SOME ASPECTS CONCERNING THE EQUIVALENCE OF THE MULTIPLE ATTRIBUTE DECISION METHODS Nicu Bizon, Emil Sofron and Radian Raducu University of Pitesti, street "Targul din Vale", Nr.1, RO-0480 Pitesti Phone: +40-48- 222949, Fax: +40-48-216448, email:
[email protected]
ABSTRACT. In this paper are presented the obtained results by the authors behind analysis of the multiple attribute decision methods. Especially, there are references about the equivalence of the obtained results with diferent decision algorithms proposed in speciality literature. Tackled equivalencies or non-equivalencies are mathematically substantiated through demonstration or counter-examples. Finally, it is analysed a decision model based on the fuzzy concept. For a practical decision problem, the model's flexibility permits of obtain the same results like those obtained using decision crisp algorithms. KEYWORDS: multiple attributes decision methods, equivalence of the result decision, fuzzy decision model, correlation, utility, flexibility.
INTRODUCTION Along the last years, by the expert systems, the information potential by the processing of the dates knows an increased of the human decision support grade. For assurance the support of the management information system, between 1960-1970 years it was generate a lot of decision methods, named Multiple Attribute Decision Methods (MADM) (see Arrow (1952), Ackoff (1974), Hwang (1981) and so). Generally, the methods where just experimentally tested, without an attentive and rigorous analysis of the algorithms rationality or results quality, based on a estimation criterions set accepted by the all specialists. Between 1970-1980 years the decision support system took a set of methods for their application in other engineering domains, too. They form the germ for the expert system which were applied in management problems (knowledge - based systems or short expert system) (see Zimmermann (1992), Negoita (1985), Klir (1988), Aluja (1995) and so). After 1980 year the expert decision systems were applied successfully in business, environment, engineering and so (see Zimmermann (1991), Zadeh (1987), Aluja (1992) and so). A selective set of the decision multiple attribute methods was include in the created expert decision system, together with new another ones. The problem of identity for the decision multiple attribute algorithms results (or the identity for the optimum solutions) was partially tackled by the researches. Some results about this problem are presented further on.
PROBLEM STATEMENT Let be V = {V1 , V 2 , ..., V m} the set of variants and C = {C1 , C2 , ..., Cn} the set of criterions in which the experts set D = { D1 , D2 , ..., Dh} estimated all variants. The problem characterised by a matrix A = (a ij) , i=1, 2, ..., m, j=1, 2, ..., n , where a ij represents the estimation of the variant Vi in criterion C j , it is called cardinal decision multiple attribute problem. If it is supplied directly hierarchies of the variants set for each criterion in part we have a ordinal multiple attribute decision problem, that it is make a ordinal scale for the consequences matrix, A, obtaining the places matrix, L = ( Lij) , i=1, 2, ..., m, j=1, 2, ..., n, where Lij = L( Vi , C j) ∈ Ν and place : VxC → Ν is a function. The criterions can be by maximal or by minimal, quantitative or qualitative and, also, they can be different as importance. The criterion coefficient vector estimates the importance of criterions p = ( p1 , p2 ,..., pn ) . Through the specific techniques (vectorial, linear, fuzzy so) the normalised matrix R = ( r ij) , i=1, 2, ..., m, j=1, 2, ..., n, r ij ∈ [0,1] it is obtained from A, with the price of the input dates distortion.
Generally, a decision algorithm applied for a practical decision problem gives the best action variant V* and the final hierarchy of the action variants by function f − place : V → Ν . We say that two methods are equivalent if they give the same best action variant for a given arbitrary situation. In the case when final hierarchies are just the same ones we'll say that those two methods are identically.
EQUIVALENCES BETWEEN THE CRISP MADM ALGORITHMS The first set of the decisional methods that was analysed is given from methods: the second Onicescu variant (O2), the simple additive weighting (SAW) and the utility function (UF). The final hierarchy in O2 method, Onicescu (1970), is give by the decreased values of the function f2: V→R n
- place ( V , C ) f 2 ( V i) = ∑ p j 2 i
j
j =1
respective the increased values of the function f: V→R for SAW method n
n
j=1
j=1
f( Vi ) = ∑ ( p j ⋅ r ij)/ ∑ p j UF method using utility notion u ij = u( Vi , C j) ,u: V x C → [0, 1], a ij )/( max a ij − min a ij ) uij = (a ij - min i i i
calculates the indicators:
∑ p j ( u k j − u t j)
* c ( V k , Vt ) =
u k j> u t j
∑ p j ( u t j − u k j)
d ( Vk , Vt ) = *
u k j< u t j
Starting from the p=1 and q=0 value, with predefined step, it is calculated indicators c* and d* until it is obtained a V* variant which succeed another ones, so * * * * c ( V , V1) ≥ p and d ( V , V1) ≤ q , ∀ i = 1, ..., m If the rate - setting for R matrix uses a linear function we can write for the criterion Cj the equivalencies: Vs Vi ⇔ usj > u ij ⇔ r sj > r ij ⇔ place( Vs , C j) < place( Vi , C j) The equivalencies for the final hierarchy are: Vs
∑ p j (usj - uij) ≥ p and d* ( Vs , Vi ) = ∑ p j ( uij - usj) ≤ q ⇒
* Vi ⇔ c ( Vs , V i ) =
u sj > u ij
⇒
u sj< u ij
n
n
j=1
j=1
∑ p j ( usj − uij) > ∑ p j ( uij − usj) ⇔ ∑ p j usj > ∑ p j uij , for UF method; u sj< u ij
u sj > u ij
Vs
n
n
j=1
j=1
Vi ⇔ ∑ p j r sj > ∑ p j r ij , for SAW method; n
Vi ⇔ ∑ p j 2
Vs
n
- place( V s , C j )
j=1
> ∑ p j 2- place( V , C ) , for O2 method. i
j
j=1
Because the statement: n
n
j=1
j=1
n
n
f ( x j )
