La Quinta Motor Inns is a mid-sized hotel chain headquartered in San Antonio, Texas. They are looking to expand to more locations, and know that selecting good sites is crucial to a hotel chain’s success. Of the four major marketing considerations (price, product, promotion, and location), location has been shown to be one of the most important for multisite firms.
Hotel chain owners who can pick good sites quickly have a distinct competitive advantage, since they are competing against other chains for the same sites. La Quinta used data on 57 existing inn locations to build a linear regression model to predict “Profitability”, computed as the operating margin, or earnings before interest and taxes divided by total revenue. They tried many independent variables, such as “Number of hotel rooms in the vicinity” and “Age of the Inn”. All independent variables were normalized to have mean zero and standard deviation 1.
The final regression model is given by:
Profitability = 39.05 - 5.41*(State Population per Inn) + 5.86*(Price of the Inn) - 3.09*(Square Root of the Median Income of the Area) + 1.75*(College Students in the Area)
The \(R^2\) of the model is 0.51.
In this problem, we’ll use this regression model together with integer optimization to select the most profitable sites for La Quinta.
The variables with positive coefficients in the regression equation positively affect profitability: Price of the Inn, and College Students in the Area.
By substituting the data for hotel 1 into the regression equation, we get that:
Profitability = 39.05 - 5.41*(-1.00) + 5.86*(-0.30) - 3.09*(-0.81) + 1.75*(-0.54) = 44.24
Hotel 2 has the highest predicted profitability of 53.38.
Hotel 8 has the lowest predicted profitability of 23.45.
We would start by buying hotel 2 (the most profitable hotel). This hotel costs $10,000,000, which is our entire budget. Thus, we would buy one hotel.
Since we just bought one hotel (hotel 2), our total predicted profitability is just the profitability of hotel 2, which is 53.38.
Now, build an optimization model in your spreadsheet to select hotels. The decision variables are whether or not a hotel is selected (binary variables). The objective is to maximize the total predicted profitability. We have two constraints: the decision variables should be binary, and the total cost should not exceed the budget of $10,000,000. Formulate and solve this model in LibreOffice.
Our formulation in LibreOffice has 16 decision variables (one for each hotel) and one contraint (the budget). The objective is the sumproduct of the decision variables with the profitability. If we formulate and solve this problem, the objective value of the solution is 269.925.
Seven hotels have decision variables equal to 1 in the solution.
Hotels 10-16 are located in South Lake Tahoe, and 6 of these have decision variables equal to 1 in the solution.
If we add a constraint to limit the number of hotels in South Lake Tahoe (SUM(I13:I19) less than or equal to 2) and resolve the problem, the objective value of the solution is 205.7.
Now, six decision variables have value 1 in the solution.
The hotels with decision variables equal to 1 are located in Eureka, Fresno, Los Angeles, and South Lake Tahoe. The only city in which we do not buy a hotel is Long Beach.
If we want to maximize the average profitability, then it is always optimal to select only the hotel that is the most profitable. Additionally, if we don’t have a budget constraint and instead have a constraint that we can only select N hotels, it is optimal to just select the N most profitable hotels. So in the first two situations, the greedy approach would perform as well as the optimization approach. In the third situation, the optimization approach would still perform much better than the greedy approach.