On combinatorial optimization and mechanism design problems arising at container ports

Sebastian MeiswinkelOn Combinatorial Optimization and Mechanism Design Problems Arising at Container Ports... The author made a major contribution to this field byimproving the understa

Sebastian Meiswinkel

On Combinatorial Optimization and Mechanism Design Problems Arising

at Container Ports

Produktion und Logistik

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Technische Universität Braunschweig

Sebastian Meiswinkel

On Combinatorial Optimization and Mechanism Design Problems Arising at Container Ports

With a foreword by Prof Dr Erwin Pesch

Produktion und Logistik

ISBN 978-3-658-22361-8 ISBN 978-3-658-22362-5 (eBook)

https://doi.org/10.1007/978-3-658-22362-5

Library of Congress Control Number: 2018944272

Dissertation University of Siegen, 2017

Springer Gabler

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Globalisation and the internet are main driving forces for the rapid increase of freighttransport over the past years, and its growth is predicted to continue in the same rate forthe next decade or more amongst others due to China’s One Belt One Road (or New SilkRoad) initiative, that is intended to intensify the trade between Europe and Asia alongthe former silk road

Intermodal transport is mostly the transport of containers in one supply chain inmore than one transport mode, e.g, by rail, road or sea While the long distances arecovered by huge container vessels that periodically operate between sea ports, or byintercontinental trains, trucks are operating on the so called last mile between customersand a rail transshipment terminal or a sea port in order to deliver or pick-up the freight.Transshipment yards and container ports are main points where the transportationmode of goods can be changed Since rail and roads are operating at their capacitylimits, an acceleration of freight transport requires faster processes in order to improvethe efficiency of cargo handling at container terminals and to cope with the increasingshipping volume

In this book “On Combinatorial Optimization and Mechanism Design Problems ing at Container Ports” the author concentrates exactly on that, analysing different trans-portation problems arising at container terminals from a quantitative point of view

Aris-In the first part, the author discusses situations where an operator of a containerterminal serves clients who can be assumed to behave selfish If the operator of a containerterminal requires private information from the clients for a socially optimal decision,

it is necessary to provide an incentive or mechanism that all clients commit their trueinformation and do not try to influence the outcome by false information Algorithmicmechanism design is a research area that deals with the construction of these methods tomake the clients to be truthful The author made a major contribution to this field byimproving the understanding of truthfulness in case of specific scheduling problems.The second part of this book deals with real world optimization problems withoutany private information that appear at container ports Efficient handling of the internal

processes is important for the terminal being competitive, since many terminals are underhigh competitive pressure and therefore need to optimize their processes.

This book should be most suitable to researchers and students of logistics and ations research In addition, the contents of this book might be very interesting to those

oper-in oper-industry who need to solve problems on the design, operation, and management ofcontainer ports

Prof Dr Erwin Pesch

This thesis represents the result of my PhD study at the Chair of Management InformationScience at the University of Siegen This time has been a challenging and long journey,but I was fortunate enough to be accompanied by many people I thank all of them,particularly the ones below

First, I would like to express my great gratitude to my supervisor Prof Dr ErwinPesch He offered me the opportunity to work at the Chair of Management InformationScience and to conduct my PhD research He was very open-minded in his supervisionand I enjoyed very much the freedom he gave me to pursue the research direction Ilike Throughout my study, he continuously supported me with his broad knowledge inthe fields of logistics and operations research His professional comments and invaluableadvices have helped me improve the scientific quality of this thesis

I would also like to thank Prof Dr Rob van Stee for serving as the second referee of

my thesis and PD Dr Sergei Chubanov for being member of the doctoral committee

My great gratitude goes also to Dr Dominik Kreß who shares an office with me Theconsiderable discussions we had and the constructive suggestions he made on my workwere very helpful I would also like to thank Dr Alena Otto for proofreading importantparts of my thesis

I am thankful to my colleagues David Müller, Xiyu Li and Roswitha Eifler for porting me and for participating in my defense My appreciation also goes to my formercolleagues Dr Jenny Nossack

sup-Last but not least, I would like to express my gratitude to my family and friends andespecially to my wife Anja for the strong support they have given me during the time Ihave been working on this thesis

Siegen, March 2018 Sebastian Meiswinkel

List of Figures xiii

1 Introduction and Preliminaries 1

1.1 Notation and Terminology 3

1.1.1 Machine Scheduling 3

1.1.2 Mechanism Design 5

1.1.3 Graph Theory 10

1.2 Outline 12

2 Mechanism Design and Machine Scheduling: Literature Review 15 2.1 Scope of Review 16

2.2 Review of Problem Categories and Features 16

2.2.1 Categories, Risk Attitude and Private Information of Agents 17

2.2.2 Models of Execution and Constraints on Committed Data 17

2.2.3 Characteristics of Payment Schemes 18

2.2.4 Other Problem Categories and Features 19

2.3 Classification Scheme 19

2.3.1 Review of Selected Elements of Graham et al (1979) 19

2.3.2 Including Mechanism Design Settings for Machine Scheduling Prob-lems 21

2.3.3 Examples 24

2.4 Literature Overview 24

2.5 Research Challenges and Conclusion 29

3 Truthful Algorithms for Job Agents 31

3.1 Related Literature 32

3.2 One-Parameter Job Agents 33

3.2.1 Problem Setting and Preliminaries 33

3.2.2 Monotonicity and List-Scheduling Algorithms 34

3.2.3 Multiple Parallel Machines 36

3.2.4 One Machine 38

3.3 Two-Parameter Job Agents 40

3.3.1 Problem Setting and Preliminaries 40

3.3.2 Incentive Compatible Mechanisms for P |priv{wj, dj}, Uj|P wjUj 43 3.3.3 Applying Our Results to an Example Algorithm for 1|priv{wj, dj}, Uj|P wjUj 51

3.4 Conclusion and Future Research 55

4 The Partitioning Min-Max Weighted Matching Problem 57 4.1 Detailed Problem Definition and Applications 58

4.2 Computational Complexity 62

4.3 Algorithms 64

4.3.1 Solving the Restricted Partitioning Problem 66

4.3.2 Solving the Min-Max Weighted Matching Problem 67

4.3.3 Partition-Match Heuristics 67

4.3.4 Match-Partition Heuristics 68

4.4 Computational Results 69

4.5 Conclusion 74

5 Straddle Carrier Routing at Container Ports with Quay Crane Buffers 77 5.1 Related Literature 78

5.2 Detailed Problem Definition 79

5.2.1 Problem Setting and Assumptions 80

5.2.2 Notation and Detailed Problem Description 82

5.2.3 A Mixed-Integer Program 85

5.3 Computational Complexity 88

5.4 Algorithms 92

5.4.1 Initial Solution 92

5.4.2 Routing Problem 94

5.4.3 Fast Heuristic for Times and Buffer Capacities 98

5.5 Computational Results 100

5.5.1 Comparison of Algorithms 100

5.5.2 Comparison with Practice 104

5.6 Conclusion 106

6 Summary and Outlook 109

List of Figures

1.1 Algorithmic mechanism design (in case of direct revelation) and scheduling

games 6

1.2 (Direct revelation) algorithmic mechanism design (Kress et al., 2017) 7

3.1 Cj(f (vs, v−j)) for an exemplary instance with n = 10 and m = 2 36

3.2 Plot of winfj (dj) 44

3.3 An arbitrary subsequence 47

3.4 Sorted lists of jobs L(w) and L(w0) with w < w0 53

3.5 Temporary schedules adand ad 0 k with d > d0 54

3.6 Temporary schedules adand ad 0 k with d < d0 55

4.1 A solution to an example instance of PMMWM 58

4.2 Potential schematic layout of a reach stacker based terminal 60

4.3 Rail-road terminal (Boysen and Fliedner, 2010b) 61

4.4 Partition-Match heuristics 65

4.5 Match-Partition heuristics 66

4.6 Overview of all heuristics 70

4.7 Comparison of M PLS, M P , and P MBP S 71

4.8 Runtimes of P MBP Sand P MREG 71

4.9 Runtimes of P MREG, M P , and M PLS 72

4.10 Quality of CPLEX 72

4.11 Quality of P MREG 73

4.12 Quality of P MBP S 73

4.13 Quality of M P 74

4.14 Quality of M PLS 74

5.1 Schematic layout of a straddle carrier based terminal 81

5.2 Schematic representation of the transformation from IPto IS 90

5.3 Creating an initial solution - GREEDY 94

5.4 Connect paths to a closed tour 95

5.5 Tricycle reference structure 96

5.6 Bicycle reference structure 96

5.7 Fixing time variables 99

5.8 2 quay cranes, 3 vehicles per crane 105

5.9 3 quay cranes, 3 vehicles per crane 106

5.10 4 quay cranes, 3 vehicles per crane 107

List of Tables

1.1 Number sets used throughout this thesis 3

1.2 Notation: machine scheduling problems 5

1.3 Notation: algorithmic mechanism design 8

2.1 Overview - Job agents 25

2.2 Overview - Unrelated machine agents 26

2.3 Overview - Uniform machine agents 27

3.1 Example with j = 7, w7= 18, t7= 2, n = 10, m = 2 37

3.2 Additional notation for v−j∈ V−jfixed 41

5.1 Notation used throughout this chapter 84

5.2 Variables used throughout this chapter 86

5.3 Simplifications used in the MILP 86

5.4 Definition of start(i), end(i), and stack(i) 87

5.5 2 quay cranes, 6 vehicles, CPLEX unlimited - Quality 101

5.6 2 quay cranes, 6 vehicles, CPLEX limited - Quality 102

5.7 2 quay cranes, 6 vehicles, CPLEX limited - Runtime in s 103

5.8 4 quay cranes, 12 vehicles, CPLEX limited - Quality 103

5.9 4 quay cranes, 12 vehicles, CPLEX limited - Runtime in s 104

Introduction and Preliminaries

Container ports are important parts of global transport chains They are used to transshipcontainers between vessels and vehicles for land transport Large container ports oftenoperate own rail-road terminals in order to connect the port with the hinterland efficiently.New challenges and perspectives for container ports, and therefore for intermodaltransport logistics, mainly arise due to the ongoing increase of the container flow and theresulting need to improve the throughput of containers One way to do so is to enlargethe port by building new berths and other necessary structures If this is not possible ortoo expensive, another way to improve the throughput of an existing port is to improvethe utilisation efficiency of the used equipment Danish Ship Finance (2016) reports thatthe world’s container market demand in 2016 has increased by 2.5 % in comparison tothe demand in 2012 It is furthermore projected to increase by another 4.7 % from 2016

to 2019 Moreover, the Institute of Shipping Economics and Logistics (2016) states a 8.4

% increase of the size of the international container fleet in 2015 and a doubling of theaverage size of new container ships since 2009 This development puts the entire logisticchain, and especially container ports and megahubs, under high pressure

In order to cope with the increasing container throughput, container ports tize processes and decision making (e.g., by using automated guided vehicles and (semi-)automated quay and stacking cranes as implemented at Container Terminal Altenwerder(CTA) in Hamburg, Germany) Automatization, however, induces the need for thoroughanalysis of all involved tasks Simulation studies and optimization techniques help to orga-nize the container throughput and to reduce the ships’ berthing time, which is consideredthe major cost driver and thus the main objective of a port’s operator (Steenken et al.,2004) However, apart from this objective, container ports also need to stay competitiveand attractive to their customers Topics such as quality of service, waiting times, sus-tainability, or waste minimization are increasingly important to the customers and have

automa-to be addressed

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2018

S Meiswinkel, On Combinatorial Optimization and Mechanism Design

Problems Arising at Container Ports, Produktion und Logistik,

https://doi.org/10.1007/978-3-658-22362-5_1

A considerable amount of literature aims at improving the efficiency of containerports from the perspective of the terminal operator The operator is assumed to haveall necessary information to make decisions Vis and de Koster (2003), Steenken et al.(2004), and Stahlbock and Voß (2008a) provide detailed surveys of recent literature inthis field In most cases, however, container ports act as service provider for customerslike shipping and logistic companies In various decisions, the port operator depends onthe information provided by these customers Problem settings that consider strategicbehavior of customers require other solution methods that combine methodologies fromdifferent research fields In this context, we will also focus on situations in this thesis,where the operator is not equipped with all relevant data related to the problem because it

is private information of selfish customers who aim to influence the solution determined bysome scheduling algorithm by submitting false information to the decision maker In somecases, the decision maker can extract the true information by designing an appropriatealgorithm that sets the right incentives for these players This in turn enables the decisionmaker to generate “fair” solutions with respect to some social criterion that considers theinterests of all players The design of such algorithms is studied in a field of research that

is usually referred to as algorithmic mechanism design (Nisan and Ronen, 2000)

In this thesis, we consider “classic” optimization problems as well as algorithmic anism design problems that arise at container ports In contrast to the problems at con-tainer terminals without direct interaction with customers’ information, there is no recentliterature overview for the considered algorithmic mechanism design problems Therefore

mech-we give a detailed overview over the recent development in this field Additionally, mech-weanalyze problems that arise at container terminals if the operators have to deal withcustomers by combining machine scheduling problems with methods from the field ofalgorithmic mechanism design As a main result, we improve the understanding of a ma-chine scheduling problem with two-parameter job agents by presenting new conditions fortruthful mechanisms that utilize the knowledge about the underlying scheduling problem

We also consider internal optimization problems without any customers involved To

be more precise, the transportation of containers between the quay and the storage area isanalyzed The arising optimization problem depends on the utilized equipment and on theconsidered tasks We consider the problem of routing reach stackers that move containersfrom a temporal storage area to the long-term storage area as well as the problem ofrouting straddle carriers that move containers from a buffer at the quay crane directly tothe storage area

1.1 Notation and Terminology

In this thesis we will denote the set of real (rational) numbers by R (Q), the set ofnonnegative real (rational) numbers by R≥0 (Q≥0), and the set of positive real (rationalnumbers by R>0 (Q>0) The number sets used throughout this thesis are presented inTable 1.1 We assume the reader to be familiar with the basic concepts of operations

Table 1.1: Number sets used throughout this thesis

R >0 , Q >0 positive real (rational) numbers

research, combinatorial optimization, complexity theory, and with general concepts ofmixed-integer programming Detailed introductions to these topics can be found, forexample, in Korte and Vygen (2008) or Schrijver (2003) However, a short introduction

to the notation and terminology concerning machine scheduling is given in Section 1.1.1.This section is based on Błażewicz et al (2007) Similar, Section 1.1.2 and Section 1.1.3are concerned with the basics of mechanism design and graph theory

1.1.1 Machine Scheduling

In this thesis we consider machine scheduling problems in the context of algorithmic gametheory Unfortunately, the notation used in the scheduling and (algorithmic) game theorycommunities is not always compatible Therefore, we will sometimes deviate from thestandard scheduling notation

Basically, a machine scheduling problem is characterized by a set J := {1, , n} of

n jobs (tasks) and a set M := {m1, , mm} of m machines Jobs and machines arecharacterized by certain parameters, e.g processing times or speeds Furthermore, thereexist different performance measures In this thesis, we restrict ourselves to schedulingproblems where all jobs are processed without preemption Furthermore each job has to

be processed once on exactly one machine and each machine can process only one job at

a time

We will now characterize the machines that are considered in this thesis We restrictourselves to parallel machines and distinguish three types depending on their speeds If allmachines in M operate on equal speeds, then we refer to them as identical (parallel) ma-chines If machines in M differ in their speeds, but the speed of a machine is independent

of the jobs, then we denote them by uniform (parallel) machines The job-independentspeed of a machine i ∈ M is denoted by si Third, if the speeds of the machines depend

on the jobs in J , then we call them unrelated (parallel) machines We refer to the speed

of machine i ∈ M for processing job j ∈ J as sij

In the following, we define the data that is used to characterize a job j ∈ J in thisthesis We denote by tijthe processing time of job j ∈ J on machine i ∈ M In case ofidentical machines we can drop the index for the machine, i.e tj= tijfor all i ∈ M Ifthe machines are uniform, then tij= tj/siwith i ∈ M , where tjis the standard processingtime Analogously, we have tij= tj/sijwith i ∈ M for unrelated machines All processingtimes are assumed to be positive numbers The arrival time or release date rjis the time

at which job j is ready for processing If the release dates are the same for all jobs j ∈ J ,then it is assumed rj = 0 for all j ∈ J The due date of the job is denoted by dj Itdescribes a time by which j should be completed The weight wjspecifies the relativeimportance of job j

Next, we present some definitions concerning the schedules and optimality criteria

A feasible schedule o is an assignment of all jobs from J to machines from M togetherwith starting and/or completion times for each job such that the following conditions aresatisfied:

• each job is processed by exactly one machine and each machine processes at mostone job at a time,

• job j ∈ J is processed in time interval [rj, ∞),

• no job is preempted, and

• all jobs are processed

We denote the set of all feasible schedules of a given machine scheduling problem by O.For a given schedule o ∈ O we denote the completion time of a job j ∈ J by Cj Itdescribes a time at which the processing of job j is completed Furthermore, we define amapping Cj: O → R≥0to describe the influence of the specific solution on the completiontimes We denote by Uja step function Uj: O → {0, 1} If job j completes strictly after

djin a schedule o ∈ O, Uj(o) is equal to 1 Otherwise Uj(o) is equal to 0 The load Liof amachine i ∈ M is defined as the sum of the processing times of all jobs that are assigned

to machine i Since the load depends on the realized schedule o, we define the mapping

Table 1.2: Notation: machine scheduling problems

• single machine or identical (parallel) machines: t ij = t j ∀ i ∈ M

• uniform (parallel) machines: tij= tj/sifor a given speed si

• unrelated (parallel) machines: tij= tj/sijfor a given speed sij

Uj unit penalty of job j ∈ J : 1 if j completes strictly after dj, 0 otherwise Uj: O → {0, 1}

notation, α|β|γ, where α describes the machine environment, β refers to job istics, and γ relates to the (global) performance measure (optimality criterion) Eachfield of the triple includes multiple elements, e.g α = α1, α2, , that represent specificproblem properties The empty symbol, ◦, denotes the default value of an element and isskipped when a triple is actually specified A detailed look in the classification scheme isgiven in Section 2.3.1 We refer to Błażewicz et al (2007) and Leung (2004) for a moredetailed discussion

character-1.1.2 Mechanism Design

In this section we give an overview over the basic notation and terminology concerninggame theory and algorithmic mechanism design as in Kress et al (2018b) The gamesconsidered throughout this thesis have three basic elements: players, strategy spaces, andutility functions Furthermore, we will restrict ourselves to considering non-cooperativegames That is, players cannot form coalitions in order to generate group decisions Inthe context of machine scheduling problems, players may be machines or jobs Moregenerally, one may also think of “owners” of multiple machines or jobs that act as singleplayers Each player has an associated strategy space that represents the options thatthe player can select from when the game is played For example, when the playerscorrespond to jobs, each job may be allowed to select a machine to be processed on

A player’s utility function assigns a utility level to every vector of strategies, i.e eachcombination of strategies that can potentially be selected by all players With respect tomachine scheduling problems, the utility level could, for instance, be the completion time

of a given job

We will consider fairly specific problem settings in the field of algorithmic game theory

for machine scheduling problems These settings are characterized by the existence of(rational and selfish) players, who are typically referred to as agents and can make asingle claim on some piece of information that may affect the final schedule Furthermore,there exists a central authority/planner, who is in charge of designing an interactionprotocol, a rewarding scheme (e.g payments among players), and a scheduling algorithmthat determines the final schedule Within this scope, there are two main streams ofliterature that differ in the type of information that the agents possess and in the waythat the information affects an instance of the considered machine scheduling problem(see Figure 1.1) In this thesis, we will focus one of these streams, which presumes that

Players/Central Authority

Agents (Machines or Jobs)

Stream “Algorithmic Mechanism Design”

announce agent characteristics (private information) select agent-specific machine-job assignment designs and controls

schedule “rewards” problem instance Public Information

Figure 1.1: Algorithmic mechanism design (in case of direct revelation) and scheduling games

the agents have private information on their own characteristics Jobs, for instance, mayprivately know their due dates or job weights The remaining data, e.g the number

of machines and jobs, is usually assumed to be publicly known The central plannerdesigns some protocol of interaction that the agents have to follow This protocol may

be fairly general We will, however, restrict ourselves to considering “direct protocols”that allow the agents to solely (but not necessarily truthfully) announce concrete valuesthat represent their private information when the game begins In terms of optimizationproblems, these agents therefore fix a subset of parameters When acting selfishly, theywill try to influence the solution determined by the scheduling algorithm by submittingfalse information However, by designing appropriate algorithms and rewarding schemesthat set the right incentives, the central planner can extract the true information ofthese players, for example, in order to generate fair solutions with respect to some socialcriterion that considers the interests of all agents In the second stream, that we do notconsider in this thesis (the interested reader may refer to Heydenreich et al., 2007), the(usually completely informed) agents, again pursuing selfish goals, commit decisions onmachine-job assignments and thus implicitly fix variables of optimization problems Wecan, for example, think of jobs that choose to be processed on specific machines

We would like to stress the fact that the aforementioned fields of research are not

always clearly separated in the literature Similarly, the terms used to identify specificproblems within these fields may differ among different articles We will follow Nisanand Ronen (2001), who define (algorithmic) mechanism design to aim at “study[ing] howprivately known preferences [ ] can be aggregated towards a ‘social choice’ ” (see alsoNisan and Ronen, 1999), which corresponds to the first stream described above Our focus

on direct protocols is usually termed direct revelation (see, for example, Nisan, 2007).Others use the term “algorithmic mechanism design” in a more general context, evenwhen there is no privately owned information (see, for instance, Immorlica et al., 2009).Problems in the second stream are sometimes referred to as (machine) scheduling games(see, for instance, Harks et al., 2011; Roughgarden and Tardos, 2007) or load balancinggames (Vöcking, 2007) These games are closely related to the categories of congestiongames (Rosenthal, 1973) and coordination mechanisms (Christodoulou et al., 2009a) Inall of these areas, one is usually interested in deciding whether (Nash) equilibria exist,how (in-)efficient these equilibria are when compared to socially optimal solutions, andhow fast algorithms can compute them (Harks et al., 2011; Roughgarden and Tardos,2007)

Algorithmic Mechanism Design

Based on the illustration in Figure 1.2, we will now describe the (direct revelation) rithmic mechanism design setting in the context of machine scheduling problems in moredetail The corresponding notation used throughout this thesis in context of mechanism

algo-1

v t 1

V1

Figure 1.2: (Direct revelation) algorithmic mechanism design (Kress et al., 2017)

design is summarized in Table 1.3

Let A denote the set of rational and selfish agents Each agent k ∈ A has a (true)valuation function vt

k : O → R, that maps every feasible schedule of the consideredscheduling problem to a real value vt

k is private information of the agent and is thus

Table 1.3: Notation: algorithmic mechanism design

v t

k (f (v)) + pk(v)

v −k vector of claimed valuation functions except vk, k ∈ A v −k = (v1, , vk−1, vk+1, , v|A|)

k, to the mechanism Each valuation function vk, k ∈ A, is element

of a publicly known set Vk We define V := V1× · · · × V|A| Furthermore, we denotethe vector of all valuation functions reported to the mechanism by v = (v1, , v|A|)and the vector of all valuation functions reported to the mechanism except of vk by

v−k= (v1, , vk−1, vk+1, , v|A|) For the sake of notational convenience, we will use vand (vk, v−k) interchangeably

The mechanism itself is designed and controlled by a central planner It is a pair(f, p), composed of a social choice function f : V → O and a vector of payment functions

p := (p1, , p|A|), with pk : V → R for all k ∈ A The mechanism (f, p) is said toimplement the social choice function f It is efficient, if f optimizes the given globalobjective function (Heydenreich et al., 2008; Mitra, 2001, 2002) As described in Section1.1.2, in the context of scheduling problems, the social choice function is an algorithmthat determines a feasible schedule based on the valuation functions reported to themechanism It is also referred to as the scheduling rule or allocation rule By controllingthe allocation rule and the payment functions, the cental planner can design mechanismswith different features

Each agent k ∈ A selfishly aims to maximize the utility function uk: V → R, which

is assumed to be quasi-linear, i.e corresponds to the sum of the agent’s valuation of theschedule (determined by the allocation rule) and the (potentially negative) correspondingpayment from the mechanism, uk(vk, v−k) := vt

k(f (vk, v−k)) + pk(vk, v−k) Sometimes it

is reasonable to focus on individually rational mechanisms (also referred to as voluntaryparticipation mechanisms, see Auletta et al., 2004a), that assume (or feature) the utilities

of each agent to always be non-negative (see, for instance, Hoeksma and Uetz, 2013; Nisan,2007).

Non-Deterministic Problem Settings

All of the above definitions assume a deterministic problem setting Unless stated wise, this will also be our standard assumption throughout the remainder of this thesis.The literature, however, also considers two main non-deterministic settings First, onecan assume the allocation rule to be non-deterministic, i.e let the scheduling algorithm’slogic employ some degree of randomness, or consider randomized payments A resultingmechanism is then referred to as a randomized mechanism (see, for example, Angel et al.,2012; Nisan, 2007) Second, one can deviate from the assumption of the agents having noinformation at all about the private information of the other agents Sometimes, it may

other-be appropriate to assume that there exists some commonly known probability tion over the private information of each player (Nisan, 2007) In both non-deterministiccases, agents are usually assumed to maximize expected utilities The definitions of thestandard setting carry over to the non-deterministic settings in a straightforward manner.Truthfulness and VCG Mechanisms

distribu-As indicated before, agents selfishly aim to maximize their (expected) utility functionsand may therefore lie about their true valuation functions To overcome this problem, thecentral planner may want to design the mechanism such that agents behave truthfully.The literature considers different concepts of truthfulness We will briefly outline theconcepts that are relevant for this thesis in this section

A mechanism is (dominant strategy) incentive compatible or truthful (Nisan, 2007) if

it guarantees that reporting the true valuation function maximizes the utility function

of a rationally acting agent for all possible vectors of claimed valuation functions of theother agents, i.e if uk(vt

k, v−k) ≥ uk(vk, v−k) for all k ∈ A, all vk∈ Vk, and all v−k∈ V−k

In case of randomized mechanisms, articles usually apply an adapted notion of fulness, referred to as truthfulness in expectation Formally, let E(uk(v)) denote theexpected value of the utility function of agent k ∈ A over the randomization of the mech-anism A mechanism is truthful in expectation if E(uk(vt

truth-k, v−k)) ≥ E(uk(vk, v−k)), for all

k ∈ A, vk ∈ Vk, and v−k∈ V−k Alternatively, one may slightly deviate from our inition in Section 1.1.2, and define a randomized mechanism to allow distributions overdeterministic mechanisms Then, a randomized mechanism is defined to be truthful inthe universal sense if every deterministic mechanism in the support is dominant strategyincentive compatible (Nisan, 2007)

def-Similarly, when considering the case of publicly known probability distributions overthe type spaces of agents (that we will denote by Φkfor agent k ∈ A) as described inbefore, one can apply a weaker notion of truthfulness, referred to as Bayes-Nash incentivecompatibility (see, for example, Duives et al., 2015; Heydenreich et al., 2008; Hoeksmaand Uetz, 2013) Here, for each agent, telling the truth must be (weakly) dominant inexpectation over the publicly known distributions over the type spaces of the other agents.One of the most important general results in the field of mechanism design is theVickrey-Clarke-Groves mechanism (VCG mechanism), that was suggested by Vickrey(1961) and generalized by Clarke (1971) and Groves (1973) A mechanism is called aVCG mechanism, if the social choice function maximizes social welfare, i.e the sum ofthe valuation functions of all agents, and if the payment functions pk(v), k ∈ A, are givenby

vk∈ Vkreported by agent k The concept of VCG mechanisms was further generalized

by Roberts (1979) to social choice functions that belong to the set of so called affine imizers A VCG mechanism is (dominant strategy) incentive compatible, but a majordrawback is the need for finding optimal solutions to the underlying problem of maximiz-ing social welfare, which may be NP-hard (see, for instance, Nisan, 2007) Hence, in thecontext of scheduling problems, VCG mechanisms are oftentimes not appropriate even

max-if the objective function of the specmax-ific scheduling problem corresponds to maximizingsocial welfare One must therefore usually make use of other theoretical results related toincentive compatibility that are suitable for approximate and heuristic algorithms Theseresults oftentimes turn out to “boil down to a certain algorithmic condition of mono-tonicity ” (Lavi and Swamy, 2009) The interested reader is referred to Heydenreich et al.(2007); Lavi and Swamy (2009)

1.1.3 Graph Theory

In this section, we introduce the notation used for graphs in this thesis The used tion follows standard combinatorial optimization books as Korte and Vygen (2008) andSchrijver (2003)

nota-A graph G = (V, E) consists of two (finite) sets V and E The set V is referred to asvertex set or node set and we define n := |V | The elements of V are called vertices ornodes The set E is called edge set and we define m := |E| The elements of E are referred

to as edges Edges can be directed or undirected An undirected edge e is represented

by a subset of V with two elements and is denoted by e = {u, v} ⊆ V The elements

of the subset are called end points of the edge A graph G = (V, E) with all edges in Eundirected is called undirected graph In contrast to an undirected edge, a directed edge

is an ordered pair of nodes (u, v) with u, v ∈ V The first node of the pair is called startnode and the second node is called target node or end node A graph G = (V, E) with alledges in E directed is called directed graph Note, that in this thesis the term graph refers

to either an undirected or directed graph Analogously the term edge refers to either anundirected or directed edge

If e = {u, v} ∈ E (or e = (u, v) ∈ E respectively), we say u and v are connected orjoined by an edge The nodes u and v are called adjacent if {u, v} ∈ E (or (u, v) ∈ E)

An edge e ∈ E and a node v ∈ e are called incident if v is either the start node or an endnode of e The degree of a node v denotes the number of edges incident with v In case

of a directed graph G = (V, E) and a node v ∈ V , we refer to all edges with v as its startnode as outgoing edges of v All edges with v as its end node are referred to as incomingedges of v

A bipartite graph G = (U, V, E) consists of two disjoint node sets U and V and anedge set E The node sets are called bipartitions of G In a bipartite graph G = (U, V, E)there is no edge e ∈ E that connects two nodes from the same bipartition In other words,all edges in E connect exactly one node in U with exactly one node in V

Given an undirected or directed graph G = (V, E), a walk is sequence of edges W =(e1, , ek) such that k ≥ 2 and ei = {v−i, vi} (or e = (v−i, vi), respectively) for i ∈{1, , k} We sometimes represent a walk by its nodes W = (v0, , vk), to ease thenotation We refer to the node v0as the start node and to vkas the end node of W If allnodes v0, , vkin a given walk W are distinct, we call the walk path A cycle is a walk

W = (e1, , ek) with v0, , vk−1distinct and v0= vk

A graph G is called edge-weighted, if each edge e ∈ E is associated with a weight

we ∈ R Usually, G is referred to as a triple G = (V, E, w) that consists of a nodeset V , an edge set E, and a weight (or cost) function w : E → R We use w(e) and

we synonymously Analogously, a node-weighted graph is a graph G = (V, E, w) withadditional weights wv∈ R for each v ∈ V ( or a mapping w : V → R, respectively).Optimization Problems on Graphs

In the following, we introduce some optimization problems that are defined on graphs andconsidered in this thesis

A matchings M on an undirected graphs G = (V, E) is a subset of E such that allelements in M are disjoint In other words, each node v ∈ V is incident to at most one

edge in M The matching with maximal cardinality is called maximum matching Theproblem of finding a matching M on a graph G = (V, E) with maximal cardinality iscalled maximum matching problem Given an edge-weighted graph G = (V, E, w), theweight w(M ) of a given matching M on G is the sum of the weights of all edges in M ,i.e w(M ) =P

e∈Mwe Given an edge-weighted bipartite graph G = (U, V, E, w), theproblem of finding a maximum matching of minimum weight is referred to as min-sumweighted matching These basic matching problems are well studied problems Furtherdetails and information on these problems can be found, for example, in Burkard et al.(2009)

Another well studied optimization problem is the (symmetric) traveling salesmanproblem (TSP) Given an edge weighted undirected graph G = (V, E, w), the problem is

to find a cycle that contains all nodes in V with minimal weight The weight of a circle

W is defined as the sum of the weights of all edges in W In context of the TSP, the circle

W is often referred to as a tour The asymmetric traveling salesman problem is definedanalogously on edge weighted directed graphs Further information on the TSP and itsvariants can be found, for example, in Gutin and Punnen (2007)

The remainder of this thesis is organized as follows In Chapter 2, we present a review ofrecent contributions in the field of machine scheduling problems in the context of algo-rithmic mechanism design We review the categories and characterizing problem features

of machine scheduling settings in the algorithmic mechanism design literature and extendthe widely accepted classification scheme of Graham et al (1979) for scheduling prob-lems to include aspects relating to mechanism design Based on this hierarchical scheme,

we give a systematic overview of recent contributions in this field of research In ter 3, we consider two machine scheduling problems that arise at container ports whendealing with customers Both problems are investigated with methods from the field ofalgorithmic mechanism design The first part of the chapter is concerned with machinescheduling problems in context of one-parameter valuation function domains We considerthe problem of minimizing the total weighted completion time of all jobs and investigatethe truthfulness of List-Scheduling algorithms for parallel machines In the case of onemachine, we come up with results for a budget-balanced VCG mechanism In the secondpart, we consider machine scheduling problems in context of two-parameter valuationfunction domains We derive a set of properties that is equivalent to the well-knowncondition of cycle monotonicity, which is a general condition for truthful mechanisms innon-convex valuation function domains Our results utilize knowledge about the underly-

Chap-ing schedulChap-ing problem, so that the resultChap-ing properties are easier to implement and verifythan the general condition of cycle monotonicity We illustrate the use of our results byanalyzing an example algorithm that has recently been proposed in the literature for thecase of one machine.

Chapter 4 and 5 are concerned with “classic” optimization problems Both chaptershave in common, that they consider optimization problems dealing with internal containermovements between the quay and the storage area of the ports In Chapter 4, we intro-duce and analyze the Partitioning Min-Max Weighted Matching Problem (PMMWM).PMMWM combines the problem of partitioning a set of vertices of a bipartite graph intodisjoint subsets of restricted size and the strongly NP-hard Min-Max Weighted Match-ing (MMWM) Problem, that has recently been introduced in the literature In contrast

to PMMWM, the latter problem assumes the partitioning to be given Applications ofthe PMMWM arise at small and midsize container ports when reach stackers are used

to transport containers from a temporary storage area to the long-term storage area.Other applications of the PMMWM arise at the rail-road terminal of a container port

We propose a MILP formulation for PMMWM and prove that the problem is NP-hard

in the strong sense Two heuristic frameworks are presented Both of them outperformstandard optimization software Our extensive computational study proves that the al-gorithms provide high quality solutions within reasonable time Chapter 5 deals with astraddle carrier routing problem that arises at container ports where containers need to

be exchanged between a storage area of the sea port and a small buffer for locally storingloaded or unloaded containers within reach of the quay cranes The problem is how toroute the container carrying straddle carriers such that a loading or unloading sequence atthe quay crane is respected and the turnaround time of the vessel is minimized The prob-lem is proven to be strongly NP-hard and we present a mixed-integer programm based

on the asymmetric traveling salesman problem with precedence constraints We proposetwo decomposition heuristics and compare the most promising one with an approach used

in practice Computational experiments are based on real-world data The thesis closeswith a summary and an outlook on future research in Chapter 6

Chapter 2

Mechanism Design and Machine

Scheduling: Literature Review

There exists a tremendous body of literature that focuses on intersections of (algorithmicaspects of) computer science and game theory (as well as economic theory) The resultingfields of intersecting disciplines are usually referred to as algorithmic game theory (anexcellent introduction and overview is given by Nisan et al., 2007) Many research articles

in this field focus on auction contexts (see Krishna, 2010) Recently, however, there hasbeen a growing interest in taking a game theoretic perspective on machine schedulingproblems, which has resulted in a fairly large amount of research articles that we aim toreview and classify in this chapter as in Kress et al (2018b)

In order to give a systematic record of the academic efforts in this field of research,

we provide a corresponding hierarchical classification scheme This scheme augments theclassification scheme by Graham et al (1979) for machine scheduling problems, which iswidely used and generally accepted in the scheduling community We are motivated bythe fact that adoptions and extensions of Graham et al (1979) have been successfullyimplemented in a variety of other problem fields (see, for example, Allahverdi et al., 2008;Boysen and Fliedner, 2010a; Boysen et al., 2007, 2009; Brucker et al., 1999; Potts andKovalyov, 2000)

The remainder of this chapter is structured as follows The scope of review is scribed in Section 2.1 In Section 2.2, we present an overview of problem categories andproblem features that characterize machine scheduling settings in the algorithmic mecha-nism design literature This will lay the foundation for our extension of the classificationscheme of Graham et al (1979) in Section 2.3 and allow a structured overview of theliterature in Section 2.4 The chapter closes with a conclusion and an illustration of re-search challenges that can be identified based on the prior classification of the literature

de-© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2018

S Meiswinkel, On Combinatorial Optimization and Mechanism Design

Problems Arising at Container Ports, Produktion und Logistik,

https://doi.org/10.1007/978-3-658-22362-5_2

in Section 2.5.

In this section, we present details on the scope of our literature review

Broadly speaking, scheduling problems are concerned with allocating scarce resourcesover time to perform a set of tasks with the objective of optimizing one or more perfor-mance measures (Błażewicz et al., 2007; Leung, 2004) Resources, tasks and performancemeasures can be of very different nature We will focus on resources that (directly) rep-resent some kind of processor or machine, i.e machine scheduling problems, and set thescope of our literature review, that complements the articles by Heydenreich et al (2007)and Christodoulou and Koutsoupias (2009), to include research on

• machine scheduling problems

• in offline-settings, where all information regarding the problem is known or has beenannounced (in contrast to online-settings) at the unique time of planning,

• in a non-cooperative, game theoretic context, where players cannot form coalitions

• and have private information on their own characteristics which they directly (butnot necessarily truthfully) announce by making a single claim,

• in the presence of a central authority that is in charge of designing a rewardingscheme and the scheduling algorithm that determines the final schedule based onthe information submitted by the players and the publicly known information

A more detailed description of the mechanism design nomenclature and therefore ofthe scope of review is given in Section 1.1.2

In addition to the classical problem categories of the classification scheme of Graham

et al (1979) and its extensions, the algorithmic mechanism design literature for machinescheduling problems (as restricted in Section 2.1) can be structured based on multiple cat-egories, that we will present in the following sections, where we will also discuss additionalproblem features that we have not yet introduced

2.2.1 Categories, Risk Attitude and Private Information of Agents

With respect to categories of agents, there exist two streams of publications The firstgroup of articles, that follows the seminal work of Nisan and Ronen (1999), presumesthat solely the machines are selfish agents (typically referred to as machine agents ), i.e

A = M Machine agents usually aim for small loads Similarly, the second stream ofpublications assumes that only the jobs are selfish agents (job agents ), i.e A = J , mostlyaiming at small completion times Prominent examples of the latter stream are, Suijs(1996) or Angel et al (2006) The literature on job agents that are to be scheduled on

a single machine sometimes analyzes the independence of irrelevant alternatives (IIA)property of allocation rules (e.g Heydenreich et al., 2008) It is satisfied if the relativeorder of any two jobs on the machine is independent of the committed types of all otherjobs

Of course, one may also think of more general settings, where subsets of jobs ormachines represent the selfish players of the considered game However, such settingsusually represent cooperative games or are at least closely related to cooperative settingsand are thus excluded from our review (see, for example, Mukherjee, 2013) Even moregeneral, there might be “owners” of multiple jobs or machines that act as single agents.However, to the best of the authors’ knowledge, this setting has not yet been considered

in the non-cooperative literature

Concerning the risk attitude, the vast majority of research articles assumes the agents

to be risk neutral Exceptions are Kovalyov and Pesch (2014) and Kovalyov et al (2016),where job agents are assumed to be “fully” risk averse

As described in Section 1.1.2, mechanism design settings can also be classified withrespect to the knowledge of agents about the private information of the other agents.The standard case is to assume that agents have no information at all about the other’sprivate information Sometimes, however, one assumes that there exists some commonlyknown distribution over the private information of each agent

2.2.2 Models of Execution and Constraints on Committed Data

When agents possess private information on the processing times of jobs, the literaturedistinguishes between different models of execution These models differ in the length ofthe actual time slots that are reserved on the machines once the schedule that has beendetermined by the scheduling algorithm is implemented There are two widely used models

of execution (see, e.g., Christodoulou et al., 2007) The strong model of execution assumesthat the schedules, which are implemented based on the computations of the scheduling

algorithms, always apply the true processing times of the jobs, no matter which valuehas been committed by the agents In contrast, in the weak model of execution, theimplemented schedules use the reported processing times A third model of execution

is used by Koutsoupias (2014) Here, the implemented processing times are defined bythe maximum of the reported and the true processing times We refer to this model ofexecution as the maximum model of execution

Additionally, one may want to constrain the data that the agents are allowed to report

to the mechanism For example, in case of private processing times (release dates), it may

be reasonable to restrict the agents to commit processing times (release dates) that arebounded from below by their true values (see, e.g., Angel et al., 2012; Christodoulou

et al., 2007) Another example is to restrict the privately known speed factors of machineagents to be natural numbers that are bounded from above by a publicly known constant(Auletta et al., 2004a)

2.2.3 Characteristics of Payment Schemes

There exist many applications, where mechanisms may be restricted to not include ments for compensation purposes, i.e where pk(v) = 0 must hold for all k ∈ A and v ∈ V(see, for example, Koutsoupias, 2014) “This constraint can arise from ethical and/orinstitutional considerations: many political decisions must be made without monetarytransfers; organ donations can be arranged by ’trade’ involving multiple needy patientsand their relatives, yet monetary compensation is illegal” (Schummer and Vohra, 2007).Other applications strive for mechanisms that have no surplus or deficit of mone-tary payments that cannot be redistributed among the agents (Suijs, 1996), i.e whereP

pay-k∈Apk(v) = 0 for all v ∈ V These payment schemes are usually referred to as budgetbalanced

Some articles consider the design of optimal mechanisms for scheduling problems (see,for instance, Duives et al., 2015; Heydenreich et al., 2008; Hoeksma and Uetz, 2013), wherethe central planner aims to design Bayes-Nash or dominant strategy incentive compatiblemechanisms that minimize the sum of the (expected) payments (see also Hartline andKarlin, 2007) Usually, an arbitrary probability distribution over the private information

of each agent is assumed to be publicly known Additionally, the articles usually focus

on individually rational mechanisms or some approximation guarantee of the schedulingalgorithms (given truthful commitments of the agents)

2.2.4 Other Problem Categories and Features

There exist some mechanism design related problem features that do not fall into thecategories of the above sections In the following, we list the features that are relevantfor this chapter

A mechanism is called anonymous if, whenever two agents switch all of their erties, these two agents also switch positions in the resulting schedule (see, for example,Ashlagi et al., 2012)

prop-In a mechanism with verification, the calculation of the payments depends on theresults of the execution of the schedule (see, e.g., Nisan and Ronen, 2001) If, for example,the processing time of a job is part of the private information of an agent, additionalinformation on the true processing time becomes available after the execution of theschedule, which depends on the model of execution

Next, a mechanism is called envy-free, if no agent is able to improve her utility functionvalue by switching both, the position in the schedule and the realized payments, withanother agent (see, for instance, Kayı and Ramaekers, 2010, 2014)

When considering a mechanism design setting with machine agents, a mechanism iscalled local decisive, if each agent can enforce her allocation by reporting very low or highvalues (see, e.g., Christodoulou et al., 2008)

We are now ready to present our extension of the classification scheme of Graham et al.(1979) The resulting scheme is extensible, i.e it allows for including more features whenneeded

2.3.1 Review of Selected Elements of Graham et al (1979)

We will first review parts of the notation introduced by Graham et al (1979) Withrespect to the machine environment α they define:

• Machine environment, α1∈ {◦, P, Q, R, }

◦ Single machine, i.e m = 1

P Identical (parallel) machines

Q Uniform (parallel) machines

R Unrelated (parallel) machines

• Number of machines, α2∈ {◦, N}

◦ m is variable

pos integer m There exists a constant number m of machines

Regarding the job characteristics β, we will only make use of a few elements:

• Release dates, β4∈ {◦, rj}

◦ No release dates are specified

rj Release dates per job are specified

• Processing times, β6∈ {◦, tj= 1, }

◦ Processing times are arbitrary

tj= 1 Each job has unit processing time

Finally, based on Graham et al (1979) and with respect to the (global) optimalitycriterion γ, i.e the objective function that the scheduling algorithm aims to optimizebased on the publicly known parameters and the values committed by the agents, wedefine:

• Global optimality criterion, γ ∈ {Cmax,P Cj,P wjCj, max wjCj,P fj(Cj),P f (Cj),

P wjf (Cj),P wjUj,P f (Li), kSkp, max min Li, }:

Cmax Minimize the makespan, i.e the maximum of the completion times

P Cj Minimize the sum of completion times

P wjCj Minimize the weighted sum of completion times

max wjCj Minimize the maximum weighted completion time

P fj(Cj) Minimize the sum of functions fjof the completion times of jobs

j ∈ J

P f (Cj) Minimize the sum of a function f of the completion times

P wjf (Cj) Minimize the weighted sum of a function f of the completion times

P wjUj Minimize the total weight of late jobs

P f (Li) Minimize the sum of a function f of the load of the machines.kSkp Minimize the lp norm of the schedule S

max min Li Maximize the minimum load over all machines

2.3.2 Including Mechanism Design Settings for Machine

Schedul-ing Problems

We can now define the additional notation needed to include mechanism design settings

As described in Section 2.2.1, the existing literature can be divided into two groups ofarticles that either presume the existence of machine agents or job agents We thereforeaugment the first two fields of the classification scheme of Graham et al (1979), that rep-resent the machine environment α and the job characteristics β, with additional elements

αmdl and βmdl, l = 1, 2, , respectively Here, an index mdlrefers to the l-th elementthat refers to mechanism design (md) characteristics in the specific field

First, in order to be able to indicate the risk attitude of the agents, we define theelements αmd 1for machine agents and βmd 1for job agents:

• Risk attitude of agents, αmd1, βmd1∈ {◦, averse, seeking, }

◦ No agents at all (no mechanism design setting) or all agents are risk

neutral

averse All agents are risk averse

seeking All agents are risk seeking

The elements αmd 2and βmd 2specify the set of parameters that are private information

of the agents and indicate whether the other agents have some common knowledge aboutthis private information

• Private information of machine agents, αmd2∈ {◦, privρ{ }, }

◦ No machine agents

privρ{ } Each element of the set { } refers to an entity of private

information of the machine agents:

– sτ

i: Machine agent i ∈ M has private information on the speedfactor si The superscript τ indicates whether there are anyadditional restrictions or assumptions Multiple entries in thesuperscript are separated by commas

- No superscript: No additional restrictions

- ∈ N: Speed factors are natural numbers

- ≤ ¯s: All speed factors are bounded from above by apublicly known constant ¯s

- div: Speed factors are divisible, i.e they belong to a set

C = {c1, c2, } such that for each i, ci+1is a multiple of

c

- c-div: Speed factors are c-divisible, i.e they are powers of

a given positive constant c

– tτ

ij: Machine agent i ∈ M has private information on theprocessing times tijfor all jobs j ∈ J The superscript τindicates whether there are any additional restrictions orassumptions Multiple entries in the superscript are separated

by commas

- No superscript: No additional restrictions

- ∈ {L, H}: There exist two publicly known values L < H(L: “low”, H:“high”) for the processing times of each job

- ∈ {Lj, Hj}: As before, but the values can be different foreach job

- ∈ {L, H}part: As before, but there exists a publicly knownpartition of the jobs into two sets for each machine Thejobs of a set have identical processing times

- max: The maximum model of execution is applied.The subscript ρ indicates whether there are any additionalrestrictions or assumptions

– No subscript: No additional restrictions

– Φ: A distribution of each job agent’s private information ispublicly known

• Private information of job agents, βmd2∈ {◦, privρ{ }, }

◦ No job agents

privρ{ } Each element of the set { } refers to an entity of private

information of the job agents:

– wj: Job agent j ∈ J has private information on its weight wj.– dj: Job agent j ∈ J has private information on its due date ordeadline dj

– fj: Job agent j ∈ J has private information on the function fj

that maps every possible completion time of its job to a realvalue

– rτ: Job agent j ∈ J has private information on its release date

rj The superscript τ indicates whether there are anyadditional restrictions or assumptions

- No superscript: No additional restrictions

- ≥: The realease date committed by job agent j ∈ J isbounded from below by the true release date

– tτ: Job agent j ∈ J has private information on its processingtime tj The superscript τ indicates whether there are anyadditional restrictions or assumptions Multiple entries in thesuperscript are separated by commas

- No superscript: No additional restrictions

- strong: The strong model of execution is applied

- weak: The weak model of execution is applied

- ≥: The processing time committed by job agent j ∈ J isbounded from below by the true processing time

– tτ

ij: Job agent j ∈ J has private information on its processingtimes tijthat may differ among machines i ∈ M Thesuperscript τ is defined as above

The subscript ρ indicates whether there are any additionalrestrictions or assumptions Multiple entries in the subscript areseparated by commas

– No subscript: No additional restrictions

– Φ: A distribution of each job agent’s private information ispublicly known

– d: The reported information is element of a publicly knowndiscrete set with a finite number of elements

Finally, we define elements αmd3and βmd3that represent the (true) valuation functions

of the agents, i.e their “local” objective functions related to the scheduling problem

• Objective of agents, αmd3∈ {Li, }, βmd3∈ {Cj, Uj }

αmd3= Li Each machine agent i ∈ M aims to minimize its load Li

βmd3= Cj Each job agent j ∈ J aims to minimize its completion time Cj

βmd 3= Uj Each job agent j ∈ J aims to complete before or at dj, i.e to

minimize the unit penalty function U

As mentioned above, our classification scheme is extensible This is indicated bydots in the above notation, which allow for including new problem settings for existingcategories, for example when considering new entities of private information of agents.Furthermore, when considering more general categories of agents or similar problem gen-eralizations or extensions, one can include new symbols to represent those settings.

2.3.3 Examples

We will now illustrate the above classification scheme by presenting two examples

P ||Cmax: We are given an arbitrary number of m parallel identical machines and aset J of n jobs The processing time tjof any job j ∈ J is independent of the machines.Each job can be processed by at most one machine at a time and each machine is capable

of processing at most one job at a time The objective is to assign each job to exactly onemachine and find non-preemptive sequences of the resulting subsets of jobs of each ma-chine, so that the makespan is minimized There is no private information; a mechanismdesign setting is not considered

P |priv{tstrong,≥j }, Cj|Cmax: The setting is in analogy to P ||Cmax, but we now consider

a mechanism design setting with job agents who aim to minimize their completion times.The processing time of each job agent is private information The processing times com-mitted to the mechanism are bounded below by their true values The strong model ofexecution is applied

Based on the classification scheme of Section 2.3, we can now present a structured overview

of the relevant literature We will do so by presenting three tables that refer to articlesthat consider problem settings with job agents (Table 2.1), machine agents and unrelatedmachines (Table 2.2), and machine agents and uniform machines (Table 2.3)

Within the tables, each article, identified by its authors and publishing year, is fied according to the extended classification scheme presented in Section 2.3 Furthermore,the tables highlight the contribution of each article by specifying whether it focuses onselected properties These selected properties slightly differ among the tables because theliterature related to each table usually takes a fairly specific perspective on mechanismdesign settings for machine scheduling problems

classi-With respect to the characteristics of the payment scheme (Paym.), we indicatewhether an article considers nonzero payments at all (∃) and whether the presentedpayment scheme is budget balanced (BB) Furthermore, we present information on the