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Project Network Summary Measures Constrained-Resource SchedulingEdward W. Davis aa Graduate School of Business Administration, University of North Carolina , ChapelHill, North Carolina , 17514Published online: 09 Jul 2007.

To cite this article: Edward W. Davis (1975) Project Network Summary Measures Constrained- Resource Scheduling, A I IE Transactions, 7:2, 132-142, DOI: 10.1080/05695557508974995

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Project Network Summary Measures Constrained- Resource Scheduling

EDWARD W. DAVIS SENIOR MEMBER, AIIE

Graduate School of Business Administration University of North Carolina

Chapel H a , North Carolina 17514

Abstract: This paper describes some summary measures which might be used in specifying the size, shape, logic, time and resource characteristics of project networks of the PERTICPM type.The results of an investigation to determine relationships between some selected such measures and project duration increase under constrained-resource scheduling are presented, and procedures for estimating project duration increase in advance of scheduling are described.

Network models of the PERT/CPM type have proved useful through more than a decade of practical experience. Yet very little formal study has been conducted on the characteristics or properties of such networks. There is not, for example, any commonly accepted method of describing how the network representing one project differs from that OF another project, except in such terms as "size" (number of activities), general appearance (e.g., "short and fat" or "long and thin") and, after an initial schedule has been obtained, critical path duration.

While general descriptions like the above are useful to management in making such decisions as whether to employ a computer for network scheduling, they are of little help in answering other important questions having to do with differences between networks and changes in the character- istics of a given network. For example, in constrained- resource scheduling situations the imposition of constraints on the amounts of resource available for use by project activities in any period often produces an increase in pro- ject duration beyond that obtained under the usual CPM analysis (which ignores such constraints). The extent of such increase is of primary importance to management and numerous heuristic and optimal solution procedures have

Received January 1974; revised March 1975.

been developed for minimizing the duration increase. Heur- istic procedures produce "good" resource-feasible schedules; optimal procedures produce true minimum-duration sched- ules.

Heuristic procedures are in wide use today, capable of solving complex problems that would be extremely diffi- cult, if not impossible, to solve manually. However, compared to "standard" CPM procedures (i.e., without consideration of resource constraints) these approaches are much more expensive in terms of computer time. In the case of very large or complex problems, for example, it is often economically feasible to obtain only one or at the most a few computer solutions. Furthermore, changes in either the characteristics of the network (through changes in precedence relations, activity durations and/or resource requirements) or the resource constraints will produce changes in the constrained-resource solution to be obtained, requiring re-solution of the problem. Thus, in the case of heuristic procedures, it would be useful to have a method for quickly estimating the effect on project duration of changes in the network and/or resource constraints.

Optimal procedures, while not yet so powerful as heur- istic approaches in terms of the complexity of problems that can be solved, have experienced considerable develop- ment in recent years. Among the more promising techniques in this area are those involving some form of implicit enumeration, such as "branch and bound." Such techniques

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June 1975, AIIE TRANSAC~ONS 133

To illustrate, Fig. 2 shows the network diagram for an actual project involving the design and implementation of software for a computer-based management information

NETWORK A

NETWORK B

Fig. 1. Similar 27-job networks.

have produced optimal solutions to constrained-resource problems of sizes much greater than was possible only a few years ago. However, an apparently common disadvantage of many of these procedures is the extreme variation in computation time experienced from problem to problem, even among seemingly similar networks. Figure 1, for example, shows two similar 27-job networks. When solved with an implicit enumeration scheme for the minimum duration under identical sets of resource requirements and availabilities, Davis [2] found that network A required nine times as much computer time as network B.' Other researchers have also reported similar phenomena [9], [7] and have attempted without success to isolate the factors responsible for the unpredictability of computation time. Thus, in the case of at least some optimization procedures a system for describing or measuring the difference among networks, or in changes to a given network problem, might conceivably be useful in helping to predict the associated computational requirements.

Focus of this Research

While the general focus of this research is on the develop- ment of a broad system of describing and categorizing project networks, this paper is concerned with the special case of constrained-resource scheduling. The objective of this particular study was to determine the feasibility of network summary measures for explaining and predicting the project duration increases encountered in constrained- resource scheduling.

'180 seconds vs. 20 seconds IBM 7094 time, as reported i n [2] .

system. Activity durations (in days) and resource require- ments are shown in Table 1. The resources required in this

* Table 1 : Activity durationsand units of each resource type required.

Activity Type 1 TY pe 2 No. Duration resource resource

1 4 0 0 2 2 1 0 3 3 1 0 4 6 0 1 5 5 1 0 6 4 1 0 7 4 1 0 8 4 0 1 9 7 0 1

10 8 1 0 11 6 1 0 12 7 0 1 13 3 1 0 14 2 0 1 15 4 0 1 16 12 1 0 17 5 1 0 18 9 0 2 19 8 0 1 20 8 0 2 21 4 1 0 22 13 1 0 23 10 0 1 24 3 1 0 25 13 0 1 26 7 0 1 27 8 o 1 28 1 1 0 29 3 1 0 30 9 o I 3 1 6 0 1 32 9 I o 33 2 I 0 34 8 1 0 35 9 0 1 36 6 0 1 37 12 1 0 38 8 0 1 39 4 1 0 40 5 0 1 41 8 0 1 42 11 0 1 43 6 0 3 44 6 1 0 45 7 0 1 46 3 0 3 47 3 1 0 48 1 1 0 49 14 0 1 50 3 1 o 51 7 0 1 52 10 2 0 53 9 2 0 54 6 1 0 55 10 0 2 56 3 2 0 57 6 2 0 58 13 0 2 59 6 0 2 60 10 0 1 61 4 I o 62 3 1 0 63 3 1 0 64 8 I o 65 8 1 0 66 12 0 3 67 7 o 1 68 6 0 1 69 7 0 3 70 13 0 o 71 5 0 3 72 5 0 3 73 4 0 0

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Fig. 2. Project network for management information system.

project are two types of computer programming personnel; PROJECT requirements are expressed in terms of the number of DURATION,

personnel of each type required per day to accomplish DAYS

110- 1 I I I I I I 1 I

activities in the given duration. If the Critical Path (CP) of the network is calculated in

the usual manner without regard to resource availabilities, a 105 - project duration of 75 days is indicated. When the amounts of resource types 1 and 2 are limited, however, the project duration may increase beyond 75 days since some activities I00 - will have to be delayed. For example, if a maximum of only five type 1 and eight type 2 programmers are avail- able each day for assignment, the expected duration will 95 - increase to 78 days. On the other hand, if only three type 1 and seven type 2 are available, the expected duration increases to 106 days. Figure 3 shows how project duration 90 - varies as a function of available resources. This curve was developed from successive solutions of the problem with a computer program capable of scheduling project networks

85 - under resource constraints. The "heuristic" scheduling rule used in the program to resolve resource conflicts was a 80 - minimum-LFT (late finish time) rule, which has been found generally effective in reducing the duration increase asso- ciated with constrained-resource scheduling [4]. This rule is 75 - - similar to ones used today in many commercially-offered I I I I I 1 I I

3.7 3,8 3,9 4.7 4.8 4,9 5,7 5,8 5.9 network scheduling programs. AVAILABLE MEN PER DAY (Type 1, Type 21

The question of interest here is, "Can resource-con- strained durations such as shown in Fig. 3 be estimated prior to actual solution?" Fig. 3. Resource-constrained project durations.

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The approach taken in this study was based on the analysis of given networks in terms of a series of experi- mental quantitative factors or summary measures. The factors used were an attempt to specify the salient char- acteristics of given networks, such as size, shape, logic, time and resources.

While the idea of network summary measures is not new, there has been little previous research on the subject and few positive results. We will briefly review the known research to date, then describe the summary measures used in this study, how they were tested, and the results obtained.

Previ us Research

Pascoe [ lo] , [ l l ] was the first to propose some summary measures for specifying interconnectedness, time, and re- source characteristics of project networks. He used these factors to control the characteristics of networks artificially generated for his study of alternative constrained-resource heuristic sequencing rules, but attempted t o draw no relationships between heuristic performance and network characteristics.

Johnson [7] proposed measures similar to Pascoe's and used them in an attempt to develop relationships with computation time of his branch-and-bound optimization procedure for constrained-resource scheduling. He con- cluded however that the measures tested showed no signifi- cant relationships and that computational difficulty (time) was "an unpredictable function of detailed problem struc- ture."

Davis [2], [3] followed Pascoe's approach in using selected summary measures to control the characteristics of projects generated for testing a "bounded enumeration" constrained-resource scheduling procedure. He also pro- posed some new measures in addition to Pascoe's and used them in an attempt to find relationships with the variability in computation time exhibited by his solution procedure. Like Johnson, he found no discernible strong relationship with computation time but concluded that his test sample of 65 problems was too small for definitive generalizations and pointed out that the evidence in most cases was not sufficient to reject a null hypothesis of no relationship.

In a somewhat different vein, Ruda [13] proposed some simple estimates of the critical path duration and number of critical paths in networks without resource constraints, under special conditions. Prideaux and Cullingford [12] proposed a measure of the "quantity" of decision-making in network scheduling under resource constraints and re- lated this to objective functions of duration and cost for small 10-activity networks. Also, Klovstad [8] presented some useful concepts involving "technological networks" developed in such fashion as to display more prominently for analysis the ordering of network activities.

There is also some pertinent prior research outside the field of project network scheduling. Geographers, for exam- ple, regularly encounter problems involving flows of some

variable or commodity through networks or channels of networks. River basins and highway systems are but two examples, and a great deal of geographical research has been conducted on the analysis and categorization of these particular types of network structures. For example, a measure of the "diameter" of a road network has been defined as the maximum over all node pairs of the shortest route between any pair, and the "shape" of a stream net- work has been defined as the ratio of total mileage in the network to its diameter. Haggett and Chorley [6] sum- marize this research, point out some of the problems in attempting summary measures, and illustrate application of some of the concepts in geographical science.

Network Summary Measures Used

The network summary measures we will describe can be divided into three general classes: (1) measures which characterize the size, shape and logic (precedence structure) of the network: (2) measures which indicate time char- acteristics and (3) measures which characterize resource demands and availabilities.

These summary measures can be further distinguished as to whether they are obtained before or after application of any of the standard network analysis procedures, such as critical path determination. In general, we are interested primarily in measures which are obtained prior to any network scheduling. But because of our focus on the constrained-resource case, the "original" critical path (i.e., without resource considerations) and its associated statistics were considered here as part of the analysis which comes prior to constrained-resource scheduling.

It should be noted that in spite of the particular focus here, some possible measures in classes (1) and (2) above could prove useful as general measures for network cate- gorization. We will describe a number of these possible general measures even though some of them were not used in these tests. We will distinguish such general measures from measures available only to resource-associated pro- blems by terming the former "general statistics" and the latter "resource statistics." Further, the resource-associated measures will be subdivided into two categories according to whether they are designed to measure: (a) resource requirements of the network or (b) relationships between resource requirements and demands.

Some summary measures in each of the three classes above are described below, and the particular ones used in this experiment noted.

Characteristics of Network Size, Shape and Logic

These characteristics of project networks are among the most difficult to measure. Haggett and Chorley, for exam- ple, in their geography studies point out that shape has proved one of the most elusive of geometric characteristics to capture in any exact quantitative fashion. They note that

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many of the common terms in use by geographers for describing networks (such as "circular," "oxbow," etc.) have turned out to be so arbitrary that misclassification is common. But, as they also suggest, the paradox lies in the fact that the more mathematical definitions fail to do justice to our intuitive notions of what constitutes shape. The general measures of network size and shape described below were not used in this study but are proposed for their possible general interest.

One possible measure of network size, as noted earlier, is the total number of nodes (activities or "jobs") contained in the network, including the necessary single beginning and single ending nodes.

Network shape can be specified on the basis of three separate factors: a measure of network length, in terms of the maximum number of consecutive nodes from beginning to end; a measure of network width, in terms of the maxi- mum number of nodes in parallel, based purely on preced- ence relations and ignoring time durations; and a measure of the relationship of length to width obtained by dividing the first measure above by the second. Measures of length and width can be developed from the concept of activity "rank" proposed by Klovstad in his discussion of "technological networks." This concept can also be utilized to develop an equivalent representation of any project network which gives a rough idea of the over-all "shape" of the network. We call this equivalent representation the "Rank Frequency Table" and illustrate its development in Appendix A.

A measure of network logic, used in this study, has been given the name "complexity" by Pascoe [l I ] . It is the ratio of number of arcs to number of nodes, i.e.,

After critical path analysis is performed, additional measures are available. The most obvious of these is the critical path duration. Another is the amount of total slack contained in the network (sum of individual job slack) and a third is the total free slack, which is always equal to or less than the total slack. Strictly speaking, each of these measures is a function of network logic also and might be included in the first class of measures, but we prefer to include them here. Finally, another measure of time, ori- ginally suggested by Pascoe, is termed "network density" and is calculated as follows:

Sum of job durations Density =

Sum of job durations + total free slack

For this measure, 0 < Density 1.0. For purposes of resource scheduling it can be seen that high values of Den- sity indicate less free slack and, consequently, less freedom to make sequencing decisions without causing further resource conflicts. In a network with Density = 1.0, a l l jobs are critical.

Of the latter four the measures described above, only the critical path duration was used in these tests. The measure of network density, while intuitively appealing, was dis- carded on the basis of previous research [2], as well as prelirninay tests which indicated little relationship with the increases in critical path duration taken as the primary objective measure of interest for this study.

Resource Characteristics of the Network

Complexity (C) = number of arcs - A number of nodes -N

Although Pascoe originally proposed this measure for use with activity-on-arrow networks, we have adopted it for use with the activity-on-node networks used in this study. For a given project of some logical configuration with N original jobs, an increase in A , the number of arcs, will increase the value of Complexity, C, which indicates an increase in "interconnectedness" of the network. For a given value of N it might be noted that the range of possible variation in C is:

Time Characteristics of The Network

One such possible measure, used in these tests, is simply the

sum of durations of all activities (jobs) in the network. Other such measures (not used) are the average job dura- tion and variance in job duration. Each of these three measures can be calculated prior to the usual critical path analysis.

The particular case we are considering is one in which each network activity has specified requirements of one or more resource types that are fixed over the duration of the activity. In this situation, some resource characteristics, such as the "profile" of requirements, are, strictly speaking, a function of the logic and time characteristics of the net- work. The profile of resource requirements, for example, is generated from a particular job ordering such as the "early start time" (EST) or "late start time" (LST) job arrange- ments obtained from standard critical path analysis, con- sidering time and precedence relationships but without regard to resource availabilities. Thus some resource mea- sures are developed using the original critical path duration and associated job arrangements as a base, while others are developed directly from the resource data itself.

As noted earlier, measures of resource characteristics fall into two groups, according to whether they measure: (1) requirements or (2) relationships between requirements and availabilities. Measures in the first group are calculated separately for each resource type, in some cases on the basis of the initial critical path analysis as noted above. Such measures include: the average resource requirement per job, which provides a gross measure of the level of requirement across nodes; the average requirement per period, providing a similar gross measure across periods of the original

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critical path; the variance by period which provides a mea- sure of the general variation of the period-by-period level and the maximum required resource level, which provides an indication of the "peak" level of the resource profile.

A gross measure of the total resource requirements of a project is provided by what we term the "total work content" of the project. This is simply the sum, over all job durations, of the per-period unit requirements of each resource type. It is also equal to the period-by-period sum of resource requirements along the original critical path. Thus,

N CP

total work content = W j = C dirij = C rjt (resource type j) i = I t =l

1 where: d = duration, job i

ri j = per-period requirements of resource type j by job i

I t =time period

I N = number of jobs

~ CP = original critical path duration

rjt = total resource requirements of resource type j in time period t .

Another measure which was used attempts to indicate the predominant location of requirements with respect to time periods of project duration. This was termed product moment. It is calculated by first finding the mid-point period of initial critical path duration, then calculating the product sum of: period resource requirements times dis- tance from the mid-point (in number of periods). Periods to the left of the mid-point are increasingly negative, to the right, positive. It is calculated as:

Product Moment = PM = "'{[( t - 1) - ( c z - l ) ] r j t } - t = l

The net value of product moment (for each resource type) indicates roughly whether the predominant influence of that resource type comes in the time periods of the first or second half of the original project duration.

Finally, with regard to the first group of resource mea- sures, it should be noted that the number of different resource types is a simple, but possibly important value.

Resource measures in the second group (which we call "Resource Constraint Statistics") were designed specifically with regard to possible relationships with the increases in project duration typically encountered in constrained- resource scheduling. Information from the initial critical path analysis was heavily used in the development of these statistics.

June 1975, AIIE TRANSACTIONS

A general measure of the magnitude of resource con- straints (for each particular resource type) is the excess resource requirements generated by the per-period differ- ences between a particular resource profile (i.e., EST or LST job arrangement) and resource availabilities. Such excess requirements may be specified in several different ways. One measure, for example, is simply the number of periods in which unit resource requirements of each type exceed unit resource availabilities. A second, associated, measure is the maximum consecutive periods ("maximum consecutive run," MCR) of such excess demand. This latter measure attempts to discriminate between the difference in resource-constrained scheduling flexibility inherent in situa- tions in which a given number of periods of excess demand is intermixed with periods of excess availability and those in which the excess demand periods fall in a continuous sequence.

Other aggregate resource-constraint statistics can be specified through operations on excess resource require- ments. For instance, the ratio of excess resource require- ments to total work content is an important measure. This was termed the "0- actor"^ and was calculated in a two- step process as follows:

Max (0; rjt - Aj) t = 1

(1) o j = wj

j = 1

where: Aj = units available per period of resource type j

M = number of different resource types

wj = total work content, type j (defined previously)

and r j t= is calculated on the basis of an all EST job arrangement.

The counterpart of 0-Factor, calculated identically but based on an all-LST job arrangement, was termed the "LC- Factor."

Two other measures incorporating the calculation of excess resource requirements, but adjusted by the number of periods (of initial CP duration) and number of different resource types, were also tested. Termed "ADOPES" and "ADOPLS" (for Adjusted Obstruction per Period, based on an EST or LST schedule, respectively), these were calcu- lated as follows:

C P

C Max (0; rj - Aj)

j = 1 M X C P

20 for "obstruction," after a somewhat similar measure first pro- posed by Pascoe [lo] .

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where rjt is based on an EST job arrangement, and

ADOPLS = same as above but with rjt

based on an LST arrangement.

In order to have a single measure representing the com- bined effect of both the last two measures above, the average of ADOPES and ADOPLS was used and termed "ADAOP" (Adjusted Average Obstruction per Period), i.e.,

ADAOP = ADOPES + ADOPLS .

2

Finally, a different resource-constraint measure tested was one originally investigated by Davis [2]. Termed "resource utilization factor" and designated f, this is the ratio, for a given resource type, of the total work content to the "total work initially available," where the latter is measured by the product of initial CP duration and units of resource available per period. To obtain a single measure for use in multiple-resource cases, the maximum value of the individual resource utilization factors (termed F 3 was used, i.e.:

wj where fj =.

CP X Aj

Experimental Investigation of Summary Measures

Investigation of the relationships between network sum- mary measures and resource-constrained project duration was made using a collection of 202 network problems. This

collection included both artificially-generated problems of the type described in [2], problems take? from the litera- ture [ I ] , and actual examples froin practice obtained from private sources. The number of activities per network in these problems ranged from 22-73, with from 1 to 6 different resource types per job and network involved. Each problem was solved successively under a range of available resource levels, producing a total of 721 different resource- constrained solutions. An idea of the characteristics of these problems is given by Table 2 which shows the range in some summary measures for problems in the collection.

The approach taken in this experiment involved factor analysis and multiple regression. The summary measures described earlier were taken as independent variables, with the dependent variable being the ratio D RATIO defined as:

where: D'= project duration resulting from scheduling under a given set of resource availabilities using a minimum-LFT heuristic.

CP =initial (resource non-constrained) Critical Path duration.

Although only 202 different network problems were involved, the aprlication of a series of different resource constraints to eazh problem produced 721 different sets of summary measures (independent variables) and an associ- ated dependent variable. While some summary measures were calculated for all 721 different observations, some resource requirement statistics, such as Product Moment, which were defined only for each resource type separately (and not aggregated as 0-Factor), were calculated only for

Table 2: Range in selected summary measures of network problems used in this study.

I -Resource 2-Resource 3-Resource 6-Resource Total 57 networks 15 networksa 1 1 5 networks 15 networksa 202 networks

309 solutions 126 solutions 226 solutions 60 solutions 721 solutions

Number of jobs 22 - 27 73 22 - 45 45 22 - 73 Complexity 1.19 - 3.01 1.30 - 3.81 1.07 - 3.87 1.36 - 3.87 1.19 -3.87 Initial CP duration 18 -37 67-113 18 -67 47 - 67 18 -113 Percent increase, resource-constrained project duration to initial CP 0 -140 0 - 745 0 - 205 0 - 100 0 - 205

Sum of durations 57 - 101 474 54 - 173 173 54 - 474

Average job duration 2.5 - 3.87 6.49 2.63 - 5.86 5.86 2.5 - 5.86

Variance, job duration 3.7 - 7.41 10.31 2.28 - 7.48 10.2 2.28 - 10.31

F * 0.58 - 1.67 0.47 - 1.06 0.39 - 1.77 0.36 - 0.86 0.39 - 1.77

0-factor 0.01 - 0.41 0.04 - 0.53 0.01 - 1.57 0.01 - 0.86 0.01 - 1.57

ADOP 0.15 - 2.39 0.04 - 1.14 0.04 - 6.66 0.04 - 0.96 0.04 - 6.66

he 15 different problems in each of these groups were created from one original 73-job or 45-job problem, respectively, by varying arc connections and resource requirements.

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the 1-resource problems. Thus separate analyses were under- taken of different groups of problems. We will describe here only the results obtained for the 1-resource problems and for the total collection of all 202 problems combined (72 1 observations).

Factor 1 : F*, 0, LC, ADAOP, ADOPES, MERD, MCR

Factor 2 : SD, CP, PM

Factor 3: : C, VAR

Analysis of I-Resource Problems

In performing this analyas, values or each or tne summary measures shown in Table 3 were calculated. The intended approach to determining the importance of these measures as determinants of the increase in project duration was to use them as independent variables in a regression on the dependent variable, D'RATIO. However, it was apparent from the manner in which certain measures were defined that a high collinearity existed among them (e.g., ADAOP is partially derived from ADOPES: the resulting correlation coefficient between these two variables has a value of 0.984).

In the face of this high collinearity factor analysis was used in an attempt to determine the relative importance of the 12 summary measures in accounting for the variance in D'RATIO. Using this technique, the 12 variables were combined into three independent combinations of variables ("Factors") as shown in Table 4. These factors were then employed in regressions against D'RATIo.~

The results of this analysis showed, among other things, ~ that Factor 1 was overwhelmingly important in explaining the variance in D'RATIO, with a calculated R~ value of (0.82). Factors 2 and 3 had R~ values of only (0.31) and (0.06), respectively. These results are not surprising in the sense that Factor 1 consists entirely of resource-associated summary measures, and might apriori have been expected to be most important. However, the relatively slight impor- tance of Factors 2 and 3 is somewhat surprising, as is the fact that the resource requirements measures PM and VAR were not included in Factor 1.

While interesting "explanatory" information, the results using Factor analysis were not considered particularly use- ful for forecasting purposes, since simpler, more effective relationships could be developed for that purpose alone by ignoring the collinearity among summary measures. Thus, numerous multiple regressions of D'RATIO against the summary measures were run, using linear combinations of both the original and transformed measures. These regres- sions showed that R values on the order of .90 were obtainable using non-transformed values of all 12 summary measures. However, similar results were produced using fewer combinations of variables which included some trans- formed variables. In general, as long as the variables C, F* and ADAOP were used, the value of R~ varied closely about .9 15 irrespective of what other variables were included. For example, Table 5 shows some of the better results obtained using only these three measures.

r

Table 5: Regression results, I-resource problems, three variables.

Equation: D'RATIO = -0.407+0.486(F+l+O.O68(C) + O . O ~ ~ ( A D A O P I ~

Regression coefficient std. deviation: 0.03 0.04 0.008 0.003

Incremental contgbution t o R : 0.028 0.019 0.867

Std. error of estimate: 0.061

R~ : 0.914

Table 3: Summaw measures calculated for 1 -resource problems

Variable No. Symbol Meaning

1. SD Sum of job durations

2. F* Max. resource utilization factor

3. C Complexity

4. 0 0-Factor

5. LC LC-Factor

6. CP Critical Path duration

7. ADAOP Adjusted Average Obstruction

8. ADOPES Adjusted Obstruction per Period,(EST)

9. MERD Maximum excess Resource Demand

10. VAR Variance, Resource Requirement per period

11. MCR Maximum Consecutive Run

12. PM Product Moment

Analysis of 721 Solutions

Analysis of the total set of observations was made in similar fashion to that described above for the 1-resource problems, but using only those 7 of the 12 summary measures which were defined in single-measure terms for multi-resource problems: SD, F*, C, 0, LC, CP, and ADAOP.

The matrix of correlation coefficients for the 7 measures showed that, in general, lower inter-correlations than for the 1-resource problems existed, but still sufficiently kigh to warrant investigation by factor analysis. The resulting application of factor analysis produced results similar to those given earlier for the 1-resource problems: one factor consisting entirely of resource-associated summary measures was predominant, and two other factors consisting of, respectively, SD, CP and C were of neghgible importance.

As with the 1-resource problems, numerous multiple 3~eta i ls of the factor analysis approach and other results are given regressions of D'RATIO against combinations of the ori- in [5 ] . ginal summary measures were run. These runs produced R~

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- Table 6: Typical regression results, all problems, few selected variable

D'RATIO = 0.46 - 1.64(F*) + 1 . 2 9 ( ~ * ) ~ + 1.04(ADAOP) - O . O ~ ~ ( A D A O P ) ~ + 0.49 (CBAR)

Regression coefficient standard deviation: 0.10 0.055 0.01 3 0.003 0.018

Incremental contgbution to R 0.031 0.080 0.049 0.081 0.674

Std. error of estimate: 0.085

: 0.915

values on the order of 0.85 with a standard error of estimate equal to 11.06 using linear combinations of all 7 original variables. These values could be improved (e.g., RZ = 0.93, std. error = 0.07) using complex combinations of trans- formed and original variables and factor scores. However results almost as good were obtainable using simpler com- binations of a few transformed and original variables. In general, R2 values on the order of 0.91 were produced with combinations of the following variables: F*, ADAOP, and "CBAR" (average of 0 and LC). Table 7 shows some typical results obtained using only these variables. If the usual assumptions regarding normality of forecast error distribution are allowed, these results suggest the possibility of forecasting values of D'RATIO for specific multi-resource problems with an accuracy of about plus-or-minus 16% two-thirds of the time.

Forecasting Project Duration With Summary Measures

The results of the various regression analyses as described above indicate the feasibility of forecasting resource- constrained project durations for specific problems with fair accuracy, given the required values of project summary measures.

An interactive computer program to implement this approach has been developed for testing purposes. The program will calculate values of required summary measures and automatically produce a forecast of expected project duration associated with specified combinations of resource limits, using imbedded regression equations of the type presented earlier (the program is currently set to select one of 4 different equations automatically, depending upon whether 1, 2, 3 or more than 3 different resource types are involved).

To illustrate, Appendix B shows a sample printout from the computer program, with values of the summary mea- sures calculated for the network data shown earlier in Fig. 2 and Table 1. As can be seen, the program first calculates general statistics, including some defined earlier but not used in the regression tests described here. Resource require- ments statistics and resource constraint statistics are cal- culated separately, if desired. In the case of the latter,

limits on each resource type must be specified. For this illustration, limits of 4 units of resource 1 and 7 units of resource 2 were specified, producing the resource-constraint statistics shown near the end of the printout. For these limits the program forecast a project duration of 90 days, with confidence limits as shown. This compares with the value of 89 days shown earlier in Fig. 3 (and repeated below in Fig. 4) obtained by application of a minimum-LFT scheduling procedure. Figure 4 also shows the forecasted values of project duration associated with other resource limits, compared to the scheduled values; these forecasted values were obtained by repeated program runs with all intermediate printout suppressed.

PROJECT DURATION, DAYS

Fig. 4. Comparison of forecast vs. scheduled project durations.

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95

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Summary [ l l ] Pascoe, Timothy, "Allocation of Resources CPM," Revue FranCai~e de Recherche Operationnelle, no. 38 (1966).

[12] Prideaux, J. D. C. and CuUingford, "The Nature of the The results of this study suggest that useful summary Decision Process Implied in Network Analysis," Project measures can be developed for describing the charaqter- Planning by Network Analysis, North Holland Press, Amster-

istics of project networks.- or the special case of constraided- dam (1969). [13] Ruda, M., "Some Estimates in Connection with the Critical

resource scheduling discussed here, summary measures based Path," Project Planning by Network Analysis, North Holland on resource requirements and availabilities appear to be Press, Amsterdam (1969). more important in explaining the duration increases en- countered than general measures such as size and com- plexity. The relationships between duration increase and summary measures that were developed can be used as the basis of a simple procedure for estimating the expected duration increases associated with scheduling particular networks under specified resource constraints. Sensitivity analyses can also readily be conducted with this procedure to evaluate the impact of changes in network parameters or resource availabilities. While the project durations that are forecast from this procedure are based on heuristically- derived solutions, a similar procedure could be developed based on optimally-derived solutions if the required data were available.

This paper has also presented some general measures for summarizing network characteristics which were not used in the tests described. These measures could conceivably be

Dr. Edward W. Davis is Associate Professor, Graduate School of Business Administration, University of North Carolina, Chapel Hill. His current research interests are in the areas of project management, production scheduling and inventory control. He was previously on the faculties of the Sloan School of Management, MIT, and Harvard Business School, and also worked with a national management consulting firm for 4 years. Dr. Davis has a BME and MSIE from Georgia Tech and PhD from Yale University, and is a licensed pro- fessional IE. In addition to membership in AIIE, he is a member of TIMS, APICS and The Project Management Institute.

used-in combination with others-as the basis for categor- izing the types of project networks encountered in practice APPENDIX A

in certain industries or in different applications within Illustration of Rank-Associated Network Measures

given industries.

References

Battersby, A., Network Analysis for Planning and Scheduling, St. Martin's Press, New York (1967). Davis, E. W., "An Algorithm for the Multiple Constrained- Resource Project Scheduling Problem," unpublished Ph.D. thesis, Yale University (1968). Davis, E. W. and Heidorn, G. E., "An Algorithm for Optimal Project Scheduling Under Multiple Resource Constraints," Management Science (August 19 7 1 ). Davis, E. W. and Patterson, J. H., "A Comparison of Heuristic and Optimal Solutions in Resource-Constrained Project Sche- duling," Management Science (February 1975). Davis, E. W., "Project Summary Measures and Resource- Constrained Scheduling," Working Paper No. HBS 73-74, Harvard Business School Division of Research (September 1973). Haggett, Peter and Chorley, R. J., Network Analysis in Geography, St. Martin's Press, New York (1969). Johnson, T. J. R., "An Algorithm for the Resource- Constrained Project Scheduling Problem," unpublished Ph.D. thesis, School of Management, MIT (August 1967). Klovstad, P., "The Technological Network; Macro and Micro Structures," Project Planning by Network Analysis, North Holland Press, Amsterdam (1969). Mueller-Mehrbach, H., "Ein Verfahren zur Planung des O p tirnalen Betriebsmitteleinsatzes bei der Terminierung von Grossprojekten," Zeitschrift fur Wirtschaftlich Fertigung, Munich (Feb., March 1967). Pascoe, Timothy, "An Experimental Comparison of Heuris- tic Methods for Allocating Resources," unpublished Ph.D. thesis, Cambridge University (1965).

In his discussion of technological networks, Klovstad [8] utilized the concept of activity "rank." Essentially, the rank of an activity is based on the maximum number of predecessor activities along a path from that activity to the single beginning activity of the project, without regard to activity durations. For example, the beginning node (acti- vity) has rank 0; rank 1 activities have only the beginning activity as predecessor, rank 2 activities have the beginning activitiy and one other activity as predecessor, etc.

Utilizing this concept the activities of a given project network can be ranked and rearranged according to rank order. From this rearrangement a "rank frequency table" showing the number of activities associated with each rank can be developed. This table gives a rough indication of network "shape," based on technological orderings. Also, it produces a measure of network "length" (the highest computed rank value plus one) and "width" (maximum number of activities associated with any given rank, mea- sured over all ranks).

To illustrate, the table below indicates the rank ordering of the network activities of Fig. 2. In the computer printout shown in Appendix B this information is typically printed out in schematic form, with an "X" representing each activity and activity numbers not shown. Turned sideways, either table indicates the technological "shape" of the network. Since the highest computed rank value is 13, the "length" can be taken as 14, and with a maximum of 12 activities at any rank the "width" can be taken as 12. Thus, the computed "length width ratio" is 1.17, as shown on the printout.

June 1975, AIIE TRANSACTIONS 141

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Table A.l: Activity rank ordering for network of Fig. 2.

Rank Activity Number

0 1

1 2 28

2 3 4 5 12 29 30 32 33

3 6 9 10 11 13 31 34 35 37 47

4 7 8 15 16 17 18 19 36 38 39 48 49

5 14 21 22 40 41 50 51

6 20 24 25 42 44 52

7 23 27 45 53 54 55 56

8 26 43 57 58 59 60

9 46 61 62

10 63 64 65 66 67 68

11 69 70

12 71 72

13 73

APPENDIX B Sample Computer Printout

PROGRAN NETCRL: CHLCULRTES SUMMHRY CHRRACTERISTICS OF PROJECT NETWORKS WITH OR WITHOUT RESOURCE REQUIRENENTS

RESPOND TO R L L QUESTIONS V=YES OR NO=N, U N L t S 3 OTHEKWI>t INSTRUCTEU

INPUT DHTH PRINTED OUT FOR DEBU6GINb PURPObES'N

INPUT F I L E NRURY

GENERRL S T H T I S T I C S 7 Y

NUNBER OF JOBS = 7 3 SUN OF OIJRRTIONS = 4 7 4 , 8W NUMBER 8 F RRCS = 1 1 8 CONPLEXITY = 1. 6 2 RVG JOB DURRr lON = 6 . 49 VHRIANCE.. JOB DURRTION = 1 0 3 1

RRNK FREQUENCY ---- +----+----+----+----+--L-+----+----+ 0 X 1 XX 2 XXXXXXXX i XXXXXXXXXX 4 XXAXXXXXXXXX 5 XXXXXXX 6 XXXXXX 7 XXXXXXX 8 XXXXXX 9 XXX

1 8 XXXXXX 11 XX 1 2 XX 1 3 X ---- +----+----+----+----+----+----*

LENGTH-WIDTH RRTIO = 1 i 7

PROJECT SCHEDULE? N

U R I T I C H L PATH = 75 TOTRL FREE SLACK = 2 8 3 TOTRL SLHCK = 6 8 8 PROJECT DENSITY = 8 76

RESOURCE HNRLYSLSTY

EHRLV STHRT RESOURCE REQUIREl l tNTS THBLE'?N

LRTE STRPT RESOURCE REQUIREMENTS TRBLE7N

REQUIRED RESOURCE S T H T I S T I C S FOR E S T d l ) , LST(2,, BOTH(3), NONE(4)91

RESIjlJRCE CONSTRHINT S T A T I S T I C S 7 Y

ENTER UNITS RVHILHBLE OF ERCH OF 1 RESOURCE TYPES 4 7

REQUIRED RESOURCE S T H T l S T l C S (EST)

R 1 R2 R3 R4 R5 R6 TOTRL WORK CONTENT. 2 1 2 3 9 5

RESIjCIRCE 1 ------------ RYO RESOURCE HEPUIRENENl PER JOB 2 YO HYl j RE5 REP PER PERlOD 2 53 VRRIHNCE BY PERIOD 6 4Y nnx REQ RES LEVEL = 8 w w PRODUCT MONENT - 2 7 8 9 EM

PERIODS OF EXCESS PEBUIRENEN,TS- ............................... TOTRL NUNBER = 2 5 MRXINUN CONSECUTIVE RUN - 2 5 STHRTING PERIOD FOR r lw RUN = 7

RV6 RESOURCE REQUIREMENT PEP JOB 5 4 1 RVG RE5 REQ PEP PkRIOD 5 2 7 VRRIHNCE BY PERIOD H 30 NHX REG. RES. LEVEL = 1 0 . 0 0 PRODUCT NONENT 1 3 7 WO

PERIODS OF EXCESS REQUIRENENTS. ............................... TUTRL NUMBER = 1 7 NHXINIJN CONSECUTIVE RUN = 7 STHRTING PERIOD FOR MRX RUN = 48

RESOURCE CONSTRHLNT S T R T I S T I C S .

0-FRCTOR = W 3 8 OPES = 1. W i RDBPES = 8. 5 1 LC-FF)CTOR = 8 5 7 OPLS = I 4 8 HDOPLS = 8 . 7 4 C-BHR = W 3 3 OPRYG = 1. 25 RDHDP = W 62 F ( N > = 8 71, 0 7 5 , F* = 0 . 7 5

FORECRSTED DIJDRTION WITH CONFIDENCE LEVELS'Y

FOR 1 RESOURCE TYPES THE PREDICTED 1,URRTlOtI = 90

U I T H 6 4 % CONFIDENCE UPPER BOUND = 97 . Lt lWtR BOUND = 8' U I T H 99% CONFIDENCE U B = 1 0 0 L B = 8U

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