Question

Billy Penny is trying to determine how many units of two types of lawn mowers to produce each day. One of these is the Standard model, while the other is the Deluxe model. The profit per unit on the Standard model is $60, while the profit per unit on the Deluxe model is $40. The Standard model requires 20 minutes of assembly time, while the Deluxe model requires 35 minutes of assembly time. The Standard model requires 10 minutes of inspection time, while the Deluxe model requires 15 minutes of inspection time. The company must fill an order for 6 Deluxe models. There are 450 minutes of assembly time and 180 minutes of inspection time available each day.

Formulate as an LP, solve using QM, then interpret your output (the optimal solution, the reduced costs, slack/surplus, dual prices, ranges of feasibility, ranges of optimality)

Answer 1

Billy Penny needs to optimize production of Standard and Deluxe lawn mowers using Linear Programming with profit maximization, subject to constraints on assembly and inspection time. Optimal production levels and dual prices can be found using QM for Windows, providing information on profit, resource usage, and sensitivity of the solution.

Step-by-step explanation:

Billy Penny needs to determine the optimal production mix of two types of lawn mowers, Standard and Deluxe, to maximize profit subject to constraints on assembly and inspection time. To formulate this as a Linear Programming (LP) problem, assign x to represent the number of Standard models and y to represent the number of Deluxe models to be produced.

Objective Function:

Maximize Profit = 60x + 40y

Constraints:

Assembly Time: 20x + 35y <= 450

Inspection Time: 10x + 15y <= 180

Deluxe Model Requirement: y >= 6

Non-negativity Conditions:

x, y >= 0

Using a tool like QM for Windows, you can solve the LP to find the optimal values of x and y. The output will include the optimal solution detailing the number of each model to produce, reduced costs indicating the cost savings per additional unit of resources, slack/surplus revealing any unused resources, dual prices or shadow prices, and ranges of feasibility and optimality for changes in constraints and objective function coefficients without altering the solution structure.

Once the optimal production levels are known, Billy can plan daily production to achieve the highest profit without exceeding the constraints of his production capacity.

Answer 2

A software like QM can be used to solve the LP problem and the result is interpreted by referring to different metrics such as the optimal solution, reduced costs, slack/surplus, dual prices, and ranges of feasibility and optimality.

Let's define Standard model units as S and Deluxe model units as D.

The objective function, representing the company's total profit, would be: Maximize P = 60S + 40D (since profits are $60 for Standard and $40 for Deluxe).

The constraints concerning available assembly and inspection time would look like this:

20S + 35D <= 450 (since each Standard model requires 20 minutes of assembly and each Deluxe model requires 35 minutes)
10S + 15D <= 180 (since each Standard model requires 10 minutes for inspection and each Deluxe model requires 15 minutes).
Additionally, the company needs to fulfill the order of 6 Deluxe models, so:

D >= 6.
And since production can't be negative:

S, D >= 0.
This would be solved using a software like QM. The optimal solution will provide you with Billy's best strategy. The reduced costs tell you the change in total profit if the right-hand side of the constraint is increased by one. The slack/surplus values show how much capacity is left or missing in a constraint. Dual prices are the improvement in the objective function if a constraint is relaxed by one unit. The ranges of feasibility and optimality help to determine how changes in the parameters of the problem will affect the solution.