Scholarly article on topic 'Grey Target Decision Method for a Variable Target Centre Based on the Decision Maker’s Preferences'

Grey Target Decision Method for a Variable Target Centre Based on the Decision Maker’s Preferences Academic research paper on "Mathematics"

CC BY
0
0
Share paper
Academic journal
Journal of Applied Mathematics
OECD Field of science
Keywords
{""}

Academic research paper on topic "Grey Target Decision Method for a Variable Target Centre Based on the Decision Maker’s Preferences"

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 572529, 6 pages http://dx.doi.org/10.1155/2014/572529

Research Article

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

Jinshan Ma1,2

1 School of Mines, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

2 School of Energy Science and Engineering, Henan Polytechnic University, Jiaozuo, Henan 454000, China

Correspondence should be addressed to Jinshan Ma; mjscumttf@163.com

Received 15 January 2014; Revised 3 April 2014; Accepted 4 April 2014; Published 17 April 2014

Academic Editor: Zhihua Zhang

Copyright © 2014 Jinshan Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 [1316] 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: Section 2 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. Let S = (S1, S2,..., S„| be an alternative set, let A = {A 1,A 2,..., Am| be an attribute set, and let Stj (i = 1,2 ,...,n,j = 1,2, ...,m) be the measure of alternative St under attribute Aj, and /+and 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 values Sjj (( = 1,2,..., n, j = 1,2,..., m) and the difference in the target centre distance calculation.

Definition 3. Let Ca = (C", C2,..., C^) be the target centre determined by the alternative measure S^ (i = 1,2,..., n, j = 1,2,..., m), where C" satisfies

max {s;j}, Sjj e J+ min {s;j}, Sjj e

i = 1,2,..., n, j = 1,2,..., m.

Figure 1: The impact of desirable attribute preference on the alternative.

Definition 4. Let Cde (fc e {1,2,..., rnj) 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 A j, which satisfies

Cf > max {S;j} ;

e /+, i = 1,2,..., n, j = 1,2,..., m,

Cf < min {Sy}, e i = 1,2,..., n, j = 1,2,..., m.

Definition 5. Let C^ (fc e {1,2, ...,rn|) 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 Cfc: given by the DM under attribute Ap which satisfies

min {Sj;-} < < max {Si;-}, i = 1,2,..., n, j = 1,2,..., m.

Definition 6. Suppose that the target centre Ca = (Ca1,Ca2,...,Carn) is decided by S;j (i = 1,2,...,n,j = 1,2,...,m) and DM's preference value under attribute Ak (fc e {1,2,...,rnj) is Cf (fc e {1,2,...,rnj). Then the target centre, determined partially by preference values, becomes C0 = (C°, C^,..., C^), the elements of which are as follows:

Caj, j = 1,2,...,m, j = fc Cf, fc e {1,2,...,rnj, j = fc.

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. Figure 1 shows the impact of desirable attribute preference on the alternatives.

In Figure 1, suppose that S^ which belongs to the benefit type attribute set is the index value under attribute Ap and Stj is

the worst value. Let C" and cj be the target centre indices under attribute A j, as determined by feasible alternatives and DM's preferences, respectively. Suppose that d1t d2, and d3 are the distances of index values S1;-, S2j-, and S3j- to C", respectively, and D1, D2, and D3 are the distances of index values S1;-, S2j, and S3j to C^, respectively, while rda is the difference between cj and C". Obviously, the target centre index Cj, 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 S1;-, S2j-, and S3j- to C" to the distances of S1;-,

S2j, and S3j to C^, such that d1, d2, and d3 changed to D1, D2, and D3, respectively. It can be seen from Figure 2 that cj was actually inferior to and S2j-, so there was no meaning attributable to either D1 or D2. From the perspective of a cluster of indices, the distances of and S2j to cj 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 C" and C^.

The impacts of target centre determined by different preferences over the alternatives are discussed as follows: assume that S^ (i = 1,2,..., n, j = 1,2,..., m) is the measure of alternative S, under attribute A; and S, ; and Sn ,1); are any two index values. Let dio and dio+1 be the distances of Sioj and S(io+1);- to C", respectively; then set dio < dio+1 without affecting the conclusions so that under attribute A j the distances of S, ; and Sfi ,1); to cf are D, and D, ,,,

>0j (J0+1)J J >0 J0 + 1

respectively: the difference between cj and C" 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

Figure 2: The impact of selection attribute preference on the alternative.

alternatives' target centre distances under some attribute for different target centres:

Ky = M'o+1___u'o = u'o+1

1 '=1 d' 1 '=1 d' 1 '=1 d'

where d' is the distance of Sj to Ca:; namely, d' = \Ca- - St|:

. __ D0 + 1 -D'o

1=1 D, 1=1 D, = 1h D, '

where Dt is the distance of Sj to Cj, which can be calculated by (11).

The target centre Ca determined by alternatives and the target centre cj determined by DM's preference value have the following relationship:

Cd s^a j - ■

So (7) can be rewritten as

rda + d.

Ti=1 (rda + d'.

rda + d '0

Ti=1 (rda + d') ntda + Ti=1 d'

Compared with (5) and (2), the conclusions maybe drawn as follows.

(1) If rda > 0, which is the DM's desirable preference, then AZd < AZa 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) If rda < 0, which is the DM's selection preference, then AZd > AZa 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 by S¡j (i = \,2,...,n,j = \,2,...,m) was Ca = (Cv Ca2,..., Cam), so the target centre combined with the DM's preferences can be calculated according to the following steps.

(1) The new index measure Ij can be obtained from Sj (i = \,2,...,n,j = 1,2, ...,m) compared with the target centre index Cj (j = l,2,...,m) under

attribute A: (j = 1,2,. ■■, m):

Sj, if

c°, if

Sjj < Cj, Sy

's,j > Cj, S,j € T

(Sjj > Cj, Sjj (s,j < Cj, S,j € T

€J~)

(2) Calculate the distance of index value Itj (i =

l,2,...,n,j = 1,2,. value Cj (j = 1,2,

. ,m) to the target centre index ., m) under attribute Aj (j =

1,2,... ,m), using the Hamming distance:

r'j = \Cj - I

i = 1,2,

,n, j = 1,2,... ,m. (11)

(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: r ■

i = 1,... ,n; j = 1,... ,m. (12)

(4) Having obtained the weight Wj under attribute A j (j = 1,2,..., m), the integrated target centre distances for all alternatives can then be calculated using (13):

wi = UjZij,

i = 1,... ,n, j = 1,...

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 theoriginal targetcentrefor 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), mortality per 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.

4.2. Decision MakingProcess

(1) Calculate the target centre determined by alternatives.

The original target centre Ca = (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 centre C° = (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 centre C°.

Use (10) and the original index matrix can be converted to a new index matrix based on target centre C : 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.

Given w = (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 > ^10 > ^7.

Given w = (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 > Sq > S3 > S7 > Sg.

4.3. Discussion. The results, considering the attribute preferences of A2, 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.

As seen in Table 5, 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.

Table 1: Safety data from coal mines.

Si A1 A 2 A 3 A 4 A 5 A 6 A 7 A 8

Si 21 6 220 12 92 1.8 0.18 381

S2 16 3.7 200 6 90 1.4 0.712 564

S3 26 9.2 180 10 88 2.7 1.34 1051.6

S4 10 4 260 8 94 1.2 0 442.5

S5 30 8.2 350 10 96 3.6 0.641 788

S6 19 5 130 12 100 2.4 0 300

s7 17 9.6 400 6 86 1.3 1.23 964.7

S8 40 14 600 6 95 2.1 1.12 885.6

s9 12 12.8 120 10 91 1.5 0.872 839.3

S10 14 5.8 155 12 89 1.7 0.426 617.2

Table 2: Index values processed based on final target centre.

Si A1 A 2 A 3 A 4 A 5 A 6 A 7 A 8

Si 21 6 220 12 92 1.8 0.2 381

S2 16 3.7 200 6 90 1.4 0.712 564

S3 26 9.2 180 10 88 2.0 1.34 1051.6

S4 10 4 260 8 94 1.2 0.2 442.5

S5 30 8.2 350 10 95 2.0 0.641 788

S6 19 5 130 12 95 2.0 0.2 300

S7 17 9.6 400 6 86 1.3 1.23 964.7

S8 40 14 600 6 95 2.0 1.12 885.6

s9 12 12.8 120 10 91 1.5 0.872 839.3

S10 14 5.8 155 12 89 1.7 0.426 617.2

Table 3: All index target centre distances.

ra A1 A 2 A 3 A 4 A 5 A 6 A 7 A 8

ru 11 6 100 0 3 0.2 0 81

6 3.7 80 6 5 0.6 0.512 264

r3j 16 9.2 60 2 7 0 1.14 751.6

f4j 0 4 140 4 1 0.8 0 142.5

^ 20 8.2 230 2 0 0 0.441 488

r6j 9 5 10 0 0 0 0 0

f7j 7 9.6 280 6 9 0.7 1.03 664.7

^ 30 14 480 6 0 0 0.92 585.6

rçj 2 12.8 0 2 4 0.5 0.672 539.3

ri0) 4 5.8 35 0 6 0.3 0.226 317.2

Table 4: Normalised index target centre distances.

A1 A 2 A 3 A 4 A 5 A 6 A 7 A 8

ZU 0.104762 0.076628 0.070671 0 0.085714 0.064516 0 0.021127

0.057143 0.047254 0.056537 0.214286 0.142857 0.193548 0.103623 0.068859

0.152381 0.117497 0.042403 0.071429 0.2 0 0.230723 0.196041

0 0.051086 0.09894 0.142857 0.028571 0.258065 0 0.037168

0.190476 0.104725 0.162544 0.071429 0 0 0.089253 0.127286

0.085714 0.063857 0.007076 0 0 0 0 0

0.066667 0.122605 0.19788 0.214286 0.257143 0.225806 0.20846 0.173374

0.285714 0.178799 0.339233 0.214286 0 0 0.186197 0.152743

Z9j 0.019048 0.163474 0 0.071429 0.114286 0.16129 0.136005 0.140666

^10) 0.038095 0.074074 0.024735 0 0.171429 0.96774 0.04574 0.082736

Table 5: Alternatives ranked either with, or without, consideration of preferences.

Si (no preferences) Ranking (no preferences) (preferences) Ranking (preferences) Ranking changes

Si 0.055198 3 0.043051 2 -1

S2 0.094923 6 0.110387 5 -1

S3 0.147292 8 0.140379 8 0

S4 0.049522 2 0.064991 3 +1

S5 0.091520 5 0.082207 4 -1

S6 0.020273 1 0.015661 1 0

s7 0.166158 9 0.189271 10 +1

S8 0.173715 10 0.146908 7 -3

s9 0.119444 7 0.124351 6 -1

Sio 0.076746 4 0.186678 9 +5

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author thanks the Key Research Project of Science and Technology of Henan Province for its support (Grant no. 13B620033), the Natural Science Foundation of the Education Department of Henan Province (Grant no. 2011B620001), and Henan Coal Mine Safety Production Technology Development Project (Grant no. H09-50). The author is also grateful to the editors and the anonymous reviewers for their comments and suggestions for improving the quality of this paper.

References

[1] J. L. Deng, Grey System Theory, Huazhong University of Science and Technology Press, Wuhan, China, 2002.

[2] Y.-M. Chen and H. Xie, "Test of the inconsistency problem on Deng's grey transformation by simulation," Systems Engineering and Electronics, vol. 29, no. 8, pp. 1285-1287, 2007.

[3] Y. G. Dang, G. F. Liu, J. P. Wang, and B. Liu, "Multi-attribute decision model of grey target considering weights," Statistics and Decision, no. 3, pp. 29-30, 2004.

[4] J. Song, Y.-G. Dang, and Z.-X. Wang, "Multi-attribute decision model of grey target based on majorant operator of "rewarding good and punishing bad"," Systems Engineering and Electronics, vol. 32, no. 6, pp. 1229-1232, 2010.

[5] Z.-X. Wang, Y.-G. Dang, and H. Yang, "Improvements on decision method of grey target," Systems Engineering and Electronics, vol. 31, no. 11, pp. 2634-2636, 2009.

[6] J. Zhu and K. W. Hipel, "Multiple stages grey target decision making method with incomplete weight based on multi-granularity linguistic label," Information Sciences, no. 212, pp. 15-32, 2012.

[7] S.-F. Liu, W.-F. Yuan, and K.-Q. Sheng, "Multi-attribute intelligent grey target decision model," Control and Decision, vol. 25, no. 8, pp. 1159-1163, 2010.

[8] Y. G. Dang, S. F. Liu, and B. Liu, "Study on the multi-attribute decision model of grey target based on interval number," Engineering Science, vol. 7, no. 8, pp. 31-35, 2005.

[9] C. G. Shen, Y. G. Dang, and L. L. Pei, "Hybrid multi-attribute decision model of grey target," Statistics and Decision, no. 12, pp. 17-20, 2010.

[10] D. Luo, "Multi-objective grey target decision model based on positive and negative clouts," Control and Decision, vol. 28, no. 2, pp. 241-246, 2013.

[11] D. Luo and X. Wang, "The multi-attribute grey target decision method for attribute value within three-parameter interval grey number," Applied Mathematical Modelling, vol. 36, no. 5, pp. 1957-1963, 2012.

[12] J. Song, Y.-G. Dang, Z.-X. Wang, and K. Zhang, "New decision model of grey target with both the positive clout and the negative clout," System Engineering—Theory & Practice, vol. 30, no. 10, pp. 1822-1827, 2010.

[13] Y. Liu, F. Jeffrey, S. F. Liu, and J. S. Liu, "Multi-objective grey target decision-making based on prospect theory," Control and Decision, vol. 28, no. 3, pp. 345-350, 2013.

[14] B. Zeng, S. F. Liu, C. Li, and J. M. Chen, "Grey target decisionmaking model of interval grey number based on cobweb area," Systems Engineering and Electronics, vol. 35, no. 11, pp. 23292334, 2013.

[15] J. J. Zhu, L. L. Zhang, Y. H. Liang, and P. Li, "Greytarget decision method based on uncertain evidence aggregation under conflict interest participants," Control and Decision, vol. 27, no. 7, pp. 1037-1046, 2012.

[16] H. H. Wang, J. Zhu, and F. Z. Geng, "Grey target cluster decision method on linguistic evaluation case-based," Systems Engineering—Theory & Practice, vol. 33, no. 12, pp. 3172-3181, 2013.

[17] S. G. Cao, A. M. Xu, Y. B. Liu, and L. Q. Zhang, "Comprehensive assessment of security in coal mines based on grey relevance analysis," Journal of Mining & Safety Engineering, vol. 24, no. 2, pp. 141-145, 2007.

Copyright of Journal of Applied Mathematics is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.