Calculate the distance of every two adjacent SK(119895)119906

rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061

minus SK(119895minus1)1199063

(41)

(5) Evaluate whether there is no overlap by

120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899

119895=1120595(119895 minus 1 119895) gt 120595max update 120595max to sum

119899

119895=1120595(119895 minus

1 119895)(7) If 120595max ge a threshold stop otherwise return to step

(2)

This procedure is a polynomial-time algorithm By repeatedapplications of this procedure one obtains an optimal sched-ule with the fewest overlaps in 119874(1198992) time

326 Step 5 Applying the New Rule to Sequence Jobs Anexample is given in Table 5The sequencing results by the twotraditional fluctuation smoothing rules are

FSMCT 7 rarr 11 rarr 16 rarr 6 rarr 13 rarr 19 rarr

4 rarr 1 rarr 3 rarr 14 rarr 2 rarr 9 rarr 10 rarr 8 rarr

12 rarr 18 rarr 15 rarr 17 rarr 5FSVCT 7 rarr 13 rarr 19 rarr 11 rarr 1 rarr 16 rarr 6 rarr

4 rarr 9 rarr 15 rarr 17 rarr 8 rarr 3 rarr 12 rarr 2 rarr 18rarr 14 rarr 5 rarr 10

Subsequently if the remaining cycle time is estimatedwith a fuzzy value instead (see Table 6) then the sequencingresults by the two fuzzy fluctuation smoothing rules are

FFSMCT 7 rarr 11 rarr 16 rarr 6 rarr 13 rarr 4 rarr


12 rarr 18 rarr 15 rarr 17 rarr 5


Table 6 The example with fuzzy remaining cycle times (120582 = 118)

119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906

SK119895119906(FFSMCT) SK

119895119906(FFSVCT)

1 102 159 881 560 (1200 1399 1458) (minus1324 minus1265 minus1066) (minus1357 minus1297 minus1099)2 756 37 227 451 (976 1127 1176) (minus1145 minus1096 minus945) (minus421 minus371 minus221)3 826 37 157 489 (1086 1223 1299) (minus1269 minus1192 minus1055) (minus474 minus397 minus261)4 652 86 331 729 (1618 1822 1976) (minus1904 minus1750 minus1546) (minus1325 minus1170 minus967)5 208 55 775 212 (455 530 557) (minus511 minus484 minus410) (minus350 minus322 minus248)6 783 84 200 816 (1742 2040 2158) (minus2088 minus1969 minus1671) (minus1376 minus1257 minus960)7 800 96 183 946 (2039 2366 2549) (minus2468 minus2285 minus1959) (minus1750 minus1566 minus1240)8 478 52 505 377 (848 942 992) (minus949 minus898 minus805) (minus515 minus464 minus371)9 469 65 514 446 (992 1116 1176) (minus1122 minus1061 minus938) (minus708 minus647 minus524)10 699 32 284 398 (853 995 1031) (minus1005 minus968 minus827) (minus333 minus296 minus155)11 836 85 147 860 (1830 2151 2311) (minus2240 minus2079 minus1759) (minus1476 minus1315 minus995)12 497 45 486 353 (794 883 918) (minus881 minus845 minus757) (minus422 minus386 minus298)13 596 101 387 819 (1700 2047 2170) (minus2086 minus1962 minus1615) (minus1575 minus1451 minus1105)14 798 34 185 458 (975 1146 1256) (minus1228 minus1118 minus948) (minus459 minus348 minus178)15 197 79 786 297 (659 743 800) (minus734 minus677 minus593) (minus604 minus546 minus463)16 804 85 179 837 (1819 2092 2318) (minus2247 minus2020 minus1748) (minus1515 minus1288 minus1016)17 163 78 820 259 (560 647 708) (minus643 minus581 minus495) (minus546 minus484 minus398)18 457 44 526 324 (685 810 839) (minus803 minus773 minus649) (minus383 minus353 minus229)19 523 100 460 740 (1547 1851 2042) (minus1958 minus1767 minus1463) (minus1520 minus1328 minus1025)

FFSVCT 7 rarr 13 rarr 19 rarr 16 rarr 11 rarr 1 rarr



Obviously after considering the uncertainty in theremaining cycle time the sequencing results are different

The four-factor fuzzy slack-diversifying fluctuation-smoothing rule and Wang et alrsquos method are also appliedto this example After 50 iterations the optimal values ofthe parameters are (120572 120573 120574 120578 120599) = (04 008 043 043 003)with 120595max = 16 That means that 16 out of 19 jobs are notoverlapping with their neighbors The slacks of the jobs areshown in Figure 6 For a comparison the slacks obtained byusing Wang et alrsquos method are shown in Figure 7 in which120590SK119895119906

= 268 Obviously Consider the following

(1) The slacks obtained by using the proposed method-ology are evenly distributed and there are very littleslack overlapping and very few ties

(2) Conversely the slacks obtained using Wang et alrsquosmethod concentrate on one or two extremes andthere are still some overlaps and ties that causedifficulties in sequencing jobs and may lead to miss-cheduling

(3) The cumulative fuzziness during the reasoning pro-cess of a fuzzy dispatching rule raises the possibilityof forming ties In this regard the proposed method-ology surpasses Wang et alrsquos method in reducing thenumber of ties The advantage is 83

(4) In addition the proposed methodology also achievesa very good performance in reducing the averageoverlapping When compared with Wang et alrsquosmethod the advantage is as high as 98 hours

00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)

Figure 6 The fuzzy slacks of the 19 jobs after optimization

00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)

Figure 7 The fuzzy slacks obtained by using Wang et alrsquos method

4 Simulation Study

Simulation is widely used to assess the effectiveness of ascheduling policy especially when the proposed policy andthe current practice are very different [35] A real waferfabrication factory located in Taichung Scientific Park ofTaiwan with a monthly capacity of about 25000 wafers was


Table 7 The performances of various approaches in the average cycle time

Average cycle time (hrs) A (normal) A (hot) A (super hot) B (normal) B (hot)FIFO 1254 400 317 1278 426EDD 1094 345 305 1433 438SRPT 948 350 308 1737 457CR 1148 355 300 1497 440FSMCT 1313 347 293 1851 470FSVCT 1014 382 315 1672 4754o-SDR 1183 347 271 1160 339The proposed methodology 921 267 253 811 253

simulatedWe used simulation to avoid disturbing the regularoperations of the wafer fabrication factory The goal wasto evaluate the effectiveness of the fuzzy slack-diversifyingfluctuation-smoothing rule for multiobjective job schedulingin the wafer fabrication factory The simulation program hasbeen validated by comparing the actual cycle times with thesimulated values and verified by analyzing the trace reports

Thewafer fabrication factory producesmore than 10 typesof memory products and has more than 500 workstations forperforming single-wafer or batch operations using 58 nmsim

110 nm technologies Jobs released into the wafer fabricationfactory are assigned three types of priorities that is ldquonormalrdquoldquohotrdquo and ldquosuper hotrdquo Jobs with the highest priorities will beprocessed firstThe large scale and the reentrant process flowsof the wafer fabrication factory exacerbate the difficulties ofjob dispatching Currently the longest average cycle timeexceeds three months with a variation of more than 300hours The wafer fabrication factory is therefore seekingbetter dispatching rules to replace FIFO and EDD in order toshorten the average cycle times and ensure on-time deliveryto its customers One hundred replications of the simulationwere successively run The time required for each simulationreplication was about 30 minutes on a PC with Intel DualE2200 22GHz CPUs and 199G RAM A horizon of twenty-four months was simulated

To assess the effectiveness of the proposed methodologyand to make comparison with some existing approachesseven methods were tested FIFO EDD shortest remainingprocessing time (SRPT) CR FSVCT FSMCT and the four-objective slack-diversifying rule (4o-SDR) [14] were appliedto schedule the simulated wafer fabrication factory Wecollected the data of 1000 jobs and then we separated thecollected data by product types and priorities

Jobs with the highest priorities are processed first WithFIFO jobs were sequenced on each machine first by theirpriorities and then by their arrival times at themachineWithEDD jobs were sequenced first by their priorities and then bytheir due dates With SRPT the remaining processing timeof each job was calculated Then jobs were sequenced firstby their priorities and then by their remaining processingtimes With CR jobs were sequenced first by their prioritiesthen by their critical ratios FSMCT and FSVCT used twostages First jobs were scheduled using FIFO in which theremaining cycle times of all jobs were recorded and averagedat each step Then FSMCTFSVCT was applied to schedule

the jobs based on the average remaining cycle times obtainedearlier In other words jobs were sequenced on eachmachinefirst by their priorities then by their slack values With 4o-SDR the remaining cycle time of a job was estimated usingthe fuzzy c-means and back propagation network (FCM-BPN) approach [28] it was a crisp value The five adjustableparameters were set to (120572 120573 120574 120578 120599) = (06 02 0 minus06 0)

after initial scenarios had been examined In the proposedmethodology the remaining cycle time of a job was estimatedusing the SA-FBPN approach The effectiveness of the SA-FBPN approach can be seen from Figure 8 The SA-FBPNapproach can generate a very precise interval of the remainingcycle time for each job thereby reducing the risk of miss-cheduling In this case the performance of the effective FBPNapproach [12] was close to that of the proposedmethodologyNevertheless in theory the proposed SA-FBPN approachwill outperform the effective FBPN approach because of itsiterative nature By contrast the FCM-BPN approach doesnot guarantee that the actual value falls within a distance ofthree times the root mean squared error (RMSE) from theestimate

The average cycle time cycle time standard deviation thenumber of tardy jobs and the maximum lateness of all caseswere calculated to assess the scheduling performance Theresults were summarized in Tables 7 8 9 and 10

According to the experimental results the followingpoints can be made

(1) For the average cycle time the fuzzy slack-dive-rsifying fluctuation-smoothing rule outperformedthe existing approaches In this respect FIFO is oftena basis for comparison The most obvious advantageof the fuzzy slack-diversifying fluctuation-smoothingrule over FIFO was about 31

(2) The fuzzy slack-diversifying fluctuation-smoothingrule also achieved a very good performance in reduc-ing the maximum lateness When compared withEDD the advantage was as high as 36 on average Inthis regard the fuzzy slack-diversifying fluctuation-smoothing rule also outperformed 4o-SDR in mostcases which was reasonable due to the uncertainty inthe remaining cycle time

(3) In addition the fuzzy slack-diversifying fluctuation-smoothing rule surpassed the FSVCTpolicy in reduc-ing cycle time standard deviation The most obvious


600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)

Center valueActual value

Upper boundLower bound

Job (FCM-BPN)

(c)

Figure 8 The performances of three approaches in estimating the remaining cycle time

Table 8 The performances of various approaches in the maximum lateness

The maximum lateness (hrs) A (normal) A (hot) A (super hot) B (normal) B (hot)FIFO 401 minus122 164 221 172EDD 295 minus181 144 336 185SRPT 584 minus142 174 718 194CR 302 minus159 138 423 192FSMCT 875 minus165 125 856 171FSVCT 706 minus112 174 686 2604o-SDR 360 minus152 118 21 94The proposed methodology 292 minus143 113 23 102


Table 9 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 51EDD 129 25 22 50 63SRPT 248 31 22 106 53CR 69 29 18 58 53FSMCT 419 33 16 129 104FSVCT 280 37 27 201 774o-SDR 71 41 22 30 29The proposed methodology 66 18 25 28 33

Table 10 The performances of various approaches in the number of tardy jobs

Number of tardy jobs A (normal) A (hot) A (super hot) B (normal) B (hot)FIFO 79 0 12 16 5EDD 71 0 12 19 5SRPT 37 0 12 19 5CR 79 0 12 19 5FSMCT 58 0 12 19 5FSVCT 56 0 12 18 54o-SDR 79 0 12 19 5The proposed methodology 37 0 12 19 5

advantage was 86when product type B with normalpriority was scheduled The fuzzy slack-diversifyingfluctuation-smoothing rule also surpassed 4o-SDRpolicy in three out of five cases with an averageadvantage of 8

(4) In reducing the number of tardy jobs the proposedmethodology outperformed the existing methods inmost cases and achieved the best performance whenproduct type A with hot priority was scheduled

(5) As expected SRPT performed well in reducing theaverage cycle times especially for product types withshort cycle times (eg product A) but gave anexceedingly bad performance with respect to cycletime standard deviation If the cycle time is longthe remaining cycle time will be much longer thanthe remaining processing time which renders SRPTineffective SRPT is similar to FSMCT Both try tomake all jobs equally early or late

(6) The performance of EDD was also satisfactory forproduct types with short cycle times If the cycle timeis long it is more likely to deviate from the prescribedinternal due date which makes EDD ineffective Thisbecomesmore serious if the percentage of the producttype is high in the product mix (eg product type A)CR has similar problems

(7) The FCM-BPN approachwas also applied to the fuzzyslack-diversifying fluctuation-smoothing rule Takingproduct type A (normal priority) as an example

the results are shown in Table 11 We noticed thatwith poorer remaining cycle time estimation theperformances of fuzzy slack-diversifying fluctuation-smoothing rule were indeed worsened Howeverincorporating the SA-FBPN approach with the fuzzyslack-diversifying fluctuation-smoothing rule couldimprove the scheduling performance significantly

(8) Wilcoxon signed-rank test [36] a commonly usednonparametric statistical hypothesis test was used inthis study for comparisons of two related samples orrepeated measurements on a single sample to assesswhether their population means differed or not Theresults were summarized in Table 12Thenull hypoth-esis119867

119886was rejected at 120572 = 0025 which showed that

the fuzzy slack-diversifying fluctuation-smoothingrule was superior to seven existing approaches inreducing the average cycle time With regard to themaximum lateness the advantage of the fuzzy slack-diversifying fluctuation-smoothing rule over FIFOSRPT and FSVCT was significant Similar resultscould be observed with cycle time standard devi-ation However the advantage of the fuzzy slack-diversifying fluctuation-smoothing rule was not sta-tistically significant for the number of tardy jobs

5 Conclusions and Directions forFuture Research

Multiobjective scheduling in a wafer fabrication factoryis a challenging but important task For such a complex


Table 11 The results of applying the FCM-BPN approach to the fuzzy slack-diversifying fluctuation-smoothing rule

Approach Average cycle time Maximum lateness Cycle time standard deviation Number of tardy jobsFCM-BPN + the proposed rule 958 307 70 37SA-FBPN + the proposed rule 921 292 66 37

Table 12 Results of the Wilcoxon sign-rank test

119867

1198860(the average cycle time) 119867

1198870(the maximum lateness) 119867

1198870(cycle time standard deviation) 119867

1198870(the number of tardy jobs)

FIFO 202lowastlowast 202lowastlowast 121 054EDD 202lowastlowast 121 175lowast 121SRPT 202lowastlowast 202lowastlowast 175lowast 067CR 202lowastlowast 148 148 121FSMCT 202lowastlowast 148 175lowast 121FSVCT 202lowastlowast 202lowastlowast 202lowastlowast 0544o-SDR 202lowastlowast minus013 067 121lowastP lt 005lowastlowastP lt 0025

production system to optimize a single objective is toughenough and to optimize four objectives at the same time isa remarkable challenge Further the uncertainty in variousproduction conditions often leads to incorrect schedulingdecisions To deal with these difficulties the mainstreamof research is still the development of dispatching rulesthrough generalization or fusion For this reason this studyhas proposed an effective fuzzy dispatching rule First toconsider the uncertainty in the fabrication process the SA-FBPN approach has been proposed to estimate the remainingcycle time of a job Compared to the existing methods theSA-FBPN approach can generate a very precise estimate ofthe remaining cycle time in an iterative mannerThe FI-GAVapproach has also been proposed to aggregate the estimationresults from various FBPNs Subsequently the fuzzy slack-diversifying fluctuation-smoothing rule has been proposedThe fuzzy remaining cycle time estimate is input to thefuzzy slack-diversifying fluctuation-smoothing rule to derivethe job slack There are five parameters in the fuzzy slack-diversifying fluctuation-smoothing rule that can be adjustedto optimize the rule To this end a systematic procedure hasbeen proposed

After a simulation study we concluded the followingpoints

(1) By considering the uncertainty in the remaining cycletime four aspects of the scheduling performancemdashthe average cycle time the maximum lateness cycletime standard deviation and the number of tardyjobsmdashcan indeed be simultaneously improved How-ever if the estimation accuracy is insufficient thescheduling process may be misled

(2) The cumulative fuzziness during a fuzzy inferenceprocess must be properly dealt with

(3) By tackling slack overlapping in a nonsubjectiveway the problem of misscheduling can be effectively

avoided which augments the performance of thefuzzy slack-diversifying fluctuation-smoothing rule

However any further assessment of the proposedmethodology requires its application to an actual waferfabrication factory In addition different objectives can befused and fuzzified in the same way Further the SA-FBPNapproach can be applied to other real production lines infuture studies

Appendix

Proof of Theorem 3 First let us compare the four-objectivenonlinear fluctuation smoothing rule and FSMCT For a faircomparison the parameters 120573 120574 and 120599 are set to 0 becausethe corresponding variables are not considered in FSMCT

When 119895120582 increases by 1 in FSMCT SK119895119906is changed by

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)

If RCTE119895119906is greater than 119895120582 then (A1) becomes

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)

Conversely in the four-objective nonlinear fluctuationsmoothing rule SK

119895119906will be changed by


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)

because 119895 ge 1 In addition since 0 le 120572 le 1

1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)

Therefore (A3) becomes

(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)

If RCTE119895119906is greater than ((119895 minus 1)

120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895

120572minus1RCTE119895119906ge (119895 minus 1)

120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895

sdot RCTE119895119906ge (119895 minus 1)

120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)

which finishes the proof The comparison between the four-objective nonlinear fluctuation smoothing rule and the otherrules can be done in similar ways

Acknowledgment

This work was supported by the National Science Council ofTaiwan

References

[1] R M Dabbas and J W Fowler ldquoA new scheduling approachusing combined dispatching criteria in wafer fabsrdquo IEEE Trans-actions on Semiconductor Manufacturing vol 16 no 3 pp 501ndash510 2003

[2] T C Chiang A C Huang and L C Fu ldquoModeling schedulingand performance evaluation for wafer fabrication a queueingcolored petri-net and GA-based approachrdquo IEEE Transactionson Automation Science and Engineering vol 3 no 3 pp 330ndash337 2006

[3] C Grimme and J Lepping ldquoCombining basic heuristics forsolving multi-objective scheduling problemsrdquo in Proceedingsof the IEEE Symposium Series on Computational Intelligence(CISched rsquo11) pp 9ndash16 April 2011

[4] H Xiong M Zhou and C N Manikopoulos ldquoSchedulingflexible manufacturing systems based on timed petri nets and


fuzzy dispatching rulesrdquo in Proceedings of the INRIAIEEESymposium on Emerging Technologies and Factory Automationvol 3 pp 309ndash315 October 1995

[5] A X Benincasa O Morandin Jr and E R R Kato ldquoReactivefuzzy dispatching rule for automated guided vehiclesrdquo in Pro-ceedings of the IEEE International Conference on Systems Manand Cybernetics vol 5 pp 4375ndash4380 October 2003

[6] E H Mamdani ldquoApplication of fuzzy logic to approximatereasoning using linguist synthesisrdquo IEEE Transactions on Com-puters C vol 26 no 12 pp 1182ndash1191 1977

[7] T Takagi and M Sugeno ldquoFuzzy identification of systems andits applications to modeling and controlrdquo IEEE Transactions onSystems Man and Cybernetics vol 15 no 1 pp 116ndash132 1985

[8] C H Cheng ldquoA new approach for ranking fuzzy numbers bydistancemethodrdquo Fuzzy Sets and Systems vol 95 no 3 pp 307ndash317 1998

[9] K K LeeWC Yoon andDH Baek ldquoGenerating interpretablefuzzy rules for adaptive job dispatchingrdquo International Journalof Production Research vol 39 no 5 pp 1011ndash1030 2001

[10] K K Tan and K Z Tang ldquoVehicle dispatching system basedon Taguchi-tuned fuzzy rulesrdquo European Journal of OperationalResearch vol 128 no 3 pp 545ndash557 2001

[11] M Dong and M Liu ldquoAn ANFIS-based dispatching rule forcomplex fuzzy job shop scheduling problemrdquo in Proceedingsof the International Conference on Information Science andTechnology (ICIST rsquo11) pp 263ndash266 March 2011

[12] H-R Tsai and T Chen ldquoA fuzzy nonlinear programmingapproach for optimizing the performance of a four-objectivefluctuation smoothing rule in a wafer fabrication factoryrdquo Jou-rnal of Applied Mathematics vol 2013 Article ID 720607 15pages 2013

[13] S C H Lu D Ramaswamy and P R Kumar ldquoEfficientscheduling policies to reduce mean and variance of cycle-timein semiconductor manufacturing plantsrdquo IEEE Transactions onSemiconductor Manufacturing vol 7 no 3 pp 374ndash388 1994

[14] T Chen ldquoThe optimized-rule-fusion and certain-rule-first app-roach for multi-objective job scheduling in a wafer fabricationfactoryrdquo International Journal of Innovative Computing Infor-mation and Control vol 9 no 6 pp 2283ndash2302 2013

[15] Y-C Wang T Chen and C-W Lin ldquoA slack-diversifyingnonlinear fluctuation smoothing rule for job dispatching ina wafer fabrication factoryrdquo Robotics amp Computer IntegratedManufacturing vol 29 no 3 pp 41ndash47 2013

[16] K Pal and S K Pal ldquoSoft computingmethods used for themod-elling and optimisation of Gas Metal Arc Welding a reviewrdquoInternational Journal of Manufacturing Research vol 6 no 1pp 15ndash29 2011

[17] T Chen Y-C Wang and H-R Tsai ldquoLot cycle time predictionin a ramping-up semiconductor manufacturing factory witha SOM-FBPN-ensemble approach with multiple buckets andpartial normalizationrdquo International Journal of Advanced Man-ufacturing Technology vol 42 no 11-12 pp 1206ndash1216 2009

[18] T Chen and Y-C Wang ldquoLong-term load forecasting by thecollaborative fuzzy-neural approachrdquo International Journal ofElectrical Power and Energy Systems vol 43 no 1 pp 454ndash4642012

[19] T Chen and R Romanowski ldquoPrecise and accurate job cycletime forecasting in a wafer fabrication factory with a fuzzy datamining approachrdquo Mathematical Problems in Engineering vol2013 Article ID 496826 14 pages 2013

[20] X L Xie and G Beni ldquoA validity measure for fuzzy clusteringrdquoIEEE Transactions on Pattern Analysis andMachine Intelligencevol 13 no 8 pp 841ndash847 1991

[21] T Chen and Y C Wang ldquoEnhancing scheduling performancefor awafer fabrication factory the biobjective slack-diversifyingnonlinear fluctuation-smoothing rulerdquo Computational Intelli-gence and Neuroscience vol 2012 Article ID 404806 12 pages2012

[22] T Chen ldquoA fuzzy rule for job dispatching in a wafer fabrica-tion factory-a simulation studyrdquo The International Journal ofAdvancedManufacturing Technology vol 67 no 1ndash4 pp 47ndash582013

[23] T Chen and R Romanowski ldquoA novel fuzzy-neural slack-dive-rsifying rule based on soft computing applications for job dis-patching in a wafer fabrication factoryrdquoMathematical Problemsin Engineering vol 2013 Article ID 980984 15 pages 2013

[24] S Taghadomi-Saberi M Omid Z Emam-Djomeh and HAhmadi ldquoEvaluating the potential of artificial neural networkand neuro-fuzzy techniques for estimating antioxidant activityand anthocyanin content of sweet cherry during ripening byusing image processingrdquo Journal of the Science of Food andAgriculture 2013

[25] D T Bui B Pradhan O Lofman I Revhaug and O B DickldquoLandslide susceptibility mapping at Hoa Binh province (Vie-tnam) using an adaptive neuro-fuzzy inference systemandGISrdquoComputers and Geosciences vol 45 pp 199ndash211 2012

[26] W Ocampo-Duque R Juraske V Kumar M Nadal J LDomingo and M Schuhmacher ldquoA concurrent neuro-fuzzyinference system for screening the ecological risk in riversrdquoEnvironmental Science and Pollution Research vol 19 no 4 pp983ndash999 2012

[27] J Nocedal and S J Wright Numerical Optimization SpringerBerlin Germany 2006

[28] T Chen and Y C Wang ldquoIncorporating the FCM-BPN appro-ach with nonlinear programming for internal due date assign-ment in a wafer fabrication plantrdquo Robotics and Computer-Integrated Manufacturing vol 26 no 1 pp 83ndash91 2010

[29] T Chen and Y-C Lin ldquoA collaborative fuzzy-neural approachfor internal due date assignment in a wafer fabrication plantrdquoInternational Journal of Innovative Computing Information andControl vol 7 no 9 pp 5193ndash5210 2011

[30] T Chen ldquoIncorporating fuzzy c-means and a back-propagationnetwork ensemble to job completion time prediction in asemiconductor fabrication factoryrdquo Fuzzy Sets and Systems vol158 no 19 pp 2153ndash2168 2007

[31] H-CWu andT Chen ldquoA fuzzy-neural ensemble and geometricrule fusion approach for scheduling a wafer fabrication factoryrdquoMathematical Problems in Engineering vol 2013 Article ID956978 14 pages 2013

[32] E S Lee and R-J Li ldquoComparison of fuzzy numbers basedon the probability measure of fuzzy eventsrdquo Computers andMathematics with Applications vol 15 no 10 pp 887ndash896 1988

[33] T Chu and C Tsao ldquoRanking fuzzy numbers with an areabetween the centroid point and original pointrdquo Computers andMathematics with Applications vol 43 no 1-2 pp 111ndash117 2002

[34] L Zhu andRXu ldquoRanking fuzzy numbers based on fuzzymeanand standard deviationrdquo in Proceedings of the 8th InternationalConference on Fuzzy Systems and Knowledge Discovery (FSKDrsquo11) pp 854ndash857 July 2011

[35] T Chen ldquoA flexible way of modelling the long-term cost com-petitiveness of a semiconductor productrdquo Robotics amp ComputerIntegrated Manufacturing vol 29 no 3 pp 31ndash40 2013


[36] F Wilcoxon ldquoIndividual comparisons by ranking methodsrdquoBiometrics Bulletin vol 1 no 6 pp 80ndash83 1945

[37] T Chen and Y-C Wang ldquoA post-classifying fuzzy-neural anddata-fusion rule for job scheduling in a wafer fab-a simulationstudyrdquo International Journal of Manufacturing Research vol 8no 2 pp 150ndash170 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Table 1 Some existing fuzzy dispatching rules

References System type Rulesinputsoutputs Number of objectivesXiong et al [4] TSK 221 2 (average cycle time lateness)Benincasa et al [5] Mamdani 2731 2 (average cycle time WIP level)Lee et al [9] Single-rule-based 241 2 (average cycle time cycle time standard deviation)Tan and Tang [10] Mamdani 841 3 (throughput average cycle time no of vehicles)Dong and Liu [11] ANFIS 1641 1 (average cycle time)

Tsai and Chen [12] Fusion 451 4 (average cycle time cycle time standard deviationmaximum lateness number of tardy jobs)

lowastWIP work-in progress

dispatching rules must be aggregated To this end there areat least three types of fuzzy inference systemsmdashMamdanirsquosfuzzy inference systems [6] Takagi-Sugeno-Kangrsquos (TSKrsquos)[7] fuzzy inference systems and adaptive neurofuzzy infer-ence systems (ANFISs) [8] These systems use differentaggregation and defuzzification methods Xiong et al [4]scheduled a flexible manufacturing system (FMS) using twofuzzy dispatching rules of the TSK type Lee et al [9]established fuzzy inference rules to select a combination ofsome existing dispatching rules for scheduling a FMS Theeffectiveness of a fuzzy inference system depends critically onthe way in which the related variables are partitioned Forthis reason Tan and Tang [10] applied Taguchirsquos design ofexperiment (DOE) techniques to improve the design of somefuzzy dispatching rules for a test facility According to Benin-casa et al [5] up to 27 rules (each with three inputs and oneoutput) were established for scheduling automated guidedvehicles According to Dong and Liu [11] the uncertainty inthe processing time was modeled with a fuzzy number andan ANFIS was established to schedule a job shop Inputs tothe ANFIS were the differences between any two jobs whilethe output from the ANFIS determined the sequence of thetwo jobs If the output was greater than zero then the first jobshould be processed before the second job Tsai andChen [12]fuzzified four traditional dispatching rules and added fiveadjustable parameters to merge the four rules A summaryof some existing fuzzy dispatching rules is shown in Table 1

The existing approaches have the following problems(1) Establishing fuzzy inference rules is a subjective

process that is not easy to optimize(2) The fuzziness of parameters causes the uncertainty in

scheduling and therefore must be controlled(3) The parameters that have the greatest relevance for

the scheduling performance must be estimated moreaccurately (such as the remaining cycle time) in orderto enhance the effectiveness of scheduling

(4) How to solve ties that is jobs with the same slackswhen the parameters are fuzzy valued has not beenfully discussed

To tackle these problems and to enhance the performanceof multiobjective job scheduling in a wafer fabrication fac-tory a fuzzy slack-diversifying fluctuation-smoothing rule isproposed in this study The unique features of the proposedmethodology include the following

(1) Considering the uncertainty in the remaining cycletime Dispatching rules that consider the remainingcycle time are more able to respond to the changesin the production environment [13] To consider theuncertainty in the remaining cycle time it is estimatedwith a fuzzy number and fed into a fuzzy dispatchingrule To this end a self-adjusted fuzzy back propaga-tion network (SA-FBPN) approach is proposed Com-pared with the existing methods this SA-FBPNapproach can generate a very precise estimate of theremaining cycle time in an efficient manner

(2) Establishing a fuzzy dispatching rule directly from theexisting rules A fuzzy slack-diversifying fluctuation-smoothing rule is proposed by fuzzifying the four-objective fluctuation-smoothing dispatching rule [14]and diversifying the job slack This rule accepts thefuzzy remaining cycle time as an input

(3) Diversifying the slackwith a new approach accordingto Wang et al [15] the slack was defuzzified by maxi-mizing the standard deviation However such a treat-ment has several drawbacks This study establishes asystematic procedure to overcome those drawbacksso that the slacks of jobs can be evenly distributedand the possibility of forming ties may be reduced asmuch as possible

(4) Solving the problem of slack overlapping without theneed to defuzzify the slacks subjectively

(5) Proposing the fuzzy intersection (FI) and generalizedaverage (GAV) approach to aggregate the estimationresults from various FBPNs

Some predictive scheduling methods have been pro-posed in recent years The differences between the pro-posed methodology and these methods were summarized inTable 2 The distinct advantages of the proposed methodol-ogy over these predictive scheduling methods include thefollowing

(1) The upper and lower bounds of the remaining cycletime used in the proposed methodology are tighter

(2) The proposed methodology provides an easier wayto optimize the scheduling rule which improves theusability of the proposed methodology

To assess the effectiveness of the proposed methodologyproduction simulation is also applied in this studyThe rest of




method



Number ofobjectives


Optimization method

















119898 119862max is






CT119895



RCT119895119906



RPT119895119906



SK119895119906



ℎ


119908

119900


119908

ℎ

119901119897


120579

ℎ


120579





119898|119903

119895 119889

119895 rent|119862 120590

119862 119871max 119873119879


Figure 1

3 Methodology







approach









119895= [119909

119895119901] 119901 = 1 sim 119875


Min119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896) (1)



119895(119896)


(1) Set 119870 = 1



119909

(119896)= 119909

(119896)119901 119901 = 1 sim 119875

119909

(119896)119901=

sum

119899

119895=1120583

119898

119895(119896)119909

119895119901

sum

119899

119895=1120583

119898

119895(119896)

120583

119895(119896)=

1

sum

119870

119902=1(119890

119895(119896)119890

119895(119902))

2(119898minus1)

119890

119895(119896)=

radic

sum

all119901(119909

119895119901minus 119909

(119896)119901)

2

(2)

where 119909(119896)





max119896

max119895

1003816

1003816

1003816

1003816

1003816

120583

(119905)

119895(119896)minus 120583

(119905minus1)

119895(119896)

1003816

1003816

1003816

1003816

1003816

lt 119889 (3)


1

2

Input layer

1

2

Hidden layer

1

Output layer

RMSE

xj1xj2

Ih1 nh1 120579h1 h1

wo1

wo2

oj

Io o

IhL nhL 120579hL hL

LP

woL

whP1

whP2

xjP

whPL

middotmiddotmiddot

middotmiddotmiddot

wh11

wh12

wh1L

aj

no



119869

119898=

119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896)

119890

2

min = min1198961

= 1198962

(sum

all119901(119909

(1198961)119901minus 119909

(1198962)119901)

2

)

119878 =

119869

119898

119899 times 119890

2

min

(4)





RCTE119895119906)) of





ℎ

119897= (ℎ

1198971 ℎ

1198972 ℎ

1198973) =

1

1 + 119890

minus119899ℎ

119897

= (

1

1 + 119890

minus119899ℎ

1198971

1

1 + 119890

minus119899ℎ

1198972

1

1 + 119890

minus119899ℎ

1198973

)

(5)

where

119899

ℎ

119897= (119899

ℎ

1198971 119899

ℎ

1198972 119899

ℎ

1198973) =

119868

ℎ

119897(minus)

120579

ℎ

119897

= (119868

ℎ

1198971minus 120579

ℎ

1198973 119868

ℎ

1198972minus 120579

ℎ

1198972 119868

ℎ

1198973minus 120579

ℎ

1198971)

119868

ℎ

119897= (119868

ℎ

1198971 119868

ℎ

1198972 119868

ℎ

1198973) = sum

all 119901119908

ℎ

119901119897sdot 119909

119895119901

= (sum

all 119901min (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901)

sum

all 119901119908

ℎ

1199011198972119909

119895119901 sum

all 119901max (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901))

(6)


ℎ



119900

119895= (119900

1198951 119900

1198952 119900

1198953) =

1

1 + 119890

minus119899119900

= (

1

1 + 119890

minus119899119900

1

1

1 + 119890

minus119899119900

2

1

1 + 119890

minus119899119900

3

)

(7)

where

119899

119900

= (119899

119900

1 119899

119900

2 119899

119900

3) =

119868

119900

(minus)

120579

119900

= (119868

119900

1minus 120579

119900

3 119868

119900

2minus 120579

119900

2 119868

119900

3minus 120579

119900

1)

(8)

119868

119900

= (119868

119900

1 119868

119900

2 119868

119900

3) = sum

all 119897119908

119900

119897(times)

ℎ

119897

cong (sum

all 119897min (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973) sum

all 119897119908

119900

1198972ℎ

1198972

sum

all 119897max (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973))

(9)


ℎ

1199011198972 120579ℎ

1198972 119908119900

1198972 and 120579

119900





ℎ

1199011198973 120579ℎ

1198973 119908119900

1198973 and 120579

119900




119900

1198952=

1

1 + 119890

minus119899119900

1198952

=

1

1 + 119890

minus(119868119900

1198952minus120579119900

2)

=

1

1 + 119890

120579119900

2minus119868119900

1198952

(10)

Therefore

ln( 1

119900

1198952

minus 1) = 120579

119900

2minus 119868

119900

1198952 (11)

So

119868

119900

1198952= 120579

119900

2minus ln( 1

119900

1198952

minus 1) (12)


119900

3minus120579

119900




119900

1198953=

1

1 + 119890

minus119899119900

1198953

(13)

where

119899

119900

1198953= 119868

119900

1198953minus 120579

119900

3= 119868

119900

1198953minus (120579

119900

2+ Δ120579

119900

) (14)


119900

1198953=

1

1 + 119890

minus(119868119900

1198953minus120579119900

2minusΔ120579119900)

(15)


119900

1198953=

1

1 + 119890

minus(120579119900

2minusln(1119900


119900

2minusΔ120579119900)

=

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900=

1

1 + 119890

Δ120579119900

(1119900

1198952minus 1)

(16)



119900

1198953ge 119873(RCT

119895119906)

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900ge 119873(RCT

119895119906) (17)

Δ120579

119900

le ln( 1

119873(RCT119895119906)


119900

1198952

minus 1) (18)


Upper bound 1

Upper bound 2

Upper bound 3


Upper bound 6

Final result

⊤

⊤⊤

⊤⊤

⊤

⊤

Actualcycle time



Δ120579

119900

le min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1)) (19)


Δ120579

119900lowast

= min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(20)


1198953in the 119905th


119900


119905

119900

1198953(119905) (21)



ℎ

1199011198971 120579ℎ

1198973 119908119900

1198973 and 120579119900



Δ120579

119900lowast

= max119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(22)



119900


119905

119900

1198951(119905) (23)


Actual cycle time

Final result


perp

perp

perpperp

perpperp

perp




2 rarr 1


RCTE119895119906= (max (RCTE

1198951199061(119896))

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)sdot RCTE

1198951199062(119896)

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)

min (RCTE1198951199063

(119896)))

(24)



32 The New Rule






Job no1

Job no2

Job no3

Job no1

Job no2 Job no3


Job no1

Job no2

Job no3






(FSMCT)

SK119895119906(FSMCT) =

119895

120582


(25)

(FSVCT)

SK119895119906(FSVCT) = 119877

119895minus RCTE

119895119906 (26)




SK119895119906(FSMCT) =

119895

120582

minus

RCTE119895119906

(27)


SK119895119906(FSVCT) = 119877

119895minus

RCTE119895119906 (28)



0minus 119886 119909

0 119909

0+ 119887) is

120583119860= 119909

0+

119887 minus 119886

3

(29)


120590119860=

radic

119886

2

+ 119886119887 + 119887

2

18

(30)






(1) 120583119860gt 120583

119861if and only if 119860 ≻

119861

(2) 120583119860lt 120583

119861if and only if 119860 ≺

119861

(3) if 120583119860= 120583

119861 then

(i) 120590119860gt 120590

119861if and only if 119860 ≺

119861(ii) 120590

119860lt 120590

119861if and only if 119860 ≻

119861(iii) 120590

119860= 120590

119861if and only if 119860 =

119861




120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)






(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)


120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)




fabrication factory



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)


SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)



119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












method



Number ofobjectives


Optimization method

















119898 119862max is






CT119895



RCT119895119906



RPT119895119906



SK119895119906



ℎ


119908

119900


119908

ℎ

119901119897


120579

ℎ


120579





119898|119903

119895 119889

119895 rent|119862 120590

119862 119871max 119873119879


Figure 1

3 Methodology







approach









119895= [119909

119895119901] 119901 = 1 sim 119875


Min119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896) (1)



119895(119896)


(1) Set 119870 = 1



119909

(119896)= 119909

(119896)119901 119901 = 1 sim 119875

119909

(119896)119901=

sum

119899

119895=1120583

119898

119895(119896)119909

119895119901

sum

119899

119895=1120583

119898

119895(119896)

120583

119895(119896)=

1

sum

119870

119902=1(119890

119895(119896)119890

119895(119902))

2(119898minus1)

119890

119895(119896)=

radic

sum

all119901(119909

119895119901minus 119909

(119896)119901)

2

(2)

where 119909(119896)





max119896

max119895

1003816

1003816

1003816

1003816

1003816

120583

(119905)

119895(119896)minus 120583

(119905minus1)

119895(119896)

1003816

1003816

1003816

1003816

1003816

lt 119889 (3)


1

2

Input layer

1

2

Hidden layer

1

Output layer

RMSE

xj1xj2

Ih1 nh1 120579h1 h1

wo1

wo2

oj

Io o

IhL nhL 120579hL hL

LP

woL

whP1

whP2

xjP

whPL

middotmiddotmiddot

middotmiddotmiddot

wh11

wh12

wh1L

aj

no



119869

119898=

119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896)

119890

2

min = min1198961

= 1198962

(sum

all119901(119909

(1198961)119901minus 119909

(1198962)119901)

2

)

119878 =

119869

119898

119899 times 119890

2

min

(4)





RCTE119895119906)) of





ℎ

119897= (ℎ

1198971 ℎ

1198972 ℎ

1198973) =

1

1 + 119890

minus119899ℎ

119897

= (

1

1 + 119890

minus119899ℎ

1198971

1

1 + 119890

minus119899ℎ

1198972

1

1 + 119890

minus119899ℎ

1198973

)

(5)

where

119899

ℎ

119897= (119899

ℎ

1198971 119899

ℎ

1198972 119899

ℎ

1198973) =

119868

ℎ

119897(minus)

120579

ℎ

119897

= (119868

ℎ

1198971minus 120579

ℎ

1198973 119868

ℎ

1198972minus 120579

ℎ

1198972 119868

ℎ

1198973minus 120579

ℎ

1198971)

119868

ℎ

119897= (119868

ℎ

1198971 119868

ℎ

1198972 119868

ℎ

1198973) = sum

all 119901119908

ℎ

119901119897sdot 119909

119895119901

= (sum

all 119901min (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901)

sum

all 119901119908

ℎ

1199011198972119909

119895119901 sum

all 119901max (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901))

(6)


ℎ



119900

119895= (119900

1198951 119900

1198952 119900

1198953) =

1

1 + 119890

minus119899119900

= (

1

1 + 119890

minus119899119900

1

1

1 + 119890

minus119899119900

2

1

1 + 119890

minus119899119900

3

)

(7)

where

119899

119900

= (119899

119900

1 119899

119900

2 119899

119900

3) =

119868

119900

(minus)

120579

119900

= (119868

119900

1minus 120579

119900

3 119868

119900

2minus 120579

119900

2 119868

119900

3minus 120579

119900

1)

(8)

119868

119900

= (119868

119900

1 119868

119900

2 119868

119900

3) = sum

all 119897119908

119900

119897(times)

ℎ

119897

cong (sum

all 119897min (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973) sum

all 119897119908

119900

1198972ℎ

1198972

sum

all 119897max (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973))

(9)


ℎ

1199011198972 120579ℎ

1198972 119908119900

1198972 and 120579

119900





ℎ

1199011198973 120579ℎ

1198973 119908119900

1198973 and 120579

119900




119900

1198952=

1

1 + 119890

minus119899119900

1198952

=

1

1 + 119890

minus(119868119900

1198952minus120579119900

2)

=

1

1 + 119890

120579119900

2minus119868119900

1198952

(10)

Therefore

ln( 1

119900

1198952

minus 1) = 120579

119900

2minus 119868

119900

1198952 (11)

So

119868

119900

1198952= 120579

119900

2minus ln( 1

119900

1198952

minus 1) (12)


119900

3minus120579

119900




119900

1198953=

1

1 + 119890

minus119899119900

1198953

(13)

where

119899

119900

1198953= 119868

119900

1198953minus 120579

119900

3= 119868

119900

1198953minus (120579

119900

2+ Δ120579

119900

) (14)


119900

1198953=

1

1 + 119890

minus(119868119900

1198953minus120579119900

2minusΔ120579119900)

(15)


119900

1198953=

1

1 + 119890

minus(120579119900

2minusln(1119900


119900

2minusΔ120579119900)

=

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900=

1

1 + 119890

Δ120579119900

(1119900

1198952minus 1)

(16)



119900

1198953ge 119873(RCT

119895119906)

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900ge 119873(RCT

119895119906) (17)

Δ120579

119900

le ln( 1

119873(RCT119895119906)


119900

1198952

minus 1) (18)


Upper bound 1

Upper bound 2

Upper bound 3


Upper bound 6

Final result

⊤

⊤⊤

⊤⊤

⊤

⊤

Actualcycle time



Δ120579

119900

le min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1)) (19)


Δ120579

119900lowast

= min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(20)


1198953in the 119905th


119900


119905

119900

1198953(119905) (21)



ℎ

1199011198971 120579ℎ

1198973 119908119900

1198973 and 120579119900



Δ120579

119900lowast

= max119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(22)



119900


119905

119900

1198951(119905) (23)


Actual cycle time

Final result


perp

perp

perpperp

perpperp

perp




2 rarr 1


RCTE119895119906= (max (RCTE

1198951199061(119896))

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)sdot RCTE

1198951199062(119896)

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)

min (RCTE1198951199063

(119896)))

(24)



32 The New Rule






Job no1

Job no2

Job no3

Job no1

Job no2 Job no3


Job no1

Job no2

Job no3






(FSMCT)

SK119895119906(FSMCT) =

119895

120582


(25)

(FSVCT)

SK119895119906(FSVCT) = 119877

119895minus RCTE

119895119906 (26)




SK119895119906(FSMCT) =

119895

120582

minus

RCTE119895119906

(27)


SK119895119906(FSVCT) = 119877

119895minus

RCTE119895119906 (28)



0minus 119886 119909

0 119909

0+ 119887) is

120583119860= 119909

0+

119887 minus 119886

3

(29)


120590119860=

radic

119886

2

+ 119886119887 + 119887

2

18

(30)






(1) 120583119860gt 120583

119861if and only if 119860 ≻

119861

(2) 120583119860lt 120583

119861if and only if 119860 ≺

119861

(3) if 120583119860= 120583

119861 then

(i) 120590119860gt 120590

119861if and only if 119860 ≺

119861(ii) 120590

119860lt 120590

119861if and only if 119860 ≻

119861(iii) 120590

119860= 120590

119861if and only if 119860 =

119861




120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)






(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)


120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)




fabrication factory



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)


SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)



119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













CT119895



RCT119895119906



RPT119895119906



SK119895119906



ℎ


119908

119900


119908

ℎ

119901119897


120579

ℎ


120579





119898|119903

119895 119889

119895 rent|119862 120590

119862 119871max 119873119879


Figure 1

3 Methodology







approach









119895= [119909

119895119901] 119901 = 1 sim 119875


Min119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896) (1)



119895(119896)


(1) Set 119870 = 1



119909

(119896)= 119909

(119896)119901 119901 = 1 sim 119875

119909

(119896)119901=

sum

119899

119895=1120583

119898

119895(119896)119909

119895119901

sum

119899

119895=1120583

119898

119895(119896)

120583

119895(119896)=

1

sum

119870

119902=1(119890

119895(119896)119890

119895(119902))

2(119898minus1)

119890

119895(119896)=

radic

sum

all119901(119909

119895119901minus 119909

(119896)119901)

2

(2)

where 119909(119896)





max119896

max119895

1003816

1003816

1003816

1003816

1003816

120583

(119905)

119895(119896)minus 120583

(119905minus1)

119895(119896)

1003816

1003816

1003816

1003816

1003816

lt 119889 (3)


1

2

Input layer

1

2

Hidden layer

1

Output layer

RMSE

xj1xj2

Ih1 nh1 120579h1 h1

wo1

wo2

oj

Io o

IhL nhL 120579hL hL

LP

woL

whP1

whP2

xjP

whPL

middotmiddotmiddot

middotmiddotmiddot

wh11

wh12

wh1L

aj

no



119869

119898=

119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896)

119890

2

min = min1198961

= 1198962

(sum

all119901(119909

(1198961)119901minus 119909

(1198962)119901)

2

)

119878 =

119869

119898

119899 times 119890

2

min

(4)





RCTE119895119906)) of





ℎ

119897= (ℎ

1198971 ℎ

1198972 ℎ

1198973) =

1

1 + 119890

minus119899ℎ

119897

= (

1

1 + 119890

minus119899ℎ

1198971

1

1 + 119890

minus119899ℎ

1198972

1

1 + 119890

minus119899ℎ

1198973

)

(5)

where

119899

ℎ

119897= (119899

ℎ

1198971 119899

ℎ

1198972 119899

ℎ

1198973) =

119868

ℎ

119897(minus)

120579

ℎ

119897

= (119868

ℎ

1198971minus 120579

ℎ

1198973 119868

ℎ

1198972minus 120579

ℎ

1198972 119868

ℎ

1198973minus 120579

ℎ

1198971)

119868

ℎ

119897= (119868

ℎ

1198971 119868

ℎ

1198972 119868

ℎ

1198973) = sum

all 119901119908

ℎ

119901119897sdot 119909

119895119901

= (sum

all 119901min (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901)

sum

all 119901119908

ℎ

1199011198972119909

119895119901 sum

all 119901max (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901))

(6)


ℎ



119900

119895= (119900

1198951 119900

1198952 119900

1198953) =

1

1 + 119890

minus119899119900

= (

1

1 + 119890

minus119899119900

1

1

1 + 119890

minus119899119900

2

1

1 + 119890

minus119899119900

3

)

(7)

where

119899

119900

= (119899

119900

1 119899

119900

2 119899

119900

3) =

119868

119900

(minus)

120579

119900

= (119868

119900

1minus 120579

119900

3 119868

119900

2minus 120579

119900

2 119868

119900

3minus 120579

119900

1)

(8)

119868

119900

= (119868

119900

1 119868

119900

2 119868

119900

3) = sum

all 119897119908

119900

119897(times)

ℎ

119897

cong (sum

all 119897min (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973) sum

all 119897119908

119900

1198972ℎ

1198972

sum

all 119897max (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973))

(9)


ℎ

1199011198972 120579ℎ

1198972 119908119900

1198972 and 120579

119900





ℎ

1199011198973 120579ℎ

1198973 119908119900

1198973 and 120579

119900




119900

1198952=

1

1 + 119890

minus119899119900

1198952

=

1

1 + 119890

minus(119868119900

1198952minus120579119900

2)

=

1

1 + 119890

120579119900

2minus119868119900

1198952

(10)

Therefore

ln( 1

119900

1198952

minus 1) = 120579

119900

2minus 119868

119900

1198952 (11)

So

119868

119900

1198952= 120579

119900

2minus ln( 1

119900

1198952

minus 1) (12)


119900

3minus120579

119900




119900

1198953=

1

1 + 119890

minus119899119900

1198953

(13)

where

119899

119900

1198953= 119868

119900

1198953minus 120579

119900

3= 119868

119900

1198953minus (120579

119900

2+ Δ120579

119900

) (14)


119900

1198953=

1

1 + 119890

minus(119868119900

1198953minus120579119900

2minusΔ120579119900)

(15)


119900

1198953=

1

1 + 119890

minus(120579119900

2minusln(1119900


119900

2minusΔ120579119900)

=

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900=

1

1 + 119890

Δ120579119900

(1119900

1198952minus 1)

(16)



119900

1198953ge 119873(RCT

119895119906)

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900ge 119873(RCT

119895119906) (17)

Δ120579

119900

le ln( 1

119873(RCT119895119906)


119900

1198952

minus 1) (18)


Upper bound 1

Upper bound 2

Upper bound 3


Upper bound 6

Final result

⊤

⊤⊤

⊤⊤

⊤

⊤

Actualcycle time



Δ120579

119900

le min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1)) (19)


Δ120579

119900lowast

= min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(20)


1198953in the 119905th


119900


119905

119900

1198953(119905) (21)



ℎ

1199011198971 120579ℎ

1198973 119908119900

1198973 and 120579119900



Δ120579

119900lowast

= max119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(22)



119900


119905

119900

1198951(119905) (23)


Actual cycle time

Final result


perp

perp

perpperp

perpperp

perp




2 rarr 1


RCTE119895119906= (max (RCTE

1198951199061(119896))

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)sdot RCTE

1198951199062(119896)

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)

min (RCTE1198951199063

(119896)))

(24)



32 The New Rule






Job no1

Job no2

Job no3

Job no1

Job no2 Job no3


Job no1

Job no2

Job no3






(FSMCT)

SK119895119906(FSMCT) =

119895

120582


(25)

(FSVCT)

SK119895119906(FSVCT) = 119877

119895minus RCTE

119895119906 (26)




SK119895119906(FSMCT) =

119895

120582

minus

RCTE119895119906

(27)


SK119895119906(FSVCT) = 119877

119895minus

RCTE119895119906 (28)



0minus 119886 119909

0 119909

0+ 119887) is

120583119860= 119909

0+

119887 minus 119886

3

(29)


120590119860=

radic

119886

2

+ 119886119887 + 119887

2

18

(30)






(1) 120583119860gt 120583

119861if and only if 119860 ≻

119861

(2) 120583119860lt 120583

119861if and only if 119860 ≺

119861

(3) if 120583119860= 120583

119861 then

(i) 120590119860gt 120590

119861if and only if 119860 ≺

119861(ii) 120590

119860lt 120590

119861if and only if 119860 ≻

119861(iii) 120590

119860= 120590

119861if and only if 119860 =

119861




120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)






(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)


120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)




fabrication factory



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)


SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)



119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119895= [119909

119895119901] 119901 = 1 sim 119875


Min119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896) (1)



119895(119896)


(1) Set 119870 = 1



119909

(119896)= 119909

(119896)119901 119901 = 1 sim 119875

119909

(119896)119901=

sum

119899

119895=1120583

119898

119895(119896)119909

119895119901

sum

119899

119895=1120583

119898

119895(119896)

120583

119895(119896)=

1

sum

119870

119902=1(119890

119895(119896)119890

119895(119902))

2(119898minus1)

119890

119895(119896)=

radic

sum

all119901(119909

119895119901minus 119909

(119896)119901)

2

(2)

where 119909(119896)





max119896

max119895

1003816

1003816

1003816

1003816

1003816

120583

(119905)

119895(119896)minus 120583

(119905minus1)

119895(119896)

1003816

1003816

1003816

1003816

1003816

lt 119889 (3)


1

2

Input layer

1

2

Hidden layer

1

Output layer

RMSE

xj1xj2

Ih1 nh1 120579h1 h1

wo1

wo2

oj

Io o

IhL nhL 120579hL hL

LP

woL

whP1

whP2

xjP

whPL

middotmiddotmiddot

middotmiddotmiddot

wh11

wh12

wh1L

aj

no



119869

119898=

119870

sum

119896=1

119899

sum

119895=1

120583

119898

119895(119896)119890

2

119895(119896)

119890

2

min = min1198961

= 1198962

(sum

all119901(119909

(1198961)119901minus 119909

(1198962)119901)

2

)

119878 =

119869

119898

119899 times 119890

2

min

(4)





RCTE119895119906)) of





ℎ

119897= (ℎ

1198971 ℎ

1198972 ℎ

1198973) =

1

1 + 119890

minus119899ℎ

119897

= (

1

1 + 119890

minus119899ℎ

1198971

1

1 + 119890

minus119899ℎ

1198972

1

1 + 119890

minus119899ℎ

1198973

)

(5)

where

119899

ℎ

119897= (119899

ℎ

1198971 119899

ℎ

1198972 119899

ℎ

1198973) =

119868

ℎ

119897(minus)

120579

ℎ

119897

= (119868

ℎ

1198971minus 120579

ℎ

1198973 119868

ℎ

1198972minus 120579

ℎ

1198972 119868

ℎ

1198973minus 120579

ℎ

1198971)

119868

ℎ

119897= (119868

ℎ

1198971 119868

ℎ

1198972 119868

ℎ

1198973) = sum

all 119901119908

ℎ

119901119897sdot 119909

119895119901

= (sum

all 119901min (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901)

sum

all 119901119908

ℎ

1199011198972119909

119895119901 sum

all 119901max (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901))

(6)


ℎ



119900

119895= (119900

1198951 119900

1198952 119900

1198953) =

1

1 + 119890

minus119899119900

= (

1

1 + 119890

minus119899119900

1

1

1 + 119890

minus119899119900

2

1

1 + 119890

minus119899119900

3

)

(7)

where

119899

119900

= (119899

119900

1 119899

119900

2 119899

119900

3) =

119868

119900

(minus)

120579

119900

= (119868

119900

1minus 120579

119900

3 119868

119900

2minus 120579

119900

2 119868

119900

3minus 120579

119900

1)

(8)

119868

119900

= (119868

119900

1 119868

119900

2 119868

119900

3) = sum

all 119897119908

119900

119897(times)

ℎ

119897

cong (sum

all 119897min (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973) sum

all 119897119908

119900

1198972ℎ

1198972

sum

all 119897max (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973))

(9)


ℎ

1199011198972 120579ℎ

1198972 119908119900

1198972 and 120579

119900





ℎ

1199011198973 120579ℎ

1198973 119908119900

1198973 and 120579

119900




119900

1198952=

1

1 + 119890

minus119899119900

1198952

=

1

1 + 119890

minus(119868119900

1198952minus120579119900

2)

=

1

1 + 119890

120579119900

2minus119868119900

1198952

(10)

Therefore

ln( 1

119900

1198952

minus 1) = 120579

119900

2minus 119868

119900

1198952 (11)

So

119868

119900

1198952= 120579

119900

2minus ln( 1

119900

1198952

minus 1) (12)


119900

3minus120579

119900




119900

1198953=

1

1 + 119890

minus119899119900

1198953

(13)

where

119899

119900

1198953= 119868

119900

1198953minus 120579

119900

3= 119868

119900

1198953minus (120579

119900

2+ Δ120579

119900

) (14)


119900

1198953=

1

1 + 119890

minus(119868119900

1198953minus120579119900

2minusΔ120579119900)

(15)


119900

1198953=

1

1 + 119890

minus(120579119900

2minusln(1119900


119900

2minusΔ120579119900)

=

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900=

1

1 + 119890

Δ120579119900

(1119900

1198952minus 1)

(16)



119900

1198953ge 119873(RCT

119895119906)

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900ge 119873(RCT

119895119906) (17)

Δ120579

119900

le ln( 1

119873(RCT119895119906)


119900

1198952

minus 1) (18)


Upper bound 1

Upper bound 2

Upper bound 3


Upper bound 6

Final result

⊤

⊤⊤

⊤⊤

⊤

⊤

Actualcycle time



Δ120579

119900

le min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1)) (19)


Δ120579

119900lowast

= min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(20)


1198953in the 119905th


119900


119905

119900

1198953(119905) (21)



ℎ

1199011198971 120579ℎ

1198973 119908119900

1198973 and 120579119900



Δ120579

119900lowast

= max119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(22)



119900


119905

119900

1198951(119905) (23)


Actual cycle time

Final result


perp

perp

perpperp

perpperp

perp




2 rarr 1


RCTE119895119906= (max (RCTE

1198951199061(119896))

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)sdot RCTE

1198951199062(119896)

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)

min (RCTE1198951199063

(119896)))

(24)



32 The New Rule






Job no1

Job no2

Job no3

Job no1

Job no2 Job no3


Job no1

Job no2

Job no3






(FSMCT)

SK119895119906(FSMCT) =

119895

120582


(25)

(FSVCT)

SK119895119906(FSVCT) = 119877

119895minus RCTE

119895119906 (26)




SK119895119906(FSMCT) =

119895

120582

minus

RCTE119895119906

(27)


SK119895119906(FSVCT) = 119877

119895minus

RCTE119895119906 (28)



0minus 119886 119909

0 119909

0+ 119887) is

120583119860= 119909

0+

119887 minus 119886

3

(29)


120590119860=

radic

119886

2

+ 119886119887 + 119887

2

18

(30)






(1) 120583119860gt 120583

119861if and only if 119860 ≻

119861

(2) 120583119860lt 120583

119861if and only if 119860 ≺

119861

(3) if 120583119860= 120583

119861 then

(i) 120590119860gt 120590

119861if and only if 119860 ≺

119861(ii) 120590

119860lt 120590

119861if and only if 119860 ≻

119861(iii) 120590

119860= 120590

119861if and only if 119860 =

119861




120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)






(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)


120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)




fabrication factory



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)


SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)



119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ

119897= (ℎ

1198971 ℎ

1198972 ℎ

1198973) =

1

1 + 119890

minus119899ℎ

119897

= (

1

1 + 119890

minus119899ℎ

1198971

1

1 + 119890

minus119899ℎ

1198972

1

1 + 119890

minus119899ℎ

1198973

)

(5)

where

119899

ℎ

119897= (119899

ℎ

1198971 119899

ℎ

1198972 119899

ℎ

1198973) =

119868

ℎ

119897(minus)

120579

ℎ

119897

= (119868

ℎ

1198971minus 120579

ℎ

1198973 119868

ℎ

1198972minus 120579

ℎ

1198972 119868

ℎ

1198973minus 120579

ℎ

1198971)

119868

ℎ

119897= (119868

ℎ

1198971 119868

ℎ

1198972 119868

ℎ

1198973) = sum

all 119901119908

ℎ

119901119897sdot 119909

119895119901

= (sum

all 119901min (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901)

sum

all 119901119908

ℎ

1199011198972119909

119895119901 sum

all 119901max (119908ℎ

1199011198971119909

119895119901 119908

ℎ

1199011198973119909

119895119901))

(6)


ℎ



119900

119895= (119900

1198951 119900

1198952 119900

1198953) =

1

1 + 119890

minus119899119900

= (

1

1 + 119890

minus119899119900

1

1

1 + 119890

minus119899119900

2

1

1 + 119890

minus119899119900

3

)

(7)

where

119899

119900

= (119899

119900

1 119899

119900

2 119899

119900

3) =

119868

119900

(minus)

120579

119900

= (119868

119900

1minus 120579

119900

3 119868

119900

2minus 120579

119900

2 119868

119900

3minus 120579

119900

1)

(8)

119868

119900

= (119868

119900

1 119868

119900

2 119868

119900

3) = sum

all 119897119908

119900

119897(times)

ℎ

119897

cong (sum

all 119897min (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973) sum

all 119897119908

119900

1198972ℎ

1198972

sum

all 119897max (119908119900

1198971ℎ

1198971 119908

119900

1198973ℎ

1198973))

(9)


ℎ

1199011198972 120579ℎ

1198972 119908119900

1198972 and 120579

119900





ℎ

1199011198973 120579ℎ

1198973 119908119900

1198973 and 120579

119900




119900

1198952=

1

1 + 119890

minus119899119900

1198952

=

1

1 + 119890

minus(119868119900

1198952minus120579119900

2)

=

1

1 + 119890

120579119900

2minus119868119900

1198952

(10)

Therefore

ln( 1

119900

1198952

minus 1) = 120579

119900

2minus 119868

119900

1198952 (11)

So

119868

119900

1198952= 120579

119900

2minus ln( 1

119900

1198952

minus 1) (12)


119900

3minus120579

119900




119900

1198953=

1

1 + 119890

minus119899119900

1198953

(13)

where

119899

119900

1198953= 119868

119900

1198953minus 120579

119900

3= 119868

119900

1198953minus (120579

119900

2+ Δ120579

119900

) (14)


119900

1198953=

1

1 + 119890

minus(119868119900

1198953minus120579119900

2minusΔ120579119900)

(15)


119900

1198953=

1

1 + 119890

minus(120579119900

2minusln(1119900


119900

2minusΔ120579119900)

=

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900=

1

1 + 119890

Δ120579119900

(1119900

1198952minus 1)

(16)



119900

1198953ge 119873(RCT

119895119906)

1

1 + 119890

ln(11199001198952minus1)+Δ120579

119900ge 119873(RCT

119895119906) (17)

Δ120579

119900

le ln( 1

119873(RCT119895119906)


119900

1198952

minus 1) (18)


Upper bound 1

Upper bound 2

Upper bound 3


Upper bound 6

Final result

⊤

⊤⊤

⊤⊤

⊤

⊤

Actualcycle time



Δ120579

119900

le min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1)) (19)


Δ120579

119900lowast

= min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(20)


1198953in the 119905th


119900


119905

119900

1198953(119905) (21)



ℎ

1199011198971 120579ℎ

1198973 119908119900

1198973 and 120579119900



Δ120579

119900lowast

= max119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(22)



119900


119905

119900

1198951(119905) (23)


Actual cycle time

Final result


perp

perp

perpperp

perpperp

perp




2 rarr 1


RCTE119895119906= (max (RCTE

1198951199061(119896))

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)sdot RCTE

1198951199062(119896)

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)

min (RCTE1198951199063

(119896)))

(24)



32 The New Rule






Job no1

Job no2

Job no3

Job no1

Job no2 Job no3


Job no1

Job no2

Job no3






(FSMCT)

SK119895119906(FSMCT) =

119895

120582


(25)

(FSVCT)

SK119895119906(FSVCT) = 119877

119895minus RCTE

119895119906 (26)




SK119895119906(FSMCT) =

119895

120582

minus

RCTE119895119906

(27)


SK119895119906(FSVCT) = 119877

119895minus

RCTE119895119906 (28)



0minus 119886 119909

0 119909

0+ 119887) is

120583119860= 119909

0+

119887 minus 119886

3

(29)


120590119860=

radic

119886

2

+ 119886119887 + 119887

2

18

(30)






(1) 120583119860gt 120583

119861if and only if 119860 ≻

119861

(2) 120583119860lt 120583

119861if and only if 119860 ≺

119861

(3) if 120583119860= 120583

119861 then

(i) 120590119860gt 120590

119861if and only if 119860 ≺

119861(ii) 120590

119860lt 120590

119861if and only if 119860 ≻

119861(iii) 120590

119860= 120590

119861if and only if 119860 =

119861




120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)






(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)


120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)




fabrication factory



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)


SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)



119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Upper bound 1

Upper bound 2

Upper bound 3


Upper bound 6

Final result

⊤

⊤⊤

⊤⊤

⊤

⊤

Actualcycle time



Δ120579

119900

le min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1)) (19)


Δ120579

119900lowast

= min119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(20)


1198953in the 119905th


119900


119905

119900

1198953(119905) (21)



ℎ

1199011198971 120579ℎ

1198973 119908119900

1198973 and 120579119900



Δ120579

119900lowast

= max119895

(ln( 1

119873(RCT119895119906)


119900

1198952

minus 1))

(22)



119900


119905

119900

1198951(119905) (23)


Actual cycle time

Final result


perp

perp

perpperp

perpperp

perp




2 rarr 1


RCTE119895119906= (max (RCTE

1198951199061(119896))

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)sdot RCTE

1198951199062(119896)

sum

119870

119896=1

2(119898minus1)radic1120583

119895(119896)

min (RCTE1198951199063

(119896)))

(24)



32 The New Rule






Job no1

Job no2

Job no3

Job no1

Job no2 Job no3


Job no1

Job no2

Job no3






(FSMCT)

SK119895119906(FSMCT) =

119895

120582


(25)

(FSVCT)

SK119895119906(FSVCT) = 119877

119895minus RCTE

119895119906 (26)




SK119895119906(FSMCT) =

119895

120582

minus

RCTE119895119906

(27)


SK119895119906(FSVCT) = 119877

119895minus

RCTE119895119906 (28)



0minus 119886 119909

0 119909

0+ 119887) is

120583119860= 119909

0+

119887 minus 119886

3

(29)


120590119860=

radic

119886

2

+ 119886119887 + 119887

2

18

(30)






(1) 120583119860gt 120583

119861if and only if 119860 ≻

119861

(2) 120583119860lt 120583

119861if and only if 119860 ≺

119861

(3) if 120583119860= 120583

119861 then

(i) 120590119860gt 120590

119861if and only if 119860 ≺

119861(ii) 120590

119860lt 120590

119861if and only if 119860 ≻

119861(iii) 120590

119860= 120590

119861if and only if 119860 =

119861




120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)






(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)


120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)




fabrication factory



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)


SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)



119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Job no1

Job no2

Job no3

Job no1

Job no2 Job no3


Job no1

Job no2

Job no3






(FSMCT)

SK119895119906(FSMCT) =

119895

120582


(25)

(FSVCT)

SK119895119906(FSVCT) = 119877

119895minus RCTE

119895119906 (26)




SK119895119906(FSMCT) =

119895

120582

minus

RCTE119895119906

(27)


SK119895119906(FSVCT) = 119877

119895minus

RCTE119895119906 (28)



0minus 119886 119909

0 119909

0+ 119887) is

120583119860= 119909

0+

119887 minus 119886

3

(29)


120590119860=

radic

119886

2

+ 119886119887 + 119887

2

18

(30)






(1) 120583119860gt 120583

119861if and only if 119860 ≻

119861

(2) 120583119860lt 120583

119861if and only if 119860 ≺

119861

(3) if 120583119860= 120583

119861 then

(i) 120590119860gt 120590

119861if and only if 119860 ≺

119861(ii) 120590

119860lt 120590

119861if and only if 119860 ≻

119861(iii) 120590

119860= 120590

119861if and only if 119860 =

119861




120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)






(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)


120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)




fabrication factory



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)


SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)



119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120590SK119895119906

=

radic

sum

119873

119894=1(SK

119895119906minus SK

119906)

2

119899 minus 1

(31)



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895RCTE

119895119906

max119895RCTE


119895RCTE

119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(32)






(Linear model)

120572 = 1 minus 2120573 minus 120574 120574 = 120599 = 120578 + 120572

(34)

(Nonlinear model)

120572 = (1 minus 2120573 minus 120574)

119906

119906 isin 119885

+

120574 = 120599 = (120578 + 120572)

V V = 1 3 5

(35)


120572 =

ln (2 minus 2120573 minus 120574)

ln 2 120574 = 120599 =

ln (15120578 + 120572 + 25)

ln 2minus 1

(36)




fabrication factory



SK119895119906= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (


119895

RCTE119895119906

max119895


119895

RCTE119895119906

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

= (SK1198951199061

SK1198951199062

SK1198951199063

)

(37)


SK1198951199061

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199061

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(38)

SK1198951199062

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199062

minusmin119895RCTE

1198951199062

max119895RCTE

1198951199062minusmin

119895RCTE

1198951199062

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(39)



119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877

119895119895 SCT

119895119906RPT

119895119906RCTE

119895119906SK

119895119906(FSMCT) SK

119895119906(FSVCT)


SK1198951199063

= (

119895 minus 1

119899 minus 1

)

120572

sdot (


119895RPT

119895119906

max119895RPT


119895RPT

119895119906

)

120573

sdot (

119877

119895minusmin

119895119877

119895

max119895119877

119895minusmin

119895119877

119895

)

120574

sdot (

RCTE1198951199063

minusmin119895RCTE

1198951199061

max119895RCTE

1198951199063minusmin

119895RCTE

1198951199061

)

120578

sdot (


119895SCT

119895119906

max119895SCT


119895SCT

119895119906

)

120599

(40)


SK119896119906


(1) Set 120595max to 0




rsquosas

119889 (

SK(119895minus1)119906

SK(119895)119906

) = SK(119895)1199061


(41)


120595 (119895 minus 1 119895) =

1 if 119889 (SK(119895minus1)119906

SK119895119906) gt 0

0 otherwise(42)

(6) If sum119899


119899

119895=1120595(119895 minus


(2)













119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119877

119895119895 SCT

119895119906RPT

119895119906

RCTE119895119906


119895119906(FFSVCT)












00102030405060708091

0 01 02 03 04 05SKju

120583(SKju)


00102030405060708091

0 2 4 6 8 10 12 14SKju

120583(SKju)


4 Simulation Study


















600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

Rem

aini

ng cy

cle ti

me (

hrs)

(SA-FBPN)

(a)

600

700

800

900

1000

1100

1200

1300

1400

1500

0 10 20 30 40Job

(Effective FBPN)

Rem

aini

ng cy

cle ti

me (

hrs)

(b)

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0 10 20 30 40

Rem

aini

ng cy

cle ti

me (

hrs)



Job (FCM-BPN)

(c)
























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























119867













Appendix



1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(1 + 1) (119895120582) minus RCTE119895119906

119895120582 minus RCTE119895119906

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895120582

119895120582 minus RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(A1)


1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

119895

119895 minus 120582 sdot RCTE119895119906

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

=

119895

120582 sdot RCTE119895119906minus 119895

(A2)




1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot

(

(1 + 1) (119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

(

(119895120582) minus (1120582)

(119899120582) minus (1120582)

)

120572

sdot(


119895(RPT

119895119906)

max119895(RPT

119895119906) minusmin

119895(RPT

119895119906)

)

0

sdot(

119877

119895minusmin

119895(119877

119895)

max119895(119877

119895) minusmin

119895(119877

119895)

)

0

sdot(


119895(RCTE

119895119906)

max119895(RCTE

119895119906) minusmin

119895(RCTE

119895119906)

)

120578

sdot(


119895(SCT

119895119906)

max119895(SCT

119895119906) minusmin

119895(SCT

119895119906)

)

0

minus 1

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

=

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

(

001119895

119895 minus 1

)

1205721003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

sdot 100

= (

001119895

119895 minus 1

)

120572

sdot 100

(A3)


1

120572

ge 1

100

1120572

ge 100

1

= 100

(A4)


(

001119895

119895 minus 1

)

120572

sdot 100 = (

001119895

119895 minus 1

)

120572

sdot (100

1120572

)

120572

ge (

001119895

119895 minus 1

)

120572

sdot (100)

120572

= (

119895

119895 minus 1

)

120572

(A5)


120572

+ 119895

120572

)120582119895

120572minus1 then

RCTE119895119906ge

(119895 minus 1)

120572

+ 119895

120572

120582119895

120572minus1

(A6)

120582119895


120572

+ 119895

120572

(A7)

120582 sdot

119895

120572

119895


120572

+ 119895

120572

(A8)

120582RCTE119895119906

119895

ge (

119895 minus 1

119895

)

120572

+ 1 (A9)

120582RCTE119895119906

119895

minus 1 ge (

119895 minus 1

119895

)

120572

(A10)

120582RCTE119895119906minus 119895

119895

ge (

119895 minus 1

119895

)

120572

(A11)

119895

120582RCTE119895119906minus 119895

le (

119895

119895 minus 1

)

120572

(A12)


Acknowledgment


References
















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a fuzzy rule for improving the performance of multiobjective...

Documents