https://file.aiquickdraw.com/mathbot/file/1727529799103fq75a923.jpg

Реши задачата пато p=5 q=8

The task in the image is a **transportation problem** where the goal is to minimize the total transportation cost. The transportation cost matrix \(C\) is provided along with supply vector \(a\) and demand vector \(b\).

The cost matrix is given by:

\[
C = \begin{pmatrix}
q & 1 & 4 \\
3 & 3 & 1 \\
1 & p & 2
\end{pmatrix}
\]

The supply vector \(a\) is:

\[
a = (30, 20, 30 + 10p)
\]

The demand vector \(b\) is:

\[
b = (20, 30, 20 + 10q)
\]

You have specified values for \(p = 5\) and \(q = 8\). Let's substitute these into the supply and demand vectors, and the cost matrix:

### Updated cost matrix \(C\):

Substituting \(p = 5\) and \(q = 8\):

\[
C = \begin{pmatrix}
8 & 1 & 4 \\
3 & 3 & 1 \\
1 & 5 & 2
\end{pmatrix}
\]

### Updated supply vector \(a\):

Substituting \(p = 5\):

\[
a = (30, 20, 30 + 10(5)) = (30, 20, 30 + 50) = (30, 20, 80)
\]

### Updated demand vector \(b\):

Substituting \(q = 8\):

\[
b = (20, 30, 20 + 10(8)) = (20, 30, 20 + 80) = (20, 30, 100)
\]

### Verifying the feasibility:

The total supply must equal the total demand:

\[
30 + 20 + 80 = 130
\]
\[
20 + 30 + 100 = 150
\]

Since total supply (130) does not equal total demand (150), this is an **unbalanced transportation problem**. We will need to add a dummy row (extra supply) with zero costs to balance the problem.

Now, I will proceed to solve the transportation problem using the updated cost matrix, supply, and demand vectors by applying an appropriate method (e.g., the Northwest Corner Method or the Least Cost Method) to minimize the total cost. Let me compute that for you.The optimal transportation plan that minimizes the total cost is:

\[
\begin{pmatrix}
0 & 30 & 0 \\
0 & 0 & 20 \\
20 & 0 & 60 \\
0 & 0 & 20 \\
\end{pmatrix}
\]

This means:

- 30 units are shipped from source 1 to destination 2.
- 20 units are shipped from source 2 to destination 3.
- 20 units are shipped from source 3 to destination 1.
- 60 units are shipped from source 3 to destination 3.
- 20 units are covered by the dummy source (to balance demand).

The **minimum total transportation cost** is **190**.

If you'd like more details or have any questions, feel free to ask!

Here are 5 follow-up questions for deeper understanding:
1. How is the transportation problem balanced when supply and demand don't match?
2. What are the steps in the Least Cost or Northwest Corner method?
3. How does the introduction of a dummy row affect the transportation cost?
4. Can this solution be visualized on a transportation network graph?
5. How would the solution change if we altered the values of \(p\) and \(q\)?

**Tip**: Always verify if your transportation problem is balanced; if not, add dummy rows/columns to avoid infeasibility.

Learn how to solve a transportation problem for cost minimization using linear programming. This problem involves substituting values p=5 and q=8 into the cost matrix and determining an optimal transportation plan, while balancing an unbalanced problem. Suitable for students in an operations research course.

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