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 (more time usually saves expense).

j Min.

Time Max.

Time Cost Min.

Cost Max.

Predecessor Tasks 1 6 12 1600 1000 None 2 8 16 2400 1800 None 3 16 24 2900 2000 2 4 14 20 1900 1300 1,2 5 4 16 3800 2000 3 6 12 16 2900 2200 3 7 2 12 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.

(a) Formulate an LP model of this time/cost planning problem using the decision variables

(j = 1,c, 7)

sj ! start time of task j 1in days2 tj ! days to complete task j Your model should have an objective function summing interpolated cost and main constraints to enforce precedence relationships and the time limit.

(b) Enter and solve your model with the class optimization software.