Operations and Supply Chain Management The Core

146 OPERATIONS AND SUPPLY CHAIN MANAGEMENT
Probability Analysis
The three-time-estimate approach introduces the ability
to consider the probability that a project will be completed
within a particular amount of time. The assumption needed
to make this probability estimate is that the activity duration
times are independent random variables. If this is true, the
central limit theorem can be used to find the mean and the
variance of the sequence of activities that form the critical
path. The central limit theorem says that the sum of a group
of independent, identically distributed random variables
approaches a normal distribution as the number of random
variables increases. In the case of project management
problems, the random variables are the actual times for the
activities in the project. (Recall that the time for each activity
is assumed to be independent of other activities
and to follow a beta statistical distribution.) For
this, the expected time to complete the critical
path activities is the sum of the activity times.
Likewise, because of the assumption of
activity time independence, the sum of the variances
of the activities along the critical path is
the variance of the expected time to complete
the path. Recall that the standard deviation is
equal to the square root of the variance.
To determine the actual probability of completing
the critical path activities within a certain
amount of time, we need to find where on our
probability distribution the time falls. Appendix E
shows the areas of the cumulative standard normal distribution
for different values of Z. Z measures the number of
standard deviations either to the right or to the left of zero
in the distribution. The values correspond to the cumulative
probability associated with each value of Z. For example,
the first value in the table, −4.00, has a G(z) equal to
0.00003. This means that the probability associated with a
Z value of −4.0 is only 0.003 percent. Similarly, a Z value of
1.50 has a G(z) equal to 0.93319 or 93.319 percent. The
Z values are calculated using equation 5.3, given in step
7b of the “Three Time Estimates” example solution. These
cumulative probabilities also can be obtained by using the
NORM.S.DIST(Z,TRUE) function built into Microsoft Excel.
Negative values of Z
Probability
of Z
0
Positive values of Z
Minimum-Cost Scheduling (Time–Cost Trade-Off) The basic assumption in minimumcost
scheduling, also known as “crashing,” is that there is a relationship between activity
completion time and the cost of a project. Crashing refers to the compression or shortening
of the time to complete the project. It costs money to expedite an activity, and these costs
are termed activity direct costs and add to the project direct cost. Some may be workerrelated,
such as scheduling overtime work, hiring more workers, and transferring workers
from other jobs; others are resource-related, such as buying or leasing additional or more
efficient equipment and drawing on additional support facilities.
The costs associated with sustaining the project are termed project indirect costs: overhead,
facilities, and resource opportunity costs, and, under certain contractual situations,
penalty costs or lost incentive payments. Because activity direct costs and project indirect
costs are opposing costs dependent on time, the scheduling problem is essentially one of
finding the project duration that minimizes their sum, or in other words, finding the optimum
point in a time–cost trade-off.
The procedure for project crashing consists of the following five steps. It is explained
by using the simple four-activity network shown in Exhibit 5.10. Assume that the indirect
costs remain constant at $10 per day if the project takes eight days or less and then
increases at the rate of $5 per day.