Student scheduling- A solution method for the conflict matrix

ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & 1970 Student scheduling: A solution method for the conflict matrix Beverly Leo Higginbotham The Univer

ScholarWorks at University of Montana

Graduate Student Theses, Dissertations, &

1970

Student scheduling: A solution method for the conflict matrix

Beverly Leo Higginbotham

The University of Montana

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A SOLUTION ME T H O D FOR THE CONFLICT MATRIX

By

B everly L Hig g i n b o t h a m B.S., C a r negie-Mellon University, I966

Presented in partial fulfillment of the requirements

for the degree of Master of Business A d m i n i s t r a t i o n

U N I V E R S I T Y OF MONTANA

1970

A pproved

'ârd of Exa m i n e r s tan

D e ^ , G r a te Schcrol

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I am indebted to Dr Bernard J Bowlen and

Mr Carl J, Schwendiman for their constructive criticism

d uring the pr e p a r a t i o n of this professional paper.

Mr, Ed w a r d A Peressini of The College Of Great Falls

contributed m u c h to the mathematical content of this paper.

I am grateful to Grace M o l e n for her un f a i l i n g and efficient assistance in typing and proof i n g this work.

My greatest debt is to Susan, my wife Without her patience and encouragement, this work would not be

possible.

Finally, I would like to thank the Strategic Air

C ommand and the Air Force Institute Of Technology of the

U n i t e d States Air Force for p r o v i d i n g the opportunity to complete this paper.

ii

I INTRODUCTION

The General Problem Setting Problems E n c o u n t e r e d in C o n s tructing the Master Schedule

Research Problem Research Objective Procedures to be Used

II RE V I E W OF THE L I T E R A T U R E 7

Manual S c heduling The Unit Record Approach Computer Scheduling

D ata C o llection and Student A s signment Programs

G enerating and E v a l u a t i n g the Master Schedule Heuristic Schedulers

A Mathematical Scheduler III RAPID A P P R O X I M A T I O N METHOD - A PROPOSED

E x p a n d i n g the Usefu l n e s s of RA M Computerized RAM

IV TESTING THE M E T H O D 51

Two Practical Tests Four Hypothetical Tests

V S U M M A R Y 63

R esults of Testing RAM Limitations of the RA M Method Conclusions

Re c o m m e ndations for Further Research

SOURCES C O N S U L T E D .

111

Figure Page

1 The Conflict Matrix 24

2 The Complete Conflict Matrix 29

3* A n Illustration of R A M 33

4a Conflict Between Three Courses No student r e questing all three 38

4h Conflict Between Three Courses One student r e questing all three 38

5» Three Dimens i o n a l Conflict Matrix 40

6 R A M Flowchart 42

7 R A M Computer P r o g r a m 44

8 R A M Input D ata 48

9 E v a l u a t i o n of R A M 57

10 E v a l u a t i o n of R A M 60

11 E v a l u a t i o n of R A M 6 l 12 E v a l u a t i o n of R A M 62

I V

rapidly expanding and changing curriculum requirements,

a shortage of qualified faculty, and a scarcity of a v a i l ­ able classroom space Nowhere are the results of the

changes felt more dramatically than in our secondary and collegiate level schools Adm i n i s t r a t o r s now find greater demands than ever before leveled u p o n their time and

talents They must make long range plans for u p g rading

facilities They are a ttempting to implement new teaching methods and desirable curriculum changes, and like it or not, they must cope wit h an ever e x panding student bod y

in physical plants w h i c h are rapi d l y becoming over-crowded and under-staffed.

As a result of these pressures, school officials are faced wit h two conflicting goals On one hand, a d m i n i s ­ trators seek expanded and revised curriculums for a grow i n g student population And on the other, they are hard

1

limiting constraints of their present facilities The

time demands of scheduling current classes seriously

detract from the time available for revis i n g and improving present curriculums The purpose of this paper is to

reduce the time r e q uired to construct a ma s t e r schedule

by providing a useful decision method, and in so doing,

permit the a d m i nistrator to tackle the more pressing p r o b ­ lem of curriculum development.

Problems E n c o u n t e r e d in Constructing

The Master Schedule All aspects of education, no matter how far removed from the actual learning situation, have a bearing

u p o n the whole of education One of the most difficult

and time consuming problems for the school admin i s t r a t o r

is e fficiently alloc a t i n g the resources at his disposal in constructing the master schedule At best, it is a tedious, frustrating, and time consuming process "For a medium

sized high school it takes upwards of 1,000 man-hours,

including a hig h percentage of expensive and scarce a d m i n ­ istrative t i m e Much of the work is sheer clerical

drudgery* the listing of subjects, rooms and instructors;

1 Judith Murphy, School S c h e d u l i n g bv Computer : The S tory of G A S P (Educational F acilities Laboratories,

N ew York, New York, 196^), p 7

the tallying of student course requests; and the m a k i n g of

a conflict matrix But, once these menial tasks have been completed, it remains for the a d m i n i s t r a t o r to fit the

pieces of the puzzle together to form the best possible

schedule In so doing, he must be careful to avoid c o n ­ flicts or resolve them for the greatest good for students and teachers.

The Research Problem Minimizing Student Conflicts in the Master Schedule

E a c h year numerous school administrators spend untold hours w r e s t l i n g w i t h the thorny task of constructing the master schedule for their schools They are c o n ­

strained by limited facilities and intimidated by ever

expanding enrollments In order to effici e n t l y allocate the resources at their disposal, they must produce a s c h e d ­ ule of classes w h i c h contains as few conflicts as possible.

A nu m b e r of different conflicts arise in a t t e m p t ­ ing to construct the master schedule The first is a s s i g n ­ ing one instructor to teach two courses during the same

period, because no other instructor is available The

second involves scheduling two courses into the same roo m for one period Obv i o u s l y these two types of conflicts can not be p e rmitted in the final master schedule.

The third type of conflict involves student conflicts A conflict arises when a student desires to

take two s i n g le-section courses w h i c h are offered during the same period For example, assume that Biol o g y and

L a t i n 2 are s i n gle-section courses If a student requested both courses, and if they were offered during the same time period, this w ould count as one conflict.

The conflict matrix is a compil a t i o n of all c o n ­ flicts for all students It lists the n u m b e r of students desiring to take any combination of two s i n gle-section

courses.

These single-section courses are usually the most difficult to fit into the master schedule When two such courses are scheduled during the same period, certain students will be denied the opportunity to enroll in both Therefore it is imperative that the master schedule be

designed to reduce such conflicts to a minimum.

In a small school w h i c h has only eight s i n g l e - section courses to be scheduled during a four period day there are one hundred and five different possible schedules

A n y one of these schedules could be the optimal solution.

To find the schedule having the fewest conflicts, the

administrator could list all possible solutions and then choose the best But in a school where there are sixteen

s i n g le-section courses to be scheduled, the number of

possible combinations is over two million Obviously the

a d m i n i s t r a t o r needs a method w h i c h will give hi m the s o l u ­ tio n more e f f i c i e n t l y tha n the en u m e r a t i o n method.

There is no method currently available w h i c h indicates w h i c h s i n gle-section courses should be scheduled concurrently in order to minimize student conflicts In order to meet student requests, many schools schedule course sections throughout the day This attempt to reduce s i n g l e ­ section conflicts results in some sections being less than full and is therefore wasteful in terms of teacher and

classroom utilization.

The problem to be attacked is min i m i z a t i o n of student conflicts for single-section courses in the final master schedule.

Research Objective

E f f i c i e n t Use of School Resources The objective of this research is to find a method to schedule classes in order to minimize the number

of student conflicts In this manner, more efficient use can be made of school resources This research will deal

w i t h only a small port i o n of the prob l e m of building an

efficient master schedule It will embrace the area of

student conflict avoidance for s i n gle-section courses in the final master schedule.

Procedures to be Used Org a n i z a t i o n of the Paoer

A review of the literature perta i n i n g to student scheduling is contained in Chapter II The problems

posed solution are presented in Chapter III In Chapter IV the proposed solution method is tested as a practical t e c h ­ nique The results of these tests and limitations on the method are p r esented in the final chapter Additionally, the conclusions reached and recommendations for further

research are outlined in Chapter V.

R E V I E W OF THE LITERATURE

The p r o b l e m of resolving sin g l e - s e c t i o n conflicts

in school scheduling is part of the larger task of d e s i g n ­ ing an acceptable master schedule As such, any d i s c u s ­ sion of conflict avoidance must n e c e s s a r i l y include a

review of various approaches to the p r o b l e m of gener a t i n g

a master schedule A d iscussion of m a n y approaches will help to crystallize the vari o u s problems encountered The survey w i l l start w i t h the earliest manual techniques and proceed c h r onologically to the most recent computerized

m athematical methods.

Manual S cheduling

Prior to the use of data p r o c e s s i n g equipment and later computers, the scheduling process was done

entirely by hand At best, this w a s a tedious and time

consuming task Before the admin i s t r a t o r can design a

master schedule, he needs inform a t i o n concerning student course requests, c l assroom space, and teacher availability.

The first step in c o llecting information is the

t a l lying of student requests in order to determine the

n u m b e r of sections r e q u i r e d for each course Next, a

conflict matrix is developed This matrix reflects the

num b e r of students desiring to take an y co m b i n a t i o n of

two courses Additionally, i n f o r mation on the number of

classrooms and the seating c a p acity of each is required.

Here, c o n s ideration must be g iven to the use of specialized classrooms which can be used only for certain classes, for example, h o memaking and woodwo r k i n g rooms Finally, the

scheduler requires information concerning the courses w h i c h individual instructors are qualified to teach.

The vast number of variables involved demands a system to insure that all items are considered One of the earliest such accounting systems involved using a s c h e d u l ­ ing board This board was divided into rectangles with

each area representing a period in the school day Wi t h i n

e ach period, hooks representing available classrooms were

a t t ached to the board Additionally, tags representing

each teacher and each class section were prepared.

Once the data and a method for handling it were developed, the a d m i nistrator was p r e pared to combine the

resources at his disposal into an initial schedule D a v i d B,

A u s t i n and Noble Gividen^ suggest the following approach:

David B A u s t i n an d Noble Gividen, The H igh School Principal and Staff D e v e l o p the Master S c h e d u l e

Bureau of Publication, Teachers College, Columbia U ,

N e w York, i9 6 0 , p 6 8 *

tags in periods w h i c h will a l l o w for the fewest conflicts among them.

2, Now locate all other class tags as to periods#

3* Adjust the section tags Hold them in the

same designated periods, but shift them among teachers and rooms so that the best teacher programs will be completed, and reasonable use of space and equipment provided.

4 Check for internal consistency In a d d i t i o n

to a reasonable assignment for each teacher,

it is well to examine the first tentative schedule, period by period, to estimate the

ad e quacy of the program relative to various student groupings.

5 Test wit h sample programs for students.

Seek conflicts and adjust the schedule to accommodate t h e m "

With this tentative schedule in hand, the a d m i n ­ istrator will begin the task of a s s igning students to the courses w hich they had requested With o u t fail this will involve great numbers of conflicts, over loaded or i m b a l ­ anced sections, and in general, an unacceptable schedule This, in turn, w o u l d necessitate revisions to the ma s t e r schedule, assignment changes, réév a l u a t i o n of the n e w

schedule, further schedule adjustments, and so it w o u l d go until a workable master schedule was finally produced.

These first attempts at preparing a master schedule were time consuming and a d m inistrators quite

ob v i o u s l y did not have time to prepare a great num b e r of

a l t e r natives in an atte m p t to find a "better" schedule.

However, this initial experience w i t h school scheduling did identify the three basic phases w hich are inherent in all student scheduling operationsi

Phase 1 Ï Coll e c t i o n and sorting of input data.

Phase 2 1 G e n e r a t i o n and e v a l u a t i o n of a master

s c h e d u l e Phase 3 * A s s i g n m e n t of students to classes.

As the art of student scheduling evolved, the old schedule boards gave wa y to data p rocessing cards and the tags were replaced by w o r d s w r i t t e n in computer m e m ­ ories At first, the improvements tended to be c o n c e n ­

trated in one or another of the three phases Then, as researchers became aware of work done in other phases,

whole systems were developed to handle the entire process from start to finish.

schedule, but once this was completed, successive sorting and collating during the a s signment phase enabled them to evaluate their schedules much more rapidly.

The major drawbacks of the unit record a p p roach were the amount of time required for the manual c o n s t r u c ­ tion of the master schedule and the n e c e s s i t y for a s e p a r ­ ate read-sort ru n each time a different set of course

2 combinations was att e m p t e d in the master schedule.

During the mid 1 9 5 0 ' s, computer memory and logic limitations pre c l u d e d its use in the complicated task of constructing a m a s t e r schedule The machine's capabilities obviously pointed toward applying its high speed p r ocessing and printing abil i t y to the data collection and assignment

p hases of student scheduling.

p Student S c h e d u l i n g Sy s t e m / 3 6 0 A p p l i c a t i o n Description, IBM C o r p , I9 6 6 , p 11.

Dat a C o l l e c t i o n and Student A s s i g n m e n t Programs

One of the first workable programs of the data-

c o l lection-assignment v a r i e t y was developed at Purdue

U n i v e r s i t y in I956 by James Blakeley.^ S h o r t l y thereafter, IBM developed a scheduling p r o g r a m called CLASS based u p o n the e x p l o r a t o r y work done at P u r d u e ^ The prog r a m assumes

a fixed schedule of classes, times, rooms, and teachers

a s s igned b y the traditional manual method Combining this input i nformation w i t h student course requests, CLASS

produces (1 ) a tally of course requests, (2 ) a conflict

matrix, and (3 ) individual student schedule printouts.

Since that time, several refinements have been made to the earl i e r assignment programs w h i c h increase

their usefulness as an aid to the manual construction of the ma s t e r schedule These newer software packages perform additional tasks such as p r i nting class lists, and class cards, g enerating grade reports, and so forth Today, there are a number of a s signment programs available IBM alone has several Three of these are their STUDENT prog r a m for the 1620 series computer, SOCRATES for the 1400 series.

^Judith Murphy, School Sche d u l i n g bv Computer* The S t o r v of G A S P E d u c a t i o n a l Facilities L a b o r a t o r i e s ,

N e w York, New York, 1964, p 3 9

^ I b i d , p 3 9

and Student Sched u l i n g System/ 3 6 0 for the 36 O series m a ­

chines Additionally, Control Data, National Cash Register and Honeywell all have assignment programs available for

their hardware systems.

The assignment programs are g e n e r a l l y suited for large schools w h i c h have up to twenty sections of a single course and as many as 20,000 students Their ability to

rearrange scheduling information into a useful format have made them invaluable to the administrator E v e n with the

aid of assignment programs, however, the administrator must still construct the master schedule by hand A l t h o u g h the unit record and assignment programs perf o r m clerical o p e r a ­ tions well, and tally information rapidly, the scheduler

must still rely on his own genius for the actual c o n s t r u c ­ tion of a master schedule Somehow he has to efficiently allocate the resources at his disposal and produce a w o r k ­ able schedule.

Generating and E v a l u a t i n g the Master Schedule

In 1 9 6 4 , N L Boyles developed a numeric code system w hich identified teachers, rooms, and time periods to the computer U s i n g these codes he devised the following set

of heuristic decision rules to aid in the construction of the

c

m aster schedule «

N L Boyles, T heoretical Rules for the C o n s t r u c ­

t io n of a S e c ondary School Master Schedule U t i l i z i n g an

E l e c t r o n i c C o m p u t e r , u n p u b l i s h e d Ph.D dissertation U n i v e r ­ sity of Tennessee, 1964.

"1 Schedule classes so that the subject code

and the r o o m code are equal.

2 Place single- s e c t i o n subjects in the schedule matrix after fixed time activities and before multiple section subjects, and place all c o n ­ flicting single- s e c t i o n subjects in the matrix

at a time period other than one opposing the

c onflicting s i n g le-section subject.

3 Schedule a single-section subject opposite a

subject being offered in multiple sections,

or opposite a subject w h i c h has not been chosen by any of the students choosing the original s i n gle-section subject.

4 Do not schedule a tw o - s e c t i o n subject opp o ­

site two single- s e c t i o n subjects.

5 Do not schedule a t h r e e-section subject opposite two single-section subjects.

6 Schedule a subject with four or more se c ­

tions opposite any subject regardless of the number of sections in the opposing subject However, do not schedule m u l ­ tiple sections of the same subject at the same time.**

Heuristic Schedulers

In the e arly 1960*s two computerized methods w h i c h simulate manual scheduling techniques were introduced These approaches do not completely take over the job of the s c h e d ­ uler Rather, they are tools w hich speed the p r o d u c t i o n of schedules.

The first of these simulation tools is GASP It was developed at Massac h u s e t t s Institute of Technology,^ and follows a set of p r e p r o g r a m m e d d e c i s i o n rules in order to arrive at a workable ma s t e r schedule.

^Judith Murphy, op c i t , p 8

First the computer is fed inform a t i o n on the

n u m b e r of sections in each course, the nu m b e r of students allowed in various sections, the number of times per wee k each section is to meet, faculty available, rooms available, and room capacity In addition, certain courses may be

specified for a particular room or time.

The p r o g r a m then attempts to a s s i g n the course sections to an appropriate c ombination of time, space and

instructor, so that no conflicts arise among these variables.

If a conflict is encountered, the program will attempt

another random assignment It will continue to loop until

a no conflict schedule is found If a particular course

cannot be scheduled in a specified number of loops, the

program goes on to the next course A n y unsche d u l e d courses are later assigned by changing their faculty or room s pec­ ification.

The prog r a m then reads a s t u d e n t ’s course requests and attempts to a s s i g n him to a section If a p a rticular

section is full, it attempts to find an alternate section

of the same course If all sections of a course are closed

or if a student cannot be scheduled due to a time conflict, that course is a s s essed "penalty points," and the p r o g r a m

goes on to the next student.

After an attempt is made to schedule all students,

p e n a l t y points for all courses are tallied This figure and the initial master schedule are stored on tape The master

schedule in core is then adjusted m a n u a l l y in an effort to alleviate some of the conflicts and another assignment run

is m a d e If the adjus t e d schedule produces fewer pena l t y points, it replaces the schedule on tape The assignment adjustment cycle is reite r a t e d until an acceptable schedule

is devised.

Stanford U n i v e r s i t y also developed a heuristic

n

schedule u nder the direction of P r ofessor Robert Oakford.^

It does not follow p r e cisely the GAS P heuristic method;

however, many elements are common to both The m a j o r d i f ­ ference involves the nu m b e r of elements of the scheduling problem each will accept G ASP can schedule as many as

4,000 courses and a n unlimited number of students compared with S t a n f o r d ’s limit of $00 courses and 3,000 students

S t a n f o r d ’s advantage lies in its use of the modular s c h e d u l ­ ing concept By b r e a k i n g the school day into twenty minute periods, rather than the traditional sixty minute length, the prog r a m is able to incorporate such innovations as

short group discussions, long l aboratory periods, and

independent student s tudy in the schedule Additionally,

S t a n f o r d ’s p r o g r a m also provides for team teaching and

specifying two a l t e r n a t i v e s for each course for each student.

D, W A l l e n and D Delay, " S t a n f o r d ’s C o m puter

S y s t e m Gives S c h e d u l i n g F r e e d o m to 26 D i s t r i c t s , ” Nation* s

S c h o o l s , Vol 77, No 3, March, 1966, pp 124-125.

Because of the a d vantages of the S t a n f o r d program» it seems

to have replaced the G A S P scheduling methods in many schools.

A Mathematical Scheduler

In the past few years, educators and computer personnel have become interested in applying linear p r o g r a m ­ ming to the task of building a master schedule Robert E

Harding developed the first feasible theoretical model® and later F T Helmers incorporated the model in a practical

r epresented teaching the ith course in the jth time period

if the element has the value 1 If the course is not

scheduled for the jth time period, the value is zero.

The scheduler is faced w ith m any restrictions whe n

he attempts to construct the schedule The linear p r o g r a m ­ ming model simulates some of these restrictions by defining

O Robert E l t o n Harding, Optional Scheduling in

E d u c a t i o n a l I n s t i t u t i o n s , A G H o l z m a n and W R Turkes, e d ,

U n i v e r s i t y of Pittsburgh, 1964, pp 248-262.

^F, T Helmers, G e n e r a t i o n of an E d u c a t i o n a l Master S c h e d u l e — A Li n e a r Programming; S y s t e m , M S Thesis, Dept, of Industrial Engineering, A D 4 3 8 -6 6 0 7 I, U n i v e r s i t y

of Pittsburgh, 1964.

constraint equations If x is defined as the value of the

X J

ith course in the jth time period (either 0 or l), then the following constraints can be constructed.

1 A g i v e n course can meet no more than once in

a g iven time period* x S 1

^ J

2 A specific course section must meet T times

J per schedule cycle* z x = T

3 A particular course is not to meet in a g iven time period* x^^ = 0

4 Cert a i n courses must meet in a particular

period* x = 1

J

5 Because of student conflicts, single-section

courses (say Xp and x c; ; ) must not meet during the

generated system to find the course-section, time period

relations (the x *s) w h i c h do not violate any of the g iven

The model wa s field tested by Helmers^*^ at the Turtle Creek High School in P e n n s ylvania and the approach yielded acceptable results However, all c *s were

assigned a value of 1 A school having one hundred courses

to be scheduled during a n eight period day w o u l d require a basic tableau containing eight hundred elements The a d m i n ­ istrator is r e q uired to assign a subjective "figure of

merit" to each of these elements Because of the large

number of x *s, it w o u l d have bee n next to impossible to

relate that a solution does not exist under the given

restrictions Second, linear p r o g r a m m i n g will provide

shadow price information With the aid of these prices, the administrator can evaluate the incremental returns

associated with contemplated changes in student course

demands, physical plant size, and instructor resourses.

In spite of these obvious advantages, the linear progra m m i n g a p p r o a c h has some significant drawbacks The schedule generated must be the same each day of the week There is also a need to easily identify and express

^^Ibid p 42.

mathematically» the interrelationships betw e e n school

policies and the resultant constraint structure Last,

and most important, the sheer size of the model has deterred schools w h i c h do not have access to a computer, from i m p l e ­ menting the method.

In 1 9 6 8 , Robert Voght devised a ne w method for expressing the conflict matrix as part of a modular sched-

11 uling system for the Florida State U n i v e r s i t y School.

The concept involves expressing conflicts in terms of

probability It u t i lizes an index number, expressed as a

per cent, to indicate the possib i l i t y of a student in a

g iven course having a conflict w i t h another course Voght then uses these index numbers to aid in the construction

of the master schedule.

E a c h of the scheduling systems discussed in this chapter deals wit h the problem of single-section courses

and student conflicts in one of two ways The heuristic

schedulers such as GASP, try vari o u s r a n d o m combinations

of courses in a n attempt to find some schedule with an

acceptable number of conflicts The alternate m e t h o d of

dealing w ith single-sections is simply to disallow any

schedule w h i c h has two conflicting single-section courses

in the same period This a p p r o a c h is incorporated in the

11 Robert Lee Voght, A Computerized Modular Schedule Model for the Florida State U n i v e r s i t y S c h o o l , un p u b l i s h e d Ph.D dissertation, Florida State University, I9 6 8 , p 88.

methods proposed by A u s t i n and Gividen, Boyles, and

Harding W h e n a t t e m p t i n g to use these methods the q u e s t i o n arises; how should single-section courses be scheduled w h e n the number of conflicting single-section courses is greater than the number of periods in the school day, i.e., whe n some single-section course must be scheduled opposite other single-section courses?

In an attempt to answer this question the author has developed an alg o r i t h m w h i c h schedules single-section courses so that the total number of conflicts in the s c h e d ­ ule tends to be minimized This algorithm is presented in the following chapter.

RAPID APPRO X I M A T I O N METHOD

A PROPOSED SOLUTION METHOD

Int r o d u c t i o n

An a l g orithm is pre s e n t e d w hich schedules single- section courses so that the total n u m b e r of student c o n ­ flicts tends to be minimized First, an illustration of the type of scheduling problem for w hich the alg o r i t h m was developed is presented Next, the traditional methods of resolving conflicts are presented together wit h a d i s c u s ­ sion of some problems encountered in their application Then, the a l g orithm developed by the author is introduced, coupled w i t h an e x p l o r a t i o n of the principles upon which

it is based Finally, a number of modifications to the algorithm w hich make it useful as a practical tool are

discussed The last section of this chapter contains a computer program w h i c h m a y be used to a pply the proposed algorithm.

S c h e d u l i n g the U n i v e r s i t y of Montana MBA Prog r a m at Mal m s t r o m Air Force Base

The method for scheduling students presented in this paper is the direct result of the unique scheduling

22

requirements for the U n i v e r s i t y of Montana MBA prog r a m at Mal m s t r o m Air Force Base The first requirement is that all courses must be single-sections Second, classes must meet once every fifth day Third, since eight courses are offered each quarter and there are only five faculty members available, some professors must teach two courses Fourth, students may enter the prog r a m at the beginning of any

quarter and be given transfer credit for all undergraduate courses and up to twelve graduate course credits This

yields a student body wit h diverse combinations of course requests for any g i v e n quarter A constraint found in

usual student scheduling problems is absent at Malmstrom Classrooms are adequate for current enrollment.

Considering the constraints of single-section courses, combined w i t h diverse student course requests, the scheduling prob l e m becomes one of minim i z i n g rather than eliminating student conflicts Since m a n y other academic institutions are faced with this same problem, the author feels that the method presented in this paper may be useful

to others.

The Conflict Matrix

It was noted in Chapter II that all scheduling systems took student conflicts into account in one w a y or another One of the simplest ways to present the number

of student conflicts is the conflict matrix.

Fig u r e 1 i l l u s t r a t e s a t y p i c a l c o n f l i c t matrix

E a c h r o w a n d e a c h c o l u m n r e p r e s e n t one course The i n t e r ­

s e c t i o n of a r o w a n d c o l u m n s h o w s the n u m b e r of c o n f l i c t s

b e t w e e n the two c ourses F o r instance, if the s c h e d u l e r

w ere c o n s i d e r i n g s c h e d u l i n g course n u m b e r s three a n d five

d u r i n g the same period, he c o u l d r e a d i l y d e t e r m i n e t h a t four s t u d e n t s w o u l d h ave a conflict.

Use of the Conflict Matrix

In order to illustrate how the conflict matrix can aid in the c o n s t ruction of a master schedule, let us assume a four period day There are eight courses, t h e r e ­ fore two courses will be taught during each period The scheduler g e n e r a l l y tries vari o u s combinations of courses until he arrives at some satisfactory schedule In the example matrix, the scheduler could determine wit h some

effort that teaching courses two and eight during one

period, three and five during a second period, four and seven in a third, and one and six during the remaining

period will result in a schedule w h i c h minimizes the total number of conflicts.

In making this determination, the scheduler could use one of two approaches The first approach

requires that every distinct combination or schedule be

evaluated, and that the one w h i c h minimizes total conflicts

be chosen In this simple example there are one hundred and five possible schedules But the scheduling of eight courses is a small p r o b l e m by most standards As the

num b e r of courses increases, the total number of distinct combinations becomes astronomical.

The nu m b e r of combinations can be expressed by the f o llowing equation*

where = nu m b e r of distinct combinations

n = the n u m b e r of courses to be scheduled

p = the number of periods in the school day.

(See Appendix A for derivation.)

To illustrate, suppose we have a school with eighteen single- s e c t i o n courses to be taught in a nine

period day U s i n g the above equation, we find that there are

or 3^*^00,000 distinct scheduling combinations to be e v a l ­ uated A task of this size w o u l d take a n IBM 1620 computer approx i m a t e l y t w e n t y - s e v e n days Obviously, complete e n u ­

m e r ation of scheduling combinations is not the answer.

The other alternative available to the scheduler

is the sampling approach This technique may involve n o t h ­ ing more than solution b y inspection, i.e., examining the conflict matrix and choosing some schedule w h i c h tends to have m i n i m u m conflicts This a p p roach is s a t i s factory w h e n

a small number of courses are involved, or w h e n only two courses are to be taught per period W h e n the scheduling

p rob l e m grows beyond these bounds, however, solution by

inspection becomes a n unmanageable task.

The f o llowing sequential sampling approach eliminates this problem (and also the problem of complete enumeration) It involves selecting a ra n d o m sample of

schedule combinations and choosing the one w ith the smallest conflict If the selected combin a t i o n indicates an a c c e p t ­ able level of total conflict, the scheduler uses that

schedule If the conflict level is unacceptable, the

scheduler determines the amount by w hich the total conflicts are expected to decrease if another set of random schedules were evaluated.^ If the value of a schedule containing

fewer conflicts is greater than the cost of increasing the sample size, another set of schedules is evaluated This process is reiterated until either an acceptable conflict level is achieved or until the cost of evaluating one more set of schedules is greater than the amount by w h i c h total

2 conflicts are e x p ected to decrease.

This a p p roach has the advantage of indicating

wh e n it is no longer "profitable'* to increase the sample

G i v e n i n f o r mation about the distribution of conflicts in schedules previ o u s l y evaluated, the a d m i n ­

istrator can make a subjective judgement concerning the

probabilities of finding a schedule containing fewer

conflicts in future samples.

2 Robert Schlaifer, I n t r o d u c t i o n to S tatistics For Business D e c i s i o n s McGraw - H i l l Book Company, Inc.,

Ne w York, I9 6 I , p 329»

size D i f f i c u l t y is encountered, however, in e m p loying the technique because there is no exact method of expressing the w o r t h of decreasing the nu m b e r of conflicts.

Because of the difficulties encountered in the above approaches to scheduling single-section courses, the author has developed an a l g o r i t h m called RAM (Rapid A p p r o x ­ imation M e t h o d ) R A M seeks a schedule of courses which

will tend to minimize the total number of conflicts for single-section courses The alg o r i t h m has the following features.

1 Given a list of s i n gle-section courses, teaching

assignments, and student course requests, R A M finds a solution set which tends to minimize total conflicts.

2, The me t h o d requires a finite number of iterations equal to the number of periods in the school day.

3, The solution set does not indicate in w h i c h period the combin a t i o n should be taught This is left to the scheduler.

4 The r e s u l t i n g schedule is not ne c e s s a r i l y optimal

in all cases Several heuristic decision rules are built into the method, however, w h i c h give it

a high pr o b a b i l i t y of finding the schedule w hich

is optimal or n e a r l y optimal

The Rapid A p p r o x i m a t i o n Method

The conflict ma t r i x is illustrated in Figure 1

in its usual form Because the matrix is symmetrical, the lower left port i o n is g e n e r a l l y omitted However, for the purpose of this discussion the lower left is included as shown in Figure 1 In this form each row or each column shows the conflict of one course w i t h all other courses.

For instance, suppose we are interested in finding the c o n ­ flicts for all courses w i t h course nu m b e r four Using

Figure 1, the conflicts of courses one, two, and three with four could be found in column four The remaining conflicts with courses five through eight can be found in row four.

By u sing Figure 2, the same can be accomplished by i n v e s t i ­ gating row or column number four.

D e f i n e d b e l o w is a system of n o t a t i o n w h i c h will

be useful in the development of RAM,

n - the number of single- s e c t i o n courses to be

tj^ = the total conflict of N wit h all other courses

w h i c h have not yet been scheduled t^* = the largest t^

C = the conflict matrix as shown in Figure 2

C = the conflict b e t w e e n course N and course N

C., = the course having the smallest conflict w i t h iN,min course N

TC = the total conflict in the schedule defined

by S

S = the s o l u t i o n set containing p members

M = a member of the solution set w h i c h defines

one c o m b i nation of courses that will minimize total conflict in the schedule.

R A M is a n iterative process E a c h of the p iterations selects one c o m b i nation of courses to be taught

in one period At the heart of R A M is a set of simple, yet powerful heuristics w h i c h enables it to find an optimal or nea r l y optimal solution set These heuristics are based on the following c o n s iderations ;

1* E a c h course conflicts to a greater or lesser degree

w i t h all other courses present in C.

2 A measure of the amount any one course conflicts with all other courses can be obtained by summing the columns of C, i.e., finding tj^ for all N.

3* By selecting a n M in iteration k, such that the greatest amount of conflict is eliminated for M*s

in iterations k + 1 through k = p, TC can be m i n ­ imized.

The R A M a l g o r i t h m operates as follows : Choosing

an N such that tj^ = t^^*, and pairing it with the course w i t h

one member of S This process is repeated exactly p times until all courses are paired Obviously, once two courses have been paired, they must be excluded from consideration

in future iterations.

Occasionally, problems will be encountered w hen there are ties for t.,* or C., W h e n these ties are

encountered at a decision point, they can be resolved

a c cording to the fol l o w i n g rules*

1 Whe n there is a tie for t^^*, there is no w a y to break the t i e In order to insure that a schedule

h aving the smallest total conflict (TC) is found, the p r o b l e m must be w o r k e d through to completion selecting first one and then the other course T hen select the S w h i c h has the smallest value of TC.

2 When there is a tie for Cj^ choose the course

w h i c h has the largest tj^.

3 When there is a tie for and the c o r r e s p o n d ­

ing tj^'s are equal, choose either course.

At this point, an example will help to illustrate the way in w hich R A M is appl i e d to a conflict matrix The problem defined by Figure 3 will be used where n = 8 , and

a s s uming that p = 4.

First I t eration (k = 1) Total the columns Then choose t^^, i.e., the course having the largest total con­ flict In Figure 3> is the largest column total and

corresponds to course num b e r two.

Some other course must n o w be paired with two.

This is acc o m p l i s h e d by scanning row two and choosing the course w i t h w h i c h course two has the least conflict ( min^* Investigating row two, it is found that course eight fits

this condition The first element of S has no w been defined: Teach courses two and eight during the same period.

Second It e r a t i o n (k - 2) Total the columns, excluding elements in rows two and eight and columns two

and eight (The e x c l u s i o n prevents courses two and eight from entering the s o l u t i o n set in later iterations.) Choose the course wit h the largest total conflict This time t^* corresponds to course five S c a n n i n g row or column five

reveals that courses three and seven both have C., = 4.

N,min

Course Number

1 c

0 2 u

r 3 s

e 4

N 5 u

m 6 b

e 7 r 8

totals for kth iteration

k = 1

2 3 4