91Ó°ÊÓ

In linear programming, the objective function is a mathematical expression that represents the goal of the optimization problem. It is essentially what you are trying to maximize or minimize.
In the context of our advertising problem, Excelsior Company aims to maximize exposure. The objective function is based on the estimated number of viewers for each time slot: morning, afternoon, and evening.

• For morning slots, each minute reaches 200,000 people.
• Afternoon slots reach 100,000 people per minute.
• Evening slots, being prime time, reach 600,000 people per minute.

These figures are combined to form the objective function:\[ Expose = 200,000 \times x + 100,000 \times y + 600,000 \times z \]This equation helps us understand how many people will be exposed to the commercials based on the time bought. The coefficients (200,000, 100,000, 600,000) reflect the potential audience in each segment. By maximizing the objective function, Excelsior can achieve the greatest possible audience reach.

Constraints are limitations or conditions that the solution must satisfy. They are crucial in linear programming because they limit how much of each resource you can use while striving to meet the objective.
In this example, several constraints define the boundaries of the problem:

• Budget Constraint: Excelsior can't spend more than $102,000 on advertising. This translates to a cost constraint: \( 3000x + 1000y + 12000z \le 102,000 \).
• Prime Time Constraint: Excelsior is limited to 6 minutes in the evening (prime time). This gives us \( z \le 6 \).
• Total Time Constraint: The total advertising time should not exceed 25 minutes across all slots, represented by \( x + y + z \le 25 \).
• Non-negativity Constraints: Time purchased in each slot must be non-negative: \( x \ge 0, y \ge 0, z \ge 0 \).

Together, these constraints create a feasible region within which Excelsior must operate. They ensure that while maximizing exposure, the company doesn't overrun its budget or the time limits imposed by Station KAOS.

Optimization in the context of linear programming involves finding the best solution within the defined constraints. It means finding values for the variables that maximize or minimize the objective function while respecting the conditions applied.
For the Excelsior advertising problem, optimization involves determining the best mix of advertising time that achieves the maximum audience exposure. Using linear programming methods, such as the Simplex Method or Solver software, you can efficiently find this optimal solution. These techniques evaluate the feasible region defined by the constraints to identify the point that gives the best outcome.
Upon solving, the optimal solution determined was:

• 0 minutes of morning advertising (\(x = 0\))
• 10 minutes of afternoon advertising (\(y = 10\))
• 6 minutes of evening advertising (\(z = 6\))

This allocation of advertising time ensures that Excelsior reaches the maximum possible audience of 3,800,000 people while staying within all constraints. Optimization thus ensures resource efficiency while achieving maximal results.