Tunnel height optimization using triangular fuzzy numbers

In this thesis, the change in the maximum value of the height function of a basic tunnel against the uncertainty that may occur in the independent variable or coefficients was examined by using fuzzy methods. A second-order height function h(x)=ax^{2}+bx+c of a tunnel to be used for a two-lane highway was considered. The solution of the problem was examined in two parts; maximum height values were obtained by applying fuzzification methods on the independent variable x, which indicates the width of the tunnel, in the first part, and on the coefficients of the function in the second part. In the first part of the Research Findings section of the thesis, the maximum height was calculated through three different methods depending on the uncertainty in the independent variable of the tunnel height function. In all three methods, the independent variable x was fuzzified with the triangular fuzzy number ~x=tfn(x-Delta_{1}, x, x+Delta_{2}). The values ~h(x) corresponding to this new triangular fuzzy variable in the function constitute non-triangular fuzzy numbers. In order to make it meaningful, defuzzification was done on the values ~h(x). In the first method, defuzzification was applied to the non-triangular value ~h(x) with the centroid method. In other methods, after linearization was performed on ~h(x) and converted into a triangular fuzzy number, it was defuzzified with the centroid method in the second method and the signed distance method in the third method. In each method, after defuzzification, a new H(x) height function depending on the initial x variable and the Delta_{1}, Delta_{2} parameters was obtained. Our initial problem was transformed into the problem maxH(x). Finally, the optimum solutions x* and the values H_{max} corresponding to some Delta_{1} and Delta_{2} parameter values in all three methods were found and compared in the tables. In each table, the change in the classical solution with the results obtained with different fuzzy methods depending on the degree of uncertainty in the variable x was observed. In the second part of the Research Findings section, a solution was sought for the problem depending on the uncertainty that may occur in the coefficients a, b and c of the function h(x). These coefficients were fuzzified with triangular fuzzy numbers in the form of ~a, ~b and ~c. The ~h(x) corresponding to these new fuzzy coefficients in the function also constitutes a triangular fuzzy number. In the defuzzification part of ~h(x), the function H(x) was obtained by using the centroid method and the signed distance method. As a result of the operations performed, the initial problem maxh(x) was turned into the problem maxH(x). In both methods, the solution of the problem maxH(x) over some values of all Delta parameters was presented in a table form. Depending on the degree of uncertainty in the coefficients a, b and c, changes in the classical solution were observed on the tables.