5. A manufacturer makes two type of products X and Y at each of the two different locations A and B. the materials used to make each of the products are steel, glass and plastic. Suppose it takes three units of steel, one unit of glass and two units of plastic to make one unit of product X; and four units of steel , one-half(1/2) unit of glass, and three units of plastic to make one unit of product Y.
Suppose further that steel, glass and plastic cost birr10, 2, and 3 per unit respectively at location A. At location B, steel, glass and plastic cost birr 9, 3, and 4 per unit respectively. Using matrix algebra, find the material cost of making one unit of each product at each of the two locations.
Let's represent the quantities of steel, glass, and plastic needed to make one unit of product X and Y as follows:
For product X:
- Steel: 3 units
- Glass: 1 unit
- Plastic: 2 units
For product Y:
- Steel: 4 units
- Glass: 1/2 unit
- Plastic: 3 units
Let's also represent the cost of steel, glass, and plastic at location A and B as a matrix:
Location A:
- Steel: birr 10
- Glass: birr 2
- Plastic: birr 3
Location B:
- Steel: birr 9
- Glass: birr 3
- Plastic: birr 4
Now, we can use matrix algebra to calculate the material cost of making one unit of each product at each location.
First, let's create matrices for the quantities of materials needed for each product:
Product X matrix:
[X_steel, X_glass, X_plastic] = [3, 1, 2]
Product Y matrix:
[Y_steel, Y_glass, Y_plastic] = [4, 1/2, 3]
Next, let's create matrices for the cost of materials at each location:
Location A matrix:
[A_steel, A_glass, A_plastic] = [10, 2, 3]
Location B matrix:
[B_steel, B_glass, B_plastic] = [9, 3, 4]
Now, we can calculate the material cost for one unit of each product at each location using matrix multiplication.
Material cost for product X at location A:
[Cost_X_A] = [X_steel, X_glass, X_plastic] ⨉ [A_steel, A_glass, A_plastic]
[Cost_X_A] = [3, 1, 2] ⨉ [10, 2, 3]
[Cost_X_A] = [30 + 2 + 6]
[Cost_X_A] = [38]
Material cost for product X at location B:
[Cost_X_B] = [X_steel, X_glass, X_plastic] ⨉ [B_steel, B_glass, B_plastic]
[Cost_X_B] = [3, 1, 2] ⨉ [9, 3, 4]
[Cost_X_B] = [27 + 3 + 8]
[Cost_X_B] = [38]
Material cost for product Y at location A:
[Cost_Y_A] = [Y_steel, Y_glass, Y_plastic] ⨉ [A_steel, A_glass, A_plastic]
[Cost_Y_A] = [4, 1/2, 3] ⨉ [10, 2, 3]
[Cost_Y_A] = [40 + 1 + 9]
[Cost_Y_A] = [50]
Material cost for product Y at location B:
[Cost_Y_B] = [Y_steel, Y_glass, Y_plastic] ⨉ [B_steel, B_glass, B_plastic]
[Cost_Y_B] = [4, 1/2, 3] ⨉ [9, 3, 4]
[Cost_Y_B] = [36 + 1.5 + 12]
[Cost_Y_B] = [49.5]
So, the material cost of making one unit of each product at each location is:
Product X at location A: birr 38
Product X at location B: birr 38
Product Y at location A: birr 50
Product Y at location B: birr 49.5