An Integer Linear Programming Solution to the Telescope Network

Scheduling Problem

Sotiria Lampoudi

1, ∗

, Eric Saunders

and Jason Eastman

Earth Research Institute, 6832 Ellison Hall, University of California, Santa Barbara, California 93106-3060, U.S.A.

Las Cumbres Observatory Global Telescope Network, 6740 Cortona Dr., Suite 102, Goleta, California 93117, U.S.A.

Keywords:

Scheduling, Integer Linear Programming, Astronomy.

Abstract:

Telescope networks are gaining traction due to their promise of higher resource utilization than single tele-

scopes and as enablers of novel astronomical observation modes. However, as telescope network sizes in-

crease, the possibility of scheduling them completely or even semi-manually disappears. In an earlier paper,

a step towards software telescope scheduling was made with the speciﬁcation of the Reservation formalism,

through the use of which astronomers can express their complex observation needs and preferences. In this

paper we build on that work. We present a solution to the discretized version of the problem of scheduling

a telescope network. We derive a solvable integer linear programming (ILP) model based on the Reservation

formalism. We show computational results verifying its correctness, and conﬁrm that our Gurobi-based im-

plementation can address problems of realistic size. Finally, we extend the ILP model to also handle the novel

observation requests that can be speciﬁed using the more advanced Compound Reservation formalism.

1 INTRODUCTION

Telescope networks have the potential to enable in-

creased resource utilization and novel observation

modalities. Historically, observation requests for sin-

gle telescopes were made in human-readable form,

and any conﬂicts between requests were resolved by

a person, often working directly with the request-

ing astronomer in a tight feedback loop. This type

of manual scheduling is not practical for general-

purpose telescope networks containing more than a

small number of telescopes, due to the large number

of competing requests received for a typical schedul-

ing interval, and the added complexity of choosing

among multiple resources. Further, in networks wish-

ing to enable the study of fast transient phenomena,

manual scheduling is infeasible due to the need for

near-real-time re-scheduling responsiveness, which is

necessary to achieve these scientiﬁc objectives.

Las Cumbres Observatory Global Telescope

(LCOGT) is a robotic telescope network that cur-

rently (in September 2014) includes two 2m and nine

1m telescopes, with plans to add a number of 0.4m

telescopes in the near future (Brown et al., 2013). The

∗

Author to whom correspondence should be addressed.

Current Afﬁliation: Liquid Robotics Inc., 1329 Moffet Park

Dr, Sunnyvale, California, 94089, USA

telescopes are robotically controlled and connected

via the Internet to LCOGT headquarters in California.

Professional astronomers, citizen scientists and edu-

cators can apply for access to the network on a bian-

nual semester basis. The scientiﬁc merit of proposals

is assessed by a Time Allocation Committee (TAC),

which assigns each accepted project some amount of

total time on the network and a scalar per-unit-time

priority. Each project then requests speciﬁc astronom-

ical observations to be conducted, not exceeding the

project’s total time allocation. The motivation for the

contributions of this paper is the design and deploy-

ment of a software telescope network scheduler for

the LCOGT network (Saunders et al., 2014).

A software solution stack for scheduling a tele-

scope network has three components: (a) methods al-

lowing the network’s users to make observation re-

quests, (b) a scheduling algorithm capable of resolv-

ing conﬂicts between users’ requests to produce a vi-

able schedule, and (c) additional control logic that

adds awareness of the state of the network, manages

schedule re-calculation (due to new input, weather,

network outages and other reasons), and deals with re-

quest completion. In (Lampoudi and Saunders, 2013)

a formalism was developed for allowing astronomers

to express their complex observation needs and pref-

erences in an unambiguous, machine-readable way

331

Lampoudi S., Saunders E. and Eastman J..

An Integer Linear Programming Solution to the Telescope Network Scheduling Problem.

DOI: 10.5220/0005207003310337

In Proceedings of the International Conference on Operations Research and Enterprise Systems (ICORES-2015), pages 331-337

ISBN: 978-989-758-075-8

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

that would allow software to arbitrate among them.

The contribution of the present paper is a solvable

integer linear programming (ILP) model for the of-

ﬂine, discretized version of the scheduling problem

expressed by this formalism.

This paper is organized as follows: in section 2 we

present the Reservation formalism thatis used to com-

municate astronomers’ requests to the scheduler. Sec-

tion 3 presents the solvable ILP model of the schedul-

ing problem. Section 4 presents computational results

conﬁrming the correctness and evaluating the perfor-

mance of the ILP solution. Section 5 extends the

model of section 3 to include the more complex Com-

pound Reservations possible in telescope networks.

Related and future work are discussed in the ﬁnal sec-

tion.

2 THE RESERVATION

FORMALISM

A request for an observation on a telescope network

contains two types of information: (a) observation-

speciﬁc information about the target of the observa-

tion, which instrument (i.e. camera or spectrograph)

to use, the exposure settings, etc, and (b) information

about when, where and for how long the observation

can occur, based on astronomical factors that can be

calculated a priori, such as visibility of the target dur-

ing local night, lack of interference by the moon, and

so on. Information contained in (a) is necessary so

that a robotic telescope or a human operator can carry

out the observation. But it is not necessary for choos-

ing which of many requests to fulﬁll, when and where.

This task, the scheduling of the telescope network, is

performed solely on the basis of the information con-

tained in (b). In our work the information contained

in (b) is encapsulated in a “Reservation”: a represen-

tation of a request for exclusive access to a resource

during one contiguous chunk of time at one or more

speciﬁc times in the future.

As formally speciﬁed in (Lampoudi and Saunders,

2013), a Reservation R is a 4-tuple (d, p, t, W) where:

• d is a scalar duration,

• p is a scalar priority,

• t is a resource (i.e. telescope),

• W is a list of “windows of opportunity”

Windows of opportunity specify the times during

which the observation is possible; that is, the entire

observation must ﬁt within a single window of oppor-

tunity.

In the vocabulary of a telescope network, a Reser-

vation R is a request by a project with priority p for

exclusive access of duration d to resourcet during one

of the windows in the list W.

Typically, however, an observation can be carried

out on one of many telescopes. According to the

above deﬁnition of Reservation, a request with multi-

ple resources (and corresponding windows of oppor-

tunity for each resource) will result in multiple “or”-

ed Reservations. Because this is such a common oc-

currence, for compactness and with no loss of gener-

ality, we simply extend the above deﬁnition to merge

multi-resourceReservations into a singleReservation.

That is, each Reservation is now permitted to include

a list of resources, instead of a single resource, and

windows of opportunity become subscripted by re-

source. The resulting deﬁnition of Reservation is the

4-tuple (d, p, T, W) where:

• d is a scalar duration,

• p is a scalar priority,

• T is a list of resources (i.e. telescopes), t

∈ T,

• W

∈ W are lists of “windows of opportunity”,

where list W

is the list of windows correspond-

ing to t

3 THE INTEGER LINEAR

PROGRAMMING MODEL

Given a list of Reservations we wish to ﬁnd a maxi-

mum priority subset, the subset of scheduled reserva-

tions, and an assignment of a speciﬁc resource and

start time for each scheduled reservation, such that

there are no overlaps between scheduled reservations.

This is an ofﬂine, multi-resource, interval schedul-

ing problem with slack. The slack is introduced by the

fact that windows can be longer than the duration of a

Reservation.

Our ILP modelis inspired by a similar approach to

a problem in truck scheduling (Lee et al., 2012). We

ﬁrst discretize time into “slots”, which can be of uni-

form or non-uniform lengths. Each slot is deﬁned by

the resource to which it corresponds and a (start time,

duration) or (start time, end time) tuple that we abbre-

viate as (timeslice) in the text that follows. We then

express the non-overlap constraints as linear inequal-

ities. Boolean variables are used to select between

the possible starting slots for each Reservation, and

the sum of the priorities of scheduled Reservations is

maximised.

The resulting ILP resembles a weighted maxi-

mum set packing problem, which is known to be NP-

complete (Garey and Johnson, 1990).

Speciﬁcally, our model formulation is as follows:

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

332

3.1 Parameters

• I: set of reservations

• T: set of slots, each speciﬁed as a tuple: (resource,

timeslice)

• S

: set of possible start slots for reservation i

• a

ikt

= 1 if starting reservation i at k ∈ S

means

that it will occupy slot t; 0 otherwise

• p

: priority of reservation i

3.2 Decision Variables

= 1 if reservation i starts at k ∈ S

; 0 otherwise

3.3 Objectives

Maximise the sum of the priorities of scheduled

Reservations.

max

∑

i∈I

∑

k∈S

(1)

3.4 Constraints

No reservation should be scheduled for more than one

start.

∑

k∈S

≤ 1, ∀i ∈ I (2)

No more than one reservation should be scheduled

in each slot.

∑

i∈I

∑

k∈S

ikt

≤ 1, ∀t ∈ T (3)

Decision variable must be binary.

∈ {0, 1}, ∀i ∈ I, ∀k ∈ S

(4)

The inequality constraint matrix contains |I| + |T|

rows, i.e. the sum of the number of reservations and

time slots.

4 COMPUTATIONAL RESULTS

4.1 Correctness

The ILP modelwe have just describedis implemented

in a software component called the “scheduling ker-

nel”. This forms the core of the software stack re-

sponsible for managing the telescope network. Our

current implementation of the kernel is in Python, and

uses the Gurobi solver (Gurobi Optimization, 2014).

The input to the kernel is a list of Reservation objects

which are direct implementations of the concept of a

Reservation. The kernel translates this list of Reserva-

tions into a model description that is passed to Gurobi,

and invokes the solver. When the solver completes its

run, the kernel uses the output to modify the Reserva-

tion objects to reﬂect whether or not they were sched-

uled, and, if they were, at what resource and starting

time.

As is common software engineering practice,

the kernel was unit tested using a variety of small

scheduling scenarios that can be solved manually. But

it was also desirable to validate that the kernel’s per-

formance is what one would expect on a larger scale.

This type of validation can be achieved by us-

ing large scheduling scenarios for which the optimal

scheduling outcome is known a priori, due to the way

in which they were constructed. When these scenarios

are run through the scheduling kernel, it is possible to

compare the experimental performance of the kernel

to this theoretically optimal and achievable outcome.

The “subscription rate” of the network is deﬁned

as the ratio of the total amount of time requested over

the total amount of time available for scheduling. The

total amount of time requested is the sum of the dura-

tions of all the reservations submitted to the scheduler.

The total amount of time available for scheduling is

the time covered by the union of all windows of op-

portunity.

The subscription rate is a property of the input

to the scheduler. To characterize the outcome of a

scheduling run we need a corresponding performance

metric. This is the “scheduled/requested” (s/r) ratio,

that is the ratio of time scheduled (the sum of the du-

rations of all scheduled reservations) over the ratio of

time requested (the sum of the durations of all reser-

vations that were submitted).

For subscription rates below 100% (“undersub-

scribed”), it is possible to construct problem instances

for which the optimal s/r ratio of 100% is achievable.

Given those problem instances, a well-functioning

scheduler should achieve s/r of nearly 100%.

For subscription rates above 100% (“oversub-

scribed”), it is possible to construct problem instances

for which the optimal s/r ratio is known. Speciﬁcally,

we constructed cases for which the optimal s/r was

the inverse of the subscription rate – another way to

express that is to say that utilization was 100%. On

those problem instances a well-functioning scheduler

should achieve close to this theoretically optimal s/r.

To produce experimental data that can be com-

pared to the optimal values we conducted 15 simu-

lations of a telescope network, spanning subscription

rates between 10% and 150%. The size of the net-

AnIntegerLinearProgrammingSolutiontotheTelescopeNetworkSchedulingProblem

333

work was chosen to be nine telescopes, and the time

slices on all telescopes were set to be 5 minutes long.

For each individual simulation run, i.e., for each

subscription rate value, we generated an ensemble of

hundreds of reservations for which we knew, by con-

struction, that an optimal or close to optimal schedule

was feasible. Then the scheduling problem was made

artiﬁcially harder in two ways: ﬁrst, all reservations

were assigned the same 24 hour window of opportu-

nity; second, all were also randomly assigned to be

possible on additional resources. This had the effect

of introducing large amounts of slack and seeming

contention to the problem.

0 0.5 1 1.5

0.2

0.4

0.6

0.8

subscription rate

scheduled/requested

experimental

theoretical

Figure 1: The scheduling kernel is able to achieve optimal

performance in cases of undersubscription, and performs

close to optimally in cases of oversubscription. The dis-

crepancy is accounted for by the time wasted due to the dis-

cretization of time into slots.

Figure 1 shows that the kernel was able to match

theoretically optimal performance very closely. The

small mismatch that begins to occur around 100%

subscription can be attributed to the small amount of

time that is wasted due to the discretization of time

into slots. In the case of the artiﬁcial scenarios used

for this test, this wasted amount of time can be calcu-

lated.

In real-world scenarios the impact of that wasted

time is an open problem. The amount of wasted time

depends on (a) the distribution of Reservation dura-

tions, and (b) the choice of slot sizes. Clearly, smaller

slots decrease the expected amount of wasted time,

but increase the size of the ILP optimization problem.

We plan to study this effect in simulation, by model-

ing the distribution of durations so that we can gener-

ate appropriate synthetic workloads, and empirically,

by running the kernel on real inputs but, in “parallel”,

using different hypothetical slot sizes.

4.2 Performance

To give an idea of what currently constitutes typical

and exceptional operating conditions for the LCOGT

scheduling kernel, in this section we report timings

from two real-world runs. The ﬁrst is a randomly cho-

sen typical recent run (date: 2014-08-29). The second

is the largest run that occurred since the scheduler be-

gan ofﬁcial operations in May 2014 (date: 2014-06-

21). They both occurred on the same server, a 16 core

Intel Xeon L5530 2.4GHz, with 24GB of RAM. The

implementation ran under Python 2.7.5 and Gurobi

5.6.2 on a Linux OS. Gurobi was conﬁgured to use 16

threads, i.e. all the available cores.

In the typical run the input included 833 possi-

ble Reservations, and seven of eleven resources were

available for scheduling. This resulted in a prob-

lem description with 20,895 rows, 138,635 columns

and 479,240 nonzeros, as reported by Gurobi. This

was reduced to 8,826 rows, 92,093 columns, 293,297

nonzeros by the Gurobi pre-solver. 650 Reservations

were ultimately scheduled. Wall clock time spent in

the kernel (which includes the translation of the prob-

lem into a format that can serve as an input to Gurobi,

a process that we have since further optimized) was

23.77 seconds; in the Gurobi pre-solver 6.56 seconds;

in the Gurobi root relaxation routine 0.21 seconds;

and in the Gurobi integer solver 11.50 seconds. In

total, 23% of the time was spent in kernel overhead,

with the remaining time spent in Gurobi.

The biggest scheduling run during the last few

months had an input of 3864 possible Reservations,

roughly four times as many as the typical run. The

same fraction (7/11) of resources were available for

scheduling at the time of this run. The ﬁnal sched-

ule included 2055 of the submitted Reservations. The

total kernel wall time was 76.54 seconds, of which

17.22 seconds was spent in the Gurobi pre-solver;

20.39 seconds were spent in the Gurobi root re-

laxation step; and 24.83 seconds were spent in the

Gurobi integer solver. The kernel overhead was thus

18% for this larger problem. These measurements are

summarized in Table 1.

Table 1: Measurements for typical and largest runs.

Typical Largest

reservation count 833 3864

wall time (s) 23.77 76.54

% kernel overhead 23 18

pre-solver (s) 6.56 17.22

root relaxation (s) 0.21 20.39

integer solution (s) 11.5 24.83

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

334

5 EXTENSION TO COMPOUND

RESERVATIONS

Astronomical observations requested by a particu-

lar project are usually part of a larger scientiﬁc pro-

gramme, so they are frequently not independent of

each other. When, as is most common, the inter-

dependence between observations is in the targets,

instruments, and exposures of observations, it does

not affect scheduling. But it is occasionally helpful

to provide a way to express a limited form of inter-

dependence between the scheduling status of obser-

vations, i.e. whether or not they were scheduled. This

is useful in situations where one of several alternative

Reservations can fulﬁll the same scientiﬁc objective,

or when sets of Reservations must be scheduled in an

“all-or-none” fashion in order to be scientiﬁcally use-

ful. We allow for this limited type of dependency be-

tween Reservations via the concept of a Compound

Reservation, ﬁrst introduced in (Lampoudi and Saun-

ders, 2013).

A Compound Reservation is a set of Reservations,

inter-connected by one of two logicaloperators: AND

and ONE-OF.

The AND operator is the traditional conjunction

operator. (r

AND r

) means simply that either both

reservations r

and r

should be scheduled, or neither

should be scheduled. (r

AND r

AND ... AND r

) is,

by extension, deﬁned as one would expect.

The ONE-OF operator is equivalent to a “one-

hot circuit” in digital circuit design. (r

ONE-OF r

)

means that either reservation r

or r

should be sched-

uled, but not both. (Because of the no-overlap con-

straint, it is possible that neither reservation can be

scheduled, making this a set packing, rather than a set

partitioning constraint.) For two arguments ONE-OF

is equivalent to XOR. The reason we use the notation

ONE-OF rather than XOR is that the implementation

of a greater-than-two input XOR is not unique. The

most common implementation (i.e. wiring diagram)

of XOR for greater than two arguments yields a par-

ity checker. What we need is a circuit that evaluates to

True when exactly one of its arguments is true. Since

the term “one-hot” is not commonly used outside dig-

ital design, we use the more intuitive label “ONE-OF”

for this operator.

Single-level Compound Reservations, as we de-

scribe them here, enable some of the novel capabil-

ities of a telescope network. They make it possi-

ble to schedule an observation so that it occurs on

one of multiple alternative resources requiring differ-

ent exposure times (using ONE-OF), potentially in-

creasing utilization by leveraging ﬂexibility. Com-

pound Reservations also make it possible to sched-

ule time-series observations in an all-or-none fashion

(using AND), decreasing time wasted obtaining par-

tial data. They make it possible to conduct concur-

rent observations of a single target, or many corre-

lated targets, from multiple resources (using AND),

which previously required human coordination. Fi-

nally, on the LCOGT network, which is global, they

enable the tracking of stationary or moving targets in

spite of the earth’s rotation, using a succession of re-

sources (AND), which has never before been possi-

ble. Importantly, although an arbitrary level of Com-

pound Reservation nesting is conceptually possible

but computationally intractable, all these capabilities

are gained using a single level of nesting, which it is

feasible to schedule.

The presence of Compound Reservations modi-

ﬁes the problem deﬁnition as follows: Given a list of

Reservations, where some are possibly grouped into

Compound Reservations, we wish to ﬁnd a maximum

priority subset of scheduled reservations, and assign

them speciﬁc resourcesand start times, such that there

are no overlaps between scheduled reservations, and

the constraints implied by the Compound Reserva-

tions (single-level ANDs and ONE-OFs) are satisﬁed.

This extension adds the following parameters to

the list of section 3.1:

• O: the set of ONE-OF constraints

• A: the set of AND constraints

The following constraints are added to those of

section 3.4:

ONE-OF constraints.

∑

i∈r

∑

k∈S

≤ 1, ∀r ∈ O

, O

∈ O (5)

AND constraints.

∑

k∈S

−

∑

k∈S

= 0, ∀i, j ∈ r, ∀r ∈ A

, A

∈ A (6)

The size of the inequality constraint matrix is

modiﬁed to be |I| + |T| + |O| rows, i.e. the sum of

the number of reservations, time slots and ONE-OF

constraints.

Finally, AND constraints introduce an equality

constraint matrix with number of rows proportional

to the number of reservations participating in AND

constraints.

6 RELATED AND FUTURE

WORK

The literature on ILP for intervalscheduling is ubiqui-

tous (see, e.g. (Schrijver, 1986), (Graham et al., 1979)

AnIntegerLinearProgrammingSolutiontotheTelescopeNetworkSchedulingProblem

335

and (Potts and Strusevich, 2009)). Our own effort

to model the telescope network scheduling problem

as an ILP problem was inspired by a similar (though

more complicated, due to the presence of multiple ob-

jectives) model for truck scheduling(Lee et al., 2012).

Early work in telescope scheduling, which was

surveyed in (Lampoudi and Saunders, 2013), was of a

highly practical and heuristic nature. In general those

early approaches were difﬁcult to evaluate for ﬁtness

of purpose, and they commonly handled complexity,

especially dynamic volatility, through direct human

intervention and decision-making.

More recently, methods adopted from the Arti-

ﬁcial Intelligence community, e.g. neural networks

(Colom´e et al., 2014), and from Operations Research,

e.g. genetic algorithms in the context of optimization

(Garcia-Piquer et al., 2014), have been making in-

roads in telescope network scheduling. A necessary

shift is occuring in the ﬁeld towards methods whose

performance can be quantitatively compared to either

theoretical models or simulation outcomes.

For completeness, it is worth noting that there

have been two previous design iterations for the

LCOGT scheduler. A randomised planning approach

was proposed in (Brown and Baliber, 2007). In

(Hawkins et al., 2010) the scheduling problem was

broken into a hierarchy of seasonal, monthly and

adaptive planning steps, but speciﬁc implementations

for those steps were not proposed. Both of these were

preliminary proposals, and were never fully imple-

mented or evaluated. They both reﬂected a desire

to steer clear of global optimisation, a stance that

was justiﬁed by citing resource availability, volatil-

ity and computational cost. Given improvements in

computational speeds, this stance was reversed, and

our present optimization approach was adopted.

Our own work is divided between, on the one

hand, efforts to gain a deeper understanding of

the structure of the ILP optimization problem, (e.g.

analysing the structure of the conﬂict graph) and, on

the other hand, evaluating several practical questions

concerningthe schedulingkernel and its performance.

One of these is the impact of the time discretization

introduced by the ILP model, as we explained in sec-

tion 4.2. Another is the choice of priority model,

or more broadly, objective functions, in order to best

match the science objectives of the network. Finally,

the possibilities implied by the compound reservation

scheduling capabilities of the kernel remain as yet un-

characterized. Such a complex feature requires both

an excellent user interface to be useful, as well as

considerable community training efforts to gain adop-

tion. We anticipate that when these conditions come

to fruition, new statistics will become available, and

inform new models of telescope utilization, driving

forward the next iteration of development and theo-

retical advances.

REFERENCES

Brown, T. M. and Baliber, N. (2007). Towards A Sched-

uler For The LCOGT Multi-telescope Network. In

American Astronomical Society Meeting Abstracts,

volume 39 of Bulletin of the American Astronomical

Society, pages 807–807.

Brown, T. M., Baliber, N., Bianco, F. B., Bowman, M.,

Burleson, B., Conway, P., Crellin, M., Depagne,

E.,

De Vera, J., Dilday, B., Dragomir, D., Dubberley, M.,

Eastman, J. D., Elphick, M., Falarski, M., Foale, S.,

Ford, M., Fulton, B. J., Garza, J., Gomez, E. L., Gra-

ham, M., Greene, R., Haldeman, B., Hawkins, E., Ha-

worth, B., Haynes, R., Hidas, M., Hjelstrom, A. E.,

Howell, D. A., Hygelund, J., Lister, T. A., Lobdill,

R., Martinez, J., Mullins, D. S., Norbury, M., Par-

rent, J., Paulson, R., Petry, D. L., Pickles, A., Pos-

ner, V., Rosing, W. E., Ross, R., Sand, D. J., Saun-

ders, E. S., Shobbrook, J., Shporer, A., Street, R. A.,

Thomas, D., Tsapras, Y., Tufts, J. R., Valenti, S., Van-

der Horst, K., Walker, Z., White, G., and Willis, M.

(2013). Las Cumbres Observatory Global Telescope

Network. PASP, 125:1031–1055.

Colom´e, J., Colomer, P., Campreci`os, J., Coiffard, T.,

de O˜na, E., Pedaletti, G., Torres, D. F., and Garcia-

Piquer, A. (2014). Artiﬁcial intelligence for the CTA

Observatory scheduler . In Proceedings of SPIE, vol-

ume 9149, pages 91490H–91490H–15.

Garcia-Piquer, A., Gu`ardia, J., Colom´e, J., Ribas, I., Gesa,

L., Morales, J. C., P´erez-Calpena, A., Seifert, W.,

Quirrenbach, A., Amado, P. J., Caballero, J. A., and

Reiners, A. (2014). CARMENES instrument control

system and operational scheduler . In Proceedings of

SPIE, volume 9152, pages 915221–915221–15.

Garey, M. R. and Johnson, D. S. (1990). Computers

and Intractability; A Guide to the Theory of NP-

Completeness. W. H. Freeman & Co., New York, NY,

USA.

Graham, R., Lawler, E., Lenstra, J., and Kan, A. (1979).

Optimization and approximation in deterministic se-

quencing and scheduling: a survey. In P.L. Ham-

mer, E. J. and Korte, B., editors, Discrete Optimiza-

tion II Proceedings of the Advanced Research Institute

on Discrete Optimization and Systems Applications of

the Systems Science Panel of NATO and of the Dis-

crete Optimization Symposium co-sponsored by IBM

Canada and SIAM Banff, Aha. and Vancouver, vol-

ume 5 of Annals of Discrete Mathematics, pages 287

– 326. Elsevier.

Gurobi Optimization, I. (2014). Gurobi optimizer reference

manual.

Hawkins, E., Baliber, N., Bowman, M., Brown, T.,

Burleson, B., Foale, S., Ford, M., Lister, T., Norbury,

M., Saunders, E., and Walker, Z. (2010). Schedul-

ing observations on the LCOGT network. In Silva,

ICORES2015-InternationalConferenceonOperationsResearchandEnterpriseSystems

336

D. R., Peck, A. B., and Soifer, B. T., editors, Observa-

tory Operations: Strategies, Processes, and Systems

III, volume 7737. SPIE.

Lampoudi, S. and Saunders, E. (2013). Telescope network

scheduling - rationale and formalisms. In Vitoriano,

B. and Valente, F., editors, ICORES’13, pages 313–

317. SciTePress.

Lee, S., Turner, J., Daskin, M. S., de Mello, T. H., and

Smilowitz, K. (2012). Improving ﬂeet utilization for

carriers by interval scheduling. European Journal of

Operational Research, 218(1):261–269.

Potts, C. N. and Strusevich, V. A. (2009). Fifty years of

scheduling: a survey of milestones. JORS, 60(S1).

Saunders, E. S., Lampoudi, S., Lister, T. A., Norbury, M.,

and Walker, Z. (2014). Novel scheduling approaches

in the era of multi-telescope networks. In Proc. SPIE,

volume 9149, pages 91490E–91490E–7.

Schrijver, A. (1986). Theory of Linear and Integer Pro-

gramming. John Wiley & Sons, Inc., New York, NY,

USA.

AnIntegerLinearProgrammingSolutiontotheTelescopeNetworkSchedulingProblem

337