Elsevier

Information Sciences

Volume 169, Issues 1–2, 6 January 2005, Pages 97-112
Information Sciences

An approach to fuzzy multiattribute decision making under uncertainty

https://doi.org/10.1016/j.ins.2003.12.007Get rights and content

Abstract

It seems that there is little investigation on fuzzy multiattribute decision making (FMADM) problems under uncertainty, which are of important to scientific researches and real life applications. FMADM problems under uncertainty are investigated in this paper. Novel mathematical programming models are constructed for FMADM problems under uncertainty, and corresponding solving methods are proposed. The approach proposed in this paper may reflect both subjective judgment and objective information. Moreover, pairwise chain comparison methods for determination of relative membership degrees and weights are also proposed. Feasibility and effectiveness of the models and approach proposed in this paper are illustrated with a numerical example.

Introduction

Fuzzy multicriteria decision making (FMCDM) has provoked great interest in Decision Science, Systems Engineering, Management Science and Operations Research. Fuzzy multiattribute decision making (FMADM) is an important type of the FMCDM [7], [8], [16]. Al-Najjar and Alsyouf [1] assessed the most popular maintenance approaches using fuzzy multiple criteria decision making evaluation methodology. Aouam et al. [2] incorporated the fuzzy set theory and the basic nature of subjectivity due to the ambiguity to achieve a flexible decision approach suitable for uncertain and fuzzy environment. Carlsson and Fuller [4], [5] investigated multiobjective programming problems in which the functional relationship between the decision variables and the objective functions is not completely known and provided a method for finding a fair solution to the problems. Mohan and Nguyen [21] presented an interactive satisfying method for solving multiobjective mixed fuzzy-stochastic programming problems. Sakawa et al. [27] proposed interactive fuzzy programming for multi-level 0–1 programming problems with fuzzy parameters through genetic algorithms. Cheng and Chau [9] proposed a practical fuzzy iteration methodology for reservoir flood control operation based on the Fuzzy Cross Iteration Algorithm [16], [18]. Cheng and Chau [10] proposed a three-person multiobjective conflict decision-making model for reservoir flood control problem, which is a special kind of multiobjective group decision making. Also a good real-life example is provided with practitioners of Operational Research. Li and Chen [18] proposed a fuzzy cross iteration algorithm for multiobjective optimization problems with incomplete information. Li and Cheng [19] proposed fuzzy multiobjective programming methods for fuzzy constrained matrix games with fuzzy numbers using ranking functions of fuzzy numbers and multiobjective programming techniques. Li and Yang [20] proposed a fuzzy linear programming technique for multiattribute group decision making under fuzzy uncertainty using vertex distance measurement between triangular fuzzy numbers and linear programming model. Li [17] proposed the average weighted programming and minimum average weighted deviation methods for fuzzy multiattribute decision making with the decision maker's preference information incompletely known. Hsu and Chen [13] discussed aggregation of fuzzy opinions under group decision making. Chen [6] extended the TOPSIS for group decision making under fuzzy environment. Lee [15] studied the group decision making using fuzzy sets theory for evaluating the rate of aggregative risk in software development. But many real FMADM problems are under uncertainty. However, there exists little research into FMADM problems under uncertainty, which are of important value to scientific researches and real applications. Bender and Simonovic [3] proposed a fuzzy compromise approach to water resource systems planning under uncertainty. Delgado et al. [11] provided a fuzzy rule based backpropagation method for training binary multilayer perceptrons. Gorzalczany and Piasta [12] proposed a neuro-fuzzy approach versus rough-set inspired methodology for intelligent decision support. Raj and Kumar [22] give a method for ranking alternatives with fuzzy weights using maximizing set and minimizing set. Ravi and Reddy [23] presented a method for ranking of Indian Coals via fuzzy multiattribute decision making. Yager [24], [25] studied modelling uncertainty using partial information and decision making under Dempster–Shafer uncertainties. Kim and Han [14] presented an interactive procedure to aggregate each group member's preferences when each group member incompletely articulates his or her preference information about utilities and attribute weights. In this paper, FMADM problems under uncertainty are mainly discussed, and novel mathematical programming models and approaches which may reflect both subjective judgment and objective information in real life decision situations are proposed. Moreover, pairwise chain comparison methods for determination of relative membership degrees and weights are proposed.

This paper is organized as follows. An approach to FMADM problems under uncertainty is proposed in next Section. The pairwise comparison qualitative ranking consistency principle and pairwise chain comparison method for determination of relative membership degrees are proposed in Section 3. Section 4 gives the pairwise chain comparison method for determination of attribute weights. A numerical example is given in Section 5. Short conclusion is made in Section 6.

Section snippets

Theoretical model and method

Suppose that there exists a set X={x1,x2,…,xn} which consists of feasible alternatives assessed on m attributes, both quantitative and qualitative. Assume without loss of generality that the attributes Ai (i=1,2,…,m1) be quantitative while the attributes Ai (i=m1+1,m2+1,…,m) be qualitative. Denote the attribute set A={A1,A2,…,Am} and state set Θ={θ1,θ2,…,θq}, where θk (k=1,2,…,q) occurs with a probability pk which satisfies ∑k=1qpk=1 and pk⩾0 (k=1,2,…,q).

Let fijk denote the value of each

Pairwise chain comparison method for determination of relative membership degrees

To determine relative membership degree rijk of any alternative xjX at each state θkΘ (k=1,2,…,q) on each qualitative attribute AiA (i=m1+1,m1+2,…,m), the pairwise comparison ranking order is determined on fuzzy concept “excellence”. While the alternative xj is compared with the alternative xl at the state θkΘ on the qualitative attribute AiA, stipulate

  • (1)

    ujlik=1 and uljik=0 if xj is superior to xl;

  • (2)

    ujlik=uljik=0.5 if xj is as good as xl;

  • (3)

    ujlik=0 and uljik=1 if xj is inferior to xl;


where ujlik

Two-dimensional comparison method for determination of weights

In a similar way of Section 3, a pairwise comparison qualitative ranking consistency scale matrix v=(vik)m×m of the attribute set A on fuzzy concept “importance” can be constructed. According to decreasing order of hi=∑k=1mvik, a qualitative consistency ranking order of A on “importance” can be determined. Without loss of generality, assume thatA1≻A2≻⋯≻Am

When attribute Ai−1 is compared with Ai on “importance”, a rational judgment score of ωi−1/ωi is given by an expert or a decision maker as

Numerical example

Assume that the alternative set X={x1,x2,x3} and the state set Θ={θ1,θ2}, where θ1 and θ2 occur with probabilities p1=2/3 and p2=1/3, respectively. Consider three attributes A1 (military value), A2 (political influence) and A3 (benefit, 10,000 yuan). A1 and A2 are qualitative attributes, while A3 is quantitative attribute.

A pairwise comparison qualitative ranking consistency scale matrix u11=(ujl11)3×3 of x1, x2 and x3 at state θ1 on attribute A1 with respect to fuzzy concept “excellence” can

Short conclusion

In the above, FMADM problems under uncertainty are investigated, corresponding mathematical programmings are constructed, and their solutions are obtained using the method of Lagrange multipliers. The approach proposed in this paper may reflect both subjective judgment and objective information for FMADM problems in real life situations. In essence, determination of weights by this approach proposed in this paper is objective and automatic. Therefore, the final decision results are relatively

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