Model Answer
0 min readIntroduction
Public health campaigns often face budgetary constraints while aiming for maximum impact. Effective resource allocation is crucial, especially during crises like pandemics. This requires a systematic approach to optimize the use of available funds to reach the largest possible audience with vital information. Linear programming, a mathematical technique, provides a powerful tool for such optimization problems. This question presents a scenario where a State Public Health Department needs to allocate funds between radio and newspaper advertisements to maximize reach, subject to budgetary and minimum advertisement constraints.
Problem Formulation
Let:
- x = Number of radio advertisements
- y = Number of newspaper advertisements
Objective Function
The objective is to maximize the total reach. The objective function is:
Maximize Z = 3000x + 7000y
Constraints
The problem is subject to the following constraints:
- Budget Constraint: 20000x + 50000y ≤ 4000000 (Simplifies to 2x + 5y ≤ 400)
- Minimum Radio Advertisements: x ≥ 10
- Minimum Newspaper Advertisements: y ≥ 10
- Non-negativity Constraints: x ≥ 0, y ≥ 0 (These are implicitly satisfied by the minimum advertisement constraints)
Graphical Solution (or Corner Point Method)
We can solve this linear programming problem graphically. First, we plot the constraints on a graph. The feasible region is the area that satisfies all constraints. The optimal solution lies at one of the corner points of the feasible region.
Let's find the corner points:
- Intersection of x = 10 and y = 10: (10, 10)
- Intersection of x = 10 and 2x + 5y = 400: 2(10) + 5y = 400 => 5y = 380 => y = 76. So, (10, 76)
- Intersection of y = 10 and 2x + 5y = 400: 2x + 5(10) = 400 => 2x = 350 => x = 175. So, (175, 10)
Now, we evaluate the objective function Z at each corner point:
| Corner Point (x, y) | Z = 3000x + 7000y |
|---|---|
| (10, 10) | Z = 3000(10) + 7000(10) = 30000 + 70000 = 100000 |
| (10, 76) | Z = 3000(10) + 7000(76) = 30000 + 532000 = 562000 |
| (175, 10) | Z = 3000(175) + 7000(10) = 525000 + 70000 = 595000 |
The maximum value of Z is 595000, which occurs at the point (175, 10).
Optimal Solution
Therefore, the Public Health Department should use 175 radio advertisements and 10 newspaper advertisements to maximize reach.
Total Reach
The total number of people reached will be:
Total Reach = 3000(175) + 7000(10) = 525000 + 70000 = 595000 people
Conclusion
In conclusion, by strategically allocating the ₹40,00,000 budget, the State Public Health Department can reach 595,000 people by utilizing 175 radio advertisements and 10 newspaper advertisements. This solution, derived through linear programming, demonstrates the power of mathematical optimization in public health resource allocation. Further analysis could consider factors like audience demographics and advertisement effectiveness to refine the campaign even further.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.