Fuzzy multicriteria decision-making in distribution of factories: an application of approximate reasoning

FUZZY

sets and systems ELSEVIER

Fuzzy Sets and Systems 71 (1995) 197 205

Fuzzy multicriteria decision-making in distribution of factories: an application of approximate reasoning Chu Feng Southeast University, Nanjing, People's Republic of China Received December 1993; revised May 1994

Abstract

With a view to seeking the optimal plan of the distribution of factories, an approach of multicriteria decision-making (Feng, 1990; Yager, 1982) is adopted. This technique can reflect the fuzzy characteristic of a person's brain for recognition and judgment in a complicated soft information system and will not miss fragmentary and imprecise information.

Keywords: Approximate reasoning; Decision-making; Multicriteria decision; Multiple criteria evaluation; Implication; Imprecision

1. Introduction

In economic management systems, the study of decision-making has gained increased attention. In this field the methods presented by specialists and used by decision-makers have their flourishing periods every now and then. However, the economic management system is a complicated soft information system. In general, the decision-makers could not receive significant results when the methodology of precise traditional mathematics is used. We might as well quote from Zadeh [6] "as the complexity of a system increases, our ability to make precise and yet significant statements about its behavior diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics." Due to the complexity of the system, the decision-making is affected by many factors and also needs many criteria. In numerous situations involving complicated soft information systems it is almost impossible that these factors and criteria are precisely stated and measured. Indeed most things in the real world do not have sharply defined boundaries. For the sake of coping with the imprecision of the real world the theory of fuzzy subsets was initially developed by Zadeh [5] in 1965. In particular, in [9,7] fuzzy logic provides a basis for approximate reasoning, that is, a mode of reasoning which is not exact nor very inexact. Such logic is far better suited for dealing with real-world problems than the traditional logical systems, because most of human reasoning - almost all of human reasoning - is imprecise [8]. In this paper the advantages of multicriteria decision-making based on approximate reasoning are sufficiently flexible and adaptable, especially more in accord with the thinking process of a human being for decision-making on a complex issue. This method proves well fit to the purpose as compared with the others. 0165-0114/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 1 6 5 - 0 1 1 4 ( 9 4 ) 0 0 2 3 8 - X

C Feng / Fuzzy Sets and Systems 71 (1995) 197 205

198

In a large corporation the head would most likely meet up with such problems of the distribution of factories. F o r example, in order to develop the products quite a few factories have to be built, removed, merged, or expanded, etc. There m a y be m a n y proposals of the distribution, but only a few of the proposals can be selected by experts. However, it is not so easy to evaluate the optimal plan from the few proposals which are a b o u t the same. It is this difficult evaluation that this paper would do.

2. Model First, we form the set of factors needed by the distribution of factories F={f~},

i = 1,2, . . . , I , I = 6 .

In the process of building, removing, merging or expanding, .fl low in funds consumption, f 2 - low cost in time, after distribution completed, f3 - convenience in traffic, f4 having little influence on environment, f5 broad prospect of development, f6 - convenience in workers' daily life. The decision-making criterion set C consists of J criteria: C={cj},

j=l,Z,...,J,

J=8.

In accordance with the decision-maker's wishes, the criteria are stated as follows: c~: If expending m o n e y and time is low then it is considered satisfactory. Cz: Apart from Cl, with convenience in traffic more satisfactory. c3: Besides cl, the factories have little influence on environment; it can be also considered m o r e satisfactory. c4: If the needs of cl, cz and c3 are met at the same time - much m o r e satisfactory. c5: Apart from c3, with a b r o a d prospect of development - also much m o r e satisfactory. c6: If the needs of c4 and c5 are met at the same time - very satisfactory. c7: If the needs of c4 and c5 are met at the same time and workers' daily life is convenient - perfect. c8: A lot of m o n e y or time would be spent, or environment would be influenced very seriously - unsatisfactory. We define the appraisal set A={Ak},

k=l,2,...,K,

K=6,

and the appraisal function A (v) is defined as A1 satisfactory: A~ = v, A2 - m o r e satisfactory: A2 = v 3/2, A3 - much m o r e satisfactory: A3 = v 2, A4 - very satisfactory: A~ = v3, 1, v = l , As-perfect:As= 0, v4: 1, A6 - unsatisfactory: A6 = 1 -- v, where v E V, V = {v~} = {0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9, i.e. V is the unit appraisal space (see Fig. 1).

1},

I = 1,2 . . . . . L, L = 11,

C. Feng / Fuzzy Sets and Systems 71 (1995) 197-205

199

Therefore eight criteria are described as: If c 1 = f l ( ~ f 2 then A1,

If c2 = flnf2nf3 then A2, If c3 =flnf2nf4 then A2, If c4 =flc~f2nf3c~f4 then A3, If c5 =flnf2c~f4nfs then A3, If c6 =flc~f2nf3nf4nfs then A4, If c7 =flnfz~f3nfanfsnf6 then As, Ifc8 =flwfzwHzf4 then a6, ,4, ( v )

A2( v )

l

.8

.8

.6 ¸

.4

.4

.2 ¸

.2

T

.6

.2

.8

A iv)

•2

.4

•2

.4

A,(V)

.8

.8

.6

.6

.4

.4.

.2

T .2

T •4

.6

.8

Fig.

1.

.6

200

C. Feng Fuzzy Sets and Systems 71 (1995) 197 205

,\5( v )

A6t'v ;,

.8

.8

.6

.6

.2

o

9 .'2

o

9 . ~

o

9 .6

o

v

9 .8

2

.4

.6

.8

9 1

•

V

Fig. 1 (Continued) where Hz is the m o o d c o n c e n t r a t o r "very" defined by H2f4~(J~) 2

and

J~ = 1 - f l ,

f2

=

1 - - j 2 , etc.

S u p p o s e the set U is built up by all the d i s t r i b u t i o n p r o p o s a l s which are to be evaluated: U=

{u,,},

m=

1,2 . . . . .

m.

The single-factor e v a l u a t i o n for p r o p o s a l u,, is a fuzzy m a p p i n g from F to U , f : F ~ , U, a n d the fuzzy m a p p i n g . f i m p l i e s a fuzzy relation which can be represented by a fuzzy matrix R=(rim)=(R1

R2

...

Ri

...

Rt)Te,J~/t×M.

W e use R as the input. Processing it a c c o r d i n g to the d e c i s i o n - m a k i n g criteria, we can get a fuzzy m a p p i n g , .f: C U, which can be described by a fuzzy m a t r i x CR=((1

(2

... /:i ..- c.I)l e , t / J × M .

N o w we can use the following a p p r o x i m a t e reasoning: If x = ( 1 then y = A 1, If x = (2 then y = A 2, If x = (3 then y = A 2 , If x = g4 then y = A3, If x = ( 5 t h e n y = A 3 , If x = (6 then y = A4, If x = g7 then y = As, If x = / : s then y = A6. Using implication as t r a n s l a t i o n rule [7], d j ( m , l ) = 1 A (1 - 6j(u,,) + Ak(Vl)).

C Feng/ FuzzySets"and Systems 71 (1995) 197-205

201

From this, we can get a fuzzy mapping from U to V, f: U--~V, which can be represented by a fuzzy matrix

Dj = (dj(m,l)) ~ J/IM×L. After that we can get the fuzzy multicriteria decision-making matrix J

D = (~ Dj = (/~1/~2 .../~.. -../~M) T e . # ~ × L . j-1

In order to avoid losing too much information we define

j=l

j=l

Also, D is a fuzzy mapping from U to V, f: U ~ V. Note: ~,, is a fuzzy subset of the unit appraisal space V, which represents the extent of the satisfaction for the proposal u,,. We now apply the following procedure for comparing these fuzzy subsets of V,/~1,/~2, ...,/~ . . . . . . /~M, to obtain the best solution. Assume E,., is the a level set of E,,, a ~ [0, 1] = I. It should be noted that the sets Era, are ordinary subsets of V. For each E,,, we can calculate the mean value of the elements in Em, as follows: ml(Em~) ~ - ~

Z Zn(a)' n=l

where a is the level of the level set, Z,(a) is the element in E,,,, Z,(a) 6 Era,, and N, is the cardinality of the finite set E,,,. Then we can calculate the point value [4] of/~,, as 1

S(m) = -

L

~ Ht(Em,)Aat,

amax l = 1

where am,x is the maximum membership grade of/~,,, and Aat = at--at

t,

ao =0.

We are interested in the point value for each fuzzy subset/~,, because it is the "satisfaction value" for each proposal u,,. And the bigger the S(m), the better the proposal. The best one is the optimal plan of the distribution of factories, which we want to seek.

3. Example

This example taken from the experience of an existing organization is a part of the work which the author successfully did for a mechanical corporation in China. There are five factories, F1, F2, F3, F4, F5, and six places, P1, P2, P3, P4, P5, P6 (see Fig. 2). Three factories, F2, F3, F5, should be merged into one factory. Three places, P3, P4, P5, should be emptied. Three proposals of the distribution of factories are selected by experts (see Fig. 2). And the optimal plan is evaluated from these proposals, see the following. Input is R=

0.7 0.8

0.9 0.8

0.7 0.5

0.7 0.9

0.9 0.8

0.71v 0.8 .

0.9

0.7

0.9

0.8

0.7

0.5

I

C. Feng / Fuzzy Sets and Systems 71 (1995) 197 205

202

Proposal 1.

P r o p o s a l 2.

P r o p o s a l 3. Fig. 2.

Processing it according to the decision-making criteria, we get

oR=I

0.630 0.441 0.441 0.309 0.397 0.278 0.194 0.490"] ~r 0.640 0.320 0.576

0.288 0.461 0.230 0.184 0.410 I 0.630 0.567 0.504 0.454 0.353 0.318 0.159 0.440/

203

C. Feng / Fuzzy Sets and Systems 71 (1995) 197-205

By dj(m, l) : 1 /k (1 - ~j(um) -4- Ak(vl)) and Dj = (di(m, I)) we get fuzzy matrices Dj as follows: I0.370

0.470

0.570

0.670

0.770

0.870

0.970

1 1 1 1-]

0.360 0.370

0.460 0.470

0.560 0.570

0.660 0.670

0.760 0.770

0.860 0.870

0.960 0.970

1 1 1 llA, 1 1 1

0.559

0.591

0.648

0.723

0.812

0.913

1

1 1 1 1-]

0.680 0.433

0.712 0.465

0.769 0.522

0.844 0.597

0.933 0.686

1 0.787

1 0.898

1 1 1 llA, 1 1 1

0.559

0.591

0.648

0.723

0.812

0.913

1

1 1 1 1-]

0.424

0.456

0.513

0.588

0.677

0.778

0.889

1

1 1 ]A

0.496

0.528

0.585

0.660

0.749

0.850

0.961

1

1 1

0.691

0.701

0.731

0.781

0.851

0.941

1

1 1 1 1-]

0.712

0.722

0.752

0.802

0.872

0.962

1

1 1 1 ]A ,

0.546

0.556

0.586

0.636

0.706

0.796

0.906

1 1 1

0.603

0.613

0.643

0.693

0.763

0.853

0.963

1 1 1 1-]

0.539 0.647

0.549 0.657

0.579 0.687

0.629 0.737

0.699 0.807

0.789 0.897

0.899 1

1 1 1 llA 1 1 1

I0.722

0.723

0.730

0.749

0.786

0.847

0.938

1 1 1 1-]

0.770 0.682

0.771 0.683

0.778 0.690

0.797 0.709

0.834 0.746

0.895 0.807

0.986 0.898

1 1 1 1

1 llA , 1

I0.806

0.806

0.806

0.806

0.806

0.806

0.806

0.806

0.806

0.806

1-7

0.816 0.841

0.816 0.841

0.816 0.841

0.816 0.841

0.816 0.841

0.816 0.841

0.816 0.841

0.816 0.841

0.816 0.841

0.816 0.841

111,

D1 =

D2=

I

°3 E D4=

I

°5 E D6=

Dr=

Ds=

1 1 1 1

1 1 0.910

0.810

0.710

0.610

0.510"]

1

1 1 1

1 1 0.990

0.890

0.790

0.690

0.590 [.

1 1 1 1

1 1 0.960

0.860

0.760

0.660

0.560]

I

Finally D, the fuzzy multicriteria decision-making matrix, is

j=1

j=1

displayed as follows:

D=

0.028 0.025 0.016

I

0.041 0.037 0.024

0.066 0.061 0.041

0.114 0.107 0.074

0.209 0.199 0.141

0.397 0.371 0.282

0.643 0.653 0.611 0.726 0.550 0.723

0.572 0.645 0.639

0.492 0.563 0.555

0.510"] 0.590[.

o.56Ol

C Feng / Fuzzy Sets and Systems 71 (1995) 197-205

204 Table 1 l

Range of ~

E,,

Ht(EI~,)

Actl

1 2 3 4 5 6 7 8 9 10 11

0 < ct ~< 0.028 0.028 < ~ ~< 0.041 0.041 < ~ ~< 0.066 0.066 < ~ ~< 0.114 0.114 < ~ ~< 0.209 0.209 < ~ ~< 0.397 0.397 < ~t ~< 0.492 0.492 < :t ~< 0.510 0.510 < z~ ~< 0.572 0.572 < z~ ~< 0.643 0.643 < :~ ~< 0.653

{0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9, 1} {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9, 1} {0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9, 1} {0.3,0.4,0.5,0.6,0.7,0.8,0.9, 1} {0.4,0.5,0.6,0.7,0.8,0.9,1} {0.5,0.6,0.7,0.8,0.9, 1} {0.6,0.7,0.8,0.9,1 } {0.6,0.7,0.8,1} {0.6, 0.7, 0.8,} {0.6,0.7} {0.7}

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.775 0.7 0.65 0.7

0.028 0.013 0.025 0.048 0.095 0.188 0.095 0.018 0.062 0.071 0.010

For proposal u,, from the first row in D we have the fuzzy subset of V, 0.028

0.041

0.066

0.114

0.209

0.397

0.643

0.653

0.572

0.492

0.510

o

According to 1

N~

=

n=l

we can calculate a series of E,~, Ht(EI~) and A~,, as shown in Table I. Then

l

S(I) =

11

~ Ht(EI,)A~I ~Zmax l = 1

1 = 0.65-----3(0.5(0.028) + 0.55(0.013) + 0.6(0.025) + 0.65(0.048) + 0.7(0.095) + 0.75(0.188)

+ 0.8(0.095) + 0.775(0.018) + 0.7(0.062) + 0.65(0.071) + 0.7(0.010)) = 0.707. At last, we have the following satisfaction values, respectively: 1. S(3) = 0.744, 2. S(2) = 0.728, 3. S(1) = 0.707. The best is u3 and next, from good to bad, u2, ua. It is u3, the optimal plan of the distribution of factories, that this paper wants to get.

4. Conclusions

This paper deals with a technique for fuzzy multicriteria decision-making. This technique has wide applicability in problems involving imprecision. It is supported by a series of software products and used successfully for decision-making and evaluation of complicated soft information systems [1-3]. The experiment showed that by using this technique the results are much more likely to represent the actual facts. Besides the model of multifactors, multicriteria decision-making in this paper, we could also build the model of multifactors, multicriteria and multistages, so as to meet the needs of the judgment and recognition of more complex systems.

C. Feng / Fuzzy Sets and Systems 71 (1995) 197-205

205

In fuzzy set theory, there are a number of operators. How to use these operators in the process of applications, including the technique in this paper, is quite an art. In order to avoid losing too much information, the author changed the Zadeh operator, k/, which is used in [3,4] for the Yager operator,

aXb~min(1, (a v +

bv)t/v), v = 1-.

v

References [1] Chu Feng, Fuzzy multicriteria decision-making in selecting the enterprise leader, J. Chongqing Inst. Traffic 4 (1985) 66-73 (in Chinese), [2] Chu Feng, Fuzzy appraisal for university teaching quality, in: Wilfried Kuhn, Ed., Didaktik der Physik, Vortr~ige, Physikertagung, Berlin, (1987) 534-543. [3] Chu Feng, Quantitative evaluation of university teaching quality - an application of fuzzy set and approximate reasoning, Fuzzy Sets and Systems 37 (1990) 1-11. [4] R.R. Yager, Multicriteria decision with soft information: an application of fuzzy set and possibility theory (I), (II), Fuzzy Math. 2(2)(3) (1982) 21-28, 7-16. [5] L.A. Zadeh, Fuzzy sets, Inform. and Control 84 (1965) 338-353. I-6] L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems Man Cybernet. SMC-3 (1973) 28-44. [7] L.A. Zadeh, A theory of approximate reasoning, in: J. Hayes, D. Michie and L.I. Mikulich, Eds., Machine Intelligence, Vol. 9 (Halstead Press, New York, 1979) 149-194. [8] L.A. Zadeh, Coping with the imprecision of the real world: an interview with Lotfi A. Zadeh, Comm. ACM 27 (1984) 304-311. [9] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning - I, II, III, in: R.R. Yager, S. Ovchinnikov, R.M. Tong, H.T. Nguyen, Eds., Fuzzy Sets and Applications: Selected Papers by L.A. Zadeh (Wiley, New York, 1987) 219-366.