Resilient Course and Instructor Scheduling in the ... · PDF fileResilient Course and Instructor Scheduling in the Mathematics ... derived from actual data from the past three

Resilient Course and Instructor Scheduling

in the Mathematics Department

at the United States Naval Academy

Stephen J. Ward1, Joseph Foraker2, and Nelson A. Uhan2

1Naval Postgraduate School, Monterey, CA, USA. [email protected] Department, United States Naval Academy, Annapolis, MD, USA.

{foraker,uhan}@usna.edu

September 2017

Abstract

In this work, we study the problem of scheduling courses and instructors in the MathematicsDepartment at the United States Naval Academy (USNA) in a resilient manner. Every semester,the department needs to schedule around 70 instructors and 150-180 course sections into 30class periods and 30 rooms. We formulate a stochastic integer linear program that schedulesthese courses, instructors, and rooms. In addition to maximizing instructor preferences androom stability, this stochastic integer linear program minimizes the expected number of changesrequired in the schedule if a disruption were to occur, given a subjective probability distributionover a finite set of possible disruption scenarios. We run our model on a number of instancesderived from actual data from the past three years, and investigate the effect of emphasizingdifferent parts of the objective function on the running time and resulting schedules.

1 Introduction

Colleges and universities continually face the problem of constructing a schedule for courses,

instructors, and students that respects various constraints and objectives such as room availability,

curriculum conflicts, and the preferences of students and faculty. Such university timetabling

problems have been widely studied for the past several decades, beginning as early as 1969 (Thornley

1969). Due to the size of many academic departments and the number of potentially conflicting

objectives and constraints that must be considered, university timetabling is a task ideally suited

for operations research techniques.

Creating schedules for courses and instructors at the United States Naval Academy (USNA), the

undergraduate college of the United States Navy, has some interesting challenges. One such challenge

1

is the uncertainty of available manpower: an instructor may or may not be able to teach in the

upcoming semester. This happens often with USNA’s military officer instructors, whose start and

end dates at USNA are sometimes uncertain for a variety of reasons, such as extended deployments

and sudden reassignments (e.g., individual augmentation). The availability of instructors, both

civilian and military, is also affected by events such as long-term illnesses and family crises.

At USNA, the course and instructor schedule for the next semester is published near the end

of the previous semester. Students (i.e., midshipmen) register for their courses around the same

time. Unfortunately, disruptions to the published schedule, such as the sudden loss of an instructor,

can occur between registration and the start of the next semester. However, in most cases, a

course cannot simply be canceled if the instructor is no longer available to teach. Through an

act of Congress, the academic program at USNA is 47 months (8 semesters) of study, and no

more (United States Naval Academy 2016). This fixed length program requires USNA to put the

highest priority on ensuring students get their required classes. This involves, among many things,

adjusting the course and instructor schedule. When a disruption occurs, the schedule must change

to guarantee that students can take the courses they need to meet their graduation requirements

on time. Generally speaking, course offerings are handled at the department level. This means

that even a minor disruption can cause widespread changes to an existing schedule, creating a

trickle-down effect that requires significant effort across multiple departments to address. As a

result, having a timetable that is resilient – one that requires a minimum number of changes in the

face of disruption – is an important consideration.

In this work, we study the problem of scheduling the courses and instructors in the Mathematics

Department at USNA in a resilient manner. Every semester, the department needs to schedule

around 70 instructors and 150-180 course sections into 30 class periods and 30 rooms. We formulate

a stochastic integer linear program that schedules these courses, instructors, and rooms. In addition

to maximizing instructor preferences and room stability, this stochastic integer linear program

minimizes the expected number of changes required in the schedule if a disruption were to occur,

given a subjective probability distribution over a finite set of possible disruption scenarios. We

run our model on a number of instances derived from actual data from the past three years, and

investigate the effect of emphasizing different parts of the objective function on the running time

and resulting schedules.

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2 Literature Review

2.1 Problem Applicability

Traditionally, university departments – including the Mathematics Department at USNA – scheduled

their courses and instructors by hand over the course of several hours or days. Sometimes a schedule

from previous years could be reused with minor modifications; however, changes happening from

year to year often required a complete reworking of the schedule. With schedules created by hand, it

was also difficult to justify the solution to those instructors who did not receive preferable schedules.

Saltzman (2009) asserts that integer programming significantly reduces the amount of time spent

constructing what is to be considered a good, feasible schedule that is on par or even preferred to a

schedule created by hand.

While it remains difficult to find a provably optimal solution to many university timetabling

problems, a near-optimal solution can usually be found in a matter of hours (Burke et al. 2008).

The goal of these relatively quick and near-optimal solutions is to provide decision makers with

several feasible solutions that can be deemed as good solutions by the stakeholders involved (Benli

and Botsalı 2004). The scheduling process often begins well in advance due to university planning

requirements (Waterer 1995). Disruptions can occur between the time a schedule is published and

the beginning of a semester, and so it would be ideal to have a first schedule that is resistant to

potential changes.

2.2 Basic Structure

Most university timetabling models include the same basic elements. Most objective functions are

centered around the idea of satisfying the preferences of individuals as much as possible and often

times contain soft constraints for various desirable but not vital schedule properties. The objective

for the model described in this paper also has these ideas, as well as others that create a resilient

course and instructor schedule.

A typical university timetabling model includes constraints that ensure fundamental restrictions:

for example, at any given time, only one section of a course can be assigned to a particular room.

More complicated constraints commonly used include those that ensure instructors teach classes in

consecutive or non-consecutive time slots, depending on the structure and rules of the university

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(Martin 2004). These “consecutivity” constraints are also implemented in the model detailed in this

paper.

The idea of “coherence” or setting a maximum number of days that instructors teach a course

at a given university is also a fairly common set of constraints within these timetabling models

(Kumar 2014). This is an important set of constraints in our model in order to give instructors time

to work on their research and other administrative tasks.

Another commonality between our formulation and other university timetabling models is

the implementation of “curriculum compactness” which strives to constrain lectures of the same

curriculum to be taught in consecutive meeting times (Cacchiani et al. 2013). Our model takes a

slightly different interpretation of this where an instructor’s lectures are constrained to be taught in

consecutive meeting times (Bettinelli et al. 2015). Another difficult, but common and beneficial set

of constraints consistently implemented in more recent formulations is referred to as “room stability”

(Lach and Lubbecke 2008). In our context, room stability translates to the idea that instructors

desire to teach in the same room throughout their day. Just as in our model, these room stability

constraints are soft constraints that can be weighted appropriately in the objective function based

on how much instructors at the institution value this idea (Lach and Lubbecke 2012).

2.3 Problem Difficulty

As previously mentioned, the topic of university timetabling has been widely studied at various

universities. The reason it continues to be studied is the problem’s difficulty as well as its applicability.

One of the main reasons university timetabling is difficult, is the formulation of constraints that

satisfy the specific rules of the university (Bakır and Aksop 2008). Perhaps the most difficult set

of constraints common across various models is the idea of consecutive periods (Daskalaki et al.

2004). This set of constraints requires multi-period courses to be taught in consecutive periods

and has been known to make the problem NP-hard (Daskalaki et al. 2004). Furthermore, the

implementation of consecutive meeting times for instructors who teach multiple sections of the same

course drastically increases the size of the model and the amount of time it takes for solvers to find

solutions (Daskalaki and Birbas 2005).

Due to the complexity and difficulty of this timetabling problem, many researchers have turned

to using heuristics instead of integer programming to find acceptable solutions (Sampson et al. 1995).

4

Sørensen et al. (2013) examined an integer linear program for a high school timetabling problem

with over 100 datasets and found that there existed large optimality gaps even after two hours of

running the solver. In fact, it is consistently observed that integer programs reach solutions much

slower than heuristics for this type of scheduling problem (Sørensen and Stidsen 2014).

2.4 Disruptions and Scheduling Persistence

Our model follows from a similar idea in Phillips et al. (2014), where the objective seeks to minimize

the number of changes required to an existing schedule as a result of some disruption to the data. The

difference is that we focus on how to overcome the inevitable changes that occur while attempting to

schedule courses and instructors, while Phillips et al. (2014) purposely perturbs an existing solution

to create infeasibility in the hopes of finding a “better” solution. In Phillips et al. (2016), the authors

briefly allude to the idea of perturbing the solution to discover the changes needed to an existing

schedule for disruptions that could occur to the original data set. The disruptions that Phillips et al.

(2016) study are room losses. On the other hand, the most common disruption to happen at USNA

within the Mathematics Department is the sudden loss of an instructor who was scheduled to teach

multiple courses in the upcoming semester. While the idea of minimizing the number of changes

required to an existing solution is present in both Phillips et al. (2014) and Phillips et al. (2016),

they do not attempt to take preventive measures in forming the original schedule to minimize these

changes.

The closest work we were able to find that describes the ideas unique in this paper are those

of persistence in Brown et al. (1997). A schedule is called persistent if it stays the same across

potential scenarios. Brown et al. (1997) details how they integrated the idea of persistence into

various case studies involving scheduling. These models would seek to minimize the number of

differences between scenarios with small input changes between them. This is similar to the goal

of our model which seeks to minimize the number of potential differences from a baseline scenario

to each disruption scenario. The major difference between the models in Brown et al. (1997) and

our formulation is that the schedule constructed in our model is based upon minimizing potential

changes to a baseline schedule given the likelihood each disruption occurs, whereas Brown et al.

(1997) seeks to minimize disruptions between scenarios and does not necessarily focus on a baseline

schedule as the center of the model.

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3 Problem description and mathematical programming formula-

tion

In this section, we describe the data, decision variables, constraints, and objective function of our

stochastic integer program that finds a resilient schedule for the courses and instructors in the

Mathematics Department at USNA.

3.1 Data

At USNA, classes are held 5 days per week, Monday through Friday, abbreviated here as M, T, W,

R, F. Each day, there are 6 periods – 4 periods in the morning, 2 periods in the afternoon – for a

total of 30 periods per week:

M1, . . . ,M6,T1, . . . ,T6,W1, . . . ,W6,R1, . . . ,R6,F1, . . . ,F6.

A meeting time is a set of periods. For example, the meeting time MWF1 consists of periods M1,

W1, and F1.

A meeting time cannot contain both morning and afternoon periods. We denote the set of

periods and possible meeting times as

P = set of periods,

M = set of meeting times,

Pm = set of periods in meeting time m for m ∈M.

For example, PMWF1 = {M1,W1,F1}. Furthermore, we denote the set of morning periods and the

set of afternoon periods as

PAM = {M1,M2,M3,M4,T1,T2,T3,T4, . . . ,F1,F2,F3,F4},

PPM = {M5,M6,T5,T6, . . . ,F5,F6},

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and the set of morning meeting times and the set of afternoon meeting times as

MAM = {m ∈M : Pm ⊆ PAM},

MPM = {m ∈M : Pm ⊆ PPM}.

There is a set S of scenarios that consists of a baseline scenario s0 and several disruption

scenarios. Each scenario represents one possible realization of the offered courses, active instructors,

and available rooms for the next semester. The baseline scenario represents the planned courses,

instructors, and rooms; students will register based on a schedule feasible for the baseline scenario.

The disruption scenarios represent contingency plans that adjust the baseline scenario plan in

the face of a disruption, such as a loss of an instructor or room. Each scenario s ∈ S has an

associated probability qs of being realized, which in practice, would be subjectively determined by

the department chair.

Due to the small class sizes at USNA, many courses are taught in multiple sections. Every

semester, the department chair needs to assign each of these sections to an instructor and a meeting

time. We define

Is = set of instructors for s ∈ S,

Cs = set of courses for s ∈ S.

Based on instructor qualifications and the needs of the department, the chair decides how many

sections of each course that each instructor must teach in each scenario:

Cs,i = set of courses assigned to instructor i in scenario s for s ∈ S, i ∈ Is,

ts,i,c = number of sections of course c assigned to

instructor i in scenario s

for s ∈ S, i ∈ Is,

c ∈ Cs,i,

ts,i =∑

c∈Cs,i

ts,i,c = total number of sections assigned to

instructor i in scenario s

for s ∈ S, i ∈ Is.

Instructors often have preferences on when they teach, often due to non-teaching duties such

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as service obligations, extra instruction sessions, and child care. We capture these preferences

numerically as

ps,i,m = instructor i’s preference for meeting time m in scenario s for s ∈ S, i ∈ Is,m ∈M

where a higher value of ps,i,m indicates that instructor i has a greater preference for teaching during

meeting time m in scenario s.

In addition, each section needs to be assigned a room for each period in its assigned meeting

time. We define

Rs = set of rooms available in scenario s for s ∈ S.

Ideally, each section would be assigned the same room for all periods in its meeting time. Due to

the limited number of rooms available to the department, however, this is often not possible. On

the other hand, sometimes it is necessary to assign the same room to multiple periods: for example,

the meeting time MF1R12 consists of a double-period class during periods R1 and R2, and so the

same room must be assigned to those two periods. To capture these requirements, we partition the

set of periods in each meeting time into period groups:

P ∗m = set of period groups for meeting time m for m ∈M,

where P ∗m is a partition of Pm, and each period group G ∈ P ∗m must be assigned to the same room.

Using preregistration data, the chair determines how many sections of each course are needed

and the student capacity of each course. In addition, each course can only be taught using certain

meeting times, based on the structure (lectures vs. labs) and the number of credits of the course.

We denote these as:

ns,c = number of sections needed for course c in scenario s for s ∈ S, c ∈ Cs,

es,c = capacity of one section of course c in scenario s for s ∈ S, c ∈ Cs,

Ms,c = set of meeting times allowed for course c in scenario s for s ∈ S, c ∈ Cs.

To ensure that senior students can take the courses they need to graduate on time, the chair also

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uses the preregistration data to determine which courses must be conflict-free, or in other words,

courses that must have sections offered during non-overlapping periods:

Ks = set of conflict-free course pairs in scenario s ⊆ Cs × Cs for s ∈ S.

Each room has a capacity, and therefore the sections of each course can only be assigned to

certain rooms. In addition, some rooms are not available in certain periods or scenarios. We denote

these quantities as:

fr = capacity of room r for r ∈ R,

Rs,c = set of rooms allowable for course c in scenario s ⊆ {r ∈ Rs : es,c ≤ fr} for s ∈ S, c ∈ Cs,

Ps,r = set of periods during which room r is not available in scenario s for s ∈ S, r ∈ Rs.

Most instructors want to teach consecutive periods whenever possible. To capture this, we

introduce the notion of distance between meeting times, which roughly measures how many periods

apart two meeting times are. For example, the distance between MWF3 and MWRF4 is 1, the

distance between TR8 (which consists of periods T1, T2, R1, and R2) and TR9 (which consists

of periods T3, T4, R3, and R4) is also 1, and the distance between MWF3 and TR9 is +∞. In

addition, to ensure that most instructors have at least one day a week to devote to research, class

preparation, and service duties, we set the distance between two meeting times that together result

in no free days as +∞. We denote these distances as

dm,m′ = distance between meeting times m and m′ for m,m′ ∈M.

3.2 Decision variables

We define the following decision variables:

xs,i,c,m =

1 if in scenario s, instructor i is assigned

to course c and meeting time m,

0 otherwise

for s ∈ S, i ∈ Is, c ∈ Cs,i,m ∈Ms,c,

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ys,i,c,m,p,r =

1 if in scenario s, instructor i is assigned

to course c and meeting time m,

with period p in room r,

0 otherwise

for s ∈ S, i ∈ Is, c ∈ Cs,i,m ∈Ms,c,

p ∈ Pm, r ∈ Rs,c,

y∗s,i,c,m,p,r =

1 if ys,i,c,m,p,r 6= ys0,i,c,m,p,r in scenario s,

0 otherwise

for s ∈ S, i ∈ Is, c ∈ Cs,i,

m ∈Ms,c, p ∈ Pm, r ∈ Rs,c,

us,i,c,m =

1 if

∑p∈Pm

∑r∈Rs,c

y∗s,i,c,m,p,r ≥ 1

0 otherwise

for s ∈ S, i ∈ Is, c ∈ Cs,i,

m ∈Ms,c,

zts,i,r =

1 if instructor i is assigned to

to room r in t in scenario s,

0 otherwise

for s ∈ S, i ∈ Is, r ∈ Rs,

t ∈ {AM, PM},

zs,i,r =

1 if instructor i is assigned to

room r in scenario s,

0 otherwise

for s ∈ S, i ∈ Is, r ∈ Rs.

Note that us,i,c,m = 1 if: (i) instructor i teaches course c during meeting time m in scenario s,

but not in the baseline scenario s0, (ii) instructor i teaches course c during meeting time m in the

baseline scenario s0, but not in scenario s, or (iii) instructor i teaches course c during meeting

time m in both scenarios s and s0, but not in the same room.

3.3 Constraints

We begin by describing some basic constraints that result in a physically feasible assignment of

instructors to courses and meeting times. For instance, each instructor must be assigned to the

right number of sections of each course in every scenario:

∑m∈Ms,c

xs,i,c,m = ts,i,c for s ∈ S, i ∈ Is, c ∈ Cs,i.

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Also, each instructor can only teach at most one course per period in every scenario:

∑c∈Cs,i

∑m∈Ms,c:p∈Pm

xs,i,c,m ≤ 1 for s ∈ S, i ∈ Is, p ∈ P.

Next, we describe some constraints that are particular to the scheduling of courses and instructors

in the Mathematics Department at USNA. First, we ensure that each instructor teaches consecutive

periods according to the distance metric d defined above in every scenario:

∑c∈Cs,i:m∈Ms,c

xs,i,c,m +∑

c′∈Cs,i:m′∈Ms,c′

xs,i,c′,m′ ≤ 1 for s ∈ S, i ∈ Is,m ∈M,

m′ ∈M : dm,m′ ≥ ts,i,∑c∈Cs,i:m∈Ms,c

xs,i,c,m ≤∑

c′∈Cs,i

∑m′∈Ms,c′ :dm,m′<ts,i

xs,i,c′,m′ for s ∈ S, i ∈ Is,m ∈M : ts,i ≥ 2.

Moreover, for the sake of convenience, we ensure that if an instructor teaches exactly two sections

of the same course in a given scenario, these sections should be in consecutive periods, either both

in the morning or both in the afternoon, but not one before lunch and one after lunch:

xs,i,c,m ≤∑

m′∈Ms,c∩MAM:dm,m′=1

xs,i,c,m′ for s ∈ S, i ∈ Is, c ∈ Cs,i : ts,i,c = 2,m ∈Ms,c ∩MAM,

xs,i,c,m ≤∑

m′∈Ms,c∩MPM:dm,m′=1

xs,i,c,m′ for s ∈ S, i ∈ Is, c ∈ Cs,i : ts,i,c = 2,m ∈Ms,c ∩MPM.

We also ensure that if the instructor teaches three or more sections of the same course in a given

scenario, these sections should be in consecutive periods:

xs,i,c,m ≤∑

m′∈Ms,c:dm,m′=1

xs,i,c,m′ for s ∈ S, i ∈ Is, c ∈ Cs,i : ts,i,c ≥ 3,m ∈Ms,c.

In addition, we constrain our assignments so that for each pair of conflict-free courses, there is at

least one section from one course that does not meet at the same time as any of the sections of the

other course:

∑i∈Is:c∈Cs,i

∑m∈Ms,c:p∈Pm

xs,i,c,m +∑

i∈Is:c′∈Cs,i

∑m∈Ms,c′ :p∈Pm

xs,i,c′,m ≤ ns,c + ns,c′ − 1

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for s ∈ S, (c, c′) ∈ Ks, p ∈ P.

We now describe constraints that assign rooms to the sections. Naturally, we ensure that each

section must be assigned one room for each of its scheduled periods in every scenario:

∑r∈Rs,c

ys,i,c,m,p,r = xs,i,c,m for s ∈ S, i ∈ Is, c ∈ Cs,i,m ∈Ms,c, p ∈ Pm.

We also ensure that each room can only be assigned at most one section per period in every scenario:

∑i∈Is

∑c∈Cs,i:r∈Rs,c

∑m∈Ms,c:p∈Pm

ys,i,c,m,p,r ≤ 1 for s ∈ S, r ∈ Rs, p ∈ P.

We ensure that each period group in a meeting time must be assigned to the same room in every

scenario:

ys,i,c,m,p,r = ys,i,c,m,p′,r for s ∈ S, i ∈ Is, c ∈ Cs,i,m ∈Ms,c, r ∈ Rs,c,

G ∈ P ∗m, p, p′ ∈ G : p 6= p′.

In addition, we ensure that sections cannot be assigned to rooms when they are not available in

every scenario:

ys,i,c,m,p,r = 0 for s ∈ S, i ∈ Is, c ∈ Cs,i,m ∈Ms,c, r ∈ Rs,c, p ∈ Pm ∩ Pr.

We also include constraints that ensure we properly track the rooms assigned to instructors – in

the morning, in the afternoon, and all day for every scenario:


∑m∈Ms,c:p∈Pm

ys,i,c,m,p,r ≤ zAMs,i,r for s ∈ S, i ∈ Is, p ∈ PAM, r ∈ Rs, (1)


∑m∈Ms,c:p∈Pm

ys,i,c,m,p,r ≤ zPMs,i,r for s ∈ S, i ∈ Is, p ∈ PPM, r ∈ Rs, (2)


∑m∈Ms,c:p∈Pm

ys,i,c,m,p,r ≤ zs,i,r for s ∈ S, i ∈ Is, p ∈ P, r ∈ Rs. (3)

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Finally, we account for every change from the baseline scenario to each of the disruption scenarios:

y∗s,i,c,m,p,r ≥ ys0,i,c,m,p,r − ys,i,c,m,p,r for s ∈ S, i ∈ Is, c ∈ Cs,i,m ∈Ms,c,


y∗s,i,c,m,p,r ≥ ys,i,c,m,p,r − ys0,i,c,m,p,r for s ∈ S, i ∈ Is, c ∈ Cs,i,m ∈Ms,c,


us,i,c,m ≥ y∗s,i,c,m,p,r for s ∈ S, i ∈ Is, c ∈ Cs,i,m ∈Ms,c,

p ∈ Pm, r ∈ Rs,c.

As alluded to earlier when defining the decision variables us,i,c,m, we define a change between the

baseline scenario and a particular disruption scenario as one of the following: (i) an instructor

teaches a particular course during a specific meeting time in the disruption scenario, but not in the

baseline scenario, (ii) an instructor teaches a particular course during a specific meeting time in

the baseline scenario, but not in the disruption scenario, or (iii) an instructor teaches a particular

course during a specific meeting time in both the baseline scenario and disruption scenario, but not

in the same room.

3.4 Objective function

First, we describe the various performance measures that constitute our objective function. The

total instructor preference score in scenario s ∈ S is

Preferencess =∑i∈Is

∑c∈Cs,i

∑m∈Ms,c

ps,i,c,mxs,i,c,m.

For each scenario s ∈ S, the total number of rooms assigned to instructors in the morning, afternoon,

and overall are

RoomsAMs =

∑i∈Is

∑r∈Rs

zAMs,i,r, RoomsPMs =

∑i∈Is

∑r∈Rs

zPMs,i,r, Roomss =∑i∈Is

∑r∈Rs

zs,i,r.

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The number of changes between the baseline scenario and a disruption scenario s ∈ S \ {s0} is

Differencess =∑i∈Is0

∑c∈Cs,i

∑m∈Ms,c

us,i,c,m.

We optimize a weighted sum of four objectives based on these performance measures:

(a) Maximize the total instructor preference score in the baseline scenario while minimizing the

number of rooms assigned to instructors in the baseline scenario. Minimizing the number of

rooms assigned to instructors is a way to achieve room stability, by attempting to have each

instructor teach all of his or her sections in the same room.

(b) Minimize the expected number of changes between the baseline scenario and the disruption

scenarios.

(c) Minimize the expected number of rooms assigned to instructors in the disruption scenarios.

As in (a), the goal of this part of the objective is to achieve room stability in the disruption

scenarios.

(d) Maximize the expected total instructor preference score in the disruption scenarios.

Let wa, wb, wc, and wd be the weights for objectives (a), (b), (c), and (d), respectively. The

objective of our stochastic integer program is

maximizewa

A

[πs0Preferencess0 −

(λAMs0 RoomsAM

s0 + λPMs0 RoomsPMs0 + λs0Roomss0

)]− wb

B

∑s∈S\{s0}

qs Differencess

− wc

C

∑s∈S\{s0}

qs

(λAMs RoomsAM

s + λPMs RoomsPMs + λsRoomss

)+wd

D

∑s∈S\{s0}

qsπsPreferencess,

where πs, λAMs , λPMs , and λs for s ∈ S are user-specified weights, and A, B, C, and D are scaling

constants chosen so that the ranges of objectives (a), (b), (c), and (d) are equal.

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3.5 Strengthening the formulation

In order to strengthen the formulation above in the hopes of faster running times, we add several

families of valid inequalities. The definitions of zs,i,r, zAMs,i,r and zPMs,i,r imply the inequalities below:

zAMs,i,r ≤ zs,i,r for s ∈ S, i ∈ Is, r ∈ Rs,

zPMs,i,r ≤ zs,i,r for s ∈ S, i ∈ Is, r ∈ Rs,

zAMs,i,r + zPMs,i,r ≥ zs,i,r for s ∈ S, i ∈ Is, r ∈ Rs.

In addition, the definitions of zs,i,r, zAMs,i,r, z

PMs,i,r, and ys,i,c,m,p,r imply the following family of inequali-

ties:

zs,i,r ≤∑

c∈Cs,i:r∈Rs,c

∑m∈Ms,c

∑p∈Pm

ys,i,c,m,p,r for s ∈ S, i ∈ Is, r ∈⋃

c∈Cs,i

Rs,c,

zAMs,i,r ≤


∑m∈Ms,c∩MAM

∑p∈Pm


c∈Cs,i

Rs,c,

zPMs,i,r ≤∑

c∈Cs,i:r∈Rs,c

∑m∈Ms,c∩MPM

∑p∈Pm


c∈Cs,i

Rs,c.

Finally, by adding constraints (1) over all periods p in PAM, and recognizing that each instructor

can teach at most

min

{ ∑c∈Cs,i

ts,i,c maxm∈Ms,c

|Pm|, 20

}

periods total in the morning, the following family of inequalities is valid:


∑m∈Ms,c

∑p∈Pm∩PAM

ys,i,c,m,p,r ≤ min

{ ∑c∈Cs,i

ts,i,c maxm∈Ms,c

|Pm|, 20

}zAMs,i,r

for s ∈ S, i ∈ Is, r ∈⋃

c∈Cs,i

Rs,c.

Similarly, starting from constraints (2) and (3), the following inequalities are also valid:


∑m∈Ms,c

∑p∈Pm∩PPM


{ ∑c∈Cs,i

ts,i,c maxm∈Ms,c

|Pm|, 10

}zPMs,i,r

15

Table 1. Test instance characteristics

Baseline scenario

Instance Number of Number of Number of Number of Number of Number ofinstructors courses sections rooms constraints variables

Fall AYE 2015 68 36 168 30 1,154,878 573,258Spring AYE 2015 66 39 159 30 1,036,803 508,632Fall AYE 2016 74 39 189 29 1,196,054 586,242Spring AYE 2016 68 40 160 29 1,284,108 648,452Fall AYE 2017 69 33 184 29 1,269,928 641,370Spring AYE 2017 62 37 144 30 923,652 449,167

Average Fall 70 36 180 29 1,206,953 600,290Average Spring 65 39 154 30 1,081,521 535,417

for s ∈ S, i ∈ Is, r ∈⋃

c∈Cs,i

Rs,c,


∑m∈Ms,c

∑p∈Pm


{ ∑c∈Cs,i

ts,i,c maxm∈Ms,c

|Pm|, 30

}zs,i,r

for s ∈ S, i ∈ Is, r ∈⋃

c∈Cs,i

Rs,c.

4 Experiment Setup

In order to test the functionality and performance of our model, we used available data from the six

most recent semesters at USNA: Fall AYE (Academic Year Ending in) 2015, Spring AYE 2015, Fall

AYE 2016, Spring AYE 2016, Fall AYE 2017, and Spring AYE 2017. Each test instance consists of

a baseline data set and two potential disruption scenario data sets. Each data set consists of a set

of instructors, courses, meeting times, and rooms for that particular scenario as well as other data

as described in Section 3.1 of this paper. When possible, we created disruption scenarios based on

actual information available at the time of the instance: for example, a disruption scenario might be

based on the fact that an instructor was a likely candidate for a change in duty station and would

not be available to teach in the upcoming semester. As seen in Table 1, the size of these problems

are rather large.

To illustrate how the disruption scenarios were formed, we describe the Fall AYE 2017 instance

in detail. We describe the rest of the instances in Appendix A. In the first disruption scenario of

this test instance, an instructor (Instructor 9) is added to the faculty at the last minute to teach

one section each of two courses (SM121 and SM122) which previously had 30 and 15 sections in the

16

baseline data set respectively. The department chair decides to take one section of SM121 from one

instructor (Instructor 2) and one section of SM122 from another instructor (Instructor 71) so that

they are now teaching a total of two sections each instead of three. The number of total sections

of each course thus remains the same. In the second disruption scenario of this test instance, an

instructor (Instructor 94) who was supposed to teach three sections of a course (SM121) with 30

sections must suddenly depart USNA. The department chair decides to give another instructor

(Instructor 64) an additional section of this course so he is now teaching a total of three sections

instead of two. Furthermore, the department chair decreases the overall number of sections from 30

down to 28, slightly increasing the number of students per section of this course.

To solve these instances, we used Gurobi 7.0.1 with its Python API. These instances were solved

on a computer with an 8-core 2.6 GHz Intel Xeon E5-2560 v2 CPU and 64 GB RAM, running

Windows Server 2008. For each of the test instances, we created a warm start solution by running

the model for the baseline and disruption scenarios individually and combining them. We allowed a

total of three hours to find a warm start solution: one hour each for the baseline and disruption

scenarios. We stopped the solver after 24 hours for each test instance. Therefore, the results below

are based on three hours of finding warm start solutions and then 24 hours of solving the full model

for a total of 27 hours of CPU time.

For the user-specified weights πs, λAMs , λPMs , and λs for s ∈ S in the objective function, we used

the following values for every instance:

πs = 2, λAMs = 2, λPMs = 2, λs = 1 for s ∈ S.

These weights were chosen based on our previous experience with finding desirable schedules for a

single (i.e. the baseline) scenario.

5 Results

We began testing the model by running it for the extreme cases of the objective function, setting the

weight for each part (a) through (d) to 1 and the rest 0. For these extreme cases, we assumed that

each scenario was equally likely: i.e., q0 = 1/3, q1 = 1/3, q2 = 1/3. Due to the nature of USNA’s

17

Table 2. Performance – Baseline only

Instance Gap after Time of best Time when gap24 hours solution (s) less than 5% (s)

Fall AYE 2015 1.72% 1519 127Spring AYE 2015 0.56% 42 91Fall AYE 2016 6.37% 26008 > 86400Spring AYE 2016 0.00% 18064 96Fall AYE 2017 3.74% 37663 16584Spring AYE 2017 0.00% 28 56

Average Fall 3.95% 21730 > 34370Average Spring 0.19% 6045 81

academic curriculum, the Fall semesters are often more difficult to schedule because of the higher

number of sections as seen in Table 1. In particular, a large number of Calculus I and II sections

must be offered in the Fall. These courses also have more restrictions on meeting times due to their

length (4 credit hours) and rooms due to their larger section sizes. In light of this disparity between

the Fall and Spring semesters, it is useful to look at the results of these tests for the Fall and Spring

semesters separately.

5.1 Baseline Only

We first tested the objective with only performance measures related to the baseline scenario: i.e.,

wa = 1, wb = 0, wc = 0, and wd = 0. As seen in Table 2, the solver is able to find a “good” solution

(i.e., a solution that is less than 5% away from optimal) in under five minutes in four of the six

instances. In fact, for two of the Spring semester instances, provably optimal solutions were found.

The satisfaction of every instructor is a value between 0 and 1 that represents proportionally how

well their preferences were met. Table 3 shows the average satisfaction of instructors in the baseline

and disruption scenarios. In the first three instances, instructor preference data was not collected so

there is no meaningful value to display. Table 4 reports the percentage of instructors with multiple

rooms in the morning, afternoon, and overall; these percentages are reported as averages over the

three scenarios. This table also shows the proportion of instructors who can expect their schedule

to change from the baseline scenario to either of the disruption scenarios. Under this “baseline

only” objective, that proportion is 100% for every instance. This behavior is expected since there is

nothing in the model driving the solution to consider these schedule changes. By using a model

without regard for potential disruptions, a complete rescheduling may be required for even simple

18

Table 3. Satisfaction metrics – Baseline only

Instance Baseline average Scenario 1 Scenario 2satisfaction of average satisfaction of average satisfaction ofeach instructor each instructor each instructor

Fall AYE 2015 N/A N/A N/ASpring AYE 2015 N/A N/A N/AFall AYE 2016 N/A N/A N/ASpring AYE 2016 0.997 0.867 0.895Fall AYE 2017 0.936 0.946 0.944Spring AYE 2017 0.979 0.966 0.971

Average Fall 0.936 0.946 0.944Average Spring 0.988 0.916 0.933

Table 4. Room and disruption metrics – Baseline only

Instance Instructors with Instructors with Instructors with Instructors whomore than more than more than could experience

1 AM room 1 PM room 1 room some disruption

Fall AYE 2015 0.67% 0.00% 3.47% 100.00%Spring AYE 2015 0.00% 0.00% 0.00% 100.00%Fall AYE 2016 15.42% 5.18% 18.21% 100.00%Spring AYE 2016 43.74% 37.57% 54.23% 100.00%Fall AYE 2017 11.88% 7.73% 21.74% 100.00%Spring AYE 2017 0.00% 0.00% 0.00% 100.00%

Average Fall 9.32% 4.30% 14.47% 100.00%Average Spring 14.58% 12.52% 18.08% 100.00%

disruptions.

5.2 Differences Only

The second objective we tested was to only minimize the number of changes between the baseline

scenario and each disruption scenario: i.e., wa = 0, wb = 1, wc = 0, and wd = 0. Table 5 shows the

performance results for this objective. On average, the optimality gap for the Spring semesters is

smaller than the optimality gap for the Fall semesters; however, none of the instances close the gap

to less than 5% in 24 hours.

Tables 6 and 7 show the satisfaction, room, and disruption metrics under this “differences

only” objective. On average, the number of instructors with multiple rooms was lower in the Fall

semesters than the Spring. The solutions we found performed as expected with this particular

weighting situation, where all but one of the instances have less than 6% of instructors expecting a

schedule change if one of the potential disruption scenarios were to occur, as compared to 100% in

all instances for the “baseline only” objective (Table 4).

19

Table 5. Performance – Differences only


Fall AYE 2015 66.67% 36750 > 86400Spring AYE 2015 50.00% 261 > 86400Fall AYE 2016 99.99% 10820 > 86400Spring AYE 2016 47.92% 282 > 86400Fall AYE 2017 56.23% 78120 > 86400Spring AYE 2017 64.25% 168 > 86400

Average Fall 74.29% 41896 > 86400Average Spring 54.06% 237 > 86400

Table 6. Satisfaction metrics – Differences only




Table 7. Room and disruption metrics – Differences only





20

Table 8. Performance – Rooms in disruption scenarios only


Fall AYE 2015 8.67% 37196 > 86400Spring AYE 2015 1.94% 43 86400Fall AYE 2016 11.59% 77998 > 86400Spring AYE 2016 3.09% 73797 24278Fall AYE 2017 22.07% 27878 > 86400Spring AYE 2017 0.00% 1742 1394


Table 9. Satisfaction metrics – Rooms in disruption scenarios only




5.3 Rooms in Disruption Scenarios Only

The third objective we tested was to only minimize the expected number of rooms assigned to the

instructors in the disruption scenarios: i.e., wa = 0, wb = 0, wc = 1, and wd = 0. Table 8 shows

that for this objective, the time for solutions to have an optimality gap less than 5% was on average

10.4 hours for Spring semester instances, and more than 24 hours for Fall semester instances. For

this objective, we were able to find a provably optimal solution for Spring AYE 2017 in the allotted

time. This is not entirely surprising, since as seen in Table 1, this instance is much smaller than the

other five.

Comparing the results in Table 10 with those for the “differences only” objective in Table 7, we

see that the percentages of instructors with more than one room are much lower; in the Fall the

overall percentage has been reduced by more than a factor of three, and in the Spring the overall

percentage has been reduced by more than a factor of five. However, once again it is apparent that

when the disruption portion of the objective function is ignored, it can be expected that 100% of

instructors will have a schedule change if one of the disruption scenarios were to occur.

21

Table 10. Room and disruption metrics – Rooms in disruption scenarios only





Table 11. Performance – Preferences in disruption scenarios only


Fall AYE 2015 0.00% 52 89Spring AYE 2015 0.00% 41 65Fall AYE 2016 0.00% 45 82Spring AYE 2016 0.00% 123 104Fall AYE 2017 0.00% 787 108Spring AYE 2017 0.00% 38 38

Average Fall 0.00% 295 93Average Spring 0.00% 67 69

5.4 Preferences in Disruption Scenarios Only

The fourth objective we tested was to maximize each instructor’s preferences in the disruption

scenarios: i.e., wa = 0, wb = 0, wc = 0, and wd = 1. As Table 11 shows, the solver was able to solve

every instance to provable optimality, and all of the instances reached optimality gaps under 5% in

less than two minutes with an average time of 1.35 minutes across instances regardless of semester.

As expected, the average instructor satisfaction values are higher in the disruption scenarios than

in the baseline scenario as seen in Table 12. Just as seen in the results for the “baseline only” and

“rooms in disruption scenarios only” objectives described in Sections 5.1 and 5.3 respectively, Table

13 shows that when we ignore the number of changes from the baseline schedule to the disruption

scenario schedules in the objective, 100% of instructors can expect to have a schedule change if one

of those disruptions were to occur.

22

Table 12. Satisfaction metrics – Preferences in disruption scenarios only




Table 13. Room and disruption metrics – Preferences in disruption scenarios only





5.5 Difficulty and Importance of Minimizing Disruptions

These tests on the extreme cases of the objective function have shown that the model with the

“baseline only,” “rooms in disruption scenarios only,” and “preferences in disruption scenarios only”

objectives can be computationally tractable on their own. The model with the “differences only”

objective, however, took significantly longer to make progress on the optimality gap, and for all

instances, the solver was not able to converge to any “good” solutions. However, when the changes

between the baseline scenario and disruption scenarios were not taken into consideration in the

objective, the obtained solutions had 100% of the instructors experience a change to their schedule if

a disruption were to occur. This means using essentially a completely new schedule to accommodate

the new scenario as opposed to having a solution that disrupts a minimal number of instructors.

5.6 Equal Weights, Scenarios Equally Likely

Our next set of tests weighted the four parts of the overall objective function equally: i.e., wa = 1,

wb = 1, wc = 1, and wd = 1. Furthermore, we continued to assign equal probabilities to each

23

Table 14. Performance – Equal weights, scenarios equally likely


Fall AYE 2015 5.63% 86400 > 86400Spring AYE 2015 1.06% 86400 561Fall AYE 2016 25.06% 86400 > 86400Spring AYE 2016 1.34% 86400 754Fall AYE 2017 5.98% 30939 > 86400Spring AYE 2017 0.16% 84764 620


Table 15. Instructor satisfaction – Equal weights, scenarios equally likely




scenario: i.e., q0 = 1/3, q1 = 1/3, q2 = 1/3. Table 14 shows that the time to reach an optimality

gap less than 5% takes more than 24 hours in the Fall semester instances. However, when compared

to the results for the “differences only” objective described in Section 5.2 (Table 5), this weighting

of the objective produces better results in much less time. This possibly implies that attempting

to maximize instructor preferences and minimize the number of rooms per instructor in every

scenario actually aids the solver in finding solutions with minimal changes between the baseline and

disruption scenarios.

Table 15 summarizes the performance of this set of test runs with regards to instructor preferences.

Compared to the corresponding results for the “baseline only” and “preferences in disruption scenarios

only” objectives (Tables 3 and 12, respectively), these values are only marginally worse (with the

exception of the baseline average satisfaction for the Fall semester instances), even though the solver

was seeking to optimize the other portions of the overall objective function simultaneously.

Table 16 summarizes the performance of these test runs with regards to room usage and the

number of changes between the baseline and disruption scenarios. Compared to the room usage

metrics for the “rooms in disruption scenarios only” objective (Table 10), the room usage metrics

24

Table 16. Room and disruption metrics – Equal weights, scenarios equally likely





from this set of test runs are largely worse, seemingly at the expense of lowering the percentages

of instructors who would experience a disruption if one of the disruption scenarios were to occur.

In every test instance except for Fall AYE 2016, the number of instructors who could be forced

to change their schedule significantly improved. Recall from Table 1 that Fall AYE 2016 has the

largest number of instructors and sections to schedule. It is possible that given more time, solving

this instance would also have resulted in a schedule with a lower number of instructors who could

experience some disruption.

5.7 Equal Weights, Baseline Scenario Likely

We further tested the model’s performance by continuing to equally weight the four parts of the

overall objective function (i.e., wa = 1, wb = 1, wc = 1, and wd = 1), but changing the probabilities

of each scenario. In this set of tests, we assumed the baseline scenario is twice as likely to occur as

either of the disruption scenarios combined: i.e., q0 = 2/3, q1 = 1/6, q2 = 1/6. The performance

of the model under these conditions is displayed in Table 17. When compared to the performance

observed for the “equal weights, scenarios equally likely” objective described in Section 5.6 (Table

14), the optimality gaps are slightly worse on the whole, but the differences are not significant. We

note that the optimality gap after 24 hours for Fall AYE 2016 is in fact smaller, but the difference

is slight.

Table 18 shows the average satisfaction of instructors in each scenario for this objective. Com-

paring these values to the results for the “equal weights, scenarios equally likely” objective found in

Table 15, the average satisfaction values are identical to the second decimal place in all instances.

In addition, the satisfaction values from this set of tests are close to the “maximum” possible values

25

Table 17. Performance – Equal weights, baseline scenario likely


Fall AYE 2015 5.81% 86400 > 86400Spring AYE 2015 2.62% 86400 1229Fall AYE 2016 24.50% 42476 > 86400Spring AYE 2016 7.48% 677 > 86400Fall AYE 2017 18.72% 1215 > 86400Spring AYE 2017 0.82% 86400 677

Average Fall 16.34% 43364 > 86400Average Spring 3.64% 57826 > 29435

Table 18. Instructor satisfaction – Equal weights, baseline scenario likely




found when optimizing only the instructors’ preferences in the baseline scenario (Table 3) and only

the instructors’ preferences in the disruption scenarios (Table 12).

The percentages of instructors with multiple rooms as well as the percentages of instructors who

can expect a schedule change in a disruption scenario are displayed in Table 19. Compared with the

“equal weights, scenarios equally likely” objective (Table 16), the room metrics are neither uniformly

better nor worse. On the other hand, the percentages of instructors who can expect a schedule

change in a disruption scenario generally do not compare favorably against those for the “equal

weights, scenarios equally likely” objective. These results might indicate that giving more credence

to the baseline scenario could result in a higher percentage of instructors experiencing a schedule

change if a disruption scenario occurs.

5.8 Equal Weights, Disruption Scenarios Likely

We continued to test the model’s performance by again equally weighting the four different parts

of the overall objective function (i.e., wa = 1, wb = 1, wc = 1, and wd = 1), but varying the

probabilities of each scenario. In this set of tests, we assumed the disruption scenarios were twice as

26

Table 19. Room and disruption metrics – Equal weights, baseline scenario likely





Table 20. Performance – Equal weights, disruption scenarios likely


Fall AYE 2015 4.08% 4352 4352Spring AYE 2015 1.03% 8109 7584Fall AYE 2016 25.39% 26221 > 86400Spring AYE 2016 1.09% 86400 3906Fall AYE 2017 7.95% 14142 > 86400Spring AYE 2017 0.28% 77565 862


likely to occur as the baseline scenario: i.e., q0 = 1/5, q1 = 2/5, q2 = 2/5. We found from these

tests that the optimality gaps and times to reach a “good” solution (Table 20) were similar to

those for the “equal weights, scenarios equally likely” objective tested in Section 5.6 (Table 14)

and largely better than those for the “equal weights, baseline scenario likely” objective tested in

Section 5.7 (Table 17). These results seem to indicate that varying the probabilities assigned to

each potential scenario could have a profound effect on the time it takes to close the optimality gap

and find solutions.

Furthermore, Table 21 shows that the average satisfaction of instructors in the Fall and Spring

semesters is extremely close to those for the “equal weights, scenarios equally likely” and “equal

weights, baseline scenario likely” objectives (Tables 15 and 18, respectively). It appears from these

results that finding schedules with close to maximum instructor preferences can be done quickly.

The results on room usage are mixed. Looking at the average values in Table 22, the results

are similar but slightly worse across the board than those for the “equal weights, scenarios equally

likely” objective (Table 16). On the other hand, these results are sometimes better and sometimes

worse than those for the “equal weights, baseline scenarios likely” objective (Table 19).

27

Table 21. Instructor satisfaction – Equal weights, disruption scenarios likely




Table 22. Room and disruption metrics – Equal weights, disruption scenarios likely





The percentages of instructors who can expect a schedule change given a disruption scenario

occurs are the same or better for this objective compared to the “equal weights, baseline scenario

likely” objective, as seen by comparing the values in Tables 22 and 19. On the other hand, the

percentages of instructors who can expect a schedule change in a disruption scenario for this objective

are sometimes better and sometimes worse than those for “equal weights, scenarios equally likely”

objective, as seen by comparing the values in Tables 22 and 16.

5.9 Importance and Difficulty of Timeliness

While Sections 5.6, 5.7, and 5.8 have shown that “good” feasible schedules resilient to known

potential disruptive events can be constructed by weighing all parts of the objective function equally,

it has also made clear that in general, these problems require a long time to obtain these schedules.

It would be ideal if the decision maker – in this case, the department chair – could quickly obtain

solutions for instances with more than two disruption scenarios.

However, each additional disruption scenario adds a large number of rows and columns to

28

the integer linear program described in Section 3. For example, the Fall AYE 2016 test instance

has 163,151 rows and 120,009 columns with only the baseline scenario, but with each additional

disruption scenario, the model for this test instance increases by approximately 500,000 rows and

230,000 columns. As previously discussed in Sections 5.2 and 5.5, minimizing the number of

changes between the baseline and disruption scenarios appears to be the most difficult portion of

the objective. Furthermore, as demonstrated in Sections 5.6, 5.7, and 5.8, optimizing this part

the objective function also seems to worsen the solution quality of the other three portions of the

objective function.

6 Future Work

Expanding on Section 5.9, the biggest issue with our stochastic integer program is that it is difficult

to solve in a reasonable amount of time. Our ultimate goal is to enable decision makers to efficiently

schedule instructors while planning for several potential likely disruptions. Possessing the ability

to compute several viable options quickly for consideration gives a decision maker control in a

stressful situation. Therefore, a key direction for future research is to discover a way to reach “good”

solutions quicker. One possible approach would be to reformulate the model in a way that enables

solvers to take advantage of any special structure in the problem. Another possible approach could

be to investigate decomposition strategies in order to reduce the size of the model that the solver

must work with.

The ideas contained in this paper about finding resilient schedules in the face of potential

disruptions can naturally be extended past university timetabling into the military and civilian

business sectors. It would be ideal for ships, submarines, aircraft, and soldiers to deploy on schedules

that are resilient to several known potential disruptions. As for civilian use, many businesses would

benefit from the ability to minimize the costs resulting from having to reschedule the transportation

of materials, goods, and services after a disruption occurs that forces major changes.

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A Description of Test Instances

A.1 Fall AYE 2015

In the first disruption scenario of this test instance, an instructor (Instructor 16) who was teaching

two sections of a course (SM122) with ten sections must suddenly depart USNA. The department

chair decides to give another instructor (Instructor 77) an additional section of this course so she is

now teaching a total of three sections instead of two. Furthermore, the department chair decreases

the overall number of sections from ten down to nine, slightly increasing the number of students per

section of this course.

31

In the second disruption scenario of this test instance, an instructor (Instructor 21) who was

teaching three sections of a course (SM223) with 20 sections must suddenly depart USNA. The

department chair decides to give two other instructors (Instructors 19 and 50) an additional section

of this course so they are now teaching a total of three sections instead of two. Furthermore, the

department chair decreases the overall number of sections from 20 down to 19, slightly increasing

the number of students per section of this course.

A.2 Spring AYE 2015


three sections of a course (SM122) with 29 sections must suddenly depart USNA. The department

chair decides to give another instructor (Instructor 78) an additional section of this course so he is

now teaching a total of three sections instead of two. Furthermore, the department chair decreases

the overall number of sections from 29 down to 27, slightly increasing the number of students per

section of this course.



department chair decides to give another instructor (Instructor 78) an additional section of this

course so he is now teaching a total of three sections instead of two. Furthermore, the department

chair decreases the overall number of sections from 29 down to 27, slightly increasing the number of

students per section of this course.

A.3 Fall AYE 2016


one section of a course (SM261) with nine sections must suddenly depart USNA. The department

chair decides to give another instructor (Instructor 59) an additional section of this course so he

is now teaching a total of two sections instead of one. The number of total sections of this course

remains the same.




32

course so he is now teaching a total of three sections instead of two. Furthermore, the department



A.4 Spring AYE 2016


two sections of a course (SM221) with ten sections must suddenly depart USNA. The department

chair decides to give two other instructors (Instructors 53 and 60) an additional section of this

course so they are now each teaching a total of three sections instead of two. Consequently, the

number of sections for this course remains the same.


teaching two sections of a course (SA305) with six sections and one section of another course

(SM212P) with nine sections must suddenly depart USNA. The department chair decides to give

another instructor (Instructor 27) an additional section of the first course (SA305) and another

instructor (Instructor 8) an additional section of the second course (SM212P) so they are now each

teaching a total of three sections instead of two. Furthermore, the department chair decreases

the overall number of sections for SA305 from six down to five, slightly increasing the number of


A.5 Spring AYE 2017


two sections of a course (SM212P) with seven sections must suddenly depart USNA. The department

chair decides to give two other instructors (Instructors 6 and 35) an additional section of this course

so Instructor 6 is now teaching a total of three sections instead of two and Instructor 35 is teaching

a total of two sections instead of one. Consequently, the number of sections for this course remains

the same.


teaching three sections of a course (SM221) with 17 sections is no longer able to teach. The


course so she is now teaching a total of three sections instead of two. Furthermore, the department

33



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