A construction company is considering building three residential buildings in th

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 , $50,000; building 2, S30,000; building 3, $40,000 Duration of project(years) Number of 30 20 20 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)

Explanation / Answer

There are three buildings to be constructed, with the contraint that only one building can start during a year, constraint of 60 workers in a year, and building 2 must be completed by the end of year 4.

Building 1 requires 30 workers whereas building 2 & 3 require 20 workers each. Therefore we can not have a year in which all the three buildings are in costruction because it requires 70 workers so maximum two buildings continue with construction in a year.

Building 1 & 2 takes 2 years whereas building 3 takes the maximum time period of three years. Once all the three buildings are constructed, the total yearly rent is constant. Therefore, the objective is to have maximum yearly rent till all buildings are completed and further we require to complete the buildings at the earliest.

Due to workers constraint (mentioned above), minimum yearsrequired for construction are four.

Therefore, building 2 may be started either in 1st, 2nd or 3rd year.

Let us define our decision variables Xij taking the value 1 if ith building is started in the jth year otherwise it takes the value zero.

Objective is to have maximum rent and it is mentioned that yearly rent for 1st building is $50,000; for 2nd building $30,000 and that for 3rd building is $40,000. Say Ri for the ith building.

Maximum rent is for building 1, therefore let us start building 1 in the first year so that we can have rent for the years 3rd and fourth. In 2nd year we can start either of the remaining two building 2 or building 3 as construction of building 3 takes 3years therefore keeping the total construction as 4 years, we should have building 3 started (also having more rent) in 2nd year.

Lastly building 2 has to be completed by the end of 4th year so start it in the 3rd year.

Building 1 in 1st year Rent for two years = $100,000

Building 3 in 2nd year   Building 2 in the 3rd year

Maximum rent from start of fifth year.

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