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Job Shop Scheduling With Flexible Energy Prices

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  • Tesla Manufacutring Brandenburg SE
Conference Paper

Job Shop Scheduling With Flexible Energy Prices

Abstract and Figures

The rising energy prices – particularly over the last decade – pose a new challenge for the manufacturing industry. Reactions to climate change, such as the advancement of renewable energies, raise the expectation of further price increases and variations. Regarding the manufacturing industry, production planning and controlling can have a significant influence on the in-plant energy consumption. In this paper, we develop a scheduling method as a linear optimization model with the objective to minimize energy costs in a job shop production system.
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JOB SHOP SCHEDULING
WITH FLEXIBLE ENERGY PRICES
Maximilian Selmair
Thorsten Claus
Marco Trost
Department of Business Science
Dresden Technical University
01062 Dresden, Germany
maximilian.selmair@mailbox.tu-dresden.de
Andreas Bley
Department of Mathematics
Kassel University
34132 Kassel, Germany
andreas.bley@uni-kassel.de
Frank Herrmann
Innovation and Competence Centre
for Production Logistics and
Factory Planning (IPF)
OTH Regensburg
93025 Regensburg, Germany
frank.herrmann@oth-regensburg.de
KeywordsJob Shop Scheduling; Flexible Energy Prices; En-
ergy Efficient Production Planning; Energy Consumption; Standby
Abstract—The rising energy prices – particularly over the last
decade – pose a new challenge for the manufacturing industry.
Reactions to climate change, such as the advancement of renew-
able energies, raise the expectation of further price increases and
variations. Regarding the manufacturing industry, production
planning and controlling can have a significant influence on the in-
plant energy consumption. In this paper, we develop a scheduling
method as a linear optimization model with the objective to
minimise energy costs in a job shop production system.
INTRODUCTION
Since the industrial revolution, the worldwide economic
prosperity depends on the reliable provision of electric en-
ergy. Yet the generation of this energy by means of fossil
fuels is, as measured by the associated CO2-emissions, the
main contributor to climate change (Finkbeiner et al. 2010).
According to the Federal Association for Energy and Water
Management, the electricity costs for private customers rose
by 85% between the years 2000 and 2010. Within the same
period, an increase of 130% was noted for the industrial sector
(Bauernhansl et al. 2013). One of the driving factors in this
distinct rise are increases in taxes and other charges, such as
the EEG reallocation charge (EEG = Erneuerbare-Energien-
Gesetz; Renewable Energies Act of Germany). The most of
the remunerated electricity under the EEG is traded at spot-
markets like the European Energy Exchange (EEX) or the
European Power Exchange (EPEX). As supply and demand
determine the price, energy tariffs are highly variable over the
day. In line with this, methodologies for price predictions for
competitive energy markets have been published by Lei and
Feng 2012 and others. The spot-markets are trading electricity
for the following day (Day-Ahead). Figure 1 shows exemplary
the hourly electricity price for the following day - in this case
for the 21st of January 2016, with a standard deviation of
20.75 (39.8%). The hourly electricity prices are used in this
research to minimise the energy costs by means of intelligent
scheduling.
1 8 16 24
0
20
40
60
80
100
Timeline [h]
Energy Price in e/MWh
Fig. 1. Hourly electricity price and average (dashed line, for information
purposes) for the following day, in this case 21th of January 2016 (Own
representation of data from www.epexspot.com)
REL ATED LITERATURE
Energy-efficient scheduling and the reduction of energy
consumption has been a very important issue over the recent
years. In this area of research, Weinert et al. 2011 introduced
a so-called energy blocks methodology, which allows for
the accurate prediction of energy consumption and integrates
energy efficiency criteria into production system planning and
scheduling. Dai et al. 2013 proposing an improved genetic
simulated annealing algorithm for energy efficient flexible flow
shop scheduling, focusing on the two objectives makespan and
energy consumption. Furthermore, Liu et al. 2014 developed
a multi-objective scheduling method in which the reduction of
the energy consumption was one of the primary objectives.
The three papers mentioned above consider only two oper-
ational machine states with respect to the energy consumption:
Idle (or standby) and processing. In 2014, Shrouf et al. 2014
extended these works by making also decisions on a machine
level, which allowed them to consider more operational-modes
of a machine. Developing a model for optimizing the total
energy costs when scheduling jobs on a single machine, they
consider the operating states Idle,Processing,Turning Up and
Turning Down.
The extension of this approach to more than one machine
complicates matters substantially. Dependencies between all
machines are unavoidable and need to be modeled when
assuming a job shop production system. Already the basic
job shop problem is known to be NP-complete and to be
computationally extremely difficult.
Concerning exact solution methods for job shop problems,
rather few methods have been published. Until 2005, the most
effective approaches have been branch-and-bound algorithms
that branch on the job orders on the machines in the so-called
disjunctive graph model. In the traditional job shop problem,
the optimal starting times of the jobs can be easily computed
once the decisions concerning the order of the jobs are made.
Aiming to avoid unnecessary branchings, these algorithms
typically also employ constraint programming techniques in
order to tighten the bounds for the job starting times and infer
job orders during the branch-and-bound process.
Motivated by the success of time-indexed models and
solution approaches for other scheduling problems (Sousa
and Wolsey 1992; Akker 1994), Martin and Shmoys 2005
eventually proposed to use time-indexed integer programming
formulations also for the job shop problem. Using such a
formulation together with effective bound tightening tech-
niques and specialized branching, they have been able to
computationally derive lower bounds that were stronger than
those obtained with disjunctive graph models and job order
based formulations.
In a time-indexed formulation, the planning horizon is dis-
cretized and binary variables are used to indicate if a job starts
at a specific time. Formulations of this type are widely used
to tackle project scheduling and dynamic planning problems
that involve complex resource, precedence, or state constraints,
as these additional constraints often can be formulated much
easier in a time-index model than in a continuous time model.
Already Ford and Fulkerson 1962 observed that dynamic flow
problems in a network with transit times on the arcs can be
modeled equivalently as static flow problems in time-expanded
networks, which is equivalent to a time-indexed formulation of
the problem.
Successful applications of time-indexed and time-expanded
problem formulations include the optimization of supply
chains (K¨
uc¸ ¨
ukyavuz 2011; Pochet and Wolsey 2006), pro-
duction planning in mining, energy production, and other
industries (Louis and Hill 2003; Chicoisne et al. 2012; Epstein
et al. 2012; Lambert et al. 2014), timetabling in transportation
(Sch¨
obel 2007; Serafini and Ukovich 1989), and many more.
In many of these cases, the time-indexed integer program-
ming formulations also lead to mathematically stronger linear
relaxation than their continuous time counterparts, which is
beneficial in branch-and-bound algorithms. This benefit typi-
cally comes at the cost of a much larger problem formulation.
However, exploiting the special structure of the time-indexed
formulations in specialized solution algorithms, the size of
the formulation that actually has to be solved often can
be reduced substantially. A discussion of the main features,
strengths, and limitations of alternative modeling and optimiza-
tion techniques, with a special focus on short-term scheduling
of chemical batch processing, can be found in the survey of
M´
endez et al. 2006.
A computational evaluation of different mixed-integer pro-
gramming formulations for parallel machine scheduling prob-
lems for job-related objective functions such as weighted
completion time, weighted tardiness, maximum lateness, and
number of tardy jobs has been published in Unlu and Mason
2010. The results of this study, as mentioned also in Berghman
et al. 2014, suggest that time-indexed formulations perform
reliably well for such problems and should be explored further
for the solution of scheduling problems with multiple ma-
chines. Time-indexed formulations are widely used to model
variable operational-modes of devices and plants in various
applications (for example in unit commitment planning for
electricity networks or in dynamic spectrum assignment in
telecommunication networks) or to model time-dependent job-
related objective functions in scheduling problems. To the best
of our knowledge, however, the use of time-indexed formu-
lations to model the job-independent ramping and switching
dynamics of the machines’ operational states in a multi-
machine scheduling problem has not yet been investigated, yet.
PROB LE M DEFINITION
When considering a common job shop production system,
each machine usually has a varying energy demand depending
on its operational state. Production systems that consist of
chipping (e.g. milling machines) or transforming tool machines
(e.g. presses or benders) typically have a vast demand of
energy (Neugebauer 2008). Further examples of high energy
consumers are industrial laser welding or laser cutting sys-
tems (Ahn et al. 2016). Note that a considerable share of
the electricity consumption of these machines in practice is
actually associated with the standby-mode, when the machines
are active but not working (Neugebauer 2008; Ahn et al. 2016).
Furthermore, peripheral systems, such as cooling and ventila-
tion, loading and unloading mechanisms, or hydraulic systems
require a significant amount of electricity even in standby-
mode. Shutting down these modules is generally refused in
industrial practice on account of the necessary process stability.
Operational states would have to be predictable and reliable
in order to initiate a safe ramp down without risking process
stability.
If one did assume that machines ramp down entirely when
not in use, an initial evaluation would exhibit short idle times
and, thus, a high level of machine capacity utilization, which
in turn saves energy. This would reduce the energy demand
during standby-mode and the machine in question could ramp
down after each processing operation. However, long idle times
are also possible, which would allow for a complete ramp
down of the machine. The feasibility of this option depends on
planning a timely and safe restart and the subsequent flawless
resumption of production.
Our research specifically addresses these questions. We
aim to develop models where the operating-modes of all
machines are planned together with the scheduling of the jobs
in a period-specific manner such that longer ramp up, ramp
down, and standby-processes are adequately considered. Thus,
periods with lower energy costs could be utilized to schedule
production processes with high energy demands and remaining
in standby-mode or even ramping down production facilities
in more expensive periods can save energy costs.
Referring to the above mentioned use case (chipping or
transforming tool machines as well as laser welding and
cutting), we have identified five crucial operational states that
should be considered: off,ramp up,setup,processing,standby
and ramp down. Ramp up and ramp down can be seen as
transitional states with a fixed duration depending on the
machine. The transition time between standby and processing
or standby and setup and vice versa is assumed to be negligible.
In industrial practice, this transition only lasts a matter of
seconds and is typically too short to affect a solution that
ranges from minutes to hours. The essential decisions related
to the machines are to decide whether a machine is switched
off and on or whether it is left in standby in a production break.
Both choices require energy and cause costs, and the first one
is only possible if the break is long enough for ramping down
and up.
To determine the processing periods for all operations and
the operational states for each machine, our proposed model
provides:
1) start period of processing each operation on the ma-
chines,
2) start period for setting up a machine for the upcoming
operation (implicitly), and
3) all operational status transitions for each machine.
FORMULATION OF THE MOD EL
All jobs and machine states are planned within a specific
time period. The planning horizon is discretized into TN
equally long intervals, called periods, and denoted by [T] =
{0, . . . , T 1}. If `represents the duration of a period, t[T]
denotes the period from time t` to time (t+1)`. In accordance
with Shrouf et al. 2014, every time period is associated with
its individual energy price described by CtR+. Note that
all durations and times are given and modeled as integers, so
only integer multiples of the period length `can be represented
exactly in this model.
The given set of vmachines is denoted by M={Mj}v
j=1
(using an arbitrary predefined order on the machines). The
considered operational machine states are described by the
set S={off, standby, processing, setup, rampup,
rampdown}. For each operational state sSand each
machine jM, a specific energy demand Pj,s Ris
given. For the two transition states ramp up and ramp down,
we are also given the transition times drampup
jNand
drampdown
jNfor ramping up machine jfrom operational
state off and for ramping it down to off, respectively.
In accordance with ¨
Ozg¨
uven et al. 2010, we let J=
{Ji}n
i=1 denote the given set of njobs.
Each job iJconsists of OiNindividual operations
(sub-tasks). The k-th operation of job iis denoted operation
(i, k). The overall set of all operations of all jobs is denoted
by O=(i, k)|iJ, k ∈ {1, . . . , Oi}. For each operation
(i, k)Owe are given
the machine setup time dsetup
i,k N0,
the operation processing time dop
i,k N, and
the associated machine mi,k M.
Furthermore, for each job iJwe have
a release time ai
a due time fi
Note: Release date aimeans job ican start from period ai
(at time ai`). Due date fimeans job imust be completed
within period fi1(not later than fi`).
Assumptions
1) Every machine can only process or setup for one oper-
ation at a time.
2) Once an operation has started to process, interruptions
are not allowed. The same applies for setup processes.
3) Every job contains operations in a linear sequence.
Consequential operation (i, k)must be completed before
operation (i, k + 1) begins.
4) No time is required for changes between operating-
modes from standby to processing and vice versa.
5) Changes between operating-modes (ramp up and ramp
down) cannot be interrupted after they have been initi-
ated.
6) A machine can be setup for operation (i, k)even if the
preceding operation of the same job (i, k 1) is still
being processed on another machine.
7) The setup of operations (i, 1) can be initiated prior to
the release time aiof job i.
8) Processing operations have to start immediately after the
related setup process.
9) Two artificial periods are added at the beginning and at
the end of the planning horizon (1and T), which are
free of any machine activity (processing, setup, ramp up
or ramp down). These only serve to describe the initial
and final states of the machines. In this paper, we assume
that all machine must be in state off in these periods.
Preprocessing
Initially, bounds ai,k and fi,k for the earliest and the latest
starting times for the individual operations (i, k), respectively,
are determined on the basis of the given parameters. This
approach reduces the solution space significantly and increases
the speed and efficiency of the model.
1) For all operations (i, k)Odetermine:
ai,k = maxai+
k1
X
q=1
dop
i,q, dr ampup
mi,k +dsetup
i,k
fi,k =fi1
Oi
X
q=k
dop
i,q
2) Determine A={(i, k, t)O×[T]|ai,k tfi,k }
of possible operations-startperiod-pairs. Thus, operation
(i, k)can only start between the periods ai,k , . . . , fi,k.
Decision Variables
We introduce two types of binary decision variables: α-
variables model the start periods of the operations and β-
variables represents the operational states for all machines in
all periods.
For each operation (i, k)and each start-period twith
(i, k, t)A(i.e., tis a permissible start time for (i, k)), we
have a binary variable αi,k,t ∈ {0,1}, which is interpreted as
αi,k,t =
1Processing of operation (i, k)
starts in period t.
0Else.
For each machine jM, each state sS, and each period
t[T]∪ {−1, T }, we have a binary variable βj,s,t ∈ {0,1},
which means
βj,s,t =
1In period tmachines j
is in operational state s.
0Else.
Objective Function
The objective function needs to determine and minimise
the energy costs. The operational state of each machine is
set by the decision variable β. Parameter Pj,s represents the
associated power demand. With Ctthe energy price per period
is provided. Thus equation (1) minimises the total energy costs.
minZ=X
jM
T1
X
t=0 X
sS
βj,s,t ·Pj,s ·Ct(1)
Constraints
Equation (2) ensure that every machine has exactly one
operational state in each period.
Equation (3) fix the specific operational state off at the
beginning (period 1) and in the end (period T) of the
planning horizon for each machine.
Equation (4) ensure that every operation will start exactly
once in its permissible horizon (depending on the release and
due date).
Inequation (5) ensure that machine jis in operational state
processing in period tif some operation of duration dstarted
between td+ 1 and tand, thus, is still running in period t
on this machine. Similarly, inequation (6) ensure that machine
jis in operational state setup in period tif some operation
with setup time dstarts between t+ 1 and t+dand, thus,
requires machine setup in period ton this machine. Moreover,
together with (2) these constraints guarantee that machine j
can be in setup-mode for or actually executing at most one
single operation at a time. Thus, operations and setups do not
overlap on any machine, the so-called parallel constraints hold.
Inequation (7) imply the so-called sequential constraints.
Enforcing for all times tthat operation (i, k)starts no later
than tdprocessing
i,k if operation (i, k + 1) starts in period t(or
earlier), these inequations imply that operation (i, k)indeed
completes running before operation (i, k + 1) starts.
Inequation (8) and (9) finally model the technical con-
straints that are related to the machine states and the duration
of ramp up and ramp down phases. The required minimum
duration of the ramp down phases is enforced via constraints
(8). These ensures that, if machine jis active (i.e. processing,
in setup, or in standby) in period t, then it cannot be off (or
even already in ramp up-mode again) in period t+drampdown
j
(or earlier): It must either remain active in processing, setup, or
standby-mode after the operation it was executing (or setting
up for) in period tor, if it decides to ramp down after this
operation, the ramp down phase cannot have ended by period
t+drampdown
jor earlier. Similarly, constraints (9) ensure that
the ramp up phases are at least as long as required. If the
energy consumption in the ramp up and ramp down states is
not lower than that in the off state and, similarly, that energy
consumption in the processing and setup state is not lower than
that in the standby state, these constraints suffice to ensure that
the machine state schedules in an optimal solution of the model
satisfy the given constraints. Otherwise, one may add further
constraints similar to (8) and (9) to ensure that ramping phases
have exactly the required lengths and that machines actually
switch to off or standby whenever possible.
X
sS
βj,s,t = 1
jM, t [T]∪ {−1, T }
(2)
βj,off,t = 1
jM, t ∈ {−1, T }(3)
X
t[T]:(i,k,t)A
αi,k,t = 1
(i, k)O
(4)
X
(i,k)O:
mi,k=j
t
X
q=tdprocessing
i,k +1
αi,k,q βj,processing,t
jM, t [T]
(5)
X
(i,k)O:
mi,k=j
t+dsetup
i,k
X
q=t+1
αi,k,q βj,setup,t
jM, t [T]
(6)
tdprocessing
i,k
X
q=0
αi,k,q
t
X
q=0
αi,k+1,q
iJ, k ∈ {1, . . . , Oi1}, t [T]
(7)
βj,off,q +βj,rampup,q
1βj,processing,t βj,setup,t βj,standby,t
(8)
jM, t [T], q ∈ {t+ 1, . . . , t +drampdow n
j}
βj,off,q +βj,rampdown,q
1βj,processing,t βj,setup,t βj,standby,t
(9)
jM, t [T], q ∈ {tdrampup
j, . . . , t 1}
COMPUTATIONAL RESULTS
This section presents an exemplary case study of a 5×5 job
shop problem to demonstrate how scheduling affects the total
energy consumption and total energy costs. The study scruti-
nizes five jobs processed on the same number of machines. The
planning horizon spans three consecutive days. It was decided
to plan by hours and every period lasts one hour with a total
of 72 periods. The proposed plans rely on the energy price
model given in Figure 1 for each day. Consequential energy
is most expensive between 8 a.m. and 8 p.m.. Our proposed
planning horizon begins and ends at midnight. All jobs and
their respective release and due dates are given in Table I.
These dates are to be strictly adhered to, as delayed jobs are not
allowed. The associated operations with all related parameters
are given in Table II.
TABLE I. JO BS
i aifi
1 0 72
2 8 72
3 16 72
4 24 72
5 48 72
TABLE II. OPERATIONS
(i, k)mi,k dsetup
i,k dprocessing
i,k
1,1 1 3 4
1,2 2 3 4
1,3 4 1 6
1,4 5 1 6
1,5 2 4 4
2,1 3 3 4
2,2 2 3 4
2,3 5 1 5
2,4 4 1 5
2,5 1 3 4
3,1 1 4 5
3,2 2 4 5
3,3 3 4 8
3,4 5 3 4
4,1 3 2 5
4,2 2 2 5
4,3 4 1 4
4,4 5 1 4
5,1 1 2 3
5,2 2 2 3
5,3 3 2 3
TABLE III. MAC HIN ES
j1 2 3 4 5
drampup
j3 3 3 2 1
drampdown
j2 2 2 1 1
Pj,off 0 0 0 0 0
Pj,rampup 18 10 5 4 2
Pj,setup 8 8 8 3 3
Pj,processing 20 20 20 6 6
Pj,standby 7 1 0.5 0.5 0.5
Pj,rampdown 5 5 5 2 2
As presented by Table III, the duration for ramping up
and down as well as the demand for energy in the different
operational states varies between machines. Machine M1, for
example, has the highest energy consumption in standby-mode.
Ramping up is also quite expensive in comparison to the other
machines of the production system. Machines M3–M5require
less energy and are comparatively cheap in standby-mode. The
highest consumption of energy for processing and setup-mode
is linked with Machines M1–M3. It is expected that our model
will schedule jobs to these machines only in periods with cheap
energy prices, if possible.
Figure 3 visualizes a schedule plan without taking either
energy consumption or energy prices into consideration. All
jobs are planned by minimising their makespan to complete
them as soon as possible. Along with the planned operational
periods, all further machine-specific operating-modes are vi-
sualized. The key can be found in Figure 5.
Figure 4 presents the energy-efficient solution of our new
model. Several things are particularly noticeable. The first
salient findings are the scheduled operational states. The
machines are not switched on continuously. In addition to the
setup and processing states, ramping up and down is planned as
well as the standby-mode. The analysis of the schedule of M1
M3was the first step. As shown in Table III, these machines
have a vast demand for energy in all operational states. M1
has the highest energy consumption in standby-mode. This is
reflected by the schedule plan: M2and M3ramp up hours
before they start to process operations. This can be explained
by the energy prices. As energy is cheap between 0 a.m. and
8 a.m., the model plans expensive processes in such periods.
Obviously the cost for the subsequent standby-mode over many
hours is lower than ramping up the machines just prior to the
job. This was also observed for the ramping down of M3.
M1ramps up just in time due to its high energy consumption
during standby-mode. Consequently the standby-mode for M1
is used very rarely. M3is in standby-mode during the more
expensive periods. In contrast, M1and M2are processing
during these expensive periods as specific due dates need to
be met. M5does not use the standby-mode. Although energy
consumption in standby-mode is very low, it is cheaper to turn
the machine off completely during the non-productive time.
The key performance indicators for both solutions are
compared in Figure 2. It is interesting to note that, with
exception of M1, the energy consumption of the optimized so-
lution remains the same or is indeed higher than its makespan
counterpart. Yet the resulting energy costs are lower owing
to the well-conceived scheduling strategy. Merely M4causes
slightly higher costs in our model compared to the minimising
makespan model.
Table IV aggregates the energy consumption and the result-
ing costs for all machines of scenario 1 (optimized makespan)
and scenario 2 (optimized energy costs). The provided signif-
icant savings are given in the last two columns.
TABLE IV. R ES ULTS
Scenario 1 Scenario 2 Savings
energy consumption 2,194 kWh 2,052 kWh 142 kWh 6.5%
energy costs e120 e93 e27 22.3%
CONCLUSIONS AND FUTURE WORK
This work proposed a model for minimising the total en-
ergy costs when scheduling a job shop production system. Con-
sidering the continuous changes of energy prices, our model
can help to organize a more efficient production schedule,
especially for high-energy production systems. Furthermore
we evaluated the significant energy price savings that could be
obtained by using this model instead of the commonly used
lead time minimisation.
For further benchmark experiments, we propose to use
the model for a continuous rolling and overlapping planning
long-term study by means of simulation. Finally, our study
M1M2M3M4M5
0
200
400
600
800
Energy Consumption in kWh
M1M2M3M4M5
0
10
20
30
40
Energy Costs in e
Minimised Makespan
Minimised Energy Costs
Fig. 2. Comparison of Schedule Plans in Terms of Energy Consumption and
Costs
is planned to be integrated as an ecological component of
a sustainable production planning concept. The hierarchical
production planning as proposed by Hax and Meal 1973
might contribute to creating an ecological and also social
environment for sustainable production planning (Trost et al.
2016).
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0 8 16 24 32 40 48 56 64 72
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AUTHOR BIOGRAPHIES
Maximilian Selmair is doctoral student at the Department
of Business Science at the Dresden Technical University.
Currently employed at the SimPlan AG, he is in charge of
projects in the area of material flow simulation. His email
address is: maximilian.selmair@mailbox.tu-dresden.de and his
website can be found at maximilian.selmair.de.
Prof. Dr. Thorsten Claus holds the professor-ship for
Production and Information Technology at the International
Institute (IHI) Zittau, a central academic unit of Dresden
Technical University. His e-mail address is: thorsten.claus@tu-
dresden.de.
Prof. Dr. Frank Herrmann holds the professor-ship for in-
formation systems in the department of informatics and math-
ematics at the Regensburg Technical University of Applied
Sciences and he is the head of the Innovation and Competence
Centre for Production Logistics and Factory Planning (IPF).
His e-mail address is: frank.herrmann@oth-regensburg.de.
Prof. Dr. Andreas Bley is professor for applied discrete
mathematics at the University of Kassel. His e-mail address
is: andreas.bley@uni-kassel.de.
Marco Trost is doctoral student at the Department of
Business Science at the Dresden Technical University and he
is sponsored by the European Social Fund (ESF). His e-mail
address is: marco.trost@mailbox.tu-dresden.de.
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