Research ArticleGrey Situation Group Decision-Making MethodBased on Prospect Theory

Na Zhang123 Zhigeng Fang3 and Xiaqing Liu3

1 Corps Financial Development Research Center Shihezi University Wujiaqu 831300 China2 Commerce College Shihezi University Wujiaqu 831300 China3 College of Economics and Management Nanjing University of Aeronautics and Astronautics Nanjing 210016 China

Correspondence should be addressed to Na Zhang zhangnanuaa163com

Received 12 June 2014 Accepted 26 July 2014 Published 12 August 2014

Academic Editor Young Bae Jun

Copyright copy 2014 Na Zhang et alThis is an open access article distributed under theCreativeCommonsAttributionLicense whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper puts forward a grey situation group decision-makingmethod on the basis of prospect theory in view of the grey situationgroup decision-making problems that decisions are oftenmade bymultiple decision experts and those experts have risk preferencesThemethod takes the positive and negative ideal situation distance as reference points defines positive and negative prospect valuefunction and introduces decision expertsrsquo risk preference into grey situation decision-making to make the final decision be morein line with decision expertsrsquo psychological behavior Based on TOPSIS method this paper determines the weight of each decisionexpert sets up comprehensive prospect value matrix for decision expertsrsquo evaluation and finally determines the optimal situationAt last this paper verifies the effectiveness and feasibility of the method by means of a specific example

1 Introduction

Grey situation decision is a method to choose the decisionwith optimal effect from multiple decisions and objectiveswith the prerequisite that decision information should havegrey elements [1ndash3] Since Professor Dengjulong [1] putforward grey decision-making theory numerous expertsand scholars at home and abroad conduct researches andapplications on the theory

In the aspect of improvement for grey situation model[4] applies grey situation decision-making to the situationof decision information as interval number on the basis ofanalysis on distance of interval number and grey relationentropy and proposes optimization methods for objectiveweight of grey situation decision-making Reference [5]defines parameter setting for intelligence algorithm as auniform design problem and makes decisions for uniformdesigned parameter combination by means of grey situationdecision-making Reference [6] brings up a multiobjectivegrey situation decision-making method based on prospect

theory in consideration of the influences of decision expertsrsquorisk attitude to multiobjective decision Reference [7] comesup with a linear transformation operator with bonus and for-feit property and gives multiobjective interval number greysituation decision-making method Reference [8] exploresthe case of assessment information as interval grey numberand sets up an objective weight optimization model formultiobjective grey situation decision-making Reference [9]proposes a weighting method for objective weight of greysituation decision-making in response to the question thatequal weight method for grey situation decision-making isunable to reflect decision expertsrsquo preference and practicalsituation of decision-making process Reference [10] analyzesthe generality of grey situation decision-making using gen-eral system theory defines grey general situation decision-making and develops its mathematical model

In recent years the grey situation model has made alot of achievements in application field too Reference [11]applies grey situation decision-making method to stockmarket Reference [12] conducts performance assessment

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 703597 7 pageshttpdxdoiorg1011552014703597

2 The Scientific World Journal

for medical industry by means of grey situation decision-making method Reference [13] assumes that all the attributeindexes are in the state of trapezoid fuzzy numbers andmakesoptimization for site selection alternatives by grey situationdecision-making method Reference [14] develops optimalmodel for artillery fire scheme on the basis of grey situationdecision-making method aiming at solving the problemsof information uncertainty and information incompletenessin the battlefield Reference [15] suggests a general mainte-nance resource integration method based on grey situationdecision-making theory on account of growing number ofmaintenance resource categories for troops equipment andincreasingly heavy security burden Reference [16] evaluatessupply chain partnership using 9 d cobweb model and solvesa variety of comprehensive evaluations with different polarityobjectives by means of grey decision-making theory

Throughout relevant researches on grey situationdecision-making it is found that the situation that multipledecision experts participate in decision-making and thosedecision experts have risk preference is rarely consideredHowever in practical decision-making process especiallyin the case that the scheme set has numerous qualitativeindexes multiple decision experts should participate indecision-making and inevitably their subjective preferencewill influence the final decision results Therefore this papermakes use of prospect theory [17 18] to deal with decisionexpert sample matrix and reveals the decision expertsrsquodecision psychology and risk attitudes that contribute to riskaversion in choices involving sure gains and risk seeking inchoices involving sure losses Meanwhile as for the questionthat it is hard to determine expertsrsquo weight in group decisionthis paper develops the optimal and the worst prospectvalue matrix for decision experts situation gets decisionexpertsrsquo weight based on TOPSIS method collects expertsopinions sorts situations of each event and finally getsthe optimal situation This model provides a feasible andpractical method for grey situation group decision-makingin consideration of decision expertsrsquo risk preference

2 Main Methods and Results

21 Problem Description In a grey situation group decision-making problem the event set is denoted as 119860 = 119886

1 1198862

119886119899 the strategy set as 119861 = 119887

1 1198872 119887

119898 the decision-

making experts as 119863 = 1198891 1198892 119889

119901 the 119896th decision

expertrsquos situation set as 119878 = 119904119896119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861

and the effect sample value of the situation 119904119896119894119895isin 119878 given

by the 119896th decision expert under the target 119897 as 120583119896(119897)119894119895(otimes)(119894 =

1 2 119899 119895 = 1 2 119898 119897 = 1 2 119904 119896 = 1 2 119901) thenthere are the objective weight vector of 119904119896

119894119895for the kth decision

expertrsquos situation as 119882119896 = 120596119896

1 120596119896

2 120596

119896

119904 which satisfies

sum119904

119897=1120596119896

119897= 1 and the weight vector of decision experts as

119882119896= 120596119896

1 120596119896

2 120596

119896

119904which satisfiessum119901

119896=1120596119896= 1This paper

mainly discusses how to determine the optimal situation forgrey decision-making in the situation that multiple decisionexperts participate in decision-making process

The effect sample value matrix of the 119896th decision expertsituation 119904119896

119894119895under the objective 119897 can be expressed in the

following way

119880119896(119897)

1(otimes) =

[[[[[[[[[

[

[120583119896(119897)

11 120583119896(119897)

11] [120583119896(119897)

12 120583119896(119897)

12] sdot sdot sdot [120583

119896(119897)

1119898 120583119896(119897)

1119898]

[120583119896(119897)

21 120583119896(119897)

21] [120583119896(119897)

22 120583119896(119897)

22] sdot sdot sdot [120583

119896(119897)

2119898 120583119896(119897)

2119898]

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot

[120583119896(119897)

1198991 120583119896(119897)

1198991] [120583119896(119897)

1198992 120583119896(119897)

1198992] sdot sdot sdot [120583

119896(119897)

119899119898 120583119896(119897)

119899119898]

]]]]]]]]]

]

(1)

Definition 1 Given the corresponding value of 120583+119896(119897)119894119895

= maxmax(120583119896(119897)

119894119895+ 120583119896(119897)

119894119895)2(1 le 119894 le 119899 1 le 119895 le 119898 119897 = 1 2

119904 119896 = 1 2 119901) as [120583+119896(119897)119894119895

120583+119896(119897)

119894119895] 120583+119896(119897)119894119895

is the positive ideal

value measure of the 119896th decision expert situation 119904119896119894119895under

the objective 119897Given the corresponding value of 120583minus119896(119897)

119894119895= min

min(120583119896(119897)119894119895+ 120583119896(119897)

119894119895)2 (119896 = 1 2 119901) as [120583minus119896(119897)

119894119895 120583minus119896(119897)

119894119895] 120583minus119896(119897)119894119895

is the negative ideal value measure of the 119896th decision expertsituation 119904119896

119894119895under the objective 119897

Definition 2 Given the value measure of the 119896th decisionexpert situation 119904119896

119894119895under the objective 119897 as 120583119896(119897)

119894119895 the positive

ideal value measure as 120583+119896(119897)119894119895

and the negative ideal valuemeasure as 120583minus119896(119897)

119894119895 then (2) and (3) are positive and negative

ideal situation distance of the 119896th decision expert situation 119904119896119894119895

under the objective 119897 respectively Consider

119889 (120583119896(119897)

119894119895 120583+119896(119897)

119894119895) = radic[120583+119896(119897)

119894119895minus 120583119896(119897)

119894119895]2+ [120583+119896(119897)

119894119895minus 120583119896(119897)

119894119895]2

(2)

119889 (120583119896(119897)

119894119895 120583minus119896(119897)

119894119895) = radic[120583minus119896(119897)

119894119895minus 120583119896(119897)

119894119895]2+ [120583119896(119897)

119894119895minus 120583119896(119897)

119894119895]2

(3)

22 Prospect Value of Grey Situation Effect Measure

221 Efficiency Index Assuming that the effect measure ofthe situation 119904119896

119894119895given by the 119896th decision expert is efficiency

index if negative ideal situation distance is taken as referencepoint the situation distance of alternatives is superior tothat of negative ideal scheme and at this time psychologicalperception of the decision expert is gains The value functionis V+[119904minus119896(119897)

119894119895] = [119889(120583

119896(119897)

119894119895 120583minus119896(119897)

119894119895)]120572

However the situation distance of alternatives cannot beworse than that of negative ideal scheme That is to say if thesituation distance of negative ideal is taken as reference pointits value function will become Vminus[119904minus119896(119897)

119894119895] = 0

If the situation distance of positive ideal is taken asreference point the situation distance of alternatives willbe worse than that of positive ideal scheme At this timepsychological perception of decision expert is risksThe valuefunction is Vminus[119904+119896(119897)

119894119895] = minus120579[minus119889(120583

119896(119897)

119894119895 120583+119896(119897)

119894119895)]120573

The Scientific World Journal 3

However the situation distance of alternatives cannot besuperior to that of positive ideal scheme That is to say if thesituation distance of positive ideal is taken as reference pointits value function will become V+[119904+119896(119897)

119894119895] = 0

222 Cost Index Assuming that the effect measure of thesituation 119904119896

119894119895given by the 119896th decision expert is cost index

if positive ideal situation distance is taken as referencepoint the situation distance of alternatives is superior tothat of positive ideal scheme and at this time psychologicalperception of decision expert is gains The value function isV+[119904+119896(119897)119894119895

] = [119889(120583119896(119897)

119894119895 120583+119896(119897)

119894119895)]120572

However the situation distance of alternatives cannot besuperior to that of positive ideal scheme That is to say if thesituation distance of positive ideal is taken as reference pointits value function will become Vminus[119904+119896(119897)

119894119895] = 0

If the situation distance of negative ideal is taken asreference point the situation distance of alternatives willbe worse than that of positive ideal scheme At this timepsychological perception of decision expert is risksThe valuefunction is Vminus[119904minus119896(119897)

119894119895] = minus120579[minus119889(120583

119896(119897)

119894119895 120583minus119896(119897)

119894119895)]120573

However the situation distance of alternatives cannot beworse than that of negative ideal scheme That is to say if thesituation distance of negative ideal is taken as reference pointits value function will become V+[119904minus119896(119897)

119894119895] = 0

No matter in the case of efficiency index or cost indexboth V+[119904+119896(119897)

119894119895] and V+[119904minus119896(119897)

119894119895] show that the psychological

perception of decision expert for alternatives measure isrisk aversion and is expressed as positive value of gainsBoth Vminus[119904+119896(119897)

119894119895] and Vminus[119904minus119896(119897)

119894119895] reveal that the psychological

perception of decision expert for alternatives measure isrisk seeking and is expressed as negative value of lossesParameters 120572 and 120573 express the concave and convex degreeof gains and losses value power function respectively and120572 120573 lt 1 shows decision expertsrsquo sensitivity is decreasing 120579expresses that the losses curve is much deeper than the gainscurve and 120579 gt 1 reveals loss aversion [17]

Nomatter in the case of efficiency index or cost index theprospect value by taking positive and negative ideal situationdistance as different reference points is positive value of gainsand negative value of risks in decision expertsrsquo psychologicalperception In order to eliminate the influences of dimensionto calculation results the prospect value gotten from positiveand negative ideal situation as different reference points isnormalized as follows

Vplusmn [119904plusmn119896(119897)119894119895

] =

Vplusmn [119904plusmn119896(119897)119894119895

]

max119894=12119899

10038161003816100381610038161003816Vplusmn [119904plusmn119896(119897)119894119895

]10038161003816100381610038161003816

(4)

Normalized prospect value Vplusmn[119904plusmn119896(119897)119894119895

] (119894 = 1 2 119899 119895 =1 2 119898 119897 = 1 2 119904 119896 = 1 2 119901) has the followingproperties (1) Vplusmn[119904plusmn119896(119897)

119894119895] is dimensionless (2) the better the

situation the bigger Vplusmn[119904plusmn119896(119897)119894119895

] and the worse the situationthe smaller the Vplusmn[119904plusmn119896(119897)

119894119895] (3) V+[119904plusmn119896(119897)

119894119895] isin [0 1] Vminus[119904plusmn119896(119897)

119894119895] isin

[minus1 0]

Proof Consider the following

Case (i) For V+[119904plusmn119896(119897)119894119895

] = V+[119904plusmn119896(119897)119894119895

]max119894=12119899

V+[119904plusmn119896(119897)119894119895

]considering that 0 le V+[119904plusmn119896(119897)

119894119895] le max

119894=12119899V+[119904plusmn119896(119897)119894119895

] thebetter the V+[119904plusmn119896(119897)

119894119895] the bigger the V+[119904plusmn119896(119897)

119894119895] and V+[119904plusmn119896(119897)

119894119895] isin

[0 1] Thus properties (1) (2) and (3) are proved

Case (ii) For Vminus[119904plusmn119896(119897)119894119895

] = minusVminus[119904plusmn119896(119897)119894119895

]min119894=12119899

Vminus[119904plusmn119896(119897)119894119895

]considering that minusmin

119894=12119899Vminus[119904plusmn119896(119897)119894119895

] le Vminus[119904plusmn119896(119897)119894119895

] le 0 theworse the Vminus[119904plusmn119896(119897)

119894119895] the smaller the Vplusmn[119904plusmn119896(119897)

119894119895] and Vminus[119904plusmn119896(119897)

119894119895] isin

[minus1 0] Thus properties (1) (2) and (3) are proved

According to the value function given by [17] assumingthat the prospect weight that decision experts have whenfacing gains and losses is 120587+(120596119896

119897) and 120587minus(120596119896

119897) respectively

the comprehensive prospect value of the scheme is thesum of positive and negative prospect values Suppose thatdecision experts take the positive and negative ideal situationdistance of the situation 119904119896

119894119895under objective 119897 as different

references points and meanwhile they think the distancebetween reference points and the positive ideal situation is asimportant as that between reference points and the negativeideal situation distance and then comprehensive prospectvalue based on positive and negative ideal situation distanceas two reference points can be defined as follows

119881119896

119894119895=

119904

sum

119897=1

V+ [119904minus119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904minus119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

+

119904

sum

119897=1

V+ [119904+119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904+119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

=

119904

sum

119897=1

V+ [119904minus119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904+119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

120587+(120596119896

119897) =

(120596119896

119897)120574

(120596119896

119897)120574+ [1 minus (120596

119896

119897)]1205741120574

120587minus(120596119896

119897) =

(120596119896

119897)120575

(120596119896

119897)120575

+ [1 minus (120596119896

119897)]120575

1120575

(5)

Therefore the comprehensive prospect based on positiveand negative ideal situation distance as two reference pointsis the sum of positive prospect value taking negative idealsituation distance as reference point and negative prospectvalue taking positive ideal situation distance as referencepoint

According to [17] the parameters of prospect valuefunction andweight function in this paper are consistent withpractical empirical data as 120572 = 120573 = 088 120579 = 225 120574 = 061and 120575 = 069

4 The Scientific World Journal

23 Determination of Decision Expertsrsquo Weight Vector

Definition 3 Assume that119881+ = 119881+1119895 119881+

2119895 119881

+

119899119895 is the opti-

mal prospect valuematrix of all the decision expertsrsquo decisionnamely the optimal situation matrix of the situations for alldecision expertsrsquo choices where 119881+

119894119895= max

119896=12119901119881119896

119894119895(119894 =

1 2 119899 119895 = 1 2 119898)Assume that119881minus = 119881minus

1119895 119881minus

2119895 119881

minus

119899119895 is the worst prospect

value matrix of all the decision expertsrsquo decision namelythe worst situation matrix of the situations for all decisionexpertsrsquo choices where 119881minus

119894119895= min

119896=12119901119881119896

119894119895(119894 = 1 2 119899

119895 = 1 2 119898)

Definition 4 Given 119881119896 = 119881119896

1119895 119881119896

2119895 119881

119896

119899119895 is the prospect

value matrix of the 119896th decision expertrsquos situation 119904119896119894119895 then

119889 (119881119896 119881+)

= radic

119898

sum

119895=1

(119881119896

1119895minus 119881+

1119895)2

+

119898

sum

119895=1

(119881119896

2119895minus 119881+

2119895)2

+ sdot sdot sdot +

119898

sum

119895=1

(119881119896

119899119895minus 119881+

119899119895)2

(6)is the distance between the prospect value matrix of the 119896thdecision expertsrsquo situation and the optimal prospect valuematrix Given 119881119896 = 119881

119896

1119895 119881119896

2119895 119881

119896

119899119895 is the prospect value

matrix of the 119896th decision expert situation 119904119896119894119895 then

119889 (119881119896 119881minus)

= radic

119898

sum

119895=1

(119881119896

1119895minus 119881minus

1119895)2

+

119898

sum

119895=1

(119881119896

2119895minus 119881minus

2119895)2

+ sdot sdot sdot +

119898

sum

119895=1

(119881119896

119899119895minus 119881minus

119899119895)2

(7)

is the distance between the prospect value matrix of the119896th decision expertsrsquo situation and the worst prospect valuematrix

In order to avoid one-sidedness of evaluation resultsmake full use of decision information and keep the con-sistency of decision results the greatest weight is given tothe decision expert whose matrix is closest with the optimalprospect value and is the most far away from the worstprospect value matrix and vice versa This paper takesthe optimal prospect value matrix as the optimal situationand the worst prospect value matrix as the worst situationand calculates the prospect value close degree of decisionexpertsrsquo situation according to traditional TOPSIS conceptand in view of the distance between prospect value matrixof multiple decision experts and the optimal and the worstprospect value matrix Consider

119862lowast

119896=

119889 (119881119896 119881+)

119889 (119881119896 119881+) + 119889 (119881119896 119881minus) (8)

Then the weight of the 119896th decision expert is

120582119896=

119862lowast

119896

sum119901

119896=1119862lowast

119896

(9)

And then the comprehensive prospect value matrixbased on the weight vector of decision experts is

119881 = [V119894119895] =

119901

sum

119896=1

120582119901sdot 119881119896

119894119895 (10)

Definition 5 Given the comprehensive prospect valuematrix119881 if max

1le119895le119898V119894119895 = V1198941198950 1198871198950is optimal strategy for the event

119886119894 if max

1le119895le119898V119894119895 = V

1198940119895 1198861198940is the corresponding optimal

event of strategy 119887119895 if max

1le119895le119898V119894119895 = V11989401198950

11990411989401198950

is the optimalsituation

24 Decision-Making Procedure To sum up the procedureof grey situation group decision-making method based onprospect theory is as follows

Step 1 According to the event set 119860 = 1198861 1198862 119886

119899 the

strategy set 119861 = 1198871 1198872 119887

119898 and decision experts set

119863 = 1198891 1198892 119889

119901 the situation set of multiple decision

experts is 119878 = 119904119896119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861

Step 2 Determine the decision objective and give corre-sponding effect samplematrix ofmultiple decision experts forobjective 119897 = 1 2 119904

Step 3 In light of Definitions 1 and 2 calculate the positiveand negative ideal situation distance Take positive and nega-tive ideal situation distances as different reference points andcalculate and normalize the positive and negative prospectvalue matrix

Step 4 Give target weight vectors of the situation given byeach decision expert plug those target weight vectors intoformula (5) and get comprehensive prospectmatrix based ontwo reference points of positive and negative ideal situationdistances

Step 5 According to Definition 3 determine the optimal andworst prospect value matrix of all decision experts Andin light of Definition 4 calculate the distances between theprospect value matrix of multiple decision experts situationand the optimal and the worst prospect value matrix

Step 6 According to formulas (8) and (9) calculate theweight of expertsrsquo decisions In light of formula (10) getcomprehensive prospect value matrix based on the matrixof decision expertsrsquo weight vector and finally determine theoptimal situation

3 Example Analysis

A new round of aid program for Xinjiang is an importantdecision and strategic deployment made by the Party CentralCommittee and the State Council to promote Xinjiangrsquosdevelopment and maintain Xinjiangrsquos social stability requir-ing that 19 provinces and cities in the mid-east region aidXinjiangrsquos development Since March 2010 new industryproduction lines in Xinjiang are increasing day by daySuppose that there are 3 production lines in an industry area

The Scientific World Journal 5

4 suppliers are shortlisted in selecting suppliers and there are3 experts in decision-making expert group

Step 1 Choose suppliersrsquo choices in 3 production lines as theevents and the event set is119860 = 119886

1 1198862 1198863 Supplier 1 supplier

2 supplier 3 and supplier 4 constitute the strategy set 119861 =

1198871 1198872 1198873 1198874 Expert 1 expert 2 and expert 3 constitute the

decision expert set119863 = 1198891 1198892 1198893The event set strategy set

and decision expert set constitute decision expertsrsquo situationset 119878 = 119904119896

119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861 119896 isin 119863 119894 = 1 2 3 119895 =

1 2 3 4 119896 = 1 2 3

Step 2 Determine decision-making objective After severalrounds of specialist researches take quality design andprice as decision-making objectives where quality and designare qualitative index Decision experts make assessment bymeans of expert scoring method The higher the score isthe more uncertain the index is Decision experts give scoresbetween 0 and 10 Price is cost index the lower the data whichthe suppliers provided the better the decision Effect samplevalue matrix given by multiple experts with regard to thequality and design of decision-making objectives is as follows

1198801(1)

1= [

[

[5 7] [6 7] [5 6] [7 9]

[3 5] [4 6] [4 6] [3 4]

[5 6] [4 6] [6 8] [5 7]

]

]

1198801(2)

1= [

[

[4 5] [5 7] [4 6] [3 5]

[5 6] [4 5] [4 7] [4 6]

[5 7] [3 6] [7 9] [4 5]

]

]

1198802(1)

1= [

[

[5 6] [6 8] [5 6] [8 9]

[3 4] [4 6] [4 5] [4 5]

[4 6] [5 7] [7 9] [5 6]

]

]

1198802(2)

1= [

[

[4 6] [6 7] [4 5] [3 6]

[5 7] [4 6] [4 7] [4 7]

[5 8] [4 6] [8 10] [5 8]

]

]

1198803(1)

1= [

[

[4 5] [6 7] [5 7] [8 9]

[3 4] [4 5] [3 4] [4 6]

[4 5] [5 6] [7 8] [3 5]

]

]

1198803(2)

1= [

[

[3 4] [5 6] [4 6] [3 4]

[5 6] [4 6] [5 7] [4 5]

[7 8] [4 7] [8 9] [6 8]

]

]

(11)

The decision objective target is an effect sample valuematrix of suppliersrsquo offers Because this is not assessed bydecision experts the sample value matrix of target effect isthe same Consider

119880123(3)

1= [

[

8 7 9 8

15 12 13 14

11 13 13 10

]

]

(12)

Step 3 Calculate the distances of the positive and the negativeideal situation according toDefinitions 1 and 2 Calculate pos-itive and negative prospect value matrix taking the distancesof the positive and the negative ideal situation as different

reference points The matrix for normalized Vplusmn[119904plusmn1(1)119894119895

] is asfollows

V+ [119904minus1(1)119894119895

] = [

[

06516 06999 05159 1

03803 04926 04926 02541

05159 04926 08482 06516

]

]

Vminus [119904minus1(1)119894119895

] = 0

Vminus [119904+1(1)119894119895

] = [

[

minus06345 minus04926 minus06516 minus04677

minus09495 minus07856 minus07856 minus1

minus06516 minus07856 minus05159 minus06345

]

]

Vminus [119904+1(1)119894119895

] = 0

(13)

Other normalized positive and negative prospect valuematrixes can be gotten in the same way

Step 4 The weight vectors of situation given by multipleexperts are 1198821 = [04 03 03] 1198822 = [05 025 025] and1198823= [05 02 03] Plug those objective weight vectors into

formula (5) and get comprehensive prospect value based ontwo reference points of the positive and the negative idealsituation distances Consider

1198811

119894119895= [

[

00598 03783 minus00239 02401

minus06592 minus03916 minus03653 minus06721

minus00976 minus04398 00679 minus01091

]

]

1198812

119894119895= [

[

00084 03999 minus01129 04052

minus07221 minus03550 minus04722 minus05486

minus01765 minus02935 02253 minus00399

]

]

1198813

119894119895= [

[

minus01744 03345 00537 03785

minus07625 04141 minus05374 minus05472

01359 minus03061 02095 minus01347

]

]

(14)

Step 5 Determine the optimal and the worst prospect valuematrix based on decision expertsrsquo judgment according toDefinition 3 and calculate the distances between the prospectvalue matrix of multiple decision expertsrsquo situation and theoptimal and the worst prospect value matrix according toDefinition 4 Consider

119889 (1198811 119881+) = 07993 119889 (119881

1 119881minus) = 01119

119889 (1198812 119881+) = 07351 119889 (119881

2 119881minus) = 01426

119889 (1198813 119881+) = 01093 119889 (119881

3 119881minus) = 08472

(15)

Step 6 Calculate the weight of expertsrsquo strategy according toformulas (8) and (9) as 120582

1= 04796 120582

2= 04579 120582

3=

00625The comprehensive prospect valuematrix of situationbased on decision experts weight vector is as follows

119881 = [

[

00216 03855 minus00598 03243

minus06945 minus03245 minus04250 minus06077

minus01191 minus03645 01488 minus00790

]

]

(16)

According to comprehensive prospect value matrixchoosing supplier 2 is the optimal strategy for production

6 The Scientific World Journal

line 1 choosing supplier 2 is the optimal strategy for supplier2 choosing supplier 3 is the optimal strategy for supplier3 providing production line 1 is the most appropriate forsupplier 1 providing production line 1 is themost appropriatefor supplier 2 providing production line 3 is the mostappropriate for supplier 3 and providing production line 1is the most appropriate for supplier 4 In a word the optimalsituation is 119878

12

According to themethods used to calculate grey situationin group decision-making in [3] assuming that the expertsrsquoknowledge is the same (ie the expertrsquos weight is equal) theweight vectors of the situation given by multiple experts areas follows1198821 = [04 03 03]1198822 = [05 025 025]1198823 =[05 02 03]

The comprehensive effect sample valuematrix of situationbased on normative approaches given by [4] is as follows

119877 = [

[0179 0269] [0235 0342] [0187 0293] [0221 0330]

[0183 0304] [0204 0348] [0197 0346] [0189 0324]

[0193 0299] [0169 0289] [0149 0379] [0189 0301]

]

(17)

According to the effect sample value matrix without thedecision makersrsquo behavioral preference the optimal situationis 11987812as well However the expertsrsquo weights are relying on the

subjective experience and judgment If the expertsrsquo weightsare different there may be different results

From the example analysis and comparison the methodstudies the group decision problem with a view to decisionexpertsrsquo risk attitude and effect sample value considers deci-sion expertsrsquo decision-making psychology takes referencepoints as the positive and the negative ideal situation groupdistances gets comprehensive prospect value matrix undermultiple expertsrsquo evaluation plugs the decision expertsrsquo riskpreference into grey situation group decision andmakes surethat the final decision conforms to the psychology behaviorof decision experts Meanwhile determine multiple decisionexpertsrsquo weight based on TOPSIS method by consideringthe close degree between expertrsquos individual evaluation andoverall assessment Compared with the former subjectivevalue methods this method is more objective and is almostnot relying on subjective experience and judgment

4 Conclusion

This paper puts forward a grey situation group decision-making method based on the prospect theory aiming atsolving such problem that the grey situation group decisionmodel does not consider the decision expertsrsquo risk pref-erence According to various types of effect sample valuedifferent prospect value functions are constructed the valuefunctions for two reference points are integrated strengthsand weaknesses are measured by means of prospect valueand the prospect value is normalized to [minus1 1] This paperdetermines multiple decision expertsrsquo weight by means ofTOPSIS method collects expertsrsquo opinion makes sequencefor each eventrsquos situation and gets the optimal situationat last This method is convenient for computer procedureoperation and provides a new method to solve problem ofgrey group situation taking decision expertsrsquo risk preference

into consideration However the parameters of the prospectvalue functions and the weight functions are usually gainedby experiments The determination of the reasonable param-eters still needs to be discussed and further researched Inview of the fact that decision experts discuss again andagain evaluation process andmodify assessment informationfurther study on multiround interactive grey situation groupdecision is needed

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work and there is no profes-sional or other personal interest of any nature or kind in anyproduct service andor company that could be construed asinfluencing the position presented in or the review of thepaper

Acknowledgments

This paper is sponsored by the National Natural ScienceFoundation of China (no 71363046 no 71271226 no71301064) the Humanistic and Social Science Foundationof Ministry of Education of China (no 10YJA790174) theHumanistic and Social Science Youth Foundation ofMinistryof Education of China (no 13YJC790198)

References

[1] J L Deng ldquoIntelligence space in grey situation decisionrdquo TheJournal of Grey System vol 10 no 3 pp 254ndash262 1998

[2] J L Deng Prediction and Grey Decision Press of HuazhongUinversity of Science amp Technology Wuhan China 2002

[3] S F Liu and Y Lin Grey Information Theory and PracticalApplications Springer London UK 2006

[4] Z XWang Y GDang andC P Song ldquoMultiobjective decisionmodel of the grey situation based on interval numberrdquo Controland Decision vol 24 no 3 pp 388ndash392 2009

[5] J Luo Y Liu and K Xu ldquoParameter establishment of intelligentalgorithmbased onuniformdesign and grey situation decisionrdquoJournal of Shenyang University of Technology vol 32 no 1 pp84ndash89 2010

[6] Y Liu J Forrest H H Zhao S F Liu and J Liu ldquoMulti-objective grey situation decision making method based onprospect theoryrdquo Systems Engineering and Electronics vol 34no 12 pp 2514ndash2519 2012

[7] Y Liu J Forrest and S F Liu ldquoMulti-objective grey situationdecision making model and application with interval numbersbased on linear transformation operator with rewarding goodand punishingrdquo Statistics amp Information Forum vol 12 no 7pp 22ndash26 2012

[8] L Zhou and D Luo ldquoStudy on decision-making method formulti-objective grey situation decisionrdquo Journal of North ChinaInstitute of Water Conservancy and Hydroelectric Power vol 31no 4 pp 150ndash153 2010

[9] Y MWang and Y G Dang ldquoNewmethod in situation decisionmaking based on entropyrdquo Systems Engineering and Electronicsvol 31 no 6 pp 1350ndash1352 2009

The Scientific World Journal 7

[10] D J Chen SH Zhang and J Y Chen ldquoResearch of gray generalsituation decision and its applicationrdquo Systems Engineering andElectronics vol 26 no 4 pp 423ndash429 2004

[11] C T Lin N H Shinh and W C Chuang ldquoStock tradingstrategy based on grey situation decision model an empiricalstudy of the electronic component industry in Taiwanrdquo Journalof Grey System vol 18 no 3 pp 239ndash250 2006

[12] C T Lin and H F Chang ldquoPerformance evaluation for medicalindustry in Taiwan applying grey situation decision makingrdquoJournal of Grey System vol 22 no 3 pp 219ndash226 2010

[13] X Shan X Z Wang and S X Liu ldquoDecision-making ofmissile flexible support sites locating in wartime with fuzzynumbers and grey situationrdquo Journal of Naval Aeronautical andAstronautical University vol 26 no 6 pp 703ndash706 2011

[14] G X Huang X B Wang and Y L Wang ldquoScheme theartillery fire plan by grey situation analysisrdquo Ordnance IndustryAutomation vol 31 no 2 pp 31ndash37 2012

[15] Y B Wang X S Jia and G Y Wang ldquoMethod of integrationfor currency maintenance resources based on grey situationdecision-makingrdquo Fire Control ampCommandControl vol 38 no4 pp 4ndash8 2013

[16] L M Chen P Zheng and S X Ji ldquoGrey situation evaluation ofsupply chain cooperative relationship based on cobweb modelrdquoLogisticstech vol 31 no 7 pp 321ndash323 2012

[17] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Economica vol 47 no 2 pp 263ndash291 1979

[18] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

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

2 The Scientific World Journal

for medical industry by means of grey situation decision-making method Reference [13] assumes that all the attributeindexes are in the state of trapezoid fuzzy numbers andmakesoptimization for site selection alternatives by grey situationdecision-making method Reference [14] develops optimalmodel for artillery fire scheme on the basis of grey situationdecision-making method aiming at solving the problemsof information uncertainty and information incompletenessin the battlefield Reference [15] suggests a general mainte-nance resource integration method based on grey situationdecision-making theory on account of growing number ofmaintenance resource categories for troops equipment andincreasingly heavy security burden Reference [16] evaluatessupply chain partnership using 9 d cobweb model and solvesa variety of comprehensive evaluations with different polarityobjectives by means of grey decision-making theory

Throughout relevant researches on grey situationdecision-making it is found that the situation that multipledecision experts participate in decision-making and thosedecision experts have risk preference is rarely consideredHowever in practical decision-making process especiallyin the case that the scheme set has numerous qualitativeindexes multiple decision experts should participate indecision-making and inevitably their subjective preferencewill influence the final decision results Therefore this papermakes use of prospect theory [17 18] to deal with decisionexpert sample matrix and reveals the decision expertsrsquodecision psychology and risk attitudes that contribute to riskaversion in choices involving sure gains and risk seeking inchoices involving sure losses Meanwhile as for the questionthat it is hard to determine expertsrsquo weight in group decisionthis paper develops the optimal and the worst prospectvalue matrix for decision experts situation gets decisionexpertsrsquo weight based on TOPSIS method collects expertsopinions sorts situations of each event and finally getsthe optimal situation This model provides a feasible andpractical method for grey situation group decision-makingin consideration of decision expertsrsquo risk preference

2 Main Methods and Results

21 Problem Description In a grey situation group decision-making problem the event set is denoted as 119860 = 119886

1 1198862

119886119899 the strategy set as 119861 = 119887

1 1198872 119887

119898 the decision-

making experts as 119863 = 1198891 1198892 119889

119901 the 119896th decision

expertrsquos situation set as 119878 = 119904119896119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861

and the effect sample value of the situation 119904119896119894119895isin 119878 given

by the 119896th decision expert under the target 119897 as 120583119896(119897)119894119895(otimes)(119894 =

1 2 119899 119895 = 1 2 119898 119897 = 1 2 119904 119896 = 1 2 119901) thenthere are the objective weight vector of 119904119896

119894119895for the kth decision

expertrsquos situation as 119882119896 = 120596119896

1 120596119896

2 120596

119896

119904 which satisfies

sum119904

119897=1120596119896

119897= 1 and the weight vector of decision experts as

119882119896= 120596119896

1 120596119896

2 120596

119896

119904which satisfiessum119901

119896=1120596119896= 1This paper

mainly discusses how to determine the optimal situation forgrey decision-making in the situation that multiple decisionexperts participate in decision-making process

The effect sample value matrix of the 119896th decision expertsituation 119904119896

119894119895under the objective 119897 can be expressed in the

following way

119880119896(119897)

1(otimes) =

[[[[[[[[[

[

[120583119896(119897)

11 120583119896(119897)

11] [120583119896(119897)

12 120583119896(119897)

12] sdot sdot sdot [120583

119896(119897)

1119898 120583119896(119897)

1119898]

[120583119896(119897)

21 120583119896(119897)

21] [120583119896(119897)

22 120583119896(119897)

22] sdot sdot sdot [120583

119896(119897)

2119898 120583119896(119897)

2119898]

sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot

[120583119896(119897)

1198991 120583119896(119897)

1198991] [120583119896(119897)

1198992 120583119896(119897)

1198992] sdot sdot sdot [120583

119896(119897)

119899119898 120583119896(119897)

119899119898]

]]]]]]]]]

]

(1)

Definition 1 Given the corresponding value of 120583+119896(119897)119894119895

= maxmax(120583119896(119897)

119894119895+ 120583119896(119897)

119894119895)2(1 le 119894 le 119899 1 le 119895 le 119898 119897 = 1 2

119904 119896 = 1 2 119901) as [120583+119896(119897)119894119895

120583+119896(119897)

119894119895] 120583+119896(119897)119894119895

is the positive ideal

value measure of the 119896th decision expert situation 119904119896119894119895under

the objective 119897Given the corresponding value of 120583minus119896(119897)

119894119895= min

min(120583119896(119897)119894119895+ 120583119896(119897)

119894119895)2 (119896 = 1 2 119901) as [120583minus119896(119897)

119894119895 120583minus119896(119897)

119894119895] 120583minus119896(119897)119894119895

is the negative ideal value measure of the 119896th decision expertsituation 119904119896

119894119895under the objective 119897

Definition 2 Given the value measure of the 119896th decisionexpert situation 119904119896

119894119895under the objective 119897 as 120583119896(119897)

119894119895 the positive

ideal value measure as 120583+119896(119897)119894119895

and the negative ideal valuemeasure as 120583minus119896(119897)

119894119895 then (2) and (3) are positive and negative

ideal situation distance of the 119896th decision expert situation 119904119896119894119895

under the objective 119897 respectively Consider

119889 (120583119896(119897)

119894119895 120583+119896(119897)

119894119895) = radic[120583+119896(119897)

119894119895minus 120583119896(119897)

119894119895]2+ [120583+119896(119897)

119894119895minus 120583119896(119897)

119894119895]2

(2)

119889 (120583119896(119897)

119894119895 120583minus119896(119897)

119894119895) = radic[120583minus119896(119897)

119894119895minus 120583119896(119897)

119894119895]2+ [120583119896(119897)

119894119895minus 120583119896(119897)

119894119895]2

(3)

22 Prospect Value of Grey Situation Effect Measure

221 Efficiency Index Assuming that the effect measure ofthe situation 119904119896

119894119895given by the 119896th decision expert is efficiency

index if negative ideal situation distance is taken as referencepoint the situation distance of alternatives is superior tothat of negative ideal scheme and at this time psychologicalperception of the decision expert is gains The value functionis V+[119904minus119896(119897)

119894119895] = [119889(120583

119896(119897)

119894119895 120583minus119896(119897)

119894119895)]120572

However the situation distance of alternatives cannot beworse than that of negative ideal scheme That is to say if thesituation distance of negative ideal is taken as reference pointits value function will become Vminus[119904minus119896(119897)

119894119895] = 0

If the situation distance of positive ideal is taken asreference point the situation distance of alternatives willbe worse than that of positive ideal scheme At this timepsychological perception of decision expert is risksThe valuefunction is Vminus[119904+119896(119897)

119894119895] = minus120579[minus119889(120583

119896(119897)

119894119895 120583+119896(119897)

119894119895)]120573

The Scientific World Journal 3

However the situation distance of alternatives cannot besuperior to that of positive ideal scheme That is to say if thesituation distance of positive ideal is taken as reference pointits value function will become V+[119904+119896(119897)

119894119895] = 0

222 Cost Index Assuming that the effect measure of thesituation 119904119896

119894119895given by the 119896th decision expert is cost index

if positive ideal situation distance is taken as referencepoint the situation distance of alternatives is superior tothat of positive ideal scheme and at this time psychologicalperception of decision expert is gains The value function isV+[119904+119896(119897)119894119895

] = [119889(120583119896(119897)

119894119895 120583+119896(119897)

119894119895)]120572

However the situation distance of alternatives cannot besuperior to that of positive ideal scheme That is to say if thesituation distance of positive ideal is taken as reference pointits value function will become Vminus[119904+119896(119897)

119894119895] = 0

If the situation distance of negative ideal is taken asreference point the situation distance of alternatives willbe worse than that of positive ideal scheme At this timepsychological perception of decision expert is risksThe valuefunction is Vminus[119904minus119896(119897)

119894119895] = minus120579[minus119889(120583

119896(119897)

119894119895 120583minus119896(119897)

119894119895)]120573

However the situation distance of alternatives cannot beworse than that of negative ideal scheme That is to say if thesituation distance of negative ideal is taken as reference pointits value function will become V+[119904minus119896(119897)

119894119895] = 0

No matter in the case of efficiency index or cost indexboth V+[119904+119896(119897)

119894119895] and V+[119904minus119896(119897)

119894119895] show that the psychological

perception of decision expert for alternatives measure isrisk aversion and is expressed as positive value of gainsBoth Vminus[119904+119896(119897)

119894119895] and Vminus[119904minus119896(119897)

119894119895] reveal that the psychological

perception of decision expert for alternatives measure isrisk seeking and is expressed as negative value of lossesParameters 120572 and 120573 express the concave and convex degreeof gains and losses value power function respectively and120572 120573 lt 1 shows decision expertsrsquo sensitivity is decreasing 120579expresses that the losses curve is much deeper than the gainscurve and 120579 gt 1 reveals loss aversion [17]

Nomatter in the case of efficiency index or cost index theprospect value by taking positive and negative ideal situationdistance as different reference points is positive value of gainsand negative value of risks in decision expertsrsquo psychologicalperception In order to eliminate the influences of dimensionto calculation results the prospect value gotten from positiveand negative ideal situation as different reference points isnormalized as follows

Vplusmn [119904plusmn119896(119897)119894119895

] =

Vplusmn [119904plusmn119896(119897)119894119895

]

max119894=12119899

10038161003816100381610038161003816Vplusmn [119904plusmn119896(119897)119894119895

]10038161003816100381610038161003816

(4)

Normalized prospect value Vplusmn[119904plusmn119896(119897)119894119895

] (119894 = 1 2 119899 119895 =1 2 119898 119897 = 1 2 119904 119896 = 1 2 119901) has the followingproperties (1) Vplusmn[119904plusmn119896(119897)

119894119895] is dimensionless (2) the better the

situation the bigger Vplusmn[119904plusmn119896(119897)119894119895

] and the worse the situationthe smaller the Vplusmn[119904plusmn119896(119897)

119894119895] (3) V+[119904plusmn119896(119897)

119894119895] isin [0 1] Vminus[119904plusmn119896(119897)

119894119895] isin

[minus1 0]

Proof Consider the following

Case (i) For V+[119904plusmn119896(119897)119894119895

] = V+[119904plusmn119896(119897)119894119895

]max119894=12119899

V+[119904plusmn119896(119897)119894119895

]considering that 0 le V+[119904plusmn119896(119897)

119894119895] le max

119894=12119899V+[119904plusmn119896(119897)119894119895

] thebetter the V+[119904plusmn119896(119897)

119894119895] the bigger the V+[119904plusmn119896(119897)

119894119895] and V+[119904plusmn119896(119897)

119894119895] isin

[0 1] Thus properties (1) (2) and (3) are proved

Case (ii) For Vminus[119904plusmn119896(119897)119894119895

] = minusVminus[119904plusmn119896(119897)119894119895

]min119894=12119899

Vminus[119904plusmn119896(119897)119894119895

]considering that minusmin

119894=12119899Vminus[119904plusmn119896(119897)119894119895

] le Vminus[119904plusmn119896(119897)119894119895

] le 0 theworse the Vminus[119904plusmn119896(119897)

119894119895] the smaller the Vplusmn[119904plusmn119896(119897)

119894119895] and Vminus[119904plusmn119896(119897)

119894119895] isin

[minus1 0] Thus properties (1) (2) and (3) are proved

According to the value function given by [17] assumingthat the prospect weight that decision experts have whenfacing gains and losses is 120587+(120596119896

119897) and 120587minus(120596119896

119897) respectively

the comprehensive prospect value of the scheme is thesum of positive and negative prospect values Suppose thatdecision experts take the positive and negative ideal situationdistance of the situation 119904119896

119894119895under objective 119897 as different

references points and meanwhile they think the distancebetween reference points and the positive ideal situation is asimportant as that between reference points and the negativeideal situation distance and then comprehensive prospectvalue based on positive and negative ideal situation distanceas two reference points can be defined as follows

119881119896

119894119895=

119904

sum

119897=1

V+ [119904minus119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904minus119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

+

119904

sum

119897=1

V+ [119904+119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904+119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

=

119904

sum

119897=1

V+ [119904minus119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904+119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

120587+(120596119896

119897) =

(120596119896

119897)120574

(120596119896

119897)120574+ [1 minus (120596

119896

119897)]1205741120574

120587minus(120596119896

119897) =

(120596119896

119897)120575

(120596119896

119897)120575

+ [1 minus (120596119896

119897)]120575

1120575

(5)

Therefore the comprehensive prospect based on positiveand negative ideal situation distance as two reference pointsis the sum of positive prospect value taking negative idealsituation distance as reference point and negative prospectvalue taking positive ideal situation distance as referencepoint

According to [17] the parameters of prospect valuefunction andweight function in this paper are consistent withpractical empirical data as 120572 = 120573 = 088 120579 = 225 120574 = 061and 120575 = 069

4 The Scientific World Journal

23 Determination of Decision Expertsrsquo Weight Vector

Definition 3 Assume that119881+ = 119881+1119895 119881+

2119895 119881

+

119899119895 is the opti-

mal prospect valuematrix of all the decision expertsrsquo decisionnamely the optimal situation matrix of the situations for alldecision expertsrsquo choices where 119881+

119894119895= max

119896=12119901119881119896

119894119895(119894 =

1 2 119899 119895 = 1 2 119898)Assume that119881minus = 119881minus

1119895 119881minus

2119895 119881

minus

119899119895 is the worst prospect

value matrix of all the decision expertsrsquo decision namelythe worst situation matrix of the situations for all decisionexpertsrsquo choices where 119881minus

119894119895= min

119896=12119901119881119896

119894119895(119894 = 1 2 119899

119895 = 1 2 119898)

Definition 4 Given 119881119896 = 119881119896

1119895 119881119896

2119895 119881

119896

119899119895 is the prospect

value matrix of the 119896th decision expertrsquos situation 119904119896119894119895 then

119889 (119881119896 119881+)

= radic

119898

sum

119895=1

(119881119896

1119895minus 119881+

1119895)2

+

119898

sum

119895=1

(119881119896

2119895minus 119881+

2119895)2

+ sdot sdot sdot +

119898

sum

119895=1

(119881119896

119899119895minus 119881+

119899119895)2

(6)is the distance between the prospect value matrix of the 119896thdecision expertsrsquo situation and the optimal prospect valuematrix Given 119881119896 = 119881

119896

1119895 119881119896

2119895 119881

119896

119899119895 is the prospect value

matrix of the 119896th decision expert situation 119904119896119894119895 then

119889 (119881119896 119881minus)

= radic

119898

sum

119895=1

(119881119896

1119895minus 119881minus

1119895)2

+

119898

sum

119895=1

(119881119896

2119895minus 119881minus

2119895)2

+ sdot sdot sdot +

119898

sum

119895=1

(119881119896

119899119895minus 119881minus

119899119895)2

(7)

is the distance between the prospect value matrix of the119896th decision expertsrsquo situation and the worst prospect valuematrix

In order to avoid one-sidedness of evaluation resultsmake full use of decision information and keep the con-sistency of decision results the greatest weight is given tothe decision expert whose matrix is closest with the optimalprospect value and is the most far away from the worstprospect value matrix and vice versa This paper takesthe optimal prospect value matrix as the optimal situationand the worst prospect value matrix as the worst situationand calculates the prospect value close degree of decisionexpertsrsquo situation according to traditional TOPSIS conceptand in view of the distance between prospect value matrixof multiple decision experts and the optimal and the worstprospect value matrix Consider

119862lowast

119896=

119889 (119881119896 119881+)

119889 (119881119896 119881+) + 119889 (119881119896 119881minus) (8)

Then the weight of the 119896th decision expert is

120582119896=

119862lowast

119896

sum119901

119896=1119862lowast

119896

(9)

And then the comprehensive prospect value matrixbased on the weight vector of decision experts is

119881 = [V119894119895] =

119901

sum

119896=1

120582119901sdot 119881119896

119894119895 (10)

Definition 5 Given the comprehensive prospect valuematrix119881 if max

1le119895le119898V119894119895 = V1198941198950 1198871198950is optimal strategy for the event

119886119894 if max

1le119895le119898V119894119895 = V

1198940119895 1198861198940is the corresponding optimal

event of strategy 119887119895 if max

1le119895le119898V119894119895 = V11989401198950

11990411989401198950

is the optimalsituation

24 Decision-Making Procedure To sum up the procedureof grey situation group decision-making method based onprospect theory is as follows

Step 1 According to the event set 119860 = 1198861 1198862 119886

119899 the

strategy set 119861 = 1198871 1198872 119887

119898 and decision experts set

119863 = 1198891 1198892 119889

119901 the situation set of multiple decision

experts is 119878 = 119904119896119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861

Step 2 Determine the decision objective and give corre-sponding effect samplematrix ofmultiple decision experts forobjective 119897 = 1 2 119904

Step 3 In light of Definitions 1 and 2 calculate the positiveand negative ideal situation distance Take positive and nega-tive ideal situation distances as different reference points andcalculate and normalize the positive and negative prospectvalue matrix

Step 4 Give target weight vectors of the situation given byeach decision expert plug those target weight vectors intoformula (5) and get comprehensive prospectmatrix based ontwo reference points of positive and negative ideal situationdistances

Step 5 According to Definition 3 determine the optimal andworst prospect value matrix of all decision experts Andin light of Definition 4 calculate the distances between theprospect value matrix of multiple decision experts situationand the optimal and the worst prospect value matrix

Step 6 According to formulas (8) and (9) calculate theweight of expertsrsquo decisions In light of formula (10) getcomprehensive prospect value matrix based on the matrixof decision expertsrsquo weight vector and finally determine theoptimal situation

3 Example Analysis

A new round of aid program for Xinjiang is an importantdecision and strategic deployment made by the Party CentralCommittee and the State Council to promote Xinjiangrsquosdevelopment and maintain Xinjiangrsquos social stability requir-ing that 19 provinces and cities in the mid-east region aidXinjiangrsquos development Since March 2010 new industryproduction lines in Xinjiang are increasing day by daySuppose that there are 3 production lines in an industry area

The Scientific World Journal 5

4 suppliers are shortlisted in selecting suppliers and there are3 experts in decision-making expert group

Step 1 Choose suppliersrsquo choices in 3 production lines as theevents and the event set is119860 = 119886

1 1198862 1198863 Supplier 1 supplier

2 supplier 3 and supplier 4 constitute the strategy set 119861 =

1198871 1198872 1198873 1198874 Expert 1 expert 2 and expert 3 constitute the

decision expert set119863 = 1198891 1198892 1198893The event set strategy set

and decision expert set constitute decision expertsrsquo situationset 119878 = 119904119896

119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861 119896 isin 119863 119894 = 1 2 3 119895 =

1 2 3 4 119896 = 1 2 3

Step 2 Determine decision-making objective After severalrounds of specialist researches take quality design andprice as decision-making objectives where quality and designare qualitative index Decision experts make assessment bymeans of expert scoring method The higher the score isthe more uncertain the index is Decision experts give scoresbetween 0 and 10 Price is cost index the lower the data whichthe suppliers provided the better the decision Effect samplevalue matrix given by multiple experts with regard to thequality and design of decision-making objectives is as follows

1198801(1)

1= [

[

[5 7] [6 7] [5 6] [7 9]

[3 5] [4 6] [4 6] [3 4]

[5 6] [4 6] [6 8] [5 7]

]

]

1198801(2)

1= [

[

[4 5] [5 7] [4 6] [3 5]

[5 6] [4 5] [4 7] [4 6]

[5 7] [3 6] [7 9] [4 5]

]

]

1198802(1)

1= [

[

[5 6] [6 8] [5 6] [8 9]

[3 4] [4 6] [4 5] [4 5]

[4 6] [5 7] [7 9] [5 6]

]

]

1198802(2)

1= [

[

[4 6] [6 7] [4 5] [3 6]

[5 7] [4 6] [4 7] [4 7]

[5 8] [4 6] [8 10] [5 8]

]

]

1198803(1)

1= [

[

[4 5] [6 7] [5 7] [8 9]

[3 4] [4 5] [3 4] [4 6]

[4 5] [5 6] [7 8] [3 5]

]

]

1198803(2)

1= [

[

[3 4] [5 6] [4 6] [3 4]

[5 6] [4 6] [5 7] [4 5]

[7 8] [4 7] [8 9] [6 8]

]

]

(11)

The decision objective target is an effect sample valuematrix of suppliersrsquo offers Because this is not assessed bydecision experts the sample value matrix of target effect isthe same Consider

119880123(3)

1= [

[

8 7 9 8

15 12 13 14

11 13 13 10

]

]

(12)

Step 3 Calculate the distances of the positive and the negativeideal situation according toDefinitions 1 and 2 Calculate pos-itive and negative prospect value matrix taking the distancesof the positive and the negative ideal situation as different

reference points The matrix for normalized Vplusmn[119904plusmn1(1)119894119895

] is asfollows

V+ [119904minus1(1)119894119895

] = [

[

06516 06999 05159 1

03803 04926 04926 02541

05159 04926 08482 06516

]

]

Vminus [119904minus1(1)119894119895

] = 0

Vminus [119904+1(1)119894119895

] = [

[

minus06345 minus04926 minus06516 minus04677

minus09495 minus07856 minus07856 minus1

minus06516 minus07856 minus05159 minus06345

]

]

Vminus [119904+1(1)119894119895

] = 0

(13)

Other normalized positive and negative prospect valuematrixes can be gotten in the same way

Step 4 The weight vectors of situation given by multipleexperts are 1198821 = [04 03 03] 1198822 = [05 025 025] and1198823= [05 02 03] Plug those objective weight vectors into

formula (5) and get comprehensive prospect value based ontwo reference points of the positive and the negative idealsituation distances Consider

1198811

119894119895= [

[

00598 03783 minus00239 02401

minus06592 minus03916 minus03653 minus06721

minus00976 minus04398 00679 minus01091

]

]

1198812

119894119895= [

[

00084 03999 minus01129 04052

minus07221 minus03550 minus04722 minus05486

minus01765 minus02935 02253 minus00399

]

]

1198813

119894119895= [

[

minus01744 03345 00537 03785

minus07625 04141 minus05374 minus05472

01359 minus03061 02095 minus01347

]

]

(14)

Step 5 Determine the optimal and the worst prospect valuematrix based on decision expertsrsquo judgment according toDefinition 3 and calculate the distances between the prospectvalue matrix of multiple decision expertsrsquo situation and theoptimal and the worst prospect value matrix according toDefinition 4 Consider

119889 (1198811 119881+) = 07993 119889 (119881

1 119881minus) = 01119

119889 (1198812 119881+) = 07351 119889 (119881

2 119881minus) = 01426

119889 (1198813 119881+) = 01093 119889 (119881

3 119881minus) = 08472

(15)

Step 6 Calculate the weight of expertsrsquo strategy according toformulas (8) and (9) as 120582

1= 04796 120582

2= 04579 120582

3=

00625The comprehensive prospect valuematrix of situationbased on decision experts weight vector is as follows

119881 = [

[

00216 03855 minus00598 03243

minus06945 minus03245 minus04250 minus06077

minus01191 minus03645 01488 minus00790

]

]

(16)

According to comprehensive prospect value matrixchoosing supplier 2 is the optimal strategy for production

6 The Scientific World Journal

line 1 choosing supplier 2 is the optimal strategy for supplier2 choosing supplier 3 is the optimal strategy for supplier3 providing production line 1 is the most appropriate forsupplier 1 providing production line 1 is themost appropriatefor supplier 2 providing production line 3 is the mostappropriate for supplier 3 and providing production line 1is the most appropriate for supplier 4 In a word the optimalsituation is 119878

12

According to themethods used to calculate grey situationin group decision-making in [3] assuming that the expertsrsquoknowledge is the same (ie the expertrsquos weight is equal) theweight vectors of the situation given by multiple experts areas follows1198821 = [04 03 03]1198822 = [05 025 025]1198823 =[05 02 03]

The comprehensive effect sample valuematrix of situationbased on normative approaches given by [4] is as follows

119877 = [

[0179 0269] [0235 0342] [0187 0293] [0221 0330]

[0183 0304] [0204 0348] [0197 0346] [0189 0324]

[0193 0299] [0169 0289] [0149 0379] [0189 0301]

]

(17)

According to the effect sample value matrix without thedecision makersrsquo behavioral preference the optimal situationis 11987812as well However the expertsrsquo weights are relying on the

subjective experience and judgment If the expertsrsquo weightsare different there may be different results

From the example analysis and comparison the methodstudies the group decision problem with a view to decisionexpertsrsquo risk attitude and effect sample value considers deci-sion expertsrsquo decision-making psychology takes referencepoints as the positive and the negative ideal situation groupdistances gets comprehensive prospect value matrix undermultiple expertsrsquo evaluation plugs the decision expertsrsquo riskpreference into grey situation group decision andmakes surethat the final decision conforms to the psychology behaviorof decision experts Meanwhile determine multiple decisionexpertsrsquo weight based on TOPSIS method by consideringthe close degree between expertrsquos individual evaluation andoverall assessment Compared with the former subjectivevalue methods this method is more objective and is almostnot relying on subjective experience and judgment

4 Conclusion

This paper puts forward a grey situation group decision-making method based on the prospect theory aiming atsolving such problem that the grey situation group decisionmodel does not consider the decision expertsrsquo risk pref-erence According to various types of effect sample valuedifferent prospect value functions are constructed the valuefunctions for two reference points are integrated strengthsand weaknesses are measured by means of prospect valueand the prospect value is normalized to [minus1 1] This paperdetermines multiple decision expertsrsquo weight by means ofTOPSIS method collects expertsrsquo opinion makes sequencefor each eventrsquos situation and gets the optimal situationat last This method is convenient for computer procedureoperation and provides a new method to solve problem ofgrey group situation taking decision expertsrsquo risk preference

into consideration However the parameters of the prospectvalue functions and the weight functions are usually gainedby experiments The determination of the reasonable param-eters still needs to be discussed and further researched Inview of the fact that decision experts discuss again andagain evaluation process andmodify assessment informationfurther study on multiround interactive grey situation groupdecision is needed

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work and there is no profes-sional or other personal interest of any nature or kind in anyproduct service andor company that could be construed asinfluencing the position presented in or the review of thepaper

Acknowledgments

This paper is sponsored by the National Natural ScienceFoundation of China (no 71363046 no 71271226 no71301064) the Humanistic and Social Science Foundationof Ministry of Education of China (no 10YJA790174) theHumanistic and Social Science Youth Foundation ofMinistryof Education of China (no 13YJC790198)

References

[1] J L Deng ldquoIntelligence space in grey situation decisionrdquo TheJournal of Grey System vol 10 no 3 pp 254ndash262 1998

[2] J L Deng Prediction and Grey Decision Press of HuazhongUinversity of Science amp Technology Wuhan China 2002

[3] S F Liu and Y Lin Grey Information Theory and PracticalApplications Springer London UK 2006

[4] Z XWang Y GDang andC P Song ldquoMultiobjective decisionmodel of the grey situation based on interval numberrdquo Controland Decision vol 24 no 3 pp 388ndash392 2009

[5] J Luo Y Liu and K Xu ldquoParameter establishment of intelligentalgorithmbased onuniformdesign and grey situation decisionrdquoJournal of Shenyang University of Technology vol 32 no 1 pp84ndash89 2010

[6] Y Liu J Forrest H H Zhao S F Liu and J Liu ldquoMulti-objective grey situation decision making method based onprospect theoryrdquo Systems Engineering and Electronics vol 34no 12 pp 2514ndash2519 2012

[7] Y Liu J Forrest and S F Liu ldquoMulti-objective grey situationdecision making model and application with interval numbersbased on linear transformation operator with rewarding goodand punishingrdquo Statistics amp Information Forum vol 12 no 7pp 22ndash26 2012

[8] L Zhou and D Luo ldquoStudy on decision-making method formulti-objective grey situation decisionrdquo Journal of North ChinaInstitute of Water Conservancy and Hydroelectric Power vol 31no 4 pp 150ndash153 2010

[9] Y MWang and Y G Dang ldquoNewmethod in situation decisionmaking based on entropyrdquo Systems Engineering and Electronicsvol 31 no 6 pp 1350ndash1352 2009

The Scientific World Journal 7

[10] D J Chen SH Zhang and J Y Chen ldquoResearch of gray generalsituation decision and its applicationrdquo Systems Engineering andElectronics vol 26 no 4 pp 423ndash429 2004

[11] C T Lin N H Shinh and W C Chuang ldquoStock tradingstrategy based on grey situation decision model an empiricalstudy of the electronic component industry in Taiwanrdquo Journalof Grey System vol 18 no 3 pp 239ndash250 2006

[12] C T Lin and H F Chang ldquoPerformance evaluation for medicalindustry in Taiwan applying grey situation decision makingrdquoJournal of Grey System vol 22 no 3 pp 219ndash226 2010

[13] X Shan X Z Wang and S X Liu ldquoDecision-making ofmissile flexible support sites locating in wartime with fuzzynumbers and grey situationrdquo Journal of Naval Aeronautical andAstronautical University vol 26 no 6 pp 703ndash706 2011

[14] G X Huang X B Wang and Y L Wang ldquoScheme theartillery fire plan by grey situation analysisrdquo Ordnance IndustryAutomation vol 31 no 2 pp 31ndash37 2012

[15] Y B Wang X S Jia and G Y Wang ldquoMethod of integrationfor currency maintenance resources based on grey situationdecision-makingrdquo Fire Control ampCommandControl vol 38 no4 pp 4ndash8 2013

[16] L M Chen P Zheng and S X Ji ldquoGrey situation evaluation ofsupply chain cooperative relationship based on cobweb modelrdquoLogisticstech vol 31 no 7 pp 321ndash323 2012

[17] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Economica vol 47 no 2 pp 263ndash291 1979

[18] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

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

The Scientific World Journal 3

However the situation distance of alternatives cannot besuperior to that of positive ideal scheme That is to say if thesituation distance of positive ideal is taken as reference pointits value function will become V+[119904+119896(119897)

119894119895] = 0

222 Cost Index Assuming that the effect measure of thesituation 119904119896

119894119895given by the 119896th decision expert is cost index

if positive ideal situation distance is taken as referencepoint the situation distance of alternatives is superior tothat of positive ideal scheme and at this time psychologicalperception of decision expert is gains The value function isV+[119904+119896(119897)119894119895

] = [119889(120583119896(119897)

119894119895 120583+119896(119897)

119894119895)]120572

However the situation distance of alternatives cannot besuperior to that of positive ideal scheme That is to say if thesituation distance of positive ideal is taken as reference pointits value function will become Vminus[119904+119896(119897)

119894119895] = 0

If the situation distance of negative ideal is taken asreference point the situation distance of alternatives willbe worse than that of positive ideal scheme At this timepsychological perception of decision expert is risksThe valuefunction is Vminus[119904minus119896(119897)

119894119895] = minus120579[minus119889(120583

119896(119897)

119894119895 120583minus119896(119897)

119894119895)]120573

However the situation distance of alternatives cannot beworse than that of negative ideal scheme That is to say if thesituation distance of negative ideal is taken as reference pointits value function will become V+[119904minus119896(119897)

119894119895] = 0

No matter in the case of efficiency index or cost indexboth V+[119904+119896(119897)

119894119895] and V+[119904minus119896(119897)

119894119895] show that the psychological

perception of decision expert for alternatives measure isrisk aversion and is expressed as positive value of gainsBoth Vminus[119904+119896(119897)

119894119895] and Vminus[119904minus119896(119897)

119894119895] reveal that the psychological

perception of decision expert for alternatives measure isrisk seeking and is expressed as negative value of lossesParameters 120572 and 120573 express the concave and convex degreeof gains and losses value power function respectively and120572 120573 lt 1 shows decision expertsrsquo sensitivity is decreasing 120579expresses that the losses curve is much deeper than the gainscurve and 120579 gt 1 reveals loss aversion [17]

Nomatter in the case of efficiency index or cost index theprospect value by taking positive and negative ideal situationdistance as different reference points is positive value of gainsand negative value of risks in decision expertsrsquo psychologicalperception In order to eliminate the influences of dimensionto calculation results the prospect value gotten from positiveand negative ideal situation as different reference points isnormalized as follows

Vplusmn [119904plusmn119896(119897)119894119895

] =

Vplusmn [119904plusmn119896(119897)119894119895

]

max119894=12119899

10038161003816100381610038161003816Vplusmn [119904plusmn119896(119897)119894119895

]10038161003816100381610038161003816

(4)

Normalized prospect value Vplusmn[119904plusmn119896(119897)119894119895

] (119894 = 1 2 119899 119895 =1 2 119898 119897 = 1 2 119904 119896 = 1 2 119901) has the followingproperties (1) Vplusmn[119904plusmn119896(119897)

119894119895] is dimensionless (2) the better the

situation the bigger Vplusmn[119904plusmn119896(119897)119894119895

] and the worse the situationthe smaller the Vplusmn[119904plusmn119896(119897)

119894119895] (3) V+[119904plusmn119896(119897)

119894119895] isin [0 1] Vminus[119904plusmn119896(119897)

119894119895] isin

[minus1 0]

Proof Consider the following

Case (i) For V+[119904plusmn119896(119897)119894119895

] = V+[119904plusmn119896(119897)119894119895

]max119894=12119899

V+[119904plusmn119896(119897)119894119895

]considering that 0 le V+[119904plusmn119896(119897)

119894119895] le max

119894=12119899V+[119904plusmn119896(119897)119894119895

] thebetter the V+[119904plusmn119896(119897)

119894119895] the bigger the V+[119904plusmn119896(119897)

119894119895] and V+[119904plusmn119896(119897)

119894119895] isin

[0 1] Thus properties (1) (2) and (3) are proved

Case (ii) For Vminus[119904plusmn119896(119897)119894119895

] = minusVminus[119904plusmn119896(119897)119894119895

]min119894=12119899

Vminus[119904plusmn119896(119897)119894119895

]considering that minusmin

119894=12119899Vminus[119904plusmn119896(119897)119894119895

] le Vminus[119904plusmn119896(119897)119894119895

] le 0 theworse the Vminus[119904plusmn119896(119897)

119894119895] the smaller the Vplusmn[119904plusmn119896(119897)

119894119895] and Vminus[119904plusmn119896(119897)

119894119895] isin

[minus1 0] Thus properties (1) (2) and (3) are proved

According to the value function given by [17] assumingthat the prospect weight that decision experts have whenfacing gains and losses is 120587+(120596119896

119897) and 120587minus(120596119896

119897) respectively

the comprehensive prospect value of the scheme is thesum of positive and negative prospect values Suppose thatdecision experts take the positive and negative ideal situationdistance of the situation 119904119896

119894119895under objective 119897 as different

references points and meanwhile they think the distancebetween reference points and the positive ideal situation is asimportant as that between reference points and the negativeideal situation distance and then comprehensive prospectvalue based on positive and negative ideal situation distanceas two reference points can be defined as follows

119881119896

119894119895=

119904

sum

119897=1

V+ [119904minus119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904minus119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

+

119904

sum

119897=1

V+ [119904+119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904+119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

=

119904

sum

119897=1

V+ [119904minus119896(119897)119894119895

] sdot 120587+(120596119896

119897) +

119904

sum

119897=1

Vminus [119904+119896(119897)119894119895

] sdot 120587minus(120596119896

119897)

120587+(120596119896

119897) =

(120596119896

119897)120574

(120596119896

119897)120574+ [1 minus (120596

119896

119897)]1205741120574

120587minus(120596119896

119897) =

(120596119896

119897)120575

(120596119896

119897)120575

+ [1 minus (120596119896

119897)]120575

1120575

(5)

Therefore the comprehensive prospect based on positiveand negative ideal situation distance as two reference pointsis the sum of positive prospect value taking negative idealsituation distance as reference point and negative prospectvalue taking positive ideal situation distance as referencepoint

According to [17] the parameters of prospect valuefunction andweight function in this paper are consistent withpractical empirical data as 120572 = 120573 = 088 120579 = 225 120574 = 061and 120575 = 069

4 The Scientific World Journal

23 Determination of Decision Expertsrsquo Weight Vector

Definition 3 Assume that119881+ = 119881+1119895 119881+

2119895 119881

+

119899119895 is the opti-

mal prospect valuematrix of all the decision expertsrsquo decisionnamely the optimal situation matrix of the situations for alldecision expertsrsquo choices where 119881+

119894119895= max

119896=12119901119881119896

119894119895(119894 =

1 2 119899 119895 = 1 2 119898)Assume that119881minus = 119881minus

1119895 119881minus

2119895 119881

minus

119899119895 is the worst prospect

value matrix of all the decision expertsrsquo decision namelythe worst situation matrix of the situations for all decisionexpertsrsquo choices where 119881minus

119894119895= min

119896=12119901119881119896

119894119895(119894 = 1 2 119899

119895 = 1 2 119898)

Definition 4 Given 119881119896 = 119881119896

1119895 119881119896

2119895 119881

119896

119899119895 is the prospect

value matrix of the 119896th decision expertrsquos situation 119904119896119894119895 then

119889 (119881119896 119881+)

= radic

119898

sum

119895=1

(119881119896

1119895minus 119881+

1119895)2

+

119898

sum

119895=1

(119881119896

2119895minus 119881+

2119895)2

+ sdot sdot sdot +

119898

sum

119895=1

(119881119896

119899119895minus 119881+

119899119895)2

(6)is the distance between the prospect value matrix of the 119896thdecision expertsrsquo situation and the optimal prospect valuematrix Given 119881119896 = 119881

119896

1119895 119881119896

2119895 119881

119896

119899119895 is the prospect value

matrix of the 119896th decision expert situation 119904119896119894119895 then

119889 (119881119896 119881minus)

= radic

119898

sum

119895=1

(119881119896

1119895minus 119881minus

1119895)2

+

119898

sum

119895=1

(119881119896

2119895minus 119881minus

2119895)2

+ sdot sdot sdot +

119898

sum

119895=1

(119881119896

119899119895minus 119881minus

119899119895)2

(7)

is the distance between the prospect value matrix of the119896th decision expertsrsquo situation and the worst prospect valuematrix

In order to avoid one-sidedness of evaluation resultsmake full use of decision information and keep the con-sistency of decision results the greatest weight is given tothe decision expert whose matrix is closest with the optimalprospect value and is the most far away from the worstprospect value matrix and vice versa This paper takesthe optimal prospect value matrix as the optimal situationand the worst prospect value matrix as the worst situationand calculates the prospect value close degree of decisionexpertsrsquo situation according to traditional TOPSIS conceptand in view of the distance between prospect value matrixof multiple decision experts and the optimal and the worstprospect value matrix Consider

119862lowast

119896=

119889 (119881119896 119881+)

119889 (119881119896 119881+) + 119889 (119881119896 119881minus) (8)

Then the weight of the 119896th decision expert is

120582119896=

119862lowast

119896

sum119901

119896=1119862lowast

119896

(9)

And then the comprehensive prospect value matrixbased on the weight vector of decision experts is

119881 = [V119894119895] =

119901

sum

119896=1

120582119901sdot 119881119896

119894119895 (10)

Definition 5 Given the comprehensive prospect valuematrix119881 if max

1le119895le119898V119894119895 = V1198941198950 1198871198950is optimal strategy for the event

119886119894 if max

1le119895le119898V119894119895 = V

1198940119895 1198861198940is the corresponding optimal

event of strategy 119887119895 if max

1le119895le119898V119894119895 = V11989401198950

11990411989401198950

is the optimalsituation

24 Decision-Making Procedure To sum up the procedureof grey situation group decision-making method based onprospect theory is as follows

Step 1 According to the event set 119860 = 1198861 1198862 119886

119899 the

strategy set 119861 = 1198871 1198872 119887

119898 and decision experts set

119863 = 1198891 1198892 119889

119901 the situation set of multiple decision

experts is 119878 = 119904119896119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861

Step 2 Determine the decision objective and give corre-sponding effect samplematrix ofmultiple decision experts forobjective 119897 = 1 2 119904

Step 3 In light of Definitions 1 and 2 calculate the positiveand negative ideal situation distance Take positive and nega-tive ideal situation distances as different reference points andcalculate and normalize the positive and negative prospectvalue matrix

Step 4 Give target weight vectors of the situation given byeach decision expert plug those target weight vectors intoformula (5) and get comprehensive prospectmatrix based ontwo reference points of positive and negative ideal situationdistances

Step 5 According to Definition 3 determine the optimal andworst prospect value matrix of all decision experts Andin light of Definition 4 calculate the distances between theprospect value matrix of multiple decision experts situationand the optimal and the worst prospect value matrix

Step 6 According to formulas (8) and (9) calculate theweight of expertsrsquo decisions In light of formula (10) getcomprehensive prospect value matrix based on the matrixof decision expertsrsquo weight vector and finally determine theoptimal situation

3 Example Analysis

A new round of aid program for Xinjiang is an importantdecision and strategic deployment made by the Party CentralCommittee and the State Council to promote Xinjiangrsquosdevelopment and maintain Xinjiangrsquos social stability requir-ing that 19 provinces and cities in the mid-east region aidXinjiangrsquos development Since March 2010 new industryproduction lines in Xinjiang are increasing day by daySuppose that there are 3 production lines in an industry area

The Scientific World Journal 5

4 suppliers are shortlisted in selecting suppliers and there are3 experts in decision-making expert group

Step 1 Choose suppliersrsquo choices in 3 production lines as theevents and the event set is119860 = 119886

1 1198862 1198863 Supplier 1 supplier

2 supplier 3 and supplier 4 constitute the strategy set 119861 =

1198871 1198872 1198873 1198874 Expert 1 expert 2 and expert 3 constitute the

decision expert set119863 = 1198891 1198892 1198893The event set strategy set

and decision expert set constitute decision expertsrsquo situationset 119878 = 119904119896

119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861 119896 isin 119863 119894 = 1 2 3 119895 =

1 2 3 4 119896 = 1 2 3

Step 2 Determine decision-making objective After severalrounds of specialist researches take quality design andprice as decision-making objectives where quality and designare qualitative index Decision experts make assessment bymeans of expert scoring method The higher the score isthe more uncertain the index is Decision experts give scoresbetween 0 and 10 Price is cost index the lower the data whichthe suppliers provided the better the decision Effect samplevalue matrix given by multiple experts with regard to thequality and design of decision-making objectives is as follows

1198801(1)

1= [

[

[5 7] [6 7] [5 6] [7 9]

[3 5] [4 6] [4 6] [3 4]

[5 6] [4 6] [6 8] [5 7]

]

]

1198801(2)

1= [

[

[4 5] [5 7] [4 6] [3 5]

[5 6] [4 5] [4 7] [4 6]

[5 7] [3 6] [7 9] [4 5]

]

]

1198802(1)

1= [

[

[5 6] [6 8] [5 6] [8 9]

[3 4] [4 6] [4 5] [4 5]

[4 6] [5 7] [7 9] [5 6]

]

]

1198802(2)

1= [

[

[4 6] [6 7] [4 5] [3 6]

[5 7] [4 6] [4 7] [4 7]

[5 8] [4 6] [8 10] [5 8]

]

]

1198803(1)

1= [

[

[4 5] [6 7] [5 7] [8 9]

[3 4] [4 5] [3 4] [4 6]

[4 5] [5 6] [7 8] [3 5]

]

]

1198803(2)

1= [

[

[3 4] [5 6] [4 6] [3 4]

[5 6] [4 6] [5 7] [4 5]

[7 8] [4 7] [8 9] [6 8]

]

]

(11)

The decision objective target is an effect sample valuematrix of suppliersrsquo offers Because this is not assessed bydecision experts the sample value matrix of target effect isthe same Consider

119880123(3)

1= [

[

8 7 9 8

15 12 13 14

11 13 13 10

]

]

(12)

Step 3 Calculate the distances of the positive and the negativeideal situation according toDefinitions 1 and 2 Calculate pos-itive and negative prospect value matrix taking the distancesof the positive and the negative ideal situation as different

reference points The matrix for normalized Vplusmn[119904plusmn1(1)119894119895

] is asfollows

V+ [119904minus1(1)119894119895

] = [

[

06516 06999 05159 1

03803 04926 04926 02541

05159 04926 08482 06516

]

]

Vminus [119904minus1(1)119894119895

] = 0

Vminus [119904+1(1)119894119895

] = [

[

minus06345 minus04926 minus06516 minus04677

minus09495 minus07856 minus07856 minus1

minus06516 minus07856 minus05159 minus06345

]

]

Vminus [119904+1(1)119894119895

] = 0

(13)

Other normalized positive and negative prospect valuematrixes can be gotten in the same way

Step 4 The weight vectors of situation given by multipleexperts are 1198821 = [04 03 03] 1198822 = [05 025 025] and1198823= [05 02 03] Plug those objective weight vectors into

formula (5) and get comprehensive prospect value based ontwo reference points of the positive and the negative idealsituation distances Consider

1198811

119894119895= [

[

00598 03783 minus00239 02401

minus06592 minus03916 minus03653 minus06721

minus00976 minus04398 00679 minus01091

]

]

1198812

119894119895= [

[

00084 03999 minus01129 04052

minus07221 minus03550 minus04722 minus05486

minus01765 minus02935 02253 minus00399

]

]

1198813

119894119895= [

[

minus01744 03345 00537 03785

minus07625 04141 minus05374 minus05472

01359 minus03061 02095 minus01347

]

]

(14)

Step 5 Determine the optimal and the worst prospect valuematrix based on decision expertsrsquo judgment according toDefinition 3 and calculate the distances between the prospectvalue matrix of multiple decision expertsrsquo situation and theoptimal and the worst prospect value matrix according toDefinition 4 Consider

119889 (1198811 119881+) = 07993 119889 (119881

1 119881minus) = 01119

119889 (1198812 119881+) = 07351 119889 (119881

2 119881minus) = 01426

119889 (1198813 119881+) = 01093 119889 (119881

3 119881minus) = 08472

(15)

Step 6 Calculate the weight of expertsrsquo strategy according toformulas (8) and (9) as 120582

1= 04796 120582

2= 04579 120582

3=

00625The comprehensive prospect valuematrix of situationbased on decision experts weight vector is as follows

119881 = [

[

00216 03855 minus00598 03243

minus06945 minus03245 minus04250 minus06077

minus01191 minus03645 01488 minus00790

]

]

(16)

According to comprehensive prospect value matrixchoosing supplier 2 is the optimal strategy for production

6 The Scientific World Journal

line 1 choosing supplier 2 is the optimal strategy for supplier2 choosing supplier 3 is the optimal strategy for supplier3 providing production line 1 is the most appropriate forsupplier 1 providing production line 1 is themost appropriatefor supplier 2 providing production line 3 is the mostappropriate for supplier 3 and providing production line 1is the most appropriate for supplier 4 In a word the optimalsituation is 119878

12

According to themethods used to calculate grey situationin group decision-making in [3] assuming that the expertsrsquoknowledge is the same (ie the expertrsquos weight is equal) theweight vectors of the situation given by multiple experts areas follows1198821 = [04 03 03]1198822 = [05 025 025]1198823 =[05 02 03]

The comprehensive effect sample valuematrix of situationbased on normative approaches given by [4] is as follows

119877 = [

[0179 0269] [0235 0342] [0187 0293] [0221 0330]

[0183 0304] [0204 0348] [0197 0346] [0189 0324]

[0193 0299] [0169 0289] [0149 0379] [0189 0301]

]

(17)

According to the effect sample value matrix without thedecision makersrsquo behavioral preference the optimal situationis 11987812as well However the expertsrsquo weights are relying on the

subjective experience and judgment If the expertsrsquo weightsare different there may be different results

From the example analysis and comparison the methodstudies the group decision problem with a view to decisionexpertsrsquo risk attitude and effect sample value considers deci-sion expertsrsquo decision-making psychology takes referencepoints as the positive and the negative ideal situation groupdistances gets comprehensive prospect value matrix undermultiple expertsrsquo evaluation plugs the decision expertsrsquo riskpreference into grey situation group decision andmakes surethat the final decision conforms to the psychology behaviorof decision experts Meanwhile determine multiple decisionexpertsrsquo weight based on TOPSIS method by consideringthe close degree between expertrsquos individual evaluation andoverall assessment Compared with the former subjectivevalue methods this method is more objective and is almostnot relying on subjective experience and judgment

4 Conclusion

This paper puts forward a grey situation group decision-making method based on the prospect theory aiming atsolving such problem that the grey situation group decisionmodel does not consider the decision expertsrsquo risk pref-erence According to various types of effect sample valuedifferent prospect value functions are constructed the valuefunctions for two reference points are integrated strengthsand weaknesses are measured by means of prospect valueand the prospect value is normalized to [minus1 1] This paperdetermines multiple decision expertsrsquo weight by means ofTOPSIS method collects expertsrsquo opinion makes sequencefor each eventrsquos situation and gets the optimal situationat last This method is convenient for computer procedureoperation and provides a new method to solve problem ofgrey group situation taking decision expertsrsquo risk preference

into consideration However the parameters of the prospectvalue functions and the weight functions are usually gainedby experiments The determination of the reasonable param-eters still needs to be discussed and further researched Inview of the fact that decision experts discuss again andagain evaluation process andmodify assessment informationfurther study on multiround interactive grey situation groupdecision is needed

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work and there is no profes-sional or other personal interest of any nature or kind in anyproduct service andor company that could be construed asinfluencing the position presented in or the review of thepaper

Acknowledgments

This paper is sponsored by the National Natural ScienceFoundation of China (no 71363046 no 71271226 no71301064) the Humanistic and Social Science Foundationof Ministry of Education of China (no 10YJA790174) theHumanistic and Social Science Youth Foundation ofMinistryof Education of China (no 13YJC790198)

References

[1] J L Deng ldquoIntelligence space in grey situation decisionrdquo TheJournal of Grey System vol 10 no 3 pp 254ndash262 1998

[2] J L Deng Prediction and Grey Decision Press of HuazhongUinversity of Science amp Technology Wuhan China 2002

[3] S F Liu and Y Lin Grey Information Theory and PracticalApplications Springer London UK 2006

[4] Z XWang Y GDang andC P Song ldquoMultiobjective decisionmodel of the grey situation based on interval numberrdquo Controland Decision vol 24 no 3 pp 388ndash392 2009

[5] J Luo Y Liu and K Xu ldquoParameter establishment of intelligentalgorithmbased onuniformdesign and grey situation decisionrdquoJournal of Shenyang University of Technology vol 32 no 1 pp84ndash89 2010

[6] Y Liu J Forrest H H Zhao S F Liu and J Liu ldquoMulti-objective grey situation decision making method based onprospect theoryrdquo Systems Engineering and Electronics vol 34no 12 pp 2514ndash2519 2012

[7] Y Liu J Forrest and S F Liu ldquoMulti-objective grey situationdecision making model and application with interval numbersbased on linear transformation operator with rewarding goodand punishingrdquo Statistics amp Information Forum vol 12 no 7pp 22ndash26 2012

[8] L Zhou and D Luo ldquoStudy on decision-making method formulti-objective grey situation decisionrdquo Journal of North ChinaInstitute of Water Conservancy and Hydroelectric Power vol 31no 4 pp 150ndash153 2010

[9] Y MWang and Y G Dang ldquoNewmethod in situation decisionmaking based on entropyrdquo Systems Engineering and Electronicsvol 31 no 6 pp 1350ndash1352 2009

The Scientific World Journal 7

[10] D J Chen SH Zhang and J Y Chen ldquoResearch of gray generalsituation decision and its applicationrdquo Systems Engineering andElectronics vol 26 no 4 pp 423ndash429 2004

[11] C T Lin N H Shinh and W C Chuang ldquoStock tradingstrategy based on grey situation decision model an empiricalstudy of the electronic component industry in Taiwanrdquo Journalof Grey System vol 18 no 3 pp 239ndash250 2006

[12] C T Lin and H F Chang ldquoPerformance evaluation for medicalindustry in Taiwan applying grey situation decision makingrdquoJournal of Grey System vol 22 no 3 pp 219ndash226 2010

[13] X Shan X Z Wang and S X Liu ldquoDecision-making ofmissile flexible support sites locating in wartime with fuzzynumbers and grey situationrdquo Journal of Naval Aeronautical andAstronautical University vol 26 no 6 pp 703ndash706 2011

[14] G X Huang X B Wang and Y L Wang ldquoScheme theartillery fire plan by grey situation analysisrdquo Ordnance IndustryAutomation vol 31 no 2 pp 31ndash37 2012

[15] Y B Wang X S Jia and G Y Wang ldquoMethod of integrationfor currency maintenance resources based on grey situationdecision-makingrdquo Fire Control ampCommandControl vol 38 no4 pp 4ndash8 2013

[16] L M Chen P Zheng and S X Ji ldquoGrey situation evaluation ofsupply chain cooperative relationship based on cobweb modelrdquoLogisticstech vol 31 no 7 pp 321ndash323 2012

[17] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Economica vol 47 no 2 pp 263ndash291 1979

[18] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

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 The Scientific World Journal

23 Determination of Decision Expertsrsquo Weight Vector

Definition 3 Assume that119881+ = 119881+1119895 119881+

2119895 119881

+

119899119895 is the opti-

mal prospect valuematrix of all the decision expertsrsquo decisionnamely the optimal situation matrix of the situations for alldecision expertsrsquo choices where 119881+

119894119895= max

119896=12119901119881119896

119894119895(119894 =

1 2 119899 119895 = 1 2 119898)Assume that119881minus = 119881minus

1119895 119881minus

2119895 119881

minus

119899119895 is the worst prospect

value matrix of all the decision expertsrsquo decision namelythe worst situation matrix of the situations for all decisionexpertsrsquo choices where 119881minus

119894119895= min

119896=12119901119881119896

119894119895(119894 = 1 2 119899

119895 = 1 2 119898)

Definition 4 Given 119881119896 = 119881119896

1119895 119881119896

2119895 119881

119896

119899119895 is the prospect

value matrix of the 119896th decision expertrsquos situation 119904119896119894119895 then

119889 (119881119896 119881+)

= radic

119898

sum

119895=1

(119881119896

1119895minus 119881+

1119895)2

+

119898

sum

119895=1

(119881119896

2119895minus 119881+

2119895)2

+ sdot sdot sdot +

119898

sum

119895=1

(119881119896

119899119895minus 119881+

119899119895)2

(6)is the distance between the prospect value matrix of the 119896thdecision expertsrsquo situation and the optimal prospect valuematrix Given 119881119896 = 119881

119896

1119895 119881119896

2119895 119881

119896

119899119895 is the prospect value

matrix of the 119896th decision expert situation 119904119896119894119895 then

119889 (119881119896 119881minus)

= radic

119898

sum

119895=1

(119881119896

1119895minus 119881minus

1119895)2

+

119898

sum

119895=1

(119881119896

2119895minus 119881minus

2119895)2

+ sdot sdot sdot +

119898

sum

119895=1

(119881119896

119899119895minus 119881minus

119899119895)2

(7)

is the distance between the prospect value matrix of the119896th decision expertsrsquo situation and the worst prospect valuematrix

In order to avoid one-sidedness of evaluation resultsmake full use of decision information and keep the con-sistency of decision results the greatest weight is given tothe decision expert whose matrix is closest with the optimalprospect value and is the most far away from the worstprospect value matrix and vice versa This paper takesthe optimal prospect value matrix as the optimal situationand the worst prospect value matrix as the worst situationand calculates the prospect value close degree of decisionexpertsrsquo situation according to traditional TOPSIS conceptand in view of the distance between prospect value matrixof multiple decision experts and the optimal and the worstprospect value matrix Consider

119862lowast

119896=

119889 (119881119896 119881+)

119889 (119881119896 119881+) + 119889 (119881119896 119881minus) (8)

Then the weight of the 119896th decision expert is

120582119896=

119862lowast

119896

sum119901

119896=1119862lowast

119896

(9)

And then the comprehensive prospect value matrixbased on the weight vector of decision experts is

119881 = [V119894119895] =

119901

sum

119896=1

120582119901sdot 119881119896

119894119895 (10)

Definition 5 Given the comprehensive prospect valuematrix119881 if max

1le119895le119898V119894119895 = V1198941198950 1198871198950is optimal strategy for the event

119886119894 if max

1le119895le119898V119894119895 = V

1198940119895 1198861198940is the corresponding optimal

event of strategy 119887119895 if max

1le119895le119898V119894119895 = V11989401198950

11990411989401198950

is the optimalsituation

24 Decision-Making Procedure To sum up the procedureof grey situation group decision-making method based onprospect theory is as follows

Step 1 According to the event set 119860 = 1198861 1198862 119886

119899 the

strategy set 119861 = 1198871 1198872 119887

119898 and decision experts set

119863 = 1198891 1198892 119889

119901 the situation set of multiple decision

experts is 119878 = 119904119896119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861

Step 2 Determine the decision objective and give corre-sponding effect samplematrix ofmultiple decision experts forobjective 119897 = 1 2 119904

Step 3 In light of Definitions 1 and 2 calculate the positiveand negative ideal situation distance Take positive and nega-tive ideal situation distances as different reference points andcalculate and normalize the positive and negative prospectvalue matrix

Step 4 Give target weight vectors of the situation given byeach decision expert plug those target weight vectors intoformula (5) and get comprehensive prospectmatrix based ontwo reference points of positive and negative ideal situationdistances

Step 5 According to Definition 3 determine the optimal andworst prospect value matrix of all decision experts Andin light of Definition 4 calculate the distances between theprospect value matrix of multiple decision experts situationand the optimal and the worst prospect value matrix

Step 6 According to formulas (8) and (9) calculate theweight of expertsrsquo decisions In light of formula (10) getcomprehensive prospect value matrix based on the matrixof decision expertsrsquo weight vector and finally determine theoptimal situation

3 Example Analysis

A new round of aid program for Xinjiang is an importantdecision and strategic deployment made by the Party CentralCommittee and the State Council to promote Xinjiangrsquosdevelopment and maintain Xinjiangrsquos social stability requir-ing that 19 provinces and cities in the mid-east region aidXinjiangrsquos development Since March 2010 new industryproduction lines in Xinjiang are increasing day by daySuppose that there are 3 production lines in an industry area

The Scientific World Journal 5

4 suppliers are shortlisted in selecting suppliers and there are3 experts in decision-making expert group

Step 1 Choose suppliersrsquo choices in 3 production lines as theevents and the event set is119860 = 119886

1 1198862 1198863 Supplier 1 supplier

2 supplier 3 and supplier 4 constitute the strategy set 119861 =

1198871 1198872 1198873 1198874 Expert 1 expert 2 and expert 3 constitute the

decision expert set119863 = 1198891 1198892 1198893The event set strategy set

and decision expert set constitute decision expertsrsquo situationset 119878 = 119904119896

119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861 119896 isin 119863 119894 = 1 2 3 119895 =

1 2 3 4 119896 = 1 2 3

Step 2 Determine decision-making objective After severalrounds of specialist researches take quality design andprice as decision-making objectives where quality and designare qualitative index Decision experts make assessment bymeans of expert scoring method The higher the score isthe more uncertain the index is Decision experts give scoresbetween 0 and 10 Price is cost index the lower the data whichthe suppliers provided the better the decision Effect samplevalue matrix given by multiple experts with regard to thequality and design of decision-making objectives is as follows

1198801(1)

1= [

[

[5 7] [6 7] [5 6] [7 9]

[3 5] [4 6] [4 6] [3 4]

[5 6] [4 6] [6 8] [5 7]

]

]

1198801(2)

1= [

[

[4 5] [5 7] [4 6] [3 5]

[5 6] [4 5] [4 7] [4 6]

[5 7] [3 6] [7 9] [4 5]

]

]

1198802(1)

1= [

[

[5 6] [6 8] [5 6] [8 9]

[3 4] [4 6] [4 5] [4 5]

[4 6] [5 7] [7 9] [5 6]

]

]

1198802(2)

1= [

[

[4 6] [6 7] [4 5] [3 6]

[5 7] [4 6] [4 7] [4 7]

[5 8] [4 6] [8 10] [5 8]

]

]

1198803(1)

1= [

[

[4 5] [6 7] [5 7] [8 9]

[3 4] [4 5] [3 4] [4 6]

[4 5] [5 6] [7 8] [3 5]

]

]

1198803(2)

1= [

[

[3 4] [5 6] [4 6] [3 4]

[5 6] [4 6] [5 7] [4 5]

[7 8] [4 7] [8 9] [6 8]

]

]

(11)

The decision objective target is an effect sample valuematrix of suppliersrsquo offers Because this is not assessed bydecision experts the sample value matrix of target effect isthe same Consider

119880123(3)

1= [

[

8 7 9 8

15 12 13 14

11 13 13 10

]

]

(12)

Step 3 Calculate the distances of the positive and the negativeideal situation according toDefinitions 1 and 2 Calculate pos-itive and negative prospect value matrix taking the distancesof the positive and the negative ideal situation as different

reference points The matrix for normalized Vplusmn[119904plusmn1(1)119894119895

] is asfollows

V+ [119904minus1(1)119894119895

] = [

[

06516 06999 05159 1

03803 04926 04926 02541

05159 04926 08482 06516

]

]

Vminus [119904minus1(1)119894119895

] = 0

Vminus [119904+1(1)119894119895

] = [

[

minus06345 minus04926 minus06516 minus04677

minus09495 minus07856 minus07856 minus1

minus06516 minus07856 minus05159 minus06345

]

]

Vminus [119904+1(1)119894119895

] = 0

(13)

Other normalized positive and negative prospect valuematrixes can be gotten in the same way

Step 4 The weight vectors of situation given by multipleexperts are 1198821 = [04 03 03] 1198822 = [05 025 025] and1198823= [05 02 03] Plug those objective weight vectors into

formula (5) and get comprehensive prospect value based ontwo reference points of the positive and the negative idealsituation distances Consider

1198811

119894119895= [

[

00598 03783 minus00239 02401

minus06592 minus03916 minus03653 minus06721

minus00976 minus04398 00679 minus01091

]

]

1198812

119894119895= [

[

00084 03999 minus01129 04052

minus07221 minus03550 minus04722 minus05486

minus01765 minus02935 02253 minus00399

]

]

1198813

119894119895= [

[

minus01744 03345 00537 03785

minus07625 04141 minus05374 minus05472

01359 minus03061 02095 minus01347

]

]

(14)

Step 5 Determine the optimal and the worst prospect valuematrix based on decision expertsrsquo judgment according toDefinition 3 and calculate the distances between the prospectvalue matrix of multiple decision expertsrsquo situation and theoptimal and the worst prospect value matrix according toDefinition 4 Consider

119889 (1198811 119881+) = 07993 119889 (119881

1 119881minus) = 01119

119889 (1198812 119881+) = 07351 119889 (119881

2 119881minus) = 01426

119889 (1198813 119881+) = 01093 119889 (119881

3 119881minus) = 08472

(15)

Step 6 Calculate the weight of expertsrsquo strategy according toformulas (8) and (9) as 120582

1= 04796 120582

2= 04579 120582

3=

00625The comprehensive prospect valuematrix of situationbased on decision experts weight vector is as follows

119881 = [

[

00216 03855 minus00598 03243

minus06945 minus03245 minus04250 minus06077

minus01191 minus03645 01488 minus00790

]

]

(16)

According to comprehensive prospect value matrixchoosing supplier 2 is the optimal strategy for production

6 The Scientific World Journal

line 1 choosing supplier 2 is the optimal strategy for supplier2 choosing supplier 3 is the optimal strategy for supplier3 providing production line 1 is the most appropriate forsupplier 1 providing production line 1 is themost appropriatefor supplier 2 providing production line 3 is the mostappropriate for supplier 3 and providing production line 1is the most appropriate for supplier 4 In a word the optimalsituation is 119878

12

According to themethods used to calculate grey situationin group decision-making in [3] assuming that the expertsrsquoknowledge is the same (ie the expertrsquos weight is equal) theweight vectors of the situation given by multiple experts areas follows1198821 = [04 03 03]1198822 = [05 025 025]1198823 =[05 02 03]

The comprehensive effect sample valuematrix of situationbased on normative approaches given by [4] is as follows

119877 = [

[0179 0269] [0235 0342] [0187 0293] [0221 0330]

[0183 0304] [0204 0348] [0197 0346] [0189 0324]

[0193 0299] [0169 0289] [0149 0379] [0189 0301]

]

(17)

According to the effect sample value matrix without thedecision makersrsquo behavioral preference the optimal situationis 11987812as well However the expertsrsquo weights are relying on the

subjective experience and judgment If the expertsrsquo weightsare different there may be different results

From the example analysis and comparison the methodstudies the group decision problem with a view to decisionexpertsrsquo risk attitude and effect sample value considers deci-sion expertsrsquo decision-making psychology takes referencepoints as the positive and the negative ideal situation groupdistances gets comprehensive prospect value matrix undermultiple expertsrsquo evaluation plugs the decision expertsrsquo riskpreference into grey situation group decision andmakes surethat the final decision conforms to the psychology behaviorof decision experts Meanwhile determine multiple decisionexpertsrsquo weight based on TOPSIS method by consideringthe close degree between expertrsquos individual evaluation andoverall assessment Compared with the former subjectivevalue methods this method is more objective and is almostnot relying on subjective experience and judgment

4 Conclusion

This paper puts forward a grey situation group decision-making method based on the prospect theory aiming atsolving such problem that the grey situation group decisionmodel does not consider the decision expertsrsquo risk pref-erence According to various types of effect sample valuedifferent prospect value functions are constructed the valuefunctions for two reference points are integrated strengthsand weaknesses are measured by means of prospect valueand the prospect value is normalized to [minus1 1] This paperdetermines multiple decision expertsrsquo weight by means ofTOPSIS method collects expertsrsquo opinion makes sequencefor each eventrsquos situation and gets the optimal situationat last This method is convenient for computer procedureoperation and provides a new method to solve problem ofgrey group situation taking decision expertsrsquo risk preference

into consideration However the parameters of the prospectvalue functions and the weight functions are usually gainedby experiments The determination of the reasonable param-eters still needs to be discussed and further researched Inview of the fact that decision experts discuss again andagain evaluation process andmodify assessment informationfurther study on multiround interactive grey situation groupdecision is needed

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work and there is no profes-sional or other personal interest of any nature or kind in anyproduct service andor company that could be construed asinfluencing the position presented in or the review of thepaper

Acknowledgments

This paper is sponsored by the National Natural ScienceFoundation of China (no 71363046 no 71271226 no71301064) the Humanistic and Social Science Foundationof Ministry of Education of China (no 10YJA790174) theHumanistic and Social Science Youth Foundation ofMinistryof Education of China (no 13YJC790198)

References

[1] J L Deng ldquoIntelligence space in grey situation decisionrdquo TheJournal of Grey System vol 10 no 3 pp 254ndash262 1998

[2] J L Deng Prediction and Grey Decision Press of HuazhongUinversity of Science amp Technology Wuhan China 2002

[3] S F Liu and Y Lin Grey Information Theory and PracticalApplications Springer London UK 2006

[4] Z XWang Y GDang andC P Song ldquoMultiobjective decisionmodel of the grey situation based on interval numberrdquo Controland Decision vol 24 no 3 pp 388ndash392 2009

[5] J Luo Y Liu and K Xu ldquoParameter establishment of intelligentalgorithmbased onuniformdesign and grey situation decisionrdquoJournal of Shenyang University of Technology vol 32 no 1 pp84ndash89 2010

[6] Y Liu J Forrest H H Zhao S F Liu and J Liu ldquoMulti-objective grey situation decision making method based onprospect theoryrdquo Systems Engineering and Electronics vol 34no 12 pp 2514ndash2519 2012

[7] Y Liu J Forrest and S F Liu ldquoMulti-objective grey situationdecision making model and application with interval numbersbased on linear transformation operator with rewarding goodand punishingrdquo Statistics amp Information Forum vol 12 no 7pp 22ndash26 2012

[8] L Zhou and D Luo ldquoStudy on decision-making method formulti-objective grey situation decisionrdquo Journal of North ChinaInstitute of Water Conservancy and Hydroelectric Power vol 31no 4 pp 150ndash153 2010

[9] Y MWang and Y G Dang ldquoNewmethod in situation decisionmaking based on entropyrdquo Systems Engineering and Electronicsvol 31 no 6 pp 1350ndash1352 2009

The Scientific World Journal 7

[10] D J Chen SH Zhang and J Y Chen ldquoResearch of gray generalsituation decision and its applicationrdquo Systems Engineering andElectronics vol 26 no 4 pp 423ndash429 2004

[11] C T Lin N H Shinh and W C Chuang ldquoStock tradingstrategy based on grey situation decision model an empiricalstudy of the electronic component industry in Taiwanrdquo Journalof Grey System vol 18 no 3 pp 239ndash250 2006

[12] C T Lin and H F Chang ldquoPerformance evaluation for medicalindustry in Taiwan applying grey situation decision makingrdquoJournal of Grey System vol 22 no 3 pp 219ndash226 2010

[13] X Shan X Z Wang and S X Liu ldquoDecision-making ofmissile flexible support sites locating in wartime with fuzzynumbers and grey situationrdquo Journal of Naval Aeronautical andAstronautical University vol 26 no 6 pp 703ndash706 2011

[14] G X Huang X B Wang and Y L Wang ldquoScheme theartillery fire plan by grey situation analysisrdquo Ordnance IndustryAutomation vol 31 no 2 pp 31ndash37 2012

[15] Y B Wang X S Jia and G Y Wang ldquoMethod of integrationfor currency maintenance resources based on grey situationdecision-makingrdquo Fire Control ampCommandControl vol 38 no4 pp 4ndash8 2013

[16] L M Chen P Zheng and S X Ji ldquoGrey situation evaluation ofsupply chain cooperative relationship based on cobweb modelrdquoLogisticstech vol 31 no 7 pp 321ndash323 2012

[17] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Economica vol 47 no 2 pp 263ndash291 1979

[18] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

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

The Scientific World Journal 5

4 suppliers are shortlisted in selecting suppliers and there are3 experts in decision-making expert group

Step 1 Choose suppliersrsquo choices in 3 production lines as theevents and the event set is119860 = 119886

1 1198862 1198863 Supplier 1 supplier

2 supplier 3 and supplier 4 constitute the strategy set 119861 =

1198871 1198872 1198873 1198874 Expert 1 expert 2 and expert 3 constitute the

decision expert set119863 = 1198891 1198892 1198893The event set strategy set

and decision expert set constitute decision expertsrsquo situationset 119878 = 119904119896

119894119895= (119886119896

119894 119887119896

119895) | 119886119896

119894isin 119860 119887

119896

119895isin 119861 119896 isin 119863 119894 = 1 2 3 119895 =

1 2 3 4 119896 = 1 2 3

Step 2 Determine decision-making objective After severalrounds of specialist researches take quality design andprice as decision-making objectives where quality and designare qualitative index Decision experts make assessment bymeans of expert scoring method The higher the score isthe more uncertain the index is Decision experts give scoresbetween 0 and 10 Price is cost index the lower the data whichthe suppliers provided the better the decision Effect samplevalue matrix given by multiple experts with regard to thequality and design of decision-making objectives is as follows

1198801(1)

1= [

[

[5 7] [6 7] [5 6] [7 9]

[3 5] [4 6] [4 6] [3 4]

[5 6] [4 6] [6 8] [5 7]

]

]

1198801(2)

1= [

[

[4 5] [5 7] [4 6] [3 5]

[5 6] [4 5] [4 7] [4 6]

[5 7] [3 6] [7 9] [4 5]

]

]

1198802(1)

1= [

[

[5 6] [6 8] [5 6] [8 9]

[3 4] [4 6] [4 5] [4 5]

[4 6] [5 7] [7 9] [5 6]

]

]

1198802(2)

1= [

[

[4 6] [6 7] [4 5] [3 6]

[5 7] [4 6] [4 7] [4 7]

[5 8] [4 6] [8 10] [5 8]

]

]

1198803(1)

1= [

[

[4 5] [6 7] [5 7] [8 9]

[3 4] [4 5] [3 4] [4 6]

[4 5] [5 6] [7 8] [3 5]

]

]

1198803(2)

1= [

[

[3 4] [5 6] [4 6] [3 4]

[5 6] [4 6] [5 7] [4 5]

[7 8] [4 7] [8 9] [6 8]

]

]

(11)

The decision objective target is an effect sample valuematrix of suppliersrsquo offers Because this is not assessed bydecision experts the sample value matrix of target effect isthe same Consider

119880123(3)

1= [

[

8 7 9 8

15 12 13 14

11 13 13 10

]

]

(12)

Step 3 Calculate the distances of the positive and the negativeideal situation according toDefinitions 1 and 2 Calculate pos-itive and negative prospect value matrix taking the distancesof the positive and the negative ideal situation as different

reference points The matrix for normalized Vplusmn[119904plusmn1(1)119894119895

] is asfollows

V+ [119904minus1(1)119894119895

] = [

[

06516 06999 05159 1

03803 04926 04926 02541

05159 04926 08482 06516

]

]

Vminus [119904minus1(1)119894119895

] = 0

Vminus [119904+1(1)119894119895

] = [

[

minus06345 minus04926 minus06516 minus04677

minus09495 minus07856 minus07856 minus1

minus06516 minus07856 minus05159 minus06345

]

]

Vminus [119904+1(1)119894119895

] = 0

(13)

Other normalized positive and negative prospect valuematrixes can be gotten in the same way

Step 4 The weight vectors of situation given by multipleexperts are 1198821 = [04 03 03] 1198822 = [05 025 025] and1198823= [05 02 03] Plug those objective weight vectors into

formula (5) and get comprehensive prospect value based ontwo reference points of the positive and the negative idealsituation distances Consider

1198811

119894119895= [

[

00598 03783 minus00239 02401

minus06592 minus03916 minus03653 minus06721

minus00976 minus04398 00679 minus01091

]

]

1198812

119894119895= [

[

00084 03999 minus01129 04052

minus07221 minus03550 minus04722 minus05486

minus01765 minus02935 02253 minus00399

]

]

1198813

119894119895= [

[

minus01744 03345 00537 03785

minus07625 04141 minus05374 minus05472

01359 minus03061 02095 minus01347

]

]

(14)

Step 5 Determine the optimal and the worst prospect valuematrix based on decision expertsrsquo judgment according toDefinition 3 and calculate the distances between the prospectvalue matrix of multiple decision expertsrsquo situation and theoptimal and the worst prospect value matrix according toDefinition 4 Consider

119889 (1198811 119881+) = 07993 119889 (119881

1 119881minus) = 01119

119889 (1198812 119881+) = 07351 119889 (119881

2 119881minus) = 01426

119889 (1198813 119881+) = 01093 119889 (119881

3 119881minus) = 08472

(15)

Step 6 Calculate the weight of expertsrsquo strategy according toformulas (8) and (9) as 120582

1= 04796 120582

2= 04579 120582

3=

00625The comprehensive prospect valuematrix of situationbased on decision experts weight vector is as follows

119881 = [

[

00216 03855 minus00598 03243

minus06945 minus03245 minus04250 minus06077

minus01191 minus03645 01488 minus00790

]

]

(16)

According to comprehensive prospect value matrixchoosing supplier 2 is the optimal strategy for production

6 The Scientific World Journal

line 1 choosing supplier 2 is the optimal strategy for supplier2 choosing supplier 3 is the optimal strategy for supplier3 providing production line 1 is the most appropriate forsupplier 1 providing production line 1 is themost appropriatefor supplier 2 providing production line 3 is the mostappropriate for supplier 3 and providing production line 1is the most appropriate for supplier 4 In a word the optimalsituation is 119878

12

According to themethods used to calculate grey situationin group decision-making in [3] assuming that the expertsrsquoknowledge is the same (ie the expertrsquos weight is equal) theweight vectors of the situation given by multiple experts areas follows1198821 = [04 03 03]1198822 = [05 025 025]1198823 =[05 02 03]

The comprehensive effect sample valuematrix of situationbased on normative approaches given by [4] is as follows

119877 = [

[0179 0269] [0235 0342] [0187 0293] [0221 0330]

[0183 0304] [0204 0348] [0197 0346] [0189 0324]

[0193 0299] [0169 0289] [0149 0379] [0189 0301]

]

(17)

According to the effect sample value matrix without thedecision makersrsquo behavioral preference the optimal situationis 11987812as well However the expertsrsquo weights are relying on the

subjective experience and judgment If the expertsrsquo weightsare different there may be different results

From the example analysis and comparison the methodstudies the group decision problem with a view to decisionexpertsrsquo risk attitude and effect sample value considers deci-sion expertsrsquo decision-making psychology takes referencepoints as the positive and the negative ideal situation groupdistances gets comprehensive prospect value matrix undermultiple expertsrsquo evaluation plugs the decision expertsrsquo riskpreference into grey situation group decision andmakes surethat the final decision conforms to the psychology behaviorof decision experts Meanwhile determine multiple decisionexpertsrsquo weight based on TOPSIS method by consideringthe close degree between expertrsquos individual evaluation andoverall assessment Compared with the former subjectivevalue methods this method is more objective and is almostnot relying on subjective experience and judgment

4 Conclusion

This paper puts forward a grey situation group decision-making method based on the prospect theory aiming atsolving such problem that the grey situation group decisionmodel does not consider the decision expertsrsquo risk pref-erence According to various types of effect sample valuedifferent prospect value functions are constructed the valuefunctions for two reference points are integrated strengthsand weaknesses are measured by means of prospect valueand the prospect value is normalized to [minus1 1] This paperdetermines multiple decision expertsrsquo weight by means ofTOPSIS method collects expertsrsquo opinion makes sequencefor each eventrsquos situation and gets the optimal situationat last This method is convenient for computer procedureoperation and provides a new method to solve problem ofgrey group situation taking decision expertsrsquo risk preference

into consideration However the parameters of the prospectvalue functions and the weight functions are usually gainedby experiments The determination of the reasonable param-eters still needs to be discussed and further researched Inview of the fact that decision experts discuss again andagain evaluation process andmodify assessment informationfurther study on multiround interactive grey situation groupdecision is needed

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work and there is no profes-sional or other personal interest of any nature or kind in anyproduct service andor company that could be construed asinfluencing the position presented in or the review of thepaper

Acknowledgments

This paper is sponsored by the National Natural ScienceFoundation of China (no 71363046 no 71271226 no71301064) the Humanistic and Social Science Foundationof Ministry of Education of China (no 10YJA790174) theHumanistic and Social Science Youth Foundation ofMinistryof Education of China (no 13YJC790198)

References

[1] J L Deng ldquoIntelligence space in grey situation decisionrdquo TheJournal of Grey System vol 10 no 3 pp 254ndash262 1998

[2] J L Deng Prediction and Grey Decision Press of HuazhongUinversity of Science amp Technology Wuhan China 2002

[3] S F Liu and Y Lin Grey Information Theory and PracticalApplications Springer London UK 2006

[4] Z XWang Y GDang andC P Song ldquoMultiobjective decisionmodel of the grey situation based on interval numberrdquo Controland Decision vol 24 no 3 pp 388ndash392 2009

[5] J Luo Y Liu and K Xu ldquoParameter establishment of intelligentalgorithmbased onuniformdesign and grey situation decisionrdquoJournal of Shenyang University of Technology vol 32 no 1 pp84ndash89 2010

[6] Y Liu J Forrest H H Zhao S F Liu and J Liu ldquoMulti-objective grey situation decision making method based onprospect theoryrdquo Systems Engineering and Electronics vol 34no 12 pp 2514ndash2519 2012

[7] Y Liu J Forrest and S F Liu ldquoMulti-objective grey situationdecision making model and application with interval numbersbased on linear transformation operator with rewarding goodand punishingrdquo Statistics amp Information Forum vol 12 no 7pp 22ndash26 2012

[8] L Zhou and D Luo ldquoStudy on decision-making method formulti-objective grey situation decisionrdquo Journal of North ChinaInstitute of Water Conservancy and Hydroelectric Power vol 31no 4 pp 150ndash153 2010

[9] Y MWang and Y G Dang ldquoNewmethod in situation decisionmaking based on entropyrdquo Systems Engineering and Electronicsvol 31 no 6 pp 1350ndash1352 2009

The Scientific World Journal 7

[10] D J Chen SH Zhang and J Y Chen ldquoResearch of gray generalsituation decision and its applicationrdquo Systems Engineering andElectronics vol 26 no 4 pp 423ndash429 2004

[11] C T Lin N H Shinh and W C Chuang ldquoStock tradingstrategy based on grey situation decision model an empiricalstudy of the electronic component industry in Taiwanrdquo Journalof Grey System vol 18 no 3 pp 239ndash250 2006

[12] C T Lin and H F Chang ldquoPerformance evaluation for medicalindustry in Taiwan applying grey situation decision makingrdquoJournal of Grey System vol 22 no 3 pp 219ndash226 2010

[13] X Shan X Z Wang and S X Liu ldquoDecision-making ofmissile flexible support sites locating in wartime with fuzzynumbers and grey situationrdquo Journal of Naval Aeronautical andAstronautical University vol 26 no 6 pp 703ndash706 2011

[14] G X Huang X B Wang and Y L Wang ldquoScheme theartillery fire plan by grey situation analysisrdquo Ordnance IndustryAutomation vol 31 no 2 pp 31ndash37 2012

[15] Y B Wang X S Jia and G Y Wang ldquoMethod of integrationfor currency maintenance resources based on grey situationdecision-makingrdquo Fire Control ampCommandControl vol 38 no4 pp 4ndash8 2013

[16] L M Chen P Zheng and S X Ji ldquoGrey situation evaluation ofsupply chain cooperative relationship based on cobweb modelrdquoLogisticstech vol 31 no 7 pp 321ndash323 2012

[17] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Economica vol 47 no 2 pp 263ndash291 1979

[18] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

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 The Scientific World Journal

line 1 choosing supplier 2 is the optimal strategy for supplier2 choosing supplier 3 is the optimal strategy for supplier3 providing production line 1 is the most appropriate forsupplier 1 providing production line 1 is themost appropriatefor supplier 2 providing production line 3 is the mostappropriate for supplier 3 and providing production line 1is the most appropriate for supplier 4 In a word the optimalsituation is 119878

12

According to themethods used to calculate grey situationin group decision-making in [3] assuming that the expertsrsquoknowledge is the same (ie the expertrsquos weight is equal) theweight vectors of the situation given by multiple experts areas follows1198821 = [04 03 03]1198822 = [05 025 025]1198823 =[05 02 03]

The comprehensive effect sample valuematrix of situationbased on normative approaches given by [4] is as follows

119877 = [

[0179 0269] [0235 0342] [0187 0293] [0221 0330]

[0183 0304] [0204 0348] [0197 0346] [0189 0324]

[0193 0299] [0169 0289] [0149 0379] [0189 0301]

]

(17)

According to the effect sample value matrix without thedecision makersrsquo behavioral preference the optimal situationis 11987812as well However the expertsrsquo weights are relying on the

subjective experience and judgment If the expertsrsquo weightsare different there may be different results

From the example analysis and comparison the methodstudies the group decision problem with a view to decisionexpertsrsquo risk attitude and effect sample value considers deci-sion expertsrsquo decision-making psychology takes referencepoints as the positive and the negative ideal situation groupdistances gets comprehensive prospect value matrix undermultiple expertsrsquo evaluation plugs the decision expertsrsquo riskpreference into grey situation group decision andmakes surethat the final decision conforms to the psychology behaviorof decision experts Meanwhile determine multiple decisionexpertsrsquo weight based on TOPSIS method by consideringthe close degree between expertrsquos individual evaluation andoverall assessment Compared with the former subjectivevalue methods this method is more objective and is almostnot relying on subjective experience and judgment

4 Conclusion

This paper puts forward a grey situation group decision-making method based on the prospect theory aiming atsolving such problem that the grey situation group decisionmodel does not consider the decision expertsrsquo risk pref-erence According to various types of effect sample valuedifferent prospect value functions are constructed the valuefunctions for two reference points are integrated strengthsand weaknesses are measured by means of prospect valueand the prospect value is normalized to [minus1 1] This paperdetermines multiple decision expertsrsquo weight by means ofTOPSIS method collects expertsrsquo opinion makes sequencefor each eventrsquos situation and gets the optimal situationat last This method is convenient for computer procedureoperation and provides a new method to solve problem ofgrey group situation taking decision expertsrsquo risk preference

into consideration However the parameters of the prospectvalue functions and the weight functions are usually gainedby experiments The determination of the reasonable param-eters still needs to be discussed and further researched Inview of the fact that decision experts discuss again andagain evaluation process andmodify assessment informationfurther study on multiround interactive grey situation groupdecision is needed

Conflict of Interests

The authors declare that they have no financial and personalrelationships with other people or organizations that caninappropriately influence their work and there is no profes-sional or other personal interest of any nature or kind in anyproduct service andor company that could be construed asinfluencing the position presented in or the review of thepaper

Acknowledgments

This paper is sponsored by the National Natural ScienceFoundation of China (no 71363046 no 71271226 no71301064) the Humanistic and Social Science Foundationof Ministry of Education of China (no 10YJA790174) theHumanistic and Social Science Youth Foundation ofMinistryof Education of China (no 13YJC790198)

References

[1] J L Deng ldquoIntelligence space in grey situation decisionrdquo TheJournal of Grey System vol 10 no 3 pp 254ndash262 1998

[2] J L Deng Prediction and Grey Decision Press of HuazhongUinversity of Science amp Technology Wuhan China 2002

[3] S F Liu and Y Lin Grey Information Theory and PracticalApplications Springer London UK 2006

[4] Z XWang Y GDang andC P Song ldquoMultiobjective decisionmodel of the grey situation based on interval numberrdquo Controland Decision vol 24 no 3 pp 388ndash392 2009

[5] J Luo Y Liu and K Xu ldquoParameter establishment of intelligentalgorithmbased onuniformdesign and grey situation decisionrdquoJournal of Shenyang University of Technology vol 32 no 1 pp84ndash89 2010

[6] Y Liu J Forrest H H Zhao S F Liu and J Liu ldquoMulti-objective grey situation decision making method based onprospect theoryrdquo Systems Engineering and Electronics vol 34no 12 pp 2514ndash2519 2012

[7] Y Liu J Forrest and S F Liu ldquoMulti-objective grey situationdecision making model and application with interval numbersbased on linear transformation operator with rewarding goodand punishingrdquo Statistics amp Information Forum vol 12 no 7pp 22ndash26 2012

[8] L Zhou and D Luo ldquoStudy on decision-making method formulti-objective grey situation decisionrdquo Journal of North ChinaInstitute of Water Conservancy and Hydroelectric Power vol 31no 4 pp 150ndash153 2010

[9] Y MWang and Y G Dang ldquoNewmethod in situation decisionmaking based on entropyrdquo Systems Engineering and Electronicsvol 31 no 6 pp 1350ndash1352 2009

The Scientific World Journal 7

[10] D J Chen SH Zhang and J Y Chen ldquoResearch of gray generalsituation decision and its applicationrdquo Systems Engineering andElectronics vol 26 no 4 pp 423ndash429 2004

[11] C T Lin N H Shinh and W C Chuang ldquoStock tradingstrategy based on grey situation decision model an empiricalstudy of the electronic component industry in Taiwanrdquo Journalof Grey System vol 18 no 3 pp 239ndash250 2006

[12] C T Lin and H F Chang ldquoPerformance evaluation for medicalindustry in Taiwan applying grey situation decision makingrdquoJournal of Grey System vol 22 no 3 pp 219ndash226 2010

[13] X Shan X Z Wang and S X Liu ldquoDecision-making ofmissile flexible support sites locating in wartime with fuzzynumbers and grey situationrdquo Journal of Naval Aeronautical andAstronautical University vol 26 no 6 pp 703ndash706 2011

[14] G X Huang X B Wang and Y L Wang ldquoScheme theartillery fire plan by grey situation analysisrdquo Ordnance IndustryAutomation vol 31 no 2 pp 31ndash37 2012

[15] Y B Wang X S Jia and G Y Wang ldquoMethod of integrationfor currency maintenance resources based on grey situationdecision-makingrdquo Fire Control ampCommandControl vol 38 no4 pp 4ndash8 2013

[16] L M Chen P Zheng and S X Ji ldquoGrey situation evaluation ofsupply chain cooperative relationship based on cobweb modelrdquoLogisticstech vol 31 no 7 pp 321ndash323 2012

[17] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Economica vol 47 no 2 pp 263ndash291 1979

[18] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

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

The Scientific World Journal 7

[10] D J Chen SH Zhang and J Y Chen ldquoResearch of gray generalsituation decision and its applicationrdquo Systems Engineering andElectronics vol 26 no 4 pp 423ndash429 2004

[11] C T Lin N H Shinh and W C Chuang ldquoStock tradingstrategy based on grey situation decision model an empiricalstudy of the electronic component industry in Taiwanrdquo Journalof Grey System vol 18 no 3 pp 239ndash250 2006

[12] C T Lin and H F Chang ldquoPerformance evaluation for medicalindustry in Taiwan applying grey situation decision makingrdquoJournal of Grey System vol 22 no 3 pp 219ndash226 2010

[13] X Shan X Z Wang and S X Liu ldquoDecision-making ofmissile flexible support sites locating in wartime with fuzzynumbers and grey situationrdquo Journal of Naval Aeronautical andAstronautical University vol 26 no 6 pp 703ndash706 2011

[14] G X Huang X B Wang and Y L Wang ldquoScheme theartillery fire plan by grey situation analysisrdquo Ordnance IndustryAutomation vol 31 no 2 pp 31ndash37 2012

[15] Y B Wang X S Jia and G Y Wang ldquoMethod of integrationfor currency maintenance resources based on grey situationdecision-makingrdquo Fire Control ampCommandControl vol 38 no4 pp 4ndash8 2013

[16] L M Chen P Zheng and S X Ji ldquoGrey situation evaluation ofsupply chain cooperative relationship based on cobweb modelrdquoLogisticstech vol 31 no 7 pp 321ndash323 2012

[17] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Economica vol 47 no 2 pp 263ndash291 1979

[18] A Tversky and D Kahneman ldquoAdvances in prospect theorycumulative representation of uncertaintyrdquo Journal of Risk andUncertainty vol 5 no 4 pp 297ndash323 1992

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