Model Answer
0 min readIntroduction
Public health campaigns often face the challenge of maximizing impact with limited resources. Effective advertising is crucial for disseminating vital information, particularly during public health crises like pandemics. The allocation of funds across different advertising channels requires careful consideration to reach the largest possible audience. This problem exemplifies a common operational research scenario where optimization techniques are employed to achieve the best possible outcome given specific constraints. This question tests the ability to apply mathematical modeling to a real-world administrative problem, a skill valuable for effective public health management.
Problem Formulation
Let:
- x = the number of radio advertisements
- y = the number of newspaper advertisements
Objective Function: Maximize the total reach (Z)
Z = 3000x + 7000y
Constraints:
- Budget Constraint: 20000x + 50000y ≤ 4000000
- Minimum Radio Advertisements: x ≥ 10
- Minimum Newspaper Advertisements: y ≥ 10
- Non-negativity Constraint: x, y ≥ 0 (implied, but good practice to state)
Simplifying the Budget Constraint
The budget constraint can be simplified by dividing by 10000:
2x + 5y ≤ 400
Graphical Solution (or Algebraic Approach)
We can solve this problem graphically or algebraically. Let's use an algebraic approach combined with understanding the feasible region.
Finding Corner Points of the Feasible Region
The feasible region is defined by the constraints. We need to find the corner points of this region.
- 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)
Evaluating the Objective Function at Corner Points
Now, we evaluate the objective function Z = 3000x + 7000y at each corner point:
- Z(10, 10) = 3000(10) + 7000(10) = 30000 + 70000 = 100000
- Z(10, 76) = 3000(10) + 7000(76) = 30000 + 532000 = 562000
- Z(175, 10) = 3000(175) + 7000(10) = 525000 + 70000 = 595000
Optimal Solution
The maximum reach (Z) is 595000, which occurs when x = 175 and y = 10.
Total Reach
Therefore, the Public Health Department should use 175 radio advertisements and 10 newspaper advertisements to reach the maximum number of people, which is 595,000.
Conclusion
In conclusion, by employing a linear programming approach, the State Public Health Department can optimize its advertising campaign for the Coronavirus pandemic. The optimal strategy involves allocating resources towards 175 radio advertisements and 10 newspaper advertisements, resulting in a reach of 595,000 people. This demonstrates the power of quantitative methods in public health decision-making, ensuring efficient resource utilization and maximizing the impact of crucial information dissemination. Further analysis could consider factors like demographic targeting 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.