A Novel Two-Stage Intutitonistic Fuzzy Approach Proposal Based on Anp and Mathematical Modelling for Risk Assessment

Intuitionistic Fuzzy Programming

IFP was first introduced by Angelov [20]. The model aims to maximize the difference between the smallest membership degree for objective, intuitionistic fuzzy constraints and the biggest non-membership degree for objective, intuitionistic fuzzy constraints.

$$\text{Max}(\alpha - \beta)$$

subject to

$$\alpha \leq \mu_o(x), a = 1, \dots, p + q$$ $$ \rho \geq v _ {a} (x), a = 1, \dots , p + q \tag {3} $$

$$ \alpha \geq \beta (4) $$

$$ \alpha + \beta \leq 1 $$

$$ \beta \geq 0 (5) $$

where; α is the smallest membership degree of objective and constraints. β is the biggest non-membership degree of objective and constraints.

The Proposed Approach

The proposed approach has two stages as ranking FMs via using ANP and utilizing intuitionistic fuzzy mathematical model to determine the most important FM/FMs considering real constraints of the firm. The steps of the proposed approach are given below.

First Stage: Determining importance weights of alternatives • Step 1: Construct the model and structure the problem, form expert group Defined criteria and alternatives are expressed as a cluster and a relationship diagram is generated between them. This structure is built on a detailed analysis of the system. The problem should be stated clearly and decomposed into a rational system similar to a network. This structure can be obtained by experts through brainstorming or other appropriate methods.

The clusters are denoted as

$$ C _ {j} = \left\{C _ {1}, C _ {2}, \dots , C _ {n} \right\}, (j = 1, \dots , n) \mathrm {a n d} $$ the criteria are indicated as { } ( )( ) 1 2 1 1 1 , , , ; 1, , 1, , t l j n n C C C C j n t l = … = … = … . FMs are also represented as { } ( ) 1 2 , , , , 1, , s p FM FM FM FM s p = … = … and r experts $$ E _ {k}; k = 1, \dots , r $$

form the expert team. Figure 1 shows the network structure.

Click to enlarge

• Step 2: Assign the weight of each expert The weight of each expert is indicated as ( ) ( ) , , ; 1, , k k k k WE v k v µ π = = … . WEk is assigned considering working experience of each expert by using Table 1. Then, WEk is transformed into crisp values as k β providing 0 k β > l $$ \sum_ {k = 1} ^ {l} \beta_ {k} = 1 $$

by using Eq.(6).

and

1 1 k k β $$ k = \frac {\left(\mu_ {k} + \pi_ {k} \left(\frac {\mu_ {k}}{\mu_ {k} + v _ {k}}\right)\right)}{\sum_ {k = 1} ^ {l} \left(\mu_ {k} + \pi_ {k} \left(\frac {\mu_ {k}}{\mu_ {k} + v _ {k}}\right)\right)} $$ $$ \beta_ {k} = \frac {\left(\mu_ {k} + \pi_ {k}\right) \left(\frac {\mu_ {k}}{\mu_ {k} + v _ {k}}\right)}{\sum_ {k = 1} ^ {l} \left(\mu_ {k} + \pi_ {k}\right) \left(\frac {\mu_ {k}}{\mu_ {k} + v _ {k}}\right)} $$ ( k k k k k k l k k k k k k v (6) ( ) v

1

• Step 3: Structure the pairwise comparison matrices of criteria in terms of risk levels that they can create Pairwise comparison matrices of criteria in terms of risk levels are formed for each expert by using intuitionistic fuzzy risk level scale given in Table 2.

The intuitionistic fuzzy evaluation matrix for each cluster is as follows. Pairwise comparison matrix of each expert for cluster is demonstrated as k W      is given in Eq.(7).

$$ \left[ W _ {i i} ^ {k} \right] = \left[ \begin{array}{c c c c} \square_ {w _ {1 1}} ^ {k} & \square_ {w _ {1 2}} ^ {k} & \dots & \square_ {w _ {1 l _ {n}}} ^ {k} \\ \square_ {w _ {2 1}} ^ {k} & \square_ {w _ {2 2}} ^ {k} & \dots & \square_ {w _ {2 l _ {n}}} ^ {k} \\ \vdots & \vdots & \vdots & \vdots \\ \square_ {w _ {l _ {n} 1}} ^ {k} & \square_ {w _ {l _ {n} 2}} ^ {k} & \dots & \square_ {w _ {l _ {n} l _ {n}}} ^ {k} \end{array} \right] $$ k k k l w w w

11 12 1 n

k k k k l ii

w w w W

21 22 2 n    

k k k l l l l w w w

1 2 n n n n (7)

where,

i refers to the number of criteria in relation.

• Step 4: Compute the consistency index of pairwise

comparison matrix of each expert

Consistency indexes (CIs) of

k W     

are computed as in Eq.(8)

$$ \begin{array}{l} [ 2 3 ]. \\ C R = \frac {\left(\lambda_ {\max } - l _ {n}\right) / \left(l _ {n} - 1\right)}{R I} (8) \\ \end{array} $$ ( ) max nl λ − () is the average value of Π(x) in pairwise comparison matrix of each expert. 𝜆max is the biggest eigen value obtained via dividing weight vector by related relative importance value. RI defines the randomness index (RI) and it can take different values according to ln. Table 3 shows the RI for different ln values [22].

• Step 5: Form the aggregated pairwise comparison matrices k W      matrices are aggregated with intuitionistic fuzzy weighted geometric average operator (IFWG) as in Eq.(9).

( ) 1 2 1 1 1 , , , ( ,1 (1 ) k k k k k k

$$ \tilde {p} _ {i} = \left(\tilde {p} _ {1}, \tilde {p} _ {2}, \dots , \tilde {p} _ {j}\right) = \prod_ {j = 1} ^ {k} p _ {j} ^ {\beta_ {k}} = \left(\prod_ {j = 1} ^ {k} \mu_ {j} ^ {\beta_ {k}}, 1 - \prod_ {j = 1} ^ {k} (1 - v\right) $$ j j j j j j j IFWG p p p p v β β β β µ (9) In this way, aggregated pairwise comparison matrix of k W      is obtained as in Eq.(10). The element of k W      is denoted as $$ \tilde {w} _ {l _ {n} l _ {n}} = \left(\mu_ {l _ {n} l _ {n}}, v _ {l _ {n} l _ {n}}, \pi_ {l _ {n} l _ {n}}\right) \cdot \tag {10} $$ • Step 6: Form the super matrix and determine limit super matrix Let the clusters of a decision system be CLi, i= 1,2,…., n and each cluster i has li elements. The local priority vectors obtained in Step 5 are grouped and located in an appropriate positions in a super matrix based on the flow of influence from a cluster to another cluster, or from a cluster to itself as in the loop. Each pairwise comparison matrix is denoted as W     . S   defined as super matrices consists of matrices W      as in Eq.(11).

W W W $$ \left[ \tilde {S} \right] = \left[ \begin{array}{c c c c} W _ {1 1} & W _ {1 2} & \dots & W _ {1 n} \\ W _ {2 1} & W _ {2 2} & \dots & W _ {2 n} \\ \vdots & \vdots & \vdots & \vdots \\ W _ {n 1} & W _ {n 2} & \dots & W _ {n n} \end{array} \right] $$ (11)

11 12 1 n W W W S

21 22 2 n     

W W W

1 2 n n nn • Step 7: Synthesize the results Raising a matrix to powers gives the long-term relative influences of the elements on each other. To achieve a convergence on the importance weights, the weighted super matrix is raised to the power of 2k-1, where k is an arbitrarily large number, and this new matrix is called the limit super matrix.

The limit super matrix has the same form as the weighted super matrix, but all the columns of the limit super matrix are the same. By normalizing each block of this super matrix, the final priorities of all the elements in the matrix can be obtained. If the super matrix formed in Step 6 covers the whole network, the priority weights of alternatives can be found in the column of alternatives in the normalized super matrix. On the other hand, if a super matrix is only composed of interrelated clusters, additional calculations must be made to obtain the overall priorities of the alternatives. The alternative with the highest overall priority should be the one selected.

Second Stage: Determine the FMs having the top priority considering real constraints In this study, the new model is advanced by adding crisp constraints into model. In this way, crisp and intuitionistic fuzzy constraints can be considered at the same time. Additionally, deviation from the goal of crisp constraints can be minimized. The developed model is given in between Eq.(12) and (19).

$$ M a x \left(\alpha - \rho - d _ {s p} ^ {-} + d _ {s p} ^ {+}\right) \tag {12} $$ subject to $$ C C _ {i} x - d _ {s p} ^ {+} + d _ {s p} ^ {-} = 1 \tag {13} $$ $$ \alpha \leq \mu_ {a} (x), a = 1, \dots , d + h \tag {14} $$ $$ \rho \geq v _ {a} (x), a = 1, \dots , d + h \tag {15} $$ $$ \alpha \geq \rho (1 6) $$ $$ \left| \alpha + \rho \leq 1 (1 7) \right| $$ $$ \rho \geq 0 (1 8) $$

0 x ≥ (19)In this context, different intuitionistic fuzzy goal programming (GP) models have been set up to prioritize FMs for the reason that real constraints are not known clearly and managers, experts have doubts about the information that they give. Since the cost and safety level constraints are the intuitionistic fuzzy values, these are defuzzified using Eq. (20) in the modeling.

For defuzzification of triangular IFN ( ) ( )( ) ' ' 1 2 3 1 2 3 , , , , R a a a a a a a =  Eq. (20) is used [24].

( ) ( )( ) ( )( ) ( ) ' ' ' ' '2 '2 3 1 2 3 1 3 1 1 2 3 3 1

$$ (\tilde {a}) = \frac {1}{3} \left[ \frac {\left(a _ {3} ^ {\prime} - a _ {1} ^ {\prime}\right) \left(a _ {2} - 2 a _ {3} ^ {\prime} - 2 a _ {1} ^ {\prime}\right) + \left(a _ {3} - a _ {1}\right) \left(a _ {1} + a _ {2} + a _ {3}\right) + 3 \left(a _ {3} ^ {\prime 2} - a _ {1} ^ {\prime 2}\right)}{a _ {3} ^ {\prime} - a _ {1} ^ {\prime} + a _ {3} - a _ {1}} \right] $$

2 2 3 1 3 a a a a a a a a a a a a R a a a a a

' ' 3 1 3 1

(20) Where ( ) R ais the defuzzfy version of .

Application of the Proposed Approach

First Stage: Determining importance weights of alternatives • Step 1: Construct the model and structure the problem, form expert group The proposed approach was utilized for RA in an assembly line of a firm in electromechanical sector producing air insulated metal shielded cells. 9 FMs { } ( ) 1 2 9 , , , , 1, ,9 s FM FM FM FM s … … depicted in Table 4 are identified by 4 experts Ek; k= 1,2,3,4 in the assembly line.

First expert E1 has an A class work safety expert certificate and he has 10 years working experiences in the firm. Second expert (E2) works as an electric electronic engineer and he has 15 years experiences in electro mechanic sector. He is responsible for the assembly line. Third expert (E3) has B class work safety expert certificate and he has working in the firm for 5 years. Fourth expert (E5) has working for 10 years in the firm as a production manager who responsible for air insulated metal shielded cells manufacturing. 5 criteria and alternative clusters Cj={C1,C2,C3,C4,C5} are identified as environmental factors (C1), administrative factors (C2), risk structure related factors (C3), cost (C4) and failure modes (FMs) respectively. Environmental factors (C1) includes five sub-criteria

1 2 5 1 1 1 1 { , ,..., } t C C C C = ;(j=1) (t= 1,…,5) as awkward working postures

11 C , noise level

2 1 C , cleanliness and arrangement of workplace

31 ( ) C , lighting

4 1 ( ) C , usability

51 C . Administrative factors (C2) covers four sub-criteria of hand tools $$ C _ {2 _ {t}} = \left\{C _ {2 _ {1}}, C _ {2 _ {2}}, C _ {2 _ {3}}, C _ {2 _ {4}} \right\}; $$

;(j=2) (t=1,…,4) as

application of work health and safety procedure

1 2 C , planned maintenance and repair 2 2 C , using machine protectors

3 2 C , using personal protectors

4 2 C . Risk structure related factor (C3) comprises four criteria $$ C _ {3 _ {t}} = \left\{C _ {3 _ {1}}, C _ {3 _ {2}}, C _ {3 _ {3}}, C _ {3 _ {4}} \right\}; [ j = 2 ] $$

(t=1,…,4) as probability

13 C , severity

2 3 C , detection

3 3 C ,

4 3 C . Cost (C4) contains two criteria impact of FM on system reliability $$ C _ {4 _ {t}} = \left\{C _ {4 _ {1}}, C _ {4 _ {2}} \right\}; $$

;(j=4) (t=1,2) as cost of FM

1 4 C and prevention cost

2 4 C .

Figure 2 shows the model construction and problem structuring.

Click to enlarge

Relationship of criteria and alternatives defined as Table 5.

Where each letter represents the relationship matrix. For example, the cost factor affects the environmental factor and it is shown by matrix. matrix indicates the importance level of cost factors according to environmental factors. The same interpretation can be made for the others.

• Step 2: Assign the weight of each expert The weight of the four experts Ek; (k=1,2,3,4)were assigned by using Table 1 and the crisp values of Ek were computed for all experts as β1= 0.25, β2=0.32, β3= 0.18 and β4= 0.25 as in Eq.(6).

• Step 3: Structure the pairwise comparison matrices of criteria in terms of risk levels that they can create k W      given in Eq.(7) is formed for each expert. Table 6 shows pairwise comparison matrix 1 W      of E1 for risk levels that can be created by criteria according to impact of FM on system reliability

4 3 ( ) C as an example. It represents the relation to . So that is

• Step 4: Compute the consistency index of criteria’s pairwise comparison matrix of each expert CIs were computed for k W      using Eq.(8). CI for

1

1 W is computed as 0.08 as an example.

2 • Step 5: Form the aggregated pairwise comparison matrices [ ] ii W is formed using Eq.(9) and it is established for criteria in Table 7.

Second Stage: Determine the FMs having the top priority considering real constraints In this stage, the most important FMs that should be prevented primarily are determined considering the priority ranking weights for 9 FMs obtained from the first and second stages and the real constraints of the firm as cost, safety level. Cost constraint defines the limit for the budget of the firm that can be used to prevent FMs. The firm where the application of the proposed approach is performed has a budget as nearly 30.000 TL. Safety level explains the level of the assembly area safety that may be occurred after preventing FMs in the assembly line. This level is estimated by experts. Four experts in the related firm want nearly % 80 safety levels after preventing FMs in the assembly area. The priority ranking weights are complementary constraints and the deviation from these weights is wanted to minimize.

In this context, different intuitionistic fuzzy GP models have been set up to prioritize FMs for the reason that real constraints are not known clearly and managers, experts have doubts about the information that they give. Since the cost and safety level constraints are the intuitionistic fuzzy values, these are defuzzified using Eq. (20) in the modeling. The targets of the mathematical model is identified as minimizing the budget and minimizing the risk level or maximizing the safety level. The objective function of the model consists of maximizing the membership degree. Decision variable is a zero-one integer decision variable defining the FM that should be prevented. These two intuitionistic fuzzy GP models consisting crisp and intuitionistic fuzzy constraints have objective functions including different units. To provide additivity of these different units, it is suggested to realize proportional normalization [25]. To perform proportional normalization, deviation variables are divided to target values of related constraint. With this new method developed, the following results have been obtained as in Table 8 [26, 27, 28].

When looking at the solution produced by the model, it is seen that x3, x5, x6 and x8 should be prevented firstly as FM3, FM5, FM6 and FM8. In other words, manual handling heavy loads (FM3), using worn hand tools (FM5), noisy assembly (FM6) area and vibration exposure (FM8) should be forestalled primarily.