Case Study: Optimizing Advertising Spend for a Product
Problem Statement
A company sells a product and wants to maximize its profit by optimally allocating its
advertising budget between two channels: online ads (O) and TV ads (T). The profit (P)
generated from the product sales depends on the amount of money spent on these two
advertising channels.
Objective
To find the optimal amounts of spending on online ads (O) and TV ads (T) that maximize the
profit (P).
Profit Function
The profit function is given by: P(O,T)=20O0.5+30T0.5−O−T
Here:
20O0.5 represents the revenue generated from online ads.
30T0.5 represents the revenue generated from TV ads.
O and T are the expenditures on online and TV ads respectively.
The terms −O and −T represent the costs associated with these expenditures.
Solution Approach
To maximize the profit, we need to find the values of O and T that maximize the function
P(O,T). This involves finding the critical points of P(O,T) and determining whether these
points are maxima, minima, or saddle points.
Detailed Steps
1. Compute the Partial Derivatives:
P’o=0 ∂P/∂O=0 10O-0.5 -1 =0 O = 100
P’T=0 ∂P/∂T=0 15T-0.5−1 = 0 T = 225
So, the stationary point is (O,T)=(100,225)
2. Second Partial Derivative Test:
P’’OO ( 100,225) = -5*100-1.5= -0.05
P’’TT ( 100,225) = -7.5*225-1.5 = -0.01
P’’OT (100,225) = 0
