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CHAPTER TWELVE
PROJECT MANAGEMENT : A NETWORK MODEL
Projects that are undertaken and completed many times over are those that best lend themselves to tight control. The time, personnel, and other resources needed for completion are fairly well understood from past experience. Projects that are one-time-only undertakings are more difficult to plan and con- trol. Many activities in the public sector are of this variety. The overall project may he a first, but the components or tasks of (^) the project have been done many times before in similar projects. Some examples are : construction of a highway or a hospital ; preparing for a municipal bond issue ; development and imple- mentation of a management information system ; transferring operations from one site to another, as from one hospital building to a new one ; and conducting a major research effort. A model that is particularly well suited to the control of such major projects is the subject of this chapter. While various diagrams, often informal, have long been used to represent sequences and interrelationships that characterize multiactivity projects, it was not until the late 1950s that formal network methods were proposed to plan, schedule, and control such projects. The two different network models that were developed concurrently and independently wound up being amazingly similar. Except for minor structural and notational differences, each model had one main ingredient that was absent in the other. The PERT model (perform- ance evaluation review technique or program evaluation review technique) originally permitted three different time estimates for each individual task, while CPM (critical path method) permitted only a single time estimate. On the 273
2MIUANT" IVE
METHODS F(iR - PUHLIC`FhSJTvrAK, ING w
other hand, CPM identified time-cost trade-off points, whereas PERT did not. During the years since their origin the two methods have essentially been merged, with the current product incorporating the advantages of each model as well as many later developments. In our discussion of project management we will make no distinction between PERT and CPM .'
CONTENTS OF PERT/CPM
To plan and to control a project require two pieces of basic information about each task :
- The length of time to complete each task
- The tasks that must be completed before another task can begin
We will present these and the other fundamental ideas of PERT/CPM within the framework of a seminar development project, a commonly occurring one in any service agency.
A Seminar Development Project : An Illustration
Suppose the state council on mental health and mental retardation has devel- oped an information system that it expects will provide the council with state-wide program information and the many county agencies with local pro- gram and client information. Realizing that such an information/evaluation system requires user understanding and cooperation, the council is planning to conduct a number of 3-day seminars at key locations throughout the state. Ann Maxall is responsible for coordinating the seminars. She has identified the activities that must be completed prior to the seminars themselves. She has also identified which activities must be completed prior to any one activity and has estimated the number of weeks needed to perform each activity. With these
Table 12-1 Activities, predecessors, and time estimates in the seminar planning project
' Actually we will be using the terminology of PERT and the network analysis of CPM.
W W M1 M , =1 =I W M) PROJECT MANAGEMENT. A NE I - WORK MODEL (^275)
Start
o s (^) IS
03 Finish
Figure 12-1 (^) PERT diagram of the seminar planning project with activity completion times.
two pieces of basic information, the list of immediate predecessors and the es- timated time for each activity, the planning project for the seminars is described as in Table 12-I. No~c (^) ~f ~~k /mss (nod~s~ 1~^ GI^ - 1o r~ t~spr,~
Project Network
The immediate predecessor of activity (^) B is activity A ; (^) this means that activity A, (^) "plan seminar content," must be completed before activity (^) B, "obtain speakers," can he started. In using the network diagram to illustrate the pro- ject, activities are represented by circles and (^) sequencing is represented by arrows Fo example, the arrow Tom activity A to activity B shows that activ- ity A (^) is the immediate predecessor of activity (^) B. (^) Once the network is complete, the estimated time of each activity can be inserted in the circle representing that activity. Figure 12-1 is the network diagram of the seminar planning pro- ject. Note in Table 12-I that activities (^) B (^) and C are both predecessors of activity D ; (^) in the diagram this is represented by arrows drawn from both (^) B (^) and C to D. Similarly, since activity (^) D (^) is the immediate predecessor of both (^) E and (^) F, arrows are drawn from D to both (^) E and F. (^) The activities represented by the "start" circle and "finish" circle are not real activities but merely signify the beginning and end of the project. The estimated time for each activity can be used to estimate the time for the whole project ; in particular the time estimates will lead us to the (^) critical purl, (^) for the project.
THE CRITICAL PATH
A series of connected activities is called (^) a pail,. (^) In the seminar planning pro- ject, one path consists of activities (^) A, B, D, E ; (^) another path consists of (^) C, D, /. There are two other paths through the entire project from start to finish (can you find them?). The length of time associated with the path is the sum of the
M) =I
Activity Description
Immediate predecessors
Estimated time (weeks)
A Plan seminar content 2 B Obtain speakers A 1 C Select seminar sites 2 D (^) Prepare and mail flyer/invitations B, C 3 E Accept reservations (^) D 3 F Notify press D I
278 QUANTITA^
N& M M TIVE METHODS^ FOR^ PUBLIC^ ULCISIOT~^ MAKING
M o trb M
PROJECT MANAGEMENT : A NLTWORK MODEL (^279)
I. The late finish time of an activity with no successors equals the project's finish time.
- The late start time of any activity is its late finish time minus the time to complete it.
- The late finish time of any other activity is the smallest of its successor's late start times.
The last activities in our network are E and F. The early finish times for those activities are 9 and 7 ; thus the earliest that the entire project can finish is 9, which becomes the finish time for the project. The LF of the last activity on any path is equal to the finish time of the project. Thus LF E = 9 and^ LF F^ =^9 The LS for any activity is its LF minus its completion time.
and so
LFE =9 LS E = LF E -^ t^ E
Similarly, LF
F = 9 LS F = LF F -^ tF =9-l
Activity D is the predecessor of more than one activity. The LF of such an activity is the smallest of the LS values of its successors. LS E = 6 and^ LS F^ =^8 LF„= Figure 12-3 gives LS and LF as well as ES and EF of all the activities in the network. Slack is the amount of time an activity (-an be delayec~_ttllhc2Y dcluyii
ent'r. For example, activity C can begin^ as^ early as week 0 ; it can also
begin as late as week I without delaying the project. Hence, activity C can be delayed I week without delaying the project ; that is, it has I week of slack. In general, the slack for an activity is found by getting the difference between its LS and its ES. (Equivalently, this is also the difference between its LF and EF .) In our illustration^ F^ has two weeks of slack and C has one week of slack. _Shared slack^ is shared with the other activities in sequence with it
. In Fig-
ure 12-5, activity B^ and activity^ D^ each have 3 days slack ; however, a delay in
activity B reduces^ D's^ slack time.^ FIK~sslack^ is the amount of time an activity can be delayed without delaying or reducing the slack of any other activity. Ex- ercise 12-5 focuses on the distinction between the two kinds of slack. Activities with 0 slack are termed criticcd activities. A, B, D,^ and E all have 0 slack, and are thus critical activities of Figure 12-3. A delay in any critical
Start
0 O
O 3 6 cAmolk
C,
Finish
Figure 12-3 PERT diagram with (^) I^ r-F
activity will delay the entire project. (^) A sequence of critical activities forms a critical nulls, The above method generates ES and LS for each activity, and so will always find the critical path. It may happen that there is more than one critical path. In Figure 12-3, if the estimated time of (^) F were 3 weeks rather than I week, then (^) ABDF (^) as well as ABDE (^) would be a critical path, and a delay in either F E^ or would delay the complete project. As one can imagine, complex projects gen- erate rather complex networks. There are many computer programs available for computing the slack and finding the critical path, provided one knows the sequence and estimated times of activities. Having found the project completion time, one can then determine when the project must start in order for it to be completed on time. Suppose in our il- lustration the seminars are to be started on Monday, May 12 ; then the planning, which requires 9 weeks, must start on Monday, March 10. If it is already March 17, then it is quite reasonable to ask which activity times can be shortened or crashed, in order to speed up the project. We now consider this aspect of pro- ject management.
CRASHING ACTIVITY TIMES
The costs of a project are of three basic kinds :
I. Direct costs of activities (^2). Indirect costs of supervision, administration, and so forth (^3). Opportunity costs including late penalties and forfeited early bonuses
Generally, as the overall length of a project increases, the direct costs decrease
i-- M^ M^ _^ M^ M^ i^ i^ i^
M- -1 _ lmiiiiiiiliiiiii` '~ 280 QUANTITATIVE^ METHODS^ FOR^ PUBLIC^ DECISION^ MAKING^ PROJECT^ MANAGEMENT :^ A^ NETWORK^ MODEL^281
Cost Total cost
Indirect cost
Opportunity cost
Direct cost
Time for minimum Project time project cost
Figure 12-4 Cost-time relationship for a typical project.
but the indirect and opportunity costs increase, as depicted in Figure 12-4. Con- sequently, crashing activities may decrease total^ project costs while increasing direct costs. Costs here include monetary outlays and nonmonetary effects . Estimated activity times are ordinarily based on normal working condi- tions. The length of time required to complete an activity can sometimes^ be shortened, at a price. In some instances that price may be a dollar cost ; for in- stance, additional personnel may have to be hired temporarily or regular per- sonnel may be asked to work overtime. In other instances, crashing the activity may lessen the quality of the activities ; for instance, the surface of a highway
Figure 12-5 PERT network with critical path AC.
Table 12-2 A simple network with crash times
may not be quite smooth if the smoothing operation has been rushed, or an agency report may be less informative if charts have to be omitted in the rush, or, in our illustration, the proper equipment may not be available if the (^) sites have to be hurriedly selected. If it is necessary to speed up the project, then it is worthwhile determining (1) which activities are worth crashing and (2) which of those activities can be crashed at least cost, either monetary or otherwise. We will first illustrate the time-cost trade-offs with a simple dollar example. Then we will return to the seminar planning illustration in which the costs may be nonmonetary. Consider the simple PERT network of Figure 12-5. The regular time and cost, and the crash time and cost, are as in Table 12-. Consider critical activities A and C ; we determine the marginal cost per day crashed so that crashing can be done as cheaply as possible. Here we assume that the cost of crashing A from 4 to 3 days is the same as the cost of crashing it from 3 to 2 days ; that is, we assume the time-cost trade-off to be (^) linear. If this were not the case, the logic would be the same but the computation would be a bit more involved. Table 12-3 focuses on the incremental cost per day for crashing critical activities. If it is necessary to crash the project by I day, then C should be crashed. Its crash time is only $100 for the day, whereas crashing A one day would cost $200. If the project has to be crashed 2 days, then C should be crashed I day, and A should he crashed I day, resulting in a crash cost of $300. Finally if the project has to be crashed 3 days, then both A and C would be crashed com- pletely. The crash cost would be $500 above the regular project cost. If the critical path were crashed as much as possible, its completion (^) time would be 4 days. Since the regular completion time of path BD is 4 days, it
Table 12-3 Crashing critical activities
Days Additional Marginal Activity shortened (^) cost cost per day
A $400 $2(N) C I I0(1 (^) I (x)
Time (days) (^) Cost (dollars)
Activity Regular (^) Crash (^) Regular Crash
A (^4 2) $ 40 0 $ 8(k) H (^2) I 300 450 C (^3 2) 15o0 (^) 6(l D (^2 11 300 ) $1500 $
i i i i i== i= = f=`== S S= S 284 QUANTITATIVE^ METHODS^ FOR^ PUBLIC^ DECISION^ MAKING
Determining time-cost trade-offs in PERT/CPM is analogous to sensitivity analysis in linear programming and, more accurately, to postoptimality analysis in goal programming. The time-cost trade-off of network management is a con- tribution specifically made by CPM. The contribution made particularly by PERT relaxes the requirement that there be a single time estimate to be treated as a constant and allows the network to be based on variable activity times. We will consider this aspect of project management in our next section.
VARIABLE ACTIVITY TIMES
The time required by most activities, especially those depending on human per- formances, is rarely a constant. Rather than make a single estimate of the activ- ity time, three estimates are made. The three different estimates and their symbols are :
From these three estimates of activity time the mean, or expected activity time, is found by getting a weighted average of the three estimates. Intuitively it seems reasonable that the most likely time in should be given a greater weight than either the optimistic time a or the pessimistic time b. The equation for the expected activity time supports this notion ; it is 2 :
t=
a+4m+h 6
then
Example If
a= m = 2k h=
PROJECT MANAGEMENT : A NETWORK
1= 6 =3 t
With a and h defined as they were, we can estimate the standard deviation of activity time. Assuming that the range b - a covers six standard deviations, 3 the equation for the standard deviation is :
h (^) a 6
v 6 7-I 6
MODEL (^285)
a The optimistic time.^ This is the shortest amount of time the activity^ Example (^) Suppose that the simple network of Figure 12-5 has multiple time would need under the most favorable circumstances short of a miracle.^ estimates as given in Table 12-6, columns headed^ "a," "in," and "h ." The probability is very small that the activity could actually be com- The mean or expected completion time of activity C is found by pleted in this amount of time. (^) _ u + 4m + h The most likely time. This is the amount of time that the activity is most^ to^6 likely to require. b The pessimistic time. This is the amount of time the activity needs under^ _^ I^ + (4)2k + 7 the most unfavorable circumstances short of an act of God or some 6 man-made catastrophe. The probability is small that this amount of time^18 would actually be needed by the activity.^ =^6
=
The standard deviation of the completion time of activity C is given by h-a
The mean and standard deviations are found in like manner for the other aCUV1UCti.
I The equations for the expected activity time and standard deviation of activity time actually come from :t hero probability distribution. The beta distribution is continuous over a linite range of values from n to h with mode tit. We simply point out that the equations used have not been pulled from a hat ; they come from the theoretical beta distribution, which repeated applications have shown to he an appropriate description of the activity time. It is beyond the scope of this hook to examine the details of the beta distribution, how well it describes actual activity times, or the derv^ ' In the nur,nal distribution 99 .73 percent of rite di ii ihution Jails within the (^) Nix-N tand :od- vation of the equations for expected activity time t and standard deviation of activity time Q.^ deviation range ; regardless of its form, at least 89 percent (^) of any distribution falls within this range.
M, =) U) m) =I (^) W) W =1 W1 W) =I^ =1^ mil)^ =I^ =1^ W1^ =I^ =I^ iI 286 QUANTITATIVE^ METHODS^ FOR^ PUBLIC^ DECISION^ MAKING
Table 12-6 Multiple time estimates
A 2 4 B l 2 C (^) I 21 D I^ II
Expected Time and Standard Deviation of a Path A path is a sequence of separate tasks or activities. The time to complete a path is the sum of the completion times of the activities on the path. To find the ex- pected completion time and the standard deviation of the completion time for a path, we use two results from probability theory :
l. The expected value of a sum of variables is the sum of the expected values
- The variance of a sum of variables is the sum of the variances, provided the variables are statistically independent.
Example The expected completion time of path^ AC^ is 1AC= 1A+ tC. =4+3= To find the standard deviation of the completion time of AC, we first find its variance. The variance of AC, assuming its activity times are inde- pendent, is given by : 0, .42c=^ 0-2^ +QC = .45 + 1 .0 = 1. The standard deviation of the path completion time is (T, (, = / 1 .45 = 1.
One finds expected completion time and standard deviation of completion time of any path in the same way. With these two pieces of information proba- bility statements become meaningful. Of particular interest are probability statements about the critical path and about the whole project.
Completion Time and Probabilities To be able to make a probability statement about a path's completion time (for example, the probability of completing the path in 4 weeks is .90), its probabil- ity distribution must be known. Since most projects have a large number of
PROJECT MANAGEMENT : A NETWORK MODEL 287
Figure 12-6 (^) Normal probability of path completion time, with (^) r,,, - 7 and u,,,. .. (^) 1 .2,
activities, we can focus on those without much loss. If a path has many activi- ties, then its completion time will be approximately normally distributed. 4
Example (^) With only two activities on the critical path the assumption of normality is a rather coarse approximation. Assuming for the moment, however, that the critical path completion time is approximately normally distributed with a mean of 7 days and a standard deviation of 1 .2 days, the critical path completion time follows the probability distribution illustrated in Figure 12-6. Assuming a normal distribution, we can find the probability that the project will be completed within any given number of days .' The probability that path AC will be completed within 8 .2 days is given by P(t AC S^ 8 .2) = P(Z < (^) 1) =.
The probability that the project will be completed within 9 days is found as follows :
P(1,,1 .s9)=P (^) (Z (^) ;
- 2 7 ) = P(ZS^ 1 .67) =^.
Finding these probabilities assumes that the critical path completion tine is normally distributed, which is more closely adhered to when the path has many activities. Determining the mean and the standard deviation from the three esti- mates assumes that the activity time is approximately beta-distributed. If either of these assumptions is not satisfied, then the network may best be analyzed by simulation rather than by a direct analysis.
' (^) According to the( enlru/ lirnil Iheorem of probability theory, the Sum of ii indcpendenI v able,, tends to he normally distributed as ii tends toward infinity ;ui '. Finding normal probabilities is described in Chapter 3. The standard normal table is Table (^) 2 of the Appendix to this text.
r - - III - - (^) _ it M M M M 290 QUANrITATIVE^ METHODS^ FOR^ PUBLIC^ DECISION^ MAKING
PERT/Cost
PERT/Cost is a cost accounting extension to project management ; it groups costs by activities rather than by organizational lines, fitting quite well with project management. PERT/Cost provides information on activity cost-to- date, which can be compared to activity completion-to-date. The mechanism allows for better control of each activity and the project as a whole. Like any model or mechanism, PERT in any form or extension has its shortcomings.
Limitations Project management is not without its problems, which include difficulties in estimating completion times, estimating regular and crash costs, the validity of statistical independence of activities, the validity of applying the beta distribu- tion to activity times and the normal distribution to project times. There are also less technical difficulties. The effort required both to develop and to up- date the network is sometimes seen as extraneous both to the project and to its operational control. Our discussion of project management will conclude with an application of a project network to a local referendum.
APPROVING A BOND ISSUE : AN APPLICATION'
Ordinary revenue is the usual source for expenses associated with day-to-day and year-to-year operations of a municipality or school district. Extraordinary expenses, arising from the construction of a new sanitary system, water purifi- cation plant, or a new school building, are often satisfied by grants from a higher authority (federal or state) or from funds invested by the public to be re- turned with interest at a later date. The latter source, through a bond issue, often requires voter approval. The process of gaining voter approval through a referendum must convince the voters of the real need for the bond issue, iden- tify the time and places of polling, and record and publicize the results of the referendum. Such a project, then, is composed of many activities. Coordinating and planning such a project may benefit from a network representation. Here we present a simplified version of the activities that would comprise gaining voter approval for a school district's bond issue. The activities listed below are identified by number in the project network in Figure 12-.
- Establish need within long-range plans.
- List resources and timed needs.
This application is an adaptation of H. W. Handy and K. M. Hussain, Network Anahvsis f r Educational Management, (^) Prentice-Hall, Inc ., Englewood Cliffs, N .J ., 1969, pp. 79-.
291
292 QUANTITATIVE^ METHODS^ FOR^ PUBLIC^ DECISION^ MAKING
Form citizens' planning committee.
Identify campaign objectives and amount of bond issue.
Seek support of board members.
Organize information committees.
Plan promotion strategy and costs.
Develop master chart of dates and events.
Activate a speakers bureau.
Arrange for speakers and audiences.
Prepare campaign literature.
Distribute materials.
Solicit volunteer workers.
Solicit aid of a volunteer worker leader.
Organize materials to be given to volunteers.
Petition school board for elections.
Set election date.
Check for conflict with other elections.
Verify sufficiency of petition.
Determine polling places and district boundaries.
Declare election (with 6-week waiting period).
Obtain group endorsements of bond issue.
Run advertising campaign in media.
Conduct PTA voter registration drive.
Select election personnel.
Print election materials.
Deliver election materials.
Assign districts to volunteers.
Hold house-to-house campaign.
Arrange for baby-sitting and transportation.
Have speakers make presentation to their groups.
Hold election.
Apply for no litigation certificate (1-week delay required).
Canvas election results.
Publicize results.
Make official record of proceedings.
Dummy activity.
The network requires a minimum of 23 .2 weeks for completion. The estimated time for each activity, in weeks, appears above the activity circle in the diagram. The critical path, indicated by a heavy line, requires 23. weeks for completion. Note the parallel critical paths from activity 20 to activ- ity 32, the election itself. A^ dunlmv ar^ ii'iiv_^ 99, has been inserted after activi- ties 18 and 19 to simplify the network. Three activities, 20, 25, and 26, depend on the completion of activities 18 and 19. In such a case, the use of a dummy activity avoids the jumble of many intersecting sequence arrows. One of the advantages of developing such a project network is that it pro-
PROJECT MANAGEMENT : A NETWORK MODEL (^293)
vides greater opportunity for identifying all the activities, and indicates when each one should start. This helps to avoid omitting a necessary or at least helpful activity, or starting it so late that the result is less than desirable. This is particularly the case in a "soft" project such as the bond issue, where the elec- tion will take place even if some of the prerequisite activities have not been completed. For example, the election will be held on the declared day even if sufficient time has not been given to the voter registration drive, and even if the advertising campaign never got into full swing, and even if house-to-house cam- paigning reached only a few voters. In a "hard" project, such as the construc- tion of a building, prerequisite activities are less easily omitted but may be started late with the ultimate effect of delaying the project.
SUMMARY
Management of projects which are rather unique combinations of activities is appropriately in the realm of a network model. A hybrid of PERT and CPM re- quires identifying the project activities, the activity times, and the sequencing information. From that, the earliest possible time an activity can start (ES) and the latest it can start without delaying the project (LS) are calculated. The dif- ference LS - (^) ES yields the slack time for each activity slack are critical activities. Activities with zero path. A sequence of critical activities forms a critical ; the completion time of the critical path is the project's completion time. Critical activities may be able to be crashed at a price. PERT/CPM helps one find the right activities to crash to keep the cost of crashing to a minimum. Es- pecially when those costs are nonmonetary, network analysis helps to assess the time-cost trade-offs in much the same way that goal programming helps to assess goal priorities. Where activity times are random variables, one makes three different time estimates :
The optimistic, u The most likely, in The pessimistic, b
From these, the expected completion time
a+4m+b t= 6 and the standard deviation of completion time
h-a
can be estimated.
296 QUANTITATIVE METHODS FOR PUBLIC DECISION MAKING
12-10 The State Department of Public Welfare has had separate contracts with data processing firms to service its needs. It has just been decided that the payroll, client status/eligibility, and cash assistance/medical assistance functions will henceforth be handled internally on the department's own new computer system. Besides the authorization to enter into a leasing agreement for the equipment, the department has also been given hiring authority to secure programmers who will develop and maintain the system. Programmers who are hired will be responsible for developing the payroll program (P) and the client status/eligibility (CS/E) program, and for testing and main- taining the cash assistance/medical assistance (CA/MA) program, that will actually be written by the management services division of the Bureau of the Budget (BOB). Certain aspects of the cash assistance/medical assistance program depend on the client status/eligibility program. The director of systems support has attempted to identify the various activities that must be executed to get the complete set of programs running on the system that will be shortly installed. The activities and their expected completion times are described in the following table.
(a) How long will it take for the three programs to be up and running and under the control of department programmers? (h) Which activities are critical in meeting the schedule of part a'? (c) How long will it be before the department can do an internal audit on client eligibility on the new computer'? 12-11 This problem, though greatly simplified, is based on the Kittleson and McCarthy (1973) ar- title, which concludes with, "it is clear that producing a play is a complex systems management problem. What was surprising to the director was the sheer usefulness of PERT in perceiving the interrelationships of the large number of activities". The Community Theatre starts its Shakespearian Festival on Monday, June 30. It has identi- tied the following tasks necessary for the production.
(a) On what date must the selection of plays begin :' (h) It is now 2 weeks after that date. Propose alternatives for crashing, presenting the cost of crashing each activity.
Symbol Activity
Time (weeks) Predecessors
A Compare computer systems ; notify manufacturer of choices. 8 B Await delivery of computer.^16 A C Hire programmers.^4 A D Flow chart P program.^6 C E Flow chart CS/E program.^10 C F Code P program.^3 1) G Code CS/E.^5 E H Consult with BOB staff on CA/MA program. 4 A Flow chart CA/MA program.^8 E, 11 J Code CA/MA program.^8 G, I K Test P and CS/E programs.^2 F, G L Debug P and CS/E programs.^2 K M Install and test new computer system.^2 B
Activity Description (^) Predecessors (^) Time (weeks)
A (^) Selection of plays (^5) H Casting (^) A (^6) C (^) Set design (^) A (^4) D (^) Costume design (^) A (^5) li (^) Set construction (^) C 6 Rehearsals (^) B 5 G (^) Dress rehearsals (^) D,E.F 4 H (^) Printing tickets and programs (^) B 8 Festival presentations (^) G, H 2
PROJECT (^) MANAGEMENT : A NETWORK (^) MODEL 297
N
0
Test P and CS/E programs computer. Prepare (^) P and CS/E
on new
program
I (^) L, M
manuals. (^4 1) K P
Q R S T
Test CA/MA program on puter. Prepare CA/MA program Implement P and CS/E programs. Implement CA/MA program. Thoroughly acquaint (^) hired grammers with CA/MA
new com-
manual.
pro- program.
3
2
3
./. M P N P
N, P
Time (days) Cost
Activity Regular^ Crash^ Regular^ Crash A 2 1 $2W $ 3W B 3 2 300 400 C 3 U^400 D I - 200 - E 4 2 700 10(X) F 2 1 300 500 G 1 k^200
298 QUANTITATIVE METHODS FOR PUBLIC DECISION MAKING
Project problems
12-12 Pick some project you are or will be involved in, such as a recruiting effort to fill some posi- tion(s), the implementation of a new system, the development of some legislation, a research paper, or other undertaking. List its components, or activities, and their predecessors, and esti- mate times. Construct a project network. Identify the critical path and any almost critical paths. 12 .13 The Calvin and Fielding (1975) article presents an application of a modification of PERT in an area that is not always considered the domain of management techniques. (a) In what way is (^) PERT modified in this application? (h) What contributions did PERT make'? (c) Can you think of another potential application from an unlikely area?
I
