The Quick Company has learned that a competitor is planning to come out with a new product with 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$$s management now wishes to rush the product out to meet competition. There are four nonoverlapping phases left to be accomplished: the remaining research $($activity $1),$ development $($activity $2),$ design of manufacturing system $($activity $3),$ and initial production and distribution $($activity $4).$ Activity $1$ can be conducted at three levels: normal, priority, and crash. Each level of activity is characterized by a different activity duration and cost. Similarly, activities $2,3,$ and $4$ can be conducted at a priority or crash levels. The times and costs at these levels are given on the project network diagram in Figure $1($in parentheses$).$ The times at the normal level for activities $2,3,$ and $4$ have been ruled out as too long. Management has allocated $$30$ million for these four phases.

The management wishes to determine at which level to conduct each of the four phases of the project to minimize the total time until the product can be marketed subject to the budget constraint of $$30$ million.