The resource transfer problem a framework for integrated scheduling and routing problems

Contributions to Management ScienceIlla Weiss The Resource Transfer Problem A Framework for Integrated Scheduling and Routing Problems... The Resource Transfer ProblemA Framework for In

Contributions to Management Science

Illa Weiss

The Resource Transfer

Problem

A Framework for Integrated Scheduling and Routing Problems

The Resource Transfer Problem

A Framework for Integrated Scheduling and Routing Problems

123

Germany

Dissertation Clausthal University of Technology, Germany, 2018, D 104

ISSN 1431-1941 ISSN 2197-716X (electronic)

Contributions to Management Science

ISBN 978-3-030-02537-3 ISBN 978-3-030-02538-0 (eBook)

https://doi.org/10.1007/978-3-030-02538-0

Library of Congress Control Number: 2018959095

© Springer Nature Switzerland AG 2019

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Due to their practical relevance and challenging intractability, scheduling andvehicle routing problems have been matters of intense research since the earlydays of operations research From the combinatorial perspective, scheduling andvehicle routing are closely related, both dealing with the allocation of resourcesand the sequencing of activities over time A large number of variants have beenconsidered in the literature In industrial applications of complex scheduling prob-lems, beyond precedence constraints and scarce renewable resources representingpotential factors, generalized precedence relations, sequence-dependent changeovertimes, and storage resources for consumable factors like materials have to be takeninto account Rich vehicle routing problems include such diverse requirements

as temporal or spatial synchronization constraints, multi-dimensional capacityrestrictions, incompatibilities between goods, restricted accessibility of roads andlocations, working time regulations, or inter-route constraints arising from limitedprocessing capacities at depots

Despite their structural similarities, scheduling and vehicle routing problemswere mostly considered separately Moreover, the overwhelming majority of modelsand methods proposed in the literature are dedicated to specific problem settings Inreal-world planning, however, scheduling and routing problems often arise jointly,and a large variety of individual requirements must be considered In supply chainoperations planning, production and transportation must be coordinated to reduceorder-to-delivery time and stocks Due to short shelf life times, aligning productionscheduling and vehicle routing is frequently of particular importance in food supplychains Multi-site scheduling of distributed project portfolios involving resourcesthat are transferred between the locations may be cited as a further example of animportant problem setting including scheduling and routing aspects In industrialpractice, scheduling and routing are generally performed sequentially This hierar-chical approach, however, may lead to substantial performance losses, which could

be avoided if scheduling and routing decisions were made simultaneously

The need for general and integrated scheduling and routing approaches was thestarting point for the research of Illa Weiss In her dissertation, she proposes theresource transfer problem (RTP) as a comprehensive reference model for complex

v

scheduling and rich vehicle routing problems The RTP model offers a compactunifying framework for modeling and solving scheduling and routing problems.Activities and haulage are represented via events among which resource units are to

be transferred, and the problem consists in allocating resource units and assigningoccurrence times to the events The model also covers multi-modal settings, whereeach event can occur in alternative modes with different resource requirements.Moreover, incompatibility and inclusions constraints can be formulated for theresource units allotted to distinct events Having devised a conceptual and amathematical programming model of the RTP, the author demonstrates the modelingpower of the framework by explaining how to express the various features ofcomplex scheduling and rich vehicle routing problems within the RTP framework.The book also presents major algorithmic achievements Generalizing classicalresults from resource-constrained project scheduling, Illa Weiss shows how forgiven occurrence modes and occurrence times of the events, a feasible assignment

of resource units can be computed efficiently using a column-generation approachthat is based on a path-based formulation of a minimum-flow problem with sideconstraints As a solver for RTP instances, she devises a time-oriented branch-and-bound algorithm, which relies on constraint propagation and takes advantage ofseveral consistency tests that she adapted to the general RTP setting An extensiveexperimental performance analysis demonstrates that the solver is able to providegood solutions to complex instances of supply chain operations planning within afew minutes of CPU time

The results obtained by Illa Weiss are highly relevant to scientists and ers in scheduling and transportation It is my hope that the ideas developed in thisexcellent thesis will achieve a wide dissemination and stimulate further research inthese vital fields

This PhD thesis was created during my time at Clausthal University of Technology,where I worked as a scientific assistant in the Operations Management Group.During this time, I had the privilege to meet many great people with whom Ienjoyed spending my time First, I would like to thank my supervisor Prof Dr.Christoph Schwindt, who taught me a lot and who was always available for supportand interesting discussions I also thank Prof Dr Jürgen Zimmermann, who kindlyagreed to be the second reviewer.

Furthermore, I would like to thank my colleagues from the Operations agement Group for excellent collaboration and a nice and inspiring workingatmosphere: Tobias Paetz, Mario Sillus, Anja Heßler, Nora Krippendorff, AstridLudwig, and Michael Krause Special thanks go to Tobias Paetz, who shared anoffice with me for several years and who always took time for any question I had

Man-I enjoy remembering this beautiful time Man-It was (and still is) a great pleasure tospend time with Mario Sillus, Anja Heßler, and Nora Krippendorff who did notonly contribute to our great working atmosphere but with whom I also had manyspecial moments outside the office Without Astrid Ludwig it would have been a lotharder to cope with all the administrative work

Finally, I would like to thank Michael Smyth and Sophie Weiss for proofreading

my thesis and Janis Kesten-Kühne who was always open for any discussion andsupported me whenever needed

April 2018

vii

1 Introduction 1

2 Elements of Scheduling and Routing Theory 3

2.1 Scheduling Problems 3

2.1.1 Machine Scheduling Problems 4

2.1.2 Project Scheduling Problems 9

2.1.3 Resource Transfers in Project Scheduling 19

2.2 Vehicle Routing Problems 20

2.2.1 Standard Vehicle Routing Problems 21

2.2.2 Pickup and Delivery Problems 25

2.2.3 Additional Constraints and Further Variants of Vehicle Routing Problems 31

2.2.4 Rich Vehicle Routing Problems 36

2.3 Integrated Scheduling and Routing Problems 41

2.4 Reformulation of Scheduling and Vehicle Routing Problems 46

3 The Resource Transfer Problem 49

3.1 Problem Description 49

3.2 Conceptual Model and Mathematical Formulation 53

3.2.1 Conceptual Model 53

3.2.2 Mathematical Formulation 59

3.3 Graph-Based Representation 63

3.3.1 Time Lag Graph 63

3.3.2 Transfer Graph 66

3.3.3 Inclusion and Incompatibility Graphs 67

4 Modeling Power of the Framework 69

4.1 Scheduling Problems as Resource Transfer Problems 69

4.1.1 Machine Scheduling Problems 69

4.1.2 Project Scheduling Problems 75

4.1.3 Resource Transfers in Project Scheduling 84

ix

4.2 Vehicle Routing Problems as Resource Transfer Problems 92

4.2.1 Standard Vehicle Routing and Pickup and Delivery Problems 93

4.2.2 Further Variants of Vehicle Routing Problems and Additional Constraints 103

4.3 Integrated Scheduling and Routing Problems as Resource Transfer Problems 117

4.4 Summary of the Building Blocks 120

5 Solution Approach 123

5.1 Allocation of Resource Units 123

5.2 Branch-and-Bound Algorithm 144

5.2.1 Enumeration Scheme 145

5.2.2 Lower Bounds for the Makespan 159

5.2.3 Preprocessing 169

5.2.4 Truncated Branch-and-Bound Algorithm 174

5.3 Consistency Tests 177

5.3.1 Consistency Tests for Renewable Resources 178

5.3.2 Consistency Tests for Storage Resources 198

5.3.3 Consistency Tests for the Mode Selection 201

6 Experimental Analysis 205

6.1 Experimental Design 205

6.2 Validation 207

6.2.1 Results of RCPSP/Max Instances 209

6.2.2 Results of MRCPSP/Max Instances 217

6.2.3 Results of 1-PDVRPTW Instances 221

6.3 Generation of Test Sets 225

6.4 Evaluation of the Results 235

7 Conclusions 267

Appendix A 269

References 301

Index 309

RCPSP/max Resource-constrained project scheduling problem with

general-ized precedence relations

Vehicle Routing

1-PDVRP One-commodity pickup and delivery vehicle routing problem1-PDVRPTW One-commodity pickup and delivery vehicle routing problem with

time windows

CVRP Capacitated vehicle routing problem

MAVRP Multi-attribute vehicle routing problems

MDVRP Multi-depot vehicle routing problem

VRPB Vehicle routing problem with backhauls

VRPDDP Vehicle routing problem with divisible delivery and pickupVRPMB Vehicle routing problem with mixed linehauls and backhaulsVRPSPD Vehicle routing problem with simultaneous pickup and deliveryVRPTW Vehicle routing problem with time windows

xi

Resource Transfer Problem

Application

Solution Approach

B&B(Γ ) Exact branch-and-bound algorithm with consistency testsB&B Exact branch-and-bound algorithm without consistency tests

FB(Γ ) Filtered beam search algorithm with consistency tests

FB Filtered beam search algorithm without consistency tests

Machine Scheduling

Sets

Model Parameters

p o , p k o Processing time of operation o (using machine k)

A k ( S) Set of all pairs of activities that may share a resource

unit of renewable resource k according to schedule S

xiii

N, N k Set of all activities (requiring renewable resource k)

¯

plus dummy activities 0 and|N| + 1

Model Parameters

ssδminij Start-to-start minimum time lag between activities i

and j

activities i and j

renewable resource k between activities i and j

ω ij , ω m ˜m

executed in modes m and ˜m)

between activities i and j

resource k

Model Variables

changed over from activity i to activity j

Vehicle Routing

Sets

¯

and|N| + 1

Model Parameters

and j

and j

and j

cus-tomer j

Model Variables

travels from customer i to customer j

Resource Transfer Problem

Model

and schedule ( t, x)

Sets

unit of renewable resource k (according to schedule

constraint (referring to renewable resource k) is

defined

unit-incompatibility constraint (referring to renewable

resource k) is defined

unit-inclusion constraint between events i and j must be

observed

¯

unit-incompatibility constraint between events i and j

must be observed

V k , V k ( x) Set of all events requiring renewable resource k

(according to mode assignmentx)

¯V k , ¯ V k ( x) Set of all events requiring renewable resource k

(according to mode assignment x) plus dummy

events 0 and n+ 1

event j occurring in mode ˜m

assignmentx

r ik , r ik m Requirement of event i (occurring in mode m) for

renewable resource k

given mode assignmentx

storage resource

given mode assignmentx

to event i

occurs in mode m

x = (x m

and event j does not occur before event i

y = (y m

z ij u Binary variable indicating whether or not resource

unit u of renewable resource k is transferred from event i to event j

event i

˜d m

p i , p i m Duration of activity i (occurring in mode m)

ij ) m∈M i , ˜m∈ M j Matrix of the transfer times referring to events i, j

given mode assignmentx

d ij ( x) Transitive time lag between events i and j for given

mode assignmentx

G L , G L ( x) Time lag graph (for mode assignmentx)

G T , G T ( x) Transfer graph (for mode assignmentx)

Vector of resource requirements

R = (R k , R ) k∈R ρ , ∈R σ Vector of resource capacities

Solution Approach

Sets

A 1,2 k , A 1,2 k (α) Arc set of graph G T k (G T k (α))

matrix B

D i , D i (α) Domain of all pairs of a time- and resource-feasible

occurrence time and an occurrence mode of event i (at node α of the search tree)

D im , D im (α) Domain of all time- and resource-feasible

occur-rence times of event i in mode m (at node α of the

search tree)

renewable resource which is required by event i∗in

mode m∗

a storage resource which is required by event i∗ in

mode m∗

¯

resource units to a subsequent event cannot be formed in parallel

per-I k1, I k1( t, x) Set of event pairs (i, j ) ∈ A k ((i, j ) ∈ A k ( t, x))

for which unit-inclusion constraint (i, j ) refers to renewable resource k

I k2, I k2( t, x) Set of event pairs (i, j ) ∈ A k ((i, j ) ∈ A k ( t, x))

for which unit-inclusion constraint (j, i) refers to renewable resource k

not satisfied by the current solution for therelaxed resource allocation problem for renewable

P k Set of forbidden paths in network G k ( t, x) with

respect to renewable resource k

path from node 0 to node i in network ¯ G k ( t, x)

¯

path from node 0 to node i in network ¯ G k ( t, x)

R ρ

allo-cation is represented by a mode selection

R ρ

1 Set of all renewable resources whose resource

allo-cation is not represented by a mode selection

T k1, T k2, T k3 Sets of occurrence times investigated by the

ener-getic reasoning consistency test for renewable

resource k

V1 ⊂ ˆV , V2⊂ ˆV Subsets of event set ˆV

V1(i) Set of events that have to take place before event i

lag to event i∗occurring in mode m∗

resource k

to a succeeding event must be completed before the

occurrence of event i

Graphs and Parameters

(B T )−1which belong to the constraints identified in

the pivot step within the column generation dure

proce-δ

performance-guarantee algorithm

Θ = (θ m ˜m

ij ) i,j ∈ ¯V ,m∈ M i ,

˜m∈ M j

Matrix of the lower bounds on the transfer times

units after event i

events i and j

event i occurring in mode m to another event

event i occurring in mode m to some event

h ∈ ˜V ⊆ ¯V

θ m ˜m

i occurring in mode m and event j occurring in

referring to the basic variables

˜c = (˜c ij ) (i,j ) ∈Ak Vector of the arc weights from network ˜G k

c

reduced network obtained from ¯G k

c k = (c ij ) i ∈V1,j ∈V2 Vector of arc weights of the arcs from network G T k

c k (α) = (c ij ) i ∈V1(α),j ∈V2(α) Vector of arc weights of the arcs from

Matrix of the lower bounds on the time lags

and event j

d m ˜m

occur-ring in mode m ∈ M i and event j occurring in

mode ˜m

ec im ( ˜ V ) Earliest completion time of a transfer of resource

units from event i occurring in mode m to some event h ∈ ˜V ⊆ ¯V

ec i ( ˜ V ) Earliest completion time of a transfer of resource

units from event i to some event h ∈ ˜V ⊆ ¯V

et im , et im (α) Earliest occurrence time of event i in mode m (at

node α of the search tree)

et i , et i (α) Earliest occurrence time of event i (at node α of the

search tree)

˜et im (α) Earliest time- and resource-feasible occurrence time

of event i in mode m at node α of the search tree

when neglecting temporal constraints with respect tounscheduled events

G T

k , G T

k (α) Weighted bipartite graph of the transportation

prob-lem (5.27) (at node α of the search tree)

¯V k ( x) and arc set A k ( t, x))

respecting the unit-inclusion constraints

length respecting the unit-inclusion constraints

LB2 Modified workload-based lower bound

lc i ( ˜ V ) Latest completion time of a transfer of resource units

from event i to some event h ∈ ˜V ⊆ ¯V

lt im , lt im (α) Latest occurrence time of event i in mode m (at

node α of the search tree)

lt i , lt i (α) Latest occurrence time of event i (at node α of the

t i ( ), ˜t i ( ) Hypothetical occurrence time of event i used in the

profile test for storage resource

occu-pied at time t

w m ik (t, ˜t) Workload required within interval

t, ˜t to transfer

all resource units of renewable resource k required

by event i if it occurs in mode m

w k i (t, ˜t) Minimum workload required within interval

t, ˜tto

transfer all resource units of renewable resource k required by event i

w k i ( ˜ V ) Minimum workload required to transfer all resource

units of renewable resource k required by events h∈

˜V ⊆ ¯V kto a succeeding event

consistency tests

x i m ( ), ˜x m

i ( ) Hypothetical binary value specifying whether or not

mode m is assigned to event i, which is used for the profile test for storage resource

constraint (5.9b)

by the truncated branch-and-bound algorithm

prob-lem (5.27)

inter-val]0, 1]

transferred along arc (i, j ) ∈ A 1,2

k ( t, x)

y = (y ij ) i ∈V1,j ∈V2 Vector of variables y ij

Generation of Test Instances

Sets

assigned to

assigned to, but do not have to

unloading activity i

P Set of loading activities for which incompatible

vehicles are defined

at tier h

Graphs and Parameters

between the locations a and ˜a

distance between two locations

plan

total number of items

assigned to a location with respect to the size of itsfleet

number of arcs that can be added to A

of any item

τ a v ˜a Travel time of vehicle v from location a to location ˜a

distributor

bound UB

˜ψmin

any item

any item

ω a Maximum deviation among the reciprocal velocities

of the vehicles belonging to the fleet of

manufactur-ing location a

the locations of the locations

manufacturing orders as well as loading, unloading,and transportation of goods

produced by alternative process plans per item

loca-tion a belonging to a supplier or an OEM

the number of resources that may be assigned to aprocess plan or a loading or unloading activity

nrmin, nrmax Minimum and maximum value of percentage nr

nrmin, nrmax Minimum and maximum value of percentage nr

resources assigned to a location of a supplier or anOEM

nsminh , nsmaxh Minimum and maximum number of locations at

tier h

supply chain

nvmin, nvmax Minimum and maximum size of the fleet of

manu-facturing location a

˜nvmin, ˜nvmax

Minimum and maximum size of the fleet of the party logistics provider

activity i carried out in mode m

loca-tion a

˜pmin, ˜pmax Minimum and maximum per-unit processing time of

all process plans

belonging to the fleet of a manufacturing location

˜

belonging to the fleet of the third-party logisticsprovider

resource

unloading activity i for resource k if i is executed

rfmin, rfmax Minimum and maximum value of the resource factor

assigned to over ns h

rpmin

location a over np h

rsmina , rsmaxa Minimum and maximum value of rs a

location a over np h

rsmina , rsmaxa Minimum and maximum value of rs a

tmin, tmax Earliest and latest feasible delivery time of order o

UB Upper bound on the time required for processing the

manufacturing orders as well as loading, unloading,and transportation of goods

resource k

set A

w = (w u ˜u ) (u, ˜u)∈A Vector of arc weights of graph G

The overwhelming majority of research dealing with scheduling or vehicle routingproblems is devoted to classical problems like the capacitated vehicle routingproblem with time windows, the pickup and delivery problem, or the resource-constrained project scheduling problem In practical applications, however, theregenerally exist more complex requirements that do not fit these basic problems.Moreover, many real-world problems arising in multi-site project management orsupply chain planning involve interdependent scheduling and routing decisions

If these requirements are not considered in the planning process, the resultingschedules may require considerable rework before being implemented or mayeven be infeasible for practical use On the other hand, adapting a scheduling

or routing procedure designed for a basic problem to a more general problem isoften very expensive and may lead to inefficient algorithms Hence, most of thesolution approaches presented in the literature cannot be easily adapted to complexreal-life instances combining several non-standard problem features However,separating the scheduling and the routing part into disjoint models and solvingthem sequentially, as it is done within a hierarchical planning approach, may lead toconsiderable performance losses Only few papers address more general conceptsthat allow for covering diverse practical requirements By developing the resourcetransfer model and a corresponding solution approach we aim at reducing the gapbetween academic problems and the requirements of scheduling and vehicle routingprofessionals

We present the resource transfer problem (RTP), which is a new modeling andsolution framework for integrated complex scheduling and rich vehicle routingproblems The RTP enables modeling a broad variety of scheduling problems,vehicle routing problems, as well as their combinations to integrated problems

We can include various specific requirements and restrictions arising in practicalscheduling and vehicle routing problems thus covering a wide range of planningsituations encountered in industrial applications

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The RTP basically consists in scheduling a set of activities that have to beperformed at different locations in a network The activities may, for example,represent manufacturing tasks, tasks of multiple projects being carried out atdifferent sites, pickups or deliveries of goods at customer locations, visits ofpatients by ambulant medical care services, or, more generally speaking, customerdemands to be served or tasks to be performed The activities require a set ofresources with given capacities, for example, vehicles, machines, handling facilities,

or personnel The resources may be transferred between the locations using a fleet

of heterogeneous vehicles or may require changeovers between the execution ofconsecutive activities, where sequence- and resource-dependent transfer times have

to be observed Moreover, we consider generalized precedence relations between theactivities, which define minimum and maximum time lags between the respectivestart or completion times

Based on the unifying framework of the resource transfer problem, we propose ageneric constraint propagation approach exploiting the specific structure of sched-uling and routing problems Both types of problem basically consist in sequencingactivities sharing common scarce resources As a solver for RTP instances wepropose a time-oriented branch-and-bound algorithm We use consistency tests inorder to reduce the search space and improve the efficiency of the solver Compared

to sophisticated and more efficient special-purpose models and algorithms, thisgeneric approach allows for considerable savings in modeling and algorithm designefforts

This monograph has been organized into five chapters In Chap.2, we give anoverview on problems and solution approaches dealing with machine and projectscheduling problems, vehicle routing problems, as well as integrated scheduling androuting problems that have been considered in the literature Chapter3introducesthe RTP and presents a conceptual and a mathematical formulation of the problem.Chapter4is devoted to the way of modeling various types of machine and projectscheduling problems, vehicle routing problems, as well as their combinations inthe RTP framework Moreover, we explain how further requirements and additionalconstraints that are relevant to these types of problem can be represented InChap.5, we propose a branch-and-bound algorithm incorporating consistency testsfor reducing the search space, which can be used to solve small and medium-sizedRTP instances Moreover, we present different truncated variants of this algorithm,which aim at finding good solutions without exploring the whole enumeration tree,thus bearing the chance to speed up the solution process The first part of Chap.6deals with an experimental analysis of the algorithms presented in Chap.5based oninstances of different types of pure scheduling and vehicle routing problems fromthe literature In the second part, we describe a generator for supply chain schedulinginstances comprising more complex requirements which arise in the context of anintegrated scheduling and routing problem Finally, we discuss the computationalresults for the instances we have generated with this generator

Elements of Scheduling and Routing

Theory

In this chapter, we review a number of problems which occur in the areas ofscheduling and routing The first part of this chapter, Sect.2.1, is devoted toscheduling problems In this section, we consider various types of machine andproject scheduling problems Further, we provide references to the related literature.Section2.2deals with the routing part of the review An introduction and overview

on different vehicle routing problems is given, along with references to the routingliterature Section2.3addresses integrated problems, which include simultaneousscheduling and vehicle routing Finally, Sect.2.4deals with reformulations of thevarious models for scheduling and vehicle routing In view of the considerablenumber of problem variants and requirements, this review is not intended to beexhaustive Instead, the focus will be on models and additional constraints that arerelevant to the purpose of this work

In the literature, two types of scheduling problem have attracted most attention:machine and project scheduling Machine scheduling problems typically ariseduring production planning, when orders, also called jobs, have to be processed byresources such as machines or manpower Project scheduling problems are moregeneral in scope, typically in project management, where activities have to beallocated to resources over time Basically, three subproblems have to be considered:the allocation of jobs or activities to resources if multiple resources are available,the determination of a sequence in which the jobs or activities are carried out by theresources, and the calculation of start and end times In the following subsections,

we introduce the problems of machine scheduling and project scheduling in moredetail In view of the limited scope of this work, we focus on deterministic problemswith non-preemptive jobs or activities, that is, those in which a job cannot be

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interrupted during its execution Section2.1.1deals with different types of machinescheduling problem In Sect.2.1.2, various kinds of project scheduling problems arepresented, and in Sect.2.1.3, we turn our attention to resource transfers in projectscheduling.

2.1.1 Machine Scheduling Problems

Machine scheduling problems typically arise during production planning The basic

assumption is that a given set N of jobs has to be carried out using a given set R μofmachines At any point in time, a job can only be processed on at most one machine,

and each machine can perform at most one job at a time A job j ∈ N may comprise several production stages Therefore each job j consists of a set of operations, where

an operation o = o sj represents the execution of the corresponding job j at the production stage s ∈ S j Each operation o is associated with an integer processing time p k o , which represents the time the corresponding job j is required to spend on

a machine k processing the job at stage s An allocation of execution time intervals

for one or more machines to jobs is called a schedule The problem consists indetermining a schedule which minimizes a given objective function, and varioustypes of function are possible

In the literature, a great variety of machine scheduling problems have beenstudied Graham et al (1979) introduce a classification scheme for machinescheduling problems, which allows for the categorization of problems according

to a three-field notation covering three main attributes The first field providesinformation on the machine environment For example, the set of machines mayconsist of either a single machine, several identical or heterogeneous machines

at a single stage, or machines at several stages The second field covers jobcharacteristics, such as precedence relations, preemption, or release dates The lastfield specifies which kind of objective function is being considered; for example,completion times, lateness, or cost In the second part of the paper, examples ofmachine scheduling problems are given and categorized according to the three-field notation Solution algorithms and complexity results are also surveyed Thesame three-field notation is used by Lawler et al (1993), who also review machinescheduling problems with different characteristics, along with solution algorithmsand complexity results The books by Brucker (2007) and Pinedo (2016) provide

a comprehensive overview on various machine scheduling problems and furtherextend the three-field classification scheme In the following, we will give anintroduction on elements frequently occurring in machine scheduling, which ismainly based on the overview presented in the book by Pinedo (2016, Chap 2).With respect to the machine environment, planning may involve a single machine

or multiple machines In a single machine environment, only one machine is

available to process all jobs Two types of scheduling problem may occur in

a multiple machine environment: parallel machine scheduling, where each job

consists of a single operation that must be executed by one of the parallel machines,

and shop scheduling, where each job runs through several machines Consequently,

each job consists of only one operation in both single and parallel machinescheduling, whereas it requires several operations in shop scheduling Parallel

machines may be identical or heterogeneous with respect to their processing speed Heterogeneous machines are either uniform, that is, they run at different machine- dependent speeds, or unrelated, which covers machine- and job-dependent speeds.

Of course, the processing speed of a machine necessarily impacts on the processing

time of an operation Processing times of the same operation o of job j ∈ N

at production stage s ∈ S j may thus differ among heterogeneous machines, that

is, processing time p k

o depends on machine k, whereas processing times among identical machines are equal, that is, p k

o = p o Note that in parallel machine

scheduling we can omit index s as only one production stage is being considered.

Shop scheduling problems can be categorized into flow shop, flexible flow shop,

job shop, flexible job shop, open shop, and flexible open shop problems In a flow shop, all jobs involve the same production stages Each is equipped with a single

machine, that is, each job is processed by each machine Moreover, the stagesare passed by the jobs in the same order and the machines are arranged in seriesaccording to the production stages Since there is only one machine at each stage,

it can also be said that all jobs follow the same machine sequence A flexible flow shop problem is a more general problem where work centers comprising a

set of identical machines in parallel are available at each production stage This issometimes also referred to as a multiprocessor flow shop problem Of course thisproblem can be generalized to fit the situation where heterogeneous machines areused for processing the jobs at each work center Example2.1illustrates a feasiblesolution to a flow shop problem

Example 2.1 We consider a flow shop problem with three jobs j = 1, 2, 3 that must

be performed at two production stages s = 1, 2 Both stages are equipped with a single machine M s and all jobs must pass stage s = 1 prior to stage s = 2 The processing times of the operations corresponding to the execution of the jobs j at stages s are given in Table2.1and a feasible solution to the problem is shown in

Table 2.1 Processing times

of the operations from

Fig 2.2 Schedule of a feasible solution to the job shop problem of Example2.2

Fig 2.3 Schedule of a feasible solution to the open shop problem of Example2.3

A job shop is a generalization of the flow shop, where the machine sequences may vary between the jobs Similarly, a flexible job shop, or multiprocessor job

shop, refers to the case where work centers with a set of identical or heterogeneousmachines in parallel are available at each stage The following example deals with

an instance of a job shop problem

Example 2.2 We consider the problem from Example2.1but assume that job j = 2

must pass stage s = 2 prior to stage s = 1, while the jobs j = 1 and j = 3 are processed at stage s = 1 first Figure2.2illustrates a feasible solution to the

In an open shop, each job must be processed by each machine, but no sequences

need to be observed If parallel machines are available to execute the operations in

the open shop, the problem is referred to as a flexible open shop or multiprocessor

open shop Example2.3illustrates this problem

Example 2.3 We return to the problem from Example2.1 Since in an open shopthe jobs do not follow prescribed sequences, the jobs can pass the production stages

in an arbitrary order Figure2.3shows a feasible solution to the problem Figure2.4provides an overview on typical machine environments considered inthe context of machine scheduling problems In brackets we give an abbreviationfor the respective problems which is frequently found in the literature

An interesting visualization of relationships among machine environments can

be established using complexity hierarchies In general, a complexity hierarchyshows which problems or characteristics are special cases of other characteristicsand thus allow the application of a solution algorithm for the more general problem

A complexity hierarchy for machine environments can, for example, be found in thebook by Pinedo (2016, Sect 2.4) In order to include all the machine environmentsdescribed above, we extended the complexity hierarchy from Pinedo (2016), which

is shown in Fig.2.5

Additional restrictions and characteristics can be defined with respect to the jobs

If a job j ∈ N is not available at the start of the planning period but becomes

Fig 2.4 Machine environments in machine scheduling problems

1 Pm

Qm Rm

F FP FQ FR

J JP JQ JR

O OP OQ OR

Fig 2.5 Complexity hierarchy of machine environments in machine scheduling problems

available later, processing must not start before this release date r j If job j should

be finished by a given date at the latest, a due date ˜d j must be taken into account

In case a deadline d j is specified, j must be completed by time d j at the latest

Whereas j may not be completed after deadline d j, exceeding a due date ˜d j is

merely penalized In some applications, a job j ∈ N is not finished before a certain time has elapsed after the last operation of job j has been completed During

this time, no machine is in operation and the time is referred to as quarantine

time q j If completion of processing job j is required before starting processing

job ˜j , a precedence relation (j, ˜ j ) ∈ A has to be taken into account, where A is

defined as the set of all precedence relations Additional requirements frequentlyassigned to the second field of the three-field notation arise from manufacturingconstraints For example, machines may require setups in order to be able to process

several jobs or groups of jobs, for example, due to cleaning In this case, setup or

changeover times must be allowed for Such changeover or setup times s k

j ˜ j may be

machine- and sequence-dependent, that is, they depend on the machine k used and both the preceding job j and the following job ˜ j An extensive literature review onmachine scheduling problems involving setup times is given by Allahverdi (2015)

Another manufacturing constraint is imposed by planned machine downtimes,

where machines are not available over specified time intervals, for example, due

completion times A popular objective is the minimization of the makespan, which

is defined as the maximum completion time among all jobs Further objectives often

considered in the literature include the total weighted completion time, where each

job is assigned a weight defining the importance of the job, or the minimization of

the maximum lateness, which is the maximum violation of the due dates and may

be negative if all jobs are finished early A similar function can be defined using the

tardiness, that is, the nonnegative part of the lateness, of jobs with respect to the due

dates For regular objective functions, the complexity hierarchy shown in Fig.2.6isbased on the hierarchy given in the book by Pinedo (2016, Sect 2.4)

More complex scheduling problems are considered in the monograph by Bruckerand Knust (2012, Chap 4), who investigate machine scheduling problems whichinclude additional requirements such as more general time lags, blocking of

machines, or transportation times Transportation times have to be taken into

account if machines are geographically dispersed and material must be carriedbetween the production sites Brucker and Knust (2012, Sect 4.6), assume thattransportation is performed by automated guided vehicles which can cope with onejob at a time Bła˙zewicz et al (1983) study machine scheduling problems whereadditional scarce resources are required for the execution of the jobs and propose

a classification scheme for such resource constraints As machine scheduling

total weighted completion time

maximum lateness total tardiness total weighted tardiness

problems can be modeled as a special case of project scheduling problems, whichare covered in the next subsection, we will not go into further detail here and referthe reader to the books by Brucker (2007), Brucker and Knust (2012), and Pinedo(2016) for mathematical problem formulations, a survey on solution algorithms, aswell as specific examples of problems arising in the field of machine scheduling

2.1.2 Project Scheduling Problems

In project management, project scheduling is an important planning step followingthe project conception phase A project consists of a set of activities that must becarried out using a set of resources, such as machines or staff, during specified timeperiods which are referred to as the activity durations Typically, some activitiesare not allowed to start before some other activities have been completed, in whichcase we speak of precedence relations Applications of project scheduling arise, forexample, in civil engineering, software development, or event management

An introduction to project management and project scheduling is, for example,given in the book by Demeulemeester and Herroelen (2002) In practical applica-tions, resources are often not available with unlimited supply This means that ifresources are scarce, capacities must be taken into account In resource-constrained

project scheduling problems, a set of activities i ∈ N must be scheduled subject

to limited availability of resources In the standard setting, activities have to beperformed by a set of renewable resourcesR ρthat represent potential factors such as

machine pools or manpower Each renewable resource k ∈ R ρcomprises a given

amount of R k resource units, where R k = ∞ means that the resource availability is

unconstrained The special case of a renewable resource k with unit capacity R k = 1

is termed a unary resource Resource units of renewable resources are used during

the execution of the activities and released afterwards; that is, the resource is said

to be constrained per time period Each activity i requires a constant amount of r ik

resource units from a renewable resource k ∈ R ρ , that is, r ik units of k are occupied during the processing time p i of activity i Once an activity has been started, it must

not be interrupted until it is completed We establish the convention that the resource

units remain allocated in a half-open time interval, the start time being inclusive andthe completion time exclusive For notational convenience, two dummy activitieswithout resource requirements and with zero duration are introduced into the model:

the project start and completion respectively Precedence relations are defined for

pairs (i, j ) ∈ A of activities where activity j must not start before i is completed.

Since the project start takes place before all activities and the project completionafter all activities, precedence relations are introduced between the project startand all initial activities in the project, as well as between all terminal activitiesand the project completion Activities with zero duration are often referred to asevents Additional events may, for example, be introduced to model milestones As

in machine scheduling, an objective function frequently considered is the makespan

C max = maxi ∈N {S i + p i } with S i denoting the start time of activity i In project

scheduling, the makespan is often also referred to as the project duration

The problem that has received the greatest attention in the project scheduling

literature since the early 1960s is the resource-constrained project scheduling problem RCPSP The RCPSP consists in finding a schedule for the execution of

the activities such that all precedence relations are observed, the resource capacitiesare not exceeded at any point, and the project duration is minimized A project can

be represented using an activity-on-node network, where each activity i is assigned

to a node of the network For simplicity, we identify each with its corresponding

activity in what follows A precedence relation (i, j ) between activities i and j is represented by an arc from node i to node j The processing time of an activity i

is represented as the weights of the arcs (i, j ) ∈ A On the basis of the formulation

of a more general problem given by Neumann et al (2003, Sect 2.1), we formulatethe following model (2.1)

Minimize C max= maxi ∈N {S i + p i} ( a)

k = 2

k = 1

Fig 2.8 Schedule of a feasible solution to the RCPSP of Example2.4

Example 2.4 We consider four activities i = 1, , 4 that have to be performed

by two renewable resources k = 1, 2 with capacities R1 = 2 and R2 = 3 Theactivity-on-node network of the problem is shown in Fig.2.7 The nodes correspond

to the activities, including the dummy activities 0 and n+ 1 = 6, while an arc

(i, j ) represents a precedence relation between activities i and j The activity

durations and resource requirements are given as the weights of the respective

node i = 1, , 4 Figure2.8 shows the schedule of a feasible solution to the

In project scheduling, there may be restrictions on resources that are not renewedfor each time period Such a resource may, for example, correspond to a budget

that must not be exceeded To model these requirements, nonrenewable resources

can be introduced into the problem The resource units of nonrenewable resourcesare consumed by the activities and cannot be used again Moreover, nonrenewableresources cannot be replenished within the planning horizon Therefore, they

are said to be constrained with respect to the total project duration A constrained resource is one with limited capacity per period and with respect to

doubly-the whole project duration This type of resource can be modeled as a combination

of one renewable and one nonrenewable resource We therefore do not consider it in

more detail here A more general type of resource are storage resources In contrast

to nonrenewable resources, there are also activities producing resource units, that is,the resource is depleted and replenished over time Such resources can be used tomodel consumable factors like inventories of raw materials, intermediate products,

or a cash balance if cash inflow and outflow are being considered Storage resources

do not only generalize nonrenewable resources, but also renewable resources Anonrenewable resource can be interpreted as a storage resource which is neverreplenished, whereas a renewable resource can be modeled using a storage resource

that is depleted by r ik resource units at the start of activity i and replenished with the same amount of resource units at the completion of i Project scheduling problems

with storage resources are, for example, considered by Neumann and Schwindt(2002), Laborie (2003), and Carlier and Moukrim (2015) In the literature, this

resource type is sometimes also referred to as a cumulative resource Partially renewable resources are a generalization of renewable or nonrenewable resources,

where the capacity refers to specified time intervals Each partially renewableresource is assigned to a set of time periods during which a given capacity mustnot be exceeded The resource types mentioned above are described in more detail

in the books by Demeulemeester and Herroelen (2002, Sect 2.2), Neumann et al.(2003, Chap 2), and Brucker and Knust (2012, Chap 1)

In practical applications, a mere consideration of basic precedence relations

is often not sufficient For instance, a waiting time must elapse between theexecution of activities, two activities must be executed immediately one afteranother, or activities must be started simultaneously Such situations can be modeled

using generalized precedence relations, which define minimum and maximum

time lags δ ij for pairs (i, j ) of activities Generally, eight types of generalized

precedence relations can be distinguished: to-start, completion-to-start, to-completion, and completion-to-completion minimum and maximum time lags,where start refers to the start time of an activity and completion to the completiontime of an activity respectively For example, a start-to-start minimum time lag

start-ssδminij means that activity j can be startedssδ ijmintime units after the start of activity

iat the earliest, whereas a start-to-completion maximum time lag scδ ijmaxrequires

activity j to be completedscδ ijmaxtime units after the start of activity i at the latest Given that a negative time lag between i and j coincides with a positive time lag between j and i and that activities must not be interrupted during execution, the

eight types of generalized precedence relations can be transformed into each other

A complexity hierarchy for machine