modelling of optimal expansion of a fuzzy competence set

Modelling of Optimal Expansion of aFuzzy Competence Set

HSIAO-FAN WANG and C. H. WANGDepartment of Industrial Engineering, National Tsing Hua University, Hsinchu, Taiwan, ROC

A Competence Set, which is a Habitual Domain (HD), is a collection of knowledge and skills. We usethese knowledge and skills to solve problems. Based on the concept of fuzzy set theory, this study pro-poses a method to ®nd an optimal process to expand one's competence set. This process presents anexpansion order and by a directed graph, the optimal path of expansion at each stage can be found bya proposed model. This model is a minimization form constrained by max-product fuzzy relationinequalities. With this model, not only the compound skills can be considered, but also the levels of theacquired skills to solve a problem can be described. Theoretical support is accompanied by an illustra-tive example. # 1998 IFORS. Published by Elsevier Science Ltd. All rights reserved

Key words: Habitual domain, fuzzy competence sets, max-product relation constraints, modelling.

1. INTRODUCTION

The concept of Habitual Domain (HD) was proposed by Yu (1990). It is a collection of allideas, knowledge and skills. All of these elements can help us to solve problems. So any elementconsidered above can be regarded as a competence, and thus a competence set is a habitualdomain of a person. When time passes by, unless one pays special e�ort to acquired new knowl-edge, one's competence set will be stabilized in a certain domain. Thus, when one faces a pro-blem that is beyond one's current competence set, how can one expand one's competence set sothat one can solve the problem e�ciently?

Yu and Zhang (1992) have employed the method of a minimum spanning tree to ®nd an opti-mal expansion path of a competence set. Then Li and Yu (1994) used a deduction graph to ®ndan optimal expansion process through a 0-1 integer programming model. However, bothmethods considered either non or absolute capability to solve a problem, which is often unrealis-tic. For instance, a person who plays basketball, can do ``very well'', ``well'', ``badly'', ``verybadly'', etc. Therefore, a description of the ability of a basketball player is, with certain levels inlinguistic terms instead of `yes' or `no', a crisp term. In such cases, employing the concept offuzzy sets initiated by Zadeh (1965) would be more appropriate. In particular, in a learning pro-cess of expanding a competence set, requiring a 100% ability for solving a problem is not onlyunrealistic, but also slows down the learning speed. Therefore, the fuzzy set approach has beensuggested by Wang (1993). However, using max±min composite operators to describe an expan-sion process can be too conservative to fully describe the acquired, and required, competencesets. Therefore, in this study, we shall analyze such expanding process details and propose amodel to suggest an optimal expansion of a competence set.

In the following sections the properties and structure of a competence set are discussed andthe expansion model is derived. Then a solution procedure is developed. The proposed modelwith the solution procedure is illustrated with a learning example. Finally, the conclusion anddiscussion are drawn.

2. CONCEPT AND STRUCTURE OF A COMPETENCE SET

From the Introduction above, one may realize that, in general, whether a person has compe-tence g or not cannot be described in precise terms. Therefore, employing the concept of fuzzy

Int. Trans. Opl Res. Vol. 5, No. 5, pp. 413±424, 1998# 1998 IFORS. Published by Elsevier Science Ltd

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413

sets to describe the level of competence a person possesses and the strength of competencewhich is required for solving a problem would be an appropriate approach.

Before we proceed further analysis, let us de®ne a fuzzy set as follows:

De®nition 1. (Zimmerman, 1996) Let X be a universal set. A fuzzy set A in X is a set of orderedpairs:

A � f�x, mA�x��jx 2 X gwhere mA(x) is the membership function or degree of truth of x in A which maps X into themembership space [0,1] denoted by mA: X 4 [0,1] with x 4 mA (x).

Then we shall have the following de®nition which describes such levels or strengths in therange of unit interval:

De®nition 2. For each competence g, its strength a is a function of a person P or an event E asde®ned by

a: {P or EvP is a person, E is an event} [0,1]

Thus a competence can be described by (g, a(P)) = (g, a)(P)0ga(P) or (g, a(E)) = (g,a)(E)0ga(E), respectively. If there is no speci®c assertion and confusion, we use ga in the fol-lowing discussions. If a = 0, we call it a pseudo competence.

The learning experience in school tells us that if we want to learn course B, we would do bet-ter to learn B's fundamental courses ®rst. A fundamental course can be compound or single. Nomatter which type it is, let us denote it as A. Thus, we can say that course A is a backgroundcourse of course B. How easy one can learn course B depends on ``how well'' one has learnedA, and ``how close'' it relates to A. This idea can be extended to a general competence relationas follows:

De®nition 3. If g1 is a background competence of competence g2, then there is a background re-lation denoted by r(g1, g2) between g1 and g2 with 0 < r(g1,g2) = r21R1. We denote the relationof g1 and g2 as g1ÿ4r21g2 or g14g2 for simplicity.

So, if r21=0, then competence g1 is not a background competence of competence g2 which isdenoted by g1-g2. Thus, we say that g1 and g2 are independent if both r12=0 and r21=0. Suchbackground relations have the following properties:

Property 1

(i) (Irre¯exive) For competence g, g- g.(ii) (Asymmetric) If g14g2, then g2-g1.(iii) (Transitive) If g14g2 and g24g3, then g1-g3.

Apparently, if one has competence ga11 , then it would be easy to obtain g2 to level a2, if botha1 and r21 are large enough. Therefore, we have

Property 2. For a person P, if g1 is the only background competence of competence g2 withstrength a1, then the background-strength (potential) of obtaining g2 is b2=a1�r21.

If there are several background competences contributing to a competence with respectivestrengths, then we observe that the learning behaviour is according to the principle of maximalsupport as stated in the following:

De®nition 4. (Principle of Maximal Support)

Suppose ga11 �P �, . . . gajj �P �, . . . ,gann �P � are the background competences of gi, for eachi = 1,2, . . .m, with respective background relations ri1, . . . ,rij, . . . ,rin, then for a person P, thebackground-strength of obtaining gi is determined by

H.-F. Wang and C. H. WangÐOptimal Expansion of a Competence Set414

bi � max1<j<n aj � rij � _nj�1aj � rij, 8i � 1,2, . . . ,m:

This is called the Principle of Maximum Support.This Principle tells that for a person P, if he/she possesses several background competences

with di�erent strength and di�erent relations of competence g, then the background-strength ofobtaining g will be at least to the level of the maximum degree among the joint e�ects of itsbackground competences and their respective relations.

Now let us consider that a person P faces a problem E which requires competence g with suf-®cient level a. If the person does not have this competence, i.e a = 0, but he/she has back-ground-strength of competence g to certain level g, then the person will obtain competence g tothe level a easily. Otherwise, the person will need extra e�ort to obtain the same level of compe-tence g. Then, such extra e�ort is a cost denoted by c and we have the following de®nition.

De®nition 5. For a person P faces a problem E, and the required competence to solve the pro-blem is g to the level a (denoted by ga(E)), then a critical level g for the background-strength bto obtain ga is the level de®ned in [0,1] of which if brg, then c = 0; Otherwise, c>0.

Normally, the value of g depends on what competence one desires as denoted g(g). From the ob-servation above, one may conclude that:

Property 3. To obtain competence g, one should have enough background-strength b, such thatbrg(g). Otherwise, one cannot enhance the strength of competence g at all, and thus a = 0.

Therefore, the issue is how to e�ectively expand one's competence set so that the total cost willbe kept at a minimum.

De®nition 6. Let b be a person P's background-strength for obtaining competence g. Then g iscalled the person's P's skill competence, if b(P)rg(g); otherwise, g is called P's non-skill compe-tence. The set of skill competences is denoted by Sk(P) and that of the non-skill competence isNSk(P).

Lemma 1. A non-skill competence ga, b(P) is a pseudo competence. That is a = 0.

Proof. Since ga, b(P) is a non-skill competence of person P, so b(P) < g(g). Based on Property3, we have a = 0. Q.E.D.q

Therefore, if a(P)>0, then competence ga(P) must be a person P's skill competence. Besides,since learning is an accumulating process, the more one has background competences of compe-tence g, the easier one learns g because the joint e�ect to learn competence g is greater than theindividual e�ect. This joint e�ect then, is at least at the lowest level of these background compe-tences. Thus, we have the following statements.

De®nition 7. If ga11 (P), . . . ,gajj (P), . . . ,gann (P) contribute to person P's competence Ga(P), then

Ga�P � � fg1M� � �gj

M� � �gngminfa1,��,aj,��,ang

where$denotes an aggregation of gj, j = 1,2, . . .n which are the competence G's componentcompetences and Ga(P) is a compound competence of gj, j= 1,2, . . . ,n.

Besides, if any of gj in G has aj= 0, which is a non-skill competence, then G will be a non-skill competence as stated in the following:

Property 4. Ga(P) is a person P's skill competence if all gj are a person P's skill competences;Otherwise, Ga(P) is a P's non-skill competence.

In general, for the same competence, a compound competence has a stronger background re-lation than its component competences. However, we shall use g to denote a competence in gen-eral unless otherwise stated.

Now, from the viewpoint of an existing problem, we observe that

International Transactions in Operational Research Vol. 5, No. 5 415

De®nition 8. Whenever a problem E is solved, there is a non-pseudo competence with respect toproblem E with its background strength b(E) = g(g), then this competence is denoted byga(E),b(E)=ga,b(E).

This de®nition tells us that a person P can solve the problem E only if b(P)rb(E) = g(g). Thatis, g must be P's skill competence, and then he/she can enhance competence g's strength to a(E)in order to solve problem E.

Suppose ga(E),b(E) is the required competence for solving problem E, we can classify it intoone of the three competence types as follows.

De®nition 9.

(i) If b(P)rb(E) and a(P)ra(E), competence ga,b(E) is called a person P's type (1) compe-tence.

(ii) If b(P)rb(E) and a(P) < a(E), competence ga,b(E) is a person P's type (2) competence.(iii) If b(P) < b(E) then ga,b(E) is person P's type (3) competence.

Property 5. For a person P, any required competence ga,b(E) for solving problem E must belongto one of the three types described above.

In case (i), person P has solved problem E already. In (ii), person P can enhance the competenceg's strength from a(P) to a(E), then solve problem E. In (iii), person P is unable to enhance thecompetence g's strength, because g is P's non-skill competence (De®nition 6). Therefore, the per-son has to enhance g's background strength ®rst, and then enhance competence g's strength.

3. AN EXPANSION PROCESS OF A COMPETENCE SET

Let a competence set be a collection of competences. Since one has a competence or has a po-tential to obtain such competence depending on the degrees of strength as described inDe®nition 2 and De®nition 3, the collection of competences will form a competence set which isa fuzzy set de®ned by

fgak,bkk g � f�gk,�ak,bk��j0RakR1; 0RbR1,8kgThen, based on Yu's concept of Habitual Domains, we may have the following de®nitions:

De®nition 10. A person P's Habitual Domain is an in®nite set de®ned by[1k�1fgak,bkk �P �g � H Da,b�P �

where gak,bkk (P) is a single or compound competence. Therefore, P's skill competence set is

de®ned by

fgakbkk �P �jbkrgk�gk�, k � 1,2 . . .g � Skx,y�P �where with xk0ak for k = 1 . . .n, x � �x1 . . . xn�Tn�1and with yk0bk for k= 1 . . .n, y �� y1 . . . yn�Tn�1 are subvectors of a and b, respectively. Thus, P's non-skill competence set is

NSkd,x�P � � HDa,b�P �=Skx,y�P �where aT=[x, d]T and bT=[y, x]T.

De®nition 11. For a problem E, a competence set that is required to solve it is called a trulyneeded competence set, and is denoted by Trc,o(E)=

Sli�1 {Tr

ci,o i

i (E)}, where c ��c1, . . . ,c2, . . .cn�Tl�1 and o � �o1, . . . ,o2, . . .on�Tl�1 are Tr(E)'s respective strength and back-ground-strength vector with oi=gi (Tri (E)) and ci=ai (Tri(E)).


From Property 5, any competence Trci,o i

i (E) must be one of the three competence types as

described in De®nition 9. Since type (1) competences of Trci,o i

i (E) have reached the required

level of solving problem E, one does not have to enhance their strength. If all Trci,o i

i (E) are P's

type (1) competences, then person P can solve the problem E. Besides, we can enhance the

strength of type (2) competence Trci,o i

i (E) without extra e�ort to let them become person P's

type (1) competences. Hence, these skill competences are the ones one should obtain ®rst.

Therefore, we have

De®nition 12. (Principle of The Least Resistence)

Whenever a Trci,o i

i (E) is person P's type (2) competence, the competence expansion process is

performed based on the Principle of The Least Resistence.

If there is an expansion order in Trc,o(E), the Principle of The Least Resistence may induce

new type (2) competences which used to be person P's type (3) competences.

This stage of expansion will upgrade all type (2) competences to type (1) competences, and

induce certain type (3) competences to become type (2) competence, repeating the above pro-

cedure until no type (3) competences can be changed into type (2) competences. Then, we call

this stage a preliminary stage of expansion.

De®nition 13. After the preliminary stage of competence expansion process, the status of person

P's skill competence set is de®ned by Skx0

(P). Skx0

(P) includes two types of competences as

Skx0

(P) = {Skjvj = 1,2, . . . ,q < n} {Skq + k k= 1,2, . . . ,nÿ q}, where Skx0

j (P) is a single compe-

tence for all j = 1,2, . . . ,q < n, and each of Skx0

q�k(P), k= 1,2, . . . ,nÿ q, is a compound compe-

tence whose component competences are some of Skj(P), for any j = 1,2, . . . ,q. Then the rest of

type (3) competences are included in a non-skill competence set denoted by NSkf,j(P)vE, whichis a subset of person P's total non-skill competence set, (i.e

NSk(P)vEWKSk(P) = HD(P)%Sk(P)), and with f and j being the respective strength and back-

ground-strength vectors.

After the preliminary stage of expansion process, there are person P's non-skill competences left

in NSkf,j(P)vE which are needed to be treated, because these competences do not reach the

levels for solving the problem E. Figure 1 shows the relationship of HD(P), Sk(P), NSk(P),

Tr(E), and NSk(P)vE as described above.

Fig. 1. The relationship of HD(P), Sk(P), NSk(P), Tr(E) and NSk(P)vE.


De®nition 14. Skx(P)Sn

j�1 {Skxjj (P)} is a potential competence set of NSkf, j(P)vE if and only if_nj�1 xj�rijrji, for all i = 1,2, . . .m, with rij being a relation between Skj(P) and NSki(P)vE.

Therefore, one can expand ones competence set only if there exists at least one potentialcompetence Skj(P) such that max1R jRn{xj�rij} = V n

j�1xj�rijrji= gi (NSki(P)vE for anyi = 1,2, . . .m.

Thus, if we have a potential competence set Skx(P), then we can obtain NSkf,j(P)vE easily.That is, without any cost to obtain NSkf,j (P)vE.

4. THE EXPANSION MODEL

Theorem 1. For a person P who faces a problem E, after the preliminary stage of expansion pro-cess, if Skx

0

(P) is person P's skill competence set, and NSkf,j(P)vE is the collection of the type(3) competences, then

R o x0<j

where ``o'' is the max-product composite operator, and R = [rij]m� n is the relation matrix with rijbeing a relation between Skj(P) and NSki(P)vE.

Proof. If R o x05j, then there exists _nj�1x0j �rijji, for at least one i. That is, the ith competence in

NSk(P)vE is a skill competence of person P. It is a contradiction. Q.E.D.q

Then, we have the following property.

Theorem 2. (Monotonicity). If �0,0, . . . ,0�Tn�1Rx0RxR�1,1, . . . ,1�Tn�1, then R o x0RR o x.

Proof. Since all x0j Exj, implies rij�xj with 0RrijR1 for all i and j. Thus, we have

_nj�1rij�x0j E_nj�1 rij�x0

j for all i = 1,2, . . . ,m. That is, R o x0RR o x. Q.E.D.q

From Theorem 2, we know that we can enhance the strength of these skill competences inSk(P) from x0 to x to upgrade the background-strength in NSk(P)vE. In other words, based onDe®nition 14, we shall ®nd a strength vector x, x0RxR�1,1, . . . ,1�Tn�1, such that Skx(P) is a poten-tial set of NSkf,j(P)vE with minimum cost.

Since the cost of the compound competences are adhered to the respective costs of their com-ponent competences, therefore, the expansion model can be formulated as follows:

4.1. The expansion model

MinXq<n

j�1cj�xj � ÿ cj

ÿx0j

� �1�

s:t: R o xrj �2�

�0,0, . . . ,0�Tn�1Rx0RxR�1,1, . . . ,1�Tn�1 �3�

xq�k � min1RjRq

�xjjj 2 Jq�k,

, 8k � 1,2, . . . ,nÿ q �4�

where R is the given relation matrix between Sk(P) and NSk(P)vE, and cj is a monotonic increas-ing cost function of the Skj(P)'s strength. Equation (4) shows the implicit constraints about therelationship between single and compound competences, where Jq + k is an index set where eachj refers to a component competence of the compound competence Skq + k(P).

Note that the proposed expansion model implies that all competences in NSk(P)vE are inde-pendent, so they can be learned simultaneously. However, if there are dependent competences, it


means that one competence has to be obtained before another. Then the expansion order shouldbe considered when one analyzes the expansion process. In the following section we shall con-sider this case to develop a series of expansion stages in which the competences obtained at eachstage have independent properties.

5. THE EXPANSION PROCESS

Because there is an expansion order along the expansion process, we may describe it by a di-rected graph.

De®nition 15. Let a directed graph G(V,E) represent the structure of a non-skill competence setNSkf,j(P)vE. Then, the set of vertices V is the set of competences as V = NSkf,j(P)vE. And theset of directed arcs, E, represents the relations of these competences.

If gi4gj belongs to G(V,E), then we say gi is gj's predecessor, and gj is gi's successor. Then weobserve the following fact.

Theorem 3. G(V,E) is acyclic.

Proof. Suppose we have a cycle g14��gk4��gm4g1 in G(V,E). From Property 1, item (iii),g14gm. Since it is cyclic, gm4g1. It contradicts to Property 1, item (ii). Therefore, g(v,e) is acyc-lic. Q.E.D.q

From the concept of Activity-On-Node (AON) network developed by Stevenson (1982), weknow that when G(V,E) is acyclic, if gi4gj belongs to G(V,E), there exists a topological (linear)sort such that gi is a predecessor of gj. Thus, the topological sort provides an expansion orderas described in the following:

Step 1. Input an AON-network: G(V,E).Step 2. i = 1; Repeat Step 2.1 to Step 2.4 until G(V,E) is empty.Step 2.1 Pick all competences which have no predecessors.Step 2.2 Output these competences to Si.Step 2.3 Delete these competences in Siand all edges directed out of these competences in Si

from the G(V,E).Step 2.4 i= i + 1.

After the above algorithm is performed, we have a sequence of competence set S1,��,Si,��,Sp

to represent the stages of the expansion process. Then they have the following properties.

Theorem 4.

(i) All competences in each Si are independent.(ii) Suppose gi belongs to Si, gj belongs to Sj with j>i, then gj-gi.

Proof

(i) If g1 and g2 in Si are not independent, that is, it is g14g2. Then when we output g1 to Si,we will not output g2 to Si. That is, Si does not contain g1 and g2 simultaneously. It is acontradiction.

(ii) Suppose gi belongs to Si,gj belongs to Sj, and gj4gi, that is, gj is a predecessor of gi. Fromthe topological sort algorithm, j < i. It is a contradiction. Q.E.D.q

Theorem 4 (ii) tells us that the sequence of S1,��,Si,��,Sp provides a natural expansion order,and the background competences precede the desired competences. From Theorem 4 (i), we


notice that each competence set Si satis®es the independence property. Thus we can obtain thesecompetences in Si by the proposed expansion model.

6. SUMMARY AND DISCUSSION

Based on the concept of Habitual Domains, we have investigated the characteristics and re-lations of HD's element-competences. Then we proposed an expansion process from thePrinciple of The Least Resistence. Figure 2 summarizes this process. Due to the fact of the

Fig. 2. Expansion process bases on The Least Resistance Principle.


learning order, the expansion process is described by the stages in a directed graph. At eachstage of expansion, the desired competences are independent. Therefore, they can be obtainedsimultaneously with the minimum expansion cost found by the proposed model. Since a compe-tence depends on how well on can obtain ones's potential in learning, which includes how goodthe background one has and how close these background competences are related to the desiredcompetence, this expansion model is to ®nd the optimal expansion path constrained by thisjoint e�ects as described by max-product fuzzy relation inequalities. In the following section, weshall present an example to illustrate the application of the model.

6.1. An illustrative example

Let us consider a freshgraduate, John of Management School, whose advisor wants him to do aresearch about ``Investment Cooperative Planning'' as his thesis. So, how to guide John to takecourses based on his knowledge and background so that he can ®nish his thesis e�ectively?

Analysis: Now, John faces a problem E = ``®nish his thesis''. Suppose John has studied thefollowing courses as his skill competence set: Skx (John)0{Linear Algebra, MathematicStatistics, Linear Programming, Economics}(0.5,0,0.4,0.7)={LA, MS, LP, Eco}(0.5,0,0.4,0.7). FromDe®nition 7 and Property 4, the compound competences such as {LP$Eco}min{0.4,0.7},{MS$LA}0, {LA$LP}0.4 are also John's skill competences.

For simplicity, we only pick up some compound competences that are signi®cant for theanalysis. Thus, we have John's skill competence set de®ned by Skx (John) = {LA, MS, LP, Eco,{LP$Eco}, {MS$LA}, {LA$LP}}(0.5,0,0.4,0.7,0.4,0,0.4).

To ®nish John's thesis, the required courses from a truly needed competence set de®ned byTrc,o(E) = {Linear Programming, Economics, Game Theorem, Multivariate Applied StatisticalAnalysis, Simulation, Multiple Criteria Decision Making}c,o0{LP, Eco, GT, MASA, Sim,MCDM}(0.5,0.7,0.8,0.6,0.7,0.6),(0.7,0,0.5,0.4,0.8,0.6,0.6).

Because in Trc,o(E), LP is John's type (2) competence, Eco is John's type (1) competence, andthe others are John's type (3) competence. So we list the background relations between John'snon-skill competence ``MASA, Sim, MCDM, GT'' and his skill competences in Table 1.

Based on the Principle of The Least Resistence, John enhances LP from LP0.4 to LP0.5, andthe expansion process is LP0.44LP0.5.

As soon as John has LP0.5, he has a compound competence {LP$Eco}0.5 which inducesGT's background-strength to 0.5� 1.0rg(GT) = 0.5. That is, John obtains a new competenceGT. Similarly, he can enhance GT from GT0 to GT0.8 by expansion process GT04GT0.8.This Preliminary Expansion Process is shown in Fig. 3. Since there is no type (3) competencewhich can be changed into a type (2) competence, the preliminary expansion stage results in aset of skill competences and a set of non-skill competences with their relations shown inTable 2.

Fig. 3. The Preliminary Expansion Process of the case.

Table 1. The Relation Matrix of John's skill and non-skill competences

LA0.5 MS0 LP0.4 Eco0.7 {LP$Eco}0.4 {MS$LA}0 {LA$LP}0.4

MASA0.6,0.8 0.5 0.6 0 0 0 1.0 0.5Sim0.7,0.6 0 0.9 0 0 0 0.9 0MCDM0.6,0.6 0.3 0 0.8 0 0.8 0.3 1.0GT0.8,0.5 0.3 0 0.3 0.6 1.0 0.3 0.4


From Table 2, it can be noted that {LP$Eco} is a redundant compound competence,

because this compound competence does not increase the background relation at all to any one

of the MASA, Sim, or MCDM component competences. That is, max{R(Eco, MASA), R(LP,

MASA)} = 0 = R({LP$Eco}, MASA), etc. In fact, we can simply ignore it.

Now we proceed to the main Expansion Stage. Because NSkf,j (John)vE={MASA(0.6,0.8),

Sim(0.7,0.6), MCDM(0.6,0.6)}, we shall make use of Skx0

(John) = {LA, MS, LP, Eco, GT,

{LP$Eco}, {MS$LA}, {LA$LP}}(0.5,0,0.5,0.7,0.8,0.5,0,0.5) to transform NSkf,j (John)vE from

John's non-skill competence set into his skill competence set e�ectively. That is, making

Skxrx 0

(John) becomes NSkf,j (John)vE's potential set (De®nition 14) with minimum cost.

That is,

Minx21 � x1 � x2 � x3 � x2

4 � x35� ÿ ��x0

1�2� x01 � x0

2 � x03 � �x0

4�2 � �x05�3�

s:t:

� 0:5 0:6 0 0 0 0 1:0 0:5

0 0:9 0 0 0 0 0:9 0

0:3 0 0:8 0 0:3 0:8 0:3 1:0

�o

x1

x2

x3

x4

x5

x6

x7

x8

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCAr� 0:8

0:6

0:6

�

�x1, x2, x3 ,x4 ,x5 ,x6 ,x7 ,x8�r�1=2, 0, 1=2, 7=10, 4=5, 1=2, 0, 1=2� �4�where the unit of xj is one year, for all j = 1,2, . . . ,5. Besides, because

LPL

Ecox 6 � LPL

Ecominfx 3,x 4g. So, we have the implicit constraint x6=min {x3, x4} so do

x7=min {x1, x2} and x8=min {x1, x3}.

Applied the solution procedure by Wang (1994), we have six index sequences: (7,3,2), (7,2,6),

(7,2,8), (7,7,3), (7,7,6), (7,7,8). We can ®nd their corresponding quasi-minimum solutions and

lower bound solutions in our example. From the ®rst index sequence (7,2,3), we have

�x�1� �x01, x

02 _ 2=3, x0

3 _ 3=4, x04, x

05, x

06, x

07 _ 4=5, x0

8��1=2, 2=3, 3=4, 7=10, 4=5, 1=2, 4=5, 1=2�, �4�

and xl�1�=(4/5, 4/5, 3/4, 7/10, 4/5, 7/10, 4/5, 3/4). Similarly, we can ®nd

xl�2� � xl

�5� � �4=5, 4=5, 3=4, 3=4, 4=5, 3=4, 4=5, 3=4�,

xl�3� � xl

�6� � �4=5, 4=5, 3=5, 7=10, 4=5, 3=5, 4=5, 3=5�,�4�

and

xl�4� � xl

�1�:

Apparently, we have xl�3� � xl

�6�Rxl�k�, k= 1,2,4,5. Therefore, xl

�3� and xl�6� are the optimal sol-

utions as shown in Fig. 4(a) and (b), respectively.

In Fig. 4 the costs are incurred on the following learning process, LA0.54LA0.8, MS04MS0.8,

LP0.54LA0.6. Thus, the total cost is (0.82 + 0.8ÿ 0.52ÿ 0.5) + (0.8ÿ 0) + (0.6ÿ 0.5) = 1.59

(year).

Table 2. The relation matrix of John's skill and non-skill competences after preliminary expansion

LA0.5 MS0 LP0.4 Eco0.7 GT0.8 {LP$Eco}0.4 {MS$LA}0 {LA$LP}0.4

MASA0.6,0.8 0.5 0.6 0 0 0 0 1.0 0.5Sim0.7,0.6 0 0.9 0 0 0 0 0.9 0MCDM0.6,0.6 0.3 0 0.8 0 0.3 0.8 0.3 1.0


Although in Fig. 4(b), ``Simulation'' needs higher level of LA 2/3 then that in Fig. 4(a),because xl

�3� � xl�6�, these two expansion processes have the same cost. In fact, in Fig. 4(a) we

notice that John learns ``Simulation'' without LA2/3. So the expansion process in Fig. 4(a) couldbe better than Fig. 4(b) in terms of e�ciency.

7. CONCLUSION

This study proposed an optimal expansion process of a competence set based on its fuzzy prop-erties. From a directed graph, the expansion stages can be identi®ed to represent the expansionorder, and at each stage, the optimal expansion path can be found by the proposed model. Thismodel suggests that a natural and e�ective way to expand one's competence set is when thePrinciple of Least Resistence is followed and based on this Principle, reaching the required levelsof competences for solving a problem depends on the levels of the background competences andtheir relations to the required competences. Since compound background competences havestronger relations to the required competences but without extra expansion costs, they are easierto obtain the required competences than the single competence. Thus, the model is formulatedin a minimization form with fuzzy max-product relation constraints for single competences andimplicit constraints for compound competences.

Whenever the level to solve a problem is set too high and one can not cope with it even whenone makes an e�ort, then the required level is unfeasible. In this case, how to explore the exter-nal competences in Fig. 1 needs further investigation.

Fig. 4. (a) The Optimal Expansion Process (the 3rd Index Sequence). (b) The Optimal Expansion Process (the 6th IndexSequence).


AcknowledgementsÐThis work was supported by the National Science Council, Taiwan, Republic of China with projectnumber NSC 82-0415-E007-01.

REFERENCES

Li, H. L. and Yu, P. L. (1994) Optimal competence set expansion using deduction graph. Journal of Optimal Theory andApplication 80, 75±91.

Stevenson, W. J. (1982). Production and Operation Management. R.D. Irwin, Homewood, Ill.Wang, H. F. (1993). A fuzzy approach to the analysis of competence set. Proceedings of the First Asia Fuzzy System

Symposium.Wang, C. H. (1994), Fuzzy Analysis of a Competence Set, Master Thesis, Department of Industrial Engineering,

National Tsing Hua University.Yu, P. L. (1990), Forming Winning StrategiesÐan Integrated Theory of Habitual Domains. Springer-Verlag,

Heidelberg.Yu, P. L. and Zhang, D. (1992) Optimal expansion for competence set and decision support. Information Systems and

Operational Research 30, 68±84.Zadeh, L. A. (1965) Fuzzy sets. Control 8, 338±353.Zimmerman, H. J. (1996), Fuzzy Set Theory and Its Applications, 3rd ed., Kluwer-Nijho�, Amsterdam.


modelling of optimal expansion of a fuzzy competence set

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