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

Contributions to Management Science

Illa Weiss

The Resource
Transfer
Problem
A Framework for Integrated Scheduling
and Routing Problems


Contributions to Management Science


More information about this series at />

Illa Weiss

The Resource Transfer
Problem
A Framework for Integrated Scheduling
and Routing Problems

123


Illa Weiss
Clausthal-Zellerfeld
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)
/>Library of Congress Control Number: 2018959095
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Foreword

Due to their practical relevance and challenging intractability, scheduling and
vehicle routing problems have been matters of intense research since the early
days of operations research. From the combinatorial perspective, scheduling and
vehicle routing are closely related, both dealing with the allocation of resources

and the sequencing of activities over time. A large number of variants have been
considered in the literature. In industrial applications of complex scheduling problems, beyond precedence constraints and scarce renewable resources representing
potential factors, generalized precedence relations, sequence-dependent changeover
times, and storage resources for consumable factors like materials have to be taken
into account. Rich vehicle routing problems include such diverse requirements
as temporal or spatial synchronization constraints, multi-dimensional capacity
restrictions, incompatibilities between goods, restricted accessibility of roads and
locations, working time regulations, or inter-route constraints arising from limited
processing capacities at depots.
Despite their structural similarities, scheduling and vehicle routing problems
were mostly considered separately. Moreover, the overwhelming majority of models
and methods proposed in the literature are dedicated to specific problem settings. In
real-world planning, however, scheduling and routing problems often arise jointly,
and a large variety of individual requirements must be considered. In supply chain
operations planning, production and transportation must be coordinated to reduce
order-to-delivery time and stocks. Due to short shelf life times, aligning production
scheduling and vehicle routing is frequently of particular importance in food supply
chains. Multi-site scheduling of distributed project portfolios involving resources
that are transferred between the locations may be cited as a further example of an
important problem setting including scheduling and routing aspects. In industrial
practice, scheduling and routing are generally performed sequentially. This hierarchical 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 the
starting point for the research of Illa Weiss. In her dissertation, she proposes the
resource transfer problem (RTP) as a comprehensive reference model for complex
v


vi

Foreword

scheduling and rich vehicle routing problems. The RTP model offers a compact
unifying 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 assigning
occurrence times to the events. The model also covers multi-modal settings, where
each event can occur in alternative modes with different resource requirements.
Moreover, incompatibility and inclusions constraints can be formulated for the
resource units allotted to distinct events. Having devised a conceptual and a
mathematical programming model of the RTP, the author demonstrates the modeling
power of the framework by explaining how to express the various features of
complex scheduling and rich vehicle routing problems within the RTP framework.
The book also presents major algorithmic achievements. Generalizing classical
results from resource-constrained project scheduling, Illa Weiss shows how for
given occurrence modes and occurrence times of the events, a feasible assignment
of resource units can be computed efficiently using a column-generation approach
that is based on a path-based formulation of a minimum-flow problem with side
constraints. As a solver for RTP instances, she devises a time-oriented branch-andbound algorithm, which relies on constraint propagation and takes advantage of
several consistency tests that she adapted to the general RTP setting. An extensive
experimental performance analysis demonstrates that the solver is able to provide
good solutions to complex instances of supply chain operations planning within a
few minutes of CPU time.
The results obtained by Illa Weiss are highly relevant to scientists and practitioners in scheduling and transportation. It is my hope that the ideas developed in this
excellent thesis will achieve a wide dissemination and stimulate further research in
these vital fields.
Clausthal-Zellerfeld, Germany

Christoph Schwindt

Acknowledgements

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 I
enjoyed 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 support
and interesting discussions. I also thank Prof. Dr. Jürgen Zimmermann, who kindly
agreed to be the second reviewer.
Furthermore, I would like to thank my colleagues from the Operations Management Group for excellent collaboration and a nice and inspiring working
atmosphere: Tobias Paetz, Mario Sillus, Anja Heßler, Nora Krippendorff, Astrid
Ludwig, and Michael Krause. Special thanks go to Tobias Paetz, who shared an
office with me for several years and who always took time for any question I had.
I enjoy remembering this beautiful time. It was (and still is) a great pleasure to
spend time with Mario Sillus, Anja Heßler, and Nora Krippendorff who did not
only contribute to our great working atmosphere but with whom I also had many
special moments outside the office. Without Astrid Ludwig it would have been a lot
harder 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 and
supported me whenever needed.
Clausthal-Zellerfeld, Germany
April 2018

Illa Weiss

vii

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

2 Elements of Scheduling and Routing Theory .. . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1 Scheduling Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1.1 Machine Scheduling Problems .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1.2 Project Scheduling Problems .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1.3 Resource Transfers in Project Scheduling .. . . . . . . . . . . . . . . . . . . .
2.2 Vehicle Routing Problems . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2.1 Standard Vehicle Routing Problems . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2.2 Pickup and Delivery Problems . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2.3 Additional Constraints and Further Variants of Vehicle
Routing Problems .. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2.4 Rich Vehicle Routing Problems .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 Integrated Scheduling and Routing Problems . . . . .. . . . . . . . . . . . . . . . . . . .
2.4 Reformulation of Scheduling and Vehicle Routing Problems . . . . . . . .

3
3
4
9
19
20
21
25

31
36
41
46

3 The Resource Transfer Problem . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2 Conceptual Model and Mathematical Formulation .. . . . . . . . . . . . . . . . . . .
3.2.1 Conceptual Model . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2.2 Mathematical Formulation . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3 Graph-Based Representation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.1 Time Lag Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.2 Transfer Graph .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.3 Inclusion and Incompatibility Graphs . . . . .. . . . . . . . . . . . . . . . . . . .

49
49
53
53
59
63
63
66
67

4 Modeling Power of the Framework .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1 Scheduling Problems as Resource Transfer Problems . . . . . . . . . . . . . . . .
4.1.1 Machine Scheduling Problems .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1.2 Project Scheduling Problems .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1.3 Resource Transfers in Project Scheduling .. . . . . . . . . . . . . . . . . . . .

69
69
69
75
84

ix


x

Contents

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1 Allocation of Resource Units . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2 Branch-and-Bound Algorithm .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.1 Enumeration Scheme . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.2 Lower Bounds for the Makespan . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.3 Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.4 Truncated Branch-and-Bound Algorithm... . . . . . . . . . . . . . . . . . . .
5.3 Consistency Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

5.3.1 Consistency Tests for Renewable Resources . . . . . . . . . . . . . . . . . .
5.3.2 Consistency Tests for Storage Resources ... . . . . . . . . . . . . . . . . . . .
5.3.3 Consistency Tests for the Mode Selection .. . . . . . . . . . . . . . . . . . . .

123
123
144
145
159
169
174
177
178
198
201

6 Experimental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1 Experimental Design .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2 Validation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.1 Results of RCPSP/Max Instances .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.2 Results of MRCPSP/Max Instances . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2.3 Results of 1-PDVRPTW Instances.. . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3 Generation of Test Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.4 Evaluation of the Results . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

205
205
207
209
217

221
225
235

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 267
Appendix A .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 269
References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 301
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 309


List of Acronyms

Project Scheduling

MRCPSP
MRCPSP/max
RCPSP
RCPSP/max

Multi-mode resource-constrained project scheduling problem
Multi-mode resource-constrained project scheduling problem
with generalized precedence relations
Resource-constrained project scheduling problem
Resource-constrained project scheduling problem with generalized precedence relations

Vehicle Routing

1-PDVRP
1-PDVRPTW
DARP

CVRP
MAVRP
MDVRP
PDP
VRP
VRPB
VRPDDP
VRPMB
VRPSPD
VRPTW

One-commodity pickup and delivery vehicle routing problem
One-commodity pickup and delivery vehicle routing problem with
time windows
Dial-a-ride problems
Capacitated vehicle routing problem
Multi-attribute vehicle routing problems
Multi-depot vehicle routing problem
Pickup and delivery problem
Vehicle routing problem
Vehicle routing problem with backhauls
Vehicle routing problem with divisible delivery and pickup
Vehicle routing problem with mixed linehauls and backhauls
Vehicle routing problem with simultaneous pickup and delivery
Vehicle routing problem with time windows

xi


xii

List of Acronyms

Resource Transfer Problem
RTP
UAP

Resource transfer problem
Unit-assignment problem

Application

OEM
MRP

Original Equipment Manufacturer
Material Requirements Planning

Solution Approach

B&B(Γ )
B&B
FB(Γ )
FB
MILP

Exact branch-and-bound algorithm with consistency tests
Exact branch-and-bound algorithm without consistency tests
Filtered beam search algorithm with consistency tests
Filtered beam search algorithm without consistency tests

Mixed-integer linear program


List of Symbols

Machine Scheduling
Sets
A
N
Rμ
Sj

Set of all precedence relations
Set of all jobs
Set of all machines
Set of all production stages of job j

Model Parameters
d˜j
dj
po , pok
qj
rj
sk ˜
jj

Due date of job j
Deadline of job j
Processing time of operation o (using machine k)
Quarantine time of job j

Release date of job j
Setup or changeover time of machine k from job j
to job j˜

Project Scheduling
Sets
A
Ak (S)

Set of all precedence relations
Set of all pairs of activities that may share a resource
unit of renewable resource k according to schedule S
xiii


xiv

N, Nk
N¯ k
Rρ

List of Symbols

Set of all activities (requiring renewable resource k)
Set of all activities requiring renewable resource k
plus dummy activities 0 and |N| + 1
Set of all renewable resources

Model Parameters
δij

ss δ min
ij
sc δ max
ij

ϑijk
ρij
σij
ωij , ωijmm˜
ωijk
di
pi , pim
Rk
ri
rik
si
sij

Time lag between activities i and j
Start-to-start minimum time lag between activities i
and j
Start-to-completion maximum time lag between
activities i and j
Setup, changeover, or transfer time of a unit from
renewable resource k between activities i and j
Minimum overlap time of activities i and j
Minimum separating time between activities i and j
Transfer time between activities i and j (being
executed in modes m and m)
˜

Transfer time of a unit from renewable resource k
between activities i and j
Deadline of activity i
Processing time of activity i (in mode m)
Capacity of renewable resource k
Release date of activity i
Resource requirement of activity i for renewable
resource k
Setup time for activity i
Setup time between activities i and j

Model Variables
φijk
Cmax
Si
S = (Si )i∈N

Number of units of renewable resource k which are
changed over from activity i to activity j
Project duration (makespan)
Start time of activity i
Schedule, vector of start times


List of Symbols

xv

Vehicle Routing
Sets

Bv
K
L⊆N
N
N¯

Set of dummy services assigned to a vehicle v
Fleet of vehicles
Set of all linehaul customers
Set of all customers
Set of all customers including dummy customers 0
and |N| + 1

Model Parameters
Δij
ρij
τij
τ max
[ai , bi ]
ai
bi
cij
w¯
d
Mij
Q
qi
qij
si

Distance between the locations of customers i and j
Minimum overlap time of services to customers i
and j
Travel time between the locations of customers i
and j
Maximum route duration
Time window of customer i
Earliest feasible start time of service to customer i
Latest feasible start time of service to customer i
Travel cost between the locations of customers i
and j
Maximum driving time without a break
Maximum driving time per period
Large constant depending on pair (i, j ) of customers
Capacity of the vehicles
Quantity to be delivered to customer i
Quantity to be transported from customer i to customer j
Service duration of customer i

Model Variables
tik
xij k

Start time of service to customer i
Binary variable encoding whether or not vehicle k
travels from customer i to customer j


xvi

List of Symbols

Resource Transfer Problem
Model
UAPk (t, x)

Unit-assignment problem for renewable resource k
and schedule (t, x)

Sets
Ak , Ak (t, x)

AL
AT
I, Ik
I¯, I¯k
Mi
Rρ
Rσ
Rij ⊆ Rρ
¯ ij ⊆ Rρ
R

Uk
V
Vk , Vk (x)
V¯
V¯k , V¯k (x)

Set of all pairs of events that may share a resource

unit of renewable resource k (according to schedule
(t, x))
Set of all pairs of events for which a time lag is
specified
Set of all pairs of events for which a transfer time is
specified
Set of all pairs of events for which a unit-inclusion
constraint (referring to renewable resource k) is
defined
Set of all pairs of events for which a unitincompatibility constraint (referring to renewable
resource k) is defined
Set of all occurrence modes of event i
Set of all renewable resources
Set of all storage resources
Subset of renewable resources k for which a unitinclusion constraint between events i and j must be
observed
Subset of renewable resources k for which a unitincompatibility constraint between events i and j
must be observed
Set of resource units of renewable resource k ∈ Rρ
Set of all real events
Set of all events requiring renewable resource k
(according to mode assignment x)
Set of all events including dummy events 0 and n+1
Set of all events requiring renewable resource k
(according to mode assignment x) plus dummy
events 0 and n + 1


List of Symbols

xvii

Model Parameters
δij , δijmm˜
δij (x)
ϑijmm˜
ϑij (x)
Mij
Rk
R
m
rik , rik
rik (x)
rim
ri (x)
Z k (t, x)

Time lag between events i and j (occurring in modes
m and m)
˜
Time lag between events i and j for given mode
assignment x
Non-Archimedean infinitesimal
Transfer time from event i occurring in mode m to
event j occurring in mode m
˜
Transfer time between events i and j for given mode
assignment x
Large constant depending on pair (i, j ) of events
Capacity of renewable resource k

Capacity of storage resource
Requirement of event i (occurring in mode m) for
renewable resource k
Requirement of event i for renewable resource k for
given mode assignment x
Requirement of event i occurring in mode m for
storage resource
Requirement of event i for storage resource for
given mode assignment x
Set of feasible unit-assignment vectors zk for problem UAPk (t, x)

Model Variables
ϕk (t, x, zk )
f (t, x)
(t, x)
ti
t = (ti )i∈V¯
Uk (i)
xim
x = (xim )i∈V¯ ,m∈Mi
yijm
y = (yijm )i,j ∈V¯ ,m∈Mi

Objective function value of problem UAPk (t, x)
Objective function
Schedule
Occurrence time of event i
Timetable, vector of occurrence times
Set of units of renewable resource k being allocated
to event i

Binary variable encoding whether or not event i
occurs in mode m
Mode assignment vector
Binary variable indicating whether or not xim = 1
and event j does not occur before event i
Vector of binary variables yijm


xviii

List of Symbols

u
zij

u)
z = (zij
i,j ∈V¯ ,k∈Rρ ,u∈Uk
u
k
z = (zij
)i,j ∈V¯ ,u∈Uk

Binary variable indicating whether or not resource
unit u of renewable resource k is transferred from
event i to event j
u
Vector of binary variables zij
Vector of the unit-assignment variables belonging to
renewable resource k ∈ Rρ

Applications
Cost function in mode assignment x
Cost function in timetable t
Tardiness cost of event i
Resource cost incurred by assigning mode m to
event i
Cost rate for maintaining a project organization
Due date of event i in mode m
Earliest time-feasible occurrence time of event i
Latest time-feasible occurrence time of event i
Duration of activity i (occurring in mode m)

c(x)
c(t)
˜
c¯i
cim
cp
d˜im
eti
lti
pi , pim

Graph-Based Representation
δ = (δ ij )i,j ∈V¯
δ ij = (δijmm˜ )m∈Mi ,m∈
˜ Mj
ϑ ij = (ϑijmm˜ )m∈Mi ,m∈
˜ Mj

θij (x)

GL , GL (x)
GT , GT (x)
GI
GI¯
m m
r = (rik
, ri )i∈V¯ ,m∈Mi ,

Time lag matrix
Matrix of the time lags referring to events i, j
Matrix of the transfer times referring to events i, j
Transitive transfer time between events i and j for
given mode assignment x
Transitive time lag between events i and j for given
mode assignment x
Time lag graph (for mode assignment x)
Transfer graph (for mode assignment x)
Inclusion graph
Incompatibility graph
Vector of resource requirements

R = (Rk , R )k∈Rρ , ∈Rσ

Vector of resource capacities

dij (x)

k∈Rρ , ∈Rσ

List of Symbols

xix

Solution Approach
Sets
Γ
Ak1,2 , Ak1,2 (α)
B
D, D(α)
Di , Di (α)
Dim , Dim (α)
Eρ
Eσ
F
F¯
Ik1 , Ik1 (t, x)
Ik2 , Ik2 (t, x)
Iˆk

Mi (α)
N
Pk
Pik

Set of all consistency tests
Arc set of graph GTk (GTk (α))
Set of indices s of the columns as included in basic

matrix B
Set of domains (at node α of the search tree)
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)
Domain of all time- and resource-feasible occurrence times of event i in mode m (at node α of the
search tree)
Set of all event-mode combinations using units of a
renewable resource which is required by event i ∗ in
mode m∗
Set of all event-mode combinations using units of
a storage resource which is required by event i ∗ in
mode m∗
Set of event pairs being in disjunction
Set of event pairs whose transfers of renewable
resource units to a subsequent event cannot be performed in parallel
Set of event pairs (i, j ) ∈ Ak ((i, j ) ∈ Ak (t, x))
for which unit-inclusion constraint (i, j ) refers to
renewable resource k
Set of event pairs (i, j ) ∈ Ak ((i, j ) ∈ Ak (t, x))
for which unit-inclusion constraint (j, i) refers to
renewable resource k
Set of all unit-inclusion constraints which are
not satisfied by the current solution for the
relaxed resource allocation problem for renewable
resource k
Set of occurrence modes of event i at mode α of the
search tree
Set of indices s of the columns as not included in
basic matrix B

Set of all paths from node 0 to node n + 1 in network
Gk (t, x)
Set of all paths from node 0 to node n + 1 in network
Gk (t, x) passing node i


xx

P

List of Symbols

k

Pi
P¯i
ρ

R1 ⊆ Rρ
ρ

ρ

R2 = Rρ \ R1
Tk1 , Tk2 , Tk3

Va
V1 ⊂ Vˆ , V2 ⊂ Vˆ
V1 (i)
V2 (i)

Vk1 ∪ Vk2 , Vk1 (α) ∪ Vk2 (α)
V¯ksucc (i)
V+
V−
V max
Vˆ ⊆ Vk
Vˆi

Set of forbidden paths in network Gk (t, x) with
respect to renewable resource k
Set of predecessors of node i on a longest feasible
¯ k (t, x)
path from node 0 to node i in network G
Set of predecessors of node i on a longest forbidden
¯ k (t, x)
path from node 0 to node i in network G
Set of all renewable resources whose resource allocation is represented by a mode selection
Set of all renewable resources whose resource allocation is not represented by a mode selection
Sets of occurrence times investigated by the energetic reasoning consistency test for renewable
resource k
Set of mode activities
Subsets of event set Vˆ
Set of events that have to take place before event i
Set of events that may occur before event i
Node set of the bipartite graph GTk (GTk (α))
Set of events receiving units of renewable resource
k from event i
Set of events replenishing storage resource
Set of events depleting storage resource
Set of unscheduled events having a maximum time

lag to event i ∗ occurring in mode m∗
Nonempty subset of events requiring renewable
resource k
Set of events for which the transfer of resource units
to a succeeding event must be completed before the
occurrence of event i

Graphs and Parameters

β
β¯
γw

δ

P

ζw

k

(w)

Beam width of the filtered beam search algorithm
Maximum beam width
Sum of the elements of the w-column in matrix
(BT )−1 which belong to the constraints identified in
the pivot step within the column generation procedure
k
Indicator whether or not w ∈ P

Maximum relative gap of a solution found by the
performance-guarantee algorithm
Reduced cost of path w


List of Symbols

xxi

m
˜
Θ = (θ m
ij )i,j ∈V¯ ,m∈Mi ,

Matrix of the lower bounds on the transfer times

m∈
˜ Mj

Minimum time required for a transfer of resource
units after event i
Minimum lower bound on the transfer time between
events i and j
Minimum lower bound on the transfer time from
event i occurring in mode m to another event
Minimum lower bound on the transfer time from
event i occurring in mode m to some event
h ∈ V˜ ⊆ V¯
Lower bound on the transfer time between event
i occurring in mode m and event j occurring in

mode m
˜
Maximum depth of the search tree of the branchand-bound algorithm
Number of violated unit-inclusion constraints referring to event i
Maximum desired number of leaves of the search
tree
Filter width of the filtered beam search algorithm
Coefficient matrix of the constraints from Problem (5.11)
sth column of matrix A
Basic matrix
Vector of the right-hand side of the constraints from
Problem (5.11)
Supply of node i ∈ Vk1 of the transportation problem (5.27)
Demand of node i ∈ Vk2 of the transportation
problem (5.27)
Vector of the coefficients of the objective function
Vector of the coefficients of the objective function
referring to the basic variables
Arc weight of arc (i, j ) from graph GTk
˜k
Arc weight of arc (i, j ) from network G
¯k
Arc weight of arc (i, j ) from network G
˜k
Vector of the arc weights from network G

θi
θ ij
θm
i

˜
θm
i (V )
m
˜
θm
ij

ν
ν(i)
σ
ϕ
A
as
B
b
bi+
bi−
c
cB
cij
c˜ij
c¯ij
c˜ = (c˜ij )(i,j )∈Ak
cij
ck = (cij )i∈V 1 ,j ∈V 2
k

k

Length of a longest path from node i to node j in the
¯k
reduced network obtained from G
Vector of arc weights of the arcs from network GTk


xxii

List of Symbols

ck (α) = (cij )i∈V 1 (α),j ∈V 2 (α)
k

max
dP
k

d max
Pk

di
d¯i
m
˜
D = (d m
ij )i,j ∈V¯ ,m∈Mi ,

k

Vector of arc weights of the arcs from network GTk (α))

Length of a longest feasible path in network
˜ k (t, x)
G
Length of a longest forbidden path in network
˜ k (t, x)
G
Length of a longest feasible path from node 0 to
¯ k (t, x)
node i in network G
Length of a longest forbidden path from node 0 to
¯ k (t, x)
node i in network G
Matrix of the lower bounds on the time lags

m∈
˜ Mj

d ij
mm
˜
d ij

ecim (V˜ )
eci (V˜ )
etim , etim (α)
eti , eti (α)
et˜ im (α)

GTk , GTk (α)
Gk , Gk (t, x)

˜ k, G
˜ k (t, x)
G
ˆ
ˆ k (t, x)
Gk , G
¯ k, G
¯ k (t, x)
G
LB
LBinit
LB0
LB1

Minimum lower bound on a time lag between event i
and event j
Lower bound on the time lag between event i occurring in mode m ∈ Mi and event j occurring in
mode m
˜
Earliest completion time of a transfer of resource
units from event i occurring in mode m to some
event h ∈ V˜ ⊆ V¯
Earliest completion time of a transfer of resource
units from event i to some event h ∈ V˜ ⊆ V¯
Earliest occurrence time of event i in mode m (at
node α of the search tree)
Earliest occurrence time of event i (at node α of the
search tree)
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 to
unscheduled events
Weighted bipartite graph of the transportation problem (5.27) (at node α of the search tree)
Network with node set V¯k and arc set Ak (node set
V¯k (x) and arc set Ak (t, x))
Network used for solving the pricing problem
Network for determining the shortest path length
respecting the unit-inclusion constraints
Modified network for determining the longest path
length respecting the unit-inclusion constraints
Lower bound on the objective function value
Initial lower bound on the objective function value
Lower bound for the makespan being equal to etn+1
Workload-based lower bound


List of Symbols

LB2
LBC
lci (V˜ )
ltim , ltim (α)
lti , lti (α)
N
<
R (i)
R < (i)
r ik
ri
r ik

ri
(t( ), x( )), (t˜( ), x(
˜ ))
ti ( ), t˜i ( )
UB
UBinit
wk (t)
m (t, t˜ )
wik

wki (t, t˜ )
wki (V˜ )
w,
¯ w˜
xim ( ), x˜im ( )

xxiii

Modified workload-based lower bound
Cost-based lower bound
Latest completion time of a transfer of resource units
from event i to some event h ∈ V˜ ⊆ V¯
Latest occurrence time of event i in mode m (at
node α of the search tree)
Latest occurrence time of event i (at node α of the
search tree)
Non-basic matrix
Upper bound on the stock level of just before the
occurrence of event i
Lower bound on the stock level of just before the

occurrence of event i
Minimum requirement of event i for renewable
resource k
Minimum requirement of event i for storage
resource
Maximum requirement of event i for renewable
resource k
Maximum requirement of event i for storage
resource
Hypothetical schedules used in the profile test for
storage resource
Hypothetical occurrence time of event i used in the
profile test for storage resource
Upper bound on the objective function value
Initial upper bound on the objective function value
Number of resource units of resource k being occupied at time 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
Minimum workload required within interval t, t˜ to
transfer all resource units of renewable resource k
required by event i
Minimum workload required to transfer all resource
units of renewable resource k required by events h ∈
V˜ ⊆ V¯k to a succeeding event
Workloads considered when applying the interval
consistency tests
Hypothetical binary value specifying whether or not
mode m is assigned to event i, which is used for the
profile test for storage resource

xxiv

List of Symbols

Variables
β˜
λi
μ
k
φw
k)
φ k = (φw
w∈P k

φ kB
f+
f∗
fTk (y)
(t, x)p
u˜
yij
y = (yij )i∈V 1 ,j ∈V 2
k

k

Integer random variable
Dual variables corresponding to the ith constraint of

constraint (5.9b)
Dual variable of constraint (5.9c)
Number of units of renewable resource k flowing
along path w
k with w ∈ P k referring
Vector of flow variables φw
to renewable resource k
Basic solution to the pricing problem for renewable
resource k
Objective function value of the best solution found
by the truncated branch-and-bound algorithm
Objective function value of the optimal solution
Objective function of the transportation problem (5.27)
Partial schedule
Uniformly distributed random variable in the interval ]0, 1]
Integer variable specifying the number of units
transferred along arc (i, j ) ∈ Ak1,2 (t, x)
Vector of variables yij

Generation of Test Instances
Sets
Ω
Ωh
Ωh1
Ωh2
A
D
Mi
O
P

Set of all manufacturing locations
Set of locations assigned to tier h
Set of locations at tier h further items must be
assigned to
Set of locations at tier h further items may be
assigned to, but do not have to
Arc set of graph G
Set of all distributor’s locations
Set of all execution modes of production, loading, or
unloading activity i
Set of all orders
Set of items, node set of graph G


List of Symbols

P
Ph
Rh
V (nt − 1)
V(a)

xxv

Set of loading activities for which incompatible
vehicles are defined
Set of items assigned to tier h
Set of renewable resources assigned to any location
at tier h

Set of all activities to be performed at tier h = nt −1
Set of vehicles assigned to manufacturing location a

Graphs and Parameters

αa a˜
α max
γv
ϑaπ
ϑ min , ϑ max
μ
ν

ν min , ν max
ρ
ρ min , ρ max
σu
σ max
τava˜
ψo
ψ˜ o
ψ˜ omin , ψ˜ omax
φua
φ min , φ max
ϕua
ϕ min , ϕ max

Percentage of increase or decrease of the distance
between the locations a and a˜
Maximum percentage of increase or decrease of the

distance between two locations
Reciprocal velocity of vehicle v
Setup time of process plan π of location a
Minimum and maximum setup time of a process
plan
Ratio of items with incompatible vehicles over the
total number of items
Mean percentage of incompatible vehicles per item
assigned to a location with respect to the size of its
fleet
Minimum and maximum values of percentages ν
Ratio of number of arcs in set A over the maximum
number of arcs that can be added to A
Minimum and maximum value of ρ
Loading space requirement of a single unit of item u
Maximum loading space requirement of a single unit
of any item
Travel time of vehicle v from location a to location a˜
Time window width for location a belonging to a
distributor
Ratio of the time window width ψo over upper
bound UB
Minimum and maximum value of ψ˜ o
Unloading time per unit of item u at location a
Minimum and maximum unloading time per unit of
any item
Loading time per unit of item u at location a
Minimum and maximum loading time per unit of
any item