A construction contractor has undertaken a job with 7 major tasks. Some of the tasks can begin at any time, but others have predecessors that must be completed first. The following table shows those predecessor task numbers, together with the minimum and maximum time (in days) allowed for each task, and the total cost that would be associated with accomplishing each task in its minimum and maximum times. Here, more time spent reduces cost, because otherwise the contractor has to pay for more expensive pre-finished products and/or pay for an assistant. Provided precedence constraints are met, the contractor can work on multiple projects in parallel (e.g. while he waits for paint to dry at one house he can install cabinets in another).
Min Max Cost Cost Predecessor j Time Time Min Max Tasks
1 6 15 1600 1000 None
2 4 12 2400 1800 None
3 5 15 3000 2000 2
4 6 14 2000 1400 1, 2
5 4 24 4000 2000 3
6 12 20 3000 2200 5
7 5 10 1300 800 4
The contractor seeks a way to complete all work in 40 days, at least total cost, assuming that the cost of each task is linearly interpolated for times between the minimum and maximum. Formulate a linear optimization model for this problem. Clearly define in words what your decision variables represent. Label each constraint with a keyword/phrase for what the constraint corresponds to. You do not need to solve the problem. You can ignore integrality.