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Using Combinations of Heuristicsto Schedule Activities of

Constrained Multiple Resource Projects

Lucy C. Morse, John O. McIntoshand Gary E. WhitehouseUniversity of Central Florida,Orlando, Florida

Project managers face ongoing challenges inmanaging projects. A project must beaccomplished in minimum time, with mini-

mum cost, and limited resources. Because of eco-nomic factors, the project manager must makethe best use of these limited resources within aspecific time frame. Project managers use a vari-ety of techniques to accomplish this scheduling.

The most common approaches to projectscheduling are two traditional techniques thatdeveloped in the late 1950s. One is the CriticalPath Method (CPM), the other, Project Evalua-tion and Review Technique (PERT). AlthoughCPM is strictly deterministic and with PERTthere is an uncertainty in the activity time esti-mates, the methods within these techniques arequite similar. Both are computer-oriented, definearrow network diagrams, and define the conceptof a critical path. For these reasons, PERT andCPM are referred to jointly in this paper asPERT/CPM.

Network analysis is a decision-making tool forall levels of management. It may be used at variousstages of project management, from initial plan-ning or analyzing of alternatives to scheduling andcontrolling activities that comprise the project.

Statement of the ProblemBasic PERT/CPM methods commonly used fornetwork analysis do not consider the availabilityof resources in the scheduling process. The con-straints used with these networks are those ofprecedence only. Commercial microcomputerproject management software packages have the

Project Management Journa

ability to consider constrained resources. Theproblem according to Johnson [10] is that nopackage consistently finds a schedule in whichthe project completion time is minimized whenresources are considered.

When resources are taken into consideration,the constraints must consider both precedenceand resource requirements. With limitedresources the beginning time of some activitiesmay be delayed beyond the unconstrained sched-ule. Thus, if a resource is available only in limitedamounts, the schedule produced without consid-ering resources has the potential to be both inac-curate and unfeasible. When resources are limit-ed, the duration of the project could be longer.

When certain tasks must be coordinated toavoid resource and timing conflicts, the projectmanager seeks guidance on which activities toschedule and which to delay [4]. Also the projectschedule may be dynamic and have a direct rela-tionship with the availability of resources [15].

The general approach for allocating con-strained resources to activities is to assign priori-ties to the competing activities in the networkaccording to some criterion. When the prece-dence conditions are satisfied and adequateresources exist, these activities are scheduledaccording to the priorities. Scheduling an activitythat requires multiple resources is more complexthan those requiring only single resources. Allresources needed for the activity must be avail-able before an activity may occur. If only oneresource of several needed is unavailable, thatparticular activity is delayed, while other activi-ties continue to be scheduled according to thepriorities. A setback in any of the critical activi-ties may delay the entire project.

According to Davis [5], existing constrained-resource scheduling procedures may be classifiedinto two major categories. One category is to use

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optimal procedures that produce true minimumduration schedules. The other is to use some pre-determined rule-of-thumb or heuristic to set pri-orities for finding the duration of the project.

Although optimal procedures produce trueminimum durations, the disadvantage of usingoptimal procedures is the lengthy and complicat-ed computational time [7][9][12]. Optimal pro-cedures using the least computational time arethose involving some form of implicit enumera-tion, such as “branch and bound” procedures.

Since for any given problem there are a largenumber of possible combinations of activity starttime, these constrained-resource problems are ina field of mathematical problems known as com-binatorial problems. Analytical methods have notproven very successful on these combinatorialproblems, thus various heuristic-based proce-dures have been developed. Within this study theaccepted definition of a heuristic is Moder’s [12]:a rule of thumb—a simple, easy to use aid inproblem solving.

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Heuristic procedures are capable of quicklysolving complex problems that would be extreme-ly difficult if not impossible to solve otherwise [3].Heuristics might lead to optimal solutions, butthey do not guarantee them. Using different heuris-tic procedures on the same constrained-resourcenetwork can easily produce different durationtimes, since rules that perform well on one prob-lem may perform poorly on another [12]. Manydifferent heuristics exist today. They include thoseoffered by commercial computer programs, elabo-rate and precise computer heuristics, and simplegeneral rules [7]. Since no single heuristic sched-uler is consistently best, commercial project man-agement packages usually do not create optimalschedules when resources are constrained [10].

Objective of the StudyThe main objective of this study is to find a sim-ple and quick procedure for scheduling activitiesof a constrained multiple-resource single-projectnetwork that will minimize project duration andlower project costs. Based upon the survey ofprevious research [1][3][7][8][13][14], ten simplerule-of-thumb heuristics are selected and appliedto 108 multiple-resource network problems.

This study then looks at determining the com-bination of heuristics giving the minimum delaytime rather than looking at the top ranking threeor four heuristics. The previous studies haveshown that no one heuristic always produces theminimum duration on every problem. Sincesome rules perform better on certain types ofproblems, a combination of heuristics shouldhave minimum durations for most types of net-work problems. The reason for using combina-tions of heuristics in this evaluation is an attemptto improve the solution of constrained multiple-resource networks.

This study evaluates constrained multiple-resource networks using combinations of two ormore heuristic methods to allocate theseresources and minimize the total project delay.This is accomplished by first determining projectdurations using a package network schedulingprogram in combination with an additional algo-rithm for allocating resources according to thedifferent heuristic methods. Then with theseresults a combination of two or more heuristicsthat give minimum delay time is determined.

Description of Network ProblemsFor evaluating the heuristic procedures selected,108 test problems from Patterson’s benchmarkproblem set [obtained from James H. Patterson,

About the AuthorsLucy Morse is director of the Central Florida Con-sortium of Higher Education and is project manag-

er for Distance Learning. Prior toserving as a program manager inthe Engineering Directorate at theNational Science Foundation, shewas a member of the IndustrialEngineering and ManagementSystems (IEMS) faculty at theUniversity of Central Florida,

where she received her Ph.D.

John McIntosh is the assistant chair of the IEMSDepartment at the University ofCentral Florida, where he teachesengineering management cours-es, with an emphasis on costengineering. He has over 11 yearsof industrial management experi-ence with Texas Instruments,General Electric, and others.

Gary E. Whitehouse is provost and vice presidentfor academic affairs at the Univer-sity of Central Florida. He previ-ously served as dean in the Col-lege of Engineering and chair ofthe IEMS Department. He earnedhis Ph.D. at Arizona State andserved on the faculty at LeighUniversity.

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Using Combinations of Heuristics to Schedule Activities of Constrained Multiple Resource Projects

problem set on disk, 1986] with optimal solu-tions and a heuristic solution were used. Accord-ing to Patterson [14], these problems representan accumulation of all readily available multipleresource problems existing in the literature. Thisproblem data set has been previously solved andanalyzed by others using various approaches[2][6][7][10][11][17]. Optimal durations includ-ed with the problem set were obtained by Stin-son’ Branch and Bound [16]. The number ofactivities included in the test problems variesbetween 7 and 51, with the number of resourcetypes required per activity varying between oneand three with only four networks having lessthan three resources. Of these problems, 89 havebetween 22 and 27 activities and ten problemsconsist of 51 activities.

In this study, ten simple priority-rule basedheuristics are used to determine priorities foractivities competing for constrained resources.They vary from simple single attribute heuristicto simple multiple attribute heuristic and includerules found effective in previous literature.

The heuristics used to determine the prioritiesand the explanation of their coding is:

1. Longest Activity First (LAF): Priority givento the activity with the longest activity.

2. Shortest Job First (SJF): Priority given tothe activity with the shortest duration.

3. First Come First Served (FCFS): Priority givento the activity with the lowest activity number.

4. Latest Finish Time (LFT): Priority given tothe activity with the earliest PERT/CPM calculat-ed late finish time.

5. Minimum Early Finish (MEF): Prioritygiven to the activity with the earliest PERT/CPMcalculated early finish time.

6. Minimum Slack First (MSF): Priority givento the activity with the least PERT/CPM calculatedslack time.

7. Maximum Slack First (Max SF): Prioritygiven to the activity with the greatest PERT/CPMcalculated slack time.

8. ACTIM: Priority given to the activity withthe maximum ACTIM value. The ACTIM valueof an activity is calculated as the maximum timethat the activity controls through the network onany one path [1].

9. ACTRES: Priority given to activity with themaximum ACTRES value. The ACTRES value iscalculated by multiplying each activity’s time bythe sum of its resources and then finding themaximum ACTRES that an activity controlsthrough the network on any one path [1].

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10. Resources Over Time (ROT): Prioritygiven to the activity with the maximum ROTvalue. The ROT value is calculated by dividingthe sum of each activity’s resources by the dura-tion of the activity and then finding the maxi-mum ROT that an activity controls through thenetwork on any one path [8].

The heuristics are used with the serial approach,with activity priority being determined during thescheduling procedure, but based on PERT/CPMcalculations obtained at the beginning of the sched-uling procedure. For all heuristics, ties are brokenby the lowest activity number first and then oncean activity is started it is not interrupted.

Formulation of AlgorithmThe project networks were scheduled using anexpanded version of “Activity-on-Node NetworkAnalysis” published in Project Management: IIEMicrosoftware [18]. Each of the 108 networkproblems was first solved independently usingthe ten heuristic methods previously described.

With a matrix of problem durations for eachof the ten scheduling heuristics, an attempt now ismade to find what combination of those heuris-tics gives the “best” results. “Best” is defined asthe minimum project duration. The combinationis a solution subset of two or more heuristics withgroup’s lowest mean, not necessarily the group ofheuristics with the lowest individual mean.

A heuristic computer algorithm was developedto find which combination of heuristics minimizesthe durations of constrained resource project net-works. An important difference between thisalgorithm and other methods used to find the topperforming heuristics for allocating constrainedresources is that the objective here is to findwhich combination or subset of heuristics mini-mizes project duration and meets this objective byfinding the average of the minimum durations.

ResultsUsing the computer algorithm to find the combi-nation of heuristics giving the best answer wasapplied to the durations of a group of 108 net-work problems. The means of the percentageincrease above optimum for each individualheuristic and each combination of heuristics isgiven in Table 1. As can be seen, the combinationof heuristics performed best. ACTIM with thelowest mean percentage increase is a 5.2 percentincrease above optimum compared to the combi-nation of two heuristics, which is 3.8 percent orthe combination of four heuristics, which is 2.9percent above optimum.

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Table 1. Percentage Increase Above Optimum for Each Individual Heuristicand for the Combination of Heuristics

Heuristic Percentage Increase Combination Size Percentage IncreaseMAX SF . . . . . . . . . . . . . 19.9. . . . . . . . . . . . . . . . . . . . 2 . . . . . . . . . . . . . . . . . . . . 3.8SJF . . . . . . . . . . . . . . . . . 15.7. . . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . 3.2LAF . . . . . . . . . . . . . . . . . 12.5. . . . . . . . . . . . . . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . 2.9ROT . . . . . . . . . . . . . . . . 12.2. . . . . . . . . . . . . . . . . . . . 5 . . . . . . . . . . . . . . . . . . . . 2.8MEF . . . . . . . . . . . . . . . . 12.1. . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . . 2.7MSF . . . . . . . . . . . . . . . . 10.9. . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . 2.7FCFS . . . . . . . . . . . . . . . . 9.9 . . . . . . . . . . . . . . . . . . . . 8 . . . . . . . . . . . . . . . . . . . . 2.7LFT . . . . . . . . . . . . . . . . . 6.5 . . . . . . . . . . . . . . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . 2.7ACTRES. . . . . . . . . . . . . . 5.4. . . . . . . . . . . . . . . . . . . . 10. . . . . . . . . . . . . . . . . . . . 2.7ACTIM . . . . . . . . . . . . . . . 5.2

Table 2. Solution Subset Results for All Problems

Subset size 1* 2 3 4 5 6 7

Heuristic ACTIM ACTIM ACTIM ACTIM ACTIM ACTIM ACTIM

LFT LFT LFT LFT LFT LFT

ROT ROT ROT ROT ROT

ACTRES ACTRES ACTRES ACTRES

MEF MEF MEF

FCFS FCFS

SJF

Percent IncreaseAbove Optimum 5.2 3.79 3.23 2.91 2.77 2.71 2.68

*Heuristic with lowest mean, not found by algorithm

A summary of the combination of heuristicsresults is given in Table 2. The size of the solu-tion subset is shown with the heuristic combi-nation for each subset. Also shown is the per-centage increase of the combination ofheuristics above the optimum duration. Thesevalues were obtained by computing the differ-ence (in time units) between the combination ofheuristics minimum duration and the optimumduration for each problem as a percentage ofthe optimum, and then taking the mean of thisdifference.

Table 2 shows, for example, that when thesolution subset size is four, the combination ofheuristics is ACTIM, LFT, ROT, and ACTRES.Using this combination the percentage increaseof the heuristics above optimum is 2.91 percent.

In comparing the combination of heuristics inthe solution subset of four to the four individualheuristics with the lowest mean, one heuristic is

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different. ACTIM, LFT, and ACTRES have thethree lowest means and are in the subset whileROT is in the solutions subset and ranks sixth onan individual comparison. ACTIM and LFT havebeen successful heuristics in previous research,and ACTRES and ROT take into considerationthe resources used.

Table 3 is a summary of the number of net-work problems that obtain optimal durationusing the different priority approaches. Theheuristics used by Patterson [14] only gave opti-mum durations twenty-seven times while thecombinations of heuristics of sizes three, four andfive gave forty-five, forty-six, and forty-eight,respectively. Looking at these results, combina-tions of simple heuristics give more optimumdurations and a lower percentage of error thanother heuristics evaluated in this study.

Johnson [10] used the same set of Patterson’snetwork problems. His study examined different

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Table 3. Summary of 108 Network Problems Obtaining Optimal DurationUsing Different Priority Approaches

Scheduling Priority Approach Number of Optimum DurationsMax SF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6SJF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8MEF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10ROT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10LAF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17FCFS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18MSF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Patterson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27LFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29ACTRES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30ACTIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Combination of 3 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Combination of 4 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Combination of 5 Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Using Combinations of Heuristics to Schedule Activities of Constrained Multiple Resource Projects

commercial software packages and found thedurations of all the network problems. A com-parison was then made to the optimal durationsby Talbot [17], similar to the current study.

Table 4 gives the best of Johnson’s results. Theresults show that none of the commercial pack-ages performed as well as the combination ofheuristics did.

ConclusionThis study presents a combination of heuristicsthat find the average of the minimum durationsfor the constrained resource problem. This com-bination not only supports previous research onheuristic methods for setting priorities for con-strained resource problems, but also producesresults that are significantly better than thoseobtained by single heuristics. Due to the variablenature of network structure and resource avail-ability, different heuristics are more applicable tocertain network problems. For this reason using acombination of heuristics gives a better chance ofapproaching the optimum duration.

Combination of Heuristics. The main objec-tive of this study was to examine constrainedmultiple-resource single-project networks usingcombinations of two or more heuristic methodsto allocate these resources and minimize thetotal project delay. Ten simple rule-of-thumbheuristics were selected and applied to a series ofnetworks. The heuristics are Shortest Job First(SJF), First Come First Served (FCFS), LatestFinish Time (LFT), Minimum Slack First (MSF),

Project Management Journa

Minimum Early Finish (MEF), Maximum SlackFirst (Max SF), Longest Activity First (LAF),ACTIM, ACTRES, and Resources Over Time(ROT). These heuristics were applied to 108multiple resource network problems that havebeen previously solved and analyzed by others.The project durations were determined using apackage network program with an additionalalgorithm for allocating resources. With theseresults, a heuristic computer algorithm deter-mined which combination or subset of heuristicsgave the best results.

When the solution subset is two, the best per-forming heuristics are ACTIM and LFT, whichwere found by Davis [7] to be the most success-ful heuristics in achieving optimality. What isdifferent is that when the solution subset size isthree and four, the additional heuristics,ACTRES and ROT, involve resources over theremaining network path. For example, whenusing a solution subset of four the percentageabove optimum is 2.9 and two of the heuristicsused consider the resources in determining thepriorities of the activities.

The four heuristics used when the subset size isfour take into consideration the remaining pathlength of the network for scheduling priorities.ACTIM is based on the value of the final criticalpath time minus the latest start time of that activ-ity. LFT is found by using a backward pass of con-ventional critical path methods. ACTRES andROT consider durations and resources over theremaining path.

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Table 4. Summary of the Most Successful of Johnson’s (1992) Study ofCommercial Software Packages

Commercial Software Package Number of Optimal DurationsHarvard TPM II/3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

SuperProject Expert 1.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

SuperProject 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

MS Project for Windows 3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Hornet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Timeline 2.0/4.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Since ROT used individually does not perform aswell as some other heuristics, the characteristics ofROT let it handle some types of network problemsmore effectively. Further investigation of resource-related heuristic performances and individual prob-lem characteristics should be considered.

The developers of commercial packages thatwere less successful in providing optimal solu-tions should examine using a combination ofheuristics for each network to achieve more opti-mal solutions.

Limitations. A possible limitation to using theresults of this study is the size of network prob-lems encountered. Practical project schedulingcould include hundreds of activities. The largestproblems used in this study had fifty-one activities,although Pascoe [13] concluded that the mosteffective heuristics for small networks were alsoeffective for larger networks and that network sizedoes not change the results.

An assumption made for this study is that totalunits of available resources remain constantthroughout the project. Once an activity is com-pleted and frees the resource, it is available for fur-ther use. On an actual project this might not be thecase. This study does not consider resource level-ing. Once personnel are hired or equipment is onsite, they are alternately used or unused.

Future Research. There are several areas forfurther research based on this study. Investigationof different heuristic performances and individ-ual problem characteristics could be consideredin the future. Although the results obtained withthe combination of heuristics in this evaluationoutperformed the other rules compared.

Another possible area for research is the tie-breaking rule. Here, the lowest-numbered activitywas always taken first. A number of differentrules could be tried with different heuristics.These include: activity with the most resources,activity with the least slack, activity with the

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shortest duration. Also, since the heuristics con-sidering resources gave better results for someproblems, more resource-related heuristics shouldbe investigated.

Although this study was developed to find thecombination of heuristics giving the average ofminimum project durations, the program couldbe changed to minimize the maximum error, ifthat is the objective.

Combinations of heuristics to schedule projectactivities give more optimal solutions than othermethods previously used. This combination ofheuristics not only supports the previous researchon successful simple heuristic methods that setthe priorities for constrained resource problems,but also produces results that are significantlybetter than those obtained by single heuristics.

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2. Brown, James Taylor. 1995. Priority Rule SearchTechnique for Resource Constrained Project Schedul-ing. Unpublished Ph.D. Dissertation, University ofCentral Florida.

3. Cooper, Dale. 1976. Heuristics for SchedulingResource-Constrained Projects: An Experimental Inves-tigation. Management Science, 22 (Jul.), 1186–1194.

4. Davis, E.W. 1974. Networks: Resource Alloca-tion. Industrial Engineering (Apr.).

5. Davis, E.W. 1973. Project Scheduling UnderResource Constraints: Historical Review and Categoriza-tion of Procedures. AIIE Transactions (Dec.), 297–313.

6. Davis, E.W., and Heidorn, G.E. 1971. OptimalProject Scheduling Under Multiple Resource Con-straints. Management Science, 17 (Aug.), 803–816.

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Pass Heuristic Decomposition Procedure with OtherResource-Constrained Project Scheduling Procedures.Management Science, 25 (Sept.), 862–872.

10. Johnson, Roger V. 1992. Resource ConstrainedScheduling Capabilities of Commercial Project Man-agement Software. Project Management Journal, 22(Dec.), 39–43.

11. Khattab, M., and Choobineh, F. 1991. NewHeuristic for Project Scheduling with a Single ResourceConstraint. Computers and Industrial Engineering,381–387.

12. Moder, Joseph J., Phillips, Cecil R., and Davis,Edward W. 1983. Project Management with CPM,PERT and Precedence Diagramming. New York: VanNostrand Reinhold.

13. Pascoe, T.L. 1965. An Experimental Compari-son of Heuristic Methods for Allocating Resources.Unpublished Ph.D. Thesis, Cambridge University.

14. Patterson, James H. 1984. Comparison ofExact Approaches for Solving the Multiple Con-strained Resource Project Scheduling Problem. Man-agement Science, 30 (Jul.), 854–867.

15. Seibert, James E., and Evans, Gerald W. 1991.Time-Constrained Resource Leveling. Journal of Con-struction Engineering and Management, 117, 3 (Sep-tember), 503–520.

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18. Whitehouse, Gary E. ed. 1979. Project Manage-ment: IIE Microsoftware Norcross, GA: IndustrialEngineering.

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