In a project there are N tasks to be performed with the following data: N=number of task (task N last) (j=1,2,..,N) pj=duration of task j (number of periods) cj=cost of performing task j per period (payment), TL.

rj=revenue (cash) that will be collected when task j is completed,TL.

Bj=set of immediate predecessors of task j

Aj=set of immediate successors of task j

K0=amount of cash available at the start of the project fc= interest earned for cash per period (lending rate)

During the project when a task j is performed in a period a cash payment of cj TL is made and a revenue of rj TL is collected when task j is completed. Available cash in any period can be invested to bring a return of fc interest next period. Management does not want to allow cash available in any period be negative. Preemption of tasks are not allowed. For parts (b)-(e) describe only the changes that are needed to be done on the model of part (a). The models you develop must be in a form (Linear Expressions) that they could be solvable by commercial programming codes (like LINDO, GAMS, etc.) and clearly state the following: i) decision variables ii) additional parameters iii) constraints and objective function iv) assumptions (if any)

a) The management wants to complete the project within Tp periods such that total cash available at the end of the project will be maximized. Formulate a mathematical model for this objective. b) Let: Available cash in any period can be invested to bring a return of fc interest and it is possible to borrow money if management is short of cash at any period. Assume that if in any period some amount is borrowed then this amount together with the interest, rate of fb, is payed in the next period. Borrowing is limited to B TL per period. fc= interest earned for cash per period (lending rate) fb= interest payed for borrowing per period (borrowing rate) (fc< fb) How would you modify the model? c) Consider the model of part a. Suppose for each task j we have two alternative durations as pj1 > pj2 and their respective unit time costs as cj1 < cj2. How would you modify the model?