Supplier risk assessment based on trapezoidal intuitionistic fuzzy numbers and ELECTRE TRI-C: a case illustration involving service suppliers

Abstract

Many companies today have embraced the concept of risk management, usually in the form of enterprise risk management or supply chain risk management. Both are based on a holistic view of risks. Hence, risks related to specific functions within a company must be considered more broadly than previously. Risks, however, involve uncertainty, and the less specific the context in which risks are viewed, the more uncertainty will be involved. One particular way to express uncertainty is through trapezoidal intuitionistic fuzzy numbers (TrIFNs). In this paper, risks that are relevant for supplier risk assessments are first collected from the literature. Then it is illustrated how the multi-criteria decision analysis method ELECTRE TRI-C can be used for sorting suppliers into risk categories, when the risks as well as some of the method’s parameters are expressed with TrIFNs. In order to do this, we make use of a small modification of an existing method for converting TrIFNs into crisp values. The approach is illustrated in a case problem based on a company that is looking for service providers (suppliers) of electrical maintenance. The problem involves 20 suppliers that are sorted into three risk categories based on evaluations from 27 criteria. Results from the case study point to two low risk suppliers. A further ad-hoc analysis suggests one of these to be less risky than the other.

Appendices

Appendix A

A.1. Decision matrix

An MCDA problem generally include a set of alternatives or actions A={a _i :i=1, 2, …, n}, which has to be evaluated on a set of criteria F={g _j :j=1, 2, …, m}. It is common to denote an evaluation or performance of alternative a _i on criterion g _j as g _j (a _i ), where g _j ( ⋅ ) is a criterion function corresponding to the scale and the possible values of evaluations on criterion g _j .

A decision matrix M _{n × m} with n rows and m columns is defined as:

Note that in standard matrix notation m is used to indicate the number of rows and n to indicate the number of columns. In MCDA problems, it is customary to use n and m to indicate the number of alternatives, respectively, criteria. Since we prefer to present alternatives in rows, we use n rows and m columns.

When analysing an MCDA problem, we need to know the direction of preferences on each criterion, i.e. if a high or a low value of g _j (a _i ) is preferred. Criteria of the former type are referred to as benefit criteria and criteria of the latter type as cost criteria.

A.2. Criteria weights

Many MCDA methods require that each criterion be given a weight, in order to indicate the relative importance of the criteria. These weights are interpreted differently in various MCDA methods (Choo et al, 1999). In ELECTRE methods, each weight is interpreted as the voting power of the corresponding criterion in a coalition in favour of an outranking (Figueira et al, 2005).

Several methods have been proposed to aid a DM in determining criteria weights (Choo et al, 1999). Some applications of such methods can be found in Massam (1991) (AHP), Milani and Shanian (2006) (Entropy method), Certa et al (2009) (ANP), and Zardari et al (2010) (Conjoint analysis). Specifically aimed at ELECTRE, Rogers and Bruen (1998) proposed the ‘resistance to change grid’, Figueira and Roy (2002) presented the revised Simos’ card procedure, and Greco et al (2002) proposed to use decision rules obtained by a dominance-based rough set approach. For ELECTRE TRI, a number of preference disaggregation procedures have been developed (see Govindan and Jepsen, 2015).

We denote a set of criteria weights as {k _j |k _j >0, ∀j∈J}.

A.3. Discrimination thresholds and types of criteria

The most general form of criteria used in ELECTRE methods are pseudo-criteria. A pseudo-criterion takes into account the imperfect nature of a DM’s evaluations of the alternatives. For a pseudo-criterion g _j ∈F, two threshold functions must be defined (Figueira et al, 2010):

• Indifference threshold q _j ( ⋅ )

• Preference threshold p _j ( ⋅ )

These two thresholds are generally referred to as discrimination thresholds. An indifference threshold indicates a zone, in which the difference of the evaluations of two alternatives does not justify a preference of one over the other. The preference threshold, on the other hand, indicates a level, such that in order for an alternative a to be regarded as better than alternative a′, the difference g _j (a)−g _j (a′) of the evaluations must exceed the given threshold level. If the difference lies in a zone between the indifference threshold and the preference threshold, then a hesitation between indifference and preference occurs (Figueira et al, 2005).

The threshold functions must satisfy the condition p _j (g _j (a))⩾q _j (g _j (a))⩾0 for all a∈A. They can either be constant or vary according to the evaluations. If a threshold is variable, then a distinction is made between a direct threshold and an inverse threshold, corresponding to whether the threshold is computed using the best or the worst evaluation, respectively, whenever two alternatives are under consideration (Figueira et al, 2010).

When using variable thresholds, further restrictions apply to the parameters of the functions. Usually the threshold functions are given as affine functions. In such a case, the function of a direct indifference threshold can be expressed⁹ as , where , such that α _qj >−1 on a benefit criterion and on a cost criterion. An inverse threshold is expressed as , where , such that on a benefit criterion and on a cost criterion. For preference thresholds, simply exchange q with p everywhere. If a threshold function is defined as direct (inverse), then the parameters necessary for the function for the inverse (direct) threshold can be obtained through a transformation, because direct and inverse thresholds are functionally linked. Note that when using variable thresholds in ELECTRE TRI, TRI-C, and TRI-nC, the threshold functions are based on the performance of a reference norm. These methods are created, such that the algorithm takes care of the transformations, because separate formulas are used if an alternative is compared to a norm or if a norm is compared to an alternative. In other ELECTRE methods with pseudo-criteria, the formulas are based on input in the form of direct thresholds; hence, if inverse thresholds are preferred, then the parameters must be transformed. For more details on variable thresholds, please refer to Dias (2011). In this paper, we only consider input in the form of constant thresholds.

Let p _j (a′) and q _j (a′) be the thresholds associated with criterion g _j ∈F and a particular alternative a′∈A. Then a pseudo-criterion g _j reduces to (Roy, 1996):

• A true criterion, if p _j (a)=q _j (a)=0 for all a∈A

• A semi-criterion, if p _j (a)=q _j (a) for all a∈A

• A pre-criterion, if q _j (a)=0 for all a∈A

In ELECTRE I and II (and the unofficial ELECTRE IV), only true criteria are used, whereas the other ELECTRE methods are based on pseudo-criteria. Any single criterion in the latter methods can still be reduced to any of the other types of criteria though.

A.4. Veto thresholds

A veto threshold v _j ( ⋅ ) can be assigned to each criterion. In the case of pseudo-criteria, it can be either constant or a function of the performances. As was the case for the discrimination thresholds, a further distinction is made between a direct and an inverse threshold if the threshold is variable. Generally, it must satisfy the condition v _j (g _j (a))⩾p _j (g _j (a)) for all a∈A and if variable thresholds are used, similar restrictions apply as the ones mentioned in Section A.3 for the indifference and preference thresholds. In the method referred to as ELECTRE IV and in applications, where veto thresholds are used in ELECTRE I and II, the thresholds are usually constant. In this paper, we only consider input in the form of constant thresholds.

A veto threshold indicates the maximum difference allowed between the performances of two alternatives on a specific criterion. A single veto threshold can invalidate an outranking regardless of what the other performances are, if the difference between two performances on a given criterion exceeds the veto thresholds (Roy, 1991).

A.5. Concordance, discordance, and credibility indices used in some ELECTRE methods

The formulas in the following subsections are used in ELECTRE III, TRI, TRI-C, and TRI-nC. The concordance formula is also used in ELECTRE IS.

A.5.1. Concordance indices

A concordance index c(x, y) for an ordered pair (x, y) is given as follows:

where c _j (x, y) is the jth partial concordance index for (x, y) and given as follows:

For a benefit criterion:

The formula for c _j (x, y) on a benefit criterion can be generalized to:

For a cost criterion:

The formula for c _j (x,y) on a cost criterion can be generalized to:

A.5.2. Partial discordance indices

A partial concordance index d _j (x, y) for an ordered pair (x, y) is given as follows:

For a benefit criterion

The formula above can be generalized to:

For a cost criterion

The formula above can be generalized to:

A.5.3. Credibility indices

A credibility index σ(x, y) for an ordered pair (x, y) is given as follows:

where

A.6. ELECTRE TRI-C

ELECTRE TRI-C was designed for the sorting problematic, but instead of using limits (profiles) to determine the categories as in ELECTRE TRI, a characteristic action must be defined for each category. A characteristic action indicates the typical performance on each criterion g _j ∈F of an action (alternative) in the corresponding category. Apart from these characteristic actions, an additional parameter, called a credibility level, needs to be determined. It is similar to the cutting level in ELECTRE TRI, but has an additional restriction. In ELECTRE TRI-C, the exploitation phase includes two assignment rules as in ELECTRE TRI, but they must be considered jointly. Hence, if the two rules do not assign a particular alternative to the same category, then the two different categories define a range of possible categories for the final assignment of the alternative.

A stepwise illustration is provided in the following sub-section. The illustration is based on the description in Dias et al (2010). Note that the notation in the illustration is based on constant thresholds. For variable thresholds, simply exchange the constant threshold with the expression for the variable threshold.

A.6.1. Stepwise illustration of ELECTRE TRI-C

The stepwise illustration below is based on the description in Dias et al (2010). Note that we interchangeably use the terms action and alternative in the illustration.

Step 1::

Create a decision matrix M and assign a weight k _j to each criterion g _j ∈F.

Step 2::

Determine indifference and preference thresholds q _j and p _j for each criterion g _j ∈F, such that p _j ⩾q _j (see section A.3).

Step 3::

For any criterion g _j ∈F, a veto threshold v _j ⩾p _j can be determined if desired (see Section A.4)

Step 4::

In this step, a set of ordered categories must be defined. For this purpose, a set of characteristic actions b _h , h=1, …, q must be defined, where q is the number of desired categories. Let g _j (b _h ) denote the performance value of b _h on criterion g _j and {C _h } _h =₁, …, _q the set of categories. A characteristic action b _h must be defined, such that g _j (b _h ) is a typical performance value on criterion g _j for an alternative assigned to category C _h .

In what follows, let:

• F ⁺={g _j ∈F|g _j is a benefit criterion}

• F ⁻={g _j ∈F|g _j is a cost criterion}

The characteristic actions must as a minimum fulfil the following conditions:

• g _j (b _h+1)⩾g _j (b _h ) for h=1, …, q−1 and for all g _j ∈F ⁺

• g _j (b _h+1)⩽g _j (b _h ) for h=1, …, q−1 and for all g _j ∈F ⁻

• At least one of the following two conditions is true:

  • ° There is at least one g _j ∈F ⁺, such that g _j (b _h+1)>g _j (b _h ) for h=1, …, q−1

  • ° There is at least one g _j ∈F ⁻, such that g _j (b _h+1)<g _j (b _h ) for h=1, …, q−1

• The set {b _h |h=1, …, q} of characteristic actions fulfills the weak separability condition (see Section A.6.2)

In addition, any of the two stronger conditions strict separability or hyper-strict separability can be applied (see Section A.6.2). The stronger the condition the set of characteristic actions fulfills, the better the categories can be distinguished by the ELECTRE TRI-C method.

Finally, two special characteristic actions b ₀ and b _q+1 must be defined, such that:

• g _j (b ₀)<g _j (a)<g _j (b _q+1) for all g _j ∈F ⁺, a∈A

• g _j (b ₀)>g _j (a)>g _j (b _q+1) for all g _j ∈F ⁻, a∈A

• g _j (b ₁)−g _j (b ₀)>0 for all g _j ∈F ⁺

• g _j (b ₀)−g _j (b ₁)>0 for all g _j ∈F ⁻

• g _j (b _q+1)−g _j (b _q )>0 for all g _j ∈F ⁺

• g _j (b _q )−g _j (b _q+1)>0 for all g _j ∈F ⁻

That is, the two special characteristic actions represents the worst and the best possible performance, respectively, of an alternative on the specific criterion.

Let B={b _h |h=0, …, q+1}.

Step 5::

Compute a concordance index c(a, b _h ) for any pair (a, b _h ), a∈A, b _h ∈B. See section A.5.1 for formulas.

Step 6::

Compute a concordance index c(b _h , a) for any pair (a, b _h ), a∈A, b _h ∈B. See Section A.5.1 for formulas.

Step 7::

Compute partial discordance indices d _j (a _i , b _h ) for any pair (b _h , a), a∈A, b _h ∈B. See section A.5.2 for formulas.

Step 8::

Compute partial discordance indices d _j (b _h , a) for any pair (b _h , a), a∈A, b _h ∈B. See Section A.5.2 for formulas.

Step 9::

Compute a credibility index σ _j (a, b _h ) for any pair (a, b _h ), a∈A, b _h ∈B. See Section A.5.3 for formulas.

Step 10::

Compute a credibility index σ _j (b _h , a) for any pair (b _h , a), a∈A, b _h ∈B. See Section A.5.3 for formulas.

Step 11::

Determine a minimum credibility level λ, 1/2⩽λ⩽1, such that

λ⩾max{σ(b _h , b _h+1)∣h=1, …, q−1}.

The credibility level is the minimum degree of credibility considered necessary to validate an outranking of one action over another. It should, in general, be within the range [0.5, 1]. If, however, the set of characteristic actions only fulfills the weak separability condition, then the minimum required credibility level is further restricted by the maximum of credibility indices σ(b _h , b _h+1), h=1, 2, …,q−1, such that in addition to the restriction λ∈[0.5, 1], also max{σ(b _h , b _h+1)∣h=1, 2, …, q−1}⩽λ⩽1 must be fulfilled (Dias et al, 2010).

Step 12::

As is the case for ELECTRE TRI, ELECTRE TRI-C uses two assignment (exploitation) procedures. In ELECTRE TRI-C, however, the resulting assignments should be considered jointly. That is, if the two procedures suggest different categories for an alternative to be placed in, then the two categories define a range of possible categories for the alternative under consideration.

Before the two procedures are illustrated in Steps 13–15, we need to define the possible preference situations between a∈A and b _h ∈B. Let aP ⁺ b _h , denote a preference of a over b _h and let aP ⁻ b _h denote the situation b _h P ⁺ a. For the two special characteristic actions b ₀ and b _q+1, it is always the case that aP ⁺ b ₀ and aP ⁻ b _q+1. Indifference is indicated as aIb _h and incomparability as aRb _h .

The preference situations between a∈A and b _h ∈B are defined as follows:

• If σ(a, b _h )⩾λ and σ(b _h , a)⩾λ, then aIb _h .

• If σ(a, b _h )⩾λ and σ(b _h , a)<λ, then aP ⁺ b _h .

• If σ(a, b _h )<λ and σ(b _h , a)⩾λ, then aP ⁻ b _h .

• If σ(a, b _h )<λ and σ(b _h , a)<λ, then aRb _h .

Note that Dias et al (2010) use the notation aI ^λ b _h (λ-indifference), aP ^λ b _h (λ-preference of a _i over b _h ) and aR ^λ b _h (λ-incomparability).

Step 13::

For the two assignment procedures, it is necessary to define a selecting function ρ, which is used to select between two consecutive categories for the assignment of an alternative in each of the procedures in Steps 14–15. Dias et al (2010) define a number of properties, the selecting function should have, and propose the use of the min selecting function defined as:

Step 14::

Apply the first assignment procedure called the descending rule.

Dias et al (2010) describe the descending rule as follows:

• Decrease h from q+1 until the first value t, such that σ(a, b _t )⩾λ

  • If t=q, select C _q as a possible category for a

  • For 0<t<q, if ρ(a, b _t )<ρ(a, b _t+1), then select C _t as a possible category for a, otherwise select C _t+1

  • If t=0, select C ₁ as a possible category for a

Step 15::

Apply the second assignment procedure called the ascending rule.

Dias et al (2010) describe the ascending rule as follows:

• Increase h from zero until the first value t, such that σ(b _t , a)⩾λ

  • ° If t=1, select C ₁ as a possible category for a

  • ° For 1<t<q+1, if ρ(a, b _t )>ρ(a, b _t+1), then select C _t as a possible category for a, otherwise select C _t−1

  • ° If t=q+1, select C _q as a possible category for a

Step 16::

For each alternative a∈A, define the range of possible categories based on the minimum and maximum categories selected in Steps 14–15.

A.6.2. Separability conditions on characteristic actions

In order to test the weak, strict and/or hyper-strict separability conditions on the set of characteristic actions {b _h |h=1, …, q}, we have to go through the procedure of calculating concordance, discordance, and credibility indices for pairs of characteristic actions (b _k , b _k+1), k=1, …, q−1.

Step 1::

Determine concordance indices c _j (b _h , b _h+1), h=1, …, q−1, j=1, …, m. See Section A.5.1 for formulas.

Step 2::

Determine partial discordance indices d _j (b _h , b _h+1), h=1, …, q−1, j=1, …, m. See Section A.5.2 for formulas.

Step 3::

Determine credibility indices σ(b _h , b _h+1), h=1, …, q−1. See Section A.5.3 for formulas.

Step 4::

If σ(b _h , b _h+1)<1 for h=1, …, q−1, then the set of characteristic actions {b _h |h=1, …, q}fulfills the weak separability condition.

Step 5::

If σ(b _h , b _h+1)<1/2 for h=1, …, q−1, then the set of characteristic actions {b _h |h=1, …, q} also fulfills the strict separability condition.

Step 6::

If σ(b _h , b _h+1)=0 for h=1, …, q−1, then the set of characteristic actions {b _h |h=1, …, q} also fulfills the hyper-strict separability condition.

Appendix B

Appendix B Intuitionistic fuzzy numbers

In this appendix, we go through some of the definitions necessary for our use of intuitionistic fuzzy numbers.

Definition 1

• (Xu and Cai, 2012): Let X be a fixed set. An intuitionistic fuzzy set is a set

  

  where μ _A is referred to as a membership function and is defined as the mapping,

  

  and v _A is referred to as the non-membership function and is defined as the mapping

  

  such that

  

  The numerical values of μ _A (x) and ν _A (x) are called the membership degree and the non-membership degree of the element x∈X in A, respectively.

  If A is an IFS in X, then the value of

  

  is called the indeterminacy (or hesitancy) degree of the element x∈X to A.

  If ν _A (x)=1−μ _A (x) for all x∈X, then A degenerates into a fuzzy set.

Definition 2

• (Li, 2014): Let w _c ∈[0, 1] and u _c ∈[0, 1] be a pair of real numbers, such that 0⩽w _c +u _c ⩽1. An intuitionistic fuzzy number c is a special intuitionistic fuzzy set on the set of real numbers , where the membership function μ _c →[0, w _c ] and the non-membership function ν _c →[u _c , 1] satisfy the following four conditions (1)–(4):

  • 1) There exist at least two real numbers , such that μ _c (x′)=w _c and ν _c (x′)=u _c

  • 2) μ _c is quasi-concave and upper semi-continuous on

  • 3) ν _c is quasi-convex and lower semi-continuous on

  • 4) The support of c is compact. Note that the support of c is the set

  The numbers w _c and u _c are called the maximum membership degree and the minimum non-membership degree, respectively.

  A general intuitionistic fuzzy number is defined by the membership function

  

  and the non-membership function

  

  where the functions and are continuous and non-decreasing, and satisfy the conditions , f _l (c _1l )=w _c , g _r (c _2r )=u _c , and the functions and are continuous and non-increasing and satisfy the conditions f _r (c _1r )=w _c , , , g _r (c _2l )=u _c .

  When , c _1l =c _2l , c _1r =c _2r and , we use the notation c=〈(c ₁, c ₂, c ₃, c ₄), w _c , u _c 〉 .

Definition 3

• (Li, 2014): Let , such that a ₁<a ₂<a ₃<a ₄. A trapezoidal intuitionistic fuzzy number (TrIFN) a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 is defined by the membership function

  

  and the non-membership function

  

  Figure B1 illustrates a TrIFN.

  Figure B1

  

  Trapezoidal intuitionistic fuzzy number.

  If a ₁<a ₂=a ₃<a ₄, then the TrIFN degenerates into a triangular intuitionistic fuzzy number (TIFN). The same formulas can be used for membership and non-membership functions if a ₂⩽x⩽a ₃ is replaced with a ₂=x=a ₃. Figure B2 illustrates a triangular intuitionistic fuzzy number.

  Figure B2

  

  Triangular intuitionistic fuzzy number.

  There are some other degenerate cases, which are not intuitionistic fuzzy numbers according to Definition 2. Nevertheless, they may be relevant in cases, where TrIFNs are used in real decision problems. For convenience, we use the notion of an intuitionistic fuzzy number also in these cases. Hence, in Definition 4, the membership and non-membership functions of what we refer to, as intuitionistic fuzzy numbers, do not fulfil all the conditions in Definition 2.

Definition 4

• We add the following definitions:

  • Case 1: If a ₁=a ₂<a ₃=a ₄ then we say that a=〈(a ₁,a ₂, a ₃, a ₄), w _a , u _a 〉 is an intuitionistic fuzzy interval number. Figure B3 illustrates the case.

    Figure B3

    

    Illustration of Case 1.

  • Case 2: If a ₁=a ₂=a ₃=a ₄ then we say that a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 is an intuitionistic crisp number. Figure B4 illustrates the case. Note that each of the two small squares are endpoint of the two lines.

    Figure B4

    

    Illustration of Case 2.

  • Case 3: If a ₁=a ₂<a ₃<a ₄, then we say that a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 is a right TrIFN. Figure B5 illustrates the case.

    Figure B5

    

    Illustration of Case 3

  • Case 4: If a ₁<a ₂<a ₃=a ₄, then we say that a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 is a left TrIFN. Figure B6 illustrates the case.

    Figure B6

    

    Illustration of Case 4.

  • Case 5: If a ₁=a ₂=a ₃<a ₄, then we say that a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 is a right triangular intuitionistic fuzzy number. Figure B7 illustrates the case.

    Figure B7

    

    Illustration of Case 5.

  • Case 6: If a ₁<a ₂=a ₃=a ₄, then we say that a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 is a left triangular intuitionistic fuzzy number. Figure B8 illustrates the case.

    Figure B8

    

    Illustration of Case 6.

Definition 5

• (Li, 2014): Let a be an intuitionistic fuzzy number as in Definition 2 and let , such that 0⩽α⩽w _a and u _a ⩽β⩽1.

  An α-cut set of a is defined as the set a ^α={x∣μ(x)⩾α, x∈}, which can be represented by a closed interval denoted as [L _a (α), R _a (α)].

  A β-cut set of a is defined as the set , which can be represented by a closed interval denoted as [L _a (β), R _a (β)].

Definition 6

• (Li, 2014): Let a be an intuitionistic fuzzy number as in Definition 2 and let , such that 0⩽α⩽w _a and u _a ⩽β⩽1.

  The value indices (or values for short) of the membership and the non-membership are defined as

  

  The ambiguity indices (or ambiguities for short) of the membership and the non-membership are defined as

  

  where f:[0, w _c ]→[0, 1] is monotonic and non-decreasing and satisfy the condition f(0)=0, and g:[u _c , 1]→[0, 1] is monotonic and non-increasing and satisfy the condition g(1)=0.

  The value indices are related to the concept of centroids of the membership and non-membership functions and the ambiguity indices can be seen as the global spreads of the membership and non-membership functions (Li, 2014).

Definition 7

• (Li, 2014): Let a be an intuitionistic fuzzy number as in Definition 2 and let V _μ (a), V _ν (a), A _μ (a) and A _ν (a) be the value indices and ambiguity indices of a. Moreover, let 0⩽λ⩽1.

  The λ-weighted value index V _λ (a) and the λ-weighted ambiguity index A _λ (a) of a are defined as

  

  The parameter λ can be used to indicate preferences between the values of the membership and non-membership functions. This can also be seen as a preference between certainty and uncertainty. Hence, if λ<1/2, then uncertainty is preferred, if λ>1/2, then certainty is preferred and if λ=1/2, then certainty and uncertainty are equally preferred. For this paper, we let λ=0.5.

  The ranking of two intuitionistic fuzzy numbers a and b can be determined using λ-weighted values and λ-weighted ambiguities as follows:

  • if V _λ (a)>V _λ (b) then a>b

  • if V _λ (a)<V _λ (b) then a<b

  • if V _λ (a)=V _λ (b)∧A _λ (a)<A _λ (b) then a>b

  • if V _λ (a)=V _λ (b)∧A _λ (a)>A _λ (b) then b>a

  • if V _λ (a)=V _λ (b)∧A _λ (a)=A _λ (b) then a=b

Definition 8

• (Li, 2014): Let a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 be a TrIFN and let , such that 0⩽α⩽w _a and u _a ⩽β⩽1. Then

  

  and

  

  If a is a TIFN with a ₁<a ₂=a ₃<a ₄, then the formulas can also be applied.

Definition 9

• Let a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 be a TrIFN and let 0⩽α⩽w _a and u _a ⩽β⩽1. Moreover, let f(α)=α/w _a and g(β)=(1−β)/(1−u _a ). Then

  
  
  
  

  Note that if a is a TIFN with a ₁<a ₂=a ₃<a ₄, then it can be shown that the formulas also hold. For this paper, value and ambiguity indices will be based on the formulas from Definition 9.

Definition 10

• (Li and Yang, 2013): Let a=〈(a ₁, a ₂, a ₃, a ₄),w _a , u _a 〉 be a TrIFN and let 0⩽α⩽w _a and u _a ⩽β⩽1. Moreover, let V _λ (a) and A _λ (a) be the corresponding λ-weighted value and ambiguity index as in Definition 7. Then the λ-weighted difference index D _λ (a) is defined as

  

  The ranking of two intuitionistic fuzzy numbers a and b is determined using λ-weighted difference indices as follows:

  • if D _λ (a)>D _λ (b) then a>b

  • if D _λ (a)<D _λ (b) then a<b

  • if D _λ (a)=D _λ (b) then a=b

Definition 11

• If a=〈(a ₁, a ₂, a ₃, a ₄), w _a , u _a 〉 is one of the degenerate intuitionistic fuzzy numbers from Definition 4, then we define

  

  and

  

  and

  

Appendix C:

Data for the case study

Table C1

Table C1 Fuzzy decision matrix

Table C2

Table C2 Fuzzy characteristic actions