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An engine company manufactures jackets and pistons. Both products require coating and lathe work.

- To make one jacket, 1 hour of lathe work and 1 kg of steel are required.
- To manufacture one piston, 1.5 hours of lathe work, 1 kg of steel, and 0.5 hours of coating are needed.

The company has the following weekly resources available:
- 750 hours of lathe work
- 200 hours of coating work
- 600 kg of steel

The company charges $3 for each jacket and $4 for each piston.

Determine the number of jackets and pistons that should be produced to maximize profit. Also, provide the dual solution to this primal problem.

Answer :

Final answer:

To maximize profit for the engine company, set up linear equations based on constraints to find the optimal quantities of jackets and pistons. The dual solution involves converting the primal problem into a dual format for further analysis.

Explanation:

Optimization Problem: To maximize profit, we need to set up linear equations based on the constraints given:

  1. Let x be the number of jackets and y be the number of pistons.
  2. Objective function: Maximize Profit = 3x + 4y
  3. Constraints: 1x + 1.5y ≤ 750 (lathe jobs), 1x + 1y ≤ 200 (coating), 1x + 1y ≤ 600 (steel)
  4. Solving these equations will give you the optimal amounts of jackets and pistons to maximize profit.

Dual Solution: The dual solution to the primal problem involves converting the primal problem into a dual format by interchanging the roles of variables and constraints. The dual variables represent the shadow prices of resources and can provide insights into the value of additional resources or constraints.

The problem aims to maximize profit by producing jackets and pistons while considering resource constraints. The dual problem seeks to minimize resource costs.

To maximize profit, we need to determine the number of jackets and pistons to produce. Let's define the decision variables as follows:

Let x = number of jackets to produce
Let y = number of pistons to produce

Now let's set up the objective function, which represents the total profit:

Total Profit = 3x (profit per jacket) + 4y (profit per piston)

Next, we need to establish the constraints based on the available resources.

1. Lathe job constraint:
The total hours of lathe jobs required for jackets and pistons should not exceed the available weekly lathe job hours.
1 hour of lathe jobs is required to make one jacket, and 1.5 hours are required for one piston.
So the constraint is: 1x (jackets) + 1.5y (pistons) ≤ 750 (available lathe job hours).

2. Coating constraint:
The total hours of coating required for pistons should not exceed the available weekly coating hours.
0.5 hours of coating is required for one piston.
So the constraint is: 0.5y (pistons) ≤ 200 (available coating hours).

3. Steel constraint:
The total amount of steel required for jackets and pistons should not exceed the available steel.
1 kg of steel is required for one jacket, and 1 kg is required for one piston.
So the constraint is: 1x (jackets) + 1y (pistons) ≤ 600 (available steel in kg).

Additionally, we have non-negativity constraints:
x ≥ 0 (number of jackets cannot be negative)
y ≥ 0 (number of pistons cannot be negative)

Now, to find the amount of jackets and pistons that maximize profit, we need to solve this linear programming problem. The optimal solution will provide the values of x and y that satisfy the constraints and maximize the objective function.

To obtain the dual solution to the primal problem, we would need to set up the dual problem and solve it using the dual simplex method or any other suitable method. However, the dual problem is not explicitly mentioned in the given question, so we cannot provide the dual solution without further information.

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