nowadays , many different extensions of fss are known : l - fuzzy sets ( l - fss ) , interval - valued fuzzy sets ( ivfss ) , vague sets ( vss ) , intuitionistic fuzzy sets ( ifss ) , interval - valued intuitionistic fuzzy sets ( ivifss ) , linguistic fuzzy sets ( lfss ) , type-2 fuzzy sets ( t2fss ) , type - n fuzzy sets ( tnfss ) , and fuzzy multisets ( fmss ) . an interesting extension of fss is the hesitant fuzzy set ( hfs )   in which the membership degree of an element is defined as a set of possible values . hfss are quite suitable to deal with the situation where we have a set of possible values , rather than a margin of error ( as in ifss ) or some possibility distribution on the possible values ( as in t2fss ) . later , a number of other extensions of hfss have been developed such as dual hesitant fuzzy sets ( dhfss ) , generalized hesitant fuzzy sets ( g - hfss ) , and hesitant fuzzy linguistic term sets ( hfltss ) . however , hfs   and its recent generalization g - hfs   have their inherent drawbacks , because they express the membership degrees of an element to a given set only by crisp numbers or ifss . in many practical decision making problems , the information provided by decision makers might often be described by fss instead of crisp numbers or by other fs extensions instead of ifss . this makes decision makers uncomfortable to provide exact crisp values or just ifss for the membership degrees . such a problem can be avoided by using the higher order hfs ( hohfs ) introduced originally by the author in   for describing membership degrees . the hohfs is fit for the situation where the decision makers have a hesitation among several possible memberships for an element . the hohfs is the actual extension of hfs encompassing not only fss , ifss , vss , t2fss , fmss , and hfss , but also the recent extension of hfss , called g - hfss . since soft set theory was first initiated by molodtsov in   as a new mathematical tool to deal efficiently with uncertainty and imprecision , many authors have become interested in applying this theory in various fields such as decision making , data analysis , and forecasting   recently . moreover , a growing number of studies focus on the combination of fs and some extensions of fss with soft set . among them are , for instance , fuzzy soft set , intuitionistic fuzzy soft set , interval - valued fuzzy soft set , vague fuzzy soft set , and multifuzzy soft set . the rationale behind this study is that on the one hand we tend to propose a more general extension of fuzzy soft set encompassing the existing ones , and on the other hand we try to model those situations where the existing kinds of fuzzy soft set are not applicable . as mentioned above and it will be discussed later , the hohfs encompasses most of the fuzzy extensions as special cases , and therefore the combination of hohfs with soft set contains most of the existing kinds of fuzzy soft set as special cases , too . one of more interesting feature of the higher order hesitant fuzzy ( hohf ) soft set which will be proposed in this contribution is that we can correspond each parameter to levels with different dimensions , the situation that can not be modeled by the other kinds of fuzzy soft sets , specially by multifuzzy soft set . to make this clear , assume that x = { x1 , x2 , x3 , x4 , x5 } is the universe set indicating the set of five kinds of colored drawing paper for engineering . suppose the parameter set a = { e1 , e2 , e3 } is a set of three criteria to evaluate the performance of these papers , where e1 stands for thickness which includes three levels : thick , average , and thin , e2 stands for color which consists of three levels : red , green , and blue , and e3 stands for ingredient which is made from three levels : cellulose , hemicellulose , and lignin . but if this decision problem is reconsidered slightly different from the original problem as e1 stands for thickness which includes three levels : thick , average , and thin , e2 stands for color which consists of three levels : red , green , and blue , and e3 stands for ingredient which is made from two levels : cellulose and hemicellulose where the level lignin is not considered , then , obviously , this situation can not be handled not only by the existing kinds of fuzzy soft set theory , but also by multifuzzy soft set one in which all the parameters have to possess levels with the same dimension . to deal with such cases an extension of hfss , which is referred to as the higher order hfs ( hohfs ) , is introduced in section 2 . section 3 presents hohf soft sets and discusses the properties of hohf soft sets . in section 4 , we apply the proposed hohf soft sets to multicriteria decision making problems involving parameters whose levels have different dimensions . this section is devoted to review the basic definitions and notions of fuzzy set ( fs ) and its new generalization which are originally referred to by farhadinia   as the higher order hesitant fuzzy set ( hohfs ) . actually , the hohfs is a generalization of hesitant fuzzy set ( hfs ) introduced by torra . an ordinary fuzzy set ( fs ) a in x is defined   as a = { x , a(x):x  x } , where a : x    and the real value a(x ) represents the degree of membership of x in a. 
 definition 1 . a generalized type of fuzzy set ( g - type fs ) on x is defined as ( 1)a~={x , a~(x):xx } , where ( 2)a~:x ( ) . here ,  ( ) denotes a family of crisp or fuzzy sets that can be defined within the universal set . a generalized type of fuzzy set ( g - type fs ) on x is defined as ( 1)a~={x , a~(x):xx } , where ( 2)a~:x ( ) . here ,  ( ) denotes a family of crisp or fuzzy sets that can be defined within the universal set . it is noteworthy that most of the existing extensions of ordinary fs are special cases of g - type fs , for instance , ( i)if  ( ) = , then the g - type fs a~ reduces to an ordinary fs;(ii)if  ( ) =  ( ) denoting the set of all closed intervals , then the g - type fs a~ reduces to an ivfs;(iii)if  ( ) =  ( ) denoting the set of all ordinary fss , then the g - type fs a~ reduces to a t2fs;(iv)if  ( ) = l denoting a partially ordered lattice , then the g - type fs a~ reduces to a l - fs . if  ( ) = , then the g - type fs a~ reduces to an ordinary fs ; if  ( ) =  ( ) denoting the set of all closed intervals , then the g - type fs a~ reduces to an ivfs ; if  ( ) =  ( ) denoting the set of all ordinary fss , then the g - type fs a~ reduces to a t2fs ; if  ( ) = l denoting a partially ordered lattice , then the g - type fs a~ reduces to a l - fs . a hesitant fuzzy set ( hfs ) on x is symbolized by ( 3)h={x , h(x):xx } , where h(x ) , referred to as the hesitant fuzzy element ( hfe ) , is a set of some values in   denoting the possible membership degree of the element x  x to the set h. 
  let x be the universe of discourse . a hesitant fuzzy set ( hfs ) on x is symbolized by ( 3)h={x , h(x):xx } , where h(x ) , referred to as the hesitant fuzzy element ( hfe ) , is a set of some values in   denoting the possible membership degree of the element x  x to the set h. 
 example 3 . if x = { x1 , x2 , x3 } is the universe of discourse , h(x1 ) = { 0.2,0.4,0.5 } , h(x2 ) = { 0.3,0.4 } , and h(x3 ) = { 0.3,0.2,0.5,0.6 } are the hfes of xi  ( i = 1,2 , 3 ) to a set h , respectively . then h can be considered as a hfs ; that is , ( 4)h={x1,{0.2,0.4,0.5},x2,{0.3,0.4},iiiiiiiix3,{0.3,0.2,0.5.0.6}}. 
  if x = { x1 , x2 , x3 } is the universe of discourse , h(x1 ) = { 0.2,0.4,0.5 } , h(x2 ) = { 0.3,0.4 } , and h(x3 ) = { 0.3,0.2,0.5,0.6 } are the hfes of xi  ( i = 1,2 , 3 ) to a set h , respectively . then h can be considered as a hfs ; that is , ( 4)h={x1,{0.2,0.4,0.5},x2,{0.3,0.4},iiiiiiiix3,{0.3,0.2,0.5.0.6}}. as can be seen from definition 2 , hfs expresses the membership degrees of an element to a given set only by several real numbers between 0 and 1 , while in many real - world situations assigning exact values to the membership degrees does not describe properly the imprecise or uncertain decision information . thus , it seems to be difficult for the decision makers to rely on hfss for expressing uncertainty of an element . to overcome the difficulty associated with expressing uncertainty of an element to a given set , the concept of higher order hesitant fuzzy set ( hohfs ) is introduced here to let the membership degrees of an element to a given set be expressed by several possible g - type fss . a higher order hesitant fuzzy set ( hohfs ) on x is defined in terms of a function that when applied to x returns a set of g - type fss . a hohfs is denoted by ( 5)h~={x , h~(x):xx } , where h~(x ) , referred to as the higher order hesitant fuzzy element ( hohfe ) , is a set of some g - type fss denoting the possible membership degree of the element x  x to the set h~. in this regards , the hohfs h~ is also represented as ( 6)h~={x,{h~(1)(x),  ,h~(lx)(x)}:xx } , where all h~(1)(x), ,h~(lx)(x ) are g - type fss on x. 
  let x be the universe of discourse . a higher order hesitant fuzzy set ( hohfs ) on x is defined in terms of a function that when applied to x returns a set of g - type fss . a hohfs is denoted by ( 5)h~={x , h~(x):xx } , where h~(x ) , referred to as the higher order hesitant fuzzy element ( hohfe ) , is a set of some g - type fss denoting the possible membership degree of the element x  x to the set h~. in this regards , the hohfs h~ is also represented as ( 6)h~={x,{h~(1)(x),  ,h~(lx)(x)}:xx } , where all h~(1)(x), if x = { x1 , x2 , x3 } is the universe of discourse , ( 7)iiih~(x1)={h~(1)(x1)=(0.2,0.4),h~(2)(x1)=(0.5,0.3)},iiih~(x2)={h~(1)(x2)=(0.3,0.4)},iiih~(x3)={h~(1)(x3)=(0.3,0.2),h~(2)(x3)=(0.1,0.3),iiiiiiiiiiiiiiiiih~(3)(x3)=(0.5,0.4 ) } , are the hohfes of xi(i = 1,2 , 3 ) to a set h~ , respectively , where g - type fss h~(k)(xi)=(ki,ki ) are intuitionistic fuzzy set ( ifs )   such that 0  ki , ki  1 and 0  ki + ki  1 for k = 1,2 ,  , then h~ can be considered as a hohfs ; that is , ( 8)h~={x1,{(0.2,0.4),(0.5,0.3)},x2,{(0.3,0.4)},iiiiiiiix3,{(0.3,0.2),(0.1,0.3),(0.5,0.4)}}. 
  if x = { x1 , x2 , x3 } is the universe of discourse , ( 7)iiih~(x1)={h~(1)(x1)=(0.2,0.4),h~(2)(x1)=(0.5,0.3)},iiih~(x2)={h~(1)(x2)=(0.3,0.4)},iiih~(x3)={h~(1)(x3)=(0.3,0.2),h~(2)(x3)=(0.1,0.3),iiiiiiiiiiiiiiiiih~(3)(x3)=(0.5,0.4 ) } , are the hohfes of xi(i = 1,2 , 3 ) to a set h~ , respectively , where g - type fss h~(k)(xi)=(ki,ki ) are intuitionistic fuzzy set ( ifs )   such that 0  ki , ki  1 and 0  ki + ki  1 for k = 1,2 ,  , lxi and i = 1,2 ,  , |x | = 3 . then h~ can be considered as a hohfs ; that is , ( 8)h~={x1,{(0.2,0.4),(0.5,0.3)},x2,{(0.3,0.4)},iiiiiiiix3,{(0.3,0.2),(0.1,0.3),(0.5,0.4)}}. in view of definition 4 , it is easily deduced that each hohfs becomes a t2fs if all its g - type fss are the same and are in the form of t2fs . that is , if h~(1)(x)==h~(lx)(x)=:h~(x ) for any x  x , then the hohfs h~={x , h~(x):xx } reduces to a t2fs . it is noteworthy that the notions of interval - valued hesitant fuzzy set ( ivhfs )   and interval type-2 fuzzy set ( it2fs )   both are special cases of hohfss . a hohfs h~={x , h~(x):xx } reduces to an ivhfs , when all g - type fss h~(1)(x),,h~(lx)(x ) for any x  x are considered as closed intervals of real numbers in . furthermore , an ivhfs h~={x,[hl~(x),hr~(x)]:xx } reduces to an it2fs , when all intervals satisfy [ h~l(1)(x),h~r(1)(x)]==[h~l(lx)(x),h~r(lx)(x)]=:[h~l(x),h~r(x ) ] for any x  x. recently , hfss are extended by intuitionistic fuzzy sets ( ifss ) and referred to them as generalized hesitant fuzzy sets ( g - hfss ) . it is stated in   that fss , ifss , and hfss are special cases of g - hfss . obviously , a g - hfs h~={x , h~(x):xx } is also an special case of hohfs where all g - type fss h~(1)(x),  ,h~(lx)(x ) for any x  x are considered as ifss . this implies that hohfss are more useful than g - hfss to deal with decision making , clustering , pattern recognition , image processing , and so forth , when experts have a hesitation among several possible memberships for an element to a set . having introduced hohfes of a hohfs h~={x , h~(x):xx } , we now turn our attention to the interpretation of hohfs h~ as the union of all hohfes ; that is , ( 9)h~=h~h~{h~ } which is of fundamental importance in the study of hohfs aggregation and the higher order hesitant fuzzy soft matrix that will be introduced within the next parts of the paper . we give the definition of operation  on hohfss based on its counterpart on hohfes as follows : ( 10)h~1h~2=h1~h1~,h2~h2~{h1~h2~}=h1~h1~,h2~h2~{hh1~h1~,hh2~h2~{hh1~hh2~ } } , where the last operation  stands for the operation on g - type fss . we give the definition of operation  on hohfss based on its counterpart on hohfes as follows : ( 10)h~1h~2=h1~h1~,h2~h2~{h1~h2~}=h1~h1~,h2~h2~{hh1~h1~,hh2~h2~{hh1~hh2~ } } , where the last operation  stands for the operation on g - type fss . note that if there is no confusion , hohfs operator , hohfe operator , and g - type fs operator are denoted by the same symbol . notice in the case that g - type fss in hohfss are reduced to crisp values , the g - type fs operator in ( 10 ) is not the usual sum between real numbers because 0.6 + 0.5 = 1.1  , and indeed the g - type fs operator is the algebraic sum where 0.6  0.5 = 0.6 + 0.5  0.6  0.5 = 0.3  . thus , the hohfs operator is performed as the well - known hfs operator [ 29 , 30 ] . if the operation  on g - type fss has a property , denoted by ( pg)~ , then the operation  on hohfes has the counterpart of property ( pg)~ , denoted by ( p)~. 
  let h~1=h~h1h~1{hh1~ } and h~2=h~h2h~2{hh2~ } be two hohfes . if the operation  on g - type fss has a property , denoted by ( pg)~ , then the operation  on hohfes has the counterpart of property ( pg)~ , denoted by ( p)~. 
 proofthe proof is obvious from definition 6 . suppose that , for example , the operation  on g - type fss is commutative ; that is , for any g - type fss hh1~h1~,hh2~h2~ , the following equality holds : ( 11)hh1~hh2~=hh2~hh1~  ( property  ( pg)~ ) . then , one can see from definition 6 that ( 12)h~1h~2=hh1~h1~,hh2~h2~{hh1~hh2~}iiiiiiiiiii=hh1~h1~,hh2~h2~{hh2~hh1~}iiiiiiiiiii = h~2h~1  ( property  ( p)~ ) . this means that the operation  on hohfes is accordingly commutative.by similar reasoning , one can easily prove that the hohfe operator inherits all the properties of the g - type fs operator . suppose that , for example , the operation  on g - type fss is commutative ; that is , for any g - type fss hh1~h1~,hh2~h2~ , the following equality holds : ( 11)hh1~hh2~=hh2~hh1~  ( property  ( pg)~ ) . then , one can see from definition 6 that ( 12)h~1h~2=hh1~h1~,hh2~h2~{hh1~hh2~}iiiiiiiiiii=hh1~h1~,hh2~h2~{hh2~hh1~}iiiiiiiiiii = h~2h~1  ( property  ( p)~ ) . , one can easily prove that the hohfe operator inherits all the properties of the g - type fs operator . as a corollary of theorem 7 , it can be observed that hohfs operators inherit subsequently all operational properties of g - type fs operators . let h~1=h~1h~1{h1~ } and h~2=h~2h~2{h2~ } be two hohfss . if the operation  on hohfes has a property , denoted by ( p)~ , then the operation  on hohfss has the counterpart of property ( p)~ , denoted by ( p)~. 
  let h~1=h~1h~1{h1~ } and h~2=h~2h~2{h2~ } be two hohfss . if the operation  on hohfes has a property , denoted by ( p)~ , then the operation  on hohfss has the counterpart of property ( p)~ , denoted by ( p)~. 
 example 9 . let h~1=h~1h~1{h1~ } and h~2=h~2h~2{h2~ } be two hohfss with hohfes in the form of ifss . it can be shown for any ifss hh1~,hh2~ that ( 13)iiiiihh1~hh2~=hh2~hh1~ ;  ( property  ( pg1)~),iiiiihh1~hh2~=hh2~hh1~ ;  ( property  ( pg2)~),(hh1~hh2~)=hh1~hh2~ ,  >0 ;  ( property  ( pg3)~),i(hh1~hh2~)=hh1~hh2~ ,  >0 ;  ( property  ( pg4)~),(1+2)hh1~=1hh1~2hh1~,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1,2>0 ;  ( property  ( pg5)~),iiiiihh1~(1+2)=hh1~1hh1~2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1,2>0 ;  ( property  ( pg6)~ ) , where ( see e.g. , ) ( 14)hh1~hh2~=hh1~,hh1~hh2~,hh2~iiiiiiiiiiiii=hh1~+hh2~hh1~hh2~,hh1~hh2~;hh1~hh2~=hh1~,hh1~hh2~,hh2~iiiiiiiiiiiii=hh1~hh2~,hh1~+hh2~hh1~hh2~;hh1~=1(1hh1~),(hh1~) ,  >0;hh1~=(hh1~),1(1hh1~) ,  >0 . accordingly , from corollary 8 , it is deduced that ( 15)iiiiiih1~h2~=h2~h1~ ;  ( property  ( p1)~),iiiiiih1~h2~=h2~h1~ ;  ( property  ( p2)~),i(h1~h2~)=h1~h2~ ,  >0 ;  ( property  ( p3)~),i(h1~h2~)=h1~h2~ ,  >0 ;  ( property  ( p4)~),(1+2)h1~=1h1~2h1~,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1,2>0 ;  ( property  ( p5)~),iiiih1~(1+2)=h1~1h1~2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1,2>0 ;  ( property  ( p6)~ ) . 
  let h~1=h~1h~1{h1~ } and h~2=h~2h~2{h2~ } be two hohfss with hohfes in the form of ifss . it can be shown for any ifss hh1~,hh2~ that ( 13)iiiiihh1~hh2~=hh2~hh1~ ;  ( property  ( pg1)~),iiiiihh1~hh2~=hh2~hh1~ ;  ( property  ( pg2)~),(hh1~hh2~)=hh1~hh2~ ,  >0 ;  ( property  ( pg3)~),i(hh1~hh2~)=hh1~hh2~ ,  >0 ;  ( property  ( pg4)~),(1+2)hh1~=1hh1~2hh1~,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1,2>0 ;  ( property  ( pg5)~),iiiiihh1~(1+2)=hh1~1hh1~2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1,2>0 ;  ( property  ( pg6)~ ) , where ( see e.g. , ) ( 14)hh1~hh2~=hh1~,hh1~hh2~,hh2~iiiiiiiiiiiii=hh1~+hh2~hh1~hh2~,hh1~hh2~;hh1~hh2~=hh1~,hh1~hh2~,hh2~iiiiiiiiiiiii=hh1~hh2~,hh1~+hh2~hh1~hh2~;hh1~=1(1hh1~),(hh1~) ,  >0;hh1~=(hh1~),1(1hh1~) ,  >0 . accordingly , from corollary 8 , it is deduced that ( 15)iiiiiih1~h2~=h2~h1~ ;  ( property  ( p1)~),iiiiiih1~h2~=h2~h1~ ;  ( property  ( p2)~),i(h1~h2~)=h1~h2~ ,  >0 ;  ( property  ( p3)~),i(h1~h2~)=h1~h2~ ,  >0 ;  ( property  ( p4)~),(1+2)h1~=1h1~2h1~,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1,2>0 ;  ( property  ( p5)~),iiiih1~(1+2)=h1~1h1~2,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii1,2>0 ;  ( property  ( p6)~ ) . in the following , we first review briefly the concepts of soft set and fuzzy soft set . to do this , it may be helpful to recall that the sets x , (x ) , and e are referred to as the initial universe set , the power set of x , and a set of parameters , respectively . is called a soft set over x , where f is a mapping given by f : a  e  (x ) . a pair ( f , a ) is called a soft set over x , where f is a mapping given by f : a  e  (x ) . indeed , a soft set over x is not a set but a parameterized family of subsets of x. 
 example 11 . suppose that x = { x1 , x2 , x3 , x4 } is a set of houses under consideration and a = { e1 , e2 , e3 } is a set of parameter , where e1 = expensive , e2 = beautiful , and e3 = in the good location . let ( 16)f(e1)={x1 } ,  f(e2)={x1,x3 } ,  f(e3)={x2,x4}. we use here the soft set ( f , a ) to describe the  attractiveness of the houses . in this regard , f(e1 ) stands for  houses ( expensive )  whose function value is the set { x1 } , f(e2 ) means  houses ( beautiful )  whose function value is the set { x1 , x3 } , and f(e3 ) implies  houses ( in the good location )  whose function value is the set { x2 , x4}. 
  suppose that x = { x1 , x2 , x3 , x4 } is a set of houses under consideration and a = { e1 , e2 , e3 } is a set of parameter , where e1 = expensive , e2 = beautiful , and e3 = in the good location . let ( 16)f(e1)={x1 } ,  f(e2)={x1,x3 } ,  f(e3)={x2,x4}. we use here the soft set ( f , a ) to describe the  attractiveness of the houses . in this regard , f(e1 ) stands for  houses ( expensive )  whose function value is the set { x1 } , f(e2 ) means  houses ( beautiful )  whose function value is the set { x1 , x3 } , and f(e3 ) implies  houses ( in the good location )  whose function value is the set { x2 , x4}. 
 definition 12 ( see ) . a pair ( f,a ) is called a fuzzy soft set over x where (x ) is the set of all fuzzy subsets of x. 
  let f:ae(x ) . a pair ( f,a ) is called a fuzzy soft set over x where (x ) is the set of all fuzzy subsets of x. 
 example 13 . we can describe the  attractiveness of the houses  under the fuzzy circumstances using the fuzzy soft set ( f,a ) , where ( 17)f(e1)={x11,x20,x30,x40},f(e2)={x11,x20,x31,x40},f(e3)={x10,x21,x30,x41}. this example shows that if we let the corresponding membership degrees of all elements e  e  a be zero , then fuzzy soft sets ( f,a ) and ( f,e ) bear the same meaning and have the unified forms . we can describe the  attractiveness of the houses  under the fuzzy circumstances using the fuzzy soft set ( f,a ) , where ( 17)f(e1)={x11,x20,x30,x40},f(e2)={x11,x20,x31,x40},f(e3)={x10,x21,x30,x41}. this example shows that if we let the corresponding membership degrees of all elements e  e  a be zero , then fuzzy soft sets ( f,a ) and ( f,e ) bear the same meaning and have the unified forms . let f~:e~(x ) . a pair ( f~,e ) is called a higher order hesitant fuzzy ( hohf ) soft set over x where ~(x ) is the set of all higher order hesitant fuzzy subsets of x. in this regard , for any e  e , we denote ( 18)f~(e):={xh~f~(e)(x):xx}iiiiiiii={x{h~f~(e)(1)(x),  ,h~f~(e)(lx)(x)}:xx } , where all h~f~(e)(1)(x), ,h~f~(e)(lx)(x ) are g - type fss on x. that is to say , that the higher order hesitant fuzzy ( hohf ) soft set over x can be denoted by ( 19)(f~,e)=xxx{h~f~(e)(1)(x),  ,h~f~(e)(lx)(x)}. we hereafter denote all hohf soft sets over x by hohfs(x ) a pair ( f~,e ) is called a higher order hesitant fuzzy ( hohf ) soft set over x where ~(x ) is the set of all higher order hesitant fuzzy subsets of x. in this regard , for any e  e , we denote ( 18)f~(e):={xh~f~(e)(x):xx}iiiiiiii={x{h~f~(e)(1)(x),  ,h~f~(e)(lx)(x)}:xx } , where all h~f~(e)(1)(x), ,h~f~(e)(lx)(x ) are g - type fss on x. that is to say , that the higher order hesitant fuzzy ( hohf ) soft set over x can be denoted by ( 19)(f~,e)=xxx{h~f~(e)(1)(x),  ,h~f~(e)(lx)(x)}. we hereafter denote all hohf soft sets over x by hohfs(x ) . suppose that x = { x1 , x2 , x3 , x4 } is a set of color cloths under consideration and a = { e1 , e2 , e3 } is a set of parameter where e1 = color which consists of red and blue , e2 = ingredient which is made from wool , cotton , and acrylic , and e3 = price which can be considered as high and low . in this situation , we should consider a higher order hesitant fuzzy soft set ( f~,e ) as follows : ( 20)f~(e1)={x1{(0.1,0.2),(0.3,0.2)},x2{(0.2,0.3),(0.3,0.1)},iiiiiiiix3{(0.3,0.3),(0.4,0.6)},x4{(0.1,0.3),(0.3,0.1)}},f~(e2)={x1{(0.5,0.2),(0.1,0.2),(0.3,0.2)},iiiiiiiiiiiiiix2{(0.2,0.4),(0.6,0.1),(0.3,0.1)},iiiiiiiiiiiiiix3{(0.3,0.3),(0.4,0.4),(0.4,0.6)},iiiiiiiiiiiiiix4{(0.1,0.3),(0.3,0.1),(0.2,0.5)}},f~(e3)={x1{(0.5,0,2),(0.4,0.1)},x2{(0.3,0.7),(0.6,0.1)},iiiiiiiix3{(0.4,0.4),(0.4,0.6)},x4{(0.8,0.2),(0.2,0.5 ) } } , where the above g - type fss , denoted by (  ,  ) , are all in the form of ifs on x. 
  suppose that x = { x1 , x2 , x3 , x4 } is a set of color cloths under consideration and a = { e1 , e2 , e3 } is a set of parameter where e1 = color which consists of red and blue , e2 = ingredient which is made from wool , cotton , and acrylic , and e3 = price which can be considered as high and low . in this situation , we should consider a higher order hesitant fuzzy soft set ( f~,e ) as follows : ( 20)f~(e1)={x1{(0.1,0.2),(0.3,0.2)},x2{(0.2,0.3),(0.3,0.1)},iiiiiiiix3{(0.3,0.3),(0.4,0.6)},x4{(0.1,0.3),(0.3,0.1)}},f~(e2)={x1{(0.5,0.2),(0.1,0.2),(0.3,0.2)},iiiiiiiiiiiiiix2{(0.2,0.4),(0.6,0.1),(0.3,0.1)},iiiiiiiiiiiiiix3{(0.3,0.3),(0.4,0.4),(0.4,0.6)},iiiiiiiiiiiiiix4{(0.1,0.3),(0.3,0.1),(0.2,0.5)}},f~(e3)={x1{(0.5,0,2),(0.4,0.1)},x2{(0.3,0.7),(0.6,0.1)},iiiiiiiix3{(0.4,0.4),(0.4,0.6)},x4{(0.8,0.2),(0.2,0.5 ) } } , where the above g - type fss , denoted by (  ,  ) , are all in the form of ifs on x. 
 definition 16 . a higher order hesitant fuzzy soft set ( f~,e ) over x is called a null higher order hesitant fuzzy soft set over x denoted by 0~ , if for any e  e(21)f~(e):={xh~f~(e)(x):xx}={x{0~,  ,0~}:xx } , where 0~ is the zero g - type fs on x. 
  a higher order hesitant fuzzy soft set ( f~,e ) over x is called a null higher order hesitant fuzzy soft set over x denoted by 0~ , if for any e  e(21)f~(e):={xh~f~(e)(x):xx}={x{0~,  ,0~}:xx } , where 0~ is the zero g - type fs on x. 
 definition 17 . a higher order hesitant fuzzy soft set ( f~,e ) over x is called an absolute higher order hesitant fuzzy soft set over x denoted by 1~ , if for any e  e(22)f~(e):={xh~f~(e)(x):xx}={x{1~,  ,1~}:xx } , where 1~ is the one g - type fs on x. 
  a higher order hesitant fuzzy soft set ( f~,e ) over x is called an absolute higher order hesitant fuzzy soft set over x denoted by 1~ , if for any e  e(22)f~(e):={xh~f~(e)(x):xx}={x{1~,  ,1~}:xx } , where 1~ is the one g - type fs on x. 
 definition 18 ( see ) . consider two hohfss h~1=h~1h~1{h1~ } and h~2=h~2h~2{h2~ } on x = { x1 , x2 , xn}. the definition of distance measure for hohfss is given by ( 23)d(h~1,h~2)=1ni=1ndh(h~1(xi),h~2(xi ) ) , where for each xi  x(24)dh(h~1(xi),h~2(xi ) )  = max{maxh~1h~1minh~2h~2d(h~1(xi),h~2(xi)),iiiiiiiiiiiiiiiminh~1h~1maxh~2h~2d(h~1(xi),h~2(xi))}. here , d( ,  ) is a distance measure defined on g - type fss . consider two hohfss h~1=h~1h~1{h1~ } and h~2=h~2h~2{h2~ } on x = { x1 , x2 , xn}. the definition of distance measure for hohfss is given by ( 23)d(h~1,h~2)=1ni=1ndh(h~1(xi),h~2(xi ) ) , where for each xi  x(24)dh(h~1(xi),h~2(xi ) )  = max{maxh~1h~1minh~2h~2d(h~1(xi),h~2(xi)),iiiiiiiiiiiiiiiminh~1h~1maxh~2h~2d(h~1(xi),h~2(xi))}. here , d( ,  ) is a distance measure defined on g - type fss . notice that to have a correct comparison between two hohfss h~1 and h~2 , the two following cases should be taken into account . the first case occurs if all g - type fss in each hohfes of hohfs can be arranged increasingly ( or decreasingly ) by the use of a partial order . another case occurs when all g - type fss in each hohfes of hohfs can not be arranged increasingly ( or decreasingly ) using a partial order . in view of the above arguments , we shall represent the definition of a higher order hesitant fuzzy soft subset as follows . ( f~,e ) is said to be a higher order hesitant fuzzy soft subset of ( g~,e ) if ( i)(in the ordered case ) for all e  e and for all x  x it holds ( 25)h~f~(e)(k)(x)h~g~(e)(k)(x ) ,  1klx ,  where  stands for a partial order defined on g - type fss;(ii)(in the disordered case ) for all e  e and for all x  x one has ( 26)dh(h~f~(e)(x),1~)dh(h~g~(e)(x),1~ ) ,  where dh( ,  ) is that given in definition 18 and 1~ is the absolute higher order hesitant fuzzy soft set over x. 
  suppose that ( f~,e),(g~,e)hohfs(x ) . ( f~,e ) is said to be a higher order hesitant fuzzy soft subset of ( g~,e ) if ( i)(in the ordered case ) for all e  e and for all x  x it holds ( 25)h~f~(e)(k)(x)h~g~(e)(k)(x ) ,  1klx ,  where  stands for a partial order defined on g - type fss;(ii)(in the disordered case ) for all e  e and for all x  x one has ( 26)dh(h~f~(e)(x),1~)dh(h~g~(e)(x),1~ ) ,  where dh( ,  ) is that given in definition 18 and 1~ is the absolute higher order hesitant fuzzy soft set over x. ( in the ordered case ) for all e  e and for all x  x it holds ( 25)h~f~(e)(k)(x)h~g~(e)(k)(x ) ,  1klx ,   where  stands for a partial order defined on g - type fss ; ( in the disordered case ) for all e  e and for all x  x one has ( 26)dh(h~f~(e)(x),1~)dh(h~g~(e)(x),1~ ) , where dh( ,  ) is that given in definition 18 and 1~ is the absolute higher order hesitant fuzzy soft set over x. the inclusion relation between ( f~,e ) and ( g~,e ) is denoted by ( f~,e)~(g~,e ) . 
 definition 20 . two higher order hesitant fuzzy soft sets ( f~,e),(g~,e ) are said to be higher order hesitant fuzzy soft equal sets if and only if ( f~,e)~(g~,e ) and ( g~,e)~(f~,e ) . two higher order hesitant fuzzy soft sets ( f~,e),(g~,e ) are said to be higher order hesitant fuzzy soft equal sets if and only if ( f~,e)~(g~,e ) and ( g~,e)~(f~,e ) . 
 definition 21 . ( fc~,e ) is said to be a higher order hesitant fuzzy soft complementary set of ( f~,e ) if for all e  e and for all x  x(27)h~fc~(e)(k)(x)=(h~f~(e)(k)(x))c ,  1klx , where ( h~f~(e)(k)(x))c means the complement of the corresponding g - type fss h~f~(e)(k)(x ) . ( fc~,e ) is said to be a higher order hesitant fuzzy soft complementary set of ( f~,e ) if for all e  e and for all x  x(27)h~fc~(e)(k)(x)=(h~f~(e)(k)(x))c ,  1klx , where ( h~f~(e)(k)(x))c means the complement of the corresponding g - type fss h~f~(e)(k)(x ) . union of ( f~,e),(g~,e)hohfs(x ) , denoted by ( f~,e)~(g~,e ) , is defined for all e  e as ( 28)(m~,e )  = ( f~,e)~(g~,e )  = xxx{(h~f~(e)(1)(x)h~g~(e)(1)(x)), ,(h~f~(e)(lx)(x)h~g~(e)(lx)(x))}. 
 
 union of ( f~,e),(g~,e)hohfs(x ) , denoted by ( f~,e)~(g~,e ) , is defined for all e  e as ( 28)(m~,e )  = ( f~,e)~(g~,e )  = xxx{(h~f~(e)(1)(x)h~g~(e)(1)(x)), intersection of ( f~,e),(g~,e)hohfs(x ) , denoted by ( f~,e)~(g~,e ) , is defined for all e  e as ( 29)(n~,e )  = ( f~,e)~(g~,e )  = xxx{(h~f~(e)(1)(x)h~g~(e)(1)(x)),  ,(h~f~(e)(lx)(x)h~g~(e)(lx)(x))}. 
 
 intersection of ( f~,e),(g~,e)hohfs(x ) , denoted by ( f~,e)~(g~,e ) , is defined for all e  e as ( 29)(n~,e )  = ( f~,e)~(g~,e )  = xxx{(h~f~(e)(1)(x)h~g~(e)(1)(x)), ( ( f~,e)~(g~,e))c=(f~,e)c~(g~,e)c:=(fc~,e)~(gc~,e ) ,   ( ( f~,e)~(g~,e))c=(f~,e)c~(g~,e)c:=(fc~,e)~(gc~,e ) .  
 proof 
  part ( 1 ) . in this regard , ( ( f~,e)~(g~,e))c=(m~,e)c:=(mc~,e ) . on the other hand , from definition 22 one gets ( 30)((f~,e)~(g~,e))c  = ( mc~,e )  = xxx{(h~f~(e)(1)(x)h~g~(e)(1)(x))c,  ,(h~f~(e)(lx)(x)h~g~(e)(lx)(x))c }  = xx(  ,((h~f~(e)(lx)(x))c(h~g~(e)(lx)(x))c)})1x(((h~f~(e)(lx)(x))c(h~g~(e)(lx)(x))c){((h~f~(e)(1)(x))c(h~g~(e)(1)(x))c),  ,iiiiiiiiiiiiiiiiiiiiiiiii((h~f~(e)(lx)(x))c(h~g~(e)(lx)(x))c)})1 )  = xxx{(h~fc~(e)(1)(x)h~gc~(e)(1)(x)),  ,(h~fc~(e)(lx)(x)h~gc~(e)(lx)(x ) ) }  = ( fc~,e)~(gc~,e )  = ( f~,e)c~(g~,e)c  [ proved ] . in this regard , ( ( f~,e)~(g~,e))c=(m~,e)c:=(mc~,e ) . on the other hand , from definition 22 one gets ( 30)((f~,e)~(g~,e))c  = ( mc~,e )  = xxx{(h~f~(e)(1)(x)h~g~(e)(1)(x))c,  ,(h~f~(e)(lx)(x)h~g~(e)(lx)(x))c }  = xx(  ,((h~f~(e)(lx)(x))c(h~g~(e)(lx)(x))c)})1x(((h~f~(e)(lx)(x))c(h~g~(e)(lx)(x))c){((h~f~(e)(1)(x))c(h~g~(e)(1)(x))c),  ,iiiiiiiiiiiiiiiiiiiiiiiii((h~f~(e)(lx)(x))c(h~g~(e)(lx)(x))c)})1 )  = xxx{(h~fc~(e)(1)(x)h~gc~(e)(1)(x)),  ,(h~fc~(e)(lx)(x)h~gc~(e)(lx)(x ) ) }  = ( fc~,e)~(gc~,e )  = ( f~,e)c~(g~,e)c  [ proved ] . part ( 2 ) . then ( 1)((f~,e)~(g~,e))~(k~,e)=(f~,e)~((g~,e)~(k~,e ) ) , ( 2)((f~,e)~(g~,e))~(k~,e)=(f~,e)~((g~,e)~(k~,e ) ) , ( 3)(f~,e)~((g~,e)~(k~,e))=((f~,e)~(g~,e))~((f~,e)~(k~,e ) ) , ( 4)(f~,e)~((g~,e)~(k~,e))=((f~,e)~(g~,e))~((f~,e)~(k~,e ) ) . then ( 1)((f~,e)~(g~,e))~(k~,e)=(f~,e)~((g~,e)~(k~,e ) ) , ( 2)((f~,e)~(g~,e))~(k~,e)=(f~,e)~((g~,e)~(k~,e ) ) , ( 3)(f~,e)~((g~,e)~(k~,e))=((f~,e)~(g~,e))~((f~,e)~(k~,e ) ) , ( 4)(f~,e)~((g~,e)~(k~,e))=((f~,e)~(g~,e))~((f~,e)~(k~,e ) ) . ( ( f~,e)~(g~,e))~(k~,e)=(f~,e)~((g~,e)~(k~,e ) ) ,   ( ( f~,e)~(g~,e))~(k~,e)=(f~,e)~((g~,e)~(k~,e ) ) ,   ( f~,e)~((g~,e)~(k~,e))=((f~,e)~(g~,e))~((f~,e)~(k~,e ) ) ,   ( f~,e)~((g~,e)~(k~,e))=((f~,e)~(g~,e))~((f~,e)~(k~,e ) ) . let ( f~,e)hohfs(x ) , let 1~ be the absolute higher order hesitant fuzzy soft set over x , and let 0~ be the null higher order hesitant fuzzy soft set over x. then ( 1)(f~,e)~(f~,e)=(f~,e ) , ( 2)(f~,e)~(f~,e)=(f~,e ) , ( 3)(f~,e)~1~=1~ , ( 4)(f~,e)~1~=(f~,e ) , ( 5)(f~,e)~0~=0~ , ( 6)(f~,e)~0~=(f~,e ) . let ( f~,e)hohfs(x ) , let 1~ be the absolute higher order hesitant fuzzy soft set over x , and let 0~ be the null higher order hesitant fuzzy soft set over x. then ( 1)(f~,e)~(f~,e)=(f~,e ) , ( 2)(f~,e)~(f~,e)=(f~,e ) , ( 3)(f~,e)~1~=1~ , ( 4)(f~,e)~1~=(f~,e ) , ( 5)(f~,e)~0~=0~ , ( 6)(f~,e)~0~=(f~,e ) . ( f~,e)~(f~,e)=(f~,e ) ,   ( f~,e)~(f~,e)=(f~,e ) ,  
 proofthe proof is straightforward . in what follows , we are interested to present an application of hohf soft set theory in a multiple criterion decision making problem . in order to solve decision making problems involving hohf soft sets , an approach is proposed based on the concept of -level higher order hesitant fuzzy soft set . | = m. the higher order hesitant fuzzy soft matrix ~ is defined in the concrete form of ( 31)f~=(e1  emx1h~f~(e1)(x1)  h~f~(em)(x1)xnh~f~(e1)(xn)  h~f~(em)(xn))ii=[h~f~(ej)(xi)]nm . we denote all higher order hesitant fuzzy soft matrix ~ over x by hohfsm(x ) . | = m. the higher order hesitant fuzzy soft matrix ~ is defined in the concrete form of ( 31)f~=(e1  emx1h~f~(e1)(x1)  h~f~(em)(x1)xnh~f~(e1)(xn)  h~f~(em)(xn))ii=[h~f~(ej)(xi)]nm . we denote all higher order hesitant fuzzy soft matrix ~ over x by hohfsm(x ) . as can be seen from definitions 27 and 14 , there exists a bijective mapping from hohfs(x ) to hohfsm(x ) , implying that any higher order hesitant fuzzy soft set ( f~,e)hohfs(x ) can be equivalent with its higher order hesitant fuzzy soft matrix ~hohfsm(x ) . this equivalent relationship allows us to identify any ( f~,e)hohfs(x ) with its ~hohfsm(x ) , interchangeably some arithmetic operations on hohfsm(x ) can be defined as follows : ( 32)f~g~=[h~f~(ej)(xi)h~g~(ej)(xi)]nm , f~g~=[h~f~(ej)(xi)h~g~(ej)(xi)]nm , iiiif~=[(h~f~(ej)(xi))]nm ,  >0,iiiif~=[(h~f~(ej)(xi))]nm ,  >0 , where the right - hand - side operations are that considered on g - type fss in definition 6 . some arithmetic operations on hohfsm(x ) can be defined as follows : ( 32)f~g~=[h~f~(ej)(xi)h~g~(ej)(xi)]nm , f~g~=[h~f~(ej)(xi)h~g~(ej)(xi)]nm , iiiif~=[(h~f~(ej)(xi))]nm ,  >0,iiiif~=[(h~f~(ej)(xi))]nm ,  >0 , where the right - hand - side operations are that considered on g - type fss in definition 6 . accordingly , from definition 28 and corollary 8 , it can be deduced for any ~,~hohfsm(x ) that ( 33)iiiiiiiif~g~=g~f~,iiiiiiiif~g~=g~f~,iii(f~g~)=f~g~ ,  >0,iii(f~g~)=f~g~ ,  >0,i(1+2)f~=1f~2f~ ,  1,2>0,iiiiiif~(1+2)=f~1f~2 ,  1,2>0 . the latter results can be employed whenever some experts are invited to conduct evaluation for one decision making problem . in such a situation , we are required to use aggregation operators on hohfsm(x ) given by experts to obtain a collective ~hohfsm(x ) . however , dealing with the aggregation methods based on multiexperts ' higher order hesitant fuzzy soft matrix is beyond the scope of this paper . before introducing the concept of the -level higher order hesitant fuzzy soft set , we define a threshold vector over parameter set e as (e ) = ( (e1 ) ,  , (em ) ) where (ej)=h~(ej )  ( j=1,  ,m ) . obviously , the hohfes h~(ej )  ( j=1,  ,m ) are changed dependently on the different parameter set e. consider again the higher order hesitant fuzzy soft matrix ~ given in the concrete form of ~=[h~f~(ej)(xi)]nm . in order to get reasonably a hohfe for each parameter ej  e from ~ we define ( 34)h~(ej)agg = agg(h~f~(ej)(x1),  ,h~f~(ej)(xn)),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiieje(j=1,  ,m ) , where agg( ) is an aggregation operator on hohfes . in view of the latter setting , the threshold vector over parameter set e is defined as ( 35)agg(e)={agg(ej ) ,  j=1,  ,m}iiiiiiiiiiii={h~(ej)agg ,  j=1, a real function i : hohfe(x )  hohfe(x) is called the inclusion measure of the hohfes h~f~(e)(x)={h~f~(e)(k)(x)}k=1lx , h~g~(e)(x)={h~g~(e)(k)(x)}k=1lx , and h~k~(e)(x)={h~k~(e)(k)(x)}k=1lx , if i( ,  ) satisfies the following properties . ( 2)i(h~f~(e)(x),h~g~(e)(x))=1 iff ( 36)h~f~(e)(x)~h~g~(e)(x )  iff  h~f~(e)(k)(x)h~g~(e)(k)(x),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiik=1,  ,lx ,  where the last inclusion operator  is that introduced on g - type fss.(3)if h~f~(e)(x)~h~g~(e)(x)~h~k~(e)(x ) , then i(h~k~(e)(x),h~f~(e)(x))i(h~g~(e)(x),h~f~(e)(x ) ) and i(h~k~(e)(x),h~f~(e)(x))i(h~k~(e)(x),h~g~(e)(x ) ) . let us now define the following two - valued function i( ,  ) on g - type fss h~f~(e)(k)(x ) and h~g~(e)(k)(x ) as ( 37)i(h~f~(e)(k)(x),h~g~(e)(k)(x ) )  = { 1,if  h~f~(e)(k)(x)  h~g~(e)(k)(x ) ,  k=1,  ,lx,;0,otherwise , and then , the following function is an inclusion measure of the hohfes h~f~(e)(x)={h~f~(e)(k)(x)}k=1lx and h~g~(e)(x)={h~g~(e)(k)(x)}k=1lx given by ( 38)i(h~f~(e)(x),h~g~(e)(x))=1lxk=1lxi(h~f~(e)(k)(x),h~g~(e)(k)(x ) ) . 
  a real function i : hohfe(x )  hohfe(x) is called the inclusion measure of the hohfes h~f~(e)(x)={h~f~(e)(k)(x)}k=1lx , h~g~(e)(x)={h~g~(e)(k)(x)}k=1lx , and h~k~(e)(x)={h~k~(e)(k)(x)}k=1lx , if i( ,  ) satisfies the following properties . ( 2)i(h~f~(e)(x),h~g~(e)(x))=1 iff ( 36)h~f~(e)(x)~h~g~(e)(x )  iff  h~f~(e)(k)(x)h~g~(e)(k)(x),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiik=1,  ,lx ,  where the last inclusion operator  is that introduced on g - type fss.(3)if h~f~(e)(x)~h~g~(e)(x)~h~k~(e)(x ) , then i(h~k~(e)(x),h~f~(e)(x))i(h~g~(e)(x),h~f~(e)(x ) ) and i(h~k~(e)(x),h~f~(e)(x))i(h~k~(e)(x),h~g~(e)(x ) ) . let us now define the following two - valued function i( ,  ) on g - type fss h~f~(e)(k)(x ) and h~g~(e)(k)(x ) as ( 37)i(h~f~(e)(k)(x),h~g~(e)(k)(x ) )  = { 1,if  h~f~(e)(k)(x)  h~g~(e)(k)(x ) ,  k=1,  ,lx,;0,otherwise , and then , the following function is an inclusion measure of the hohfes h~f~(e)(x)={h~f~(e)(k)(x)}k=1lx and h~g~(e)(x)={h~g~(e)(k)(x)}k=1lx given by ( 38)i(h~f~(e)(x),h~g~(e)(x))=1lxk=1lxi(h~f~(e)(k)(x),h~g~(e)(k)(x ) ) . i(h~f~(e)(x),h~g~(e)(x))=1 iff ( 36)h~f~(e)(x)~h~g~(e)(x )  iff  h~f~(e)(k)(x)h~g~(e)(k)(x),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiik=1,  ,lx , where the last inclusion operator  is that introduced on g - type fss . if h~f~(e)(x)~h~g~(e)(x)~h~k~(e)(x ) , then i(h~k~(e)(x),h~f~(e)(x))i(h~g~(e)(x),h~f~(e)(x ) ) and i(h~k~(e)(x),h~f~(e)(x))i(h~k~(e)(x),h~g~(e)(x ) ) . 
 definition 30 . we define the (e)-level higher order hesitant fuzzy soft set of ~hohfsm(x ) , denoted by l(~,(e ) ) , as follows : ( 39)l(f~,(ej))={xix    i((ej),h~f~(ej)(xi))>0},iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiieje(j=1, ,m ) , where i( ,  ) is the inclusion measure introduced by ( 38 ) . we define the (e)-level higher order hesitant fuzzy soft set of ~hohfsm(x ) , denoted by l(~,(e ) ) , as follows : ( 39)l(f~,(ej))={xix    i((ej),h~f~(ej)(xi))>0},iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiieje(j=1, ,m ) , where i( ,  ) is the inclusion measure introduced by ( 38 ) . as a result from definition 30 and ( 35 ) , the agg(e)-level higher order hesitant fuzzy soft set of ~hohfsm(x ) is characterized by ( 40)l(f~,agg(ej ) )  = { xix    i(h~(ej)agg , h~f~(ej)(xi))>0}. once the (e)-level higher order hesitant fuzzy soft set of ~ is achieved , we calculate the inclusion measure values i((ej),h~f~(ej)(xi ) ) for ej  ( j = 1 ,  , m ) and xi  ( i = 1 ,  , n ) given by ( 38 ) to rank all the alternatives . the optimal decision is then selected as xk if ( 41)j=1mi((ej),h~f~(ej)(xk ) )  = maxi=1,  ,n{j=1mi((ej),h~f~(ej)(xi))}. it is noteworthy to say that if hohfes h~f~(ej)(xi ) reduce to ifss , then the inclusion measure values i((ej),h~f~(ej)(xi ) ) are coincided to choice values ci of xi  ( i = 1 ,  , now we are in a position to present an algorithm for solving higher order hesitant fuzzy soft set based decision making problem by using (e)-level higher order hesitant fuzzy soft sets initiated in this study . input the higher order hesitant fuzzy soft matrix ~ over a finite initial universe x and a finite parameter set e. 
  input the higher order hesitant fuzzy soft matrix ~ over a finite initial universe x and a finite parameter set e. 
 step 2 . compute the threshold vector (e ) = ( (e1 ) ,  , (em ) ) corresponding to ~. 
  compute the threshold vector (e ) = ( (e1 ) ,  , (em ) ) corresponding to ~. 
 step 3 . calculate inclusion measure value of all alternatives to determine their ranking and then selecting the optimal decision by the use of ( 41 ) . 
  calculate inclusion measure value of all alternatives to determine their ranking and then selecting the optimal decision by the use of ( 41 ) . to illustrate the idea of algorithm 31 , let us take into consideration the following example which addresses example 15 with more details . suppose that x = { x1 , x2 , x3 , x4 } is a set of color cloths under consideration and a = { e1 , e2 , e3 } is a set of parameter where e1 = color which consists of red and blue , e2 = ingredient which is made from wool , cotton , and acrylic , and e3 = price which can be considered as high and low . in this situation , we should consider a higher order hesitant fuzzy soft set ( f~,e ) as follows : ( 42)f~(e1)={x1{(0.1,0.2),(0.3,0.2)},x2{(0.2,0.3),(0.3,0.1)},iiiiiiiiiiiiix3{(0.3,0.3),(0.4,0.6)},x4{(0.1,0.3),(0.3,0.1)}},f~(e2)={x1{(0.5,0.2),(0.1,0.2),(0.3,0.2)},iiiiiiiiiiiiix2{(0.2,0.4),(0.6,0.1),(0.3,0.1)},iiiiiiiiiiiiix3{(0.3,0.3),(0.4,0.4),(0.4,0.6)},iiiiiiiiiiiiix4{(0.1,0.3),(0.3,0.1),(0.2,0.5)}},f~(e3)={x1{(0.5,0,2),(0.4,0.1)},x2{(0.3,0.7),(0.6,0.1)},iiiiiiiiiiiiix3{(0.4,0.4),(0.4,0.6)},x4{(0.8,0.2),(0.2,0.5 ) } } , where the above g - type fss , denoted by (  ,  ) , are all in the form of ifs on x. 
  suppose that x = { x1 , x2 , x3 , x4 } is a set of color cloths under consideration and a = { e1 , e2 , e3 } is a set of parameter where e1 = color which consists of red and blue , e2 = ingredient which is made from wool , cotton , and acrylic , and e3 = price which can be considered as high and low . in this situation , we should consider a higher order hesitant fuzzy soft set ( f~,e ) as follows : ( 42)f~(e1)={x1{(0.1,0.2),(0.3,0.2)},x2{(0.2,0.3),(0.3,0.1)},iiiiiiiiiiiiix3{(0.3,0.3),(0.4,0.6)},x4{(0.1,0.3),(0.3,0.1)}},f~(e2)={x1{(0.5,0.2),(0.1,0.2),(0.3,0.2)},iiiiiiiiiiiiix2{(0.2,0.4),(0.6,0.1),(0.3,0.1)},iiiiiiiiiiiiix3{(0.3,0.3),(0.4,0.4),(0.4,0.6)},iiiiiiiiiiiiix4{(0.1,0.3),(0.3,0.1),(0.2,0.5)}},f~(e3)={x1{(0.5,0,2),(0.4,0.1)},x2{(0.3,0.7),(0.6,0.1)},iiiiiiiiiiiiix3{(0.4,0.4),(0.4,0.6)},x4{(0.8,0.2),(0.2,0.5 ) } } , where the above g - type fss , denoted by (  ,  ) , are all in the form of ifs on x. the aim here is to select one of color cloths depending on its performance evaluated with respect to all parameters . the tabular representation of the above higher order hesitant fuzzy soft matrix ~ is given in table 1 . to perform the second step of algorithm 31 , we need to compute a proper threshold vector over parameter set e. first of all we evaluate the efficiency of two well - known aggregation operators which are called the arithmetic average operator ( aao ) and the geometric average operator ( gao ) . based on the aggregation aao , the threshold vector over parameter set e is defined as ( 43)aao(e)={aao(ej ) ,  j=1,  ,m}iiiiiiiiiiii={h~(ej)aao ,  j=1,  ,m } , where ( 44)h~(ej)aao=1ni=1nh~f~(ej)(xi)iiiiiiiiiiiiiiiii={1ni=1nh~f~(ej)(1)(xi),  ,1ni=1nh~f~(ej)(lx)(xi)}. needless to say that the operator  on h~f~(ej)(k)(xi )  ( k=1,  ,lx ) is that defined on g - type fss . in this example , operator  considered on ifss is given for ifss h~f~(ej)(k)(xi)=(ik , j,ik , j ) ,  i=1,  ,n by   ( 45)1ni=1nh~f~(ej)(k)(xi):=(k , j,k , j)iiiiiiiiiiiiiiiiiiiiiiii=(1i=1n(1ik , j),i=1nik , j ) . furthermore , based on the aggregation gao , we have ( 46)h~(ej)gao  = ( i=1nh~f~(ej)(xi))1/n  = { ( i=1nh~f~(ej)(1)(xi))1/n,  ,(i=1nh~f~(ej)(lx)(xi))1/n  } , where for ifss h~f~(ej)(k)(xi)=(ik , j,ik , j ) ,  i=1,  ,n , we define ( 47)(i=1nh~f~(ej)(k)(xi))1/n:=(k , j,k , j)iiiiiiiiiiiiiiiiiiiiiiiiiiiiii=(i=1nik , j,1i=1n(1ik , j ) ) . assume that ( 48)h~(ej)(k) aao = 1ni=1nh~f~(ej)(k)(xi):=(k , j,k , j)iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=(1i=1n(1ik , j),i=1nik , j),h~(ej)(k) gao = ( i=1nh~f~(ej)(k)(xi))1/n:=(k , j,k , j)iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=(i=1nik , j,1i=1n(1ik , j ) ) . then , for any xi  x(49)i(h~(ej) aao , h~f~(ej)(xi))=0,i(h~(ej) gao , h~f~(ej)(xi))=1 , where i( ,  ) is the inclusion measure given by ( 38 ) . 
  assume that ( 48)h~(ej)(k) aao = 1ni=1nh~f~(ej)(k)(xi):=(k , j,k , j)iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=(1i=1n(1ik , j),i=1nik , j),h~(ej)(k) gao = ( i=1nh~f~(ej)(k)(xi))1/n:=(k , j,k , j)iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii=(i=1nik , j,1i=1n(1ik , j ) ) . then , for any xi  x(49)i(h~(ej) aao , h~f~(ej)(xi))=0,i(h~(ej) gao , h~f~(ej)(xi))=1 , where i( ,  ) is the inclusion measure given by ( 38 ) . 
 proofthe proof is straightforward from the fact that for any indices i , j , k(50)(ik , j,ik , j)(1i=1n(1ik , j),i=1nik , j),(ik , j,ik , j)(i=1nik , j,1i=1n(1ik , j ) ) . k(50)(ik , j,ik , j)(1i=1n(1ik , j),i=1nik , j),(ik , j,ik , j)(i=1nik , j,1i=1n(1ik , j ) ) . the above proposition demonstrates that the threshold vectors over parameter set e defined in accordance with the aggregations aao and gao are not efficient , because indifferent inclusion measure values for the whole alternatives xi  x do not allow us to have a promising judgment about optimal decision . in order to overcome adverse effect of indifferent inclusion measure values , we use , but we are not limited to , median operator   instead of aao and gao . let us denote any ifs h~f~(ej)(k)(xi ) by h~f~(ej)(k)(xi)=(ik , j,ik , j ) for 1  i  n , 1  j  m , 1  k  lx . , nonmembership degree ) of all alternatives in x is arranged in increasing order , and then x(i ) is referred to as the alternative with ith largest membership degree ( resp . , nonmembership degree ) . in this regards , we define ( 51)h~(ej)(k)med:=(medk , j,medk , j ) , where ( 52)medk , j={((m+1)/2)k , j , if  m  is  odd;12((m/2)k , j+(m/2 + 1)k , j),if m  is  even,medk , j={((m+1)/2)k , j ,  if  m  is  odd ;  12((m/2)k , j+(m/2 + 1)k , j ) ,  if  m  is  even . now , the threshold vector med(e ) can be obtained as represented in tabular form in table 2 . by applying the third step of algorithm 31 this paper proposed a more general extension of fuzzy soft set that not only encompasses the existing ones as special cases , but also can be applied to model those situations where the existing kinds of fuzzy soft set are not applicable . the proposed higher order hesitant fuzzy soft set provides us with a multicriteria decision making method to deal with problems involving parameters with different - dimensional levels . as future work , we consider the study of aggregation methods based on multiexperts ' higher order hesitant fuzzy soft matrix where some experts are invited to conduct evaluation for one decision making problem .