# Grey Target Decision Method for a Variable Target Centre Based on the Decision Maker’s Preferences

Authors: Jinshan Ma
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

In multiattribute grey target decision making, the decision maker (DM) may have certain preferences for some attributes. The impacts of two types of DM’s preferences for some attribute values on alternatives were studied. To deal with the attribute preferences of a DM, a generalised grey target decision method was presented. The proposed method required that the index values of all alternatives were not normalised. The target centre index values can be obtained by substituting DM’s preference values for some of the original target centre index values as determined by the alternatives themselves. Following this, the proposed generalised method was used to calculate the target centre distances. A case study showed that this method of handling DM’s preferences for some attributes was effective.


---

## Body

## 1. Introduction

In multiattribute decision making, the relative optimality of one parameter can be obtained using a grey target decision method by comparison with feasible alternatives without recourse to other standard modes. The grey target decision method has been widely used in many fields since it was proposed by Deng [1]. Over the past few years, many scholars have made progress in this area. Chen and Xie tested the incontinency problem of Deng’s grey transformation by simulation [2]. Dang et al. improved the calculation operators of the grey target decision method [3, 4]. Some scholars also studied its weight determination [5–7]. The grey target decision method for mixed attributes has also been studied [8–14]. Furthermore, some other theories and methods were introduced to the grey target decision method [13–16] which enrich its potential. However, the consideration of the DM’s preferences was seldom studied apart from a limited contribution by Zhu and Hipel [6, 16]. This work expanded the target centre as determined by the alternatives themselves to some indices of the target centre replaced by the DM’s preferences. There are two types of preferences: some attribute values were expected to reach their desired levels; however, some other attribute values were regarded as excellent only if they reached some specified values without acquiring the optimal solution. This work assessed the effects of a variable target centre determined partially by the DM’s preferences over the available alternatives and presented a new generalised grey target decision method to deal with this problem.The remainder of this paper is organised as follows: Section2 introduces the concepts, Section 3 discusses the proposed method, Section 4 presents a case study, and Section 5 is the conclusion.


## 2. Preliminaries

Definition 1.
LetS={S1,S2,…,Sn} be an alternative set, let A={A1,A2,…,Am} be an attribute set, and let Sij(i=1,2,…,n,j=1,2,…,m) be the measure of alternative Si under attribute Aj, and J+and J- are benefit type attribute, and cost type attribute, sets, respectively: these form the basic elements of multiattribute decision making.Remark 2.
Based on the theory of grey target decision making, however the method differed from the classical version (the generalised grey target method). Compared to the traditional model, the generalised grey target method had two differences: no need to normalise the index valuesSij(i=1,2,…,n,j=1,2,…,m) and the difference in the target centre distance calculation.Definition 3.
LetCa=(C1a,C2a,…,Cma) be the target centre determined by the alternative measure Sij(i=1,2,…,n,j=1,2,…,m), where Cja satisfies
(1)Cja={max⁡{Sij},Sij∈J+i=1,2,…,n,j=1,2,…,m.min⁡{Sij},Sij∈J-Definition 4.
LetCkde(k∈{1,2,…,m}) be the DM’s desirable preference value, such that the DM’s preference value is better than or equal to the optimal index value of alternatives under attribute Aj, which satisfies
(2)nnnnnnnnnnnnnnnnnCkde≥max⁡{Sij},Sij∈J+,i=1,2,…,n,j=1,2,…,m,
or
(3)nnnnnnnnnnnnnnnnnCkde≤min⁡{Sij},Sij∈J-,i=1,2,…,n,j=1,2,…,m.Definition 5.
LetCkds(k∈{1,2,…,m}) be the DM’s selection preference value, such that some index value is regarded as excellent only when it is better than or equal to the value Ckds given by the DM under attribute Aj, which satisfies
(4)min⁡{Sij}<Ckds<max⁡{Sij},i=1,2,…,n,j=1,2,…,m.Definition 6.
Suppose that the target centreCa=(C1a,C2a,…,Cma) is decided by Sij(i=1,2,…,n,j=1,2,…,m) and DM’s preference value under attribute Ak(k∈{1,2,…,m}) is Ckd(k∈{1,2,…,m}). Then the target centre, determined partially by preference values, becomes C0=(C10,C20,…,Cm0), the elements of which are as follows:
(5)Cj0={Cja,j=1,2,…,m,j≠kCkd,k∈{1,2,…,m},j=k.


## 3. Grey Target Decision Making Method for Variable Target Centre

### 3.1. The Impacts of Variable Target Centre on Alternatives

Desirable preferences and selection preferences are two types of attribute preferences for the DM. Different DM’s attribute preferences may cause different impacts on alternatives with respect to any grey target decision model. Figure1 shows the impact of desirable attribute preference on the alternatives. In Figure 1, suppose that Sij which belongs to the benefit type attribute set is the index value under attribute Aj, and Stj is the worst value. Let Cja and Cjd be the target centre indices under attribute Aj, as determined by feasible alternatives and DM’s preferences, respectively. Suppose that d1, d2, and d3 are the distances of index values S1j, S2j, and S3j to Cja, respectively, and D1, D2, and D3 are the distances of index values S1j, S2j, and S3j to Cjd, respectively, while rda is the difference between Cjd and Cja. Obviously, the target centre index Cjd, determined by the desirable attribute preference value, expanded the distances from d1, d2, and d3 to D1, D2, and D3, respectively. Figure 2 shows the impact of selection attribute preference on alternatives (the meaning of the parameters in Figure 2 matches that in Figure 1). The target centre index value determined by DM’s selection preference is inferior to that of the alternatives, which changes the distances of S1j, S2j, and S3j to Cja to the distances of S1j, S2j, and S3j to Cjd, such that d1, d2, and d3 changed to D1, D2, and D3, respectively. It can be seen from Figure 2 that Cjd was actually inferior to S1j and S2j, so there was no meaning attributable to either D1 or D2. From the perspective of a cluster of indices, the distances of S1j and S2j to Cjd can be regarded as excellent indices with target centre distances of zero. Only D3 denoted the real target centre distances, but its value is less than d3 and the reduced value is rda which is the difference between Cja and Cjd.Figure 1
The impact of desirable attribute preference on the alternative.Figure 2
The impact of selection attribute preference on the alternative.The impacts of target centre determined by different preferences over the alternatives are discussed as follows: assume thatSij(i=1,2,…,n,j=1,2,…,m) is the measure of alternative Si under attribute Aj and Si0j and S(i0+1)j are any two index values. Let di0 and di0+1 be the distances of Si0j and S(i0+1)j to Cja, respectively; then set di0<di0+1 without affecting the conclusions so that under attribute Aj the distances of Si0j and S(i0+1)j to Cjd are Di0 and Di0+1, respectively: the difference between Cjd and Cja is rda. For comparison, the target centre distances of all indices under some attribute must be normalised. The linear method is used to normalise these target centre distances using (12). The following equations are the difference between the two alternatives’ target centre distances under some attribute for different target centres:
(6)ΔZa=di0+1∑i=1ndi-di0∑i=1ndi=di0+1-di0∑i=1ndi,
where di is the distance of Sij to Cja; namely, di=|Cja-Sij|:
(7)ΔZd=Di0+1∑i=1nDi-Di0∑i=1nDi=Di0+1-Di0∑i=1nDi,
where Di is the distance of Sij to Cjd, which can be calculated by (11).The target centreCja determined by alternatives and the target centre Cjd determined by DM’s preference value have the following relationship:
(8)rda=Cjd-Cja.
So (7) can be rewritten as
(9)ΔZd=rda+di0+1∑i=1n(rda+di)-rda+di0∑i=1n(rda+di)=di0+1-di0nrda+∑i=1ndi.
Compared with (5) and (2), the conclusions may be drawn as follows.(1)
Ifrda>0, which is the DM’s desirable preference, then ΔZd<ΔZa means the difference of the two alternatives’ target centre distances decreased, which implied that the target centre, as determined by desirable preference, can reduce the difference in index values for each alternative.(2)
Ifrda<0, which is the DM’s selection preference, then ΔZd>ΔZa means the difference of the two alternatives’ target centre distances increased, which implied that the target centre, as determined by selection preference, can enlarge the difference in index values for each alternative. However, note that some indices’ target centre distances were zero when they were superior to the target centre index implying an indifference to the value of these indices. Therefore, the target centre index, as determined by selection preference, had the potential to act as a “rewarding good and punishing bad” function.This discussion was based on benefit-type indices; however, the same conclusions may be drawn from consideration of cost-type indices.


### 3.2. Variable Target Centre Determination

To obtain the target centre combined with the DM’s preferences, the target centre, as decided by alternatives, must first be determined. The final target centre was determined by substituting some preference values for the predetermined target centre index values. Note that the predetermined target centre originated from the nonnormalised index matrix. The target centre combined with DM’s preferences can be obtained using (1) and (5).


### 3.3. Target Centre Distance Calculation

In grey target decision making, the optimal alternative is determined by the minimum of all integrated target centre distances. The target centre determined only by the DM’s desirable preferences is easy to deal with; however, the target centre combined with the DM’s selection preferences may be more complicated. Some index values may be superior to the target centre index values determined by selection preferences, so their index target centre distances were zero, as were all those regarded as excellent values. A new generalised grey target method will be used to solve this problem.Suppose that the target centre determined bySij(i=1,2,…,n,j=1,2,…,m) was Ca=(C1a,C2a,…,Cma), so the target centre combined with the DM’s preferences can be calculated according to the following steps.(1)
The new index measureIij can be obtained from Sij(i=1,2,…,n,j=1,2,…,m) compared with the target centre index Cj0(j=1,2,…,m) under attribute Aj(j=1,2,…,m):(10)Iij={Sij,if(Sij<Cj0,Sij∈J+)or(Sij>Cj0,Sij∈J-)Cj0,if(Sij≥Cj0,Sij∈J+)or(Sij≤Cj0,Sij∈J-).(2)
Calculate the distance of index valueIij(i=1,2,…,n,j=1,2,…,m) to the target centre index value Cj0(j=1,2,…,m) under attribute Aj(j=1,2,…,m), using the Hamming distance:
(11)rij=|Cj0-Iij|,i=1,2,…,n,j=1,2,…,m.(3)
Normalise the index target centre distances of all alternatives for comparability, and the linear normalised method was then used to retain the indices’ own characteristics:(12)zij=rij∑i=1nrij,i=1,…,n;j=1,…,m.(4)
Having obtained the weightωj under attribute Aj(j=1,2,…,m), the integrated target centre distances for all alternatives can then be calculated using (13):
(13)wi=ωjzij,i=1,…,n,j=1,…,m.


### 3.4. Weight Determination

The attribute weights can be determined by: subjective method, objective method, or comprehensive method. There are many articles contributing to weight determination: the interested reader is referred to the relevant literature [2–4, 10, 12].


### 3.5. Algorithm of Grey Target Decision Making Method Based on the DM’s Preferences

(1)
Give the DM’s attribute preferences.
(2)
Calculate the original target centre for nonnormalised alternatives’ matrix of index values.
(3)
Achieve the target centre combined with the DM’s attribute preferences.
(4)
Deal with the index values of all alternatives by the final target centre.
(5)
Calculate the distances of all index values to their target centre index values.
(6)
Determine the weights of all attributes.
(7)
Integrate all of the normalised target centre distances under all attributes for all alternatives, and rank the alternatives according to their integrated target centre distances in ascending order.


## 3.1. The Impacts of Variable Target Centre on Alternatives

Desirable preferences and selection preferences are two types of attribute preferences for the DM. Different DM’s attribute preferences may cause different impacts on alternatives with respect to any grey target decision model. Figure1 shows the impact of desirable attribute preference on the alternatives. In Figure 1, suppose that Sij which belongs to the benefit type attribute set is the index value under attribute Aj, and Stj is the worst value. Let Cja and Cjd be the target centre indices under attribute Aj, as determined by feasible alternatives and DM’s preferences, respectively. Suppose that d1, d2, and d3 are the distances of index values S1j, S2j, and S3j to Cja, respectively, and D1, D2, and D3 are the distances of index values S1j, S2j, and S3j to Cjd, respectively, while rda is the difference between Cjd and Cja. Obviously, the target centre index Cjd, determined by the desirable attribute preference value, expanded the distances from d1, d2, and d3 to D1, D2, and D3, respectively. Figure 2 shows the impact of selection attribute preference on alternatives (the meaning of the parameters in Figure 2 matches that in Figure 1). The target centre index value determined by DM’s selection preference is inferior to that of the alternatives, which changes the distances of S1j, S2j, and S3j to Cja to the distances of S1j, S2j, and S3j to Cjd, such that d1, d2, and d3 changed to D1, D2, and D3, respectively. It can be seen from Figure 2 that Cjd was actually inferior to S1j and S2j, so there was no meaning attributable to either D1 or D2. From the perspective of a cluster of indices, the distances of S1j and S2j to Cjd can be regarded as excellent indices with target centre distances of zero. Only D3 denoted the real target centre distances, but its value is less than d3 and the reduced value is rda which is the difference between Cja and Cjd.Figure 1
The impact of desirable attribute preference on the alternative.Figure 2
The impact of selection attribute preference on the alternative.The impacts of target centre determined by different preferences over the alternatives are discussed as follows: assume thatSij(i=1,2,…,n,j=1,2,…,m) is the measure of alternative Si under attribute Aj and Si0j and S(i0+1)j are any two index values. Let di0 and di0+1 be the distances of Si0j and S(i0+1)j to Cja, respectively; then set di0<di0+1 without affecting the conclusions so that under attribute Aj the distances of Si0j and S(i0+1)j to Cjd are Di0 and Di0+1, respectively: the difference between Cjd and Cja is rda. For comparison, the target centre distances of all indices under some attribute must be normalised. The linear method is used to normalise these target centre distances using (12). The following equations are the difference between the two alternatives’ target centre distances under some attribute for different target centres:
(6)ΔZa=di0+1∑i=1ndi-di0∑i=1ndi=di0+1-di0∑i=1ndi,
where di is the distance of Sij to Cja; namely, di=|Cja-Sij|:
(7)ΔZd=Di0+1∑i=1nDi-Di0∑i=1nDi=Di0+1-Di0∑i=1nDi,
where Di is the distance of Sij to Cjd, which can be calculated by (11).The target centreCja determined by alternatives and the target centre Cjd determined by DM’s preference value have the following relationship:
(8)rda=Cjd-Cja.
So (7) can be rewritten as
(9)ΔZd=rda+di0+1∑i=1n(rda+di)-rda+di0∑i=1n(rda+di)=di0+1-di0nrda+∑i=1ndi.
Compared with (5) and (2), the conclusions may be drawn as follows.(1)
Ifrda>0, which is the DM’s desirable preference, then ΔZd<ΔZa means the difference of the two alternatives’ target centre distances decreased, which implied that the target centre, as determined by desirable preference, can reduce the difference in index values for each alternative.(2)
Ifrda<0, which is the DM’s selection preference, then ΔZd>ΔZa means the difference of the two alternatives’ target centre distances increased, which implied that the target centre, as determined by selection preference, can enlarge the difference in index values for each alternative. However, note that some indices’ target centre distances were zero when they were superior to the target centre index implying an indifference to the value of these indices. Therefore, the target centre index, as determined by selection preference, had the potential to act as a “rewarding good and punishing bad” function.This discussion was based on benefit-type indices; however, the same conclusions may be drawn from consideration of cost-type indices.


## 3.2. Variable Target Centre Determination

To obtain the target centre combined with the DM’s preferences, the target centre, as decided by alternatives, must first be determined. The final target centre was determined by substituting some preference values for the predetermined target centre index values. Note that the predetermined target centre originated from the nonnormalised index matrix. The target centre combined with DM’s preferences can be obtained using (1) and (5).


## 3.3. Target Centre Distance Calculation

In grey target decision making, the optimal alternative is determined by the minimum of all integrated target centre distances. The target centre determined only by the DM’s desirable preferences is easy to deal with; however, the target centre combined with the DM’s selection preferences may be more complicated. Some index values may be superior to the target centre index values determined by selection preferences, so their index target centre distances were zero, as were all those regarded as excellent values. A new generalised grey target method will be used to solve this problem.Suppose that the target centre determined bySij(i=1,2,…,n,j=1,2,…,m) was Ca=(C1a,C2a,…,Cma), so the target centre combined with the DM’s preferences can be calculated according to the following steps.(1)
The new index measureIij can be obtained from Sij(i=1,2,…,n,j=1,2,…,m) compared with the target centre index Cj0(j=1,2,…,m) under attribute Aj(j=1,2,…,m):(10)Iij={Sij,if(Sij<Cj0,Sij∈J+)or(Sij>Cj0,Sij∈J-)Cj0,if(Sij≥Cj0,Sij∈J+)or(Sij≤Cj0,Sij∈J-).(2)
Calculate the distance of index valueIij(i=1,2,…,n,j=1,2,…,m) to the target centre index value Cj0(j=1,2,…,m) under attribute Aj(j=1,2,…,m), using the Hamming distance:
(11)rij=|Cj0-Iij|,i=1,2,…,n,j=1,2,…,m.(3)
Normalise the index target centre distances of all alternatives for comparability, and the linear normalised method was then used to retain the indices’ own characteristics:(12)zij=rij∑i=1nrij,i=1,…,n;j=1,…,m.(4)
Having obtained the weightωj under attribute Aj(j=1,2,…,m), the integrated target centre distances for all alternatives can then be calculated using (13):
(13)wi=ωjzij,i=1,…,n,j=1,…,m.


## 3.4. Weight Determination

The attribute weights can be determined by: subjective method, objective method, or comprehensive method. There are many articles contributing to weight determination: the interested reader is referred to the relevant literature [2–4, 10, 12].


## 3.5. Algorithm of Grey Target Decision Making Method Based on the DM’s Preferences

(1)
Give the DM’s attribute preferences.
(2)
Calculate the original target centre for nonnormalised alternatives’ matrix of index values.
(3)
Achieve the target centre combined with the DM’s attribute preferences.
(4)
Deal with the index values of all alternatives by the final target centre.
(5)
Calculate the distances of all index values to their target centre index values.
(6)
Determine the weights of all attributes.
(7)
Integrate all of the normalised target centre distances under all attributes for all alternatives, and rank the alternatives according to their integrated target centre distances in ascending order.


## 4. Case Study

### 4.1. Background and Data

To evaluate ten coal mines’ comprehensive safety performances, eight indices including seam dip (°), methane emission rate (m3/t), water inflow (m3/h), spontaneous combustion period (month), ventilating structures qualification rate (%), equivalent orifice (m2), mortalityper million tons (person/106 t), and accident economic loss (105 Yuan) [17] are denoted by A1 to A10, and alternatives are denoted by S1 to S10. The data are shown in Table 1, the benefit-type attributes are A4 to A6, and the others are cost-type attributes. The DM’s attribute preferences are A2, A5, A6, and A7 with their values set to 0, 95, 2.0, and 0.2, respectively.Table 1
Safety data from coal mines.
S
i
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
S
1
21
6
220
12
92
1.8
0.18
381
S
2
16
3.7
200
6
90
1.4
0.712
564
S
3
26
9.2
180
10
88
2.7
1.34
1051.6
S
4
10
4
260
8
94
1.2
0
442.5
S
5
30
8.2
350
10
96
3.6
0.641
788
S
6
19
5
130
12
100
2.4
0
300
S
7
17
9.6
400
6
86
1.3
1.23
964.7
S
8
40
14
600
6
95
2.1
1.12
885.6
S
9
12
12.8
120
10
91
1.5
0.872
839.3
S
10
14
5.8
155
12
89
1.7
0.426
617.2


### 4.2. Decision Making Process

(1)
Calculate the target centre determined by alternatives.
The original target centreCa = (10, 3.7, 120, 12, 100, 3.6, 0, 300) is obtained using (1).
(2)
Determine the target centre combined with the DM’s preferences.
The final target centreC0 = (10, 0, 120, 12, 95, 2.0, 0.2, 300) combined with the DM’s preferences can be determined using (5).
(3)
Deal with the index matrix based on target centreC0.
Use (10) and the original index matrix can be converted to a new index matrix based on target centre C0: the results are shown in Table 2.
(4)
Calculate all index target centre distances.
Using (11), all index target centre distances can be calculated as listed in Table 3.
(5)
Normalise all index target centre distances.
All index target centre distances can be normalised using (12) with the results shown in Table 4.
(6)
Integrate the normalised index target centre distances.Table 2
Index values processed based on final target centre.
S
i
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
S
1
21
6
220
12
92
1.8
0.2
381
S
2
16
3.7
200
6
90
1.4
0.712
564
S
3
26
9.2
180
10
88
2.0
1.34
1051.6
S
4
10
4
260
8
94
1.2
0.2
442.5
S
5
30
8.2
350
10
95
2.0
0.641
788
S
6
19
5
130
12
95
2.0
0.2
300
S
7
17
9.6
400
6
86
1.3
1.23
964.7
S
8
40
14
600
6
95
2.0
1.12
885.6
S
9
12
12.8
120
10
91
1.5
0.872
839.3
S
10
14
5.8
155
12
89
1.7
0.426
617.2Table 3
All index target centre distances.
r
i
j
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
r
1
j
11
6
100
0
3
0.2
0
81
r
2
j
6
3.7
80
6
5
0.6
0.512
264
r
3
j
16
9.2
60
2
7
0
1.14
751.6
r
4
j
0
4
140
4
1
0.8
0
142.5
r
5
j
20
8.2
230
2
0
0
0.441
488
r
6
j
9
5
10
0
0
0
0
0
r
7
j
7
9.6
280
6
9
0.7
1.03
664.7
r
8
j
30
14
480
6
0
0
0.92
585.6
r
9
j
2
12.8
0
2
4
0.5
0.672
539.3
r
10
j
4
5.8
35
0
6
0.3
0.226
317.2Table 4
Normalised index target centre distances.
Z
i
j
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
Z
1
j
0.104762
0.076628
0.070671
0
0.085714
0.064516
0
0.021127
Z
2
j
0.057143
0.047254
0.056537
0.214286
0.142857
0.193548
0.103623
0.068859
Z
3
j
0.152381
0.117497
0.042403
0.071429
0.2
0
0.230723
0.196041
Z
4
j
0
0.051086
0.09894
0.142857
0.028571
0.258065
0
0.037168
Z
5
j
0.190476
0.104725
0.162544
0.071429
0
0
0.089253
0.127286
Z
6
j
0.085714
0.063857
0.007076
0
0
0
0
0
Z
7
j
0.066667
0.122605
0.19788
0.214286
0.257143
0.225806
0.20846
0.173374
Z
8
j
0.285714
0.178799
0.339233
0.214286
0
0
0.186197
0.152743
Z
9
j
0.019048
0.163474
0
0.071429
0.114286
0.16129
0.136005
0.140666
Z
10
j
0.038095
0.074074
0.024735
0
0.171429
0.96774
0.04574
0.082736Givenω = (0.06, 0.15, 0.03, 0.08, 0.12, 0.13, 0.27, 0.14), the integrated target centre distances w = (0.043051, 0.110387, 0.140379, 0.064991, 0.082207, 0.015661, 0.189271, 0.146908, 0.124351, 0.186678) can be obtained by (13). So the alternatives, in rank order, were S6≻S1≻S4≻S5≻S2≻S9≻S8≻S3≻S10≻S7.Givenω = (0.06, 0.15, 0.03, 0.08, 0.12, 0.13, 0.27, 0.14) without considering the preferences, then w = (0.055198, 0.094923, 0.147292, 0.049522, 0.091520, 0.020273, 0.166158, 0.173715, 0.119444, 0.076746) can be obtained by (13). So the alternatives in rank order were S6≻S4≻S1≻S10≻S5≻S2≻S9≻S3≻S7≻S8.


### 4.3. Discussion

The results, considering the attribute preferences ofA2, A5, A6, and A7 with values 0, 95, 2.0, and 0.2, respectively, and the results without considering attribute preferences are shown in Table 5.Table 5
Alternatives ranked either with, or without, consideration of preferences.
S
i
Z
i
j (no preferences)
Ranking (no preferences)
Z
i
j (preferences)
Ranking (preferences)
Ranking changes
S
1
0.055198
3
0.043051
2
−1
S
2
0.094923
6
0.110387
5
−1
S
3
0.147292
8
0.140379
8
0
S
4
0.049522
2
0.064991
3
+1
S
5
0.091520
5
0.082207
4
−1
S
6
0.020273
1
0.015661
1
0
S
7
0.166158
9
0.189271
10
+1
S
8
0.173715
10
0.146908
7
−3
S
9
0.119444
7
0.124351
6
−1
S
10
0.076746
4
0.186678
9
+5As seen in Table5, the integrated target centre distances and alternative ranking would change when considering the DM’s preferences. With respect to the ranking of the alternatives, most of them changed except for S3 and S6. Alternative S10 changed its ranking from fourth to ninth when not considering preferences and considering preferences: the magnitude of this change indicated that the DM’s attributes influenced the decision making with regard to the available alternatives.


## 4.1. Background and Data

To evaluate ten coal mines’ comprehensive safety performances, eight indices including seam dip (°), methane emission rate (m3/t), water inflow (m3/h), spontaneous combustion period (month), ventilating structures qualification rate (%), equivalent orifice (m2), mortalityper million tons (person/106 t), and accident economic loss (105 Yuan) [17] are denoted by A1 to A10, and alternatives are denoted by S1 to S10. The data are shown in Table 1, the benefit-type attributes are A4 to A6, and the others are cost-type attributes. The DM’s attribute preferences are A2, A5, A6, and A7 with their values set to 0, 95, 2.0, and 0.2, respectively.Table 1
Safety data from coal mines.
S
i
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
S
1
21
6
220
12
92
1.8
0.18
381
S
2
16
3.7
200
6
90
1.4
0.712
564
S
3
26
9.2
180
10
88
2.7
1.34
1051.6
S
4
10
4
260
8
94
1.2
0
442.5
S
5
30
8.2
350
10
96
3.6
0.641
788
S
6
19
5
130
12
100
2.4
0
300
S
7
17
9.6
400
6
86
1.3
1.23
964.7
S
8
40
14
600
6
95
2.1
1.12
885.6
S
9
12
12.8
120
10
91
1.5
0.872
839.3
S
10
14
5.8
155
12
89
1.7
0.426
617.2


## 4.2. Decision Making Process

(1)
Calculate the target centre determined by alternatives.
The original target centreCa = (10, 3.7, 120, 12, 100, 3.6, 0, 300) is obtained using (1).
(2)
Determine the target centre combined with the DM’s preferences.
The final target centreC0 = (10, 0, 120, 12, 95, 2.0, 0.2, 300) combined with the DM’s preferences can be determined using (5).
(3)
Deal with the index matrix based on target centreC0.
Use (10) and the original index matrix can be converted to a new index matrix based on target centre C0: the results are shown in Table 2.
(4)
Calculate all index target centre distances.
Using (11), all index target centre distances can be calculated as listed in Table 3.
(5)
Normalise all index target centre distances.
All index target centre distances can be normalised using (12) with the results shown in Table 4.
(6)
Integrate the normalised index target centre distances.Table 2
Index values processed based on final target centre.
S
i
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
S
1
21
6
220
12
92
1.8
0.2
381
S
2
16
3.7
200
6
90
1.4
0.712
564
S
3
26
9.2
180
10
88
2.0
1.34
1051.6
S
4
10
4
260
8
94
1.2
0.2
442.5
S
5
30
8.2
350
10
95
2.0
0.641
788
S
6
19
5
130
12
95
2.0
0.2
300
S
7
17
9.6
400
6
86
1.3
1.23
964.7
S
8
40
14
600
6
95
2.0
1.12
885.6
S
9
12
12.8
120
10
91
1.5
0.872
839.3
S
10
14
5.8
155
12
89
1.7
0.426
617.2Table 3
All index target centre distances.
r
i
j
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
r
1
j
11
6
100
0
3
0.2
0
81
r
2
j
6
3.7
80
6
5
0.6
0.512
264
r
3
j
16
9.2
60
2
7
0
1.14
751.6
r
4
j
0
4
140
4
1
0.8
0
142.5
r
5
j
20
8.2
230
2
0
0
0.441
488
r
6
j
9
5
10
0
0
0
0
0
r
7
j
7
9.6
280
6
9
0.7
1.03
664.7
r
8
j
30
14
480
6
0
0
0.92
585.6
r
9
j
2
12.8
0
2
4
0.5
0.672
539.3
r
10
j
4
5.8
35
0
6
0.3
0.226
317.2Table 4
Normalised index target centre distances.
Z
i
j
A
1
A
2
A
3
A
4
A
5
A
6
A
7
A
8
Z
1
j
0.104762
0.076628
0.070671
0
0.085714
0.064516
0
0.021127
Z
2
j
0.057143
0.047254
0.056537
0.214286
0.142857
0.193548
0.103623
0.068859
Z
3
j
0.152381
0.117497
0.042403
0.071429
0.2
0
0.230723
0.196041
Z
4
j
0
0.051086
0.09894
0.142857
0.028571
0.258065
0
0.037168
Z
5
j
0.190476
0.104725
0.162544
0.071429
0
0
0.089253
0.127286
Z
6
j
0.085714
0.063857
0.007076
0
0
0
0
0
Z
7
j
0.066667
0.122605
0.19788
0.214286
0.257143
0.225806
0.20846
0.173374
Z
8
j
0.285714
0.178799
0.339233
0.214286
0
0
0.186197
0.152743
Z
9
j
0.019048
0.163474
0
0.071429
0.114286
0.16129
0.136005
0.140666
Z
10
j
0.038095
0.074074
0.024735
0
0.171429
0.96774
0.04574
0.082736Givenω = (0.06, 0.15, 0.03, 0.08, 0.12, 0.13, 0.27, 0.14), the integrated target centre distances w = (0.043051, 0.110387, 0.140379, 0.064991, 0.082207, 0.015661, 0.189271, 0.146908, 0.124351, 0.186678) can be obtained by (13). So the alternatives, in rank order, were S6≻S1≻S4≻S5≻S2≻S9≻S8≻S3≻S10≻S7.Givenω = (0.06, 0.15, 0.03, 0.08, 0.12, 0.13, 0.27, 0.14) without considering the preferences, then w = (0.055198, 0.094923, 0.147292, 0.049522, 0.091520, 0.020273, 0.166158, 0.173715, 0.119444, 0.076746) can be obtained by (13). So the alternatives in rank order were S6≻S4≻S1≻S10≻S5≻S2≻S9≻S3≻S7≻S8.


## 4.3. Discussion

The results, considering the attribute preferences ofA2, A5, A6, and A7 with values 0, 95, 2.0, and 0.2, respectively, and the results without considering attribute preferences are shown in Table 5.Table 5
Alternatives ranked either with, or without, consideration of preferences.
S
i
Z
i
j (no preferences)
Ranking (no preferences)
Z
i
j (preferences)
Ranking (preferences)
Ranking changes
S
1
0.055198
3
0.043051
2
−1
S
2
0.094923
6
0.110387
5
−1
S
3
0.147292
8
0.140379
8
0
S
4
0.049522
2
0.064991
3
+1
S
5
0.091520
5
0.082207
4
−1
S
6
0.020273
1
0.015661
1
0
S
7
0.166158
9
0.189271
10
+1
S
8
0.173715
10
0.146908
7
−3
S
9
0.119444
7
0.124351
6
−1
S
10
0.076746
4
0.186678
9
+5As seen in Table5, the integrated target centre distances and alternative ranking would change when considering the DM’s preferences. With respect to the ranking of the alternatives, most of them changed except for S3 and S6. Alternative S10 changed its ranking from fourth to ninth when not considering preferences and considering preferences: the magnitude of this change indicated that the DM’s attributes influenced the decision making with regard to the available alternatives.


## 5. Conclusions

This research proposed a grey target decision method with a variable target centre considering DM’s desirable preferences and selection preferences. The study indicated that the target centre determined by desirable preferences could reduce the difference between index values for each alternative, which resulted in indicial clustering. However, the target centre, as determined by selection preference, had the potential to act in a “rewarding good and punishing bad” role. When some index values were superior to the target centre index, these indices were rewarded as excellent values; when some index values were inferior to the target centre index, these indices were punished with a larger difference therefrom. A case study illustrated that the generalised grey target decision method could effectively solve the problem for a target centre determined partially by the DM’s preferences.



---

*Source: 572529-2014-04-17.xml*