Abstract
A new method of multiattribute decision making is proposed based on the max-min and min-max gray association degree. The gray association coefficients between the alternative, positive ideal solution and negative ideal solution are defined. Here, we construct a bi-objective programming model that maximizes the minimum gray association degree and minimizes the maximum gray association degree simultaneously. By using the linear weighted summation method, the bi-objective programming model is transformed into a linear programming model. Thus, we can obtain the weight vector of attributes by solving the linear programming model. The ranking order of alternatives is generated according to the relative closeness. A numerical example is examined to demonstrate the applicability and implementation process of the decision method proposed in this paper.
1 Introduction
Multiattribute decision making (MADM), whose theories and methods have been used in many fields, such as systemic optimization, economic evaluation, urban planning, and so on, is taking on important status in modern decision-making science [1–8, 10–23]. In the process of MADM analysis, one crucial problem is to assess the relative importance or weights of the attributes (or weights thereafter) because weights influence the results of rankings of alternatives [11]. Many methods for solving MADM problems require definitions of quantitative weights for the attributes [1, 3–7, 10, 13–15, 17, 18, 20, 21, 24–26].
Several approaches have been proposed to finish this work. Most of them can be classified into subjective approaches and objective approaches depending on the information provided. The subjective approaches select weights based on the preference information of attributes given by the decision maker (DM). They include the eigenvector method [12], weighted least square method [2], Delphi method [4], etc. The objective approaches determine weights based on the objective information (e.g., decision matrix). They include principal element analysis [6], entropy method [5], multiple objective programming model [6, 7], maximizing (or minimizing) deviation method [22, 23], least squares distance method [19], linear programming method [8, 24], etc.
Weights determined by subjective approaches reflect the subjective judgment or intuition of the DM, however, analytical results or rankings of alternatives based on the weights can be influenced by the DM owing 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 judgment information of the DM [10]. As these two kinds of approaches have those drawbacks, some researchers have proposed integrated approaches to determine weights for solving MADM problems in recent years [3, 10, 18, 21].
The main focus of this paper is on the objective approaches. Because of the complexity of objective things and the incomplete or inadequate information, there appear many uncertainties in the problems of decision-making systems, so the attribute values given by DM have a certain gray degree [12]. The decision system is a typical gray system. Technique for order preference by similarity to ideal solution (TOPSIS) [5] is one of the well-known methods for multiple-criteria decision-making (MCDM). The TOPSIS algorithm, by considering positive and negative ideal solutions, helps as an objective approach to determining weights for solving MADM problems. The new objective approach proposed in this paper integrates the theory of gray association analysis and the algorithm of TOPSIS.
The rest of this paper is arranged as follows: Section 2 presents the MADM problem. Section 3 proposes a new approach to determining attribute weights. Section 4 gives the decision method for the MADM problem. In Section 5, an example is used to illustrate the proposed method. The last section concludes the work of this paper.
2 Max-Min and Min-Max Gray Association Degree-Based Method for MADM
2.1 Representation of the MADM Problem
This section describes the MADM problem with numerical attribute values. To facilitate representation and analysis, the following notations are used throughout the paper.
Let S = {s1, s2, …, sn} be a finite set of alternatives, a = {a1, a2, …, am} be a set of attributes, and W = {w1, w2, …, wm}T be the vector of attribute weights, satisfying that
In the general case, there are benefit attribute values and cost attribute values in the MADM [10], and different attribute values may be different dimensions. For the convenience of decision making, all the attributes need to be dealt with in dimensionless units and each attribute value should be normalized. The following equations can be used to normalize the matrix A = (xij)n×m into the matrix R = (rij)n×m, where
2.2 Approach to Determining Attribute Weights
This section combines the gray association analysis theory with the algorithm of TOPSIS to determine the attribute weights. First, we give the pertinent definitions.
Definition 1. The positive ideal solution (PIS) and the negative ideal solution (NIS) are defined as
Definition 2. According to the gray system theory [9], PIS is considered as the reference sequence, whereas the alternatives are considered as the comparative sequences. The gray association coefficient between the alternative si and PIS s+ with respect to attribute aj is defined as follows:
where ρ∈[0, 1] is the distinguishing coefficient, usually taken as the value of 0.5. The smaller ρ is, the higher the distinguishing ability of the gray association coefficient is.
Similarly, we have the following definition for NIS s–:
Definition 3. The gray association coefficient between the alternative si and NIS s– with respect to attribute aj is defined as
From Definitions 2 and 3, we can get the matrices of gray association coefficient between all alternatives, PIS s+, and NIS s– as follows:
Definition 4. The gray association degree between the alternative si and PIS s+ is defined as
Definition 5. The gray association degree between the alternative si and NIS s– is defined as
The gray association degree reflects the similarity between the alternative and ideal solution. The larger the gray association degree γ(si, s+) is, the greater the similarity between the alternative and PIS. Therefore, the gray association degree can be used to discriminate how close the alternatives are to the ideal solution. The relative closeness of the alternative si with respect to PIS s+ can be expressed as
where the index value of ci lies between 0 and 1.
The basic idea of TOPSIS is to choose the alternative that is both close to PIS, and at the same time far from NIS as the satisfied solution. From Definitions 4 and 5, obviously the bigger the value of γ(si, s+) is, the closer the alternative si to PIS s+; the smaller the value of γ(si, s–) is, the farther the alternative si from NIS s–.
Because the gray association degree is used to measure the gray distance between the alternative and ideal solution, we can replace the weighted Euclidean distance in TOPSIS [5] by the gray association degree, and improve the method of TOPSIS. Therefore, this paper constructs a new bi-objective programming to objectively determine the weights for MADM problems. That is, first select the minimum gray association degree between all alternatives and PIS s+, and select the maximum gray association degree between all alternatives and NIS s–; then, maximize the minimum gray association degree and minimize the maximum gray association degree simultaneously. Therefore, we can construct the following bi-objective programming:
Eq. (5) is not only in accordance with the basic ideal of TOPSIS [5] but also can make each gray association degree between all alternatives and PIS s+ maximized and, simultaneously, make each gray association degree between all alternatives and NIS s– minimized.
To solve Eq. (5), suppose that
The linear weighted summation method in multiple objective programming analysis is applied to solve the problem. The procedure is described as follows:
where β∈[0, 1] represents the relative importance of the two objects. β = 0.5 indicates the indifference between the two objects; β = 1 indicates that DM absolutely prefers the former to the latter; and β < 0.5 indicates that DM prefers the latter to the former.
It should be pointed out that Eq. (6) is a classical linear programming problem and can be solved by the simplex method of linear programming.
2.3 The Decision Method for the MADM Problem
On the basis of the above discussion, the detailed steps of the MADM method can be summarized as follows:
Establish the decision matrix according to the MADM problem.
Normalize the decision matrix by using Eqs. (1) and (2).
Determine PIS and NIS by using Definition 1.
Calculate the matrices of gray association coefficient between all the alternatives and PIS as well as NIS according to Eq. (3), respectively.
Solve Eq. (6) to derive the weights of attributes.
Calculate the gray association degrees between each alternative and PIS as well as NIS by Definitions 4 and 5, respectively.
Calculate the relative closeness of each alternative with respect to PIS by Eq. (4) and rank the alternatives according to the non-increasing order of ci (i = 1, 2, …, n). The larger the relative closeness ci, the better the corresponding alternative si.
3 Illustrative Example
In this section, a problem of evaluating robots [10] is used to illustrate the decision method proposed in this paper.
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. A robot user intends to select a robot, and there are four alternatives {s1, s2, s3, s4} for him/her to choose. When making a decision, the attributes considered include a1: cost ($10,000), a2: velocity (m/s), a3: repeatability (mm), and a4: load capacity (kg). Among these four attributes, a2 and a4 are of benefit types; a1 and a3 are of cost types. The decision information about robots is listed in Table 1.
Decision Information about Robots.
| a1 | a2 | a3 | a4 | |
|---|---|---|---|---|
| s1 | 3.0 | 1.0 | 1.0 | 70 |
| s2 | 2.5 | 0.8 | 0.8 | 50 |
| s3 | 1.8 | 0.5 | 2.0 | 110 |
| s4 | 2.2 | 0.7 | 1.2 | 90 |
According to Table 1, Eq. (1), and Eq. (2), the normalized decision matrix is obtained as
By Definition 1, we have PIS and NIS as s+ = {1, 1, 1, 1} and s– = {0, 0, 0, 0}, respectively.
From Eq. (3), the matrices of gray association coefficient between all the alternatives and PIS as well as NIS are, respectively, obtained as follows:
Solving Eq. (6) with different β, the computation results for attribute weights, the relative closeness, and the ranking results of alternatives are listed in Table 2.
Attribute Weights, Relative Closeness, and Rankings for Selecting Robots with Different β.
| β | w1 | w2 | w3 | w4 | s1 | s2 | s3 | s4 | Rankings |
|---|---|---|---|---|---|---|---|---|---|
| 0.1 | 0.2237 | 0 | 0.4342 | 0.3421 | 0.4901 | 0.5101 | 0.5219 | 0.5556 | s4≻s3≻s2≻s1 |
| 0.3 | 0.2712 | 0.1628 | 0.2777 | 0.2882 | 0.4962 | 0.4962 | 0.5198 | 0.5412 | s4≻s3≻s2=s1 |
| 0.5 | 0.2712 | 0.1628 | 0.2777 | 0.2882 | 0.4962 | 0.4962 | 0.5198 | 0.5412 | s4≻s3≻s2=s1 |
| 0.7 | 0 | 0.0596 | 0.4575 | 0.4830 | 0.5361 | 0.4975 | 0.4943 | 0.5503 | s4≻s1≻s2≻s3 |
| 0.9 | 0 | 0.0596 | 0.4575 | 0.4830 | 0.5361 | 0.4975 | 0.4943 | 0.5503 | s4≻s1≻s2≻s3 |
Table 2 shows that the obtained weights are different with different values of β, which results in different rankings. For example, if β = 0.7, then the relative closeness of all alternatives are, respectively, as follows:
It is worth noticing that the ranking order of alternatives obtained by using the objective approach [10] is s4≻s1≻s2≻s3, which is the same as that obtained by the proposed method in this paper with β ≥ 0.7. This observation shows that the ranking order obtained by the objective approach [10] is just a special case of that obtained by the proposed method in this paper.
To compare with the conventional TOPSIS [5], we assume that the weights of attributes are equal and adopt the method of conventional TOPSIS; then, the relative closeness of all alternatives are, respectively, as follows:
Therefore, the rankings obtained by the above conventional TOPSIS is s3≻s2≻s1≻s4, which is remarkably different from the results obtained by the proposed method in this paper. It can be seen from Table 2 that the best alternative obtained by this paper is s4 for different values of parameter β, while the best alternative obtained by the conventional TOPSIS is s3, which is different from the results obtained by this paper.
Compared with the objective approach [10] and the conventional TOPSIS, the main advantages of the proposed method in this paper are summarized as follows:
The proposed method in this paper has better distinguishing capability than the conventional TOPSIS. For example, when β = 0.7, the relative closeness degrees for the best alternative and the second optimal alternative are 0.5503 and 0.5361, respectively. The difference between them is 0.5503–0.5361 = 0.0142. Whereas, applying the conventional TOPSIS, the relative closeness degrees for the best alternative and the second optimal alternative are 0.5000 and 0.4973, respectively. The difference between them is 0.5000–0.4973 = 0.0027. The relative closeness 0.5503 > 0.5000 indicates that the relative closeness degree of this paper is larger than that of TOPSIS, and the difference 0.0142 > 0.0027 implies that the distinguishing degree between the best and second optimal alternatives of this paper’s method is more obvious than that of TOPSIS.
The proposed method in this paper has better flexibility than the objective approach [10] and the conventional TOPSIS. Table 2 indicates that when DM selects different values of parameter β, the ranking orders of alternatives are also different. In other words, the proposed method in this paper is able to provide DM with more choices during the process of decision making, which verifies the flexibility of the proposed method.
4 Conclusion
This paper proposes a new objective approach to determining attribute weights and further gives the decision method for solving MADM problems. Compared with the existing literature, the method has the following advantages: it not only can avoid the subjective randomness of selecting the attribute weights through solving bi-objective programming of max-min and min-max gray association degree, but can also overcome the defects of the conventional TOPSIS. The proposed approach enriches the theories and methods of MADM problems. It can also be extended to support the group multiple-attribute comprehensive evaluation situation where the evaluation information is given by multiple experts.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (nos. 71061006, 61263018, and 11461030), Humanities Social Science Programming Project of Ministry of Education of China (no. 09YGC630107), the Natural Science Foundation of Jiangxi Province of China (nos. 20114BAB201012 and 20142BAB201011), “Twelve Five” Programming Project of Jiangxi Province Social Science (2013) (no. 13GL17), and Excellent Young Academic Talent Support Program of Jiangxi University of Finance and Economics.
Bibliography
[1] B. Al-Kloub, T. Al-Shemmeri and A. Pearman, The role of weights in multi-criteria decision aid, and the ranking of water projects in Jordan, Eur. J. Oper. Res.99 (1997), 278–288.10.1016/S0377-2217(96)00051-3Search in Google Scholar
[2] A. T. W. Chu, R. E. Kalaba and K. Spingarn, A comparison of two methods for determining the weights of belonging to fuzzy sets, J. Optimiz. Theory App.27 (1979), 531–538.10.1007/BF00933438Search in Google Scholar
[3] Y. Gong and S. Chen, Multi-attribute decision-making based on subjective and objective integrated eigenvector method, J. Southeast Univ. (English Edition) 23 (2007), 144–147.Search in Google Scholar
[4] C. L. Hwang and M. J. Lin, Group decision making under multiple criteria: methods and applications, Springer, Berlin, 1987.10.1007/978-3-642-61580-1Search in Google Scholar
[5] C. L. Hwang and K. Yoon, Multiple attribute decision making methods and applications, Springer, Berlin, 1981.10.1007/978-3-642-48318-9Search in Google Scholar
[6] D. F. Li, TOPSIS-based nonlinear-programming methodology for multiattribute decision making with interval-valued intuitionistic fuzzy sets, IEEE Trans. Fuzzy Syst.18 (2010), 299–231.Search in Google Scholar
[7] D. F. Li, Closeness coefficient based nonlinear programming method for interval-valued intuitionistic fuzzy multiattribute decision making with incomplete preference information, Appl. Soft Comput.11 (2011), 3402–3418.10.1016/j.asoc.2011.01.011Search in Google Scholar
[8] D. F. Li, G. H. Chen and Huang Z. G., Linear programming method for multiattribute group decision making using IF sets, Inform. Sci.180 (2010), 1591–1609.10.1016/j.ins.2010.01.017Search in Google Scholar
[9] S. F. Liu, The grey system theory and application, Science Press, Beijing, 2004 (in Chinese).Search in Google Scholar
[10] J. Ma, Z. P. Fan and L. H. Huang, A subjective and objective integrated approach to determine attribute weights, Eur. J. Oper. Res.112 (1999), 397–4041.10.1016/S0377-2217(98)00141-6Search in Google Scholar
[11] B. Mareschal, Weight stability intervals in multicriteria decision aid, Eur. J. Oper. Res.33 (1988), 54–64.10.1016/0377-2217(88)90254-8Search in Google Scholar
[12] C. J. Rao, X. P. Xiao and J. Peng, Novel combinatorial algorithm for the problems of fuzzy grey multi-attribute group decision making, J. Syst. Eng. Electron.18 (2007), 774–780.10.1016/S1004-4132(08)60019-5Search in Google Scholar
[13] T. L. Saaty, A scaling method for priorities in hierarchical structures, J. Math. Psychol15 (1997), 234–281.10.1016/0022-2496(77)90033-5Search in Google Scholar
[14] S. P. Wan, Power average operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making, Appl. Math. Model.37 (2013), 4112–4126.10.1016/j.apm.2012.09.017Search in Google Scholar
[15] S. P. Wan, 2-Tuple linguistic hybrid arithmetic aggregation operators and application to multi-attribute group decision making, Knowl.-Based Syst.45 (2013) 31–40.10.1016/j.knosys.2013.02.002Search in Google Scholar
[16] S. P. Wan and J. Y. Dong, A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making, J. Comput. Syst. Sci.80 (2014), 237–256.10.1016/j.jcss.2013.07.007Search in Google Scholar
[17] S. P. Wan and D. F. Li, Fuzzy LINMAP approach to heterogeneous MADM considering the comparisons of alternatives with hesitation degrees, Omega Int. J. Manage. Sci.41 (2013), 925–940.10.1016/j.omega.2012.12.002Search in Google Scholar
[18] Y. M. Wang and C. Parkan, A general multiple attribute decision-making approach for integrating subjective preferences and objective information, Fuzzy Set. Syst.157 (2006), 1333–1345.10.1016/j.fss.2005.11.017Search in Google Scholar
[19] Y. M. Wang and C. Parkan, Two new approaches for assessing the weights of fuzzy opinions in group decision analysis, Inform. Sci.176 (2006), 3538–3555.10.1016/j.ins.2005.12.011Search in Google Scholar
[20] S. P. Wan, Q. Y. Wang and J. Y. Dong, The extended VIKOR method for multi-attribute group decision making with triangular intuitionistic fuzzy numbers, Knowl.-Based Syst.52 (2013), 65–77.10.1016/j.knosys.2013.06.019Search in Google Scholar
[21] X. Z. Xu, A note on the subjective and objective integrated approach to determine attribute weights, Eur. J. Oper. Res.56 (2004), 530–532.10.1016/S0377-2217(03)00146-2Search in Google Scholar
[22] Z. S. Xu, A deviation-based approach to intuitionistic fuzzy multiple attribute group decision making, Group Decis. Negot.19 (2010), 57–76.10.1007/s10726-009-9164-zSearch in Google Scholar
[23] Y. J. Xu and Q. L. Da, Standard and mean deviation methods for linguistic group decision making and their applications, Expert Syst. Appl.37 (2010), 5905–5912.10.1016/j.eswa.2010.02.015Search in Google Scholar
[24] X. L. Zhang and Z. S. Xu, Deriving experts’ weights based on consistency maximization in intuitionistic fuzzy group decision making, J. Intell. Fuzzy Syst.27 (2014), 221–233.10.3233/IFS-130991Search in Google Scholar
[25] X. L. Zhang and Z. S. Xu, The TODIM analysis approach based on novel measured functions under hesitant fuzzy environment, Knowl.-Based Syst.61 (2014), 48–58.10.1016/j.knosys.2014.02.006Search in Google Scholar
[26] X. L. Zhang and Z. S. Xu, Interval programming method for hesitant fuzzy multi-attribute group decision making with incomplete preference over alternatives, Comput. Ind. Eng.75 (2014), 217–229.10.1016/j.cie.2014.07.002Search in Google Scholar
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