uncertain multiattribute decision-making based on interval...

Research ArticleUncertain Multiattribute Decision-Making Based on IntervalNumber with Extension-Dependent Degree and Regret Aversion

Libo Xu Xingsen Li and Chaoyi Pang

Laboratory of Intelligent Information Processing and Data Decision Zhejiang University Ningbo Institute of TechnologyZhejiang Ningbo 315000 China

Correspondence should be addressed to Libo Xu xu libo163com

Received 6 May 2018 Revised 29 July 2018 Accepted 13 August 2018 Published 27 August 2018

Academic Editor Eric Lefevre

Copyright copy 2018 Libo Xu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In view of the uncertaintymultiattribute decision-making problem with attribute values and weights both being interval number anew solution based on regret theory and extension-dependent degree is proposed It can define pass value of each attribute whichmeans decision-makerrsquos acceptance for the scheme under the pass value will decline quickly Then according to traditional regrettheory themethod defines an extension-dependent function based on pass valuewhich can improve the flexibility of the traditionalutility function and the ability to describe the risk aversion actions from decision-makers Then the extension-dependent functionfor interval number is built and the perceived utility value of each scheme is obtained based on the intervalrsquos optimal value Themethod can also reflect the decision-makerrsquos reference to high or low evaluation score by setting attitude coefficients At last anexample is presented to examine the feasibility effectiveness and stability of our method

1 Introduction

Multiattribute decision-making (MADM) is aimed at findingthemost desirable solution according to the evaluation resultsto several given attributes which is widely used in variousbusiness applications such as online auction investmentdecision and e-commence platform evaluation But due tothe increasing complexity of socioeconomic environmentand inherent knowledge restrictions of decision-makers [1]those attributesrsquo environment information always exhibitsfuzzy and uncertain features which are difficult to describeby exact numerical values Several technologies are proposedto deal with the above problem such as interval numberfuzzy number connection number and grey number [2ndash5]Among them because of outstanding simple and intuition-istic features interval number is proved to be one of themost frequently used described frameworks for uncertaininformation in many practical applications [6]

Early decision-makingmethods belong to complete ratio-nal decisions based on expected utility theory They oftenuse the expected utility value of each scheme to performsorting But many experimental studies have shown that inthe decision-making process decision-makers usually have

some psychological and behavior characteristics such asreference dependence loss aversion diminishing sensitivityand expected results comparison [7]The traditional decisionmethods cannot describe these psychological characteristicsSo the finite rational decision methods based on decision-makerrsquos behavior obtain more and more attentions [8] Therepresentatives are prospect theory and regret theory Theregret theory is one of the attractive research hotspots [910] By rejoice-regret utility function regret theory not onlyconsiders the utility value of the alternative to be selectedbut also considers the results of others alternative andcan describe those psychological characteristics of decision-makers Literature [11] calculates collective utilities of eachattribute in hesitant fuzzy set according to regret theoryand then pursues matching result by aggregating collectiveutilities with relative weights Literature [12] builds regretvalue matrix of comparisons between couple alternatives byregret theory and then obtains sorting results by countingtotal regret values between each alternative and all the otheralternatives Literature [13] defines grey perception utilityfunction based on regret theory and then builds multiob-jective optimization model which produces the maximumgrey overall perceived utility values of alternatives so that the

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6508636 10 pageshttpsdoiorg10115520186508636

2 Mathematical Problems in Engineering

optimal grey perceived utility value of each scheme is rankedby the sorting rules of interval grey numbers In literature[14] the rejoice-regret value function based on the positiveand negative ideal solution is constructed and the positiveand negative ideal bullrsquos eyes are obtained So amultiobjectivegrey target decision model is completed Literature [15]uses regret theory to obtain interval comprehensive utilityvalue of alternatives in three-parameter interval numberdecision-making In literature [16] an improved feedbackadjustment mechanism based on regret theory is employed asthe consistency model which complements prospect theoryin multiattribute group decision problems Literature [17]combines regret theory toTOPSISmethod and applies to greystochastic multicriteria decision-making Literature [12 15]point out that among those decision strategies based ondecision-makersrsquo behavior prospect theory has the strongcapacity to describe decision-makersrsquo behavior but has moreparameters and more computational complexity By contrastregret theory has fewer parameters and is easy to implementbut has weaker describing capacity Literature [18] concludesthat regret theory has more advantages in practical applica-tion than prospect theory

Although various methods adopting regret theory havebeen widely applied to uncertain decision-making there aresome problems for the researchers Firstly most of existingmethods are built on the traditional basic model of regret the-ory So the ability of these methods for describing decision-makersrsquo behavior is insufficient That is when the risk evasioncoefficient is determined the curve of the utility function inregret theory is fixed It has nothing to do with the range anddistribution of the corresponding attribute values so it lacksflexibility However in many decision-making applicationsattribute values usually have a special value called pass pointbelow which the acceptance of decision-makers will declinerapidly It means that decision-makers often show distinctrisk chasing psychology The basic model of regret theory isnot sufficient to describe these behavior features and psychol-ogy features of the people Secondly many frameworks basedon the basic model of regret theory often appear too definitewhich means it is difficult to perform any stability test oruncertainty analysis for final decision results So we do notknow whether a tiny change of the psychology or preferenceof decision makes will bring different decision results Thatwill influence the credibility of the results So a good methodfor decision-making should not only produce a sorting resultbut also bring some stability test processes

To solve the above problems the paper attempts topropose an uncertain multiattribute decision-making modelbased on regret theory and extension-dependent degreeUntil the present the related research has not been seenIn this model the pass point value of the attribute can beset by decision-maker or other methods and the extension-dependent degree of the attribute value can be obtained byusing extension-dependent function based on side distanceActually it represents a new utility function expression andis more flexible than the basic utility function of regrettheory It is also more in line with peoples decision-makinghabits Then the regret-rejoice value of each attribute valueis obtained by combining its dependent degree and regret

theory According to regret-rejoice values the comprehensiveutility value of each alternative will be calculated Lastly inour model through setting the different value of preferenceattitude coefficient the final ranking results can be givenand analyzed further which reflects the different attitudeof decision-makers towards the upper and lower bounds ofevaluation and reflects the ability of uncertainty analysis

2 Regret Theory Model withExtension-Dependent Degree

21 Extension-Dependent Function with Side Distance Inextenics extension distance is defined to describe the dis-tance between a point and an interval which is able toaccurately describe the relative position between the pointand the interval For the definition of distance classicalmathematics utilizes qualitative descriptions of ldquobelongingrdquoand ldquonot belongingrdquo while extenics utilizes quantitativedescriptions In classical mathematics there are elements thatin the same domain are homogenous and but heterogeneousin different domains while in extenics elements in the samedomain they can be further classified into different layers andcan be given quantitative description [19 20]

Definition 1 (see [19 21]) Suppose 119877 the set of real number119883 = [119886 119887] is a finite interval in 119877 and its optimal point is119898 isin 119883 then forall119909 isin 119877 according to the different location of119898in119883 extension distance 120588(119909119898119883) is

(1)when 119898 = 119886 + 1198872 120588 (119909119898119883) = |119909 minus 119898| minus 119887 minus 1198862 (1)

(2)when 119898 isin [119886 119886 + 1198872 ) 120588 (119909 119898119883)

=

119886 minus 119909 119909 le 119886119887 minus 119898119886 minus 119898 (119909 minus 119886) 119886 lt 119909 lt 119898119909 minus 119887 119909 ge 119898

(2)

(3)when 119898 isin (119886 + 1198872 119887] 120588 (119909 119898119883)

=


(3)

In addition in this paper we agree that 120588(119886 119886 119883) =120588(119887 119887 119883) = 119886 minus 119887Definition 2 (see [19 22]) Suppose an interval coveringis composed of standard positive interval X0 and positiveinterval X 1198830 = [119886 119887] 119883 = [119888 119889] 1198830 sub 119883 and optimalpoint is 1198980 isin 1198830 for any point 119909 isin 119883 extension-dependentfunction 119896(119909) is

Mathematical Problems in Engineering 3

119896 (119909) =

120588(1199091198980 119883)120588 (119909 1198980 119883) minus 120588 (1199091198980 1198830) 120588 (1199091198980 1198830) = 120588 (119909 1198980 119883) 119909 notin 1198830

120588 (1199091198980 119883) + 119886 minus 119887120588 (119909 1198980 119883) minus 120588 (1199091198980 1198830) + 119886 minus 119887 119909 isin 1198830120588 (119909 1198980 119883)

119886 minus 119887 120588 (1199091198980 1198830) = 120588 (119909 1198980 119883) 119909 notin 1198830

(4)

then 119896(119909) satisfies following properties(1) 119909 isin 1198830 and 119909 = 119886 119887 lArrrArr 119896(119909) gt 1(2) 119909 = 119886 or 119909 = 119887 lArrrArr 119896(119909) = 1(3) 119909 notin 1198830 119909 isin 119883 or 119909 = 119886 119887 119888 119889 lArrrArr 0 lt 119896(119909) lt 1(4) 119909 = 119888 or 119909 = 119889 lArrrArr 119896(119909) = 0(5) When 119909 = 1198980 119896(119909) reaches its maximum value

Extension-dependent function describes the dependentdegree between point 119909 and interval covering 1198830 119883 Itscalculation is based on the given range of values of a certainfeature and it does not need to rely on subjective judgmentor empirical value from decision-makers so it is convenientto quantitatively describe the nature of things

22 Regret Theory with Extension-Dependent FunctionRegret theory [9] is one of the most important behavioraldecision theories in behavioral economics Its basic idea isthat decision-makers in the decision-making process are con-cerned not onlywith the outcomeof the options they considerchoosing but also with the possible impact of choosing otheroptions Decision-makers may have the expectation of regretor delight from their decision result and try to avoid choosingthe alternative that may bring regret perception and tendto choose the alternative that brings delight perception Theperceived utility function of decision-makers consists of twoparts the utility function of the current selected alternativeand the rejoice-regret function for comparing with the otherschemes

In many practical decision-making problems decision-makers are usually risk aversion So for the benefit attributesthe utility function V(119909) is shown as a monotone increasingconcave function and satisfies V1015840(119909) gt 0 and usually employsnegative exponential form [12]

V (119909) = 1 minus exp (minus120573119909)120573 0 lt 120573 lt 1 (5)

Here 120573 is risk aversion coefficientThe greater the 120573 valuethe greater the risk aversion of decision-makers as shownin Figure 1 For cost attributes the utility function takes thefollowing form [12]

V (119909) = 1 minus exp (120573119909) 0 lt 120573 lt 1 (6)

Here the greater the 120573 value the greater the risk aversionof decision-makers

The utility function reflects the risk evasion attitude ofthe decision-maker It can be seen that the curve slope ofutility value under small risk aversion coefficient is largerthan the curve slope of utility value under large risk aver-sion coefficient which expresses the high sensitivity of thedecision-maker for increasing utility value near the marginalpoint But the traditional utility function in regret theory isnot enough to describe the psychological behavior of riskaversion Firstly when the value of risk aversion coefficientis determined (for example [7] suggests 002) the curve ofutility function is fixed in thewhole decision-making processand it cannot be adjusted according to the value of differentattributes Secondly in several practical decision-makingapplications every attribute always contains a psychologicalpass value under which the decision-makers acceptance tothe alternative presents a rapid decline trend So this valuecan be regarded as the pass reference value of an alternativeunder the attribute For example there are two attributes withpercentage ratings and their weights are the same Supposetheir pass values both are 60 below which it is consideredto be unqualified Here alternative A is evaluated as (5080)and alternative B is evaluated as (6070) Due to the distinctdifference of the acceptance from decision-makers to theevaluations above the pass value and below the pass valueit shows decision-makers evasion attitude to select the alter-native containing the evaluation score below the pass valueSo although the total scores of alternatives A and B bothare 130 the decision-maker will choose alternative B becausealternative A contains the unqualified score Therefore theevaluation below the pass value represents a higher riskwhich decision-makers will tend to avoid But the traditionalutility function in regret theory cannot describe this psy-chology of risk aversion Combined with the above ideasand extension-dependent function in extenics theory thispaper adopts extension-dependent function as a new utilityfunction Without increasing the number of parameters ourmethod can describe the behavior of decision-makers moreflexibly and reasonably and enhance the ability to capture riskaversion behavior

Definition 3 Suppose the value range of a beneficial attributeis 119883 = [119888 119889] and its pass value is a (119888 lt 119886 lt 119889) Then119883 = [119888 119889] can be seen as positive interval and 1198830 = [119886 119889]as standard positive interval 1198830 sub 119883 optimal value is daccording to Definition 2 for any point 119909 isin 119883 extension-dependent function 119896(119909 1198830 119883) is


= 005

= 010

= 015 = 020

0 x

v(x)

105

8

4

= 025

Figure 1 Utility function

119896 (119909 1198830 119883) =

120588 (119909 119889 119883)120588 (119909 119889 119883) minus 120588 (119909 119889 1198830) =

119888 minus 119909119888 minus 119909 minus (119886 minus 119909) =

119888 minus 119909119888 minus 119886 119909 notin 1198830

120588 (119909 119889 119883) + 119886 minus 119889120588 (119909 119889 119883) minus 120588 (119909 119889 1198830) + 119886 minus 119889 =

119888 minus 119909 + 119886 minus 119889119888 minus 119909 minus (119886 minus 119909) + 119886 minus 119889 =

119886 minus 119909119888 minus 119889 + 1 119909 isin 1198830

(7)

The curve graph of 119896(1199091198830 119883) is shown in Figure 2 and hasthe following properties

(1) 119909 isin 1198830 and 119909 = 119886 lArrrArr 119896(1199091198830 119883) gt 1(2) 119909 = 119886 lArrrArr 119896(1199091198830 119883) = 1(3) 119909 notin 1198830 119909 isin 119883 and 119909 = 119886 119888 119889 lArrrArr 0 lt 119896(1199091198830 119883) lt1(4) 119909 = 119888 lArrrArr 119896(1199091198830 119883) = 0(5) When 119909 = 119889 119896(119909 1198830 119883)reaches the maximum value

(119886 minus 119889)(119888 minus 119889) + 1(6) The slope of the line segment 119896(119909 1198830 119883)(119909 lt 119886)must

be larger than that of 119896(119909 1198830 119883)(119909 gt 119886)As shown in Figure 2 with the pass value 119886 as the

boundary point 119896(119909 1198830 119883) consists of two segments of theline The shape of the line segment changes adaptively withthe value range of the attribute or the position of the passvalue 119886 In Figure 2 when pass value is changed from ato 1198861015840 the slope of the line segment under the pass value isautomatically adjusted When the value range of the attributeis changed from [119888 119889] to [119888 1198891015840] the slope of the line segmentabove the pass value is also automatically adjusted When therange of value and pass value are both changed the whole lineshape will change So it is more flexible and adaptive than thetraditional utility function in regret theory In addition when119909 is less than the pass value 119886 119896(119909 1198830 119883) will decrease at afaster speed It means that the acceptance of decision-makerswill decline rapidly when the evaluation of an attribute isunder the pass value which distinctly reflects a higher riskaversion attitude fromdecision-makers It can be proved thatfor any pass value 119886 isin (119888 119889) the slope of the line segment

119896(1199091198830 119883)(119909 lt 119886)must be larger than that of 119896(119909 1198830 119883)(119909 gt119886)Theorem 4 In Definition 3 the slope of the line segment119896(1199091198830 119883)(119909 lt 119886)must be larger than that of 119896(119909 1198830 119883)(119909 gt119886)Proof As shown in Figure 2 forall119886 isin (119888 119889) the slope of theline segment of 119896(119909 1198830 119883)(119909 lt 119886) is 1(119886 minus 119888) and that of119896(1199091198830 119883)(119909 gt 119886) is ((119886 minus 119889)(119888 minus 119889))(119889 minus 119886) = 1(119889 minus 119888)Due to 119888 lt 119886 lt 119889 there must be 1(119886 minus 119888) gt 1(119889 minus 119888)

After getting the dependent degree of the attribute valuethe regret value of each attribute value and its optimalvalue can be calculated for each alternative Suppose theregret-rejoice function is 119877(Δ119896) due to decision-makersrsquo riskaversion attribute to regret function119877(Δ119896) is amonotonouslyincrementally concave function and satisfies 119877(Δ119896)1015840 gt 0119877(Δ119896) can be expressed as [12]

119877 (Δ119896) = 1 minus exp (minus120575Δ119896) (8)

Here 120575 is regret aversion coefficient and satisfies 120575 gt 0The setting of 120575 is decided by the experience value and thepsychology of decision-makers and usually taken in 0sim1 Asshown in Figure 3 the larger the value 120575 the greater the extentof regret evasion from the decision-maker When people hashigher risk awareness and regret evasion trend he shouldtake a greater value of 120575 If people has medium regret evasionpsychology 120575 is often taken as 03 [12 15] Δk denotes thedifference of dependent degree of the same attribute undertwo alternatives When 119877(Δ119896) gt 0 119877(Δ119896) denotes rejoicevalue and when 119877(Δ119896) lt 0 119877(Δ119896) denotes regret value


c0

a d

1

k(x)

x

a minus d

c minus d+ 1

a d

Figure 2 Extension-dependent function

= 030 = 020 = 010

0 Δk

R(Δk)

63

1

= 00505

minus05

minus1

minus3

Figure 3 Regret-rejoice function

According to formula (8) when Δ119896 gt 0 there is |119877(minusΔ119896)| gt119877(Δ119896) It indicates that decision-makers are more sensitiveto the psychological perception of minusΔ119896 than to Δ119896 that isdecision-makers are always regret aversion

For simplifying the calculating process andwithout losingits validity we obtain regret value through comparing theattribute value of each alternative to the optimal value ofthe attribute Here according formula (8) for a beneficialattribute with its positive interval 119883 = [119888 119889] and standardpositive interval 1198830 = [119886 119889] clearly its ideal value is 119889 thenthe regret value of attribute value 119909 is

119877 (1199091198830 119883)= 1 minus exp (minus120575 (119896 (119909 1198830 119883) minus 119896 (119889 1198830 119883)))

(9)

The perception utility value of x is

119906 (1199091198830 119883) = 119896 (1199091198830 119883) + 119877 (1199091198830 119883) (10)

23 Interval Extension-Dependent Degree In many decision-making applications due to the complexity and uncertaintyof objective things and the vague and finite features ofhuman thinking people often cannot give the definite value

of attributes of things but can only give an interval rangeIt is called uncertain decision-making problem When thedefinite values of attributes are generalized to intervals thecomputational model based on definite values will be trans-formed into interval-based computing models So accordingto Definition 3 here we give a computing method of intervalextension-dependent degree under beneficial attribute

Definition 5 Suppose the value range of a beneficial attributeis 119883 = [119888 119889] and its pass value is a (119888 lt 119886 lt 119889) Then119883 = [119888 119889] can be seen as positive interval and 1198830 = [119886 119889]as standard positive interval 1198830 sub 119883 optimal value is d forany interval 119876 = [119902minus 119902+] and 119876 sube 119883 extension-dependentfunction 119896(1198761198830 119883) of interval 119876 and interval covering 1198830119883 is

119896 (1198761198830 119883) = 120572119896 (119902minus 1198830 119883) + (1 minus 120572) 119896 (119902+ 1198830 119883) 120572 isin [0 1] (11)

Here 120572 is the preference attitude coefficient whichreflects the preference extent of decision-makers to thedependent degree of upper and lower bounds of the attributeindex That is to say whether decision-makers attach moreattention to the high evaluation of the attribute or to thelow evaluation When 120572 = 0 according to formula (11) itmeans that decision-makers only consider the upper boundsof evaluations and vice versa When 120572 = 05 it means thatdecision-makers pay equal attentions to the upper and lowerbounds of evaluations So the value of 120572 is decided by thepreference attitude of decision-makers to the two bounds119896(1198761198830 119883) satisfies the following properties

(1) When 119902minus = 119902+ 119896(1198761198830 119883) will degenerate intoDefinition 3

(2) When 119902minus = 119902+ = 119889 119896(1198761198830 119883) has the maximumvalue (119886minus119889)(119888minus119889)+1When 119902minus = 119902+ = 119888 119896(1198761198830 119883)reaches the minimum value 0

(3) When 119876 sub 1198830 1 lt 119896(1198761198830 119883) lt (119886 minus 119889)(119888 minus 119889) + 1When 119876 sub 119883 and 119876 sub 1198830 0 lt 119896(1198761198830 119883) lt 1

24 Pass Value Selecting Pass value selecting of attributes canadopt subjective choosing or objective choosing Subjectivechoosing is generally based on the experience of experts orsome industry standards Objective choosing can be carriedout by choosing mean value median value expectation valueand two-eight principle

25WeightCalculating Based onMaximumDeviation ofDevi-ation Degree In view of sorting alternatives if the differenceof attribute value of all alternatives under a certain attributeis smaller then the distinguishing effect of the attribute toall alternatives is smaller So the smaller the weight shouldbe given to the attribute On the contrary the greater thedifference of the attribute value of each alternative under acertain attribute the greater the distinguishing effect of theattribute the greater the weight that should be given to theattribute Here weight calculating model based on maximumdeviation of interval deviation degree is built Suppose the


evaluation matrix is (119876119894119895)119898times119899 (119894 = 1 2 119898 119895 = 1 2 119899)its evaluation value is interval value 119876119894119895 = [119902minus119894119895 119902+119894119895] 119882 isweight vector of attributes and Φ is the range constraint ofattribute weight values The single object optimization modelwill be built

max 119863 (119882) =119898

sum119894=1

119899

sum119895=1

119898

sum119896=1

(10038161003816100381610038161003816119902minus119894119895 minus 119902minus11989611989510038161003816100381610038161003816 + 10038161003816100381610038161003816119902+119894119895 minus 119902+11989611989510038161003816100381610038161003816) 119908119895

st 119882 isin Φ

st119899

sum119895=1

119908119895 = 1(12)

Solving this model the optimal weight vector 119882 = (1199081 1199082 119908119899) is obtained3 Process of Decision Algorithm

Suppose alternative set of multiattribute decision-making is119878 = 1199041 1199042 119904119898 attribute set is 119880 = 1199061 1199062 119906119899the evaluation value of alternative si to attribute uj is intervalvalue 119875119894119895 = [119901minus119894119895 119901+119894119895] then evaluation matrix is (119875119894119895)119898times119899 (119894 = 12 119898 119895 = 1 2 119899)Step 1 By normalization of evaluation matrix (119875119894119895)119898times119899 thenewnormalized evaluation matrix (119876119894119895)119898times119899 (119876119894119895 = [119902minus119894119895 119902+119894119895]) isobtained The standardized formula for the benefit attributesis 119876119894119895 = [119902minus119894119895 = 119901minus119894119895max119894(119901+119894119895) 119902+119894119895 = 119901+119894119895max119894(119901+119894119895)] 119894 =1 2 119898 The standardized formula for the cost attributesis 119876119894119895 = [119902minus119894119895 = (min119894(119901minus119894119895))119901+119894119895 119902+119894119895 = (min119894(119901minus119894119895))119901minus119894119895] 119894 =1 2 119898 After the normalization each attribute well betransformed into benefit attributewith its value range is [0 1]Step 2 Get the value range of each attribute uj and its passvalue 119902lowast119895 and then get the positive interval Xj and the stan-dard positive interval Xj0 of each attribute Due to normali-zation of evaluation matrix the value range of each attributeis mapped to [0 1] which is noted as the positive intervalXj Clearly the ideal point of Xj is 1 The pass value 119902lowast119895 ofeach attribute uj is decided as required by decision-makersThen standard positive interval Xj0 of each attribute is [119902lowast119895 1]For formula (7) the extension-dependent degree 119896(119902minus119894119895 1198831198950119883119895) and 119896(119902+119894119895 1198831198950 119883119895) of each attribute valueQij are obtainedThen for formula (11) the interval-dependent degree119896(119876119894119895 1198831198950 119883119895) of each evaluation valueQij is calculatedHerethe preference attitude coefficient 120572 should be decided asrequired by decision-makers

Step 3 According to formula (9) and 119896(119876119894119895 1198831198950 119883119895) fromStep 2 for each alternative the regret value R(Qij Xj0 Xj) ofeach attribute value Qij is given Here 119877(119876119894119895 1198830 119883) = 1 minusexp(minus120575(119896(119876119894119895 1198830 119883)minus119896(11198830 119883))) because the ideal value is1 and the setting of regret aversion coefficient 120575 is decided bythe regret psychology of decision-makers Combining k(QijXj0Xj)R(QijXj0Xj) and formula (10) the perception utilityvalue uij (uij = u(Qij Xj0 Xj)) of each attribute value Qij is

obtained Finally the perception utility value matrix (119906119894119895)119898times119899is obtained

Step 4 Counting deviation matrix (119863119894119895)119898times119899 of interval devia-tion degree from the normalized evaluation matrix (119876119894119895)119898times119899in which 119863119894119895 = sum119898119896=1(|119902minus119894119895 minus 119902minus119896119895| + |119902+119894119895 minus 119902+119896119895|) Then solvingformula (12) to get optimal weight vector 119882 = (1199081 1199082 119908119899)Step 5 From Steps 3 and 4 the optimal weight vector W ofattributes and the perception utility value matrix (119906119894119895)119898times119899 areobtained Calculating the comprehensive perception utilityvalue U(si) of each alternative si by formula 119880(119904119894) =sum119899119895=1119908119895119906119894119895 and then getting sorting results from U(si)

Step 6 Performing uncertain analysis by different setting ofpreference attitude coefficient 120572 and regret aversion coeffi-cient 120575 Here we recommend the setting ranges of 120572 and 120575 are02sim08 120572 reflects the preference extent of decision-makersto the upper and lower evaluation bounds of the attribute120575 represents the psychology of regret evasion from decision-makers

4 Example Analysis

For easy of comparison and illustration the example is basedon the data of [23] An electronic commerce website intendsto evaluate the different brands of air-conditionings soldonline and to determine the ranking of brands recommendedto users The website examines five brands (s1 s2 s3 s4 s5)The evaluation contains eight attributes that is fixed cost u1operation cost u2 performance u3 noise u4 maintainabilityu5 reliability u6 flexibility u7 and security u8 Among themthe evaluations of attributes u3 u5 u7 u8 come from websiteusers and the scoring range is from 1 to 10 In addition u1u2 u4 are cost attributes and the others are benefit attributesBased on attribute weights and evaluation matrix shownin Table 1 brand sorting will need to be determined Theexample is performed by MATLAB program

(1) According to Step 1 normalize the original evaluationmatrix and get the standardized evaluation matrix as shownin Table 2The process not only eliminates the dimension dif-ferences of each attribute but also transforms each attributeinto benefit attribute

(2) Getting the value interval of each attribute and itspass value after standardization evaluation value interval ischanged to [0 1] so that the positive interval of each attributevalue also is [0 1] Here we take the median value as thepass value of each attribute which represents the decision-makerrsquos aversion attitude towards the evaluation below themedian value as shown in Table 3 According to Step 2 theinterval extension-dependent degree of each evaluation valueis calculated and the interval-dependent degree matrix isobtained as shown in Table 4

(3) According to Step 3 the regret value and perceptionutility value of each attribute value and its optimal value arecalculated and the perception utility matrix is obtained asshown in Table 5 The regret aversion coefficient 120575 here takesa recommended value of 03 [12]


Table 1 Evaluation matrix

u1 u2 u3 u4 u5 u6 u7 u8s1 [37 47] [59 69] [8 10] [30 40] [3 5] [90 100] [3 5] [6 8]s2 [15 25] [47 57] [4 6] [65 75] [3 5] [70 80] [7 9] [4 6]s3 [30 40] [42 52] [4 6] [60 70] [7 9] [80 90] [7 9] [5 7]s4 [35 45] [45 55] [7 9] [35 45] [8 10] [85 95] [6 8] [7 9]s5 [25 35] [50 60] [6 8] [50 60] [5 7] [85 95] [4 6] [8 10]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]

Table 2 Standardized evaluation matrix

u1 u2 u3 u4 u5 u6 u7 u8s1 [0319 0405] [0609 0712] [0800 1000] [0750 1000] [0300 0500] [0900 1000] [0333 0556] [0600 0800]s2 [0600 1000] [0737 0894] [0400 0600] [0400 0462] [0300 0500] [0700 0800] [0778 1000] [0400 0600]s3 [0375 0500] [0808 1000] [0400 0600] [0429 0500] [0700 0900] [0800 0900] [0778 1000] [0500 0700]s4 [0333 0429] [0764 0933] [0700 0900] [0667 0857] [0800 1000] [0850 0950] [0667 0889] [0700 0900]s5 [0429 0600] [0700 0840] [0600 0800] [0500 0600] [0500 0700] [0850 0950] [0444 0667] [0800 1000]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]

Table 3 Positive interval and pass value of each attribute

u1 u2 u3 u4 u5 u6 u7 u8Positive interval [0 1] [0 1] [0 1] [0 1] [0 1] [0 1] [0 1] [0 1]Pass value 0429 0764 0600 0500 0500 0850 0667 0700Standard positive interval [0429 1] [0764 1] [0600 1] [0500 1] [0500 1] [0850 1] [0667 1] [0700 1]

Table 4 Interval-dependent degree matrix

u1 u2 u3 u4 u5 u6 u7 u8s1 08438 08645 13000 13750 08000 11000 06664 09786s2 13710 10473 08333 08620 08000 08824 12220 07143s3 09726 11400 08333 09290 13000 09956 12220 08571s4 08881 10845 12000 12620 14000 10500 11110 11000s5 10855 09961 11000 10500 11000 10500 08328 12000

Table 5 Perception utility matrix

u1 u2 u3 u4 u5 u6 u7 u8s1 06000 07466 12695 13368 05663 10849 04450 08773s2 13092 09891 06480 06511 05663 07987 11881 05222s3 07759 11108 06480 07422 12382 09482 11881 07151s4 06608 10380 11382 11880 13695 10195 10421 10382s5 09287 09215 10058 09055 09725 10195 06709 11695

(4) According to Step 4 deviation matrix of deviationdegree can be counted For each attribute the deviation valueof the attribute value of each alternative and that of the otheralternatives is obtained as shown in Table 6

According to formula (12) and deviation matrix a singleobject optimization model is established

max 119863 (119882)= 78431199081 + 45241199082 + 88001199083 + 94841199084+ 112001199085 + 36001199086 + 97801199087+ 80001199088

st 0042 le 1199081 le 0049 0084 le 1199082 le 0098 0121le 1199083 le 0137 0121 le 1199084 le 01370168 le 1199085 le 0182 0214 le 1199086 le 0229 0040le 1199087 le 0046 0159 le 1199088 le 0171

st8

sum119895=1

119908119895 = 1(13)


Table 6 Deviation matrix of deviation degree

u1 u2 u3 u4 u5 u6 u7 u8s1 1370 1392 2200 2585 2200 0800 2667 1200s2 3010 0644 1800 1855 2200 1200 1778 2000s3 1143 1043 1800 1654 2200 0600 1778 1400s4 1256 0710 1600 1907 2800 0500 1556 1400s5 1297 0735 1400 1483 1800 0500 2001 2000

Table 7 Comprehensive perceived utility value sorting

s1 s2 s3 s4 s5 Sorting result119880(119904119894) 09389 07271 09123 11129 09893 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042

Table 8 Sorting results under different attitude coefficient 120572120572 1199041 1199042 1199043 1199044 1199045 Sorting result03 10002 07936 09668 11594 10334 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 119904204 09696 07604 09396 11362 10113 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 119904205 09389 07271 09123 11129 09893 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 119904206 09082 06937 08850 10896 09671 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 119904207 08773 06603 08576 10663 09450 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042

Table 9 Sorting results under different regret aversion coefficient 120575120575 1199041 1199042 1199043 1199044 1199045 Sorting result02 09746 07814 09496 11325 10195 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 119904203 09389 07271 09123 11129 09893 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 119904205 08615 06088 08324 10723 09254 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 119904207 07751 04762 07449 10292 08567 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 119904208 07281 04040 06980 10068 08204 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042

The optimal weight vector W is obtained by solving thismodel

119882= (0042 0084 0136 0137 0182 0214 0046 0159) (14)

(5) According to Step 5 calculating the comprehensiveperceived utility value of each alternative and sorting by thevalues the result is shown in Table 7

(6) Performing uncertain analysis the attitude coefficient120572 represents the decision-makers preference towards theupper and lower bounds of attribute evaluation Table 8 showsthe decision sorting results under five different 120572 settingsIt shows that the sorting results under different attitudecoefficients are still the same which indicates that the currentsorting does not change with the decision-makers preferenceto the evaluation bounds Table 9 shows the decision resultsunder different regret aversion coefficient It can be seen thatalternatives sorting remains unchanged which illustrates thecurrent sorting has higher stability

(7) Comparison of decision results under different algo-rithms Table 10 shows the decision results under five algo-rithms In our algorithm the pass value is median valueand mean value respectively attitude coefficient is 05 and

regret aversion coefficient is 03 In regret theory risk aversioncoefficient is 002 and regret aversion coefficient is 03 In setpair analysis the uncertain number 119894 is 0 and 05 respectivelyThe sorting results of these algorithms are consistent Theresults of TOPSIS and relativemembership degree algorithmsare slightly different mainly on the sorting of alternatives s1s3 s5 It can be seen that for sorting of s3 and s5 TOPSISis consistent with our method and the conclusion of relativemembership degreemethod are different But for sorting of s1and s3 relative membership degree method and our methodare consistent and are disagreement with TOPSIS So in thewhole the sorting result of our method is more reasonable

(8) Comparison with the traditional regret theory theline shape of utility function in regret theory is uniquelydetermined by the risk aversion coefficient 120573 and is notflexible enough Moreover its setting is difficult due to oftenlack sufficient basis By comparison the extension-dependentfunction of our algorithm can be adjusted automaticallyaccording to the pass value and the value range of attributeand so it is more flexible In many cases setting pass valueis more in line with peoples habit in decision-making It isalso easy to find a certain basis for its value taking whetherit follows subjective method or objective method Pass valuesetting will have an impact on the result of the decision In


Table 10 Sorting results under different algorithms

s1 s2 s3 s4 s5 Sorting resultTOPSIS [24] 04176 02671 04188 05777 04498 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042elative membership degree [25] 04502 00327 03262 09167 03381 1199044 ≻ 1199041 ≻ 1199045 ≻ 1199043 ≻ 1199042set pair analysis (i = 0) [23] 06036 04781 05827 06974 06203 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042set pair analysis (i = 05) [23] 07270 05916 06958 08227 07354 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042regret theory 06268 04465 05873 07579 06369 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042regret-extension (median value) 09389 07271 09123 11129 09893 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042regret-extension (mean value) 08663 06450 08353 10474 09052 1199044 ≻ 1199045 ≻ 1199041 ≻ 1199043 ≻ 1199042

this case assuming that the decision-maker has a subjectivesetting for pass value of 04 04 and 06 for the attributes u3u4 u5 in Table 3 then alternatives s1 and s3 in the decisionresult will be reversed But in traditional regret theory thereversion will not happen

5 Conclusion

In summary the main work of the paper is as follows (1) Foruncertain multiattribute decision-making a new extension-dependent degree decision-making method based on regrettheory is proposed (2) Aiming at the problem that theutility function in traditional regret theory is not strongenough to describe risk aversion behavior an extension-dependent degree function combining pass value is given(3) For quantitative describing uncertain attribute valueinformation the expression of interval extension-dependentdegree function is given The function can reflect the degreeof decision-makers preference towards the upper and lowerbounds of evaluation interval through preference attitudecoefficient setting (4) For attribute weight being taken asinterval number an objective weight assignment methodbased on themaximum deviation of interval deviation degreeis given (5) An example is given to verify the feasibilityand applicability of our method and the uncertain analysisfor the decision result is carried out The future researchwork is very rich The regret extension-dependent decisionmethod in the paper is considered to be applied to variousdecision scenarios such as mixed value type incompleteinformation dynamic stochastic fuzzy set and in furtherstudy its extension transformation [26] in the dynamicdecision model

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was supported by the National ScienceFoundation Project of China (no 71271191) Zhejiang Science

Foundation Project of China (nos Y16G010010 andLY18F020001) and Ningbo Innovative Team The IntelligentBig Data Engineering Application for Life and Health (Grantno 2016C11024)

References

[1] MGuo J-B Yang K-S Chin H-WWang andX-B Liu ldquoEvi-dential reasoning approach for multiattribute decision analysisunder both fuzzy and Interval uncertaintyrdquo IEEE Transactionson Fuzzy Systems vol 17 no 3 pp 683ndash697 2009

[2] L Yang X Li and D A Ralescu ldquoApplications of decisionmaking with uncertain informationrdquo International Journal ofIntelligent Systems vol 30 no 1 pp 1-2 2015

[3] Z Xu and H Liao ldquoA survey of approaches to decision makingwith intuitionistic fuzzy preference relationsrdquoKnowledge-BasedSystems vol 80 pp 131ndash142 2015

[4] P Wang P Meng J-Y Zhai and Z-Q Zhu ldquoA hybridmethodusing experiment design and grey relational analysis formultiple criteria decision making problemsrdquo Knowledge-BasedSystems vol 53 no 11 pp 100ndash107 2013

[5] H Garg and K Kumar ldquoDistance measures for connectionnumber sets based on set pair analysis and its applications todecision-making processrdquo Applied Intelligence vol 1 pp 1ndash142018

[6] Y Wu H Xu C Xu and K Chen ldquoUncertain multi-attributesdecisionmakingmethod based on interval numberwith proba-bility distributionweightedoperators and stochastic dominancedegreerdquo Knowledge-Based Systems vol 113 pp 199ndash209 2016

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

[8] W-Q Jiang ldquoInterval multi-criteria decision-making approachbased on prospect theory and statistic deductionrdquo Control andDecision vol 30 no 2 pp 375ndash379 2015

[9] D E Bell ldquoDisappointment in decision making under uncer-taintyrdquo Operations Research vol 33 no 5 pp 961ndash981 1982

[10] G Loomes andR Sugden ldquoRegret theory an alternative theoryof rational choice under uncertaintyrdquoTheEconomic Journal vol92 no 368 pp 805ndash824 1982

[11] Y Lin Y-M Wang and S-Q Chen ldquoHesitant fuzzy multiat-tribute matching decision making based on regret theory withuncertain weightsrdquo International Journal of Fuzzy Systems vol19 no 4 pp 955ndash966 2017

[12] X Zhang Z-P Fan and F-D Chen ldquoRisky Multiple AttributeDecision Making with Regret Aversionrdquo Journal of Systems ampManagement vol 23 no 1 pp 111ndash117 2014


[13] L-L Qian S-F Liu Z-G Fang and Y Liu ldquoMethod forgrey-stochastic multi-criteria decision-making based on regrettheoryrdquo Control and Decision vol 32 no 6 pp 1069ndash1074 2017

[14] S-DGuo S-F Liu andZ-G Fang ldquoMulti-objective grey targetdecision model based on regret theoryrdquo Control and Decisionvol 30 no 9 pp 1635ndash1640 2015

[15] ZW Cheng X FWang andY J Shao ldquoregret theory approachto multiple attribute decision making with three-parameterinterval numberrdquoControl Theory Applications vol 33 no 9 pp1214ndash1224 2016

[16] Y Song H Yao S Yao D Yu and Y Shen ldquoRisky MulticriteriaGroup Decision Making Based on Cloud Prospect Theory andRegret Feedbackrdquo Mathematical Problems in Engineering vol2017 Article ID 9646303 12 pages 2017

[17] H Zhou J-Q Wang and H-Y Zhang ldquoGrey stochastic multi-criteria decision-making based on regret theory and TOPSISrdquoInternational Journal of Machine Learning amp Cybernetics vol 8no 2 pp 1ndash14 2015

[18] C G Chorus ldquoRegret theory-based route choices and trafficequilibriardquo Transportmetrica vol 8 no 4 pp 291ndash305 2012

[19] C Y Yang W Cai and Extenics Extenics Theory Methodand Application Science Press Columbus The EducationalPublisher Beijing China 2013

[20] M Wang X Xu J Li J Jin and F Shen ldquoA novel model of setpair analysis coupled with extenics for evaluation of surround-ing rock stabilityrdquo Mathematical Problems in Engineering vol2015 Article ID 892549 9 pages 2015

[21] L B Xu X S Li and Y Guo ldquoGauss-core extension dependentprediction algorithm for collaborative filtering recommenda-tionrdquo Cluster Computing 2018

[22] Y An and Z H Zou ldquoStudy on modified water quality assess-ment based on matter element and correspondence analysisrdquoMathematics in practice and theory vol 44 no 13 pp 160ndash1662014

[23] X M Liu and K Q Zhao ldquoMultiple attribute decision makingof interval numbers based on analysis of the uncertainty ofconnection numbersrdquo Fuzzy Systems and Mathematics vol 24no 5 pp 141ndash148 2010

[24] T-H You and Z-P Fan ldquoTOPSIS method for multiple attributedecisionmaking with intervalsrdquo Journal of Northeastern Univer-sity vol 23 no 9 pp 840ndash843 2002

[25] W-H Liu and X-B Yao ldquoRelative membership degree methodfor multiple attribute decision-making with intervalsrdquo SystemsEngineering amp Electronics vol 26 no 7 pp 903ndash905 2004

[26] X Li Y Tian F Smarandache and R Alex ldquoAn extensioncollaborative innovation model in the context of big datardquoInternational Journal of Information Technology amp DecisionMaking vol 14 no 1 pp 69ndash91 2015

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


optimal grey perceived utility value of each scheme is rankedby the sorting rules of interval grey numbers In literature[14] the rejoice-regret value function based on the positiveand negative ideal solution is constructed and the positiveand negative ideal bullrsquos eyes are obtained So amultiobjectivegrey target decision model is completed Literature [15]uses regret theory to obtain interval comprehensive utilityvalue of alternatives in three-parameter interval numberdecision-making In literature [16] an improved feedbackadjustment mechanism based on regret theory is employed asthe consistency model which complements prospect theoryin multiattribute group decision problems Literature [17]combines regret theory toTOPSISmethod and applies to greystochastic multicriteria decision-making Literature [12 15]point out that among those decision strategies based ondecision-makersrsquo behavior prospect theory has the strongcapacity to describe decision-makersrsquo behavior but has moreparameters and more computational complexity By contrastregret theory has fewer parameters and is easy to implementbut has weaker describing capacity Literature [18] concludesthat regret theory has more advantages in practical applica-tion than prospect theory

Although various methods adopting regret theory havebeen widely applied to uncertain decision-making there aresome problems for the researchers Firstly most of existingmethods are built on the traditional basic model of regret the-ory So the ability of these methods for describing decision-makersrsquo behavior is insufficient That is when the risk evasioncoefficient is determined the curve of the utility function inregret theory is fixed It has nothing to do with the range anddistribution of the corresponding attribute values so it lacksflexibility However in many decision-making applicationsattribute values usually have a special value called pass pointbelow which the acceptance of decision-makers will declinerapidly It means that decision-makers often show distinctrisk chasing psychology The basic model of regret theory isnot sufficient to describe these behavior features and psychol-ogy features of the people Secondly many frameworks basedon the basic model of regret theory often appear too definitewhich means it is difficult to perform any stability test oruncertainty analysis for final decision results So we do notknow whether a tiny change of the psychology or preferenceof decision makes will bring different decision results Thatwill influence the credibility of the results So a good methodfor decision-making should not only produce a sorting resultbut also bring some stability test processes

To solve the above problems the paper attempts topropose an uncertain multiattribute decision-making modelbased on regret theory and extension-dependent degreeUntil the present the related research has not been seenIn this model the pass point value of the attribute can beset by decision-maker or other methods and the extension-dependent degree of the attribute value can be obtained byusing extension-dependent function based on side distanceActually it represents a new utility function expression andis more flexible than the basic utility function of regrettheory It is also more in line with peoples decision-makinghabits Then the regret-rejoice value of each attribute valueis obtained by combining its dependent degree and regret

theory According to regret-rejoice values the comprehensiveutility value of each alternative will be calculated Lastly inour model through setting the different value of preferenceattitude coefficient the final ranking results can be givenand analyzed further which reflects the different attitudeof decision-makers towards the upper and lower bounds ofevaluation and reflects the ability of uncertainty analysis

2 Regret Theory Model withExtension-Dependent Degree

21 Extension-Dependent Function with Side Distance Inextenics extension distance is defined to describe the dis-tance between a point and an interval which is able toaccurately describe the relative position between the pointand the interval For the definition of distance classicalmathematics utilizes qualitative descriptions of ldquobelongingrdquoand ldquonot belongingrdquo while extenics utilizes quantitativedescriptions In classical mathematics there are elements thatin the same domain are homogenous and but heterogeneousin different domains while in extenics elements in the samedomain they can be further classified into different layers andcan be given quantitative description [19 20]

Definition 1 (see [19 21]) Suppose 119877 the set of real number119883 = [119886 119887] is a finite interval in 119877 and its optimal point is119898 isin 119883 then forall119909 isin 119877 according to the different location of119898in119883 extension distance 120588(119909119898119883) is

(1)when 119898 = 119886 + 1198872 120588 (119909119898119883) = |119909 minus 119898| minus 119887 minus 1198862 (1)

(2)when 119898 isin [119886 119886 + 1198872 ) 120588 (119909 119898119883)

=


(2)

(3)when 119898 isin (119886 + 1198872 119887] 120588 (119909 119898119883)

=


(3)

In addition in this paper we agree that 120588(119886 119886 119883) =120588(119887 119887 119883) = 119886 minus 119887Definition 2 (see [19 22]) Suppose an interval coveringis composed of standard positive interval X0 and positiveinterval X 1198830 = [119886 119887] 119883 = [119888 119889] 1198830 sub 119883 and optimalpoint is 1198980 isin 1198830 for any point 119909 isin 119883 extension-dependentfunction 119896(119909) is


119896 (119909) =

120588(1199091198980 119883)120588 (119909 1198980 119883) minus 120588 (1199091198980 1198830) 120588 (1199091198980 1198830) = 120588 (119909 1198980 119883) 119909 notin 1198830


119886 minus 119887 120588 (1199091198980 1198830) = 120588 (119909 1198980 119883) 119909 notin 1198830

(4)







V (119909) = 1 minus exp (120573119909) 0 lt 120573 lt 1 (6)





= 005

= 010

= 015 = 020

0 x

v(x)

105

8

4

= 025


119896 (119909 1198830 119883) =

120588 (119909 119889 119883)120588 (119909 119889 119883) minus 120588 (119909 119889 1198830) =





119886 minus 119909119888 minus 119889 + 1 119909 isin 1198830

(7)











c0

a d

1

k(x)

x

a minus d

c minus d+ 1

a d


= 030 = 020 = 010

0 Δk

R(Δk)

63

1

= 00505

minus05

minus1

minus3





(9)


119906 (1199091198830 119883) = 119896 (1199091198830 119883) + 119877 (1199091198830 119883) (10)




119896 (1198761198830 119883) = 120572119896 (119902minus 1198830 119883) + (1 minus 120572) 119896 (119902+ 1198830 119883) 120572 isin [0 1] (11)









max 119863 (119882) =119898

sum119894=1

119899

sum119895=1

119898

sum119896=1


st 119882 isin Φ

st119899

sum119895=1

119908119895 = 1(12)







4 Example Analysis







u1 u2 u3 u4 u5 u6 u7 u8s1 [37 47] [59 69] [8 10] [30 40] [3 5] [90 100] [3 5] [6 8]s2 [15 25] [47 57] [4 6] [65 75] [3 5] [70 80] [7 9] [4 6]s3 [30 40] [42 52] [4 6] [60 70] [7 9] [80 90] [7 9] [5 7]s4 [35 45] [45 55] [7 9] [35 45] [8 10] [85 95] [6 8] [7 9]s5 [25 35] [50 60] [6 8] [50 60] [5 7] [85 95] [4 6] [8 10]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]


u1 u2 u3 u4 u5 u6 u7 u8s1 [0319 0405] [0609 0712] [0800 1000] [0750 1000] [0300 0500] [0900 1000] [0333 0556] [0600 0800]s2 [0600 1000] [0737 0894] [0400 0600] [0400 0462] [0300 0500] [0700 0800] [0778 1000] [0400 0600]s3 [0375 0500] [0808 1000] [0400 0600] [0429 0500] [0700 0900] [0800 0900] [0778 1000] [0500 0700]s4 [0333 0429] [0764 0933] [0700 0900] [0667 0857] [0800 1000] [0850 0950] [0667 0889] [0700 0900]s5 [0429 0600] [0700 0840] [0600 0800] [0500 0600] [0500 0700] [0850 0950] [0444 0667] [0800 1000]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]




u1 u2 u3 u4 u5 u6 u7 u8s1 08438 08645 13000 13750 08000 11000 06664 09786s2 13710 10473 08333 08620 08000 08824 12220 07143s3 09726 11400 08333 09290 13000 09956 12220 08571s4 08881 10845 12000 12620 14000 10500 11110 11000s5 10855 09961 11000 10500 11000 10500 08328 12000


u1 u2 u3 u4 u5 u6 u7 u8s1 06000 07466 12695 13368 05663 10849 04450 08773s2 13092 09891 06480 06511 05663 07987 11881 05222s3 07759 11108 06480 07422 12382 09482 11881 07151s4 06608 10380 11382 11880 13695 10195 10421 10382s5 09287 09215 10058 09055 09725 10195 06709 11695



max 119863 (119882)= 78431199081 + 45241199082 + 88001199083 + 94841199084+ 112001199085 + 36001199086 + 97801199087+ 80001199088


st8

sum119895=1

119908119895 = 1(13)



u1 u2 u3 u4 u5 u6 u7 u8s1 1370 1392 2200 2585 2200 0800 2667 1200s2 3010 0644 1800 1855 2200 1200 1778 2000s3 1143 1043 1800 1654 2200 0600 1778 1400s4 1256 0710 1600 1907 2800 0500 1556 1400s5 1297 0735 1400 1483 1800 0500 2001 2000






119882= (0042 0084 0136 0137 0182 0214 0046 0159) (14)










5 Conclusion


Data Availability




Acknowledgments



References



































Journal of












Journal of







Volume 2018






Volume 2018









119896 (119909) =

120588(1199091198980 119883)120588 (119909 1198980 119883) minus 120588 (1199091198980 1198830) 120588 (1199091198980 1198830) = 120588 (119909 1198980 119883) 119909 notin 1198830


119886 minus 119887 120588 (1199091198980 1198830) = 120588 (119909 1198980 119883) 119909 notin 1198830

(4)







V (119909) = 1 minus exp (120573119909) 0 lt 120573 lt 1 (6)





= 005

= 010

= 015 = 020

0 x

v(x)

105

8

4

= 025


119896 (119909 1198830 119883) =

120588 (119909 119889 119883)120588 (119909 119889 119883) minus 120588 (119909 119889 1198830) =





119886 minus 119909119888 minus 119889 + 1 119909 isin 1198830

(7)











c0

a d

1

k(x)

x

a minus d

c minus d+ 1

a d


= 030 = 020 = 010

0 Δk

R(Δk)

63

1

= 00505

minus05

minus1

minus3





(9)


119906 (1199091198830 119883) = 119896 (1199091198830 119883) + 119877 (1199091198830 119883) (10)




119896 (1198761198830 119883) = 120572119896 (119902minus 1198830 119883) + (1 minus 120572) 119896 (119902+ 1198830 119883) 120572 isin [0 1] (11)









max 119863 (119882) =119898

sum119894=1

119899

sum119895=1

119898

sum119896=1


st 119882 isin Φ

st119899

sum119895=1

119908119895 = 1(12)







4 Example Analysis







u1 u2 u3 u4 u5 u6 u7 u8s1 [37 47] [59 69] [8 10] [30 40] [3 5] [90 100] [3 5] [6 8]s2 [15 25] [47 57] [4 6] [65 75] [3 5] [70 80] [7 9] [4 6]s3 [30 40] [42 52] [4 6] [60 70] [7 9] [80 90] [7 9] [5 7]s4 [35 45] [45 55] [7 9] [35 45] [8 10] [85 95] [6 8] [7 9]s5 [25 35] [50 60] [6 8] [50 60] [5 7] [85 95] [4 6] [8 10]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]


u1 u2 u3 u4 u5 u6 u7 u8s1 [0319 0405] [0609 0712] [0800 1000] [0750 1000] [0300 0500] [0900 1000] [0333 0556] [0600 0800]s2 [0600 1000] [0737 0894] [0400 0600] [0400 0462] [0300 0500] [0700 0800] [0778 1000] [0400 0600]s3 [0375 0500] [0808 1000] [0400 0600] [0429 0500] [0700 0900] [0800 0900] [0778 1000] [0500 0700]s4 [0333 0429] [0764 0933] [0700 0900] [0667 0857] [0800 1000] [0850 0950] [0667 0889] [0700 0900]s5 [0429 0600] [0700 0840] [0600 0800] [0500 0600] [0500 0700] [0850 0950] [0444 0667] [0800 1000]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]




u1 u2 u3 u4 u5 u6 u7 u8s1 08438 08645 13000 13750 08000 11000 06664 09786s2 13710 10473 08333 08620 08000 08824 12220 07143s3 09726 11400 08333 09290 13000 09956 12220 08571s4 08881 10845 12000 12620 14000 10500 11110 11000s5 10855 09961 11000 10500 11000 10500 08328 12000


u1 u2 u3 u4 u5 u6 u7 u8s1 06000 07466 12695 13368 05663 10849 04450 08773s2 13092 09891 06480 06511 05663 07987 11881 05222s3 07759 11108 06480 07422 12382 09482 11881 07151s4 06608 10380 11382 11880 13695 10195 10421 10382s5 09287 09215 10058 09055 09725 10195 06709 11695



max 119863 (119882)= 78431199081 + 45241199082 + 88001199083 + 94841199084+ 112001199085 + 36001199086 + 97801199087+ 80001199088


st8

sum119895=1

119908119895 = 1(13)



u1 u2 u3 u4 u5 u6 u7 u8s1 1370 1392 2200 2585 2200 0800 2667 1200s2 3010 0644 1800 1855 2200 1200 1778 2000s3 1143 1043 1800 1654 2200 0600 1778 1400s4 1256 0710 1600 1907 2800 0500 1556 1400s5 1297 0735 1400 1483 1800 0500 2001 2000






119882= (0042 0084 0136 0137 0182 0214 0046 0159) (14)










5 Conclusion


Data Availability




Acknowledgments



References



































Journal of












Journal of







Volume 2018






Volume 2018









= 005

= 010

= 015 = 020

0 x

v(x)

105

8

4

= 025


119896 (119909 1198830 119883) =

120588 (119909 119889 119883)120588 (119909 119889 119883) minus 120588 (119909 119889 1198830) =





119886 minus 119909119888 minus 119889 + 1 119909 isin 1198830

(7)











c0

a d

1

k(x)

x

a minus d

c minus d+ 1

a d


= 030 = 020 = 010

0 Δk

R(Δk)

63

1

= 00505

minus05

minus1

minus3





(9)


119906 (1199091198830 119883) = 119896 (1199091198830 119883) + 119877 (1199091198830 119883) (10)




119896 (1198761198830 119883) = 120572119896 (119902minus 1198830 119883) + (1 minus 120572) 119896 (119902+ 1198830 119883) 120572 isin [0 1] (11)









max 119863 (119882) =119898

sum119894=1

119899

sum119895=1

119898

sum119896=1


st 119882 isin Φ

st119899

sum119895=1

119908119895 = 1(12)







4 Example Analysis







u1 u2 u3 u4 u5 u6 u7 u8s1 [37 47] [59 69] [8 10] [30 40] [3 5] [90 100] [3 5] [6 8]s2 [15 25] [47 57] [4 6] [65 75] [3 5] [70 80] [7 9] [4 6]s3 [30 40] [42 52] [4 6] [60 70] [7 9] [80 90] [7 9] [5 7]s4 [35 45] [45 55] [7 9] [35 45] [8 10] [85 95] [6 8] [7 9]s5 [25 35] [50 60] [6 8] [50 60] [5 7] [85 95] [4 6] [8 10]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]


u1 u2 u3 u4 u5 u6 u7 u8s1 [0319 0405] [0609 0712] [0800 1000] [0750 1000] [0300 0500] [0900 1000] [0333 0556] [0600 0800]s2 [0600 1000] [0737 0894] [0400 0600] [0400 0462] [0300 0500] [0700 0800] [0778 1000] [0400 0600]s3 [0375 0500] [0808 1000] [0400 0600] [0429 0500] [0700 0900] [0800 0900] [0778 1000] [0500 0700]s4 [0333 0429] [0764 0933] [0700 0900] [0667 0857] [0800 1000] [0850 0950] [0667 0889] [0700 0900]s5 [0429 0600] [0700 0840] [0600 0800] [0500 0600] [0500 0700] [0850 0950] [0444 0667] [0800 1000]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]




u1 u2 u3 u4 u5 u6 u7 u8s1 08438 08645 13000 13750 08000 11000 06664 09786s2 13710 10473 08333 08620 08000 08824 12220 07143s3 09726 11400 08333 09290 13000 09956 12220 08571s4 08881 10845 12000 12620 14000 10500 11110 11000s5 10855 09961 11000 10500 11000 10500 08328 12000


u1 u2 u3 u4 u5 u6 u7 u8s1 06000 07466 12695 13368 05663 10849 04450 08773s2 13092 09891 06480 06511 05663 07987 11881 05222s3 07759 11108 06480 07422 12382 09482 11881 07151s4 06608 10380 11382 11880 13695 10195 10421 10382s5 09287 09215 10058 09055 09725 10195 06709 11695



max 119863 (119882)= 78431199081 + 45241199082 + 88001199083 + 94841199084+ 112001199085 + 36001199086 + 97801199087+ 80001199088


st8

sum119895=1

119908119895 = 1(13)



u1 u2 u3 u4 u5 u6 u7 u8s1 1370 1392 2200 2585 2200 0800 2667 1200s2 3010 0644 1800 1855 2200 1200 1778 2000s3 1143 1043 1800 1654 2200 0600 1778 1400s4 1256 0710 1600 1907 2800 0500 1556 1400s5 1297 0735 1400 1483 1800 0500 2001 2000






119882= (0042 0084 0136 0137 0182 0214 0046 0159) (14)










5 Conclusion


Data Availability




Acknowledgments



References



































Journal of












Journal of







Volume 2018






Volume 2018









c0

a d

1

k(x)

x

a minus d

c minus d+ 1

a d


= 030 = 020 = 010

0 Δk

R(Δk)

63

1

= 00505

minus05

minus1

minus3





(9)


119906 (1199091198830 119883) = 119896 (1199091198830 119883) + 119877 (1199091198830 119883) (10)




119896 (1198761198830 119883) = 120572119896 (119902minus 1198830 119883) + (1 minus 120572) 119896 (119902+ 1198830 119883) 120572 isin [0 1] (11)









max 119863 (119882) =119898

sum119894=1

119899

sum119895=1

119898

sum119896=1


st 119882 isin Φ

st119899

sum119895=1

119908119895 = 1(12)







4 Example Analysis







u1 u2 u3 u4 u5 u6 u7 u8s1 [37 47] [59 69] [8 10] [30 40] [3 5] [90 100] [3 5] [6 8]s2 [15 25] [47 57] [4 6] [65 75] [3 5] [70 80] [7 9] [4 6]s3 [30 40] [42 52] [4 6] [60 70] [7 9] [80 90] [7 9] [5 7]s4 [35 45] [45 55] [7 9] [35 45] [8 10] [85 95] [6 8] [7 9]s5 [25 35] [50 60] [6 8] [50 60] [5 7] [85 95] [4 6] [8 10]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]


u1 u2 u3 u4 u5 u6 u7 u8s1 [0319 0405] [0609 0712] [0800 1000] [0750 1000] [0300 0500] [0900 1000] [0333 0556] [0600 0800]s2 [0600 1000] [0737 0894] [0400 0600] [0400 0462] [0300 0500] [0700 0800] [0778 1000] [0400 0600]s3 [0375 0500] [0808 1000] [0400 0600] [0429 0500] [0700 0900] [0800 0900] [0778 1000] [0500 0700]s4 [0333 0429] [0764 0933] [0700 0900] [0667 0857] [0800 1000] [0850 0950] [0667 0889] [0700 0900]s5 [0429 0600] [0700 0840] [0600 0800] [0500 0600] [0500 0700] [0850 0950] [0444 0667] [0800 1000]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]




u1 u2 u3 u4 u5 u6 u7 u8s1 08438 08645 13000 13750 08000 11000 06664 09786s2 13710 10473 08333 08620 08000 08824 12220 07143s3 09726 11400 08333 09290 13000 09956 12220 08571s4 08881 10845 12000 12620 14000 10500 11110 11000s5 10855 09961 11000 10500 11000 10500 08328 12000


u1 u2 u3 u4 u5 u6 u7 u8s1 06000 07466 12695 13368 05663 10849 04450 08773s2 13092 09891 06480 06511 05663 07987 11881 05222s3 07759 11108 06480 07422 12382 09482 11881 07151s4 06608 10380 11382 11880 13695 10195 10421 10382s5 09287 09215 10058 09055 09725 10195 06709 11695



max 119863 (119882)= 78431199081 + 45241199082 + 88001199083 + 94841199084+ 112001199085 + 36001199086 + 97801199087+ 80001199088


st8

sum119895=1

119908119895 = 1(13)



u1 u2 u3 u4 u5 u6 u7 u8s1 1370 1392 2200 2585 2200 0800 2667 1200s2 3010 0644 1800 1855 2200 1200 1778 2000s3 1143 1043 1800 1654 2200 0600 1778 1400s4 1256 0710 1600 1907 2800 0500 1556 1400s5 1297 0735 1400 1483 1800 0500 2001 2000






119882= (0042 0084 0136 0137 0182 0214 0046 0159) (14)










5 Conclusion


Data Availability




Acknowledgments



References



































Journal of












Journal of







Volume 2018






Volume 2018










max 119863 (119882) =119898

sum119894=1

119899

sum119895=1

119898

sum119896=1


st 119882 isin Φ

st119899

sum119895=1

119908119895 = 1(12)







4 Example Analysis







u1 u2 u3 u4 u5 u6 u7 u8s1 [37 47] [59 69] [8 10] [30 40] [3 5] [90 100] [3 5] [6 8]s2 [15 25] [47 57] [4 6] [65 75] [3 5] [70 80] [7 9] [4 6]s3 [30 40] [42 52] [4 6] [60 70] [7 9] [80 90] [7 9] [5 7]s4 [35 45] [45 55] [7 9] [35 45] [8 10] [85 95] [6 8] [7 9]s5 [25 35] [50 60] [6 8] [50 60] [5 7] [85 95] [4 6] [8 10]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]


u1 u2 u3 u4 u5 u6 u7 u8s1 [0319 0405] [0609 0712] [0800 1000] [0750 1000] [0300 0500] [0900 1000] [0333 0556] [0600 0800]s2 [0600 1000] [0737 0894] [0400 0600] [0400 0462] [0300 0500] [0700 0800] [0778 1000] [0400 0600]s3 [0375 0500] [0808 1000] [0400 0600] [0429 0500] [0700 0900] [0800 0900] [0778 1000] [0500 0700]s4 [0333 0429] [0764 0933] [0700 0900] [0667 0857] [0800 1000] [0850 0950] [0667 0889] [0700 0900]s5 [0429 0600] [0700 0840] [0600 0800] [0500 0600] [0500 0700] [0850 0950] [0444 0667] [0800 1000]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]




u1 u2 u3 u4 u5 u6 u7 u8s1 08438 08645 13000 13750 08000 11000 06664 09786s2 13710 10473 08333 08620 08000 08824 12220 07143s3 09726 11400 08333 09290 13000 09956 12220 08571s4 08881 10845 12000 12620 14000 10500 11110 11000s5 10855 09961 11000 10500 11000 10500 08328 12000


u1 u2 u3 u4 u5 u6 u7 u8s1 06000 07466 12695 13368 05663 10849 04450 08773s2 13092 09891 06480 06511 05663 07987 11881 05222s3 07759 11108 06480 07422 12382 09482 11881 07151s4 06608 10380 11382 11880 13695 10195 10421 10382s5 09287 09215 10058 09055 09725 10195 06709 11695



max 119863 (119882)= 78431199081 + 45241199082 + 88001199083 + 94841199084+ 112001199085 + 36001199086 + 97801199087+ 80001199088


st8

sum119895=1

119908119895 = 1(13)



u1 u2 u3 u4 u5 u6 u7 u8s1 1370 1392 2200 2585 2200 0800 2667 1200s2 3010 0644 1800 1855 2200 1200 1778 2000s3 1143 1043 1800 1654 2200 0600 1778 1400s4 1256 0710 1600 1907 2800 0500 1556 1400s5 1297 0735 1400 1483 1800 0500 2001 2000






119882= (0042 0084 0136 0137 0182 0214 0046 0159) (14)










5 Conclusion


Data Availability




Acknowledgments



References



































Journal of












Journal of







Volume 2018






Volume 2018










u1 u2 u3 u4 u5 u6 u7 u8s1 [37 47] [59 69] [8 10] [30 40] [3 5] [90 100] [3 5] [6 8]s2 [15 25] [47 57] [4 6] [65 75] [3 5] [70 80] [7 9] [4 6]s3 [30 40] [42 52] [4 6] [60 70] [7 9] [80 90] [7 9] [5 7]s4 [35 45] [45 55] [7 9] [35 45] [8 10] [85 95] [6 8] [7 9]s5 [25 35] [50 60] [6 8] [50 60] [5 7] [85 95] [4 6] [8 10]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]


u1 u2 u3 u4 u5 u6 u7 u8s1 [0319 0405] [0609 0712] [0800 1000] [0750 1000] [0300 0500] [0900 1000] [0333 0556] [0600 0800]s2 [0600 1000] [0737 0894] [0400 0600] [0400 0462] [0300 0500] [0700 0800] [0778 1000] [0400 0600]s3 [0375 0500] [0808 1000] [0400 0600] [0429 0500] [0700 0900] [0800 0900] [0778 1000] [0500 0700]s4 [0333 0429] [0764 0933] [0700 0900] [0667 0857] [0800 1000] [0850 0950] [0667 0889] [0700 0900]s5 [0429 0600] [0700 0840] [0600 0800] [0500 0600] [0500 0700] [0850 0950] [0444 0667] [0800 1000]w [0042 0049] [0084 0098] [0121 0137] [0121 0137] [0168 0182] [0214 0229] [0040 0046] [0159 0171]




u1 u2 u3 u4 u5 u6 u7 u8s1 08438 08645 13000 13750 08000 11000 06664 09786s2 13710 10473 08333 08620 08000 08824 12220 07143s3 09726 11400 08333 09290 13000 09956 12220 08571s4 08881 10845 12000 12620 14000 10500 11110 11000s5 10855 09961 11000 10500 11000 10500 08328 12000


u1 u2 u3 u4 u5 u6 u7 u8s1 06000 07466 12695 13368 05663 10849 04450 08773s2 13092 09891 06480 06511 05663 07987 11881 05222s3 07759 11108 06480 07422 12382 09482 11881 07151s4 06608 10380 11382 11880 13695 10195 10421 10382s5 09287 09215 10058 09055 09725 10195 06709 11695



max 119863 (119882)= 78431199081 + 45241199082 + 88001199083 + 94841199084+ 112001199085 + 36001199086 + 97801199087+ 80001199088


st8

sum119895=1

119908119895 = 1(13)



u1 u2 u3 u4 u5 u6 u7 u8s1 1370 1392 2200 2585 2200 0800 2667 1200s2 3010 0644 1800 1855 2200 1200 1778 2000s3 1143 1043 1800 1654 2200 0600 1778 1400s4 1256 0710 1600 1907 2800 0500 1556 1400s5 1297 0735 1400 1483 1800 0500 2001 2000






119882= (0042 0084 0136 0137 0182 0214 0046 0159) (14)










5 Conclusion


Data Availability




Acknowledgments



References



































Journal of












Journal of







Volume 2018






Volume 2018










u1 u2 u3 u4 u5 u6 u7 u8s1 1370 1392 2200 2585 2200 0800 2667 1200s2 3010 0644 1800 1855 2200 1200 1778 2000s3 1143 1043 1800 1654 2200 0600 1778 1400s4 1256 0710 1600 1907 2800 0500 1556 1400s5 1297 0735 1400 1483 1800 0500 2001 2000






119882= (0042 0084 0136 0137 0182 0214 0046 0159) (14)










5 Conclusion


Data Availability




Acknowledgments



References



































Journal of












Journal of







Volume 2018






Volume 2018












5 Conclusion


Data Availability




Acknowledgments



References



































Journal of












Journal of







Volume 2018






Volume 2018






























Journal of












Journal of







Volume 2018






Volume 2018








uncertain multiattribute decision-making based on interval...

Documents