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 (eg. while he waits for paint to dry at one house he can install cabinets in another).

The contractor seeks a way to complete 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.