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Effective Staff Selection Tool:

Fuzzy Numbers and Memetic Algorithm Based

Approach

Mohamed Zaki Ramadan

Associate Professor

Industrial Engineering Department, Engineering Faculty, King Saud University, P.O. Box 800, Riyadh, 11421, Saudi Arabia

Tel.: +966-1-4676713, Fax.: +966-1-4678657 mramadan1@ksu.edu.sa

Abstract-- Evaluating worker’s suitability for a job is an important tool for Human Resources Managers (HRMs) to select the better candidates under various e valuation criteria. A problem of workers' assignment is studied in this paper in order to find the best assignment of workers to vacancies ensuring assigning a specified worker in a specified job. The objective s might be minimizing the total time to complete a set of tasks, minimizing the cost of assignments, and maximizing skill ratings. The problem is not so simple to quantify all those measures in one tangible variable. Therefore, i n this paper the use of verbal information for representing the vague knowledge in terms of natural linguistic labels is proposed. It allows the problem to be recognized as it is in a real life. For such types of problems, an analysis using the fuzzy number approach promises to be potentially effective. The fuzzy suitability evaluation is executed coupled with the memetic algorithm. Also, real case study is presented. The results demonstrate that the workers' assignment problem can be solved effectively for the multiple-criteria decision-making processes.

Index Term-- Decision Making, Fuzzy Numbers, Memetic Algorithm, S taff Assignment.

I. INTRODUCTION

Fro m a p ractical point of vie w, h iring new staff for specified vacancies represents a crucia l decision, due to the fact that the survival of the whole enterprise can depend upon the appropriate selection process. With high level of business competition, it is vital to have fle xib le staffs that are able to adapt themselves with work circu mstances. Thus, the most suitable choice of the personnel has a greater influence over the company’s future development [1].

In general, the staff selection problem is very difficu lt to be solved, even when it is tackled in a simp lified version containing only a single criterion and a ho mogeneous skill. In fact, the proble m has been known to be NP-co mplete [2]. When multip le criteria and various skills are involved, the problem beco mes much more d ifficu lt. So, it will be hard, if not impossible, to apply used mathematica l techniques or traditional progra mming to solve the problem. In most cases of staff selection proble ms the information that is available could not be precise or e xact. Even more, the imp recise informat ion could be represented as linguistic information in terms of variables such as opinions, thoughts, feelings, believes, etc. These variables refer to professional knowledge,

leadership, sense of responsibility, re lationships and cooperation with the other team members in the work group. In this respect, it is very c lear that HRMs and those who are in charge of determin ing the levels attained by each job candidate in the skills needed for the vacancy prefer to use natural language. Whatever the tests used (e.g., questionnaire, interview, aptitude test, evaluation and other methods), it is quite hard fro m the reality to e xpress these evaluations in numerical values. Therefo re, the goal of th is paper is to develop a methodology that has a capability to deal with staff selection in conditions of uncertainty.

Fro m the previous discussion, it arouses interest in applying the Fuzzy Sets Theory [3-5]. With the aim of being able to handle the uncertainty which is a characteristic of the decision ma king processes in staff selection problems. A selection process is necessary to obtain the best solution out of all available ones. Therefore , to optimize the selection process, there is a need for a good tool that is able to grasp all the comple xity of vague information in o rder to reach an e xcellent visible solution for such types of problems. Thus, for the sake of this paper a me metic algorith m is imp le mented. The me met ic algorithm is characterized by its use of a fitness function that allows the evaluation of linguistic information. To demonstrate the capability of the proposed methodology, a series of e xperiments have been carried out, and the results obtained showed that the proposed approach is very effective in finding desirable solutions. In addition, this methodology was actually imp le mented in an illustrative e xa mp le to show how the suggested approach is easy to be implemented. This paper is organized as follows. First, section 2 p resents an introduction to linguistic information operators that are very helpful for solving such types of problems. Then, section 3 shows descriptive materials of the staff selection proble m and the proposed methodology. Thereafter, the me metic a lgorithm is designed to achieve good reasonable solutions. Section 5 shows a rea l case study as a practica l e xa mp le of how the staff selection problem should be solved. Finally, the last section concludes some concluding remarks.

II. LINGUISTIC INFORM ATION & ASSIGNM ENT -SELECTION PROBLEM S

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problems, some of the dec ision data can be prec isely assessed while others cannot deal with HRMs’ a mbiguit ies, uncertainties and vagueness, which cannot be handled by crisp values [6]. When working in qualitative areas such as personnel selection, the information cannot be set out in a precise numerica l way. Thus, it would be a mo re realistic approach to use linguistic info rmation instead of numbers . This way of looking at things can be applied to a wide range of problems, informat ion retrieval [7], c lin ical diagnosis [8], education [9], and decision making [10-13].

Fuzzy assignment proble ms have rece ived great attention. For instance, Chen [14] proposed a fuzzy assignment mode l that did not consider the differences a mong indiv iduals. Wang [15] solved a similar mode l by graph theory. The author first presented his model on a network of which the arc values are fuzzy nu mbers, and then proposed a solution procedure to solve this network proble m. Dubois and Fortemps [16] proposed a fle xib le assignment proble m, which co mb ines with fuzzy theory, mu ltiple criteria dec ision -making and constraint-directed methodology. They also demonstrated and solved an exa mp le of fu zzy assignment proble m. Sa kawa et al. [17] dealt with actual proble ms on production and work force assignment of a housing materia l manufacturer, and formulated two-level linear and linear fractional progra mming problems according to profit and profitability ma ximizat ion, respectively.

The purpose of the Analytical Hierarchy Process (AHP) is to capture the expert’s knowledge; the conventional AHP cannot reflect the human thinking style. Therefore, fuzzy AHP and fuzzy e xtensions of AHP a re developed to solve hierarch ical fuzzy p roble ms. Saaty [18] proposed a method to give mean ing to both fuzziness in perception and fuzziness in mean ing. This method measures the relativity of fuzziness by structuring the functions of a system hierarch ically in a mu ltip le attribute fra me work. Buckley [19] e xtended Saaty’s AHP method in wh ich decision make rs can express their preference in fuzzy ratios instead of crisp values. The Van Laarhoven and Pedrycz’s study [20] used fuzzy scores for alternatives as well as sensitivities. Lee et a l. [21] developed a FAHP that is e mp loyed to generate the weighting of the four perspective of the balanced scorecard and the weighting of the performance of the in formation technology indicators. Also, Kahra man et al. [22] used FAHP in order to select the most appropriate industrial robotic systems. Chang [23] developed a fuzzy e xtent analysis for AHP which was re latively easier in computation than the other fuzzy AHP approaches and had similar steps of Saaty’s crisp AHP [18]. Kahraman et al. [24, 25] and Bo zdag, et al. [26] used Chang’s [23] e xtent analysis for the selection of the best catering firm, the best facility location, and the best co mputer-integrated manufacturing system, respectively.

III. FUZZY ANALYTICAL HIERARCHY PROCESS

(FAHP)

In this section, a methodology including the mathemat ical representation of the linguistic assignment-selection problem to evaluate the possible solutions using the concepts of

triangular fuzzy numbers (TFN) and linguistic variab les is proposed in order to evaluate the workers' suitability. In addition, a search method for finding a good solution by means of me met ic a lgorith m is presented later. General speaking, the evaluation criteria may be classified into three factors:

1) Socia l factors include co mmunication skill, professional knowledge, cooperation, leadership, sense of responsibility, relationship to other members, etc;

2) Performance factors include speed, quality, attendance condition, late coming, overtime, and experience, etc; and 3) Mental factors inc lude intelligence, proble m solving ability,

creativity, self-confidence, etc.

To the best author’s knowledge, there is no commercial software of FAHP that is currently available . Therefo re, t he outlines of the extent analysis method on fuzzy AHP are given in the following section, and are simila r to those outlines used in Bozdag et al. [26], Kah ra man et al. [24, 25], Kula k and Kahra man [27], and Bo zbura et al. [28]. This method is imple mented to staff selection proble m in a form of case study.

Let

X

x

x

,...,

x

n

2

,

1

be an object set, and G =

      m g g g ..., , 2 , 1

be a goal set. According to the method of

Chang’s extent analysis [23], each object is taken and extent analysis for each goal gi is performed respectively. Therefore, m e xtent ana lysis values for each object can be obtained, with the follo wing signs: 1

,

2

,

m

,

i

1,2,

...

,

n

gi

M

gi

M

gi

M

where all the

j

(j

1,2,

...

,

m)

gi

M

are TFNs whose

parameters a re l, m, and u. They are the least possible value (or rating), the most possible value (or rating), and the largest possible value (or rating), respectively. A TFN is represented as (l, m, u).

Step1.The value of fu zzy synthetic extent with respect to the ith object is defined as:

1

-M

1

*

m gij 1

1 

      

j n j gi m

j

M

i

i

S

(1)

As it is known, the mult iplication of two TFNs does not result in a TFN. In this step of the extent analysis, two TFNs are mu ltip lied as in Eq . (1). The result will not be a TFN. However, in this paper, it will be assumed that the non -linear combination of TFNs approximates to a TFN [29].

To obtain j gi M m

i j

 perform the fu zzy addit ion operation of

m extent analysis values for a particular matrix such that

n

,

...

2,

1,

i

,

,

,

j

1 1 m 1 j

            m j m j j u

j

m

l

M

gij

i j

(3)

And to obtain Mj 1 gi m 1 j 1         n i

, perform the fuzzy addit ion

operation of

M

j

(

j

1,

2,

...

,

m

)

gi

values such that

             

n

m

M

gi

j

l

ij

m

ij

u

ij

i

i j i j i j

m 1 n 1 m n m 1 n

1

,

1

1

,

1

j

1

(3)

And then compute the inverse of the vector in Eq. (4) such that

1

j

M

gi 1 1

        m j n i =          

,

1

1

,

1

1

,

1

1

i

l

n

i

i

m

n

i

i

u

n

i

, where

i i i

m

u

l

,

,

> 0. (4)

Step 2. The degree of possibility of:

M2 = (l2, m2, u2)

M1 = (l1, m1, u1) is defined as

)]

(

),

(

[min(

sup

)

(

2 1 1

x

2

y

x

y

M

M

V

M

M

(5)

And can be equivalently expressed as follows:

            

otherwise l m u m u l

l

if

m

if

M

M

hgt

M

M

V

) 1 1 ( ) 2 2 ( 2 1

2

u

1

0,

1

m

2

1,

(d)

2

)

2

M

1

(

)

1

2

(

(6)

Where d is the ordinate of the highest intersection point d

between

1

M

and

2

M

.

To compare M1and M2, we need both the values of V (M1

M2) and V (M2

M1).

Step 3.

The degree of possibility for a conve x fuzzy nu mber M to be greater than k convex fu zzy nu mbers MI (I=1, 2, … , k) can be

defined by V(M

M1 , M2 , … , Mk) = V [(M

M1 )

and M

M2) and ,… , and (M

Mk )=

Min V [(M

Mi ), i= 1, 2, 3 , … , k. (7) Assume that

d (Ai )= Min V (Si

Sk ) (8)

For k =1, 2, …, n; k = i. Then the weight vector is given by W’ = (d’ (A1), d’ (A’2)…, d’ (An)) (9)

Where Ai (i = 1, 2,…, n ) are n elements.

Step 4. Via normalization, the normalized weight vectors are: W’ = (d’ (A1), d’ (A’2)… , d’ (An)T) (10)

Step 5. Let W’js, and W’cs, are the norma lized we ight vectors of jobs/skills and candidates/skills. Those weight vectors, then,

are ready to be employed in the me metic a lgorith m. The me metic algorith m is discussed in more details in the next section. Figure 1 illustrates the hierarchy of the considered problem.

The imple mentation of the above mentioned steps is some what difficult. There fore, a co mputerized tool is developed to support this methodology and to simplify its imp le mentation. In addition, the co mputerization a lso provides the results with high accuracy. The developed tool is programmed in Visual Basic language (version #6).

IV. MEM ETIC ALGORITHM

Although the me metic a lgorith m (MA) is general in the sense that it can be used for every combinatoria l optimization problem, some co mponents are proble m-specific. The creation of the initial population, as well as the local search and the genetic operators, is specific to the assignment selection problem, and will thus be described in the following. Furthermore, the selection mechanis ms and the restart technique used in this algorithm are discussed.

1. Creating the Initial Population:

To create the init ial population of the MA, a desired number of solutions (candidates are assigned to jobs) are purely generated randomly without using any heuristic informat ion. Then, a local search procedure is applied. A variant of the 2-opt heuristic [30], a lso known as the pairwise interchange heuristic is employed as the local search method of the proposed model.

2. The Recombination Operator:

The reco mbination operator DPX (distance preserving crossover) that one is introduced in Merz and Freisleben [31] which relies on the notion of a d istance between solutions. The basic idea behind DPX can be described as follo ws. A ll informat ion contained in both parents is transferred to the offspring in the sense that all genes in the parents are identical in the offspring that has the same d istance to each of its parents, and this distance is equal to the distance between th e parents themselves. The all other genes of the developed offspring change. The reco mbination operator works as follows:

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Fig. 1. T he hierarchy of the worker assignment problem.

3. The Mutation Operator:

As shown in Fig. 3, the mutation operator used in the MA approach exchanges a sequence of candidates in the solution until the offspring has a predefined distance to it s parent. To ensure that the offspring has the predefined distance, in each step, the second candidate chos en is e xchanged again in the succeeding step, such that the resulting distance between parent and offspring is one plus the number of e xchanges. To illustrate the mutation operator, consider the e xa mp le shown in Fig. 3. In the first step, candidates 4 and 9 are e xchanged, then candidate 1 and 6, and in the last step candidate 7 is e xchanged with 3. Thus, the resulting (mutation ju mp) d istance between parent and offspring is 6.

4. Selection and Diversification:

Selection occurs two times in the main progra m of the MA. Selection for reproduction is performed before a genetic operator can be applied, and selection for survival is performed after the offspring o f a new generation have been created to reduce the population to its original size. Selection fo r reproduction is performed on a purely random basis without bias to fitter individuals, while selection for survival is achieved by choosing the best individuals fro m the pool of parents and children using tournament selection.

Fig. 2. DPX recombination operator for the assignment problem

.

Fig. 3. M utation operator for the assignment problem.

5. Tournament selection:

After performing offspring, a population of size 2p are performed (p parents generate p children). Tourna ment is performed considering the entire population (i.e., we use

(m + l) selection). Tournaments consist of c confrontations per indiv idual, with the c opponents randomly chosen fro m the entire population. When th e tournaments finish, the p individuals with the larger

3

5

7

9

8

1

2

4

3

5

7

4

8

6

1

2

9

3

5

7

4

8

1

6

2

9

7

5

3

4

8

1

6

2

9

6

3

5

7

9

8

6

1

2

4

4

1

4

9

8

3

5

2

7

6

8

3

8

8

9

5

5

5

5

5

5

5

5

7

7

2

1

1

The best Selection and

assignment of workers

Job-1

Job-2

Job-3

Job-4

Job-J

Skill-1

Skill-2

Skill-3

Skill-4

Skill-5

Skill-6

Skill

-S

Cand.-1

Cand.-2

Cand.-3

Cand.-4

Cand.-C

FAHP

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number of victories are selected to form the following generation. The tournament rules adopted for the current paper are very similar to those adopted by Deb [32] in his penalty approach based on feasibility. The ne w tourna ment rules for th is approach are as follo ws: 1) Re move the worst fa mily me mbers. Th is is the rule applied in Drezner [33]; 2) ca lculate the distance between all pa irs of e xisting 2p population members. Then, create a list of all population me mbe r whose closest distance to another population me mbe r a mong all pairs. The population me mbers in the list with the worst value of the objective function are selected for re moval with probability of 0.3; otherwise rule #1 is selected.

V. EXPERIM ENTAL EVALUATION

A common set of eight benchmark proble ms having sizes n = 5, 6, 7, 8, 12, 15, 20, and 30 fac ilities has been e xa mined to test the proposed MA [34]. Fo r each proble m, fifty runs were conducted using diffe rent initia l assignments. The results of the proposed algorithm generated better solutions in all e xa mined proble ms. The average costs and the best final costs are compared with other algorith ms in Tables 1 and 2, respectively. For n = 5, 6, 7, 8, and 12, the optima l costs were found in all those fifty runs. For h igher n, The MA converged to one of the sub-optima l costs. Tables 1 and 2 indicate that the proposed algorith m is the best among the listed algorithms.

TABLE II

Comparison of the best final costs obtained.

TABLE I

Comparison of the average costs obtained.

n H63 H63-66 Biased CRAFT FRAT GESA MA Sampling

5 27.6 29.4 26.8 28.2 29 25 25

6 44.2 44.2 43.6 44.2 44.8 43 43

7 78.8 78.4 74.8 79.6 79.8 74 74

8 114.4 110.2 107 113.4 111.0 107 107

12 317.4 310.2 293 296.2 301.6 289.36 289

15 632.6 600.2 580.2 606 601.4 575.18 575.2 20 1400.4 1345 1313 1339 1335.6 287.38 1286.3 30 3267.2 3206.8 3189.6 3189.6 3160.2 3079.32 3072.6 VI. ILLUSTRATIVE EXAM PLE A multinational co mpany of agricultural equip ment wants to hire staff me mbe rs for one of its local branch . For simp lic ity, the mid manager decided to h ire four candidates from the candidates’ applications to four available positions. All have high school degrees. The informat ion for the proble m is as follo ws: (a ) the worke rs are 4 persons that are identified by nu merical nu mbers (e.g., 1, 2, 3, and 4); (b) the available vacancies are 4. Their na mes and codes are identified as shown in Table 3; (c) there are 8 evaluation criteria. The e ight evaluation criteria are categorized in Table 4; (d) the human res ource manager of the co mpany ranked the required skills to the proposed jobs according to the ranking scale (e.g., very high "V.H", high "H", low high “LH”, h igh mediu m "H.M", mediu m “M”, lo w med iu m “ L.M.” , high low "H.L", low “ L” , and very low " V.L"). They are presented in Tab le 5; (e) each worker should be assigned to only one job; (f) the HRM as we ll as five e xpe rts fro m the co mpany provided the skills of the applied candidates according to the same ran king score very high "V.H", h igh "H", med iu m "M", low "L", and very low " V.L", as shown in Table 6; (g) the HRM co mpared each skill re lative to the other for each job, based on the intensity importance fro m Table 7. TABLE III . Job name and its code. Position name Position code Human Resources Specialist. Purchasing Specialist. Inventory Supervisor Spare Parts Seller. HRS PS IS SPS TABLE VI Candidates' skills levels TABLE IV T he workers’ evaluation criteria. n H63 H63-66 Biased CRAFT FRAT GESA MA Best Sampling Known 5 25 29 25 25 25 25 25 25

6 43 43 43 43 43 43 43 43

7 78 77 74 74 74 74 74 74

8 109 107 107 107 107 107 107 107

12 301 304 289 289 295 289 289 289

15 617 578 575 583 575 575 575 575

20 1384 1319 1304 1324 1300 1287 1285 1285

30 3244 3161 3093 3148 3129 3079.32 3062 3062

Required skill

Candidate #

1 2 3 4

1 M L VH VH

2 H H VH VH

3 V.H M L LM

4 V.H H M LM

5 V.H LM M H

6 H HM L VL

7 H M H VL

8 H H VH H

1. Foreign language knowledge (i.e., English) 2. Computer application skills (i.e., office software). 3. Report writing skill.

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TABLE V TABLE VII

Skill Levels required for each Job. T he membership functions of fuzzy number.

Using the above data and the suggested approach, the candidates are assigned to their jobs as follows. Based on linguistic variables mentioned in Tables 5 and 6, the manager has capability to determine the FTNs and their me mbe rship functions fro m Tab le 7. At this stage the comparison of each skill required for a job is compared to the other skills using the five steps that are mentioned in section #3. These processes are also employed for all jobs, as shown in Tables 8 and 9.

Ne xt is to apply the same procedures again for the candidates and the s kills they have in which each skill fo r the candidate is compared to the same skill their co mpetitive has based on the skill required for the job. Table 10 shows the starting point for this process. This table is based on "If Then statement rule". For e xa mp le, If the candidate skill is “High” and the job skill required is “High” Then their relationship between them is TFN “ 9~ ”. According to the rest candidate qualifications and the job skills required, the TFNs are assigned to the candidates’ skills at their jobs’ skills, as shown in Tables 11 and 12. Finally, adding the weights for each skill associated with each job and candidate mu ltip lied by the weights of the corresponding skill as shown in Table 13. A final score is obtained for each candidate at each job assigned. Table 14 illustrates those final scores.

Then the obtained scores fro m the FA HP are automat ically treated using the me metic algorith m progra mmed in Visual Basic language. The final output is presented on the computer screen as we ll as in a stored text file . The results showed that candidate 1 was assigned to job 2, candidate 2 was assigned to job 4, candidate 3 was assigned to job 3, and candidate 4 was assigned to job 1.

VII. DISCUSSION AND CONCLUSION

Assignment problem is of great use in decision-ma king, e.g. resource allocation problems, such as assigning personnel to jobs, tasks to machines, etc. As the actual case, the managers hope not only to raise the quality of each job, task, etc. when assigned to a specific candidate, but also to minimize the

total cost used. However, the cost of each job, depending on the quality, is not a deterministic nu mber, nor the planned total cost. In this paper, a new approach is presented to solve such types of problems in the industrial environment. A fuzzy AHP lin ked with an optimization tool (me met ic algorith m) model is proposed in order to overco me such an uncertain environment in the real world application and propose a new algorith m to solve it. Co mputational e xperience shows that the proposed algorithm performs satisfactorily. For the proposed methodology, one may consider the fu zzy A HP where the costs are triangular, trapezoidal or other types of me mbership functions. The proposed methodology was demonstrated in a real proble m fro m the business world. It is applied on a mu lt inational industrial company. The results proved that the candidates’ assignment is one of the key issues and our proposal methodology is effective to dea l with the verbal terms fo r solving selecting staff problems.

In this paper, d iffe rent leve ls of linguistic values for the jobs’ skill requirements and needs are designated in the rating and we ighting scales. However, linguistic values could be adjusted based on the needs of both detailed evaluation and available data characteristics. To simp lify and generalize the proposed approach, a computerized tool is developed using Visual Basic Software . Th is tool ca rries out many steps of calculations for different steps of the suggested method. In addition, this tool provides simp le lin kage with the optimization me met ic algorith m software which is utilized to solve and carry out the final stage of the worke r assignment proble m. An important thing to mention is that this developed software not only can solve the wo rke r assignment proble m, it can also be used for solving general multi-criteria decision-making problems.

Required skills

Jobs

HRS PS IS SPS

1 M H H H

2 H H V.H V.H

3 LM L V.H V.H

4 HM H V.H V.H

5 HL V.H H H

6 HM M M M

7 LM L L L

8 LH H H H

Fuzzy number

Definition for ratio scale

Membership function

Reciproc al Fuzzy

number

Membership function

1 ~

2 ~

3

~

4 ~

5

~

6

~

7 ~

8

~

9 ~

Very Low (V.L).

Low.

High Low (H.L).

Low medium (LM).

Medium (M).

High Medium (HM).

Low High (LH).

High (H).

Very High (VH).

(1, 1, 2)

(1, 2, 3)

(2, 3, 4)

(3, 4, 5)

(4, 5, 6)

(5, 6, 7)

(6, 7, 8)

(7, 8, 9)

(8, 9, 9) 1 ~

2 / ~ 1

3

/

~

1

4

/

~

1

5 / ~ 1

6

/

~

1

7 / ~ 1

8

/

~

1

9

/

~

1

(1, 1, 2)

(1/3, 1/2, 1)

(1/4, 1/3, 1/2)

(1/5, 1/4, 1/3)

(1/6, 1/5, 1/4)

(1/7, 1/6, 1/5)

(1/8, 1/7, 1/6)

(1/9, 1/8, 1/7)

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TABLE IX

Evaluation of skill requirements to each job in terms of T FN.

T ABLE VIII

Evaluation of comparing skill requirements to each job

.

TABLE XIV

T he final scores for the given example.

Job 1 Job 2 Job 3 Job 4 Candidate 1 0.263 0.257 0.289 0.283

Candidate 2 0.285 0.254 0.235 0.237

Candidate 3 0.231 0.228 0.206 0.216

Candidate 4 0.221 0.260 0.270 0.263

JOB 1 Skills 1 2 3 4 5 6 7 8

1 (1,1,2) (.44,.63,.86) (.80,1.25,2.0) (.57,.83,1.2) (1.0,1.67,3.0) (.57,.83,1.2) (.80,1.25,2.0) (.50,.71,1.0) 2 (1.17,1.6,2.25) (1,1,2) (1.4,2.0,3.0) (1.0,1.33,1.8) (1.75,2.67,4.5) (1.0,1.33,1.8) (.80,1.25,2) (.88,1.14,1.5)

3 (.50,.80,1.25) (.33,.50,71) (1,1,2) (.43,.67,1.0) (.75,1.33,2.5) (.43,.67,1.0) (1,1,2) (.37,.57,.83) 4 (.83,1.2,1.75) (.56.75,1.0) (1.0,1.5,2.33) (1,1,2) (1.25,2.0,3.5) (1,1,2) (1.0,1.5,2.33) (.63,.86,1.17) 5 (.33,.6,1.0 ) (.22,.37,.57) (.40,.75,1.33) (.29,.50,.80) (1,1,2) (.29,.50,.80) (.40,.75,1.33) (.25,.43,.67) 6 (.83,1.2,1.75 ) (.55,.75,1.0 ) (1.0,1.5,2.33) (1,1,2) (1.25,2.0,3.5) (1,1,2) (1.0,1.5,2.33) (.63,.86,1.17) 7 (.5,.8,1.25) (.33,.50,.71) (1,1,2) (.43,.67,1.0) (.75,1.33,2.5) (.43,.67,1.0 ) (1,1,2) (.37,.57,.83) 8 (1.0,1.4,2.0) (.67,.88,1.14) (1.2,1.75,2.67) (.86,1.17,1.6) (1.5,2.33,4.0) (.86,1.17,1.6) (1.2,1.75,2.67 ) (1,1,2)

JOB 1 Skills 1 2 3 4 5 6 7 8

1 (1,1,2) (4,5,6)/(7,8,9) (4,5,6)/(3,4,5) (4,5,6)/(5,6,7) (4,5,6)/(2,3,4) (4,5,6)/(5,6,7) (4,5,6)/(6,7,8) (4,5,6)/(6,7,8) 2 (7,8,9)/(4,5,6) (1,1,2) (7,8,9)/(3,4,5) (7,8,9)/(5,6,7) (4,5,6)/(2,3,4) (7,8,9)/(5,6,7) (7,8,9)/(3,4,5) (7,8,9)/(6,7,8)

3 (3,4,5)/(4,5,6) (3,4,5)/(7,8,9) (1,1,2) (3,4,5)/(5,6,7) (3,4,5)/(2,3,4) (3,4,5)/(5,6,7) (1,1,2) (3,4,5)/(6,7,8) 4 (5,6,7)/(4,5,6) (5,6,7)/(7,8,9) (5,6,7)/(3,4,5) (1,1,2) (5,6,7)/(2,3,4) (1,1,2) (5,6,7)/(3,4,5) (5,6,7)/(6,7,8) 5 (2,3,4)/(4,5,6) (2,3,4)/(7,8,9) (2,3,4)/(3,4,5) (2,3,4)/(5,6,7) (1,1,2) (2,3,4)/(5,6,7) (2,3,4)/(3,4,5) (2,3,4)/(6,7,8) 6 (5,6,7)/(4,5,6) (5,6,7)/(7,8,9) (5,6,7)/(3,4,5) (1,1,2) (5,6,7)/(2,3,4) (1,1,2) (5,6,7)/(3,4,5) (5,6,7)/(6,7,8) 7 (3,4,5)/(4,5,6) (3,4,5)/(7,8,9) (1,1,2) (3,4,5)/(5,6,7) (3,4,5)/(2,3,4) (3,4,5)/(5,6,7) (1,1,2) (3,4,5)/(6,7,8) 8 (6,7,8)/(4,5,6) (6,7,8)/(7,8,9) (6,7,8)/(3,4,5) (6,7,8)/(5,6,7) (6,7,8)/(2,3,4) (6,7,8)/(5,6,7) (6,7,8)/(3,4,5) (1,1,2)

JOB 2 Skills 1 2 3 4 5 6 7 8 1 (1,1,2) (7,8,9)/(6,7,8) (7,8,9)/(2,3,4) (7,8,9)/(6,7,8) (7,8,9)/(2,3,4) (7,8,9)/(6,7,8) (1,1,2) (7,8,9)/(6,7,8) 2 (6,7,8)/(7,8,9) (1,1,2) (6,7,8)/(2,3,4) (1,1,2) (6,7,8)/(2,3,4) (1,1,2) (6,7,8)/(7,8,9) (1,1,2) 3 (2,3,4)/(7,8,9) (2,3,4)/(6,7,8) (1,1,2) (2,3,4)/(6,7,8) (1,1,2) (2,3,4)/(6,7,8) (2,3,4)/(7,8,9) (2,3,4)/(6,7,8) 4 (6,7,8)/(7,8,9) (1,1,2) (6,7,8)/(2,3,4) (1,1,2) (6,7,8)/(2,3,4) (1,1,2) (6,7,8)/(7,8,9) (1,1,2) 5 (2,3,4)/(7,8,9) (2,3,4)/(6,7,8) (1,1,2) (2,3,4)/(6,7,8) (1,1,2) (2,3,4)/(6,7,8) (2,3,4)/(7,8,9) (2,3,4)/(6,7,8) 6 (6,7,8)/(7,8,9) (1,1,2) (6,7,8)/(2,3,4) (1,1,2) (6,7,8)/(2,3,4) (1,1,2) (6,7,8)/(7,8,9) (1,1,2) 7 (1,1,2) (7,8,9)/(6,7,8) (7,8,9)/(2,3,4) (7,8,9)/(6,7,8) (7,8,9)/(2,3,4) (7,8,9)/(6,7,8) (1,1,2) (7,8,9)/(6,7,8) 8 (6,7,8)/(7,8,9) (1,1,2) (6,7,8)/(2,3,4) (1,1,2) (6,7,8)/(2,3,4) (1,1,2) (6,7,8)/(7,8,9) (1,1,2)

JOB 3 Skills 1 2 3 4 5 6 7 8 1 (1,1,2) (4,5,6)/(6,7,8) (4,5,6)/(6,7,8) (1,1,2) (4,5,6)/(1,2,3) (4,5,6)/(2,3,4) (4,5,6)/(5,6,7) (4,5,6)/(2,3,4) 2 (6,7,8)/(4,5,6) (1,1,2) (1,1,2) (6,7,8)/(4,5,6) (6,7,8)/(1,2,3) (6,7,8)/(2,3,4) (6,7,8)/(5,6,7) (6,7,8)/(2,3,4) 3 (6,7,8)/(4,5,6) (1,1,2) (1,1,2) (6,7,8)/(4,5,6) (6,7,8)/(1,2,3) (6,7,8)/(2,3,4) (6,7,8)/(5,6,7) (6,7,8)/(2,3,4) 4 (1,1,2) (4,5,6)/(6,7,8) (4,5,6)/(6,7,8) (1,1,2) (4,5,6)/(1,2,3) (4,5,6)/(2,3,4) (4,5,6)/(5,6,7) (4,5,6)/(2,3,4) 5 (1,2,3)/(4,5,6) (1,2,3)/(6,7,8) (1,2,3)/(6,7,8) (1,2,3)/(4,5,6) (1,1,2) (1,2,3)/(2,3,4) (1,2,3)/(5,6,7) (1,2,3)/(2,3,4) 6 (2,3,4)/(4,5,6) (2,3,4)/(6,7,8) (2,3,4)/(6,7,8) (2,3,4)/(4,5,6) (2,3,4)/(1,2,3) (1,1,2) (2,3,4)/(5,6,7) (1,1,2) 7 (5,6,7)/(4,5,6) (5,6,7)/(6,7,8) (5,6,7)/(6,7,8) (5,6,7)/(4,5,6) (5,6,7)/(1,2,3) (5,6,7)/(2,3,4) (1,1,2) (5,6,7)/(2,3,4) 8 (2,3,4)/(4,5,6) (2,3,4)/(6,7,8) (2,3,4)/(6,7,8) (2,3,4)/(4,5,6) (2,3,4)/(1,2,3) (1,1,2) (2,3,4)/(5,6,7) (1,1,2)

JOB 4 Skills 1 2 3 4 5 6 7 8 1 (1,1,2) (8,9,9)/(7,8,9) (8,9,9)/(5,6,7) (1,1,2) (8,9,9)/(2,3,4) (8,9,9)/(6,7,8) (8,9,9)/(5,6,7)

(8,9,9)/(6,7,8)

(8)

TABLE X

T he relationship based rules in terms of T FNs between each job skill needs and the candidate's skill.

Candidate qualification

Qualification that are needed for the job

VL L HL LM M HM LH H VH VL 9~

7

~

5

~

3

~

3

~

3

~

1

~

1

~

1

~

L

8

~

9~

7 ~

5 ~

3 ~

3 ~

3 ~

1 ~

1 ~

HL ~7 ~8 ~9 7~ ~5 3~ ~3 ~3 ~1 LM

6

~

7

~

8

~

9

~

7

~

5

~

3

~

3

~

3 ~

M

5

~

6

~

~7

8

~

9

~

7

~

5

~

3

~

3

~

HM 4~ 5~ 6~ 7~ 8~ 9~ 7~ 5~ 3~ LH 3~

4

~

5~

6 ~

7

~ 8~

9 ~

7

~

5

~

H

2

~

3

~

4

~

5

~

6

~

7

~

8

~

9

~

7

~

VH

1

~

2

~

3 ~

4

~

5 ~

6 ~

7 ~

8 ~

9 ~

TABLE XII

T he candidate fuzzy evaluation matrix with respect to the goals of jobs 1, 2, 3, and 4. JOB1 SKILL1 Candidate 1 2 3 4

1 (1,1,2) (2,3,9/2) (4/3,9/5,9/4) (4/3,9/5,9/4) 2 (2/9,1/3,1/2) (1,1,2) (1/3,3/5,1) (1/3,3/5,1) 3 (4/9,5/9,3/4) (1,5/3,3) (1,1,2) (1,1,2) 4 (4/9,5/9,3/4) (1,5/3,3) (1,1,2) (1,1,2) JOB1 SKILL2 Candidate 1 2 3 4

1 (1,1,2) (1,1,2) (8/9,9/8,9/7) (8/9,9/8,9/7) 2 (1,1,2) (1,1,2) (8/9,9/8,9/7) (8/9,9/8,9/7) 3 (7/9,8/9,9/8) (7/9,8/9,9/8) (1,1,2) (1,1,2) 4 (7/9,8/9,9/8) (7/9,8/9,9/8) (1,1,2) (1,1,2) JOB1 SKILL3 Candidate 1 2 3 4

1 (1,1,2) (1/3,1/2,5/7) (1/2,4/5,5/4) (3,4,5)/(8,9,9) 2 (7/5,2,3) (1,1,2) (7/6,8/5,9/4) (7,8,9)/(8,9,9) 3 (4/5,5/4,2) (4/9,5/8,6/7) (1,1,2) (4/9,5/9,3/4) 4 (8/5,9/4,3) (8/9,9/8,9/7) (4/3,9/5,9/4) (1,1,2) JOB1 SKILL4 Candidate 1 2 3 4

1 (1,1,2) (5/8,6/7,7/6) (5/8,6/7,7/6) (5/6,6/5,7/4) 2 (6/7,7/6,8/5) (1,1,2) (1,1,2) (1,7/5,2) 3 (6/7,7/6,8/5) (1,1,2) (1,1,2) (1,7/5,2) 4 (4/7,5/6,6/5) (1/2,5/7,1) (1/2,5/7,1) (1,1,2) JOB1 SKILL5 Candidate 1 2 3 4

1 (1,1,2) (2/9,3/8,4/7) (1/4,3/7,2/3) (2/5,3/4,4/3) 2 (7/4,8/3,9/2) (1,1,2) (7/8,8/7,3/2) (7/5,2,3) 3 (3/2,7/3,4) (2/3,7/8,8/7) (1,1,2) (6/5,7/4,8/3) 4 (3/4,4/3,5/2) (1/3,1/2,5/7) (3/8,4/7,5/6) (1,1,2) JOB1 SKILL6 Candidate 1 2 3 4

1 (1,1,2) (2/3,7/9,1) (3/2,8/3,4) (3/2,7/3,4) 2 (1,9/7,3/2) (1,1,2) (2,3,9/2) (2,3,9/2) 3 (1/4,3/7,2/3) (2/9,1/3,1/2) (1,1,2) (1,1,2) 4 (1/4,3/7,2/3) (2/9,1/3,1/2) (1,1,2) (1,1,2) JOB1 SKILL7 Candidate 1 2 3 4

1 (1,1,2) (4/9,5/8,6/7) (1,1,2) (1,5/3,3) 2 (7/6,8/5,9/4) (1,1,2) (7/6,8/5,9/4) (7/4,8/3,9/2) 3 (1,1,2) (4/9,5/8,6/7) (1,1,2) (1,5/3,3) 4 (1/3,3/5,1) (2/9,3/8,4/7) (1/3,3/5,1) (1,1,2) JOB1 SKILL8 Candidate 1 2 3 4

(9)

TABLE XIII

T he FAHP results for the given example.

JO B 1

Skills 1 2 3 4 5 6 7 8 Normalization

Weight 0.116 0.162 0.097 0.135 0.067 0.149 0.112 0.162

Candidate 1 0.379 0.263 0.149 0.243 0.146 0.331 0.256 0.258 0.263

Candidate 2 0.121 0.263 0.311 0.275 0.337 0.389 0.346 0.258 0.285

Candidate 3 0.250 0.237 0.201 0.275 0.312 0.140 0.256 0.226 0.231

Candidate 4 0.250 0.237 0.340 0.207 0.205 0.140 0.143 0.258 0.221

JO B 2

Skills 1 2 3 4 5 6 7 8 Normalization

Weight 0.155 0.143 0.060 0.143 0.072 0.155 0.142 0.131

Candidate 1 0.196 0.269 0.120 0.336 0.366 0.295 0.194 0.261 0.257

Candidate 2 0.053 0.269 0.267 0.358 0.158 0.343 0.295 0.261 0.254

Candidate 3 0.376 0.231 0.324 0.153 0.158 0.181 0.194 0.218 0.228

Candidate 4 0.376 0.231 0.289 0.153 0.318 0.181 0.318 0.261 0.260

JO B 3

Skills 1 2 3 4 5 6 7 8 Normalization

Weight 0.134 0.158 0.158 0.134 0.063 0.096 0.155 0.101

Candidate 1 0.196 0.210 0.472 0.366 0.336 0.295 0.194 0.261 0.289

Candidate 2 0.053 0.210 0.234 0.318 0.153 0.343 0.295 0.261 0.235

Candidate 3 0.376 0.290 0.061 0.158 0.153 0.181 0.194 0.218 0.206

Candidate 4 0.376 0.290 0.234 0.158 0.358 0.181 0.318 0.261 0.270

JO B 4

Skills 1 2 3 4 5 6 7 8 Normalization

Weight 0.156 0.143 0.112 0.156 0.035 0.129 0.112 0.156

Candidate 1 0.196 0.210 0.472 0.366 0.336 0.295 0.194 0.261 0.283

Candidate 2 0.053 0.210 0.234 0.318 0.153 0.343 0.295 0.261 0.237

Candidate 3 0.376 0.290 0.061 0.158 0.153 0.181 0.194 0.218 0.216

Candidate 4 0.376 0.290 0.234 0.158 0.358 0.181 0.318 0.261 0.263

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Fuzzy AHP,” European Journal of Operations Research, vol. 95, pp.649-655, 1996.

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Figure

Fig. 1. The hierarchy of the worker assignment problem.
TABLE VI Candidates' skills levels
TABLE VIII Evaluation of comparing skill requirements to each job

References

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