George and Harry Haygood are building contractors who specialize in the construction of private home dwellings, storage warehouses, and small businesses $($less than $20,000$ sq$.$ ft$.$ of floor space$).$ Both George and Harry entered a carpenter union$$s apprenticeship program in the early $1990$s and, upon completion of the apprenticeship, became skilled craftsmen in $1996.$ Before going into business for themselves, they worked for several local building contractors in the Detroit area.

Typically, the Haygood Brothers submit competitive bids for the construction of proposed dwellings. Whenever their bids are accepted, various aspects of the construction $($electrical wiring, plumbing, brick laying, painting, and so forth$)$ are subcontracted. George and Harry, however, perform all carpentry work. In addition, they plan and schedule all construction operations, frequently arrange interim financing, and supervise all construction activities.

The philosophy under which the Haygood Brothers have always operated can be simply stated: $$Time is money.$$ Delays in construction increase the costs of interim financing and postpone the initiation of their building projects. Consequently, they deal with all bottlenecks promptly and avoid all delays whenever possible. To minimize the time consumed in a construction project, the Haygood Brothers use PERT.

First, all construction activities and events are itemized and properly arranged $($in parallel and sequential combinations$)$ in a network. Then time estimates for each activity are made, the expected time for completing each activity is determined, and the critical $($longest$)$ path is calculated. Finally, earliest times, latest times, and slack values are computed. Having made these calculations, George and Harry can place their resources in the critical areas to minimize the time of completing the project.

The following are the activities that constitute an upcoming project $($home dwelling$)$ of the Haygood Brothers:

Arrange financing $($A$)$

Let subcontracts $($B$)$

Set and pour foundations $($C$)$

Plumbing $($D$)$

Framing $($E$)$

Roofing $($F$)$

Electrical wiring $($G$).$

Installation of windows and doors $($H$)$

Ductwork and insulation $($including heating and cooling units$)($I$)$

Sheetrock, paneling, and paper hanging $($J$)$

Bricking $($K$)$

Installation of cabinets $($L$)$

Inside trim $($including fixtures$)($M$)$

Outside trim $($N$)$

Painting $($O$)$

Flooring $($P$)$

The immediate predecessors and the optimistic $($a$),$ most likely $($m$),$ and pessimistic $($b$)$ time estimates, are shown in Table $1.$

TABLE $1.$ Haygood Brothers Construction Co$.$

$1.$ What is the time length of the critical path? What is the significance of the critical path?

$2.$ Compute the amount of time that the completion of each activity can be delayed without affecting the overall project.

$3.$ The project was begun August $1.$ What is the probability that the project can be completed by September $30?($Note: Scheduled completion time $=60$ days.$)$