1.School of Management,Guizhou University,Guiyang 550025,China;2.School of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;3.School of Business,Slippery Rock University,Slippery Rock PA 16057,USA;4.Centre of Computational Intelligence,De Montfort University,Leicester LE19BH,UK

Fundamental definition and calculation rules of grey mathematics:a review work

Qiaoxing Li1,2,*,Sifeng Liu2,4,and Jeffrey Yi-Lin Forrest2,3

1.School of Management,Guizhou University,Guiyang 550025,China;
2.School of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China;
3.School of Business,Slippery Rock University,Slippery Rock PA 16057,USA;
4.Centre of Computational Intelligence,De Montfort University,Leicester LE19BH,UK

Grey mathematics is the mathematical foundation of the grey system theory.Recently,some important results have been achieved.In order to accelerate the development of grey mathematics,the results are summarized and redefined This paper includes the fundamental definition and calculation rules of the grey hazy set,grey number,grey matrix and grey function.Grey mathematics includes four types of operation,i.e.the grey operation,the whitened operation,the covered operation and the only potential true operation.According to its intrinsic quality,the covered operation,which differs from the interval one,is called as the whole-proximate calculation that means the proximate calculation spreads through the whole range of the covered set of every grey number,and we confi m that it may be a new branch of computational or applied mathematics.The overview should develop the grey system theory and grey mathematics.

grey mathematics,grey hazy set,grey number,covered operation,grey system theory.

1.Introduction

Duringthe process of humanactivities,we often encounter cognitiveuncertainty,i.e.the accuratevalueof theresearch object can not be obtained because of poor(missing or inadequate)information.Then we turn to get its boundary by the obtained information from the system that the research object lies in.The inadequate information plays an important role.We usually divide the system into white, black and grey boxes,respectively,according to the information.That is to say,the system with completely known information is a white box,that with absolutely unknown information is a black one,and that with partly known and partly unknown information is a grey one.In 1982,Julong Deng,a Chinese scholar,originated the grey theory to resolve the phenomenon[1,2].

The grey system theory aims at studying the cognitive problem of uncertainty with poor information[3,4].On the basis of the obtained information,the cognitive range of the research object is known but its connotation is unknown.For example,the sum of people in the world will be between 85 and 100 hundred millions in 2020,which means that the true number,i.e.the connotation,must be within the interval[85,100],but it is unknown.Since the grey theory was proposed,it has been widely employed in many field of scientifi study,producing a huge amount of theoretic and applied consequences[1–25].

Grey mathematics is the mathematical foundation of the grey theory,which encounters a tortuous development course.Before the 21st century,grey mathematics had been viewed as the extension of fuzzy mathematics[5,6], and others viewed the grey number as an interval[7–9]. In 2002,Prof.Deng pointed out that these opinions do not correctly reflec the true meaning of the grey theory,and re-define greyhazy set and the grey number.He proposed covered operation and explained the difference between the intervaland the number-coveredset of the greynumber [2].Based on these results,the authorgave rigorousdefini tions of the grey hazy set and the grey number[4,10],and also proposedthe definition andcalculationsof greyfunctionsandgreymatrices[10–17].However,theresultshave not been a systemic work and many standpoints should be revised.In order to develop the grey mathematics,we uniform its definitio system and operational rules in this paper.

This paperis organizedas follows.In Section 2,we give the definitio system of the grey hazy set.In Section 3, weproposethefundamentaldefinition andtheoperationalrules of the grey number and also point out the difference between the covered set of a grey number and the interval.In Section 4,we propose the definitio system of grey matrices and stress the operationalrules.In Section 5, some definition of grey functions are obtained.Then,we uniform the definition of grey mathematics and propose whole-proximate calculation in Section 6.Finally,a conclusion is given.

2.Grey hazy set

The grey hazy set is the set-theory foundation of the grey theory[2].Its research object is the cognitive problem of connotationof objective things,which lies in a certain system,and the cognitive process of connotation comes from the information about objective things[3,4].Because the system is complex and the cognitive ability of human beingis limited,we couldnotobtaintheaccurateconnotation by using poor(missing or partially known)information. On the basis of the connotation-cognitionproblem and obtained information,we defin the grey hazy set[4]as follows.

Definitio1The connotation-cognition problem means that people want to obtain the specifi connotation of objective things during productive and management activities,and we denote it asP(c).

Proposition 1The connotation-cognition problem should include two requisites:the cognitive object and the cognitivesituation.Ifthe cognitivesituationis obvious,we usually omit it in the connotation-cognitionproblem.

Proposition 2The cognitive object is the special connotationof a certain objective thing,and the cognitivesituation is the external environment that the objective thing lies in.

Apparently,the cognitive object includes the objective things and its connotation.

Example 1The phrase“the sum of Chinese people in 2010”is a connotation-cognitionproblem,where“the sum of people”is the cognitive objective,and“China in 2010”is the cognitivesituation,and“the sum”is connotationand“people”is the objective thing.

Definitio2The information utilized to obtain the connotation of the objective thing is the cognitive information,and it includes data,knowledge,signal,and news.

Proposition 3The cognitive information comes from the objective thing and the cognitive situation,and it represents the description and reflectio of a certain state and the modeof theobjectivething,whichcan beobservedand felt by observers.

In order to get the cognitive objective under the cognitive situation,that is to say,to realize the connotationcognition problem,we should collect cognitive information.

Definitio3All cognitive information aboutP(c)is called the cognition-information set ofP(c),and it is denoted asI(c).

For the collected information,some can represent the connotation,and others may not.

Definitio4Suppose thatI(c)is the cognitioninformation set ofP(c)and satisfie the following conditions:

(i)Propositional.InformationinI(c)should include the cognitive objective and the certain characteristic(or the combination of some characteristics)of the objective;

(ii)Complete.Information inI(c)can be converged to the connotation of the objective thing;

(iii)Objective.Information inI(c)describes correctly and reflect appropriatelythe state and the mode of the objective thing;

(iv)True.Information inI(c)exists in reality and can be felt by observers;

(v)Explicit or non-explicit.Information inI(c)represents clearly or implies the connotation of the objective things.

Then we callI(c)as the cognition-information fiel ofP(c).

Apparently,the cognition-informationfiel is partofthe cognition-informationset.

Proposition 4The connotation comes from the cognitive information.

Definitio5The representation form of the connotation that comes from the information of the cognitioninformation set is called the similar connotation.

Definitio6The representation form of the connotation that comes from the information of the cognitioninformation fiel is called as the approximate connotation.

Definitio7The approximate connotation is called authentic connotation if it represents completely the connotation of the objective thing.

Definitio8The set that is composed by the similar connotations ofP(c)is called the connotation-cognition set ofP(c).

Definitio9The set that is composed by the approximate connotations ofP(c)is called the connotationcognition fiel ofP(c).

Proposition 5The authentic connotation must be in the connotation-cognitionfield

Example 2The phrase“the stature of one man”is a cognitive problem.Please note that we can omit the cognitive situation if it is obvious or not ambiguous.The sentences such as“the person is an adult”,“his stature is normal”,and so on,are cognitive information,and they belong to the cognition-information set.Furthermore,itis a cognition-information fielif the set satisfie Defi nition 4.Suppose that the man’s stature is 178 cm and we get the interval[170 cm,182 cm]from the cognitioninformationfield then 178 cm is an authentic connotation, the connotation-cognition fiel is[170 cm,182 cm],and 178 cm∈[170 cm,182 cm]holds true.

Definitio10If we can not justify that the approximate connotation is an authentic one by the cognitioninformation field then the fiel is the poor information.

Proposition 6The cognitive information owns the 4INproperties below:

(i)Intention,i.e.the purpose to collect information is to get the cognitive objective;

(ii)Interim,i.e.the obtained information is temporary and it can be processed by using other approaches and methods;

(iii)Intangible,i.e.the informationmay wronglyor correctly represent the connotation of the cognitive problem;

(iv)Inkling,i.e.it is difficul to describe completely the cognitive objective by using the information.

Proposition 7The cognition-information set,the cognition-information fieland the poor information satisfy the 4INproperties.

Definitio11The connotation-cognition set that comes from the cognition-information set is called a hazy set.

Definitio12The connotation-cognition fielthat comes from the poor information is called the kernelled hazy set.

Definitio13The kernelled hazy set includes four types below:

(i)Embryo set,i.e.the obtained information does not represent the cognitive objective;

(ii)Growing set,i.e.although the obtained information representsthecognitiveobjective,the boundaryof theconnotation has not been obtained yet;

(iii)Mature set,i.e.the obtained information can justify the boundary of connotation;

(iv)Evidence set,i.e.the obtained information can get the authentic connotation.

Theevolutionrouteofthekernelledhazyset canbeseen in Fig.1.

Fig.1 The evolution route of the grey hazy set

Proposition 8As the cognitive information is substantially supplemented,the four types of the kernelled hazy set evolve with the following order:Embryo set→Growing set→Mature set→Evidence set.

Definitio14The four types of the kernelled hazy set are called the grey hazy set,that is to say,Grey hazy set={Embryo set,Growing set,Mature set,Evidence set}.

In fact,the embryo set and the growing set are black boxes,the evidence set is a white box,and the mature set is a grey box.It is apparent that the grey hazy set is dynamic and the evolution order is fi ed.On the other hand, the embryo set,the growing set and the evidence set are special types of the grey hazy set,and we usually view the mature set as a grey hazy set.

Example 3We guess the stature of someone and suppose that it is 178 cm.Because the stature of human being is between 0 cm to 300 cm,we have that[0,300]is the embryo set.If the person is male and he is an adult,we also get that his stature is between 0 cm and 300 cm,and [0,300]is a growingset.Please note that the latter is different from the former because we know that the stature may be close to the larger number.On the other hand,we can not narrow the interval because many men are undersized. When knowing other information,for example,his stature is normal,we get that[170 cm,180 cm]is a mature set; finall,we measure him and get his true stature,then{178 cm}is an evidence set.

3.Grey number

The grey number,which was initially proposed by Deng [2],is the basic element of grey mathematics.Based on his work,we modify the definitio of grey number and get some important operational rules[10,11].In this section, we summarize them by using the definitio system of the grey hazy set above.

3.1Definitio of grey number

Definitio 15Suppose that⊙is an operator andNis the set of naturalnumber.Letxi(i∈N)be a number(such as real number,imaginary number,fuzzy number,and so on) andδbe a real number.Ifxi⊙xi=δholds for alli∈N, then⊙is the common operation.

Proposition 9The operators?and÷are the common operations.

Definitio 16A number is called an absolute conception number(ACN),if it only exists in our mind and we need not pay attention to its specifi value when doing operations,such as∞andε.It is a relative conception number(RCN),if it belongs to the conception in mind.The value of RCN really exists but we do not know it,and we may or may not pay attention to its value when doing operations.

Definitio 17The ACN and the RCN are integrated byconception numbers.

Definitio 18The RCN is a potential number if it satisfie the common operation,and we denote it asd°.It means thatd°?d°=0 andd°÷d°=1(d°/=0)hold even if we do not know the value ofd°.

Proposition 10The conception number and the potential number are uncertain.

Proposition 11Suppose that the value of the cognitive object exists,then it is sole.

Definitio 19Suppose that the connotation of the cognitiveobjectcanbequantified thenwecall thequantitative value as the only true number of the cognitive object.

Definitio20Suppose that the connotation of the cognitive object can be quantifie and the cognitioninformation fiel is poor,then the quantitative value is the only potential true number of the connotative object.

Definitio 21For a connotation-cognition problem,if we have no reason to deny that the quantitative valueaof the cognitive object is true,then we callaas the acquiescing number.

On the basis of the definition above,we defin the grey number below.

Definitio22LetP(c)be a connotation-cognition problem that can be quantified andI(c)be the cognitioninformationfiel ofP(c).Ris the fiel of real numberandDis a subset ofR.Suppose that?is an uncertain number aboutP(c),d°is the only potential true number of the uncertain number?and the acquiescing number??,and?satisfie the following condition,

(i)?isthegreynumberabouttheconnotation-cognition problemP(c);

(ii)Dis the number-coveredset of?;

(iv)I(c)is the information background of?;

(v)?,D,andd°are unifie by the grey-number element ofP(c).

Definitio23Suppose thatDis the number-covered set of?,x+andx?are the maximum and the minimum numbers ofD,respectively,and Δ(D)=x+?x?,then Δ(D)is the chaos of?.

Remark 1The grey number is greatly different from the variable of the function,and the number-covered set is also different from the domain of the variable(see Section 5).

Example 4There are two expression forms below:

(i)D=[1,2]is a grey number;

(ii)D=[1,2]is the number-coveredset ofgreynumber?.

The expression(ii)is correct but the other is wrong.Because the expression(i)does not imply any connotationcognition problem,we will be confused without it.For example,there are two truncheons whose lengths are similar but not the same.We can estimate that the length of both is between 1.1 m and 1.12 m.That is to say,the length of the truncheons is uncertain and can be denoted by the respective grey numbers?1and?2.Their number-covered sets are the same.However,we can not distinguish?1from?2if they are only denoted as grey number[1.1,1.12].

Definitio 24Suppose thatD1and?are the numbercovered sets of?,which are based on the different(or same)cognition-information fields andD1?D2holds, thenD1is the relatively superior number-coveredset of?and?is the relatively inferior one.

Definitio 25LetDbe the number-coveredset of grey number?.

(i)IfDis discrete,then it is called the discrete numbercovered set(DNCS).

(ii)IfDis continuous,then it is called the continuous number-coveredset(CNCS).

(iii)IfDis the unity of the continuous sets and the discrete ones,then it is called the mixed number-covered set (MNCS).

Definitio 25meansthat therearethreetypesofcovered set for a grey number.

3.2Operation of grey-number element

Definitio 26Suppose that?iand?jare grey numbers, and thatDiandDjas well asare whitened numbers,number-covered sets and the only potential true numbersof?iand?j,respectively.Let?ijbe the result between?iand?jby the operator?,andDijis the number-covered set of?ij.If all of the following conditions

Definitio27The operation between grey numbers is called grey-numberoperation,that between the numbercoveredsets is number-coveredoperation,that betweenthewhitened numbers is number-whitened operation,and that between the only potential true numbers is the numberpotential operation.

Theorem 1For any grey number?,we have???= 0;and if 0 is not the only potential true number(or the true number)of?,then?÷?=1.

ProofOmit.

Suppose that?iand?jare two grey numbers,and their number-covered sets areDiandDj,respectively,?ij=?i??jandDij?Di?Dj(or?ji=?j??iandDji?Dj?Di),where?∈{+,?,×,÷},then one of the specifi forms of number-covered operation is shown below.

Definitio28IfDi={dik|k=1,2,...,n}andDj={djl|l=1,2,...,m}are the discrete numbercovered sets,thenDijis the discrete number-covered set of?ijbelow:

(i)Dij=Di+Dj={dik+djl|k=1,2,...,n;l= 1,2,...,m};

(ii)Dij=Di?Dj={dik?djl|k=1,2,...,n;l= 1,2,...,m};

(iii)Dij=Di×Dj={dik×djl|k=1,2,...,n;l= 1,2,...,m};

(iv)Dij=Di÷Dj={dik÷djl|k=1,2,...,n;l= 1,2,...,m},where 0/∈Dj.

For the other forms of number-covered operation,our readers can see[10–12].

Example 5Suppose that there are some water in a cup and someone does not know how much it is,then the volumeof the water is a greynumber?.On the basis of some information,he can determine that the volume is between 10 ml and 13 ml,so the number-covered set of?is[10, 13].If he pours the water,then the volume of the water in the cup is the grey number???and its number-covered set is[10,13]?[10,13]=[?3,3].However,we certainly know that there is no water in the cup,that is to say,we get the true number 0 from Theorem 1.Because the true number appears,the number-covered set[?3,3]evolves into evidence set{0}.

Theorem 2Suppose that?1and?2are greynumbers, andthatare theironlypotentialtruenumbers,respectively,then

ProofOmit.

Theorem 3Suppose thatDis the number-covered set of the grey number?,and the operator?∈{+,?,×,÷}, then for any real numbera,a?is a grey number andis its number-covered set,where 0/∈Dwhen the operator is“÷”.

ProofOmit.

Theorem 4Suppose that?k=?×?×···×?andk?=?+?+···+?,wherekis the sum of?,then for any grey number?,we have

and if 0 is not the only potential true number(or the true number)of?,then

ProofOmit.

Theorem 5Suppose that?is a grey number andd°is its only potential true number,and thatd°∈Dholds, whereDis a set of real number,thenDis the numbercovered set of?.

ProofOmit.

Theorem 6Suppose that?1,?2and?3are three grey numbers,and the operation?∈{+,×},then

(i)the commutative law of addition and multiplication, i.e.?1+?2=?2+?1and?1×?2=?2×?1;

(ii)the combination law of addition and multiplication, i.e.

and(?1×?2)×?3=?1×(?2×?3);

(iii)the distribution law,i.e.

hold.

ProofOmit.

Definitio29Let?1and?2be grey numbers.If?1=?2holds,then?1and?2are called identically grey numbers.Otherwise,they are non-identical ones.

Property 1For any grey number?,the set of all real numbersR=(?∞,+∞)is a number-covered set,and it is the embryo set.

Property 2If there are two number-covered setsD1andD2for a grey number?,then

(i)D∩=D1∩D2/=φ,whereφis the vacant set;

(ii)D∩=D1∩D2andD∪=D1∪D2arealso numbercovered sets of?.

Property3LetD1andD2be two sets of real numbers and?be a greynumber.IfD1∪D2is the number-covered set of?and the equationD1∩D2=φholds,then one and only one betweenD1andD2is the number-covered set of?.

Property 4For any grey number?,there must exist an interval[a,b]which is the number-coveredset of?and the chaosb?acan be as small as possible.

Property 5Suppose that?1and?2are non-identical grey numbers,then the chaos of?1+?2and?1??2are larger than the chaos of both?1and?2.

ProofOmit.

Property 6Suppose that?1and?2are non-identical grey numbers,then the chaos of?1×?2and?1÷?2may be larger or smaller than the chaos of?1(or?2).

Example 6LetD1=[0.1,0.5],D2=[0.7,0.9]andD3=[3,5]be the number-coveredsets of?1,?2and?3, respectively.Calculate the chaos of?1×?2,?1×?3,?1÷?2and?1÷?3.

Property 5 and Property 6 show that the chaos of the computational result among grey numbers can not be justifie when we carry out the number-coveredoperation.In order to make the chaos of grey results smaller,the operational order should be the grey-number operation,the number-covered operation,and the qualitative analysis to the number-coveredset of the grey result.

3.3Difference between number-covered set and interval number

The number-covered set of the grey number is greatly different from the interval number and we point out the difference below.

Definitio30If each number in the interval[a,b]is true,then[a,b]is an interval number.

Definitio 31Suppose thatD1=[a?,a+]andD2= [b?,b+]are two interval numbers,and?is an operator, thenD=D1?D2={a?b|?a∈D1and?b∈D2}is the interval operation.

Theorem 7Suppose that[a,b]is an interval number, then

(i)[a,b]?[a,b]=[a?b,b?a]is an interval number;

(ii)[a,b]÷[a,b]=[min{a÷b;b÷a},max{a÷b,b÷a] is an interval,where 0/∈[a,b].

ProofOmit.

Example 7There are two problems.

Problem 1The stature of the adults is between 140 cm and 300 cm.

Problem 2The stature of someone is between 140 cm and 300 cm.

Because the information about the stature from Problem 1 is not complete,it does not belong to the cognitioninformation fiel from Definitio 6.In fact,we certainly know that the stature of any adult may be a number in [140,300],then all numbers are true,and[140,300]is an interval number.On the other hand,the stature information from Problem 2 is complete because every estimated number about his/her stature can be convergent to the true numberby the cognition-informationfield then[140,300] is a number-coveredset and the true number is only one.

The difference between the number-covered set of the grey number and the interval number is shown below:

(i)Information.The grey number exists for the connotation-cognition problem and the number-covered set is based on the information background,i.e.the cognition-information field When information added,the chaos of the grey number becomes smaller.If all information appears,the grey number vanishes and becomes a certain number,and the number-covered set evolves into the evidence set.

Theintervalnumberhasnoinformationbackgroundand it is immovable.

(ii)The special form of the operation.The result that a grey number subtracts itself is 0,and that divides itself is 1 if 0 is not the only potential true number.That is to say,

hold true even if the grey number has two different number-covered setsD1andD2.However,the result that an interval number subtracts or divides itself is also an interval number.

(iii)The true number.It is only one within the numbercovered set of the grey number.However,for the interval number,all elements are true.

(iv)Form.The number-covered set of the grey number has many expressive forms.Before the true number appears,it can be expressed by different intervals.Furthermore,it also has the discrete form.

On the basis of the obtained information,the interval form(if the number-covered set is continuous)is dynamic and evolutionary.The form of the interval number is sole and does not change.

(v)The sealed property.The operation of the grey number does not seal for the second aspect and the interval number is adverse.

(vi)The operational form.The operational form of the continuous number-covered set is the same as the one of the interval number.

To sum up,the numbers of the grey system theory include the conception number,the potential number,the true number,the only potential true number,the grey number,the number-coveredset and the whitened number.

4.The grey matrix

The matrix is very important not only for mathematics but also for other subjects.The grey matrix theory also plays an important role for grey mathematics, and it may greatly promote the development of applied mathematics and benefi other fields such as optimization problem,complex network,and so on.In this section,we summarize thebasic definition and operation rules of the grey matrix [10,12].

4.1Definitio of grey matrix

Definitio32Suppose thatAis a matrix,and there at least exists a grey number among the elements ofA,then we call it as a grey matrix and denote it asA(?).

Definitio33SupposeA(?)=(?ij)m×nis a grey matrix,andDijis the number-covered set of?ij(i= 1,...,m;j=1,...,n),thenA(D)=(Dij)m×nis the matrix-coveredset ofA(?).

Definitio34SupposeA(?)=(?ij)m×nis a grey matrix,andis the only potential true number of?ij(i=1,2,...,m;j=1,2,...,n),thenA°=is the only potential true matrix ofA(?).

Definitio35SupposeA(?)= (?ij)m×nis a grey matrix,andis a whitened number of?ij(i= 1,2...,m;j=1,2,...,n),thenis a whitened matrix ofA(?).

Apparently,A°,hold.

Definitio36If there at least exists a grey number among the elements of a vectorX,then it is a grey vector and denoted asX(?).

Definitio 37LetX(?)=(?1,...,?m)Tbe a grey vector,where T denotes the transpose of a matrix or a vector.Suppose thatare the only potential true number,the whitened number and the numbercovered set of?j(j=1,2,...,m),respectively,thenandX(D)=(D1,D2,...,Dm)Tare the only potential true vector,the whitened vector and the vector-covered set ofX(?),respectively,andhold.

In fact,a grey vector is a special grey matrix.

Definitio 38The grey matrix,the matrix-coveredset, the whitened matrix and the only potential true matrix are unifie by grey-matrix elements.

Definitio 39LetA(D)be the matrix-covered set of grey matrixX(?),

(i)if there is no mixed number-covered set and at least a discrete one among the elements ofA(D)exists,thenA(D)is a discrete matrix-covered set ofX(?);

(ii)if all elements inA(D)are continuous numbercovered sets and real numbers,thenA(D)is the continuous matrix-coveredset ofX(?);

(iii)if there at least exists a mixed number-covered set among the elements ofA(D),thenA(D)is a mixed matrix-coveredset ofX(?).

Definitio40IfA(?)=(?ij)m×nis a grey matrix and ΔDijis the chaos of?ij(i=1,2,...,m;j= 1,2,...,n),then the chaos ofA(?)is ΔD= max1≤i≤m;1≤j≤n{ΔDij}.

Definitio41Ifholds and it meansare different matrix-covered sets ofthenA(D)1is the relatively superior matrix-covered set ofA(?)andA(D)2the relatively inferior one.

Theorem 8Suppose thatis the only potential true matrix of grey matrixA(?)=(?ij)m×nandA(D)=(Dij)m×nis a set of real matrices,andA°∈A(D)holds,thenA(D)is the matrix-covered set ofA(?).

ProofFrom Theorem 5 and Definitio 33,DijandDijis the number-covered set of?ij(i= 1,...,m;j=1,...,n)because ofA°∈A(D),and thenA(D)is the matrix-coveredset ofA(?).

Example 8Let?11,?12,?21and?22be four grey numbers,andD11=[0,2]orD11={0,2},D12={3},D21={?1}andD22={0}be their number-covered sets,respectively.If the number 1 is the only potential true numberof?11,then we get grey matrixA(?),the matrixcoveredsetA(D)orA(D′),the only potential true matrixA°and one of the whitened matricesas follows:

From Definitio 39,A(D)andA(D′)are continuous and discrete sets,respectively,andA(D′)is superior thanA(D).

4.2Operation of grey-matrix elements

Definitio 42Letas well asA(D)andB(D)be the whitened matrices,the only potential true matrices and the matrix-covered sets of grey matricesA(?)andB(?),respectively.C(?)is the result betweenA(?)andB(?)by the operator?,andC(D)is the matrix-covered set ofC(?).If the equationsA(?)?B(?)=C(?),A(D)?B(D)=C(D),ofA(?) andhold, whereofandC?=A??B?;ifC?occurs,thenC°=C?,and allC(??)andC°vanish at the same time},then?is the operationof grey-matrixelements betweenA(?)andB(?), and it is denoted asA(?)?B(?)?A(D)?B(D).

Definitio43The operations between grey matrices,matrix-covered sets,whitened and only potential true matrices are called grey-matrix operation,matrix-covered operation,matrix-whitened operation and matrix-potential operation,respectively.

The specifi forms of matrix-covered operation are as follows:

Definitio44SupposeA(D)= (Dij)m×nandare the matrix-covered sets of grey matricesA(?)=(?ij)m×nandB(?)=respectively,andA(D)?B(D)is the matrix-coveredset ofA(?)?B(?),where?∈{+,?},then we have

(i)A(D)+B(D)=(Dij+)m×n;

(ii)A(D)?B(D)=(Dij?)m×n

whereDij+(i=1,2,...,m;j= 1,2,...,n)are the operation between the number-covered sets.

Definitio45SupposeA(?)= (?ij)m×landB(?)=are grey matrices,andA(D)= (Dij)m×nandB(D)=are the respective matrix-covered sets ofA(?)andB(?),then the matrixcovered setA(D)×B(D)ofA(?)×B(?)is as below:

whereDik×(i=1,...,m;k= 1,...,l;j= 1,...,n)is the operation between number-coveredsets.

On the other hand,the only potential true matrices ofA(?)+B(?),A(?)?B(?)andA(?)×B(?)can be easily obtained.

Similar to the operation of grey-number elements,we shouldcarryout the grey-matrixoperationat first and then the matrix-covered operation.Finally,we analyze the result of the matrix-coveredset.

Theorem 9Suppose thatA(?)is a grey matrix and0is the zero matrix,thenA(?)?A(?)=0holds.

ProofOmit.

SupposeA(?)is a square grey matrix,andA(?)k=A(?)×A(?)×···×A(?)andkA(?)=A(?)+A(?)+···+A(?),wherekis the sum ofA(?),thenA(D)k=A(D)×A(D)×···×A(D)andkA(D)=A(D)+A(D)+···+A(D)are the matrix-coveredsets ofA(?)kandkA(?),respectively.

Theorem 10SupposeA(?)is a square grey matrix, thenA(?)k?A(?)k=0andkA(?)?kA(?)=0hold.

ProofOmit.

Theorem11SupposethatA(?)=(?ij)m×nisagrey matrix,and thatA(D)=(Dij)m×nandA°=are the matrix-covered set and the only potential true matrix ofA(?),respectively,and thatdis a real number,thendA(?)=(d?ij)m×nis a grey matrix,anddA(D)= (dDij)m×nanddA°=are the matrix-covered set and the only potential true matrix ofdA(?),respectively.

ProofOmit.

Theorem12SupposethatA(?)=(?ij)m×nis agrey matrix,and thatA(D)=(Dij)m×nandA°=are the matrix-covered set and the only potential true matrix ofA(?),respectively,and that?is a grey number,and thatDandd°are the number-covered set and the only potential true number of?,respectively,then?A(?)is a grey matrix,andd°A°=andDA(D)=(DDij)m×nare the only potential true matrix and the matrix-covered set of?A(?),respectively.

ProofOmit.

Theorem 13Suppose thatA(?)= (?ij)m×nandB(?)=are grey matrices,andA°=are the only potential true matrices ofA(?)andB(?),respectively,then

ProofFromA(?)=B(?),we have

and getA°=B°from Theorem 2.

Theorem 14Suppose thatA1(?),A2(?)andA3(?) are grey matrices,then we have

(i)the commutative law of addition,i.e.

(ii)the combination law of addition and multiplication, i.e.

(iii)the distribution law,i.e.

ProofOmit.

Example 9Let[A],[B]and[C]be matrix-covered sets ofA(?),B(?)andC(?),respectively,where

Calculate the matrix-covered sets of the following grey matrices,

We only give results below:

4.3Inverse of the grey matrix

The inverse of matrices plays an important role in the matrix theory,so does the inverse of the grey matrix in the grey matrix theory and we propose it below[10,13].

Thefollowinglemmasareavailableinmanymathematical textbooks.

Definitio 46For a given vectorX=(x1,...,xn)T,the equation

is called thep-norm ofX,where the real numberpsatisfie 1≤p＜+∞.

Definitio47For a given matrixA=(aij)n×nand any vectorX=(x1,...,xn)T,the equation

is called thep-norm ofA,where0is a zero matrix/vector.

Lemma 1Suppose thatAis a square matrix andholds,thenI?Ais nonsingular and its inverse(I?A)?1is

whereIis the identity matrix.

Lemma 2Suppose thatAis nonsingular andholds,whereAandBare square matrices,thenA?Bis nonsingular and its inverse(A?B)?1is

Lemma 3If the matricesthenholds.

Lemma 4For any matrix

The general formula to calculate the matrix-covered set of the inverse grey matrix is proposed below.

Suppose thatQ(?)=(qij(?))n×nis a grey matrix, and thatare the matrix-covered set,the whitened matrix and the only potential true matrix ofQ(?),respectively, whereare the number-covered set,the whitened number and the only potential true number of grey numberqij(?)(i,j=1,...,n),respectively,and that grey matrixQ(?)is nonsingular,i.e.the only potential true matrixQ°is nonsingular.Then the problem is how to obtain the matrix-covered set[Q]?1of the inverseQ(?)?1.

Theorem 15For the given[B]andabove,ifis nonsingular and satisfie

ProofOmit.

Theorem 16If(1)holds,then the matrix-covered setis

ProofLetthen from Theorem 15,we have

and(2)is the matrix-covered set ofQ(?)?1.

Theorem 17Suppose thatare as define above,and thatandand that(1)holds,then for allε＞0,there must exist an integer

where int(x)is the largest integer that is not greater thanx,and we get the matrix-covered set[Q]?1ofQ(?)?1below:

ProofSuppose that1,2,...),then we have

From(1)and Lemma 4,we have

holds for all(i,j=1,2,...,n).Obtaining the integerK0from(3),we have

Then(4)holds from(2).

And(5)is also true from(2).

Theorem 18Suppose that(1)holds and that

and thatB1≥0holds,then the matrix-covered set[Q]?1ofQ(?)?1is

ProofOmit.

Some examples to calculate the matrix-covered set of the inverse grey matrix can be seen in[13]and the computational rules of the inverse grey triangular matrix was proposed in[14].

4.4Differencebetweenthe matrix-coveredset andthe interval matrix

Justas thefactthattheintervalnumberis differentfromthe number-coveredset of the greynumber,the interval matrix is also greatly different from the matrix-covered set of the grey matrix.

Definitio 48If[A]=([aij,bij])m×n,where[aij,bij] is an interval numberthat lies in rowiand columnjof[A] (i=1,2,...,m;j=1,2,...,n),then[A]is an interval matrix.

(i)[A]is a real matrix if and only ifholds?i∈{1,...,m}and?j∈{1,...,n};

(ii)[A]=[B]holds if and only ifhold for alli∈{1,2,...,m}andj∈{1,2,...,n}.

Theorem 19If[A]=([aij,bij])m×nis an interval matrix,then the result of

is also an interval matrix.

ProofOmit.

Example 10Suppose that there are two sectors in a regional economy,and that the uncertain matrix[A]is shown below:

then justify[A]by the following problems:

Problem 3For anyone of some regions,every inputoutput value between sectors is within corresponding interval of[A].

Problem4Foragivenregion,everyinput-outputvalue between sectors is within correspondinginterval of[A].

The information about the values in Problem 3 is not complete,then[A]is an interval matrix.The information in Problem 4 is complete and[A]is a matrix-covered set. The difference between the matrix-covered set of the grey matrix and the interval matrix is shown below:

(i)Information.Thenumber-coveredset ofveryelement in a grey matrix exists for the corresponding cognitioninformation fieland the matrix-covered set of the grey matrix is dynamic and evolutive.As the information is added,thechaosofthegreymatrixbecomessmaller.When allinformationappears,thematrix-coveredsetevolvesinto the evidence set and the true matrix is obtained.The interval matrix has no information background.

(ii)The special operation form.For grey matrixA(?) andits matrix-coveredsetsA(D)1andA(D)2,thematrixcovered setA(D)1?A(D)2(orA(D)2?A(D)1)ofA(?)?A(?)must evolve into{O},i.e.A(D)1?A(D)2?{O}andA(?)?A(?)=Oholds.But the resultthat anintervalmatrixsubtractstoitself is aninterval one.

(iii)The true matrix.It is onlyone in the matrix-covered set,but all matrices within the interval matrix are true.

(iv)Form.The matrix-covered set of the grey matrix has many forms.On the basis of the cognition-information field of the grey elements within grey matrix,the matrixcovered set can be expressed by different interval matrices (if the number-coveredsets of all elements are continuous) before the true matrix appears.Furthermore,there exists the discrete matrix-covered set.The matrix-covered set is dynamic and it must evolve into the evidence set when all informationis obtained.But the formof the intervalmatrix is sole and fi ed.

(v)The sealed property.The matrix-covered operation does not seal for the second aspect and the interval matrix is adverse.

(vi)The operational form.For continuous matrixcovered set,the operational form is the same as the one of the interval matrix.

To sum up,the matrices of grey system theory include the true matrix,the only potential true matrix,the grey matrix,the whitened matrix,and the matrix-covered set.

5.The grey function

The grey function is very useful for constructing grey mathematics and we propose the general definitio [10,12,15].

Definitio49Suppose thatx1,x2,...,xnare variables,and thata1,a2,...,amare constant numbers,then the following equation

is called as the real function,wherea1,a2,...,amare parameters of the function,and the rangeofxi(i=1,2,...,n)is the domain of the function.

Definitio50If there at least exists a grey number amonga1,a2,...,am,that is to say,the following equation

is a grey function,wherea1(?),a2(?),...,am(?)are grey parameters.

From Definitio 50,the real function is grey because the constant parameters become grey.Please note that the variablesx1,x2,...,xnare not grey.

On the basis of the cognition-information field we get the number-covered set[ai]of grey parameterai(?)(i= 1,2,...,m),then the following equations

are the function-coveredset,the whitened function and the only potential true function ofy(?),respectively,whereare the whitened number and the only potential true numberofai(?)(i=1,2,...,m),respectively,andhold.

Definitio 51The grey function,the function-covered set,the whitened function and the only potential true function are unifie by grey-functionelements.

To completely understand the grey function,we should differentiate the following aspects:

(i)The parameters can be grey but the variables are not. Because we could not get the exact value of the parameteraiby the poor information,it is grey and we denote it asai(?)(i=1,2,...,m).Although we do not know the true value,it must exist and is called the only potential true numberFrom the cognition-information field we can getthenumber-coveredset[ai]andholds.Onthe other hand,the variables are not grey.xj(j=1,2,...,n) is a variable and can be changed.The value ofycomes fromxj(j=1,2,...,n).Because of the greyness ofai(?)(i=1,2,...,m),yis grey and we denote it asy(?).

Definitio 52Suppose that the equation

holds,then we call it grey equation.

Apparently,the grey equation is the special type of the grey function.

From Definitio 50,we get basic grey elementary functions below:

(i)Grey constant function:y(?)=c(?),wherec(?) is a grey constant.

On the basis of the cognition-information field we get the number-covered set[c]ofc(?),thenandy°=c°are the function-covered set,the whitened function and the only potential true function ofy(?),respectively,wherehold.

(ii)Grey power function:y(?)=xα(?),whereα(?) is a grey number.

If[α]is the number-covered set ofα(?),then[y]=are the function-covered set, the whitened function and the only potential true function ofy(?),respectively,where

(iii)Grey exponential function:y(?)=a(?)x,wherea(?)is grey and satisfiea(?)＞0 and

If[a]isthenumber-coveredsetofa(?),then[y]=[a]x,are the function-covered set, the whitened function and the only potential true function ofy(?),respectively,where[a]satisfie[a]?(0,1)or

(iv)Grey logarithmic function:y(?)= loga(?)x,wherea(?)is grey and satisfiea(?)＞0 anda(?)/=1.

If[a]is the number-covered set ofa(?),then[y]=are the functioncovered set,the whitened function and the only potential true function ofy(?),respectively,where[a]satisfie

(v)Grey trigonometricfunctions,which include the following specifi ones: Grey sine formula:

Grey cosine formula:

Grey tangent formula:

If[a]and[b]are the number-covered sets ofa(?) andb(?),respectively,thenare the functioncovered set,the whitened function and the only potential true function ofy(?)=sin(a(?)x+b(?)),respectively,andandy°=cos(a°x+b°)are the correspondingones ofy(?)= cos(a(?)x+b(?)),respectively.[y]=tan([a]x+[b]),are the corresponding ones ofy(?)=tan(a(?)x+b(?)),respectively,wherehold for all grey trigonometric functions.

(vi)Grey anti-trigonometric functions,which include the following specifi ones: Grey anti-sine formula:

Grey anti-cosine formula:

Grey anti-tangent formula:

If[a]and[b]are the number-covered sets ofa(?)andb(?),respectively,thenare thefunctioncovered set,the whitened function and the only potential true function ofy(?)=arcsin(a(?)x+b(?)),respectively,andandy°=arccos(a°x+b°)are the corresponding ones ofy(?)=arcsin(a(?)x+b(?)),respectively.[y]=arctan(a°x+b°)are the corresponding ones ofy(?)= arctan(a(?)x+b(?)),respectively,wherehold for all grey anti-trigonometric functions.

(vii)Grey polynomial function:

If[ai]is the number-covered set ofai(?)(i= 0,1,...,m),then[f(x)]=[am]xm+···+[a1]x+[a0],are the function-covered set,the whitened function and the only potential true function off(?)(x), respectively.

Basic grey elementary function plays an important role in grey mathematics.However,we only give their defini tions in this section.It is very important to get computable formulas of the function-covered set,and some of them have been discussed[16,17].Operational rules of the grey function are based on the rules of both real function and grey number.

Definitio 53The operations between grey functions, the function-covered sets,the whitened functions and the only potential true functions are called grey-function operation,function-covered operation,function-whitened operation and function-potential operation,respectively,and theyareunifie bytheoperationofgrey-functionelements.

Furthermore, grey derived function, grey linear programming, grey polynomial function, grey complex number, grey complex network, grey input-outputanalysis, grey determinantand grey digitalelevation modelwere also proposed and their computational rules of covered operation were obtained[15,18–25],and we omit them here.

Example 11Suppose thatf1(?)(x)=a3(?)x3+a2(?)x2+a1(?)x+a0(?)andf2(?)(x)=a2(?)x2?a1(?)x?2a0(?)are grey polynomial functions,and the number-covered sets ofa0(?),a1(?),a2(?)anda3(?) are[3,3.5],[?6,2],[0,2]and[?1,3],respectively,then calculatethefunction-coveredset off1(?)(x)?f2(?)(x).

Because off1(?)(x)?f2(?)(x) =a3(?)x3+a1(?)x+3a0(?),the function-coveredset off1(?)(x)?f2(?)(x)is

6.Grey elements

For the sake of simplicity,some important definition of grey mathematics can be unified and we propose them below.

Definitio 54Thebasicelementofgreymathematicsis unifie by the grey elements that include the grey-number element,thegrey-matrixelement,andthegrey-functionelement,and the grey part of the grey elements is called grey unit that includes grey number,grey matrix and grey function.And the set part of the grey elements is called the covered set of grey unit that includes the number-covered set,matrix-covered set and function-covered set,and anyonewithinthecoveredset isthewhitenedunitthatincludes whitened number,whitened matrix and whitened function, while the true value that is unknown is called the only potential true unit of grey unit,which also includes the only potential true number,matrix,and function.

Apparently,the grey elements include the grey unit, coveredset,whitenedunitandonlypotentialtrueunit.Furthermore,the whitened unit and the only potentialtrue unit must be within the covered set.

Definitio55The operation between grey units is called grey operation,the one between the covered sets is covered operation,the one between the whitened units is whitened operation,and the one between the only potential true units is potential operation.

Definitio 56The grey,covered,whitened and potential true operations are unifie by grey-element operation. When doing grey-element operation,we should do grey operation at first and then covered operation.Finally,we may analyze the operational result.That is to say,the greyelement operationis the process of both quantitativecalculation and qualitative analysis.

Definitio57Whole-proximate calculation means that the proximate calculation spreads through the whole range of every number.When doing covered operation, all numbers within the covered sets are participated into the calculation process.Then the covered operation is the whole-proximate calculation.The covered operation of grey mathematics is a new computation approach,and it may be an important part of applied mathematics.

7.Conclusions

Themathematicalfoundationofgreytheorynamedas grey mathematics has encountered a tortuous process of development.Based on the recent results,we systematize the definition and the operational rules.Although there are four types of operation forms,the covered operation is the coreofgreymathematicsinfact,andtheothersareonlythe subordinate operations.When we want to get the result of greyoperation,thecoveredformshouldbedetermined.We also point out that the covered operation is actually an approximate calculation through the numerical ranges which is different from the approximate one in classical mathematics.The work may play an important role in accelerating the development of grey mathematics.Grey system theory should be one of the uncertain theories,and it may possess a scientifi position that is the same as possibility theory and fuzzy system theory.

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Biographies

Qiaoxing Liwas born in 1973.In 2007,he received his Ph.D.degree from the College of Economics and Management,Nanjing University of Aeronautics and Astronautics,Nanjing, Jiangsu,China.He finishe one of his postdoctorate researches at School of Business Administration,South China Unversity of Technology,Guangzhou,China,and the other at College of Earth and Environ-mental Sciences,Lanzhou University,Lanzhou, Gansu,China.He was a teacher at School of Management,Lanzhou University between 2009 and 2014.He is now an associate professor in Guizhou University.His research interests are system science,management science and technology,management complexity.

E-mail:liqiaoxing@eyou.com

Sifeng Liuwas born in 1955.He received his Ph.D. degree in systems engineering from Huazhong University of Science and Technology,China,in 1998. He joined Nanjing University of Aeronautics and Astronautics in 2000 as a distinguished professor and led the College of Economics and Management from 2001 until 2012.He also serves as the founding director of Institute for Grey Systems Studies, the founding chair of IEEE SMC Technical Committee on Grey Systems, the president of Grey Systems Society of China,the founding Editor-in-Chief ofGrey Systems Theory and Application(Emerald)and the Editorin-Chief ofThe Journal of Grey Systems(Research Information).He is an SMIEEE,an Honorary Fellow of WorldOrganization of Systemsand Cybernetics,and the Fellow of Marie Curie International Incoming Program (FP7-PEOPLE-2013-IIF)of the 7th Research Framework of European Commission.At present,he is a research professor at Centre of Computational Intelligence,De Montfort University,UK.

E-mail:sifeng.liu@dmu.ac.uk

Jeffrey Yi-Lin Forrestwas born in 1959.He received his Ph.D.degree in 1988 in the area of mathematics from Auburn University,Alabama,and did one-year post-doctoral research in Carnegie Mellon University in the area of statistics.He is now a professor of mathematics,economics,finance and systems science.Currently,he is a Ph.D.program supervisor with the School of Economics and Management,Nanjing University of Aeronautics and Astronautics,and on the business faculty of Slippery Rock University of Pennsylvania.His research interests cover systems research,economics,and some other topics.

E-mail:Jeffrey.forrest@sru.edu

10.1109/JSEE.2015.00138

Manuscript received September 01,2014.

*Corresponding author.

This work was supported by the China Postdoctoral Science Foundation(200902321)and a Marie Curie International Incoming Fellowship within the 7th European Community Framework Program(FP7-PIIFGA-2013-629051).