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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

Keywords—Job 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 T∈N

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 Ct∈R+. 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 s∈Sand each

machine j∈M, 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

j∈Nand

drampdown

j∈Nfor 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 i∈Jconsists of Oi∈Nindividual 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)|i∈J, 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 i∈Jwe 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 fi−1(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+

k−1

X

q=1

dop

i,q, dr ampup

mi,k +dsetup

i,k

fi,k =fi−1−

Oi

X

q=k

dop

i,q

2) Determine A={(i, k, t)∈O×[T]|ai,k ≤t≤fi,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 j∈M, each state s∈S, 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

j∈M

T−1

X

t=0 X

s∈S

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 t−d+ 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 t−dprocessing

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

s∈S

βj,s,t = 1

∀j∈M, t ∈[T]∪ {−1, T }

(2)

βj,off,t = 1

∀j∈M, 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=t−dprocessing

i,k +1

αi,k,q ≤βj,processing,t

∀j∈M, t ∈[T]

(5)

X

(i,k)∈O:

mi,k=j

t+dsetup

i,k

X

q=t+1

αi,k,q ≤βj,setup,t

∀j∈M, t ∈[T]

(6)

t−dprocessing

i,k

X

q=0

αi,k,q ≥

t

X

q=0

αi,k+1,q

∀i∈J, k ∈ {1, . . . , Oi−1}, t ∈[T]

(7)

βj,off,q +βj,rampup,q ≤

1−βj,processing,t −βj,setup,t −βj,standby,t

(8)

∀j∈M, t ∈[T], q ∈ {t+ 1, . . . , t +drampdow n

j}

βj,off,q +βj,rampdown,q ≤

1−βj,processing,t −βj,setup,t −βj,standby,t

(9)

∀j∈M, t ∈[T], q ∈ {t−drampup

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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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.