Question

The Hustle Hut is a newly establish fitness gym in Kanata and the owners have set aside a monthly budget of up to $1000 to be used for advertising. The money can be allocated between using social media influencers or search engine optimization to promote the gym. The owners would like for at least 25% of the amount spent to be used on each form of advertising. Furthermore, the amount of money spent on the social media influencer must be at least twice the amount spent on search engine optimization to support local members of the fitness community. A marketing consultant has evaluated the impact of each advertising campaign and determined that each dollar spent on the social media influencer will result in 5 people visiting the gym’s website. Also, each dollar spent on search engine optimization yields 8 people visiting the website. How should the gym’s management team allocate their budget for this coming month in order to maximize the number of people who will visit the gym website?
1. Formulate a linear programming model that can be used to help management decide on how they should allocate their advertising budget.
2. Solve the problem using the graphical method (use the corner point method) and show all of your work.
3. Solve the problem using Excel Solver and show all of your work (i.e., include the Excel formulation and answer report in your submission).
4. Provide a managerial statement that summarizes your solution for management.

Answer 1

1.

We have two options, social media influencers and search engine optimization. Let the money allocated to social media influencers be X1 and the money allocated to search engine optimization be X2.

Our objective is to maximize the number of people visiting the gym’s website. Now, the problem has given the number of people that visit for every dollar spent on each of those options. We can formulate this as

Maximize Z where Z = 5X1 + 8X2

However, we have a few constraints.

The first constraint is that the budget is 1000. This can be written as X1 + X2 <= 1000

The next constraint is that at least 25% of the fund should be allocated to each of them. This means X1 >= 0.25(X1 + X2) and X2 >= 0.25(X1 + X2).

The final constraint is that the social media budget must be at least 2 times the search engine optimization budget. This means X1 >= 2X2

Our implicit constraint here is that the budget allocation cannot be negative. Thus we need to mention X1, X2 >= 0

This finalizes the formulation and to simplify and summarize we can write

Maximize 5X1 + 8X2

ST

X1 + X2 <= 1000

0.75X1 – 0.25X2 >= 0

-0.25X1 + 0.75X2 >= 0

X1 – 2X2 >= 0

X1, X2 >= 0

2.

In the graphical method, we need to draw the constraints and determine the feasible area. Next, draw the corner points on the feasible region. Finally substitute the coordinates for the corner points in the objective function to obtain the maximum value.

The corner point with highest objective function value is the optimal solution.

The diagram below shows the graph.

The corner points are

Corner point 1 = (0,0)

Corner point 2 = Cross between X1 + X2 = 1000 and X1 – 2X2 = 0. It will be (666.67, 333.33)

Corner point 3 = Cross between X1 + X2 = 1000 and -0.25x1 + 0.75X2 = 0. It will be (750,250)

Placing them at the objective function value, we get

Corner point 1 = 5*0 + 8*0 = 0

Corner point 2 = 5*666.67 + 8*333.33 = 6000 (approx.)

Corner point 3 = 5*750 + 250*8 = 5750

The solution is 666.67, 333.33 and the value is 6000 audiences.

3.

The excel model is shown below.

The formulas are shown below.

The solver parameters are shown below.

The result is shown below.

4.

Recommendation to the management: the management should allocate $666.67 to the social media influencer and $333.33 to the search engine optimization. The maximum possible audience that can be achieved is 6000