A brand manager for ColPal Products must decide how to allocate time and financial resources between radio advertising and television advertising for the next month.
The goal is to maximize audience exposure based on market research estimates for each medium. The following details are provided:
Budget: The total budget available for advertising is $25,000.
Advertising Costs:
Cost per minute of radio advertising is known.
Cost per minute of television advertising is known.
Audience Exposure:
Audience exposure per minute for radio advertising is provided.
Audience exposure per minute for television advertising is provided.
Constraints:
At least 70% of the total advertising time must be allocated to television advertising, as it has been found to be much more effective than radio advertising.
Task:
Determine the optimal allocation of time and financial resources between radio and television advertising to maximize audience exposure while staying within the budget and meeting the constraints
The Correct Answer and Explanation is:
To determine the optimal allocation between radio and television advertising, we must formulate a linear programming (LP) problem. Since the actual costs per minute and audience exposure per minute are mentioned but not provided in your message, I will proceed using placeholder values. You can substitute the actual values where needed.
Assumptions (Example values):
- Cost per minute (radio): $100
- Cost per minute (TV): $500
- Audience exposure per minute (radio): 1,000 people
- Audience exposure per minute (TV): 8,000 people
- Total budget: $25,000
- Constraint: At least 70% of total time must be TV
Decision Variables:
Let:
- RR = minutes of radio advertising
- TT = minutes of television advertising
Objective Function:
Maximize total exposure: Maximize Z=1000R+8000T\text{Maximize } Z = 1000R + 8000T
Constraints:
- Budget constraint:
100R+500T≤25,000100R + 500T \leq 25,000
- Time allocation constraint (TV ≥ 70% of total time):
T≥0.7(R+T)⇒0.3T≥0.7R⇒3T≥7RT \geq 0.7(R + T) \Rightarrow 0.3T \geq 0.7R \Rightarrow 3T \geq 7R
- Non-negativity:
R≥0, T≥0R \geq 0,\ T \geq 0
Solving the LP:
Let’s solve the system graphically or via substitution:
From the budget: R=25,000−500T100=250−5TR = \frac{25,000 – 500T}{100} = 250 – 5T
Substitute into the time constraint: 3T≥7R=7(250−5T)=1750−35T⇒3T+35T≥1750⇒38T≥1750⇒T≥46.05 minutes3T \geq 7R = 7(250 – 5T) = 1750 – 35T \Rightarrow 3T + 35T \geq 1750 \Rightarrow 38T \geq 1750 \Rightarrow T \geq 46.05\ \text{minutes}
Plug back: R=250−5(46.05)=19.75 minutesR = 250 – 5(46.05) = 19.75\ \text{minutes}
Optimal Allocation:
- Television: 46.05 minutes
- Radio: 19.75 minutes
- Cost: 100×19.75+500×46.05=1,975+23,025=25,000100 \times 19.75 + 500 \times 46.05 = 1,975 + 23,025 = 25,000 ✅
- Exposure: 1,000×19.75+8,000×46.05=19,750+368,400=388,1501,000 \times 19.75 + 8,000 \times 46.05 = 19,750 + 368,400 = 388,150 people
Conclusion
To maximize audience exposure given a $25,000 advertising budget, a brand manager must optimally allocate minutes to radio and television based on cost and effectiveness. Assuming radio costs $100 per minute with 1,000 audience reach, and television costs $500 per minute with 8,000 audience reach, television offers significantly higher exposure per dollar. However, cost constraints and strategic balance necessitate an optimized mix.
Using linear programming, we maximize the total exposure: Z=1000R+8000TZ = 1000R + 8000T, subject to a budget constraint 100R+500T≤25,000100R + 500T \leq 25,000, and a strategic constraint requiring at least 70% of advertising time to be on television, i.e., 3T≥7R3T \geq 7R.
Solving the system yields the optimal allocation: approximately 46.05 minutes of television and 19.75 minutes of radio. This mix exactly utilizes the entire $25,000 budget and satisfies all constraints. The total expected audience exposure with this allocation is approximately 388,150 people.
This result demonstrates that while radio is cheaper per minute, television’s much higher exposure rate makes it the dominant medium, justifying the strategic emphasis. By carefully balancing cost and reach and adhering to strategic constraints, ColPal can achieve the highest return on its advertising investment. This approach showcases how data-driven decisions can guide optimal marketing resource allocation.
