Integrating Real-Time Dynamic Electricity Price Forecast into Job Shop Production Scheduling Model with Multiple Machine Environments

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Integrating Real-Time Dynamic Electricity Price Forecast into Job
Shop Production Scheduling Model with Multiple Machine
Environments
Stefan Krstevski
Institute for Manufacturing,
Department of Engineering,
University of Cambridge
Cambridge, United Kingdom
sk2204@cam.ac.uk
Omid Fatahi Valilai
School of Business, Social, & Decision
Sciences,
Constructor University Bremen
Bremen, Germany
ofatahivalilai@constructor.university
Hendro Wicaksono∗
School of Business, Social & Decision
Sciences,
Constructor University Bremen
Bremen, Germany
hwicaksono@constructor.university
ABSTRACT
One of the challenges in the transition towards green electricity
is the intermittence of power generated by renewable sources.
Thus, power consumers, including the manufacturing industry,
must adapt their activities and processes to green electricity supply.
Real-time dynamic pricing is an approach to encourage electricity
consumers to change their consumption patterns by lowering prices
when the availability of green electricity in the grid is high. Due
to the introduction of real-time electricity pricing, manufacturing
companies must adapt their production planning by integrating
dynamic price information into their production scheduling. Our
research focuses on extending the basic production scheduling
mathematical model by introducing real-time power pricing in the
model. The prices are built based on the current proportion of green
electricity in the grid represented in the green electricity index (GEI)
with one-hour intervals. This paper also illustrates a scenario of
how to use the model. Our future research will further extend the
model addressing the exibility of manufacturing shop oors (e.g.
adding buer, retooling, and setup time) and validate the model in
two small and medium manufacturing enterprises.
CCS CONCEPTS
•Computing methodologies
→
Planning and scheduling;
Supervised learning by regression;•Theory of computation
→
Mixed discrete-continuous optimization;•Mathematics of
computing →Regression analysis.
KEYWORDS
job shop production scheduling, dynamic electricity price, mixed
integer linear programming, production planning and control, de-
mand side management
∗Corresponding Author
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ICIEA-EU 2023, January 09–11, 2023, Rome, Italy
©2023 Copyright held by the owner/author(s). Publication rights licensed to ACM.
ACM ISBN 978-1-4503-9852-7/23/01. . . $15.00
https://doi.org/10.1145/3587889.3587905
ACM Reference Format:
Stefan Krstevski, Omid Fatahi Valilai, and Hendro Wicaksono. 2023. Integrat-
ing Real-Time Dynamic Electricity Price Forecast into Job Shop Production
Scheduling Model with Multiple Machine Environments. In 2023 The 10th In-
ternational Conference on Industrial Engineering and Applications (ICIEA-EU
2023), January 09–11, 2023, Rome, Italy. ACM, New York, NY, USA, 9 pages.
https://doi.org/10.1145/3587889.3587905
1 INTRODUCTION
The necessity for a society that can full the present demands with-
out sacricing the capacity of future generations to meet their own
needs has long been recognised [
34
]. The demand for sustainabil-
ity has led to the development of new regulations and identied
emission objectives in the industrial sector [
34
]. Furthermore, due
to increased supply, the transformation of the energy system ne-
cessitates adopting a strategy that allows for the incorporation of
increasing volumes of renewable energy. Thus, industrial processes
need to be more exible, and Demand Response (DR) as a type of De-
mand Side Management (DSM) is one method of accomplishing this
change [
2
]. Consequently, renewable energy is essential to achiev-
ing a low-carbon future. Dynamic pricing is one type of DSM/DR
which might be critical in integrating additional renewable energy
sources into the power grid [9, 32].
In most countries, the most frequent power price charging method
has been a set fee per kWh combined with a base rate, otherwise
known as static electricity pricing [
9
]. In this case, the price per
kWh stays consistent for longer time. For example, in Germany,
the price remains the same for around one year [
9
]. Furthermore,
the so-called day and night rates in this pricing model feature re-
duced kWh pricing at night and are often oered in conjunction
with minimum yearly power use [
19
]. Taris such as the previously
mentioned ones were designed to reect reality, i.e., represent a
more optimal and realistic pricing scenario.
As demand has become more dynamic, requiring a more ecient
pricing model, dynamic pricing schemes have been devised to aect
demand consistent with the current supply [
5
]. According to the
denition of dynamic pricing, the price per kWh might change
depending on the time of day or the amount of electricity used
[
8
]. As seen in Figure 1, when the renewable energy supply is
higher, the price per kWh tends to be lower. The renewable energy
is produced based on wind (light blue) and solar power (orange).
Thus, the dynamic price (dark blue) is based on the available sources
mentioned above. Based on the graph interpretation and assuming
an ideal scenario, price uctuations would lead to the most ecient
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ICIEA-EU 2023, January 09–11, 2023, Rome, Italy Krstevski, Fatahi Valilai, and Wicaksono
use of power-producing capacity and, as a result, lower system
costs for the end user.
Due to its way of working, dynamic pricing attracts much atten-
tion. However, it has much room for price eciency improvement
[
8
]. For example, when the European power market began to liber-
alise, dynamic pricing was one of the most remarkable illustrations
of its introduction. An ecient and deregulated European Union-
wide power market was the goal of this liberalisation, resulting in
the establishment of the European Electricity Exchange (EEX) [
15
].
The market’s ultimate purpose is to make better use of the resource
and, with the support of open competition, to be able to ensure
acceptable power pricing for customers [
30
]. With the help of the
EEX, the manufacturing sector has been oered a better way of
energy procurement. Through the EPEX SPOT (European Power
Exchange), the companies can settle on a day-ahead and intraday
electricity contracts [
16
]. That said, it is possible to purchase quan-
tities of electricity at prices specic for certain parts of the day,
resulting in a possibility of being protable due to prior knowledge
of the daily energy price uctuations. This feature benets many
industries, including the production/manufacturing industry [3].
Speaking of the manufacturing sector, the uncontrolled rapid
industrial growth accompanied by the intensive depletion of nonre-
newable and limited raw resources is a signicant global problem
[
11
].As the industry is expanding and demand for products is rising,
the pressure put on manufacturers to increase their output while
decreasing their carbon footprint is high [
11
]. In other words, it
is crucial to stabilise production while being ecient. In this con-
text, energy eciency means producing an identical output while
decreasing electricity consumption [27].
Approximately 42% of the world’s energy consumption is used
in manufacturing facilities [
7
]. Workshop production scheduling is
essential to achieve the manufacturing sector’s operational energy
eciency benets [
20
]. The severe market rivalry in the business
allows little opportunity for operational ineciencies since costs
and performance deterioration may rapidly grow. According to
[
11
], manufacturers’ energy usage signicantly reduces operating
costs. Consequently, energy consumption is at the core of man-
ufacturing and production planning. On top of energy eciency,
manufacturers must consider two other aspects so that production
falls within the scope of the Production Planning and Control (PPC)
triangle of objectives, namely low production costs, short lead time
in production and order processing, and high product quality.
A possible method of incorporating all three objectives on top of
dynamic electricity prices is the method of production scheduling. It
is possible to describe production scheduling as allocating available
resources across time to achieve specic goals [
14
]. Scheduling
problems often require a collection of activities to be completed.
The criteria may include trade-os between early and late order
completion, keeping inventory for the task, and frequent production
changeovers.
In this paper, we point out one of the main critical problems the
manufacturing industry faces nowadays – increased demand and
energy costs. We introduce real-time dynamic electricity prices in
production scheduling problems by extending the base scheduling
model developed by Blazewicz et al. [4].
2 LITERATURE REVIEW
2.1 Scheduling Model Problems
Ten publications related to our research topic and aims were anal-
ysed to understand state of the art. In this section, specically Table
1, the publications are compared over three dierent characteristics.
First, the number of machines used in each model is noted, which
can either be single (one machine) or multiple (more than one ma-
chine). Second, the dynamic price interval used in each model is
shown. As seen in Table 1, the most common one is the one-hour or
60 minutes interval, which is also used in the model of this research.
Lastly, the scheduling model is outlined, either a ow or job shop
model. As shown in Table 1, our research focuses on job shop sched-
uling with multiple machine environments and real-time dynamic
prices with 60 minutes intervals, reecting real-world settings in
the German manufacturing environment.
2.2 Mathematical Models and Solution
Approaches
Another comparison approach was developed further to analyse
the mathematical models and solution approaches of the related
works. First, it presents each research’s optimisation model and its
objective function, decision variables, and constraints. This infor-
mation explains how each model is composed and relates to our
model. Table 2 provides the comparison of the optimisation models
and solution approaches developed in the related works.
3 PROBLEM STATEMENT AND SCHEDULING
MODEL
The wide range of scheduling issues necessitates the creation of a
standard notation on which a categorisation system may be built
[
4
]. Furthermore, using this nomenclature of problem kinds would
substantially aid in presenting and discussing scheduling issues.
Consequently, in Section 3.1, the base scheduling model is outlined
according to the already published model by [
4
]. Moreover, since
this research tackles implementing real-time dynamic electricity
prices in the scheduling model, the mathematical model of schedul-
ing is extended, and additional constraints and a decision variable
are added. Figure 2 depicts the steps to formulate the production
scheduling model in this research.
3.1 Scheduling Problem Class
Blazewicz et al
.
created a schema for categorising scheduling prob-
lems. It is necessary to add several parameters to Blazewicz et al
.
’s
schema since it does not cover all of the problem scope addressed in
our research. The following notations describe scheduling problems
based on the machine environment, task and resources, and the
objective function.
𝛼={𝛼1, 𝛼2}(1)
𝛽={𝛽1, 𝛽2, 𝛽3, 𝛽 4, 𝛽5, 𝛽6, 𝛽7, 𝛽8}(2)
𝜒=𝑜𝑏 𝑗𝑒 𝑐𝑡𝑖 𝑣𝑒 𝑓 𝑢𝑛𝑐𝑡 𝑖𝑜𝑛 (3)
𝛼
denotes the machine environment and
𝛽
describes tasks and
resources. Based on the notation above and the scheduling problem
formulation and structure in [
31
], the scheduling problem in our
research is described in Table 3 and Table 4.
99

Integrating Real-Time Dynamic Electricity Price Forecast into Job Shop Production Scheduling Model ICIEA-EU 2023, January 09–11, 2023, Rome, Italy
Figure 1: The Impact of Renewable Energy Sources on the Price of Electricity [23]
Table 1: Scheduling Models Comparison
Author Year Number of Machines Dynamic Price Interval Shop Type
Dellnitz et al. [6] 2020 multiple 60 minutes ow shop
Emec et al. [10] 2013 single 60 minutes job shop
Gong et al. [13] 2015 single 60 minutes job shop
Kawaguchi and Fukuyama [18] 2017 multiple 10 minutes job shop
Schulz et al. [24] 2020 multiple 10 hours ow shop
Selmair et al. [25] 2016 multiple 60 minutes job shop
Sharma et al. [26] 2015 multiple 8 hours ow shop
Thornton et al. [29] 2017 multiple 8 hours job shop
Wichmann et al. [33] 2018 single 8 hours job shop
Yusta et al. [35] 2010 single 60 minutes job shop
Our research 2022 multiple 60 minutes job shop
Table 2: Mathematical Models and Solution Approaches Comparison
Author
Year
Opt.
Model
Objective Function Decision Variables
Dellnitz et al. [6]
2020 MIP
𝑚𝑖𝑛 4 equations several state, time
Emec et al. [10]
2013 MIP
𝑚𝑖𝑛 1 equation several state, time
Gong et al. [13]
2015 MILP
1 equation
several state, start time, end time, task sequence
Kawaguchi and Fukuyama [18]
2017 COP
2 objective functions N of Factory, Operation End, Interval
Schulz et al. [24]
2020 MIP
2 objective functions several state, time, speed
Selmair et al. [25]
2016
/ 1 objective function several state, start time
Sharma et al. [26]
2015
OR 1 objective function several state, start time
Thornton et al. [29]
2017
OR 2 objective functions several state, start time
Wichmann et al. [33]
2018
OR 1 objective function several state, start time
Yusta et al. [35]
2010
OR 2 objective functions several state, start time
Our research
2022 MILP
𝑚𝑖𝑛 1 equation start time end time
3.2 Dynamic Pricing Model and Forecast
To achieve the main aim of the research, a real-time dynamic pricing
model is incorporated into the mathematical model available in
Section 3.3.
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ICIEA-EU 2023, January 09–11, 2023, Rome, Italy Krstevski, Fatahi Valilai, and Wicaksono
Table 3: Machine Environment Based on Blazewicz’s Scheduling Problem Framework
Variable Meaning Possible Values Our Value Description
𝛼1Characteristic of shop oor ∅Single Processor 𝐽
𝑂Open shop model
𝐹Flow shop model
𝐽Job shop model
𝛼2Number of machines ∅Number of machines is variable 𝑘The number of machines/
workstations is predened
and greater than zero
𝑘∈NNumber of machines equals to 𝑘
𝑛𝑜 −𝑤𝑎𝑖𝑡 After nishing task on a
machine, the task on the next
machine is immediately started.
Table 4: Blazewicz’s Scheduling Framework for Tasks and Resources
Variable Meaning Possible Values Our Value Description
𝛽1Pre-emption of tasks ∅Pre-emption is not allowed. ∅No pre-emption allowed
𝑝𝑚𝑡𝑛 Pre-emption is allowed.
𝛽2Resource constraints ∅Resource constraint does not exist. ∅The model assumes
unlimited resources.𝑟𝑒𝑠 Resource constraint exists.
𝛽3Precedence constraints
of the task
∅The tasks are independent. 𝑝𝑟𝑒𝑐 A task has to be nished
before the next task
starts if both tasks
belong to the same order.
𝑟𝑒𝑠 Resource constraint exists.
𝑝𝑟𝑒𝑐 General precedence constraints
𝑢𝑎𝑛 Unconnected activity network
𝑡𝑟 𝑒𝑒 Tree precedence constraints
𝑐ℎ𝑎𝑖𝑛𝑠 Chain precedence constraints
𝛽4Ready times ∅All ready times are zero ∅No ready times
𝑟𝑗Ready times dier per task.
𝛽5Task processing time ∅Arbitrary processing time ∅The process time is arbitrary
depending on the machine,
task, and order.
𝑝𝑗=𝑝All tasks have process time 𝑝units.
𝑎≤𝑝𝑗≤𝑏All tasks have process time
between 𝑎and 𝑏
𝛽6Due dates of tasks ∅No due date is assumed,
but due dates may be dened.
between 𝑎and 𝑏
∅The model considers
the due dates of orders.
˜
𝑑Due dates are dened for tasks.
𝛽7The maximal number
of tasks for an order
in case 𝑎1=𝐽
∅The maximal number of tasks
for an order is arbitrary.
∅There is no xed
denition of number of tasks.
𝑛𝑗≤𝑘The number of tasks of each order
is less than or equals to 𝑘.
𝛽8Time buers
between machines
∅Buers between machines
are unlimited.
∅No buers between machines
are considered.
𝑛𝑜-𝑤𝑎𝑖𝑡 After nishing a task on a machine,
the task on the next machine
is immediately started.
The type of dynamic pricing that our research deals with is
the Real-Time Pricing model with an interval of 1 hour. This is
because the website (https://gruenstromindex.de/) from which the
GEI data is obtained provides data hourly. Consequently, this is
the optimal and most ecient way to incorporate dynamic pricing
in the mathematical model. The obtained values for the price are
numerical. Therefore, the price variable in the model is a
numerical
value. However, as seen in Figure 3 and Table 6, the price could
also be classied into three categories: grey, yellow, and green.
Therefore, it is also possible to add an additional
categorical
price variable.
Since the website computes an accurate power forecast for 36
hours, the time horizon of the power forecast in the mathematical
101

Integrating Real-Time Dynamic Electricity Price Forecast into Job Shop Production Scheduling Model ICIEA-EU 2023, January 09–11, 2023, Rome, Italy
Figure 2: Model Formulation Procedure
model is set to 36 hours. The short-term forecast used by the web-
site provides a more accurate output as short-term forecasts are
known to be more accurate than a relatively long-term forecast [
12
].
Therefore, the forecast data used for the scenario in this research
is highly reliable. Furthermore, forecasting renewable energy 48
hours in advance has become common nowadays [12].
The calculation of the actual electricity price is based on the
GEI. The GEI indicates the proportion of electricity coming from
renewable sources, e.g., wind, biomass, and solar. The higher GEI,
the higher proportion of renewable energy, and the lower the elec-
tricity price. An example of GEI throughout 24 hours is shown in
Figure 3.The data was acquired from https://gruenstromindex.de/
for Bremen, Germany, on May 11, 2022.
As discussed above, the higher the GEI, the lower the electricity
price. Consequently, looking at Figure 3, it is evident that the lowest
electricity price would be for the period within 16:00, with a GEI of
63.
The mathematical representation for the GEI in this research is
as follows:
𝐺𝑡=(𝑔𝑡+1, 𝑔𝑡+2, .. .,𝑔𝑡+36 )(4)
where
𝐺𝑡
represents a set of GEI with start time
𝑡
. Furthermore,
to achieve the main aim of the research, the electricity price, based
on the GEI, has to be incorporated into the mathematical model.
𝐸𝑡=(𝑒𝑡+1, 𝑒𝑡+2, . .., 𝑒𝑡+36)(5)
where
𝐸𝑡
is a set of green electricity prices that are derived from
the green electricity indices with the start time 𝑡.
3.3 Mathematical Model
The expanded mathematical model, which is based on the initial
model provided by [4], is based on the assumptions listed below.
•
To complete one order, several tasks that are carried out on
dierent machines are required;
•
The power consumption of each task is dened by the aver-
age power consumption of the task over the task duration;
•
The start time of one task is only possible at a specic time
𝑡, where 𝑡={1,2, ..., 36};
•
There is no xed order release time, which means that each
order’s rst task can be started at any time.
All of the variables of our model are portrayed in Table 5.
Table 5: Variables in the Mathematical Model
Variable Explanation
𝑖Orders, 𝑖={1,2, ...,𝑚}
𝐷𝑖Due Date of the order 𝑖
𝑗Machine, 𝑗={1,2, ..., 𝑛 }
𝑛Number of machines in the shop oor
𝑘Position of the task in the corresponding order 𝑖
𝑇𝑖𝑗 𝑘 Task at the position 𝑘of order 𝑖on machine 𝑗
𝑑𝑇𝑖𝑗
Duration of task
𝑇
belonging to order
𝑖
on machine
𝑗
𝑃𝑇𝑖𝑗
Power consumption of task
𝑇
of order
𝑖
at machine
𝑗
𝑃={𝑝𝑠, 𝑝𝑠+1, 𝑝𝑠+2, . .., 𝑝𝑠+𝑡, ..., 𝑝𝑠+𝑑}
𝑆𝑇𝑖𝑗𝑘
Start time of task
𝑇
at the position
𝑘
of order
𝑖
on
machine 𝑗
3.3.1 Decision Variables. The decision variables used in the model
is to decide whether task
𝑇𝑖𝑗
is running at
𝑡
. The binary variable
has two outcomes:
•
The task (
𝑇𝑖𝑗 𝑘
) does not run (0) if
𝑡
is less than the start
time (
𝑆𝑇𝑖𝑗𝑘
) of the task (
𝑇
)OR if
𝑡
is more than the end time
(𝑆𝑇𝑖 𝑗𝑘 +𝑑𝑇𝑖𝑗 ) of the task (𝑇);
•
The task (
𝑇𝑖𝑗 𝑘
) runs (1) if
𝑡
is in between the start time (
𝑆𝑇𝑖𝑗𝑘
)
and end time (𝑆𝑇𝑖 𝑗𝑘 +𝑑𝑇𝑖𝑗 ) of the task (𝑇).
The mathematical representation of the binary decision variable
model is represented as:
𝑟𝑇𝑖𝑗 𝑡=(0,if 𝑡<𝑆𝑇𝑖𝑗𝑘 or 𝑡>𝑆𝑇𝑖𝑗 𝑘 +𝑑𝑇𝑖𝑗
1,if 𝑆𝑇𝑖 𝑗𝑘 ≤𝑡≤𝑆𝑇𝑖𝑗𝑘 +𝑑𝑇𝑖𝑗
(6)
3.3.2 Objective Function. The objective function focuses on the 36
GEI periods available from the GEI website, the orders and machines.
The objective function of the model is to minimise the electricity
cost through three aspects/variables:
•how much electricity does the task (𝑇) consume (𝑃𝑇𝑖𝑗𝑘 );
•
whether the task (
𝑇
) is running at
𝑡
- the objective function
(𝑟𝑇𝑖 𝑗 𝑡), and
•the dynamic electricity price at 𝑡- (𝑒𝑡).
Minimise
36

𝑡
𝑚

𝑖
𝑛

𝑗
𝑃𝑇𝑖𝑗 ×𝑟𝑇𝑖𝑗 𝑡×𝑒𝑡(7)
3.3.3 Constraints. The start time of the task with the position
𝑘+
1
is greater or equal to the start time of the task with the position
𝑘
,
i.e., the previous task of the sequence, with its duration. The start
of task
𝑘+
1has to wait until task
𝑘
is nished. Task
𝑘
and
𝑘+
1
belong to the same order.
𝑆𝑇𝑖𝑗𝑘 +𝑑𝑇𝑖𝑗 ≤𝑆𝑇𝑖𝑗 𝑘+1(8)
Furthermore, it is essential that all tasks belonging to the same
order have to nish before or at the same time as the due date of
the order.
𝑆𝑇𝑖𝑗𝑘 +𝑑𝑇𝑖𝑗 ≤𝐷𝑖∀𝑖, 𝑗 (9)
Lastly, each machine has to run one task at a given time; thus, the
sum of running indicators for all machines has to be less than or
102

ICIEA-EU 2023, January 09–11, 2023, Rome, Italy Krstevski, Fatahi Valilai, and Wicaksono
Figure 3: 24-hour GEI Distribution Example (Source: gruenstromindex.de)
equal to the number of machines on the shop oor.
𝑛

𝑗=1
𝑟𝑇𝑖𝑗 𝑡≤𝑛∀𝑡(10)
4 SCENARIO EXAMPLE FOR USING THE
MODEL
4.1 GEI
One crucial addition to the base mathematical scheduling model
outlined by [
4
] is the introduction of dynamic electricity prices,
i.e., GEI. Therefore, collecting data related to renewable energy and
converting it to the price per kWh is essential.
Table 6 shows the collected data from https://gruenstromindex.
de/ valid for May 2, 2022. The data contains 36 periods (
𝑡
), the actual
GEI per period
𝑡
(
𝑔𝑡
) and the coloured category. There are, in total,
three categories to which the GEI could be assigned:
•
Grey: the amount of renewable energy produced is low; thus,
the GEI is lower and the price higher;
•
Yellow: the amount of renewable energy produced is mediocre,
and the price and GEI can vary;
•
Green: the amount of renewable energy produced is high;
thus, the GEI is higher and the price lower.
4.2 GEI to Electricity Price Conversion
The website from which the GEI data was acquired does not have a
GEI to price (in €) conversion. Therefore, this subsection focuses on
developing an approach for converting the given GEI to price per
kWh (in €). In 2021, at 14.84 cents per kWh, Cyprus had the highest
industrial electricity rates in the EU for users using between 20,000
and 70,000 MWh annually [
28
]. Germany had the highest electricity
price, at 18.13 cents per kWh, for annual use between 500 and 2,000
megawatts. The cost per kWh in Cyprus was 15.15 cents [28].
In order to convert the GEI into an actual price, two important
constants were created. The rst one is the highest price, which
was rounded up to 20 cents, and the second one is the average of all
the GEI available in Table 6, which approximates 53.83. Thereafter,
a formula was created in order to convert the GEI into an actual
price with the aim of getting a realistic output. The formula used
to calculate the price is as follows:
𝑃𝑟 𝑖𝑐𝑒𝑡=𝐶𝑜𝑛𝑠𝑡 .𝑚𝑎𝑥 + (𝐶𝑜𝑛𝑠𝑡.𝑎𝑣𝑔 −𝑔𝑡)×𝐶𝑜𝑛𝑠𝑡 .𝑚𝑎𝑥 ÷𝐶𝑜𝑛𝑠𝑡 .𝑎𝑣𝑔 (11)
The formula output produced realistic prices, which are available
in Table 7.
Table 6: GEI Data and 36-hour Forecast for May 2, 2022
𝑡 𝑔𝑡Category 𝑡 𝑔𝑡Category
1 48 Yellow 19 48 Yellow
2 52 Yellow 20 49 Yellow
3 56 Green 21 52 Yellow
4 58 Green 22 57 Green
5 60 Green 23 59 Green
6 61 Green 24 61 Green
7 58 Green 25 63 Green
8 55 Yellow 26 66 Green
9 51 Yellow 27 67 Green
10 47 Yellow 28 70 Green
11 44 Grey 29 70 Green
12 43 Grey 30 68 Green
13 43 Grey 31 65 Green
14 43 Grey 32 58 Green
15 43 Grey 33 52 Yellow
16 44 Grey 34 47 Yellow
17 46 Yellow 35 44 Grey
18 47 Yellow 36 43 Grey
Source: https://gruenstromindex.de/
It is evident from Table 7 that each of the categories falls within
a specic range of pricing points. The most expensive one, i.e., grey,
has a constant price of 24 cents. The prices in the yellow category
vary from 20 to 23 cents, while the green category has the lowest
price range from 14 to 19 cents. The outcome of the formula is
expected and realistic as the grey category has the lowest GEI while
the green one has the highest GEI.
4.3 Machine Characteristics
The example focuses on three machines with two dierent charac-
teristics to make the scenario somewhat realistic. The rst one is
related to the power consumption of each machine, which can be
low, medium or high. The latter one represents the variance which
can either be low or high. That said, the machines were classied
as follows:
•Machine 1: low power consumption with a high variance;
•
Machine 2: medium power consumption with a low variance;
•Machine 3: high power consumption with a high variance.
103

Integrating Real-Time Dynamic Electricity Price Forecast into Job Shop Production Scheduling Model ICIEA-EU 2023, January 09–11, 2023, Rome, Italy
Table 7: GEI to Price Calculation
𝑡 𝑔𝑡Category 𝑒𝑡𝑡 𝑔𝑡Category 𝑒𝑡
1 48 Yellow 22 19 48 Yellow 22
2 52 Yellow 21 20 49 Yellow 22
3 56 Green 19 21 52 Yellow 21
4 58 Green 18 22 57 Green 19
5 60 Green 18 23 59 Green 18
6 61 Green 17 24 61 Green 17
7 58 Green 18 25 63 Green 17
8 55 Yellow 20 26 66 Green 15
9 51 Yellow 21 27 67 Green 15
10 47 Yellow 23 28 70 Green 14
11 44 Grey 24 29 70 Green 14
12 43 Grey 24 30 68 Green 15
13 43 Grey 24 31 65 Green 16
14 43 Grey 24 32 58 Green 18
15 43 Grey 24 33 52 Yellow 21
16 44 Grey 24 34 47 Yellow 23
17 46 Yellow 23 35 44 Grey 24
18 47 Yellow 23 36 43 Grey 24
4.4 Tasks and Orders
The scenario deals with ve tasks with three tasks, i.e., one task
per machine. Furthermore, to increase complexity and make the
scenario more realistic, each task must be performed at a specic
machine in a specied sequence. Moreover, each task has dier-
ent energy consumption and duration. The detailed scenario is
portrayed in Table 8.
Table 8: Tasks Description
𝑖 𝑗 =1𝑗=2𝑗=3
1𝑘=1𝑘=2𝑘=3
𝑑𝑇11 = 3 𝑑𝑇12 = 1 𝑑𝑇13 = 2
𝑝𝑇11 = 4 𝑝𝑇12 = 2 𝑝𝑇13 = 2
2𝑘=2𝑘=1𝑘=3
𝑑𝑇21 = 2 𝑑𝑇22 = 4 𝑑𝑇23 = 3
𝑝𝑇21 = 3 𝑝𝑇22 = 2 𝑝𝑇23 = 1
3𝑘=2𝑘=3𝑘=1
𝑑𝑇31 = 4 𝑑𝑇32 = 1 𝑑𝑇33 = 1
𝑝𝑇31 = 3 𝑝𝑇32 = 2 𝑝𝑇33 = 1
4𝑘=1𝑘=2𝑘=3
𝑑𝑇41 = 3 𝑑𝑇42 = 1 𝑑𝑇43 = 2
𝑝𝑇41 = 4 𝑝𝑇42 = 2 𝑝𝑇43 = 2
5𝑘=3𝑘=2𝑘=1
𝑑𝑇51 = 3 𝑑𝑇52 = 4 𝑑𝑇53 = 3
𝑝𝑇51 = 5 𝑝𝑇52 = 2 𝑝𝑇53 = 2
Furthermore, on top of the details provided in Table 8, the sce-
nario was further expanded by adding two more essential char-
acteristics. The rst one is the order due date (
𝐷𝑖
), meaning that
each order (
𝑖
) has a predetermined/given due date that has to be
met in order to keep the production plan adherence high and the
customers satised. The second is the sequence in which the order
is assigned to each machine. This means that the orders do not
follow a sequence of being assigned to Machine 1, then Machine 2,
and, lastly, Machine 3. Instead, this model is a job shop, where the
orders might have a random machining sequence. The details of
these two characteristics are provided in Table 9.
Table 9: Due Date and Machine Sequence of Each Order
Order (𝑖) Due Date (𝐷𝑖) Machine Sequence
1 24 1 - 2 - 3
2 24 2 - 1 - 3
3 18 2 - 3 - 1
4 12 1 - 2 - 3
5 36 3 - 2 - 1
5 DISCUSSION
5.1 Extension of the Base Scheduling Model
Compared to the base scheduling mathematical model, the extended
model of this research mainly diers due to the introduction of
dynamic electricity price (
𝑒
) - real-time pricing - in the model. The
extended model follows a job shop scheduling model. Consequently,
the decision variable, objective functions, and model constraints are
appropriately adjusted to t the main aim of the model. Initially, a
new binary-coded decision variable (
𝑟𝑇𝑖𝑗 𝑡
) is introduced (Subsection
3.3.1). The decision variable aims to check whether the task (
𝑇
)
runs or does not run; the decision is made based on one task’s start
and end time (
𝑆𝑇𝑖𝑗𝑘 +𝑑𝑇𝑖𝑗
). Furthermore, a new objective function
(Subsection 3.3.2) is introduced with the aim of minimising the
electricity, i.e., production cost. The new objective function focuses
on three main variables:
•electricity consumption of the running task;
•the outcome of the binary decision variable;
•the dynamic electricity price of the task at a certain period.
As for the constraints, there are three main dierences (Subsec-
tion 3.3.3). First, if the tasks belong to the same order, task two
might start only after task one has been successfully done. That
said, task (
𝑘+
1) must wait for the task (
𝑘
) to nish before being
processed on a machine. Furthermore, the extended model focuses
on delivery reliability as much as on production plan adherence
because delivery reliability is a crucial factor for the success of any
manufacturing company [
22
]. Therefore, one of the constraints
states that the whole order (all three tasks) must be completed be-
fore a given due date (
𝐷
) that is specic to the order (
𝑖
). Finally, each
machine is only allowed to process one task per time. Therefore,
the sum of tasks running at a specic time should not exceed 3, i.e.,
the number of available machines on the shop oor. This constraint
prevents parallel task running, enabling the machines to focus on
one task at a time, thus ensuring quality [21].
104

ICIEA-EU 2023, January 09–11, 2023, Rome, Italy Krstevski, Fatahi Valilai, and Wicaksono
5.2 Limitations
This research, due to time constraints, has several limitations. Ini-
tially, the forecast for the GEI is limited to 36 hours, meaning that
it is only possible to work with data that is up to a day and a half
in the future. Even though a short-term forecast is more accurate
than a long-term forecast [
12
], its absolute correctness is of utmost
importance. Since the time frame is usually in hours, if there is a
mistake in the forecast, there is little to no time to act on it and
prevent further complications.
Realistically speaking, an unexpected machine breakdown at
any time. Therefore, planning for the unexpected, i.e., buer time,
is required. By using conventional project management procedures,
just 44% of projects are completed on schedule [
17
]. However, in
our model, it is assumed that all of the machines perform at their
best, without any breakdowns and with perfect production plan
adherence, which is another limitation of this research. Besides
buer time, setup time for each machine is another realistic aspect
that has to be considered. Modern industrial and service settings de-
mand that dependable goods and services be provided on schedule.
Therefore scheduling with setup periods or setup charges is critical
[
1
]. Scheduling activities depend heavily on the time and expenses
necessary to prepare the facility to undertake the activities. Since
our model does not tackle setup time, this is considered a limitation
of the extended mathematical scheduling model.
6 CONCLUSIONS
This research mathematically extends the base scheduling model
by introducing real-time dynamic electricity pricing. The pricing
model was dened to follow the real-time pricing technique, and
the power forecast was decided to be up to 36 hours in advance.
We expressed the extended model as a mixed integer linear pro-
gramming model and explained the decision variables, objective
function and constraints. Information on GEI and conversion of
GEI to price was given, and the logic behind the conversion was
explained. Furthermore, the number of machines was dened, and
the machine characteristics in the sample scenario were explained.
A sample scenario was created to prepare the model for further
improvement and testing. In our future work, we will extend the
model by adding setup time, buer time, and retooling, and in-
troducing a longer price forecast time horizon. We will validate
the model using real-world manufacturing settings in two German
small-medium enterprises.
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