Question

Recall the production model from Section 1.3: Max 10x s.t. 5x s 40 x20 Suppose the firm in this example considers a second product that has a unit profit of $5 and requires 2 hours for each unit produced. Assume total production capacity remains 40 units. Use y as the number of units of product 2 produced

a. Show the mathematical model when both products are considered simultaneously
b. Identify the controllable and uncontrollable inputs for this model.
c. Draw the flowchart of the input-output process for this model (see Figure 1.5).
d. What are the optimal solution values of x and y?
e. Is this model a deterministic or a stochastic model? Explain.

Answer 1

a. The mathematical model is Maximize 10x + 5y subject to
`\(5x + 2y \leq 40\) and \(x, y \geq 0\).`

b. Controllable inputs: x and y; Uncontrollable input: Total production capacity (40 units).

Step-by-step explanation:

The mathematical model integrates both products, and the objective is to maximize the total profit, given the unit profit for each product and the production constraints. The objective function is 10x + 5y representing the total profit from products x and y. The constraints
`5x + 2y \leq 40\) and \(x, y \geq 0\)` reflect the production capacity and non-negativity requirements.

The controllable inputs are the decision variables x and y, as the firm can adjust the quantities produced to maximize profit. The uncontrollable input is the total production capacity of 40 units, which is a fixed parameter.

Drawing the flowchart involves visually representing the decision variables, objective function, constraints, and the output, which is the maximized profit. This graphical representation aids in understanding the structure of the mathematical model.

The optimal solution values of x and y can be obtained by solving the linear programming problem. These values indicate the ideal production quantities that yield the maximum profit while satisfying the given constraints. Finally, the model is deterministic since all parameters and coefficients are known and fixed, and there is no element of randomness in the decision variables or constraints.