CRITICAL PATH METHOD FOR THE ANALYSIS OF BITUMINOUS ROAD TRANSPORT NETWORK UNDER FUZZY ENVIRONMENT OF VARIOUS FUZZY QUANTITIES

: Determining the critical path in bituminous road transport network in the past period depended only on time and cost, whereas in current period the identifying the critical path has become complicated because of multi-criteria: time, distance, security, etc. We have proposed Integrated FAHP – FTOPSIS Methodology for determining the optimal critical path based on trapezoidal, hexagonal and octagonal fuzzy numbers. The aim of this research work is for transport department and society should have through the effective methodology to avoid critical path and for safe journey. Moreover, this methodology will carry for future course of action through Ministry of Road Transport to make safe roadways which is helpful and useful for effective transportations. We are exhibiting the proposed methodology by giving a numerical example and comparing trapezoidal, hexagonal and octagonal fuzzy numbers.


INTRODUCTION
To reach from one place to another place, one medium is needed for carrying goods, animals or humans which is called transport. The day was there people utilized horse and oxen or they used to go everywhere they defined on foot and return from the destination in the same way as they were. This method and system for transport was felt difficult for the people especially for the village people. Days go by, transport also has been developed due to technology growth. In general sense, three modes of transportations air, water and land have got enormous growth.
Among three modes of transportations, land is considered as an important and inevitable mode. Because air and water are utilized only by big shots and bearocracy people but land, people who are not sound in financial position, influenced persons, can utilize. Railways and airways cannot enter into nook and corner of the village and rural areas whereas roadways can execute service door to door even in a remote place. Rail and airways services can be utilized for a long distance. Shortest distance coverage, personal service, flexible and for all purposes of service reaching only possible and acceptable mode is Road Transportations which is always under easy moving and covered budget. Hence Road Transport plays an important role in all developments which are seen for the betterment of national growth in the world arena.
Therefore, roadways are considered to be the most important for communication. It is clear that efficient management of road network never choose or desire the critical route because of the barriers and disturbances. They are nature of the road, poor maintenance, interior places without infrastructure resulted often accidents, delaying in moving as well as reaching the destination, congestion, etc. In the past years, the selection of the routes were calculated by using numerical methods as attributes of road transporting were considered as crisp values. Now, in modern time, technology has been highly improved. In that juncture, sometimes attributes might have given vague values are presented for which we cannot use or utilize numerical methods. Therefore, we use Fuzzy concept to solve the previous problematic experiences. When attributes  Number).  )  ;  ,  ,  ,  ,  ,  ,  ,  (   8  7  6  5  4  3  2 and it is normal when Then, the tri-section, penta-section and hepta-section of I Thus, the trapezoidal, hexagonal and octagonal fuzzy numbers will be taken as

Goal
The main aim of this proposed integrated methodology is to identify the critical path of the road transport network and an index based model by taking both qualitative and quantitative aspects is used to calculate the critical path.

Identification of Criteria and Sub-Criteria
We have presented the criteria and their sub-criteria in Table 1.

Numbers [9]
Ordering fuzzy numbers is very important in optimization and decision making problems under uncertain environment. We have proposed centroid based ranking method to order trapezoidal fuzzy numbers for critical path selection. First, We divide the trapezoid into three triangles ABR, ARD and RCD and subsequently G1, G2 and G3 are incenters of these three triangles. Join these three incenters and find the centroid G which is a point of inference for defining the ranking function to order trapezoidal fuzzy numbers which is shown in Fig. 1 Let the trapezoidal fuzzy number be ) The incenters of  ABR,  ARD and  RCD are These incenters G1, G2 and G3 can form a triangle because they are not in the same line.
The centroid of Thus, the ranking function of the trapezoidal fuzzy number Ã is

Proposed Ranking Method based on Centroid of Incenters for Hexagonal Fuzzy Numbers
Ordering fuzzy numbers is very important in optimization and decision making problems under uncertain environment. We have proposed a centroid based ranking method to order hexagonal fuzzy numbers for critical path selection. First, hexagon is divided into two triangles ABQ, SEF and two trapezoids BQSE, BCDE. Again, Trapezoid BQSE is divided into three triangles BQR, BRE and RES and trapezoid BCDE which is divided into three triangles BCU, BUE and UDE. Hence, We find the incenters of two triangles ABQ and SEF are G1 and G2. For trapezoid BQSE, the centroid of incenters of BQR, BRE and RES is 3 G and for trapezoid BCDE, the centroid of incenters of BCU, BUE and UDE is 3 G   and the centroid of 3 G and 3 G   is G3. Join these three points G1, G2 and G3 and find the centroid G which is a point of inference Hence, the centroid of incenters I1 , I2 and I3 is Hence, the centroid of incenters I4, I5 and I6 is ( ) Therefore, the ranking function of the hexagonal fuzzy number Ã is

Proposed Ranking Method based on Centroid of Incenters for Octagonal Fuzzy Numbers
Ordering fuzzy numbers is very important in optimization and decision making problems under uncertain environment. We have proposed a centroid based ranking method to order octagonal fuzzy numbers for critical path selection. First, We divide the octagon into four trapezoids ABCR, TFHI, CRTF and CDEF. Again, We divide ABCR into three triangles ABY, AYR and YCR and the centroid of their incenters is G1 ; Divide the trapezoid into three triangles TFW, TWI and WHI and centroid of their incenters is G2 ; Divide the trapezoid CRTF into three triangles CRS, CSF and STF and the centroid of their incenters is 3 G ; Divide the trapezoid CDEF into three triangles CDX, CXF and XEF and the centroid of their incenters is 3 G   . Then 4731 ANALYSIS OF BITUMINOUS ROAD TRANSPORT NETWORK • the centroid of 3 G and 3 G   is G3. Join these three points G1, G2 and G3 and find the centroid G which is a point of inference for defining the ranking function to order octagonal fuzzy numbers shown in Fig. 3

FIGURE 3. Centroid of incenters for octagonal fuzzy number
Let the octagonal fuzzy number be ) The incenters of  ABY,  AYR and  YCR are The centroid of incenters I1, I2 and I3 is The centroid of incenters I7, I8 and I9 is ( )   Therefore, the ranking function of the octagonal fuzzy number Ã is

Proposed Integrated FAHP -FTOPSIS Methodology
There are various techniques for Multi-criteria decision making in which two important familiar techniques are AHP and TOPSIS. Saaty [11] in 1980's introduced AHP and in 1996, Chang [2] was the first to introduce FAHP using triangular fuzzy numbers and also extent analysis method was used to compute criteria's weight. In 1992 [3], FTOPSIS was introduced by Chen and Hwang for decision making problems with ambiguity and used to evaluate the alternative with respect to criteria. In the proposed methodology, We integrate Fuzzy AHP and Fuzzy TOPSIS to rank the alternatives based on chosen criteria and their sub-criteria.
The steps of our proposed integrated FAHP and FTOPSIS method for determining the critical path are given below.  and -no. of experts and all elements of K X is taken as the trapezoidal fuzzy numbers and also used the pairwise comparison presented for comparing itself in Table 2.
(or) the comparison matrix is where all elements of K X will be taken as the hexagonal and octagonal fuzzy numbers and also used the pairwise comparison presented in Tables 3 and 4 for comparing itself.  Step 3: The arithmetic operation for aggregation is Similarly, We calculate synthetic value of sub-criteria ( ) ij C SE Step 5: We have used the degree of possibility for finding comparison between criteria (or between sub-criteria) which is defined as k  j  i  h  g  C  SE  or  C  SE  and  f  e  d  c  b  a  C  SE  or  C  SE   ik  j  ij  i   ,  ,  ,  ,  ,  ,  , , Step 6: Using FAHP, the weights of criteria & sub-criteria are found as follows. Let Thus, the criteria's and sub-criteria's weight vectors are (20) Step 7: Normalize the weight vectors of criteria and sub-criteria For criteria and sub-criteria,

DIS PC
Step 16: All possible paths are ranked based on closeness coefficient which is given in eq. (29) and the path with the highest rank will be considered as the fuzzy critical path

NUMERICAL EXAMPLE
Consider six states namely, U, V, W, X, Y and Z and suppose a bituminous road is laid among these six states shown in Fig. 4. In this, U and Z are the origin and terminus. This road to 4744 A. HARI GANESH, A. HELEN SHOBANA be laid represents the connection between one state to another state and there are four ways to reach F. Using our proposed methodology, find out which of these four ways is the critical one.

Calculation for Trapezoidal Fuzzy Numbers
First, FAHP is used to find the weights of criteria and their sub-criteria

Weights of Criteria and their Sub-Criteria
Decision makers give the comparative judgments between criteria and also between sub-criteria using linguistic variables presented in Tables 8 -13.   TABLE 8 Secondly, We use FTOPSIS to determine the critical path based on closeness co-efficient.

Closeness Co-efficient
Fuzzy decision matrix is formed using alternatives and sub-criteria presented in Table 14.
Using Table 5 and using eq. (24) -(29), closeness co-efficient is computed presented in Table 15  and the path U -V -Z with maximum closeness co-efficient is the optimal fuzzy critical path

Weights of Criteria and their Sub-Criteria
Using FAHP method, We find

Closeness Co-efficient
We find the closeness co-efficient for hexagonal fuzzy numbers using FTOPSIS presented in Table 16. and the path U -V -Z with maximum closeness co-efficient is the optimal fuzzy critical path

Weights of Criteria and their Sub-Criteria
Using FAHP method, We find

Closeness Co-efficient
We calculate the closeness co-efficient for octagonal fuzzy numbers using FTOPSIS presented in Table 17. and the path U -V -Z with maximum closeness co-efficient is the optimal fuzzy critical path.

RESULTS AND DISCUSSION
The comparison among trapezoidal, hexagonal and octagonal fuzzy numbers is presented in Table 17 and Fig. 5.

CONCLUSION
In this paper, critical path in bituminous road transport is identified using proposed Integrated FAHP -FTOPSIS and more criteria and sub -criteria are used to identify it.