To better illustrate how the presented definitions can be used, this Section will presents some examples, both numerical and practical, that demonstrate their ben- efits.
Numerical Example
To illustrate the presented concept, we will calculate the aggregation of triangular- shaped IVFN’s (the upper and lower fuzzy numbers are triangular fuzzy numbers). The upper and lower triangular fuzzy numbers can be represented as AL= (a,α,β)
and AU= (a,θ,τ) respectively, where a stands for the center, (α,β) and (θ,τ) de-
notes the left and right width of the fuzzy numbers. The mean value of a trapezoidal IVFN can be expressed as [61]
E(A) = a +β −α 12 +
τ −θ 12 .
AL 1= (4,1,3), AU1 = (4,3,4), AL 2= (3,2,5), AU2 = (3,4,6), AL 3= (9,2,2), AU3 = (9,4,4),
The corresponding order inducing variables and weights are: u1=3,u2=2,u3=6.
W = (0.2,0.4,0.4). The aggregation can be calculated as:
IVFN-IOWA(h3,A1i,h2,A2i,h6,A3i) = 0.2 ∗ A3+0.4 ∗ A1+0.4 ∗ A2.
Using the arithmetic of IVFN’s, the aggregated value, A, is obtained as: the lower fuzzy number AL= (4,1.2,2.6) and
the upper fuzzy number AU= (4,2.8,3.6).
Project selection utilizing the IVFN-IOWA operator
According to Lee et al. [183], research and development (R&D) projects are im- portant factors in the field of information technology (IT). This is due to the fact that innovations are crucial to ensure the profitability of a company in the IT sec- tor; selecting the best R&D projects is therefore an important part of the decision making processes within companies.
For this example, we implement the IVFN-IOWA operator for the purpose of project selection. A multi-attribute decision making problem with multiple experts can be processed using the following steps:
• Step 1. The selection of criteria and alternatives: the appropriate set of se- lection criteria C = {c1, . . . ,cm} and the set of potential alternative solutions
A = {a1, . . . ,an} are defined.
• Step 2. Defining the evaluation measure: in this example the experts ex- press their opinion concerning to what extent an alternative satisfies a crite- rion by using linguistic labels represented by interval-valued fuzzy numbers. One possible representation using IVFN’s is presented in Table 5.1, utilizing trapezoidal shaped upper and lower fuzzy numbers.
• Step 3. The experts specify their opinion: every expert Ej,j ∈ {1,...,l}
provides his/her evaluation in the form of a matrix (Aabj )n×m where Aabj ∈ {very low, low, medium, high, very high}.
• Step 4. The opinions are aggregated into one decision matrix: every expert Ej is associated with a weight wj such that wj ∈ [0,1],∑lj=1wj =1. The
weights define the importance of the experts within the group. Using these weights, the individual evaluations are aggregated using the arithmetic oper- ations of IVFN’s; the result is denoted by (Aab)n×m.
Upper fuzzy number Lower fuzzy number Very Low (0,0,0.2,0.4) (0,0,0.15,0.3) Low (0,0.2,0.4.,0.6) (0.1,0.25,0.35,0.5) Medium (0.2,0.4,0.6,0.8) (0.3,0.45,0.5,0.7) High (0.4,0.6,0.8,1) (0.5,0.65,0.75,0.9) Very High (0.6,0.8,1,1) (0.7,0.85,0.95,1) Table 5.1: Linguistic labels represented by trapezoidal IVFN’s
• Step 5. Individually aggregating every alternative: using the matrix of ag- gregated payoffs, the overall evaluation for every alternative is obtained in- dividually by employing the IVFN-IOWA operator. The alternative with the highest value will be selected and an ordering for the alternatives is estab- lished.
E1 C1 C2 C3 C4 C5
A1 low medium very low low medium
A2 high low very high low very low
A3 medium medium medium medium high
A4 low very high high low medium
E2 C1 C2 C3 C4 C5
A1 high medium low very high medium
A2 low very low high medium high
A3 low very high medium very low high
A4 medium very high low very low medium
E3 C1 C2 C3 C4 C5
A1 medium high very low high medium
A2 medium medium very high very low medium
A3 low high very high low medium
A4 very high medium high medium medium
Table 5.2: Preference relations of the 4 alternatives constructed by 3 experts In this example the decision-makers should consider the following 5 criteria when evaluating the possible R&D projects available:
1. Competitiveness of technology 2. The potential size of market 3. Environmental and safety benefits
4. Return on development cost
5. Opportunity of project result implementation
The preferences are expressed using trapezoidal IVFN’s as described in Table 5.1. The preference matrices specified by the 3 experts for the 4 alternative projects considered are listed in Table 5.2. The weights used in this example for represent- ing the importance of the experts are: S = (0.1,0.6,0.3). The aggregated payoff matrices for the four alternatives can be produced using the importance weights of the experts and arithmetical operation on IVFN’s (weighted average).
Lower C1 C2 C3 C4 C5
A1 4 3 1 5 2
A2 2 4 3 1 5
A3 4 3 1 2 5
A4 5 4 3 1 2
Table 5.3: Order Inducing Values
Alternative Lower fuzzy number Upper fuzzy number Mean value Ranking A1 (0.31,0.45,0.55,0.70) (0.22,0.40,0.60,0.80) 0.501 3
A2 (0.32,0.45,0.56,0.69) (0.23,0.40,0.60,0.78) 0.502 2
A3 (0.30,0.42,0.54,0.67) (0.22,0.38,0.58,0.75) 0.482 4
A4 (0.34,0.48,0.59,0.72) (0.24,0.43,0.63,0.80) 0.529 1
Table 5.4: The obtained evaluation of the 4 alternatives
The weights of the IVFN-IOWA are W = (0.1,0.15,0.25,0.35,0.15). Consid- ering the possibility that the R&D projects have different characteristics, induced ordering variables are introduced to emphasize different criteria for different alter- natives (see Table 5.3). The final results of the project selection process are shown in Table 5.4. The mean value is employed to obtain the final ordering. The alterna- tive with the best payoff is A4, and the order is the following: A4 A2 A1 A3.
Special cases: numerical comparison
To further illustrate the advantages of the proposed aggregation operators the fol- lowing problem is approached: selecting a wine which is similar to (i) a set of preferences or (ii) an ideal wine. The Fuzzy Wine Ontology 4.1 is implemented as the main source of knowledge.
4 different wines are considered as alternatives as well as 3 criteria (Alcohol, Acidity, Price) with preferences specified in a specific context. Decision-makers are asked to specify to what degree they think a given wine satisfies a predefined criterion level. For example, if it is required that the wine should fit the context of
E1 Alcohol Acidity Price
A1 low medium very low
A2 high very low very high
A3 medium medium medium
A4 low very high high
Table 5.5: The preference relation of the wines
"Business Dinner", we can estimate how the alcohol level of an alternative wine is similar to the alcohol level of an ideal wine which would be the most suitable for that context.
Using the matrix of payoffs, we obtain the overall evaluation for every alterna- tive individually by employing the IVFN-IOWA operator. The alternative with the highest value will be selected and an ordering for the alternatives is established.
Upper fuzzy number Lower fuzzy number U1 (0.5,1,1.5,2) (0.75,1.2,1.4,1.75)
U2 (0.25,0.75,1,1.25) (0.5,0.8,0.9,1.1)
U3 (1,1.25,1.5,2.25) (1.2,1.3,1.4,2)
Table 5.6: Order Inducing Values
The preference matrix specified for choosing one of four wines based on three criteria is listed in Table 5.5. The order inducing variables are presented in Table 5.6. Using different special cases of the generalised IVFN-IHOWA, we can obtain the different rankings of the wines in Table 5.7. The weights of the different OWA operators are specified as W = (0.5,0.3,0.2), for the heavy OWA we use W = (0.4,0.9,0.2).
Notably, employing different operators results in different rankings. For ex- ample, the new operators result in different rankings than the fuzzy average. The order inducing variables and the weights in the heavy OWA provide more freedom to the decision maker to express the preferences regarding the importance of dif- ferent attributes with respect to each other and it becomes possible to incorporate this into the weights of the operator.
For the second case, when a wine is chosen based on its similarity to an ideal
Mean Fuzzy min Fuzzy max Fuzzy average IVFN-IOWA IVFN-IHOWA A1 0.12 (3) 0.50 (3) 0.31 (4) 0.28 (4) 0.59 (3)
A2 0.12 (3) 0.88 (1) 0.57 (2) 0.51 (2) 0.56 (4)
A3 0.50 (1) 0.50 (3) 0.50 (3) 0.45 (3) 0.75 (2)
A4 0.30 (2) 0.88 (1) 0.63 (1) 0.56 (1) 1.05 (1)
Table 5.7: The obtained evaluation of the 4 alternatives (the rankings are indicated in the parenthesis after the mean values)
wine, we assume that there is a wine which has been chosen on a previous occasion as the best for the context but it is not available at the moment. Using the same four alternatives as in the previous case one can estimate which one of these four is the most similar to the ideal wine by applying the IVFN-IOWAD operator. Also in this case, Table 5.5 is utilized even though the interpretation of the labels is different: for example, if the ideal wine has a high alcohol level it means that A1 and A4
are similar to the ideal wine to a low degree, whereas A2 can be considered to be
similar to a high degree. Using the order inducing variable from Table 5.6, we obtain that the most similar wine is A4, followed by A2, A3, and A1, as: 0.56
0.51 0.45 0.28.