Optimal solution of full fuzzy transportation problems using total integral ranking

. Full fuzzy transportation problem (FFTP) is a transportation problem where transport costs, demand, supply and decision variables are expressed in form of fuzzy numbers. To solve fuzzy transportation problem, fuzzy number parameter must be converted to a crisp number called defuzzyfication method. In this new total integral ranking method with fuzzy numbers from conversion of trapezoidal fuzzy numbers to hexagonal fuzzy numbers obtained result of consistency defuzzyfication on symmetrical fuzzy hexagonal and non symmetrical type 2 numbers with fuzzy triangular numbers. To calculate of optimum solution FTP used fuzzy transportation algorithm with least cost method. From this optimum solution, it is found that use of fuzzy number form total integral ranking with index of optimism gives different optimum value. In addition, total integral ranking value using hexagonal fuzzy numbers has an optimal value better than the total integral ranking value using trapezoidal fuzzy numbers.


Introduction
Application of transportation model in real life is used in distributing a product from supplier to demand. The problem to be solved by transportation model is determination of item distribution of that will minimize distribution total cost with item quantity to be produced does not exceed limit capacity of production and still meet demand minimum number on each destination. Some parameters can be used in the transport model, ie transportation costs, demand and inventory value (both product and supply capacity). There are transportation problems, namely value unpredictability of transportation cost coefficient, inventory amount and demand number caused by uncontrollable factors of erorr situations in real life. In addition, in some cases supply amount is not equal to demand number. Therefore, to handle these uncertainties in making decisions, Bellman and Zadeh introduce fuzziness concept. The fuzzy transportation problem (FTP) is transportation problem where transportation cost, supply quantity and demand quantity that is fuzzyness [1].
The purpose of FTP is determine delivery schedules that minimize total fuzzy costs while still meeting inventory quantities and fuzzy supply limits [2]. To solve FTP, fuzzy number parameter previously, must be converted to a number parameter in crisp number or called defuzzyfication. Total integral ranking is one of defuzzyfication method with a convex combination based on right and left integral values through an optimism index [3]. The left integral value is used to reflect the pessimistic viewpoint and the right integral value is used to reflect the optimistic viewpoint of the decision maker [3]. Previous discussion of the fuzzy number defuzzyfication method has been found that total integral ranking is superior to previously proposed defuzzyfication methods, such as Gupta have used trapezoidal numbers for defuzzyfication of fuzzy numbers using total integral rankings with results indicating consistency in the ranking between the two fuzzy numbers. Therefore in this case it will discuss to use of total integral ranking with other fuzzy number. Base on graphic membership function of trapezoidal numbers, can still be done addition of point parameters in the interval fuzzy number. Therefore, in this peper shows the use of new fuzzy numbers on total integral ranking by adding parameter point of trapezoidal fuzzy number which has 4 parameter points to 6 point parameters known as hexagonal fuzzy number. In the use of a total integral ranking with hexagonal fuzzy numbers on FTP is expected to produce better optimum value fuzzy triangular and trapezoid numbers. Also in this paper, to determine the optimal solution of FTP using fuzzy transportation algorithm least cost method.

Preliminaries
This section presents the total integral ranking.

Difinition 3 [3]. If
A is fuzzy number with membership function A f , difined as in (Eq. 1), then the total integral ranking value with index of optimism is difined as,

are the right and left integral value of
A , respectively, and

The Total Integral Rangking Value of Triangular, Trapezoidal and Hexagonal Fuzzy Number Difinition 7 [8]. The fuzzy number is a hexagonal fuzzy number, if its membership function
Suppose on introduction that has been described above that in this discussion using hexagonal fuzzy number. The hexagonal fuzzy number is generated by adding two parameter points obtained from middle value between fuzzy numbers a and b along fuzzy number c and e . The graph of membership functions can be illustrated in Figure 1.
with membership function A f expressed as follows: (2)

Comparison Consistency Defuzzyfication on Total Integral Ranking Based on Fuzzy Number Numbers Type
Defuzzyfication of fuzzy numbers using total Integral Ranking with hexagonal fuzzy number has a different consistency with fuzzy triangular number when viewed based on fuzzy number type. Therefore, the following consistency comparison of defuzzyfication based on hexagonal type is symmetrical, non symmetrical type 1 and non symmetrical type 2 with fuzzy triangular number showed in Table 1.    Table  2. From the table shows that fuzzy hexagonal type of symmetrical and non-symmetrical type 2 can be used for defuzzyfication process in solving fuzzy transportation problems that shown in Table 2.

Conclusion
This paper provides the latest form of fuzzy membership function derived from conversion of trapezoidal fuzzy numbers to fuzzy hexagonal numbers that can be used for defuzzyfication using the total integral ranking method. In addition, in the case example a symmetric fuzzy number is generated which results in an optimal solution different from the total integral ranking value using trapezoidal fuzzy numbers. Result of the total integral ranking that uses hexagonal fuzzy number has an optimal value better than the total integral ranking value using trapezoidal fuzzy number.

Reference
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