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Project scheduling under resource constraints anextension of Brooks' Algorithm.James R. Brown

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PROJECT SCHEDULING UNDER RESOURCE CONSTRAINTS

AN EXTENSION OF BROOKS' ALGORITHM

by

James R. Brown

A Thesis

Presented to the Graduate Committee

of Lehigh University

in Candidacy for the Degree of

Master of Science

in

Industrial Engineering

Lehigh University

1976

ProQuest Number: EP76319

All rights reserved

INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted.

In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed,

a note will indicate the deletion.

uest

ProQuest EP76319

Published by ProQuest LLC (2015). Copyright of the Dissertation is held by the Author.

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CERTIFICATE OF APPROVAL

This thesis is accepted and approved in partial ful-

fillment of the requirements for the degree of Master of

Science.

Date?

Chairman of the Department of Industrial Engineering

11

ACKNOWLEDGEMENTS

I am deeply indebted to Doctor Gary E. Whitehouse,

my thesis advisor, for his ideas, patience and continued

support. I am also grateful to Doctor Ben L. Wechsler,

my minor advisor, and Mrs. Faith Newhall, who typed the

manuscript. Last but not least, I am indebted to my

parents who always understand.

iii

TABLE OF CONTENTS

Page

Abstract 1

CHAPTER 1 - INTRODUCTION 2

1.1 Project Management - The Activity Network 2 1.2 Resource Considerations - General 4 1.3 Resource Considerations - Procedures 8

CHAPTER 2 - PROJECT SCHEDULING UNDER 10 RESOURCE CONSTRAINTS

2.1 Heuristic Methods 10 2.2 Brooks' Algorithm - ACTIM 10 2.3 ACTRES and TIMRES 15 2.4 A Modification of the TIMRES Criteria 20 2.5 Experimentation 20

CHAPTER 3 - ANALYSIS OF RESULTS 23

3.1 Network Number 5 23 3.2 Network Number 10 26 3.3 Network Number 12 26 3.4 Network Number 14 34

CHAPTER 4 - CONCLUSIONS AND AREAS FOR ; 38 FURTHER STUDY /'

t

4.1 Conclusions 38 4.2 Areas for Further Study 38

BIBLIOGRAPHY 4 0

Appendix A 43

VITA 76

xv

LIST OF TABLES

Page

Table 2.1 Brooks' Algorithm Solution to 12 Network #1 with Three Units of Resource

Table 2.2 Resource Allocation by ACTIM for 17 Network #3 with Total Resources Limited to Five Units

Table 2.3 Resource Allocation by ACTRES for 18 Network #3 with Total Resources Limited to Five Units

Table 2.4 Resource Allocation by TIMRES for 19 Network #3 with Total Resources Limited to Five Units

LIST OF FIGURES

Page

Figure 1.1 Arrow diagram for servicing a car 6

Figure 1.2 Revised arrow diagram for servicing 7 a car

Figure 2.1 Network 1 13


Figure 2.3 Flow Chart of GENRES Search Model 21


Figure 3.2 Duration graph for Network 5 25





Figure 3.7 Gantt Chart and resource loading 32 diagram for Network 12 with constrained resource of 10 men

Figure 3.8 Gantt Chart and resource loading 33 diagram for Network 12 with constrained resource of 11 men



vi

PROJECT SCHEDULING UNDER RESOURCE CONSTRAINTS:

AN EXTENSION OF BROOKS' ALGORITHM

by: James R. Brown

Abstract

In project scheduling by network analysis, traditional

critical path methods fail to include resource considera-

tions. Other methods must be used to allow for resource

considerations. This thesis explores one area of resource

considerations; project scheduling under resource con-

straints. The specific case investigated is the single

resource, single project schedule. A model, entitled the

GENRES search model, is developed. The model utilizes

Brooks' Algorithm (BAG) to generate the project schedule.

The criteria used are various weighted combinations of

ACTIM and ACTRES (Bedworth, Industrial Systems). The best

project schedule is that which gives the least project

duration.

The GENRES model was found effective in finding project

durations equal to or less than that of ACTIM, ACTRES or

TIMRES (the combination of ACTIM and ACTRES with each

given equal weight). The research also found that when

the project completion time found by the algorithm

approaches the critical path duration, resource levelling

procedures may be preferred.

CHAPTER 1

INTRODUCTION

1.1 Project Management - The Activity Network

The term project management encompasses a large field

and may mean different things to different people. One

thing is certain, however, that the "project" approach has

come to occupy an important position in business and

industry. The wealth of literature concerning project

management appearing in recent years is evidence of the

increasing development and use of project management

techniques.

The basis of most of the more successful project

management techniques is the activity network or project

network model. The advent of these techniques began in

the late fifties with the development of PERT (Program

Evaluation and Review Technique) and CPM (Critical Path

Method). These techniques are commonly called critical path

methods and both require the preparation of an activity

network. The activity network is a physical portrayal of

the plan for carrying out a project. The network shows the

precedence relationships of the project's elements which

lead to the project's completion.

Whitehouse states that the basic activity-network

procedure consists of three phases: the planning phase,

the scheduling phase, and the control phase. Moder and

Phillips suggest a further classification within the

scheduling phase. In total, the procedure involves six

steps:

A. The planning phase

Step 1. Project Planning. The activities making

up the project are defined, and their

technological dependencies upon one

another are shown explicitly in the form

of a network diagram.

B. The scheduling phase

Step 2. Time and Resource Estimation. Estimates

of the time required to perform each of

the network activities are made. These

time estimates are based on a stated

resource level (manpower, machinery, etc.)

for each activity.

Step 3. Basic Scheduling. The basic scheduling

computations are commonly called Forward-

Pass Rules and Backward-Pass Rules. These

computations give the earliest and latest

allowable start and finish times for each

activity and identify the critical path

through the network. The amount of slack

or float associated with activities on

the non-critical paths is also determined.

Step 4. Time-cost Trade-offs. Here the time-cost

trade-offs of activity performance times

may be considered if the analyst is

interested in determining the cost of

reducing the project completion time.

Step 5. Resource Allocation. The feasibility of

each schedule must be checked with respect

to resource requirements and availability.

The details of this step are considered

later in this paper.

C. The control phase

Step 6. Project Control. When the network plan

and schedule have been developed to a

satisfactory extent, they are prepared in

final form for use in the field. The

project is controlled, or monitored, by a

comparison of the actual status of the

project against the prevailing schedule.

This monitoring allows for frequent review

and, when necessary, revision of the

project plan.

1.2 Resource Considerations - General

The basic critical-path method provides plans and

schedules that are technologically feasible. However, they

may not always be practical when resource requirements and

availability are considered. The equipment, manpower and

shop space requirements may exceed availability or fluctuate

violently over the course of the project life. Similarly,

a schedule may require that money be spent faster than it

can be raised or may tie up funds that could be used

profitably elsewhere. Thus, some means is required for plan-

ning and scheduling a project that accounts for not only the

work methods employed but also the availability of resources.

Whitehouse presents an activity network for

servicing a car. In the preparation of this network,

Figure 1.1, two service station attendants were assumed to

be available to perform the project. Thus, after the car is

hoisted, only two activities can occur: remove drain plug

and grease underside fittings. If four attendants were

available, two other activities could also begin: inspect

tires and exhaust and check differential and transmission.

After the car is lowered only two activities may begin

under the assumption of only two attendants. However, four

events are now technologically feasible: grease upper

fittings, oil generator and distributor, check radiator and

battery, and refill crankcase. A revised network is pre-.

sented in Figure 1.2. This network considers only the

technological dependencies of the events. This is the

project planning step or Step 1 suggested in Section 1.1.

The next step is to estimate the time and resources required

to perform each activity. Step 3 would be the performance

of the basic critical path scheduling algorithm. Assuming 5

c/ratAf>/"ftr?\ a/)c/ e«./)4<J±/

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Figure 1.1 Arrow diagram for servicing a car

si

Figure 1.2 Revised arrow diagram for servicing a car

the analyst is satisfied with the resulting project dura-

tion, Step 5 can be performed. If only two service station

attendants are available, then only two activities can

commence from nodes 2 or 8 upon their realization. Some

procedure must now be used to allow for this constraint on

the available resource, manpower.

1.3 Resource Considerations - Procedures

(8 9) Davis ' has categorized the procedures of project

scheduling under resource considerations. Based on the

type of resource allocation problem, three distinct cate-

gories are suggested: (1) time/cost trade-off problems,

(2) resource leveling problems, and (3) constrained resource

problems.

Time/cost trade-off procedures are directed at deter-

mining the least-cost schedule for any given project

duration. These are usually under the assumption of

unlimited resources. The traditional CPM method is such a

time/cost trade-off procedure.

Resource leveling procedures attempt to' reduce any

fluctuations in the level of resource usage while maintain-

ing a given project duration. This project duration is

normally that determined by critical path procedures. Jobs

are then rescheduled within their available slack to give

the most acceptable profile(s) of resource usage over time.

The acceptable profile is judged according to some pre-

8

determined criteria such as maximum utilization of resources,

In the resource leveling procedures, the project

duration is not allowed to increase. This is in contrast

to the constrained resource problem where, out of necessity,

the project duration may exceed that determined by tradi-

tional critical path methods. The constrained resource

problem arises when the amount of resource available during

a project is not sufficient to satisfy the demands of

concurrent activities. To satisfy this constraint,

sequencing decisions are required which often cause an

increase in the critical path duration. The most common

objective of these procedures is to minimize the total

project duration. The procedures available are of three

(2) types: analytical, analogue, or heuristic . The pro-

cedure examined by this thesis is of the heuristic type.

CHAPTER 2

PROJECT SCHEDULING UNDER RESOURCE CONSTRAINTS

2.1 Heuristic Methods

A heuristic method is based on a set of formal decision

rules. These rules derive from assumptions which appear to

be reasonable. Use of an heuristic does not guarantee an

optimum solution. They attempt to provide a solution which

is near-optimal, and they are utilized when the optimal

solution cannot be reached either because no suitable

analytical method is known, or, if known, it is not

technically feasible.

(25 26 27) Analytic methods are available ' ' ' to obtain an

optimal solution to the constrained resource problem. These

include enumerative and mathematical programming techniques.

Both areas are still being researched . However, most

success to date has been found in the application of the

heuristic methods.

2.2 Brooks' Algorithm - ACTIM

One of the available heuristic procedures is the

Brooks' Algorithm (BAG) presented in Chapter 6 of

(3) Bedworth . The algorxthm presents a rule for determining

which activities should receive limited resources first.

The algorithm considers the single project, single resource

case and is best described by example. The example is taken (2)

from Bedworthx ' .

10

The steps required to assign the single resource with

BAG are as follows. (Table 2.1 gives the tabular results

of these steps for Network 1 in Figure 2.1 with three

resources available.)

1. Develop the project network as with the critical

path procedure identifying activities, their required times

and required resources.

2. Determine for each activity the maximum time it

controls through the network on any one path. This is

equivalent to the critical path time minus the latest start

time of the starting node of each activity. These times

are then scaled from 0 to 100. This scaled control time is

designated ACTIM.

3. Rank these in decreasing ACTIM sequence, as in

Table 2.1. The duration and resources required for each

activity are those determined in the first step. The rows

TEARL, TSCHED, TFIN and TNOW need explanation:

a. TEARL is the earliest start time of an

activity determined by traditional

critical path calculations.

b. TSCHED is the actual scheduled starting

time of an activity as determined by BAG.

c. TFIN is the completion time of each activity.

d. TNOW is the time at which resource assign-

ments are now being considered.

11

Activity 1-5 1-2 1-3 3-4 2-4 3-5 •>

4-5

Duration (days) 16 5 5 7 4 ^ 4

ACTIM 16 16 16 11 8 8 4

ACTIM (scaled) 100 100 100 69 50 50 25

Resources required 1 1 1 1 1 1 1

TEARL 0 0 0 5 5 5 12

TSCHED 0 0 0 5 5 9 12

TFIN 16 5 5 12 9 © 16

TNOW 0 5 9 12

Resources available #ji/?Q ^^° ^*"o x*^0

ACT. ALLOW. J>>*f,JJ*<fi'.^)yrf' j^),J^C), (3-5) {^) (4-5)

Iteration no.

Table 2.1. Brooks' Algorithm Solution to Network #1 with Three Units of Resource.

12

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13

4. Set TNOW at 0. The allowable activities (ACT.

ALLOW.) to be considered for scheduling at TNOW of zero

are those activities with TEARL of 0. These are 1-2, 1-3,

1-5. These are placed in the ACT. ALLOW, row in decreasing

ACTIM order. Ties are scheduled by scheduling the activity

of longest duration first. The number of resources initially

available, 3, are placed in the resources available column.

5. Determine if the first activity in ACT. ALLOW., 1-5,

can be assigned. Activity 1-5 requires only one resource

and three are available, so 1-5 can be assigned. A line

is struck through 1-5 to indicate assignment and the number

of resources available is decreased by one. TSCHED and

TFIN are then set for activity 1-5. This same process is

repeated for the remainder of the ACT. ALLOW, activities

until the resources available are depleted.

6. TNOW is raised to the next TFIN time of 5 which

occurs at the completion of both activities 1-2 and 1-3.

The resources available are now 2. ACT. ALLOW, includes

those activities not assigned at the previous TNOW (in

this case, none) and those new activities whose predecessors

have been completed (2-4, 3-4, 3-5).

7. Repeat this assignment process until all activities

have been scheduled. The latest TFIN gives the duration of

the project, which is 17 days for this example.

14

2.3 ACTRES and TIMRES

The Brooks' Algorithm could be used with any number

of criteria other than ACTIM. A number of these criteria

are discussed by Patterson . Bedworth presents two

other possible control criteria, ACTRES and TIMRES.

ACTRES incorporates both activity time and resource

level in the control criteria. ACTRES is computed by

taking the value of the activity time multiplied by the

number of resource units for an activity plus the maximum

of the ACTRES values following this activity. Again,

after the ACTRES value is calculated for all of the net-

work' s activities, they are appropriately scaled from 0 to

100.

The TIMRES criteria suggested by Bedworth is a

combination of ACTIM and ACTRES. It is calculated by

adding ACTIM and ACTRES. Therefore, ACTIM and ACTRES are

given equal weight in the TIMRES criteria. (To keep

TIMRES on the 0-100 scale, TIMRES is calculated

0.5 (ACTIM) + 0.5 (ACTRES).)

The Resource Allocation Schedule for Network 3 of

Figure 2.2 is shown using ACTIM, ACTRES and TIMRES in

Tables 2.2, 2.3 and 2.4 respectively. Note that each gives

a different project schedule.

15

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Activity 1-4 »

4-5 1-2 1-3 2-5 3-5

Duration (days) 1 7 3 1 3 2

ACTIM 8 7 6 3 3 2

ACTIM (scaled) 100 88 75 38 38 25

Resources required 1 1 1 1 2 4

TEARL 0 1 0 0 3 1

TSCHED 0 1 0 0 3 6

TFIN 1 8 3 1 6 © TNOW 0 13 6

Resources available /8/*'/2 y^3 /C 2 ^f0

ACT. ALLOW. ^jf,(^KO^tf ^4^), (3-5) (^$1,(3-5) p^5)

Iteration no.

Table 2.2. Resource Allocation by ACTIM for Network #3 with Total Resources Limited to Five Units.

17

Activity 1-2 1-3 1-4 3-5 4-5 2-5


ACTRES 9 9 8 8 7 6

ACTRES (scaled) 100 100 89 89 78 67


TEARL 0 0 0 1 1 3

TSCHED 0 0 0 1 3 3

TFIN 3 1 1 3 © 6

TNOW 0 1 3

Resources available /// 2 Xo X/2 ACT. ALLOW. pSZ) ,p^i) , (>*£) ^), (4-5) \yi) ,psi) Iteration no. 1 2 3

Table 2.3. Resource Allocation by ACTRES for Network #3 with Total Resources Limited to Five Units.

18

Activity 1-4 1-2 4-5 1-3 3-5 2-5


TIMRES 16 15 14 12 10 9

TIMRES (scaled) 100 94 88 75 63 56


TEARL 0 0 1 0 1 3

TSCHED 0 0 1 0 3 5

TFIN 1 3 8 1 5 © TNOW 0 13 5

Resources available XXX2 X3 X® X1

ACT. ALLOW. gX) , (3^) , d-3) (4^) , (3-5) (3^) , (2-5) (2-

Iteration no. 1 2 3 4

Table 2.4. Resource Allocation by TIMRES for Network #3 with Total Resources Limited to Five Units.

5)

19

2.4 A Modification of the TIMRES Criteria

Particularly for large networks, ACTIM, ACTRES and

TIMRES can each give a different activity sequence which

will result in three possible project schedules. The best

project schedule is chosen as the one resulting in the

least total project duration. The premise of this thesis

is that further project schedules can be generated using

combinations of ACTIM and ACTRES which are not equally

weighted. Various weightings would be tried and the best

project schedule is selected as before. This search

technique lends itself to computer application. Con-

sidering the decreasing cost of computer time, this extended

search appears feasible. For convenience the search model

will be called GENRES. A flow chart" of the GENRES search

model is presented in Figure 2.3. A computer program was

constructed from this model.

2.5 Experimentation

Experimentation with the GENRES search model was per-

formed to determine if it does suggest a useful concept.

Sixteen different networks were used in the experimentation.

All of the networks were of the single resource, single

project category. Various levels of the constrained

resource were tested for each network. (For purposes of

brevity, the terms resources and men will be used inter-

changeably.) These levels ranged from the minimum resource

20

r READ NETWORK DATA AND

INITIALIZE VARIABLES

1

PERFORM TRADITIONAL CRITICAL PATH CALCULATIONS

1

CALCULATE ACTIM & ACTRES

FOR EACH ACTIVITY

I INITIALIZE ACTRES WEIGHT (W) = 0

I CTMBP5 — IW\ far-TTJTTC^ J. ( 1 -W\ lTi(-"VTM\

-1

PERFORM BAG USING CURRENT TIMRES

1 INCREMENT ACTRES WEIGHT (W)

THROUGH W = 1

• r

SEARCH POSSIBLE SCHEDULES FOR ONE WITH LEAST PROJECT DURATION

•> r

OUTPUT BEST SCHEDULE

Figure 2.3 Flow Chart of GENRES Search Model

21

level acceptable, i.e., the maximum number of resources

required by any one activity, to that level of resource

which allowed a project duration equal to the critical path

duration.

The sixteen networks are presented in Appendix A

together with their experimental results. The experimental

results consist of a plot of total project duration versus

ACTRES weight. The results are presented for various

levels of the constrained resource. The critical path

duration is also shown. These plots will be termed

"duration graphs", and will be repeated in the body when-

ever a specific example is cited.

For all of the networks, ACTRES weights were tried

at increments of 0.1. For certain networks smaller incre-

ments were also tried. These specific cases are discussed

in the next chapter.

f

CHAPTER 3

ANALYSIS OF RESULTS

This chapter discusses four selected networks. These

networks are representative of the studied networks and

are typical of the networks which lead to the conclusions

of Chapter 4. Other networks which exhibit similar

characteristics as these will be noted.

3.1 Network Number 5

Figures 3.1 and 3.2 show Network Number 5 and its

resultant duration graph. For a resource level of 6, 1,

8 or 9 men the changing ACTRES weighting does not affect

the total project duration. However, for 5 men a change in

the ACTRES weight does cause a change in the project's

total duration. The ACTIM duration is 77 days while the

ACTRES and TIMRES (ACTRES weight of 0.5) duration is 87

days. Bedworth has suggested that if ACTIM is significantly

better than ACTRES, then TIMRES should automatically pick

it up. In this case ACTIM gives better than a 10% reduction

in total project time from ACTRES. This 10 day reduction

(2 work weeks) would be considered by many project managers

to be significant. However, the TIMRES did not pick it up.

The basic question here is how does one define a signifi-

cant difference in this context. Most projects modelled by

activity networks are one-time projects. Each network

model is unique. Any attempt to generalize about this

process appears dangerous. \ 23

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Figure 3.2 Duration graph for Network 5

25

Many of the studied networks indicate similar

characteristics. However, these are overshadowed by other

observations discussed below.


The network discussed above had a maximum of two possi-

ble project completion times. Network Number 10 (Figure

3.3) is different in this respect. Take for example the

possible durations for a resource level of 9 men depicted

in the duration graph of Figure 3.4. Six possible dura-

tions exist. The best schedule of 122 days results using

an ACTRES weighting of 0.1. This is six days shorter than

using ACTIM, fifteen days shorter than ACTRES and nine

days shorter than TIMRES. A similar result is noted for

the other manpower levels with the exception of 13 men.

Note that for manpower levels of 7, 11, and 14 neither

ACTIM, ACTRES, or TIMRES produce the minimum makespan.

The duration graphs of Networks 7, 8, 11, 12, 13,

14, 15 and 16 all exhibit at least some of these character-

istics of the duration graph of Network 10.


A change in the ACTRES weighting does not change the

project duration of Network 12 (Figure 3.5) for resource

levels of 7, 10 or 11 men. The results of the algorithm

for 10 and 11 men are disturbing (see Figure 3.6). The

resulting project duration for 11 men is thirty days or

26

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Figure 3.6 Duration graph for Network 12

30

one day longer than that for 10 men. Normally, the decision

to add men to a project is made to reduce the project

duration. The Brooks' Algorithm does not produce this

result.

Examination of Figures 3.7 and 3.8 explains this dis-

turbance. The Gantt chart and resource loading diagram

for both 10 and 11 men are the same up until day 15. On

day 15 two activities may start. These are (3-5) and

(4-12), with 3-5 having higher priority. Six men are

already tied up on day 15 by activities (6-7), (6-9) and

(10-11). Therefore with 10 men only activity (4-12),

requiring 3 men versus the five required by (3-5), can

begin. However with 11 men activity (3-5) can begin.

The resulting resource loading diagram is different from

day 15 on for the two levels of constrained resource. For

the 10 men constraint the total labor force of 10 men is

utilized for 6 days (day 20 to day 26). The total labor

force of 11 men is utilized for only two days after day 15

(day 19 to day 21). In essence the resulting resource

utilization is poorer for the 11 men case than for that

of 10 men.

Similar observations were made for Networks 14, 15

and 16. An increase in the level of manpower resulted in

either no decrease in project duration or an increase such

as in Network 12. In all cases this occurred where the

duration was approaching the critical path duration.

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33

This suggests that the problem is one of resource leveling

rather than one of scheduling under resource constraints.


The duration graph (Figure 3.10) for Network 14

(Figure 3.9) exhibits many of the characteristics already

discussed. Note that for each level of manpower the project

duration does not vary the same with ACTRES weighting.

For 8 men, there are two local maxima and two local minima.

For 9 men, there are also two local maxima and minima each.

However, these do not occur at the same ACTRES weighting.

Analysis of this and most of the other networks shows that

the best ACTRES weight, i.e., that which yields the least

project duration, can vary from resource level to resource

level for any given network.

The duration graphs presented in this thesis were

produced using 0.10 increments of ACTRES weight. The

GENRES search model was also used with ACTRES weight

increments of 0.05 and 0.01 for Networks 14 and 10.

These finer increments did not produce shorter schedules

for Network 10 at any resource level. Using increments of

0.05 did produce a shorter schedule for Network 14 at a

resource level of 10 men. At an ACTRES weight of 0.05,

the project duration was 171 in contrast with the least

project duration of 172 generated using 0.10 increments.

Using increments of 0.01 did not produce a further

34

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reduction in the project duration. The conclusion to be

drawn by this is that the analyst using the model must

decide on the size of the increment in testing the

different ACTRES weightings.

C&

37

CHAPTER 4

CONCLUSIONS AND AREAS FOR FURTHER STUDY

4.1 Conclusions

The following conclusions can be drawn from the

research:

1. The GENRES search model is effective in finding

project schedules with a duration equal to or less than

that of ACTIM, ACTRES or the traditional TIMRES.

2. When the project completion time found by the

Brooks' Algorithm approaches the critical path time, resource

leveling procedures may be preferred.

3. The characteristics of the duration graph for a

given network can vary with resource level.

4. That the decision to use a particular size

increment to test the ACTRES weighting in the search model

must be based on the judgment of the network analyst.

4*2 Areas for Further Study

Three general areas for further study of the GENRES

search model exist. The first is mentioned in the con-

clusions; the adaptation of resource leveling techniques

when the project duration approaches the critical path

duration. This might involve setting further guidelines

for the network analyst.

The second general area is concerned with conclusion

number 4. Further study is required to establish guidelines

38

to help the analyst choose an appropriate increment for

the change in ACTRES weighting.

The other general area is the extension of this

technique to solve other problems of scheduling under

resource constraints. These include the multi-resource,

multi-project and combined multi-resource/multi-project

(3) cases. Bedworth discusses the use of BAG in the multi-

resource case but does not consider its use in the multi-

project case. In these cases different ACTRES weightings

could be used for each resource within each project.

Study in this area would require the preparation of guide-

lines to decide on how many combinations of these weightings

should be tried.

39

BIBLIOGRAPHY

1. Balas, Egon, "Project Scheduling with Resource Constraints," Applications of Mathematical Program- ming Techniques"E. M. L. Beale (ed.), London, England: The English Universities Press Ltd., 1970.

2. Battersby, Albert, Network Analysis for Planning and Scheduling, New York, N. Y.: St. Martin's Press, 1967.

3. Bedworth, David D., Industrial Systems: Planning, Analysis and Control, New York, N. Y.: The Ronald Press Co., 1973.

4. Bennington, G. E. and G. F. McGinnis, "A Critique of Project Planning with Constrained Resources," Proceedings of Symposium on the Theory of Scheduling and Its Applications, New York, N. Y., Springer- Verlag, 1973.

5. Conway, R. W. et. al., Theory of Scheduling, Reading Mass.: Addison-Wesley Publishing Co., 1967.

6. Davis, Edward W., "Networks: Resource Allocation," Journal of Industrial Engineering, April 1974, pp 22-32.

7. Davis, Edward W., (ed.) Project Management: Techniques Applications and Managerial Issues, Norcross, Georgia: American Institute of Industrial Engineers, 1976.

8. Davis, Edward W., "Project Scheduling under Resource Constraints - Historical Review and Categorization of Procedures," AIIE Transactions, December 1973.

9. Davis, Edward W., "Resource Allocation in Project Network Models - A Survey," Journal of Industrial Engineering, April 1966, pp 177-188.

10. Davis, Edward W. and George E. Heidorn, "An Algorithm for Optimal Project Scheduling under Multiple Resource Constraints," Management Science, August 1971, pp B-803-816.

11. Davis, Edward W. and James H. Patterson, "A Comparison of Heuristic and Optimum Solutions in Resource Constrained Project Scheduling, Management Science, April 1975, pp 944-955.

40

12. Fendley, Larry G., "The Development of a Complete Multiproject Scheduling System Using a Forecasting and Sequencing Technique," Proceedings of the 18th Annual A. I.I.E. Conference and Convention, Atlanta, Ga.: A.I.I.E., 1967.

13. Gillett, Billy E., Introduction to Operations Research A Computer-Oriented" Algorithmic Approach, New York, N. Y.: McGraw-Hill, Inc., 1976.

14. Gorenstein, Samuel, "An Algorithm for Project (Job) Sequencing with Resource Constraints," Operations Research, Jul-Aug 1972, pp 835-850.

15. Lambourn, S., "Resource Allocation and Multi-Project Scheduling (RAMPS) - A New Tool in Planning and Control," The Computer Journal, January 1963, pp 300- 304.

16. Lardi, Peter, "Project Planning in Pharmaceutical Research and Development with the Aid of the Critical Path Method," The Practical Application of Project Planning by Network Techniques, Vol. II, pp 207-218, Mars Ogander (ed.), INTERNET 72 Congress, New York, N. Y.: Halsted Press, 1972.

17. Lockyer, K. G., An Introduction to Critical Path Analysis, New York, N. Y.: Pitman Publishing Corp., 1969.

18. Maggard, Michael J. et. al., "Network Analysis with GERTS III QR," Journal of Industrial Engineering, May 1974, pp 24-29.

19. Martino, R. L., Project Management and Control: Vol. Ill, Allocating and Scheduling Resources,~~ New York, N. Y.: American Management Association, 1965.

20. Moder, Joseph J. and Cecil R. Phillips, Project Management with CPM and PERT, New York, N. Y.: Van Nostrand Remhold Co., 1970.

21. Moshman, J. et. al., "RAMPS - A Technique for Resource Allocation and Multi-Project Scheduling," Proceedings, 1963 Spring Joint Computer Conference, Baltimore, Md.: Spartan Books, Inc., 1963.

22. Muth, John F. and Gerald L. Thompson, Industrial Scheduling, Englewood Cliffs, N. J.: Prentice Hall, Inc., 1963.

41

23. O'Brien, J. J. (ed.), Scheduling Handbook, New York, N. Y.: McGraw-Hill Book Co., 1969.

24. Patterson, James H., "Alternate Methods of Project Scheduling with Limited Resources," Naval Research Logistics Quarterly, December 1973, pp 767-784.

25. Pritsker, A. Alan B. et. al., "Multiproject Scheduling with Limited Resources: A Zero-One Programming Approach," Management Science, Sept. 1969, pp 93-108.

26. Schrage, Linus, "Solving Resource-Constrained Network Problems by Implicit Enumeration - Non-preemptive Case," Operations Research, March-April 1970, pp 263-278.

27. Schrage, Linus, "Solving Resource-Constrained Network Problems by Implicit Enumeration - Preemptive Case," Operations Research, May-June 1972, pp 668-677.

28. Shaffer, L. R. et. al., The Critical Path Method, New York, N. Y.: McGraw-Hill, Inc., 1964.

29. Whitehouse, Gary E., Info-Cision - A Network Technique for Analyzing Decision Systems, Bethlehem, Pa., Department of Industrial Engineering, Lehigh University, 1972.

30. Whitehouse, Gary E., Systems Analysis and Design Using Network Techniques, Englewood Cliffs, N. J.: Prentice Hall, Inc., 1973.

31. Wiest, J. D., "A Heuristic Model for Scheduling Large Projects with Limited Resources," Management Science, Feb. 1967, pp B-359-377.

32. Wiest, J. D., "Some Properties of Schedules for Large Projects with Limited Resources," Operations Research, May-June 1964, pp 395-418.

33. Woodgate, H. S., "Planning Networks and Resource Allocation," Datamation, Jan. 1968, pp 36-43.

42

Appendix A

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VITA

The author was born in Norristown, Pennsylvania on

October 12, 1951, the son of John W. and Mary R. Brown.

He attended secondary school in Ramsey, New Jersey

and received his high school diploma in 1969 from

Don Bosco High School, Ramsey, New Jersey. He continued

his studies at Lehigh University, where he received the

Bachelor of Science Degree in Civil Engineering

in May 1973.

From June 1973 to January 1975, he was employed by

Yerkes Associates, Inc., Bryn Mawr, Pennsylvania as a

Civil Engineer. During this time, he did graduate work

in Civil Engineering at Villanova University.

In January 1975, he began work toward the Master of

Science in Industrial Engineering at Lehigh University.

During that time, he held positions as Research Assistant

and Teaching Assistant in the Department of Industrial

Engineering. ;

76