Cost-Time Minimization In A Transportation Problem With Fuzzy Parameters A Case Study

$$\begin{aligned} A_1 =\left( {A_1^U ,A_1^L } \right)= & {} \left( \left( {a_{11}^U ,a_{12}^U ,a_{13}^U,a_{14}^U ;H_1 \left( {A_1^U } \right) \!, H_2 \left( {A_1^U } \right) } \right) \!, \right. \nonumber \\&\quad \left. \left( {a_{11}^L ,a_{12}^L ,a_{13}^L,a_{14}^L ;H_1 \left( {A_1^L } \right) \!, H_2 \left( {A_1^L } \right) } \right) \right) \end{aligned}$$

(3)

$$\begin{aligned} A_2 =\left( {A_2^U ,A_2^L } \right)= & {} \left( \left( {a_{21}^U ,a_{22}^U,a_{23}^U ,a_{24}^U ;H_1 \left( {A_2^U } \right) \!, H_2 \left( {A_2^U } \right) } \right) \!, \right. \nonumber \\&\quad \left. \left( {a_{21}^L ,a_{22}^L ,a_{23}^L,a_{24}^L ;H_1 \left( {A_2^L } \right) \!, H_2 \left( {A_2^L } \right) } \right) \right) \end{aligned}$$

(4)

$$\begin{aligned}&(1)\quad A_1 +A_2 =\left( {A_1^U ,A_1^L } \right) +\left( {A_2^U ,A_2^L } \right) \\&\qquad \quad =\left( {{ \begin{array}{l} \left( a_{11}^U +a_{21}^U,a_{12}^U +a_{22}^U,a_{13}^U +a_{23}^U,a_{14}^U +a_{24}^U ;\min \left( H_1 \left( A_1^U \right) \!, H_1 \left( A_2^U \right) \right) \right. ,\\ \quad \left. {\min \left( H_2 \left( {A_1^U } \right) \!, H_2 \left( {A_2^U } \right) \right) }\right) , \\ \left( a_{11}^L +a_{21}^L,a_{12}^L +a_{22}^L,a_{13}^L +a_{23}^L,a_{14}^L +a_{24}^L ;\min \left( {H_1 \left( {A_1^L } \right) , H_1 \left( {A_2^L } \right) }\right) \right. ,\\ \quad \left. {\min \left( H_2 \left( A_1^L \right) \!, H_2 \left( A_2^L \right) \right) }\right) \\ \end{array} }} \right) .\\&(2)\quad A_1 -A_2 =\left( {A_1^U,A_1^L } \right) -\left( {A_2^U,A_2^L} \right) \\&\qquad \quad =\left( {{\begin{array}{l} {\left( {a_{11}^U -a_{21}^U,a_{12}^U -a_{22}^U,a_{13}^U -a_{23}^U,a_{14}^U -a_{24}^U;}\right. }\\ \quad {\left. {\min \left( {H_1 \left( {A_1^U } \right) ,H_1 \left( {A_2^U } \right) } \right) \!, \min \left( {H_2 \left( {A_1^U } \right) \!, H_2 \left( {A_2^U } \right) } \right) } \right) ,} \\ {\left( {a_{11}^L -a_{21}^L,a_{12}^L -a_{22}^L,a_{13}^L -a_{23}^L,a_{14}^L -a_{24}^L;}\right. }\\ \quad {\left. {\min \left( {H_1 \left( {A_1^L } \right) ,H_1 \left( {A_2^L} \right) } \right) ,\min \left( {H_2 \left( {A_1^L } \right) \!, H_2 \left( {A_2^L } \right) } \right) } \right) } \\ \end{array} }} \right) .\\&(3)\quad A_1 \times A_2 =\left( {A_1^U,A_1^L} \right) \times \left( {A_2^U,A_2^L } \right) \\&\qquad \quad =\left( {{\begin{array}{l} {\left( {a_{11}^U \times a_{21}^U,a_{12}^U \times a_{22}^U,a_{13}^U \times a_{23}^U,a_{14}^U \times a_{24}^U ;}\right. }\\ \quad {\left. {\min \left( {H_1 \left( {A_1^U } \right) , H_1 \left( {A_2^U } \right) } \right) ,\min \left( {H_2 \left( {A_1^U } \right) , H_2 \left( {A_2^U } \right) } \right) } \right) ,} \\ {\left( {a_{11}^L \times a_{21}^L,a_{12}^L \times a_{22}^L,a_{13}^L \times a_{23}^L,a_{14}^L \times a_{24}^L ;}\right. }\\ \quad {\left. {\min \left( {H_1 \left( {A_1^L } \right) \!, H_1 \left( {A_2^L } \right) } \right) \!, \min \left( {H_2 \left( {A_1^L } \right) ,H_2 \left( {A_2^L } \right) } \right) } \right) } \\ \end{array} }} \right) .\\&(4)\quad kA_1 =\left( { \left( {ka_{11}^U,ka_{12}^U,ka_{13}^U,ka_{14}^U;H_1 \left( {A_2^U } \right) , H_2 \left( {A_2^U } \right) } \right) }\right. ,\\&\qquad \qquad \qquad \quad \quad \left. {\left( {ka_{11}^L,ka_{12}^L,ka_{13}^L,ka_{14}^L;H_1 \left( {A_2^L } \right) ,H_2 \left( {A_2^L } \right) } \right) } \right) .\\&(5)\quad \frac{1}{k}A_1 =\left( { \left( {\frac{1}{k}a_{11}^U,\frac{1}{k}a_{12}^U,\frac{1}{k}a_{13}^U,\frac{1}{k}a_{14}^U;H_1 \left( {A_1^U} \right) ,H_2 \left( {A_1^U } \right) } \right) }\right. ,\\&\qquad \qquad \quad \quad \quad \left. {\left( {\frac{1}{k}a_{11}^L,\frac{1}{k}a_{12}^L,\frac{1}{k}a_{13}^L,\frac{1}{k}a_{14}^L ;H_1 \left( {A_1^L } \right) ,H_2 \left( {A_1^L } \right) } \right) } \right) . \end{aligned}$$

Several drawbacks are found in the above arithmetic operations as shown below:

Please, wait while we are validating your browser

0 comments

Leave a Reply

Your email address will not be published. Required fields are marked *