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 that had been scheduled to come to market in 20 months. However, 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, and these are the only levels that will be considered for the last three phases. The times required at these levels are given in the following table. (The times in parentheses at the normal level have been ruled out as too long.)

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 $50 million.
(a) Formulate this problem as a shortest-path problem.
(b) Use the algorithm described in Sec. 10.3 to solve this shortestpath problem.

Time Initiate Design of Production Remaining Manufacturingand Level Research Development System Distribution Normal 5 months(4 months) (7 months)(4 months) Priority 4 months3 months 5 months 2 months Crash 2 months2 months 3 months 1 month Cost Initiate Design of Production System Distribution Priority 6mn$6 million 9 mon $3 million Remaining Manufacturingand Level Research Development Normal $3 million Crash 9 mlin9 million 12 million 6 million