Question

A toy factory manufactures two types of wooden toys: soldiers and trains. A soldier sells for R27 and uses R10 worth of raw material and R14 worth labour. A train sells for R21 and uses R9 worth of raw material and R10 worth of labour. The manufacture of each toy requires two types of labour: carpentry and finishing. A soldier requires two hours of finishing labour and one hour of carpentry labour. A train requires one hour of finishing labour and one hour of carpentry labour. Each week only 100 hours of finishing labour and 80 hours of carpentry labour are available. All the trains can be sold, but at most 40 soldiers can be sold each week. Answer the following questions to ultimately determine how many soldiers and trains should be produced each week to maximize profit if R520 is budgeted for raw material and R650 is budgeted for labour costs. 1. State the objective function and clearly indicate if it is a maximization or a minimization problem. (1) 2. State all the constraints. (3) 3. Sketch a graph that clearly shows all constraints and shade the solution space if it exists. Also clearly label everything on the diagram including the axes. (7) 4. Sketch the isoprofit(isocost) line on the diagram in question 3 when the factory makes R420 of profit. Then sketch the isoprofit(isocost) line on the diagram in question 3 when the factory makes the most money from having optimally produced and sold toy trains and toy soldiers. (2) 5. What is the optimal profit that the factory makes? (1) 6. How many soldiers and trains leads to the result in question

Answer 1

Explanation:

1:The objective function in this case is to maximize profit.

2:Constraints:

Raw material budget constraint: 10x + 9y ≤ 520, where x is the number of soldiers produced and y is the number of trains produced.

Labour cost budget constraint: 14x + 10y ≤ 650.

Finishing labour constraint: 2x + y ≤ 100.

Carpentry labour constraint: x + y ≤ 80.

Maximum number of soldiers constraint: x ≤ 40.

3. graph

4. Sketching isoprofit (isocost) lines:

When the factory makes R420 of profit, the isoprofit line would be parallel to the profit axis and intersecting the feasible region. The slope of the isoprofit line would be -420 (negative slope).

When the factory makes the most money from optimally producing and selling toy soldiers and toy trains, the isoprofit line would be tangent to the highest possible profit line within the feasible region.

5.The optimal profit that the factory makes can be determined at the point of intersection between the highest possible profit line and the feasible region on the graph.

6.The number of soldiers and trains that lead to the optimal profit can be read from the coordinates of the point of intersection on the graph.