Linear Programming Problem for Optimal Production Plan

What are the constraints and objectives in formulating a linear programming problem for optimal production planning?

Given the data provided, what is the best production plan to maximize profit?

Optimal Production Plan

The company's best production plan to maximize profit is to produce 90 deluxe ink pens and 15 budget ink pens per week.

To formulate the linear programming problem for optimal production planning, we need to consider the constraints and objectives involved. In this case, the company wishes to produce deluxe and budget ink pens while maximizing profit within certain limitations.

Constraints:

1. Labor Availability:

The company has 38.25 hours of labor available per week. Each deluxe ink pen requires 9 minutes, and each budget ink pen requires 4 minutes to produce. The total labor constraint can be expressed as:

9D + 4B ≤ 2295 (where D is the number of deluxe ink pens and B is the number of budget ink pens)

2. Material Availability:

The company has 1950 units of material available per week. Deluxe ink pens use 8 units of material, and budget ink pens use 6 units. The total material constraint can be formulated as:

8D + 6B ≤ 1950

3. Production Ratio Constraint:

The company wants the number of deluxe ink pens produced to be at least 6 times the number of budget ink pens produced, which can be represented as:

D ≥ 6B

4. Non-Negativity Constraint:

Both the number of deluxe ink pens (D) and budget ink pens (B) should be greater than or equal to zero.

Objective:

The objective of the linear programming problem is to maximize profit. The profit for deluxe ink pens is $2.40 per pen and for budget ink pens is $1.75 per pen. The objective function can be defined as:

Maximize Profit = $2.40D + $1.75B

By formulating the constraints and objective function, the company can determine the optimal production plan that maximizes profit while respecting the labor and material availability constraints.