The difficulty of figuring out an optimal solution of a multiple objective engineering or a business problem lies in the competing conflicts that may exist between the optimal solutions for the different objectives. The best solution for a given objective might be undesirable for another.
Both functions 1 and 2 are maximized at two different point of feasible region. A feasible solution is considered efficient or Pareto optimal if there are no other feasible solution with all the objective functions.
Adword Spend Optimization using goal programming
A common inbound marketing budget requirement is to generate maximum number of qualified leads as cheaply as possible. Excel Solver and Matlab can be used as effectively to allocate a marketing budget among various internet marketing vehicles. A Marketing Research Analyst collected the following Ad-word conversion rate across different social media and search engine vehicles for a client in financial industry.
| Internet Mktg. Vehicle | Min. Lead Conversion Rate | Max. Lead Conversion Rate | Expected Lead Conversion Rate |
| Google Search | 3.50% | 8.00% | 7.32% |
| Google Display | 1.65% | 5.82% | 4.97% |
| Bing Search | 2.10% | 6.32% | 5.25% |
| Bing Display | 1.52% | 6.95% | 5.21% |
| Yahoo Search | 1.89% | 5.82% | 4.97% |
| Yahoo Display | 1.45% | 5.92% | 5.10% |
| 4.50% | 8.50% | 6.25% | |
| 3.89% | 7.20% | 6.30% | |
| 3.23% | 4.80% | 4.15% |
- The client has stipulated these conditions,
- All $1M is to be invested
- At most $600,000 of ad-word spend can be in Search and Display Network
- At least $650,000 of the total ad spend should have the maximum potential of qualified lead conversion rate to be 4% or greater
The client requirement stipulates to set up a goal based programming where the minimum lead conversion rate is assumed to be to twice as important as the expected lead conversion rate and three times as important as Ad-words on Display network. Particularly, the following goals are set on each objective:
1. The minimum overall lead conversion rate must be at least 3.5%
2. The expected lead conversion rate must be at least 5.5%
3. At most, $200,000 can be allocated in Display networks (Bing, Google, and Yahoo)
4. At minimum, $250,000 must be allocated towards Social Media networks (Facebook, LinkedIn, and Twitter)
Mathematical Background:
For a set of objective functions, we need to repose multi-objective problem as a single objective, fp(x) by placing limits on remaining objectives, and keep hard constraints.
subject to
gj(x) ≤ 0 (inequality constraints)
hk(x) = 0 (Hessian Matrix)
xiL ≤ xi ≤ xiU
The above problem can be formulated as
minimize fp(x)
subject to fl(x) ≤ εl l = 1,2,…,nobj (l ≠ p)
gj(x) ≤ 0
hk(x) = 0
xiL ≤ xi ≤ xiU
Problem Formulation
In this goal attainment problem, the following variables need to be defined for under and overachieving goals.
U1 – Under achievement of goal 1 which is the minimum lead conversion rate
O1 – Over achievement of goal 1
U2 – Under achievement of goal 2 which is expected lead conversion rate
O2 – Over achievement of goal 2
U3 – Under achievement of goal 3
O3 – The planned budget $200,000 allocated towards Display Network advertising should not be exceeded. O3 needs to be minimized.
U4 – At least $250,00 must be allocated towards Social Media advertising. Under achieving U4 is undesirable and thus needs to be minimized
O4 – Over achievement of goal 4
The objective function must be constructed in a way that the sum of undesirable deviations from the predetermined goals is minimized.
Goal 1 – is to have 3.5% minimum lead conversion over the number of clicks. Underachievement of the goal of 3.5% is not desirable. Hence, the variable to minimize is therefore U1.
Goal 2 – is to have 5.5% expected lead conversion over the number of clicks. Under achievement of 5.5% goal is undesirable. U2 therefore needs to be minimized.
Goal 3 – Budget allocation towards online display network should be restricted within $200,000. Over achievement of this goal is undesirable. This variable has to be minimized.
The objective function f(x), can be written as
f(x) = 3U1 + 1.5U2 + 1O3
Since, minimum lead conversion rate is assumed to be to twice as important as the expected lead conversion rate and three times as important as Ad-words on Display network.
Decision Variables – Let GS represent the Google Search Network, GD represent Google display network, similar abbreviations for other search engines, FB represents FaceBook, TW represents the Twitter, and LI for LinkedIn.
There are few soft and hard constraints in this problem. Soft constraints give more scope for relaxing the restrictions depending on the preference of the constraint being satisfied in the objective.
Solver results interpretation:
1. Goal 1 is over achieved since O1 = 40. The client wanted to limit the minimum overall lead conversion to $35,000. Anyone would love higher returns than expected as long as other hard constraints have been satisfied !
2. Goal 2 is also over achieved since the slack variable O2 = $9,700. This actually a good news since the expected lead conversion rate exceeds the target $55,000 (5%). The actual conversion is $55,000 + $9,700 = $64,700 or 6.47% !
3. Goal 3 is fully achieved as both U3 and O3 are zero.
4. Goal 4 is over achieved.
The client’s ad budget allocation should be as:
$400,00 towards Google Search Ad-words
$0 for Google Display Ads
$0 for Bing Search Ad-words
$200,000 towards Bing Display Ads
$0 for both Yahoo Search and Display
$400,000 towards Facebook
$0 for both Twitter and LinkedIn
to achieve the ad-spend allocation optimally.
Note that the goal setting approach to multi-objective optimization can bring inefficient solutions. Although, they can satisfy all the goals, but cannot be considered as optimal.
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