Research ArticleEngineering Mathematical Analysis Method forProductivity Rate in Linear Arrangement Serial StructureAutomated Flow Assembly Line

Tan Chan Sin1 Ryspek Usubamatov1 M A Fairuz2

Mohd Fidzwan B Md Amin Hamzas1 and Low Kin Wai1

1School of Manufacturing Engineering Universiti Malaysia Perlis 02600 Arau Perlis Malaysia2Faculty of Engineering Technology Universiti Malaysia Perlis 02600 Arau Perlis Malaysia

Correspondence should be addressed to Tan Chan Sin tcs5077gmailcom

Received 29 June 2014 Revised 29 October 2014 Accepted 13 November 2014

Academic Editor Qing-WenWang

Copyright copy 2015 Tan Chan Sin et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Productivity rate (Q) or production rate is one of the important indicator criteria for industrial engineer to improve the systemand finish good output in production or assembly line Mathematical and statistical analysis method is required to be appliedfor productivity rate in industry visual overviews of the failure factors and further improvement within the production lineespecially for automated flow line since it is complicated Mathematical model of productivity rate in linear arrangement serialstructure automated flow line with different failure rate and bottleneck machining time parameters becomes the basic model forthis productivity analysis This paper presents the engineering mathematical analysis method which is applied in an automotivecompany which possesses automated flow assembly line in final assembly line to produce motorcycle in Malaysia DCASengineering and mathematical analysis method that consists of four stages known as data collection calculation and comparisonanalysis and sustainable improvement is used to analyze productivity in automated flow assembly line based on particularmathematical model Variety of failure rate that causes loss of productivity and bottleneck machining time is shown specifically inmathematic figure and presents the sustainable solution for productivity improvement for this final assembly automated flow line

1 Introduction

Manufacturing or production process is a crucial departmentin primary and secondary industry which usually performautomated flow line for mass and batch while manual oper-ated line for small scale production Automated flow line orsome publication considered to be automated transfer line iscategorized into two groups with regard to the arrangementwhich are linear type and rotor type automated flow lineLinear type is arranged in straight flow conveyor or othertransport system while rotor type is set up by circular orround transport rotors [1 2] For each group type of arrange-ment automated flow line is divided into three subgroupswhich are serial parallel and serial parallel that perform theproduction flow in different actionThis research finding onlyfocuses on linear type arrangement automated flow line withserial action since this type of flow line is generally applied

for assembly line in automotive industry as shown in Figure1

Productivity of the automated flow line is an importantindicator to measure the profit and performance for produc-tion line To measure the productivity rate for linear arrange-ment with serial action automated flow line productivitytheory is required for analysis use In theory of productivitythere is a basic mathematical model to express productivityto output finished goods per input variables which is shownin (1) [4] Output finished goods stand for the output productor service that is produced through a certain process whileinput variables mean any of the inputs including humancost time and raw material that help to produce the outputfinished goods during production time

Productivity = Output Finished GoodsInput Variables

(1)

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 592061 10 pageshttpdxdoiorg1011552015592061

2 Mathematical Problems in Engineering

1 2 3 middot middot middot i q minus 1 qmiddot middot middot

(a)

Lift table

Vac feederRev coater

UV ovenConveyor

UnifinisherCoater

UnwindLaminator

Edge wrapAuto-cutoff

Stacker

(b)

Figure 1 (a) Schematic diagram (b) Example of real plant layout for linear arrangement with serial action automated flow line with 119902 stations[3]

Based on the model above presented in (1) it can be gen-erally used to describe the productivity in a company Fromthis basic model the variables of input and output can bespecified and adjustedwith regard to different variable differ-ent perspective different framework and different applica-tion [5] For automated flow line engineer is required toemphasise more on the time used and output of productssince both of these variables are affected by technical andtechnological factors of production line Due to this condi-tion productivity rate (119876) or production rate is developed asshown in (2) which is specifically developed from base modelof productivity to fulfill the need of engineer in a productionline to calculate and analyze [3]

Productivity Rate (119876) =number of parts produced (119911)

time used (120579)

(2)

Equation (2) presents productivity rate (119876) into numberof parts produced (119911) per time used to produce 119911 parts(120579) which produce the yield of 119876 with unit of partsminuteor partshour eventually Output of this productivity ratemodel is very helpful for production engineers to go throughand analyze the productivity level of the automated flowline for further improvement Since 119876 is a useful indicatorfor automated flow line this model is elaborated based onreliability theory and developed a mathematical model ofproductivity rate with average level failure rate of workstation(119876afr) that is shown in (3) [6] There are a few parametersincluded which are as follows 119905

119898119900

is total machining time119905119886

is auxiliary time 119902 is total number of serial stations 119898119903

ismean time to repair 120582s119894 is average failure rate per station inautomated flow line 120582

119888

is failure rate of control system and120582tr is failure rate of transport system in the line

Productivity Rate 119876afr

=

1

(119905119898119900

119902 + 119905119886

) [1 + 119898119903

(119902120582s119894 + 120582119888 + 120582tr)]

(3)

Because of the mathematical model of productivity ratewith average level failure rate of workstation (119876afr) does

not consider a few of the important parameters that willaffect productivity rate for automated flow lineMathematicalmodel of productivity rate with different failure rate of station(119876dfr) has been developed due to consideration of severalimportant parameters which are different workstation failurerate with theory of probability bottleneck machining timecorrection factor with mathematic theory and defect partsfailure rate with reliability theory [3] The mathematicalmodel of productivity rate with different station failure rateis shown in (4) with several of parameters that are the samewith (3) which are stated above and a few new parametersadded in (4) which are as follows 119891

119898

is the correction factordue to bottleneck machining time in the line sum119902

119894=1

120582s119894 isrepresenting the summation of total different failure rate ofeach station and 120582

119889

is failure rate of defect raw materialand finished goods This mathematical model consists of twoparts which are technical part multiplied with reliability partas shown in

Productivity Rate 119876dfr

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

119902

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

(4)

Equation (4) presents themathematicalmodel of produc-tivity with different stations failure rates which theoreticallyshows that it is providing higher accuracy result for forecast-ing and analysing the actual productivity ratewhen comparedto (3) since (4) covers up mostly all parameters which willaffect productivity rate Since a new robust mathematicalmodel of productivity rate for linear arrangement serialaction automated flow line is developed it is complex andinconvenient to apply in real industry by engineers

In engineering field mathematic theory and applicationbecome a basic tool for design and analysis work such asnetwork biomedical and manufacturing engineering as wellas with mathematical model or mathematical analysis [78] Mathematical method generally combines with statisticalmethod in analysis stage which is graphical presented with

Mathematical Problems in Engineering 3

Figure 2 Final assembly line in ABC Company which produces Xtype motorcycle

support of mathematical calculation This method is con-sidered mathematical and statistical analysis method sinceit involves mathematical model application and statisticalanalysis method [9 10] Due to the high complexity of apply-ing the mathematical model in (4) in industry the demandfor less complicated convenient and effective engineeringmathematical and statistical analysis method is stronglyneeded for the automated flow line productivity rate [11]

This research paper presents the real industry validationfor this mathematical model in (4) by using mathematicalanalysis method with help of statistical method in onecompany which obtained an assembly automated flow lineThis paper will also present a set of engineering mathe-matical methods to apply the mathematical model into realindustry condition systematically and conveniently whichincluded the analysis and improvement of productivity ratein particular automated flow line Consequently this researchpresents the mathematical model of productivity rate whichis assisted by engineering mathematical method to solve theproductivity rate analysis problem in industry

2 Methodology

For this research a company is required to be attachedto the mathematical analysis by using mathematical modeland engineering mathematical method application for vali-dation purpose ABC Company is an automotive companyin Malaysia which produces the motorcycle with automatedflow line with serial action in linear arrangement in finalassembly stage This company is selected to apply the mathe-matical analysis method of productivity with mathematicalmodel because it is suitable and the company is willing tocollaborate with us for this research The final assembly lineof ABC Company is shown in Figure 2 which is using theautomated flow line in linear arrangement serial actionThereare several types of products that are assembled by using thisline For this particular validation X type of motorcycle isselected for detail analysis and accurate data collection

After selection of company and type for mathematicalanalysis method validation purpose next step of research isto select and design the most appropriate methods to obtainbetter and high trustworthiness results from ABC CompanyEngineeringmethod is considered a set of methods guidelineto solve engineering problem with support of engineering

Data collection

Calculationand comparison

Analysis forfailure rate

Sustainableimprovement

Figure 3 Concept of DCAS engineering mathematical analysismethod to apply the mathematical model analysis in real industry

and mathematic theory generally [12ndash14] A set of engineer-ing mathematical analysis methods are designed specificallyto solve the problem of real application of complicatedmathematicalmodel of productivity which is stated in (1) (2)and (3) in industry (ABCCompany)The termof engineeringis applied in this method because this problem is relatedto manufacturing engineering and industrial engineeringproblem This method as shown in Figure 3 consists of datacollection calculation and comparison analysis for failurerate and sustainable improvement

Engineering mathematical analysis method which isshown in Figure 3 consists of four stages that are data collec-tion calculation and comparison analysis for failure rate andsustainable improvement which have been summarized asDCAS The main purpose of this methods set is to overcomethe inconvenient problem of real application of mathematicalmodel of productivity in industry so that the problem can besolved by applying and following DCAS method which con-sists of all important waymethod analysis and improvementto applymathematicalmodel Each of the stages ofmethods isdiscussed clearly below which start from data collection (D)until the last stage sustainable improvement (S)

(D) Data Collection The first stage of the engineering math-ematical analysis method is data collection to obtain highenough trustworthiness data for productivity and failure rateanalysis There are three categories of data required to becollected for this method usage which are actual productiondata technical data and reliability data Detail of all data isshown clearly in Table 1

Thefirst category is the actual productivity data content ofnumber of parts produced and time used using retrospectivestudy Retrospective study is a data collection method thatrefers to previous or historical data Number of parts pro-duced and time used are required to measure and calculateactual productivity (2) in automated flow line which presentsthe final and real productivity in automated flow line asguideline for mathematical model comparison use

Technical data is a set of data regarding the technical workin automated flow line such as machining time and auxiliarytime Number of station and correction factor of bottleneckmachining time are also included in technical data setbecause both parameters technically affect the productivityrate and these parameters can be improved via technicalwork Observational study method such as time study or

4 Mathematical Problems in Engineering

Table 1 Overall detail data list required for engineering mathematical analysis method

Data category Collection method Data parameter Symbol Unit

Actual productivity Retrospective study Number of parts produced 119911 PartsTime used 120579 min or sec

Technical data Observational study

Total machining time 119905119898119900

min or secAuxiliary time 119905

119886

min or secNumber of stations 119902 mdash

Correction factor of bottleneck machining time 119891119898

mdash

Reliability data Retrospective study

Stations failure rate 120582s119894 1min or 1secControl system failure rate 120582

119888

1min or 1secTransport system failure rate 120582tr 1min or 1sec

Defect parts failure rate 120582119889

1min or 1secMean time to repair 119898

119903

min or sec

chronometric analysis [15] interview and focus group isapplied to obtain high confidence level data set

Retrospective study is applied to collect reliability datasuch as failure rate of stations control system transportsystem and defected parts which is related to the reliabilityparameters specified in failure rate of automated flow lineMean time to repair is also considered which is calculated bytotal failure time divided by total frequency of failure rate Forretrospective study data at least 90 of data trustworthinessis required for engineering data [16] The deviation of 10 iscalculated due to the total of plusmn5 from the mean value and itis calculated using

120575119894

=

sum

119899

119894=1

119911119894

minus 119911av

119911av (5)

Deviation 120575119894

is presented in (5) which is the yield ofnumber of motorcycle produced in 119894 shift (119873

119894

) minus averageor mean of motorcycle produce in 119894 shift (119911av) and divided bymean (119911av) For example average or mean of motorcycle pro-duce in 119894 shift (119911av) is 40 motorcycles In order to reach 90of statistical trustworthiness the deviation (120575

119894

) is requiredto be in the range of plusmn5 For this plusmn5 119911

119894

becomes plusmn2motorcycles as upper and lower limit Consequently to reach90 of statistical trustworthiness the 119911

119894

value is requiredto be converted to average mean cumulative number sincethe production of motorcycle is usually highly fluctuatedreaching the range of 40 plusmn 2 motorcycles Since the datacollection includes failure rate parameter it is found that 160times of failure are required to obtain 90 of probabilitytrustworthiness So to obtain 90 of statistical and probabil-ity trustworthiness the retrospective data should consist of160 times of failures or breakdowns of automated flow lineand plusmn5 deviation of motorcycle quantity produced 95 ofstatistical and probability trustworthiness which consists ofplusmn25 deviation will also be determined in this research

(C) Calculation and Comparison The second stage of engi-neeringmathematical analysis method is to calculate the datacollected using mathematical model above and compare theyield of the calculation Actual productivity data set is usedto calculate the real or actual productivity rate of X type ofmotorcycle in final assembly automated flow line by using (2)

Both sets of technical and reliability data are applied in (3) and(4) which ismathematicalmodel of productivity with averagelevel and different level of failure rateThe yields of (3) and (4)are compared to the actual productivity rate in final X typemotorcycle assembly automated flow line and the comparisonis done using percentage error shown in (6) The purpose ofcomparing with (6) is to show which of the mathematicalmodels of productivity can obtain more accurate result (lessthan 10 error) when compared to actual productivity rate

Percentage error =1003816100381610038161003816

119876dfr minus 119876ac1003816100381610038161003816

119876actimes 100 Or

Percentage error =1003816100381610038161003816

119876afr minus 119876ac1003816100381610038161003816

119876actimes 100

(6)

(A) Analysis for Failure Rate In analysis of failure ratestage the tool to apply for failure rate parameter analysis isproductivity losses diagram analysis tools [16] This analysisspecifically presents the losses of productivity due to differentfailure rate in term of graphical based on mathematicalcalculated result used in (4) Through this analysis impactof parameters causing losses of productivity in final assemblyautomated flow line for each of the failure rate parameter isdetected with regard to high impact or low impact whichis presented by mathematical numbering Pareto diagramanalysis is applied to arrange and analyze the parameter thatcauses losses of productivity in automated flow line from thehighest impact to the lowest impact parameter

(S) Sustainable Improvement The final stage of engineeringmathematical analysis method is sustainable improvementThis stage presents the meaning of the improvement step orsolution that can be applied for long term application andlong term improvement [17] Since there are a lot of param-eters causing the losses of productivity rate in final assemblyautomated line PACE prioritization matrix is introduced toselect which parameters are preferred to be improved andwhich (parameters) are suggested to be maintained [18]There are four levels in the PACE prioritization matrix whichare priority action consider and eliminate Priority group

Mathematical Problems in Engineering 5

100120140160180200220240260280300

(pro

duct

s)M

otor

cycle

pro

duce

d (z

)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Number of shift (n)

Figure 4 Motorcycle produced (119911) versus number of shift (119899) finalassembly automated line

parameters are preferred to proceed with improvement sinceit is highly affect to the productivity losses and easier toovercome when compare to others parameter After prioritygroup the next preferable group for further improvement toincrease the productivity of automated line is action groupfollowed by consider group and last but not least is eliminategroup The grouping discussion is done by closed group andthe solution is executed by the engineers

3 Results and Discussions

ABC Company is selected to apply and validate the math-ematical model of productivity by using engineering math-ematical analysis method Productivity rate and failure rateparameter that is causing loss of productivity in final assemblyautomated flow line in ABC Company are calculated andanalyzed based on DCAS method The result for each stageof engineering mathematical method is presented separatelyas below

31 Data Collection There are a total of three sets of data thatis required to be collected in this stage for calculation andanalysis useThe first set of data is actual data which includesnumber of shift production time and motorcycle producedThe actual data set for two months which are February andMarch 2014 in total consists of 31 shifts Within 31 shifts ofproduction for X type motorcycle the total production is7103 products with 14760 minutes of production time Toanalyze the trustworthiness of this retrospective study datait is required to perform the data in form of line chart toshow the trend of actual products produced regarding shiftas shown in Figure 4

Figure 4 presents the motorcycle produced (119911) in the 31shifts for two-month duration Regarding Figure 4 the prod-ucts per shift consist of high fluctuation and are required to beconverted into average mean cumulativeness of motorcycleproduced 119911

119888

To obtained 90 of statistical and probabilitytrustworthiness the deviation (120575

119894

) is required in the range ofplusmn5 Since the average or mean of motorcycle produced in 119894shift (119911av) is 22913 the deviation (120575

119894

) for plusmn5 is plusmn1146 wherethe upper limit is 24059 and lower limit is 21767 while forplusmn25 it is plusmn573 where upper limit is 23486 and lower limitis 2234 Figure 5 presents the average mean cumulativenessfor motorcycle produced according to the number of shift

180190200210220230240250260270280

Aver

age m

ean

cum

ulat

iven

ess (zc)

Upper limit (+5) Upper limit (+25)Mean (zav)

Lower limit (minus5) Lower limit (minus25)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Number of shift (n)

Figure 5 Average mean cumulativeness of motorcycle produced(119911119888

) versus number of shift (119899)

Table 2 Technical data for final assembly flow line in 31 workingshifts

Title DataTotal machining time 119905

119898119900

(min) 2655Auxiliary time 119905

119886

(min) 052Number of stations 119902 37Correction factor for machining time of thebottleneck station 119891

119898

125

From Figure 5 the number of motorcycle produced inaverage mean cumulative form (119911

119888

) falls into the toleranceof 90 at shift number 7 and the fluctuation of 119911

119888

does notfluctuate out of the range limit any more meaning that itis stable after shift number 7 So by collecting data fromshift 7 and onwards it consists of at least 90 of statisticaltrustworthiness For higher requirement of trustworthinesswhich is 95 the deviation isplusmn25where the trend droppedwithin shifts 13-14 Within 31 shifts of motorcycle producedthere are a total 160 times of failure or breakdown occurringin the final assemblyHence to obtain the data set with at least95 of statistical and 90 of probability trustworthiness theretrospective data collection is done until shift 31 in finalassembly automated flow line Another two sets of data whichare technical and reliability within 31 shifts are presented inTables 2 and 3

32 Calculation and Comparison After data collection stagethe data collected is required to proceed to the calculationstage by using (2) (3) and (4) Equation (2) is applied tocalculate actual productivity rate for final assembly auto-mated flow line and the calculation is shown in (7) For (3)it presents the mathematical model for productivity rate withaverage stations failure rate By applying the data in Table 3into (3) the result is shown in (8) Equation (4) presentsthe mathematical model for productivity rate with differentstations failure and (9) is the result after substituting the dataabove into (4)

Actual productivity rate is

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (7)

6 Mathematical Problems in Engineering

Productivity rate with average station failure rate is

119876afr =1

(119905119898119900

119902 + 119905119886

) [1 + 119898119903

(119902120582savg + 120582119888 + 120582tr)]

= 1 ((

2655

37

+ 052)

times [1 + 1124

times (37 times 49 times 10

minus4

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)])

minus1

= 06641 productsminute

(8)

Productivity rate with different station failure rate is

119876dfr =1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 ([1 + 1124

times (72 times 10

minus3

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

= 04911 productsminute

(9)

Percentage error for 119876dfr with 119876ac

=

|04911 minus 04751|

04751

times 100 = 337(10)

Percentage error for 119876afr with 119876ac

=

|06641 minus 04751|

04751

times 100 = 3978(11)

From the productivity rate result in (8) and (9) (7) whichis the actual productivity rate is used as guideline to checkwhich mathematical model can provide more accurate resultto actual result and the percentage error is calculated in(10) and (11) Regarding result of (10) and (11) it presentsthe percentage error when compared to actual productivityrate result Productivity rate with different station failure rate119876dfr is just 337 of percentage error while productivity ratewith average station failure rate 119876afr obtained is 3978 ofpercentage error when compared to actual productivity rate119876ac The reason 119876dfr is not exactly same to actual is due tounexpected factors that occurred in line Hence it is provenand validated that mathematical model of productivity ratewith different station failure rate yields the most accurateresult which is less than 10 error when compared to the

Table 3 Reliability data for final assembly flow line in 31 workingshifts

Title 119902 120582s119894 (1min)

Failure rate of the stations 119902 120582s

1 1355 times 10minus4

2 8130 times 10minus4

3 4743 times 10minus4

4 8130 times 10minus4

5 00006 5420 times 10minus4

7 00008 00009 6775 times 10minus5

10 4065 times 10minus4

11 6775 times 10minus5

12 000013 000014 000015 000016 000017 4065 times 10minus4

18 6775 times 10minus5

19 000020 000021 6775 times 10minus5

22 000023 000024 000025 7453 times 10minus4

26 4065 times 10minus4

27 5420 times 10minus4

28 5420 times 10minus4

29 6775 times 10minus5

30 000031 8130 times 10minus4

32 000033 1355 times 10minus4

34 000035 000036 6775 times 10minus5

37 6775 times 10minus5

Total stations failure rate sum37119894=1

120582s119894 7200 times 10minus3

Average station failure rate 120582savg 4900 times 10minus4

Failure rate of the control system 120582119888

9485 times 10minus4

Failure rate of the transport system 120582tr 2033 times 10minus4

Failure rate of the defect parts 120582119889

3048 times 10minus2

Mean repair time119898119903

(min) 1124

actual result Hence mathematical model of productivity ratewith different station failure rate can be further used in moreapplications in the same area Figure 6 presents the differ-ences between 119876dfr and 119876afr with respect to the number ofstation to prove that bothmodels produce high differences InFigure 6 there are three stages which are A B and C which

Mathematical Problems in Engineering 7

0

01

02

03

04

05

06

07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

A

B

C

Prod

uctiv

ity (Q

)

Qafr

Qdfr

Number of station (q)

Figure 6 Comparison of different station failure rate productivity(119876dfr) with average station failure rate productivity (119876afr) withrespect to number of station

present difference between119876dfr and119876afr each 10 stations Instage A which is productivity station number 10 the differ-ence between both is 01078 But when going to stage B whichis 20 stations the difference is increased to 01537 and keepsincreasing to 01725 at 30 stations in stage C In this analysis itis proven that the difference of119876dfr and119876afr increases whenthe number of station increases In conclusion the higherthe quantity of stations in the automated line the higher thedeviation or inaccuracy of 119876afr to actual productivity

33 Analysis for Failure Rate In this analysis for failure ratestage each of the parameters that contributes to the lossesof productivity rate in final assembly automated flow line islisted down and the value of productivity rate losses due toparticular parameter is calculated Equation below presentsthe calculation to determine the losses in productivity ratevalue and Figure 6 shows the total productivity losses dia-gram of final assembly automated flow line

(a) Potential productivity

119870 =

1

(119905119898119900

119902)

=

1

(265537)

= 13936 productsminute

(12)

(b) Cyclic productivity

1198761

=

1

(119905119898119900

119902 + 119905119886

)

=

1

(265537 + 052)

= 08080 productsminute(13)

(c) Bottleneck parameter productivity

1198762

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

=

1

[(265537) (125) + 052]

= 07057 productsminute(14)

(d) Station failure rate productivity

1198763

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894)]

=

1

[(265537) (125) + 052]

times

1

1 + 1124 (72 times 10

minus3

)

= 06529 productsminute

(15)

(e) Control system failure rate productivity

1198764

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

)]

= 06462 productsminute

(16)

(f) Transport system failure rate productivity

1198765

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)]

= 06451 productsminute(17)

(g) Defect failure rate productivity

1198766

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 (1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

+ 3048 times 10

minus2

))

minus1

= 04911 productsminute(18)

8 Mathematical Problems in Engineering

K = 13936 Q1 = 08080 Qac = 04751

L1=05856

L2=01023

L3=00528

L4=00067

L5=00011

L6=01540

L7=00160

Figure 7 Total productivity losses diagram of final assemblyautomated flow line

(h) Actual productivity

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (19)

(i) Losses of productivity

1198711

= 119870 minus 1198761

= 13936 minus 08080

= 05856 productsminute

1198712

= 1198761

minus 1198762

= 08080 minus 07057

= 01023 productsminute

1198713

= 1198762

minus 1198763

= 07057 minus 06529

= 00528 productsminute

1198714

= 1198763

minus 1198764

= 06529 minus 06462

= 00067 productsminute

1198715

= 1198764

minus 1198765

= 06462 minus 06451

= 00011 productsminute

1198716

= 1198765

minus 1198766

= 06451 minus 04911

= 01540 productsminute

1198717

= 1198766

minus 119876ac = 04911 minus 04751

= 00160 productsminute

(20)

From total productivity losses diagram in Figure 7 and(10)ndash(18) all of the productivity losses parameter are pre-sented in mathematic calculation form and show in num-bering result 119871

1

is the losses of productivity due to auxiliarytime which is 05856 productsminute and this parameter isthe highest factor that contributes to the loss of productivityin final assembly automated flow line 119871

2

is caused bybottleneck machining time in line which contributes 01023productsminute 119871

3

is the loss of 00528 productsminutedue to station failure rate 119871

4

is the factor of control systemwhich is 00067 productsminute 119871

5

is the lowest contributor

factor to productivity losses due to transport problem 1198716

is 01540 productsminute which contribute to the secondhighest of losses of productivity due to the defect in rawmaterial and finish goods and last is 119871

7

which is causedby unexpected factor such as electrical breakdown thatcontributes 00160 productsminute

After complete analyzing of the overall failure rate thatcauses loss of productivity workstation within the finalassembly automated flow line is required to determine theideal number of station to obtainmaximumproductivity rateFor productivity with station calculation (19) applied themathematical model with different station failure rate but theparameter of station failure is changed to average failure ratedue to lack of data of failure for the 37 stations above Formaximum productivity the ideal number of station is deter-mined by using (20) All of calculations are present in Figure7The range of 119902 station is from 10 stations until 300 stations

119876119902

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(120582savg + 120582119888 + 120582tr + 120582119889)]

=

1

[(2655119902) (125) + 052]

times 1 ([1 + 1124

times (49 times 10

minus4

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

119902opt = radic119905119898119900

119891119898

(1119898119903

) + 120582119888

+ 120582tr + 120582119889119905119886

120582119904

= ((2655 times 125

times [(11124) + 9485 times 10

minus4

+ 3048 times 10

minus2

])

times (052 times 49 times 10

minus4

)

minus1

)

12

= 12533 = 126 stations

(21)

From Figure 8 stage A shows that the current pro-ductivity is 04715 productsminute at station 37 Throughcalculation in (20) and particular figure the maximumproductivity that can be produced by 126 stations is 06228productsminute which presents in stage CThe productivitystarts to drop after station 126 even with the increment ofstation From the trend of the productivity rate productivityis increasing drastically from stage A until stage B whenthe number of station is increased Even from stage B to Cthe productivity keeps on increasing but the increment isonly improved for around 5 productivity with incrementof 50 stations From profit consideration it is not worth thatincrement of around 50 stations in automated flow line foronly 5 increment of productivity Thus the adjustment of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

070 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

A B C

Prod

uctiv

ity ra

te (Q

)

Number of station (q)

Figure 8 Station analysis with productivity rate versus number ofstation

Ease

of i

mpl

emen

tatio

nD

ifficu

ltEa

sy

BenefitsHigh Low

Priority Action

Consider Eliminate

Defect parts Bottleneck

Control and transport system

Stations failure rateAuxiliary time

Unexpected factor

Figure 9 PACE prioritization matrix for final assembly automatedflow line

stations from 37 stations can increase up to 80 stations toimprove the productivity profitably

34 Sustainable Improvement Final stage inDCAS engineer-ing mathematical analysis method is sustainable improve-ment Improvement process is selected by PACE prioritiza-tion matrix as shown in Figure 9 and this is the result from aclosed group discussion with Production Manager and Engi-neer in ABC Company Defect part parameter is preferred tobe improved due to its high impact on productivity losses andbecause it is easier to improve than others

4 Conclusion

After applying DCAS engineering mathematical analysismethod in ABC Company with the help of mathematicalmodel of productivity rate with different reliability all of theproductivity rate losses parameters are calculated measuredand analyzed with mathematical figures The losses andproductivity of final assembly automated flow line can beanalyzed visually clearly systematically and effectively whichis much better than just simply analyzing previous productiv-ity In conclusion mathematical model of productivity rate

with different station failure rate is proven and validatedto be useful and more accurate than previous productivitymathematical model which obtains 337 error or more than90 accuracy It is proved that new mathematical modelproduces higher accuracy result for productivity rate fore-casting DCAS engineering mathematical analysis method isalso proven to be systematic and effective to be applied inlinear arrangement serial action automated flow line whichis the final assembly line in ABC Company to calculateand analyze the productivity rate in that particular line Theproductivity losses parameters are determined and improvedsystematically with the help of mathematical model of pro-ductivity rate with different station failure rate In the futurethis mathematical model of productivity rate with differentstation failure rate and DCAS engineering mathematicalanalysis method can be applied to more field of sector suchas food processing or machining automated line since thismathematical model and method can be applied generally inlinear arrangement serial structure automated flow line

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Tan Chan Sin the main author of this paper would liketo thank the PhD degree supervisor Professor RyspekUsubamatov for his excellent guidance One of the authorswould also like to express appreciation to the authorrsquosresearch partners Mr Mohd Fidzwan B Md Amin HamzasM A Fairuz and Low Kin Wai who always help in solvingsome technical problems

References

[1] T Freiheit M Shpitalni and S J Hu ldquoProductivity of pacedparallel-serial manufacturing lines with and without crossoverrdquoJournal of Manufacturing Science and Engineering vol 126 no2 pp 361ndash367 2004

[2] R Usubamatov M Z Abdulmuin A M Nor and M NMurad ldquoProductivity rate of rotor-type automated lines andoptimisation of their structurerdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 222 no 11 pp 1561ndash1566 2008

[3] R Usubamatov T C Sin andM F BMd AminHamzas ldquoPro-ductivity theory for industrial automated linesrdquo in Proceedingsof the Mechanical Engineering Congress amp Exhibition pp 1ndash11ASME 2013

[4] S Saari ldquoTheory and measurement in businessrdquo in Proceedingsof the International Conference on Concurrent Engineering inConstruction no 110ndash1010 Espoo Finland 2006

[5] S Saari ldquoProduction and productivity as souces of well-beingrdquo2011

[6] R Usubamatov K A Ismail and J M Sah ldquoMathematicalmodels for productivity and availability of automated linesrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 59ndash69 2013

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Figure 2 Final assembly line in ABC Company which produces Xtype motorcycle

support of mathematical calculation This method is con-sidered mathematical and statistical analysis method sinceit involves mathematical model application and statisticalanalysis method [9 10] Due to the high complexity of apply-ing the mathematical model in (4) in industry the demandfor less complicated convenient and effective engineeringmathematical and statistical analysis method is stronglyneeded for the automated flow line productivity rate [11]

This research paper presents the real industry validationfor this mathematical model in (4) by using mathematicalanalysis method with help of statistical method in onecompany which obtained an assembly automated flow lineThis paper will also present a set of engineering mathe-matical methods to apply the mathematical model into realindustry condition systematically and conveniently whichincluded the analysis and improvement of productivity ratein particular automated flow line Consequently this researchpresents the mathematical model of productivity rate whichis assisted by engineering mathematical method to solve theproductivity rate analysis problem in industry

2 Methodology

For this research a company is required to be attachedto the mathematical analysis by using mathematical modeland engineering mathematical method application for vali-dation purpose ABC Company is an automotive companyin Malaysia which produces the motorcycle with automatedflow line with serial action in linear arrangement in finalassembly stage This company is selected to apply the mathe-matical analysis method of productivity with mathematicalmodel because it is suitable and the company is willing tocollaborate with us for this research The final assembly lineof ABC Company is shown in Figure 2 which is using theautomated flow line in linear arrangement serial actionThereare several types of products that are assembled by using thisline For this particular validation X type of motorcycle isselected for detail analysis and accurate data collection

After selection of company and type for mathematicalanalysis method validation purpose next step of research isto select and design the most appropriate methods to obtainbetter and high trustworthiness results from ABC CompanyEngineeringmethod is considered a set of methods guidelineto solve engineering problem with support of engineering

Data collection

Calculationand comparison

Analysis forfailure rate

Sustainableimprovement

Figure 3 Concept of DCAS engineering mathematical analysismethod to apply the mathematical model analysis in real industry

and mathematic theory generally [12ndash14] A set of engineer-ing mathematical analysis methods are designed specificallyto solve the problem of real application of complicatedmathematicalmodel of productivity which is stated in (1) (2)and (3) in industry (ABCCompany)The termof engineeringis applied in this method because this problem is relatedto manufacturing engineering and industrial engineeringproblem This method as shown in Figure 3 consists of datacollection calculation and comparison analysis for failurerate and sustainable improvement

Engineering mathematical analysis method which isshown in Figure 3 consists of four stages that are data collec-tion calculation and comparison analysis for failure rate andsustainable improvement which have been summarized asDCAS The main purpose of this methods set is to overcomethe inconvenient problem of real application of mathematicalmodel of productivity in industry so that the problem can besolved by applying and following DCAS method which con-sists of all important waymethod analysis and improvementto applymathematicalmodel Each of the stages ofmethods isdiscussed clearly below which start from data collection (D)until the last stage sustainable improvement (S)

(D) Data Collection The first stage of the engineering math-ematical analysis method is data collection to obtain highenough trustworthiness data for productivity and failure rateanalysis There are three categories of data required to becollected for this method usage which are actual productiondata technical data and reliability data Detail of all data isshown clearly in Table 1

Thefirst category is the actual productivity data content ofnumber of parts produced and time used using retrospectivestudy Retrospective study is a data collection method thatrefers to previous or historical data Number of parts pro-duced and time used are required to measure and calculateactual productivity (2) in automated flow line which presentsthe final and real productivity in automated flow line asguideline for mathematical model comparison use

Technical data is a set of data regarding the technical workin automated flow line such as machining time and auxiliarytime Number of station and correction factor of bottleneckmachining time are also included in technical data setbecause both parameters technically affect the productivityrate and these parameters can be improved via technicalwork Observational study method such as time study or

4 Mathematical Problems in Engineering

Table 1 Overall detail data list required for engineering mathematical analysis method

Data category Collection method Data parameter Symbol Unit

Actual productivity Retrospective study Number of parts produced 119911 PartsTime used 120579 min or sec

Technical data Observational study

Total machining time 119905119898119900

min or secAuxiliary time 119905

119886

min or secNumber of stations 119902 mdash

Correction factor of bottleneck machining time 119891119898

mdash

Reliability data Retrospective study

Stations failure rate 120582s119894 1min or 1secControl system failure rate 120582

119888

1min or 1secTransport system failure rate 120582tr 1min or 1sec

Defect parts failure rate 120582119889

1min or 1secMean time to repair 119898

119903

min or sec

chronometric analysis [15] interview and focus group isapplied to obtain high confidence level data set

Retrospective study is applied to collect reliability datasuch as failure rate of stations control system transportsystem and defected parts which is related to the reliabilityparameters specified in failure rate of automated flow lineMean time to repair is also considered which is calculated bytotal failure time divided by total frequency of failure rate Forretrospective study data at least 90 of data trustworthinessis required for engineering data [16] The deviation of 10 iscalculated due to the total of plusmn5 from the mean value and itis calculated using

120575119894

=

sum

119899

119894=1

119911119894

minus 119911av

119911av (5)

Deviation 120575119894

is presented in (5) which is the yield ofnumber of motorcycle produced in 119894 shift (119873

119894

) minus averageor mean of motorcycle produce in 119894 shift (119911av) and divided bymean (119911av) For example average or mean of motorcycle pro-duce in 119894 shift (119911av) is 40 motorcycles In order to reach 90of statistical trustworthiness the deviation (120575

119894

) is requiredto be in the range of plusmn5 For this plusmn5 119911

119894

becomes plusmn2motorcycles as upper and lower limit Consequently to reach90 of statistical trustworthiness the 119911

119894

value is requiredto be converted to average mean cumulative number sincethe production of motorcycle is usually highly fluctuatedreaching the range of 40 plusmn 2 motorcycles Since the datacollection includes failure rate parameter it is found that 160times of failure are required to obtain 90 of probabilitytrustworthiness So to obtain 90 of statistical and probabil-ity trustworthiness the retrospective data should consist of160 times of failures or breakdowns of automated flow lineand plusmn5 deviation of motorcycle quantity produced 95 ofstatistical and probability trustworthiness which consists ofplusmn25 deviation will also be determined in this research

(C) Calculation and Comparison The second stage of engi-neeringmathematical analysis method is to calculate the datacollected using mathematical model above and compare theyield of the calculation Actual productivity data set is usedto calculate the real or actual productivity rate of X type ofmotorcycle in final assembly automated flow line by using (2)

Both sets of technical and reliability data are applied in (3) and(4) which ismathematicalmodel of productivity with averagelevel and different level of failure rateThe yields of (3) and (4)are compared to the actual productivity rate in final X typemotorcycle assembly automated flow line and the comparisonis done using percentage error shown in (6) The purpose ofcomparing with (6) is to show which of the mathematicalmodels of productivity can obtain more accurate result (lessthan 10 error) when compared to actual productivity rate

Percentage error =1003816100381610038161003816

119876dfr minus 119876ac1003816100381610038161003816

119876actimes 100 Or

Percentage error =1003816100381610038161003816

119876afr minus 119876ac1003816100381610038161003816

119876actimes 100

(6)

(A) Analysis for Failure Rate In analysis of failure ratestage the tool to apply for failure rate parameter analysis isproductivity losses diagram analysis tools [16] This analysisspecifically presents the losses of productivity due to differentfailure rate in term of graphical based on mathematicalcalculated result used in (4) Through this analysis impactof parameters causing losses of productivity in final assemblyautomated flow line for each of the failure rate parameter isdetected with regard to high impact or low impact whichis presented by mathematical numbering Pareto diagramanalysis is applied to arrange and analyze the parameter thatcauses losses of productivity in automated flow line from thehighest impact to the lowest impact parameter

(S) Sustainable Improvement The final stage of engineeringmathematical analysis method is sustainable improvementThis stage presents the meaning of the improvement step orsolution that can be applied for long term application andlong term improvement [17] Since there are a lot of param-eters causing the losses of productivity rate in final assemblyautomated line PACE prioritization matrix is introduced toselect which parameters are preferred to be improved andwhich (parameters) are suggested to be maintained [18]There are four levels in the PACE prioritization matrix whichare priority action consider and eliminate Priority group

Mathematical Problems in Engineering 5

100120140160180200220240260280300

(pro

duct

s)M

otor

cycle

pro

duce

d (z

)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Number of shift (n)

Figure 4 Motorcycle produced (119911) versus number of shift (119899) finalassembly automated line

parameters are preferred to proceed with improvement sinceit is highly affect to the productivity losses and easier toovercome when compare to others parameter After prioritygroup the next preferable group for further improvement toincrease the productivity of automated line is action groupfollowed by consider group and last but not least is eliminategroup The grouping discussion is done by closed group andthe solution is executed by the engineers

3 Results and Discussions

ABC Company is selected to apply and validate the math-ematical model of productivity by using engineering math-ematical analysis method Productivity rate and failure rateparameter that is causing loss of productivity in final assemblyautomated flow line in ABC Company are calculated andanalyzed based on DCAS method The result for each stageof engineering mathematical method is presented separatelyas below

31 Data Collection There are a total of three sets of data thatis required to be collected in this stage for calculation andanalysis useThe first set of data is actual data which includesnumber of shift production time and motorcycle producedThe actual data set for two months which are February andMarch 2014 in total consists of 31 shifts Within 31 shifts ofproduction for X type motorcycle the total production is7103 products with 14760 minutes of production time Toanalyze the trustworthiness of this retrospective study datait is required to perform the data in form of line chart toshow the trend of actual products produced regarding shiftas shown in Figure 4

Figure 4 presents the motorcycle produced (119911) in the 31shifts for two-month duration Regarding Figure 4 the prod-ucts per shift consist of high fluctuation and are required to beconverted into average mean cumulativeness of motorcycleproduced 119911

119888

To obtained 90 of statistical and probabilitytrustworthiness the deviation (120575

119894

) is required in the range ofplusmn5 Since the average or mean of motorcycle produced in 119894shift (119911av) is 22913 the deviation (120575

119894

) for plusmn5 is plusmn1146 wherethe upper limit is 24059 and lower limit is 21767 while forplusmn25 it is plusmn573 where upper limit is 23486 and lower limitis 2234 Figure 5 presents the average mean cumulativenessfor motorcycle produced according to the number of shift

180190200210220230240250260270280

Aver

age m

ean

cum

ulat

iven

ess (zc)

Upper limit (+5) Upper limit (+25)Mean (zav)

Lower limit (minus5) Lower limit (minus25)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Number of shift (n)

Figure 5 Average mean cumulativeness of motorcycle produced(119911119888

) versus number of shift (119899)

Table 2 Technical data for final assembly flow line in 31 workingshifts

Title DataTotal machining time 119905

119898119900

(min) 2655Auxiliary time 119905

119886

(min) 052Number of stations 119902 37Correction factor for machining time of thebottleneck station 119891

119898

125

From Figure 5 the number of motorcycle produced inaverage mean cumulative form (119911

119888

) falls into the toleranceof 90 at shift number 7 and the fluctuation of 119911

119888

does notfluctuate out of the range limit any more meaning that itis stable after shift number 7 So by collecting data fromshift 7 and onwards it consists of at least 90 of statisticaltrustworthiness For higher requirement of trustworthinesswhich is 95 the deviation isplusmn25where the trend droppedwithin shifts 13-14 Within 31 shifts of motorcycle producedthere are a total 160 times of failure or breakdown occurringin the final assemblyHence to obtain the data set with at least95 of statistical and 90 of probability trustworthiness theretrospective data collection is done until shift 31 in finalassembly automated flow line Another two sets of data whichare technical and reliability within 31 shifts are presented inTables 2 and 3

32 Calculation and Comparison After data collection stagethe data collected is required to proceed to the calculationstage by using (2) (3) and (4) Equation (2) is applied tocalculate actual productivity rate for final assembly auto-mated flow line and the calculation is shown in (7) For (3)it presents the mathematical model for productivity rate withaverage stations failure rate By applying the data in Table 3into (3) the result is shown in (8) Equation (4) presentsthe mathematical model for productivity rate with differentstations failure and (9) is the result after substituting the dataabove into (4)

Actual productivity rate is

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (7)

6 Mathematical Problems in Engineering

Productivity rate with average station failure rate is

119876afr =1

(119905119898119900

119902 + 119905119886

) [1 + 119898119903

(119902120582savg + 120582119888 + 120582tr)]

= 1 ((

2655

37

+ 052)

times [1 + 1124

times (37 times 49 times 10

minus4

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)])

minus1

= 06641 productsminute

(8)

Productivity rate with different station failure rate is

119876dfr =1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 ([1 + 1124

times (72 times 10

minus3

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

= 04911 productsminute

(9)

Percentage error for 119876dfr with 119876ac

=

|04911 minus 04751|

04751

times 100 = 337(10)

Percentage error for 119876afr with 119876ac

=

|06641 minus 04751|

04751

times 100 = 3978(11)

From the productivity rate result in (8) and (9) (7) whichis the actual productivity rate is used as guideline to checkwhich mathematical model can provide more accurate resultto actual result and the percentage error is calculated in(10) and (11) Regarding result of (10) and (11) it presentsthe percentage error when compared to actual productivityrate result Productivity rate with different station failure rate119876dfr is just 337 of percentage error while productivity ratewith average station failure rate 119876afr obtained is 3978 ofpercentage error when compared to actual productivity rate119876ac The reason 119876dfr is not exactly same to actual is due tounexpected factors that occurred in line Hence it is provenand validated that mathematical model of productivity ratewith different station failure rate yields the most accurateresult which is less than 10 error when compared to the

Table 3 Reliability data for final assembly flow line in 31 workingshifts

Title 119902 120582s119894 (1min)

Failure rate of the stations 119902 120582s

1 1355 times 10minus4

2 8130 times 10minus4

3 4743 times 10minus4

4 8130 times 10minus4

5 00006 5420 times 10minus4

7 00008 00009 6775 times 10minus5

10 4065 times 10minus4

11 6775 times 10minus5

12 000013 000014 000015 000016 000017 4065 times 10minus4

18 6775 times 10minus5

19 000020 000021 6775 times 10minus5

22 000023 000024 000025 7453 times 10minus4

26 4065 times 10minus4

27 5420 times 10minus4

28 5420 times 10minus4

29 6775 times 10minus5

30 000031 8130 times 10minus4

32 000033 1355 times 10minus4

34 000035 000036 6775 times 10minus5

37 6775 times 10minus5

Total stations failure rate sum37119894=1

120582s119894 7200 times 10minus3

Average station failure rate 120582savg 4900 times 10minus4

Failure rate of the control system 120582119888

9485 times 10minus4

Failure rate of the transport system 120582tr 2033 times 10minus4

Failure rate of the defect parts 120582119889

3048 times 10minus2

Mean repair time119898119903

(min) 1124

actual result Hence mathematical model of productivity ratewith different station failure rate can be further used in moreapplications in the same area Figure 6 presents the differ-ences between 119876dfr and 119876afr with respect to the number ofstation to prove that bothmodels produce high differences InFigure 6 there are three stages which are A B and C which

Mathematical Problems in Engineering 7

0

01

02

03

04

05

06

07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

A

B

C

Prod

uctiv

ity (Q

)

Qafr

Qdfr

Number of station (q)

Figure 6 Comparison of different station failure rate productivity(119876dfr) with average station failure rate productivity (119876afr) withrespect to number of station

present difference between119876dfr and119876afr each 10 stations Instage A which is productivity station number 10 the differ-ence between both is 01078 But when going to stage B whichis 20 stations the difference is increased to 01537 and keepsincreasing to 01725 at 30 stations in stage C In this analysis itis proven that the difference of119876dfr and119876afr increases whenthe number of station increases In conclusion the higherthe quantity of stations in the automated line the higher thedeviation or inaccuracy of 119876afr to actual productivity

33 Analysis for Failure Rate In this analysis for failure ratestage each of the parameters that contributes to the lossesof productivity rate in final assembly automated flow line islisted down and the value of productivity rate losses due toparticular parameter is calculated Equation below presentsthe calculation to determine the losses in productivity ratevalue and Figure 6 shows the total productivity losses dia-gram of final assembly automated flow line

(a) Potential productivity

119870 =

1

(119905119898119900

119902)

=

1

(265537)

= 13936 productsminute

(12)

(b) Cyclic productivity

1198761

=

1

(119905119898119900

119902 + 119905119886

)

=

1

(265537 + 052)

= 08080 productsminute(13)

(c) Bottleneck parameter productivity

1198762

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

=

1

[(265537) (125) + 052]

= 07057 productsminute(14)

(d) Station failure rate productivity

1198763

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894)]

=

1

[(265537) (125) + 052]

times

1

1 + 1124 (72 times 10

minus3

)

= 06529 productsminute

(15)

(e) Control system failure rate productivity

1198764

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

)]

= 06462 productsminute

(16)

(f) Transport system failure rate productivity

1198765

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)]

= 06451 productsminute(17)

(g) Defect failure rate productivity

1198766

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 (1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

+ 3048 times 10

minus2

))

minus1

= 04911 productsminute(18)

8 Mathematical Problems in Engineering

K = 13936 Q1 = 08080 Qac = 04751

L1=05856

L2=01023

L3=00528

L4=00067

L5=00011

L6=01540

L7=00160

Figure 7 Total productivity losses diagram of final assemblyautomated flow line

(h) Actual productivity

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (19)

(i) Losses of productivity

1198711

= 119870 minus 1198761

= 13936 minus 08080

= 05856 productsminute

1198712

= 1198761

minus 1198762

= 08080 minus 07057

= 01023 productsminute

1198713

= 1198762

minus 1198763

= 07057 minus 06529

= 00528 productsminute

1198714

= 1198763

minus 1198764

= 06529 minus 06462

= 00067 productsminute

1198715

= 1198764

minus 1198765

= 06462 minus 06451

= 00011 productsminute

1198716

= 1198765

minus 1198766

= 06451 minus 04911

= 01540 productsminute

1198717

= 1198766

minus 119876ac = 04911 minus 04751

= 00160 productsminute

(20)

From total productivity losses diagram in Figure 7 and(10)ndash(18) all of the productivity losses parameter are pre-sented in mathematic calculation form and show in num-bering result 119871

1

is the losses of productivity due to auxiliarytime which is 05856 productsminute and this parameter isthe highest factor that contributes to the loss of productivityin final assembly automated flow line 119871

2

is caused bybottleneck machining time in line which contributes 01023productsminute 119871

3

is the loss of 00528 productsminutedue to station failure rate 119871

4

is the factor of control systemwhich is 00067 productsminute 119871

5

is the lowest contributor

factor to productivity losses due to transport problem 1198716

is 01540 productsminute which contribute to the secondhighest of losses of productivity due to the defect in rawmaterial and finish goods and last is 119871

7

which is causedby unexpected factor such as electrical breakdown thatcontributes 00160 productsminute

After complete analyzing of the overall failure rate thatcauses loss of productivity workstation within the finalassembly automated flow line is required to determine theideal number of station to obtainmaximumproductivity rateFor productivity with station calculation (19) applied themathematical model with different station failure rate but theparameter of station failure is changed to average failure ratedue to lack of data of failure for the 37 stations above Formaximum productivity the ideal number of station is deter-mined by using (20) All of calculations are present in Figure7The range of 119902 station is from 10 stations until 300 stations

119876119902

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(120582savg + 120582119888 + 120582tr + 120582119889)]

=

1

[(2655119902) (125) + 052]

times 1 ([1 + 1124

times (49 times 10

minus4

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

119902opt = radic119905119898119900

119891119898

(1119898119903

) + 120582119888

+ 120582tr + 120582119889119905119886

120582119904

= ((2655 times 125

times [(11124) + 9485 times 10

minus4

+ 3048 times 10

minus2

])

times (052 times 49 times 10

minus4

)

minus1

)

12

= 12533 = 126 stations

(21)

From Figure 8 stage A shows that the current pro-ductivity is 04715 productsminute at station 37 Throughcalculation in (20) and particular figure the maximumproductivity that can be produced by 126 stations is 06228productsminute which presents in stage CThe productivitystarts to drop after station 126 even with the increment ofstation From the trend of the productivity rate productivityis increasing drastically from stage A until stage B whenthe number of station is increased Even from stage B to Cthe productivity keeps on increasing but the increment isonly improved for around 5 productivity with incrementof 50 stations From profit consideration it is not worth thatincrement of around 50 stations in automated flow line foronly 5 increment of productivity Thus the adjustment of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

070 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

A B C

Prod

uctiv

ity ra

te (Q

)

Number of station (q)

Figure 8 Station analysis with productivity rate versus number ofstation

Ease

of i

mpl

emen

tatio

nD

ifficu

ltEa

sy

BenefitsHigh Low

Priority Action

Consider Eliminate

Defect parts Bottleneck

Control and transport system

Stations failure rateAuxiliary time

Unexpected factor

Figure 9 PACE prioritization matrix for final assembly automatedflow line

stations from 37 stations can increase up to 80 stations toimprove the productivity profitably

34 Sustainable Improvement Final stage inDCAS engineer-ing mathematical analysis method is sustainable improve-ment Improvement process is selected by PACE prioritiza-tion matrix as shown in Figure 9 and this is the result from aclosed group discussion with Production Manager and Engi-neer in ABC Company Defect part parameter is preferred tobe improved due to its high impact on productivity losses andbecause it is easier to improve than others

4 Conclusion

After applying DCAS engineering mathematical analysismethod in ABC Company with the help of mathematicalmodel of productivity rate with different reliability all of theproductivity rate losses parameters are calculated measuredand analyzed with mathematical figures The losses andproductivity of final assembly automated flow line can beanalyzed visually clearly systematically and effectively whichis much better than just simply analyzing previous productiv-ity In conclusion mathematical model of productivity rate

with different station failure rate is proven and validatedto be useful and more accurate than previous productivitymathematical model which obtains 337 error or more than90 accuracy It is proved that new mathematical modelproduces higher accuracy result for productivity rate fore-casting DCAS engineering mathematical analysis method isalso proven to be systematic and effective to be applied inlinear arrangement serial action automated flow line whichis the final assembly line in ABC Company to calculateand analyze the productivity rate in that particular line Theproductivity losses parameters are determined and improvedsystematically with the help of mathematical model of pro-ductivity rate with different station failure rate In the futurethis mathematical model of productivity rate with differentstation failure rate and DCAS engineering mathematicalanalysis method can be applied to more field of sector suchas food processing or machining automated line since thismathematical model and method can be applied generally inlinear arrangement serial structure automated flow line

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Tan Chan Sin the main author of this paper would liketo thank the PhD degree supervisor Professor RyspekUsubamatov for his excellent guidance One of the authorswould also like to express appreciation to the authorrsquosresearch partners Mr Mohd Fidzwan B Md Amin HamzasM A Fairuz and Low Kin Wai who always help in solvingsome technical problems

References

[1] T Freiheit M Shpitalni and S J Hu ldquoProductivity of pacedparallel-serial manufacturing lines with and without crossoverrdquoJournal of Manufacturing Science and Engineering vol 126 no2 pp 361ndash367 2004

[2] R Usubamatov M Z Abdulmuin A M Nor and M NMurad ldquoProductivity rate of rotor-type automated lines andoptimisation of their structurerdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 222 no 11 pp 1561ndash1566 2008

[3] R Usubamatov T C Sin andM F BMd AminHamzas ldquoPro-ductivity theory for industrial automated linesrdquo in Proceedingsof the Mechanical Engineering Congress amp Exhibition pp 1ndash11ASME 2013

[4] S Saari ldquoTheory and measurement in businessrdquo in Proceedingsof the International Conference on Concurrent Engineering inConstruction no 110ndash1010 Espoo Finland 2006

[5] S Saari ldquoProduction and productivity as souces of well-beingrdquo2011

[6] R Usubamatov K A Ismail and J M Sah ldquoMathematicalmodels for productivity and availability of automated linesrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 59ndash69 2013

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 1 Overall detail data list required for engineering mathematical analysis method

Data category Collection method Data parameter Symbol Unit

Actual productivity Retrospective study Number of parts produced 119911 PartsTime used 120579 min or sec

Technical data Observational study

Total machining time 119905119898119900

min or secAuxiliary time 119905

119886

min or secNumber of stations 119902 mdash

Correction factor of bottleneck machining time 119891119898

mdash

Reliability data Retrospective study

Stations failure rate 120582s119894 1min or 1secControl system failure rate 120582

119888

1min or 1secTransport system failure rate 120582tr 1min or 1sec

Defect parts failure rate 120582119889

1min or 1secMean time to repair 119898

119903

min or sec

chronometric analysis [15] interview and focus group isapplied to obtain high confidence level data set

Retrospective study is applied to collect reliability datasuch as failure rate of stations control system transportsystem and defected parts which is related to the reliabilityparameters specified in failure rate of automated flow lineMean time to repair is also considered which is calculated bytotal failure time divided by total frequency of failure rate Forretrospective study data at least 90 of data trustworthinessis required for engineering data [16] The deviation of 10 iscalculated due to the total of plusmn5 from the mean value and itis calculated using

120575119894

=

sum

119899

119894=1

119911119894

minus 119911av

119911av (5)

Deviation 120575119894

is presented in (5) which is the yield ofnumber of motorcycle produced in 119894 shift (119873

119894

) minus averageor mean of motorcycle produce in 119894 shift (119911av) and divided bymean (119911av) For example average or mean of motorcycle pro-duce in 119894 shift (119911av) is 40 motorcycles In order to reach 90of statistical trustworthiness the deviation (120575

119894

) is requiredto be in the range of plusmn5 For this plusmn5 119911

119894

becomes plusmn2motorcycles as upper and lower limit Consequently to reach90 of statistical trustworthiness the 119911

119894

value is requiredto be converted to average mean cumulative number sincethe production of motorcycle is usually highly fluctuatedreaching the range of 40 plusmn 2 motorcycles Since the datacollection includes failure rate parameter it is found that 160times of failure are required to obtain 90 of probabilitytrustworthiness So to obtain 90 of statistical and probabil-ity trustworthiness the retrospective data should consist of160 times of failures or breakdowns of automated flow lineand plusmn5 deviation of motorcycle quantity produced 95 ofstatistical and probability trustworthiness which consists ofplusmn25 deviation will also be determined in this research

(C) Calculation and Comparison The second stage of engi-neeringmathematical analysis method is to calculate the datacollected using mathematical model above and compare theyield of the calculation Actual productivity data set is usedto calculate the real or actual productivity rate of X type ofmotorcycle in final assembly automated flow line by using (2)

Both sets of technical and reliability data are applied in (3) and(4) which ismathematicalmodel of productivity with averagelevel and different level of failure rateThe yields of (3) and (4)are compared to the actual productivity rate in final X typemotorcycle assembly automated flow line and the comparisonis done using percentage error shown in (6) The purpose ofcomparing with (6) is to show which of the mathematicalmodels of productivity can obtain more accurate result (lessthan 10 error) when compared to actual productivity rate

Percentage error =1003816100381610038161003816

119876dfr minus 119876ac1003816100381610038161003816

119876actimes 100 Or

Percentage error =1003816100381610038161003816

119876afr minus 119876ac1003816100381610038161003816

119876actimes 100

(6)

(A) Analysis for Failure Rate In analysis of failure ratestage the tool to apply for failure rate parameter analysis isproductivity losses diagram analysis tools [16] This analysisspecifically presents the losses of productivity due to differentfailure rate in term of graphical based on mathematicalcalculated result used in (4) Through this analysis impactof parameters causing losses of productivity in final assemblyautomated flow line for each of the failure rate parameter isdetected with regard to high impact or low impact whichis presented by mathematical numbering Pareto diagramanalysis is applied to arrange and analyze the parameter thatcauses losses of productivity in automated flow line from thehighest impact to the lowest impact parameter

(S) Sustainable Improvement The final stage of engineeringmathematical analysis method is sustainable improvementThis stage presents the meaning of the improvement step orsolution that can be applied for long term application andlong term improvement [17] Since there are a lot of param-eters causing the losses of productivity rate in final assemblyautomated line PACE prioritization matrix is introduced toselect which parameters are preferred to be improved andwhich (parameters) are suggested to be maintained [18]There are four levels in the PACE prioritization matrix whichare priority action consider and eliminate Priority group

Mathematical Problems in Engineering 5

100120140160180200220240260280300

(pro

duct

s)M

otor

cycle

pro

duce

d (z

)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Number of shift (n)

Figure 4 Motorcycle produced (119911) versus number of shift (119899) finalassembly automated line

parameters are preferred to proceed with improvement sinceit is highly affect to the productivity losses and easier toovercome when compare to others parameter After prioritygroup the next preferable group for further improvement toincrease the productivity of automated line is action groupfollowed by consider group and last but not least is eliminategroup The grouping discussion is done by closed group andthe solution is executed by the engineers

3 Results and Discussions

ABC Company is selected to apply and validate the math-ematical model of productivity by using engineering math-ematical analysis method Productivity rate and failure rateparameter that is causing loss of productivity in final assemblyautomated flow line in ABC Company are calculated andanalyzed based on DCAS method The result for each stageof engineering mathematical method is presented separatelyas below

31 Data Collection There are a total of three sets of data thatis required to be collected in this stage for calculation andanalysis useThe first set of data is actual data which includesnumber of shift production time and motorcycle producedThe actual data set for two months which are February andMarch 2014 in total consists of 31 shifts Within 31 shifts ofproduction for X type motorcycle the total production is7103 products with 14760 minutes of production time Toanalyze the trustworthiness of this retrospective study datait is required to perform the data in form of line chart toshow the trend of actual products produced regarding shiftas shown in Figure 4

Figure 4 presents the motorcycle produced (119911) in the 31shifts for two-month duration Regarding Figure 4 the prod-ucts per shift consist of high fluctuation and are required to beconverted into average mean cumulativeness of motorcycleproduced 119911

119888

To obtained 90 of statistical and probabilitytrustworthiness the deviation (120575

119894

) is required in the range ofplusmn5 Since the average or mean of motorcycle produced in 119894shift (119911av) is 22913 the deviation (120575

119894

) for plusmn5 is plusmn1146 wherethe upper limit is 24059 and lower limit is 21767 while forplusmn25 it is plusmn573 where upper limit is 23486 and lower limitis 2234 Figure 5 presents the average mean cumulativenessfor motorcycle produced according to the number of shift

180190200210220230240250260270280

Aver

age m

ean

cum

ulat

iven

ess (zc)

Upper limit (+5) Upper limit (+25)Mean (zav)

Lower limit (minus5) Lower limit (minus25)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Number of shift (n)

Figure 5 Average mean cumulativeness of motorcycle produced(119911119888

) versus number of shift (119899)

Table 2 Technical data for final assembly flow line in 31 workingshifts

Title DataTotal machining time 119905

119898119900

(min) 2655Auxiliary time 119905

119886

(min) 052Number of stations 119902 37Correction factor for machining time of thebottleneck station 119891

119898

125

From Figure 5 the number of motorcycle produced inaverage mean cumulative form (119911

119888

) falls into the toleranceof 90 at shift number 7 and the fluctuation of 119911

119888

does notfluctuate out of the range limit any more meaning that itis stable after shift number 7 So by collecting data fromshift 7 and onwards it consists of at least 90 of statisticaltrustworthiness For higher requirement of trustworthinesswhich is 95 the deviation isplusmn25where the trend droppedwithin shifts 13-14 Within 31 shifts of motorcycle producedthere are a total 160 times of failure or breakdown occurringin the final assemblyHence to obtain the data set with at least95 of statistical and 90 of probability trustworthiness theretrospective data collection is done until shift 31 in finalassembly automated flow line Another two sets of data whichare technical and reliability within 31 shifts are presented inTables 2 and 3

32 Calculation and Comparison After data collection stagethe data collected is required to proceed to the calculationstage by using (2) (3) and (4) Equation (2) is applied tocalculate actual productivity rate for final assembly auto-mated flow line and the calculation is shown in (7) For (3)it presents the mathematical model for productivity rate withaverage stations failure rate By applying the data in Table 3into (3) the result is shown in (8) Equation (4) presentsthe mathematical model for productivity rate with differentstations failure and (9) is the result after substituting the dataabove into (4)

Actual productivity rate is

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (7)

6 Mathematical Problems in Engineering

Productivity rate with average station failure rate is

119876afr =1

(119905119898119900

119902 + 119905119886

) [1 + 119898119903

(119902120582savg + 120582119888 + 120582tr)]

= 1 ((

2655

37

+ 052)

times [1 + 1124

times (37 times 49 times 10

minus4

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)])

minus1

= 06641 productsminute

(8)

Productivity rate with different station failure rate is

119876dfr =1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 ([1 + 1124

times (72 times 10

minus3

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

= 04911 productsminute

(9)

Percentage error for 119876dfr with 119876ac

=

|04911 minus 04751|

04751

times 100 = 337(10)

Percentage error for 119876afr with 119876ac

=

|06641 minus 04751|

04751

times 100 = 3978(11)

From the productivity rate result in (8) and (9) (7) whichis the actual productivity rate is used as guideline to checkwhich mathematical model can provide more accurate resultto actual result and the percentage error is calculated in(10) and (11) Regarding result of (10) and (11) it presentsthe percentage error when compared to actual productivityrate result Productivity rate with different station failure rate119876dfr is just 337 of percentage error while productivity ratewith average station failure rate 119876afr obtained is 3978 ofpercentage error when compared to actual productivity rate119876ac The reason 119876dfr is not exactly same to actual is due tounexpected factors that occurred in line Hence it is provenand validated that mathematical model of productivity ratewith different station failure rate yields the most accurateresult which is less than 10 error when compared to the

Table 3 Reliability data for final assembly flow line in 31 workingshifts

Title 119902 120582s119894 (1min)

Failure rate of the stations 119902 120582s

1 1355 times 10minus4

2 8130 times 10minus4

3 4743 times 10minus4

4 8130 times 10minus4

5 00006 5420 times 10minus4

7 00008 00009 6775 times 10minus5

10 4065 times 10minus4

11 6775 times 10minus5

12 000013 000014 000015 000016 000017 4065 times 10minus4

18 6775 times 10minus5

19 000020 000021 6775 times 10minus5

22 000023 000024 000025 7453 times 10minus4

26 4065 times 10minus4

27 5420 times 10minus4

28 5420 times 10minus4

29 6775 times 10minus5

30 000031 8130 times 10minus4

32 000033 1355 times 10minus4

34 000035 000036 6775 times 10minus5

37 6775 times 10minus5

Total stations failure rate sum37119894=1

120582s119894 7200 times 10minus3

Average station failure rate 120582savg 4900 times 10minus4

Failure rate of the control system 120582119888

9485 times 10minus4

Failure rate of the transport system 120582tr 2033 times 10minus4

Failure rate of the defect parts 120582119889

3048 times 10minus2

Mean repair time119898119903

(min) 1124

actual result Hence mathematical model of productivity ratewith different station failure rate can be further used in moreapplications in the same area Figure 6 presents the differ-ences between 119876dfr and 119876afr with respect to the number ofstation to prove that bothmodels produce high differences InFigure 6 there are three stages which are A B and C which

Mathematical Problems in Engineering 7

0

01

02

03

04

05

06

07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

A

B

C

Prod

uctiv

ity (Q

)

Qafr

Qdfr

Number of station (q)

Figure 6 Comparison of different station failure rate productivity(119876dfr) with average station failure rate productivity (119876afr) withrespect to number of station

present difference between119876dfr and119876afr each 10 stations Instage A which is productivity station number 10 the differ-ence between both is 01078 But when going to stage B whichis 20 stations the difference is increased to 01537 and keepsincreasing to 01725 at 30 stations in stage C In this analysis itis proven that the difference of119876dfr and119876afr increases whenthe number of station increases In conclusion the higherthe quantity of stations in the automated line the higher thedeviation or inaccuracy of 119876afr to actual productivity

33 Analysis for Failure Rate In this analysis for failure ratestage each of the parameters that contributes to the lossesof productivity rate in final assembly automated flow line islisted down and the value of productivity rate losses due toparticular parameter is calculated Equation below presentsthe calculation to determine the losses in productivity ratevalue and Figure 6 shows the total productivity losses dia-gram of final assembly automated flow line

(a) Potential productivity

119870 =

1

(119905119898119900

119902)

=

1

(265537)

= 13936 productsminute

(12)

(b) Cyclic productivity

1198761

=

1

(119905119898119900

119902 + 119905119886

)

=

1

(265537 + 052)

= 08080 productsminute(13)

(c) Bottleneck parameter productivity

1198762

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

=

1

[(265537) (125) + 052]

= 07057 productsminute(14)

(d) Station failure rate productivity

1198763

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894)]

=

1

[(265537) (125) + 052]

times

1

1 + 1124 (72 times 10

minus3

)

= 06529 productsminute

(15)

(e) Control system failure rate productivity

1198764

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

)]

= 06462 productsminute

(16)

(f) Transport system failure rate productivity

1198765

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)]

= 06451 productsminute(17)

(g) Defect failure rate productivity

1198766

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 (1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

+ 3048 times 10

minus2

))

minus1

= 04911 productsminute(18)

8 Mathematical Problems in Engineering

K = 13936 Q1 = 08080 Qac = 04751

L1=05856

L2=01023

L3=00528

L4=00067

L5=00011

L6=01540

L7=00160

Figure 7 Total productivity losses diagram of final assemblyautomated flow line

(h) Actual productivity

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (19)

(i) Losses of productivity

1198711

= 119870 minus 1198761

= 13936 minus 08080

= 05856 productsminute

1198712

= 1198761

minus 1198762

= 08080 minus 07057

= 01023 productsminute

1198713

= 1198762

minus 1198763

= 07057 minus 06529

= 00528 productsminute

1198714

= 1198763

minus 1198764

= 06529 minus 06462

= 00067 productsminute

1198715

= 1198764

minus 1198765

= 06462 minus 06451

= 00011 productsminute

1198716

= 1198765

minus 1198766

= 06451 minus 04911

= 01540 productsminute

1198717

= 1198766

minus 119876ac = 04911 minus 04751

= 00160 productsminute

(20)

From total productivity losses diagram in Figure 7 and(10)ndash(18) all of the productivity losses parameter are pre-sented in mathematic calculation form and show in num-bering result 119871

1

is the losses of productivity due to auxiliarytime which is 05856 productsminute and this parameter isthe highest factor that contributes to the loss of productivityin final assembly automated flow line 119871

2

is caused bybottleneck machining time in line which contributes 01023productsminute 119871

3

is the loss of 00528 productsminutedue to station failure rate 119871

4

is the factor of control systemwhich is 00067 productsminute 119871

5

is the lowest contributor

factor to productivity losses due to transport problem 1198716

is 01540 productsminute which contribute to the secondhighest of losses of productivity due to the defect in rawmaterial and finish goods and last is 119871

7

which is causedby unexpected factor such as electrical breakdown thatcontributes 00160 productsminute

After complete analyzing of the overall failure rate thatcauses loss of productivity workstation within the finalassembly automated flow line is required to determine theideal number of station to obtainmaximumproductivity rateFor productivity with station calculation (19) applied themathematical model with different station failure rate but theparameter of station failure is changed to average failure ratedue to lack of data of failure for the 37 stations above Formaximum productivity the ideal number of station is deter-mined by using (20) All of calculations are present in Figure7The range of 119902 station is from 10 stations until 300 stations

119876119902

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(120582savg + 120582119888 + 120582tr + 120582119889)]

=

1

[(2655119902) (125) + 052]

times 1 ([1 + 1124

times (49 times 10

minus4

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

119902opt = radic119905119898119900

119891119898

(1119898119903

) + 120582119888

+ 120582tr + 120582119889119905119886

120582119904

= ((2655 times 125

times [(11124) + 9485 times 10

minus4

+ 3048 times 10

minus2

])

times (052 times 49 times 10

minus4

)

minus1

)

12

= 12533 = 126 stations

(21)

From Figure 8 stage A shows that the current pro-ductivity is 04715 productsminute at station 37 Throughcalculation in (20) and particular figure the maximumproductivity that can be produced by 126 stations is 06228productsminute which presents in stage CThe productivitystarts to drop after station 126 even with the increment ofstation From the trend of the productivity rate productivityis increasing drastically from stage A until stage B whenthe number of station is increased Even from stage B to Cthe productivity keeps on increasing but the increment isonly improved for around 5 productivity with incrementof 50 stations From profit consideration it is not worth thatincrement of around 50 stations in automated flow line foronly 5 increment of productivity Thus the adjustment of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

070 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

A B C

Prod

uctiv

ity ra

te (Q

)

Number of station (q)

Figure 8 Station analysis with productivity rate versus number ofstation

Ease

of i

mpl

emen

tatio

nD

ifficu

ltEa

sy

BenefitsHigh Low

Priority Action

Consider Eliminate

Defect parts Bottleneck

Control and transport system

Stations failure rateAuxiliary time

Unexpected factor

Figure 9 PACE prioritization matrix for final assembly automatedflow line

stations from 37 stations can increase up to 80 stations toimprove the productivity profitably

34 Sustainable Improvement Final stage inDCAS engineer-ing mathematical analysis method is sustainable improve-ment Improvement process is selected by PACE prioritiza-tion matrix as shown in Figure 9 and this is the result from aclosed group discussion with Production Manager and Engi-neer in ABC Company Defect part parameter is preferred tobe improved due to its high impact on productivity losses andbecause it is easier to improve than others

4 Conclusion

After applying DCAS engineering mathematical analysismethod in ABC Company with the help of mathematicalmodel of productivity rate with different reliability all of theproductivity rate losses parameters are calculated measuredand analyzed with mathematical figures The losses andproductivity of final assembly automated flow line can beanalyzed visually clearly systematically and effectively whichis much better than just simply analyzing previous productiv-ity In conclusion mathematical model of productivity rate

with different station failure rate is proven and validatedto be useful and more accurate than previous productivitymathematical model which obtains 337 error or more than90 accuracy It is proved that new mathematical modelproduces higher accuracy result for productivity rate fore-casting DCAS engineering mathematical analysis method isalso proven to be systematic and effective to be applied inlinear arrangement serial action automated flow line whichis the final assembly line in ABC Company to calculateand analyze the productivity rate in that particular line Theproductivity losses parameters are determined and improvedsystematically with the help of mathematical model of pro-ductivity rate with different station failure rate In the futurethis mathematical model of productivity rate with differentstation failure rate and DCAS engineering mathematicalanalysis method can be applied to more field of sector suchas food processing or machining automated line since thismathematical model and method can be applied generally inlinear arrangement serial structure automated flow line

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Tan Chan Sin the main author of this paper would liketo thank the PhD degree supervisor Professor RyspekUsubamatov for his excellent guidance One of the authorswould also like to express appreciation to the authorrsquosresearch partners Mr Mohd Fidzwan B Md Amin HamzasM A Fairuz and Low Kin Wai who always help in solvingsome technical problems

References

[1] T Freiheit M Shpitalni and S J Hu ldquoProductivity of pacedparallel-serial manufacturing lines with and without crossoverrdquoJournal of Manufacturing Science and Engineering vol 126 no2 pp 361ndash367 2004

[2] R Usubamatov M Z Abdulmuin A M Nor and M NMurad ldquoProductivity rate of rotor-type automated lines andoptimisation of their structurerdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 222 no 11 pp 1561ndash1566 2008

[3] R Usubamatov T C Sin andM F BMd AminHamzas ldquoPro-ductivity theory for industrial automated linesrdquo in Proceedingsof the Mechanical Engineering Congress amp Exhibition pp 1ndash11ASME 2013

[4] S Saari ldquoTheory and measurement in businessrdquo in Proceedingsof the International Conference on Concurrent Engineering inConstruction no 110ndash1010 Espoo Finland 2006

[5] S Saari ldquoProduction and productivity as souces of well-beingrdquo2011

[6] R Usubamatov K A Ismail and J M Sah ldquoMathematicalmodels for productivity and availability of automated linesrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 59ndash69 2013

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

100120140160180200220240260280300

(pro

duct

s)M

otor

cycle

pro

duce

d (z

)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Number of shift (n)

Figure 4 Motorcycle produced (119911) versus number of shift (119899) finalassembly automated line

parameters are preferred to proceed with improvement sinceit is highly affect to the productivity losses and easier toovercome when compare to others parameter After prioritygroup the next preferable group for further improvement toincrease the productivity of automated line is action groupfollowed by consider group and last but not least is eliminategroup The grouping discussion is done by closed group andthe solution is executed by the engineers

3 Results and Discussions

ABC Company is selected to apply and validate the math-ematical model of productivity by using engineering math-ematical analysis method Productivity rate and failure rateparameter that is causing loss of productivity in final assemblyautomated flow line in ABC Company are calculated andanalyzed based on DCAS method The result for each stageof engineering mathematical method is presented separatelyas below

31 Data Collection There are a total of three sets of data thatis required to be collected in this stage for calculation andanalysis useThe first set of data is actual data which includesnumber of shift production time and motorcycle producedThe actual data set for two months which are February andMarch 2014 in total consists of 31 shifts Within 31 shifts ofproduction for X type motorcycle the total production is7103 products with 14760 minutes of production time Toanalyze the trustworthiness of this retrospective study datait is required to perform the data in form of line chart toshow the trend of actual products produced regarding shiftas shown in Figure 4

Figure 4 presents the motorcycle produced (119911) in the 31shifts for two-month duration Regarding Figure 4 the prod-ucts per shift consist of high fluctuation and are required to beconverted into average mean cumulativeness of motorcycleproduced 119911

119888

To obtained 90 of statistical and probabilitytrustworthiness the deviation (120575

119894

) is required in the range ofplusmn5 Since the average or mean of motorcycle produced in 119894shift (119911av) is 22913 the deviation (120575

119894

) for plusmn5 is plusmn1146 wherethe upper limit is 24059 and lower limit is 21767 while forplusmn25 it is plusmn573 where upper limit is 23486 and lower limitis 2234 Figure 5 presents the average mean cumulativenessfor motorcycle produced according to the number of shift

180190200210220230240250260270280

Aver

age m

ean

cum

ulat

iven

ess (zc)

Upper limit (+5) Upper limit (+25)Mean (zav)

Lower limit (minus5) Lower limit (minus25)

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31Number of shift (n)

Figure 5 Average mean cumulativeness of motorcycle produced(119911119888

) versus number of shift (119899)

Table 2 Technical data for final assembly flow line in 31 workingshifts

Title DataTotal machining time 119905

119898119900

(min) 2655Auxiliary time 119905

119886

(min) 052Number of stations 119902 37Correction factor for machining time of thebottleneck station 119891

119898

125

From Figure 5 the number of motorcycle produced inaverage mean cumulative form (119911

119888

) falls into the toleranceof 90 at shift number 7 and the fluctuation of 119911

119888

does notfluctuate out of the range limit any more meaning that itis stable after shift number 7 So by collecting data fromshift 7 and onwards it consists of at least 90 of statisticaltrustworthiness For higher requirement of trustworthinesswhich is 95 the deviation isplusmn25where the trend droppedwithin shifts 13-14 Within 31 shifts of motorcycle producedthere are a total 160 times of failure or breakdown occurringin the final assemblyHence to obtain the data set with at least95 of statistical and 90 of probability trustworthiness theretrospective data collection is done until shift 31 in finalassembly automated flow line Another two sets of data whichare technical and reliability within 31 shifts are presented inTables 2 and 3

32 Calculation and Comparison After data collection stagethe data collected is required to proceed to the calculationstage by using (2) (3) and (4) Equation (2) is applied tocalculate actual productivity rate for final assembly auto-mated flow line and the calculation is shown in (7) For (3)it presents the mathematical model for productivity rate withaverage stations failure rate By applying the data in Table 3into (3) the result is shown in (8) Equation (4) presentsthe mathematical model for productivity rate with differentstations failure and (9) is the result after substituting the dataabove into (4)

Actual productivity rate is

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (7)

6 Mathematical Problems in Engineering

Productivity rate with average station failure rate is

119876afr =1

(119905119898119900

119902 + 119905119886

) [1 + 119898119903

(119902120582savg + 120582119888 + 120582tr)]

= 1 ((

2655

37

+ 052)

times [1 + 1124

times (37 times 49 times 10

minus4

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)])

minus1

= 06641 productsminute

(8)

Productivity rate with different station failure rate is

119876dfr =1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 ([1 + 1124

times (72 times 10

minus3

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

= 04911 productsminute

(9)

Percentage error for 119876dfr with 119876ac

=

|04911 minus 04751|

04751

times 100 = 337(10)

Percentage error for 119876afr with 119876ac

=

|06641 minus 04751|

04751

times 100 = 3978(11)

From the productivity rate result in (8) and (9) (7) whichis the actual productivity rate is used as guideline to checkwhich mathematical model can provide more accurate resultto actual result and the percentage error is calculated in(10) and (11) Regarding result of (10) and (11) it presentsthe percentage error when compared to actual productivityrate result Productivity rate with different station failure rate119876dfr is just 337 of percentage error while productivity ratewith average station failure rate 119876afr obtained is 3978 ofpercentage error when compared to actual productivity rate119876ac The reason 119876dfr is not exactly same to actual is due tounexpected factors that occurred in line Hence it is provenand validated that mathematical model of productivity ratewith different station failure rate yields the most accurateresult which is less than 10 error when compared to the

Table 3 Reliability data for final assembly flow line in 31 workingshifts

Title 119902 120582s119894 (1min)

Failure rate of the stations 119902 120582s

1 1355 times 10minus4

2 8130 times 10minus4

3 4743 times 10minus4

4 8130 times 10minus4

5 00006 5420 times 10minus4

7 00008 00009 6775 times 10minus5

10 4065 times 10minus4

11 6775 times 10minus5

12 000013 000014 000015 000016 000017 4065 times 10minus4

18 6775 times 10minus5

19 000020 000021 6775 times 10minus5

22 000023 000024 000025 7453 times 10minus4

26 4065 times 10minus4

27 5420 times 10minus4

28 5420 times 10minus4

29 6775 times 10minus5

30 000031 8130 times 10minus4

32 000033 1355 times 10minus4

34 000035 000036 6775 times 10minus5

37 6775 times 10minus5

Total stations failure rate sum37119894=1

120582s119894 7200 times 10minus3

Average station failure rate 120582savg 4900 times 10minus4

Failure rate of the control system 120582119888

9485 times 10minus4

Failure rate of the transport system 120582tr 2033 times 10minus4

Failure rate of the defect parts 120582119889

3048 times 10minus2

Mean repair time119898119903

(min) 1124

actual result Hence mathematical model of productivity ratewith different station failure rate can be further used in moreapplications in the same area Figure 6 presents the differ-ences between 119876dfr and 119876afr with respect to the number ofstation to prove that bothmodels produce high differences InFigure 6 there are three stages which are A B and C which

Mathematical Problems in Engineering 7

0

01

02

03

04

05

06

07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

A

B

C

Prod

uctiv

ity (Q

)

Qafr

Qdfr

Number of station (q)

Figure 6 Comparison of different station failure rate productivity(119876dfr) with average station failure rate productivity (119876afr) withrespect to number of station

present difference between119876dfr and119876afr each 10 stations Instage A which is productivity station number 10 the differ-ence between both is 01078 But when going to stage B whichis 20 stations the difference is increased to 01537 and keepsincreasing to 01725 at 30 stations in stage C In this analysis itis proven that the difference of119876dfr and119876afr increases whenthe number of station increases In conclusion the higherthe quantity of stations in the automated line the higher thedeviation or inaccuracy of 119876afr to actual productivity

33 Analysis for Failure Rate In this analysis for failure ratestage each of the parameters that contributes to the lossesof productivity rate in final assembly automated flow line islisted down and the value of productivity rate losses due toparticular parameter is calculated Equation below presentsthe calculation to determine the losses in productivity ratevalue and Figure 6 shows the total productivity losses dia-gram of final assembly automated flow line

(a) Potential productivity

119870 =

1

(119905119898119900

119902)

=

1

(265537)

= 13936 productsminute

(12)

(b) Cyclic productivity

1198761

=

1

(119905119898119900

119902 + 119905119886

)

=

1

(265537 + 052)

= 08080 productsminute(13)

(c) Bottleneck parameter productivity

1198762

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

=

1

[(265537) (125) + 052]

= 07057 productsminute(14)

(d) Station failure rate productivity

1198763

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894)]

=

1

[(265537) (125) + 052]

times

1

1 + 1124 (72 times 10

minus3

)

= 06529 productsminute

(15)

(e) Control system failure rate productivity

1198764

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

)]

= 06462 productsminute

(16)

(f) Transport system failure rate productivity

1198765

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)]

= 06451 productsminute(17)

(g) Defect failure rate productivity

1198766

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 (1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

+ 3048 times 10

minus2

))

minus1

= 04911 productsminute(18)

8 Mathematical Problems in Engineering

K = 13936 Q1 = 08080 Qac = 04751

L1=05856

L2=01023

L3=00528

L4=00067

L5=00011

L6=01540

L7=00160

Figure 7 Total productivity losses diagram of final assemblyautomated flow line

(h) Actual productivity

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (19)

(i) Losses of productivity

1198711

= 119870 minus 1198761

= 13936 minus 08080

= 05856 productsminute

1198712

= 1198761

minus 1198762

= 08080 minus 07057

= 01023 productsminute

1198713

= 1198762

minus 1198763

= 07057 minus 06529

= 00528 productsminute

1198714

= 1198763

minus 1198764

= 06529 minus 06462

= 00067 productsminute

1198715

= 1198764

minus 1198765

= 06462 minus 06451

= 00011 productsminute

1198716

= 1198765

minus 1198766

= 06451 minus 04911

= 01540 productsminute

1198717

= 1198766

minus 119876ac = 04911 minus 04751

= 00160 productsminute

(20)

From total productivity losses diagram in Figure 7 and(10)ndash(18) all of the productivity losses parameter are pre-sented in mathematic calculation form and show in num-bering result 119871

1

is the losses of productivity due to auxiliarytime which is 05856 productsminute and this parameter isthe highest factor that contributes to the loss of productivityin final assembly automated flow line 119871

2

is caused bybottleneck machining time in line which contributes 01023productsminute 119871

3

is the loss of 00528 productsminutedue to station failure rate 119871

4

is the factor of control systemwhich is 00067 productsminute 119871

5

is the lowest contributor

factor to productivity losses due to transport problem 1198716

is 01540 productsminute which contribute to the secondhighest of losses of productivity due to the defect in rawmaterial and finish goods and last is 119871

7

which is causedby unexpected factor such as electrical breakdown thatcontributes 00160 productsminute

After complete analyzing of the overall failure rate thatcauses loss of productivity workstation within the finalassembly automated flow line is required to determine theideal number of station to obtainmaximumproductivity rateFor productivity with station calculation (19) applied themathematical model with different station failure rate but theparameter of station failure is changed to average failure ratedue to lack of data of failure for the 37 stations above Formaximum productivity the ideal number of station is deter-mined by using (20) All of calculations are present in Figure7The range of 119902 station is from 10 stations until 300 stations

119876119902

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(120582savg + 120582119888 + 120582tr + 120582119889)]

=

1

[(2655119902) (125) + 052]

times 1 ([1 + 1124

times (49 times 10

minus4

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

119902opt = radic119905119898119900

119891119898

(1119898119903

) + 120582119888

+ 120582tr + 120582119889119905119886

120582119904

= ((2655 times 125

times [(11124) + 9485 times 10

minus4

+ 3048 times 10

minus2

])

times (052 times 49 times 10

minus4

)

minus1

)

12

= 12533 = 126 stations

(21)

From Figure 8 stage A shows that the current pro-ductivity is 04715 productsminute at station 37 Throughcalculation in (20) and particular figure the maximumproductivity that can be produced by 126 stations is 06228productsminute which presents in stage CThe productivitystarts to drop after station 126 even with the increment ofstation From the trend of the productivity rate productivityis increasing drastically from stage A until stage B whenthe number of station is increased Even from stage B to Cthe productivity keeps on increasing but the increment isonly improved for around 5 productivity with incrementof 50 stations From profit consideration it is not worth thatincrement of around 50 stations in automated flow line foronly 5 increment of productivity Thus the adjustment of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

070 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

A B C

Prod

uctiv

ity ra

te (Q

)

Number of station (q)

Figure 8 Station analysis with productivity rate versus number ofstation

Ease

of i

mpl

emen

tatio

nD

ifficu

ltEa

sy

BenefitsHigh Low

Priority Action

Consider Eliminate

Defect parts Bottleneck

Control and transport system

Stations failure rateAuxiliary time

Unexpected factor

Figure 9 PACE prioritization matrix for final assembly automatedflow line

stations from 37 stations can increase up to 80 stations toimprove the productivity profitably

34 Sustainable Improvement Final stage inDCAS engineer-ing mathematical analysis method is sustainable improve-ment Improvement process is selected by PACE prioritiza-tion matrix as shown in Figure 9 and this is the result from aclosed group discussion with Production Manager and Engi-neer in ABC Company Defect part parameter is preferred tobe improved due to its high impact on productivity losses andbecause it is easier to improve than others

4 Conclusion

After applying DCAS engineering mathematical analysismethod in ABC Company with the help of mathematicalmodel of productivity rate with different reliability all of theproductivity rate losses parameters are calculated measuredand analyzed with mathematical figures The losses andproductivity of final assembly automated flow line can beanalyzed visually clearly systematically and effectively whichis much better than just simply analyzing previous productiv-ity In conclusion mathematical model of productivity rate

with different station failure rate is proven and validatedto be useful and more accurate than previous productivitymathematical model which obtains 337 error or more than90 accuracy It is proved that new mathematical modelproduces higher accuracy result for productivity rate fore-casting DCAS engineering mathematical analysis method isalso proven to be systematic and effective to be applied inlinear arrangement serial action automated flow line whichis the final assembly line in ABC Company to calculateand analyze the productivity rate in that particular line Theproductivity losses parameters are determined and improvedsystematically with the help of mathematical model of pro-ductivity rate with different station failure rate In the futurethis mathematical model of productivity rate with differentstation failure rate and DCAS engineering mathematicalanalysis method can be applied to more field of sector suchas food processing or machining automated line since thismathematical model and method can be applied generally inlinear arrangement serial structure automated flow line

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Tan Chan Sin the main author of this paper would liketo thank the PhD degree supervisor Professor RyspekUsubamatov for his excellent guidance One of the authorswould also like to express appreciation to the authorrsquosresearch partners Mr Mohd Fidzwan B Md Amin HamzasM A Fairuz and Low Kin Wai who always help in solvingsome technical problems

References

[1] T Freiheit M Shpitalni and S J Hu ldquoProductivity of pacedparallel-serial manufacturing lines with and without crossoverrdquoJournal of Manufacturing Science and Engineering vol 126 no2 pp 361ndash367 2004

[2] R Usubamatov M Z Abdulmuin A M Nor and M NMurad ldquoProductivity rate of rotor-type automated lines andoptimisation of their structurerdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 222 no 11 pp 1561ndash1566 2008

[3] R Usubamatov T C Sin andM F BMd AminHamzas ldquoPro-ductivity theory for industrial automated linesrdquo in Proceedingsof the Mechanical Engineering Congress amp Exhibition pp 1ndash11ASME 2013

[4] S Saari ldquoTheory and measurement in businessrdquo in Proceedingsof the International Conference on Concurrent Engineering inConstruction no 110ndash1010 Espoo Finland 2006

[5] S Saari ldquoProduction and productivity as souces of well-beingrdquo2011

[6] R Usubamatov K A Ismail and J M Sah ldquoMathematicalmodels for productivity and availability of automated linesrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 59ndash69 2013

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Productivity rate with average station failure rate is

119876afr =1

(119905119898119900

119902 + 119905119886

) [1 + 119898119903

(119902120582savg + 120582119888 + 120582tr)]

= 1 ((

2655

37

+ 052)

times [1 + 1124

times (37 times 49 times 10

minus4

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)])

minus1

= 06641 productsminute

(8)

Productivity rate with different station failure rate is

119876dfr =1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 ([1 + 1124

times (72 times 10

minus3

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

= 04911 productsminute

(9)

Percentage error for 119876dfr with 119876ac

=

|04911 minus 04751|

04751

times 100 = 337(10)

Percentage error for 119876afr with 119876ac

=

|06641 minus 04751|

04751

times 100 = 3978(11)

From the productivity rate result in (8) and (9) (7) whichis the actual productivity rate is used as guideline to checkwhich mathematical model can provide more accurate resultto actual result and the percentage error is calculated in(10) and (11) Regarding result of (10) and (11) it presentsthe percentage error when compared to actual productivityrate result Productivity rate with different station failure rate119876dfr is just 337 of percentage error while productivity ratewith average station failure rate 119876afr obtained is 3978 ofpercentage error when compared to actual productivity rate119876ac The reason 119876dfr is not exactly same to actual is due tounexpected factors that occurred in line Hence it is provenand validated that mathematical model of productivity ratewith different station failure rate yields the most accurateresult which is less than 10 error when compared to the

Table 3 Reliability data for final assembly flow line in 31 workingshifts

Title 119902 120582s119894 (1min)

Failure rate of the stations 119902 120582s

1 1355 times 10minus4

2 8130 times 10minus4

3 4743 times 10minus4

4 8130 times 10minus4

5 00006 5420 times 10minus4

7 00008 00009 6775 times 10minus5

10 4065 times 10minus4

11 6775 times 10minus5

12 000013 000014 000015 000016 000017 4065 times 10minus4

18 6775 times 10minus5

19 000020 000021 6775 times 10minus5

22 000023 000024 000025 7453 times 10minus4

26 4065 times 10minus4

27 5420 times 10minus4

28 5420 times 10minus4

29 6775 times 10minus5

30 000031 8130 times 10minus4

32 000033 1355 times 10minus4

34 000035 000036 6775 times 10minus5

37 6775 times 10minus5

Total stations failure rate sum37119894=1

120582s119894 7200 times 10minus3

Average station failure rate 120582savg 4900 times 10minus4

Failure rate of the control system 120582119888

9485 times 10minus4

Failure rate of the transport system 120582tr 2033 times 10minus4

Failure rate of the defect parts 120582119889

3048 times 10minus2

Mean repair time119898119903

(min) 1124

actual result Hence mathematical model of productivity ratewith different station failure rate can be further used in moreapplications in the same area Figure 6 presents the differ-ences between 119876dfr and 119876afr with respect to the number ofstation to prove that bothmodels produce high differences InFigure 6 there are three stages which are A B and C which

Mathematical Problems in Engineering 7

0

01

02

03

04

05

06

07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

A

B

C

Prod

uctiv

ity (Q

)

Qafr

Qdfr

Number of station (q)

Figure 6 Comparison of different station failure rate productivity(119876dfr) with average station failure rate productivity (119876afr) withrespect to number of station

present difference between119876dfr and119876afr each 10 stations Instage A which is productivity station number 10 the differ-ence between both is 01078 But when going to stage B whichis 20 stations the difference is increased to 01537 and keepsincreasing to 01725 at 30 stations in stage C In this analysis itis proven that the difference of119876dfr and119876afr increases whenthe number of station increases In conclusion the higherthe quantity of stations in the automated line the higher thedeviation or inaccuracy of 119876afr to actual productivity

33 Analysis for Failure Rate In this analysis for failure ratestage each of the parameters that contributes to the lossesof productivity rate in final assembly automated flow line islisted down and the value of productivity rate losses due toparticular parameter is calculated Equation below presentsthe calculation to determine the losses in productivity ratevalue and Figure 6 shows the total productivity losses dia-gram of final assembly automated flow line

(a) Potential productivity

119870 =

1

(119905119898119900

119902)

=

1

(265537)

= 13936 productsminute

(12)

(b) Cyclic productivity

1198761

=

1

(119905119898119900

119902 + 119905119886

)

=

1

(265537 + 052)

= 08080 productsminute(13)

(c) Bottleneck parameter productivity

1198762

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

=

1

[(265537) (125) + 052]

= 07057 productsminute(14)

(d) Station failure rate productivity

1198763

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894)]

=

1

[(265537) (125) + 052]

times

1

1 + 1124 (72 times 10

minus3

)

= 06529 productsminute

(15)

(e) Control system failure rate productivity

1198764

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

)]

= 06462 productsminute

(16)

(f) Transport system failure rate productivity

1198765

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)]

= 06451 productsminute(17)

(g) Defect failure rate productivity

1198766

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 (1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

+ 3048 times 10

minus2

))

minus1

= 04911 productsminute(18)

8 Mathematical Problems in Engineering

K = 13936 Q1 = 08080 Qac = 04751

L1=05856

L2=01023

L3=00528

L4=00067

L5=00011

L6=01540

L7=00160

Figure 7 Total productivity losses diagram of final assemblyautomated flow line

(h) Actual productivity

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (19)

(i) Losses of productivity

1198711

= 119870 minus 1198761

= 13936 minus 08080

= 05856 productsminute

1198712

= 1198761

minus 1198762

= 08080 minus 07057

= 01023 productsminute

1198713

= 1198762

minus 1198763

= 07057 minus 06529

= 00528 productsminute

1198714

= 1198763

minus 1198764

= 06529 minus 06462

= 00067 productsminute

1198715

= 1198764

minus 1198765

= 06462 minus 06451

= 00011 productsminute

1198716

= 1198765

minus 1198766

= 06451 minus 04911

= 01540 productsminute

1198717

= 1198766

minus 119876ac = 04911 minus 04751

= 00160 productsminute

(20)

From total productivity losses diagram in Figure 7 and(10)ndash(18) all of the productivity losses parameter are pre-sented in mathematic calculation form and show in num-bering result 119871

1

is the losses of productivity due to auxiliarytime which is 05856 productsminute and this parameter isthe highest factor that contributes to the loss of productivityin final assembly automated flow line 119871

2

is caused bybottleneck machining time in line which contributes 01023productsminute 119871

3

is the loss of 00528 productsminutedue to station failure rate 119871

4

is the factor of control systemwhich is 00067 productsminute 119871

5

is the lowest contributor

factor to productivity losses due to transport problem 1198716

is 01540 productsminute which contribute to the secondhighest of losses of productivity due to the defect in rawmaterial and finish goods and last is 119871

7

which is causedby unexpected factor such as electrical breakdown thatcontributes 00160 productsminute

After complete analyzing of the overall failure rate thatcauses loss of productivity workstation within the finalassembly automated flow line is required to determine theideal number of station to obtainmaximumproductivity rateFor productivity with station calculation (19) applied themathematical model with different station failure rate but theparameter of station failure is changed to average failure ratedue to lack of data of failure for the 37 stations above Formaximum productivity the ideal number of station is deter-mined by using (20) All of calculations are present in Figure7The range of 119902 station is from 10 stations until 300 stations

119876119902

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(120582savg + 120582119888 + 120582tr + 120582119889)]

=

1

[(2655119902) (125) + 052]

times 1 ([1 + 1124

times (49 times 10

minus4

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

119902opt = radic119905119898119900

119891119898

(1119898119903

) + 120582119888

+ 120582tr + 120582119889119905119886

120582119904

= ((2655 times 125

times [(11124) + 9485 times 10

minus4

+ 3048 times 10

minus2

])

times (052 times 49 times 10

minus4

)

minus1

)

12

= 12533 = 126 stations

(21)

From Figure 8 stage A shows that the current pro-ductivity is 04715 productsminute at station 37 Throughcalculation in (20) and particular figure the maximumproductivity that can be produced by 126 stations is 06228productsminute which presents in stage CThe productivitystarts to drop after station 126 even with the increment ofstation From the trend of the productivity rate productivityis increasing drastically from stage A until stage B whenthe number of station is increased Even from stage B to Cthe productivity keeps on increasing but the increment isonly improved for around 5 productivity with incrementof 50 stations From profit consideration it is not worth thatincrement of around 50 stations in automated flow line foronly 5 increment of productivity Thus the adjustment of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

070 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

A B C

Prod

uctiv

ity ra

te (Q

)

Number of station (q)

Figure 8 Station analysis with productivity rate versus number ofstation

Ease

of i

mpl

emen

tatio

nD

ifficu

ltEa

sy

BenefitsHigh Low

Priority Action

Consider Eliminate

Defect parts Bottleneck

Control and transport system

Stations failure rateAuxiliary time

Unexpected factor

Figure 9 PACE prioritization matrix for final assembly automatedflow line

stations from 37 stations can increase up to 80 stations toimprove the productivity profitably

34 Sustainable Improvement Final stage inDCAS engineer-ing mathematical analysis method is sustainable improve-ment Improvement process is selected by PACE prioritiza-tion matrix as shown in Figure 9 and this is the result from aclosed group discussion with Production Manager and Engi-neer in ABC Company Defect part parameter is preferred tobe improved due to its high impact on productivity losses andbecause it is easier to improve than others

4 Conclusion

After applying DCAS engineering mathematical analysismethod in ABC Company with the help of mathematicalmodel of productivity rate with different reliability all of theproductivity rate losses parameters are calculated measuredand analyzed with mathematical figures The losses andproductivity of final assembly automated flow line can beanalyzed visually clearly systematically and effectively whichis much better than just simply analyzing previous productiv-ity In conclusion mathematical model of productivity rate

with different station failure rate is proven and validatedto be useful and more accurate than previous productivitymathematical model which obtains 337 error or more than90 accuracy It is proved that new mathematical modelproduces higher accuracy result for productivity rate fore-casting DCAS engineering mathematical analysis method isalso proven to be systematic and effective to be applied inlinear arrangement serial action automated flow line whichis the final assembly line in ABC Company to calculateand analyze the productivity rate in that particular line Theproductivity losses parameters are determined and improvedsystematically with the help of mathematical model of pro-ductivity rate with different station failure rate In the futurethis mathematical model of productivity rate with differentstation failure rate and DCAS engineering mathematicalanalysis method can be applied to more field of sector suchas food processing or machining automated line since thismathematical model and method can be applied generally inlinear arrangement serial structure automated flow line

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Tan Chan Sin the main author of this paper would liketo thank the PhD degree supervisor Professor RyspekUsubamatov for his excellent guidance One of the authorswould also like to express appreciation to the authorrsquosresearch partners Mr Mohd Fidzwan B Md Amin HamzasM A Fairuz and Low Kin Wai who always help in solvingsome technical problems

References

[1] T Freiheit M Shpitalni and S J Hu ldquoProductivity of pacedparallel-serial manufacturing lines with and without crossoverrdquoJournal of Manufacturing Science and Engineering vol 126 no2 pp 361ndash367 2004

[2] R Usubamatov M Z Abdulmuin A M Nor and M NMurad ldquoProductivity rate of rotor-type automated lines andoptimisation of their structurerdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 222 no 11 pp 1561ndash1566 2008

[3] R Usubamatov T C Sin andM F BMd AminHamzas ldquoPro-ductivity theory for industrial automated linesrdquo in Proceedingsof the Mechanical Engineering Congress amp Exhibition pp 1ndash11ASME 2013

[4] S Saari ldquoTheory and measurement in businessrdquo in Proceedingsof the International Conference on Concurrent Engineering inConstruction no 110ndash1010 Espoo Finland 2006

[5] S Saari ldquoProduction and productivity as souces of well-beingrdquo2011

[6] R Usubamatov K A Ismail and J M Sah ldquoMathematicalmodels for productivity and availability of automated linesrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 59ndash69 2013

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0

01

02

03

04

05

06

07

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

A

B

C

Prod

uctiv

ity (Q

)

Qafr

Qdfr

Number of station (q)

Figure 6 Comparison of different station failure rate productivity(119876dfr) with average station failure rate productivity (119876afr) withrespect to number of station

present difference between119876dfr and119876afr each 10 stations Instage A which is productivity station number 10 the differ-ence between both is 01078 But when going to stage B whichis 20 stations the difference is increased to 01537 and keepsincreasing to 01725 at 30 stations in stage C In this analysis itis proven that the difference of119876dfr and119876afr increases whenthe number of station increases In conclusion the higherthe quantity of stations in the automated line the higher thedeviation or inaccuracy of 119876afr to actual productivity

33 Analysis for Failure Rate In this analysis for failure ratestage each of the parameters that contributes to the lossesof productivity rate in final assembly automated flow line islisted down and the value of productivity rate losses due toparticular parameter is calculated Equation below presentsthe calculation to determine the losses in productivity ratevalue and Figure 6 shows the total productivity losses dia-gram of final assembly automated flow line

(a) Potential productivity

119870 =

1

(119905119898119900

119902)

=

1

(265537)

= 13936 productsminute

(12)

(b) Cyclic productivity

1198761

=

1

(119905119898119900

119902 + 119905119886

)

=

1

(265537 + 052)

= 08080 productsminute(13)

(c) Bottleneck parameter productivity

1198762

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

=

1

[(265537) (125) + 052]

= 07057 productsminute(14)

(d) Station failure rate productivity

1198763

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894)]

=

1

[(265537) (125) + 052]

times

1

1 + 1124 (72 times 10

minus3

)

= 06529 productsminute

(15)

(e) Control system failure rate productivity

1198764

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

)]

= 06462 productsminute

(16)

(f) Transport system failure rate productivity

1198765

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr)]

=

1

[(265537) (125) + 052]

times

1

[1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

)]

= 06451 productsminute(17)

(g) Defect failure rate productivity

1198766

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(sum

37

119894=1

120582s119894 + 120582119888 + 120582tr + 120582119889)]

=

1

[(265537) (125) + 052]

times 1 (1 + 1124 (72 times 10

minus3

+ 9485 times 10

minus4

+ 2033 times 10

minus4

+ 3048 times 10

minus2

))

minus1

= 04911 productsminute(18)

8 Mathematical Problems in Engineering

K = 13936 Q1 = 08080 Qac = 04751

L1=05856

L2=01023

L3=00528

L4=00067

L5=00011

L6=01540

L7=00160

Figure 7 Total productivity losses diagram of final assemblyautomated flow line

(h) Actual productivity

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (19)

(i) Losses of productivity

1198711

= 119870 minus 1198761

= 13936 minus 08080

= 05856 productsminute

1198712

= 1198761

minus 1198762

= 08080 minus 07057

= 01023 productsminute

1198713

= 1198762

minus 1198763

= 07057 minus 06529

= 00528 productsminute

1198714

= 1198763

minus 1198764

= 06529 minus 06462

= 00067 productsminute

1198715

= 1198764

minus 1198765

= 06462 minus 06451

= 00011 productsminute

1198716

= 1198765

minus 1198766

= 06451 minus 04911

= 01540 productsminute

1198717

= 1198766

minus 119876ac = 04911 minus 04751

= 00160 productsminute

(20)

From total productivity losses diagram in Figure 7 and(10)ndash(18) all of the productivity losses parameter are pre-sented in mathematic calculation form and show in num-bering result 119871

1

is the losses of productivity due to auxiliarytime which is 05856 productsminute and this parameter isthe highest factor that contributes to the loss of productivityin final assembly automated flow line 119871

2

is caused bybottleneck machining time in line which contributes 01023productsminute 119871

3

is the loss of 00528 productsminutedue to station failure rate 119871

4

is the factor of control systemwhich is 00067 productsminute 119871

5

is the lowest contributor

factor to productivity losses due to transport problem 1198716

is 01540 productsminute which contribute to the secondhighest of losses of productivity due to the defect in rawmaterial and finish goods and last is 119871

7

which is causedby unexpected factor such as electrical breakdown thatcontributes 00160 productsminute

After complete analyzing of the overall failure rate thatcauses loss of productivity workstation within the finalassembly automated flow line is required to determine theideal number of station to obtainmaximumproductivity rateFor productivity with station calculation (19) applied themathematical model with different station failure rate but theparameter of station failure is changed to average failure ratedue to lack of data of failure for the 37 stations above Formaximum productivity the ideal number of station is deter-mined by using (20) All of calculations are present in Figure7The range of 119902 station is from 10 stations until 300 stations

119876119902

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(120582savg + 120582119888 + 120582tr + 120582119889)]

=

1

[(2655119902) (125) + 052]

times 1 ([1 + 1124

times (49 times 10

minus4

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

119902opt = radic119905119898119900

119891119898

(1119898119903

) + 120582119888

+ 120582tr + 120582119889119905119886

120582119904

= ((2655 times 125

times [(11124) + 9485 times 10

minus4

+ 3048 times 10

minus2

])

times (052 times 49 times 10

minus4

)

minus1

)

12

= 12533 = 126 stations

(21)

From Figure 8 stage A shows that the current pro-ductivity is 04715 productsminute at station 37 Throughcalculation in (20) and particular figure the maximumproductivity that can be produced by 126 stations is 06228productsminute which presents in stage CThe productivitystarts to drop after station 126 even with the increment ofstation From the trend of the productivity rate productivityis increasing drastically from stage A until stage B whenthe number of station is increased Even from stage B to Cthe productivity keeps on increasing but the increment isonly improved for around 5 productivity with incrementof 50 stations From profit consideration it is not worth thatincrement of around 50 stations in automated flow line foronly 5 increment of productivity Thus the adjustment of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

070 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

A B C

Prod

uctiv

ity ra

te (Q

)

Number of station (q)

Figure 8 Station analysis with productivity rate versus number ofstation

Ease

of i

mpl

emen

tatio

nD

ifficu

ltEa

sy

BenefitsHigh Low

Priority Action

Consider Eliminate

Defect parts Bottleneck

Control and transport system

Stations failure rateAuxiliary time

Unexpected factor

Figure 9 PACE prioritization matrix for final assembly automatedflow line

stations from 37 stations can increase up to 80 stations toimprove the productivity profitably

34 Sustainable Improvement Final stage inDCAS engineer-ing mathematical analysis method is sustainable improve-ment Improvement process is selected by PACE prioritiza-tion matrix as shown in Figure 9 and this is the result from aclosed group discussion with Production Manager and Engi-neer in ABC Company Defect part parameter is preferred tobe improved due to its high impact on productivity losses andbecause it is easier to improve than others

4 Conclusion

After applying DCAS engineering mathematical analysismethod in ABC Company with the help of mathematicalmodel of productivity rate with different reliability all of theproductivity rate losses parameters are calculated measuredand analyzed with mathematical figures The losses andproductivity of final assembly automated flow line can beanalyzed visually clearly systematically and effectively whichis much better than just simply analyzing previous productiv-ity In conclusion mathematical model of productivity rate

with different station failure rate is proven and validatedto be useful and more accurate than previous productivitymathematical model which obtains 337 error or more than90 accuracy It is proved that new mathematical modelproduces higher accuracy result for productivity rate fore-casting DCAS engineering mathematical analysis method isalso proven to be systematic and effective to be applied inlinear arrangement serial action automated flow line whichis the final assembly line in ABC Company to calculateand analyze the productivity rate in that particular line Theproductivity losses parameters are determined and improvedsystematically with the help of mathematical model of pro-ductivity rate with different station failure rate In the futurethis mathematical model of productivity rate with differentstation failure rate and DCAS engineering mathematicalanalysis method can be applied to more field of sector suchas food processing or machining automated line since thismathematical model and method can be applied generally inlinear arrangement serial structure automated flow line

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Tan Chan Sin the main author of this paper would liketo thank the PhD degree supervisor Professor RyspekUsubamatov for his excellent guidance One of the authorswould also like to express appreciation to the authorrsquosresearch partners Mr Mohd Fidzwan B Md Amin HamzasM A Fairuz and Low Kin Wai who always help in solvingsome technical problems

References

[1] T Freiheit M Shpitalni and S J Hu ldquoProductivity of pacedparallel-serial manufacturing lines with and without crossoverrdquoJournal of Manufacturing Science and Engineering vol 126 no2 pp 361ndash367 2004

[2] R Usubamatov M Z Abdulmuin A M Nor and M NMurad ldquoProductivity rate of rotor-type automated lines andoptimisation of their structurerdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 222 no 11 pp 1561ndash1566 2008

[3] R Usubamatov T C Sin andM F BMd AminHamzas ldquoPro-ductivity theory for industrial automated linesrdquo in Proceedingsof the Mechanical Engineering Congress amp Exhibition pp 1ndash11ASME 2013

[4] S Saari ldquoTheory and measurement in businessrdquo in Proceedingsof the International Conference on Concurrent Engineering inConstruction no 110ndash1010 Espoo Finland 2006

[5] S Saari ldquoProduction and productivity as souces of well-beingrdquo2011

[6] R Usubamatov K A Ismail and J M Sah ldquoMathematicalmodels for productivity and availability of automated linesrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 59ndash69 2013

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

K = 13936 Q1 = 08080 Qac = 04751

L1=05856

L2=01023

L3=00528

L4=00067

L5=00011

L6=01540

L7=00160

Figure 7 Total productivity losses diagram of final assemblyautomated flow line

(h) Actual productivity

119876ac =119911

120579

=

7013

14760

= 04751 productsminute (19)

(i) Losses of productivity

1198711

= 119870 minus 1198761

= 13936 minus 08080

= 05856 productsminute

1198712

= 1198761

minus 1198762

= 08080 minus 07057

= 01023 productsminute

1198713

= 1198762

minus 1198763

= 07057 minus 06529

= 00528 productsminute

1198714

= 1198763

minus 1198764

= 06529 minus 06462

= 00067 productsminute

1198715

= 1198764

minus 1198765

= 06462 minus 06451

= 00011 productsminute

1198716

= 1198765

minus 1198766

= 06451 minus 04911

= 01540 productsminute

1198717

= 1198766

minus 119876ac = 04911 minus 04751

= 00160 productsminute

(20)

From total productivity losses diagram in Figure 7 and(10)ndash(18) all of the productivity losses parameter are pre-sented in mathematic calculation form and show in num-bering result 119871

1

is the losses of productivity due to auxiliarytime which is 05856 productsminute and this parameter isthe highest factor that contributes to the loss of productivityin final assembly automated flow line 119871

2

is caused bybottleneck machining time in line which contributes 01023productsminute 119871

3

is the loss of 00528 productsminutedue to station failure rate 119871

4

is the factor of control systemwhich is 00067 productsminute 119871

5

is the lowest contributor

factor to productivity losses due to transport problem 1198716

is 01540 productsminute which contribute to the secondhighest of losses of productivity due to the defect in rawmaterial and finish goods and last is 119871

7

which is causedby unexpected factor such as electrical breakdown thatcontributes 00160 productsminute

After complete analyzing of the overall failure rate thatcauses loss of productivity workstation within the finalassembly automated flow line is required to determine theideal number of station to obtainmaximumproductivity rateFor productivity with station calculation (19) applied themathematical model with different station failure rate but theparameter of station failure is changed to average failure ratedue to lack of data of failure for the 37 stations above Formaximum productivity the ideal number of station is deter-mined by using (20) All of calculations are present in Figure7The range of 119902 station is from 10 stations until 300 stations

119876119902

=

1

((119905119898119900

119902) 119891119898

+ 119905119886

)

times

1

[1 + 119898119903

(120582savg + 120582119888 + 120582tr + 120582119889)]

=

1

[(2655119902) (125) + 052]

times 1 ([1 + 1124

times (49 times 10

minus4

+ 9485 times 10

minus4

+2033 times 10

minus4

+ 3048 times 10

minus2

)])

minus1

119902opt = radic119905119898119900

119891119898

(1119898119903

) + 120582119888

+ 120582tr + 120582119889119905119886

120582119904

= ((2655 times 125

times [(11124) + 9485 times 10

minus4

+ 3048 times 10

minus2

])

times (052 times 49 times 10

minus4

)

minus1

)

12

= 12533 = 126 stations

(21)

From Figure 8 stage A shows that the current pro-ductivity is 04715 productsminute at station 37 Throughcalculation in (20) and particular figure the maximumproductivity that can be produced by 126 stations is 06228productsminute which presents in stage CThe productivitystarts to drop after station 126 even with the increment ofstation From the trend of the productivity rate productivityis increasing drastically from stage A until stage B whenthe number of station is increased Even from stage B to Cthe productivity keeps on increasing but the increment isonly improved for around 5 productivity with incrementof 50 stations From profit consideration it is not worth thatincrement of around 50 stations in automated flow line foronly 5 increment of productivity Thus the adjustment of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

070 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

A B C

Prod

uctiv

ity ra

te (Q

)

Number of station (q)

Figure 8 Station analysis with productivity rate versus number ofstation

Ease

of i

mpl

emen

tatio

nD

ifficu

ltEa

sy

BenefitsHigh Low

Priority Action

Consider Eliminate

Defect parts Bottleneck

Control and transport system

Stations failure rateAuxiliary time

Unexpected factor

Figure 9 PACE prioritization matrix for final assembly automatedflow line

stations from 37 stations can increase up to 80 stations toimprove the productivity profitably

34 Sustainable Improvement Final stage inDCAS engineer-ing mathematical analysis method is sustainable improve-ment Improvement process is selected by PACE prioritiza-tion matrix as shown in Figure 9 and this is the result from aclosed group discussion with Production Manager and Engi-neer in ABC Company Defect part parameter is preferred tobe improved due to its high impact on productivity losses andbecause it is easier to improve than others

4 Conclusion

After applying DCAS engineering mathematical analysismethod in ABC Company with the help of mathematicalmodel of productivity rate with different reliability all of theproductivity rate losses parameters are calculated measuredand analyzed with mathematical figures The losses andproductivity of final assembly automated flow line can beanalyzed visually clearly systematically and effectively whichis much better than just simply analyzing previous productiv-ity In conclusion mathematical model of productivity rate

with different station failure rate is proven and validatedto be useful and more accurate than previous productivitymathematical model which obtains 337 error or more than90 accuracy It is proved that new mathematical modelproduces higher accuracy result for productivity rate fore-casting DCAS engineering mathematical analysis method isalso proven to be systematic and effective to be applied inlinear arrangement serial action automated flow line whichis the final assembly line in ABC Company to calculateand analyze the productivity rate in that particular line Theproductivity losses parameters are determined and improvedsystematically with the help of mathematical model of pro-ductivity rate with different station failure rate In the futurethis mathematical model of productivity rate with differentstation failure rate and DCAS engineering mathematicalanalysis method can be applied to more field of sector suchas food processing or machining automated line since thismathematical model and method can be applied generally inlinear arrangement serial structure automated flow line

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Tan Chan Sin the main author of this paper would liketo thank the PhD degree supervisor Professor RyspekUsubamatov for his excellent guidance One of the authorswould also like to express appreciation to the authorrsquosresearch partners Mr Mohd Fidzwan B Md Amin HamzasM A Fairuz and Low Kin Wai who always help in solvingsome technical problems

References

[1] T Freiheit M Shpitalni and S J Hu ldquoProductivity of pacedparallel-serial manufacturing lines with and without crossoverrdquoJournal of Manufacturing Science and Engineering vol 126 no2 pp 361ndash367 2004

[2] R Usubamatov M Z Abdulmuin A M Nor and M NMurad ldquoProductivity rate of rotor-type automated lines andoptimisation of their structurerdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 222 no 11 pp 1561ndash1566 2008

[3] R Usubamatov T C Sin andM F BMd AminHamzas ldquoPro-ductivity theory for industrial automated linesrdquo in Proceedingsof the Mechanical Engineering Congress amp Exhibition pp 1ndash11ASME 2013

[4] S Saari ldquoTheory and measurement in businessrdquo in Proceedingsof the International Conference on Concurrent Engineering inConstruction no 110ndash1010 Espoo Finland 2006

[5] S Saari ldquoProduction and productivity as souces of well-beingrdquo2011

[6] R Usubamatov K A Ismail and J M Sah ldquoMathematicalmodels for productivity and availability of automated linesrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 59ndash69 2013

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

0

01

02

03

04

05

06

070 10 20 30 40 50 60 70 80 90 100

110

120

130

140

150

160

170

180

190

200

210

220

230

240

250

A B C

Prod

uctiv

ity ra

te (Q

)

Number of station (q)

Figure 8 Station analysis with productivity rate versus number ofstation

Ease

of i

mpl

emen

tatio

nD

ifficu

ltEa

sy

BenefitsHigh Low

Priority Action

Consider Eliminate

Defect parts Bottleneck

Control and transport system

Stations failure rateAuxiliary time

Unexpected factor

Figure 9 PACE prioritization matrix for final assembly automatedflow line

stations from 37 stations can increase up to 80 stations toimprove the productivity profitably

34 Sustainable Improvement Final stage inDCAS engineer-ing mathematical analysis method is sustainable improve-ment Improvement process is selected by PACE prioritiza-tion matrix as shown in Figure 9 and this is the result from aclosed group discussion with Production Manager and Engi-neer in ABC Company Defect part parameter is preferred tobe improved due to its high impact on productivity losses andbecause it is easier to improve than others

4 Conclusion

After applying DCAS engineering mathematical analysismethod in ABC Company with the help of mathematicalmodel of productivity rate with different reliability all of theproductivity rate losses parameters are calculated measuredand analyzed with mathematical figures The losses andproductivity of final assembly automated flow line can beanalyzed visually clearly systematically and effectively whichis much better than just simply analyzing previous productiv-ity In conclusion mathematical model of productivity rate

with different station failure rate is proven and validatedto be useful and more accurate than previous productivitymathematical model which obtains 337 error or more than90 accuracy It is proved that new mathematical modelproduces higher accuracy result for productivity rate fore-casting DCAS engineering mathematical analysis method isalso proven to be systematic and effective to be applied inlinear arrangement serial action automated flow line whichis the final assembly line in ABC Company to calculateand analyze the productivity rate in that particular line Theproductivity losses parameters are determined and improvedsystematically with the help of mathematical model of pro-ductivity rate with different station failure rate In the futurethis mathematical model of productivity rate with differentstation failure rate and DCAS engineering mathematicalanalysis method can be applied to more field of sector suchas food processing or machining automated line since thismathematical model and method can be applied generally inlinear arrangement serial structure automated flow line

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Tan Chan Sin the main author of this paper would liketo thank the PhD degree supervisor Professor RyspekUsubamatov for his excellent guidance One of the authorswould also like to express appreciation to the authorrsquosresearch partners Mr Mohd Fidzwan B Md Amin HamzasM A Fairuz and Low Kin Wai who always help in solvingsome technical problems

References

[1] T Freiheit M Shpitalni and S J Hu ldquoProductivity of pacedparallel-serial manufacturing lines with and without crossoverrdquoJournal of Manufacturing Science and Engineering vol 126 no2 pp 361ndash367 2004

[2] R Usubamatov M Z Abdulmuin A M Nor and M NMurad ldquoProductivity rate of rotor-type automated lines andoptimisation of their structurerdquo Proceedings of the Institution ofMechanical Engineers Part B Journal of Engineering Manufac-ture vol 222 no 11 pp 1561ndash1566 2008

[3] R Usubamatov T C Sin andM F BMd AminHamzas ldquoPro-ductivity theory for industrial automated linesrdquo in Proceedingsof the Mechanical Engineering Congress amp Exhibition pp 1ndash11ASME 2013

[4] S Saari ldquoTheory and measurement in businessrdquo in Proceedingsof the International Conference on Concurrent Engineering inConstruction no 110ndash1010 Espoo Finland 2006

[5] S Saari ldquoProduction and productivity as souces of well-beingrdquo2011

[6] R Usubamatov K A Ismail and J M Sah ldquoMathematicalmodels for productivity and availability of automated linesrdquoInternational Journal of Advanced Manufacturing Technologyvol 66 no 1ndash4 pp 59ndash69 2013

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[7] W M Baek J H Yoon and C Kim ldquoModeling and analysis ofmobilitymanagement inmobile communicationnetworksrdquoTheScientific World Journal vol 2014 Article ID 250981 11 pages2014

[8] R Bighamian and J-O Hahn ldquoRelationship between strokevolume and pulse pressure during blood volume perturbationa mathematical analysisrdquo BioMed Research International vol2014 Article ID 459269 10 pages 2014

[9] J You and T Ando ldquoA statistical modeling methodology forthe analysis of term structure of credit risk and its dependencyrdquoExpert Systems with Applications vol 40 no 12 pp 4897ndash49052013

[10] J Guo L Monas and E Gill ldquoStatistical analysis andmodellingof small satellite reliabilityrdquo Acta Astronautica vol 98 no 1 pp97ndash110 2014

[11] C D Senanayake and V Subramaniam ldquoEstimating customerservice levels in automated multiple part-type production linesan analytical methodrdquo Computers and Industrial Engineeringvol 64 no 1 pp 109ndash121 2013

[12] G Muller ldquoSystems engineering research methodsrdquo ProcediaComputer Science vol 16 pp 1092ndash1101 2013

[13] T P Thyvalikakath M P Dziabiak R Johnson et al ldquoAdvanc-ing cognitive engineering methods to support user interfacedesign for electronic health recordsrdquo International Journal ofMedical Informatics vol 83 no 4 pp 292ndash302 2014

[14] T Ruiz-Lopez M Noguera M J Rodrıguez J L Garrido andL Chung ldquoREUBI a Requirements Engineering method forubiquitous systemsrdquo Science of Computer Programming vol 78no 10 pp 1895ndash1911 2013

[15] G D Kirk B S Linas R P Westergaard et al ldquoThe exposureassessment in current time study implementation feasibilityand acceptability of real-time data collection in a communitycohort of illicit drug usersrdquo AIDS Research and Treatment vol2013 Article ID 594671 10 pages 2013

[16] R Usubamatov A R Riza and M N Murad ldquoA methodfor assessing productivity in unbuffered assembly processesrdquoJournal of Manufacturing Technology Management vol 24 no1 pp 123ndash139 2013

[17] H-Z Quan and H Kasami ldquoExperimental study on durabilityimprovement of fly ash concrete with durability improvingadmixturerdquo The Scientific World Journal vol 2014 11 pages2014

[18] G S Kaplan ldquoValue stream mappingmdashPACE matrixrdquo inAdvanced Lean Thinking Proven Methods to Reduce Waste andImprove Quality in Health Care H M Fry Ed p 29 JointComission Resource 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of