approaches for planning the iss cosmonaut trainingwerner/preprints/p15-12.pdfย ยท approaches for...

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Approaches for planning the ISS cosmonaut training * Sergey Bronnikov , Alexandre Dolgui , Alexander Lazarev โ€  , Nikolay Morozov , Aleksey Petrov , Ruslan Sadykov , Alexander Sologub , Frank Werner โ„Ž , Denis Yadrentsev , Elena Musatova , Nail Khusnullin Rocket and Space Corporation Energia after S.P. Korolev, 4A Lenin Street, Korolev, Moscow Region, 141070, Russian Federation (e-mail: [email protected]) Ecole Nationale Superieure des Mines, LIMOS-UMR CNRS 6158, F-42023 Saint-Etienne, France (e-mail: [email protected]) V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences, Profsoyuznaya street 65, 117997 Moscow, Russian Federation; Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation; Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russian Federation; International Laboratory of Decision Choise and Analysis, National Research University Higher School of Economics, 20 Myasnsnitskaya street, 101000 Moscow, Russian Federation (e-mail: [email protected]) Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russian Federation; V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences, Profsoyuznaya street 65, 117997 Moscow, Russian Federation (e-mail: [email protected]) V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences, Profsoyuznaya street 65, 117997 Moscow, Russian Federation; * The work was supported by the grants DAAD (A/14/00328), RFBR (13-01-12108, 15-07-07489, 15-07-03141), HSE Faculty of Economic Sciences. โ€  Corresponding author. E-mail address: [email protected]. 1

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Page 1: Approaches for planning the ISS cosmonaut trainingwerner/preprints/p15-12.pdfย ยท Approaches for planning the ISS cosmonaut training * Sergey Bronnikov , Alexandre Dolgui , Alexander

Approaches for planning the ISS cosmonauttraining *

Sergey Bronnikov๐‘Ž, Alexandre Dolgui๐‘, Alexander Lazarev๐‘โ€ ,Nikolay Morozov๐‘‘, Aleksey Petrov๐‘’, Ruslan Sadykov๐‘“ ,

Alexander Sologub๐‘”, Frank Wernerโ„Ž, Denis Yadrentsev๐‘–,Elena Musatova๐‘—, Nail Khusnullin๐‘˜

๐‘Ž Rocket and Space Corporation Energia after S.P. Korolev, 4A Lenin Street,Korolev, Moscow Region, 141070, Russian Federation

(e-mail: [email protected])๐‘ Ecole Nationale Superieure des Mines, LIMOS-UMR CNRS 6158, F-42023

Saint-Etienne, France(e-mail: [email protected])

๐‘ V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences,Profsoyuznaya street 65, 117997 Moscow, Russian Federation;

Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991,Russian Federation;

Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny,Moscow Region, 141700, Russian Federation;

International Laboratory of Decision Choise and Analysis, National ResearchUniversity Higher School of Economics, 20 Myasnsnitskaya street, 101000

Moscow, Russian Federation(e-mail: [email protected])

๐‘‘ Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991,Russian Federation;

V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences,Profsoyuznaya street 65, 117997 Moscow, Russian Federation

(e-mail: [email protected])๐‘’V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences,

Profsoyuznaya street 65, 117997 Moscow, Russian Federation;

*The work was supported by the grants DAAD (A/14/00328), RFBR (13-01-12108,15-07-07489, 15-07-03141), HSE Faculty of Economic Sciences.

โ€ Corresponding author. E-mail address: [email protected].

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International Laboratory of Decision Choise and Analysis, National ResearchUniversity Higher School of Economics, 20 Myasnsnitskaya street, 101000

Moscow, Russian Federation(e-mail: [email protected])

๐‘“ Project-team RealOpt, INRIA Bordeaux - Sud-Ouest, 351 cours de laLiberation, 33405 Talence, France(e-mail: [email protected])

๐‘” Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991,Russian Federation;

V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences,Profsoyuznaya street 65, 117997 Moscow, Russian Federation

(e-mail: [email protected])โ„Ž Fakultat fur Mathematik, Otto-von-Guericke-Universitat Magdeburg, PSF 4120,

39016 Magdeburg, Germany(e-mail: [email protected])

๐‘– YU. A. Gagarin Research & Test Cosmonaut Training Center, Star City,141160 Moscow Region, Russian Federation

(e-mail: [email protected])๐‘—V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences,

Profsoyuznaya street 65, 117997 Moscow, Russian Federation(e-mail: [email protected])

๐‘˜V.A. Trapeznikov Institute of Control Science of Russian Academy of Sciences,Profsoyuznaya street 65, 117997 Moscow, Russian Federation

(e-mail: [email protected])

September 8, 2015

Abstract

We consider the problem of planning the ISS cosmonaut trainingwith different objectives. A pre-defined set of minimum qualificationlevels should be distributed between the crew members with minimumtraining time differences, training expenses or a maximum of the train-ing level with a limitation of the budget.

First, a description of the cosmonaut training process is given.Then four models are considered for the volume planning problem.The objective of the first model is to minimize the differences betweenthe total time of the preparation of all crew members, the objective ofthe second one is to minimize the training expenses with a limitation ofthe training level, and the objective of the third one is to maximize thetraining level with a limited budget. The fourth model considers theproblem as an ๐‘›-partition problem. Then two models are consideredfor the calendar planning problem.

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For the volume planning problem, two algorithms are presented.The first one is a heuristic with a complexity of ๐‘‚(๐‘›) operations. Thesecond one consists of a heuristic and exact parts, and it is based onthe ๐‘›-partition problem approach.

Contents1 Relevance of the problem 4

2 Description of the cosmonaut training 52.1 General space training . . . . . . . . . . . . . . . . . . . . . . 62.2 Training in groups, separated by the type of manned space-

craft or areas of specialization . . . . . . . . . . . . . . . . . . 72.3 Training in crews . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Mathematical models 113.1 Volume planning problem . . . . . . . . . . . . . . . . . . . . 11

3.1.1 Model 1: Minimizing the differences. . . . . . . . . . . 113.1.2 Model 2: Minimizing the expenses. . . . . . . . . . . . 133.1.3 Model 3: Maximizing the training level. . . . . . . . . . 153.1.4 Model 4: ๐พ-partition formulation. . . . . . . . . . . . . 16

3.2 Calendar planning problem . . . . . . . . . . . . . . . . . . . . 173.2.1 Model 5. . . . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 Model 6. . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Approaches 264.1 Algorithm-3.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Partition algorithm . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2.1 ๐’ฉ๐’ซ-completeness . . . . . . . . . . . . . . . . . . . . . 284.2.2 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Integer programming . . . . . . . . . . . . . . . . . . . . . . . 31

5 Comparison of the algorithms 31

6 Conclusion 32

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1 Relevance of the problemIn Fig. 1, one can see the logotype of the

Figure 1: ISS logotype

International Space Station.The International Space Station (ISS) is an

artificial satellite, on a low Earth orbit. Itis made of many modules. The first one waslaunched in 1998. Nowadays it is the biggestartificial object on an orbit, often it could beseen from the earth with a naked eye.

The ISS has the potential to conduct a widespectrum of scientific researches. The experi-ments could last for decades with the possibil-ity of a careful control of the humans. The ISSmaintains an orbit with an altitude between

330 km (205 mi) and 435 km (270 mi) by means of reboot maneuvers usingthe engines of the Zvezda module or the visiting spacecraft. It completes15.51 orbits per day. The length is 72.8 m (239 ft), and the width is 108.5m (356 ft).

In Fig. 2, one can see the ISS when it would be placed on the Red square.

10

ะ ะฐะทะผะตั€ั‹ ะœะšะก ะฝะฐ ั„ะพะฝะต ะšั€ะฐัะฝะพะน ะฟะปะพั‰ะฐะดะธ

10

Figure 2: ISS in comparison with the Red square

Among all the cosmonautic problems, particular attention is dedicated tothe planning problems. For scheduling the operations during the flight andfor scheduling the trainings before, it is necessary to maximize the efficiency.Due to the date, it takes lots of human, time and material resources.

The proper preparation of cosmonauts is a long, expensive and sophisti-cated process. In order to maintain reliability of a flight, the crew membersare obligated to be trained for different types of situations and operations,

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to obtain required skills and knowledge before the launch. Hence, the Yu.A. Gagarin Research & Test Cosmonaut Training Center (CTC) must planand schedule a list of trainings for every cosmonaut.

In general, three crew qualification levels are defined; a user level, anoperator level and a specialist level. For a given flight program, for everyonboard complex, a pre-defined set of minimum qualifications is needed tosafely operate and maintain the system (e.g. one specialist, one operator andone user).

All this has to fit into an overall integrated training schedule, which isa challenge of its own โ€“ remember that all astronauts and cosmonauts haveindividually tailored training plans.

Each crew member, while being a specialist for some systems, will bean operator or only a user for other systems. Consequently, the trainingprogram for each crew member is individually tailored to his or her set oftasks and pre-defined qualification levels.

Then the Mission Control Center (MCC) has to distribute the flight op-erations between the crew members and the MCC controllers.

Due to the fact that the potential of the ISS is limited, it is extremelyimportant to maximize the efficiency of use, according to a budget constraint.

Nowadays, scheduling is performed manually without using any mathe-matical approach, based only on the experience of the employees. Besides,errors cumulate during the planning process and cause huge time and finan-cial expenses. We hope that the considered approaches and models have thepotential to reduce these expenses.

In this paper, the following problems are considered: The developmentof training plans for a crew of three cosmonauts, which is determined as thedistribution of a pre-defined set of minimum qualification levels among themembers of a crew, using the following criteria:

โˆ™ minimizing the differences between the total time of the preparation ofall crew members;

โˆ™ minimizing the training expenses;

โˆ™ maximizing the training level with a limited budget.

In Section 2, the cosmonaut training process is described. Mathematicalmodels are given in Section 3, and approaches are considered in Section 4.

2 Description of the cosmonaut trainingThe sequence of the training program is based on four training phases:

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1. General space training (GST) of candidates for cosmonauts;

2. Training in groups, separated by the type of manned spacecraft (MSC)or areas of specialization;

3. Training in approved crews for a specific space flight on MSC;

Passing the sequence of the stages of the training is mandatory for allRussian cosmonauts. The GST is performed for every candidate only once.The other stages can be performed repeatedly. Usually, the first three phaseslast 2, 2 and 2.5 years, repspectively. In order to increase the efficiency andto decrease the expenses, the training time should be as short as possible.

2.1 General space training

The general space training provides the candidate cosmonauts with basicknowledge on space technology and science, basic medical skills and basicskills related to their future operational tasks, including those related to thestation systems and operations.

The GST objectives are:

โˆ™ to provide with knowledge and skills related to

โ€“ theoretical foundations of cosmonautics;

โ€“ principles of the design and the basis of the MSC, its service sys-tems, scientific and special equipment;

โ€“ operation of the MSC, its serving systems, scientific and specialequipment;

โ€“ theoretical foundations of scientific research and experiments car-rying out at the MSC;

โ€“ systems of the manned orbital station;

โ€“ foreign MSC;

โ€“ the objects of the ground space infrastructure;

โ€“ interaction with the ground;

โ€“ working on a personal computer;

โ€“ conducting testing, research and experimentation on MSC;

โ€“ influence of dynamic factors of a space flight;

โ€“ working in space suits;

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โ€“ Extra Vehicular Activity (EVA) in a hydrosphere and short-termweightlessness on flying laboratories;

โ€“ initial implementation of maintenance operations and maintenance(MOM), assembly and dismantling operations (ADO), handlingoperations (HO);

โ€“ scuba diving (see Fig. 3);

โ€“ landing under extreme conditions of various climatic zones (CZ)(seeFig. 4);

โ€“ flight and parachute;

โ€“ functions and responsibilities of crew members of the MSC;

โ€“ safety of space flight, including medical support;

โ€“ international and space law;

โ€“ English language, minimally required to prepare on the bases ofthe ISS program partners;

โˆ™ to develop individual neuro-psychological resistance to adverse factorsof a space flight and skills when working under difficult conditions ofexistence;

โˆ™ to control and improve health;

โˆ™ to identify individual psychophysiological characteristics of each candi-date.

This training phase is a candidacy period and upon completion, successfulcandidates are certified as being career cosmonaut-test pilot or cosmonaut-researcher. The GST has a duration of up to two years.

2.2 Training in groups, separated by the type of mannedspacecraft or areas of specialization

The main purpose of this training phase is to study the MSC elementsmore in-depth. The cosmonauts learn to service and operate the differentmodules, systems and subsystems, and to fly and dock transport vehiclesand an unmanned cargo carrier.

The objectives of this phase are to acquire a better knowledge and skillsrelated to

โˆ™ design, layout, on-board service systems, scientific and specialized equip-ment of a specific MSC;

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Figure 3: Underwater low-gravity training

โˆ™ work with on-board systems and scientific equipment;

โˆ™ flight procedures and mission;

โˆ™ typical operations on EVA in the Neutral Buoyancy Laboratory (NBL)and short-term weightlessness on flying laboratories and other technicalfacilities of the Cosmonaut Training (TFCT);

โˆ™ active behavior in real stress;

โˆ™ equipment inventory, MOM, ADO, HO;

โˆ™ physical condition and functional capacity of the organism, high effi-ciency in the performance of professional tasks;

โˆ™ on-board documentation;

โˆ™ work with the Lead Operations Management Group of MCC;

โˆ™ safety of the manned missions;

โˆ™ operation and control of the MSC;

โˆ™ typical accidents and emergency situations;

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โˆ™ English language (to a level that allows to participate in a trainingprogram at the facilities of the partner countries).

During this training phase, the cosmonauts ought to pass exams andtests in the relevant disciplines. The training in groups can be performedeven after the formation of crews during the third training phase.

2.3 Training in crews

During this training phase, the cosmonauts learn everything they need toknow for their mission. All crew members, prime and backup crews, selectedfor the space flight will train together.

Figure 4: Winter forest landing training

This is important not only because the crew members have to becomeknown to each other (later they will spend about half a year together inthe enclosed environment of the ISS), but they also learn to work efficientlytogether as a team and according to the distributed roles and responsibilitiesfor which they are assigned to.

The crew tasks on the ISS are individually tailored, always consideringthe particular experience of the astronauts and the professional background.

The objectives of this phase are:

โˆ™ to acquire knowledge and skills related to

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โ€“ features and maintenance rules of the concrete MSC;

โ€“ program of the upcoming flight, on-board documentation and doc-uments governing the rules of interaction between the crew mem-bers themselves and with the management teams and provide theflight crew with the code of conduct;

โ€“ control and operation of the MSC in regular modes and in case ofemergencies;

โ€“ scientific research and experiments included into the flight pro-gram;

โ€“ functional duties in a crew;

โ€“ personal equipment (rescue suits, EVA suits, spacecraft chairs andothers);

โ€“ concrete flight EVA tasks;

โ€“ rules and methods to ensure flight safety on a specific MSC;

โˆ™ to form a psychological compatibility in the crew;

โˆ™ to improve the interaction between the crew members, between thecrews, and between the crews and the control groups;

โˆ™ to ensure a good health, a high performance and the readiness to per-form a biomedical section of the flight program;

โˆ™ to conduct a pre-launch preparation with the crew;

โˆ™ to increase the level of English up to the one needed to perform a spaceflight as a part of the international crew of the ISS.

There exist the following crew functions:

โˆ™ on an MSC:

โ€“ commander,

โ€“ onboard engineer,

โ€“ onboard engineer-2,

โ€“ space flight participant.

โˆ™ on an ISS:

โ€“ commander,

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โ€“ onboard engineer,

โ€“ cosmonaut-researcher,

โ€“ space flight participant.

The training program contains a knowledge-based classroom training, aswell as โ€™hands-onโ€™ using flight-like training mock-ups and simulators.

3 Mathematical modelsThe whole planning of the ISS cosmonaut training can be logically divided

into two stages: the problem of volume planning and the problem of calendarplanning.

The data for the volume planning problem is a set of onboard complexesand the required number of cosmonauts of different qualifications of eachonboard complex. The aim is to distribute the training in qualifications ofonboard complexes between the cosmonauts so that the total time of trainingbecomes minimal.

The next important step of the planning process is the calendar planning.Once solved the volume problem of planning for each cosmonautโ€™s definedset of onboard complexes for which it is necessary to be trained, it also raisedthe necessary qualifications for these onboard complexes. It is necessary toplan the training to minimize the time of preparation of the first crew, butto comply with the resource constraints and deadlines of the preparation ofthe other crews.

3.1 Volume planning problem

3.1.1 Model 1: Minimizing the differences.

Notations

โˆ™ ๐พ โ€“ number of cosmonauts;

โˆ™ ๐’ฆ={1, . . . , ๐พ} โ€“ set of cosmonauts;

โˆ™ ๐ฝ โ€“ number of onboard complexes;

โˆ™ ๐’ฅ={1, . . . , ๐ฝ} โ€“ set of onboard complexes;

โˆ™ ๐‘„ โ€“ number of qualifications;

โˆ™ ๐’ฌ={1, . . . , ๐‘„} โ€” set of qualifications;

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โˆ™ ๐’ช๐‘— โ€“ set of tasks on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐’ช๐‘—๐‘ž โ€“ set of tasks on the onboard complex ๐‘— โˆˆ ๐’ฅ , available for acosmonaut with qualification ๐‘ž โˆˆ ๐’ฌ;

โˆ™ ๐‘๐‘—,๐‘ž,๐‘’ , ๐‘’ โˆˆ {0, 1} โ€“ amount of time needed to train an experienced(๐‘’ = 1) or an inexperienced (๐‘’ = 0) cosmonaut to qualification level๐‘ž โˆˆ ๐’ฌ on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐‘›๐‘—,๐‘ž โ€“ required number of cosmonauts with qualification level ๐‘ž โˆˆ ๐’ฌ onthe onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐ท โ€“ maximum time of the training plan (constrained by the initialdata);

Data A set of cosmonauts ๐’ฆ should be trained. The cosmonauts could beexperienced (๐‘’ = 1) or inexperienced (๐‘’ = 0). We have ๐ฝ onboard complexes๐‘— โˆˆ ๐’ฅ . The number of qualifications can range from ๐‘„ = 2 to ๐‘„ = 4. Weconsider the case of the following ๐‘„ = 3 qualifications ๐‘ž โˆˆ ๐’ฌ: user (๐‘ž = 1),operator (๐‘ž = 2) and specialist (๐‘ž = 3). It is assumed that all amounts ๐‘๐‘—,๐‘ž,๐‘’and all numbers ๐‘›๐‘—,๐‘ž are known.Besides we know that

๐’ช๐‘—1

โ‹‚๐’ช๐‘—2 = รธ, ๐‘—1 = ๐‘—2, ๐‘—1, ๐‘—2 โˆˆ ๐’ฅ

Variables

โˆ™ ๐‘ฅ๐‘˜,๐‘—,๐‘ž โˆˆ {0, 1} โ€” Boolean variable: We have ๐‘ฅ๐‘˜๐‘—๐‘ž = 1 if cosmonaut๐‘˜ โˆˆ ๐’ฆ should have the qualification level ๐‘ž โˆˆ ๐’ฌ on the onboard complex๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐œ๐‘˜ โ€” total time of the training plan for cosmonaut ๐‘˜ โˆˆ ๐’ฆ;

In our notation, the total training time of cosmonaut ๐‘˜ can be representedas the sum of the training times, assigned to the cosmonaut:

๐œ๐‘˜ =โˆ‘๐‘žโˆˆ๐’ฌ

โˆ‘๐‘—โˆˆ๐’ฅ

๐‘๐‘—๐‘ž๐‘’๐‘ฅ๐‘˜๐‘—๐‘ž.

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Objective function

(max๐‘˜

๐œ๐‘˜ โˆ’ min๐‘˜

๐œ๐‘˜) โ†’ min, ๐‘˜ โˆˆ ๐’ฆ, (1)

max๐‘˜

๐œ๐‘˜ โ†’ min, ๐‘˜ โˆˆ ๐’ฆ, (2)

min๐‘˜

๐œ๐‘˜ โ†’ max, ๐‘˜ โˆˆ ๐’ฆ. (3)

Constraints โˆ‘๐‘˜โˆˆ๐’ฆ

๐‘ฅ๐‘˜,๐‘—,๐‘ž = ๐‘›๐‘—,๐‘ž, ๐‘— โˆˆ ๐’ฅ , ๐‘ž โˆˆ ๐’ฌ, (4)

โˆ‘๐‘žโˆˆ๐’ฌ

๐‘ฅ๐‘˜,๐‘—,๐‘ž โ‰ค 1, ๐‘— โˆˆ ๐’ฅ , ๐‘˜ โˆˆ ๐’ฆ, (5)

โˆ‘๐‘žโˆˆ๐’ฌ

๐‘›๐‘—,๐‘ž โ‰ค ๐‘„, ๐‘— โˆˆ ๐’ฅ . (6)

๐œ๐‘˜ โ‰ค ๐ท, ๐‘˜ โˆˆ ๐’ฆ (7)

In [10], it was shown that for this type of problem it is possible to usethree different objective functions (1), (2), (3). Constraint (5) forbids thata cosmonaut has two different qualification levels on the same onboard com-plex. Constraint (4) requires that the number of cosmonauts, trained foreach onboard complex, should be equal to the required number.

3.1.2 Model 2: Minimizing the expenses.

Notations

โˆ™ ๐พ โ€” number of cosmonauts;

โˆ™ ๐’ฆ={1, . . . , ๐พ} โ€” set of cosmonauts;

โˆ™ ๐ฝ โ€” number of onboard complexes;

โˆ™ ๐’ฅ={1, . . . , ๐ฝ} โ€” set of onboard complexes;

โˆ™ ๐‘„ โ€” number of qualifications;

โˆ™ ๐’ฌ={1, . . . , ๐‘„} โ€” set of qualifications;

โˆ™ ๐’ช โ€” set of all tasks provided on the ISS;

โˆ™ ๐’ช๐‘— โ€” set of tasks on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

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โˆ™ ๐’ช๐‘—๐‘ž โ€” set of tasks on the onboard complex ๐‘— โˆˆ ๐’ฅ , available for acosmonaut with qualification ๐‘ž โˆˆ ๐’ฌ;

โˆ™ ๐‘๐‘—,๐‘ž,๐‘’ , ๐‘’ โˆˆ {0, 1}โ€” amount of time needed to train an experienced(๐‘’ = 1) or an inexperienced (๐‘’ = 0) cosmonaut to qualification level๐‘ž โˆˆ ๐’ฌ on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐‘›๐‘—,๐‘ž โ€” required amount of cosmonauts with qualification level ๐‘ž โˆˆ ๐’ฌon the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐ท โ€” maximum time of the training plan;

โˆ™ ๐‘ฅ๐‘˜,๐‘—,๐‘ž โˆˆ {0, 1} โ€” Boolean variable: We have ๐‘ฅ๐‘˜๐‘—๐‘ž = 1 if cosmonaut๐‘˜ โˆˆ ๐’ฆ should have the qualification level ๐‘ž โˆˆ ๐’ฌ on the onboard complex๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐œ๐‘˜ โ€” total time of the training plan for cosmonaut ๐‘˜ โˆˆ ๐’ฆ;

โˆ™ ๐‘๐‘˜,๐‘—,๐‘ž โ€” the cost of training cosmonaut ๐‘˜ โˆˆ ๐’ฆ to qualification level๐‘ž โˆˆ ๐’ฌ on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐‘Š๐‘— โ€” required training level on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐‘“๐‘ž โ€” training level of a cosmonaut with the qualification level ๐‘ž โˆˆ ๐’ฌ.

Objective function โˆ‘๐‘˜โˆˆ๐พ

โˆ‘๐‘—โˆˆ๐’ฅ

โˆ‘๐‘žโˆˆ๐’ฌ

๐‘๐‘˜,๐‘—,๐‘ž๐‘ฅ๐‘˜,๐‘—,๐‘ž โ†’ min . (8)

Constraints โˆ‘๐‘žโˆˆ๐’ฌ

โˆ‘๐‘˜โˆˆ๐’ฆ

๐‘“๐‘ž๐‘ฅ๐‘˜,๐‘—,๐‘ž โ‰ฅ ๐‘Š๐‘—, (9)

(5): โˆ‘๐‘žโˆˆ๐’ฌ

๐‘ฅ๐‘˜,๐‘—,๐‘ž โ‰ค 1, ๐‘— โˆˆ ๐’ฅ , ๐‘˜ โˆˆ ๐’ฆ,

(4): โˆ‘๐‘˜โˆˆ๐’ฆ

๐‘ฅ๐‘˜,๐‘—,๐‘ž = ๐‘›๐‘—,๐‘ž, ๐‘— โˆˆ ๐’ฅ , ๐‘ž โˆˆ ๐’ฌ,

(7):๐œ๐‘˜ โ‰ค ๐ท, ๐‘˜ โˆˆ ๐’ฆ.

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3.1.3 Model 3: Maximizing the training level.

Notations

โˆ™ ๐พ โ€“ number of cosmonauts;

โˆ™ ๐’ฆ={1, . . . , ๐พ} โ€“ set of cosmonauts;

โˆ™ ๐ฝ โ€“ number of onboard complexes;

โˆ™ ๐’ฅ={1, . . . , ๐ฝ} โ€“ set of onboard complexes;

โˆ™ ๐‘„ โ€“ number of qualifications;

โˆ™ ๐’ฌ={1, . . . , ๐‘„} โ€“ set of qualifications;

โˆ™ ๐’ช โ€“ set of all tasks provided on ISS;

โˆ™ ๐’ช๐‘— โ€“ set of tasks on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐’ช๐‘—๐‘ž โ€“ set of tasks on the onboard complex ๐‘— โˆˆ ๐’ฅ , available for acosmonaut with qualification ๐‘ž โˆˆ ๐’ฌ;

โˆ™ ๐‘๐‘—,๐‘ž,๐‘’ , ๐‘’ โˆˆ {0, 1} โ€“ amount of time needed to train an experienced(๐‘’ = 1) or an inexperienced (๐‘’ = 0) cosmonaut to qualification level๐‘ž โˆˆ ๐’ฌ on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐‘›๐‘—,๐‘ž โ€“ required number of cosmonauts with qualification level ๐‘ž โˆˆ ๐’ฌ onthe onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐ท โ€“ maximum time of the training plan;

โˆ™ ๐‘ฅ๐‘˜,๐‘—,๐‘ž โˆˆ {0, 1} โ€“ Boolean variable: We have ๐‘ฅ๐‘˜๐‘—๐‘ž = 1 if cosmonaut๐‘˜ โˆˆ ๐’ฆ should have the qualification level ๐‘ž โˆˆ ๐’ฌ on the onboard complex๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐œ๐‘˜ โ€“ total time of the training plan for cosmonaut ๐‘˜ โˆˆ ๐’ฆ;

โˆ™ ๐‘๐‘˜,๐‘—,๐‘™ โ€“ the cost of training cosmonaut ๐‘˜ to qualification level ๐‘ž โˆˆ ๐’ฌ onthe onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐‘Š๐‘— โ€“ required training level on the onboard complex ๐‘— โˆˆ ๐’ฅ ;

โˆ™ ๐‘“๐‘ž โ€“ training level of a cosmonaut with the qualification level ๐‘ž โˆˆ ๐’ฌ.

โˆ™ ๐ต โ€“ limit of the budget of a whole training process.

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Objective function โˆ‘๐‘˜โˆˆ๐’ฆ

โˆ‘๐‘—โˆˆ๐ฝ

โˆ‘๐‘žโˆˆ๐’ฌ

๐‘“๐‘ž๐‘ฅ๐‘˜,๐‘—,๐‘ž โ†’ max . (10)

Constraints โˆ‘๐‘˜โˆˆ๐’ฆ

โˆ‘๐‘—โˆˆ๐’ฅ

โˆ‘๐‘žโˆˆ๐’ฌ

๐‘๐‘˜,๐‘—,๐‘ž๐‘ฅ๐‘˜,๐‘—,๐‘ž โ‰ค ๐ต. (11)

(5): โˆ‘๐‘žโˆˆ๐’ฌ

๐‘ฅ๐‘˜,๐‘—,๐‘ž โ‰ค 1, ๐‘— โˆˆ ๐’ฅ , ๐‘˜ โˆˆ ๐’ฆ,

(4): โˆ‘๐‘˜โˆˆ๐’ฆ

๐‘ฅ๐‘˜,๐‘—,๐‘ž = ๐‘›๐‘—,๐‘ž, ๐‘— โˆˆ ๐’ฅ , ๐‘ž โˆˆ ๐’ฌ,

(7):๐œ๐‘˜ โ‰ค ๐ท, ๐‘˜ โˆˆ ๐’ฆ.

3.1.4 Model 4: ๐พ-partition formulation.

Problem Consider now the special case when only one qualification levelexists (๐‘„ = 1) and each onboard complex can be assigned only to one cos-monaut. Then

๐‘๐‘’๐‘˜,๐‘—,๐‘ž โ†’ ๐‘๐‘’๐‘˜,๐‘—,

๐‘›๐‘—,๐‘ž โ†’ ๐‘›๐‘—,

๐‘›๐‘— โ‰ค ๐พ,

๐‘’ = {0, 1}, ๐‘˜ โˆˆ ๐’ฆ, ๐‘— โˆˆ ๐’ฅ

Let ๐’ฅ be the set of onboard complexes and ๐’ฅ๐‘˜ be the subset of onboardcomplexes assigned to cosmonaut ๐‘˜:โ‹ƒ

๐‘˜

๐’ฅ๐‘˜ = ๐’ฅ , ๐’ฅ โ€ฒ๐‘˜ โˆฉ ๐’ฅ๐‘˜ = รธ (12)

๐œ๐‘˜ โ†’ ๐œ๐‘˜ =โˆ‘๐‘—โˆˆ๐’ฅ๐‘˜

๐‘๐‘’๐‘˜,๐‘—,

๐‘’ = {0, 1}, ๐‘˜, ๐‘˜โ€ฒ โˆˆ ๐’ฆ, ๐‘˜ = ๐‘˜โ€ฒ, ๐‘— โˆˆ ๐’ฅ .

The major goal is to find a partition of the set ๐’ฅ , which minimizes thedifference between the total time of the preparation of all crew members.

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Objective function Consider the same objective function as in the bot-tleneck assignment problem [10]:

max๐‘˜

๐œ๐‘˜ โ†’ min๐’ฅ๐‘—

. (13)

In [2], algorithms for the "exact" solution of this problem were presentedwith a constraint on the number of jobs assigned to each cosmonaut. In[12], it was showen that these algorithms have a pseudo-polynomial timecomplexity. It was also proven that the problem is strongly ๐’ฉ๐’ซ-hard forgeneral ๐‘š. Thus, the algorithms in [2] cannot guarantee an optimal solutionunless ๐’ซ = ๐’ฉ๐’ซ , although they may be used as good heuristics [4].

From the statement of the problem, it can be seen that it is possible touse the objective function of one of the multi-way partition problems. In [10],it was shown that there were at least three of them: minimizing the largestsubset sum (as in (13)), maximizing the smallest subset sum, and minimizingthe difference between the largest and smallest subset sums.

We will use the third objective function:

๐›ฟ = (max๐‘˜

๐œ๐‘˜ โˆ’ min๐‘˜

๐œ๐‘˜) โ†’ min๐’ฅ๐‘—

. (14)

3.2 Calendar planning problem

3.2.1 Model 5.

Notations

โˆ™ ๐’ž = {1, . . . ,๐ถ} โ€“ set of crews, where the crews are sorted according toa non-decreasing order of their due dates;

โˆ™ ๐’ฆ๐‘={1, . . . , ๐พ๐‘} โ€“ set of cosmonauts in crew ๐‘ โˆˆ ๐’ž;

โˆ™ ๐’ฆ=โ‹ƒ

๐‘ ๐’ฆ๐‘ โ€“ complete set of cosmonauts;

โˆ™ ๐’ฅ๐‘˜ โ€“ set of tasks of cosmonaut ๐‘˜, which are required for the implemen-tation of the training plan;

โˆ™ ๐’ฅ=โ‹ƒ

๐‘˜ ๐’ฅ๐‘˜ โ€” set of all tasks;

โˆ™ ๐’ฏ ={1, . . . , ๐‘‡} โ€“ set of the time moments (planning horizon);

โˆ™ ๐‘๐‘— โ€“ execution time of the operation ๐‘— โˆˆ ๐’ฅ ;

โˆ™ โ„› = {1, . . . ,๐‘…} โ€“ set of resources;

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โˆ™ ๐‘Ÿ๐‘๐‘—๐‘Ÿ โ€“ amount of the resource ๐‘Ÿ needed to perform the task ๐‘—;

โˆ™ ๐‘Ÿ๐‘Ž๐‘Ÿ๐‘ก โ€“ amount of resource ๐‘Ÿ accessible during the time moment ๐‘ก;

โˆ™ ๐ท๐‘ โ€“ due date of the crew ๐‘ โˆˆ ๐’ž;

โˆ™ ๐บ = (๐ฝ,ฮ“) โ€“ graph of the precedence relationships between the tasks:We have (๐‘—,๐‘—โ€ฒ) โˆˆ ฮ“ if task ๐‘— must be performed before task ๐‘—โ€ฒ.

โˆ™ ๐ป = (๐ฝ,โ„‹) โ€“ the graph of the strict precedence relationships betweenthe tasks: We have (๐‘—,๐‘—โ€ฒ) โˆˆ โ„‹ if task ๐‘—โ€ฒ must be performed immediatelyafter the task ๐‘—.

Variables

โˆ™ ๐‘ฅ๐‘—๐‘ก โˆˆ {0, 1} โ€“ Boolean variable: We have ๐‘ฅ๐‘–๐‘—๐‘ก = 1 if and only if task ๐‘—starts the execution at time moment ๐‘ก;

โˆ™ ๐‘†๐‘ โˆˆ ๐‘† โ€“ set of moments at which the execution of tasks from the set๐ฝ๐‘ starts.

Formulation of the problem

The optimization criterion is to minimize the total training time of thefirst crew:

๐ถ๐‘š๐‘Ž๐‘ฅ(๐‘†1) โ†’ min, (15)

where๐ถ๐‘š๐‘Ž๐‘ฅ(๐‘†1) = max

๐‘—โˆˆ๐’ฅ 1{๐‘†๐‘— + ๐‘๐‘—}.

Each task must be performed during the planning horizon:

๐’ฏโˆ‘๐‘ก=1

๐‘ฅ๐‘—๐‘ก = 1, ๐‘— โˆˆ ๐’ฅ . (16)

The resource limits must be respected:

โˆ‘๐‘—โˆˆ๐’ฅ

๐‘กโˆ‘๐‘กโ€ฒ=๐‘กโˆ’๐‘๐‘—+1

๐‘Ÿ๐‘๐‘—๐‘Ÿ๐‘ฅ๐‘—๐‘กโ€ฒ โ‰ค ๐‘Ÿ๐‘Ž๐‘Ÿ๐‘ก, โˆ€๐‘ก โˆˆ ๐’ฏ , โˆ€๐‘Ÿ โˆˆ โ„›. (17)

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The precedence relationships between the tasks must be respected:

๐‘‡โˆ‘๐‘กโ€ฒ=๐‘กโˆ’๐‘๐‘—+1

๐‘ฅ๐‘—๐‘กโ€ฒ +๐‘กโˆ‘

๐‘กโ€ฒ=1

๐‘ฅ๐‘—โ€ฒ๐‘กโ€ฒ โ‰ค 1, โˆ€(๐‘—,๐‘—โ€ฒ) โˆˆ ฮ“,โˆ€๐‘ก โˆˆ ๐’ฏ . (18)

The strict precedence relationships between the tasks must be respected:

๐‘ฅ๐‘—๐‘ก โˆ’ ๐‘ฅ๐‘—โ€ฒ(๐‘ก+๐‘๐‘—) = 0, โˆ€(๐‘—,๐‘—โ€ฒ) โˆˆ โ„‹,โˆ€๐‘ก โˆˆ ๐’ฏ . (19)

The completion of the training time of the remaining crews may notexceed the due dates:

๐ถ๐‘š๐‘Ž๐‘ฅ(๐‘†๐‘) โ‰ค ๐ท๐‘ ๐‘ = 2, . . . , ๐ถ. (20)

Formulation as an integer programming problem

The formulation below is a RCPSP. As it is known, such a problem canbe represented as an integer programming problem.

We introduce the formal tasks:

โˆ™ zero task ๐‘—0, ๐‘0 = 0;

โˆ™ final task for the first crew ๐‘— = ๐ฝ + 1 that should be performed afterall tasks of the first crew, ๐‘๐ฝ+1 = 0;

โˆ™ . . . ;

โˆ™ final task for the crew ๐‘ โˆˆ ๐’ž, ๐‘— = ๐ฝ + ๐‘ that should be performed afterall tasks of the crew ๐‘ โˆˆ ๐’ž, ๐‘๐ฝ+๐‘ = 0;

Let ๐‘’๐‘ ๐‘— and ๐‘™๐‘ ๐‘— be the earliest and the latest moments at which task๐‘— โˆˆ ๐’ฅ๐‘˜ can be performed.

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Then we get the following optimization problem:๐‘™๐‘ ๐ฝ+1โˆ‘

๐‘ก=๐‘’๐‘ ๐ฝ+1

๐‘ก๐‘ฅ(๐ฝ+1)๐‘ก โ†’ min; (21)

๐‘™๐‘ ๐‘—โˆ‘๐‘ก=๐‘’๐‘ ๐‘—

๐‘ฅ๐‘—๐‘ก = 1 ๐‘— โˆˆ ๐’ฅ ; (22)

๐‘™๐‘ ๐‘—โˆ‘๐‘ก=๐‘’๐‘ ๐‘—

๐‘ก๐‘ฅ๐‘—๐‘ก โˆ’๐‘™๐‘ ๐‘–โˆ‘

๐‘ก=๐‘’๐‘ ๐‘–

๐‘ก๐‘ฅ๐‘–๐‘ก โ‰ฅ ๐‘๐‘– (๐‘–, ๐‘—) โˆˆ ๐บ = (๐ฝ,ฮ“); (23)

๐‘™๐‘ ๐‘—โˆ‘๐‘ก=๐‘’๐‘ ๐‘—

๐‘ก๐‘ฅ๐‘—๐‘ก โˆ’๐‘™๐‘ ๐‘–โˆ‘

๐‘ก=๐‘’๐‘ ๐‘–

๐‘ก๐‘ฅ๐‘–๐‘ก = ๐‘๐‘– (๐‘–, ๐‘—) โˆˆ ๐ป = (๐ฝ,โ„‹); (24)

โˆ‘๐‘—โˆˆ๐’ฅ

๐‘Ÿ๐‘๐‘—๐‘Ÿ

๐‘กโˆ‘๐œ=๐œŽ(๐‘ก,๐‘—)

๐‘ฅ๐‘—๐œ โ‰ค๐‘Ÿ๐‘Ž๐‘Ÿ๐‘ก ๐‘ก โˆˆ ๐’ฏ , ๐‘Ÿ โˆˆ โ„›, (25)

with ๐œŽ(๐‘ก, ๐‘—) = max(0, ๐‘กโˆ’ ๐‘๐‘— + 1);

๐‘™๐‘ ๐ฝ+๐‘โˆ‘๐‘ก=๐‘’๐‘ ๐ฝ+๐‘

๐‘ก๐‘ฅ(๐ฝ+๐‘)๐‘ก โ‰ค ๐ท๐‘ ๐‘ = 2, . . . , ๐ถ. (26)

where (21) is the price function that minimizes the total training time ofthe first crew, constraint (22) means that each task must be performed, con-straints (23) and (24) describe the precedence and strict precedence relation-ships, respectively, constraint (25) is a resource constraint, and constraint(26) means that the completion of the training time of the remaining crewsmay not exceed the due dates.

Given the structure of the constraints as well as the size of the inputdata, one can observe that this problem is quite complex for modern solversimplementing standard algorithms of integer programming. Therefore, it ismore likely, that an optimal solution cannot be obtained within a reasonabletime. For this reason, we will develop heuristic algorithms for solving thisproblem.

3.2.2 Model 6.

Notations First, we introduce some time intervals:

โˆ™ ๐‘Š = {1, . . . ,|๐‘Š |} โ€” set of weeks in the planning period. The maximumis |๐‘Š | = 130 weeks (2.5 years). Because of the time previously givento specific operations, this set can be significantly reduced.

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1 2 3 4 5 6 7

week 1 week 2

. ..

. . .123...hours

days

1 2 3 4 5... t

Figure 5: Time intervals

โˆ™ ๐ท๐‘ค = {1,2,3,4,5,6,7} โ€” set of days per week, ๐‘ค โˆˆ ๐‘Š . If required, theoperator can change this set, increasing or decreasing it (e.g., holidays,etc.).

โˆ™ ๐ป๐‘ค๐‘‘ = {1, . . . ,18,19} โ€” set of half-hour intervals of the day ๐‘‘ โˆˆ ๐ท๐‘ค ofweek ๐‘ค โˆˆ ๐‘Š .

It is assumed that the first interval begins at 9.00 a.m. and ends latestat 6.00 p.m., followed by the dinner. Due to the dinner, it is necessary todivide the days in the model. In some cases, according to the informationavailable to us, a typical schedule may increase the duration of the day for1 hour (two intervals). Probably, the following process is possible: a feasibleschedule with the current set of intervals cannot be developed, the programindicates where the conflict is, the operator decides to extend the workingday, and to re-develop the schedule. As in a typical schedule there are veryfew operations that take no more than half an hour, perhaps the partition ofthe working day by hours (i.e., not by half of an hour) can be done. Thenwe have ๐ป๐‘ค๐‘‘ = {1, . . . , 9} and a significantly smaller dimension.

It will be convenient to work with restrictions such as โ€œnot more than 2times a weekโ€, โ€œin the morningโ€, etc. On the other hand, for the calculationof the duration of the steps, it is necessary to have a linear decompositionof the planning horizon. To do this, let us arrange all triples (๐‘ค, ๐‘‘, โ„Ž) inlexicographical order and to each triple, we associate its number: (๐‘ค, ๐‘‘, โ„Ž) โ†’๐‘ก(๐‘ค, ๐‘‘, โ„Ž), where ๐‘ก โˆˆ ๐‘‡ = {1, . . . , |๐‘‡ |}, |๐‘‡ | is the number of triples (see Fig. 5).

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t

cosmonaut 1cosmonaut 2cosmonaut 3

6 months

Crew 1

Due date of crew 1

Figure 6: Training schedule for all crews.

We denote the set of all triples (๐‘ค,๐‘‘,โ„Ž) as ๐‘Œ :

๐‘Œ = {(๐‘ค,๐‘‘,โ„Ž)|๐‘ค โˆˆ ๐‘Š,๐‘‘ โˆˆ ๐ท๐‘ค, โ„Ž โˆˆ ๐ป๐‘ค๐‘‘}.

The crews start the training at different moments (see. Fig. 6). There-fore, over the period of 2.5 years, some cosmonauts have already mastered apart of the operations and thus, each cosmonaut has its own set of currentoperations.

Denote by ๐‘Œ (๐‘˜,๐‘—) the set of all possible time intervals for performing task๐‘— by cosmonaut ๐‘˜. In this case, we do not consider the days when cosmonaut๐‘˜ is on vacation and consider time constraints (limits ๐‘’๐‘— and ๐‘™๐‘—).

Next, we introduce the basic notations.

โˆ™ ๐ถ = {1, . . . ,|๐ถ|} โ€” set of crews.

โˆ™ ๐พ๐‘ โ€” set of cosmonauts in the crew ๐‘ โˆˆ ๐ถ. Usually, ๐พ๐‘ = {1,2,3}.

โˆ™ ๐พ โ€” the complete set of cosmonauts.

โˆ™ ๐ฝ๐‘ โ€” set of tasks of the crew ๐‘ โˆˆ ๐ถ.

โˆ™ ๐ฝ๐‘˜ โ€” set of tasks of cosmonaut ๐‘˜, which are required for the imple-mentation of the training plan. We divide this set into the followingsubsets:

โ€“ ๐ฝ๐‘‡๐‘˜ โ€” set of technical tasks of the cosmonaut ๐‘˜, all tasks with

onboard complexes are contained in it. Denote all tasks for theonboard complexes as ๐ฝ๐ต

๐‘˜ .

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โ€“ ๐ฝ๐น๐‘˜ โ€” set of physical training tasks of the cosmonaut ๐‘˜ (which last

2 hours, i.e., we have 4 intervals).โ€“ ๐ฝ๐ด

๐‘˜ โ€” set of administrative tasks of the cosmonaut ๐‘˜.โ€“ ๐ฝ๐ฟ

๐‘˜ โ€” set of language lessons of the cosmonaut ๐‘˜ (which last 2hours, i.e., we have 4 intervals).

We distinguish subsets in the set ๐ฝ๐ต๐‘˜ which contain the tasks of the

onboard complexes ๐ฝ๐ต1๐‘˜ , ๐ฝ๐ต2

๐‘˜ , . . . , ๐ฝ๐ต๐‘š๐‘˜๐‘˜ , where ๐‘š๐‘˜ denotes the number

of onboard complexes which should examine cosmonaut ๐‘˜.

โˆ™ ๐‘๐‘— โ€” ๐‘— โˆˆ ๐ฝ execution time.

โˆ™ ๐‘… = {1, . . . ,|๐‘…|} โ€” set of resources. Each cosmonaut is also a resourcethat is available in amount 1 at any time.

โˆ™ ๐‘Ÿ๐‘๐‘—๐‘Ÿ โ€” amount of the resource ๐‘Ÿ needed to perform the task ๐‘—.

โˆ™ ๐‘Ÿ๐‘Ž๐‘Ÿ๐‘ค๐‘‘โ„Ž โ€” amount of the resource ๐‘Ÿ accessible during time interval โ„Ž ofa day ๐‘‘, week ๐‘ค.

โˆ™ ๐‘’๐‘—, ๐‘™๐‘— โ€” the earliest and the latest moments at which task ๐‘— โˆˆ ๐ฝ canbe performed.

โˆ™ ๐ฝ ๐‘๐‘œ๐‘ข๐‘›๐‘‘๐‘˜ โ€” set of tasks for which time constraints are defined. Due dates

can also be described using these boundaries.

โˆ™ ๐ฝ๐‘‘๐‘Ž๐‘ฆ๐‘˜ โ€” set of tasks that should be performed during one day.

โˆ™ ๐ฝ๐‘ค๐‘’๐‘’๐‘˜๐‘˜ โ€” set of tasks that should be performed during one week.

โˆ™ ๐ฝ123๐‘ โ€” set of tasks that should be performed by all cosmonauts of the

crew ๐‘ โˆˆ ๐ถ. Similarly, we define the sets ๐ฝ12๐‘ , ๐ฝ13

๐‘ , ๐ฝ23๐‘ .

โˆ™ ๐บ = (๐ฝ,ฮ“) โ€“ the graph of the precedence relations between the tasks:We have (๐‘—,๐‘—โ€ฒ) โˆˆ ฮ“ if task ๐‘— must be performed before the task ๐‘—โ€ฒ.

โˆ™ ๐ป = (๐ฝ,โ„‹) โ€“ the graph of the strict precedence relations between thetasks: We have (๐‘—,๐‘—โ€ฒ) โˆˆ โ„‹ if task ๐‘—โ€ฒ must be performed immediatelyafter the task ๐‘—.

We can divide the operations that take more than one day into one-dayoperations. For these operations, we can introduce the graph of the "almoststrict" precedence relations ๐‘†๐ป. If (๐‘—1,๐‘—2) โˆˆ ๐‘†๐ป, then operation ๐‘—2 should beperformed after operation ๐‘—1 and there should be one time interval betweenthem. So, we get a sequence of one-day operations divided by the dinnerinstead of a multi-day operation.

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Variables

โˆ™ ๐‘ฅ๐‘˜๐‘—๐‘ค๐‘‘โ„Ž โ€” Boolean variable: We have ๐‘ฅ๐‘˜๐‘—๐‘ค๐‘‘โ„Ž = 1 if and only if thecosmonaut ๐‘˜ starts the task ๐‘— from the interval โ„Ž of the day ๐‘‘ of week๐‘ค;

โˆ™ ๐‘ฆ๐‘–๐‘˜๐‘ค โ€” Boolean variable: We have ๐‘ฆ๐‘–๐‘˜๐‘ค = 1 if and only if the cosmonaut๐‘˜ trains for the onboard complex ๐‘– during the week ๐‘ค.

Constraints The following relations between the variables have to be sat-isfied: โˆ‘

๐‘—โˆˆ๐ฝ๐ต๐‘–๐‘˜

๐‘ฅ๐‘˜๐‘—๐‘ค๐‘‘โ„Ž โ‰ค ๐‘ฆ๐‘–๐‘˜๐‘ค, (27)

โˆ€๐‘˜ โˆˆ ๐พ, โˆ€๐‘– โˆˆ {1, . . . ,๐‘š๐‘˜}, โˆ€(๐‘ค,๐‘‘,โ„Ž) โˆˆ ๐‘Œ.

The resource limits have to be respected:โˆ‘๐‘˜โˆˆ๐พ

โˆ‘๐‘—โˆˆ๐ฝ

๐‘Ÿ๐‘๐‘—๐‘Ÿโˆ‘

(๐‘คโ€ฒ,๐‘‘โ€ฒ,โ„Žโ€ฒ) โˆˆ ๐‘Œ๐‘ก(๐‘ค,๐‘‘,โ„Ž) โˆ’ ๐‘๐‘— + 1 โ‰ค ๐‘ก(๐‘คโ€ฒ,๐‘‘โ€ฒ,โ„Žโ€ฒ) โ‰ค ๐‘ก(๐‘ค,๐‘‘,โ„Ž)

๐‘ฅ๐‘˜๐‘—๐‘คโ€ฒ๐‘‘โ€ฒโ„Žโ€ฒ โ‰ค ๐‘Ÿ๐‘Ž๐‘Ÿ๐‘ค๐‘‘โ„Ž,

โˆ€๐‘Ÿ โˆˆ ๐‘…, โˆ€(๐‘ค,๐‘‘,โ„Ž) โˆˆ ๐‘Œ. (28)

In this inequality, for each (๐‘ค,๐‘‘,โ„Ž) โˆˆ ๐‘Œ , we consider only the operations thatare performed at this interval, i.e., which started in the interval [๐‘ก(๐‘ค,๐‘‘,โ„Ž) โˆ’๐‘๐‘— + 1, ๐‘ก(๐‘ค,๐‘‘,โ„Ž)].

Each cosmonaut should perform all required tasks:โˆ‘(๐‘ค,๐‘‘,โ„Ž)โˆˆ๐‘Œ (๐‘˜,๐‘—)

๐‘ฅ๐‘˜๐‘—๐‘ค๐‘‘โ„Ž = 1, โˆ€๐‘˜ โˆˆ ๐พ, โˆ€๐‘— โˆˆ ๐ฝ๐‘–. (29)

Each cosmonaut must have 4 hours (2 tasks for 2 hours) of physicaltraining per week:โˆ‘

๐‘—โˆˆ๐ฝ๐น๐‘˜

โˆ‘๐‘‘โˆˆ๐ท๐‘ค

โˆ‘โ„Žโˆˆ๐ป๐‘ค๐‘‘

๐‘ฅ๐‘˜๐‘—๐‘ค๐‘‘โ„Ž โ‰ค 2, โˆ€๐‘˜ โˆˆ ๐พ,โˆ€๐‘ค โˆˆ ๐‘Š. (30)

Similarly, we can set constraints on the language study. Each cosmonautmust have 4 hours of language lessons per week at the beginning of the wholetraining: โˆ‘

๐‘—โˆˆ๐ฝ๐ฟ๐‘˜

โˆ‘๐‘‘โˆˆ๐ท๐‘ค

โˆ‘โ„Žโˆˆ๐ป๐‘ค๐‘‘

๐‘ฅ๐‘˜๐‘—๐‘ค๐‘‘โ„Ž โ‰ค 2, โˆ€๐‘˜ โˆˆ ๐พ,โˆ€๐‘ค โˆˆ ๐‘Š. (31)

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Similarly, we have constraints for the administrative tasks:โˆ‘๐‘—โˆˆ๐ฝ๐ด

๐‘˜

โˆ‘๐‘‘โˆˆ๐ท๐‘ค

โˆ‘โ„Žโˆˆ๐ป๐‘ค๐‘‘

๐‘ฅ๐‘˜๐‘—๐‘ค๐‘‘โ„Ž โ‰ค 4, โˆ€๐‘˜ โˆˆ ๐พ,โˆ€๐‘ค โˆˆ ๐‘Š. (32)

It is forbidden to plan more than 4 hours of training for one onboardcomplex per day: โˆ‘

๐‘—โˆˆ๐ฝ๐ต๐‘–๐‘˜

โˆ‘โ„Žโˆˆ๐ป๐‘ค๐‘‘

๐‘๐‘—๐‘ฅ๐‘˜๐‘—๐‘ค๐‘‘โ„Ž โ‰ค 8, (33)

โˆ€๐‘˜ โˆˆ ๐พ, โˆ€๐‘– โˆˆ {1, . . . ,๐‘š๐‘˜},โˆ€๐‘ค โˆˆ ๐‘Š,โˆ€๐‘‘ โˆˆ ๐ท๐‘ค.

It is forbidden to plan the training for more than two onboard complexesper week:

๐‘š๐‘˜โˆ‘๐‘˜=1

๐‘ฆ๐‘–๐‘˜๐‘ค โ‰ค 2, โˆ€๐‘˜ โˆˆ ๐พ, โˆ€๐‘ค โˆˆ ๐‘Š. (34)

There are time limits for some tasks:

๐‘ฅ๐‘˜๐‘—๐‘คโ„Ž๐‘‘ = 0, (35)

โˆ€๐‘˜ โˆˆ ๐พ, โˆ€๐‘— โˆˆ ๐ฝ ๐‘๐‘œ๐‘ข๐‘›๐‘‘๐‘˜ ,โˆ€(๐‘ค,๐‘‘,โ„Ž) โˆˆ ๐‘Œ : ๐‘ก(๐‘ค,๐‘‘,โ„Ž) โ‰ค ๐‘’๐‘— โˆ’ 1,

๐‘ฅ๐‘˜๐‘—๐‘คโ„Ž๐‘‘ = 0, (36)

โˆ€๐‘˜ โˆˆ ๐พ, โˆ€๐‘— โˆˆ ๐ฝ ๐‘๐‘œ๐‘ข๐‘›๐‘‘๐‘˜ ,โˆ€(๐‘ค,๐‘‘,โ„Ž) โˆˆ ๐‘Œ : ๐‘ก(๐‘ค,๐‘‘,โ„Ž) โ‰ฅ ๐‘™๐‘— + 1.

The precedence relations (if (๐‘—1,๐‘—2) โˆˆ ฮ“, then task ๐‘—1 must be performedbefore task ๐‘—2) must be respected:โˆ‘

(๐‘ค,๐‘‘,โ„Ž)โˆˆ๐‘Œ (๐‘˜,๐‘—2)

๐‘ก(๐‘ค,๐‘‘,โ„Ž)๐‘ฅ๐‘˜๐‘—2๐‘ค๐‘‘โ„Ž โˆ’โˆ‘

(๐‘ค,๐‘‘,โ„Ž)โˆˆ๐‘Œ (๐‘˜,๐‘—1)

๐‘ก(๐‘ค,๐‘‘,โ„Ž)๐‘ฅ๐‘˜๐‘—1๐‘ค๐‘‘โ„Ž โ‰ค ๐‘๐‘—1 , (37)

โˆ€๐‘˜ โˆˆ ๐พ, โˆ€(๐‘—1,๐‘—2) โˆˆ ฮ“.

The strict precedence relations must be respected:โˆ‘(๐‘ค,๐‘‘,โ„Ž)โˆˆ๐‘Œ (๐‘˜,๐‘—2)

๐‘ก(๐‘ค,๐‘‘,โ„Ž)๐‘ฅ๐‘˜๐‘—2๐‘ค๐‘‘โ„Ž โˆ’โˆ‘

(๐‘ค,๐‘‘,โ„Ž)โˆˆ๐‘Œ (๐‘˜,๐‘—1)

๐‘ก(๐‘ค,๐‘‘,โ„Ž)๐‘ฅ๐‘˜๐‘—1๐‘ค๐‘‘โ„Ž = ๐‘๐‘—1 , (38)

โˆ€๐‘˜ โˆˆ ๐พ, โˆ€(๐‘—1,๐‘—2) โˆˆ โ„‹.

We must consider that ๐ป โŠ† ๐บ.The "almost strict" precedence relations must be respected:โˆ‘

(๐‘ค,๐‘‘,โ„Ž)โˆˆ๐‘Œ (๐‘˜,๐‘—2)

๐‘ก(๐‘ค,๐‘‘,โ„Ž)๐‘ฅ๐‘˜๐‘—2๐‘ค๐‘‘โ„Ž โˆ’โˆ‘

(๐‘ค,๐‘‘,โ„Ž)โˆˆ๐‘Œ (๐‘˜,๐‘—1)

๐‘ก(๐‘ค,๐‘‘,โ„Ž)๐‘ฅ๐‘˜๐‘—1๐‘ค๐‘‘โ„Ž = ๐‘๐‘—1 + 1, (39)

โˆ€๐‘˜ โˆˆ ๐พ, โˆ€(๐‘—1,๐‘—2) โˆˆ ๐’ฎโ„‹.

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Remark 3.1. If the date of the vacation of the cosmonaut is not defined andmust be determined during the planning phase, it can be considered as anadditional task of appropriate length.

Remark 3.2. The lunch time can be strictly fixed or it can be considered asa task with time constraints (for example, from 12.00 a.m. to 3.00 p.m.).

Objective function Since the first crew starts before the others, it hasthe priority in the planning phase, and a possible formulation of the problemincludes the minimization of its total training time. To do this, we introducean additional variable ๐‘ก๐‘“ and constraints on the additional last task ๐‘ก๐‘“๐‘˜ foreach cosmonaut ๐‘˜ โˆˆ ๐พ1:

๐‘ก(๐‘ค,๐‘‘,โ„Ž)๐‘ฅ๐‘˜๐‘—๐‘“๐‘˜๐‘ค๐‘‘โ„Ž โ‰ค ๐‘ก๐‘“ , โˆ€๐‘˜ โˆˆ ๐พ1, โˆ€(๐‘ค,๐‘‘,โ„Ž) โˆˆ ๐‘Œ. (40)

In this case, the objective function is:

min ๐‘ก๐‘“ . (41)

4 Approaches

4.1 Algorithm-3.0

Consider the model 1.We combine the variables ๐‘ฅ๐‘˜,๐‘—,๐‘ž of the same qualification level and onboard

complex into a vector ๏ฟฝ๏ฟฝ๐‘—,๐‘ž = {๐‘ฅ1,๐‘—,๐‘ž, ๐‘ฅ2,๐‘—,๐‘ž, ๐‘ฅ3,๐‘—,๐‘ž}.

Step 1 First, we are going to find all onboard complexes which require allthree cosmonauts or none of them have the same qualification level. Thenthere exists only one option of the training plan satisfying this condition:

๏ฟฝ๏ฟฝ๐‘—๐‘ž = {1, 1, 1}, โˆ€๐‘—,๐‘ž : ๐‘›๐‘—,๐‘ž = 3,

๏ฟฝ๏ฟฝ๐‘—๐‘ž = {0, 0, 0}, โˆ€๐‘—,๐‘ž : ๐‘›๐‘—,๐‘ž = 0.

Step 2 Let ๐ฝ โ€ฒ be the number of onboard complexes left after the previ-ous step. Consider this problem as ๐ฝ โ€ฒ independent subproblems. For eachsubproblem, we will find the minimum separately.

It can be interpreted as ๐ฝ โ€ฒ boxes (Fig. 7) with

๐ถ๐‘— = ๐ถ๐‘›๐‘—,๐‘†

3 ยท ๐ถ๐‘›๐‘—,๐‘‚

3โˆ’๐‘›๐‘—,๐‘†ยท ๐ถ๐‘›๐‘—,๐‘ˆ

3โˆ’๐‘›๐‘—,๐‘†โˆ’๐‘›๐‘—,๐‘‚, ๐‘— = 1, ๐ฝ โ€ฒ,

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Figure 7: Each box is an onboard complex, and each item is an option forthe distribution.

objects in each of them. It is allowed to take only one of them from eachbox.

Considering all subproblems as independent ones reduces the number ofvariations from

๐ฝ โ€ฒโˆ๐‘—=1

๐ถ๐‘— โ‰ค 6๐ฝ โ€ฒ

to๐ฝ โ€ฒโˆ‘๐‘—=1

๐ถ๐‘— โ‰ค 6๐ฝ โ€ฒ.

Step 3 For each required qualification level on each onboard complex, acosmonaut should be determined. Due to this fact, the problem is less com-plicated than that where a subset of qualification levels should be chosen onwhich the best value is distinguished like in a knapsack problem.

At this step, we sort the values ๐‘1,๐‘—,๐‘ž in non-increasing order to cover thedifference among the total time of the training plans of each cosmonaut atthe next steps of the algorithm.

Step 4 Perform ๐ฝ โ€ฒ iterations.At iteration ๐‘—, find ๏ฟฝ๏ฟฝ๐‘—,๐‘†, ๏ฟฝ๏ฟฝ๐‘—,๐‘‚, ๏ฟฝ๏ฟฝ๐‘—,๐‘ˆ such that

min๏ฟฝ๏ฟฝ๐‘—,๐‘† , ๏ฟฝ๏ฟฝ๐‘—,๐‘‚, ๏ฟฝ๏ฟฝ๐‘—,๐‘ˆ

(๐‘…๐‘—(๏ฟฝ๏ฟฝ๐‘—,๐‘†, ๏ฟฝ๏ฟฝ๐‘—,๐‘‚, ๏ฟฝ๏ฟฝ๐‘—,๐‘ˆ)) = min๏ฟฝ๏ฟฝ๐‘—,๐‘† , ๏ฟฝ๏ฟฝ๐‘—,๐‘‚, ๏ฟฝ๏ฟฝ๐‘—,๐‘ˆ

(3โˆ‘

๐‘˜โ€ฒ=1

โˆ‘๐‘˜โ€ฒ>๐‘˜

| ๐œ๐‘˜,๐‘— โˆ’ ๐œ๐‘˜โ€ฒ,๐‘— |

),

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๐œ๐‘˜,0 = 0,

๐œ๐‘˜,๐‘— = ๐œ๐‘˜,๐‘—โˆ’1 +โˆ‘

๐‘ž=๐‘†,๐‘‚,๐‘ˆ

๐‘๐‘’๐‘˜,๐‘—,๐‘ž๐‘ฅ๐‘˜,๐‘—,๐‘ž,

๐‘˜ = 1, 3, ๐‘— = 1, ๐ฝ โ€ฒ, ๐‘’๐‘˜ = 1 if and only if cosmonaut ๐‘˜ is experienced and๐‘ฅ๐‘˜,๐‘—,๐‘ž satisfies the constraints (4) and (5).

If ๐‘›๐‘—,๐‘ž = 1, then

๏ฟฝ๏ฟฝ๐‘—,๐‘ž = {1, 0, 0} or {0, 1, 0} or {0, 0, 1}.

If ๐‘›๐‘—,๐‘ž = 2, then

๏ฟฝ๏ฟฝ๐‘—,๐‘ž = {1, 1, 0} or {0, 1, 1} or {1, 0, 1}.

Complexity At each iteration, we have to calculate the R maximum of

๐ถ๐‘— = ๐ถ๐‘›๐‘—,๐‘†

3 ยท ๐ถ๐‘›๐‘—,๐‘‚

3โˆ’๐‘›๐‘—,๐‘†ยท ๐ถ๐‘›๐‘—,๐‘ˆ

3โˆ’๐‘›๐‘—,๐‘†โˆ’๐‘›๐‘—,๐‘‚= 6

times with ๐‘›๐‘—,๐‘† = ๐‘›๐‘—,๐‘‚ = ๐‘›๐‘—,๐‘† = 1.Due to this fact, if there are ๐‘› onboard complexes, the number of opera-

tions is equal to ๐‘‚(๐‘›).

4.2 Partition algorithm

Consider the model 4.

4.2.1 ๐’ฉ๐’ซ-completeness

The first goal in the analysis is an ๐’ฉ๐’ซ-completeness proof for the problemwith the criterion (14) subject to (12). Obviously, the problem is ๐’ฉ๐’ซ-hard.By a local replacement [5], it is possible to show that the problem is also๐’ฉ๐’ซ-complete. Suppose that all cosmonauts have an equal training timefor every job. Then the problem reduces to the multiway-partition problemwhich is indeed ๐’ฉ๐’ซ-complete. So, the cosmonaut assignment problem is๐’ฉ๐’ซ-complete as well.

4.2.2 Algorithms

Heuristic algorithm As a first approximation, a greedy algorithm basedon heuristic considerations can be used. At each step, the algorithm fixes ajob and makes an assignment to the cosmonaut, which will have the min-imum objective function value (14). The complexity of such an algorithm

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is ๐‘‚(๐‘š2๐‘›), where ๐‘š is the number of cosmonauts and ๐‘› is the number of jobs.

Example (Tables 1-4)

Table 1: InitializationJob โ†’ First Second Third

Commander 1 2 3Engineer 2 4 6

User 2 1 2

Table 2: Assignment of the first jobJob โ†’ First Second Third

Commander 1 2 3Engineer 2 4 6

User 2 1 2

Table 3: Assignment of the second jobJob โ†’ First Second Third

Commander 1 2 3Engineer 2 4 6

User 2 1 2

Table 4: Assignment of the third jobJob โ†’ First Second Third

Commander 1 2 3Engineer 2 4 6

User 2 1 2

Here we have ๐›ฟ = 3, while the optimal objective function value is 0.

Observation It is possible to find an instance which will have an arbitraryerror as it is shown in Table 5.

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Table 5: "Bad" instance[h] Job โ†’ First Second Third

Commander 0 A AEngineer A A A

User A A A

Here A is a random number. So the error of the heuristic algorithm willbe equal to A, but the optimal objective function value is equal to 0.

According to the proposed algorithm, a program was written and testswere carried out on real data provided by the Yu. A. Gagarin research & testcosmonaut training center. In spite of the above observation, the error of thegreedy algorithm does not exceed 10% of the optimum value.

Exact algorithm Assume that we have found an optimal solution. Thenany permutation of the work of one cosmonaut would lead to the fact that theobjective function will increase or remain the same. If there is a permutation,leading to a decrease in the objective function, it can be argued that thechosen solution is not optimal. Let a solution be obtained, for example byusing the heuristic algorithm described earlier. Then we can find out whetherit is optimal using the following lemma.

Lemma 4.1. Let ๐’ฅ โ€ฒ๐‘˜ = {๐’ฅ โ€ฒ

1, . . . ,๐’ฅ โ€ฒ๐‘š} be the feasible subsets of the set ๐’ฅ .

Moreover, let all cosmonauts be sorted in non-increasing order of the keys๐‘ฆ๐‘˜. Then it is possible to check whether an optimal solution is obtained with๐‘‚(|๐’ฅ โ€ฒ

1|(๐‘š +โˆ‘

2โ‰ค๐‘ โ‰ค๐‘š

|๐’ฅ โ€ฒ๐‘  |)) operations in the worst case.

Proof An exhaustive search can be avoided if we consider that the objec-tive function can only increase or remain the same, if the cosmonauts withmaximum or minimum time of the training are not involved into the permu-tation.

Algorithm

1. Find a solution by the heuristic algorithm.

2. Sort the cosmonauts in non-increasing order of the keys ๐‘ฆ๐‘˜.

3. Check whether the solution is optimal using Lemma 4.1.

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4. If the solution is not optimal, apply a permutation that leads to adecrease of ๐›ฟ.

5. Repeat step 2.

4.3 Integer programming

The software package CPLEX has been used for the solution of the prob-lem.

1. The integer constraints were relaxed, and then the linear programmingproblem was solved.

2. The original problem was solved by the branch-and-bound method us-ing the solution of the relaxed problem as a lower bound.

3. Constraints on the one of the variables were added, and then the algo-rithm iterates again.

5 Comparison of the algorithmsThe two proposed algorithms were tested on real data and compared

with an integer programming technique. All results are shown in Table 6.The running time of the exact branch-and-bound method, implemented onCPLEX, was limited to 15 minutes.

Table 6: Numerical ExperimentsSample Experience Alg.-3.0 Partition alg. Integer Progr.(CPLEX)

max min ๐›ฟ max min ๐›ฟ max min ๐›ฟ

minSQRM

3 Inexperienced 883.25 881.00 2.25 887.8 886.75 1.05 888.05 887.75 0.33 Experienced 570.00 568.50 1.5 570.5 569 1.5 570 569.5 0.51 Exp 2 Inexp 697.25 695.25 22 Exp 1 Inexp 616.5 612.75 3.75

6months

3 Inexperienced 266.25 265 1.25 265.75 265.2 0.553 Experienced 234.2 233 1.2 233.75 233.25 0.51 Exp 2 Inexp 244.45 244 0.452 Exp 1 Inexp 233.75 233.25 0.5

2years

3 Inexperienced 661.25 657.5 3.75 659.85 659.75 0.13 Experienced 353.5 353.05 0.45 353.5 353 0.51 Exp 2 Inexp 484.05 481.75 2.32 Exp 1 Inexp 393.5 382.5 1

STFL

3 Inexperienced 925.75 922.25 3.5 925 924.8 0.23 Experienced 587 586.5 0.5 587 586.5 0.51 Exp 2 Inexp 731.5 730.75 0.752 Exp 1 Inexp 628.75 628 0.75

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It can be observed that the third algorithm based on integer program-ming techniques has the best accuracy, but it still cannot be guaranteed thatthe values obtained are indeed optimal. For example, it can be seen whencompared with the second algorithm for the data "2 years" for three experi-enced cosmonauts. In this case, the greedy algorithm gives a better solution.However, among all data, this is the only case.

The essential difference between the first and second algorithms on oneside and the third algorithm on the other side is that two qualification levelswith the highest training time would be assigned to two different cosmonauts,whereas this is not mandatory in the third algorithm. It can be found thatobvious correlations in the distribution of the work with medium and smalldurations in the second and third algorithms cannot be observed.

From our experiments, it can be seen that the solution quality of the firsttwo algorithms is similar.

6 ConclusionThis article described the process of the training of cosmonauts and the

relevance of the planning phase. Some models and methods for solving thetraining scheduling problem were suggested. For the problem of volumeplanning, several models and three algorithms were presented. To solve thecalendar problem, integer programming methods problems and RCPSP al-gorithms will be used. In the future, we plan to develop an automated work-place (AWP), which allows to automate the process of planning the trainingof the ISS crews.

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[3] Bronnikov, S.V., Development of requirements for training crews of thespace station, Space Technics and Technology, โ„–1(8), p.81-87, 2015.

[4] Burkard, R., Dellโ€™Amico, M., and Martello, S. (2009). Assignment Prob-lems. SIAM e-books. Society for Industrial and Applied Mathematics(SIAM, 3600 Mar- ket Street, Floor 6, Philadelphia, PA 19104).

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[5] Garey, M. and Johnson, D. (1979). Computers and Intractability: AGuide to the Theory of NP-Completeness. A Series of books in the math-ematical sciences. W. H. Freeman.

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[7] Karmarkar, N., R.M. Karp, G.S. Lueker, and A.M. Odlyzko, Probabilisticanalysis of optimum partitions, Journal of Applied Probability, Vol.23,1986, 626-645.

[8] Kolokolceva, O. M., Belyaev A. M., Kozlechkov, A. G., Budnikova, T. S.,Stanilovskaya, V. I., Potockiy, S. I., Automated flight planning system onrussian segment of the international space station, Software and Systems,Vol. 3, 2013, 48โ€“54.

[9] Korf, R.E., A complete anytime algorithm for number partitioning, Ar-tificial Intelligence, 106, 181โ€“203, 1998

[10] Korf, R.E. (2010). Objective functions for multi- way number partition-ing. In Proceedings of the Third Annual Symposium on CombinatorialSearch, SOCS 2010, Stone Mountain, Atlanta, Georgia, USA, July 8-10,2010.

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[12] Punnen, A. (2004). On bottleneck assignment problems under catego-rization. Computers & Oper. Res., 31, 151โ€“ 154.

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