Question

Optimal Profit A manufacturer produces two models of elliptical cross-training exercise machines. The times for assembling, finishing, and packaging model \(\mathrm{A}\) are 3 hours, 3 hours, and \(0.8\) hour, respectively. The times for model B are 4 hours, \(2.5\) hours, and \(0.4\) hour. The total times available for assembling, finishing, and packaging are 6000 hours, 4200 hours, and 950 hours, respectively. The profits per unit are \(\$ 300\) for model \(A\) and $$\$ 375$$ for model \(B\). What is the optimal production level for each model? What is the optimal profit?

Short answer

The optimal production levels for Model A and Model B, and the optimal profit are dependent on the solution obtained from the LP problem in Step 4.

Step-by-step solution

Define Variables

First, the variables need to be defined. Let \(x_A\) be the number of Model A machines produced and \(x_B\) be the number of Model B machines produced.

Define Constraints

The constraints are defined by the resource limitations of the machine shop. These are the maximum time available for each step of the work.\n\n1. Assembly: \(3x_A + 4x_B \leq 6000\)\n2. Finishing: \(3x_A + 2.5x_B \leq 4200\)\n3. Packaging: \(0.8x_A + 0.4x_B \leq 950\)\nAlso, \(x_A \geq 0\) and \(x_B \geq 0\) because the number of produced machines cannot be negative.

Define Objective Function

The objective function is defined by what needs to be optimized. In this case, it's the total profit which is to be maximized. The profit contributions per unit are $300 for model A and $375 for model B. Thus, the objective function: Maximize \(300x_A + 375x_B\)

Solve LP Problem

Using a LP solving method (such as the simplex method, graphical method, use of software etc.), find the values of \(x_A\) and \(x_B\) that maximize the objective function subject to the constraints. This step generally requires some knowledge of mathematics or usage of a software and will result in the optimal solution.

Answer the Question

The optimal production level for each model is given by the solution to the LP problem (in Step 4). Substituting \(x_A\) and \(x_B\) into the objective function will provide the optimal profit.

Key concepts

Linear Programming

When tackling real-world optimization problems, like maximizing the profit of producing different products, we turn to a powerful mathematical method known as Linear Programming (LP). This technique is used to find the best possible outcome in a given mathematical model whose requirements are represented by linear relationships. At its core, linear programming involves two main components: the objective function and the system of linear inequalities representing constraints.

In the case of our Optimal Profit Problem, the manufacturer aims to determine the number of two models of elliptical cross-training exercise machines to produce in order to achieve the highest possible profit. The constraints come from the limited hours available for different stages such as assembly, finishing, and packaging. Linear programming allows the manufacturer to systematically evaluate all possible production combinations within the constraints to find the optimal solution. It's a balancing act between the potential profit of each model and the finite resources - time, in this case - available to produce them.

Systems of Inequalities

When managing resources in production, businesses must consider various limits, like machine capacity or labor hours. When constructing a linear programming model, these limitations are mathematically described using systems of inequalities. Each inequality corresponds to a resource or a condition that cannot be exceeded.

In our exercise problem, there are inequalities for the assembly, finishing, and packaging steps, signifying the maximum time that can be spent on these processes for the production of models A and B. Furthermore, non-negativity constraints are added (\(x_A \text{\geq\ 0}\) and \text{\(x_B \geq\ 0}\text{\)}) to reflect the reality that you can't produce a negative number of machines. Solving these systems of inequalities helps determine the feasible region, a set of all possible combinations of \text{\(x_A}\text{\)} and \text{\(x_B}\text{\)} that satisfy the constraints. The best solution - that is, the optimal production levels - must lie within this feasible region.

Objective Function

At the heart of every linear programming problem is the objective function, the mathematical representation of the goal – whether it's to maximize profits, minimize costs, or achieve the best balance of different variables. For the Optimal Profit Problem, the objective function is designed to maximize the profit from producing and selling the exercise machines.

The function \text{\(300x_A + 375x_B}\text{\)} directly translates the profit per unit made from models A and B into an equation. The values of \text{\(x_A}\text{\)} and \text{\(x_B}\text{\)}, which represent the number of each machine to produce, are what we are trying to find in order to maximize this function. The objective function thus acts as a guide, pointing towards the most profitable production levels within the constraints imposed by resources.

Simplex Method

The Simplex Method is a widely-used algorithm that efficiently solves linear programming problems. Developed by George Dantzig in 1947, it systematically checks the vertices of the feasible region to locate the maximum or minimum value of the objective function. It's particularly useful when dealing with multiple variables and constraints, where graphical methods become unwieldy or impossible to use.

In essence, the Simplex Method moves from one vertex of the feasible region to an adjacent vertex, choosing the move that increases (or decreases, for minimization problems) the value of the objective function. It continues this process until it reaches a vertex where no further improvement can be made – the optimal solution. Applying the Simplex Method to our problem would involve iterating through potential production level combinations of \text{\(x_A}\text{\)} and \text{\(x_B}\text{\)}, constrained by the assembly, finishing, and packaging capacities, to discover the one that maximizes the profit according to the objective function.