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Research ArticleFuzzy Similar Priority Method for Mixed Attributes
Jinshan Ma12 Changsheng Ji2 and Jing Sun3
1 School of Energy Science and Engineering Henan Polytechnic University Jiaozuo Henan 454000 China2 School of Mines China University of Mining and Technology Xuzhou Jiangsu 221116 China3Department of Financial Management Jiaozuo Branch of Bank of China Ltd Jiaozuo Henan 454000 China
Correspondence should be addressed to Jinshan Ma mjscumttf163com
Received 13 June 2014 Accepted 22 August 2014 Published 3 September 2014
Academic Editor Olivier Bahn
Copyright copy 2014 Jinshan Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Fuzzy similar priority ratio method is to select the most suitable one to the specific object from feasible alternatives However thismethod considering only the index values of real number has its disadvantages of inaccuracy in result and complexity in calculationSo this method was extended to handle mixed attributes including real number interval number triangular fuzzy number andtrapezoidal fuzzy number The proposed method decides the optimal alternative by the minimum of integrated nearness degreescalculated by all index vectors and the fixed index vectors based on the theory of similarity The improved method can not onlyaddress mixed attributes but also simplify the calculation and improve the accuracy of result A case study illustrated this method
1 Introduction
Knowing some index values of the specific object themost suitable one can be determined from some feasiblealternatives using the method of fuzzy similar priority ratiowhich is of fuzzy mathematic method [1] Being of simpleprinciple and clear meaning this method has been widelyused in many fields However the classical method onlyconcerns the indices of real numbers Furthermore it hasits disadvantages of inaccuracy in result and complexity incalculation In decision making the indices of an alternativemay be characterized as interval number triangular fuzzynumber and trapezoid fuzzy number But the traditionalmethod fails to deal with the mixed types of numbers Article[2] presented a novel method in grey target decision makingfor handling mixed attributes The method in [2] introducedthe binary connection number provided in set pair analysis(SPA) [3 4] to transform all types of indices into connectionnumber vectors regarded as in the determinacy-uncertaintyspace [5 6] Then all the nearness degrees of index vectorsand the given index vectors could be calculated based on thetheory of similarity And the decision making was decidedby the nearness degrees The novel method addressed somealternatives for mixed attribute values but its accuracy insolving the uncertain terms of index vectors needs improving
Thus an improved fuzzy similarity priority method is pro-posed to handle the mixed attributes based on the previousstudy of decision making by the nearness degrees Theimproved fuzzy similar priority method has some featuresFirst it can deal withmixed attributes Second its calculationis simple compared with the traditional method Third itimproves the accuracy of the nearness degrees provided in [2]through optimizing the uncertain terms of the index vectors
The remainder of the organization is as follows Sec-tion 2 presents some basic concepts Section 3 discusses theproposed method A case study is given in Section 4 AndSection 5 concludes this work
2 Basic Concepts
In this section some concepts are presented
Definition 1 Let 119877 be a set of real numbers then 119909 is called afuzzy number if it has one of the following forms [6ndash11]
(1) if 119909 = [119909119871 119909119880] then 119909 is called interval number
where119909119871119909119880 isin 119877119909119871 lt 119909119880119909119871 and119909119880 are the intervalnumberrsquos lower limits and upper limits respectively
(2) if 119909 = [119909119871 119909119872 119909119880] then 119909 is called triangular fuzzy
number where 119909119871 119909119872 119909119880 isin 119877 0 lt 119909119871lt 119909119872
lt
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 304202 7 pageshttpdxdoiorg1011552014304202
2 Journal of Applied Mathematics
119909119880 119909119871 and 119909
119880 are the triangular fuzzy numberrsquoslower limits and upper limits respectively while 119909119872is medium value
(3) if 119909 = [119909119871 119909119872 119909119873 119909119880] then 119909 is called trapezoidal
fuzzy number where 0 lt 119909119871lt 119909119872lt 119909119873lt 119909119880isin 119877
119909119871 and 119909119880 are the trapezoidal fuzzy numberrsquos lower
limits and upper limits respectively while [119909119872 119909119873]is medium interval
Definition 2 Assume 119909 is a fuzzy number with its parameterswhich can be seen as 119899 (119899 ⩾ 2) observed values [2] andthen the fuzzy numberrsquos average value standard deviationand maximum deviation are as follows
119909 =1
119899
119899
sum
119905=1
119909119905 (1)
119904 = radic1
(119899 minus 1)
119899
sum
119905=1
(119909119905minus 119909)2 (2)
119898119904 = max 10038161003816100381610038161003816119909119871minus 119909
1003816100381610038161003816100381610038161003816100381610038161003816119909119880minus 119909
10038161003816100381610038161003816 (3)
where 119909119905(119905 = 1 119899) is the parameter of the fuzzy number
119909 is the average value of the parameters 119904 is the standarddeviation of the parameters119898119904 is the maximum deviation ofthe parameters and 119909119871 and 119909119880 are the fuzzy numberrsquos lowerlimits and upper limits respectively
Definition 3 Let 119877 be a set of real numbers 119860 + 119861119894 is calledbinary connection number where 119860 119861 isin 119877 119894 isin [minus1 1]119860 denotes the deterministic term 119861 denotes the uncertainterm and 119894 is a variable term unifying the determinacy anduncertainty of a set pair So the determinacy-uncertaintyvector based on connection number can be constructed asfollows [2]
119906 (119909 V) = 119860 + 119861119894 = 119909 + V119894 (119894 isin [minus1 1]) (4)
where 119909 and V are the concentrated value and the diversevalue of 119899 parameter values of the same fuzzy number 119909respectively 119909 can be calculated using (1) and V can beobtained by the following equation
V = min 119904 119898119904 (5)
Article [12] provided the idea of using maximum devia-tion as the uncertain term of an index vector only involvingthe upper limits in trapezoidal numberHowever consideringboth the lower limits and upper limits of a fuzzy number ismore reasonable
Remark 4 A real number can be regarded as the special formof a fuzzy number thus the real number can also be convertedinto a binary connection number with the form 119906(119909 V) = 119860+0119894 where 119860 is the real number itself and the uncertain termis 0
Definition 5 The mutual interaction of the average value119909 and the diverse value (standard deviation or maximum
D
U
O
r
v E
x
u = x +
Figure 1 Determinacy-uncertainty space
deviation) V of the connection number 119906(119909 V) can bemappedto the binary determinacy-uncertainty space (D-U space)based on SPA If 119906(119909 119904) = 119909 + V119894 represents the vector of D-U space then the ldquo119894rdquo only denotes the signal of the uncertainterm without representing the changeable value [2]
Figure 1 is a D-U space The 119880 axis and the 119863 axisrepresent the relative uncertain measure and the relativedeterministic measure respectively In Figure 1 the 119909 andthe V interact with each other and the space reflection isthe vector 119874119864 from 119874 to 119864 and the degree of interactionmeans the module of the vector 119874119864 denoted by 119903 [6]Measures between vectors can use distance method [13] suchas Hamming distance and Euclidean distance or similaritymethod [14] such as cosine value and projection value
Definition 6 Let119883 = (1199091 1199092 119909
119899) be a vector then
|119883| = radic
119899
sum
119905=1
1199092
119905(6)
is called the module of119883
Definition 7 Let119883 = (1199091 1199092 119909
119899) and 119884 = (119910
1 1199102 119910
119899)
be two vectors then
Prj119884(119883) =
sum119899
119905=1119909119905119910119905
radicsum119899
119905=11199092
119905sum119899
119905=11199102
119905
radic
119899
sum
119905=1
1199092
119905=sum119899
119905=1119909119905119910119905
radicsum119899
119905=11199102
119905
(7)
is the projection of vector119883 on vector119884 Generally speakingthe bigger the value of Prj
119884(119883) the more the similarity
between119883 and 119884 [15]
Definition 8 Assume 119903 is the module of vector 119884 and 119901119903119895is
the projection value of vector119883 on vector 119884 then
1198891198990=10038161003816100381610038161003816119903 minus 119901119903119895
10038161003816100381610038161003816(8)
is the distance of projection value 119901119903119895and the module 119903 is
called nearness degree When the two index vectors are alltransformed from real numbers the 119889
1198990in (8) is actually
degraded to Hamming distance [2]
Journal of Applied Mathematics 3
Remark 9 Measuring the similarity of the two index vectorswith nearness degree has its advantages over cosine valueand projection value First it can specifically reflect thesimilarity of the two vectors whether they transformed fromreal number or fuzzy number Second it is the generalizeddistance especially for that it is degraded to Hammingdistance when the two index vectors are converted from realnumbersThedetailed discussion can be seen in the paper [2]
3 Fuzzy Similar Priority Method forMixed Attributes
31 Brief Introduction to the Method of FuzzySimilar Priority Ratio
311 Fuzzy Preference Relation
Definition 10 Let119866 = (1198921 1198922 119892
119898) be a set of alternatives
if 119892119894and 119892
119895(119894 119895 = 1 2 119898) are compared with the alter-
native 1198920 then the fuzzy preference relation 119877 is described as
follows119877 = (119903
119894119895)119898times119898
119903119894119895isin [0 1] (119894 119895 = 1 2 119898) 119877 119860 times 119860 rarr
[0 1] and 119903119894119895satisfies the following relations
(1) 119903119894119894= 0 119894 = 1 2 119898 (2) 119903
119894119895+ 119903119895119894= 1 (119894 = 119895 119894 119895 =
1 2 119898)Equation (1) indicates that 119892
119894has indifference with itself
so 119903119894119894= 0 And (2) indicates that if 119892
119894is preferred to 119892
119895
denoted by 119903119894119895 then 119892
119895is preferred to 119892
119894 denoted by 119903
119895119894=
1 minus 119903119894119895 especially and 119903
119894119895= 1 indicates that 119892
119894is absolutely
preferred to 119892119895with respect to 119892
0[16ndash18]
312 Algorithm of Fuzzy Similar Priority Ratio Let 1198830=
(11990901 11990902 119909
0119899) be index vector of the specific object and
let the index value of the 119894th feasible alternative be 119909119894119895(119894 =
1 2 119898 119895 = 1 2 119899) so the algorithm of fuzzy similarpriority ratio is as follows [16 19]
Step 1 With respect to the attribute 119860119896 calculate the Ham-
ming distances of 119909119894119896and 119909
119895119896with 119909
0119896 respectively
119889119894119896=1003816100381610038161003816119909119894119896 minus 1199090119896
1003816100381610038161003816 119889119895119896=10038161003816100381610038161003816119909119894119895119896minus 1199090119896
10038161003816100381610038161003816 (9)
Step 2 Construct the fuzzy preference relation
119903119894119895=
119889119895119896
119889119894119896+ 119889119895119896
(10)
so the fuzzy similar matrix for attribute 119860119896is
119877119896= (
11990311
1199031119898
d
1199031198981
sdot sdot sdot 119903119898119898
) 119896 = 1 2 119899 (11)
Step 3 A series of120582-sectional sets can be determined throughoperating 119877119896 with the 120582 ascending in [0 1] To get the 120582-sectional set the elements of119877119896must be comparedwith the120582The elements of 119877119896 will be rewritten as 1 if they are no less
than the 120582 value Under attribute 119860119896 the most similarity
alternative to the specific object is the one whose row firstgets all ldquo1rdquo except the diagonal elements and the alternativeacquires the order value ldquo1rdquo Then the impact of the alterna-tive must be eliminated by removing the row correspondingto the alternative and the same column in 119877119896 Repeating thesame work a series of similarity alternatives can be obtainedand assigned the order values 2 3 119898 And the order valuethat the alternative gains under attribute 119860
119896is denoted by
119862119894119896(119894 = 1 2 119898 119896 = 1 2 119899)
Step 4 Integrate the order values under different attributesfor every alternative denoted by TC
119894with the following
equation
TC119894=
119899
sum
119896=1
119862119894119896 119894 = 1 2 119898 119896 = 1 2 119899 (12)
Step 5 Rank the alternativeswithTC119894in ascending order and
the optimal alternative to the specific object is the one withthe minimum value TC
119894
32 Fuzzy Similar Priority Method for Mixed Attributes
321 Methodology of the Proposed Method Let 1198830= (11990901
11990902 119909
0119895 119909
0119899) be index vector of the specific object and
let the index value of the 119894th feasible alternative be 119909119894119895(119894 =
1 2 119898 119895 = 1 2 119899) where119909119894119895and1199090119895can be expressed
as fuzzy numbers or real numbers The algorithm of theimproved fuzzy similar priority method can be concluded asfollows [20]
Step 1 Calculate the average values standard deviations andmaximum deviations of all indices (119909
119894119895and 119909
0119895) by using
Equations from (1) to (3)
Step 2 Translate all index values into the binary 119860 + 119861119894 con-nection number vectors using (4) and (5) So the convertedindex vectors can be expressed as
119906119904119905= 119909119904119905+ V119904119905119894 (119894 isin [minus1 1]) 119904 = 0 119898 119905 = 1 119899
(13)
Step 3 Calculate the nearness degrees of all index vectors119906119904119905(119904 = 1 119898 119905 = 1 119899) and the fixed index vectors
119906119904119905(119904 = 0 119905 = 1 119899) denoted by 119902
119894119895(119894 = 1 119898 119895 =
1 119899) using Equations from (6) to (8) and (13)
Step 4 Normalize the nearness degrees of all index vectorsand the fixed index vector by
119911119894119895=
119902119894119895
sum119898
119894=1119902119894119895
119894 = 1 119898 119895 = 1 119899 (14)
where 119911119894119895is the normalized nearness degree
Step 5 Determine the weights of all attributes denoted as120596119895(119895 = 1 119899) by objective method or subjective method
Step 6 Integrate the normalized nearness degrees consider-ing weights of all attributes using the following equation
119908119894= 120596119895119911119894119895 119894 = 1 119898 119895 = 1 119899 (15)
4 Journal of Applied Mathematics
Table 1 Safety data from coal mines
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
[18 20 25] [4 6 8] [180 260] 12 [90 91 93 94] 181198782
[12 16 20] [28 4 43] [190 210] 6 [88 89 91 92] 141198783
[22 25 31] [88 92 96] [160 200] 10 [84 87 89 92] 271198784
[6 11 13] [36 4 44] [240 280] 8 [91 93 95 97] 121198785
[24 32 34] [73 83 9] [340 360] 10 [94 95 97 98] 361198786
[14 16 18] [94 96 98] [360 440] 6 [82 84 88 90] 131198787
[33 42 45] [13 14 15] [550 650] 6 [93 94 96 97] 211198788
[9 12 15] [12 13 134] [100 140] 10 [89 90 91 94] 151198780
[8 12 22] [52 6 62] [150 160] 12 [87 88 90 91] 17
Step 7 Rank the feasible alternatives by the integrated nor-malized nearness degrees 119908
119894(119894 = 1 119898)
322 Merits of the Proposed Method The improved fuzzysimilar method has its advantages over the traditionalmethod Above all the proposed method can address themixed numbers based on the similarity of the improveddeterminacy-uncertainty index vectors through optimizingthe uncertain terms of them Moreover it can simplify thecalculation by avoiding the traditional tedious computingprocedure such that Steps 2 and 3 in the traditional methodare reduced or avoided Nevertheless it determines theoptimal alternative by the minimum of integrated nearnessdegrees instead of the minimum order values and canimprove the accuracy of results Finally it can also reflectthe decision makersrsquo preferences by considering attributeweights
4 Cases Study
41 Background A coal enterprise is to select suitable safetymanagement model from some coal enterprise with differentadvanced safety management level The safety managementmodel of a coal enterprise depends on some indices includingseam dip (∘) methane emission rate (m3t) water inflow(m3h) spontaneous combustion period (month) ventilatingstructures qualification rate () and equivalent orifice (m2)[21 22] denoted by 119860
1to 1198606respectively Thus some coal
enterprises with different index values can be regarded asdifferent alternatives Here 119878
0represents the specific coal
enterprise wants to learn advanced safety management expe-rience And the eight feasible alternatives (coal enterprises)are denoted as 119878
1to 1198788respectively Besides the index values
are given as mixed attribute values including real numbersinterval numbers triangle fuzzy numbers and trapezoid fuzzynumbers The aim of the decision makers is to seek for themost suitable enterprise whose safety management experiececan be learnt So the proposed method can address thisproblem easily The data is shown in Table 1
42 Process of Decision Making
421 Calculate the Parameters of Index Vectors All averagevalues standard deviations and maximum deviations of all
indices can be calculated using Equations from (1) to (3) andthe results are shown in Table 2
422 Translate All Index Values into Connection Number Vec-tors Having obtained the deterministic terms and uncertainterms of the connection numbers above all index values canbe transformed into index vectors using (4) Table 3 shows theconnection numbers converted from all indices
423 Calculate Nearness Degrees of All Index Vectors with theFixed Index Vectors All nearness degrees of all index vectorswith the fixed index vectors can be calculated as shown inTable 4 using Equations from (6) to (8)
424 Normalize the Nearness Degrees The normalized near-ness degrees shown in Table 5 can be calculated using (14)
425 Integrate the Normalized Nearness Degrees With-out considering all attribute weights the integrated near-ness degrees of all alternatives can be calculated as 119882 =(0272792 0427914 0610936 0666333 1191736 0754849and 1557908 0517532) So the alternatives ranking is 119878
1≻
1198782≻ 1198788≻ 1198783≻ 1198784≻ 1198786≻ 1198785≻ 1198787 In brief 119878
1is the best
selection for 1198780
If the attributes 120596 = (011 026 007 012 021 023) aregiven then the weighted integrated nearness degrees of allalternatives can be calculated as 119882 = (0042693 00694550112996 0110897 0210185 0116780 0232191 and 0104803)So the alternatives ranking is 119878
1≻ 1198782≻ 1198788≻ 1198784≻ 1198783≻ 1198786≻
1198785≻ 1198787
43 Discussion A comparison between the order valuemethod and the nearness degree method is performed toillustrate the advantage of the proposedmethod to determinethe optimal alternative Table 6 shows the order values ofall indices near the fixed indices determined by nearnessdegrees The order values of all alternatives under someattribute are given by their index nearness degrees in termsof the principle of the smaller the better
So the total integrated order values can be calculated asW= (16 16 21 26 34 27 39 19) using (12) And the alternativesranking is 119878
1= 1198782≻ 1198788≻ 1198783≻ 1198784≻ 1198786≻ 1198785≻ 1198787
Journal of Applied Mathematics 5
Table 2 Average values standard deviations and maximum deviations of all indices
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
2136055514 622 220565685440 1200 9218257422 18001198782
1644 37079372509 200141421410 600 9018257422 14001198783
2645825765 920404 180282842720 1000 8833665024 27001198784
1036055514 40404 260282842720 800 9425819893 12001198785
3052915036 820854409 350141421410 1000 9618257422 36001198786
1622 960202 400565685440 600 8636514844 13001198787
4062449987 1411 600707106850 600 9518257422 21001198788
1233 12807211106 120282842720 1000 9121602473 15001198780
1472111038 5805291506 15570710685 1200 8918257422 1700Note Each term with the form ldquo119886119887119888rdquo in Table 2 denotes ldquoaverage valuestandard deviationmaximum deviationrdquo for example the term ldquo2136055514rdquorepresents 119886 = 21 119887 = 3605551 and 119888 = 4
Table 3 Index vectors transformed from index values
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
21 + 3605551119894 6 + 2119894 220 + 40119894 12 + 0119894 92 + 1825742119894 18 + 0119894
1198782
16 + 4119894 37 + 0793725119894 200 + 10119894 6 + 0119894 90 + 1825742119894 14 + 0119894
1198783
26 + 4582576119894 92 + 04119894 180 + 20119894 10 + 0119894 88 + 3366502119894 27 + 0119894
1198784
10 + 3605551119894 4 + 04119894 260 + 20119894 8 + 0119894 94 + 2581989119894 12 + 0119894
1198785
30 + 5291503119894 82 + 08544119894 350 + 10119894 10 + 0119894 96 + 182574119894 36 + 0119894
1198786
16 + 2119894 96 + 02119894 400 + 40119894 6 + 0119894 86 + 3651484119894 13 + 0119894
1198787
40 + 6244998119894 14 + 1119894 600 + 50119894 6 + 0119894 95 + 1825742119894 21 + 0119894
1198788
12 + 3119894 128 + 06119894 120 + 20119894 10 + 0119894 91 + 2160247119894 15 + 0119894
1198780
14 + 7211103119894 58 + 052915119894 155 + 5119894 12 + 0119894 89 + 1825742119894 17 + 0119894
Table 4 Nearness degrees of all index vectors with the fixed index vectors
119902119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1199021119895
45720046 0332808 66094653 0 2999369 011199022119895
03076204 2067276 45137812 6 099979 031199023119895
94643938 3374204 25470622 2 0968189 11199024119895
5207005 1804289 10542903 4 5014459 051199025119895
13345019 2419625 19505983 2 6998528 191199026119895
06081906 3754378 24600107 6 2961924 041199027119895
22671639 8208865 44621951 6 5998738 041199028119895
37062887 6977486 34498185 2 200644 02
Table 5 Normalized nearness degrees of all index vectors with the fixed index vectors
119885119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1198851119895
0076350 00115 00567867 0 0107322 00208331198852119895
0005137 0071436 00387812 02143 0035774 006251198853119895
0158050 0116597 00218837 00714 0034643 02083331198854119895
0086954 0062348 00905817 01429 0179425 01041671198855119895
0222855 0083611 016759 00714 0250418 03958331198856119895
0010156 0129735 02113573 02143 0105982 00833331198857119895
0378604 0283662 03833795 02143 0214644 00833331198858119895
0061893 0241111 00296399 00714 0071793 0041667
6 Journal of Applied Mathematics
Table 6 Order values of index vectors with fixed index vectors
119862119894119895
1198601
1198602
1198603
1198604
1198605
1198606
sum119862119894119895
1198621119895
4 1 4 1 5 1 161198622119895
1 3 3 4 2 3 161198623119895
6 5 1 2 1 6 211198624119895
5 2 5 3 6 5 261198625119895
7 4 6 2 8 7 341198626119895
2 6 7 4 4 4 271198627119895
8 8 8 4 7 4 391198628119895
3 7 2 2 3 2 19
Table 7 Alternatives ranking with two types of method
119878119894
Order values Nearness degrees (no weights) Nearness degrees (weights)119862119894
Ranking 119882119894
Ranking 119882119894
Ranking1198781
16 1 0272792 1 0042693 11198782
16 1 0427914 2 0069455 21198783
21 3 0610936 4 0112996 51198784
26 4 0666333 5 0110897 41198785
34 6 1191736 7 0210185 71198786
27 5 0754849 6 0116780 61198787
39 7 1557908 8 0232191 81198788
19 2 0517532 3 0104803 3
Table 7 is the comparison of two types of calculationmethod one by integrated order values and one by integratednearness degrees However decision making by integratednearness degrees includes considering the attribute weightsor not
Seen from Table 7 alternatives ranking by order values isrough with the alternatives 119878
1and 119878
2in the same position
While the results by nearness degrees are accuracy with thealternatives 119878
1and 119878
2in different position Furthermore
the proposed method can also use the attribute weights toreflect the decision makersrsquo preferences There is somewhatdifference between the decisionmaking considering attributeweights or not
5 Conclusions
An improved fuzzy similar priority method was proposedto handle alternatives for mixed attributes The proposedmethod solves the mixed attributes using the nearnessdegrees as derived from the similarity of the index vectorsAnd it improves the uncertain term of the index vectorby selecting the minimum of the standard deviation or theminimum deviation of the parameters of the index and willcontribute to the accuracy in result Furthermore the newmethod can reflect the decision makerrsquos preferences throughconsidering attribute weights Another feature of thismethodis that its procedure and calculation are simple A case studyillustrated that it has advantages over the classical method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the Key Research Project of Scienceand Technology of Henan province (Grant no 13B620033)Natural Science Foundation of the Education Department ofHenan Province (Grant no 2011B620001) and Henan CoalMine Safety Production Technology Development Project(Grant noH09-50) for the support
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] J S Ma and C S Ji ldquoGeneralized grey target decision methodfor mixed attributes based on connection numberrdquo Journal ofAppliedMathematics vol 2014 Article ID 763543 8 pages 2014
[3] K Q Zhao and A L Xuan ldquoSet pair theorymdasha new theorymethod of non-define and its applicationsrdquo Systems Engineer-ing vol 14 no 1 pp 18ndash23 1996
[4] K Q Zhao Set Pair Analysis and Its Prelimiary ApplicationZhejiang Press of Science and Technology Hangzhou China2000
[5] K Q Zhao ldquoThe theoretical basis and basic algorithm of binaryconnection A+Bi and its application in AIrdquo CAA I Transactionson Intelligent Systems vol 3 no 6 pp 476ndash486 2008
Journal of Applied Mathematics 7
[6] K Q Zhao ldquoDecision making algorithm based on set pairanalysis for use when facing multiple uncertain in attributesrdquoCAAI Transactions on Intelligent Systems vol 5 no 1 pp 41ndash502010
[7] T-P Lo and S-J Guo ldquoEffective weighting model based onthe maximum deviation with uncertain informationrdquo ExpertSystems with Applications vol 37 no 12 pp 8445ndash8449 2010
[8] Z Yue ldquoAn extended TOPSIS for determining weights of deci-sion makers with interval numbersrdquo Knowledge-Based Systemsvol 24 no 1 pp 146ndash153 2011
[9] L Liu J-H Chen G-M Wang and D-Z Lao ldquoMulti-attributed decision making for mining methods based ongrey system and interval numbersrdquo Journal of Central SouthUniversity vol 20 no 4 pp 1029ndash1033 2013
[10] A I Ban and L Coroianu ldquoNearest interval triangular andtrapezoidal approximation of a fuzzy number preserving ambi-guityrdquo International Journal of Approximate Reasoning vol 53no 5 pp 805ndash836 2012
[11] D Luo and X Wang ldquoThe multi-attribute grey target decisionmethod for attribute value within three-parameter interval greynumberrdquo Applied Mathematical Modelling vol 36 no 5 pp1957ndash1963 2012
[12] W XWu ldquoUtilization of the interval number in multi-attributedecision-making of trapezoidal fuzzy numbersrdquoMathematics inPractice andTheory vol 43 no 1 pp 160ndash166 2013
[13] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011
[14] J Ye ldquoCosine similarity measures for intuitionistic fuzzy setsand their applicationsrdquoMathematical and Computer Modellingvol 53 no 1-2 pp 91ndash97 2011
[15] G W Wei Theory and Methods of Multiple Attribute DecisionMaking Based on Fuzzy Information China Economic Publish-ing House Beijing China 2010
[16] H Q He K B Wen Z X Chen and W J Ye ldquoCase-basedreasoning and fuzzy analogy preferred ratio for effective depthof collapsible loess treated with dynamic consolidationrdquo Journalof Engineering Geology vol 17 no 11 pp 88ndash93 2009
[17] Y L Jiang ldquoAn approach to group decision making based oninterval fuzzy preference relationsrdquo Journal of Systems Scienceand Systems Engineering vol 16 no 1 pp 113ndash120 2007
[18] Y XuQDa and L Liu ldquoNormalizing rank aggregationmethodfor priority of a fuzzy preference relation and its effectivenessrdquoInternational Journal of Approximate Reasoning vol 50 no 8pp 1287ndash1297 2009
[19] S H Wang ldquoForecasting the bid of Engineering projects withfuzzy similar-priority comparisonrdquo Journal of Southwest JiaoTong University vol 36 no 1 pp 96ndash99 2001
[20] J Y Zhou S D Song and Z F Yuan ldquoMethod of syntheticdistance priority in fuzzy decesionrdquo Acta Universitatis Agri-culturalis Boreali-Occidentalis Acta Universitatis AgriculturalisBoreali-Occidentalis vol 24 no 2 pp 71ndash74 1996
[21] J SMa ldquoGrey target decisionmethod for a variable target centrebased on the decision makerrsquos preferencesrdquo Journal of AppliedMathematics vol 2014 Article ID 572529 6 pages 2014
[22] S G Cao AM Xu Y B Liu and L Q Zhang ldquoComprehensiveassessment of security in coal mines based on grey relevanceanalysisrdquo Journal of Mining amp Safety Engineering vol 24 no 2pp 141ndash145 2007
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Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
119909119880 119909119871 and 119909
119880 are the triangular fuzzy numberrsquoslower limits and upper limits respectively while 119909119872is medium value
(3) if 119909 = [119909119871 119909119872 119909119873 119909119880] then 119909 is called trapezoidal
fuzzy number where 0 lt 119909119871lt 119909119872lt 119909119873lt 119909119880isin 119877
119909119871 and 119909119880 are the trapezoidal fuzzy numberrsquos lower
limits and upper limits respectively while [119909119872 119909119873]is medium interval
Definition 2 Assume 119909 is a fuzzy number with its parameterswhich can be seen as 119899 (119899 ⩾ 2) observed values [2] andthen the fuzzy numberrsquos average value standard deviationand maximum deviation are as follows
119909 =1
119899
119899
sum
119905=1
119909119905 (1)
119904 = radic1
(119899 minus 1)
119899
sum
119905=1
(119909119905minus 119909)2 (2)
119898119904 = max 10038161003816100381610038161003816119909119871minus 119909
1003816100381610038161003816100381610038161003816100381610038161003816119909119880minus 119909
10038161003816100381610038161003816 (3)
where 119909119905(119905 = 1 119899) is the parameter of the fuzzy number
119909 is the average value of the parameters 119904 is the standarddeviation of the parameters119898119904 is the maximum deviation ofthe parameters and 119909119871 and 119909119880 are the fuzzy numberrsquos lowerlimits and upper limits respectively
Definition 3 Let 119877 be a set of real numbers 119860 + 119861119894 is calledbinary connection number where 119860 119861 isin 119877 119894 isin [minus1 1]119860 denotes the deterministic term 119861 denotes the uncertainterm and 119894 is a variable term unifying the determinacy anduncertainty of a set pair So the determinacy-uncertaintyvector based on connection number can be constructed asfollows [2]
119906 (119909 V) = 119860 + 119861119894 = 119909 + V119894 (119894 isin [minus1 1]) (4)
where 119909 and V are the concentrated value and the diversevalue of 119899 parameter values of the same fuzzy number 119909respectively 119909 can be calculated using (1) and V can beobtained by the following equation
V = min 119904 119898119904 (5)
Article [12] provided the idea of using maximum devia-tion as the uncertain term of an index vector only involvingthe upper limits in trapezoidal numberHowever consideringboth the lower limits and upper limits of a fuzzy number ismore reasonable
Remark 4 A real number can be regarded as the special formof a fuzzy number thus the real number can also be convertedinto a binary connection number with the form 119906(119909 V) = 119860+0119894 where 119860 is the real number itself and the uncertain termis 0
Definition 5 The mutual interaction of the average value119909 and the diverse value (standard deviation or maximum
D
U
O
r
v E
x
u = x +
Figure 1 Determinacy-uncertainty space
deviation) V of the connection number 119906(119909 V) can bemappedto the binary determinacy-uncertainty space (D-U space)based on SPA If 119906(119909 119904) = 119909 + V119894 represents the vector of D-U space then the ldquo119894rdquo only denotes the signal of the uncertainterm without representing the changeable value [2]
Figure 1 is a D-U space The 119880 axis and the 119863 axisrepresent the relative uncertain measure and the relativedeterministic measure respectively In Figure 1 the 119909 andthe V interact with each other and the space reflection isthe vector 119874119864 from 119874 to 119864 and the degree of interactionmeans the module of the vector 119874119864 denoted by 119903 [6]Measures between vectors can use distance method [13] suchas Hamming distance and Euclidean distance or similaritymethod [14] such as cosine value and projection value
Definition 6 Let119883 = (1199091 1199092 119909
119899) be a vector then
|119883| = radic
119899
sum
119905=1
1199092
119905(6)
is called the module of119883
Definition 7 Let119883 = (1199091 1199092 119909
119899) and 119884 = (119910
1 1199102 119910
119899)
be two vectors then
Prj119884(119883) =
sum119899
119905=1119909119905119910119905
radicsum119899
119905=11199092
119905sum119899
119905=11199102
119905
radic
119899
sum
119905=1
1199092
119905=sum119899
119905=1119909119905119910119905
radicsum119899
119905=11199102
119905
(7)
is the projection of vector119883 on vector119884 Generally speakingthe bigger the value of Prj
119884(119883) the more the similarity
between119883 and 119884 [15]
Definition 8 Assume 119903 is the module of vector 119884 and 119901119903119895is
the projection value of vector119883 on vector 119884 then
1198891198990=10038161003816100381610038161003816119903 minus 119901119903119895
10038161003816100381610038161003816(8)
is the distance of projection value 119901119903119895and the module 119903 is
called nearness degree When the two index vectors are alltransformed from real numbers the 119889
1198990in (8) is actually
degraded to Hamming distance [2]
Journal of Applied Mathematics 3
Remark 9 Measuring the similarity of the two index vectorswith nearness degree has its advantages over cosine valueand projection value First it can specifically reflect thesimilarity of the two vectors whether they transformed fromreal number or fuzzy number Second it is the generalizeddistance especially for that it is degraded to Hammingdistance when the two index vectors are converted from realnumbersThedetailed discussion can be seen in the paper [2]
3 Fuzzy Similar Priority Method forMixed Attributes
31 Brief Introduction to the Method of FuzzySimilar Priority Ratio
311 Fuzzy Preference Relation
Definition 10 Let119866 = (1198921 1198922 119892
119898) be a set of alternatives
if 119892119894and 119892
119895(119894 119895 = 1 2 119898) are compared with the alter-
native 1198920 then the fuzzy preference relation 119877 is described as
follows119877 = (119903
119894119895)119898times119898
119903119894119895isin [0 1] (119894 119895 = 1 2 119898) 119877 119860 times 119860 rarr
[0 1] and 119903119894119895satisfies the following relations
(1) 119903119894119894= 0 119894 = 1 2 119898 (2) 119903
119894119895+ 119903119895119894= 1 (119894 = 119895 119894 119895 =
1 2 119898)Equation (1) indicates that 119892
119894has indifference with itself
so 119903119894119894= 0 And (2) indicates that if 119892
119894is preferred to 119892
119895
denoted by 119903119894119895 then 119892
119895is preferred to 119892
119894 denoted by 119903
119895119894=
1 minus 119903119894119895 especially and 119903
119894119895= 1 indicates that 119892
119894is absolutely
preferred to 119892119895with respect to 119892
0[16ndash18]
312 Algorithm of Fuzzy Similar Priority Ratio Let 1198830=
(11990901 11990902 119909
0119899) be index vector of the specific object and
let the index value of the 119894th feasible alternative be 119909119894119895(119894 =
1 2 119898 119895 = 1 2 119899) so the algorithm of fuzzy similarpriority ratio is as follows [16 19]
Step 1 With respect to the attribute 119860119896 calculate the Ham-
ming distances of 119909119894119896and 119909
119895119896with 119909
0119896 respectively
119889119894119896=1003816100381610038161003816119909119894119896 minus 1199090119896
1003816100381610038161003816 119889119895119896=10038161003816100381610038161003816119909119894119895119896minus 1199090119896
10038161003816100381610038161003816 (9)
Step 2 Construct the fuzzy preference relation
119903119894119895=
119889119895119896
119889119894119896+ 119889119895119896
(10)
so the fuzzy similar matrix for attribute 119860119896is
119877119896= (
11990311
1199031119898
d
1199031198981
sdot sdot sdot 119903119898119898
) 119896 = 1 2 119899 (11)
Step 3 A series of120582-sectional sets can be determined throughoperating 119877119896 with the 120582 ascending in [0 1] To get the 120582-sectional set the elements of119877119896must be comparedwith the120582The elements of 119877119896 will be rewritten as 1 if they are no less
than the 120582 value Under attribute 119860119896 the most similarity
alternative to the specific object is the one whose row firstgets all ldquo1rdquo except the diagonal elements and the alternativeacquires the order value ldquo1rdquo Then the impact of the alterna-tive must be eliminated by removing the row correspondingto the alternative and the same column in 119877119896 Repeating thesame work a series of similarity alternatives can be obtainedand assigned the order values 2 3 119898 And the order valuethat the alternative gains under attribute 119860
119896is denoted by
119862119894119896(119894 = 1 2 119898 119896 = 1 2 119899)
Step 4 Integrate the order values under different attributesfor every alternative denoted by TC
119894with the following
equation
TC119894=
119899
sum
119896=1
119862119894119896 119894 = 1 2 119898 119896 = 1 2 119899 (12)
Step 5 Rank the alternativeswithTC119894in ascending order and
the optimal alternative to the specific object is the one withthe minimum value TC
119894
32 Fuzzy Similar Priority Method for Mixed Attributes
321 Methodology of the Proposed Method Let 1198830= (11990901
11990902 119909
0119895 119909
0119899) be index vector of the specific object and
let the index value of the 119894th feasible alternative be 119909119894119895(119894 =
1 2 119898 119895 = 1 2 119899) where119909119894119895and1199090119895can be expressed
as fuzzy numbers or real numbers The algorithm of theimproved fuzzy similar priority method can be concluded asfollows [20]
Step 1 Calculate the average values standard deviations andmaximum deviations of all indices (119909
119894119895and 119909
0119895) by using
Equations from (1) to (3)
Step 2 Translate all index values into the binary 119860 + 119861119894 con-nection number vectors using (4) and (5) So the convertedindex vectors can be expressed as
119906119904119905= 119909119904119905+ V119904119905119894 (119894 isin [minus1 1]) 119904 = 0 119898 119905 = 1 119899
(13)
Step 3 Calculate the nearness degrees of all index vectors119906119904119905(119904 = 1 119898 119905 = 1 119899) and the fixed index vectors
119906119904119905(119904 = 0 119905 = 1 119899) denoted by 119902
119894119895(119894 = 1 119898 119895 =
1 119899) using Equations from (6) to (8) and (13)
Step 4 Normalize the nearness degrees of all index vectorsand the fixed index vector by
119911119894119895=
119902119894119895
sum119898
119894=1119902119894119895
119894 = 1 119898 119895 = 1 119899 (14)
where 119911119894119895is the normalized nearness degree
Step 5 Determine the weights of all attributes denoted as120596119895(119895 = 1 119899) by objective method or subjective method
Step 6 Integrate the normalized nearness degrees consider-ing weights of all attributes using the following equation
119908119894= 120596119895119911119894119895 119894 = 1 119898 119895 = 1 119899 (15)
4 Journal of Applied Mathematics
Table 1 Safety data from coal mines
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
[18 20 25] [4 6 8] [180 260] 12 [90 91 93 94] 181198782
[12 16 20] [28 4 43] [190 210] 6 [88 89 91 92] 141198783
[22 25 31] [88 92 96] [160 200] 10 [84 87 89 92] 271198784
[6 11 13] [36 4 44] [240 280] 8 [91 93 95 97] 121198785
[24 32 34] [73 83 9] [340 360] 10 [94 95 97 98] 361198786
[14 16 18] [94 96 98] [360 440] 6 [82 84 88 90] 131198787
[33 42 45] [13 14 15] [550 650] 6 [93 94 96 97] 211198788
[9 12 15] [12 13 134] [100 140] 10 [89 90 91 94] 151198780
[8 12 22] [52 6 62] [150 160] 12 [87 88 90 91] 17
Step 7 Rank the feasible alternatives by the integrated nor-malized nearness degrees 119908
119894(119894 = 1 119898)
322 Merits of the Proposed Method The improved fuzzysimilar method has its advantages over the traditionalmethod Above all the proposed method can address themixed numbers based on the similarity of the improveddeterminacy-uncertainty index vectors through optimizingthe uncertain terms of them Moreover it can simplify thecalculation by avoiding the traditional tedious computingprocedure such that Steps 2 and 3 in the traditional methodare reduced or avoided Nevertheless it determines theoptimal alternative by the minimum of integrated nearnessdegrees instead of the minimum order values and canimprove the accuracy of results Finally it can also reflectthe decision makersrsquo preferences by considering attributeweights
4 Cases Study
41 Background A coal enterprise is to select suitable safetymanagement model from some coal enterprise with differentadvanced safety management level The safety managementmodel of a coal enterprise depends on some indices includingseam dip (∘) methane emission rate (m3t) water inflow(m3h) spontaneous combustion period (month) ventilatingstructures qualification rate () and equivalent orifice (m2)[21 22] denoted by 119860
1to 1198606respectively Thus some coal
enterprises with different index values can be regarded asdifferent alternatives Here 119878
0represents the specific coal
enterprise wants to learn advanced safety management expe-rience And the eight feasible alternatives (coal enterprises)are denoted as 119878
1to 1198788respectively Besides the index values
are given as mixed attribute values including real numbersinterval numbers triangle fuzzy numbers and trapezoid fuzzynumbers The aim of the decision makers is to seek for themost suitable enterprise whose safety management experiececan be learnt So the proposed method can address thisproblem easily The data is shown in Table 1
42 Process of Decision Making
421 Calculate the Parameters of Index Vectors All averagevalues standard deviations and maximum deviations of all
indices can be calculated using Equations from (1) to (3) andthe results are shown in Table 2
422 Translate All Index Values into Connection Number Vec-tors Having obtained the deterministic terms and uncertainterms of the connection numbers above all index values canbe transformed into index vectors using (4) Table 3 shows theconnection numbers converted from all indices
423 Calculate Nearness Degrees of All Index Vectors with theFixed Index Vectors All nearness degrees of all index vectorswith the fixed index vectors can be calculated as shown inTable 4 using Equations from (6) to (8)
424 Normalize the Nearness Degrees The normalized near-ness degrees shown in Table 5 can be calculated using (14)
425 Integrate the Normalized Nearness Degrees With-out considering all attribute weights the integrated near-ness degrees of all alternatives can be calculated as 119882 =(0272792 0427914 0610936 0666333 1191736 0754849and 1557908 0517532) So the alternatives ranking is 119878
1≻
1198782≻ 1198788≻ 1198783≻ 1198784≻ 1198786≻ 1198785≻ 1198787 In brief 119878
1is the best
selection for 1198780
If the attributes 120596 = (011 026 007 012 021 023) aregiven then the weighted integrated nearness degrees of allalternatives can be calculated as 119882 = (0042693 00694550112996 0110897 0210185 0116780 0232191 and 0104803)So the alternatives ranking is 119878
1≻ 1198782≻ 1198788≻ 1198784≻ 1198783≻ 1198786≻
1198785≻ 1198787
43 Discussion A comparison between the order valuemethod and the nearness degree method is performed toillustrate the advantage of the proposedmethod to determinethe optimal alternative Table 6 shows the order values ofall indices near the fixed indices determined by nearnessdegrees The order values of all alternatives under someattribute are given by their index nearness degrees in termsof the principle of the smaller the better
So the total integrated order values can be calculated asW= (16 16 21 26 34 27 39 19) using (12) And the alternativesranking is 119878
1= 1198782≻ 1198788≻ 1198783≻ 1198784≻ 1198786≻ 1198785≻ 1198787
Journal of Applied Mathematics 5
Table 2 Average values standard deviations and maximum deviations of all indices
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
2136055514 622 220565685440 1200 9218257422 18001198782
1644 37079372509 200141421410 600 9018257422 14001198783
2645825765 920404 180282842720 1000 8833665024 27001198784
1036055514 40404 260282842720 800 9425819893 12001198785
3052915036 820854409 350141421410 1000 9618257422 36001198786
1622 960202 400565685440 600 8636514844 13001198787
4062449987 1411 600707106850 600 9518257422 21001198788
1233 12807211106 120282842720 1000 9121602473 15001198780
1472111038 5805291506 15570710685 1200 8918257422 1700Note Each term with the form ldquo119886119887119888rdquo in Table 2 denotes ldquoaverage valuestandard deviationmaximum deviationrdquo for example the term ldquo2136055514rdquorepresents 119886 = 21 119887 = 3605551 and 119888 = 4
Table 3 Index vectors transformed from index values
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
21 + 3605551119894 6 + 2119894 220 + 40119894 12 + 0119894 92 + 1825742119894 18 + 0119894
1198782
16 + 4119894 37 + 0793725119894 200 + 10119894 6 + 0119894 90 + 1825742119894 14 + 0119894
1198783
26 + 4582576119894 92 + 04119894 180 + 20119894 10 + 0119894 88 + 3366502119894 27 + 0119894
1198784
10 + 3605551119894 4 + 04119894 260 + 20119894 8 + 0119894 94 + 2581989119894 12 + 0119894
1198785
30 + 5291503119894 82 + 08544119894 350 + 10119894 10 + 0119894 96 + 182574119894 36 + 0119894
1198786
16 + 2119894 96 + 02119894 400 + 40119894 6 + 0119894 86 + 3651484119894 13 + 0119894
1198787
40 + 6244998119894 14 + 1119894 600 + 50119894 6 + 0119894 95 + 1825742119894 21 + 0119894
1198788
12 + 3119894 128 + 06119894 120 + 20119894 10 + 0119894 91 + 2160247119894 15 + 0119894
1198780
14 + 7211103119894 58 + 052915119894 155 + 5119894 12 + 0119894 89 + 1825742119894 17 + 0119894
Table 4 Nearness degrees of all index vectors with the fixed index vectors
119902119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1199021119895
45720046 0332808 66094653 0 2999369 011199022119895
03076204 2067276 45137812 6 099979 031199023119895
94643938 3374204 25470622 2 0968189 11199024119895
5207005 1804289 10542903 4 5014459 051199025119895
13345019 2419625 19505983 2 6998528 191199026119895
06081906 3754378 24600107 6 2961924 041199027119895
22671639 8208865 44621951 6 5998738 041199028119895
37062887 6977486 34498185 2 200644 02
Table 5 Normalized nearness degrees of all index vectors with the fixed index vectors
119885119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1198851119895
0076350 00115 00567867 0 0107322 00208331198852119895
0005137 0071436 00387812 02143 0035774 006251198853119895
0158050 0116597 00218837 00714 0034643 02083331198854119895
0086954 0062348 00905817 01429 0179425 01041671198855119895
0222855 0083611 016759 00714 0250418 03958331198856119895
0010156 0129735 02113573 02143 0105982 00833331198857119895
0378604 0283662 03833795 02143 0214644 00833331198858119895
0061893 0241111 00296399 00714 0071793 0041667
6 Journal of Applied Mathematics
Table 6 Order values of index vectors with fixed index vectors
119862119894119895
1198601
1198602
1198603
1198604
1198605
1198606
sum119862119894119895
1198621119895
4 1 4 1 5 1 161198622119895
1 3 3 4 2 3 161198623119895
6 5 1 2 1 6 211198624119895
5 2 5 3 6 5 261198625119895
7 4 6 2 8 7 341198626119895
2 6 7 4 4 4 271198627119895
8 8 8 4 7 4 391198628119895
3 7 2 2 3 2 19
Table 7 Alternatives ranking with two types of method
119878119894
Order values Nearness degrees (no weights) Nearness degrees (weights)119862119894
Ranking 119882119894
Ranking 119882119894
Ranking1198781
16 1 0272792 1 0042693 11198782
16 1 0427914 2 0069455 21198783
21 3 0610936 4 0112996 51198784
26 4 0666333 5 0110897 41198785
34 6 1191736 7 0210185 71198786
27 5 0754849 6 0116780 61198787
39 7 1557908 8 0232191 81198788
19 2 0517532 3 0104803 3
Table 7 is the comparison of two types of calculationmethod one by integrated order values and one by integratednearness degrees However decision making by integratednearness degrees includes considering the attribute weightsor not
Seen from Table 7 alternatives ranking by order values isrough with the alternatives 119878
1and 119878
2in the same position
While the results by nearness degrees are accuracy with thealternatives 119878
1and 119878
2in different position Furthermore
the proposed method can also use the attribute weights toreflect the decision makersrsquo preferences There is somewhatdifference between the decisionmaking considering attributeweights or not
5 Conclusions
An improved fuzzy similar priority method was proposedto handle alternatives for mixed attributes The proposedmethod solves the mixed attributes using the nearnessdegrees as derived from the similarity of the index vectorsAnd it improves the uncertain term of the index vectorby selecting the minimum of the standard deviation or theminimum deviation of the parameters of the index and willcontribute to the accuracy in result Furthermore the newmethod can reflect the decision makerrsquos preferences throughconsidering attribute weights Another feature of thismethodis that its procedure and calculation are simple A case studyillustrated that it has advantages over the classical method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the Key Research Project of Scienceand Technology of Henan province (Grant no 13B620033)Natural Science Foundation of the Education Department ofHenan Province (Grant no 2011B620001) and Henan CoalMine Safety Production Technology Development Project(Grant noH09-50) for the support
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] J S Ma and C S Ji ldquoGeneralized grey target decision methodfor mixed attributes based on connection numberrdquo Journal ofAppliedMathematics vol 2014 Article ID 763543 8 pages 2014
[3] K Q Zhao and A L Xuan ldquoSet pair theorymdasha new theorymethod of non-define and its applicationsrdquo Systems Engineer-ing vol 14 no 1 pp 18ndash23 1996
[4] K Q Zhao Set Pair Analysis and Its Prelimiary ApplicationZhejiang Press of Science and Technology Hangzhou China2000
[5] K Q Zhao ldquoThe theoretical basis and basic algorithm of binaryconnection A+Bi and its application in AIrdquo CAA I Transactionson Intelligent Systems vol 3 no 6 pp 476ndash486 2008
Journal of Applied Mathematics 7
[6] K Q Zhao ldquoDecision making algorithm based on set pairanalysis for use when facing multiple uncertain in attributesrdquoCAAI Transactions on Intelligent Systems vol 5 no 1 pp 41ndash502010
[7] T-P Lo and S-J Guo ldquoEffective weighting model based onthe maximum deviation with uncertain informationrdquo ExpertSystems with Applications vol 37 no 12 pp 8445ndash8449 2010
[8] Z Yue ldquoAn extended TOPSIS for determining weights of deci-sion makers with interval numbersrdquo Knowledge-Based Systemsvol 24 no 1 pp 146ndash153 2011
[9] L Liu J-H Chen G-M Wang and D-Z Lao ldquoMulti-attributed decision making for mining methods based ongrey system and interval numbersrdquo Journal of Central SouthUniversity vol 20 no 4 pp 1029ndash1033 2013
[10] A I Ban and L Coroianu ldquoNearest interval triangular andtrapezoidal approximation of a fuzzy number preserving ambi-guityrdquo International Journal of Approximate Reasoning vol 53no 5 pp 805ndash836 2012
[11] D Luo and X Wang ldquoThe multi-attribute grey target decisionmethod for attribute value within three-parameter interval greynumberrdquo Applied Mathematical Modelling vol 36 no 5 pp1957ndash1963 2012
[12] W XWu ldquoUtilization of the interval number in multi-attributedecision-making of trapezoidal fuzzy numbersrdquoMathematics inPractice andTheory vol 43 no 1 pp 160ndash166 2013
[13] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011
[14] J Ye ldquoCosine similarity measures for intuitionistic fuzzy setsand their applicationsrdquoMathematical and Computer Modellingvol 53 no 1-2 pp 91ndash97 2011
[15] G W Wei Theory and Methods of Multiple Attribute DecisionMaking Based on Fuzzy Information China Economic Publish-ing House Beijing China 2010
[16] H Q He K B Wen Z X Chen and W J Ye ldquoCase-basedreasoning and fuzzy analogy preferred ratio for effective depthof collapsible loess treated with dynamic consolidationrdquo Journalof Engineering Geology vol 17 no 11 pp 88ndash93 2009
[17] Y L Jiang ldquoAn approach to group decision making based oninterval fuzzy preference relationsrdquo Journal of Systems Scienceand Systems Engineering vol 16 no 1 pp 113ndash120 2007
[18] Y XuQDa and L Liu ldquoNormalizing rank aggregationmethodfor priority of a fuzzy preference relation and its effectivenessrdquoInternational Journal of Approximate Reasoning vol 50 no 8pp 1287ndash1297 2009
[19] S H Wang ldquoForecasting the bid of Engineering projects withfuzzy similar-priority comparisonrdquo Journal of Southwest JiaoTong University vol 36 no 1 pp 96ndash99 2001
[20] J Y Zhou S D Song and Z F Yuan ldquoMethod of syntheticdistance priority in fuzzy decesionrdquo Acta Universitatis Agri-culturalis Boreali-Occidentalis Acta Universitatis AgriculturalisBoreali-Occidentalis vol 24 no 2 pp 71ndash74 1996
[21] J SMa ldquoGrey target decisionmethod for a variable target centrebased on the decision makerrsquos preferencesrdquo Journal of AppliedMathematics vol 2014 Article ID 572529 6 pages 2014
[22] S G Cao AM Xu Y B Liu and L Q Zhang ldquoComprehensiveassessment of security in coal mines based on grey relevanceanalysisrdquo Journal of Mining amp Safety Engineering vol 24 no 2pp 141ndash145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Remark 9 Measuring the similarity of the two index vectorswith nearness degree has its advantages over cosine valueand projection value First it can specifically reflect thesimilarity of the two vectors whether they transformed fromreal number or fuzzy number Second it is the generalizeddistance especially for that it is degraded to Hammingdistance when the two index vectors are converted from realnumbersThedetailed discussion can be seen in the paper [2]
3 Fuzzy Similar Priority Method forMixed Attributes
31 Brief Introduction to the Method of FuzzySimilar Priority Ratio
311 Fuzzy Preference Relation
Definition 10 Let119866 = (1198921 1198922 119892
119898) be a set of alternatives
if 119892119894and 119892
119895(119894 119895 = 1 2 119898) are compared with the alter-
native 1198920 then the fuzzy preference relation 119877 is described as
follows119877 = (119903
119894119895)119898times119898
119903119894119895isin [0 1] (119894 119895 = 1 2 119898) 119877 119860 times 119860 rarr
[0 1] and 119903119894119895satisfies the following relations
(1) 119903119894119894= 0 119894 = 1 2 119898 (2) 119903
119894119895+ 119903119895119894= 1 (119894 = 119895 119894 119895 =
1 2 119898)Equation (1) indicates that 119892
119894has indifference with itself
so 119903119894119894= 0 And (2) indicates that if 119892
119894is preferred to 119892
119895
denoted by 119903119894119895 then 119892
119895is preferred to 119892
119894 denoted by 119903
119895119894=
1 minus 119903119894119895 especially and 119903
119894119895= 1 indicates that 119892
119894is absolutely
preferred to 119892119895with respect to 119892
0[16ndash18]
312 Algorithm of Fuzzy Similar Priority Ratio Let 1198830=
(11990901 11990902 119909
0119899) be index vector of the specific object and
let the index value of the 119894th feasible alternative be 119909119894119895(119894 =
1 2 119898 119895 = 1 2 119899) so the algorithm of fuzzy similarpriority ratio is as follows [16 19]
Step 1 With respect to the attribute 119860119896 calculate the Ham-
ming distances of 119909119894119896and 119909
119895119896with 119909
0119896 respectively
119889119894119896=1003816100381610038161003816119909119894119896 minus 1199090119896
1003816100381610038161003816 119889119895119896=10038161003816100381610038161003816119909119894119895119896minus 1199090119896
10038161003816100381610038161003816 (9)
Step 2 Construct the fuzzy preference relation
119903119894119895=
119889119895119896
119889119894119896+ 119889119895119896
(10)
so the fuzzy similar matrix for attribute 119860119896is
119877119896= (
11990311
1199031119898
d
1199031198981
sdot sdot sdot 119903119898119898
) 119896 = 1 2 119899 (11)
Step 3 A series of120582-sectional sets can be determined throughoperating 119877119896 with the 120582 ascending in [0 1] To get the 120582-sectional set the elements of119877119896must be comparedwith the120582The elements of 119877119896 will be rewritten as 1 if they are no less
than the 120582 value Under attribute 119860119896 the most similarity
alternative to the specific object is the one whose row firstgets all ldquo1rdquo except the diagonal elements and the alternativeacquires the order value ldquo1rdquo Then the impact of the alterna-tive must be eliminated by removing the row correspondingto the alternative and the same column in 119877119896 Repeating thesame work a series of similarity alternatives can be obtainedand assigned the order values 2 3 119898 And the order valuethat the alternative gains under attribute 119860
119896is denoted by
119862119894119896(119894 = 1 2 119898 119896 = 1 2 119899)
Step 4 Integrate the order values under different attributesfor every alternative denoted by TC
119894with the following
equation
TC119894=
119899
sum
119896=1
119862119894119896 119894 = 1 2 119898 119896 = 1 2 119899 (12)
Step 5 Rank the alternativeswithTC119894in ascending order and
the optimal alternative to the specific object is the one withthe minimum value TC
119894
32 Fuzzy Similar Priority Method for Mixed Attributes
321 Methodology of the Proposed Method Let 1198830= (11990901
11990902 119909
0119895 119909
0119899) be index vector of the specific object and
let the index value of the 119894th feasible alternative be 119909119894119895(119894 =
1 2 119898 119895 = 1 2 119899) where119909119894119895and1199090119895can be expressed
as fuzzy numbers or real numbers The algorithm of theimproved fuzzy similar priority method can be concluded asfollows [20]
Step 1 Calculate the average values standard deviations andmaximum deviations of all indices (119909
119894119895and 119909
0119895) by using
Equations from (1) to (3)
Step 2 Translate all index values into the binary 119860 + 119861119894 con-nection number vectors using (4) and (5) So the convertedindex vectors can be expressed as
119906119904119905= 119909119904119905+ V119904119905119894 (119894 isin [minus1 1]) 119904 = 0 119898 119905 = 1 119899
(13)
Step 3 Calculate the nearness degrees of all index vectors119906119904119905(119904 = 1 119898 119905 = 1 119899) and the fixed index vectors
119906119904119905(119904 = 0 119905 = 1 119899) denoted by 119902
119894119895(119894 = 1 119898 119895 =
1 119899) using Equations from (6) to (8) and (13)
Step 4 Normalize the nearness degrees of all index vectorsand the fixed index vector by
119911119894119895=
119902119894119895
sum119898
119894=1119902119894119895
119894 = 1 119898 119895 = 1 119899 (14)
where 119911119894119895is the normalized nearness degree
Step 5 Determine the weights of all attributes denoted as120596119895(119895 = 1 119899) by objective method or subjective method
Step 6 Integrate the normalized nearness degrees consider-ing weights of all attributes using the following equation
119908119894= 120596119895119911119894119895 119894 = 1 119898 119895 = 1 119899 (15)
4 Journal of Applied Mathematics
Table 1 Safety data from coal mines
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
[18 20 25] [4 6 8] [180 260] 12 [90 91 93 94] 181198782
[12 16 20] [28 4 43] [190 210] 6 [88 89 91 92] 141198783
[22 25 31] [88 92 96] [160 200] 10 [84 87 89 92] 271198784
[6 11 13] [36 4 44] [240 280] 8 [91 93 95 97] 121198785
[24 32 34] [73 83 9] [340 360] 10 [94 95 97 98] 361198786
[14 16 18] [94 96 98] [360 440] 6 [82 84 88 90] 131198787
[33 42 45] [13 14 15] [550 650] 6 [93 94 96 97] 211198788
[9 12 15] [12 13 134] [100 140] 10 [89 90 91 94] 151198780
[8 12 22] [52 6 62] [150 160] 12 [87 88 90 91] 17
Step 7 Rank the feasible alternatives by the integrated nor-malized nearness degrees 119908
119894(119894 = 1 119898)
322 Merits of the Proposed Method The improved fuzzysimilar method has its advantages over the traditionalmethod Above all the proposed method can address themixed numbers based on the similarity of the improveddeterminacy-uncertainty index vectors through optimizingthe uncertain terms of them Moreover it can simplify thecalculation by avoiding the traditional tedious computingprocedure such that Steps 2 and 3 in the traditional methodare reduced or avoided Nevertheless it determines theoptimal alternative by the minimum of integrated nearnessdegrees instead of the minimum order values and canimprove the accuracy of results Finally it can also reflectthe decision makersrsquo preferences by considering attributeweights
4 Cases Study
41 Background A coal enterprise is to select suitable safetymanagement model from some coal enterprise with differentadvanced safety management level The safety managementmodel of a coal enterprise depends on some indices includingseam dip (∘) methane emission rate (m3t) water inflow(m3h) spontaneous combustion period (month) ventilatingstructures qualification rate () and equivalent orifice (m2)[21 22] denoted by 119860
1to 1198606respectively Thus some coal
enterprises with different index values can be regarded asdifferent alternatives Here 119878
0represents the specific coal
enterprise wants to learn advanced safety management expe-rience And the eight feasible alternatives (coal enterprises)are denoted as 119878
1to 1198788respectively Besides the index values
are given as mixed attribute values including real numbersinterval numbers triangle fuzzy numbers and trapezoid fuzzynumbers The aim of the decision makers is to seek for themost suitable enterprise whose safety management experiececan be learnt So the proposed method can address thisproblem easily The data is shown in Table 1
42 Process of Decision Making
421 Calculate the Parameters of Index Vectors All averagevalues standard deviations and maximum deviations of all
indices can be calculated using Equations from (1) to (3) andthe results are shown in Table 2
422 Translate All Index Values into Connection Number Vec-tors Having obtained the deterministic terms and uncertainterms of the connection numbers above all index values canbe transformed into index vectors using (4) Table 3 shows theconnection numbers converted from all indices
423 Calculate Nearness Degrees of All Index Vectors with theFixed Index Vectors All nearness degrees of all index vectorswith the fixed index vectors can be calculated as shown inTable 4 using Equations from (6) to (8)
424 Normalize the Nearness Degrees The normalized near-ness degrees shown in Table 5 can be calculated using (14)
425 Integrate the Normalized Nearness Degrees With-out considering all attribute weights the integrated near-ness degrees of all alternatives can be calculated as 119882 =(0272792 0427914 0610936 0666333 1191736 0754849and 1557908 0517532) So the alternatives ranking is 119878
1≻
1198782≻ 1198788≻ 1198783≻ 1198784≻ 1198786≻ 1198785≻ 1198787 In brief 119878
1is the best
selection for 1198780
If the attributes 120596 = (011 026 007 012 021 023) aregiven then the weighted integrated nearness degrees of allalternatives can be calculated as 119882 = (0042693 00694550112996 0110897 0210185 0116780 0232191 and 0104803)So the alternatives ranking is 119878
1≻ 1198782≻ 1198788≻ 1198784≻ 1198783≻ 1198786≻
1198785≻ 1198787
43 Discussion A comparison between the order valuemethod and the nearness degree method is performed toillustrate the advantage of the proposedmethod to determinethe optimal alternative Table 6 shows the order values ofall indices near the fixed indices determined by nearnessdegrees The order values of all alternatives under someattribute are given by their index nearness degrees in termsof the principle of the smaller the better
So the total integrated order values can be calculated asW= (16 16 21 26 34 27 39 19) using (12) And the alternativesranking is 119878
1= 1198782≻ 1198788≻ 1198783≻ 1198784≻ 1198786≻ 1198785≻ 1198787
Journal of Applied Mathematics 5
Table 2 Average values standard deviations and maximum deviations of all indices
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
2136055514 622 220565685440 1200 9218257422 18001198782
1644 37079372509 200141421410 600 9018257422 14001198783
2645825765 920404 180282842720 1000 8833665024 27001198784
1036055514 40404 260282842720 800 9425819893 12001198785
3052915036 820854409 350141421410 1000 9618257422 36001198786
1622 960202 400565685440 600 8636514844 13001198787
4062449987 1411 600707106850 600 9518257422 21001198788
1233 12807211106 120282842720 1000 9121602473 15001198780
1472111038 5805291506 15570710685 1200 8918257422 1700Note Each term with the form ldquo119886119887119888rdquo in Table 2 denotes ldquoaverage valuestandard deviationmaximum deviationrdquo for example the term ldquo2136055514rdquorepresents 119886 = 21 119887 = 3605551 and 119888 = 4
Table 3 Index vectors transformed from index values
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
21 + 3605551119894 6 + 2119894 220 + 40119894 12 + 0119894 92 + 1825742119894 18 + 0119894
1198782
16 + 4119894 37 + 0793725119894 200 + 10119894 6 + 0119894 90 + 1825742119894 14 + 0119894
1198783
26 + 4582576119894 92 + 04119894 180 + 20119894 10 + 0119894 88 + 3366502119894 27 + 0119894
1198784
10 + 3605551119894 4 + 04119894 260 + 20119894 8 + 0119894 94 + 2581989119894 12 + 0119894
1198785
30 + 5291503119894 82 + 08544119894 350 + 10119894 10 + 0119894 96 + 182574119894 36 + 0119894
1198786
16 + 2119894 96 + 02119894 400 + 40119894 6 + 0119894 86 + 3651484119894 13 + 0119894
1198787
40 + 6244998119894 14 + 1119894 600 + 50119894 6 + 0119894 95 + 1825742119894 21 + 0119894
1198788
12 + 3119894 128 + 06119894 120 + 20119894 10 + 0119894 91 + 2160247119894 15 + 0119894
1198780
14 + 7211103119894 58 + 052915119894 155 + 5119894 12 + 0119894 89 + 1825742119894 17 + 0119894
Table 4 Nearness degrees of all index vectors with the fixed index vectors
119902119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1199021119895
45720046 0332808 66094653 0 2999369 011199022119895
03076204 2067276 45137812 6 099979 031199023119895
94643938 3374204 25470622 2 0968189 11199024119895
5207005 1804289 10542903 4 5014459 051199025119895
13345019 2419625 19505983 2 6998528 191199026119895
06081906 3754378 24600107 6 2961924 041199027119895
22671639 8208865 44621951 6 5998738 041199028119895
37062887 6977486 34498185 2 200644 02
Table 5 Normalized nearness degrees of all index vectors with the fixed index vectors
119885119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1198851119895
0076350 00115 00567867 0 0107322 00208331198852119895
0005137 0071436 00387812 02143 0035774 006251198853119895
0158050 0116597 00218837 00714 0034643 02083331198854119895
0086954 0062348 00905817 01429 0179425 01041671198855119895
0222855 0083611 016759 00714 0250418 03958331198856119895
0010156 0129735 02113573 02143 0105982 00833331198857119895
0378604 0283662 03833795 02143 0214644 00833331198858119895
0061893 0241111 00296399 00714 0071793 0041667
6 Journal of Applied Mathematics
Table 6 Order values of index vectors with fixed index vectors
119862119894119895
1198601
1198602
1198603
1198604
1198605
1198606
sum119862119894119895
1198621119895
4 1 4 1 5 1 161198622119895
1 3 3 4 2 3 161198623119895
6 5 1 2 1 6 211198624119895
5 2 5 3 6 5 261198625119895
7 4 6 2 8 7 341198626119895
2 6 7 4 4 4 271198627119895
8 8 8 4 7 4 391198628119895
3 7 2 2 3 2 19
Table 7 Alternatives ranking with two types of method
119878119894
Order values Nearness degrees (no weights) Nearness degrees (weights)119862119894
Ranking 119882119894
Ranking 119882119894
Ranking1198781
16 1 0272792 1 0042693 11198782
16 1 0427914 2 0069455 21198783
21 3 0610936 4 0112996 51198784
26 4 0666333 5 0110897 41198785
34 6 1191736 7 0210185 71198786
27 5 0754849 6 0116780 61198787
39 7 1557908 8 0232191 81198788
19 2 0517532 3 0104803 3
Table 7 is the comparison of two types of calculationmethod one by integrated order values and one by integratednearness degrees However decision making by integratednearness degrees includes considering the attribute weightsor not
Seen from Table 7 alternatives ranking by order values isrough with the alternatives 119878
1and 119878
2in the same position
While the results by nearness degrees are accuracy with thealternatives 119878
1and 119878
2in different position Furthermore
the proposed method can also use the attribute weights toreflect the decision makersrsquo preferences There is somewhatdifference between the decisionmaking considering attributeweights or not
5 Conclusions
An improved fuzzy similar priority method was proposedto handle alternatives for mixed attributes The proposedmethod solves the mixed attributes using the nearnessdegrees as derived from the similarity of the index vectorsAnd it improves the uncertain term of the index vectorby selecting the minimum of the standard deviation or theminimum deviation of the parameters of the index and willcontribute to the accuracy in result Furthermore the newmethod can reflect the decision makerrsquos preferences throughconsidering attribute weights Another feature of thismethodis that its procedure and calculation are simple A case studyillustrated that it has advantages over the classical method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the Key Research Project of Scienceand Technology of Henan province (Grant no 13B620033)Natural Science Foundation of the Education Department ofHenan Province (Grant no 2011B620001) and Henan CoalMine Safety Production Technology Development Project(Grant noH09-50) for the support
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] J S Ma and C S Ji ldquoGeneralized grey target decision methodfor mixed attributes based on connection numberrdquo Journal ofAppliedMathematics vol 2014 Article ID 763543 8 pages 2014
[3] K Q Zhao and A L Xuan ldquoSet pair theorymdasha new theorymethod of non-define and its applicationsrdquo Systems Engineer-ing vol 14 no 1 pp 18ndash23 1996
[4] K Q Zhao Set Pair Analysis and Its Prelimiary ApplicationZhejiang Press of Science and Technology Hangzhou China2000
[5] K Q Zhao ldquoThe theoretical basis and basic algorithm of binaryconnection A+Bi and its application in AIrdquo CAA I Transactionson Intelligent Systems vol 3 no 6 pp 476ndash486 2008
Journal of Applied Mathematics 7
[6] K Q Zhao ldquoDecision making algorithm based on set pairanalysis for use when facing multiple uncertain in attributesrdquoCAAI Transactions on Intelligent Systems vol 5 no 1 pp 41ndash502010
[7] T-P Lo and S-J Guo ldquoEffective weighting model based onthe maximum deviation with uncertain informationrdquo ExpertSystems with Applications vol 37 no 12 pp 8445ndash8449 2010
[8] Z Yue ldquoAn extended TOPSIS for determining weights of deci-sion makers with interval numbersrdquo Knowledge-Based Systemsvol 24 no 1 pp 146ndash153 2011
[9] L Liu J-H Chen G-M Wang and D-Z Lao ldquoMulti-attributed decision making for mining methods based ongrey system and interval numbersrdquo Journal of Central SouthUniversity vol 20 no 4 pp 1029ndash1033 2013
[10] A I Ban and L Coroianu ldquoNearest interval triangular andtrapezoidal approximation of a fuzzy number preserving ambi-guityrdquo International Journal of Approximate Reasoning vol 53no 5 pp 805ndash836 2012
[11] D Luo and X Wang ldquoThe multi-attribute grey target decisionmethod for attribute value within three-parameter interval greynumberrdquo Applied Mathematical Modelling vol 36 no 5 pp1957ndash1963 2012
[12] W XWu ldquoUtilization of the interval number in multi-attributedecision-making of trapezoidal fuzzy numbersrdquoMathematics inPractice andTheory vol 43 no 1 pp 160ndash166 2013
[13] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011
[14] J Ye ldquoCosine similarity measures for intuitionistic fuzzy setsand their applicationsrdquoMathematical and Computer Modellingvol 53 no 1-2 pp 91ndash97 2011
[15] G W Wei Theory and Methods of Multiple Attribute DecisionMaking Based on Fuzzy Information China Economic Publish-ing House Beijing China 2010
[16] H Q He K B Wen Z X Chen and W J Ye ldquoCase-basedreasoning and fuzzy analogy preferred ratio for effective depthof collapsible loess treated with dynamic consolidationrdquo Journalof Engineering Geology vol 17 no 11 pp 88ndash93 2009
[17] Y L Jiang ldquoAn approach to group decision making based oninterval fuzzy preference relationsrdquo Journal of Systems Scienceand Systems Engineering vol 16 no 1 pp 113ndash120 2007
[18] Y XuQDa and L Liu ldquoNormalizing rank aggregationmethodfor priority of a fuzzy preference relation and its effectivenessrdquoInternational Journal of Approximate Reasoning vol 50 no 8pp 1287ndash1297 2009
[19] S H Wang ldquoForecasting the bid of Engineering projects withfuzzy similar-priority comparisonrdquo Journal of Southwest JiaoTong University vol 36 no 1 pp 96ndash99 2001
[20] J Y Zhou S D Song and Z F Yuan ldquoMethod of syntheticdistance priority in fuzzy decesionrdquo Acta Universitatis Agri-culturalis Boreali-Occidentalis Acta Universitatis AgriculturalisBoreali-Occidentalis vol 24 no 2 pp 71ndash74 1996
[21] J SMa ldquoGrey target decisionmethod for a variable target centrebased on the decision makerrsquos preferencesrdquo Journal of AppliedMathematics vol 2014 Article ID 572529 6 pages 2014
[22] S G Cao AM Xu Y B Liu and L Q Zhang ldquoComprehensiveassessment of security in coal mines based on grey relevanceanalysisrdquo Journal of Mining amp Safety Engineering vol 24 no 2pp 141ndash145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
Table 1 Safety data from coal mines
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
[18 20 25] [4 6 8] [180 260] 12 [90 91 93 94] 181198782
[12 16 20] [28 4 43] [190 210] 6 [88 89 91 92] 141198783
[22 25 31] [88 92 96] [160 200] 10 [84 87 89 92] 271198784
[6 11 13] [36 4 44] [240 280] 8 [91 93 95 97] 121198785
[24 32 34] [73 83 9] [340 360] 10 [94 95 97 98] 361198786
[14 16 18] [94 96 98] [360 440] 6 [82 84 88 90] 131198787
[33 42 45] [13 14 15] [550 650] 6 [93 94 96 97] 211198788
[9 12 15] [12 13 134] [100 140] 10 [89 90 91 94] 151198780
[8 12 22] [52 6 62] [150 160] 12 [87 88 90 91] 17
Step 7 Rank the feasible alternatives by the integrated nor-malized nearness degrees 119908
119894(119894 = 1 119898)
322 Merits of the Proposed Method The improved fuzzysimilar method has its advantages over the traditionalmethod Above all the proposed method can address themixed numbers based on the similarity of the improveddeterminacy-uncertainty index vectors through optimizingthe uncertain terms of them Moreover it can simplify thecalculation by avoiding the traditional tedious computingprocedure such that Steps 2 and 3 in the traditional methodare reduced or avoided Nevertheless it determines theoptimal alternative by the minimum of integrated nearnessdegrees instead of the minimum order values and canimprove the accuracy of results Finally it can also reflectthe decision makersrsquo preferences by considering attributeweights
4 Cases Study
41 Background A coal enterprise is to select suitable safetymanagement model from some coal enterprise with differentadvanced safety management level The safety managementmodel of a coal enterprise depends on some indices includingseam dip (∘) methane emission rate (m3t) water inflow(m3h) spontaneous combustion period (month) ventilatingstructures qualification rate () and equivalent orifice (m2)[21 22] denoted by 119860
1to 1198606respectively Thus some coal
enterprises with different index values can be regarded asdifferent alternatives Here 119878
0represents the specific coal
enterprise wants to learn advanced safety management expe-rience And the eight feasible alternatives (coal enterprises)are denoted as 119878
1to 1198788respectively Besides the index values
are given as mixed attribute values including real numbersinterval numbers triangle fuzzy numbers and trapezoid fuzzynumbers The aim of the decision makers is to seek for themost suitable enterprise whose safety management experiececan be learnt So the proposed method can address thisproblem easily The data is shown in Table 1
42 Process of Decision Making
421 Calculate the Parameters of Index Vectors All averagevalues standard deviations and maximum deviations of all
indices can be calculated using Equations from (1) to (3) andthe results are shown in Table 2
422 Translate All Index Values into Connection Number Vec-tors Having obtained the deterministic terms and uncertainterms of the connection numbers above all index values canbe transformed into index vectors using (4) Table 3 shows theconnection numbers converted from all indices
423 Calculate Nearness Degrees of All Index Vectors with theFixed Index Vectors All nearness degrees of all index vectorswith the fixed index vectors can be calculated as shown inTable 4 using Equations from (6) to (8)
424 Normalize the Nearness Degrees The normalized near-ness degrees shown in Table 5 can be calculated using (14)
425 Integrate the Normalized Nearness Degrees With-out considering all attribute weights the integrated near-ness degrees of all alternatives can be calculated as 119882 =(0272792 0427914 0610936 0666333 1191736 0754849and 1557908 0517532) So the alternatives ranking is 119878
1≻
1198782≻ 1198788≻ 1198783≻ 1198784≻ 1198786≻ 1198785≻ 1198787 In brief 119878
1is the best
selection for 1198780
If the attributes 120596 = (011 026 007 012 021 023) aregiven then the weighted integrated nearness degrees of allalternatives can be calculated as 119882 = (0042693 00694550112996 0110897 0210185 0116780 0232191 and 0104803)So the alternatives ranking is 119878
1≻ 1198782≻ 1198788≻ 1198784≻ 1198783≻ 1198786≻
1198785≻ 1198787
43 Discussion A comparison between the order valuemethod and the nearness degree method is performed toillustrate the advantage of the proposedmethod to determinethe optimal alternative Table 6 shows the order values ofall indices near the fixed indices determined by nearnessdegrees The order values of all alternatives under someattribute are given by their index nearness degrees in termsof the principle of the smaller the better
So the total integrated order values can be calculated asW= (16 16 21 26 34 27 39 19) using (12) And the alternativesranking is 119878
1= 1198782≻ 1198788≻ 1198783≻ 1198784≻ 1198786≻ 1198785≻ 1198787
Journal of Applied Mathematics 5
Table 2 Average values standard deviations and maximum deviations of all indices
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
2136055514 622 220565685440 1200 9218257422 18001198782
1644 37079372509 200141421410 600 9018257422 14001198783
2645825765 920404 180282842720 1000 8833665024 27001198784
1036055514 40404 260282842720 800 9425819893 12001198785
3052915036 820854409 350141421410 1000 9618257422 36001198786
1622 960202 400565685440 600 8636514844 13001198787
4062449987 1411 600707106850 600 9518257422 21001198788
1233 12807211106 120282842720 1000 9121602473 15001198780
1472111038 5805291506 15570710685 1200 8918257422 1700Note Each term with the form ldquo119886119887119888rdquo in Table 2 denotes ldquoaverage valuestandard deviationmaximum deviationrdquo for example the term ldquo2136055514rdquorepresents 119886 = 21 119887 = 3605551 and 119888 = 4
Table 3 Index vectors transformed from index values
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
21 + 3605551119894 6 + 2119894 220 + 40119894 12 + 0119894 92 + 1825742119894 18 + 0119894
1198782
16 + 4119894 37 + 0793725119894 200 + 10119894 6 + 0119894 90 + 1825742119894 14 + 0119894
1198783
26 + 4582576119894 92 + 04119894 180 + 20119894 10 + 0119894 88 + 3366502119894 27 + 0119894
1198784
10 + 3605551119894 4 + 04119894 260 + 20119894 8 + 0119894 94 + 2581989119894 12 + 0119894
1198785
30 + 5291503119894 82 + 08544119894 350 + 10119894 10 + 0119894 96 + 182574119894 36 + 0119894
1198786
16 + 2119894 96 + 02119894 400 + 40119894 6 + 0119894 86 + 3651484119894 13 + 0119894
1198787
40 + 6244998119894 14 + 1119894 600 + 50119894 6 + 0119894 95 + 1825742119894 21 + 0119894
1198788
12 + 3119894 128 + 06119894 120 + 20119894 10 + 0119894 91 + 2160247119894 15 + 0119894
1198780
14 + 7211103119894 58 + 052915119894 155 + 5119894 12 + 0119894 89 + 1825742119894 17 + 0119894
Table 4 Nearness degrees of all index vectors with the fixed index vectors
119902119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1199021119895
45720046 0332808 66094653 0 2999369 011199022119895
03076204 2067276 45137812 6 099979 031199023119895
94643938 3374204 25470622 2 0968189 11199024119895
5207005 1804289 10542903 4 5014459 051199025119895
13345019 2419625 19505983 2 6998528 191199026119895
06081906 3754378 24600107 6 2961924 041199027119895
22671639 8208865 44621951 6 5998738 041199028119895
37062887 6977486 34498185 2 200644 02
Table 5 Normalized nearness degrees of all index vectors with the fixed index vectors
119885119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1198851119895
0076350 00115 00567867 0 0107322 00208331198852119895
0005137 0071436 00387812 02143 0035774 006251198853119895
0158050 0116597 00218837 00714 0034643 02083331198854119895
0086954 0062348 00905817 01429 0179425 01041671198855119895
0222855 0083611 016759 00714 0250418 03958331198856119895
0010156 0129735 02113573 02143 0105982 00833331198857119895
0378604 0283662 03833795 02143 0214644 00833331198858119895
0061893 0241111 00296399 00714 0071793 0041667
6 Journal of Applied Mathematics
Table 6 Order values of index vectors with fixed index vectors
119862119894119895
1198601
1198602
1198603
1198604
1198605
1198606
sum119862119894119895
1198621119895
4 1 4 1 5 1 161198622119895
1 3 3 4 2 3 161198623119895
6 5 1 2 1 6 211198624119895
5 2 5 3 6 5 261198625119895
7 4 6 2 8 7 341198626119895
2 6 7 4 4 4 271198627119895
8 8 8 4 7 4 391198628119895
3 7 2 2 3 2 19
Table 7 Alternatives ranking with two types of method
119878119894
Order values Nearness degrees (no weights) Nearness degrees (weights)119862119894
Ranking 119882119894
Ranking 119882119894
Ranking1198781
16 1 0272792 1 0042693 11198782
16 1 0427914 2 0069455 21198783
21 3 0610936 4 0112996 51198784
26 4 0666333 5 0110897 41198785
34 6 1191736 7 0210185 71198786
27 5 0754849 6 0116780 61198787
39 7 1557908 8 0232191 81198788
19 2 0517532 3 0104803 3
Table 7 is the comparison of two types of calculationmethod one by integrated order values and one by integratednearness degrees However decision making by integratednearness degrees includes considering the attribute weightsor not
Seen from Table 7 alternatives ranking by order values isrough with the alternatives 119878
1and 119878
2in the same position
While the results by nearness degrees are accuracy with thealternatives 119878
1and 119878
2in different position Furthermore
the proposed method can also use the attribute weights toreflect the decision makersrsquo preferences There is somewhatdifference between the decisionmaking considering attributeweights or not
5 Conclusions
An improved fuzzy similar priority method was proposedto handle alternatives for mixed attributes The proposedmethod solves the mixed attributes using the nearnessdegrees as derived from the similarity of the index vectorsAnd it improves the uncertain term of the index vectorby selecting the minimum of the standard deviation or theminimum deviation of the parameters of the index and willcontribute to the accuracy in result Furthermore the newmethod can reflect the decision makerrsquos preferences throughconsidering attribute weights Another feature of thismethodis that its procedure and calculation are simple A case studyillustrated that it has advantages over the classical method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the Key Research Project of Scienceand Technology of Henan province (Grant no 13B620033)Natural Science Foundation of the Education Department ofHenan Province (Grant no 2011B620001) and Henan CoalMine Safety Production Technology Development Project(Grant noH09-50) for the support
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] J S Ma and C S Ji ldquoGeneralized grey target decision methodfor mixed attributes based on connection numberrdquo Journal ofAppliedMathematics vol 2014 Article ID 763543 8 pages 2014
[3] K Q Zhao and A L Xuan ldquoSet pair theorymdasha new theorymethod of non-define and its applicationsrdquo Systems Engineer-ing vol 14 no 1 pp 18ndash23 1996
[4] K Q Zhao Set Pair Analysis and Its Prelimiary ApplicationZhejiang Press of Science and Technology Hangzhou China2000
[5] K Q Zhao ldquoThe theoretical basis and basic algorithm of binaryconnection A+Bi and its application in AIrdquo CAA I Transactionson Intelligent Systems vol 3 no 6 pp 476ndash486 2008
Journal of Applied Mathematics 7
[6] K Q Zhao ldquoDecision making algorithm based on set pairanalysis for use when facing multiple uncertain in attributesrdquoCAAI Transactions on Intelligent Systems vol 5 no 1 pp 41ndash502010
[7] T-P Lo and S-J Guo ldquoEffective weighting model based onthe maximum deviation with uncertain informationrdquo ExpertSystems with Applications vol 37 no 12 pp 8445ndash8449 2010
[8] Z Yue ldquoAn extended TOPSIS for determining weights of deci-sion makers with interval numbersrdquo Knowledge-Based Systemsvol 24 no 1 pp 146ndash153 2011
[9] L Liu J-H Chen G-M Wang and D-Z Lao ldquoMulti-attributed decision making for mining methods based ongrey system and interval numbersrdquo Journal of Central SouthUniversity vol 20 no 4 pp 1029ndash1033 2013
[10] A I Ban and L Coroianu ldquoNearest interval triangular andtrapezoidal approximation of a fuzzy number preserving ambi-guityrdquo International Journal of Approximate Reasoning vol 53no 5 pp 805ndash836 2012
[11] D Luo and X Wang ldquoThe multi-attribute grey target decisionmethod for attribute value within three-parameter interval greynumberrdquo Applied Mathematical Modelling vol 36 no 5 pp1957ndash1963 2012
[12] W XWu ldquoUtilization of the interval number in multi-attributedecision-making of trapezoidal fuzzy numbersrdquoMathematics inPractice andTheory vol 43 no 1 pp 160ndash166 2013
[13] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011
[14] J Ye ldquoCosine similarity measures for intuitionistic fuzzy setsand their applicationsrdquoMathematical and Computer Modellingvol 53 no 1-2 pp 91ndash97 2011
[15] G W Wei Theory and Methods of Multiple Attribute DecisionMaking Based on Fuzzy Information China Economic Publish-ing House Beijing China 2010
[16] H Q He K B Wen Z X Chen and W J Ye ldquoCase-basedreasoning and fuzzy analogy preferred ratio for effective depthof collapsible loess treated with dynamic consolidationrdquo Journalof Engineering Geology vol 17 no 11 pp 88ndash93 2009
[17] Y L Jiang ldquoAn approach to group decision making based oninterval fuzzy preference relationsrdquo Journal of Systems Scienceand Systems Engineering vol 16 no 1 pp 113ndash120 2007
[18] Y XuQDa and L Liu ldquoNormalizing rank aggregationmethodfor priority of a fuzzy preference relation and its effectivenessrdquoInternational Journal of Approximate Reasoning vol 50 no 8pp 1287ndash1297 2009
[19] S H Wang ldquoForecasting the bid of Engineering projects withfuzzy similar-priority comparisonrdquo Journal of Southwest JiaoTong University vol 36 no 1 pp 96ndash99 2001
[20] J Y Zhou S D Song and Z F Yuan ldquoMethod of syntheticdistance priority in fuzzy decesionrdquo Acta Universitatis Agri-culturalis Boreali-Occidentalis Acta Universitatis AgriculturalisBoreali-Occidentalis vol 24 no 2 pp 71ndash74 1996
[21] J SMa ldquoGrey target decisionmethod for a variable target centrebased on the decision makerrsquos preferencesrdquo Journal of AppliedMathematics vol 2014 Article ID 572529 6 pages 2014
[22] S G Cao AM Xu Y B Liu and L Q Zhang ldquoComprehensiveassessment of security in coal mines based on grey relevanceanalysisrdquo Journal of Mining amp Safety Engineering vol 24 no 2pp 141ndash145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Table 2 Average values standard deviations and maximum deviations of all indices
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
2136055514 622 220565685440 1200 9218257422 18001198782
1644 37079372509 200141421410 600 9018257422 14001198783
2645825765 920404 180282842720 1000 8833665024 27001198784
1036055514 40404 260282842720 800 9425819893 12001198785
3052915036 820854409 350141421410 1000 9618257422 36001198786
1622 960202 400565685440 600 8636514844 13001198787
4062449987 1411 600707106850 600 9518257422 21001198788
1233 12807211106 120282842720 1000 9121602473 15001198780
1472111038 5805291506 15570710685 1200 8918257422 1700Note Each term with the form ldquo119886119887119888rdquo in Table 2 denotes ldquoaverage valuestandard deviationmaximum deviationrdquo for example the term ldquo2136055514rdquorepresents 119886 = 21 119887 = 3605551 and 119888 = 4
Table 3 Index vectors transformed from index values
119878119894
1198601
1198602
1198603
1198604
1198605
1198606
1198781
21 + 3605551119894 6 + 2119894 220 + 40119894 12 + 0119894 92 + 1825742119894 18 + 0119894
1198782
16 + 4119894 37 + 0793725119894 200 + 10119894 6 + 0119894 90 + 1825742119894 14 + 0119894
1198783
26 + 4582576119894 92 + 04119894 180 + 20119894 10 + 0119894 88 + 3366502119894 27 + 0119894
1198784
10 + 3605551119894 4 + 04119894 260 + 20119894 8 + 0119894 94 + 2581989119894 12 + 0119894
1198785
30 + 5291503119894 82 + 08544119894 350 + 10119894 10 + 0119894 96 + 182574119894 36 + 0119894
1198786
16 + 2119894 96 + 02119894 400 + 40119894 6 + 0119894 86 + 3651484119894 13 + 0119894
1198787
40 + 6244998119894 14 + 1119894 600 + 50119894 6 + 0119894 95 + 1825742119894 21 + 0119894
1198788
12 + 3119894 128 + 06119894 120 + 20119894 10 + 0119894 91 + 2160247119894 15 + 0119894
1198780
14 + 7211103119894 58 + 052915119894 155 + 5119894 12 + 0119894 89 + 1825742119894 17 + 0119894
Table 4 Nearness degrees of all index vectors with the fixed index vectors
119902119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1199021119895
45720046 0332808 66094653 0 2999369 011199022119895
03076204 2067276 45137812 6 099979 031199023119895
94643938 3374204 25470622 2 0968189 11199024119895
5207005 1804289 10542903 4 5014459 051199025119895
13345019 2419625 19505983 2 6998528 191199026119895
06081906 3754378 24600107 6 2961924 041199027119895
22671639 8208865 44621951 6 5998738 041199028119895
37062887 6977486 34498185 2 200644 02
Table 5 Normalized nearness degrees of all index vectors with the fixed index vectors
119885119894119895
1198601
1198602
1198603
1198604
1198605
1198606
1198851119895
0076350 00115 00567867 0 0107322 00208331198852119895
0005137 0071436 00387812 02143 0035774 006251198853119895
0158050 0116597 00218837 00714 0034643 02083331198854119895
0086954 0062348 00905817 01429 0179425 01041671198855119895
0222855 0083611 016759 00714 0250418 03958331198856119895
0010156 0129735 02113573 02143 0105982 00833331198857119895
0378604 0283662 03833795 02143 0214644 00833331198858119895
0061893 0241111 00296399 00714 0071793 0041667
6 Journal of Applied Mathematics
Table 6 Order values of index vectors with fixed index vectors
119862119894119895
1198601
1198602
1198603
1198604
1198605
1198606
sum119862119894119895
1198621119895
4 1 4 1 5 1 161198622119895
1 3 3 4 2 3 161198623119895
6 5 1 2 1 6 211198624119895
5 2 5 3 6 5 261198625119895
7 4 6 2 8 7 341198626119895
2 6 7 4 4 4 271198627119895
8 8 8 4 7 4 391198628119895
3 7 2 2 3 2 19
Table 7 Alternatives ranking with two types of method
119878119894
Order values Nearness degrees (no weights) Nearness degrees (weights)119862119894
Ranking 119882119894
Ranking 119882119894
Ranking1198781
16 1 0272792 1 0042693 11198782
16 1 0427914 2 0069455 21198783
21 3 0610936 4 0112996 51198784
26 4 0666333 5 0110897 41198785
34 6 1191736 7 0210185 71198786
27 5 0754849 6 0116780 61198787
39 7 1557908 8 0232191 81198788
19 2 0517532 3 0104803 3
Table 7 is the comparison of two types of calculationmethod one by integrated order values and one by integratednearness degrees However decision making by integratednearness degrees includes considering the attribute weightsor not
Seen from Table 7 alternatives ranking by order values isrough with the alternatives 119878
1and 119878
2in the same position
While the results by nearness degrees are accuracy with thealternatives 119878
1and 119878
2in different position Furthermore
the proposed method can also use the attribute weights toreflect the decision makersrsquo preferences There is somewhatdifference between the decisionmaking considering attributeweights or not
5 Conclusions
An improved fuzzy similar priority method was proposedto handle alternatives for mixed attributes The proposedmethod solves the mixed attributes using the nearnessdegrees as derived from the similarity of the index vectorsAnd it improves the uncertain term of the index vectorby selecting the minimum of the standard deviation or theminimum deviation of the parameters of the index and willcontribute to the accuracy in result Furthermore the newmethod can reflect the decision makerrsquos preferences throughconsidering attribute weights Another feature of thismethodis that its procedure and calculation are simple A case studyillustrated that it has advantages over the classical method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the Key Research Project of Scienceand Technology of Henan province (Grant no 13B620033)Natural Science Foundation of the Education Department ofHenan Province (Grant no 2011B620001) and Henan CoalMine Safety Production Technology Development Project(Grant noH09-50) for the support
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] J S Ma and C S Ji ldquoGeneralized grey target decision methodfor mixed attributes based on connection numberrdquo Journal ofAppliedMathematics vol 2014 Article ID 763543 8 pages 2014
[3] K Q Zhao and A L Xuan ldquoSet pair theorymdasha new theorymethod of non-define and its applicationsrdquo Systems Engineer-ing vol 14 no 1 pp 18ndash23 1996
[4] K Q Zhao Set Pair Analysis and Its Prelimiary ApplicationZhejiang Press of Science and Technology Hangzhou China2000
[5] K Q Zhao ldquoThe theoretical basis and basic algorithm of binaryconnection A+Bi and its application in AIrdquo CAA I Transactionson Intelligent Systems vol 3 no 6 pp 476ndash486 2008
Journal of Applied Mathematics 7
[6] K Q Zhao ldquoDecision making algorithm based on set pairanalysis for use when facing multiple uncertain in attributesrdquoCAAI Transactions on Intelligent Systems vol 5 no 1 pp 41ndash502010
[7] T-P Lo and S-J Guo ldquoEffective weighting model based onthe maximum deviation with uncertain informationrdquo ExpertSystems with Applications vol 37 no 12 pp 8445ndash8449 2010
[8] Z Yue ldquoAn extended TOPSIS for determining weights of deci-sion makers with interval numbersrdquo Knowledge-Based Systemsvol 24 no 1 pp 146ndash153 2011
[9] L Liu J-H Chen G-M Wang and D-Z Lao ldquoMulti-attributed decision making for mining methods based ongrey system and interval numbersrdquo Journal of Central SouthUniversity vol 20 no 4 pp 1029ndash1033 2013
[10] A I Ban and L Coroianu ldquoNearest interval triangular andtrapezoidal approximation of a fuzzy number preserving ambi-guityrdquo International Journal of Approximate Reasoning vol 53no 5 pp 805ndash836 2012
[11] D Luo and X Wang ldquoThe multi-attribute grey target decisionmethod for attribute value within three-parameter interval greynumberrdquo Applied Mathematical Modelling vol 36 no 5 pp1957ndash1963 2012
[12] W XWu ldquoUtilization of the interval number in multi-attributedecision-making of trapezoidal fuzzy numbersrdquoMathematics inPractice andTheory vol 43 no 1 pp 160ndash166 2013
[13] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011
[14] J Ye ldquoCosine similarity measures for intuitionistic fuzzy setsand their applicationsrdquoMathematical and Computer Modellingvol 53 no 1-2 pp 91ndash97 2011
[15] G W Wei Theory and Methods of Multiple Attribute DecisionMaking Based on Fuzzy Information China Economic Publish-ing House Beijing China 2010
[16] H Q He K B Wen Z X Chen and W J Ye ldquoCase-basedreasoning and fuzzy analogy preferred ratio for effective depthof collapsible loess treated with dynamic consolidationrdquo Journalof Engineering Geology vol 17 no 11 pp 88ndash93 2009
[17] Y L Jiang ldquoAn approach to group decision making based oninterval fuzzy preference relationsrdquo Journal of Systems Scienceand Systems Engineering vol 16 no 1 pp 113ndash120 2007
[18] Y XuQDa and L Liu ldquoNormalizing rank aggregationmethodfor priority of a fuzzy preference relation and its effectivenessrdquoInternational Journal of Approximate Reasoning vol 50 no 8pp 1287ndash1297 2009
[19] S H Wang ldquoForecasting the bid of Engineering projects withfuzzy similar-priority comparisonrdquo Journal of Southwest JiaoTong University vol 36 no 1 pp 96ndash99 2001
[20] J Y Zhou S D Song and Z F Yuan ldquoMethod of syntheticdistance priority in fuzzy decesionrdquo Acta Universitatis Agri-culturalis Boreali-Occidentalis Acta Universitatis AgriculturalisBoreali-Occidentalis vol 24 no 2 pp 71ndash74 1996
[21] J SMa ldquoGrey target decisionmethod for a variable target centrebased on the decision makerrsquos preferencesrdquo Journal of AppliedMathematics vol 2014 Article ID 572529 6 pages 2014
[22] S G Cao AM Xu Y B Liu and L Q Zhang ldquoComprehensiveassessment of security in coal mines based on grey relevanceanalysisrdquo Journal of Mining amp Safety Engineering vol 24 no 2pp 141ndash145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Table 6 Order values of index vectors with fixed index vectors
119862119894119895
1198601
1198602
1198603
1198604
1198605
1198606
sum119862119894119895
1198621119895
4 1 4 1 5 1 161198622119895
1 3 3 4 2 3 161198623119895
6 5 1 2 1 6 211198624119895
5 2 5 3 6 5 261198625119895
7 4 6 2 8 7 341198626119895
2 6 7 4 4 4 271198627119895
8 8 8 4 7 4 391198628119895
3 7 2 2 3 2 19
Table 7 Alternatives ranking with two types of method
119878119894
Order values Nearness degrees (no weights) Nearness degrees (weights)119862119894
Ranking 119882119894
Ranking 119882119894
Ranking1198781
16 1 0272792 1 0042693 11198782
16 1 0427914 2 0069455 21198783
21 3 0610936 4 0112996 51198784
26 4 0666333 5 0110897 41198785
34 6 1191736 7 0210185 71198786
27 5 0754849 6 0116780 61198787
39 7 1557908 8 0232191 81198788
19 2 0517532 3 0104803 3
Table 7 is the comparison of two types of calculationmethod one by integrated order values and one by integratednearness degrees However decision making by integratednearness degrees includes considering the attribute weightsor not
Seen from Table 7 alternatives ranking by order values isrough with the alternatives 119878
1and 119878
2in the same position
While the results by nearness degrees are accuracy with thealternatives 119878
1and 119878
2in different position Furthermore
the proposed method can also use the attribute weights toreflect the decision makersrsquo preferences There is somewhatdifference between the decisionmaking considering attributeweights or not
5 Conclusions
An improved fuzzy similar priority method was proposedto handle alternatives for mixed attributes The proposedmethod solves the mixed attributes using the nearnessdegrees as derived from the similarity of the index vectorsAnd it improves the uncertain term of the index vectorby selecting the minimum of the standard deviation or theminimum deviation of the parameters of the index and willcontribute to the accuracy in result Furthermore the newmethod can reflect the decision makerrsquos preferences throughconsidering attribute weights Another feature of thismethodis that its procedure and calculation are simple A case studyillustrated that it has advantages over the classical method
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors thank the Key Research Project of Scienceand Technology of Henan province (Grant no 13B620033)Natural Science Foundation of the Education Department ofHenan Province (Grant no 2011B620001) and Henan CoalMine Safety Production Technology Development Project(Grant noH09-50) for the support
References
[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965
[2] J S Ma and C S Ji ldquoGeneralized grey target decision methodfor mixed attributes based on connection numberrdquo Journal ofAppliedMathematics vol 2014 Article ID 763543 8 pages 2014
[3] K Q Zhao and A L Xuan ldquoSet pair theorymdasha new theorymethod of non-define and its applicationsrdquo Systems Engineer-ing vol 14 no 1 pp 18ndash23 1996
[4] K Q Zhao Set Pair Analysis and Its Prelimiary ApplicationZhejiang Press of Science and Technology Hangzhou China2000
[5] K Q Zhao ldquoThe theoretical basis and basic algorithm of binaryconnection A+Bi and its application in AIrdquo CAA I Transactionson Intelligent Systems vol 3 no 6 pp 476ndash486 2008
Journal of Applied Mathematics 7
[6] K Q Zhao ldquoDecision making algorithm based on set pairanalysis for use when facing multiple uncertain in attributesrdquoCAAI Transactions on Intelligent Systems vol 5 no 1 pp 41ndash502010
[7] T-P Lo and S-J Guo ldquoEffective weighting model based onthe maximum deviation with uncertain informationrdquo ExpertSystems with Applications vol 37 no 12 pp 8445ndash8449 2010
[8] Z Yue ldquoAn extended TOPSIS for determining weights of deci-sion makers with interval numbersrdquo Knowledge-Based Systemsvol 24 no 1 pp 146ndash153 2011
[9] L Liu J-H Chen G-M Wang and D-Z Lao ldquoMulti-attributed decision making for mining methods based ongrey system and interval numbersrdquo Journal of Central SouthUniversity vol 20 no 4 pp 1029ndash1033 2013
[10] A I Ban and L Coroianu ldquoNearest interval triangular andtrapezoidal approximation of a fuzzy number preserving ambi-guityrdquo International Journal of Approximate Reasoning vol 53no 5 pp 805ndash836 2012
[11] D Luo and X Wang ldquoThe multi-attribute grey target decisionmethod for attribute value within three-parameter interval greynumberrdquo Applied Mathematical Modelling vol 36 no 5 pp1957ndash1963 2012
[12] W XWu ldquoUtilization of the interval number in multi-attributedecision-making of trapezoidal fuzzy numbersrdquoMathematics inPractice andTheory vol 43 no 1 pp 160ndash166 2013
[13] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011
[14] J Ye ldquoCosine similarity measures for intuitionistic fuzzy setsand their applicationsrdquoMathematical and Computer Modellingvol 53 no 1-2 pp 91ndash97 2011
[15] G W Wei Theory and Methods of Multiple Attribute DecisionMaking Based on Fuzzy Information China Economic Publish-ing House Beijing China 2010
[16] H Q He K B Wen Z X Chen and W J Ye ldquoCase-basedreasoning and fuzzy analogy preferred ratio for effective depthof collapsible loess treated with dynamic consolidationrdquo Journalof Engineering Geology vol 17 no 11 pp 88ndash93 2009
[17] Y L Jiang ldquoAn approach to group decision making based oninterval fuzzy preference relationsrdquo Journal of Systems Scienceand Systems Engineering vol 16 no 1 pp 113ndash120 2007
[18] Y XuQDa and L Liu ldquoNormalizing rank aggregationmethodfor priority of a fuzzy preference relation and its effectivenessrdquoInternational Journal of Approximate Reasoning vol 50 no 8pp 1287ndash1297 2009
[19] S H Wang ldquoForecasting the bid of Engineering projects withfuzzy similar-priority comparisonrdquo Journal of Southwest JiaoTong University vol 36 no 1 pp 96ndash99 2001
[20] J Y Zhou S D Song and Z F Yuan ldquoMethod of syntheticdistance priority in fuzzy decesionrdquo Acta Universitatis Agri-culturalis Boreali-Occidentalis Acta Universitatis AgriculturalisBoreali-Occidentalis vol 24 no 2 pp 71ndash74 1996
[21] J SMa ldquoGrey target decisionmethod for a variable target centrebased on the decision makerrsquos preferencesrdquo Journal of AppliedMathematics vol 2014 Article ID 572529 6 pages 2014
[22] S G Cao AM Xu Y B Liu and L Q Zhang ldquoComprehensiveassessment of security in coal mines based on grey relevanceanalysisrdquo Journal of Mining amp Safety Engineering vol 24 no 2pp 141ndash145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
[6] K Q Zhao ldquoDecision making algorithm based on set pairanalysis for use when facing multiple uncertain in attributesrdquoCAAI Transactions on Intelligent Systems vol 5 no 1 pp 41ndash502010
[7] T-P Lo and S-J Guo ldquoEffective weighting model based onthe maximum deviation with uncertain informationrdquo ExpertSystems with Applications vol 37 no 12 pp 8445ndash8449 2010
[8] Z Yue ldquoAn extended TOPSIS for determining weights of deci-sion makers with interval numbersrdquo Knowledge-Based Systemsvol 24 no 1 pp 146ndash153 2011
[9] L Liu J-H Chen G-M Wang and D-Z Lao ldquoMulti-attributed decision making for mining methods based ongrey system and interval numbersrdquo Journal of Central SouthUniversity vol 20 no 4 pp 1029ndash1033 2013
[10] A I Ban and L Coroianu ldquoNearest interval triangular andtrapezoidal approximation of a fuzzy number preserving ambi-guityrdquo International Journal of Approximate Reasoning vol 53no 5 pp 805ndash836 2012
[11] D Luo and X Wang ldquoThe multi-attribute grey target decisionmethod for attribute value within three-parameter interval greynumberrdquo Applied Mathematical Modelling vol 36 no 5 pp1957ndash1963 2012
[12] W XWu ldquoUtilization of the interval number in multi-attributedecision-making of trapezoidal fuzzy numbersrdquoMathematics inPractice andTheory vol 43 no 1 pp 160ndash166 2013
[13] Z Xu andMXia ldquoDistance and similaritymeasures for hesitantfuzzy setsrdquo Information Sciences vol 181 no 11 pp 2128ndash21382011
[14] J Ye ldquoCosine similarity measures for intuitionistic fuzzy setsand their applicationsrdquoMathematical and Computer Modellingvol 53 no 1-2 pp 91ndash97 2011
[15] G W Wei Theory and Methods of Multiple Attribute DecisionMaking Based on Fuzzy Information China Economic Publish-ing House Beijing China 2010
[16] H Q He K B Wen Z X Chen and W J Ye ldquoCase-basedreasoning and fuzzy analogy preferred ratio for effective depthof collapsible loess treated with dynamic consolidationrdquo Journalof Engineering Geology vol 17 no 11 pp 88ndash93 2009
[17] Y L Jiang ldquoAn approach to group decision making based oninterval fuzzy preference relationsrdquo Journal of Systems Scienceand Systems Engineering vol 16 no 1 pp 113ndash120 2007
[18] Y XuQDa and L Liu ldquoNormalizing rank aggregationmethodfor priority of a fuzzy preference relation and its effectivenessrdquoInternational Journal of Approximate Reasoning vol 50 no 8pp 1287ndash1297 2009
[19] S H Wang ldquoForecasting the bid of Engineering projects withfuzzy similar-priority comparisonrdquo Journal of Southwest JiaoTong University vol 36 no 1 pp 96ndash99 2001
[20] J Y Zhou S D Song and Z F Yuan ldquoMethod of syntheticdistance priority in fuzzy decesionrdquo Acta Universitatis Agri-culturalis Boreali-Occidentalis Acta Universitatis AgriculturalisBoreali-Occidentalis vol 24 no 2 pp 71ndash74 1996
[21] J SMa ldquoGrey target decisionmethod for a variable target centrebased on the decision makerrsquos preferencesrdquo Journal of AppliedMathematics vol 2014 Article ID 572529 6 pages 2014
[22] S G Cao AM Xu Y B Liu and L Q Zhang ldquoComprehensiveassessment of security in coal mines based on grey relevanceanalysisrdquo Journal of Mining amp Safety Engineering vol 24 no 2pp 141ndash145 2007
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of