A Ranking Approach for Intuitionistic Fuzzy Numbers and its Application | Journal of Applied Research and Technology. JART

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In many actual problems, the cost, capacities, supplies and demand parameters may be imprecise. To deal quantitatively with imprecise information the concept and techniques of probability can be employed. There are articles discussing the network flow problems where arc parameters are random variable [<a class="elsevierStyleCrossRef" href="#bib0055">11</a>, <a class="elsevierStyleCrossRef" href="#bib0105">21</a>, <a class="elsevierStyleCrossRef" href="#bib0110">22</a>], however, in order to construct, probability distributions require either a priori predictable regularity or a posteriori frequency distribution. Moreover, the premise that imprecision can be equated with randomness is still questionable. As an alternative, uncertain values can be represented by membership functions of the fuzzy set theory <a class="elsevierStyleCrossRef" href="#bib0130">[26]</a>.The main advantages of methodologies based on fuzzy theory are that they do not require prior predictable regularities or posterior frequency distributions and they can deal with imprecise input information containing feelings and emotions quantified based on the decision-maker’s subjective judgment. From this point of view, Shih and Lee <a class="elsevierStyleCrossRef" href="#bib0115">[23]</a> proposed a fuzzy version of MCF problem using multilevel linear programming problem. But, they did not use the nice structure of network constraints.Ghatee and Hashemi <a class="elsevierStyleCrossRef" href="#bib0035">[7]</a> proposed a method to find the fuzzy optimal solution of balanced fully fuzzy MCF problems. Ghatee and Hashemi [<a class="elsevierStyleCrossRef" href="#bib0040">8</a>, <a class="elsevierStyleCrossRef" href="#bib0045">9</a>] and Ghatee et al. <a class="elsevierStyleCrossRef" href="#bib0050">[10]</a> applied the existing method <a class="elsevierStyleCrossRef" href="#bib0035">[7]</a> for solving real-life problems.The most common concept used in all those studies is ranking of fuzzy numbers. Ranking of fuzzy numbers is an important issue in the study of fuzzy set theory. In order to rank fuzzy numbers, one fuzzy number needs to be compared with the others but it is difficult to determine clearly which of them is larger or smaller. Numerous methods have been proposed in previous studies to rank fuzzy numbers. There is not a unique method for comparing fuzzy numbers.A membership function of a classical fuzzy set assigns to each element of the universe of discourse a number from the unit interval to indicate the degree of belongingness to the set under consideration. The degree of nondecision−maker’s belongingness is just automatically the complement to one of the membership degree. Nonetheless, a human being who expresses the <a name="p382"></a>degree of membership of a given element in a fuzzy set very often does not express the corresponding degree of nonmembership as the complement to 1. This reflects a well-known psychological fact that the linguistic negation do not always identifies with logical negation. Thus, Atanassov <a class="elsevierStyleCrossRef" href="#bib0010">[2]</a> introduced the concept of an IF set which is characterized by two functions expressing the degree of belongingness and the degree of nonbelongingness respectively. This idea, which is a natural generalization of usual fuzzy set, seems to be useful when modeling many real-life situations.Concerning ranking IF numbers some work has been reported in the literature. Grzegorzewski <a class="elsevierStyleCrossRef" href="#bib0060">[12]</a> defined two families of metrics in the space of IF numbers and proposed a method for comparing IF numbers based on these metrics. Mitchell <a class="elsevierStyleCrossRef" href="#bib0075">[15]</a> extended the natural ordering of real numbers to triangular intuitionistic fuzzy (TrIF) numbers by adopting a statistical view point and interpreting each IF number as ensemble of ordinary fuzzy numbers. Nayagam et al. <a class="elsevierStyleCrossRef" href="#bib0095">[19]</a> introduced TrIF numbers of special type and described a method to compare them. Although their ranking method appears to be attractive, the definition of TrIF number seems unrealistic. This is because the triangular nonmembership function is defined to geometrically behave in an identical manner as the membership function. Nan and Li <a class="elsevierStyleCrossRef" href="#bib0080">[16]</a> proposed a method for comparing TrIF number using lexicographic technique. Nehi <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a> proposed a new method for comparing IF numbers in which two characteristic values for IF numbers are defined by the integral of the inverse fuzzy membership and nonmembership functions multiplied by the grade with powered parameter. Almost in parallel, Li <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a> introduced a new definition of the TrIF number which has an appealing and logically reasonable interpretation. He defined two concepts of the value and the ambiguity of a TrIF number similar to those for a fuzzy number introduced by Delgado et al. <a class="elsevierStyleCrossRef" href="#bib0020">[4]</a>. Dubey and Mehra <a class="elsevierStyleCrossRef" href="#bib0025">[5]</a> defined a TrIF number which is more general than the one defined in [<a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRef" href="#bib0080">16</a>]. They extended the definitions of the value and the ambiguity index given by Li <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a> to the newly defined TrIF numbers and proposed an approach to handle linear programming problems with data as IF numbers.In this paper, the limitations of the existing methods [<a class="elsevierStyleCrossRef" href="#bib0025">5</a>, <a class="elsevierStyleCrossRef" href="#bib0060">12</a>, <a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRef" href="#bib0075">15</a>, <a class="elsevierStyleCrossRef" href="#bib0090">18</a>, <a class="elsevierStyleCrossRef" href="#bib0095">19</a>, <a class="elsevierStyleCrossRef" href="#bib0120">24</a>, <a class="elsevierStyleCrossRef" href="#bib0125">25</a>] as well as their shortcomings [<a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRef" href="#bib0085">17</a>, <a class="elsevierStyleCrossRef" href="#bib0100">20</a>], are pointed out. Furthermore, to overcome such limitations and shortcomings, a new ranking approach—by modifying an existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a>—is proposed for comparing IF numbers. Also—with the help of the projected ranking approach—a new method is proposed to find the optimal solution of such unbalanced MCF problems in which all the parameters are represented by IF numbers.This paper is organized as follows. In <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a>, some basic definitions and arithmetic operations of trapezoidal intuitionistic fuzzy (TIF) numbers are presented. In <a class="elsevierStyleCrossRef" href="#sec0025">Sections 3</a> and <a class="elsevierStyleCrossRef" href="#sec0030">4</a>, the limitations and shortcomings of the existing methods [<a class="elsevierStyleCrossRef" href="#bib0025">5</a>, <a class="elsevierStyleCrossRef" href="#bib0060">12</a>, <a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRef" href="#bib0075">15</a>, <a class="elsevierStyleCrossRef" href="#bib0090">18</a>, <a class="elsevierStyleCrossRef" href="#bib0095">19</a>, <a class="elsevierStyleCrossRef" href="#bib0120">24</a>, <a class="elsevierStyleCrossRef" href="#bib0125">25</a>] and [<a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRef" href="#bib0085">17</a>, <a class="elsevierStyleCrossRef" href="#bib0100">20</a>] respectively, are pointed out. In <a class="elsevierStyleCrossRef" href="#sec0035">Section 5</a>, a ranking approach is proposed for comparing IF numbers is presented. In <a class="elsevierStyleCrossRef" href="#sec0045">Section 6</a>, a method for solving MCF problems in IF environment is proposed and to illustrate the proposed method, a numerical example is solved. In <a class="elsevierStyleCrossRef" href="#sec0080">Section 7</a>, the results are compared. Finally, in <a class="elsevierStyleCrossRef" href="#sec0085">Section 8</a>, the conclusion is discussed.2PreliminariesIn this section, some basic definitions and arithmetic operations are presented <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a>.2.1Basic definitionsDefinition 2.1. An IF set Ã={x, μÃ(x), υÃ(x)) | x ∈ X} on the universal set X is characterized by a truth membership function μÃ, μÃ,: X → [0,1] and a false membership function υÃ,υÃ,: X → [0,1]. The values μÃ(x) and υÃ(x) represent the degree of membership and the degree of nonmembership for x ∈ X and always satisfies the condition μÃ(x)+υÃ(x)≤1 ∀ x∈X. The value (1−μÃ(x)−υÃ(x)) represents the degree of hesitation for x∈X.Definition 2.2. Let Ã be an IF set. Then, Ãα={x∈X | μÃ(x)≥α,υÃ(x)≤(1−α)} is said to be an α−cut of Ã.<a name="p383"></a>Definition 2.3. An IF set Ãα={(x, μÃ(x), υÃ(x) | x ∈ X} is called IF-normal, if there are at least two points x0, x1 ∈ X such that μÃ(x0)=1, υÃ(x1)=1.Definition 2.4. An IF set Ã={(x, μÃ(x),υÃ(x)) | x ∈ X} is called IF-convex, if ∀x1, x2 ∈ X, λ ∈ [0,1] μÃ(λx1+(1 − λ)x2)≥min(μÃ(x1), μÃ(x2))υÃ(λx1+(1−λ)x2)≤max(υÃ(x1),υÃ(x2))Definition 2.5. An IF set Ã={(x, μÃ(x),υÃ(x)) | x ∈ X} of the real line is called IF number if<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005">(a)Ã is IF-normal</li><li class="elsevierStyleListItem" id="lsti0010">(b)Ã is IF-convex</li><li class="elsevierStyleListItem" id="lsti0015">(c)μÃ is upper semi continuous andυÃ is lower semi continuous</li><li class="elsevierStyleListItem" id="lsti0020">(d)Ã={x∈X | υÃ(x)<1} is bounded</li></ul>Definition 2.6. An IF number Ã, defined on the universal set of real numbers R, denoted as Ã={a’1,a1,a’2,a2,a3,a’3,a4,a’4}, where a’1≤a1≤a’2≤a2≤a3≤a’3≤a4≤a’4 is said to be TIF number, if the degree of membership μÃ(x) and the degree of non-membership υÃ(x) are given by:<elsevierMultimedia ident="eq0005"></elsevierMultimedia><elsevierMultimedia ident="eq0010"></elsevierMultimedia>If in a TIF number Ã, let a2=a3 (and hence a’2=a’3), then it gives a TrIF number with parameters a’1≤a1≤a’2 (a2=a3=a’3)≤a4≤a’4 and is denoted by Ã={a’1,a1,a’2,a4,a’4}.<elsevierMultimedia ident="f0015"></elsevierMultimedia>Definition 2.7. Two TIF numbers Ã={a’1,a1,a’2,a2,a3,a’3,a4,a’4}, B˜={b'1,b1,b'2,b2,b3,b'3,b4,b'4} are said to be equal i.e., A˜=B˜ if and only if a’1=b’1, a1=b1, a’2=b’2, a2=b2, a3=b3, a’3=b’3, a4=b4 and a’4=b’4.Definition 2.8. A TIF number Ã={a’1,a1,a’2,a2,a3,a’3,a4,a’4} is said to be non-negative TIF number if and only if a’1≥0.Remark 1. In the existing methods <a class="elsevierStyleCrossRef" href="#bib0025">[5]</a> the TrIF number Ã={a’1,a1,a’2,a4,a’4} is represented by Ã={(a1,a’2,a4;1),(a’1,a’2,a’4;1)}.<a name="p384"></a>Remark 2. In the existing methods <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a> the TrIF number Ã=〈(a1,a2,a3);1,1〉 is used which is obtained by the TrIF number Ã={a’1,a1,a’2,a4,a’4} by assuming a’1=a1,a’2,a4=a’4.2.2Arithmetic operations between TIF numbersIn this section, some arithmetic operations between TIF numbers, defined on a universal set of real numbers R, are presented.(i) Let Ã={a’1,a1,a’2,a2,a3,a’3,a4,a’4} and B˜={b'1,b1,b'2,b2,b3,b'3,b4,b'4} be two TIF numbers. Then, A˜⊕B˜=(a'1+b'1,a1+b1,a'2+b'2,a2+b2,a3+b3,a'3+b'3,a4+b4,a'4+b'4)(ii) Let Ã={a’1,a1,a’2,a2,a3,a’3,a4,a’4} and B˜={b'1,b1,b'2,b2,b3,b'3,b4,b'4} be two TIF numbers. Then, A˜ΘB˜=(a'1-b'4,a1-b4,a'2-b'3,a2-b3,a3-b2,a'3-b'2,a4-b1,a'4-b'1}(iii) Let Ã={a’1,a1,a’2,a2,a3,a’3,a4,a’4} be any TIF number. Then,<elsevierMultimedia ident="eq0015"></elsevierMultimedia>(iv) Let Ã={a’1,a1,a’2,a2,a3,a’3,a4,a’4} and B˜={b'1,b1,b'2,b2,b3,b'3,b4,b'4} be two non-negative TIF numbers. Then, A˜⊗B˜=(a'1b'1,a1b1,a'2b'2,a2b2,a3b3,a'3b'3,a4b4,a'4b'4)3Limitations of the existing methodsIn this section the limitations of the existing methods [<a class="elsevierStyleCrossRef" href="#bib0025">5</a>, <a class="elsevierStyleCrossRef" href="#bib0060">12</a>, <a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRef" href="#bib0075">15</a>, <a class="elsevierStyleCrossRef" href="#bib0090">18</a>, <a class="elsevierStyleCrossRef" href="#bib0095">19</a>, <a class="elsevierStyleCrossRef" href="#bib0120">24</a>, <a class="elsevierStyleCrossRef" href="#bib0125">25</a>] for comparing IF numbers are pointed out.<ul class="elsevierStyleList" id="lis0010"><li class="elsevierStyleListItem" id="lsti0025">1The existing methods [<a class="elsevierStyleCrossRef" href="#bib0060">12</a>, <a class="elsevierStyleCrossRef" href="#bib0075">15</a>, <a class="elsevierStyleCrossRef" href="#bib0120">24</a>, <a class="elsevierStyleCrossRef" href="#bib0125">25</a>] can be used only for comparing IF set, however, none of the existing methods [<a class="elsevierStyleCrossRef" href="#bib0060">12</a>, <a class="elsevierStyleCrossRef" href="#bib0075">15</a>, <a class="elsevierStyleCrossRef" href="#bib0120">24</a>, <a class="elsevierStyleCrossRef" href="#bib0125">25</a>] can be used for comparing IF numbers.</li><li class="elsevierStyleListItem" id="lsti0030">2Nayagam and Sivaraman <a class="elsevierStyleCrossRef" href="#bib0090">[18]</a> pointed out the shortcomings of the existing method <a class="elsevierStyleCrossRef" href="#bib0095">[19]</a> and proposed a method for comparing such TrIF numbers {(a,b,c),(e,f,g)} for which either the conditionse≥b andf≥c or the conditionsf≤a andg≤b are satisfied, however, the existing method <a class="elsevierStyleCrossRef" href="#bib0090">[18]</a> cannot be used for comparing such TrIF numbers for which neither the conditionse≥b andf≥c nor the conditionsf≤a andg≤b are satisfied.</li><li class="elsevierStyleListItem" id="lsti0035">3Dubey and Mehra <a class="elsevierStyleCrossRef" href="#bib0025">[5]</a> pointed out the shortcomings of the existing methods <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a> and proposed a method for comparing such TrIF sets a˜={(a_;μ,a,a¯μ;Wa~),(a_;υ,a,a¯υ;ua~)} for which the condition Vμ(ã)≤Vυ(ã) is satisfied, where Vμa˜=((a_;μ+4a+a¯μ)wa˜)/6 and Vυa˜=((a_;υ+4a+a¯υ)(1−ua˜))/6, however, the existing method <a class="elsevierStyleCrossRef" href="#bib0090">[18]</a> cannot be used for comparing such TrIF sets for which the condition Vμ(ã)≤Vυ(ã) is not satisfied.</li></ul>4Shortcomings of the existing methodsIn this section the shortcomings of the existing methods [<a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRef" href="#bib0085">17</a>, <a class="elsevierStyleCrossRef" href="#bib0095">19</a>, <a class="elsevierStyleCrossRef" href="#bib0100">20</a>] for comparing IF numbers are pointed out.<ul class="elsevierStyleList" id="lis0015"><li class="elsevierStyleListItem" id="lsti0040">1Li <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a> proposed the following method for comparing TrIF sets. LetÃ=〈(a1,a2,a3;w1,u1)〉 and B˜=〈(a1,a2,a3;w2,u2)〉 be two TrIF sets.<ul class="elsevierStyleList" id="lis0020"><li class="elsevierStyleListItem" id="lsti0045">(i)A˜<˜B˜ifLTλ(A˜)<LTλ(B˜)</li><li class="elsevierStyleListItem" id="lsti0050">(ii)A˜>˜B˜ifLTλ(A˜)>LTλ(B˜)</li><li class="elsevierStyleListItem" id="lsti0055">(iii)A˜≅B˜ifLTλ(A˜)=LTλ(B˜)</li></ul></li></ul>Where<elsevierMultimedia ident="eq0020"></elsevierMultimedia><a name="p385"></a>It is not genuine to apply this method due to the following reasons. It is obvious from the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a> that if (a1+4a2+a3) ≠ 0 then the comparison of Ã and B˜ will depend upon the values of w1,u1,w2,u2 and if (a1+4a2+a3)=0 then A˜≅B˜ for all values of w1u1,w2 and u2 i.e., according to the existing approach <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a>, in the first case comparison of TrIF sets depends upon the degree of membership and the degree of nonmembership of IF sets whereas in the second case comparison of TrIF sets does not depend upon the degree of membership and the degree of nonmembership of IF sets which is a contradiction.Example 4.1. Let Ã=〈(1,1,1);w1,u1〉 and B˜=〈(1,1,1);w2,u2〉 be two TrIF sets then according to the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a> values of Ã and B˜ will depend upon the values of w1,u1,w2 and u2 i.e., the ordering of Ã and B˜ will depend upon the values of w1,u1,w2 and u2.Example 4.2. Let Ã=〈(−8,1,4);w1,u1〉 and B˜=〈(-8,1,4);w2,u2〉 be two TrIF sets. Then, according to the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0065">[13]</a>, LTλ(A˜)= LTλ(B˜)=0⇒A˜≅B˜ i.e., in this case the ordering of Ã and B˜ is independent from the values of w1,u1,w2 and u2.<ul class="elsevierStyleList" id="lis0025"><li class="elsevierStyleListItem" id="lsti0060">2Nayagam and Sivaraman <a class="elsevierStyleCrossRef" href="#bib0085">[17]</a> proposed the following method for comparing interval valued IF sets:LetÃ={x : [a1,b1],[c1,d1] |x ∈X} and B˜={x:[a2,b2],[c2,d2]|x∈X} be two interval valued IF sets defined on a universal setX. Then<ul class="elsevierStyleList" id="lis0030"><li class="elsevierStyleListItem" id="lsti0065">(i)A˜<˜B˜ifLG(A˜)<LG(B˜)</li><li class="elsevierStyleListItem" id="lsti0070">(ii)A˜>˜B˜ifLG(A˜)>LG(B˜)</li><li class="elsevierStyleListItem" id="lsti0075">(iii)A˜≅B˜ifLG(A˜)=LG(B˜)</li></ul>where LG(A˜)=(a1+b1)(1-δ)+δ(2-(c1+d1)2<elsevierMultimedia ident="eq0025"></elsevierMultimedia>and δ ∈ [0,1].Nayagam and Sivaraman <a class="elsevierStyleCrossRef" href="#bib0085">[17]</a> have used the same method for comparing TIF numbers<elsevierMultimedia ident="eq0030"></elsevierMultimedia>Nonetheless, it is not genuine to use this method for comparing TIF numbers due to the following reasonsIn the IF setÃ={x : [a1,b1],[c1,d1] |x ∈X}a1,b1 andc1,d1 represents the infimum and supremum values of membership degree and nonmembership degree corresponding to pointsX. Although, in the TIF numberÃ={(a1,a1,b1,b1),(c1,c1,d1,d1)}a1 andb1 represent those points of universal setX corresponding to which the membership degree is 1. Similarly,c1 andd1 represent those points of universal setX corresponding to which the nonmembership degree is 1.</li><li class="elsevierStyleListItem" id="lsti0080">3Given that, IF numbers are the generalization of fuzzy numbers, hence, the approach which can be used for comparing IF numbers can also be used for comparing fuzzy numbers. To show the shortcomings of the existing approach <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a>, two fuzzy numbers are compared by using such approaches [<a class="elsevierStyleCrossRef" href="#bib0030">6</a>,<a class="elsevierStyleCrossRef" href="#bib0070">14</a>,<a class="elsevierStyleCrossRef" href="#bib0100">20</a>] and it was demonstrated that the ordering of fuzzy numbers obtained by using Nehi’s approach <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a> and Liou and Wang approach <a class="elsevierStyleCrossRef" href="#bib0070">[14]</a> is the same one and contradicts the ordering of fuzzy numbers obtained by using the Garcia and Lamata’s approach<a class="elsevierStyleCrossRef" href="#bib0030">[6]</a>, because, Garcia and Lamata’s approach <a class="elsevierStyleCrossRef" href="#bib0030">[6]</a> pointed out that their approach is better than Liou and Wang’s approach <a class="elsevierStyleCrossRef" href="#bib0070">[14]</a>. Thus, the ordering of IF numbers given by Nehi’s is not genuine.<a name="p386"></a></li></ul>Example 4.3. Let us consider two trapezoidal fuzzy numbers Ã=(1,4,4,5) and B˜=(2,3,3,6). The intuitionistic representation of these trapezoidal fuzzy numbers is Ã=(1,1,4,4,4,4,5,5) and B˜=(2,2,3,3,3,3,6,6).The ordering of these numbers obtained by using the existing ranking approaches [<a class="elsevierStyleCrossRef" href="#bib0030">6</a>, <a class="elsevierStyleCrossRef" href="#bib0070">14</a>, <a class="elsevierStyleCrossRef" href="#bib0100">20</a>] are shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>.<elsevierMultimedia ident="tbl0005"></elsevierMultimedia>It is obvious from the results, shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a> that the ordering of fuzzy numbers Ã and B˜, obtained by using the existing approach <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a>, is the same as the one obtained by the existing approach <a class="elsevierStyleCrossRef" href="#bib0070">[14]</a> whereas it is different from the ordering obtained by using the existing approach <a class="elsevierStyleCrossRef" href="#bib0030">[6]</a>.5Proposed ranking approach for comparing two IF numbersGarcia and Lamata <a class="elsevierStyleCrossRef" href="#bib0030">[6]</a> pointed out that although trapezoidal fuzzy number Ã=(a1,a2,a3,a4), is defined by four points, in the existing formula <a class="elsevierStyleCrossRef" href="#bib0070">[14]</a>, ℜA˜=λ∫01(a1+a2−a1α+1−λ∫01a4+a3−a4α) where, λ ∈ [0,1] the central points a2 and a3 are taken into account in an indirect way and proposed the following modified formula ℜA˜=β[∫01(λa2+(1−λ)a3)dα]+1−β[λ∫01(a1+(a2−a1)α)dα+(1−λ)∫01(a4+(a3−a4)α)dα]). Because in the existing formula <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a>, CμkA˜=k+12∫01rk[a1+(a2−a1)r+a4+(a3−a4)r]dr, CνkA˜=k+12∫01rk[a'1+(a'2−a'1)r+a'4+(a'3−a'4)r]dr where, k ∈ [0,8) the central points a2, a3 and a’2, a’3 are also taken into account in an indirect way hence, to overcome the shortcomings of existing approach <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a>, pointed out in <a class="elsevierStyleCrossRef" href="#sec0075">Section 4</a>, a new ranking approach, by modifying the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a>, is proposed for comparing IF numbers Ã=(a’1,a1,a’2,a2, a3, a’3, a4,a’4) and B˜=(b'1,b1,b'2,b2,b3,b'3,b4,b'4).The steps of the proposed ranking approach are as follows:Step 1. Calculate, Mμβ,kA˜=(β[k+12∫01rk(a2+a3)dr]+(1−β)[k+12∫00rk[a1+(a2−a1)r+a4+(a3−a4)r]dr]),Mμβ,kB˜=(β[k+12∫01rk(b2+b3)dr]+(1−β)[k+12∫00rk[b1+(b2−b1)r+b4+(b3−b4)r]dr]), and check Mμβ,k(A˜)>Mμβ,k(B˜) or Mμβ,k(A˜)<Mμβ,k(B˜) or Mμβ,k(A˜)=Mμβ,k(B˜).Case (i) If Mμβ,k(A˜)>Mμβ,k(B˜) then A˜≻B˜ i.e., minimum (A˜,B˜)=B˜.Case (ii) If Mμβ,k(A˜)<Mμβ,k(B˜) then A˜≺B˜ i.e., minimum (A˜,B˜)=A˜.Case (iii) If Mμβ,k(A˜)=Mμβ,k(B˜) then go to step 2.Step 2. Calculate,<a name="p387"></a>Mνβ,kA˜=βk+12 ∫01rka'2+a'3dr+1−βK+12 ∫00rka'1+a'2−a'1r+a'4+a'3−a'4rdrMνβ,kB˜=βk+12 ∫01rkb'2+b'3dr+1−βK+12 ∫00rkb'1+b'2−b'1r+b'4+b'3−b'4rdr, and check -Mυβ,k(A˜)>-Mυβ,k(B˜) or -Mυβ,k(A˜)<-Mυβ,k(B˜) or -Mυβ,k(A˜)=-Mυβ,k(B˜).Case (i) If -Mυβ,k(A˜)>-Mυβ,k(B˜) then A˜≻B˜ i.e., minimum (A˜,B˜)=B˜.Case (ii) If -Mυβ,k(A˜)<-Mυβ,k(B˜) then A˜≺B˜ i.e., minimum (A˜,B˜)=A˜.Case (iii) If Mυβ,k(A˜)=Mυβ,k(B˜) then A˜≅B˜.Remark 3. For β=1/3 and k=Q the index for membership and nonmembership functions Mμ1/3,0(A˜i) and Mυ1/3,0(A˜i), are as follows:<elsevierMultimedia ident="eq0035"></elsevierMultimedia>5.1Advantages of the proposed ranking approachIn this section, the advantages of the proposed ranking approach, over the existing ranking approaches for comparing fuzzy numbers as well as IF numbers [<a class="elsevierStyleCrossRef" href="#bib0025">5</a>, <a class="elsevierStyleCrossRef" href="#bib0030">6</a>, <a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRef" href="#bib0090">18</a>-<a class="elsevierStyleCrossRef" href="#bib0100">20</a>] are discussed.<ul class="elsevierStyleList" id="lis0040"><li class="elsevierStyleListItem" id="lsti0085">1The existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0030">[6]</a> can be used only for comparing fuzzy numbers but cannot be used for comparing IF numbers. Whereas the proposed ranking approach can be used for comparing fuzzy numbers as well as IF numbers and the ordering of fuzzy numbers, obtained by using the proposed ranking approach, is the same as that one obtained by using the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0030">[6]</a>.</li><li class="elsevierStyleListItem" id="lsti0090">2Although the existing ranking approaches [<a class="elsevierStyleCrossRef" href="#bib0025">5</a>, <a class="elsevierStyleCrossRef" href="#bib0065">13</a>, <a class="elsevierStyleCrossRefs" href="#bib0085">17-20</a>] can be used for comparing fuzzy numbers as well as IF numbers, due to shortcomings pointed out in <a class="elsevierStyleCrossRef" href="#sec0030">Section 4</a>, it is not genuine to apply these approaches. Whereas by using the proposed ranking approach, all the shortcomings, occurring in the results due to the use of existing approaches [<a class="elsevierStyleCrossRef" href="#bib0025">5</a>,<a class="elsevierStyleCrossRef" href="#bib0065">13</a>,<a class="elsevierStyleCrossRefs" href="#bib0085">17-20</a>], are resolved.</li></ul>6MCF problems in IF environmentThe aim of the MCF problems is to find minimum cost for transporting the product from a given point to certain destinations with the assumption that there is at least one node, called intermediate node, at which the product can be stored in the case of excess availability of the product and the stored product can be supplied in the case of excess demand of the product.Ghatee and Hashemi <a class="elsevierStyleCrossRef" href="#bib0035">[7]</a> modified the existing linear programming formulation of balanced MCF problems by representing all of its parameters as fuzzy numbers instead of crisp numbers and proposed a method to find the fuzzy optimal solution of such balanced MCF problems.On the same direction, in this section the existing linear programming formulation of MCF problems is further generalized by representing all the parameters as TIF numbers and on the basis of the ranking approach, proposed in <a class="elsevierStyleCrossRef" href="#sec0035">Section 5</a>, a new method is proposed to find the IF optimal solution of such MCF problems.6.1Linear programming formulations of balanced MCF problems in crisp and IF environment6.1.1In this subsection, the linear programming formulation of balanced MCF problems in crisp environment is presented <a class="elsevierStyleCrossRef" href="#bib0005">[1]</a>.If the set of source nodes, destination nodes, such intermediate nodes from which some quantity of the product is supplied, such intermediate nodes at which some quantity of the product is stored and such intermediate nodes from which the product is neither supplied nor stored, of a directed and connected network, are represented by NS,ND, NIS,NID and NIT respectively. Then, the MCF problem can be formulated as<a name="p388"></a>Minimize ∑(i,j)∈A(cijxij)Subject to<elsevierMultimedia ident="eq0040"></elsevierMultimedia>Subject toWhere A : The set of arcs (i, y), xij : Decision variable denoting the transportation amount of the product from ith node to jth node, Cij : Cost for transporting one unit quantity of the product from ith node to jth node, ai, : Supply of the product at ith source node, ei, : Supply of the product at ith intermediate source node, bj Demand of the product at jth destination node, dj : Demand of the product at jth intermediate destination node, Iij: Minimum quantity of the product that can be transported from ith node to jth node, uij: Maximum quantity of the product that can be transported from ith node to jth node.6.1.2Linear programming formulation of balanced MCF problems in IF environmentIn this section the linear programming formulation of balanced MCF problems in IF environment is presented.Suppose all the parameters cij,xij,ai,ei,bj,dj,lij,uij are represented by non-negative TIF numbers c˜ij,x˜ij,a˜i,e˜i,b˜j,d˜j,l˜ij,u˜ij. Then (R1) in IF environment is:inimize ∑(i,j)∈A(c˜ij⊗x˜ij)Subject to<elsevierMultimedia ident="eq0045"></elsevierMultimedia>6.2Proposed methodIn this section, a new method is proposed for finding the exact IF optimal solution of such MCF problems in which all the parameters are represented by IF numbers. If the supply of product at ith source node and ith intermediate source node are a˜i=(a'1i,a1i,a'2i,a2i,a3i,a'3i,a4i,a'4i) and e˜i=(e'1i,e1i,e'2i,e2i,e3i,e'3i,e4i,e'4i) respectively and the demand of the product at jth destination node and jth intermediate destination node are b˜j=(b'1j,b1j,b'2j,b2j,b3j,b'3j,b4j,b'4j) and d˜j=(d'1j,d1j,d'2j,d2j,d3j,d'3j,d4j,d'4j) respectively, then the exact IF optimal solution of IF MCF problems can be obtained by using the following stepsStep 1. Find the total fuzzy supply ∑i∈NSa˜i⊕∑i∈NISe˜i and the total IF demand ∑j∈NDb˜j⊕∑j∈NIDd˜j. Let ∑i∈NSa˜i⊕∑i∈NISe˜i=(m'1,m1,m'2,m2,m3,m'3,m4,m'4) and ∑j∈NDb˜j⊕∑j∈NIDd˜j=(n'1,n1,n'2,n2,n3,n'3,n4,n'4).<a name="p389"></a>Examine whether the problem is balanced or not, i.e., ∑i∈NSa˜i⊕∑i∈NISe˜i=∑j∈NDb˜j⊕∑j∈NIDd˜j or ∑i∈NSa˜i⊕∑i∈NISe˜i≠∑j∈NDb˜j⊕∑j∈NIDd˜j.Case (1) If the problem is balanced i.e., ∑i∈NSa˜i⊕∑i∈NISe˜i=∑j∈NDb˜j⊕∑j∈NIDd˜j, then go to step 2.Case (2) If the problem is unbalanced i.e., ∑i∈NSa˜i⊕∑i∈NISe˜i≠∑j∈NDb˜j⊕∑j∈NIDd˜j tnen convert the unbalanced problem into balanced problem as followsCase (2a) Introduce a dummy source node with IF supply (max{0, (n’1–m’1)}, max{0, (n’1–m’1}+max{0,(n’1– n’1) – (m1–m’1)},max{0,(n’1–m’1)}+max{0,(n1–n’1) – (m1 – m’1)}+max{0, (n’2–n1) – (m’2–m1)},max {0,(n’1–m’1)}+max{0,(n1–n’1)–(m1–m’1)}+max{0, (n’2–n1)–(m’2–m1)}+max{0,(n2–n’2)–(m2–m’2)}, max{0,(n’1–m’1)}+max{0,(n1–n’1)–(m1–m’1)}+max {0,(n’2–n1)–(m’2–m1)}+max{0,(n2–n’2)–(m2–m’2)}+{0, (n3– n2) – (m3 – m2)}, max{0, (n’1–m’1)}+max{ 0,(n1–n’1)–(m1, –m’1)}+max{0,(n’2–n1)–(m’2–m1)}+max{0,(n2 –n’2)–(m2–m’2)}+max{0,(n3 –n2)–(m3 –m2)}+max{0,(n’3–n3)– (m’3–m3)},max{0,(n’1– m’1)}+max{0,(n1 –n’1)–(m1 –m’1)}+max{0,(n’2–n’1)–(m’2–m1)}+max{0,(n2–n’2)–(m2–m’2)}+max {0,(n3–n2)–(m3–m2)}+max{0,(n’3–n3)–(m’3–m3)}+max{0, (n4– n’3) – (m4 – m’3)}, max{0, (n’1–m’1)}+max{0,(n1 –n’1)–(m1, –m’1)}+max{0,(n’2–n’1)–(m’2–m1)}+max{0,(n2–n’2)– (m2–m’2)}+max{0,(n3 –n2)–(m3–m2)}+max{0,(n’3–n3)–(m’3–m3)}+max{0,(n4 –n’3) – (m4 –m’3)}+max{0,(n’4–n4)–((m’4–m4)}) and also introduced a dummy destination node with IF demand(max{0,(m’1–n’1)}, max{0,(m’1–n’1}+ max{0, (m1–m’1) – (n1–n’1)}, max{0,(m’1–n’1)}+max{0,(m1–m’1)–(n1 – n’1)}+max{0, (m’2–m1) – (n’2–n1)},max{0,(m’1–n’1)}+max {0,(m1 –m’1) – (n1 –n’1)} +max{0, (m’2–m1) – (n’2 – n1)}+max{0,(m2– m’2) – (n2–n’2)},max{0,(m’1–n’1)}+max{0,(m1–m’1)–(n1–n’1)}+max{0,(m’2 –m1) – (n’2 –n1)}+max{0, (m2 – m’2) – (n2– n’2)+max{0, (m3–m2) – (n3– n2)},max{0,(m’1–n’1)}+max{0,(m1 – m’1) – (n1 – n’1)}+max{0, (m’2 –m1) – (n’2 –n1)}+max {0,(m2 –m’2) – (n2 –n’2)}+max{0,(m3–m2)–(n3– n2)}+ max{0, (m’3–m3) – (n’3–n3)}, max{0,(m’1 –n’1)}+max{0,(m1–m’1)–(n1–n’1)}+max{0,(m’2 –m’1) – (n’2 –n1}+max{ 0,(m2 –m’2) – (n2–n’2)}+max {0,(m3 –m2)–(n3 –n2)}+max{0,(m’3 –m3) – (n’3 –n3)}+max{0,(m4 –m’3) – (n4–n’3)},max{0,(m’1–n’1)}+max{0,(m1 –m’1) – (n1 –n’1)}+max{0,(m’2–m’1) – (n’2–n1)}+max{0,(m2 –m’2) – (n2 –n’2)}+max{0,(m3–m2) – (n3 –n2)}+max{0,(m’3–m3) – (n’3–n3)}+max {0,(m4 –m’3) – (n4–n’3)}+max{0,(m’4–m4) – (n’4– n4)}). Assume the IF cost for transporting one unit quantity of the product from the introduced dummy source node to all intermediate nodes, existing destination nodes and introduced dummy destination node as zero IF number. Similarly, assume the IF cost for transporting one unit quantity of the product from all intermediate nodes, existing source nodes and introduced dummy source node to the introduced dummy destination node as zero IF number and go to Step 2.Step 2 Assuming c˜ij=(c'1ij,c1ij,c'2ij,c2ij,c3ij,c'3ij,c4ij,c'4ij),x˜ij=(x'1ij,x1ij,x'2ij,x2ij,x3ij,x'3ij,x4ij,x'4ij),a˜i=(a'1i,a1i,a'2i,a2i,a3i,a'3i,a4i,a'4i),e˜i=(e'1i,e1i,e'2i,e2i,e3i,e'3i,e4i,e'4i),b˜j=(b'1j,b1j,b'2j,b2j,b3j,b'3j,b4j,b'4j),d˜j=(d'1j,d1j,d'2j,d2j,d3j,d'3j,d4j,d'4j),,l˜ij=(l'1ij,l1ij,l'2ij,l2ij,l3ij,l'3ij,l4ij,l'4ij) u˜ij=(u'1ij,u1ij,u'2ij,u2ij,u3ij,u'3ij,u4ij,u'4ij) using the arithmetic operations of IF numbers, defined in Sub<a class="elsevierStyleCrossRef" href="#sec0020">section 2.2</a> and using the IF linear programming formulation (P2) of the balanced IF MCF problem, the IF linear programming formulation of balanced IF MCF problem obtained in Step 1 can be written as Minimize ∑(i,j)∈A(c'1ijx'1ij,c1ijx1ij,c'2ijx'2ij,c2ijx2ij,c3ijx3ij,c'3ijx'3ij,c4ijx4ij,c'4ijx'4ij),Subject to<a name="p390"></a><elsevierMultimedia ident="eq0050"></elsevierMultimedia>is a non-negative IF number ∀(i,j) ∈ AStep 3. Using Definition 2.7 and Definition 2.8, the IF linear programming problem (P3), can be written asMinimize ∑(i,j)∈A(c'1ijx'1ij,c1ijx1ij,c'2ijx'2ij,c2ijx2ij,c3ijx3ij,c'3ijx'3ij,c4ijx4ij,c'4ijx'4ij)Subject to<elsevierMultimedia ident="eq0055"></elsevierMultimedia><a name="p391"></a><elsevierMultimedia ident="eq0060"></elsevierMultimedia>Step 4. Suppose the IF linear programming problem (P4) have f basic feasible solutions and {(x'1ij)w,(x1ij)w,(x'2ij)w,(x2ij)w,(x3ij)w,(x'3ij)w,(x4ij)w,(x'4ij)w} be the wth basic feasible solution then the goal is to find such a basic feasible solution corresponding to which the value of the objective function is minimum i.e.,minimum(∑(i,j)∈A(c′1ij(x′1ij)w,c1ij(x1ij)w,c′2ij(x′2ij)w,c2ij(x2ij)w,c3ij(x3ij)w,c′3ij(x′3ij)w,c4ij(x4ij)w,c′4ij(x′4ij)w) which can be obtained by using the ranking approach proposed in <a class="elsevierStyleCrossRef" href="#sec0035">Section 5</a> i.e., the IF optimal solution of the IF linear programming problem, (P4), can be obtained by solving the following crisp linear programming problem<elsevierMultimedia ident="eq0065"></elsevierMultimedia>Subject to<elsevierMultimedia ident="eq0070"></elsevierMultimedia>As well as all the constraints of problem (P4) except (C1)Case (i) If there does not exist any alternative optimal solution then put the values of x'1ij,x1ij,x'2ij,x2ij,x3ij,x'3ij,x4ij and x'4ij in x˜ij=(x'1ij,x1ij,x'2ij,x2ij,x3ij,x'3ij,x4ij,x'4ij) to find the IF optimal solution {x˜ij} and find the IF optimal value ∑(i,j)∈A(c˜ij⊗x˜ij) by putting the value of x˜ij.Case (ii) If alternative solution exist then go to Step 5.Step 5. Solve the crisp linear programming problem (P6) to find the optimal solution {x'1ij,x1ij,x'2ij,x2ij,x3ij,x'3ij,x3ij,x'4ij}.<elsevierMultimedia ident="eq0075"></elsevierMultimedia>Subject to<elsevierMultimedia ident="eq0080"></elsevierMultimedia>As well as all the constraints of problem (P4) except (C1) where, a is the optimal value of the crisp linear programming problem (P5).Step 6. Put the values of x'1ij,x1ij,x'2ij,x2ij,x3ij,x'3ij,x4ij and x'4ij in x˜ij=(x'1ij,x1ij,x'2ij,x2ij,x3ij,x'2ij,x2ij,x3ij,x'3ij,x4ij,x'4ij) to find the IF optimal solution {x˜ij}.Step 7.Put the values of x˜ij, obtained from step 6, in∑(i,j)∈A(c˜ij⊗x˜ij), to find the minimum total IF transportation cost.<a name="p392"></a>6.3Advantages of the proposed methodThe existing method <a class="elsevierStyleCrossRef" href="#bib0035">[7]</a> can be used only to find the optimal solution of such MCF problems in which the parameters are represented by fuzzy numbers, however, the existing method <a class="elsevierStyleCrossRef" href="#bib0035">[7]</a> cannot be used to find the optimal solution for the same type of problems in which the parameters are represented by IF numbers, whereas the proposed method can be used to find the optimal solution for both types of problems.6.4Illustrative exampleIn this section, to illustrate the proposed method the IF MCF problem, chosen in Example 6.1, cannot be solved by the existing method <a class="elsevierStyleCrossRef" href="#bib0035">[7]</a>, it can be solved by using the proposed method.Example 6.1. Find the minimum IF transportation cost for the IF MCF problem with three nodes and three arcs as shown in <a class="elsevierStyleCrossRef" href="#f0005">Figure 2</a>. The IF data for the chosen IF MCF problem is summarized in <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>.<elsevierMultimedia ident="f0005"></elsevierMultimedia><elsevierMultimedia ident="tbl0010"></elsevierMultimedia>Solution. The optimal solution of IF MCF problem, chosen in Example 6.1, was obtained as followsStep 1. Total IF supply=(50,100,150,200,250,270,300,350) and total |F demand=(0,50,100,150,250,270,300,400). Given that the Total IF supply ≠ total IF demand, hence, it is an unbalanced IF MCF problem. Now, as described in the proposed method (using Case (2a) of Step 1 of the proposed method), the unbalanced IF MCF problem can be converted into a balanced IF MCF problem, by introducing a dummy source node 4 with IF supply (0,0,0,0,50,50,50,100) and a dummy destination node 5 with IF demand (50,50,50,50,50,50,50,50) as shown in <a class="elsevierStyleCrossRef" href="#f0010">Figure 3</a>, hence, that total IF supply=total IF demand i.e., (50,100,150,200,250,270,300,350)⊕(0,0,0,0, 50,50,50,100)=(0,50,100,150,250,270,300,400)⊕ (50,50,50,50,50,50,50,50).<elsevierMultimedia ident="f0010"></elsevierMultimedia>Assuming the IF cost for transporting one unit quantity of the product from the introduced dummy source node 4 to the intermediate nodes 2 and existing destination node 3 and introduced dummy destination node 5 as zero IF number. Similarly, assuming the IF cost for transporting one unit quantity of the product from intermediate nodes 2 and existing source node 1 and introduced dummy source node 4 to the introduced dummy destination node 5 as zero IF number i.e.,c˜42=c˜43=c˜45=c˜15=c˜25=(0,0,0,0,0,0,0,0) the IF linear programming formulation of balanced IF MCF problem, can be written asMinimize((5,10,50,100,1000,5500,10000,15000)⊗x˜12 ⊕(20,40,200,400,4000,22000,40000,60000)⊗ x˜13 ⊕(15,30,150,300,3000,16500,30000,45000) ⊗ x˜23 ⊕(0, 0,0,0,0,0,0,0) ⊗ x˜42 ⊕(0,0,0,0,0,0,0,0) ⊗ x˜43 ⊕(0,0,0,0,0, <a name="p393"></a>0,0,0) ⊗ x˜45 ⊕(0,0,0,0,0,0,0,0) ⊗ x˜15 ⊕(0,0,0,0,0,0,0,0) ⊗ x˜25)Subject to<elsevierMultimedia ident="eq0085"></elsevierMultimedia>x˜12,x˜13,x˜15,x˜23,x˜25,x˜42,x˜43,x˜45 are non-negative IF numbers.Step 2. Assuming x˜ij=(x'1ij,x1ij,x'2ij,x2ij,x3ij,x'3ij,x4ij,x'4ij) and using the arithmetic operations, the IF linear programming problem (P7) can be written as<elsevierMultimedia ident="eq0090"></elsevierMultimedia>Subject to<elsevierMultimedia ident="eq0095"></elsevierMultimedia>x˜12,x˜13,x˜15,x˜23,x˜25,x˜42,x˜43,x˜45 are non-negative IF numbers.Step 3. Using the Definition 2.7 and Definition 2.8, the IF linear programming problem (P8), can be written as<elsevierMultimedia ident="eq0100"></elsevierMultimedia><a name="p394"></a>Subject to<elsevierMultimedia ident="eq0105"></elsevierMultimedia><elsevierMultimedia ident="eq0110"></elsevierMultimedia>Step 4. Using Step 4 of the proposed method and assuming β=1/3,k=0, the IF optimal solution of the IF linear programming problem (P9), can be obtained by solving the following crisp linear programming problem<elsevierMultimedia ident="eq0115"></elsevierMultimedia>Subject to<elsevierMultimedia ident="eq0120"></elsevierMultimedia>As well as all the constraints of the problem (P9) except (C2)’Given that on solving the crisp linear programming problem (P10), alternative optimal solutions exist i.e., case (ii) of Step 4 of the proposed method is satisfied and the optimal value of the crisp linear programming problem (P10) is 4761400/6 hence, by using Step 5 of the proposed method the IF optimal solution of the chosen IF MCF problem can be obtained by solving the following crisp linear programming problem:<elsevierMultimedia ident="eq0125"></elsevierMultimedia><a name="p395"></a>Subject to<elsevierMultimedia ident="eq0130"></elsevierMultimedia>As well as all the constraints of problem (P9) except (C2)’Step 5. Solving the crisp linear programming problem, obtained in Step 4, the optimal values of x112,x'212,x212,x312,x'312,x412,x'412,x'213,x213,x313,x'313,x413,x'413,x'115,x115,x'215,x215,x'315,x415,x'415,x123,x'223,x223,x323,x'323,x423,x'423,x343,x'343,x'443 and x'443 are 50,90,130,180,200,230,280,10,20,20,20,20,20,50,50,50,50,50,50,50,50,30,30,30,30,30,30,30,50,50,50,and 100 respectively.Step 6. Put the values of x'1ij,x1ij,x'2ij,x2ij,x3ij,x'3ij,x4ij and x'4ij in x˜ij=(x'1ij,x1ij,x'2ij,x2ij,x3ij,x'2ij,x2ij,x3ij,x'3ij,x4ij,x'4ij) the IF optimal solution is x˜12=(0,50,90,130,180,200,230,280), x˜13=(0,0, 10,20,20,20,20,20), x˜15=(50,50,50,50,50,50,50,50), x˜23=(0,30,30,30,30,30,30,30), x˜43=(0,0,0,0,50,50,50,100) and the remaining x˜ij are ((0,0,0,0,0,0,0,0).Step 7. Putting the values of x˜ij=(x'1ij,x1ij,x'2ij,x2ij,x3ij,x'2ij,x2ij,x3ij,x'3ij,x4ij,x'4ij) in ((5,10,50,100,1000,5500,10000,15000) ⊗ x˜12 ⊕(20,40,200,400, 4000,22000,40000,60000) ⊗ x˜13 ⊕ (15,30,150,300, 3000,16500,30000,45000) ⊗ x˜23 ⊕ (0,0,0,0,0,0,0,0) ⊗ x˜42 ⊕ (0,0,0,0,0,0,0,0) ⊗ x˜43 ⊕ (0,0,0,0,0,0,0,0) ⊗ x˜45 ⊕ (0,0,0,0,0,0,0,0) ⊗ x˜15 ⊕ (0,0,0,0,0,0,0,0) ⊗ x˜25) the minimum total IF transportation cost is (0,1400,11000,20000,350000,2035000,4000000,6750000)7Comparative studyThe results of the existing fuzzy MCF problem (Example 3.5 <a class="elsevierStyleCrossRef" href="#bib0040">[8]</a>, pp. 2498) and the IF MCF problem, chosen in Example 6.1, obtained by using the existing method and the method proposed in Sub<a class="elsevierStyleCrossRef" href="#sec0065">section 6.2</a>, are shown in <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a>.<elsevierMultimedia ident="tbl0015"></elsevierMultimedia>Given that in the existing MCF problem (Example 3.5 <a class="elsevierStyleCrossRef" href="#bib0040">[8]</a>, pp. 2498) all the parameters are represented by fuzzy numbers hence, as discussed in Sub<a class="elsevierStyleCrossRef" href="#sec0070">section 6.3</a>, it can be solved by using the existing method <a class="elsevierStyleCrossRef" href="#bib0040">[8]</a> as well as by the proposed method, whereas in the MCF problem, chosen in Example 6.1, all the parameters are represented by IF numbers. Thus, as discussed in Sub<a class="elsevierStyleCrossRef" href="#sec0070">section 6.3</a>, it can be solved by the proposed method but cannot be solved by using the existing method <a class="elsevierStyleCrossRef" href="#bib0040">[8]</a>.8ConclusionBased on the proposed study, it can be concluded that it is better to use the proposed ranking approach for comparing fuzzy and IF numbers as compared to existing ranking approaches. Also, It can be concluded that it is not possible to find the IF optimal solution of IF MCF problems by using any of the existing methods. Only the proposed method can be used for the same problems." "textoCompletoSecciones" => array:1 [ "secciones" => array:12 [ 0 => array:3 [ "identificador" => "xres498774" "titulo" => "Abstract" "secciones" => array:1 [ 0 => array:1 [ "identificador" => "abst0005" ] ] ] 1 => array:2 [ "identificador" => "xpalclavsec520304" "titulo" => "Key words" ] 2 => array:2 [ "identificador" => "sec0005" "titulo" => "Introduction" ] 3 => array:3 [ "identificador" => "sec0010" "titulo" => "Preliminaries" "secciones" => array:2 [ 0 => array:2 [ "identificador" => "sec0015" "titulo" => "Basic definitions" ] 1 => array:2 [ "identificador" => "sec0020" "titulo" => "Arithmetic operations between TIF numbers" ] ] ] 4 => array:2 [ "identificador" => "sec0025" "titulo" => "Limitations of the existing methods" ] 5 => array:2 [ "identificador" => "sec0030" "titulo" => "Shortcomings of the existing methods" ] 6 => array:3 [ "identificador" => "sec0035" "titulo" => "Proposed ranking approach for comparing two IF numbers" "secciones" => array:1 [ 0 => array:2 [ "identificador" => "sec0040" "titulo" => "Advantages of the proposed ranking approach" ] ] ] 7 => array:3 [ "identificador" => "sec0045" "titulo" => "MCF problems in IF environment" "secciones" => array:4 [ 0 => array:3 [ "identificador" => "sec0050" "titulo" => "Linear programming formulations of balanced MCF problems in crisp and IF environment" "secciones" => array:2 [ 0 => array:2 [ "identificador" => "sec0055" "titulo" => "In this subsection, the linear programming formulation of balanced MCF problems in crisp environment is presented [1]." ] 1 => array:2 [ "identificador" => "sec0060" "titulo" => "Linear programming formulation of balanced MCF problems in IF environment" ] ] ] 1 => array:2 [ "identificador" => "sec0065" "titulo" => "Proposed method" ] 2 => array:2 [ "identificador" => "sec0070" "titulo" => "Advantages of the proposed method" ] 3 => array:2 [ "identificador" => "sec0075" "titulo" => "Illustrative example" ] ] ] 8 => array:2 [ "identificador" => "sec0080" "titulo" => "Comparative study" ] 9 => array:2 [ "identificador" => "sec0085" "titulo" => "Conclusion" ] 10 => array:2 [ "identificador" => "xack161135" "titulo" => "Acknowledgements" ] 11 => array:1 [ "titulo" => "References" ] ] ] "pdfFichero" => "main.pdf" "tienePdf" => true "PalabrasClave" => array:1 [ "en" => array:1 [ 0 => array:4 [ "clase" => "keyword" "titulo" => "Key words" "identificador" => "xpalclavsec520304" "palabras" => array:6 [ 0 => "manufacturing system" 1 => "replenishment lot size" 2 => "discontinuous issuing policy" 3 => "algebraic approach" 4 => "imperfect EPQ model" 5 => "rework" ] ] ] ] "tieneResumen" => true "resumen" => array:1 [ "en" => array:2 [ "titulo" => "Abstract" "resumen" => "To the best of our knowledge very few methods have been proposed in previous studies for comparing intuitionistic fuzzy (IF) numbers. In this paper, the limitations and the shortcomings of all these existing methods are pointed out. In order to overcome these limitations and shortcomings a new ranking approach—by modifying an existing ranking approach—is proposed for comparing IF numbers. Thus, with the help of proposed the ranking approach, a new method is proposed to find the optimal solution of such unbalanced minimum cost flow (MCF) problems in which all the parameters are represented by IF numbers." ] ] "multimedia" => array:32 [ 0 => array:7 [ "identificador" => "f0005" "etiqueta" => "Figure 2" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr2.jpeg" "Alto" => 374 "Ancho" => 490 "Tamanyo" => 27332 ] ] "descripcion" => array:1 [ "en" => "A network with three nodes and three arcs." ] ] 1 => array:7 [ "identificador" => "f0010" "etiqueta" => "Figure 3" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => true "mostrarDisplay" => false "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr3.jpeg" "Alto" => 420 "Ancho" => 762 "Tamanyo" => 40931 ] ] "descripcion" => array:1 [ "en" => "A modified network with a dummy source and a dummy destination." ] ] 2 => array:7 [ "identificador" => "tbl0005" "etiqueta" => "Table 1" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="center" valign="top" scope="col" style="border-bottom: 2px solid black">Approaches \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="top" scope="col" style="border-bottom: 2px solid black">Ordering \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="center" valign="top">Liou and Wang <a class="elsevierStyleCrossRef" href="#bib0070">[14]</a> \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">A˜≅B˜ \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="center" valign="top">Nehi <a class="elsevierStyleCrossRef" href="#bib0100">[20]</a> \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">A˜≅B˜ \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="center" valign="top">Garcia and Lamata <a class="elsevierStyleCrossRef" href="#bib0030">[6]</a> \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">A˜≻B˜ \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab796286.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Comparing results." ] ] 3 => array:7 [ "identificador" => "tbl0010" "etiqueta" => "Table 2" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="center" valign="top" scope="col">Node no \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="top" scope="col">Intuitionistic fuzzy supply \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="top" scope="col">Intuitionistic fuzzy demand \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="top" scope="col">c˜ij \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="top" scope="col">u˜ij \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="center" valign="top">1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">(50, 100, 150, 200, 250, 270, 300, 350) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">c˜12=(5, 10, 50, 100, 1000, 5500, 10000, 15000) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">u˜12=(20, 50, 100, 150, 220, 240, 260, 300) \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="center" valign="top">2 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">(0, 20, 60, 100, 150, 170, 200, 250) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">c˜13=(20, 40, 200, 400, 4000, 22000, 40000, 60000) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">u˜13=(4, 8, 13, 50, 55, 60, 65, 71) \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="center" valign="top">3 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">– \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">(0, 30, 40, 50, 100, 100, 100, 150} \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">c˜23=(15, 30, 150, 300, 3000, 16500, 30000, 45000) \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">u˜23=(15, 30, 60, 65, 70, 75, 81, 85) \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab796287.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Parameters for IF MCF problem (3 nodes and 3 arcs)." ] ] 4 => array:7 [ "identificador" => "tbl0015" "etiqueta" => "Table 3" "tipo" => "MULTIMEDIATABLA" "mostrarFloat" => true "mostrarDisplay" => false "tabla" => array:1 [ "tablatextoimagen" => array:1 [ 0 => array:2 [ "tabla" => array:1 [ 0 => """ <table border="0" frame="\n \t\t\t\t\tvoid\n \t\t\t\t" class=""><thead title="thead"><tr title="table-row"><th class="td" title="table-head " align="center" valign="top" scope="col">Example \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="top" scope="col">Existing method <a class="elsevierStyleCrossRef" href="#bib0040">[8]</a> \t\t\t\t\t\t\n \t\t\t\t</th><th class="td" title="table-head " align="center" valign="top" scope="col">Proposed method \t\t\t\t\t\t\n \t\t\t\t</th></tr></thead><tbody title="tbody"><tr title="table-row"><td class="td" title="table-entry " align="center" valign="top">3.5 [8, pp. 2498] \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">(1924000,1903300,7299800)LR \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">(1924000,1903300,7299300)LR \t\t\t\t\t\t\n \t\t\t\t</td></tr><tr title="table-row"><td class="td" title="table-entry " align="center" valign="top">6.1 \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">Not applicable \t\t\t\t\t\t\n \t\t\t\t</td><td class="td" title="table-entry " align="center" valign="top">(0,1400,11000,20000,350000, 2035000,4000000, 6750000) \t\t\t\t\t\t\n \t\t\t\t</td></tr></tbody></table> """ ] "imagenFichero" => array:1 [ 0 => "xTab796285.png" ] ] ] ] "descripcion" => array:1 [ "en" => "Comparative study." ] ] 5 => array:7 [ "identificador" => "f0015" "etiqueta" => "Figure 1" "tipo" => "MULTIMEDIAFIGURA" "mostrarFloat" => false "mostrarDisplay" => true "figura" => array:1 [ 0 => array:4 [ "imagen" => "gr1.jpeg" "Alto" => 471 "Ancho" => 919 "Tamanyo" => 43645 ] ] "descripcion" => array:1 [ "en" => "Trapezoidal intuitionistic fuzzy number." ] ] 6 => array:5 [ "identificador" => "eq0005" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "μA˜(x)=x-a1a2-a1if  a1≤x<a21if  a2≤x≤a3x-a1a3-a4if  a3<x≤a40otherwise" "Fichero" => "si12.jpeg" "Tamanyo" => 6015 "Alto" => 117 "Ancho" => 252 ] ] 7 => array:5 [ "identificador" => "eq0010" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "υA˜(x)=a'2-xa'2-a'1if  a'1≤x<a'20if  a'2≤x≤a'3a'3-xa'3-a'4if  a'3≤x≤a'41otherwise" "Fichero" => "si13.jpeg" "Tamanyo" => 6430 "Alto" => 128 "Ancho" => 266 ] ] 8 => array:5 [ "identificador" => "eq0015" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "λA˜=(λa'1,λa1,λa'2,λa2,λa3,λa'3,λa4,λa'4), λ≥0(λa'4,λa4,λa'3,λa3,λa2,λa'2,λa1,λa'1), λ≤0" "Fichero" => "si20.jpeg" "Tamanyo" => 6714 "Alto" => 49 "Ancho" => 385 ] ] 9 => array:5 [ "identificador" => "eq0020" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "LTλ(A˜)=Vλ(A˜)1+Aλ(A˜),Vλ(A˜)=Vμ(A˜)+λ(Vυ(A˜)-Vμ(A˜)),Aλ(A˜)=Aμ(A˜)-λ(Aυ(A˜)-Aμ(A˜)),Vμ(A˜)=w16(a1+2a2+a3),Aμ(A˜)=w13(a3-a1),Aυ(A˜)=(1-u1)3(a3-a1)" "Fichero" => "si30.jpeg" "Tamanyo" => 11824 "Alto" => 80 "Ancho" => 809 ] ] 10 => array:5 [ "identificador" => "eq0025" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "LG(B˜)=(a2+b2)(1-δ)+δ(2-(c2+d2)2" "Fichero" => "si44.jpeg" "Tamanyo" => 2010 "Alto" => 22 "Ancho" => 217 ] ] 11 => array:5 [ "identificador" => "eq0030" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "A˜={(a1,a1,b1,b1),(c1,c1,d1,d1)}andB˜={(a2,a2,b2,b2),(c2,c2,d2,d2)}." "Fichero" => "si45.jpeg" "Tamanyo" => 4486 "Alto" => 43 "Ancho" => 297 ] ] 12 => array:5 [ "identificador" => "eq0035" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Mμ1/3,0(A˜i)=13(a1+2a2+2a3+a42)andMυ1/3,0(A˜i)=13(a'1+2a'2+2a'3+a'42)" "Fichero" => "si79.jpeg" "Tamanyo" => 5993 "Alto" => 74 "Ancho" => 305 ] ] 13 => array:6 [ "identificador" => "eq0040" "etiqueta" => "(P1)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∑j:(i,j)∈Axij=ai∀i∈NS,∑j:(i,j)∈Axij=∑j:(j,i)∈Axji+ei∀i∈NIS,∑i:(i,j)∈Axij=bj∀j∈ND,∑i:(i,j)∈Axij=∑i:(j,i)∈Axji+dj∀j∈NID,∑j:(i,j)∈Axij=∑j:(j,i)∈Axji∀i∈NIT,0≤lij≤xij≤uij∀(i,j)∈A" "Fichero" => "si81.jpeg" "Tamanyo" => 13549 "Alto" => 249 "Ancho" => 274 ] ] 14 => array:6 [ "identificador" => "eq0045" "etiqueta" => "(P2)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∑j:(i,j)∈Ax˜ij=a˜i∀i∈NS,∑j:(i,j)∈Ax˜ij=∑j:(j,i)∈Ax˜ji⊕e˜i∀i∈NIS,∑i:(i,j)∈Ax˜ij=b˜j∀j∈ND,∑i:(i,j)∈Ax˜ij=∑i:(j,i)∈Ax˜ji⊕d˜j∀j∈NID,∑j:(i,j)∈Ax˜ij=∑j:(j,i)∈Ax˜ji∀i∈NIT,0˜≤˜l˜ij≤˜x˜ij≤˜u˜ij∀(i,j)∈A" "Fichero" => "si84.jpeg" "Tamanyo" => 13795 "Alto" => 249 "Ancho" => 270 ] ] 15 => array:6 [ "identificador" => "eq0050" "etiqueta" => "(P3)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "(∑j:(i,j)∈Ax' 1ij,∑j:(i,j)∈Ax1ij,∑j:(i,j)∈Ax' 2ij,∑j:(i,j)∈Ax2ij,∑j:(i,j)∈Ax3ij,∑j:(i,j)∈Ax' 3ij,∑j:(i,j)∈Ax4ij,∑j:(i,j)∈Ax' 4ij)=(a' 1i,a1i,a' 2i,a2i,a3i,a' 3i,a4i,a' 4i), ∀i∈NS(∑j:(i,j)∈Ax' 1ij,∑j:(i,j)∈Ax1ij,∑j:(i,j)∈Ax' 2ij,∑j:(i,j)∈Ax2ij,∑j:(i,j)∈Ax3ij,∑j:(i,j)∈Ax' 3ij,∑j:(i,j)∈Ax4ij,∑j:(i,j)∈Ax' 4ij)=(∑j:(j,i)∈Ax' 1ji+e' 1i,∑j:(j,i)∈Ax1ji+e1i∑j:(j,i)∈Ax' 2ji+e' 2i,∑j:(j,i)∈Ax2ij+e2i,∑j:(j,i)∈Ax3ji+e3,∑j:(j,i)∈Ax' 3ji+e' 3i,∑j:(j,i)∈Ax4ji+e4i,∑j:(j,i)∈Ax' 4ij+e' 4i), ∀i∈NIS(∑j:(i,j)∈Ax' 1ij,∑j:(i,j)∈Ax 1ij,∑j:(i,j)∈Ax' 2ij,∑j:(i,j)∈Ax2ij,∑j:(i,j)∈Ax3ij,∑j:(i,j)∈Ax' 3ij,∑j:(i,j)∈Ax4ij,∑j:(i,j)∈Ax' 4ij)=(b' 1i,b1j,b' 2i,b2j,b3j,b' 3i,b4j,b' 4j), ∀j∈ND(∑i:(i,j)∈Ax' 1ij,∑i:(i,j)∈Ax1ij,∑i:(i,j)∈Ax' 2ij,∑i:(i,j)∈Ax2ij,∑i:(i,j)∈Ax3ij,∑i:(i,j)∈Ax' 3ij,∑i:(i,j)∈Ax4ij,∑i:(i,j)∈Ax' 4ij)=(∑i:(j,i)∈Ax' 1ij+d' 1ij,∑i:(j,i)∈Ax1ji+d1j∑i:(j,i)∈Ax' 2ij+d' 2j,∑i:(j,i)∈Ax2ji+d2j,∑i:(j,i)∈Ax3ji+d3,∑i:(j,i)∈Ax' 3ji+d' 3j,∑i:(j,i)∈Ax4ji+d4j,∑i:(j,i)∈Ax' 4ji+d' 4j), ∀j∈NID(∑j:(i,j)∈Ax' 1ij,∑j:(i,j)∈Ax1ij,∑j:(i,j)∈Ax' 2ij,∑j:(i,j)∈Ax2ij,∑j:(i,j)∈Ax3ij,∑j:(i,j)∈Ax' 3ij,∑j:(i,j)∈Ax4ij,∑j:(i,j)∈Ax' 4ij)=(∑j:(i,j)∈Ax' 1ji,∑j:(i,j)∈Ax1ji,∑j:(i,j)∈Ax' 2ji,∑j:(i,j)∈Ax2ji,∑j:(i,j)∈Ax3ji,∑j:(i,j)∈Ax' 3ji,∑j:(i,j)∈Ax4ji,∑j:(i,j)∈Ax' 4ji), ∀i∈NIT(l'1ij,l1ij,l'2ij,l2ij,l3ij,l'3ij,l4ij,l' 4ij)≤˜(x' 1ij,x1ij,x' 2ij,x2ij,x3ij,x' 3ij,x4ij,x' 4ij)≤˜(u'1ij,u1ij,u' 2ij,u2ij,u3ij,u' 3ij,u4ij,u' 4ij)(x'1ij,x1ij,x' 2ij,x2ij,x3ij,x' 3ij,x4ij,x' 4ij) ∀(i,j)∈A" "Fichero" => "si100.jpeg" "Tamanyo" => 56053 "Alto" => 250 "Ancho" => 809 ] ] 16 => array:6 [ "identificador" => "eq0055" "etiqueta" => "(P4)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "∑j:(i,j)∈Ax' 1ij=a' 1i,∑j:(i,j)∈Ax1ij=a1i,∑j:(i,j)∈Ax' 2ij=a' 2i,∑j:(i,j)∈Ax2ij=a2i,∑j:(i,j)∈Ax3ij=a3i,∑j:(i,j)∈Ax' 3ij=a' 3i,∑j:(i,j)∈Ax4ij=a4i,∑j:(i,j)∈Ax' 4ij=a' 4i; ∀i∈NS∑j:(i,j)∈Ax' 1ij=∑j:(j,i)∈Ax' 1ij+e' 1i,∑j:(i,j)∈Ax1ij=∑j:(j,i)∈Ax1ji+e1i,∑j:(i,j)∈Ax' 2ij=∑j:(j,i)∈Ax' 2ij+e' 2i,∑j:(i,j)∈Ax2ij=∑j:(j,i)∈Ax2ji+e2i,∑j:(i,j)∈Ax3ij=∑j:(j,i)∈Ax3ji+e3i,∑j:(i,j)∈Ax' 3ij=∑j:(j,i)∈Ax' 3ij+e' 3i,∑j:(i,j)∈Ax4ij=∑j:(j,i)∈Ax4ji+e4i,∑j:(i,j)∈Ax' 3ij=∑j:(j,i)∈Ax' 4ij+e' 4i, ∀i∈NIS∑j:(i,j)∈Ax' 1ij= b' 1j,∑j:(i,j)∈Ax1ij=b1j,∑j:(i,j)∈Ax' 2ij= b' 2j,∑j:(i,j)∈Ax2ij= b2j,∑j:(i,j)∈Ax3ij=b3j,∑j:(i,j)∈Ax' 3ij= b' 3j,∑j:(i,j)∈Ax4ij=b4j,∑j:(i,j)∈Ax' 4ij=b' 4i, ∀j∈ND∑i:(i,j)∈Ax' 1ij=∑i:(j,i)∈Ax' 1ji+d' 1j,∑i:(i,j)∈Ax1ij=∑i:(j,i)∈Ax1ji+d1j,∑i:(i,j)∈Ax' 2ij=∑i:(j,i)∈Ax' 2ji+d' 2j,∑i:(i,j)∈Ax2ij=∑i:(j,i)∈Ax2ji+d2j,∑i:(i,j)∈Ax3ij=∑i:(j,i)∈Ax3ji+d3j,∑i:(i,j)∈Ax' 3ij=∑i:(j,i)∈Ax' 3ij+d' 3j,∑i:(i,j)∈Ax4ij=∑i:(j,i)∈Ax4ji+d4j, ∑i:(i,j)∈Ax' 4ij=∑i:(j,i)∈Ax' 4ji+d' 4j ∀j∈NID∑j:(i,j)∈Ax' 1ij=∑j:(j,i)∈Ax' 1ji,∑j:(i,j)∈Ax1ij=∑j:(j,i)∈Ax1ji,∑j:(i,j)∈Ax' 2ij=∑j:(j,i)∈Ax' 2ji,∑j:(i,j)∈Ax2ij=∑j:(j,i)∈Ax2ji,∑j:(i,j)∈Ax3ij=∑j:(j,i)∈Ax3ji,∑j:(i,j)∈Ax' 3ij=∑j:(j,i)∈Ax' 3ji,∑j:(i,j)∈Ax4ij=∑j:(j,i)∈Ax4ji,∑j:(i,j)∈Ax' 4ij=∑j:(j,i)∈Ax' 4ji ∀i∈NITx1ij-x' 1ij≥0,x' 2ij-x1ij≥0,x2ij-x' 2ij≥0,x3ij-x2ij≥0,x' 3ij-x3ij≥0,x4ij-x' 3ij≥0,x' 4ij-x4ij≥0,x' 1ij≥0" "Fichero" => "si102.jpeg" "Tamanyo" => 51178 "Alto" => 250 "Ancho" => 810 ] ] 17 => array:6 [ "identificador" => "eq0060" "etiqueta" => "(C1)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "(l'1ij,l1ij,l'2ij,l2ij,l3ij,l'3ij,l4ij,l'4ij)≤˜(x'1ij,x1ij,x'2ij,x2ij,x3ij,x'3ij,x4ij,x'4ij)≤˜(u'1ij,u1ij,u'2ij,u2ij,u3ij,u'3ij,u4ij,u'4ij)}" "Fichero" => "si103.jpeg" "Tamanyo" => 6830 "Alto" => 69 "Ancho" => 329 ] ] 18 => array:5 [ "identificador" => "eq0065" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Minimize∑(i,j)∈AMμβ,k(c′1ijx′1ij,c1ijx1ij,c′2ijx′2ij,c2ijx2ij,c3ijx4ij,c′3ijx′3ij,c4ijx4ij,c′4ijx′4ij)" "Fichero" => "si106.jpeg" "Tamanyo" => 5869 "Alto" => 40 "Ancho" => 561 ] ] 19 => array:6 [ "identificador" => "eq0070" "etiqueta" => "(P5)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Mμβ,k(l′1ij,l1ij,l′2ij,l2ij,l3ij,l′3ij,l4ij,l′4ij)≤Mμβ,k(x′1ij,x1ij,x′2ij,x2ij,x3ij,x′3ij,x4ij,x′4ij)≤Mμβ,k(u′1ij,u1ij,u′2ij,u2ij,u3ij,u′3ij,u4ij,u′4ij)" "Fichero" => "si107.jpeg" "Tamanyo" => 8527 "Alto" => 54 "Ancho" => 589 ] ] 20 => array:5 [ "identificador" => "eq0075" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Minimize∑(i,j)∈AMυβ,k(c'1ijx'1ij,c1ijx1ij,c'2ijx'2ij,c2ijx2ij,c3ijx4ij,c'3ijx'3ij,c4ijx4ij,c'4ijx'4ij)" "Fichero" => "si115.jpeg" "Tamanyo" => 5895 "Alto" => 40 "Ancho" => 595 ] ] 21 => array:6 [ "identificador" => "eq0080" "etiqueta" => "(P6)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Mυβ,k(l′1ij,l1ij,l′2ij,l2ij,l3ij,l′3ij,l4ij,l′4ij)≤Mυβ,k(x′1ij,x1ij,x′2ij,x2ij,x3ij,x′3ij,x4ij,x′4ij)≤Mυβ,k(u′1ij,u1ij,u′2ij,u2ij,u3ij,u′3ij,u4ij,u′4ij)∑(i,j)∈AMμβ,k(c′1ijx′1ij,c1ijx1ij,c′2ijx′2ij,c2ijx2ij,c3ijx4ij,c′3ijx′3ij,c4ijx4ij,c′4ijx′4ij)=a" "Fichero" => "si116.jpeg" "Tamanyo" => 12530 "Alto" => 71 "Ancho" => 809 ] ] 22 => array:6 [ "identificador" => "eq0085" "etiqueta" => "(P7)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "x˜12⊕x˜13⊕x˜15=(50,100,150,200,250,270,300,350)x˜12⊕x˜42=x˜23⊕x˜25⊕(0,20,60,100,150,170,200,250)x˜13⊕x˜23⊕x˜43=(0,30,40,50,100,100,100,150)x˜42⊕x˜43⊕x˜45=(0,0,0,0,50,50,50,100)x˜15⊕x˜25⊕x˜45=(50,50,50,50,50,50,50,50)(0,0,0,0,0,0,0,0)≤˜x˜12≤˜(20,50,100,150,220,240,260,300)(0,0,0,0,0,0,0,0)≤˜x˜13≤˜(4,8,13,50,55,60,65,71)(0,0,0,0,0,0,0,0)≤˜x˜23≤˜(15,30,60,65,70,75,81,85)" "Fichero" => "si132.jpeg" "Tamanyo" => 25468 "Alto" => 186 "Ancho" => 425 ] ] 23 => array:5 [ "identificador" => "eq0090" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "minimize(5x'112,+20x'113+15x'123+10x112+40x113+30x123,50x'212+200x'223+150x'223,100x212+400x223+300x223,1000x312+4000x313+3000x323,5500x'312+22000x'313+16500x'323,10000x412+40000x413+30000x423,15000x'412+60000x'413+45000x'423)" "Fichero" => "si135.jpeg" "Tamanyo" => 14529 "Alto" => 56 "Ancho" => 719 ] ] 24 => array:6 [ "identificador" => "eq0095" "etiqueta" => "(P8)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "(x'112+x'113+x'115,x112+x113+x115,x'212+x'213+x'215,x212+x213+x215,x312+x313+x315,x'312+x'313+x'315,x412+x413+x415,x'412+x'413+x'415)=(50,100,150,200,250,270,300,350)(x'112+x'142,x112+x142,x'212+x'242,x212+x242,x312+x342,x'312+x'342,x412+x442,x'412+x'442)=(x'123+x'125+0,x123+x125+20,x'223+x'225+60,x223+x225+100,x323+x325+150,x'323+x'325+170,x423+x425+200,x'423+x'425+250)(x'113+x'123+x'143,x113+x123x143,x'213+x'223+x'243,x213+x223+x243,x313+x323+x343,x'313+x'323+x'343,x413+x423+x443,x'413+x'423+x'443)=(0,30,40,50,100,100,100,150)(x'142+x'143+x'145,x142+x143+x145,x'242+x'243+x'245,x242+x243+x245,x342+x343+x345,x'342+x'343+x'345,x442+x443+x445,x'442+x'443+x'445)=(0,0,0,0,50,50,50,100)(x'115+x'125+x'145,x115+x125+x145,x'215+x'225+x'245,x215+x225+x245,x315+x325+x345,x'315+x'325+x'345,x415+x425+x445,x'415+x'425+x'445)=(50,50,50,50,50,50,50,50)(0,0,0,0,0,0,0,0)≤˜(x'112,x112,x'212,x212,x312,x412,x'412)≤˜(20,50,100,150,220,240,260,300)(0,0,0,0,0,0,0,0)≤˜(x'113,x113,x'213,x213,x313,x'313,x412,x'413)≤˜(4,8,13,50,55,60,65,71)(0,0,0,0,0,0,0,0,0)≤˜(x'123,x123,x'223,x223,x323,x'323,x423,x'423)≤˜(15,30,60,65,70,75,81,85)" "Fichero" => "si136.jpeg" "Tamanyo" => 48670 "Alto" => 187 "Ancho" => 809 ] ] 25 => array:5 [ "identificador" => "eq0100" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "Minimize(5x'112+20x'113+15x'123,10x112+40x113+30x123,50x'212+200x'223+150x'223,100x212+400x223+300x223,1000x312+4000x313+3000x323,5500x'312+22000x'313+16500x'323,10000x412+40000x413+30000x423,15000x'412+60000x'413+45000x'423)" "Fichero" => "si138.jpeg" "Tamanyo" => 14810 "Alto" => 56 "Ancho" => 708 ] ] 26 => array:6 [ "identificador" => "eq0105" "etiqueta" => "(P9)" "tipo" => "MULTIMEDIAFORMULA" "mostrarFloat" => false "mostrarDisplay" => true "Formula" => array:5 [ "Matematica" => "x'112+x'113+x'115=50,x112+x113+x115=100,x'212+x'213+x'215=150,x212+x213+x215=200,x312+x313+x315=250,x'312+x'313+x'315=270,x412+x413+x415=300,x'412+x'413+x'415=350x'112+x'142-x'123-x'125=0,x112+x142-x123-x125=20,x'212+x'242-x'223-x'225=60,x212+x242-x223-x225=100,x312+x342-x323-x325=150,x'312+x'342-x'323-x'325=170,x412+x442-x423-x425=200,x'412+x'442-x'423-x'425=250x'113+x'123+x'143=0,x113+x123+x143=30,x'213+x'223+x'243=40,x213+x223+x243=50,x313+x323+x343=100,x'313+x'323+x'343=100,x413+x423+x443=100,x'413+x'423+x'443=150x'142+x'143+x'145=0,x142+x143+x145=0,x'242+x'243+x'245=0,x242+x243+x245=0,x342+x343+x345=50,x'342+x'343+x'345=50,x442+x443+x445=50,x'442+x'443+x'445=100x'115+x'125+x'145=50,x115+x125+x145=50,x'215+x'225+x'245=50,x215+x225+x245=50,x315+x325+x345=50,x'315+x'325+x'345=50,x415+x425+x445=50,x'415+x'425+x'445=50x1ij-x'1ij≥0,x'2ij-x1ij≥0,x2ij-x'2ij≥0,x3ij-x2ij≥0,x'3ij-x3ij≥0,x4ij-x'3ij≥0,x'4ij-x4ij≥0,x'1ij≥0; ∀(i,j)∈A" "Fichero" => "si139.jpeg" "Tamanyo" => 36284 "Alto" => 140 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I believe that Mehar is an angel for me and without Mehar’s blessing it would not be possible to mature the idea proposed in this paper. Mehar is a lovely daughter of Parmpreet Kaur (Research Scholar under my supervision). The authors also acknowledge the <a name="p396"></a>financial support given by the University Grant Commission, Govt, of India for completing the Major Research. Project (39-40/2010(SR))." "vista" => "all" ] ] ] "idiomaDefecto" => "en" "url" => "/16656423/0000001100000003/v2_201505081636/S1665642313715487/v2_201505081636/en/main.assets" "Apartado" => null "PDF" => "https://static.elsevier.es/multimedia/16656423/0000001100000003/v2_201505081636/S1665642313715487/v2_201505081636/en/main.pdf?idApp=UINPBA00004N&text.app=https://www.elsevier.es/" "EPUB" => "https://multimedia.elsevier.es/PublicationsMultimediaV1/item/epub/S1665642313715487?idApp=UINPBA00004N"]

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A Ranking Approach for Intuitionistic Fuzzy Numbers and its Application

Amit Kumar, Manjot Kaur

School of Mathematics and Computer Applications Thapar University, Patiala-147004, India

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    "textoCompleto" => "<span class="elsevierStyleSections"><span id="sec0005" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">1</span><span class="elsevierStyleSectionTitle" id="sect0015">Introduction</span><p id="par0010" class="elsevierStylePara elsevierViewall">MCF problem&#44; which is an important problem in combinatorial optimization and network flows&#44; has many applications in practical problems such as transportation&#44; communication&#44; urban design and job scheduling models &#91;<a class="elsevierStyleCrossRef" href="#bib0005">1</a>&#44;<a class="elsevierStyleCrossRef" href="#bib0015">3</a>&#93;&#46;</p><p id="par0015" class="elsevierStylePara elsevierViewall">In its classical form the MCF problem minimizes the cost of transporting a product that is available at some sources and required at certain destinations&#46; In many actual problems&#44; the cost&#44; capacities&#44; supplies and demand parameters may be imprecise&#46; To deal quantitatively with imprecise information the concept and techniques of probability can be employed&#46; There are articles discussing the network flow problems where arc parameters are random variable &#91;<a class="elsevierStyleCrossRef" href="#bib0055">11</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0105">21</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0110">22</a>&#93;&#44; however&#44; in order to construct&#44; probability distributions require either a priori predictable regularity or a posteriori frequency distribution&#46; Moreover&#44; the premise that imprecision can be equated with randomness is still questionable&#46; As an alternative&#44; uncertain values can be represented by membership functions of the fuzzy set theory <a class="elsevierStyleCrossRef" href="#bib0130">&#91;26&#93;</a>&#46;</p><p id="par0020" class="elsevierStylePara elsevierViewall">The main advantages of methodologies based on fuzzy theory are that they do not require prior predictable regularities or posterior frequency distributions and they can deal with imprecise input information containing feelings and emotions quantified based on the decision-maker&#8217;s subjective judgment&#46; From this point of view&#44; Shih and Lee <a class="elsevierStyleCrossRef" href="#bib0115">&#91;23&#93;</a> proposed a fuzzy version of MCF problem using multilevel linear programming problem&#46; But&#44; they did not use the nice structure of network constraints&#46;</p><p id="par0025" class="elsevierStylePara elsevierViewall">Ghatee and Hashemi <a class="elsevierStyleCrossRef" href="#bib0035">&#91;7&#93;</a> proposed a method to find the fuzzy optimal solution of balanced fully fuzzy MCF problems&#46; Ghatee and Hashemi &#91;<a class="elsevierStyleCrossRef" href="#bib0040">8</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0045">9</a>&#93; and Ghatee et al&#46; <a class="elsevierStyleCrossRef" href="#bib0050">&#91;10&#93;</a> applied the existing method <a class="elsevierStyleCrossRef" href="#bib0035">&#91;7&#93;</a> for solving real-life problems&#46;</p><p id="par0030" class="elsevierStylePara elsevierViewall">The most common concept used in all those studies is ranking of fuzzy numbers&#46; Ranking of fuzzy numbers is an important issue in the study of fuzzy set theory&#46; In order to rank fuzzy numbers&#44; one fuzzy number needs to be compared with the others but it is difficult to determine clearly which of them is larger or smaller&#46; Numerous methods have been proposed in previous studies to rank fuzzy numbers&#46; There is not a unique method for comparing fuzzy numbers&#46;</p><p id="par0035" class="elsevierStylePara elsevierViewall">A membership function of a classical fuzzy set assigns to each element of the universe of discourse a number from the unit interval to indicate the degree of belongingness to the set under consideration&#46; The degree of nondecision&#8722;maker&#8217;s belongingness is just automatically the complement to one of the membership degree&#46; Nonetheless&#44; a human being who expresses the <a name="p382"></a>degree of membership of a given element in a fuzzy set very often does not express the corresponding degree of nonmembership as the complement to 1&#46; This reflects a well-known psychological fact that the linguistic negation do not always identifies with logical negation&#46; Thus&#44; Atanassov <a class="elsevierStyleCrossRef" href="#bib0010">&#91;2&#93;</a> introduced the concept of an IF set which is characterized by two functions expressing the degree of belongingness and the degree of nonbelongingness respectively&#46; This idea&#44; which is a natural generalization of usual fuzzy set&#44; seems to be useful when modeling many real-life situations&#46;</p><p id="par0040" class="elsevierStylePara elsevierViewall">Concerning ranking IF numbers some work has been reported in the literature&#46; Grzegorzewski <a class="elsevierStyleCrossRef" href="#bib0060">&#91;12&#93;</a> defined two families of metrics in the space of IF numbers and proposed a method for comparing IF numbers based on these metrics&#46; Mitchell <a class="elsevierStyleCrossRef" href="#bib0075">&#91;15&#93;</a> extended the natural ordering of real numbers to triangular intuitionistic fuzzy &#40;TrIF&#41; numbers by adopting a statistical view point and interpreting each IF number as ensemble of ordinary fuzzy numbers&#46; Nayagam et al&#46; <a class="elsevierStyleCrossRef" href="#bib0095">&#91;19&#93;</a> introduced TrIF numbers of special type and described a method to compare them&#46; Although their ranking method appears to be attractive&#44; the definition of TrIF number seems unrealistic&#46; This is because the triangular nonmembership function is defined to geometrically behave in an identical manner as the membership function&#46; Nan and Li <a class="elsevierStyleCrossRef" href="#bib0080">&#91;16&#93;</a> proposed a method for comparing TrIF number using lexicographic technique&#46; Nehi <a class="elsevierStyleCrossRef" href="#bib0100">&#91;20&#93;</a> proposed a new method for comparing IF numbers in which two characteristic values for IF numbers are defined by the integral of the inverse fuzzy membership and nonmembership functions multiplied by the grade with powered parameter&#46; Almost in parallel&#44; Li <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a> introduced a new definition of the TrIF number which has an appealing and logically reasonable interpretation&#46; He defined two concepts of the value and the ambiguity of a TrIF number similar to those for a fuzzy number introduced by Delgado et al&#46; <a class="elsevierStyleCrossRef" href="#bib0020">&#91;4&#93;</a>&#46; Dubey and Mehra <a class="elsevierStyleCrossRef" href="#bib0025">&#91;5&#93;</a> defined a TrIF number which is more general than the one defined in &#91;<a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0080">16</a>&#93;&#46; They extended the definitions of the value and the ambiguity index given by Li <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a> to the newly defined TrIF numbers and proposed an approach to handle linear programming problems with data as IF numbers&#46;</p><p id="par0045" class="elsevierStylePara elsevierViewall">In this paper&#44; the limitations of the existing methods &#91;<a class="elsevierStyleCrossRef" href="#bib0025">5</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0060">12</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0075">15</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0090">18</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0095">19</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0120">24</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0125">25</a>&#93; as well as their shortcomings &#91;<a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0085">17</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0100">20</a>&#93;&#44; are pointed out&#46; Furthermore&#44; to overcome such limitations and shortcomings&#44; a new ranking approach&#8212;by modifying an existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0100">&#91;20&#93;</a>&#8212;is proposed for comparing IF numbers&#46; Also&#8212;with the help of the projected ranking approach&#8212;a new method is proposed to find the optimal solution of such unbalanced MCF problems in which all the parameters are represented by IF numbers&#46;</p><p id="par0050" class="elsevierStylePara elsevierViewall">This paper is organized as follows&#46; In <a class="elsevierStyleCrossRef" href="#sec0010">Section 2</a>&#44; some basic definitions and arithmetic operations of trapezoidal intuitionistic fuzzy &#40;TIF&#41; numbers are presented&#46; In <a class="elsevierStyleCrossRef" href="#sec0025">Sections 3</a> and <a class="elsevierStyleCrossRef" href="#sec0030">4</a>&#44; the limitations and shortcomings of the existing methods &#91;<a class="elsevierStyleCrossRef" href="#bib0025">5</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0060">12</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0075">15</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0090">18</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0095">19</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0120">24</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0125">25</a>&#93; and &#91;<a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0085">17</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0100">20</a>&#93; respectively&#44; are pointed out&#46; In <a class="elsevierStyleCrossRef" href="#sec0035">Section 5</a>&#44; a ranking approach is proposed for comparing IF numbers is presented&#46; In <a class="elsevierStyleCrossRef" href="#sec0045">Section 6</a>&#44; a method for solving MCF problems in IF environment is proposed and to illustrate the proposed method&#44; a numerical example is solved&#46; In <a class="elsevierStyleCrossRef" href="#sec0080">Section 7</a>&#44; the results are compared&#46; Finally&#44; in <a class="elsevierStyleCrossRef" href="#sec0085">Section 8</a>&#44; the conclusion is discussed&#46;</p></span><span id="sec0010" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">2</span><span class="elsevierStyleSectionTitle" id="sect0020">Preliminaries</span><p id="par0055" class="elsevierStylePara elsevierViewall">In this section&#44; 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&#124; <span class="elsevierStyleItalic">x</span> &#8712; <span class="elsevierStyleItalic">X</span>&#125; on the universal set <span class="elsevierStyleItalic">X</span> is characterized by a truth membership function <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span><span class="elsevierStyleItalic">&#44; &#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span><span class="elsevierStyleItalic">&#44;&#58; X &#8594;</span> &#91;0&#44;1&#93; and a false membership function &#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#44;&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#44;&#58; <span class="elsevierStyleItalic">X</span> &#8594; &#91;0&#44;1&#93;&#46; The values <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41; 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The value &#40;1<span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span>&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;&#41; represents the degree of hesitation for <span class="elsevierStyleItalic">x</span><span class="elsevierStyleHsp" style=""></span>&#8712;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">X&#46;</span></p><p id="par0065" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Definition 2&#46;2&#46;</span> Let <span class="elsevierStyleItalic">&#195;</span> be an IF set&#46; Then&#44; <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#945;</span></span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleHsp" style=""></span>&#8712;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">X &#124; &#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#945;</span>&#44;&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span>&#40;1<span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#945;</span>&#41;&#125; is said to be an <span class="elsevierStyleItalic">&#945;</span><span class="elsevierStyleHsp" style=""></span>&#8722;cut of <span class="elsevierStyleItalic">&#195;&#46;</span><a name="p383"></a></p><p id="par0070" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Definition 2&#46;3&#46;</span> An IF set <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleInf">&#945;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;&#40;<span class="elsevierStyleItalic">x</span>&#44; <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;&#44; &#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41; &#124; <span class="elsevierStyleItalic">x &#8712; X</span>&#125; is called IF-normal&#44; if there are at least two points <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">0</span>&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span> &#8712; <span class="elsevierStyleItalic">X</span> such that <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">0</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#44; &#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#46;</p><p id="par0075" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Definition 2&#46;4&#46;</span> An IF set <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;&#40;<span class="elsevierStyleItalic">x&#44; &#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;&#44;&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;&#41; &#124; <span class="elsevierStyleItalic">x</span> &#8712; <span class="elsevierStyleItalic">X</span>&#125; is called IF-convex&#44; if &#8704;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">2</span> &#8712; <span class="elsevierStyleItalic">X&#44; &#955;</span> &#8712; &#91;0&#44;1&#93; <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;&#955;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>&#40;1 &#8722; &#955;&#41;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span>min&#40;<span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf">&#195;</span>&#40;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span>&#41;&#44; <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf">&#195;</span>&#40;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">2</span>&#41;&#41;</p><p id="par0080" class="elsevierStylePara elsevierViewall">&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">&#955;x</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>&#40;1<span class="elsevierStyleHsp" style=""></span>&#8722;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#955;</span>&#41;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span>max&#40;&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">1</span>&#41;&#44;&#965;<span class="elsevierStyleInf">&#195;</span>&#40;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf">2</span>&#41;&#41;</p><p id="par0085" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Definition 2&#46;5&#46;</span> An IF set <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;&#40;<span class="elsevierStyleItalic">x</span>&#44; <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;&#44;&#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;&#41; &#124; <span class="elsevierStyleItalic">x</span> &#8712; <span class="elsevierStyleItalic">X</span>&#125; of the real line is called IF number if<ul class="elsevierStyleList" id="lis0005"><li class="elsevierStyleListItem" id="lsti0005"><span class="elsevierStyleLabel">&#40;a&#41;</span><p id="par0090" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">&#195;</span> is IF-normal</p></li><li class="elsevierStyleListItem" id="lsti0010"><span class="elsevierStyleLabel">&#40;b&#41;</span><p id="par0095" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">&#195;</span> is IF-convex</p></li><li class="elsevierStyleListItem" id="lsti0015"><span class="elsevierStyleLabel">&#40;c&#41;</span><p id="par0100" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span> is upper semi continuous and<span class="elsevierStyleItalic">&#965;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span> is lower semi continuous</p></li><li class="elsevierStyleListItem" id="lsti0020"><span class="elsevierStyleLabel">&#40;d&#41;</span><p id="par0105" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">x</span><span class="elsevierStyleHsp" style=""></span>&#8712;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">X</span> &#124; &#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#60;<span class="elsevierStyleHsp" style=""></span>1&#125; is bounded</p></li></ul></p><p id="par0110" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Definition 2&#46;6&#46;</span> An IF number <span class="elsevierStyleItalic">&#195;&#44;</span> defined on the universal set of real numbers <span class="elsevierStyleItalic">R&#44;</span> denoted as <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a&#8217;</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2&#44;</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3&#44;</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125;&#44; where <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span> is said to be TIF number&#44; if the degree of membership <span class="elsevierStyleItalic">&#956;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41; and the degree of non-membership &#965;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#195;</span></span>&#40;<span class="elsevierStyleItalic">x</span>&#41; are given by&#58;</p><p id="par0115" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0005"></elsevierMultimedia></p><p id="par0120" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0010"></elsevierMultimedia></p><p id="par0125" class="elsevierStylePara elsevierViewall">If in a TIF number <span class="elsevierStyleItalic">&#195;</span>&#44; let <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span> &#40;and hence <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3</span>&#41;&#44; then it gives a TrIF number with parameters <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span> &#40;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleHsp" style=""></span>&#8804;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span> and is denoted by <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125;&#46;</p><p id="par0130" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="f0015"></elsevierMultimedia></p><p id="par0135" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Definition 2&#46;7&#46;</span> Two TIF numbers <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2&#44;</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3&#44;</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125;&#44; B&#732;&#61;&#123;b&#39;1&#44;b1&#44;b&#39;2&#44;b2&#44;b3&#44;b&#39;3&#44;b4&#44;b&#39;4&#125; are said to be equal i&#46;e&#46;&#44; A&#732;&#61;B&#732; if and only if <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span>&#8217;<span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span>&#8217;<span class="elsevierStyleInf">2</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">2</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">3</span>&#44; <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span>&#8217;<span class="elsevierStyleInf">3</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">4</span> and <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span>&#8217;<span class="elsevierStyleInf">4</span>&#46;</p><p id="par0140" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleItalic">Definition 2&#46;8&#46;</span> A TIF number <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2&#44;</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3&#44;</span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125; is said to be non-negative TIF number if and only if <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span>0&#46;</p><p id="par0145" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Remark 1&#46;</span> In the existing methods <a class="elsevierStyleCrossRef" href="#bib0025">&#91;5&#93;</a> the TrIF number <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleItalic">&#44;a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125; is represented by <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;&#40;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#59;1&#41;&#44;&#40;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#59;1&#41;&#125;&#46;<a name="p384"></a></p><p id="par0150" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Remark 2&#46;</span> In the existing methods <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a> the TrIF number <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#12296;&#40;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#41;&#59;1&#44;1&#12297; is used which is obtained by the TrIF number <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125; by assuming <span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>a&#8217;<span class="elsevierStyleInf">4</span>&#46;</p></span><span id="sec0020" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">2&#46;2</span><span class="elsevierStyleSectionTitle" id="sect0030">Arithmetic operations between TIF numbers</span><p id="par0155" class="elsevierStylePara elsevierViewall">In this section&#44; some arithmetic operations between TIF numbers&#44; defined on a universal set of real numbers <span class="elsevierStyleItalic">R&#44;</span> are presented&#46;</p><p id="par0160" class="elsevierStylePara elsevierViewall">&#40;i&#41; Let <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125; and B&#732;&#61;&#123;b&#39;1&#44;b1&#44;b&#39;2&#44;b2&#44;b3&#44;b&#39;3&#44;b4&#44;b&#39;4&#125; be two TIF numbers&#46; Then&#44; A&#732;&#8853;B&#732;&#61;&#40;a&#39;1&#43;b&#39;1&#44;a1&#43;b1&#44;a&#39;2&#43;b&#39;2&#44;a2&#43;b2&#44;a3&#43;b3&#44;a&#39;3&#43;b&#39;3&#44;a4&#43;b4&#44;a&#39;4&#43;b&#39;4&#41;</p><p id="par0165" class="elsevierStylePara elsevierViewall">&#40;ii&#41; Let <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125; and B&#732;&#61;&#123;b&#39;1&#44;b1&#44;b&#39;2&#44;b2&#44;b3&#44;b&#39;3&#44;b4&#44;b&#39;4&#125; be two TIF numbers&#46; Then&#44; A&#732;&#920;B&#732;&#61;&#40;a&#39;1-b&#39;4&#44;a1-b4&#44;a&#39;2-b&#39;3&#44;a2-b3&#44;a3-b2&#44;a&#39;3-b&#39;2&#44;a4-b1&#44;a&#39;4-b&#39;1&#125;</p><p id="par0170" class="elsevierStylePara elsevierViewall">&#40;iii&#41; Let <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125; be any TIF number&#46; Then&#44;</p><p id="par0175" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0015"></elsevierMultimedia></p><p id="par0180" class="elsevierStylePara elsevierViewall">&#40;iv&#41; Let <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">3</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">4</span>&#44;<span class="elsevierStyleItalic">a</span>&#8217;<span class="elsevierStyleInf">4</span>&#125; and B&#732;&#61;&#123;b&#39;1&#44;b1&#44;b&#39;2&#44;b2&#44;b3&#44;b&#39;3&#44;b4&#44;b&#39;4&#125; be two non-negative TIF numbers&#46; Then&#44; A&#732;&#8855;B&#732;&#61;&#40;a&#39;1b&#39;1&#44;a1b1&#44;a&#39;2b&#39;2&#44;a2b2&#44;a3b3&#44;a&#39;3b&#39;3&#44;a4b4&#44;a&#39;4b&#39;4&#41;</p></span></span><span id="sec0025" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">3</span><span class="elsevierStyleSectionTitle" id="sect0035">Limitations of the existing methods</span><p id="par0185" class="elsevierStylePara elsevierViewall">In this section the limitations of the existing methods &#91;<a class="elsevierStyleCrossRef" href="#bib0025">5</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0060">12</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0075">15</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0090">18</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0095">19</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0120">24</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0125">25</a>&#93; for comparing IF numbers are pointed out&#46;<ul class="elsevierStyleList" id="lis0010"><li class="elsevierStyleListItem" id="lsti0025"><span class="elsevierStyleLabel">1</span><p id="par0190" class="elsevierStylePara elsevierViewall">The existing methods &#91;<a class="elsevierStyleCrossRef" href="#bib0060">12</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0075">15</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0120">24</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0125">25</a>&#93; can be used only for comparing IF set&#44; however&#44; none of the existing methods &#91;<a class="elsevierStyleCrossRef" href="#bib0060">12</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0075">15</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0120">24</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0125">25</a>&#93; can be used for comparing IF numbers&#46;</p></li><li class="elsevierStyleListItem" id="lsti0030"><span class="elsevierStyleLabel">2</span><p id="par0195" class="elsevierStylePara elsevierViewall">Nayagam and Sivaraman <a class="elsevierStyleCrossRef" href="#bib0090">&#91;18&#93;</a> pointed out the shortcomings of the existing method <a class="elsevierStyleCrossRef" href="#bib0095">&#91;19&#93;</a> and proposed a method for comparing such TrIF numbers &#123;&#40;<span class="elsevierStyleItalic">a&#44;b&#44;c</span>&#41;&#44;&#40;<span class="elsevierStyleItalic">e&#44;f&#44;g</span>&#41;<span class="elsevierStyleItalic">&#125;</span> for which either the conditions<span class="elsevierStyleItalic">e</span><span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span> and<span class="elsevierStyleItalic">f</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#8805;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">c</span> or the conditions<span class="elsevierStyleItalic">f</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#8804;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span> and<span class="elsevierStyleItalic">g</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#8804;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span> are satisfied&#44; however&#44; the existing method <a class="elsevierStyleCrossRef" href="#bib0090">&#91;18&#93;</a> cannot be used for comparing such TrIF numbers for which neither the conditions<span class="elsevierStyleItalic">e</span><span class="elsevierStyleHsp" style=""></span>&#8805;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span> and<span class="elsevierStyleItalic">f</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#8805;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">c</span> nor the conditions<span class="elsevierStyleItalic">f</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#8804;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span> and<span class="elsevierStyleItalic">g</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#8804;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">b</span> are satisfied&#46;</p></li><li class="elsevierStyleListItem" id="lsti0035"><span class="elsevierStyleLabel">3</span><p id="par0200" class="elsevierStylePara elsevierViewall">Dubey and Mehra <a class="elsevierStyleCrossRef" href="#bib0025">&#91;5&#93;</a> pointed out the shortcomings of the existing methods <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a> and proposed a method for comparing such TrIF sets a&#732;&#61;&#123;&#40;a&#95;&#59;&#956;&#44;a&#44;a&#175;&#956;&#59;Wa~&#41;&#44;&#40;a&#95;&#59;&#965;&#44;a&#44;a&#175;&#965;&#59;ua~&#41;&#125; for which the condition <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#956;</span></span>&#40;<span class="elsevierStyleItalic">&#227;</span>&#41;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#8804;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#965;</span></span>&#40;<span class="elsevierStyleItalic">&#227;</span>&#41; is satisfied&#44; where V&#956;a&#732;&#61;&#40;&#40;a&#95;&#59;&#956;&#43;4a&#43;a&#175;&#956;&#41;wa&#732;&#41;&#47;6 and V&#965;a&#732;&#61;&#40;&#40;a&#95;&#59;&#965;&#43;4a&#43;a&#175;&#965;&#41;&#40;1&#8722;ua&#732;&#41;&#41;&#47;6&#44; however&#44; the existing method <a class="elsevierStyleCrossRef" href="#bib0090">&#91;18&#93;</a> cannot be used for comparing such TrIF sets for which the condition <span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#956;</span></span>&#40;<span class="elsevierStyleItalic">&#227;</span>&#41;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#8804;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">V</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">&#965;</span></span>&#40;<span class="elsevierStyleItalic">&#227;</span>&#41; is not satisfied&#46;</p></li></ul></p></span><span id="sec0030" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">4</span><span class="elsevierStyleSectionTitle" id="sect0040">Shortcomings of the existing methods</span><p id="par0205" class="elsevierStylePara elsevierViewall">In this section the shortcomings of the existing methods &#91;<a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0085">17</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0095">19</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0100">20</a>&#93; for comparing IF numbers are pointed out&#46;<ul class="elsevierStyleList" id="lis0015"><li class="elsevierStyleListItem" id="lsti0040"><span class="elsevierStyleLabel">1</span><p id="par0210" class="elsevierStylePara elsevierViewall">Li <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a> proposed the following method for comparing TrIF sets&#46; Let<span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span>&#12296;&#40;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf">3</span>&#59;<span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span>&#41;&#12297; and B&#732;&#61;&#12296;&#40;a1&#44;a2&#44;a3&#59;w2&#44;u2&#41;&#12297; be two TrIF sets&#46;<ul class="elsevierStyleList" id="lis0020"><li class="elsevierStyleListItem" id="lsti0045"><span class="elsevierStyleLabel">&#40;i&#41;</span><p id="par0215" class="elsevierStylePara elsevierViewall">A&#732;&#60;&#732;B&#732;ifLT&#955;&#40;A&#732;&#41;&#60;LT&#955;&#40;B&#732;&#41;</p></li><li class="elsevierStyleListItem" id="lsti0050"><span class="elsevierStyleLabel">&#40;ii&#41;</span><p id="par0220" class="elsevierStylePara elsevierViewall">A&#732;&#62;&#732;B&#732;ifLT&#955;&#40;A&#732;&#41;&#62;LT&#955;&#40;B&#732;&#41;</p></li><li class="elsevierStyleListItem" id="lsti0055"><span class="elsevierStyleLabel">&#40;iii&#41;</span><p id="par0225" class="elsevierStylePara elsevierViewall">A&#732;&#8773;B&#732;ifLT&#955;&#40;A&#732;&#41;&#61;LT&#955;&#40;B&#732;&#41;</p></li></ul></p></li></ul></p><p id="par0230" class="elsevierStylePara elsevierViewall">Where</p><p id="par0235" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0020"></elsevierMultimedia><a name="p385"></a></p><p id="par0240" class="elsevierStylePara elsevierViewall">It is not genuine to apply this method due to the following reasons&#46; It is obvious from the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a> that if &#40;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>4<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#41; &#8800; 0 then the comparison of <span class="elsevierStyleItalic">&#195;</span> and B&#732; will depend upon the values of <span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">2</span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">2</span> and if &#40;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>4<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0 then A&#732;&#8773;B&#732; for all values of <span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">2</span> i&#46;e&#46;&#44; according to the existing approach <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a>&#44; in the first case comparison of TrIF sets depends upon the degree of membership and the degree of nonmembership of IF sets whereas in the second case comparison of TrIF sets does not depend upon the degree of membership and the degree of nonmembership of IF sets which is a contradiction&#46;</p><p id="par0245" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Example 4&#46;1&#46;</span> Let <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#12296;&#40;1&#44;1&#44;1&#41;&#59;<span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span>&#12297; and B&#732;&#61;&#12296;&#40;1&#44;1&#44;1&#41;&#59;w2&#44;u2&#12297; be two TrIF sets then according to the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a> values of &#195; and B&#732; will depend upon the values of <span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#44;w</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">2</span> i&#46;e&#46;&#44; the ordering of <span class="elsevierStyleItalic">&#195;</span> and B&#732; will depend upon the values of <span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span>&#46;</p><p id="par0250" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Example 4&#46;2&#46;</span> Let <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#12296;&#40;&#8722;8&#44;1&#44;4&#41;&#59;<span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span>&#12297; and B&#732;&#61;&#12296;&#40;-8&#44;1&#44;4&#41;&#59;w2&#44;u2&#12297; be two TrIF sets&#46; Then&#44; according to the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0065">&#91;13&#93;</a>&#44; LT&#955;&#40;A&#732;&#41;&#61;&#8201;LT&#955;&#40;B&#732;&#41;&#61;0&#8658;A&#732;&#8773;B&#732; i&#46;e&#46;&#44; in this case the ordering of <span class="elsevierStyleItalic">&#195;</span> and B&#732; is independent from the values of <span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span> and <span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span>&#46;<ul class="elsevierStyleList" id="lis0025"><li class="elsevierStyleListItem" id="lsti0060"><span class="elsevierStyleLabel">2</span><p id="par0255" class="elsevierStylePara elsevierViewall">Nayagam and Sivaraman <a class="elsevierStyleCrossRef" href="#bib0085">&#91;17&#93;</a> proposed the following method for comparing interval valued IF sets&#58;</p><p id="par0260" class="elsevierStylePara elsevierViewall">Let<span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">x</span> &#58; &#91;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span>&#93;&#44;&#91;<span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">1</span>&#93; &#124;<span class="elsevierStyleItalic">x</span> &#8712;<span class="elsevierStyleItalic">X</span>&#125; and B&#732;&#61;&#123;x&#58;&#91;a2&#44;b2&#93;&#44;&#91;c2&#44;d2&#93;&#124;x&#8712;X&#125; be two interval valued IF sets defined on a universal set<span class="elsevierStyleItalic">X&#46;</span> Then<ul class="elsevierStyleList" id="lis0030"><li class="elsevierStyleListItem" id="lsti0065"><span class="elsevierStyleLabel">&#40;i&#41;</span><p id="par0265" class="elsevierStylePara elsevierViewall">A&#732;&#60;&#732;B&#732;ifLG&#40;A&#732;&#41;&#60;LG&#40;B&#732;&#41;</p></li><li class="elsevierStyleListItem" id="lsti0070"><span class="elsevierStyleLabel">&#40;ii&#41;</span><p id="par0270" class="elsevierStylePara elsevierViewall">A&#732;&#62;&#732;B&#732;ifLG&#40;A&#732;&#41;&#62;LG&#40;B&#732;&#41;</p></li><li class="elsevierStyleListItem" id="lsti0075"><span class="elsevierStyleLabel">&#40;iii&#41;</span><p id="par0275" class="elsevierStylePara elsevierViewall">A&#732;&#8773;B&#732;ifLG&#40;A&#732;&#41;&#61;LG&#40;B&#732;&#41;</p></li></ul></p><p id="par0280" class="elsevierStylePara elsevierViewall">where LG&#40;A&#732;&#41;&#61;&#40;a1&#43;b1&#41;&#40;1-&#948;&#41;&#43;&#948;&#40;2-&#40;c1&#43;d1&#41;2</p><p id="par0285" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0025"></elsevierMultimedia></p><p id="par0290" class="elsevierStylePara elsevierViewall">and <span class="elsevierStyleItalic">&#948;</span> &#8712; &#91;0&#44;1&#93;&#46;</p><p id="par0295" class="elsevierStylePara elsevierViewall">Nayagam and Sivaraman <a class="elsevierStyleCrossRef" href="#bib0085">&#91;17&#93;</a> have used the same method for comparing TIF numbers</p><p id="par0300" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0030"></elsevierMultimedia></p><p id="par0305" class="elsevierStylePara elsevierViewall">Nonetheless&#44; it is not genuine to use this method for comparing TIF numbers due to the following reasons</p><p id="par0310" class="elsevierStylePara elsevierViewall">In the IF set<span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;<span class="elsevierStyleItalic">x</span> &#58; &#91;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span>&#93;&#44;&#91;<span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">1</span>&#93; &#124;<span class="elsevierStyleItalic">x</span> &#8712;<span class="elsevierStyleItalic">X</span>&#125;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span> and<span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">1</span> represents the infimum and supremum values of membership degree and nonmembership degree corresponding to points<span class="elsevierStyleItalic">X&#46;</span> Although&#44; in the TIF number<span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#123;&#40;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span>&#41;&#44;&#40;<span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1&#44;</span><span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">1</span>&#44;<span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">1</span> and<span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf">1</span> represent those points of universal set<span class="elsevierStyleItalic">X</span> corresponding to which the membership degree is 1&#46; Similarly&#44;<span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf">1</span> and<span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf">1</span> represent those points of universal set<span class="elsevierStyleItalic">X</span> corresponding to which the nonmembership degree is 1&#46;</p></li><li class="elsevierStyleListItem" id="lsti0080"><span class="elsevierStyleLabel">3</span><p id="par0315" class="elsevierStylePara elsevierViewall">Given that&#44; IF numbers are the generalization of fuzzy numbers&#44; hence&#44; the approach which can be used for comparing IF numbers can also be used for comparing fuzzy numbers&#46; To show the shortcomings of the existing approach <a class="elsevierStyleCrossRef" href="#bib0100">&#91;20&#93;</a>&#44; two fuzzy numbers are compared by using such approaches &#91;<a class="elsevierStyleCrossRef" href="#bib0030">6</a>&#44;<a class="elsevierStyleCrossRef" href="#bib0070">14</a>&#44;<a class="elsevierStyleCrossRef" href="#bib0100">20</a>&#93; and it was demonstrated that the ordering of fuzzy numbers obtained by using Nehi&#8217;s approach <a class="elsevierStyleCrossRef" href="#bib0100">&#91;20&#93;</a> and Liou and Wang approach <a class="elsevierStyleCrossRef" href="#bib0070">&#91;14&#93;</a> is the same one and contradicts the ordering of fuzzy numbers obtained by using the Garcia and Lamata&#8217;s approach<a class="elsevierStyleCrossRef" href="#bib0030">&#91;6&#93;</a>&#44; because&#44; Garcia and Lamata&#8217;s approach <a class="elsevierStyleCrossRef" href="#bib0030">&#91;6&#93;</a> pointed out that their approach is better than Liou and Wang&#8217;s approach <a class="elsevierStyleCrossRef" href="#bib0070">&#91;14&#93;</a>&#46; Thus&#44; the ordering of IF numbers given by Nehi&#8217;s is not genuine&#46;<a name="p386"></a></p></li></ul></p><p id="par0320" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Example 4&#46;3&#46;</span> Let us consider two trapezoidal fuzzy numbers <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span>&#40;1&#44;4&#44;4&#44;5&#41; and B&#732;&#61;&#40;2&#44;3&#44;3&#44;6&#41;&#46; The intuitionistic representation of these trapezoidal fuzzy numbers is <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span>&#40;1&#44;1&#44;4&#44;4&#44;4&#44;4&#44;5&#44;5&#41; and B&#732;&#61;&#40;2&#44;2&#44;3&#44;3&#44;3&#44;3&#44;6&#44;6&#41;&#46;</p><p id="par0325" class="elsevierStylePara elsevierViewall">The ordering of these numbers obtained by using the existing ranking approaches &#91;<a class="elsevierStyleCrossRef" href="#bib0030">6</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0070">14</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0100">20</a>&#93; are shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a>&#46;</p><elsevierMultimedia ident="tbl0005"></elsevierMultimedia><p id="par0330" class="elsevierStylePara elsevierViewall">It is obvious from the results&#44; shown in <a class="elsevierStyleCrossRef" href="#tbl0005">Table 1</a> that the ordering of fuzzy numbers <span class="elsevierStyleItalic">&#195;</span> and B&#732;&#44; obtained by using the existing approach <a class="elsevierStyleCrossRef" href="#bib0100">&#91;20&#93;</a>&#44; is the same as the one obtained by the existing approach <a class="elsevierStyleCrossRef" href="#bib0070">&#91;14&#93;</a> whereas it is different from the ordering obtained by using the existing approach <a class="elsevierStyleCrossRef" href="#bib0030">&#91;6&#93;</a>&#46;</p></span><span id="sec0035" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">5</span><span class="elsevierStyleSectionTitle" id="sect0045">Proposed ranking approach for comparing two IF numbers</span><p id="par0335" class="elsevierStylePara elsevierViewall">Garcia and Lamata <a class="elsevierStyleCrossRef" href="#bib0030">&#91;6&#93;</a> pointed out that although trapezoidal fuzzy number <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#40;a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">4</span></span><span class="elsevierStyleItalic">&#41;</span>&#44; is defined by four points&#44; in the existing formula <a class="elsevierStyleCrossRef" href="#bib0070">&#91;14&#93;</a>&#44; &#8476;A&#732;&#61;&#955;&#8747;01&#40;a1&#43;a2&#8722;a1&#945;&#43;1&#8722;&#955;&#8747;01a4&#43;a3&#8722;a4&#945;&#41; where&#44; <span class="elsevierStyleItalic">&#955;</span> &#8712; &#91;0&#44;1&#93; the central points <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span> and <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span> are taken into account in an indirect way and proposed the following modified formula &#8476;A&#732;&#61;&#946;&#91;&#8747;01&#40;&#955;a2&#43;&#40;1&#8722;&#955;&#41;a3&#41;d&#945;&#93;&#43;1&#8722;&#946;&#91;&#955;&#8747;01&#40;a1&#43;&#40;a2&#8722;a1&#41;&#945;&#41;d&#945;&#43;&#40;1&#8722;&#955;&#41;&#8747;01&#40;a4&#43;&#40;a3&#8722;a4&#41;&#945;&#41;d&#945;&#93;&#41;&#46; Because in the existing formula <a class="elsevierStyleCrossRef" href="#bib0100">&#91;20&#93;</a>&#44; C&#956;kA&#732;&#61;k&#43;12&#8747;01rk&#91;a1&#43;&#40;a2&#8722;a1&#41;r&#43;a4&#43;&#40;a3&#8722;a4&#41;r&#93;dr&#44; C&#957;kA&#732;&#61;k&#43;12&#8747;01rk&#91;a&#39;1&#43;&#40;a&#39;2&#8722;a&#39;1&#41;r&#43;a&#39;4&#43;&#40;a&#39;3&#8722;a&#39;4&#41;r&#93;dr where&#44; <span class="elsevierStyleItalic">k</span> &#8712; &#91;0&#44;8&#41; the central points <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">2</span>&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf">3</span> and a&#8217;<span class="elsevierStyleInf">2</span>&#44; a&#8217;<span class="elsevierStyleInf">3</span> are also taken into account in an indirect way hence&#44; to overcome the shortcomings of existing approach <a class="elsevierStyleCrossRef" href="#bib0100">&#91;20&#93;</a>&#44; pointed out in <a class="elsevierStyleCrossRef" href="#sec0075">Section 4</a>&#44; a new ranking approach&#44; by modifying the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0100">&#91;20&#93;</a>&#44; is proposed for comparing IF numbers <span class="elsevierStyleItalic">&#195;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#40;a&#8217;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">1</span></span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#44;a&#8217;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#44; a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span><span class="elsevierStyleItalic">&#44; a&#8217;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span><span class="elsevierStyleItalic">&#44; a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">4</span></span><span class="elsevierStyleItalic">&#44;a&#8217;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">4</span></span><span class="elsevierStyleItalic">&#41;</span> and B&#732;&#61;&#40;b&#39;1&#44;b1&#44;b&#39;2&#44;b2&#44;b3&#44;b&#39;3&#44;b4&#44;b&#39;4&#41;&#46;</p><p id="par0340" class="elsevierStylePara elsevierViewall">The steps of the proposed ranking approach are as follows&#58;</p><p id="par0345" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 1&#46;</span> Calculate&#44; M&#956;&#946;&#44;kA&#732;&#61;&#40;&#946;&#91;k&#43;12&#8747;01rk&#40;a2&#43;a3&#41;dr&#93;&#43;&#40;1&#8722;&#946;&#41;&#91;k&#43;12&#8747;00rk&#91;a1&#43;&#40;a2&#8722;a1&#41;r&#43;a4&#43;&#40;a3&#8722;a4&#41;r&#93;dr&#93;&#41;&#44;M&#956;&#946;&#44;kB&#732;&#61;&#40;&#946;&#91;k&#43;12&#8747;01rk&#40;b2&#43;b3&#41;dr&#93;&#43;&#40;1&#8722;&#946;&#41;&#91;k&#43;12&#8747;00rk&#91;b1&#43;&#40;b2&#8722;b1&#41;r&#43;b4&#43;&#40;b3&#8722;b4&#41;r&#93;dr&#93;&#41;&#44; and check M&#956;&#946;&#44;k&#40;A&#732;&#41;&#62;M&#956;&#946;&#44;k&#40;B&#732;&#41; or M&#956;&#946;&#44;k&#40;A&#732;&#41;&#60;M&#956;&#946;&#44;k&#40;B&#732;&#41; or M&#956;&#946;&#44;k&#40;A&#732;&#41;&#61;M&#956;&#946;&#44;k&#40;B&#732;&#41;&#46;</p><p id="par0350" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;i&#41;</span> If M&#956;&#946;&#44;k&#40;A&#732;&#41;&#62;M&#956;&#946;&#44;k&#40;B&#732;&#41; then A&#732;&#8827;B&#732; i&#46;e&#46;&#44; minimum &#40;A&#732;&#44;B&#732;&#41;&#61;B&#732;&#46;</p><p id="par0355" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;ii&#41;</span> If M&#956;&#946;&#44;k&#40;A&#732;&#41;&#60;M&#956;&#946;&#44;k&#40;B&#732;&#41; then A&#732;&#8826;B&#732; i&#46;e&#46;&#44; minimum &#40;A&#732;&#44;B&#732;&#41;&#61;A&#732;&#46;</p><p id="par0360" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;iii&#41;</span> If M&#956;&#946;&#44;k&#40;A&#732;&#41;&#61;M&#956;&#946;&#44;k&#40;B&#732;&#41; then go to step 2&#46;</p><p id="par0365" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 2&#46;</span> Calculate&#44;<a name="p387"></a>M&#957;&#946;&#44;kA&#732;&#61;&#946;k&#43;12&#8201;&#8747;01rka&#39;2&#43;a&#39;3dr&#43;1&#8722;&#946;K&#43;12&#8201;&#8747;00rka&#39;1&#43;a&#39;2&#8722;a&#39;1r&#43;a&#39;4&#43;a&#39;3&#8722;a&#39;4rdrM&#957;&#946;&#44;kB&#732;&#61;&#946;k&#43;12&#8201;&#8747;01rkb&#39;2&#43;b&#39;3dr&#43;1&#8722;&#946;K&#43;12&#8201;&#8747;00rkb&#39;1&#43;b&#39;2&#8722;b&#39;1r&#43;b&#39;4&#43;b&#39;3&#8722;b&#39;4rdr&#44; and check -M&#965;&#946;&#44;k&#40;A&#732;&#41;&#62;-M&#965;&#946;&#44;k&#40;B&#732;&#41; or -M&#965;&#946;&#44;k&#40;A&#732;&#41;&#60;-M&#965;&#946;&#44;k&#40;B&#732;&#41; or -M&#965;&#946;&#44;k&#40;A&#732;&#41;&#61;-M&#965;&#946;&#44;k&#40;B&#732;&#41;&#46;</p><p id="par0370" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;i&#41;</span> If -M&#965;&#946;&#44;k&#40;A&#732;&#41;&#62;-M&#965;&#946;&#44;k&#40;B&#732;&#41; then A&#732;&#8827;B&#732; i&#46;e&#46;&#44; minimum &#40;A&#732;&#44;B&#732;&#41;&#61;B&#732;&#46;</p><p id="par0375" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;ii&#41;</span> If -M&#965;&#946;&#44;k&#40;A&#732;&#41;&#60;-M&#965;&#946;&#44;k&#40;B&#732;&#41; then A&#732;&#8826;B&#732; i&#46;e&#46;&#44; minimum &#40;A&#732;&#44;B&#732;&#41;&#61;A&#732;&#46;</p><p id="par0380" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;iii&#41;</span> If M&#965;&#946;&#44;k&#40;A&#732;&#41;&#61;M&#965;&#946;&#44;k&#40;B&#732;&#41; then A&#732;&#8773;B&#732;&#46;</p><p id="par0385" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Remark 3&#46;</span> For <span class="elsevierStyleItalic">&#946;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span>1&#47;3 and <span class="elsevierStyleItalic">k</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#61;</span><span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">Q</span> the index for membership and nonmembership functions M&#956;1&#47;3&#44;0&#40;A&#732;i&#41; and M&#965;1&#47;3&#44;0&#40;A&#732;i&#41;&#44; are as follows&#58;</p><p id="par0390" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0035"></elsevierMultimedia></p><span id="sec0040" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">5&#46;1</span><span class="elsevierStyleSectionTitle" id="sect0050">Advantages of the proposed ranking approach</span><p id="par0395" class="elsevierStylePara elsevierViewall">In this section&#44; the advantages of the proposed ranking approach&#44; over the existing ranking approaches for comparing fuzzy numbers as well as IF numbers &#91;<a class="elsevierStyleCrossRef" href="#bib0025">5</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0030">6</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0090">18</a>-<a class="elsevierStyleCrossRef" href="#bib0100">20</a>&#93; are discussed&#46;<ul class="elsevierStyleList" id="lis0040"><li class="elsevierStyleListItem" id="lsti0085"><span class="elsevierStyleLabel">1</span><p id="par0400" class="elsevierStylePara elsevierViewall">The existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0030">&#91;6&#93;</a> can be used only for comparing fuzzy numbers but cannot be used for comparing IF numbers&#46; Whereas the proposed ranking approach can be used for comparing fuzzy numbers as well as IF numbers and the ordering of fuzzy numbers&#44; obtained by using the proposed ranking approach&#44; is the same as that one obtained by using the existing ranking approach <a class="elsevierStyleCrossRef" href="#bib0030">&#91;6&#93;</a>&#46;</p></li><li class="elsevierStyleListItem" id="lsti0090"><span class="elsevierStyleLabel">2</span><p id="par0405" class="elsevierStylePara elsevierViewall">Although the existing ranking approaches &#91;<a class="elsevierStyleCrossRef" href="#bib0025">5</a>&#44; <a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44; <a class="elsevierStyleCrossRefs" href="#bib0085">17-20</a>&#93; can be used for comparing fuzzy numbers as well as IF numbers&#44; due to shortcomings pointed out in <a class="elsevierStyleCrossRef" href="#sec0030">Section 4</a>&#44; it is not genuine to apply these approaches&#46; Whereas by using the proposed ranking approach&#44; all the shortcomings&#44; occurring in the results due to the use of existing approaches &#91;<a class="elsevierStyleCrossRef" href="#bib0025">5</a>&#44;<a class="elsevierStyleCrossRef" href="#bib0065">13</a>&#44;<a class="elsevierStyleCrossRefs" href="#bib0085">17-20</a>&#93;&#44; are resolved&#46;</p></li></ul></p></span></span><span id="sec0045" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6</span><span class="elsevierStyleSectionTitle" id="sect0055">MCF problems in IF environment</span><p id="par0410" class="elsevierStylePara elsevierViewall">The aim of the MCF problems is to find minimum cost for transporting the product from a given point to certain destinations with the assumption that there is at least one node&#44; called intermediate node&#44; at which the product can be stored in the case of excess availability of the product and the stored product can be supplied in the case of excess demand of the product&#46;</p><p id="par0415" class="elsevierStylePara elsevierViewall">Ghatee and Hashemi <a class="elsevierStyleCrossRef" href="#bib0035">&#91;7&#93;</a> modified the existing linear programming formulation of balanced MCF problems by representing all of its parameters as fuzzy numbers instead of crisp numbers and proposed a method to find the fuzzy optimal solution of such balanced MCF problems&#46;</p><p id="par0420" class="elsevierStylePara elsevierViewall">On the same direction&#44; in this section the existing linear programming formulation of MCF problems is further generalized by representing all the parameters as TIF numbers and on the basis of the ranking approach&#44; proposed in <a class="elsevierStyleCrossRef" href="#sec0035">Section 5</a>&#44; a new method is proposed to find the IF optimal solution of such MCF problems&#46;</p><span id="sec0050" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;1</span><span class="elsevierStyleSectionTitle" id="sect0060">Linear programming formulations of balanced MCF problems in crisp and IF environment</span><span id="sec0055" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;1&#46;1</span><span class="elsevierStyleSectionTitle" id="sect0065">In this subsection&#44; the linear programming formulation of balanced MCF problems in crisp environment is presented <a class="elsevierStyleCrossRef" href="#bib0005">&#91;1&#93;</a>&#46;</span><p id="par0425" class="elsevierStylePara elsevierViewall">If the set of source nodes&#44; destination nodes&#44; such intermediate nodes from which some quantity of the product is supplied&#44; such intermediate nodes at which some quantity of the product is stored and such intermediate nodes from which the product is neither supplied nor stored&#44; of a directed and connected network&#44; are represented by <span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">S</span></span><span class="elsevierStyleItalic">&#44;N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">D</span></span><span class="elsevierStyleItalic">&#44; N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">IS</span></span><span class="elsevierStyleItalic">&#44;N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ID</span></span> and <span class="elsevierStyleItalic">N</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">IT</span></span> respectively&#46; Then&#44; the MCF problem can be formulated as<a name="p388"></a></p><p id="par0430" class="elsevierStylePara elsevierViewall">Minimize &#8721;&#40;i&#44;j&#41;&#8712;A&#40;cijxij&#41;</p><p id="par0435" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0440" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0040"></elsevierMultimedia></p><p id="par0445" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0450" class="elsevierStylePara elsevierViewall">Where <span class="elsevierStyleItalic">A</span> &#58; The set of arcs &#40;<span class="elsevierStyleItalic">i</span>&#44; y&#41;&#44; <span class="elsevierStyleItalic">x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span> &#58; Decision variable denoting the transportation amount of the product from <span class="elsevierStyleItalic">ith</span> node to <span class="elsevierStyleItalic">jth</span> node&#44; <span class="elsevierStyleItalic">C</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span> &#58; Cost for transporting one unit quantity of the product from <span class="elsevierStyleItalic">ith</span> node to <span class="elsevierStyleItalic">jth</span> node&#44; <span class="elsevierStyleItalic">a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#44; &#58; Supply of the product at <span class="elsevierStyleItalic">ith</span> source node&#44; <span class="elsevierStyleItalic">e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span>&#44; &#58; Supply of the product at <span class="elsevierStyleItalic">ith</span> intermediate source node&#44; <span class="elsevierStyleItalic">b</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> Demand of the product at <span class="elsevierStyleItalic">jth</span> destination node&#44; <span class="elsevierStyleItalic">d</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span> &#58; Demand of the product at <span class="elsevierStyleItalic">jth</span> intermediate destination node&#44; <span class="elsevierStyleItalic">I</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span>&#58; Minimum quantity of the product that can be transported from <span class="elsevierStyleItalic">ith</span> node to <span class="elsevierStyleItalic">jth</span> node&#44; <span class="elsevierStyleItalic">u</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span>&#58; Maximum quantity of the product that can be transported from <span class="elsevierStyleItalic">ith</span> node to <span class="elsevierStyleItalic">jth</span> node&#46;</p></span><span id="sec0060" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;1&#46;2</span><span class="elsevierStyleSectionTitle" id="sect0070">Linear programming formulation of balanced MCF problems in IF environment</span><p id="par0455" class="elsevierStylePara elsevierViewall">In this section the linear programming formulation of balanced MCF problems in IF environment is presented&#46;</p><p id="par0460" class="elsevierStylePara elsevierViewall">Suppose all the parameters <span class="elsevierStyleItalic">c</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span><span class="elsevierStyleItalic">&#44;x</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span><span class="elsevierStyleItalic">&#44;a</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleItalic">&#44;e</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">i</span></span><span class="elsevierStyleItalic">&#44;b</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span><span class="elsevierStyleItalic">&#44;d</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">j</span></span><span class="elsevierStyleItalic">&#44;l</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span><span class="elsevierStyleItalic">&#44;u</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">ij</span></span> are represented by non-negative TIF numbers c&#732;ij&#44;x&#732;ij&#44;a&#732;i&#44;e&#732;i&#44;b&#732;j&#44;d&#732;j&#44;l&#732;ij&#44;u&#732;ij&#46; Then &#40;<span class="elsevierStyleItalic">R</span><span class="elsevierStyleInf">1</span>&#41; in IF environment is&#58;</p><p id="par0465" class="elsevierStylePara elsevierViewall">inimize &#8721;&#40;i&#44;j&#41;&#8712;A&#40;c&#732;ij&#8855;x&#732;ij&#41;</p><p id="par0470" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0475" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0045"></elsevierMultimedia></p></span></span><span id="sec0065" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;2</span><span class="elsevierStyleSectionTitle" id="sect0075">Proposed method</span><p id="par0480" class="elsevierStylePara elsevierViewall">In this section&#44; a new method is proposed for finding the exact IF optimal solution of such MCF problems in which all the parameters are represented by IF numbers&#46; If the supply of product at <span class="elsevierStyleItalic">i</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">th</span></span> source node and <span class="elsevierStyleItalic">i</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">th</span></span> intermediate source node are a&#732;i&#61;&#40;a&#39;1i&#44;a1i&#44;a&#39;2i&#44;a2i&#44;a3i&#44;a&#39;3i&#44;a4i&#44;a&#39;4i&#41; and e&#732;i&#61;&#40;e&#39;1i&#44;e1i&#44;e&#39;2i&#44;e2i&#44;e3i&#44;e&#39;3i&#44;e4i&#44;e&#39;4i&#41; respectively and the demand of the product at <span class="elsevierStyleItalic">j</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">th</span></span> destination node and <span class="elsevierStyleItalic">j</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">th</span></span> intermediate destination node are b&#732;j&#61;&#40;b&#39;1j&#44;b1j&#44;b&#39;2j&#44;b2j&#44;b3j&#44;b&#39;3j&#44;b4j&#44;b&#39;4j&#41; and d&#732;j&#61;&#40;d&#39;1j&#44;d1j&#44;d&#39;2j&#44;d2j&#44;d3j&#44;d&#39;3j&#44;d4j&#44;d&#39;4j&#41; respectively&#44; then the exact IF optimal solution of IF MCF problems can be obtained by using the following steps</p><p id="par0485" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 1&#46;</span> Find the total fuzzy supply &#8721;i&#8712;NSa&#732;i&#8853;&#8721;i&#8712;NISe&#732;i and the total IF demand &#8721;j&#8712;NDb&#732;j&#8853;&#8721;j&#8712;NIDd&#732;j&#46; Let &#8721;i&#8712;NSa&#732;i&#8853;&#8721;i&#8712;NISe&#732;i&#61;&#40;m&#39;1&#44;m1&#44;m&#39;2&#44;m2&#44;m3&#44;m&#39;3&#44;m4&#44;m&#39;4&#41; and &#8721;j&#8712;NDb&#732;j&#8853;&#8721;j&#8712;NIDd&#732;j&#61;&#40;n&#39;1&#44;n1&#44;n&#39;2&#44;n2&#44;n3&#44;n&#39;3&#44;n4&#44;n&#39;4&#41;&#46;<a name="p389"></a></p><p id="par0490" class="elsevierStylePara elsevierViewall">Examine whether the problem is balanced or not&#44; i&#46;e&#46;&#44; &#8721;i&#8712;NSa&#732;i&#8853;&#8721;i&#8712;NISe&#732;i&#61;&#8721;j&#8712;NDb&#732;j&#8853;&#8721;j&#8712;NIDd&#732;j or &#8721;i&#8712;NSa&#732;i&#8853;&#8721;i&#8712;NISe&#732;i&#8800;&#8721;j&#8712;NDb&#732;j&#8853;&#8721;j&#8712;NIDd&#732;j&#46;</p><p id="par0495" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;1&#41;</span> If the problem is balanced i&#46;e&#46;&#44; &#8721;i&#8712;NSa&#732;i&#8853;&#8721;i&#8712;NISe&#732;i&#61;&#8721;j&#8712;NDb&#732;j&#8853;&#8721;j&#8712;NIDd&#732;j&#44; then go to step 2&#46;</p><p id="par0500" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;2&#41;</span> If the problem is unbalanced i&#46;e&#46;&#44; &#8721;i&#8712;NSa&#732;i&#8853;&#8721;i&#8712;NISe&#732;i&#8800;&#8721;j&#8712;NDb&#732;j&#8853;&#8721;j&#8712;NIDd&#732;j tnen convert the unbalanced problem into balanced problem as follows</p><p id="par0505" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;2a&#41;</span> Introduce a dummy source node with IF supply &#40;max&#123;0&#44; &#40;<span class="elsevierStyleItalic">n&#8217;</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m&#8217;</span><span class="elsevierStyleInf">1</span>&#41;&#125;&#44; max&#123;0&#44; &#40;<span class="elsevierStyleItalic">n&#8217;</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m&#8217;</span><span class="elsevierStyleInf">1</span>&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n&#8217;</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211; n&#8217;</span><span class="elsevierStyleInf">1</span>&#41; <span class="elsevierStyleItalic">&#8211;</span> &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m&#8217;</span><span class="elsevierStyleInf">1</span>&#41;&#125;&#44;max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n&#8217;</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m&#8217;</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n&#8217;</span><span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span> &#8211; <span class="elsevierStyleItalic">m&#8217;</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#41; <span class="elsevierStyleItalic">&#8211;</span> &#40;<span class="elsevierStyleItalic">m&#8217;</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#8211;m</span><span class="elsevierStyleInf">1</span>&#41;&#125;&#44;max &#123;0&#44;&#40;<span class="elsevierStyleItalic">n&#8217;</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n&#8217;</span><span class="elsevierStyleInf">1</span>&#41;<span class="elsevierStyleItalic">&#8211;</span>&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span><span class="elsevierStyleInf">1</span>&#41;<span class="elsevierStyleItalic">&#8211;</span>&#40;<span class="elsevierStyleItalic">m&#8217;</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;m</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#43;</span><span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleItalic">&#8211;</span>&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;&#44; max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n&#8217;</span><span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m&#8217;</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max &#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>&#123;0&#44; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span><span class="elsevierStyleItalic">&#8211; n</span><span class="elsevierStyleInf">2</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span> &#8211; <span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span>&#41;&#125;&#44; max&#123;0&#44; &#40;<span class="elsevierStyleItalic">n&#8217;</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m&#8217;</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123; 0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#41;<span class="elsevierStyleItalic">&#8211;</span>&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#8211;m</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#43;</span><span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">3</span> &#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span> &#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">3</span>&#41;&#8211; &#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">&#8211;m</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span>&#41;&#125;&#44;max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#8211; <span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#43;</span><span class="elsevierStyleHsp" style=""></span>max&#123;<span class="elsevierStyleItalic">0&#44;</span>&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleItalic">&#8211;</span>&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#43;</span><span class="elsevierStyleHsp" style=""></span>max &#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">3</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">4</span></span><span class="elsevierStyleItalic">&#8211; n</span>&#8217;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">4</span> &#8211; <span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span>&#41;&#125;&#44; max&#123;0&#44; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#44; <span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleItalic">&#8211;</span> &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span> &#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">3</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">4</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">3</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">4</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">4</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">4</span>&#41;&#8211;&#40;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">4</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">4</span>&#41;&#125;&#41; and also introduced a dummy destination node with IF demand&#40;max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;&#44; max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#125;<span class="elsevierStyleItalic">&#43;</span> max&#123;0&#44; &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41; <span class="elsevierStyleItalic">&#8211;</span> &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;&#44; max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleItalic">&#43;</span><span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;<span class="elsevierStyleItalic">&#8211;</span>&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span> &#8211; <span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44; &#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span><span class="elsevierStyleInf">1</span>&#41;&#125;&#44;max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max &#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125; &#43;max&#123;0&#44; &#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span> &#8211; <span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211; m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41; <span class="elsevierStyleItalic">&#8211;</span> &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;&#44;max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44; &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span> &#8211; <span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211; n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44; &#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span><span class="elsevierStyleItalic">&#8211;m</span><span class="elsevierStyleInf">2</span>&#41; <span class="elsevierStyleItalic">&#8211;</span> &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span><span class="elsevierStyleItalic">&#8211; n</span><span class="elsevierStyleInf">2</span>&#41;&#125;&#44;max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span> &#8211; <span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span> &#8211; <span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44; &#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max &#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">&#8211;m</span><span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleItalic">&#8211;</span>&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span><span class="elsevierStyleItalic">&#8211; n</span><span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleItalic">&#43;</span> max&#123;0&#44; &#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">&#8211;m</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span>&#41; <span class="elsevierStyleItalic">&#8211;</span> &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">&#8211;n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span>&#41;&#125;&#44; max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123; 0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max &#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span> &#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span>&#41;&#8211;&#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">3</span> &#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span> &#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">3</span> &#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">3</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">4</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">3</span>&#41;&#125;&#44;max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">1</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span><span class="elsevierStyleItalic">&#8211;m</span>&#8217;<span class="elsevierStyleInf">1</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">1</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">2</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span> &#8211;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span><span class="elsevierStyleItalic">&#8211;m</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">2</span></span>&#41; <span class="elsevierStyleItalic">&#8211;</span> &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span> &#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">2</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">3</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">3</span>&#8211;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">3</span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max &#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">4</span> &#8211;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">3</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span><span class="elsevierStyleInf">4</span><span class="elsevierStyleItalic">&#8211;n</span>&#8217;<span class="elsevierStyleInf"><span class="elsevierStyleItalic">3</span></span>&#41;&#125;<span class="elsevierStyleHsp" style=""></span>&#43;<span class="elsevierStyleHsp" style=""></span>max&#123;0&#44;&#40;<span class="elsevierStyleItalic">m</span>&#8217;<span class="elsevierStyleInf">4</span>&#8211;<span class="elsevierStyleItalic">m</span><span class="elsevierStyleInf">4</span>&#41; &#8211; &#40;<span class="elsevierStyleItalic">n</span>&#8217;<span class="elsevierStyleInf">4</span><span class="elsevierStyleItalic">&#8211; n</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">4</span></span>&#41;&#125;&#41;&#46; Assume the IF cost for transporting one unit quantity of the product from the introduced dummy source node to all intermediate nodes&#44; existing destination nodes and introduced dummy destination node as zero IF number&#46; Similarly&#44; assume the IF cost for transporting one unit quantity of the product from all intermediate nodes&#44; existing source nodes and introduced dummy source node to the introduced dummy destination node as zero IF number and go to Step 2&#46;</p><p id="par0510" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 2</span> Assuming c&#732;ij&#61;&#40;c&#39;1ij&#44;c1ij&#44;c&#39;2ij&#44;c2ij&#44;c3ij&#44;c&#39;3ij&#44;c4ij&#44;c&#39;4ij&#41;&#44;x&#732;ij&#61;&#40;x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij&#44;x&#39;4ij&#41;&#44;a&#732;i&#61;&#40;a&#39;1i&#44;a1i&#44;a&#39;2i&#44;a2i&#44;a3i&#44;a&#39;3i&#44;a4i&#44;a&#39;4i&#41;&#44;e&#732;i&#61;&#40;e&#39;1i&#44;e1i&#44;e&#39;2i&#44;e2i&#44;e3i&#44;e&#39;3i&#44;e4i&#44;e&#39;4i&#41;&#44;b&#732;j&#61;&#40;b&#39;1j&#44;b1j&#44;b&#39;2j&#44;b2j&#44;b3j&#44;b&#39;3j&#44;b4j&#44;b&#39;4j&#41;&#44;d&#732;j&#61;&#40;d&#39;1j&#44;d1j&#44;d&#39;2j&#44;d2j&#44;d3j&#44;d&#39;3j&#44;d4j&#44;d&#39;4j&#41;&#44;&#44;l&#732;ij&#61;&#40;l&#39;1ij&#44;l1ij&#44;l&#39;2ij&#44;l2ij&#44;l3ij&#44;l&#39;3ij&#44;l4ij&#44;l&#39;4ij&#41;&#8201;u&#732;ij&#61;&#40;u&#39;1ij&#44;u1ij&#44;u&#39;2ij&#44;u2ij&#44;u3ij&#44;u&#39;3ij&#44;u4ij&#44;u&#39;4ij&#41; using the arithmetic operations of IF numbers&#44; defined in Sub<a class="elsevierStyleCrossRef" href="#sec0020">section 2&#46;2</a> and using the IF linear programming formulation &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">2</span>&#41; of the balanced IF MCF problem&#44; the IF linear programming formulation of balanced IF MCF problem obtained in Step 1 can be written as Minimize &#8721;&#40;i&#44;j&#41;&#8712;A&#40;c&#39;1ijx&#39;1ij&#44;c1ijx1ij&#44;c&#39;2ijx&#39;2ij&#44;c2ijx2ij&#44;c3ijx3ij&#44;c&#39;3ijx&#39;3ij&#44;c4ijx4ij&#44;c&#39;4ijx&#39;4ij&#41;&#44;</p><p id="par0515" class="elsevierStylePara elsevierViewall">Subject to<a name="p390"></a></p><p id="par0520" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0050"></elsevierMultimedia></p><p id="par0525" class="elsevierStylePara elsevierViewall">is a non-negative IF number &#8704;&#40;<span class="elsevierStyleItalic">i</span>&#44;<span class="elsevierStyleItalic">j</span>&#41; &#8712; <span class="elsevierStyleItalic">A</span></p><p id="par0530" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 3&#46;</span> Using Definition 2&#46;7 and Definition 2&#46;8&#44; the IF linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">3</span>&#41;&#44; can be written as</p><p id="par0535" class="elsevierStylePara elsevierViewall">Minimize &#8721;&#40;i&#44;j&#41;&#8712;A&#40;c&#39;1ijx&#39;1ij&#44;c1ijx1ij&#44;c&#39;2ijx&#39;2ij&#44;c2ijx2ij&#44;c3ijx3ij&#44;c&#39;3ijx&#39;3ij&#44;c4ijx4ij&#44;c&#39;4ijx&#39;4ij&#41;</p><p id="par0540" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0545" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0055"></elsevierMultimedia><a name="p391"></a></p><p id="par0550" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0060"></elsevierMultimedia></p><p id="par0555" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 4&#46;</span> Suppose the IF linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">4</span>&#41; have <span class="elsevierStyleItalic">f</span> basic feasible solutions and &#123;&#40;x&#39;1ij&#41;w&#44;&#40;x1ij&#41;w&#44;&#40;x&#39;2ij&#41;w&#44;&#40;x2ij&#41;w&#44;&#40;x3ij&#41;w&#44;&#40;x&#39;3ij&#41;w&#44;&#40;x4ij&#41;w&#44;&#40;x&#39;4ij&#41;w&#125; be the <span class="elsevierStyleItalic">w</span><span class="elsevierStyleInf"><span class="elsevierStyleItalic">th</span></span> basic feasible solution then the goal is to find such a basic feasible solution corresponding to which the value of the objective function is minimum i&#46;e&#46;&#44;</p><p id="par0560" class="elsevierStylePara elsevierViewall">minimum&#40;&#8721;&#40;i&#44;j&#41;&#8712;A&#40;c&#8242;1ij&#40;x&#8242;1ij&#41;w&#44;c1ij&#40;x1ij&#41;w&#44;c&#8242;2ij&#40;x&#8242;2ij&#41;w&#44;c2ij&#40;x2ij&#41;w&#44;c3ij&#40;x3ij&#41;w&#44;c&#8242;3ij&#40;x&#8242;3ij&#41;w&#44;c4ij&#40;x4ij&#41;w&#44;c&#8242;4ij&#40;x&#8242;4ij&#41;w&#41; which can be obtained by using the ranking approach proposed in <a class="elsevierStyleCrossRef" href="#sec0035">Section 5</a> i&#46;e&#46;&#44; the IF optimal solution of the IF linear programming problem&#44; &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">4</span>&#41;&#44; can be obtained by solving the following crisp linear programming problem</p><p id="par0565" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0065"></elsevierMultimedia></p><p id="par0570" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0575" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0070"></elsevierMultimedia></p><p id="par0580" class="elsevierStylePara elsevierViewall">As well as all the constraints of problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">4</span>&#41; except &#40;<span class="elsevierStyleItalic">C</span><span class="elsevierStyleInf">1</span>&#41;</p><p id="par0585" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;i&#41;</span> If there does not exist any alternative optimal solution then put the values of x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij and x&#39;4ij in x&#732;ij&#61;&#40;x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij&#44;x&#39;4ij&#41; to find the IF optimal solution &#123;x&#732;ij&#125; and find the IF optimal value &#8721;&#40;i&#44;j&#41;&#8712;A&#40;c&#732;ij&#8855;x&#732;ij&#41; by putting the value of x&#732;ij&#46;</p><p id="par0590" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Case &#40;ii&#41;</span> If alternative solution exist then go to Step 5&#46;</p><p id="par0595" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 5&#46;</span> Solve the crisp linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">6</span>&#41; to find the optimal solution &#123;x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x3ij&#44;x&#39;4ij&#125;&#46;</p><p id="par0600" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0075"></elsevierMultimedia></p><p id="par0605" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0610" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0080"></elsevierMultimedia></p><p id="par0615" class="elsevierStylePara elsevierViewall">As well as all the constraints of problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">4</span>&#41; except &#40;<span class="elsevierStyleItalic">C</span><span class="elsevierStyleInf">1</span>&#41; where&#44; a is the optimal value of the crisp linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">5</span>&#41;&#46;</p><p id="par0620" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 6&#46;</span> Put the values of x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij and x&#39;4ij in x&#732;ij&#61;&#40;x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij&#44;x&#39;4ij&#41; to find the IF optimal solution &#123;x&#732;ij&#125;&#46;</p><p id="par0625" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 7&#46;</span>Put the values of x&#732;ij&#44; obtained from step 6&#44; in&#8721;&#40;i&#44;j&#41;&#8712;A&#40;c&#732;ij&#8855;x&#732;ij&#41;&#44; to find the minimum total IF transportation cost&#46;<a name="p392"></a></p></span><span id="sec0070" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;3</span><span class="elsevierStyleSectionTitle" id="sect0080">Advantages of the proposed method</span><p id="par0630" class="elsevierStylePara elsevierViewall">The existing method <a class="elsevierStyleCrossRef" href="#bib0035">&#91;7&#93;</a> can be used only to find the optimal solution of such MCF problems in which the parameters are represented by fuzzy numbers&#44; however&#44; the existing method <a class="elsevierStyleCrossRef" href="#bib0035">&#91;7&#93;</a> cannot be used to find the optimal solution for the same type of problems in which the parameters are represented by IF numbers&#44; whereas the proposed method can be used to find the optimal solution for both types of problems&#46;</p></span><span id="sec0075" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">6&#46;4</span><span class="elsevierStyleSectionTitle" id="sect0085">Illustrative example</span><p id="par0635" class="elsevierStylePara elsevierViewall">In this section&#44; to illustrate the proposed method the IF MCF problem&#44; chosen in Example 6&#46;1&#44; cannot be solved by the existing method <a class="elsevierStyleCrossRef" href="#bib0035">&#91;7&#93;</a>&#44; it can be solved by using the proposed method&#46;</p><p id="par0640" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Example 6&#46;1&#46;</span> Find the minimum IF transportation cost for the IF MCF problem with three nodes and three arcs as shown in <a class="elsevierStyleCrossRef" href="#f0005">Figure 2</a>&#46; The IF data for the chosen IF MCF problem is summarized in <a class="elsevierStyleCrossRef" href="#tbl0010">Table 2</a>&#46;</p><elsevierMultimedia ident="f0005"></elsevierMultimedia><elsevierMultimedia ident="tbl0010"></elsevierMultimedia><p id="par0645" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Solution&#46;</span> The optimal solution of IF MCF problem&#44; chosen in Example 6&#46;1&#44; was obtained as follows</p><p id="par0650" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 1&#46;</span> Total IF supply<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#40;50&#44;100&#44;150&#44;200&#44;250&#44;270&#44;300&#44;350&#41; and total &#124;F demand<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#40;0&#44;50&#44;100&#44;150&#44;250&#44;<span class="elsevierStyleHsp" style=""></span><span class="elsevierStyleHsp" style=""></span>270&#44;300&#44;400&#41;&#46; Given that the Total IF supply &#8800; total IF demand&#44; hence&#44; it is an unbalanced IF MCF problem&#46; Now&#44; as described in the proposed method &#40;using Case &#40;2a&#41; of Step 1 of the proposed method&#41;&#44; the unbalanced IF MCF problem can be converted into a balanced IF MCF problem&#44; by introducing a dummy source node 4 with IF supply &#40;0&#44;0&#44;0&#44;0&#44;50&#44;50&#44;50&#44;100&#41; and a dummy destination node 5 with IF demand &#40;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#41; as shown in <a class="elsevierStyleCrossRef" href="#f0010">Figure 3</a>&#44; hence&#44; that total IF supply<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>total IF demand i&#46;e&#46;&#44; &#40;50&#44;100&#44;150&#44;200&#44;250&#44;270&#44;300&#44;350&#41;<span class="elsevierStyleSup">&#8853;</span>&#40;0&#44;0&#44;0&#44;0&#44; 50&#44;50&#44;50&#44;100&#41;<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#40;0&#44;50&#44;100&#44;150&#44;250&#44;270&#44;300&#44;400&#41;<span class="elsevierStyleSup">&#8853;</span> &#40;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#41;&#46;</p><elsevierMultimedia ident="f0010"></elsevierMultimedia><p id="par0655" class="elsevierStylePara elsevierViewall">Assuming the IF cost for transporting one unit quantity of the product from the introduced dummy source node 4 to the intermediate nodes 2 and existing destination node 3 and introduced dummy destination node 5 as zero IF number&#46; Similarly&#44; assuming the IF cost for transporting one unit quantity of the product from intermediate nodes 2 and existing source node 1 and introduced dummy source node 4 to the introduced dummy destination node 5 as zero IF number i&#46;e&#46;&#44;c&#732;42&#61;c&#732;43&#61;c&#732;45&#61;c&#732;15&#61;c&#732;25&#61;&#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; the IF linear programming formulation of balanced IF MCF problem&#44; can be written as</p><p id="par0660" class="elsevierStylePara elsevierViewall">Minimize&#40;&#40;5&#44;10&#44;50&#44;100&#44;1000&#44;5500&#44;10000&#44;15000&#41;&#8855;x&#732;12 &#8853;&#40;20&#44;40&#44;200&#44;400&#44;4000&#44;22000&#44;40000&#44;60000&#41;&#8855; x&#732;13 &#8853;&#40;15&#44;30&#44;150&#44;300&#44;3000&#44;16500&#44;30000&#44;45000&#41; &#8855; x&#732;23 &#8853;&#40;0&#44; 0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;42 &#8853;&#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;43 &#8853;&#40;0&#44;0&#44;0&#44;0&#44;0&#44; <a name="p393"></a>0&#44;0&#44;0&#41; &#8855; x&#732;45 &#8853;&#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;15 &#8853;&#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;25&#41;</p><p id="par0665" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0670" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0085"></elsevierMultimedia></p><p id="par0675" class="elsevierStylePara elsevierViewall">x&#732;12&#44;x&#732;13&#44;x&#732;15&#44;x&#732;23&#44;x&#732;25&#44;x&#732;42&#44;x&#732;43&#44;x&#732;45 are non-negative IF numbers&#46;</p><p id="par0680" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 2&#46;</span> Assuming x&#732;ij&#61;&#40;x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij&#44;x&#39;4ij&#41; and using the arithmetic operations&#44; the IF linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">7</span>&#41; can be written as</p><p id="par0685" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0090"></elsevierMultimedia></p><p id="par0690" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0695" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0095"></elsevierMultimedia></p><p id="par0700" class="elsevierStylePara elsevierViewall">x&#732;12&#44;x&#732;13&#44;x&#732;15&#44;x&#732;23&#44;x&#732;25&#44;x&#732;42&#44;x&#732;43&#44;x&#732;45 are non-negative IF numbers&#46;</p><p id="par0705" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 3&#46;</span> Using the Definition 2&#46;7 and Definition 2&#46;8&#44; the IF linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">8</span>&#41;&#44; can be written as</p><p id="par0710" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0100"></elsevierMultimedia><a name="p394"></a></p><p id="par0715" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0720" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0105"></elsevierMultimedia></p><p id="par0725" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0110"></elsevierMultimedia></p><p id="par0730" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 4&#46;</span> Using Step 4 of the proposed method and assuming <span class="elsevierStyleItalic">&#946;</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>1&#47;3&#44;<span class="elsevierStyleItalic">k</span><span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>0&#44; the IF optimal solution of the IF linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">9</span>&#41;&#44; can be obtained by solving the following crisp linear programming problem</p><p id="par0735" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0115"></elsevierMultimedia></p><p id="par0740" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0745" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0120"></elsevierMultimedia></p><p id="par0750" class="elsevierStylePara elsevierViewall">As well as all the constraints of the problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">9</span>&#41; except &#40;<span class="elsevierStyleItalic">C</span><span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleItalic">&#8217;</span></p><p id="par0755" class="elsevierStylePara elsevierViewall">Given that on solving the crisp linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">10</span>&#41;&#44; alternative optimal solutions exist i&#46;e&#46;&#44; case &#40;ii&#41; of Step 4 of the proposed method is satisfied and the optimal value of the crisp linear programming problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">10</span>&#41; is 4761400&#47;6 hence&#44; by using Step 5 of the proposed method the IF optimal solution of the chosen IF MCF problem can be obtained by solving the following crisp linear programming problem&#58;</p><p id="par0760" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0125"></elsevierMultimedia><a name="p395"></a></p><p id="par0765" class="elsevierStylePara elsevierViewall">Subject to</p><p id="par0770" class="elsevierStylePara elsevierViewall"><elsevierMultimedia ident="eq0130"></elsevierMultimedia></p><p id="par0775" class="elsevierStylePara elsevierViewall">As well as all the constraints of problem &#40;<span class="elsevierStyleItalic">P</span><span class="elsevierStyleInf">9</span>&#41; except &#40;<span class="elsevierStyleItalic">C</span><span class="elsevierStyleInf">2</span>&#41;<span class="elsevierStyleItalic">&#8217;</span></p><p id="par0780" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 5&#46;</span> Solving the crisp linear programming problem&#44; obtained in Step 4&#44; the optimal values of x112&#44;x&#39;212&#44;x212&#44;x312&#44;x&#39;312&#44;x412&#44;x&#39;412&#44;x&#39;213&#44;x213&#44;x313&#44;x&#39;313&#44;x413&#44;x&#39;413&#44;x&#39;115&#44;x115&#44;x&#39;215&#44;x215&#44;x&#39;315&#44;x415&#44;x&#39;415&#44;x123&#44;x&#39;223&#44;x223&#44;x323&#44;x&#39;323&#44;x423&#44;x&#39;423&#44;x343&#44;x&#39;343&#44;x&#39;443 and x&#39;443 are 50&#44;90&#44;130&#44;180&#44;200&#44;230&#44;280&#44;10&#44;20&#44;20&#44;20&#44;20&#44;20&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;30&#44;30&#44;30&#44;30&#44;30&#44;30&#44;30&#44;50&#44;50&#44;50&#44;and 100 respectively&#46;</p><p id="par0785" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 6&#46;</span> Put the values of x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij and x&#39;4ij in x&#732;ij&#61;&#40;x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij&#44;x&#39;4ij&#41; the IF optimal solution is x&#732;12<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#40;0&#44;50&#44;90&#44;130&#44;180&#44;200&#44;230&#44;280&#41;&#44; x&#732;13<span class="elsevierStyleHsp" style=""></span>&#61;&#40;0&#44;0&#44; 10&#44;20&#44;20&#44;20&#44;20&#44;20&#41;&#44; x&#732;15<span class="elsevierStyleHsp" style=""></span>&#61;&#40;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#44;50&#41;&#44; x&#732;23<span class="elsevierStyleHsp" style=""></span>&#61;<span class="elsevierStyleHsp" style=""></span>&#40;0&#44;30&#44;30&#44;30&#44;30&#44;30&#44;30&#44;30&#41;&#44; x&#732;43<span class="elsevierStyleHsp" style=""></span>&#61;&#40;0&#44;0&#44;0&#44;0&#44;50&#44;50&#44;50&#44;100&#41; and the remaining x&#732;ij are &#40;&#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41;&#46;</p><p id="par0790" class="elsevierStylePara elsevierViewall"><span class="elsevierStyleBold">Step 7&#46;</span> Putting the values of x&#732;ij&#61;&#40;x&#39;1ij&#44;x1ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;2ij&#44;x2ij&#44;x3ij&#44;x&#39;3ij&#44;x4ij&#44;x&#39;4ij&#41; in &#40;&#40;5&#44;10&#44;50&#44;100&#44;1000&#44;5500&#44;10000&#44;15000&#41; &#8855; x&#732;12 &#8853;&#40;20&#44;40&#44;200&#44;400&#44; 4000&#44;22000&#44;40000&#44;60000&#41; &#8855; x&#732;13 &#8853; &#40;15&#44;30&#44;150&#44;300&#44; 3000&#44;16500&#44;30000&#44;45000&#41; &#8855; x&#732;23 &#8853; &#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;42 &#8853; &#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;43 &#8853; &#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;45 &#8853; &#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;15 &#8853; &#40;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#44;0&#41; &#8855; x&#732;25&#41; the minimum total IF transportation cost is &#40;0&#44;1400&#44;11000&#44;20000&#44;350000&#44;2035000&#44;4000000&#44;6750000&#41;</p></span></span><span id="sec0080" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">7</span><span class="elsevierStyleSectionTitle" id="sect0090">Comparative study</span><p id="par0795" class="elsevierStylePara elsevierViewall">The results of the existing fuzzy MCF problem &#40;Example 3&#46;5 <a class="elsevierStyleCrossRef" href="#bib0040">&#91;8&#93;</a>&#44; pp&#46; 2498&#41; and the IF MCF problem&#44; chosen in Example 6&#46;1&#44; obtained by using the existing method and the method proposed in Sub<a class="elsevierStyleCrossRef" href="#sec0065">section 6&#46;2</a>&#44; are shown in <a class="elsevierStyleCrossRef" href="#tbl0015">Table 3</a>&#46;</p><elsevierMultimedia ident="tbl0015"></elsevierMultimedia><p id="par0800" class="elsevierStylePara elsevierViewall">Given that in the existing MCF problem &#40;Example 3&#46;5 <a class="elsevierStyleCrossRef" href="#bib0040">&#91;8&#93;</a>&#44; pp&#46; 2498&#41; all the parameters are represented by fuzzy numbers hence&#44; as discussed in Sub<a class="elsevierStyleCrossRef" href="#sec0070">section 6&#46;3</a>&#44; it can be solved by using the existing method <a class="elsevierStyleCrossRef" href="#bib0040">&#91;8&#93;</a> as well as by the proposed method&#44; whereas in the MCF problem&#44; chosen in Example 6&#46;1&#44; all the parameters are represented by IF numbers&#46; Thus&#44; as discussed in Sub<a class="elsevierStyleCrossRef" href="#sec0070">section 6&#46;3</a>&#44; it can be solved by the proposed method but cannot be solved by using the existing method <a class="elsevierStyleCrossRef" href="#bib0040">&#91;8&#93;</a>&#46;</p></span><span id="sec0085" class="elsevierStyleSection elsevierViewall"><span class="elsevierStyleLabel">8</span><span class="elsevierStyleSectionTitle" id="sect0095">Conclusion</span><p id="par0805" class="elsevierStylePara elsevierViewall">Based on the proposed study&#44; it can be concluded that it is better to use the proposed ranking approach for comparing fuzzy and IF numbers as compared to existing ranking approaches&#46; Also&#44; It can be concluded that it is not possible to find the IF optimal solution of IF MCF problems by using any of the existing methods&#46; Only the proposed method can be used for the same problems&#46;</p></span></span>"
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        "resumen" => "<span id="abst0005" class="elsevierStyleSection elsevierViewall"><p id="spar0020" class="elsevierStyleSimplePara elsevierViewall">To the best of our knowledge very few methods have been proposed in previous studies for comparing intuitionistic fuzzy &#40;IF&#41; numbers&#46; In this paper&#44; the limitations and the shortcomings of all these existing methods are pointed out&#46; In order to overcome these limitations and shortcomings a new ranking approach&#8212;by modifying an existing ranking approach&#8212;is proposed for comparing IF numbers&#46; Thus&#44; with the help of proposed the ranking approach&#44; a new method is proposed to find the optimal solution of such unbalanced minimum cost flow &#40;MCF&#41; problems in which all the parameters are represented by IF numbers&#46;</p></span>"
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        "texto" => "<p id="par0810" class="elsevierStylePara elsevierViewall">I&#44; Dr&#46; Amit Kumar&#44; want to acknowledge the adolescent inner blessings of Mehar&#46; I believe that Mehar is an angel for me and without Mehar&#8217;s blessing it would not be possible to mature the idea proposed in this paper&#46; Mehar is a lovely daughter of Parmpreet Kaur &#40;Research Scholar under my supervision&#41;&#46; The authors also acknowledge the <a name="p396"></a>financial support given by the University Grant Commission&#44; Govt&#44; of India for completing the Major Research&#46; Project &#40;39-40&#47;2010&#40;SR&#41;&#41;&#46;</p>"
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Article information

ISSN: 16656423
Original language: English

DOI:10.1016/S1665-6423(13)71548-7

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Year/Month	Html	Pdf	Total

2024 October	9	0	9
2024 September	13	3	16
2024 August	28	5	33
2024 July	20	6	26
2024 June	20	7	27
2024 May	21	1	22
2024 April	34	3	37
2024 March	23	9	32
2024 February	26	4	30
2024 January	20	3	23
2023 December	21	6	27
2023 November	40	13	53
2023 October	55	8	63
2023 September	39	2	41
2023 August	40	5	45
2023 July	43	7	50
2023 June	46	2	48
2023 May	79	8	87
2023 April	49	4	53
2023 March	64	10	74
2023 February	32	8	40
2023 January	48	16	64
2022 December	43	5	48
2022 November	49	7	56
2022 October	40	12	52
2022 September	30	8	38
2022 August	54	16	70
2022 July	34	10	44
2022 June	39	17	56
2022 May	31	16	47
2022 April	27	7	34
2022 March	35	12	47
2022 February	37	6	43
2022 January	61	11	72
2021 December	63	17	80
2021 November	74	17	91
2021 October	83	16	99
2021 September	82	27	109
2021 August	87	17	104
2021 July	47	22	69
2021 June	19	8	27
2021 May	43	14	57
2021 April	106	27	133
2021 March	58	9	67
2021 February	53	9	62
2021 January	37	14	51
2020 December	27	7	34
2020 November	55	7	62
2020 October	29	11	40
2020 September	24	11	35
2020 August	22	11	33
2020 July	21	2	23
2020 June	18	2	20
2020 May	109	11	120
2020 April	18	3	21
2020 March	18	16	34
2020 February	29	5	34
2020 January	20	5	25
2019 December	15	2	17
2019 November	9	2	11
2019 October	9	2	11
2019 September	12	0	12
2019 August	10	2	12
2019 July	17	14	31
2019 June	87	6	93
2019 May	111	15	126
2019 April	40	4	44
2019 March	4	2	6
2019 February	7	1	8
2019 January	9	4	13
2018 December	7	0	7
2018 November	1	1	2
2018 October	8	0	8
2018 September	7	4	11
2018 August	4	0	4
2018 July	4	2	6
2018 June	8	0	8
2018 May	5	2	7
2018 April	5	5	10
2018 March	2	3	5
2018 February	1	1	2
2018 January	8	2	10
2017 December	4	3	7
2017 November	4	1	5
2017 October	6	2	8
2017 September	8	2	10
2017 August	6	2	8
2017 July	8	2	10
2017 June	9	2	11
2017 May	16	10	26
2017 April	16	1	17
2017 March	10	15	25
2017 February	8	9	17
2017 January	8	0	8
2016 December	10	6	16
2016 November	8	1	9
2016 October	13	5	18
2016 September	9	2	11
2016 August	2	0	2
2016 July	5	1	6
2016 June	3	6	9
2016 May	4	14	18
2016 April	3	1	4
2016 March	3	2	5
2016 February	5	6	11
2016 January	4	6	10
2015 December	7	0	7
2015 November	7	1	8
2015 October	16	3	19
2015 September	15	3	18
2015 August	3	1	4
2015 July	5	1	6
2015 June	1	0	1
2015 May	4	2	6
2015 April	2	1	3

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