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Задача 2. Транспортная компания организует перевозку картофеля со складов в то...

May 15, 2024

Solution by Steps

step 1

Define the variables and parameters for the transportation problem. Let

x_{ij}

be the amount of potatoes transported from warehouse

i

to store

j

. The supply at warehouses I, II, and III are 40, 150, and 100 tons respectively. The demand at stores 1, 2, 3, 4, and 5 are 20, 80, 90, 60, and 40 tons respectively

step 2

Set up the cost matrix for transportation. The cost of transporting one ton of potatoes from warehouse

i

to store

j

is given as follows:

\begin{array}{c|ccccc} &amp; 1 &amp; 2 &amp; 3 &amp; 4 &amp; 5 \\ \hline I &amp; 7 &amp; 3 &amp; 5 &amp; 4 &amp; 2 \\ II &amp; 6 &amp; 2 &amp; 3 &amp; 1 &amp; 7 \\ III &amp; 3 &amp; 5 &amp; 2 &amp; 6 &amp; 4 \\ \end{array}

step 3

Formulate the objective function to minimize the total transportation cost. The objective function is:

\text{Minimize } Z = 7x_{11} + 3x_{12} + 5x_{13} + 4x_{14} + 2x_{15} + 6x_{21} + 2x_{22} + 3x_{23} + 1x_{24} + 7x_{25} + 3x_{31} + 5x_{32} + 2x_{33} + 6x_{34} + 4x_{35}

step 4

Set up the constraints for supply and demand. The supply constraints are:

\begin{aligned} x_{11} + x_{12} + x_{13} + x_{14} + x_{15} &amp;\leq 40 \\ x_{21} + x_{22} + x_{23} + x_{24} + x_{25} &amp;\leq 150 \\ x_{31} + x_{32} + x_{33} + x_{34} + x_{35} &amp;\leq 100 \\ \end{aligned}

The demand constraints are:

\begin{aligned} x_{11} + x_{21} + x_{31} &amp;\geq 20 \\ x_{12} + x_{22} + x_{32} &amp;\geq 80 \\ x_{13} + x_{23} + x_{33} &amp;\geq 90 \\ x_{14} + x_{24} + x_{34} &amp;\geq 60 \\ x_{15} + x_{25} + x_{35} &amp;\geq 40 \\ \end{aligned}

step 5

Solve the linear programming problem using the simplex method or any other suitable optimization technique to find the values of

x_{ij}

that minimize the total cost

Z

Answer

The optimal transportation plan and the minimum transportation cost can be determined by solving the linear programming problem formulated above.

Key Concept

Linear programming is used to find the optimal solution for minimizing transportation costs in a supply chain.

Explanation

By setting up the cost matrix, objective function, and constraints, we can use optimization techniques to determine the most cost-effective way to transport goods from warehouses to stores.