Project network models past, present, and future
University of Utah
Over twenty years have passed now since the I.E. DuPont Company assigned a research team to look for new ways to schedule large maintenance projects. Walker of that company and Kelley of Remington Rand produced in 1957 the Critical Path Method (CPM), which was first described in public literature a couple of years later.1 Almost simultaneously, the Special Projects Office of the U.S. Navy assigned a team of researchers to develop an integrated planning and control system for the Polaris Weapons System program. Malcolm, Rose-boom, Clark, and Fazar of that team subsequently published a paper describing PERT, the product of their research effort.2 Born in parallel labor, these two essentially similar network techniques for planning, scheduling, and controlling large projects were evolutionary rather than revolutionary. They had their roots in related and well-established procedures, such as flow diagrams, Gantt charts, Line-of-Balance technique, and milestone method.
The research which gave birth to PERT and CPM was not unique in this period. Similar investigations were going on concurrently elsewhere in the U.S. and abroad. John Fondahl of Stanford University developed in 1958 a “circle-and-connecting-line” project diagram,3 which was the forerunner of Precedence Diagramming or Activity-on-Node (AON) charts. At the same time, Bernard Roy in France began working on his “Method of Potentials,”4 a similar AON graphic representation of projects. Even earlier (1955), in England, Andrew reported using the “controlling sequence duration” method for plant maintenance scheduling5; and the Central Electricity Generating Board devised a “minimum irreducible sequence” method of maintenance scheduling in 1957.6 It is not unlikely that others developed similar network techniques that were unreported or overshadowed by the great publicity accompanying the introduction of PERT and CPM.
Whatever was the first or best became moot when the U.S. Government, in the arly 60’s, began requiring network reporting procedures on certain contracts and specified PERT or CPM be used in cost and schedule estimating practices. Particularly was this true in the Office of Defense. In practically every research and development program, prime contractors and major subcontractors were required to make use of network-based progress reporting, usually PERT. Such requirements from an organization controlling as many billions of dollars as did the DOD had predictable results. There was an enormous flood of briefings, training programs, seminars, short courses, crash programs, and the like for teaching PERT to contractors, managers, and technicians who suddenly found they had to learn and practice a new technique in order to stay in business. Books, articles, and speeches were written by the hundreds, outlining the new tools and describing the experiences of practitioners. Academic respectability was afforded by learned papers appearing in professional journals. Dozens of computer programs were developed with evergrowing capabilities, until projects with tens of thousands of activities could be accommodated and endless printouts could be ordered through various permutations of data sorts. Since the network models were basically simple and easily learned, instant experts abounded selling their services; and consulting firms specializing in network planning techniques blossomed everywhere.
It was a hectic period, and it was hard to tell whether the enormous interest in PERT was generated on the demand or supply side — whether the clamor for information by practitioners who were forced by the DOD to use PERT gave rise to the market explosion, or whether the keen scent of hucksters for an exploitable product led to a massive supersales job. No doubt there were elements of both; they fed and gave support to each other. There was an initial bandwagon effect when everyone was clamoring to get into the action. Not only DOD contractors but many other firms started using CPM and PERT.
The OR-oriented academic community found a great new area for research and publications. Many students won master’s and doctoral degrees with theses based on project scheduling models. Modifications and extensions of the techniques multiplied. The time-only orientation of PERT, early recognized as a shortcoming, was corrected with the introduction of PERT/COST, developed under DOD and NASA sponsorship in 1962.7 More explicit recognition of resource requirements and the effect of limited resources on project scheduling led to extensive research into both analytical and heuristic approaches to resource allocation in projects.8 Interest was kindled in AON representations of project graphs, and ideas for expanding the kinds of precedence relations allowed in a network led to precedence networking and PDM (Precedence Diagramming Method).9 More efficient algorithms for time-cost trade-off problems were developed, and the methodology was extended to include other than continuous linear relationships.10 Greater emphasis on the planning aspects of project management prompted the development of Decision CPM, a method in which alternative project strategies are considered explicitly within the network model.11 An extensive modification of CPM methodology to incorporate uncertainties in the outcomes of activities as well as in activity durations led to the GERT model.12 We will come back to some of these developments later, but suffice it to say that interest was high and publications in professional journals abounded.
As interest in PERT and CPM skyrocketed in early 60’s, a wave of criticisms also welled up. Some of them were aimed at the theoretical basis of the model. Typical is the oft-cited article by MacCrimmon and Ryavec13 who examined the statistical assumptions of the PERT model, questioning the appropriateness of the Beta distribution for activity times, exploring the errors introduced by the simple formulas for expected activity times and their variances, and noting the overly optimistic results obtained in basing project duration calculations on the path with the largest sum of expected activity times.
Some observers questioned the appropriateness of a fixed network model for certain types of projects. Eyring,14 for example, argued that PERT may be dysfunctional as a planning model for R and D managers in that the mere attempt to list those activities necessary to complete a project may tend to reduce the flexibility a manager needs to shift directions and procedures as new problems and opportunities arise during the course of the project. The skillful manager, he claimed, is not one who can set down a plan and follow it faithfully but one who can rapidly respond to new information as it develops by shifting resources to more promising avenues of endeavor, even if it means scrapping an original plan. Many projects to which PERT was applied in its early years were of this type.
Other criticisms of PERT were more pragmatic. Users complained about the necessity to obtain not just one but three estimates of activity durations, a horrendous data collection job for projects with tens of thousands of activities. And there was open skeptism about the validity of many estimates that were produced (especially those that exhibited a symmetry, like “2-4-6” or (“10-15-20”). For many managers with no background in statistics, the probabilistic features of PERT had little meaning or value — even if the calculations were based on reliable data. CPM users found that time-cost tradeoff curves were hard to come by and frequently not of the simple forms assumed in the model. The network model, especially when applied to large projects, was prone to massive data-handling problems — gathering the original data and periodically updating it, drawing huge networks (“wall-to-wall PERT”), and trying to assimilate and intelligently use reams of computer output based on programs developed by imaginative and enthusiastic programmers. Many managers floundered in the PERT flood.
There were other problems that can best be described as psychological. The government’s insistence that PERT be used by DOD contractors led to a lot of unwilling and unhappy converts. To many managers it was an affront, an officious criticism of their current management techniques, a questioning of their ability to manage. Given that mental set, it is not surprising that PERT was regarded as useless or worse by many managers, and that, as has been acknowledged informally, a number of companies maintained a dual managerial system — PERT for generating reports the government required, and their own system which they actually used for managing their projects. The old maxim about leading a horse to water was once again verified. There were a lot of non-drinking horses at the PERT well.
As might have been expected, the peak in interest and applications built by intensive promotions and government pressures could not be maintained. Inevitably there followed a period of decline, a slump of interest, a falling-out of unwilling converts. The problems and criticisms mentioned above were part of it. Some users became disillusioned because they were promised too much, or expected too much, and the models didn’t solve all their project management problems. The government eventually relaxed its requirements for PERT usage by contractors, and the reluctant ones abandoned it gladly. Some voluntary users who adopted PERT/CPM when they became fashionable dropped them just as abruptly when fashions changed. Academics who were quick to ride the ascending star of network models were just as quick to heap scorn during their eclipse.15 In some circles, to admit the use of PERT became an embarrassment.
That was a remarkable turn of events, as PERT had become well entrenched in the language of project management, not only in its original form but as a verb (“the project was perted”), an adjective (pertable and pertorial), an adverb (pertwise), and in various nonderivatives (pertor, pertee, pertocracy, etc.).16 Now, it appeared, PERT had become just another four-letter word, not used in sophisticated circles. By the close of the 60’s, it seemed to have disappeared from the language of the R and D community which had launched it with such hoopla just a few years earlier. One looks in vain for references to PERT in literature describing the greatest technological and managerial project of our time — perhaps of all time; but PERT was not a part of the vocabulary of the Apolo managers.
In perspective, the reaction to the excessive promotion of PERT and CPM was probably inevitable. The “bandwagon effect” applies to both getting on and getting off. But while an adjustment in management’s attitude was necessary and desirable to bring expectations in line with realities, there has probably been an overreaction. Like a car with bad shock absorbers, an excessive “high” has been followed by an excessive “low.” As balance and perspective are gradually restored, managers will be able to view more objectively what network models can and cannot contribute to project management. Separated from wall-to-wall PERT charts, massive computer printouts, clever gadgets, and perhaps even from the stereotyped PERT and CPM acronyms, the basic, simple models can significantly assist managers in some though not all aspects of project management. As Vazsonyi describes in the “post-PERT” phase,
Project Network Publications Profile
To give some quantitative form to the reported rise and fall of PERT and to the casual impressions gained from years of PERT-watching in the literature, I turned to the International Abstracts in Operations Research, which for 15 years has documented and categorized articles appearing in OR-type journals world-wide. As a research survey technique, this approach has its weaknesses, to be sure. The set of journals covered by the IAOR has changed over the years; not all relevant articles are included; and titles and short abstracts are sometimes vague as to whether project network techniques are a part of the study reported. Nevertheless, the IAOR seemed more appropriate than any other source for finding serious articles on the subject. And the number of articles on a given subject appearing in OR-type journals is at least one measure of interest in the subject held by serious operations researchers.
Summarizing the number of articles which dealt with project network models either primarily or secondarily, I obtained the frequency distributions shown in Figure 1. The pattern is clear and pretty much confirms casual impressions. There was a great initial burst of interest and articles published, which reached a peak in the middle 60’s. Since 1966 there has been a steady decline in articles cited.
If the totals are separated according to country of origin, another interesting pattern emerges (Figure 2). Published articles peaked several years earlier in the U.S. than in the rest of the world, taken as a whole. The high point was 1963, followed by two years of rapid decline and then a more or less constant stream of five to ten articles a year. Interest abroad was slower in coming and reached its maximum in 1966, when foreign articles cited in IAOR outnumbered U.S. articles by almost four to one. Of particular note is the great surge of publications on PERT and derivative techniques that appeared in communist countries (mostly Russia) beginning in 1964. For the rest of the decade, more articles on the subject are cited by IAOR from these countries than from all other foreign sources combined — 54 articles in all. Germany comes next with 23, the U.K. with 20 (most in the 70’s), Japan with 16, and France with 14, including some pioneering studies by Roy in the early 60’s.
As a further check on trends in the 70’s (which might be distorted by the IAOR figures, given the two- to three-year lag in citation of some articles), I conducted a computer search using S.D.C. ’s International Search Service. The survey was performed on three different bases: Compendex, which covers a broad range of professional journals and magazines of interest to engineers — over FIGURE 3 3,500 titles in all; Inform, a business-oriented base which includes some 300 titles in all; and NTIS, which catalogues a great variety of technical reports available from the National Technical Information Service. The results of a search on such key words as PERT, CPM, Critical Path, and Project Networks are shown in Figure 3. Unfortunately, the bases are limited for the most part to articles appearing in the 1970’s. Nevertheless, the results are interesting. With these bases, I detect no obvious decline — at least in the last two or three years — in the number of publications on network models, especially when considering that the 1976 figures cover less than half a year. Apparently the topic is still capable of generating interest among researchers and practitioners.
Of course, a quantitative count gives only a single dimension and says nothing about the content of such articles nor the direction of current research. Examination of the titles and abstracts of articles listed in IAOR leads to the following, mostly qualitative, observations. Not surprisingly, a large number of early articles were descriptive in nature, introducing the models and documenting applications. Following the expository articles in the early 60’s were a number of studies aimed at the theoretical basis of PERT/CPM — criticizing its assumptions,18 extending the range of applications,19 modifying and generalizing the models to take into account costs, resources, uncertain activity outcomes, and so forth.20 By the middle of the decade, the topic which attracted the most attention, if we can judge by the number of published articles, was the resource allocation problem: how best to allocate limited resources to activities in a project network. Over one third of all articles listed in IAOR from 1966 to 1972 dealt in some way with this topic. Both analytic and heuristic approaches were reported, and articles on the subject continue to appear. Russian researchers, especially, made this a topic for intensive study, publishing 17 IAOR-referenced articles on various aspects of the problem within a six-year period.
Despite many reports to the effect that PERT and CPM are dead, the literature cited earlier seems to indicate that there is still some life in the corpses. Articles continue to appear at a rather steady pace, though at a lower level than in the peak years. Applications continue to be documented. And while there has been no definitive study of current usage reported, several recent surveys that have been performed point to continued — even increasing — use of network models. We cite three of them.
The first, reported by Edward Davis in 1974, was conducted among the top 400 construction firms in the U.S.21 He found over 80 percent of the 235 respondent firms using critical path methods, up from a 47 percent usage rate among the same firms prior to 1965. The present level of usage, he noted, and the importance with which CPM is regarded by these users, do not fulfill predictions made a decade ago that CPM would be “necessary for survival” of construction companies in the future.22 But 80 percent usage hardly supports the view that project network models are dead.
The primary use reported among user companies was for project planning rather than control, and not all of the users were highly enthusiastic about CPM. About 16 percent reported they had been unsuccessful in achieving the various advantages attributed to CPM, compared to 61 percent who reported “moderate success” and 15 percent who said they had been “very successful.” In his article, Davis explores the reasons for these differences (chief among them being the degree of support from top management and from people who must use the system), and closes with a remark by one of the executives in his survey which seems to pinpoint a key to successful application of network methods (or, for that matter, of any management tool):
It is not surprising, perhaps, that critical path techniques should be used so extensively in construction, a project-oriented industry. But in another survey, this one among manufacturing firms, Norman Gaither found that PERT and CPM ranked above all other operations research techniques based on percentage of firms who used them.23 He sampled 500 companies from among all manufacturing firms with 250 or more employees in a group of seven south-central states to determine the extent to which O.R. techniques are used by such firms. Just under one half of the replying firms reported using one or more O.R. techniques. Ranked according to percentage of firms that use them, the first five techniques were as follows:
|O.R. Technique||Percent of Firms Using the Technique||Percent of O.R.-Using Firms Who Use the Technique|
|Exponential Smoothing and Regression|
Thus PERT and CPM were easily the most-cited techniques in use, and the above items were well ahead of any other techniques in the list. Also noted in the study was a positive correlation between usage of O.R. techniques and both size of firms and investment per employee, though these correlations were not broken down according to technique used.
The third study we cite was directed at PERT/CPM users themselves, and it gives us some further insight into reasons for success or failure in use of these techniques. David Murphy, who conducted the extensive survey,24 commented on the dangers of management becoming overly fond of PERT/CPM:
He noted with concern the increasing emphasis within the DOD and some corporations on the creation of
Another key, then, to successful use of project network models, and an equally obvious one, is that managers must recognize the models’ limitations, and not be overly reliant on them for managerial purposes. As Murphy concluded,
Managers, like anyone else, like to find simple solutions to complex problems. PERT and CPM are essentially simple tools, which explains in part the tendency of some project managers to overrely on them, with the inevitable disillusionment that follows.
Before leaving our view of the present, we should comment on the development and current availability of some rather powerful computer-based project management systems. Research in the 60’s aimed at correcting PERT’s neglect of resource considerations has borne fruit in a number of heuristic resource allocation programs. The first large commercially available system was RAMPS, marketed by CEIR beginning in 1962.25 A number of companies developed programs for internal use, but programs for sale soon multiplied. The reader interested in an excellent historical review of the development of resource scheduling methods and models is referred to Edward Davis’s comprehensive study.26 Programs are currently available for handling large multiple projects (with essentially unlimited activities in some cases), more resource types than one can imagine needing (250 or more), multiple resource types per job, fixed or variable heuristic scheduling rules, a great variety of user-specified features, and a wide array of report options.
While no firm figures are available on usage of these project management programs, several companies to whom I wrote inquiring about their systems responded that usage is “increasing.” Also of interest is the experience of those companies which offer precedence diagramming options. MCAUTO, whose MSCS program allows either PDM or the arrow diagramming system, stated that about half of their users prefer PDM, up from 20 percent three years ago; and that a majority of new users, generally not previously exposed to networking, choose PDM. IBM reports a somewhat different experience, indicating that about 80 to 90 percent of their customers in the U.S. prefer arrow diagrams, while PDM is relatively more popular overseas. Preference for PDM among IBM’s customers ranges from about 40 percent in England to 70 or 80 percent in Germany and France.
Prognosticating can be hazardous in a world traumatized by “future shock,” and I claim no special prescience; but I’ll risk a few guesses about the future of network models in project management.
First, and this one seems fairly safe, I expect that users of basic network models like PERT and CPM will become more discriminating, using the models more for projects which satisfy the basic assumptions (predictable set of activities, durations, and precedence relationships) and less for projects which are not as well structured and which are characterized by more uncertainty, such as R and D projects. Expectations will be more realistic. The models will be used as aids for planning and controlling rather than as the primary managerial structure. While disappointed managers will continue to drop from the ranks of users, I predict that overall usage will continue to grow — more slowly than in its heyday and more appropriate for a management technique in the “mature” phase of its growth curve.
The special differences between PERT and CPM have already disappeared for the most part; most applications will make use of single time estimates, without provisions for time-cost trade-offs. This tendency, plus the embarrassment mentioned earlier with regard to the name PERT, will weaken the established preference for arrow diagrams associated with PERT, and more users will turn to the Precedence Diagramming Method, which is a simple but substantial improvement on the original arrow diagram. An algorithm for its use and a list of its advantages over ordinary arrow diagramming has been well described by Keith Crandall in an earlier issue of PMQ.27 While the expanded set of precedence relationships in PDM increases the flexibility of project networking, it also increases its complexity, both in terms of interpretation and computation. The PDM algorithm is not nearly so simple as that for ordinary CPM, especially if job splits are allowed; and the concepts of critical path and slack must be modified. Thus, the interpretation and use of a precedence diagram must be done with more care than is necessary with the simple arrow diagram. But PDM’s special advantages have proved quite useful in certain specialized applications.
In the same vein, I believe that other specialized models — modifications or extensions of the original project network model — will come into greater use. A number of such models have already been proposed, such as PERT/LOB,28 a marriage of project network and line-of-balance concepts, and Decision CPM (DCPM),29 which attempts to connect more closely the planning and scheduling phases of project management. DCPM networks contain not only jobs which must be performed, but also subnets of alternative (and mutually exclusive) jobs with their durations and precedence relationships. The selection from among alternative job sets in effect constitutes the planning phase of project management — deciding on which jobs or job methods to use and which to discard. If there are very many decision modes, then there can be a very large number of alternative networks to consider. Crowston and Thompson, who developed DCPM, have described a heuristic procedure for finding “good” solutions with reasonable computational effort. While the model sounds promising, it has yet to be widely used and tested; but it is a good example of how project networks can be extended imaginatively to permit analysis of a wider class of project management problems — in this case, the planning process itself which, in ordinary PERT/CPM, is assumed to have occurred prior to drawing the network.
Though a wide variety of resource allocating programs have already been developed, as we have seen, all commercially available ones are based on heuristic scheduling rules and thus cannot guarantee an optimal solution in the usual sense. Researchers continue to work on analytic approaches to the problem, using such methods as dynamic programming, branch and bound techniques, integer programming, etc. While some interesting advances have been made in recent years, no methods have been reported which can generate optimum schedules for other than small projects (50 to 100 activities) — a far cry from the 32,000 jobs and more than heuristic programs are capable of handling. Even with the quantum increases in computer size and speed of the last decade, and even better mathematical approaches, the size of projects which can be optimally handled has not increased very much. The difficulty is that the scheduling problem grows exponentially with the number of jobs considered, so that a modestly larger project requires vastly more computational effort.30 Thus, I see little hope of an imminent breakthrough in analytic approaches to resource allocation problems, and expect that heuristic models such as are currently available will be the only practical methods for dealing with large projects for many years to come.
One promising avenue of current research aimed at overcoming some of the shortcomings of existing resource allocation methods involves the development of man-machine interactive systems for scheduling projects. One of the vexing problems noted in studies comparing the effectiveness of various heuristics used in resource allocation programs is that no one heuristic is superior to others for all projects and in all circumstances, and it is difficult to anticipate on the basis of easily discernible project characteristics which heuristics would be best in specific situations. Experienced schedulers or imaginative researchers sometimes do better at project scheduling — at least on small networks — than the computer; but to reduce their problem-solving procedures (which are often intuitive and subconsciously used) into a step-by-step computer program has proved to be an elusive problem. The purpose of the man-machine interactive systems mentioned above is to combine man’s heuristic problem-solving skills with a computer’s speed and power, interfacing the two by means of computer graphics devices (CRT’s, light pen and tablet, “joysticks,” etc.) which facilitate communication and interaction between the two. An early effort of this kind was suggested by Leahy31 and has since been extended by Wiest,32 Crandall,33 Paulson,34 and Dressier.35 As an example, one preliminary man-machine interactive systems is described in which
At present, such systems permit the rapid investigation of alternative courses of action, an operationally useful result for today’s project managers. But they also have implications for much more imaginative uses of man and the computer together than has previously been attained.
Perhaps the most ambitious and interesting extension of project network models is GERT (Graphical Evaluation and Review Technique),36 attributed mainly to A.A.B. Pritsker but having its roots in work by Eisner, Elmaghraby, and others. While PERT allows an element of uncertainty to exist with respect to activity durations, GERT extends this possibility of uncertainty to the network itself. As a result of unforeseen circumstances or outcomes, jobs may change, tests might lead back to “redesign” with looping paths in the network (a circumstance prohibited in ordinary networks), or failures might occur that prevent a job from being completed (PERT/CPM networks require that all jobs be completed). Thus, GERT attempts to model a much wider range of project situations, where activities may have several unpredictable outcomes and the nature of the network which follows depends heavily on the particular outcome that occurs. Examples include bidding situations, test programs, feasibility studies, research programs in which multiple approaches are attempted, missile countdown procedures, and so forth.
Additional flexibility in GERT networks is gained by defining several types of nodes, each with an input and output side. The output side determines the type of branching that occurs from the node, whether deterministic (all emanating activities are undertaken) or probabilistic (only one of several emanating activities is to be performed, with stated probabilities). The input sides of nodes specify the conditions that require nodal release, or the point when branching from a node can begin. In general, the input side specifies how many incoming activity completions are required for the first release as well as for subsequent releases (cycles or feedback loops may exist in a GERT network). Symbols for the input and output sides, along with some possible combination of both sides, are described in reference 36 for the most recent version of GERT.
An impressive body of literature concerning GERT and its applications has already accumulated (a recent bibliography lists over 80 articles and papers, plus two books that deal extensively with GERT) and is growing. Applications include the modeling of construction operations, air cargo terminals, check cashing processes, transportation systems, criminal justice systems, and other equally diverse operations.
I have ended up on GERT not only because I think it is one of the most promising models to be derived from project network concepts, but also because it well illustrates the direction that is being taken in this field. More specialized models are being developed (and have been) that overcome some of the shortcomings of the original models, relax some of their stringent assumptions, and meet specific needs. The largest problem in their use will continue to be their misuse, the mismatch of tool and problem, the gap between expectation and potential. But gradually — and this is the fervent hope of everyone who has tried to teach present and future managers how to solve quantitative problems — managers are becoming more knowledgeable about quantitative decision-making techniques, more sophisticated in applying them, and more successful in their efforts.
1. James E. Kelley, Jr., and Mrogan R. Walker, “Critical Path Planning and Scheduling,” Proceedings of the Eastern Joint Computer Conference, Boston, Massachusetts (1959).
2. Donald G. Malcolm, John H. Roseboom, Charles E. Clark, and Willard Fazar, “Applications of a Technique for Research and Development Program Evaluation,” Operations Research, 7, 5 (September-October 1959).
3. John W. Fondahl, “A Non-Computer Approach to the Critical Path Method for the Construction Industry,” Technical Report No. 9, Department of Civil Engineering, Stanford University (1962).
4. B. Roy, Revue Francaise de Recherche Operationelle, 6, 323 (1962).
5. S.P.S. Andrew, “A Job Planning System for the Rapid Overhaul of Large Units of Plant,” Report G &P/SPSA/JD, January 5, 1956, I.C.I., Billingham Division.
6. Anon., “Maintenance Intensification,” Memo No. GO/18/OR, Central Electricity Generating Board, London (1957).
7. Management Systems Inc. of Cambridge, Massachusetts, headed by J. Sterling Liningston of the Harvard Business School, was consultant on the project which led to the cost system described in DOD and NASA Guide, PERT Cost Systems Design, Washington, D.C., Government Printing Office (June 1962).
8. “Resource Allocation in Project Network Models — A Survey,” Industrial Eng., XVII, 4 (April 1966). See also Reference 26.
9. For an early discussion of the Precedence Method, see John W. Fondahl, “Methods for Extending the Range of Non-Computer Critical Path Applications,” Technical Report No. 47, Department of Civil Engineering, Stanford University (1964). Additional precedence relationships are described in IBM’s Project Control System/360 Program Description and Operations Manual, H20-0376-0 (1967) and in Keith C. Crandall, “Project Planning with Precedence Lead/Lag Factors,” Project Management Quarterly, IV, 3 (September 1973).
10. For example, see W. L. Meyer and L. R. Shaffer, “Extensions of the Critical Path Method Through the Application of Integer Programming,” Department of Civil Engineering, University of Illinois (July 1963); and E. B. Berman, “Resource Allocation in a PERT Network under Continuous Activity Time-Cost Functions,” Management Science, 10, 4 (July 1964).
11. W. Crowston and G. L. Thompson, “Decision CPM: A Method for Simultaneous Planning, Scheduling, and Control of Projects,” Operations Research, 15, 3 (May-June 1967).
12. A. A. B. Pritsker and W. W. Happ, “GERT: Graphical Evaluation and Review Technique, Part I: Fundamentals,” Industrial Eng., XVII, 5 (May 1966).
13. K. R. MacCrimmon and C. A. Ryavec, “An Analytical Study of the PERT Assumptions,” Operations Research, 12, 1 (January-February 1964). See also H. O. Hartley and A. W. Wortham, “A Statistical Theory for PERT Critical Path Analysis,” Management Science, 12, 10 (June 1966).
14. Henry B. Eyring, “Evaluation of Planning Models for Research and Development Projects,” D.B,A. thesis, Harvard University (June 1963).
15. For an example of such change of heart, forthrightly told, see A. Vazsonyi’s “L’Histoire de Grandeur et de la Decadence de la Methode PERT,” Management Science, 16, 8 (April 1970).
16. M. Krakowski, “PERT and Parkinson’s Law,” Interfaces, 5, 1 (November 1974).
17. Vazsonyi, Ref. 15, p. 455.
18. For example, see MacCrimmon and Ryavec, op. cit.; D. J. A. Welsh, “Errors Introduced by a PERT Assumption,” Operations Research, 13, 1 (January-February 1965); J. Lukaszewicz, “On the Estimation of Errors Introduced by Standard Assumptions Concerning the Distribution of Activity Duration in PERT Calculations,” Operations Research, 13, 2 (March-April 1965).
19. Among many examples may be cited L. Glasser and R. Young, “Critical Path Planning and Scheduling: Application to Engineering and Construction,” Chem. Eng. Progress, 57 (November 1961); W. Gorham, “An Application of a Network Flow Model to Personnel Planning,” IRE Trans. Eng. Mgt., 10, 3 (September 1963); Y. Wong, “Critical Path Analysis for New Product Planning,” Marketing, 28, 4 (October 1964); D. G. Malcolm, “Reliability Maturity Index (RMI) — An Extension of PERT into Reliability Management,” J. Indust. Eng., XIV (January-February 1963).
20. Early example include the DOD/NASA PERT/Cost Guide cited earlier; J. D. Wiest, “The Scheduling of Large Projects with Limited Resources,” Ph.D. thesis, Carnegie Institute of Technology (1963); S. Lambourn, “Resource Allocation and Multi-project Scheduling (RAMPS), A New Tool in Planning and Control,” Computer Journal, 5, 4 (January 1963); S. E. Elmaghraby, “An Algebra for the Analysis of Generalized Activity Networks,” Management Science, 10, 3 (April 1964); R. Freeman, “A Generalized Network Approach to Project Activity Sequencing,” IRE Trans. Eng. Mgt., 7, 3 (September 1960); J. E. Kelley, “The Critical Path Method: Resources Planning and Scheduling,” Chapt. 21 in Industrial Scheduling, eds. J. F. Muth and G. L. Thompson, Prentice-Hall, Inc. (1963); M. W. Paige, “How PERT-COST Helps the General Manager,” Harvard Business Review, 41,6 (November-December 1963).
21. Edward W. Davis, “CPM Use in Top 400 Construction Firms,” of the Construction Division, ASCE, 100, COl (March 1974).
22. “CPM and Survival,” editorial, Engineering News Record, 88 (July 19, 1962).
23. Norman Gaither, “The Adoption of Operations Research Techniques by Manufacturing Organizations,” of Decision Sciences, 6, 4 (October 1975).
24. David Murphy, Determinants of Project Success, NASA (1974); also personal correspondence.
25. RAMPS Users Guide, CEIR, Inc., Arlington, Virginia (1962).
26. Edward W. Davis, “Project Scheduling under Resource Constraints — Historical Review and Categorization of Procedures,” AIIE Transactions, 5, 4 (December 1973).
27. Keith C. Crandall, “Project Planning with Precedence Lead/Lag Factors,” Project Management Quarterly, IV, 3 (September 1973).
28. Lester A. Digman, “PERT/LOB: Life-Cycle Technique,” Industrial Eng., XVIII, 2 (February 1967).
29. Crowston and Thompson, op. cit. (Ref. 11).
30. For an interesting analysis of the problem and a summary of analytic methods thus far attempted, see Davis’s article cited above (Ref. 26).
31. John P. Leahy, “A Man-Machine CPM System for Decision Making in the Construction Industry,” Construction Research Series No. 9, Department of Civil Engineering, University of Illinois (1967).
32. Jerome D. Wiest, “Computer Graphics for Project Management,” Proceedings of the Seminar-Symposium, Project Management Institute, St. Louis (1970).
33. Boyd C. Paulson, Jr., “Man-Computer Concepts for Project Management,” Technical Report No. 148, Department of Civil Engineering, Stanford University, Stanford (1971).
34. Joachim Dressier, “Development of an Interactive Computer Program for Resource Allocation,” Technical Report No. 189, Department of Civil Engineering, Stanford University, Stanford (1974).
35. Pritsker and Happ, op. cit. (Ref. 12).
36. A. A. B. Pritsker and C. E. Sigal, The GERT IIIZ User’s Manual, Pritsker and Associates, Inc., West Lafayette, Indiana (August 1974).