The Quick Company has learned that a competitor is planning to come out

with a new kind of product with a great sales potential. Quick has been

working on a similar product. Research is nearly complete and Quick's

management now wishes to rush the product out to meet the competition.

There are four nonoverlapping phases left to be accomplished, including

the remaining research that currently is being conducted at a normal pace.

However, each phase can instead be conducted at a priority or crash level

to expedite completion. The times required at these levels are given in

the following table.

Management has allocated $$30$ million for these four phases. The cost of

each phase at the different levels under consideration is as follows:

Management wishes to determine at which level to conduct each of the

four phases to minimize the total time until the product can be marketed

subject to the budget restriction of $$30$ million. Formulate this problem

as a shortest$-$path problem.