A logistics company optimizes its delivery routes between three warehouses and three stores. The transportation of goods must meet specific demand requirements at each store, while respecting the supply capacities of each warehouse. Let \(a\), \(b\), and \(c\) represent the quantity of goods transported from warehouse 1, warehouse 2, and warehouse 3 respectively. The system of equations that models the transportation constraints is given by:
\(a + b + c = 100\) (Total demand constraint)
\(2a + 3b + c = 150\) (Store 1 specific demand)
\(a + 4b + 2c = 200\) (Store 2 specific demand)
Solve the following parts to determine the optimal quantities to transport from each warehouse.
Question
Express the system of equations in matrix form \(AX = B\), where \(X = \begin{pmatrix} a \\ b \\ c \end{pmatrix}\). Identify and write down the coefficient matrix \(A\) and the constant matrix \(B\).
Hint
Hint
Hint
Carefully extract the coefficients of \(a\), \(b\), and \(c\) from each equation to form matrix \(A\), and the constants on the right side to form matrix \(B\).
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Complete Answer
\(A = \begin{pmatrix} 1 & 1 & 1 \\ 2 & 3 & 1 \\ 1 & 4 & 2 \end{pmatrix}\), \(B = \begin{pmatrix} 100 \\ 150 \\ 200 \end{pmatrix}\)
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Answer Explanation
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Question
Solve the matrix equation \(AX = B\) using a GDC or matrix methods to find the values of \(a\), \(b\), and \(c\). State your answers to three significant figures if necessary.
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Hint
Hint
To solve for \(X\), you can compute the inverse of matrix \(A\) and then calculate \(X = A^{-1}B\), or use row reduction on the augmented matrix \([A|B]\).
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Complete Answer
\(a = 25,\, b = 12.5,\, c = 62.5\)
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Question
Interpret the values of \(a\), \(b\), and \(c\) in the context of the logistics company’s transportation network. What do these values signify for the company’s operations?
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Hint
Hint
Consider what \(a\), \(b\), and \(c\) represent in terms of quantities transported from each warehouse and how this allocation might affect logistics and resource distribution.
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Complete Answer
The solution indicates that the company should transport 25 units of goods from warehouse 1, 12.5 units from warehouse 2, and 62.5 units from warehouse 3 to meet the demand constraints. These quantities represent the optimal allocation from each warehouse under the given system of equations.
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