Research Article A New Approach for Wage Management System

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 207032 11 pageshttpdxdoiorg1011552013207032

Research ArticleA New Approach for Wage Management System Using FuzzyBrackets in Industry

Erguumln Eraslan1 and Kumru Didem Atalay2

1 Department of Industrial Engineering Yıldırım Beyazıt University Cicek Sokak No 3 Ulus 06050 Ankara Turkey2Department of Medical Education Baskent University Bahcelievler 06490 Ankara Turkey

Correspondence should be addressed to Ergun Eraslan erguneraslangmailcom

Received 25 April 2013 Accepted 23 September 2013

Academic Editor Yuri Vladimirovich Mikhlin

Copyright copy 2013 E Eraslan and K D Atalay This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Job evaluation is used to determine the relative importance of each job in a company in order to structure an accurate wagemanagement system Job evaluation can be also defined as a multicriteria decision-making problem However according to thediversity of managersrsquo assessment the evaluation processes are often resulting in pay inequity This outcome can be circumventedby utilizing a fuzzy job evaluation system In this study one of the more robust multicriteria decision-making methods FuzzyAnalytic Hierarchy Process (FAHP) is performed in job evaluation system in order to rank predetermined 13 criteria The fuzzywage brackets are developed and inserted into the process which is obtained from the results of mathematical model to designatethe bounds for predefined 86 jobs Eventually an accurate payment system is proposed for a company in steel industry by usingFuzzy Regression Analysis (FRA)

1 Introduction

Job evaluation is one of the most important functions ofHuman Resources Management and a systematic techniquethat enables the design establishment and improvement ofhuman resources [1] These processes include establishinga payment system and managing personnel contracts andagreements related to their jobs In an organizational struc-ture different kinds of jobs exist and each differs in theirspecifications and contributions to the company The worthof each job has to differ from one to another in respect ofthese features The subject of job evaluation consists of thisrelative worth determination [2]

A job evaluation system is comprised of twomajor issues(1) evaluation of the job specification and (2) equal pay forequal work These issues provide the labor productivity jobroyalty and job satisfaction for employees It is also functionalfor the managers in order to do select appropriate personnelto create the division of the labor and organizations to securethe occupational health and safety and so forth

There are a few job evaluation techniques in the literatureThe techniques are divided into two categoriesOne of them is

nonanalytical techniques which are the Sequencing Methodand the Classification Method These techniques are per-formed when there are few numbers of jobs to be evaluatedfor the easiness of the application and in the developmentphase of works To avoid the subjective assessment in thecompanies where there are a large number of workers theanalytical methods should be preferred which are the FactorComparison Method and the Point Method [3]

The methods used in the job evaluation include rankingclassification factor comparison job components and pointsystem [4] The well-known and maybe the most reliablemethod is the Point Method This method is widely used indifferent areas because it achieves more accurate results [5]In this method used in this study jobs are rated on a set offactors and the total scores are calculated for all the jobs [6]In the wage brackets the seniority success and some socialexpenses are included to designate the real payment system[3] The Point Method follows some steps below [7]

(i) construction of the expert evaluation group(ii) preparation of the work plan and coverage(iii) preliminary designs

2 Mathematical Problems in Engineering

(iv) educational studies(v) work analysis(vi) determination of the point plan table(vii) calculation of the factor values

Some researchers have considered the pay inequity as beingthe most important term of job evaluation that is equal payfor equal work [8] Smith et al [9] put forth a critique aboutthe effects of job description content on the job evaluationjudgments In the studies over recent decades Spyridakoset al [1] studied the multicriteria job evaluation for largeorganizations and Das and Garcia-Diaz [7] proposed acomputerized statistical procedure in factor selection guide-lines for the job evaluation Gupta and Ahmed [10] andDagdeviren et al [11] used a goal programming approach andXing et al [5] used the Analytic Hierarchy Process (AHP)evaluation for the problem of factor weights determinationTherefore game theory is the traditional solution to the manyproblems (Perc et al) [12 13]

The job evaluation could be defined as a managerialdecision-making problem with its multiple objectives [14]Scoring these objectives is a difficult task for the decisionmakers because many different scores can emerge within agroup of managers Therefore the decision-making processshould be considered as a fuzzy process in naturally With afuzzy approach varying opinions of the managers could beconsidered in the evaluation systemThereby the subjectivityof the decision makers could be prevented

The fuzzy theory allows bothmathematical operators andfuzzy programming to apply the fuzzy domain Fuzzy set isa class of objects which uses a membership (characteristic)function and it assigns a grade of membership rangingbetween zero-one to each object There are many rankingmethods developed in the literature for the fuzzy numbersThese methods may generate different ranking results andmost of them require complexmathematical calculations [15]

In this study the fuzzyAHPmethod (FAHP) that ismoreappropriate due to its nature of fuzziness was used for the jobevaluation systems [16] During the evaluation processes ofeach job fuzziness was obtained by using the fuzzy pairwisecomparisons in the judgmentmatrices Hence more accurateweights could be obtained in order to use for factor scoringTo improve themodel the fuzzywage brackets are consideredand embedded into the decision process with the result ofmathematical model which sets the upper and lower boundsThen by using Fuzzy Regression Analysis (FRA) an accuratepayment system is proposed for the first time for a steelcompany

2 The Techniques Used in This StudyFAHP and FRA

The FAHP is a provided systematic approache to the alter-native selection and justification problem by using the con-cepts of fuzzy set theory and hierarchical structure analysisDecisionmakers are usuallymore confident in giving intervaljudgments rather than crisp judgments due to the fuzzynature of the comparison process

The earliest work in FAHP appeared in Van Laarhovenand Pedrycz [17] which compared fuzzy ratios described bytriangular membership functions Buckley [18] determinedfuzzy priorities of comparison ratios whose membershipfunctions were trapezoidal Chang [19] introduced a newapproach for handling FAHP with the use of triangular fuzzynumbers for pairwise comparison scale of FAHP and the useof the extent analysis method for the synthetic extent valuesof pairwise comparisons The outcome of Chengrsquos algorithm[20] can finally be defuzzified by forming the surface centre ofgravity of any fuzzy set Kahraman et al [21ndash23] used a fuzzyobjective and subjective method obtaining the weights fromAHP and producing a fuzzy weighted evaluation RecentlyBuyukozkan et al [24] Tolga et al [25] and Kulak andKahraman [26] applied the same approach to some selectionproblems In this study FAHP method is used to calculatelocal weights using detailed information about Chang et alrsquos[27 28] extent analysis (see the appendix)

FRA was first introduced by Tanaka et al in 1982 [29]In their study a regression problem with fuzzy output andcrisp input was formulated as a mathematical program-ming problem The objective was to minimize the totalspreads of the fuzzy parameters subject to the constraintthat the regression model needed to satisfy a prespecifiedmembership value in estimating the fuzzy outputs Differentresearchers used Tanakarsquos approach to minimize the totalspread of the output [30 31] Sakawa and Yano [32] Kimand Bishu [33] and Peters [31] considered the possibility andnecessity conditions for fuzzy equality Tanaka and Lee [34]proposed an interval regression analysis based on quadraticprogramming approach Azadeh et al [35] applied FRA forenergy forecasting problems Tseng and Hu [36] proposeda quadratic-interval Bass model that combines quadratic-interval regression with the Bass innovation diffusion modelto solve a fuzzy relationship between explanatory and outputvariables and to provide forecasts of sales to decision makers

Fuzzy regression models have been successfully appliedto various engineering and technological problems [30 37ndash42] It can also be applied for the estimation of the fuzzyjob evaluation process in the manufacturing companiesAdditionally instead of the crisp values of wage bracketsobtaining a fuzzy interval is more relevant to the real lifeproblems which are why the linguistic variables are usedcommonly FRA has been shown to be more flexible andrigorous than the traditional (nonfuzzy) approach [43 44]

The aim of regression analysis is to find an appropri-ate mathematical model and to determine the best fittingcoefficients of the model from the given data This classicalregression technique is useful in a nonfuzzy environmentwhere the relationship among variables is crisply definedHowever it is very difficult to obtain an exact relationThe use of statistical regression is restricted by some strictassumptions and statistical properties about the given dataFuzzy regression does not require normal distributed datastability tests and large samples It can be applied to manyreal life problems in which the strict assumptions of statisticalregression analysis cannot be met FRA considers the use offuzzy numbers that involves the modeling of problems wherethe output variable is affected by imprecisionThe goal of FRA

Mathematical Problems in Engineering 3

is to find a regression model that fits all observed fuzzy datawithin a specified fitting criterion

The basic model that is developed by Tanaka et al [29]model assumes a fuzzy linear regression function as

119894= 11986001198830+ 11986011198831+ 11986021198832+ sdot sdot sdot + 119860

119899119883119899= 119860119883 (1)

where 119883 = [1198830 1198831 119883

119899]119879 is an input vector 119860 =

[1198600 1198601 119860

119899] is a vector of fuzzy coefficients in the formof

symmetric triangular fuzzy number denoted by119860119895= (120572119895 119888119895)

where 120572119895is its central value and 119888

119895is the spread value and

119888119895ge 0 Therefore (1) can be rewritten as

119894= (1205720 1198880)1198830+ (1205721 1198881)1198831+ sdot sdot sdot + (120572

119899 119888119899)119883119899 (2)

The estimated output 119894can be obtained by using the

extension principle [45] The derived membership functionof fuzzy number

119894can be stated as

120583119894(119910119894) =

1 minus

10038161003816100381610038161003816119884119894minus 119883119879120572

10038161003816100381610038161003816

119888119879 |119883| 119883 = 0

1 119883 = 0 119884119894= 0

0 119883 = 0 119884119894= 0

(3)

where 119888119879 = (1198880 1198881 119888

119899) and 120572 = (120572

0 1205721 120572

119899) The

above FRAassumes that input and output data are crisp whilethe relation between the input and output data is defined byfuzzy function of which the distribution of the parameter is apossibility function [46]

To determine the optimal fuzzy coefficients 119860119895= (120572119895 119888119895)

of the fuzzy regression model (2) the sum of spreads of theestimated outputs was used as an objective function in linearprogramming (LP) problems [29 46] that has an objectivefunction as

min 119869 = 119888119879 |119883|

min 119869 =119899

sum119895=1

(119888119895

119898

sum119894=1

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816)

(4)

where the 119898 collected input data are defined as 119909119894119895(119894 =

1 119898 119895 = 1 119899) of 119899 independent variables Thisapproach is called the LP approach in fuzzy regressionanalysisTheobjective function given in (4) is tominimize thetotal spread of the fuzzy number

119894 The constraints require

that each observation 119884119894has at least ℎ degree of belonging to

119894as 120583119894(119910119894) ge ℎ (119894 = 1 119898) which is equivalent to

1 minus

10038161003816100381610038161003816119884119894minus 119883119879120572

10038161003816100381610038161003816

119888119879 |119883|ge ℎ forall119894 = 1 2 119898 (5)

The above analysis leads to the following LP problem [29]

min 119869 =119899

sum119895=1

(119888119895

119898

sum119894=1

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816)

119899

sum119895=0

120572119895119909119894119895+ (1 minus ℎ)

119899

sum119895=0

119888119895

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816ge 119910119894 119894 = 1 119898

119899

sum119895=0

120572119895119909119894119895minus (1 minus ℎ)

119899

sum119895=0

119888119895

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816le 119910119894 119894 = 1 119898

119888119895ge 0 119895 = 0 119899 0 le ℎ le 1

(6)

3 A Proposed Job EvaluationModel with Fuzzy Wage BracketsAn Experimental Study

In this section a new job evaluation model is proposed forsteel industry and its steps are identified The usability of theproposed fuzzy model was tested in a steel manufacturingcompany For the application a decision committee wasestablished comprised of seven service managers and numer-ous responsible chiefs of the company each from differentdepartments The proposed job evaluation system for steelindustry using FAHP method follows the steps of Figure 1

Step 1 (identifying the factors and subfactors to be used in themodel) It was decided to use the factors and subfactors out-lined by the Turkish Trade Union of Metal Industrialists withsome revisions consistent with the related steel companyrsquosstructure A total of 13 subfactors are given below organizedinto 4 areas mastery responsibility workload and physicaland working conditionsMastery factors (MF)

(i) education and basic knowledge (EK)(ii) experience (EX)(iii) skill (SK)(iv) predisposition (PD)

Responsibility factors (RF)

(i) machineequipment responsibility (MR)(ii) materialproduct responsibility (PR)(iii) production continuity responsibility (CR)(iv) security providing responsibility (SR)

Workload factors (WF)

(i) mental effort (ME)(ii) physical effort (PE)

Physical and working conditions factors (PF)

(i) job risks (JR)(ii) working environment (WE)(iii) protective material necessity (PN)


Start

Determine the local weights of factorssubfactors

Calculate the global weights and normalized global weights

Determine the factors and subfactorsfor the job evaluation system

(pairwise comparison matrices)

Use the grade scale for the subfactors(geometric progression)

Evaluate the jobs with respect to the grade scale

(grade point and scale values)

Determine the point and wage intervals(fuzzy regression analysis)

Structure the wage brackets

Stop

Structure the FAHP model hierarchy

Setup goal

Figure 1 The steps of the algorithm of proposed job evaluation system

In this regard the proposed job evaluation system operatingwithin a wider framework is tested on contributing factors inthe evaluation studies

Step 2 (structuring the FAHPmodel hierarchically (objectivefactors subfactors)) The FAHP model is composed of 3stages In the first stage there is an objective of evaluating thejobs There are the factors and the subfactors related to themin second and third stages respectively Then the evaluationof the jobs with FRA and the structuring of the fuzzy wagebrackets are involved in the fourth and the fifth stages Thishierarchy is shown in Figure 2

Step 3 (determination of the local weights of the factors andsubfactors by using fuzzy pairwise comparison matrices)The FAHP scale regarding relative importance tomeasure therelative weights of all the factors is given in Table 1

In this step local weights of the factors and subfactorswhich take part in the second and third levels of FAHPmodelprovided in Figure 2 are calculated Pairwise comparisonmatrices are formed by the decision committee by using thescale given in Table 1These matrices are analyzed by Changrsquosextend analysis method and local weights are calculated asseen in examples of Table 2 for main factors and Table 3 formastery subfactors

Step 4 (calculating the global weights for the subfactors)Global weights of subfactors are calculated bymultiplying thelocal weight of the sub-factor with the local weight of the

Table 1 The linguistic scale for determination of importance

Linguistic scale for importance Triangular fuzzy scaleEqual importance (EI) (1 1 1)Moderate importance (MI) (1 3 5)Strong importance (SI) (3 5 7)Very strong importance (VSI) (5 7 9)Demonstrated importance (DI) (7 9 11)

factor which belongs to (Table 4) In order to evaluate accord-ing to total 1000 points the global weights are normalized andgiven in the last column

According to the weights given in Table 4 the factorsof experience predisposition job risks and working envi-ronment are in the foreground among the 13 subfactorsrespectively

Step 5 (graduation of the subfactors for point system)Utilizing the normalized global weights the grades of thesubfactors in Table 5 are calculated using geometric progres-sionThe coefficient is used as 5 and progression multiplier iscalculated as 119899minus1radic119875max119875min where 119899 = 5 and119875max and119875min arethe maximum and the minimum points of jobs respectively

Step 6 (evaluation of the jobs by using the grade points ofsubfactors) The determined 86 jobs of steel company areevaluated using the grades obtained from Table 5 and the


Wei

ghin

g th

e fac

tors

in jo

b ev

alua

tion

Responsibility factors

Effort factors

Educationknowledge

Experience

Skill

Predisposition

Security providing

Goal

Factors

Scale

I

II

III

IV

V

Jobs

Job risks

Machineequipment

Materialproduct

Production continuity

Mental effort

Physical effort

Working environment

Mastery factors

Subfactors

Job 1

Job 2

Job 3

Job 86Physical and workingconditions factors

Protective materialnecessity

Figure 2 FAHP hierarchy for fuzzy job evaluation system

scale values and the total points of each job are calculated andgiven in Table 6 thereafter

Step 7 (determination of the point and wage intervals for theevaluated jobs using FRA) The total points are sequencedin ascending order the minimum of them is 241 and themaximum is 592 The range is divided into 5 intervals whichhave 5-point (119883

1 1198832 1198833 1198834 1198835) and 5-wage (119884

1 1198842 1198843

1198844 1198845) variables for using in fuzzy linear regression analysis

(FRA) taking into consideration the views of the decisioncommittee and the features of the steel sector The wageintervals of the jobs are decided with respect to the equivalentsteel companies in the market which are 600ndash1000 for 119883

1

1000ndash1500 for 1198832 1500ndash2000 for 119883

3 2000ndash3000 for 119883

4 and

3000ndash5000 for1198835

The calculated fuzzy linear regression models obtainedfrom the intervals for ℎ = 01 are given as follows where1198830= (1 1 1)

119879

1= (minus5993443 minus2076899)1198830 + (505964 02037769)1198831

2=(minus1432457 minus1391500)1198830 + (7688034 05175689)1198832

3= (minus1060654 2363248)1198830 + (675 002136752)1198833

4= (minus3749089 6432341)1198830 + (1298214 minus1130952)1198834

5=(minus1176960 minus9434264)1198830 + (2805211 05567617)1198835

(7)

Accordingly the fuzzy regressions for the levels of ℎ = 05

and ℎ = 09 are calculated as well Utilizing the estimatedfuzzy regression models the upper and lower bounds aregiven in Table 7 for three different ℎ-levels As an examplethe dispersions (the center the lower and the upper values)are diagrammatically demonstrated for the first interval inFigure 3

When Figure 3 is analyzed it is evident that theminimumdispersion is at level ℎ = 01 and increases towards ℎ = 09Therefore the managers of the steel companies are givenoptions to determine which plan is convenient for theirwaging policies

Step 8 (structuring the fuzzy wage brackets) Using theproposed fuzzy model it is seen that the range interval of jobpoints are formed between 241 and 592Then determined 86


Table 2 Local weights and pairwise comparison matrix of main factors

Main factors MF RF WF PF Local weightsMF (1 1 1) (1 3 5) (3 5 7) (1 3 5) 0423RF (15 13 1) (1 1 1) (1 3 5) (1 1 1) 0255WF (17 15 13) (15 13 1) (1 1 1) (15 13 1) 0067PF (15 13 1) (1 1 1) (1 3 5) (1 1 1) 0255

Table 3 Weights and pairwise comparison matrix of mastery subfactors

Mastery factors EK EX SK PD Local weightsEK (1 1 1) (17 15 13) (1 3 5) (15 13 1) 0207EX (3 5 7) (1 1 1) (3 5 7) (1 3 5) 0444SK (15 13 1) (17 15 13) (1 1 1) (15 13 1) 0032PD (1 3 5) (15 13 1) (1 3 5) (1 1 1) 0318

Table 4 Calculated global weights of subfactors

Main factors and local weights Subfactors Local weights Global weights Global weights (normalized)

Mastery factors (MF) (0423)

EK 0207 0088 88EX 0444 0188 188SK 0032 0014 14PD 0318 0135 135

Responsibility factors (RF) (0255)

MR 0150 0038 38PR 0150 0038 38CR 0350 0089 89SR 0350 0089 89

Workload factors (WF) (0067) ME 0300 0020 20PE 0700 0047 47

Physical and working conditionsfactors (PF) (0255)

JR 0409 0104 104WE 0409 0104 104PN 0182 0046 46

Total 100 1000

Table 5 Graduation of subfactors

Subfactors Grade pointsI II III IV V

EK 18 26 39 59 88EX 38 56 84 126 188SK 3 4 6 9 14PD 27 40 60 90 135MR 8 11 17 25 38PR 8 11 17 25 38CR 18 27 40 60 89SR 18 27 40 60 89ME 4 6 9 13 20PE 9 14 21 31 47JR 21 31 47 70 104WE 21 31 47 70 104PN 9 14 21 31 46


Table 6 Evaluation of 86 different jobs with scale values

Subfactors Job 1 Job Job 86Scale Points Scale Points Scale Points

EK IV 59 II 26EX IV 56 II 56SK II 4 II 4PD II 40 I 27MR IV 25 I 8PR IV 25 I 8CR IV 60 I 18SR V 89 I 18ME V 20 II 6PE III 21 I 9JR IV 70 I 21WE III 47 II 31PN IV 31 I 9Total points mdash 547 mdash mdash 241

Table 7 Upper and lower bounds for ℎ-levels (ℎ = 01 ℎ = 05 and ℎ = 09)

ℎ-level Point intervals241ndash310 311ndash380 381ndash450 451ndash520 521ndash592

ℎ = 01L 579 987 1486 1987 2980U 990 1505 2003 3033 5073

ℎ = 05L 567 967 1461 1881 2818U 1038 1549 2030 3079 5261

ℎ = 09L 363 783 1232 930 1360U 1342 1945 2269 3492 6955

jobs are evaluated in respect to FRA and the wage groups aredetermined on the basis of both the total points and wageswhich are shown in Figure 4 Additionally the acceptablewage scales and the wage management policies in similarcompanies are taken into account in the evaluation systemIn the FRA instead of crisp wages fuzzy mathematicalestimation model is run to estimate the upper and lowerbounds of wages in the brackets In the conventional systemthe managers accepted that the upper bound is over 10and the lower bound is below 10 of the estimated averagevalues The FRA-based wage brackets model is offered to themanagers for an appropriate wage management system

In the adaptation process to the new evaluation systemsome jobs will be stated as overpaid (point A) some of themstated as underpaid (point B) or somewhich can remain at thesame level (point C) as seen in Figure 4 Jobs A and B need tomove to the desired level in the direction of their points JobA can remain stable whenever the bracket reaches this wagelevel and job B can be paidmore to reach the desired intervalconversely In this manner conversion can be performedHere there is no reason to reconcile with job C since the jobis in the desired level However the adaptation process couldbe different when considering the ℎ-levels (Table 8)

4 Discussion and Conclusion

At present decisions are made within increasingly complexenvironments and structures especially in steel industry Inmost cases the use of expertsrsquo view is quite necessary andthe different measurement systems need to be taken intoaccount In many of those decision-making processes thetheory of fuzzy decision-making can easily be performedUsing the fuzzy group decision-making can overcome thedifficulty of complex environments and systems In generalmany concepts tools and techniques can be used to improvehuman consistency and implementation of the complexmodels These decision tools significantly contribute to moreaccurate decision-making

In the job evaluation the available information requiredfor proficient decision-making is vague and uncertain inmost cases It is very difficult to obtain an exact or preciseassessment data for example in description of the jobs and inconstruction of the payment system To address this concernthe concepts of fuzzy numbers can be used to evaluate thedetermined factors In this study the job evaluation processis built as a multicriteria decision-making problem underthe fuzzy environment where the imprecise decision-makerrsquosjudgments are represented as triangular fuzzy numbers


Table 8 Adaptation process of related jobs

Jobs Points Wages Status Desired wage levelsℎ = 01 ℎ = 05 ℎ = 09

A 350 2150 Overpaid 987ndash1505 967ndash1549 783ndash1945B 410 910 Underpaid 1486ndash2003 1461ndash2030 1232ndash2269C 485 2450 Desired 1987ndash3033 1881ndash3079 930ndash3492

0

500

1000

1500

241 249 262 272 282 297 309

h = 01 X1

(a)

0

500

1000

1500

241 249 262 272 282 297 309

h = 05 X1

(b)

0

500

1000

1500

241 249 262 272 282 297 309

Center LowerUpper

h = 09 X1

(c)

Figure 3 Dispersions of the first interval (1198831) for (a) ℎ = 01 (b) ℎ = 05 and (c) ℎ = 09

The FAHP method is used for the assessment of weightsand the points of each job are calculated afterwards Toimprove the model FRA-based wage brackets are consideredand embedded into these processes by calculating the resultof mathematical model which sets the upper and lowerbounds With such an approach the subjectiveness of thecommittee is reduced and the most appropriate model forthe steel companies according to their membership degrees isformed The experimental study in the steel company showsthe advantages and the usability of the proposed model Itdistinguishes the jobswith a sensitivemanner anddeterminesthe wage brackets as well

Changrsquos FAHP method and its defuzzification processare used in this study The fuzzy decision matrix is used atthe beginning of FAHP method Crisp weights are given tothis method in order to be used as the coefficients of the

objective function in the model Although the inputs aregiven crisp the outputs of upper and lower bounds of thebrackets are obtained in fuzzy variables Therefore by usingFLRA method there is no longer need to use the statisticalregression and its strict assumptions and prerequisites

This model offers a valuable support for the steel industryin the decision-making and this approach can be applied toany company using the companyrsquos necessities in determiningthe factors and job variety On the basis of the calculatedjob points 5 different wage groups are constituted betweenthe lower wage bound of TRY600 and upper wage boundof TRY5000 The payment structure is constituted for eachworker in respect of the seniority in the brackets Using thetotal points the job groups are structured by the FRA TheFRA-based wage brackets model provides the ability to themanagers to structure a flexible wage management system


h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)

Figure 4 The calculated fuzzy wage brackets for 86 jobs

Furthermore the enhanced structural adjustment process tothe new system is obtained In this way a new paymentmanagement system would be beneficial to the sectors whereit can be applicable

As an extension to this study bearing in mind therelationships between the factors and subfactors in each levelone may consider the Analytical Network Process (ANP)which is a recently proposed method Another extension isto reversion of the fuzzy scale from triangular to trapezoidalnumbers whichmdashto the best of our knowledgemdashhas not beenapplied to this kind of study so far It is believed that both ofthe methods (FAHP and FRA) could be easily used to obtainmore accurate results in the job evaluation and paymentsystems

Appendix

The steps of Changrsquos [19ndash27] extent analysis approach areas follows let 119883 = 119909

1 1199092 119909

119899 be an object set and let

119880 = 1199061 1199062 119906

119898 be a goal set According to the method

of [27] extent analysis each object is taken and extent analysisfor each goal 119892

119894 is performed respectively Therefore 119898

extent analysis values for each object can be obtained withthe following signs

1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)

where all the119872119895119892119894(119895 = 1 2 119898) are triangular fuzzy num-

bersThe steps of Changrsquos extent analysis can be given as in the

following

Step 1 The value of fuzzy synthetic extent with respect to the119894th object is defined as

119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)

To obtain sum119898119895=1

119872119895119892119894 perform the fuzzy addition operation of

m extent analysis values for a particular matrix such that

119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)

And to obtain [sum119899119894=1

sum119898

119895=1119872119895

119892119894]minus1 perform the fuzzy addition

operation of119872119895119892119894(119895 = 1 2 119898) values such that

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)

and then compute the inverse of the vector in (A4) such that

[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)

Step 2 The degree of possibility of1198722= (1198972 1198982 1199062) ge 119872

1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)

and can be equivalently expressed as follows

119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898

1minus 1198971) otherwise

(A7)

where 119889 is the ordinate of the highest intersection point 119863between 120583

1198721and 1205831198722

To compare1198721and119872

2 we need both

the values of 119881(1198721ge 1198722) and 119881(119872

2ge 1198721)

Step 3 The degree possibility for a convex fuzzy number tobe greater than 119896 convex fuzzy numbers 119872

119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)

For 119896 = 1 2 119899 119896 = 119894 Then the weight vector is given by

1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)

where 119861119894(119894 = 1 2 119899) are 119899 elements


Step 4 Via normalization the normalized weight vectors are

1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)

where119882 is a nonfuzzy number

References

[1] A Spyridakos Y Siskos D Yannacopoulos and A SkourisldquoMulticriteria job evaluation for large organizationsrdquo EuropeanJournal of Operational Research vol 130 pp 375ndash387 2009

[2] M Dagdeviren D Akay and M Kurt ldquoAnalytical hierarchyprocess for job evaluation and applicationrdquo Journal of theFaculty of Engineering and Architecture of Gazi University vol19 pp 131ndash138 2004

[3] E Eraslan and B Arıkan ldquoDetermination of wages using thepoint method seniority and success an application in internalproduction department of a manufacturing facilityrdquo Journal ofthe Faculty of Engineering and Architecture of Gazi Universityvol 19 pp 139ndash150 2004

[4] E M McCormick Job Analysis Methods and ApplicationAMACOM New York NY USA 1979

[5] L Xing Y Li and L Ma ldquoStudy on the application of PointMethod in Job Evaluationrdquo in Proceedings of the 7th WuhanInternational Conference on E-Business vol 1ndash3 pp 2835ndash28392008

[6] D Doverspike AM Carlisi G V Garrett and R A AlexanderldquoGeneralizability analysis of a point-method job evaluationinstrumentrdquo Journal of Applied Psychology vol 68 pp 476ndash4831983

[7] B Das and A Garcia-Diaz ldquoFactor selection guidelines for jobevaluation a computerized statistical procedurerdquo Computers ampIndustrial Engineering vol 40 pp 259ndash272 2001

[8] D M Figart ldquoEqual pay for equal work the role of jobevaluation in an evolving social normrdquo Journal of EconomicIssues vol 34 pp 1ndash19 2000

[9] B N Smith P G Benson and J S Hornsby ldquoThe effects ofjob description content on job evaluation judgmentsrdquo Journalof Applied Psychology vol 75 pp 301ndash309 1990

[10] J N D Gupta and N U Ahmed ldquoA goal programmingapproach to job evaluationrdquo Computers amp Industrial Engineer-ing vol 14 pp 147ndash152 1988

[11] M Dagdeviren D Akay and M Kurt ldquoJob evaluation useof goal programming technique at determine factor degreepointsrdquo Journal of the Faculty of Engineering and Architectureof Gazi University vol 19 pp 89ndash95 2004

[12] M Perc and A Szolnoki ldquoCoevolutionary gamesmdasha minireviewrdquo BioSystems vol 99 pp 109ndash125 2010

[13] M Perc J Gomez-Gardenes A Szolnoki L M Floria andY Moreno ldquoEvolutionary dynamics of group interactions onstructured populationsmdasha reviewrdquo Journal of the Royal SocietyInterface vol 10 no 80 2013

[14] S Gupta andM Chakraborty ldquoJob evaluation in fuzzy environ-mentrdquo Fuzzy Sets and Systems vol 100 pp 71ndash76 1998

[15] C Kahraman D Ruan and E Tolga ldquoCapital budgetingtechniques using discounted fuzzy versus probabilistic cashflowsrdquo Information Sciences vol 42 pp 57ndash76 2002

[16] T L Saaty The Analytic Hierarchy Process McGraw-Hill NewYork NY USA 1980

[17] P J M Van Laarhoven and W Pedrycz ldquoA fuzzy extension ofSaatyrsquos priority theoryrdquo Fuzzy Sets and Systems vol 11 no 3 pp229ndash241 1983

[18] J J Buckley ldquoFuzzy hierarchical analysisrdquo Fuzzy Sets andSystems vol 17 no 3 pp 233ndash247 1985

[19] D Y Chang ldquoApplications of the extent analysis method onfuzzy AHPrdquo European Journal of Operational Research vol 95pp 649ndash655 1996

[20] C H Cheng ldquoEvaluating naval tactical missile systems byfuzzy AHP based on the grade value of membership functionrdquoEuropean Journal of Operational Research vol 96 pp 343ndash3501997

[21] C Kahraman Z Ulukan and E Tolga ldquoA fuzzy weightedevaluation method using objective and subjective measuresrdquo inProceedings of the International ICSC Symposium onEngineeringof Intelligent Systems vol 1 pp 57ndash63 1998

[22] C Kahraman D Ruan and I Dogan ldquoFuzzy group decision-making for facility location selectionrdquo Information Sciences vol157 pp 135ndash153 2003

[23] C KahramanU Cebeci andD Ruan ldquoMulti-attribute compar-ison of catering service companies using fuzzy AHP the case ofTurkeyrdquo International Journal Production Economics vol 87 pp171ndash184 2004

[24] G Buyukozkan T Ertay C Kahraman and D Ruan ldquoDeter-mining the importance weights for the design requirements inthe house of quality using the fuzzy analytic network approachrdquoInternational Journal of Intelligent Systems vol 19 pp 443ndash4612004

[25] E TolgaM L Demircan and C Kahraman ldquoOperating systemselection using fuzzy replacement analysis and analytic hierar-chy processrdquo International Journal of Production Economics vol97 pp 89ndash117 2005

[26] O Kulak and C Kahraman ldquoFuzzy multi-attribute selectionamong transportation companies using axiomatic design andanalytic hierarchy processrdquo Information Sciences vol 170 pp191ndash210 2005

[27] D Y Chang and L L Zhang ldquoExtent analysis and syntheticdecisionrdquo in Optimization Techniques and Applications vol 1pp 352ndash359 World Scientific Singapore 1992

[28] P T Chang E S Lee and S A Konz ldquoApplying fuzzy linearregression to VDT legibilityrdquo Fuzzy Sets and Systems vol 80pp 197ndash204 1996

[29] H Tanaka S Uejima and K Asai ldquoLinear regression analysiswith fuzzy modelrdquo IEEE Transactions of Systems Man Cybernetvol 12 pp 903ndash907 1982

[30] P-T Chang and E S Lee ldquoA generalized fuzzy weighted least-squares regressionrdquo Fuzzy Sets and Systems vol 82 no 3 pp289ndash298 1996

[31] G Peters ldquoFuzzy linear regression with fuzzy intervalsrdquo FuzzySets and Systems vol 63 no 1 pp 45ndash55 1994

[32] M Sakawa and H Yano ldquoFuzzy linear regression analysis forfuzzy input-output datardquo Information Sciences vol 63 no 3 pp191ndash206 1992

[33] B Kim and R R Bishu ldquoEvaluation of fuzzy linear regressionmodels by comparing membership functionsrdquo Fuzzy Sets andSystems vol 100 pp 343ndash352 1998

[34] H Tanaka andH Lee ldquoInterval regression analysis by quadraticprogramming approachrdquo IEEE Transaction on Fuzzy Systemsvol 6 pp 473ndash481 1998

[35] A Azadeh M Khakestani and M Saberi ldquoA flexible fuzzyregression algorithm for forecasting oil consumption estima-tionrdquo Energy Policy vol 37 pp 5567ndash5579 2009

[36] F M Tseng and Y C Hu ldquoQuadratic-interval bass model fornew product sales diffusionrdquo Expert Systems with Applicationsvol 36 pp 8496ndash8502 2009


[37] Z Sener and E E Karsak ldquoA combined fuzzy linear regressionand fuzzy multiple objective programming approach for settingtarget levels in quality function deploymentrdquo Expert Systemswith Applications vol 38 pp 3015ndash3022 2011

[38] L Bing Z Meilin and X Kai ldquoA practical engineering methodfor fuzzy reliability analysis of mechanical structuresrdquo Reliabil-ity Engineering amp System Safety vol 67 pp 311ndash315 2000

[39] B Heshmaty and A Kandel ldquoFuzzy linear regression and itsapplication to forecasting in uncertain environmentrdquo Fuzzy Setsand Systems vol 15 pp 159ndash191 1985

[40] Y J Lai and S I Chang ldquoA fuzzy approach for multi-responseoptimization an off-line quality engineering problemrdquo FuzzySets and Systems vol 63 pp 117ndash129 1994

[41] T Y Chou G S Liang and T C Han ldquoApplication of fuzzyregression on air cargo forecastrdquo Quality amp Quality vol 47 pp897ndash908 2013

[42] A Azadeh P Sohrabi andV Ebrahimipour ldquoA fuzzy regressionapproach for improvement of gasoline consumption estimationwith uncertain datardquo International Journal of Industrial AdSystems Engineering vol 13 pp 92ndash109 2013

[43] N F Pan C H Ko M D Yang and K C Hsu ldquoPavement per-formance prediction through fuzzy regressionrdquo Expert Systemswith Applications vol 38 pp 10010ndash10017 2011

[44] K Chen M J Rys and E S Lee ldquoModeling of thermalcomfort in air conditioned rooms by fuzzy regression analysisrdquoMathematical and Computer Modelling vol 43 pp 809ndash8192006

[45] L A Zadeh ldquoThe concept of a linguistic variable and itsapplication to approximate reasoning Irdquo vol 8 pp 199ndash2491975

[46] H Lee and H Tanaka ldquoFuzzy regression analysis by quadraticprogramming reflecting central tendencyrdquo Behaviormetrikavol 25 pp 65ndash86 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(iv) educational studies(v) work analysis(vi) determination of the point plan table(vii) calculation of the factor values

Some researchers have considered the pay inequity as beingthe most important term of job evaluation that is equal payfor equal work [8] Smith et al [9] put forth a critique aboutthe effects of job description content on the job evaluationjudgments In the studies over recent decades Spyridakoset al [1] studied the multicriteria job evaluation for largeorganizations and Das and Garcia-Diaz [7] proposed acomputerized statistical procedure in factor selection guide-lines for the job evaluation Gupta and Ahmed [10] andDagdeviren et al [11] used a goal programming approach andXing et al [5] used the Analytic Hierarchy Process (AHP)evaluation for the problem of factor weights determinationTherefore game theory is the traditional solution to the manyproblems (Perc et al) [12 13]

The job evaluation could be defined as a managerialdecision-making problem with its multiple objectives [14]Scoring these objectives is a difficult task for the decisionmakers because many different scores can emerge within agroup of managers Therefore the decision-making processshould be considered as a fuzzy process in naturally With afuzzy approach varying opinions of the managers could beconsidered in the evaluation systemThereby the subjectivityof the decision makers could be prevented

The fuzzy theory allows bothmathematical operators andfuzzy programming to apply the fuzzy domain Fuzzy set isa class of objects which uses a membership (characteristic)function and it assigns a grade of membership rangingbetween zero-one to each object There are many rankingmethods developed in the literature for the fuzzy numbersThese methods may generate different ranking results andmost of them require complexmathematical calculations [15]

In this study the fuzzyAHPmethod (FAHP) that ismoreappropriate due to its nature of fuzziness was used for the jobevaluation systems [16] During the evaluation processes ofeach job fuzziness was obtained by using the fuzzy pairwisecomparisons in the judgmentmatrices Hence more accurateweights could be obtained in order to use for factor scoringTo improve themodel the fuzzywage brackets are consideredand embedded into the decision process with the result ofmathematical model which sets the upper and lower boundsThen by using Fuzzy Regression Analysis (FRA) an accuratepayment system is proposed for the first time for a steelcompany

2 The Techniques Used in This StudyFAHP and FRA

The FAHP is a provided systematic approache to the alter-native selection and justification problem by using the con-cepts of fuzzy set theory and hierarchical structure analysisDecisionmakers are usuallymore confident in giving intervaljudgments rather than crisp judgments due to the fuzzynature of the comparison process

The earliest work in FAHP appeared in Van Laarhovenand Pedrycz [17] which compared fuzzy ratios described bytriangular membership functions Buckley [18] determinedfuzzy priorities of comparison ratios whose membershipfunctions were trapezoidal Chang [19] introduced a newapproach for handling FAHP with the use of triangular fuzzynumbers for pairwise comparison scale of FAHP and the useof the extent analysis method for the synthetic extent valuesof pairwise comparisons The outcome of Chengrsquos algorithm[20] can finally be defuzzified by forming the surface centre ofgravity of any fuzzy set Kahraman et al [21ndash23] used a fuzzyobjective and subjective method obtaining the weights fromAHP and producing a fuzzy weighted evaluation RecentlyBuyukozkan et al [24] Tolga et al [25] and Kulak andKahraman [26] applied the same approach to some selectionproblems In this study FAHP method is used to calculatelocal weights using detailed information about Chang et alrsquos[27 28] extent analysis (see the appendix)

FRA was first introduced by Tanaka et al in 1982 [29]In their study a regression problem with fuzzy output andcrisp input was formulated as a mathematical program-ming problem The objective was to minimize the totalspreads of the fuzzy parameters subject to the constraintthat the regression model needed to satisfy a prespecifiedmembership value in estimating the fuzzy outputs Differentresearchers used Tanakarsquos approach to minimize the totalspread of the output [30 31] Sakawa and Yano [32] Kimand Bishu [33] and Peters [31] considered the possibility andnecessity conditions for fuzzy equality Tanaka and Lee [34]proposed an interval regression analysis based on quadraticprogramming approach Azadeh et al [35] applied FRA forenergy forecasting problems Tseng and Hu [36] proposeda quadratic-interval Bass model that combines quadratic-interval regression with the Bass innovation diffusion modelto solve a fuzzy relationship between explanatory and outputvariables and to provide forecasts of sales to decision makers

Fuzzy regression models have been successfully appliedto various engineering and technological problems [30 37ndash42] It can also be applied for the estimation of the fuzzyjob evaluation process in the manufacturing companiesAdditionally instead of the crisp values of wage bracketsobtaining a fuzzy interval is more relevant to the real lifeproblems which are why the linguistic variables are usedcommonly FRA has been shown to be more flexible andrigorous than the traditional (nonfuzzy) approach [43 44]

The aim of regression analysis is to find an appropri-ate mathematical model and to determine the best fittingcoefficients of the model from the given data This classicalregression technique is useful in a nonfuzzy environmentwhere the relationship among variables is crisply definedHowever it is very difficult to obtain an exact relationThe use of statistical regression is restricted by some strictassumptions and statistical properties about the given dataFuzzy regression does not require normal distributed datastability tests and large samples It can be applied to manyreal life problems in which the strict assumptions of statisticalregression analysis cannot be met FRA considers the use offuzzy numbers that involves the modeling of problems wherethe output variable is affected by imprecisionThe goal of FRA




119894= 11986001198830+ 11986011198831+ 11986021198832+ sdot sdot sdot + 119860

119899119883119899= 119860119883 (1)

where 119883 = [1198830 1198831 119883


[1198600 1198601 119860






119894= (1205720 1198880)1198830+ (1205721 1198881)1198831+ sdot sdot sdot + (120572

119899 119888119899)119883119899 (2)




120583119894(119910119894) =

1 minus

10038161003816100381610038161003816119884119894minus 119883119879120572

10038161003816100381610038161003816

119888119879 |119883| 119883 = 0

1 119883 = 0 119884119894= 0

0 119883 = 0 119884119894= 0

(3)

where 119888119879 = (1198880 1198881 119888

119899) and 120572 = (120572

0 1205721 120572

119899) The




min 119869 = 119888119879 |119883|

min 119869 =119899

sum119895=1

(119888119895

119898

sum119894=1

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816)

(4)






1 minus

10038161003816100381610038161003816119884119894minus 119883119879120572

10038161003816100381610038161003816

119888119879 |119883|ge ℎ forall119894 = 1 2 119898 (5)


min 119869 =119899

sum119895=1

(119888119895

119898

sum119894=1

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816)

119899

sum119895=0

120572119895119909119894119895+ (1 minus ℎ)

119899

sum119895=0

119888119895

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816ge 119910119894 119894 = 1 119898

119899

sum119895=0

120572119895119909119894119895minus (1 minus ℎ)

119899

sum119895=0

119888119895

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816le 119910119894 119894 = 1 119898

119888119895ge 0 119895 = 0 119899 0 le ℎ le 1

(6)












Start










Stop


Setup goal














Wei

ghin

g th

e fac

tors

in jo

b ev

alua

tion


Effort factors

Educationknowledge

Experience

Skill

Predisposition

Security providing

Goal

Factors

Scale

I

II

III

IV

V

Jobs

Job risks

Machineequipment

Materialproduct


Mental effort

Physical effort

Working environment

Mastery factors

Subfactors

Job 1

Job 2

Job 3






1 1198832 1198833 1198834 1198835) and 5-wage (119884

1 1198842 1198843



1


3 2000ndash3000 for 119883

4 and

3000ndash5000 for1198835


119879

1= (minus5993443 minus2076899)1198830 + (505964 02037769)1198831

2=(minus1432457 minus1391500)1198830 + (7688034 05175689)1198832

3= (minus1060654 2363248)1198830 + (675 002136752)1198833

4= (minus3749089 6432341)1198830 + (1298214 minus1130952)1198834

5=(minus1176960 minus9434264)1198830 + (2805211 05567617)1198835

(7)













EK 0207 0088 88EX 0444 0188 188SK 0032 0014 14PD 0318 0135 135


MR 0150 0038 38PR 0150 0038 38CR 0350 0089 89SR 0350 0089 89



JR 0409 0104 104WE 0409 0104 104PN 0182 0046 46

Total 100 1000










ℎ = 01L 579 987 1486 1987 2980U 990 1505 2003 3033 5073

ℎ = 05L 567 967 1461 1881 2818U 1038 1549 2030 3079 5261

ℎ = 09L 363 783 1232 930 1360U 1342 1945 2269 3492 6955










0

500

1000

1500

241 249 262 272 282 297 309

h = 01 X1

(a)

0

500

1000

1500

241 249 262 272 282 297 309

h = 05 X1

(b)

0

500

1000

1500

241 249 262 272 282 297 309

Center LowerUpper

h = 09 X1

(c)







h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)




Appendix


1 1199092 119909


119880 = 1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)



following


119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)




119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)


sum119898

119895=1119872119895



119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)


[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)


119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(A7)


1198721and 1205831198722


2 we need both


2ge 1198721)


119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)


1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)




1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894= 11986001198830+ 11986011198831+ 11986021198832+ sdot sdot sdot + 119860

119899119883119899= 119860119883 (1)

where 119883 = [1198830 1198831 119883


[1198600 1198601 119860






119894= (1205720 1198880)1198830+ (1205721 1198881)1198831+ sdot sdot sdot + (120572

119899 119888119899)119883119899 (2)




120583119894(119910119894) =

1 minus

10038161003816100381610038161003816119884119894minus 119883119879120572

10038161003816100381610038161003816

119888119879 |119883| 119883 = 0

1 119883 = 0 119884119894= 0

0 119883 = 0 119884119894= 0

(3)

where 119888119879 = (1198880 1198881 119888

119899) and 120572 = (120572

0 1205721 120572

119899) The




min 119869 = 119888119879 |119883|

min 119869 =119899

sum119895=1

(119888119895

119898

sum119894=1

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816)

(4)






1 minus

10038161003816100381610038161003816119884119894minus 119883119879120572

10038161003816100381610038161003816

119888119879 |119883|ge ℎ forall119894 = 1 2 119898 (5)


min 119869 =119899

sum119895=1

(119888119895

119898

sum119894=1

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816)

119899

sum119895=0

120572119895119909119894119895+ (1 minus ℎ)

119899

sum119895=0

119888119895

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816ge 119910119894 119894 = 1 119898

119899

sum119895=0

120572119895119909119894119895minus (1 minus ℎ)

119899

sum119895=0

119888119895

10038161003816100381610038161003816119909119894119895

10038161003816100381610038161003816le 119910119894 119894 = 1 119898

119888119895ge 0 119895 = 0 119899 0 le ℎ le 1

(6)












Start










Stop


Setup goal














Wei

ghin

g th

e fac

tors

in jo

b ev

alua

tion


Effort factors

Educationknowledge

Experience

Skill

Predisposition

Security providing

Goal

Factors

Scale

I

II

III

IV

V

Jobs

Job risks

Machineequipment

Materialproduct


Mental effort

Physical effort

Working environment

Mastery factors

Subfactors

Job 1

Job 2

Job 3






1 1198832 1198833 1198834 1198835) and 5-wage (119884

1 1198842 1198843



1


3 2000ndash3000 for 119883

4 and

3000ndash5000 for1198835


119879

1= (minus5993443 minus2076899)1198830 + (505964 02037769)1198831

2=(minus1432457 minus1391500)1198830 + (7688034 05175689)1198832

3= (minus1060654 2363248)1198830 + (675 002136752)1198833

4= (minus3749089 6432341)1198830 + (1298214 minus1130952)1198834

5=(minus1176960 minus9434264)1198830 + (2805211 05567617)1198835

(7)













EK 0207 0088 88EX 0444 0188 188SK 0032 0014 14PD 0318 0135 135


MR 0150 0038 38PR 0150 0038 38CR 0350 0089 89SR 0350 0089 89



JR 0409 0104 104WE 0409 0104 104PN 0182 0046 46

Total 100 1000










ℎ = 01L 579 987 1486 1987 2980U 990 1505 2003 3033 5073

ℎ = 05L 567 967 1461 1881 2818U 1038 1549 2030 3079 5261

ℎ = 09L 363 783 1232 930 1360U 1342 1945 2269 3492 6955










0

500

1000

1500

241 249 262 272 282 297 309

h = 01 X1

(a)

0

500

1000

1500

241 249 262 272 282 297 309

h = 05 X1

(b)

0

500

1000

1500

241 249 262 272 282 297 309

Center LowerUpper

h = 09 X1

(c)







h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)




Appendix


1 1199092 119909


119880 = 1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)



following


119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)




119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)


sum119898

119895=1119872119895



119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)


[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)


119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(A7)


1198721and 1205831198722


2 we need both


2ge 1198721)


119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)


1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)




1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start










Stop


Setup goal














Wei

ghin

g th

e fac

tors

in jo

b ev

alua

tion


Effort factors

Educationknowledge

Experience

Skill

Predisposition

Security providing

Goal

Factors

Scale

I

II

III

IV

V

Jobs

Job risks

Machineequipment

Materialproduct


Mental effort

Physical effort

Working environment

Mastery factors

Subfactors

Job 1

Job 2

Job 3






1 1198832 1198833 1198834 1198835) and 5-wage (119884

1 1198842 1198843



1


3 2000ndash3000 for 119883

4 and

3000ndash5000 for1198835


119879

1= (minus5993443 minus2076899)1198830 + (505964 02037769)1198831

2=(minus1432457 minus1391500)1198830 + (7688034 05175689)1198832

3= (minus1060654 2363248)1198830 + (675 002136752)1198833

4= (minus3749089 6432341)1198830 + (1298214 minus1130952)1198834

5=(minus1176960 minus9434264)1198830 + (2805211 05567617)1198835

(7)













EK 0207 0088 88EX 0444 0188 188SK 0032 0014 14PD 0318 0135 135


MR 0150 0038 38PR 0150 0038 38CR 0350 0089 89SR 0350 0089 89



JR 0409 0104 104WE 0409 0104 104PN 0182 0046 46

Total 100 1000










ℎ = 01L 579 987 1486 1987 2980U 990 1505 2003 3033 5073

ℎ = 05L 567 967 1461 1881 2818U 1038 1549 2030 3079 5261

ℎ = 09L 363 783 1232 930 1360U 1342 1945 2269 3492 6955










0

500

1000

1500

241 249 262 272 282 297 309

h = 01 X1

(a)

0

500

1000

1500

241 249 262 272 282 297 309

h = 05 X1

(b)

0

500

1000

1500

241 249 262 272 282 297 309

Center LowerUpper

h = 09 X1

(c)







h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)




Appendix


1 1199092 119909


119880 = 1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)



following


119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)




119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)


sum119898

119895=1119872119895



119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)


[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)


119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(A7)


1198721and 1205831198722


2 we need both


2ge 1198721)


119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)


1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)




1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Wei

ghin

g th

e fac

tors

in jo

b ev

alua

tion


Effort factors

Educationknowledge

Experience

Skill

Predisposition

Security providing

Goal

Factors

Scale

I

II

III

IV

V

Jobs

Job risks

Machineequipment

Materialproduct


Mental effort

Physical effort

Working environment

Mastery factors

Subfactors

Job 1

Job 2

Job 3






1 1198832 1198833 1198834 1198835) and 5-wage (119884

1 1198842 1198843



1


3 2000ndash3000 for 119883

4 and

3000ndash5000 for1198835


119879

1= (minus5993443 minus2076899)1198830 + (505964 02037769)1198831

2=(minus1432457 minus1391500)1198830 + (7688034 05175689)1198832

3= (minus1060654 2363248)1198830 + (675 002136752)1198833

4= (minus3749089 6432341)1198830 + (1298214 minus1130952)1198834

5=(minus1176960 minus9434264)1198830 + (2805211 05567617)1198835

(7)













EK 0207 0088 88EX 0444 0188 188SK 0032 0014 14PD 0318 0135 135


MR 0150 0038 38PR 0150 0038 38CR 0350 0089 89SR 0350 0089 89



JR 0409 0104 104WE 0409 0104 104PN 0182 0046 46

Total 100 1000










ℎ = 01L 579 987 1486 1987 2980U 990 1505 2003 3033 5073

ℎ = 05L 567 967 1461 1881 2818U 1038 1549 2030 3079 5261

ℎ = 09L 363 783 1232 930 1360U 1342 1945 2269 3492 6955










0

500

1000

1500

241 249 262 272 282 297 309

h = 01 X1

(a)

0

500

1000

1500

241 249 262 272 282 297 309

h = 05 X1

(b)

0

500

1000

1500

241 249 262 272 282 297 309

Center LowerUpper

h = 09 X1

(c)







h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)




Appendix


1 1199092 119909


119880 = 1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)



following


119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)




119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)


sum119898

119895=1119872119895



119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)


[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)


119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(A7)


1198721and 1205831198722


2 we need both


2ge 1198721)


119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)


1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)




1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















EK 0207 0088 88EX 0444 0188 188SK 0032 0014 14PD 0318 0135 135


MR 0150 0038 38PR 0150 0038 38CR 0350 0089 89SR 0350 0089 89



JR 0409 0104 104WE 0409 0104 104PN 0182 0046 46

Total 100 1000










ℎ = 01L 579 987 1486 1987 2980U 990 1505 2003 3033 5073

ℎ = 05L 567 967 1461 1881 2818U 1038 1549 2030 3079 5261

ℎ = 09L 363 783 1232 930 1360U 1342 1945 2269 3492 6955










0

500

1000

1500

241 249 262 272 282 297 309

h = 01 X1

(a)

0

500

1000

1500

241 249 262 272 282 297 309

h = 05 X1

(b)

0

500

1000

1500

241 249 262 272 282 297 309

Center LowerUpper

h = 09 X1

(c)







h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)




Appendix


1 1199092 119909


119880 = 1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)



following


119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)




119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)


sum119898

119895=1119872119895



119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)


[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)


119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(A7)


1198721and 1205831198722


2 we need both


2ge 1198721)


119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)


1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)




1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















ℎ = 01L 579 987 1486 1987 2980U 990 1505 2003 3033 5073

ℎ = 05L 567 967 1461 1881 2818U 1038 1549 2030 3079 5261

ℎ = 09L 363 783 1232 930 1360U 1342 1945 2269 3492 6955










0

500

1000

1500

241 249 262 272 282 297 309

h = 01 X1

(a)

0

500

1000

1500

241 249 262 272 282 297 309

h = 05 X1

(b)

0

500

1000

1500

241 249 262 272 282 297 309

Center LowerUpper

h = 09 X1

(c)







h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)




Appendix


1 1199092 119909


119880 = 1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)



following


119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)




119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)


sum119898

119895=1119872119895



119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)


[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)


119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(A7)


1198721and 1205831198722


2 we need both


2ge 1198721)


119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)


1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)




1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0

500

1000

1500

241 249 262 272 282 297 309

h = 01 X1

(a)

0

500

1000

1500

241 249 262 272 282 297 309

h = 05 X1

(b)

0

500

1000

1500

241 249 262 272 282 297 309

Center LowerUpper

h = 09 X1

(c)







h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)




Appendix


1 1199092 119909


119880 = 1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)



following


119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)




119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)


sum119898

119895=1119872119895



119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)


[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)


119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(A7)


1198721and 1205831198722


2 we need both


2ge 1198721)


119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)


1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)




1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










h = 09

h = 05

h = 01

7000

6000

5000

4000

3000

2000

1000

240 310 380 450 520 592

(A)

(B)

(C)

Point intervals

Wag

es (T

RY)




Appendix


1 1199092 119909


119880 = 1199061 1199062 119906





1198721

1198921198941198722

119892119894 119872

119898

119892119894 119894 = 1 2 119899 (A1)



following


119878119894=

119898

sum119895=1

119872119895

119892119894otimes [

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

(A2)




119898

sum119895=1

119872119895

119892119894= (

119898

sum119895=1

119897119895

119898

sum119895=1

119898119895

119898

sum119895=1

119906119895) (A3)


sum119898

119895=1119872119895



119899

sum119894=1

119898

sum119895=1

119872119895

119892119894= (

119899

sum119894=1

119897119894

119899

sum119894=1

119898119894

119899

sum119894=1

119906119894) (A4)


[

[

119899

sum119894=1

119898

sum119895=1

119872119895

119892119894

]

]

minus1

= (1

sum119899

119894=1119906119894

1

sum119899

119894=1119898119894

1

sum119899

119894=1119897119894

) (A5)


1=

(1198971 1198981 1199061) is defined as

119881 (1198722ge 1198721) = sup [min (120583

1198721(119909) 1205831198722 (119910))] (A6)


119881 (1198722ge 1198721) = hgt (119872

1cap1198722) = 1205831198722

(119889)

=

1 if 1198982ge 1198981

0 if 1198971ge 1199062

1198971minus 1199062

(1198982minus 1199062) minus (119898


(A7)


1198721and 1205831198722


2 we need both


2ge 1198721)


119894(119894 = 1 2 119896)

can be defined by

119881 (119872 ge 11987211198722 119872

119896)

= 119881 [(119872 ge 1198721) (119872 ge 119872

2) (119872 ge 119872

119896)]

= min119881 (119872 ge 119872119894) 119894 = 1 2 119896

(A8)

Assume that

1198891015840(119861119894) = min119881 (119878

119894ge 119878119896) (A9)


1198821015840= (1198891015840(1198611) 1198891015840(1198612) 119889

1015840(119861119899))119879

(A10)




1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198821015840= (119889 (119861

1) 119889 (119861

2) 119889 (119861

119899))119879 (A11)


References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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