7. MILLILITERS COLLECTED = LITERS PER HECTARE BEING APPLIED
2.1. Systems Engineering, Operations Research,
2.1.2. Time and Project Management
One of the critical engineering/management decisions associated with a project that consists of numerous tasks is the selection of which course of action to take when a project must be completed in the shortest time possible. Decisions that result in an inefficient use of resources or a delay in the project completion can be costly. The two systems tools that are utilized for time and project management are the critical path method (CPM) and project evaluation and review technique (PERT).
CPM
The initial step in utilizing CPM is a detailed table with descriptions of the jobs or tasks that must be accomplished to complete the project, with the associated durations and predecessors for each job. The process of developing this project description must incorporate knowledge of the project and experience with the anticipated durations of each task, taking into account the resources available. The next step is the development of a graphical representation of the tabular project description. A network-flow model is a graphical representation of the initial tabular description, durations, and chronological order or sequence of jobs that must be completed for the project.
Example
A farmer desires a grain dryer and bin to store 10,000 bushels of grain. He needs this system prior to harvest time. He has contacted a systems engineer to perform the work.
The engineer has developed a table (Table 2.2) of jobs associated with this project.
Once the project has been described with durations and predecessors, a graphical description (project network) of the project is formulated. This network consists of a series of nodes connected by directed lines. Each line connecting a node represents a
Table 2.2.
Activity (Jobs) Description Predecessors Duration (wk)
a Excavate and prepare sites for — 2
concrete slabs
b Order and get delivery of bin — 7
and dryer
c Pour and finish concrete slabs a 4
d Erect grain bin b, c 3
e Install grain dryer b, c 5
f Provide electrical requirements b 3
g Install conveying equipment d, e, f 5
h Test the system g 2
i Clean the site h 1
Figure 2.1.
specific job and time duration. When constructing this network there are several rules that must be followed. The first is that when analyzing concurrent activities, two lines cannot be connected to the same node. To overcome this, a new node and a dummy line should be used, as indicated by Fig. 2.1.
In Fig. 2.1, the new line, d1, has a duration of 0. The second rule is that no job or jobs leaving a node may start until all jobs entering that node are completed. For example, referring to Fig. 2.1, job c, which leaves from node 2, cannot start until both jobs a and b are completed. So, if job a has a duration of 7 weeks, and b, 5 weeks, job c cannot start until after 7 weeks. When looking at an overall project network, the longest path from the start of the entire project to its completion is referred to as the critical path. Conti- nuing with the previous example, the critical path would be 1–3–2–4 with a duration of 7 weeks plus the duration of job c. Another important term is activity slack. Activity slack is the amount of time a specific job can be delayed without affecting the overall project duration. Again, using the above example, because we know that activity 2–4, or job c, cannot start until 7 weeks, and activity 1–2, or job b, has a duration of 5 weeks, job b can be delayed, starting 2 weeks after the start of job a, and still be completed by 7 weeks so that job c can start at 7 weeks. This means that activity 1–2 has a slack time of 2 weeks.
Following the steps outlined above, a nodal network for the example project has been developed, as shown in Fig. 2.2.
The next step, once the project nodal network has been constructed, is to determine the critical path of the overall project. This allows a manager to determine what key jobs must be targeted in order to insure that the project is completed in the least time. There are several software packages on the market that allow the user to input the different jobs, their predecessors, and the job durations and output project duration, critical path, and slack times associated with the different jobs. For smaller projects, this analysis can be done by hand. This is accomplished by the use of a CPM table. This table contains
Figure 2.2.
Table 2.3.
Activity/Job Duration ES EF LS LF Float
1–2/a 2 0 2 1 3 1
1–3/b 7 0 7 0 7 0
2–4/c 4 2 6 3 7 1
3–4/d1 0 7 7 7 7 0
3–5/f 3 7 10 9 12 2
4–5/e 5 7 12 7 12 0
4–6/d 3 7 10 9 12 2
5–6/d2 0 12 12 12 12 0
6–7/g 5 12 17 12 17 0
7–8/h 2 17 19 17 19 0
8–9/i 1 19 20 19 20 0
the following column headings: activity, duration, early start (ES), early finish (EF), late start (LS), late finish (LF), and float or slack (F or S). ES and EF are the earliest times that a job can be started and completed. LS and LF are the latest times that a job can be started and finished. The float or slack is the activity slack associated with a specific job. It is equal to LF minus EF, or LS minus ES. (If the analysis is performed correctly LF−EF=LS−ES.) Activities that have a float of zero are on the critical path. Table 2.3 is the result of analysis for the example described previously.
Thus the critical path is 1–3–4–5–6–7–8–9, with a duration of 20 weeks.
PERT
PERT follows the same procedure as CPM analysis with the exception of the addition of probabilistic times for the job durations. For each job, three different times are given, a, m, and b, with a an optimistic time, m the most likely time, and b the pessimistic time.
In order to determine the expected time for the job, the following relationship is used:
timeexpected=t= a+4m+b
6 (2.20)
In addition, the variance associated with the duration of each job can be determined using the following relationship:
variance=σ2= àb−a
6
ả2
(2.21) Using the same table as for CPM analysis, and using the expected time, t , calculated from a, m, and b, the critical path and the expected duration of the critical path can be determined. In order to calculate the variance of the critical path, the variances of the jobs in the critical path are summed. The probability of the project being completed in a certain time can now be determined using the normal distribution. This is accomplished by determining the number of standard deviations away from the mean (à, duration of
critical path) of the Z value.
Z = x−à
σ (2.22)
Using this Z value, a normal distribution table, critical path duration (à), and standard deviationσ (σ =sqrtσ2), one can determine the probability that a certain project will be completed in duration X .