Extensions of LINMAP model for multi criteria decision making with grey numbers.

Hajiagha, Seyed Hossein Razavi ; Hashemi, Shide Sadat ; Zavadskas, Edmundas Kazimieras 等

1. Introduction

Decision makers always seek a criterion to appraise their decisions. In this context, decision-making methods arise when decision maker simultaneously envisages various criteria for evaluating his or her decisions favorite (Kuo et al. 2008). Such a problem is the subject of multiple criteria decision making (MCDM) methods (Zavadskas, Turskis 2011; Peng et al. 2011; Kou et al. 2012). This class is further divided into multi objective decision making (MODM) and multi attribute decision making (MADM) (Climaco 1997). A formalized definition of MADM problems can be stated as follows.

Let we have a nonempty and finite set of decision alternatives, i.e. [A.sub.1],[A.sub.2],...,[A.sub.m], and there are a finite set of goals, attributes or criteria, i.e. [C.sub.1],[C.sub.2],...,[C.sub.n], according to which the desirability of an alternative is to be judged. The aim of MADM is to determine the optimal alternative with the highest degree of desirability with respect to all relevant goals (Zimmerman 1987). An optimal alternative in MADM problems can be defined as an alternative [A.sup.*] that has the highest value in all decision-making criteria. Usually, MADM problems do not have an optimal solution in practice and current methods seek an alternative with the highest degree of satisfaction for decision makers. In last decades, MADM techniques have a wide application in different areas that concern with selection.

Multi attribute decision making methods require decision makers judgments and evaluations about alternatives performance regard to multiple attributes. These judgments are a subject of uncertainty. Indeed, decision makers do not have complete information about alternatives or their conditions regard to a certain attribute. Therefore, it will be so difficult for them to express their evaluations based on exact numbers. Hence, uncertainty contexts are widely applied in MADM problems.

Fuzzy set theory (FST) was developed by Zadeh (Zadeh 1965) as a generalized form of the classical set theory that assigns a membership degree to each element of a given set in a universe. FST is one of the well-known paradigms in studying systems with uncertainty. Bellman and Zadeh (Bellman, Zadeh 1970) have introduced the concept of decision making under fuzzy environment. Afterward, MADM techniques have been extended under fuzzy environment (Aouam et al. 2003; Yazdani et al. 2011; Xu 2004; Li, Yang 2004; Wang, Chuu 2004; Hu et al. 2004; Antucheviciene et al. 2011; Kersuliene, Turskis 2011; Brauers et al. 2011; Fouladgar et al. 2011; Balezentis, Balezentis 2011).

Another paradigm of uncertainty is developed such that the crisp numbers are substituted with grey numbers (Deng 1982). Interval numbers also have a wide application in decision making field with TOPSIS (Chen, Tzeng 2004; Jahanshahloo et al. 2006; Lin et al. 2008; Zavadskas et al. 2010a; Tsaur 2011; Yue 2011), with PROMETHEE (Le Teno, Mareschal 1998), with ELECTRE (Vahdani et al. 2010; Ozcan et al. 2011), with COPRAS-G (Zavadskas et al. 2010b; Hashemkhani Zolfani et al. 2011), with ARAS-G (Turskis, Zavadskas 2010; Zavadskas et al. 2010c), with SAW-G (Zavadskas et al. 2010a), with MOORA (Stanujkic et al. 2012). An interval number can be considered as a number whose exact value is unknown, but a range within which the value lies is known (Moore 1966).

The Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) was developed by Srinivasan and Schocker (Srinivasan, Shocker 1973) as one of the MADM techniques that determines the preference order among a set of alternatives by determination of weight vector w and positive ideal solution (PIS) vector. However, the LINMAP can only deal with MADM problems in crisp environments. Xia et al. (Xia et al. 2006) developed LINMAP method for MADM problems under fuzzy environment. Li and Sun (Li, Sun 2007) developed LINMAP method for MADM problems with linguistic variables and incomplete preference information. Li (2008) also developed this method under intuitionistic fuzzy environment. Considering the simplicity and clearness of grey numbers in expressing the uncertainty and lack of knowledge, in this paper, a LINMAP method is extended with grey data. The rest of the paper is organized as follows: Section 2 briefly introduces the grey numbers and their operations. MADM problems with grey data are expressed in section 3. The extended grey LINMAP model and proposed decision process is introduced in section 4. The proposed method is illustrated with an example in section 5. Finally, the paper is concluded in section 6.

2. Grey numbers

As stated in previous section, a grey number can be indicated as a range. In fact, a number x is called an grey number when its exact value is unknown and only it is known that x [member of] [[x.bar],[bar.x]], where [x.bar] is the lower bound and [bar.x] is the upper bound, such that [x.bar] [??] [bar.x]. Arithmetic operations on interval numbers are introduced by Moore (Moore 1966). If x = [[x.bar],[bar.x]] and y = [[y.bar],[bar.y]] are two grey numbers, then:

x + y = [[x.bar] + [y.bar], [bar.x] + [bar.y]], (1)

x - y = [[x.bar] + [bar.y], [bar.x] - [y.bar]], (2)

x x y = [min ([x.bar][y.bar],[x.bar][bar.y],[bar.x][y.bar],[bar.x][bar.y]), max ([x.bar][y.bar],[x.bar][bar.y],[bar.x][y.bar],[bar.x][bar.y])] (3)

x/y = [[x.bar],[bar.x]] x [[1/[bar.y]]/[1/y]] (4)

The center, [x.sub.C], and width, [x.sub.W] of a grey number x are defined as follows (Ishibuchi, Tanaka 1990):

[x.sub.C] = 1/2([x.bar] + [bar.x]), (5)

[x.sub.W] = 1/2([bar.x] - [x.bar]). (6)

3. Grey MADM problem definition

Consider an MADM problem that consist evaluation of a set of m alternatives regard to a set of n attributes. In classic form, ratings of alternatives regard to attributes are stated with crisp data. However in many situations, and due to uncertainty or lack of knowledge, the crispness of ratings is an unfair assumption. Therefore, suppose that the ratings values are expressed in form of grey numbers. If [A.sub.1],[A.sub.2],...,[A.sub.m] are m possible alternatives and [C.sub.1],[C.sub.2],...,[C.sub.n] criteria over which alternatives performance are measured, and [x.sub.ij] [member of] [[[x.bar].sub.ij],[[bar.x].sub.ij]] is the rating of alternative [A.sub.i], with respect to criterion [C.sub.j], then a grey MADM problem can be concisely defined in a decision matrix as follows.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In addition, the weigh vector of criteria is defined as W = ([w.sub.1],[w.sub.2], ..., [w.sub.n]) that [w.sub.j] is the importance weight of criterion j. The problem here is to rank the alternative set's elements.

4. Grey LINMAP

In this section, the proposed method of LINMAP-G (Srinivasan, Shocker 1973; Hwang, Yoon 1981) with grey data is developed. It is noted that the LINMAP-G method seeks a positive ideal solution (PIS) and a weight vector w that minimizes the distance of a set of preference relations among alternatives that are expressed priori by decision makers from unknown PIS.

4.1. Normalization of Grey decision matrix

An intrinsic aspect of MADM problems is that different attributes have different dimensions that make their comparison impossible. Therefore, an initial step before the decision making process, is to normalize the grey decision matrix, defined in previous section. Different procedures are introduced to normalize grey decision matrix. In this paper the method proposed in Jahanshahloo et al. (Jahanshahloo et al. 2006) is applied by modifications. If attribute j is as profit (maximization) type, its normalized values are calculated as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Otherwise, if attribute j is as cost (minimization) type, its normalized values are calculated as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

4.2. Grey LINMAP modeling process

The main idea of LINMAP-G method is to determine an unknown PIS vector, like PIS = [A.sup.*] = ([x.sup.*.sub.1],[x.sup.*.sub.2],...,[x.sup.*.sub.n]), where [x.sup.*.sub.i] = [[[x.bar].sup.*.sub.i]],[[[bar.x].sup.*.sub.i]]. Then, the best alternative is chosen as the nearest one to this PIS vector. A note here is to define the distance between two grey vectors.

If [A.sub.i] =([x.sub.i1],[x.sub.i2],...,[x.sub.in]) and [A.sup.*] =([x.sup.*.sub.1],[x.sup.*.sub.2],...,[x.sup.*.sub.n]) are two grey vectors, and W = ([w.sub.1],[w.sub.2],...,[w.sub.n]) is a weight vector, the weighted grey numbers Euclidean distance is defined as follows:

[d.sub.i] = [square root of [n.summation over (j=1)][w.sub.j][[([[x.bar].sub.ij] - [[x.bar].sup.*.sub.j]).sup.2] + [([[bar.x].sub.ij] - [[bar.x].sup.*.sub.j]).sup.2]]]. (9)

Now, the variable [s.sub.i] = [d.sup.2.sub.i] is defined. Suppose that decision maker specified an order relations set between alternatives as [OMEGA], where each (k,l) [member of] [OMEGA] means that decision maker preferred alternative [A.sub.k] to alternative [A.sub.l].

For a given PIS and weight vector w, alternative [A.sub.k] is closer to PIS than alternative [A.sub.l], if [s.sub.k] [less than or equal to] [s.sub.l]. In this case, the ranking obtained by (w, PIS) is consistent with decision maker's preference. Otherwise, if [s.sub.k] [??] [s.sub.l], then the ranking obtained by (w, PIS) is inconsistent with decision maker's preference.

The inconsistency between alternatives [A.sub.k] and [A.sub.l], ranking based on [s.sub.k] and [s.sub.l], with preference relations that are determined by decision maker is measured by an inconsistency index [([s.sub.l] - [s.sub.k]).sup.-]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

In fact, the alternatives [A.sub.k] and [A.sub.l] ranking is consistent with decision maker's preferences if [s.sub.k] [less than or equal to] [s.sub.l] and [([s.sub.l] - [s.sub.k]).sup.-] will be equal to zero. Otherwise, if [s.sub.k] [??] [s.sub.l] the rankings are not consistent and their inconsistency will be equal to [([s.sub.l] - [s.sub.k]).sup.-] = [s.sub.k] - [s.sub.l]. Therefore:

[([s.sub.l] - [s.sub.k]).sup.-] = max(0,[s.sub.k] - [s.sub.l]) (11)

Now, the total inconsistency index is defined as follows:

I = [summation over ((k,j) [member of] [OMEGA])][([s.sub.k] - [s.sub.l]).sup.-] = [summation over ((k,j) [member of] [OMEGA])] max(0,[s.sub.k] - [s.sub.l]). (12)

Similarly, the consistency index is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Eq. (13) can be written as follows:

[([s.sub.l] - [s.sub.k]).sup.+] = max(0,[s.sub.l] - [s.sub.k]) (14)

Therefore, total consistency index is defined as follows:

C = [summation over ((k,l) [member of] [OMEGA])][([s.sub.l] - [s.sub.k]).sup.+] = [summation over ((k,l) [member of] [OMEGA])] max(0,[s.sub.l] - [s.sub.k]). (15)

Note that whether [s.sub.k] [less than or equal to] [s.sub.l] or [s.sub.k] [??] [s.sub.l], the following relation is hold:

[([s.sub.l] - [s.sub.k]).sup.+] - [([s.sub.l] - [s.sub.k]).sup.-] = [s.sub.l] - [s.sub.k]. (16)

The grey LINMAP-G model to determine the PIS [A.sup.*] = ([x.sup.*.sub.1],[x.sup.*.sub.2],...,[x.sup.*.sub.n]) and weight vector W = ([w.sub.1],[w.sub.2],...,[w.sub.n]) can be constructed as follows:

Max C

S.T. C - B [greater than or equal to] h

[w.sub.j] [greater than or equal to] [epsilon] j = 1,2,...,n, (17)

where, h is a constant that is determined by decision maker. Also, [epsilon] [??] 0 is a sufficiently small real value that guarantees that obtained weights are greater than zero. The objective of model (17) is to maximize the consistency index C, while it will be greater than I at least as pre-determined value of h.

Using Eq. (15) and (16), the model (17) is translated to:

Max [summation over ((k,l) [member of] [OMEGA])] max(0,[s.sub.l] - [s.sub.k])

S.T. [summation over ((k,l) [member of] [OMEGA])] max(0,[s.sub.l] - [s.sub.k]) [greater than or equal to] h

[w.sub.j] [greater than or equal to] [epsilon] j = 1,2,...,n, (18)

The variable [[lambda].sub.kl] is introduced as follows:

[[lambda].sub.kl] = max(0,[s.sub.l] - [s.sub.k]) (19)

For each pair (k,l) [member of] [OMEGA] the following relations are hold:

[[lambda].sub.kl] [greater than or equal to] 0, (20)

and

[[lambda].sub.kl] [greater than or equal to] [s.sub.l] - [s.sub.k] (21)

Substituting Eq. (19)-(21), the model (18) is transformed to the following model:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

The final LINMAP-G model is achieved by acquisition of a corresponding relation for [s.sub.k] - [s.sub.l]. Using Eq. (9) and the definition of variable [s.sub.i], this relation is obtained as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The extended form of the above equation after calculation of squares and factorization is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The model (22) is now transformed into the following model, which is called grey LINMAP-G model.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

where,

[[v.bar].sub.j] = [w.sub.j][[x.bar].sup.*.sub.j], (25)

and

[[bar.v].sub.j] = [w.sub.j][[bar.x].sup.*.sub.j], (26)

Note that constraints [[v.bar].sub.j] [less than or equal to] [[bar.v].sub.j] j = 1,2,...,n are added to guarantee the grey property of obtained PIS. By solving the model (24), the optimal values of [[v.bar].sub.j], [[bar.v].sub.j] and [w.sub.j] are determined. Then, the PIS solution [A.sup.*] = ([x.sup.*.sub.1],[x.sup.*.sub.2],...,[x.sup.*.sub.n]) and weight vector W = ([w.sub.1],[w.sub.2],...,[w.sub.n]) are determined. The optimal weights of attributes can be determined after normalization of weight vector W. Finally, the ranking of alternatives are specified by calculation of [s.sub.i] variables for all alternatives and ascending sort of these values.

Figure 1 shows an algorithm about the decision making process with LINMAP-G method. It is possible that decision maker has some viewpoints regard to weight vector, such that he/ she do not want none of the attribute's weights be greater than other's weights. This set of constraints can be added to model as [u.sub.kj] [less than or equal to] [w.sub.k]/[w.sub.j] [less than or equal to] [l.sub.kj], k,j = 1,2,...,n,k [not equal to] j.

[FIGURE 1 OMITTED]

5. Numerical example

In this section, two numerical examples are solved by proposed LINMAP-G decision making process.

5.1. Ranking of constructing projects

The first study is done on a relatively small instance. This example is about a company that wants to rank its target market sectors. The company's market is divided into five different sectors A, B, C, D, and E that are evaluated based on four attributes: three attributes include (1) market size, (2) market growth and (3) consistency with company's mission as profit attributes and a (4) structural risk attribute as cost attributes. The grey decision matrix is constructed as follows (see Table 1).

Assume that decision makers have specified their preferences between alternatives as [OMEGA] = {(2,l), (3,2), (4,3), (5,4)}. The first step is to normalize the decision matrix. Attributes 1-3 are normalized based on Eq. (7) and the attribute 4 by Eq. (8). The normalized decision matrix is shown in Table 2.

In the next step, the grey LINMAP-G method is developed according to Eq. (24) as follows. Assume that decision maker wants that none of the attributes weights be more than three times greater than the others. Note that h = 1 and [epsilon] = 0.001.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The optimal solution of the above model is as follows:

[W.sup.*] = ([w.sub.1], [w.sub.2], [w.sub.3], [w.sub.4]) = (0.35,1.05,1.05,0.35),

[V.sup.*] = ((0.299,0.299),(0,0),(0,3.54),(0.713,0.713)).

Now, the PIS [A.sup.*] can be derived as [A.sup.*] = [V.sup.*]/[W.sup.*]

[A.sup.*] = ((0.854,0.854),(0,0),(0,3.37),(2.037,2.037)).

Now, the square distances [s.sub.i] from PIS [A.sup.*] are [s.sub.1] = 11.813, [s.sub.2] = 11.816, [s.sub.3] = 11.813, [s.sub.4] = 10.810 and [s.sub.5] = 10.813. Therefore, the ranking order of alternatives are D [??] E [??] A [??] = C [??] B.

5.2. Contractor ranking

This example is solved in (Jahanshahloo et al. 2006) through grey TOPSIS method. The problem is to rank 15 bank branches based on four financial attributes. Grey decision matrix is shown in Table 3. Assume that the decision maker determines his/her preferences between branches as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The next step according to Figure 1 is to normalize the grey decision matrix. Table 4 shows the normalized decision matrix (while all attributes are as maximizing type, Eq. (7) is used here). Then, the grey model (24) is constructed and solved. In this model h = 1 and [epsilon] = 0.01 are considered. The obtained solution is as follows:

[W.sup.*] = (0.086,0.086,0.173,0.173), [V.sup.*] = ((0,0),(0.1332,0.385),(0,0.1),(0,0)).

Now, the PIS [A.sup.*] can be derived as [A.sup.*] = ((0,0),(1.54,4.47),(0.0.578),(0,0)). Table 5 shows the square distances and ranking of alternatives by proposed.

6. Conclusions

Constructing project selection is an important issue, according to high risks and costs of mistakes. Therefore decision making based on multiple attributes eases it to prevent these likely problems. In construction evaluation problems, the decision maker selects some alternatives among different ones and it is necessary to consider different qualitative and quantitative criteria.

Evaluation of a set of alternatives regard to a set of quantitative or qualitative attributes is the main concentration of multi attribute decision making problems. In crisp MADM algorithms, subjective judgments and qualitative measures are translated into crisp numbers. This transformation means that decision maker ignores the uncertainty and ambiguity of his/her thinking and believes. Therefore some frameworks are presented to handle these uncertainties. According to uncertainty in the real world, we tried to calculate these parameters by uncertain data and in this paper, a new version of LINMAP method, originally presented in (Srinivasan, Shocker 1973), where decision maker's judges are expressed as grey number is proposed. The proposed method ranks alternatives by solving a linear programming that determines the attributes weight vector and an ideal solution. Then, the alternatives are ranked regard to their distances from PIS by specified weights. Application of the developed method is shown in two constructing examples that one of them was about ranking a set of various constructing projects for a developer company and another example was about ranking a set of constructing contractors. This suggests that the proposed method can be applied in different multi attribute decision making problems which contain uncertainty and ill-defined data and decision maker has not determined attributes weights priori.

doi: 10.3846/20294913.2012.740518

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Zavadskas, E. K.; Turskis, Z.; Tamosaitiene, J. 2010b. Risk assessment of construction projects, Journal of Civil Engineering and Management 16(1): 33-46. http://dx.doi.org/10.3846/jcem.2010.03

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Zavadskas, E. K.; Turskis, Z. 2011. Multiple criteria decision making (MCDM) methods in economics: an overview, Technological and Economic Development of Economy 17(2): 397-427. http://dx.doi.org/10.3846/20294913.2011.593291

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Seyed Hossein Razavi Hajiagha (1), Shide Sadat Hashemi (2) Edmundas Kazimieras Zavadskas (3), Hadi Akrami (4)

(1,4) Institute for Trade Studies and Research, Tehran, Iran (2) Faculty of management and accounting, Allame Tabatabaei University, Tehran, Iran (3) Vilnius Gediminas Technical University, Institute of Internet and Intelligent Technologies, Sauletekio al. 11, LT-10223 Vilnius, Lithuania

E-mail: (1) sh.razavi@st.atu.ac.ir; (2) shide_hashemi@yahoo.com; (3) edmundas.zavadskas@vgtu.lt (corresponding author); (4) hadi.akrami@gmail.com

Received 03 January 2012; accepted 06 August 2012

Seyed Hossein RAZAVI HAJIAGHA received his BSc in Industrial Engineering at Islamic Azad University, 2005 and his M.A. in Industrial management at Allame Tabatabaei University, 2007. He has worked as a researcher at institute for Trade Studies and Researches since 2008. He has received his Ph.D at the same university, 2012. His research interest spans the fields of Operation research and Decision making methods under uncertainty. He has taught operation research at Allame University since 2009 and he has published some papers in these areas.

Shide Sadat HASHEMI has received her BSc in Industrial management, at Damavand University, 2004 and her M.A. in Industrial management at Allame Tabatabaei University, 2008. She has worked as a researcher since 2009. Her research interests include operation management and Data envelopment analysis.

Edmundas Kazimieras ZAVADSKAS. Prof., Head of the Department of Construction Technology and Management at Vilnius Gediminas Technical University, Vilnius, Lithuania. He has a PhD in Building Structures (1973) and Dr Sc. (1987) in Building Technology and Management. He is a member of the Lithuanian and several foreign Academies of Sciences. He is Doctore Honoris Causa at Poznan, Saint-Petersburg, and Kiev universities as well as a member of international organizations; he has been a member of steering and programme committees at many international conferences. E. K. Zavadskas is a member of editorial boards of several research journals. He is the author and co-author of more than 400 papers and a number of monographs in Lithuanian, English, German and Russian. Research interests are: building technology and management, decision-making theory, automation in design and decision support systems.

Hadi AKRAMI has received his BSc in Industrial Engineering, at Sharif University, 2003 and his M.Sc. in Industrial Engineering at Amirkabir University, 2007. He has worked as a researcher since 2005. His research interests include operation research and decision making.

Table 1. Grey decision matrix (example 1)

       1         2         3        4

A    [6, 7]    [3, 4]    [4, 5]   [6, 7]

B    [4, 5]    [5, 6]    [5, 6]   [6, 7]

C    [5, 7]   [6, 7.5]   [4, 5]   [3, 4]

D    [7, 8]    [4, 5]    [6, 8]   [5, 6]

E    [6, 8]    [5, 6]    [7, 9]   [7, 8]

Table 2. Normalized grey decision matrix (example 1)

           1                2                3

A    [0.295, 0.344]   [0.179, 0.239]   [0.207, 0.259]

B    [0.197, 0.246]   [0.299, 0.358]   [0.259, 0.311]

C    [0.246, 0.344]   [0.358, 0.448]   [0.207, 0.259]

D    [0.344, 0.394]   [0.299, 0.311]   [0.311, 0.414]

E    [0.295, 0.394]   [0.358, 0.362]   [0.362, 0.466]

           1                4

A    [0.295, 0.344]   [0.649, 0.699]

B    [0.197, 0.246]   [0.598, 0.649]

C    [0.246, 0.344]   [0.799, 0.849]

D    [0.344, 0.394]   [0.699, 0.749]

E    [0.295, 0.394]    [0.598, 649]

Table 3. Grey decision matrix (example 2)

                 [C.sub.1]            [C.sub.2]          [C.sub.3]

[A.sub.1]     [500.37, 961.37]    [2696995, 3126798]   [26364, 38254]

[A.sub.2]     [873.7, 1775.5]     [1027546, 1061260]   [3791, 50308]

[A.sub.3]     [95.93, 196.39]     [1145235, 1213541]   [22964, 26846]

[A.sub.4]    [848.07, 1752.66]     [390902, 395241]     [492, 1213]

[A.sub.5]     [58.69, 120.47]      [144906, 165818]    [18053, 18061]

[A.sub.6]     [464.39, 955.61]     [408163, 416416]    [40539, 48643]

[A.sub.7]     [155.29, 342.89]     [335070, 410427]    [33797, 44933]

[A.sub.8]    [1752.31, 3629.54]    [700842, 768593]     [1437, 1519]

[A.sub.9]     [244.34, 495.78]     [641680, 696338]    [11418, 24108]

[A.sub.10]   [730.27, 1417.11]     [453170, 481943]     [2719, 2955]

[A.sub.11]    [454.75, 931.24]     [309670, 642598]     [2016, 2617]

[A.sub.12]    [303.58, 630.01]     [286149, 317186]    [14918, 27070]

[A.sub.13]   [658.81, 1345.58]     [321435, 347848]     [6616, 8045]

[A.sub.14]    [420.18, 860.79]     [618105, 835839]    [24425, 40457]

[A.sub.15]    [144.68, 292.15]     [119948, 120208]     [1494, 1749]

                 [C.sub.1]            [C.sub.4]

[A.sub.1]     [500.37, 961.37]    [965.97, 6957.33]

[A.sub.2]     [873.7, 1775.5]      [2285.03, 3174]

[A.sub.3]     [95.93, 196.39]      [207.98, 510.93]

[A.sub.4]    [848.07, 1752.66]      [63.32, 92.3]

[A.sub.5]     [58.69, 120.47]      [176.58, 370.81]

[A.sub.6]     [464.39, 955.61]    [4654.71, 5882.53]

[A.sub.7]     [155.29, 342.89]    [560.26, 2506.67]

[A.sub.8]    [1752.31, 3629.54]     [58.89, 86.86]

[A.sub.9]     [244.34, 495.78]    [1070.81, 2283.08]

[A.sub.10]   [730.27, 1417.11]     [375.07, 559.85]

[A.sub.11]    [454.75, 931.24]    [936.62, 1468.45]

[A.sub.12]    [303.58, 630.01]    [1203.79, 4335.24]

[A.sub.13]   [658.81, 1345.58]     [200.36, 399.8]

[A.sub.14]    [420.18, 860.79]    [2781.24, 4555.42]

[A.sub.15]    [144.68, 292.15]     [282.73, 471.22]

Table 4. Normalized grey decision matrix (example 2)

                [C.sub.1]          [C.sub.2]          [C.sub.3]

[A.sub.1]    [0.0856, 0.1645]   [0.5176, 0.6001]   [0.1974, 0.2865]

[A.sub.2]    [0.1495, 0.3038]   [0.1972, 0.2037]   [0.0283, 0.3768]

[A.sub.3]    [0.0164, 0.0336]   [0.2198, 0.2329]   [0.1720, 0.2010]

[A.sub.4]    [0.1451, 0.2999]   [0.0750, 0.0758]   [0.0036, 0.0090]

[A.sub.5]    [0.0100, 0.0206]   [0.0278, 0.0318]   [0.1352, 0.1352]

[A.sub.6]    [0.0794, 0.1635]   [0.0783, 0.0799]   [0.3036, 0.3643]

[A.sub.7]    [0.0265, 0.0586]   [0.0643, 0.0787]   [0.2531, 0.3365]

[A.sub.8]    [0.2999, 0.6211]   [0.1345, 0.1475]   [0.0107, 0.0113]

[A.sub.9]    [0.0418, 0.0848]   [0.1231, 0.1336]   [0.0855, 0.1805]

[A.sub.10]   [0.1249, 0.2425]   [0.0869, 0.0925]   [0.0203, 0.0221]

[A.sub.11]   [0.0788, 0.1593]   [0.0594, 0.0657]   [0.0151, 0.0196]

[A.sub.12]   [0.0519, 0.1078]   [0.0549, 0.0608]   [0.1117, 0.2027]

[A.sub.13]   [0.1127, 0.2302]   [0.0616, 0.0667]   [0.0495, 0.0602]

[A.sub.14]   [0.0719, 0.1473]   [0.1186, 0.1604]   [0.1829, 0.3030]

[A.sub.15]   [0.0247, 0.0500]   [0.0230, 0.0230]   [0.0111, 0.0131]

                [C.sub.1]          [C.sub.4]

[A.sub.1]    [0.0856, 0.1645]   [0.0706, 0.5086]

[A.sub.2]    [0.1495, 0.3038]   [0.1670, 0.2320]

[A.sub.3]    [0.0164, 0.0336]   [0.0152, 0.0373]

[A.sub.4]    [0.1451, 0.2999]   [0.0046, 0.0067]

[A.sub.5]    [0.0100, 0.0206]   [0.0129, 0.0271]

[A.sub.6]    [0.0794, 0.1635]   [0.3403, 0.4300]

[A.sub.7]    [0.0265, 0.0586]   [0.0409, 0.1832]

[A.sub.8]    [0.2999, 0.6211]   [0.0043, 0.0063]

[A.sub.9]    [0.0418, 0.0848]   [0.0782, 0.1669]

[A.sub.10]   [0.1249, 0.2425]   [0.0274, 0.0409]

[A.sub.11]   [0.0788, 0.1593]   [0.0684, 0.1073]

[A.sub.12]   [0.0519, 0.1078]   [0.0880, 0.3169]

[A.sub.13]   [0.1127, 0.2302]   [0.0146, 0.0292]

[A.sub.14]   [0.0719, 0.1473]   [0.2033, 0.3330]

[A.sub.15]   [0.0247, 0.0500]   [0.0206, 0.0344]

Table 5. Square distances and
ranking of alternatives
(example 2)

             [S.sub.i]   ranking

[A.sub.1]     1.4479        1
[A.sub.2]     1.7515        3
[A.sub.3]     1.7240        2
[A.sub.4]     1.9107       11
[A.sub.5]     1.9279       14
[A.sub.6]     1.9199       13
[A.sub.7]     1.8733        7
[A.sub.8]     1.8732        6
[A.sub.9]     1.8251        5
[A.sub.10]    1.8899        8
[A.sub.11]    1.9162       12
[A.sub.12]    1.9081       9.5
[A.sub.13]    1.9081       9.5
[A.sub.14]    1.8185        4
[A.sub.15]    1.9544       15