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1. CONFERENCE PROCEEDINGS
16th
International Conference on Project
Management and Scheduling
Rome, April 17-20, 2018
pms2018.ing.uniroma2.it

3. Proceedings of the 16th International Conference on Project Management and
Scheduling
PMS 2018 - April 17-20, 2018 - Rome, Italy
ISBN: 9788894982022
Editors:
Massimiliano Caramia
Lucio Bianco
Stefano Giordani
Address of the Editors:
University of Rome “Tor Vergata”
Dipartimento di Ingegneria dell’Impresa
Via del Politecnico, 1
00133 Roma - Italy
Tel. +39 06 72597360
Fax. +39 06 72597305
email: caramia@dii.uniroma2.it
Published by:
TexMat
Via di Tor Vergata, 93-95
00133 Roma
Tel. +39 06 2023572
www.texmat.it
e-mail: info@texmat.it
Place and Date of Publication:
Rome (Italy), March 31, 2018
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4. ii

5. PMS 2018 Preface
Preface
This volume contains the papers that will be presented at PMS 2018, the 16th
International
Conference on Project Management and Scheduling to be held on April, 17-20 2018 in Rome
- Italy.
The EURO Working Group on Project Management and Scheduling was established by Pro-
fessors Luı́s Valadares Tavares and Jan Weglarz during the EURO VIII Conference, Lisbon, in
September 1986. It was decided to organize a workshop every two years. Gathering the most
promising theoretical and applied advances in Project Management and Scheduling, and assess-
ing both the state-of-the-art of this field and its potential to support management systems are
the main objectives of these workshops.
76 extended abstracts have been submitted to the PMS 2018 Confenrece. These valuable con-
tributions were reviewed by 2 referees who are members of the International Program Committee
and distinguished researchers of the associated fields. The proceedings at hand contain the 65
papers that were finally accepted for presentation at the conference. There papers involves 165
authors from 23 different countries.
The 16th
edition of PMS has four plenary speakers: Professor Jacques Carlier (Univer-
sité de Technologie de Compiègne) will present the talk “Comparing event-node graphs with
nonrenewable resources and activity-node graphs with renewable resources”, Professor Erwin
Pesch (University of Siegen) will discuss on “Optimization problems in intermodal transport”,
Professor Ruben Ruiz (University of Valencia) and Professor Erik Demeulemeester (Katholic
University of Leuven) will delight us talking on “Simple metaheuristics for flowshop scheduling:
all you need is local search” and “On the construction of optimal policies for the RCPSP with
stochastic activity durations”, respectively.
The scientific program and the social events will give to all the participants an opportunity
to share research ideas and debate on recent advances on project managment and scheduling. I
am sure that together we will contribute to make PMS 2018 a great success.
Welcome in Rome and have an enjoyable stay!
Rome, 17th April 2018 Massimiliano Caramia (Conference Chair)
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6. PMS 2018
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7. PMS 2018 Committees
Organizing Committee
Massimiliano Caramia (Chairman) Università di Roma “Tor Vergata” (Italy)
Lucio Bianco Università di Roma “Tor Vergata” (Italy)
Stefano Giordani Università di Roma “Tor Vergata” (Italy)
Program Committee
Alessandro Agnetis Università di Siena (Italy)
Ali Allahverdi Kuwait University (Kuwait)
Christian Artigues LAAS-CNRS (France)
Francisco Ballestrin Universitat de València (Spain)
Fayez Boctor Université Laval (Canada)
Jazek Blażewicz Poznań University of Technology (Poland)
Massimiliano Caramia Università di Roma “Tor Vergata” (Italy)
Jacques Carlier Université de Technologie de Compiègne (France)
Erik Demeulemeester Katholieke Universiteit Leuven (Belgium)
Joanna Józefowska Poznań University of Technology (Poland)
Sigrid Knust Universität Osnabrück (Germany)
Rainer Kolisch Technische Universität München (Germany)
Mikhail Kovalyov National Academy of Sciences of Belarus (Belarus)
Wieslaw Kubiak Memorial University (Canada)
Erwin Pesch Universität Siegen (Germany)
Chris Potts University of Southampton (United Kingdom)
Rubén Ruiz Universitat Politècnica de València (Spain)
Avraham Shtub Technion - Israel Institute of Technology (Israel)
Funda Sivrikaya Şerifoğlu Istanbul Bilgi Üniversitesi (Turkey)
Vincent T’Kindt Université François Rabelais Tours (France)
Norbert Trautmann Universität Bern (Switzerland)
Mario Vanhoucke Ghent University (Belgium)
Jan Weglarz Poznań University of Technology (Poland)
Jürgen Zimmermann Technische Universität Clausthal (Germany)
Linet Özdamar Yeditepe Üniversitesi (Turkey)
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8. PMS 2018
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9. PMS 2018 Table of Contents
Table of Contents
Scheduling energy-consuming jobs on parallel machines with piecewise-linear costs and
storage resources: A lot-sizing and scheduling perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Nabil Absi, Christian Artigues, Safia Kedad-Sidhoum, Sandra Ulrich Ngueveu, Janik
Rannou and Omar Saadi
The truck scheduling problem at cross docking terminals: Formulations and valid
inequalities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Alessandro Agnetis, Lotte Berghman and Cyril Briand
The Price of Fairness in a Two-Agent Single-Machine Scheduling Problem . . . . . . . . . . . . . . . . 9
Alessandro Agnetis, Bo Chen, Gaia Nicosia and Andrea Pacifici
A MILP formulation for an operating room scheduling problem under sterilizing
activities constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Hasan Al Hasan, Christelle Guéret, David Lemoine and David Rivreau
Modeling and solving a two-stage assembly scheduling problem with buffers . . . . . . . . . . . . . . 17
Carlos Andrés and Julien Maheut
A new polynomial-time algorithm for calculating upper bounds on resource usage for
RCPSP problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Dmitry Arkhipov, Olga Battaı̈a and Alexander Lazarev
Assembly Flowshops Scheduling Problem to Minimize Maximum Tardiness with Setup
Times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Asiye Aydilek, Harun Aydilek and Ali Allahverdi
No-Wait Flowshop Scheduling Problem to Minimize Total Tardiness Subject to Makespan 32
Harun Aydilek, Asiye Aydilek and Ali Allahverdi
A Robust Optimization Model for the Multi-mode Resource Constrained Project
Scheduling Problem with Uncertain Activity Durations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Noemie Balouka and Izack Cohen
Scheduling data gathering with limited base station memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Joanna Berlinska
A Chance Constrained Optimization Approach for Resource Unconstrained Project
Scheduling with Uncertainty in Activity Execution Intensity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Lucio Bianco, Massimiliano Caramia and Stefano Giordani
Single-machine capacitated lot-sizing and scheduling with delivery dates and quantities . . . 50
Fayez Boctor
Single machine scheduling with m:n relations between jobs and orders: Minimizing the
sum of completion times and its application in warehousing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Nils Boysen, Konrad Stephan and Felix Weidinger
A MILP formulation for multi-robot pick-and-place scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Cyril Briand, Jeanine Codou Ndiaye and Rémi Parlouar
Minimizing resource management costs in a portfolio with resource transfer possibilities . . 62
Jerome Bridelance and Mario Vanhoucke
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10. PMS 2018 Table of Contents
Vehicle sequencing at transshipment terminals with handover relations . . . . . . . . . . . . . . . . . . . 66
Dirk Briskorn, Malte Fliedner and Martin Tschöke
Synchronous flow shop scheduling with pliable jobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Matthias Bultmann, Sigrid Knust and Stefan Waldherr
Computation of the project completion time distribution in Markovian PERT networks. . . 74
Jeroen Burgelman and Mario Vanhoucke
Comparing event-node graphs with nonrenewable resources and activity-node graphs
with renewable resources (Plenary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Jacques Carlier
Synchronizing Heterogeneous Vehicles in a Routing and Scheduling Context . . . . . . . . . . . . . . 79
Marc-Antoine Coindreau, Olivier Gallay and Nicolas Zufferey
On the construction of optimal policies for the RCPSP with stochastic activity durations . 83
Erik Demeulemeester
A B&B Approach to Schedule a No-wait Flow Shop to Minimize the Residual Work
Content Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Simone Dolceamore and Marcello Urgo
On Index Policies in Stochastic Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Franziska Eberle, Felix Fischer, Jannik Matuschke and Nicole Megow
Unrelated Parallel Machine Scheduling at a TV Manufacturer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Ali Ekici, Okan Ozener and Merve Burcu Sarikaya
A new set of benchmark instances for the Multi-Mode Resource Investment Problem . . . . . 100
Patrick Gerhards
A simheuristic for stochastic permutation flow shop problem considering quantitative
and qualitative decision criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Eliana Marı́a Gonzalez-Neira and Jairo R. Montoya-Torres
An Algorithm for Schedule Delay Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Pier Luigi Guida and Giovanni Sacco
Minimizing the total weighted completion time in single machine scheduling with
non-renewable resource constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Peter Gyorgyi and Tamas Kis
The Cyclic Job Shop Problem with uncertain processing times. . . . . . . . . . . . . . . . . . . . . . . . . . . .119
Idir Hamaz, Laurent Houssin and Sonia Cafieri
Modeling techniques for the eS-graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Máté Hegyháti
Scheduling Multiple Flexible Projects with Different Variants of Genetic Algorithms . . . . . . 128
Luise-Sophie Hoffmann and Carolin Kellenbrink
A comparison of neighborhoods for the blocking job-shop problem with total tardiness
minimization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Julia Lange
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11. PMS 2018 Table of Contents
A parallel machine scheduling problem with equal processing time jobs, release dates
and eligibility constraints to minimize total completion time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Kangbok Lee and Juntaek Hong
A new grey-box approach to solve challenging workforce planning and activities
scheduling problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Ludovica Maccarrone and Stefano Lucidi
Scheduling Identical Parallel Machines with Delivery Times to Minimize Total Weighted
Tardiness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Söhnke Maecker and Liji Shen
Modelling and Solving the Hotspot Problem in Air Traffic Control. . . . . . . . . . . . . . . . . . . . . . . . 149
Carlo Mannino, Giorgio Sartor and Patrick Schittekat
A proactive-reactive approach to schedule an automotive assembly line (Plenary) . . . . . . . . . 152
Massimo Manzini, Erik Demeulemeester and Marcello Urgo
Applying a cost, resource or risk perspective to improve tolerance limits for project
control: an empirical validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Annelies Martens and Mario Vanhoucke
A Metamodel Approach to Projects Risk Management: outcome of an empirical testing
on a set of similar projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
Federico Minelle, Franco Stolfi, Roberto Di Gioacchino and Stefano Santini
A column generation scheme for the Periodically Aggregated Resource-Constrained
Project Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Pierre-Antoine Morin, Christian Artigues and Alain Haı̈t
Development of a Schedule Cost Model for a Resource Constrained Project that
incorporates Idleness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Babatunde Omoniyi Odedairo and Victor Oluwasina Oladokun
Optimization problems in intermodal transport (Plenary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Erwin Pesch
The Stakeholder Perspective: how management of KPIs can support value generation to
increase the success rate of complex projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Massimo Pirozzi
Multi-skill project scheduling in a nuclear research facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Oliver Polo Mejia, Marie-Christine Anselmet, Christian Artigues and Pierre Lopez
Scheduling Vehicles with spatial conflicts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Atle Riise, Carlo Mannino, Oddvar Kloster and Patrick Schittekat
On some approach to solve a scheduling problem with a continuous doubly-constrained
resource. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Rafal Rozycki and Grzegorz Waligóra
Simple metaheuristics for Flowshop Scheduling: All you need is local search (Plenary) . . . . 194
Rubén Ruiz
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12. PMS 2018 Table of Contents
Exact Methods for Large Unrelated Parallel Machine Scheduling Problems with
Sequence Dependent Setup Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Rubén Ruiz, Luis Fanjul-Peyró and Federico Perea
Power usage minimization in server problems of scheduling computational jobs on a
single processor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Rafal Różycki, Grzegorz Waligóra and Jan Weglarz
Scheduling resource-constrained projects with makespan-dependent revenues and costly
overcapacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Andre Schnabel and Carolin Kellenbrink
On the complexity of scheduling start time dependent asymmetric convex processing times209
Helmut A. Sedding
Resource-constrained project scheduling with alternative project structures . . . . . . . . . . . . . . . 213
Tom Servranckx and Mario Vanhoucke
A New Pre-Processing Procedure for the Multi-Mode Resource-Constrained Project
Scheduling Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Christian Stürck
An O∗
(1.41n
)-time algorithm for a single machine just-in-time scheduling problem with
common due date and symmetric weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Vincent T’Kindt, Lei Shang and Federico Della Croce
Finding a specific permutation of jobs for a single machine scheduling problem with
deadlines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
Thanh Thuy Tien Ta and Jean-Charles Billaut
Minimizing makespan on parallel batch processing machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Karim Tamssaouet, Stéphane Dauzère-Pérès and Claude Yugma
Order Acceptance and Scheduling Problem with Batch Delivery . . . . . . . . . . . . . . . . . . . . . . . . . . 233
İstenç Tarhan and Ceyda Oğuz
Energy Conscious Scheduling of Robot Moves in Dual-Gripper Robotic Cells . . . . . . . . . . . . . 237
Nurdan Tatar, Hakan Gültekin and Sinan Gürel
A continuous-time assignment-based MILP formulation for the resource-constrained
project scheduling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
Norbert Trautmann, Tom Rihm, Nadine Saner and Adrian Zimmermann
A heuristic procedure to solve the integration of personnel staffing in the project
scheduling problem with discrete time/resource trade-offs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Mick Van Den Eeckhout, Mario Vanhoucke and Broos Maenhout
Production and distribution planning for smoothing supply-chain variability . . . . . . . . . . . . . . 250
Marie-Sklaerder Vié, Nicolas Zufferey and Leandro Coelho
Modeling Non-preemptive Parallel Scheduling Problem with Precedence Constraints . . . . . . 255
Tianyu Wang and Odile Bellenguez-Morineau
A Branch-and-Bound Procedure for the Resource-Constrained Project Scheduling
Problem with Partially Renewable Resources and Time Windows. . . . . . . . . . . . . . . . . . . . . . . . . 259
Kai Watermeyer and Jürgen Zimmermann
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13. PMS 2018 Table of Contents
Fixed interval multiagent scheduling problem with rejected costs . . . . . . . . . . . . . . . . . . . . . . . . . 263
Boukhalfa Zahout, Ameur Soukhal and Patrick Martineau
Integrating case-based analysis and fuzzy programming for decision support in project
risk response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Yao Zhang, Fei Zuo and Xin Guan
Multi-Level Tabu Search for Job Scheduling in a Variable-Resource Environment . . . . . . . . . 272
Nicolas Zufferey
Scheduling a forge with due dates and die deterioration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .276
Olivér Ősz, Balázs Ferenczi and Máté Hegyháti
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14. PMS 2018
xii

15. PMS 2018 Author Index
Author Index
Absi, Nabil 1
Agnetis, Alessandro 5, 9
Al Hasan, Hasan 13
Allahverdi, Ali 26, 32
Andrés, Carlos 17
Anselmet, Marie-Christine 181
Arkhipov, Dmitry 22
Artigues, Christian 1, 165, 181
Aydilek, Asiye 26, 32
Aydilek, Harun 26, 32
Balouka, Noemie 37
Battaı̈a, Olga 22
Bellenguez-Morineau, Odile 255
Berghman, Lotte 5
Berlinska, Joanna 42
Bianco, Lucio 46
Billaut, Jean-Charles 225
Boctor, Fayez 50
Boysen, Nils 55
Briand, Cyril 5, 58
Bridelance, Jerome 62
Briskorn, Dirk 66
Bultmann, Matthias 70
Burgelman, Jeroen 74
Cafieri, Sonia 119
Caramia, Massimiliano 46
Carlier, Jacques 78
Chen, Bo 9
Codou Ndiaye, Jeanine 58
Coelho, Leandro 250
Cohen, Izack 37
Coindreau, Marc-Antoine 79
Dauzère-Pérès, Stéphane 229
Della Croce, Federico 221
Demeulemeester, Erik 83, 152
Di Gioacchino, Roberto 160
Dolceamore, Simone 88
Eberle, Franziska 92
Ekici, Ali 96
xiii

16. PMS 2018 Author Index
Fanjul-Peyró, Luis 195
Ferenczi, Balázs 276
Fischer, Felix 92
Fliedner, Malte 66
Gallay, Olivier 79
Gerhards, Patrick 100
Giordani, Stefano 46
Gonzalez-Neira, Eliana Marı́a 104
Guan, Xin 267
Guida, Pier Luigi 110
Guéret, Christelle 13
Gyorgyi, Peter 115
Gültekin, Hakan 237
Gürel, Sinan 237
Hamaz, Idir 119
Haı̈t, Alain 165
Hegyháti, Máté 123, 276
Hoffmann, Luise-Sophie 128
Hong, Juntaek 136
Houssin, Laurent 119
Kedad-Sidhoum, Safia 1
Kellenbrink, Carolin 128, 205
Kis, Tamas 115
Kloster, Oddvar 185
Knust, Sigrid 70
Lange, Julia 132
Lazarev, Alexander 22
Lee, Kangbok 136
Lemoine, David 13
Lopez, Pierre 181
Lucidi, Stefano 141
Maccarrone, Ludovica 141
Maecker, Söhnke 145
Maenhout, Broos 246
Maheut, Julien 17
Mannino, Carlo 149, 185
Manzini, Massimo 152
Martens, Annelies 156
Martineau, Patrick 263
Matuschke, Jannik 92
Megow, Nicole 92
Minelle, Federico 160
Montoya-Torres, Jairo R. 104
xiv

17. PMS 2018 Author Index
Morin, Pierre-Antoine 165
Ngueveu, Sandra Ulrich 1
Nicosia, Gaia 9
Odedairo, Babatunde Omoniyi 169
Oladokun, Victor Oluwasina 169
Ozener, Okan 96
Oğuz, Ceyda 233
Pacifici, Andrea 9
Parlouar, Rémi 58
Perea, Federico 195
Pesch, Erwin 175
Pirozzi, Massimo 176
Polo Mejia, Oliver 181
Rannou, Janik 1
Rihm, Tom 242
Riise, Atle 185
Rivreau, David 13
Rozycki, Rafal 189
Ruiz, Rubén 194, 195
Różycki, Rafal 199
Saadi, Omar 1
Sacco, Giovanni 110
Saner, Nadine 242
Santini, Stefano 160
Sarikaya, Merve Burcu 96
Sartor, Giorgio 149
Schittekat, Patrick 149, 185
Schnabel, Andre 205
Sedding, Helmut A. 209
Servranckx, Tom 213
Shang, Lei 221
Shen, Liji 145
Soukhal, Ameur 263
Stephan, Konrad 55
Stolfi, Franco 160
Stürck, Christian 217
T’Kindt, Vincent 221
Ta, Thanh Thuy Tien 225
Tamssaouet, Karim 229
Tarhan, İstenç 233
Tatar, Nurdan 237
Trautmann, Norbert 242
xv

18. PMS 2018 Author Index
Tschöke, Martin 66
Urgo, Marcello 88, 152
Van Den Eeckhout, Mick 246
Vanhoucke, Mario 62, 74, 156, 213, 246
Vié, Marie-Sklaerder 250
Waldherr, Stefan 70
Waligóra, Grzegorz 189, 199
Wang, Tianyu 255
Watermeyer, Kai 259
Weidinger, Felix 55
Weglarz, Jan 199
Yugma, Claude 229
Zahout, Boukhalfa 263
Zhang, Yao 267
Zimmermann, Adrian 242
Zimmermann, Jürgen 259
Zufferey, Nicolas 79, 250, 272
Zuo, Fei 267
Ősz, Olivér 276
xvi

19. PMS 2018 Keyword Index
Keyword Index
Activities scheduling 169
Air traffic control 149
Aircraft assembly 88
Alternative project structure 213
Analytical tolerance limits 156
Approximation algorithm 92
Assembly flowshop 17, 26
Assembly line 152
Assignment and scheduling 58
Bargaining problems 9
Batch delivery 233
Batching 229
Benchmark instances 100
Benders’ reformulation 149
Bicriteria optimization 237
Blocking 17, 132
Branch and bound 259
Budgeted uncertainty set 119
Buffer monitoring 156
Buffer size 17
Carpooling 79
Chance constrained optimization 46
Characterization 225
Claim management 110
Column generation 165, 233
Computational complexity 66, 209
Conditional Value at Risk 88
Conflict resolution 185
Constraint programming 22
Context based risk analysis 160
Continuous resource 189
Continuous time mixed integer linear
programming 242
Crossdocking 5
Cyclic scheduling 119
Data gathering network 42
Deadline 225
Decision support system 267
Decomposition 246
Delivery times 145
Deterioration 276
xvii

20. PMS 2018 Keyword Index
Discrete continuous scheduling 189
Discrete time/resource trade-off 246
Distribution planning 250
Doubly constrained resource 189
Dual gripper 237
Due dates 276
Dynamic programming 1, 136, 233
E-government 160
Earliness 104
Eligibility constraints 136
Energy 199
Energy optimization 237
Equal processing time jobs 136
eS-graph 123
Exact methods 195
Exponential time algorithm 221
Fairness 9
Flexible project 128
Flow shop 42, 70, 152
Forge 276
Fuzzy mathematical programming 267
Genetic algorithm 128, 205
Grey box optimization 141
Handover relation 66
Heuristics 50, 145, 205, 263
Hotspot problem 149
Idleness cost 169
Integer programming 5, 50, 149, 255
Iterated local search 246
Job shop scheduling 132, 185, 272
Just in time 221
KPI 176
Limited memory 42
Linear algebra 74
Linear programming 185
Local search 205, 229
Lot sizing 1
Lot sizing and scheduling 50
Makespan 32
xviii

21. PMS 2018 Keyword Index
Maximum tardiness 26
Memetic algorithm 145
Metaheuristics 145, 233
Mixed integer linear programming 1, 13, 58, 141, 145, 165, 242, 263, 276
MMLIB 217
Mode reduction 217
Modeling technique 123
Multi skill 181
Multi-mode resource availability cost
problem 100
Multi-mode resource constrained project
scheduling problem 217
Multi-mode resource investment problem 100
Multi-project experimental outcome 160
Multi-project scheduling 128
Multi-robot pick-and-place 58
Multiagent scheduling 9, 263
New objective function 225
No wait 32
Non-renewable resources 115
Nuclear laboratory 181
Operating rooms in health services 13
Optimal policies 83
Optimization 181
Order acceptance 233
Order consolidation 55
Overcapacity 205
Parallel machine scheduling 5, 96, 145, 189, 195, 229, 255
Pareto optimization 263
Partially renewable resources 259
Periodical aggregation 165
PERT 46, 74
Piecewise linear convex processing time 209
Planning 165
Pliability 70
Polynomial algorithms 22
Portfolio management 62
Power 199
Precedence constraints 255
Preemptive scheduling 181
Preprocessing 217
Proactive reactive scheduling 152
Production planning 250
Project 165
Project control 156
xix

22. PMS 2018 Keyword Index
Project management 110, 169
Project network 123
Project planning 22
Project risk management 160, 267
Project scheduling 37, 46, 74, 100, 141, 213, 259
Project staffing 246
Propagator 22
Qualitative criteria 104
RCMPSP-PS 128
Release dates 136
Resource availability 62
Resource constrained project scheduling
problem 83, 169, 181, 205, 242
Resource transfers 62
Risk response action 267
Robotic cell scheduling 237
Robust optimization 37, 119
Robustness 104
Schedule delay analysis 110
Scheduling under energy constraints
and costs 1
Semiconductor manufacturing 229
Sequence dependent setup times 96, 233
Server problem 199
Setup times 26, 195
Simulated annealing 26, 132
Single machine scheduling 55, 115, 221, 225, 263
Single processor 199
Stakeholder 176
Sterilization unit 13
Stochastic activity durations 83
Stochastic permutation flow shop 104
Stochastic scheduling 88, 92
Success 176
Supply chain management 250
Synchronization 79
Synchronous movement 70
Tabu search 213, 272
Tardiness 104
Time dependent scheduling 209
Total completion time 92, 136
Total tardiness 32, 132
Total weighted completion time 115
Transshipment terminals 66
xx

23. PMS 2018 Keyword Index
Truck scheduling 5
Uncertainty 37
Unrelated machine scheduling 96
Variable resources 272
Vehicle routing and scheduling 79
Vehicle sequencing 66
Warehousing 55
Weighted tardiness 145
Workforce management 141
xxi

24. PMS 2018
xxii

25. Scheduling energy-consuming jobs on parallel machines
with piecewise-linear costs and storage resources
A lot-sizing and scheduling perspective
Nabil Absi1
, Christian Artigues2
, Safia Kedad-Sidhoum3
, Sandra U. Ngueveu2
, Janik
Rannou2
, et Omar Saadi3
1
Mines Saint-Etienne and UMR CNRS 6158 LIMOS, Gardanne, France
absi@emse.fr
2
LAAS-CNRS, Université de Toulouse, CNRS, INP, Toulouse, France
sungueve,artigues@laas.fr
3
Sorbonne Université, UPMC, UMR 7606, LIP6, Paris, F-75005, France
safia.kedad-sidhoum@lip6.fr
Keywords: Scheduling under energy constraints and costs, Mixed-integer programming,
Lot-sizing, Dynamic programming
1 Problem formulation and complexity
In this paper, a scheduling and energy source assignment problem is studied. The
problem is abstracted from several applications including data centers (Guérout et al. 2017),
smart buildings (Desdouits et al. 2016), hybrid vehicles (Caux et al. 2017, Ngueveu et
al. 2017) and manufacturing (Haouassi et al. 2016). A set of jobs J has to be scheduled
on a set of energy consuming machines M. The energy consumed by a machine k has a
fixed part Dk depending on whether the machine is switched on or off and a variable part
depending on the tasks that are currently in process. Each task j requires an amount Djk
of energy at each time period it is processed on machine k. We are interested in optimizing
the total energy cost induced by energy production required to satisfy the total energy
demand of a given schedule over a fixed discrete horizon T = {1, . . . , |T|}. At each time
period, the energy required by the schedule can be supplied by two energy sources. One
source is reversible, such as batteries and super-capacitors. Such a source is able not only
to produce energy but also to retrieve it assuming a limited capacity Q. Such a resource is
equivalent to a continuously-divisible storage resource in the scheduling terminology. The
second source is a non-reversible source, only able to produce energy, such as the external
power grid (assuming here that energy cannot be sold to the network). This source is
assumed of infinite capacity, but its usage comes with a cost (see below). In this paper, we
consider a parallel machine environment, such that pj units of each task j must be scheduled
preemptively inside a time window [rj, dj] and have to be assigned at each time period to
one and only one machine. Job units also require resources from a set R (e.g. CPU, RAM)
on their assigned machines. A job requires a non-negative amount cjr on each resource
r ∈ R. On each machine k, resource r is available in a limited amount Ckr. The energy
cost for a period of the scheduling horizon is a piecewise linear (PWL) function ft, t ∈ T, of
the required amount of energy on the non-reversible source. The PWL function is assumed
to be time dependent. This allows to model time dependent electricity prices as well as
previsions of photovoltaic production since the cost can be zero up to a required amount
of energy corresponding to the expected photovoltaic production on the considered time
period. We introduce variables yjkt ∈ {0, 1} indicating whether one unit of job j is assigned
to machine k at time t and zkt ∈ {0, 1} indicating whether machine k is switched on at time
t. Continuous variables are used for energy amounts: xt ≥ 0 gives the amount of energy
used on the non-reversible source at time t, st ≥ 0 is the level of energy remaining in the
1

26. non-reversible source at time t (s0 is a constant indicating the initial energy level). Based
on these variables, we define a MILP formulation of the problem aiming at minimizing the
total energy cost.
minimize
P
t∈T ft(xt) (1)
subject to P
k∈M yjkt ≤ 1, j ∈ J , t ∈ T (2)
P
k∈M
P
t∈T yjkt = pj, j ∈ J (3)
P
j∈J cjryjkt ≤ Ckr, k ∈ M, r ∈ R, t ∈ T (4)
zkt − yjkt ≥ 0, k ∈ M, j ∈ J , t ∈ T (5)
xt + st − st−1 −
P
j∈J
P
k∈M Djkyjkt −
P
k∈M Dkzkt = 0, t ∈ T (6)
s|T | − s0 ≥ 0, (7)
st − Q ≤ 0, t ∈ T (8)
st ≥ 0 t ∈ T (9)
xt ≥ 0 t ∈ T (10)
yjkt ∈ {0, 1} j ∈ J , k ∈ M, t ∈ T (11)
zkt ≥ 0 k ∈ M, t ∈ T (12)
The total energy cost minimization objective (1) is considered. Constraints (2) state that
a job may be in process on only one machine at a given time. Constraints (3) enforce each
unit of a job to be scheduled on one machine. Constraints (4) are the resource constraints.
Constraints (5) enforce a machine to be switched on at each time it processes at least one
job. Constraints (6) are the energy balance constraints between the schedule demand, the
energy provided by the non-reversible source (xt) and the energy taken from or provided to
the reversible source (st − st−1, that can be positive or negative). Constraint (7) enforces
the final energy level in the reversible source to be at least the initial one. Constraints
(8) are the reversible source capacity constraints (storage limit). Scheduling preemptive
jobs with PWL energy costs is NP-hard, even with an unlimited number of machines and
single non-reversible source (Ngueveu et al. 2016). Therefore the proposed MILP becomes
intractable as the problem size increases.
2 A lot-sizing and scheduling matheuristic
We propose a natural decomposition of the problem. Let dt =
P
j∈J
P
k∈M Djkyjkt
+
P
k∈M Dkzkt denote the total energy demand of a fixed schedule at time t. Then con-
straints (6) can be rewritten
xt + st − st−1 − dt = 0, t ∈ T (13)
Now observe that for fixed dt, problem LSP: min
P
t∈T ft(xt) s.t. (7–10), (13) is a single-
item (continuous) lot sizing problem with PWL production costs where dt is the demand
for period t, xt is the production variable for period t, and st is the variable giving the
amount of inventory at the end of period t. In Absi et al. (2017), the problem is shown to
be NP-hard but for integer inventory levels, a pseudo-polynomial dynamic programming
(DP) algorithm of complexity O(T2
qd) where d is the average demand and q is the average
number of breakpoints of the PWL functions ft is given, generalizing the results of Shaw
and Wagelmans (1998). On the other hand, if the variables st are fixed, by performing
change of variables xt ← xt − st + st−1 for all t, we obtain
xt =
X
j∈J
X
k∈M
Djkyjkt +
X
k∈M
Dkzkt, t ∈ T (14)
2

27. and problem MSP: min
P
t∈T f′
t(xt) =
P
t∈T ft(xt − st + st−1) s.t. (2–5), (10–12), (14),
which is a parallel machine scheduling problem with PWL costs and a single non-reversible
source, NP-hard in the strong sense as shown in Ngueveu et al. (2016).
A matheuristic is obtained by solving alternatively MSP and LSP. Starting with initial
reversible source transferred amounts st − st−1 = 0 for all t, MSP is solved and the output
energy demand (dt)t∈T is used as input of LSP. The output inventory levels are used to
update the PWL functions f′
t. Then, MSP is solved again, and so on until no improvement
is observed in the objective function.
We first compare the MILP (solved with Cplex) and the pseudo polynomial DP algo-
rithm with fixed demands (only LSP), see table 1. The merits and the drawback of the
two approaches are illustrated on 4 instances with T = 1000, q = 10 breakpoints, and
varying average maximal capacities Q and demands d. Under a 300s time limit for the
MILP, the CPU times (in seconds) and obtained costs are compared. It appears that no
algorithm dominates the other one in terms of CPU time, while the DP is more impacted
by the maximal available capacity for the reversible source. However the MILP shows a
more erratic and unpredictable behavior. Note that due to the integrity requirement of the
inventory levels, the DP costs are higher than the MILP costs.
Table 1. Comparison of MILP and DP on the lot sizing problem (LSP)
T q Q d MILP cost MILP CPU DP cost DP CPU
1000 10 1000 100 6661 300 6916 14.16
1000 10 10000 100 4396 2.90 4508 279
1000 10 100 2000 975523 0.57 975617 1.31
1000 10 1000 2000 940165 300 937886 15
Finally we compare the MSP/LSP decomposition matheuristic (MH) with the full MILP
to solve the global problem. We also illustrate the cost and CPU time differences on 4
instances with varying horizon, number of machines, number of jobs (see table 2). On 3
instances (marked with a ∗) the full MILP reached the time limit. The matheuristic is
only slightly faster than the MILP except on the last instance, where it is much faster.
The costs can be close to the MILP ones although important gaps can also be observed.
In parenthesis, the maximal CPU time per iteration and the iteration number assigned to
MH is indicated. A closer analysis of the CPU times between MSP and LSP reveals that
90% of the CPU time is spent on solving the scheduling problem MSP.
Table 2. Comparison of full MILP and matheuristic on the global problem
T |M| |J | MILP cost MILP CPU MH cost MH CPU
30 2 50 11280 450∗
11280 154 (150× 3)
60 4 150 25966 2000∗
26183 1893 (1000× 2)
120 2 150 5944 2000∗
6972 1855 (1000× 2)
120 1 150 11255 1353 11265 228 (200× 10)
3

28. 3 Conclusion
We have proposed an original lot sizing and scheduling decomposition approach to
solve an energy management and scheduling problem on parallel machines. The lot sizing
subproblem can be solved considerably faster than the scheduling subproblem and conse-
quently, further research on the problem should focus on improvement of the scheduling
solution procedure. To improve the decomposition heuristic, optimality cuts issued from
lot sizing could be designed for the scheduling problem. Another interesting issue is to
consider a non ideal yield of the reversible source. In practice, due to energy conversion
and losses only a fraction of st − st−1 is available to fulfill the demand and a possibly non
linear efficiency function g(st − st − 1) has to be used to compute the obtained energy. It
remains to know whether efficient lot sizing procedure can be devised with such efficiency
functions.
Acknowledgements
This research benefited from the support of the FMJH Program PGMO and from the
support of EDF, Thales, Orange. It also benefited from funding from the Cellule Energie
of the CNRS.
References
Absi N., C. Artigues, S. Kedad-Sidhoum, S.U. Ngueveu and O. Saadi, 2017, “Lot-sizing models
for energy management”. LAAS report.
Caux S., Y. Gaoua, and P. Lopez, 2017, “A combinatorial optimisation approach to energy man-
agement strategy for a hybrid fuel cell vehicle”. Energy, Vol. 133, pp.219-230.
Desdouits C., M. Alamir, R. Giroudeau and C. Le Pape, 2016, “The Sourcing Problem - Energy
Optimization of a Multisource Elevator”, ICINCO, Vol. 1, pp. 19-30.
Guérout T., Y. Gaoua, C. Artigues, G. Da Costa, P. Lopez and T. Monteil, 2017, “Mixed in-
teger linear programming for quality of service optimization in Clouds” Future Generation
Computer Systems 71: 1–17.
Haouassi M., C. Desdouits, R. Giroudeau and C. Le Pape, 2016, “Production scheduling with a
piecewise-linear energy cost function” IEEE Symposium Series on Computational Intelligence
(SSCI), pp. 1-8.
Ngueveu S. U., C. Artigues and P. Lopez, 2016, “Scheduling under a non-reversible energy source:
An application of piecewise linear bounding of non-linear demand/cost functions”, Discrete
Applied Mathematics, Vol. 208, pp. 98-113.
Ngueveu S.U., S. Caux, F. Messine and M. Guemri, 2017, “Heuristics and lower bound for energy
management in hybrid-electric vehicles”, 4OR, to appear.
Shaw D. X., A. P. Wagelmans, 1998, “An algorithm for single-item capacitated economic lot sizing
with piecewise linear production costs and general holding costs”, Management Science, Vol.
44, pp. 831-838.
4

29. The truck scheduling problem at cross docking
terminals: Formulations and valid inequalities
A. Agnetis1
, L. Berghman2
and C. Briand3
1
A. Agnetis, Università degli Studi di Siena, DIISM, Siena, Italy
agnetis@diism.unisi.it
2
L. Berghman, Université de Toulouse - Toulouse Business School,
20 BD Lascrosses – BP 7010, 31068 Toulouse Cedex 7, France
l.berghman@tbs-education.fr
3
C. Briand, LAAS-CNRS, Université de Toulouse, UPS, Toulouse, France
briand@laas.fr
Keywords: crossdocking, truck scheduling, parallel machine scheduling, integer linear pro-
gramming.
1 Introduction
Crossdocking is a warehouse management concept in which items delivered to a ware-
house by inbound trucks are immediately sorted out, reorganized based on customer de-
mands and loaded into outbound trucks for delivery to customers, without requiring ex-
cessive inventory at the warehouse (J. van Belle et al. 2012). If any item is held in storage,
it is usually for a brief period of time that is generally less than 24 hours. Advantages
of crossdocking can accrue from faster deliveries, lower inventory costs, and a reduction
of the warehouse space requirement (U.M. Apte and S. Viswanathan 2000, N. Boysen et
al. 2010). Compared to traditional warehousing, the storage as well as the length of the
stay of a product in the warehouse is limited, which requires an appropriate coordination
of inbound and outbound trucks (N. Boysen 2010, W. Yu and P.J. Egbelu 2008).
The truck scheduling problem, which decides on the succession of truck processing at the
dock doors, is especially important to ensure a rapid turnover and on-time deliveries. The
problem studied concerns the operational level: trucks are allocated to the different docks
so as to minimize the storage usage during the product transfer. The internal organization
of the warehouse (scanning, sorting, transporting) is not explicitly taken into consideration.
We also do not model the resources that may be needed to load or unload the trucks, which
implies the assumption that these resources are available in sufficient quantities to ensure
the correct execution of an arbitrary docking schedule.
In this abstract, we present a time-indexed formulation, a network formulation and some
valid inequalities. Experimental results will be presented during the talk at the conference.
2 Detailed problem statement
We examine a crossdocking warehouse where incoming trucks i ∈ I need to be unloaded
and outgoing trucks o ∈ O need to be loaded (where I is the set containing all inbound
trucks while O is the set containing all outbound trucks). The warehouse features n docks
that can be used both for loading and unloading. The processing time of truck j ∈ I ∪ O
equals pj. This processing time includes the loading or unloading but also the transporta-
tion of goods inside the crossdock and other handling operations between dock doors. It
is assumed that there is sufficient workforce to load/unload all docked trucks at the same
time. Hence, a truck assigned to a dock does not wait for the availability of a material
handler.
5

30. The products on the trucks are packed on unit-size pallets, which move collectively as
a unit: re-packing inside the terminal is to be avoided. Each pallet on an inbound truck
i needs to be loaded on an outbound truck o, which gives rise to a start-start precedence
constraint (i, o) ∈ P ⊂ I × O, with P the set containing all couples of inbound trucks i
and outbound trucks o that share a precedence constraint. Each truck j has a release time
rj (planned arrival time) and a deadline ˜
dj (its latest departure time).
Products can be transshipped directly from an inbound to an outbound truck if the
outbound truck is placed at a dock. Otherwise, the products are temporarily stored and
will be loaded later on. Each couple (i, o) ∈ P has a weight wio, representing the number of
pallets that go from inbound truck i to outbound truck o. The problem aims at determining
time-consistent start times si and so of unload and load tasks i and o so as to minimize
the weighted sum of sojourn times of the pallets stocked in the warehouse. Remark that
the time spent by a pallet in the storage area is equal to the flow time of the pallet: the
difference between the start of loading the outbound trailer and the start of unloading the
inbound trailer (i.e., so − si).
Our problem can be modeled as a parallel machine scheduling problem with release
dates, deadlines, and precedence constraints, denoted by Pm|ri, ˜
di, prec|−. As this problem
is a generalization of the 1|rj, ˜
dj|− problem which is NP-complete (J.K. Lenstra et al. 1977),
even finding a feasible solution for the problem is NP-complete.
3 Time-indexed formulation
A time-indexed formulation discretizes the continuous time space into periods τ ∈
T of a fixed length. Let period τ be the interval [t − 1, t[. It is well known that time-
indexed formulations perform well for scheduling problems because the linear programming
relaxations provide strong lower bounds (M. E. Dyer and L. A. Wolsey 1990).
For all inbound trucks i ∈ I and for all time periods τ ∈ Ti, we have
xiτ =





1 if the unloading of inbound truck i is
started during time period τ,
0 otherwise,
(1)
with Ti = {ri + 1, ri + 2, . . . , ˜
di − pi + 1}, the relevant time window for inbound truck i.
Additionally, for all outbound trucks o ∈ O and for all time periods τ ∈ To, we have
yoτ =





1 if the loading of outbound truck o is
started during time period τ,
0 otherwise,
(2)
with To = {ro + 1, ro + 2, . . . , ˜
do − po + 1}, the relevant time window for outbound truck o.
A time-indexed formulation for the considered truck scheduling problem is the following:
min z =
X
(i,o)∈P
X
τ∈T
wioτ (yoτ − xiτ ) (3)
6

31. subject to
X
τ∈Ti
xiτ = 1 ∀i ∈ I (4)
X
τ∈To
yoτ = 1 ∀o ∈ O (5)
X
τ∈T
τ (xiτ − yoτ ) ≤ 0 ∀(i, o) ∈ P (6)
X
i∈I
τ
X
u=τ−pi+1
xiu +
X
o∈O
τ
X
u=τ−po+1
you ≤ n ∀τ ∈ T (7)
xiτ ∈ {0, 1} ∀i ∈ I; ∀τ ∈ Ti (8)
yoτ ∈ {0, 1} ∀o ∈ O; ∀τ ∈ To (9)
The objective function (3) minimizes the total weighted usage of the storage area. Con-
straints (4) and (5) demand each truck to be assigned to exactly one gate. Constraints (6)
ensure that if there exists a precedence constraint between inbound truck i and outbound
truck o, then o cannot be processed before i. Constraints (7) enforce the capacity of the
docks for any period τ ∈ T .
4 Network formulation
The formulation below makes use of the well-known concept of a critical set (see e.g. (M.
Lombardi and M. Milano 2012)), i.e., a set of tasks which cannot all be performed in
parallel. We introduce a pair of disjunctive precedence constraints for each task pair (u, v) ∈
(I ∪ O)2
with [ru, du] ∩ [rv, dv] 6= ∅, belonging to a critical set (the set of these task pairs is
further referred as C). We let E be the set of all critical sets. Additionally, we also refer to ek
as a specific critical set of k elements and to Em
⊂ E as the set of all minimal critical sets.
To model the disjunction, binary variables αuv are introduced such that αuv = 1 ≡ u ≺ v.
Our problem can be modelled as follows:
min
X
o∈O
sopo −
X
i∈I
sipi (10)
subject to
so − si ≥ 0 ∀(i, o) ∈ P (11)
sv − su + αuv(Muv − pu) ≥ Muv ∀(u, v) ∈ C (12)
sv − s0 ≥ rv ∀v ∈ I ∪ O (13)
s0 − su ≥ pu − du ∀u ∈ I ∪ O (14)
X
(u,v)∈en+1
αuv ≥ 1 ∀en+1 ∈ Em
(15)
αuv ∈ {0, 1} ∀(u, v) ∈ C (16)
su ∈ R ∀u ∈ I ∪ O (17)
with Muv = pu − du + rv and 0 a dummy vertex, which is introduced to represent the
time origin s0 = 0. Note that obviously αuv + αvu ≤ 1, even though this constraint is not
mandatory for the formulation accuracy.
Remark that constraints (15) express the limited capacity of the crossdocking terminal.
Their number is exponential, as the number of minimal critical sets is exponential. Even
7

32. though including only minimal critical sets is sufficient, we can also consider the non-
minimal critical sets, generalizing (15) as:
X
(u,v)∈ek
αuv ≥ k − n ∀ek ∈ E (18)
with n + 1 ≤ k ≤ I ∪ O .
We will show that this family of constraints can be strengthened by augmenting the
right-and-side, so that it can be replaced by:
X
(u,v)∈ek
αuv ≥
(k − n)(k − n + 1)
2
∀ek ∈ E (19)
5 Solving methodology framework
Intuitively, only a small number of constraints (19) may be required into the formulation
to obtain a feasible (optimal) solution. Consequently, we consider the following cutting-
plane method which consists in introducing progressively constraints of type (19). First, the
problem is solved without any constraint of type (19) using a MILP solver. Then, violated
constraints of type (19) are added for some k > n involving a critical set ek and the solver
is launched again. Now, each time a feasible solution is found by the solver in course of
the branch-and-cut process, violated constraints of type (19) are added on-the-fly. Note
that if such a solution is feasible with respect to the resource capacity, then it is an upper
bound of the initial problem. When the MILP solver ends up with an optimal solution
also capacity-feasible, it is also optimal. Otherwise, violated constraints of type (19) can
be added again and another MILP is ran. Within various computational time limitation
assumptions, the above methodology will be compared in terms of performance (quality of
the upper and lower bounds) with the time-indexed linear programming approach on a set
of artificial problem instances.
References
U.M. Apte and S. Viswanathan, Effective cross docking for improving distribution efficiencies,
International Journal of Logistics: Research and Applications, 3 (3), 291–302.
N. Boysen, Truck scheduling at zero-inventory cross docking terminals, Computers & Operations
Research, 37, 32–41.
N. Boysen and M. Fliedner and A. Scholl, Scheduling inbound and outbound trucks at cross
docking terminals, OR Spectrum, 32, 135–161.
M. E. Dyer and L. A. Wolsey, Formulating the single machine sequencing problem with release
dates as a mixed integer problem, Discrete Applied Mathematics, 26, 255–270.
J.K. Lenstra and A.H.G. Rinnooy Kan and P. Brucker, Complexity of machine scheduling prob-
lems, Annals of Discrete Mathematics, 1, 343–362.
M. Lombardi and M. Milano, A min-flow algorithm for Minimal Critical Set detection in Resource
Constrained Project Scheduling, Artificial Intelligence, 182-183, 58–67.
J. van Belle and P. Valckenaers and D. Cattrysse, Cross docking: State of the art, Omega,40 (6),
827–846.
W. Yu and P.J. Egbelu, Scheduling of inbound and outbound trucks in cross docking systems with
temporary storage, International Journal of Production Economics„ 184, 377–396.
8

33. ❚❤❡ Pr✐❝❡ ♦❢ ❋❛✐r♥❡ss ✐♥ ❛ ❚✇♦✲❆❣❡♥t ❙✐♥❣❧❡✲▼❛❝❤✐♥❡
❙❝❤❡❞✉❧✐♥❣ Pr♦❜❧❡♠
❆❧❡ss❛♥❞r♦ ❆❣♥❡t✐s1
✱ ❇♦ ❈❤❡♥2
✱ ●❛✐❛ ◆✐❝♦s✐❛3
✱ ❡t ❆♥❞r❡❛ P❛❝✐✜❝✐4
1
❯♥✐✈❡rs✐t② ♦❢ ❙✐❡♥❛✱ ■t❛❧②
❛❣♥❡t✐s❅❞✐✐s♠✳✉♥✐s✐✳✐t
2
❯♥✐✈❡rs✐t② ♦❢ ❲❛r✇✐❝❦✱ ❯❑
❇♦✳❈❤❡♥❅✇❜s✳❛❝✳✉❦
3
❯♥✐✈❡rs✐t② ♦❢ ❘♦♠❛ ❚r❡✱ ■t❛❧②
♥✐❝♦s✐❛❅❞✐❛✳✉♥✐r♦♠❛✸✳✐t
4
❯♥✐✈❡rs✐t② ♦❢ ❘♦♠❛ ❚♦r ❱❡r❣❛t❛✱ ■t❛❧②
♣❛❝✐❢✐❝✐❅❞✐s♣✳✉♥✐r♦♠❛✷✳✐t
❑❡②✇♦r❞s✿ ♠✉❧t✐❛❣❡♥t s❝❤❡❞✉❧✐♥❣✱ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠s✱ ❢❛✐r♥❡ss✳
✶ ■♥tr♦❞✉❝t✐♦♥
❋❛✐r♥❡ss ✐ss✉❡s ❛r✐s❡ ✐♥ s❡✈❡r❛❧ r❡❛❧✲✇♦r❧❞ ❝♦♥t❡①ts ❛♥❞ ❛r❡ ✐♥✈❡st✐❣❛t❡❞ ✐♥ ❞✐✛❡r❡♥t
r❡s❡❛r❝❤ ❛r❡❛s ♦❢ ♠❛t❤❡♠❛t✐❝s✱ ❣❛♠❡ t❤❡♦r② ❛♥❞ ♦♣❡r❛t✐♦♥s r❡s❡❛r❝❤✳ ■♥ ❝❧❛ss✐❝❛❧ t✇♦✲♣❧❛②❡r
❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠s✱ t❤❡ ♥♦t✐♦♥ ♦❢ ❢❛✐r♥❡ss ❤❛s ❜❡❡♥ ✐♥tr♦❞✉❝❡❞ t♦ ❝♦♠♣❛r❡ t❤❡ ✉t✐❧✐t② ♦❢
♦♥❡ ❛❣❡♥t t♦ t❤❡ ♦t❤❡r ❛❣❡♥t✬s✳
❍❡r❡ ✇❡ ❛❞❞r❡ss ❢❛✐r♥❡ss ❝♦♥❝❡♣ts ✐♥ t❤❡ ❝♦♥t❡①t ♦❢ ❝❧❛ss✐❝❛❧ s✐♥❣❧❡✲♠❛❝❤✐♥❡ s❝❤❡❞✉❧✐♥❣✳
❚❤❡r❡ ❛r❡ t✇♦ ❛❣❡♥ts✱ ❝❛❧❧❡❞ A ❛♥❞ B✱ ❡❛❝❤ ♦✇♥✐♥❣ ❛ s❡t ♦❢ ❥♦❜s✱ ✇❤✐❝❤ ♠✉st ❜❡ s❝❤❡❞✉❧❡❞
♦♥ ❛ ❝♦♠♠♦♥ ♣r♦❝❡ss✐♥❣ r❡s♦✉r❝❡✳ ❊❛❝❤ s❝❤❡❞✉❧❡ ✐♠♣❧✐❡s ❛ ❝❡rt❛✐♥ ✉t✐❧✐t② ❢♦r ❡❛❝❤ ❛❣❡♥t✳ ❲❡
❛❞♦♣t t❤❡ s✉♠ ♦❢ t❤❡ ❛❣❡♥ts✬ ✉t✐❧✐t✐❡s ❛s ❛♥ ✐♥❞❡① ♦❢ ❝♦❧❧❡❝t✐✈❡ s❛t✐s❢❛❝t✐♦♥ ✭s②st❡♠ ✉t✐❧✐t②✮
❛♥❞ ✇❡ r❡❢❡r t♦ ❛♥② s♦❧✉t✐♦♥ ♠❛①✐♠✐③✐♥❣ s②st❡♠ ✉t✐❧✐t② ❛s ❛ s②st❡♠ ♦♣t✐♠✉♠✳ ❊✈❡♥ ✐❢ ✐t
♠❛①✐♠✐③❡s s②st❡♠ ✉t✐❧✐t②✱ ❛ s②st❡♠ ♦♣t✐♠✉♠ ♠❛② ✇❡❧❧ ❜❡ ❤✐❣❤❧② ✉♥❜❛❧❛♥❝❡❞ ❛♥❞ t❤❡r❡❢♦r❡
♣♦ss✐❜❧② ✉♥❛❝❝❡♣t❛❜❧❡ ❜② t❤❡ ✇♦rs❡✲♦✛ ❛❣❡♥t✳ ❘❛t❤❡r✱ ❛ s♦❧✉t✐♦♥ t❤❛t ✐♥❝♦r♣♦r❛t❡s s♦♠❡
❝r✐t❡r✐♦♥ ♦❢ ❢❛✐r♥❡ss ♠❛② ❜❡ ♠♦r❡ ❛❝❝❡♣t❛❜❧❡✳ ❚❤❡ ♣r♦❜❧❡♠ ✇❡ ✐♥✈❡st✐❣❛t❡ ✐s ❤♦✇ ♠✉❝❤
s②st❡♠ ✉t✐❧✐t② ♠✉st ❜❡ s❛❝r✐✜❝❡❞ ✐♥ ♦r❞❡r t♦ r❡❛❝❤ ❛ ❢❛✐r s♦❧✉t✐♦♥✳ ❚❤❡ q✉❛♥t✐t② t❤❛t ❝❛♣t✉r❡s
t❤✐s ❝♦♥❝❡♣t ✐s ❦♥♦✇♥ ❛s ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss ✭PoF✮✳ ●✐✈❡♥ ❛♥ ✐♥st❛♥❝❡ ♦❢ ❛ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠✱
❛♥❞ ❛ ❝❡rt❛✐♥ ❞❡✜♥✐t✐♦♥ ♦❢ ❢❛✐r s♦❧✉t✐♦♥✱ PoF ✐s t❤❡ r❡❧❛t✐✈❡ ❧♦ss ✐♥ ♦✈❡r❛❧❧ ✉t✐❧✐t② ♦❢ ❛ ❢❛✐r
s♦❧✉t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ s②st❡♠ ♦♣t✐♠✉♠✳ ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ s♣❡❝✐✜❝ ♣r♦❜❧❡♠ s❡tt✐♥❣
❛♥❞ ❛❧s♦ ♦♥ t❤❡ ❛❣❡♥t ♣❡r❝❡♣t✐♦♥ ♦❢ ✇❤❛t ❛ ❢❛✐r s♦❧✉t✐♦♥ ✐s✱ ❛ss♦rt❡❞ ❞❡✜♥✐t✐♦♥s ♦❢ ❢❛✐r
s♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥ t❤❡ s❝✐❡♥t✐✜❝ ❧✐t❡r❛t✉r❡✳ ■♥ ♦✉r st✉❞② ✇❡ ❢♦❝✉s ♦♥ t✇♦ ♦❢ t❤❡ ♠♦st
♣♦♣✉❧❛r ❢❛✐r♥❡ss ❞❡✜♥✐t✐♦♥s✳
❈❛r❛❣✐❛♥♥✐s ❡t ❛❧✳ ✭❈❛r❛❣✐❛♥♥✐s ❡t ❛❧✳ ✷✵✶✷✮ ✐♥tr♦❞✉❝❡❞ t❤❡ ❝♦♥❝❡♣t ♦❢ PoF ✐♥ t❤❡ ❝♦♥t❡①t
♦❢ ❢❛✐r ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠s✿ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡② ❝♦♠♣❛r❡ t❤❡ ✈❛❧✉❡ ♦❢ t♦t❛❧ ❛❣❡♥ts✬ ✉t✐❧✐t② ✐♥ ❛
s②st❡♠ ♦♣t✐♠✉♠ ✇✐t❤ t❤❡ ♠❛①✐♠✉♠ t♦t❛❧ ✉t✐❧✐t② ♦❜t❛✐♥❡❞ ♦✈❡r ❛❧❧ ❢❛✐r s♦❧✉t✐♦♥s ✭t❤❡② ♠❛❦❡
✉s❡ ♦❢ s❡✈❡r❛❧ ♥♦t✐♦♥s ♦❢ ❢❛✐r♥❡ss✱ ♥❛♠❡❧② ♣r♦♣♦rt✐♦♥❛❧✐t②✱ ❡♥✈②✲❢r❡❡♥❡ss ❛♥❞ ❡q✉✐t❛❜✐❧✐t②✮✳ ■♥
✭❇❡rts✐♠❛s ❡t ❛❧✳ ✷✵✶✶✮✱ ❇❡rts✐♠❛s ❡t ❛❧✳ ❢♦❝✉s ♦♥ ♣r♦♣♦rt✐♦♥❛❧ ❢❛✐r♥❡ss ❛♥❞ ♠❛①✲♠✐♥ ❢❛✐r♥❡ss
❛♥❞ ♣r♦✈✐❞❡ ❛ t✐❣❤t ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ PoF ❢♦r ❛ ❜r♦❛❞ ❢❛♠✐❧② ♦❢ ❛❧❧♦❝❛t✐♦♥ ♣r♦❜❧❡♠s ✇✐t❤
❝♦♠♣❛❝t ❛♥❞ ❝♦♥✈❡① ❛❣❡♥ts✬ ✉t✐❧✐t② s❡ts✳ ■♥ ✭◆✐❝♦s✐❛ ❡t ❛❧✳ ✷✵✶✼✮ t❤❡ ❛✉t❤♦rs ♣r♦✈❡ ❛ ♥✉♠❜❡r
♦❢ ♣r♦♣❡rt✐❡s ♦♥ t❤❡ ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss ✇❤✐❝❤ ❤♦❧❞ ❢♦r ❛♥② ❣❡♥❡r❛❧ ♠✉❧t✐✲❛❣❡♥t ♣r♦❜❧❡♠ ✇✐t❤♦✉t
❛♥② s♣❡❝✐❛❧ ❛ss✉♠♣t✐♦♥ ♦♥ t❤❡ ❛❣❡♥ts✬ ✉t✐❧✐t✐❡s✱ ❢♦❝✉s✐♥❣ ♦♥ ♠❛①✲♠✐♥✱ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦✐
❛♥❞ ♣r♦♣♦rt✐♦♥❛❧ ❢❛✐r♥❡ss✳ ❙✐t✉❛t✐♦♥s ✐♥ ✇❤✐❝❤ t❤❡ ❛❣❡♥ts ♣✉rs✉❡ t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡✐r
❝♦sts ✭r❛t❤❡r t❤❛♥ t❤❡ ♠❛①✐♠✐③❛t✐♦♥ ♦❢ t❤❡✐r ✉t✐❧✐t②✮ ❤❛✈❡ ❜❡❡♥ ❞❡❛❧t ✇✐t❤ ❜② ❊rt♦❣r❛❧ ❛♥❞
❲✉ ✭❊rt♦❣r❛❧ ❛♥❞ ❲✉ ✷✵✵✵✮✱ ✇❤♦ ❞❡r✐✈❡ ❛ ♠❡❛s✉r❡ ♦❢ ❢❛✐r♥❡ss ❛♠♦♥❣ ❛ s❡t ♦❢ s✉♣♣❧② ❝❤❛✐♥
♠❡♠❜❡rs✳ ❆♥♦t❤❡r ❡①❛♠♣❧❡ ♦❢ ❢❛✐r♥❡ss ✐♥ t❤❡ ♣r❡s❡♥❝❡ ♦❢ ❝♦st ❛❧❧♦❝❛t✐♦♥s ❝❛♥ ❜❡ ❢♦✉♥❞ ✐♥
9

34. ✭❇♦❤♠ ❛♥❞ ▲❛rs❡♥ ✶✾✾✹✮✳ ❖✉r ✈✐❡✇ ♦❢ ❢❛✐r♥❡ss ✐s r❡❧❛t❡❞ t♦ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠s✱ ❜✉t ✐t
✐s ✇♦rt❤ ♦❜s❡r✈✐♥❣ t❤❛t ❢❛✐r♥❡ss ✐ss✉❡s ❛r✐s❡ ✐♥ ♦t❤❡r ❝♦♥t❡①ts✱ s✉❝❤ ❛s ❢❛✐r r❡♣r❡s❡♥t❛t✐♦♥
♣r♦❜❧❡♠s ✭❇❛❧✐♥s❦✐ ❛♥❞ ❨♦✉♥❣ ✷✵✵✶✮✱ ♦r t❤❡ ❛♣♣♦rt✐♦♥♠❡♥t ♣r♦❜❧❡♠ ✭▲✉❝❛s ✶✾✽✸✮✳ ❚❤❡s❡
♥❡❡❞ t♦ ❜❡ ❞❡❛❧t ✇✐t❤ ❜② ❞✐✛❡r❡♥t ♠❡t❤♦❞s ❛♥❞ ❛❧❣♦r✐t❤♠s t❤❛♥ t❤♦s❡ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s
t❛❧❦✳
✷ ❙❝❤❡❞✉❧✐♥❣ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠s
❇❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠s ❛❞❞r❡ss s✐t✉❛t✐♦♥s ✐♥ ✇❤✐❝❤ t✇♦ ♣❧❛②❡rs ✭❛❣❡♥ts✮ ❛r❡ ❢❛❝❡❞ ✇✐t❤ ❛
s❡t ♦❢ ♣♦ss✐❜❧❡ ❛❣r❡❡♠❡♥ts ✭r❡s♦✉r❝❡ s❡t✮✱ ❛♥❞ ♠✉st r❡❛❝❤ ❛ ❝♦♠♣r♦♠✐s❡ ♦✈❡r ♦♥❡ ♦❢ t❤❡♠✳
❍❡r❡ ✇❡ ❛r❡ ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ s❝❤❡❞✉❧✐♥❣ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠✳ ❚❤❡r❡ ❛r❡ t✇♦
❛❣❡♥ts✱ ♥❛♠❡❧② A ❛♥❞ B✳ ❊❛❝❤ ❛❣❡♥t ❤❛s ❛ s❡t ♦❢ ❥♦❜s✱ ✇❤✐❝❤ ❤❛✈❡ t♦ ❜❡ ♣r♦❝❡ss❡❞ ❜② ❛
s✐♥❣❧❡ ♠❛❝❤✐♥❡✳ ❆❣❡♥ts A ✭B✮ ❤❛s ❥♦❜s JA
1 , . . . , JA
nA
✭JB
1 , . . . , JB
nB
✮✱ ♦❢ ❧❡♥❣t❤ pA
1 , . . . , pA
nA
✭pB
1 , . . . , pB
nB
✮✳ ▲❡t PA
=
PnA
j=1 pA
j ✭PB
=
PnB
j=1 pB
j ✮✳ ❏♦❜s ❝❛♥♥♦t ❜❡ ♣r❡❡♠♣t❡❞✱ ❛♥❞ t❤❡
♠❛❝❤✐♥❡ ❝❛♥ ♦♥❧② ♣r♦❝❡ss ♦♥❡ ❥♦❜ ❛t ❛ t✐♠❡✳ ❲❡ ✉s❡ t❤❡ t❡r♠s A✲❥♦❜s ❛♥❞ B✲❥♦❜s t♦ r❡❢❡r
t♦ t❤❡ t✇♦ ❛❣❡♥ts✬ r❡s♣❡❝t✐✈❡ ❥♦❜s✳
●✐✈❡♥ ❛ ❢❡❛s✐❜❧❡ s❝❤❡❞✉❧❡ σ✱ ✇❡ ❧❡t fA
(σ) ❛♥❞ fB
(σ) ❞❡♥♦t❡ t❤❡ ❝♦st ✈❛❧✉❡s ❢♦r t❤❡
t✇♦ ❛❣❡♥ts✳ ❍❡r❡ ✇❡ ❝♦♥s✐❞❡r ❛s r❡s♦✉r❝❡ s❡t t❤❡ s❡t ΣP ♦❢ P❛r❡t♦ ♦♣t✐♠❛❧ s❝❤❡❞✉❧❡s✱
❛s t❤❡② ✐♥❝❧✉❞❡ ❛❧❧ s❡♥s✐❜❧❡ ❝♦♠♣r♦♠✐s❡ s❝❤❡❞✉❧❡s✳ ❋♦r ❡❛❝❤ σ ∈ ΣP ✱ ✇❡ ✇❛♥t t♦ ❞❡✜♥❡
✉t✐❧✐t② ✈❛❧✉❡s uA
(σ) ❛♥❞ uB
(σ)✱ s♦ t❤❛t✱ ❢♦r i = A, B✱ ui
(σ) ≥ 0 ❛♥❞ ui
(σ) ✐♥❝r❡❛s❡s
❛s fi
(σ) ❞❡❝r❡❛s❡s✳ ❋♦r t❤✐s ♣✉r♣♦s❡✱ ✇❡ ♣r♦♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡✜♥✐t✐♦♥ ♦❢ ✉t✐❧✐t②✳ ▲❡t
fi
∞ = max{fi
(σ)|σ ∈ ΣP } ✭❢♦r r❡❣✉❧❛r ❢✉♥❝t✐♦♥s fi
(σ)✱ t❤✐s ✐s t❤❡ ♠✐♥✐♠✉♠ ❝♦st t❤❡ ❛❣❡♥t
i ❜❡❛rs ✐❢ ✐ts ❥♦❜s ❛r❡ s❝❤❡❞✉❧❡❞ ❛❢t❡r ❛❧❧ t❤❡ ❥♦❜s ♦❢ t❤❡ ♦t❤❡r ❛❣❡♥t✮✳ ❚❤❡♥
ui
(σ) = fi
∞ − fi
(σ) i = A, B
■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ❝♦♥s✐❞❡r t❤❛t ❛♥ ❛❣❡♥t✬s ✉t✐❧✐t② ✐s t❤❡ s❛✈✐♥❣ ❛❝❤✐❡✈❡❞ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡
✇♦rst s❝❤❡❞✉❧❡ ❢♦r t❤❛t ❛❣❡♥t✳ ❲❡ ❛❧s♦ ❧❡t U(σ) = uA
(σ) + uB
(σ) ❞❡♥♦t❡ t❤❡ ♦✈❡r❛❧❧ ✉t✐❧✐t②
♦❢ s❝❤❡❞✉❧❡ σ ❛♥❞ ❧❡t σ∗
❞❡♥♦t❡ t❤❡ s❝❤❡❞✉❧❡ t❤❛t ♠❛①✐♠✐③❡s U(σ) ✭s②st❡♠ ♦♣t✐♠✉♠✮✱ ✐✳❡✳
U(σ∗
) = max
σ∈ΣP
{U(σ)}.
■♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❡ ❛❧s♦ ❧❡t fi∗
= min{fi
(σ)|σ ∈ ΣP }✳
❆s ❢♦r t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❢❛✐r s♦❧✉t✐♦♥✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❝♦♥❝❡♣ts✳
✶✳ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦② s♦❧✉t✐♦♥ ✭❑❛❧❛✐ ❛♥❞ ❙♠♦r♦❞✐♥s❦② ✶✾✼✺✮✳ ●✐✈❡♥ σ ∈ ΣP ✱ ❧❡t
ūi
(σ) =
ui
(σ)
fi
∞ − fi∗
❜❡ t❤❡ ♥♦r♠❛❧✐③❡❞ ✉t✐❧✐t② ♦❢ σ ❢♦r ❛❣❡♥t i✳ ❆ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦② s❝❤❡❞✉❧❡ σKS ✭❜r✐❡✢②✱
❑❙ s❝❤❡❞✉❧❡✮ ✐s ❞❡✜♥❡❞ ❛s
σKS = arg max
σ
min
i=A,B
{ūi
(σ)}.
❚❤❡ ✐❞❡❛ ✐s t❤❛t ✐♥ σKS t❤❡ ♥♦r♠❛❧✐③❡❞ ✉t✐❧✐t② ♦❢ t❤❡ ❛❣❡♥t ✇❤♦ ✐s ✇♦rs❡✲♦✛ ✐s ♠❛①✐✲
♠✐③❡❞✳ ❙♦✱ ✐♥ σKS t❤❡ t✇♦ ❛❣❡♥ts✬ ♥♦r♠❛❧✐③❡❞ ✉t✐❧✐t② ✈❛❧✉❡s ❛r❡ t②♣✐❝❛❧❧② q✉✐t❡ ❝❧♦s❡✳
❖❜✈✐♦✉s❧②✱ ✉♥❞❡r t❤✐s ❞❡✜♥✐t✐♦♥ ❛ ❑❙ s❝❤❡❞✉❧❡ ❛❧✇❛②s ❡①✐sts✳ ■♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ❛♥✲
♦t❤❡r ❞❡✜♥✐t✐♦♥ ♦❢ ❢❛✐r s♦❧✉t✐♦♥ ✐s t❤❡ ♠❛①✲♠✐♥ s♦❧✉t✐♦♥ ✭❇❡rts✐♠❛s ❡t ❛❧✳ ✷✵✶✶✮✳ ❑❛❧❛✐✲
❙♠♦r♦❞✐♥s❦② ❛♥❞ ♠❛①✲♠✐♥ s♦❧✉t✐♦♥s ❝♦✐♥❝✐❞❡ ✐❢ fA
∞ − fA∗
= fB
∞ − fB∗
✳
10

35. ✷✳ Pr♦♣♦rt✐♦♥❛❧❧② ❢❛✐r s♦❧✉t✐♦♥✳ ❆ s❝❤❡❞✉❧❡ σP F ✐s ♣r♦♣♦rt✐♦♥❛❧❧② ❢❛✐r ✐❢✱ ❢♦r ❛♥② ♦t❤❡r
P❛r❡t♦ ♦♣t✐♠❛❧ s❝❤❡❞✉❧❡ σ✱ ✐t ❤♦❧❞s
uA
(σ) − uA
(σP F )
uA(σP F )
+
uB
(σ) − uB
(σP F )
uB(σP F )
≤ 0. ✭✶✮
❚❤❡ ✐❞❡❛ ❜❡❤✐♥❞ s✉❝❤ ❞❡✜♥✐t✐♦♥ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❲❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ♠♦✈✐♥❣ ❢r♦♠ s❝❤❡❞✲
✉❧❡ σP F t♦ ❛♥② ♦t❤❡r s❝❤❡❞✉❧❡ σ✱ t❤❡ r❡❧❛t✐✈❡ ❜❡♥❡✜t t❤❛t ♦♥❡ ❛❣❡♥t ♠❛② ♦❜t❛✐♥ ✐s
❜❛❧❛♥❝❡❞ ❜② ❛ ♥♦t s♠❛❧❧❡r ✉t✐❧✐t② ❞❡❝r❡❛s❡ ❢♦r t❤❡ ♦t❤❡r ❛❣❡♥t✳ ❚❤✐s ✐s ❛❝t✉❛❧❧② t❤❡
s❛♠❡ r❛t✐♦♥❛❧❡ ❜❡❤✐♥❞ t❤❡ ❝♦♥❝❡♣t ♦❢ ◆❛s❤ s♦❧✉t✐♦♥ ✭◆❛s❤ ✶✾✺✵✮✳ ❍♦✇❡✈❡r✱ t❤❡ ◆❛s❤
s♦❧✉t✐♦♥ ✇❛s ✐♥tr♦❞✉❝❡❞ ♦♥❧② ✇✐t❤ r❡s♣❡❝t t♦ ❝♦♠♣❛❝t ❛♥❞ ❝♦♥✈❡① r❡s♦✉r❝❡ s❡ts✱ ✇❤✐❧❡
❉❡✜♥✐t✐♦♥ ✭✶✮ ✐s ♠♦r❡ ❣❡♥❡r❛❧✳ ■♥ ❢❛❝t✱ ✇❤✐❧❡ ❛ ◆❛s❤ s♦❧✉t✐♦♥ ❛❧✇❛②s ❡①✐sts✱ ❛ ♣r♦♣♦r✲
t✐♦♥❛❧❧② ❢❛✐r s♦❧✉t✐♦♥ ♠❛② ♥♦t ❡①✐st✳ ■❢ ✐t ❞♦❡s ❡①✐st✱ t❤❡♥ ✐t ❝♦✐♥❝✐❞❡s ✇✐t❤ t❤❡ s♦❧✉t✐♦♥
♠❛①✐♠✐③✐♥❣ t❤❡ ♣r♦❞✉❝t ♦❢ ✉t✐❧✐t✐❡s✱ ❛♥❞ ❤❡♥❝❡✱ ✐❢ t❤❡ r❡s♦✉r❝❡ s❡t ✐s ❝♦♠♣❛❝t ❛♥❞
❝♦♥✈❡①✱ ✇✐t❤ t❤❡ ◆❛s❤ s♦❧✉t✐♦♥✳
❖✉r st✉❞② ✐♥✈❡st✐❣❛t❡s ❤♦✇ ♠✉❝❤ ✭❣❧♦❜❛❧✮ ✉t✐❧✐t② s❤♦✉❧❞ ❜❡ ❣✐✈❡♥ ✉♣ ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ❛
❢❛✐r s♦❧✉t✐♦♥✳ ❚❤✐s ✐s ❝❛♣t✉r❡❞ ❜② t❤❡ ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss✱ ❞❡✜♥❡❞ ❛s t❤❡ r❡❧❛t✐✈❡ ❧♦ss ♦❢ ✉t✐❧✐t②
✐♥ ❛ ❢❛✐r s♦❧✉t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ♠❛①✐♠✉♠ ✉t✐❧✐t②✳ ◆♦t✐❝❡ t❤❛t t❤❡r❡ ♠❛② ❜❡ ♠♦r❡ t❤❛♥
♦♥❡ ❢❛✐r s♦❧✉t✐♦♥✱ ❞✐✛❡r✐♥❣ ✐♥ t❡r♠s ♦❢ ❣❧♦❜❛❧ ✉t✐❧✐t②✳ ❍❡r❡ ✇❡ ❛❞♦♣t t❤❡ s❛♠❡ ✈✐❡✇♣♦✐♥t ❛s
✐♥ ✭❑❛rs✉ ❛♥❞ ▼♦rt♦♥ ✷✵✶✺✱ ◆❛❧❞✐ ❡t ❛❧✳ ✷✵✶✻✮✱ ✐✳❡✳✱ ✇❤❡♥❡✈❡r t❤✐s ♦❝❝✉rs✱ ✇❡ ♠❡❛s✉r❡ t❤❡
♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❜❡st ❢❛✐r s♦❧✉t✐♦♥✳ ■♥ ❢♦r♠❛❧ t❡r♠s✱ ❧❡tt✐♥❣ I ❞❡♥♦t❡ t❤❡
s❡t ♦❢ ❛❧❧ ✐♥st❛♥❝❡s ♦❢ ❛ ❣✐✈❡♥ ♣r♦❜❧❡♠✱ I ♦♥❡ ♦❢ t❤❡♠✱ σ∗
(I) t❤❡ s②st❡♠ ♦♣t✐♠✉♠✱ ΣF t❤❡
s❡t ♦❢ ❛❧❧ ❢❛✐r s❝❤❡❞✉❧❡s ❛♥❞ σF (I) ♦♥❡ ♦❢ t❤❡♠✱ ✇❡ ❞❡✜♥❡ t❤❡ ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss ❛s✿
PoF = sup
I∈I
min
σF ∈ΣF
U(σ∗
(I)) − U(σF (I))
U(σ∗(I))
. ✭✷✮
◆♦t✐❝❡ t❤❛t t❤✐s ✐s ❛ s✐♠✐❧❛r ❞❡✜♥✐t✐♦♥ t♦ t❤❡ ✇❡❧❧✲❦♥♦✇♥ Pr✐❝❡ ♦❢ ❙t❛❜✐❧✐t② ✭❆♥s❤❡❧❡✈✐❝❤
❡t ❛❧✳ ✷✵✵✹✮✱ r❡♣❧❛❝✐♥❣ t❤❡ r♦❧❡ ♦❢ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ✇✐t❤ ❢❛✐r♥❡ss✳ ❍❡r❡❛❢t❡r✱ ✇❡ ✐♥❞✐❝❛t❡
✇✐t❤ PoFKS ❛♥❞ PoFP F t❤❡ ♣r✐❝❡ ♦❢ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦② ❢❛✐r♥❡ss ❛♥❞ ♣r♦♣♦rt✐♦♥❛❧ ❢❛✐r♥❡ss✱
r❡s♣❡❝t✐✈❡❧②✳
✸ ❙❝❡♥❛r✐♦ ❛❞❞r❡ss❡❞
■♥ t❤✐s t❛❧❦ ✇❡ ❛❞❞r❡ss t❤❡ ✈❛❧✉❡ ♦❢ PoF ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ s❝❤❡❞✉❧✐♥❣ s❝❡♥❛r✐♦✳ ❆❣❡♥t
A ♣✉rs✉❡s t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ ✐ts ❥♦❜s✬ ❝♦♠♣❧❡t✐♦♥ t✐♠❡s✱ ✇❤✐❧❡ ❛❣❡♥t B ✐s
✐♥t❡r❡st❡❞ ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ♠❛①✐♠✉♠ t❛r❞✐♥❡ss ♦❢ ✐ts ❥♦❜s ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❝♦♠♠♦♥
❞✉❡ ❞❛t❡ d✳ ❋♦❧❧♦✇✐♥❣ t❤❡ ✉s✉❛❧ ●r❛❤❛♠✬s ♥♦t❛t✐♦♥✱ ✇❡ ❞❡♥♦t❡ t❤✐s s❝❡♥❛r✐♦ ❛s 1|dB
j =
d|
P
CA
j , TB
max✳ ◆♦t❡ t❤❛t t❤✐s s❝❡♥❛r✐♦ ✐♥❝❧✉❞❡s t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ B ✇❛♥ts t♦ ♠✐♥✐♠✐③❡
✐ts ❥♦❜s✬ ♠❛❦❡s♣❛♥✱ ♦❜t❛✐♥❡❞ ❢♦r d = 0✳ ◆♦t❡ t❤❛t ✐♥ t❤✐s s❝❡♥❛r✐♦✱ ✐♥ ❛♥② P❛r❡t♦ ♦♣t✐♠❛❧
s♦❧✉t✐♦♥ ❛❧❧ B✲❥♦❜s ❛r❡ s❝❤❡❞✉❧❡❞ ❝♦♥s❡❝✉t✐✈❡❧②✱ s♦ ♦♥❡ ❝❛♥ ❛ss✉♠❡ t❤❛t B ♦✇♥s ❛ s✐♥❣❧❡
❥♦❜ ♦❢ ❧❡♥❣t❤ PB
=
PnB
j=1 pB
j ✳ ■♥ t❤✐s s❝❡♥❛r✐♦✱ t❤❡ ✈❛❧✉❡s fA∗
, fA
∞, fB∗
, fB
∞ ❝❛♥ ❛❧❧ ❜❡
❡❛s✐❧② ❝♦♠♣✉t❡❞✳ ■♥ ❢❛❝t✱ fA∗
✐s t❤❡ t♦t❛❧ ❝♦♠♣❧❡t✐♦♥ t✐♠❡ ♦❢ t❤❡ A✲❥♦❜s ✇❤❡♥ t❤❡② ❛r❡
s❡q✉❡♥❝❡❞ ✐♥ ❙P❚ ♦r❞❡r ❛♥❞ fA
∞ = fA∗
+ nAPB
✱ ✇❤✐❧❡ fB∗
= max{0, PB
− d} ❛♥❞ fB
∞ =
max{0, PA
+ PB
− d}✳
❖✉r ❝♦♥tr✐❜✉t✐♦♥s t♦ t❤✐s s❝❡♥❛r✐♦ ❛r❡ s✉♠♠❛r✐③❡❞ ✐♥ ❚❛❜❧❡ ✶✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ s❤♦✇
t❤❛t PoFKS = 2/3 ❛♥❞ PoFP F = 1/2✳ ▼♦r❡♦✈❡r✱ ✇❡ s❤♦✇ t❤❛t✱ ✐❢ t❤❡ A✲❥♦❜s ❛r❡ ❛❧r❡❛❞②
♦r❞❡r❡❞ ❜② ♥♦♥❞❡❝r❡❛s✐♥❣ ❧❡♥❣t❤✱ ✐♥ t✐♠❡ O(log nA) ❛ ♣r♦♣♦rt✐♦♥❛❧❧② ❢❛✐r s♦❧✉t✐♦♥ ❝❛♥ ❜❡
❝♦♠♣✉t❡❞ ♦r ♣r♦✈❡❞ t❤❛t ✐t ❞♦❡s ♥♦t ❡①✐st✳
11

36. ❚❛❜❧❡ ✶✳ ❘❡s✉❧ts ❢♦r s❝❡♥❛r✐♦ 1|dB
j = d|
P
CA
j , TB
max (∗
) ✐❢ A✲❥♦❜s ❛r❡ ❣✐✈❡♥ ✐♥ ❙P❚ ♦r❞❡r✳
Pr♦♣♦rt✐♦♥❛❧❧② ❢❛✐r s♦❧✉t✐♦♥ ❑❛❧❛✐✲❙♠♦r♦❞✐♥s❦② s♦❧✉t✐♦♥
P♦❋ ✈❛❧✉❡ ✶✴✷ ✷✴✸
❊①✐st❡♥❝❡ ❊st❛❜❧✐s❤❡❞ ✐♥ O(log nA)∗
✭❛❧✇❛②s ❡①✐sts✮
❘❡❢❡r❡♥❝❡s
❆♥s❤❡❧❡✈✐❝❤✱ ❊✳✱ ❉❛s❣✉♣t❛✱ ❆✳✱ ❑❧❡✐♥❜❡r❣✱ ❏✳✱ ❚❛r❞ös✱ ❊✳✱ ❲❡①❧❡r✱ ❚✳ ❛♥❞ ❚✳ ❘♦✉❣❤❣❛r❞❡♥ ✭✷✵✵✹✮✱
❚❤❡ Pr✐❝❡ ♦❢ ❙t❛❜✐❧✐t② ❢♦r ◆❡t✇♦r❦ ❉❡s✐❣♥ ✇✐t❤ ❋❛✐r ❈♦st ❆❧❧♦❝❛t✐♦♥✱ ✐♥ ✹✺t❤ ❆♥♥✉❛❧ ■❊❊❊
❙②♠♣♦s✐✉♠ ♦♥ ❋♦✉♥❞❛t✐♦♥s ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ✭❋❖❈❙✮✱ ♣❛❣❡s ✺✾✲✼✸✳
❇❛❧✐♥s❦✐✱ ▼✳▲✳✱ ❨♦✉♥❣ ❍✳P✳ ✭✷✵✵✶✮✱ ❋❛✐r ❘❡♣r❡s❡♥t❛t✐♦♥✿ ▼❡❡t✐♥❣ t❤❡ ■❞❡❛❧ ♦❢ ❖♥❡ ▼❛♥✱ ❖♥❡ ❱♦t❡✱
❇r♦♦❦✐♥❣s ■♥st✐t✉t✐♦♥ Pr❡ss✱ ❲❛s❤✐♥❣t♦♥✳
❇❡rts✐♠❛s ❉✳✱ ❱✳ ❋❛r✐❛s✱ ◆✳ ❚r✐❝❤❛❦✐s ✭✷✵✶✶✮✱ ❚❤❡ ♣r✐❝❡ ♦❢ ❢❛✐r♥❡ss✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤✱ ✺✾✭✶✮✱
✶✼✕✸✶✳
❇♦❤♠✱ P✳✱ ❇✳ ▲❛rs❡♥ ✭✶✾✾✹✮✱ ❋❛✐r♥❡ss ✐♥ ❛ tr❛❞❡❛❜❧❡✲♣❡r♠✐t tr❡❛t② ❢♦r ❝❛r❜♦♥ ❡♠✐ss✐♦♥s r❡❞✉❝t✐♦♥s
✐♥ ❊✉r♦♣❡ ❛♥❞ t❤❡ ❢♦r♠❡r ❙♦✈✐❡t ❯♥✐♦♥✱ ❊♥✈✐r♦♥♠❡♥t❛❧ ❛♥❞ ❘❡s♦✉r❝❡ ❊❝♦♥♦♠✐❝s✱ ✹✱ ✷✶✾✕✷✸✾✳
❈❛r❛❣✐❛♥♥✐s ■✳✱ ❈✳ ❑❛❦❧❛♠❛♥✐s✱ P✳ ❑❛♥❡❧❧♦♣♦✉❧♦s✱ ▼✳ ❑②r♦♣♦✉❧♦✉ ✭✷✵✶✷✮✱ ❚❤❡ ❡✣❝✐❡♥❝② ♦❢ ❢❛✐r
❞✐✈✐s✐♦♥✱ ✷✵✶✷✱ ❚❤❡♦r② ♦❢ ❈♦♠♣✉t✐♥❣ ❙②st❡♠s✱ ✺✵✭✹✮✱ ✺✽✾✕✻✶✵✳
❊rt♦❣r❛❧ ❑✳✱ ❉✳ ❲✉ ✭✷✵✵✵✮✱ ✏❆✉❝t✐♦♥✲t❤❡♦r❡t✐❝ ❝♦♦r❞✐♥❛t✐♦♥ ♦❢ ♣r♦❞✉❝t✐♦♥ ♣❧❛♥♥✐♥❣ ✐♥ t❤❡ s✉♣♣❧②
❝❤❛✐♥✑✱ ■■❊ ❚r❛♥s❛❝t✐♦♥s✱ ✸✷✭✶✵✮✱ ✾✸✶✲✾✹✵✳
❑❛❧❛✐ ❊✳✱ ▼✳ ❙♠♦r♦❞✐♥s❦② ✭✶✾✼✺✮✱ ❖t❤❡r s♦❧✉t✐♦♥s t♦ ◆❛s❤ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠✱❊❝♦♥♦♠❡tr✐❝❛✱ ✹✸✱
✺✶✸✕✺✶✽✳
❑❛rs✉ Ö✳✱ ❆✳ ▼♦rt♦♥ ✭✷✵✶✺✮✱ ■♥❡q✉✐t② ❛✈❡rs❡ ♦♣t✐♠✐③❛t✐♦♥ ✐♥ ♦♣❡r❛t✐♦♥❛❧ r❡s❡❛r❝❤✱ ❊✉r♦♣❡❛♥ ❏♦✉r✲
♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ✷✹✺✭✷✮✱ ✸✹✸✕✸✺✾✳
▲✉❝❛s✱ ❲✳❋✳✱ ✭✶✾✽✸✮✱ ❚❤❡ ❆♣♣♦rt✐♦♥♠❡♥t Pr♦❜❧❡♠✱ ✐♥✿ ❇r❛♠s ❙✳❏✳✱ ▲✉❝❛s ❲✳❋✳✱ ❙tr❛✣♥ P✳❉✳
✭❡❞s✮✱ P♦❧✐t✐❝❛❧ ❛♥❞ ❘❡❧❛t❡❞ ▼♦❞❡❧s✱ ▼♦❞✉❧❡s ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✳ ❙♣r✐♥❣❡r✱ ◆❡✇ ❨♦r❦✱
◆❨✱ ✸✺✽✲✸✾✻✳
◆❛❧❞✐ ▼✳✱ ●✳ ◆✐❝♦s✐❛✱ ❆✳ P❛❝✐✜❝✐✱ ❯✳ P❢❡rs❝❤② ✭✷✵✶✻✮✱ ▼❛①✐♠✐♥ ❋❛✐r♥❡ss ✐♥ Pr♦❥❡❝t ❇✉❞❣❡t ❆❧❧♦✲
❝❛t✐♦♥✱ ❊❧❡❝tr♦♥✐❝ ◆♦t❡s ✐♥ ❉✐s❝r❡t❡ ▼❛t❤❡♠❛t✐❝s✱ ✺✺✱ ✻✺✕✻✽✳
◆❛s❤✱ ❏✳ ✭✶✾✺✵✮✱ ❚❤❡ ❜❛r❣❛✐♥✐♥❣ ♣r♦❜❧❡♠✱ ❊❝♦♥♦♠❡tr✐❝❛✱ ✶✽✭✷✮✱ ♣♣✳ ✶✺✺✕✶✻✷✳
◆✐❝♦s✐❛ ●✳✱ ❆✳ P❛❝✐✜❝✐✱ ❯✳ P❢❡rs❝❤②✱ ✷✵✶✼✱ Pr✐❝❡ ♦❢ ❋❛✐r♥❡ss ❢♦r ❛❧❧♦❝❛t✐♥❣ ❛ ❜♦✉♥❞❡❞ r❡s♦✉r❝❡✱
❊✉r♦♣❡❛♥ ❏♦✉r♥❛❧ ♦❢ ❖♣❡r❛t✐♦♥❛❧ ❘❡s❡❛r❝❤✱ ✷✺✼✱ ♣♣✳ ✾✸✸✲✾✹✸✳
12

37. ❆ ▼■▲P ❢♦r♠✉❧❛t✐♦♥ ❢♦r ❛♥ ♦♣❡r❛t✐♥❣ r♦♦♠ s❝❤❡❞✉❧✐♥❣
♣r♦❜❧❡♠ ✉♥❞❡r st❡r✐❧✐③✐♥❣ ❛❝t✐✈✐t✐❡s ❝♦♥str❛✐♥ts
❍✳ ❆❧ ❍❛s❛♥1,2,3
✱ ❈✳ ●✉ér❡t1
✱ ❉✳ ▲❡♠♦✐♥❡2
❛♥❞ ❉✳ ❘✐✈r❡❛✉3
1
❯♥✐✈❡rs✐té ❞✬❆♥❣❡rs✱ ▲❆❘■❙✱ ❆♥❣❡rs✱ ❋r❛♥❝❡
❝❤r✐st❡❧❧❡✳❣✉❡r❡t❅✉♥✐✈✲❛♥❣❡rs✳❢r
2
■▼❚ ❆t❧❛♥t✐q✉❡✱ ▲❙✷◆✱ ◆❛♥t❡s✱ ❋r❛♥❝❡
❞❛✈✐❞✳❧❡♠♦✐♥❡❅✐♠t✲❛t❧❛♥t✐q✉❡✳❢r
3
❯♥✐✈❡rs✐té ❈❛t❤♦❧✐q✉❡ ❞❡ ❧✬❖✉❡st✱ ▲❆❘■❙✱ ❆♥❣❡rs✱ ❋r❛♥❝❡
❛❧❤❛s❛♥✱ r✐✈r❡❛✉❅✉❝♦✳❢r
❑❡②✇♦r❞s✿ ♦♣❡r❛t✐♥❣ r♦♦♠s ✐♥ ❤❡❛❧t❤ s❡r✈✐❝❡s✱ st❡r✐❧✐③❛t✐♦♥ ✉♥✐t✱ s❝❤❡❞✉❧✐♥❣✱ ▼■▲P
✶ ■♥tr♦❞✉❝t✐♦♥
❖♣❡r❛t✐♥❣ r♦♦♠s ❤❛✈❡ ❜❡❡♥ r❡❝♦❣♥✐③❡❞ t♦ ❜❡ t❤❡ ♠❛✐♥ ✐♥❝♦♠❡ s♦✉r❝❡ ❢♦r ❤♦s♣✐t❛❧s
❛s ✐t ❣❡♥❡r❛t❡s ❛r♦✉♥❞ ✻✵✪ ♦❢ ❤♦s♣✐t❛❧ r❡✈❡♥✉❡s ✭▼❛❝❛r✐♦ ❡t✳ ❛❧✳ ✶✾✾✺✮ ❜✉t ✐t ❝♦✉♥ts ❢♦r
❛r♦✉♥❞ ✹✵✪ ♦❢ ❤♦s♣✐t❛❧ ❝♦sts ✭❏❛❝❦s♦♥ ✷✵✵✷✮ t❤r♦✉❣❤♦✉t t❤❡ ✉s❡ ♦❢ ❢❛❝✐❧✐t✐❡s ✭♦♣❡r❛t✐♥❣
r♦♦♠s✱ ❡t❝✳✮ ❛♥❞ t❤❡ ♣❡rs♦♥♥❡❧ ❝♦sts✳ ❚❤✐s ❤✉❣❡ ✜♥❛♥❝✐❛❧ ❢❛❝t♦r ♠❛❦❡s t❤❡ ♦♣❡r❛t✐♥❣ r♦♦♠s
♠❛♥❛❣❡♠❡♥t ❛ ♣r✐♦r✐t② ❢♦r ❤♦s♣✐t❛❧ ♠❛♥❛❣❡rs ✐♥ ♦r❞❡r t♦ ❛❝❤✐❡✈❡ ❛♥ ❡✣❝✐❡♥t ❛♥❞ ❡✛❡❝t✐✈❡
✉s❡ ♦❢ t❤❡ ♦♣❡r❛t✐♥❣ r♦♦♠s✳ ❊①❤❛✉st✐✈❡ ❧✐t❡r❛t✉r❡ r❡✈✐❡✇s ♦♥ t❤❡ ❙✉r❣✐❝❛❧ ❈❛s❡ ❙❝❤❡❞✉❧✐♥❣
✭❙❈❙✮ ♣r♦❜❧❡♠ ❛r❡ r❡♣♦rt❡❞ ✐♥ ✭❈❛r❞♦❡♥ ❡t✳ ❛❧✳ ✷✵✶✵✱ ●✉❡rr✐❡r♦ ❡t✳ ❛❧✳ ✷✵✶✶✮✳
■♥ t❤✐s ♣❛♣❡r✱ ✇❡ st✉❞② ❛ r❡❛❧ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✇❤✐❝❤ ❝♦♥s✐sts ✐♥ s❝❤❡❞✉❧✐♥❣ ❛ s❡t
♦❢ ❡❧❡❝t✐✈❡ s✉r❣✐❝❛❧ ❝❛s❡s ✇❤✐❝❤ r❡q✉✐r❡ s✉r❣✐❝❛❧ ✐♥str✉♠❡♥ts ❛♥❞ t♦♦❧s t♦ s❡✈❡r❛❧ ♦♣❡r❛t✐♥❣
r♦♦♠s ✇✐t❤ t❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ ♠✐♥✐♠✐③✐♥❣ t❤❡ ♦♣❡r❛t✐♥❣ ❝♦sts ✇❤✐❧❡ t❛❦✐♥❣ ✐♥t♦ ❛❝❝♦✉♥t
t❤❡ ❛❝t✐✈✐t✐❡s ♦❢ t❤❡ st❡r✐❧✐③✐♥❣ ✉♥✐t✳ ❚♦ t❤❡ ❜❡st ♦❢ ♦✉r ❦♥♦✇❧❡❞❣❡ t❤❡r❡ ❛r❡ ✈❡r② ❢❡✇
❧✐t❡r❛t✉r❡ ♦♥ s✉❝❤ ♣❛rt✐❝✉❧❛r ♣r♦❜❧❡♠✳ ❋♦r ✐♥st❛♥❝❡ ✭❇❡r♦✉❧❡ ❡t✳ ❛❧✳ ✷✵✶✻✮ st✉❞② ❛♥ ♦♣❡r❛t✐♥❣
r♦♦♠ s❝❤❡❞✉❧✐♥❣ ♣r♦❜❧❡♠ ✐♥❝❧✉❞✐♥❣ ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s st❡r✐❧✐③❛t✐♦♥ ❜✉t ✇✐t❤ t❤❡ ♦❜❥❡❝t✐✈❡ ♦❢
r❡❞✉❝✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s ♥❡❡❞❡❞ ❛t t❤❡ s❛♠❡ t✐♠❡✳ ❆♥❞ ❛ ❜r❛♥❝❤✲❛♥❞✲♣r✐❝❡
t❡❝❤♥✐q✉❡ ✐s ❛♣♣❧✐❡❞ ✐♥ ✭❈❛r❞♦❡♥ ❡t✳ ❛❧✳ ✷✵✵✾✮ t♦ ✜♥❞ t❤❡ ❜❡st ♦r❞❡r ❢♦r s✉r❣❡r✐❡s ✐♥ ❛ ❞❛②
❝❛r❡ ❝❡♥t❡r ✐♥ ♦r❞❡r t♦ ♦♣t✐♠✐③❡ s❡✈❡r❛❧ ♦❜❥❡❝t✐✈❡s ✭♣❡❛❦ ✉s❡ ♦❢ r❡❝♦✈❡r② ❜❡❞s✱ ♦❝❝✉rr❡♥❝❡
♦❢ r❡❝♦✈❡r② ♦✈❡rt✐♠❡✱ ✳✳✳✮ ✇❤✐❧❡ s❛t✐s❢②✐♥❣ t❤❡ ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s st❡r✐❧✐③✐♥❣ ❝♦♥str❛✐♥ts✳
❚❤✐s r❡s❡❛r❝❤ ✇❛s ♣❡r❢♦r♠❡❞ ✐♥ ❝♦❧❧❛❜♦r❛t✐♦♥ ✇✐t❤ t❤❡ ❯♥✐✈❡rs✐t② ❍♦s♣✐t❛❧ ♦❢ ❆♥❣❡rs
✐♥ ❋r❛♥❝❡ ✭❈❍❯ ❆♥❣❡rs✮✱ ✇❤✐❝❤ ❤❛s ❛❧s♦ ♣r♦✈✐❞❡❞ ❤✐st♦r✐❝❛❧ ❞❛t❛ ❢♦r t❤❡ ❡①♣❡r✐♠❡♥ts✳ ❲❡
♣r♦♣♦s❡ ❛ ♠✐①❡❞ ✐♥t❡❣❡r ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ♠♦❞❡❧ ❢♦r t❤❡ ♣r♦❜❧❡♠ ✇❤✐❝❤ ✐s s♦❧✈❡❞ ✐♥
❛ ❧❡①✐❝♦❣r❛♣❤✐❝ ✇❛②✳ ❲❡ s❤♦✇ t❤❛t ♦✉r s♦❧✉t✐♦♥s ♣r♦✈✐❞❡ ❝♦♠♣❡t✐t✐✈❡ r❡s✉❧ts ✐♥ t❡r♠s ♦❢
♥✉♠❜❡r ♦❢ ♦♣❡r❛t✐♥❣ r♦♦♠s✱ ❛♥❞ s✐❣♥✐✜❝❛♥t❧② ✐♠♣r♦✈❡ t❤♦s❡ ♦♣❡r❛t✐♦♥❛❧❧② ✐♠♣❧❡♠❡♥t❡❞ ✐♥
t❡r♠s ♦❢ ♦✈❡rt✐♠❡ ❛♥❞ ❡♠❡r❣❡♥❝✐❡s ❛t t❤❡ st❡r✐❧✐③✐♥❣ ✉♥✐t✳
✷ Pr♦❜❧❡♠ ❞❡s❝r✐♣t✐♦♥
❚❤❡ ❈❍❯ ❆♥❣❡rs ✐♥❝❧✉❞❡s s❡✈❡r❛❧ ❜❧♦❝❦s ❛♥❞ ❛ st❡r✐❧✐③✐♥❣ ✉♥✐t ✇❤✐❝❤ ❝❡♥tr❛❧✐③❡s ❛❧❧
t❤❡ st❡r✐❧✐③✐♥❣ ❛❝t✐✈✐t✐❡s✳ ■♥ t❤✐s st✉❞②✱ ✇❡ ❢♦❝✉s ♦♥❧② ♦♥ t❤❡ ❛❝t✐✈✐t✐❡s ♦❢ t❤❡ ❖rt❤♦♣❡❞✐❝
❙✉r❣❡r② ❇❧♦❝❦ ✭❖❙❇✮ ❛♥❞ t❤❡ ❙t❡r✐❧✐③✐♥❣ ❯♥✐t ✭❙❯✮✳ ❚❤❡r❡ ❛r❡ ❛r♦✉♥❞ ✷✺✵✵ s✉r❣❡r✐❡s t❤❛t
❛r❡ ♣❡r❢♦r♠❡❞ ❛t t❤❡ ❖❙❇ ♣❡r ②❡❛r ✐♥ ✐ts ✸ ♦♣❡r❛t✐♥❣ r♦♦♠s✳ ❚❤❡ ♦♣❡♥✐♥❣ ❤♦✉rs ❢♦r t❤❡s❡
✸ ❖❘s ❛r❡ ❞✐✛❡r❡♥t ❛s✿ r♦♦♠ ✶ ❛♥❞ ✷ ❛r❡ ♦♣❡♥ ✺ ❞❛②s ❛ ✇❡❡❦ ❢r♦♠ ✽✿✶✺ t♦ ✶✼✿✵✵✱ ❛♥❞ r♦♦♠
✸ ✐s ♦♣❡♥ ♦♥❧② ✹ ❞❛②s ❛ ✇❡❡❦ ❢r♦♠ ✽✿✶✺ t♦ ✶✹✿✸✵✳ ❇❡t✇❡❡♥ ✶✵ ❛♥❞ ✶✹ s✉r❣❡♦♥s s❤❛r❡ t❤❡s❡
r♦♦♠s ❛❝❝♦r❞✐♥❣ t♦ ❛ ♣❧❛♥♥✐♥❣ ✐♥❞✐❝❛t✐♥❣ t❤❡ ❞❛②s ✇❤❡♥ t❤❡② ♦♣❡r❛t❡✱ ❛♥❞ t❤❡ ❧✐st ♦❢ r♦♦♠s
t❤❛t ❡❛❝❤ s✉r❣❡♦♥ ❝❛♥ ✉s❡ ❡❛❝❤ ❞❛②✳
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38. ❊❛❝❤ s✉r❣❡♦♥ ❤❛s t♦ ♣❡r❢♦r♠ ❛ ❧✐st ♦❢ s✉r❣❡r✐❡s ♦♥ ❛♥ ❤♦r✐③♦♥ ♦❢ ♦♥❡ ♠♦♥t❤ ✿ s♦♠❡ ♦❢
t❤❡♠ ❝❛♥ ❜❡ s❝❤❡❞✉❧❡❞ ❛♥②t✐♠❡ ❞✉r✐♥❣ t❤❡ ♦♣❡♥✐♥❣ ❤♦✉rs ♦❢ t❤❡ r♦♦♠s✱ ✇❤❡r❡❛s ♦t❤❡rs
✭❛♠❜✉❧❛t♦r② s✉r❣❡r✐❡s✮ ❤❛✈❡ t♦ ❜❡ ❝♦♠♣❧❡t❡❞ ❜❡❢♦r❡ ✶✺✿✵✵ t♦ ❛❧❧♦✇ t❤❡ ♣❛t✐❡♥t t♦ ❣♦ ❤♦♠❡
❛t t❤❡ ❡♥❞ ♦❢ t❤❡ ❞❛②✳ ❊❛❝❤ ♦❢ t❤❡s❡ s✉r❣❡r✐❡s ✐s ❝❤❛r❛❝t❡r✐③❡❞ ❜② ❛♥ ❡st✐♠❛t❡❞ ❞✉r❛t✐♦♥
t✐♠❡ ❛♥❞ r❡q✉✐r❡s ❛ ❧✐st ♦❢ s✉r❣✐❝❛❧ ✐♥str✉♠❡♥ts ✇❤✐❝❤ ❛r❡ ♦r❣❛♥✐③❡❞ ✐♥ s♠❛❧❧ ❜♦①❡s ❝❛❧❧❡❞
❦✐ts✳ ❚❤❡s❡ ❦✐ts ❛r❡ ❛✈❛✐❧❛❜❧❡ ✐♥ ❧✐♠✐t❡❞ q✉❛♥t✐t✐❡s✳ ❆❢t❡r ❡❛❝❤ s✉r❣❡r②✱ t❤❡ ✉s❡❞ ❦✐ts ❛r❡ ❦❡♣t
✐♥t♦ ✇❛t❡r ❢♦r ✸✵ ♠✐♥✉t❡s ❢♦r ♣r❡✲❞✐s✐♥❢❡❝t✐♦♥✳ ❚❤❡♥ t❤❡② ❛r❡ ❝♦❧❧❡❝t❡❞ ❛t t❤❡ ♣r❡❞❡✜♥❡❞
♣❡r✐♦❞s ❣✐✈❡♥ ✐♥ t❛❜❧❡ ✶ ❛♥❞ s❡♥t t♦ t❤❡ ❙❯ ❢♦r st❡r✐❧✐③❛t✐♦♥✳
❚❛❜❧❡ ✶✳ ❙❯✬s ♣✐❝❦✉♣s ❛♥❞ ❞❡❧✐✈❡r✐❡s t♦ t❤❡ ❖❙❇✳
P✐❝❦✲✉♣ ✵✼✿✵✵ ✶✶✿✸✵ ✶✸✿✵✵ ✶✹✿✸✵ ✶✻✿✵✵ ✶✼✿✸✵ ✶✽✿✸✵
❉❡❧✐✈❡r② ✵✼✿✵✵ ✲ ✲ ✶✹✿✸✵ ✲ ✶✼✿✸✵ ✲
❆t t❤❡ ❙❯✱ t❤❡ st❡r✐❧✐③❛t✐♦♥ ♣r♦❝❡ss ✐s ❜❡✐♥❣ ♣❡r❢♦r♠❡❞ ✐♥ s❡✈❡r❛❧ st❡♣s ✿ t❤❡ ✐♥str✉♠❡♥ts
❛r❡ ✜rst ❝❧❡❛♥❡❞ ❜② ❛✉t♦♠❛t✐❝ ✇❛s❤❡rs✱ t❤❡♥ r❡❛ss✐❣♥❡❞ ✐♥ t❤❡✐r ❝♦rr❡s♣♦♥❞✐♥❣ ❦✐t ❜❡❢♦r❡
❜❡✐♥❣ ♣r♦❝❡ss❡❞ t❤r♦✉❣❤ st❡r✐❧✐③❛t✐♦♥ ♠❛❝❤✐♥❡s✳ ❋✐♥❛❧❧②✱ t❤❡ ❦✐ts ❛r❡ ❦❡♣t ❛t t❤❡ ❙❯ t♦ ❝♦♦❧
♦✛ ❜❡❢♦r❡ ❜❡✐♥❣ r❡t✉r♥❡❞ t♦ t❤❡ ❜❧♦❝❦✳ ❖♥ ❛✈❡r❛❣❡✱ ✇❤❡♥ ❛ ❦✐t ❛rr✐✈❡s ❛t t❤❡ ❙❯✱ t❤❡ ✇❤♦❧❡
st❡r✐❧✐③❛t✐♦♥ ♣r♦❝❡ss t❛❦❡s ❛r♦✉♥❞ ✹❤✸✵✳ ❋r♦♠ t❤❡s❡ ❞❡❧✐✈❡r②✴❝♦❧❧❡❝t ❤♦✉rs ✐♥ ❚❛❜❧❡ ✶ ❛♥❞
❢r♦♠ t❤❡ ❛✈❡r❛❣❡ ❦✐ts ♣r♦❝❡ss✐♥❣ t✐♠❡ ❛t t❤❡ ❙❯✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s✐t✉❛t✐♦♥s✿
✶✳ ❆ ❦✐t ❝♦❧❧❡❝t❡❞ ❛t ✶✶✿✸✵✱ ✶✸✿✵✵ ♦r ✶✹✿✸✵ ♦♥ ❞❛② ✭t✮ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ ♠♦r♥✐♥❣ ♦♥ ❞❛②
✭t + 1✮ ✐❢ ✐t ✐s tr❡❛t❡❞ ❛s ❛ ♣r✐♦r✐t② ❛t t❤❡ ❙❯ ✭♣r✐♦r✐t② ❦✐t✱ ❝❛s❡ ✶✮✳ ■❢ ✐t ✐s ♥♦t tr❡❛t❡❞
❛s ❛ ♣r✐♦r✐t②✱ ✐t ✐s ❝♦♥s✐❞❡r❡❞ t❤❛t ✐t ❝❛♥♥♦t ❜❡ ✉s❡❞ ❜❡❢♦r❡ ✶✹✿✸✵ ♦♥ ❞❛② ✭t + 1✮✳
✷✳ ❆ ❦✐t ❝♦❧❧❡❝t❡❞ ❛t ✶✻✿✵✵ ♦♥ ❞❛② ✭t✮ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ t❤❡ ♠♦r♥✐♥❣ ♦♥ ❞❛② ✭t + 1✮ ✐❢ ✐t ✐s
tr❡❛t❡❞ ✉r❣❡♥t❧② ❛t t❤❡ ❙❯ ✭✉r❣❡♥t ❦✐t✮✳ ■❢ ✐t ✐s ♥♦t tr❡❛t❡❞ ✉r❣❡♥t❧②✱ ✐t ❝❛♥ ❜❡ ✉s❡❞
❢r♦♠ ✶✹✿✸✵ ♦♥ ❞❛② ✭t + 1✮✳
✸✳ ❆ ❦✐t ❝♦❧❧❡❝t❡❞ ❛t ✶✼✿✸✵✱ ✶✽✿✸✵ ♦♥ ❞❛② ✭t✮ ♦r ✼✿✵✵ ♦♥ ❞❛② ✭t + 1✮ ❝❛♥ ❜❡ ✉s❡❞ ♦♥ ❞❛②
✭t + 1✮ ❢r♦♠ ✶✹✿✸✵ ✐❢ ✐t ✐s tr❡❛t❡❞ ❛s ❛ ♣r✐♦r✐t② ❛t t❤❡ ❙❯ ✭♣r✐♦r✐t② ❦✐t✱ ❝❛s❡ ✷✮✳ ■❢ ✐t ✐s
♥♦t tr❡❛t❡❞ ❛s ❛ ♣r✐♦r✐t②✱ ✐t ✇✐❧❧ ❜❡ ❛✈❛✐❧❛❜❧❡ ♦♥ ❞❛② ✭t + 1✮ ❢r♦♠ ✶✼✿✸✵✳
■♥ t❤❡ ❝✉rr❡♥t ❞❡❝✐s✐♦♥ ♣r♦❝❡ss ❛t t❤❡ ❈❍❯✱ ✐t ✐s ♦♥❧② ❝❤❡❝❦❡❞ ✇❤❡t❤❡r t❤❡ s✉r❣❡r✐❡s
s❝❤❡❞✉❧❡❞ ❡❛❝❤ ❞❛② ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ♥✉♠❜❡r ♦❢ ❦✐ts ♦✇♥❡❞ ❜② t❤❡ ❜❧♦❝❦ ✭t❤❡ st❡r✲
✐❧✐③✐♥❣ ❝♦✉rs❡s ❢♦r ❦✐ts ❛r❡ ♥❡❣❧❡❝t❡❞ ❞✉r✐♥❣ t❤❡ ❛ss✐❣♥♠❡♥t ♦❢ s✉r❣❡r✐❡s t♦ t❤❡ s❤✐❢ts✮✳
❈♦♥s❡q✉❡♥t❧②✱ t❤❡ ♥✉♠❜❡r ♦❢ ✉r❣❡♥t ❛♥❞ ♣r✐♦r✐t② ❦✐ts ✐♥ ❙❯ t♦ ♣r♦❝❡ss r❡♠❛✐♥s s✉❜st❛♥t✐❛❧✳
❚❤❡ ✐♠♣❛❝t ♦♥ t❤❡ ❛❝t✐✈✐t② ❢♦r t❤❡ ❙❯ ✐s ✐♠♠❡❞✐❛t❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❡❛❝❤ ✉r❣❡♥t ❦✐t ✐♠♣❧✐❡s
t♦ st♦♣ ❛ ♠❛❝❤✐♥❡ ✐♥ ♦r❞❡r t♦ ♣r♦❝❡ss ✐t ✐♠♠❡❞✐❛t❡❧② ✭✐♥❞✉❝✐♥❣ ❛ ♥❡❡❞ t♦ r❡✲♣r♦❝❡ss r❡♠♦✈❡❞
❦✐ts ❛❢t❡r✇❛r❞✮✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ♣r✐♦r✐t② ❦✐ts ❛r❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ✉♣♦♥ t❤❡✐r ❛rr✐✈❛❧
❜② ♠♦✈✐♥❣ t❤❡♠ t♦ t❤❡ st❛rt ♦❢ t❤❡ q✉❡✉❡ t♦ ❜❡ tr❡❛t❡❞ ✜rst✱ ❛❣❛✐♥st t❤❡ ✜rst✲✐♥ ✜rst ♦✉t
❝❧❛ss✐❝❛❧ ♣♦❧✐❝② ♦❢ ❙❯✳ ❯❧t✐♠❛t❡❧②✱ ✐♥ r❛r❡ ❝❛s❡s✱ s♦♠❡ ❦✐ts ♠❛② ❡✈❡♥ ❜❡ r❡q✉❡st❡❞ ♦✉ts✐❞❡
t❤❡ ❞❡❧✐✈❡r② ❤♦✉rs ✭✈✐♦❧❛t❡❞ ❦✐ts✮✳ ■♥ t❤❛t ❝❛s❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ r❡q✉❡st ❛ s♣❡❝✐❛❧ s❤✉tt❧❡✳
❚❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ✇♦r❦ ✐s t♦ s❝❤❡❞✉❧❡ ❛❧❧ s✉r❣❡r✐❡s ❛t t❤❡ ❖❙❇ ✇❤✐❧❡ t❛❦✐♥❣ ✐♥t♦
❛❝❝♦✉♥t t❤❡ st❡r✐❧✐③✐♥❣ ♣r♦❝❡ss ✐♥ ♦r❞❡r t♦ r❡❞✉❝❡ t❤❡ ♣r❡ss✉r❡ ♦♥ t❤❡ ❙❯ st❛✛✳
■♥ t❡r♠s ♦❢ ♦❜❥❡❝t✐✈❡s✱ ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❈❍❯✱ t❤❡ ✜rst ♣r✐♦r✐t② r❡♠❛✐♥s t♦ s❝❤❡❞✉❧❡ ❛❧❧
s✉r❣❡r✐❡s ✐♥ ♦r❞❡r t♦ ♠✐♥✐♠✐③❡ t❤❡ t♦t❛❧ ♦✈❡rt✐♠❡ ♦❢ t❤❡ st❛✛ ♠❡♠❜❡rs ♦❢ t❤❡ ❖❙❇✳ ❚❤❡
s❡❝♦♥❞ ♣r✐♦r✐t② ❝♦♥s✐sts ✐♥ ♠✐♥✐♠✐③✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✉s❡❞ ♦♣❡r❛t✐♥❣ r♦♦♠s✳ ❋✐♥❛❧❧②✱ t❤❡
t❤✐r❞ ♦❜❥❡❝t✐✈❡ ✐s t♦ ❦❡❡♣ t❤❡ ♥✉♠❜❡r ♦❢ ✉r❣❡♥t ❛♥❞ ♣r✐♦r✐t② ❦✐ts ❛s ❧♦✇ ❛s ♣♦ss✐❜❧❡✳
■♥ t❤❡ ♥❡①t s❡❝t✐♦♥✱ ✇❡ ❜r✐❡✢② s❦❡t❝❤ t❤❡ ❜❛s✐s ♦❢ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧ ❢♦r s♦❧✈✐♥❣ t❤✐s
✐♥t❡❣r❛t❡❞ ❖❙❇✲❙❯ ♣r♦❜❧❡♠✳
14

39. ✸ ▼❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥
■♥ ♦r❞❡r t♦ ♠♦❞❡❧ t❤✐s ♣r♦❜❧❡♠✱ ✇❡ ♣r♦♣♦s❡ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛t✐♦♥ ❜❛s❡❞ ♦♥ t❤❡
❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ t❤❡ ❞❛② ✐♥ ❢♦✉r ♣❡r✐♦❞s ✿ ♣❡r✐♦❞ ✶ ❢r♦♠ ✽✿✶✺ t♦ ✶✹✿✵✵✱ ♣❡r✐♦❞ ✷ ❢r♦♠ ✶✹✿✵✵
t♦ ✶✹✿✸✵✱ ♣❡r✐♦❞ ✸ ❢r♦♠ ✶✹✿✸✵ t♦ ✶✺✿✸✵ ❛♥❞ ♣❡r✐♦❞ ✹ ❢r♦♠ ✶✺✿✸✵ t♦ ✶✼✿✵✵✳ ❚❤❡ st❛rt✐♥❣ ❛♥❞
❡♥❞✐♥❣ ❤♦✉rs ♦❢ t❤❡s❡ ♣❡r✐♦❞s ❛r❡ ♦❜t❛✐♥❡❞ ❜② t❤❡ ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❝r✐t✐❝❛❧ ♣✐❝❦✉♣ ❛♥❞
❞❡❧✐✈❡r② ❤♦✉rs ❛t t❤❡ ❖❙❇✱ t❤❡ ♦♣❡♥✐♥❣ ❛♥❞ ❝❧♦s✐♥❣ ❤♦✉rs ♦❢ t❤❡ ♦♣❡r❛t✐♥❣ r♦♦♠s✱ ❛♥❞ t❤❡
❢❛❝t t❤❛t s✉r❣❡r✐❡s ♠✉st ❡♥❞ ✸✵ ♠✐♥✉t❡s ❜❡❢♦r❡ t❤❡ ❝♦❧❧❡❝t ♦❢ t❤❡✐r ♠❡❞✐❝❛❧ ❞❡✈✐❝❡s✳
❲❡ t❤❡♥ ✐♥tr♦❞✉❝❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s✿
witr ❜✐♥❛r② ✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦ ✶ ✐❢ ♦♣❡r❛t✐♦♥ i ✐s s❝❤❡❞✉❧❡❞ ❛t ❞❛② t ✐♥ r♦♦♠ r
xbf
itr ❜✐♥❛r② ✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦ ✶ ✐❢ s✉r❣❡r② i ❜❡❣✐♥s ❛t ♣❡r✐♦❞ b ❛♥❞ ✜♥✐s❤❡s ❛t f✱ ♦♥ ❞❛② t✱ ✐♥ r♦♦♠ r
εtr ✐♥t❡❣❡r ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ t♦t❛❧ ♦✈❡rt✐♠❡ ✐♥ r♦♦♠ r ❛t ❞❛② t
Ltr ❜✐♥❛r② ✈❛r✐❛❜❧❡ ❡q✉❛❧ t♦ ✶ ✐❢ r♦♦♠ r ✐s ✉s❡❞ ❛t ❞❛② t
Etk ✐♥t❡❣❡r ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ t♦t❛❧ ✉r❣❡♥t ❦✐ts ♦❢ t②♣❡ k ❛t ❞❛② t
Y 1
tk ✭Y 2
tk✮ ✐♥t❡❣❡r ✈❛r✐❛❜❧❡ r❡♣r❡s❡♥t✐♥❣ t❤❡ t♦t❛❧ ♣r✐♦r✐t② ❦✐ts ♦❢ ❝❛s❡ ✶ ✭r❡s♣✳ ❝❛s❡ ✷✮ ♦❢ t②♣❡ k ❛t ❞❛② t
■♥ ❛❞❞✐t✐♦♥ t♦ t❤❡ r❡❣✉❧❛r s❝❤❡❞✉❧✐♥❣ ❝♦♥str❛✐♥ts ✭❛❧❧ s✉r❣❡r✐❡s ❤❛✈❡ t♦ ❜❡ s❝❤❡❞✉❧❡❞ ✐♥ t❤❡
❤♦r✐③♦♥✱ ❡t❝✮✱ ♦✉r ♠♦❞❡❧ ✐♥❝❧✉❞❡s ♦t❤❡r ❝♦♥str❛✐♥ts s✉❝❤ ❛s✿
✕ t❤❡ ❡①♣r❡ss✐♦♥s ♦❢ ✉r❣❡♥t ✭✶✮ ❛♥❞ ♣r✐♦r✐t② ❦✐ts ✭❜♦t❤ ❝❛s❡s✮ ✭✷✮✲✭✸✮✿
PO
i=1
PR
r=1 qik.
PJ
f=2
Pf
b=1 xbf
itr +
P2
b=1
PJ
f=b xbf
i(t+1)r
− Qk ≤ Etk ∀t ∈ {1, .., T}, ∀k ∈ {1, .., K} ✭✶✮
PO
i=1
PR
r=1 qik.
PJ
f=1
Pf
b=1 xbf
itr +
P2
b=1
PJ
f=b xbf
i(t+1)r
− Qk − Etk ≤ Y 1
tk ∀t ∈ {1, . . . , T}, ∀k ∈ {1, . . . , K} ✭✷✮
PO
i=1
PR
r=1 qik.
PJ
f=2
Pf
b=1 xbf
itr +
PJ
b=1
PJ
f=b xbf
i(t+1)r
− Qk − Etk ≤ Y 2
tk ∀t ∈ {1, .., T}, ∀k ∈ {1, .., K} ✭✸✮
✇❤❡r❡ R ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❖❘s✱ qik ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❦✐ts ♦❢ t②♣❡ k ∈ {1..K}
t❤❛t s✉r❣❡r② i ∈ {1..O} r❡q✉✐r❡s✱ J r❡♣r❡s❡♥ts t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ♣❡r✐♦❞s ✭✹✮✱ Qk ✐s
t❤❡ t♦t❛❧ ❛✈❛✐❧❛❜❧❡ q✉❛♥t✐t② ♦❢ ❦✐t k ❛♥❞ T ✐s t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❞❛②s ✐♥ t❤❡ ❤♦r✐③♦♥✳
✕ t❤❡ ❝♦♥tr♦❧ ♦❢ t❤❡ ✇♦r❦❧♦❛❞ ♦❢ s✉r❣❡r✐❡s ❢♦r ❡❛❝❤ ✐♥t❡r✈❛❧ ♦❢ t❤❡ ❞❛② ❛♥❞ ❡❛❝❤ r♦♦♠ ✭✹✮✿
O
X
i=1
γ
X
b=β
γ
X
f=b
pi · xbf
itr +
O
X
i=1
β−1
X
b=1
J
X
f=γ+1
dβγ
rt · xbf
itr ≤ dβγ
rt · Ltr + uγ · εtr
∀β ∈ {1, . . . , J}, ∀γ ∈ {β, . . . , J}, ∀t ∈ {1, . . . , T}, ∀r ∈ {1, . . . , R}
✭✹✮
✇❤❡r❡ pi ✐s t❤❡ ❞✉r❛t✐♦♥ ♦❢ s✉r❣❡r② i✱ dβγ
rt ✐s t❤❡ ❞✉r❛t✐♦♥ ❢r♦♠ ♣❡r✐♦❞ β t♦ γ ✐♥ r♦♦♠
r ♦♥ ❞❛② t✱ uγ ✐s ❛ ❜✐♥❛r② ♣❛r❛♠❡t❡r ❡q✉❛❧ t♦ ✶ ✐❢ γ = J✳
❚❤❡ ♠✉❧t✐♣❧❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s ❛r❡ t❛❦❡♥ ✐♥t♦ ❛❝❝♦✉♥t ❜② ✉s✐♥❣ ❛ ❧❡①✐❝♦❣r❛♣❤✐❝ ♠❡t❤♦❞
✭✺✮ ❛♥❞ ❛❢t❡r ❡❛❝❤ ♦❜❥❡❝t✐✈❡ ✐s s♦❧✈❡❞✱ ✐ts ✈❛❧✉❡ ✐s ❛❞❞❡❞ ❛s ❛♥ ✉♣♣❡r ❜♦✉♥❞ t♦ t❤❡ ♠♦❞❡❧✳
❋✐rst✱ f1 ♠✐♥✐♠✐③❡s t❤❡ t♦t❛❧ ♦✈❡r t✐♠❡ ❛♥❞ t❤❡♥ f2 ♠✐♥✐♠✐③❡s t❤❡ ♥✉♠❜❡r ♦❢ ✉s❡❞ r♦♦♠s
❛♥❞ ✜♥❛❧❧② f3 ♠✐♥✐♠✐③❡s t❤❡ t♦t❛❧ ♣❡♥❛❧t② ❝♦st ♦❢ ✉r❣❡♥t ✭cu✮ ❛♥❞ ♣r✐♦r✐t② ✭cp✮ ❦✐ts✳
▼✐♥✐♠✐③❡ Lex
f1 :
PT
t=1
PR
r=1 εtr ; f2 :
PT
t=1
PR
r=1 Ltr ; f3 :
PT
t=1
PK
k=1
cu · Etk + cp · (Y 1
tk + Y 2
tk)
✭✺✮
✹ ❊①♣❡r✐♠❡♥t❛❧ r❡s✉❧ts
❚♦ t❡st ❛♥❞ ✈❛❧✐❞❛t❡ ♦✉r ♠♦❞❡❧✱ ✇❡ ✉s❡❞ ❛ ✶✵ ✐♥st❛♥❝❡s ❜❡♥❝❤♠❛r❦ ♣r♦✈✐❞❡❞ ❜② t❤❡ ❈❍❯✳
❊❛❝❤ ✐♥st❛♥❝❡ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❛❝t✐✈✐t② ♦❢ t❤❡ ❖❙❇ ❞✉r✐♥❣ ♦♥❡ ♠♦♥t❤✳ ❚❤❡ ♥✉♠❜❡r ♦❢
s✉r❣❡r✐❡s ✐♥ t❤❡s❡ ✐♥st❛♥❝❡s ✈❛r✐❡s ❢r♦♠ ✶✻✹ t♦ ✷✷✵✳ ❲❡ ✉s❡❞ ❈P❧❡① ✶✷✳✻✳✶ t♦ s♦❧✈❡ t❤❡
♠♦❞❡❧ ❛♥❞ ❛ t✐♠❡ ❧✐♠✐t ♦❢ ✸✻✵✵ s❡❝♦♥❞s ✇❛s s❡t ❢♦r ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥s✳
❆s s❤♦✇♥ ✐♥ t❛❜❧❡ ✷✱ ✇❤✐❝❤ ❝♦♠♣❛r❡s t❤❡ s❝❤❡❞✉❧❡ ♦❢ t❤❡ ❈❍❯ ✇✐t❤ t❤❡ s❝❤❡❞✉❧❡ ❢r♦♠
t❤❡ ▼■▲P✱ ♦✉r ♠♦❞❡❧ ♠❛♥❛❣❡❞ t♦ ❡❧✐♠✐♥❛t❡ t❤❡ ❛♠❜✉❧❛t♦r② s✉r❣❡r✐❡s t❤❛t ✜♥✐s❤ ❛❢t❡r ✶✺✿✵✵
15