Methodology .1 Linguistic Variable

Introduction

I: DC J: Depot

5.3 Methodology .1 Linguistic Variable

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Sea depth is crucial for the vessels’ keels, hulls, and propellers. Hard ice may damage the vessel physically and stop the maneuverability of the vessel. Engaging an SB is another navigational challenge in ice-covered waters. This maneuvering technique requires special skill and experience (see Figure 5.6). The motor vessel’s speed should be reduced to half 5 cables ahead (185 × 5 m), and the vessel should be steered to the left and then given the command of full astern.

5.3 Methodology

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route seleC tion Problem in the arC tiC

as human decision processes, artificial intelligence, pattern recogni- tion, law, medical realms, economy, and related areas (Zadeh 1975).

5.3.2 Fuzzy Sets and Triangular Fuzzy Numbers

A fuzzy set was first developed by Zadeh (1965) and introduced by Bellman and Zadeh (1977). A triangular fuzzy number is a convex and normalized fuzzy set A and µA( )x is the continuous linear func- tion, which is a membership function of A.

The definition of a triangular fuzzy number A =( , , )l m u is

μ_A x

x l x l

m l l x m

x m u x

u m m x u

u x ( )

, ,

( )

( ), ,

, ,

( )

( ), ,

,

=

<

−

− ≤ <

=

−

− < ≤

<

0

1

0 ..

⎧

⎨

⎪⎪

⎪⎪

⎩

⎪⎪

⎪⎪

(5.1)

where

l and u are, respectively, the lower (smallest possible value) and upper (most promising value) bounds of the fuzzy number A and m is the midpoint

Figure 5.6 Engaging a sharp bend.

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A triangular fuzzy number is shown in Figure 5.7.

Consider two positive triangular fuzzy numbers (l1,m1,u1) and (l₂,m₂,u₂); then

l m u1, ,1 1 l m u2, 2, 2 l1 l m2, 1 m u2, 1 u2

( )

⁺

( )

⁼

(

^{+ + +}

)

l m u1, ,1 1 l m u2, 2, 2 l l m m u u1 2, 1 2, 1 2

( )

^⋅

( )

⁼

(

^{⋅ ⋅ ⋅}

)

l m u

u m l

1 1 1 1

1 1 1

1 1 1

, , , ,

( )

^≈⎛

⎝⎜ ⎞

⎠⎟

−

l m u1, ,1 1 k l1 k,m1 k,u1 k

( )

^{⋅ =}

(

^{⋅ ⋅ ⋅}

)

where k is a positive number.

The vertex method is used to calculate the distance between two triangular fuzzy numbers (Chen 2000):

d m nv

(

,

)

^{= 1}₃^⎡_⎣

(

l¹⁻l²

)

²⁺

(

m¹⁻m²

)

²⁺

(

u¹⁻u²

)

^2⎤_⎦

5.3.3 Fuzzy Analytic Hierarchy Process

Saaty (1980) proposed the first AHP as a decision-making tool. This method is widely used by researchers (Weck et al. 1997; Lee et al. 1999;

Leung and Cao 2000). The main purpose of AHP is to use the experts’

knowledge; however, the classical AHP does not reflect the human thinking style (Chang 1996) because it uses the exact values when

M2 M1

u₂ u1

d V(M₂ ≥ M1)

l2 m2l1 m1

0

Figure 5.7 The intersection between M1 and M2.

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route seleC tion Problem in the arC tiC

comparing the criteria with alternatives (Cakir and Canpolat 2008).

There has been a lot of criticism regarding the classical AHP because of its unbalanced scale, uncertainty, and imprecision of pairwise com- parisons (Kahraman et al. 2004). F-AHP is more accurate and has been developed to handle these shortcomings. Laarhoven and Pedrycz (1983) proposed the first F-AHP by the comparisons of fuzzy ratios.

Buckley (1985) worked on trapezoidal fuzzy numbers to evaluate the alternatives with respect to the criteria. For the pairwise comparisons, Chang (1996) used the extent analysis method to calculate the syn- thetic extent values.

The steps of the extent synthesis method are as follows.

Let X = {x1, x2, …, xn} be an object set and G = {g1, g2, …, gn} be a goal set. Each object is taken, and an extent analysis is performed for each goal. Therefore, m extent analysis values for each object can be obtained:

M Mg g Mgm i n

i i i

1 , 2, ,… , =1 2, , ,… (5.2) where all Mgj ( j = 1, 2, …, m) are triangular fuzzy numbers.

Step 1: For the ith object, the value of the fuzzy synthetic extent is defined as

Si Mgj M

j m

gj j

m

i n

i i

= = =

−

∑

^⊗⎡

∑ ∑

⎣

⎢⎢

⎤

⎦

⎥⎥

1 1 1

1

(5.3)

Obtaining M_g^j

j m

= i

∑

¹ the fuzzy addition operation of m extent analysis values for a particular matrix is performed:

M_g^j l m u

j m

j j j

j m

j m

j m i

= = = =

∑

^=⎛

∑ ∑ ∑

⎝

⎜⎜

⎞

⎠

⎟⎟

1 1 1 1

, , (5.4)

The fuzzy addition operation of M_g^j_i (j = 1, 2, …, m) values is performed:

M_g^j l m u

j m

i n

j j j

j m

j m

j m i

=

=

∑

= = =

∑

^=⎛

∑ ∑ ∑

⎝

⎜⎜

⎞

⎠

⎟⎟

1

1 1 1 1

, , (5.5)

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The inverse of the vector in Equation 5.3 is computed:

M

u m l

gj j

m

i n

i i n

i i n

i i i n

=

=

−

= = =

∑

∑ ∑ ∑ ∑

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥ =

⎛

⎝

⎜⎜

⎜

⎞

1 ⎠

1

1

1 1 1

1 , 1 , 1 ⎟⎟

⎟⎟ (5.6)

Step 2: The height of a fuzzy set hgt(A) is the maximum of the membership grades of A, hgt(A) = supx∈X μ_A(x).

The degree of possibility of M2 = (l2, m2, u2) ≥ M1 = (l1, m1, u1) is defined as follows:

V M M x y

y x

M M

2 ≥ 1 1 2

( )

⁼ _⎢⎣

( )

_⎥⎦

≥

sup min μ ( ),μ ( ) (5.7) and can also be expressed as

V M M hgt M M

d

m m

l u l u

m u

M

2 1 1 2

2 1

1 2

1 2

2 2

2

1 0

(

≥

)

⁼

(

^∩

)

= =

≥

≥

−

− −

μ ( ) ,

,

( )

if if (( ),

m1−l1

⎧

⎨

⎪⎪⎪

⎩

⎪⎪

⎪ otherwise

(5.8) Figure 5.8 illustrates that d is the y-axis value of the highest

intersection point D between µM1 and µM2.

Both V(M1 ≥ M2) and V(M2 ≥ M1) should be known for the comparison of M1 and M2.

1 1

μ~A(x)

A^~₁ A^~₂ A^~₃ A^~₄ A^~₅

3 5 7 9

Figure 5.8 Fuzzy number of linguistic variable set.

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route seleC tion Problem in the arC tiC

Step 3: The degree of possibility for a convex fuzzy number to be greater than k convex fuzzy numbers Mi (i = 1, 2, …) can be defined by

V(M ≥ M1, M2, …, Mk) = V[(M ≥ M1), (M ≥ M2), …,

(M ≥ Mk)] = min V(M ≥ Mi), i = 1, 2, 3, …, k. (5.9) Assume that d(Ai) = min V(Si ≥ Sk) for k = 1, 2, …, n; k ≠ i. Then

the weight vector is given by

ʹ =

(

ʹ

( )

^ʹ

( )

^{… ʹ}

( ) )

W d A d A d An T

1 , 2 , , (5.10)

where Ai (i = 1, 2, …, n) are n elements.

Step 4: Normalization and normalized weight vectors are W d A d A d An

=

( ( ) ( )

^{1 , 2 , ,…}

( ) )

T ^(5.11)

where W is a nonfuzzy number.

The nonnumerical values are expressed as fuzzy linguistic variables, which help the DM to describe the pairwise comparison of each cri- terion with its alternative, as reflected in Saaty’s (1977) nine-point fundamental scale (see Figure 5.9).

The assigned linguistic comparison terms (Chiclana 1998; Chan et al. 2000; Cakir and Canpolat 2008; Gumus 2009) and their equiv- alent fuzzy numbers considered in this chapter are given in Table 5.1.

For solving the current problem, an individual aggregation matrix is conducted by expert prioritization, which is called the lambda coefficient.

Let A = (aij)n × n, where aij > 0 and aij × aji = 1, be a judgment matrix. The prioritization method denotes the process of acquiring a priority vector.

w = (w1, w2, …, wn)^T where wi ≥ 0 and wi i

n =

∑

=^{1 1, from the}

judgment matrix A.

Let D = {d1, d2, …, dm} be the set of experts, and λ = {λ1, λ2, …, λm} be the weight vector of the DMs, where λk > 0, k = 1, 2, …, m, and

λk k

m =

∑

−^{1 1.}

Let E = {e₁, e₂, …, e_m} be the set of the experience in the professional career (in years for this chapter) for each expert, and λk for each expert is defined by

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Decision making for Arctic route

selection

Definition of the objective

Determination of the alternatives for the Arctic route selection

problem

Determination of the criteria for the Arctic route selection problem

Structure of the decision hierarchy for the Arctic route

selection problem

Data collection and pairwise comparisons

Is the model acceptable?

CCI < 0.37

Consistency check loop

Yes

Data analysis and evaluation of the alternative

Selection of the best alternative for the Arctic route problem GF-AHP

AHP Expert

consultation by survey method

No

Figure 5.9 GF-AHP procedure.

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route seleC tion Problem in the arC tiC

λk k k k m

e e

=

∑

=^{1 (5.12)}

Let A k aijk

( ) (= ^{( )})n n× be the judgment matrix that is gathered by the DM dk.

wi( )k is the priority vector of criteria for each expert calculated by

w

a a

ik ij

j

n n

j ij

n n

i n ( )

/

= /

⎛

⎝⎜ ⎞

⎠⎟

⎛

⎝⎜ ⎞

⎠⎟

=

=

=

∏

∑ ∏

1 1

1 1 1

(5.13)

The individual priority aggregation is defined by

w w

iw ikw

k m

ik k m i n

k

k ( )

( )

=

( )

( )

( )

=

=

=

∏ ∏

∑

λ

λ 1

1 1

(5.14)

where wi( )w is the aggregated weight vector. Then the extent synthesis method (Chang 1996) is applied for the consequent selection. A pair- wise comparison between the alternatives i and j for criterion C is defined by

a A

ij A

C ri

rj

= (5.15)

where Ari is the rank valuation set of alternative i. By the final con- sistency control, the procedure of generic fuzzy AHP (GF-AHP) is

Table 5.1 Membership Function of Linguistic Scale FUZZY

NUMBER LINGUISTIC SCALES

MEMBERSHIP

FUNCTION INVERSE A1 Equally important (1, 1, 1) (1, 1, 1) A2 Moderately important (1, 3, 5) (1/5, 1/3, 1) A3 More important (3, 5, 7) (1/7, 1/5, 1/3) A4 Strongly important (5, 7, 9) (1/9, 1/7, 1/5) A5 Extremely important (7, 9, 9) (1/9, 1/9, 1/7)

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achieved. Consistency control and centric consistency index (CCI) for F-AHP applications are described in the following section.

5.3.4 Centric Consistency Index

According to Saaty’s approach, all DMs’ matrix should be consistent to analyze the selection problem (Saaty and Vargas 1987). For the consistency control of the F-AHP method, Duru et al. (2012) pro- posed a CCI based on the geometric consistency index (Crawford and Williams 1985; Aguarón and Moreno-Jimenez 2003). The calculation of the CCI algorithm is as follows:

CCI A

n n

a a a

W W W

i j

Lij Mij Uij

Li Mi

( )

⁼ _{− −} ^{⎛ + +}

⎝⎜

− + +

∑

<

2

1 2 3

( )( ) log

log ^UUi W^Lj W^Mj W^Uj

3 3

2

log

+ + + ⎞

⎠⎟ (5.16) When CCI(A) is 0, A is fully consistent. Aguarón also expresses the thresholds (GCI) as (GCI) = 0.31 for n = 3, (GCI) = 0.35 for n = 4, and (GCI) = 0.37 for n > 4. When CCI(A) < (GCI), it means that this matrix is sufficiently consistent.