A construction company is considering building three residential buildings in th
ID: 366687 • Letter: A
Question
A construction company is considering building three residential buildings in the city. The time required to complete each building and the number of workers required to be on the job at all times are provided in table below. Once a building is completed, it brings in the following amount of rent per year: building 1, $50,000; building 2, $30,000; building 3, $40,000.
The construction company faces the following constraints:
1. During each year, 60 workers are available
2. At most, one building can be started during any year
3. Building 2 must be completed by the end of year 4
Formulate an IP that will maximize the total rent earned by company through the end of year 4. (Do not solve) (Hint: Define your decision variables as which buildings to start in each year and as binary )
Building Duration # of workers required 1 2 30 2 2 20 3 3 20Explanation / Answer
Decision variables: Xij (binary variable) = 1, if building i is in year j, where both i and j = 1,2,3
Objective: Maximize X11*2*50000 + X12*1*50000+X21*2*30000+X22*1*30000+X31*40000
(explanation: If building 1 is started in year 1, then it will be completed in 2 years and it will earn rental income for the remaining 2 years until the end of year 4, similarly, if it is started in year 2, then it will earn rental income for only 1 year until the end of year 4. Similarly rental is calculated for building 2. However, building will earn rental for only 1 year, if it is started in year 1 and it will earn no rental income in the four years, if it is not started in the first year, because then it will not be completed by end of year 4)
s.t.
Xi1 <= 1
Xi2 <= 1
Xi3 <= 1
30X11+20X12+20X13 <= 60
30X11+30X21+20X12+20X22+20X13+20X23 <= 60
30X21+30X31+20X22+20X32+20X13+20X23+20X33 <= 60
X11+X21+X31 = 1
X12+X22 = 1 this represents the constraint that building must be started during either 1 or year 2, so that it can be completed by the end of year 4.
X13+X23+X33 = 1
Xij = binary
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