Tom Smith is the manager of a discount store and wants to maximize the total revenue from two furniture items: a bedroom set and a living room set.

The constraints are as follows:
- The showroom space of 40,000 square feet must be completely filled with these sets.
- Bedroom sets occupy an average of 400 square feet each.
- Living room sets occupy an average of 500 square feet each.
- It requires approximately 14 minutes to unload a bedroom set.
- It requires approximately 7 minutes to unload a living room set.
- There are 980 minutes available for unloading.
- The living room set will produce revenue of $500 each.
- The bedroom set will produce revenue of $1000 each.
- A minimum of 20 living room sets should be ordered.

Tasks:
a. Formulate the problem in a linear programming format.
b. Determine the optimal solution.

Linear programming is a mathematical optimization technique used to maximize or minimize a linear objective function, subject to a set of linear constraints. It involves formulating a problem in a specific mathematical format that can be solved using linear programming algorithms.

To formulate the problem in linear programming format, we need to define the decision variables, objective function, and constraints.

Decision Variables:

Let:

x be the number of bedroom sets to be ordered.
y be the number of living room sets to be ordered.

Objective Function:

The objective is to maximize the total revenue. Since the bedroom sets produce revenue of $1000 each and the living room sets produce revenue of $500 each, the objective function can be defined as follows:

Maximize: Revenue = 1000x + 500y

Constraints:

1. Showroom Space Constraint: The total showroom space of 40,000 square feet should be completely filled with these sets, taking into account the average space occupied by each set:

400x + 500y ≤ 40,000

2. Time Constraint: The available time for unloading is 980 minutes. The time required to unload a bedroom set is 14 minutes, and the time required to unload a living room set is 7 minutes:

14x + 7y ≤ 980

3. Minimum Order Constraint: A minimum of 20 living room sets should be ordered:

y ≥ 20

4. Non-negativity Constraint: The number of sets ordered cannot be negative:

x, y ≥ 0

Now, let's determine the optimal solution by solving this linear programming problem.

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