Theory and MethodologyA subjective and objective integrated approach to determine attribute weights
Introduction
Multiple attribute decision making (MADM) refers to the problem of selecting among alternatives associated with multiple attributes. It is a problem with extensive theoretical and practical backgrounds 1, 11, 12. The following notations are used to represent a MADM problem [11]:
S={S1,S2,…,Sm}: a discrete set of m possible alternatives.
P={P1,P2,…,Pn}: a set of n attributes. The attributes are objective and additively independent.
w=(w1,w2,…,wn)T: the vector with attribute weights (or weights thereafter), where ∑nj=1wj=1,wj⩾0 ∀j.
A=[aij]m×n: the decision matrix where aij is the consequence with a numerical value for alternative i with respect to attribute j, i=1,…,m,j=1,…,n.
In decision matrix A, every aij is an objective value between 0 and 1. It allows each attribute to have the same range of measurement. This is achieved by normalising every element in matrix A=[aij]m×n into a corresponding element in matrix B=[bij]m×n using the following formulas:
The decision maker (DM) is to choose M (<m) most preferred alternatives or the most preferred alternative from the set S, .
The simple additive weighting method is one of the best known and the most widely used MADM methods [11]. The overall value of an alternative in this method can be expressed as
The DM chooses an alternative such that
One crucial problem in MADM is to assess the relative importance or weights of different attributes. Many methods for solving MADM problems require definitions of quantitative weights for the attributes 1, 2, 3, 4, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21. It is often difficult to obtain weights because the definition of weights itself is not precise, nor are the values given by a DM. The weights do not have a clear economic significance, but they influence the results of the analysis (e.g., rankings of alternatives) [13].
Several approaches have been proposed to determine weights 10, 11, 15. Majority of them can be classified into subjective approaches and objective approaches depending on the information provided. The subjective approaches select weights based on preference information of attributes given by the DM, they include eigenvector method [16], weighted least square method [3], and Delphi method [10] etc. The objective approaches determine weights based on the objective information (e.g. decision matrix A), they include principal element analysis [4], entropy method [11], and multiple objective programming model 2, 4 etc.
Weights determined by subjective approaches reflect the subjective judgement or intuition of the DM, but analytical results or rankings of alternatives based on the weights can be influenced by the DM due to his/her lack of knowledge or experience. Objective approaches often determine weights by making use of the mathematical models, but they neglect the subjective judgement information of the DM.
This paper proposes an integrated approach to determine weights for solving MADM problems, where the pairwise comparison matrix on attributes is given by a DM. It is to determine weights by solving a mathematical programming model and to reflect both the subjective considerations of the DM and the objective information. The integrated approach is based on the subjective approach, i.e. weighted least square method, of Chu et al. [3] and the objective approach of Fan [4]. Section 2introduces the subjective approach and objective approach respectively. Section 3proposes an integrated approach for determining weights. To demonstrate the applicability of the proposed approach, an example of selecting a robot is presented in Section 4.
Section snippets
Subjective approach
Chu et al. [3] propose the weighted least square method to obtain weights. The method involves the solution of a set of simultaneous linear algebraic equations and is conceptually easy to understand. A brief description of this method is given below.
To determine weights, suppose the DM gives his/her pairwise comparison matrix D=[dkj]n×n (i.e. the Satty's matrix [16]) on the attribute set P. The elements of matrix D satisfywhere dkj denotes the relative weight of
Integrated approach to determine weights
This paper considers the MADM problem where the pairwise comparison matrix on attributes is given by a DM, and all the attributes are objective. In the attribute pairwise comparison matrix D=[dkj]n×n, the weights determined by the above subjective approach reflect the subjective consideration of the DM. In the objective decision matrix A=[aij]m×n, the weights determined by the above objective approach reflect the objective information. In order to make weights reflect both subjective and
Application of the integrated approach
Robots are widely used in many industries. A potential robot user is faced with many options. The decision of which robot to select is very complicated because robot performance is specified by many parameters for which there are no industry standards 8, 12. In this section, a MADM problem of selecting a robot is used to illustrate the proposed approach.
A robot user intends to select a robot and there are four alternatives for him/her to choose. When making a decision, the attributes considered
Summary
This paper proposes a subjective and objective integrated approach to determine attribute weights in MADM problems. The approach determines weights by solving a mathematical programming model and takes into consideration both subjective and objective factors. It overcomes the shortages which occurs in either a subjective approach or an objective approach. The proposed approach requires the preference information of a single DM. It can also be extended to support the situation where the
Acknowledgements
This research was partly supported by the National Natural Science Foundation of China (Project No. 79600006), the Earmarked Grant for Research, Hong Kong (Project No. 9040232), and the Quality Enhancement Grant of City University of Hong Kong (Project No. 87105001). Two anonymous referees made valuable comments and suggestions on the content and presentation of the paper.
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