Final answer:
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.