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Solving Non-Quadratic Matrices In Assignment Problems With An Improved Version Of Vogel's Approximation Method

Authors:

Abstract

The efficient allocation of tasks to vehicles in a fleet of self-driving vehicles (SDV) becomes challenging for large-scale systems (e. g. more than hundred vehicles). Operations research provides different methods that can be applied to solve such assignment problems. Integer Linear Programming (ILP), the Hungarian Method (HM) or Vogel's Approximation Method (VAM) are frequently used in related literature (Paul 2018; Dinagar and Keerthivasan 2018; Nahar et al. 2018; Ahmed et al. 2016; Koruko˘ glu and Ballı 2011; Balakrishnan 1990). The underlying paper proposes an adapted version of VAM which reaches better solutions for non-quadratic matrices, namely Vogel's Approximation Method for non-quadratic Matrices (VAM-nq). Subsequently, VAM-nq is compared with ILP, HM and VAM by solving matrices of different sizes in computational experiments in order to determine the proximity to the optimal solution and the computation time. The experimental results demonstrated that both VAM and VAM-nq are five to ten times faster in computing results than HM and ILP across all tested matrix sizes. However, we proved that VAM is not able to generate optimal solutions in large quadratic matrices constantly (starting at approx. 15 × 15) or small non-quadratic matrices (starting at approx. 5 × 6). In fact, we show that VAM produces insufficient results especially for non-quadratic matrices. The result deviate further from the optimum if the matrix size increases. Our proposed VAM-nq is able to provide similar results as the original VAM for quadratic matrices, but delivers much better results in non-quadratic instances often reaching an optimum solution. This is especially important for practical use cases since quadratic matrices are rather rare.
SOLVING NON-QUADRATIC MATRICES IN
ASSIGNMENT PROBLEMS WITH AN
IMPROVED VERSION OF VOGEL’S
APPROXIMATION METHOD
Maximilian Selmair
Alexander Swinarew
BMW GROUP
80788 Munich, Germany
maximilian.selmair@bmw.de
Klaus-J¨
urgen Meier
University of Applied Sciences Munich
80335 Munich, Germany
klaus-juergen.meier@hm.edu
Yi Wang
University of Plymouth
PL4 8AA Plymouth, United Kingdom
yi.wang@plymouth.ac.uk
KeywordsVogel’s Approximation Method; Hungarian
Method; Generalised Assignment Problem
Abstract—The efficient allocation of tasks to vehicles in a
fleet of self-driving vehicles (SDV) becomes challenging for large-
scale systems (e. g. more than hundred vehicles). Operations
research provides different methods that can be applied to solve
such assignment problems. Integer Linear Programming (ILP),
the Hungarian Method (HM) or Vogel’s Approximation Method
(VAM) are frequently used in related literature (Paul 2018;
Dinagar and Keerthivasan 2018; Nahar et al. 2018; Ahmed et al.
2016; Koruko˘
glu and Ballı 2011; Balakrishnan 1990). The under-
lying paper proposes an adapted version of VAM which reaches
better solutions for non-quadratic matrices, namely Vogel’s
Approximation Method for non-quadratic Matrices (VAM-nq).
Subsequently, VAM-nq is compared with ILP, HM and VAM by
solving matrices of different sizes in computational experiments
in order to determine the proximity to the optimal solution and
the computation time. The experimental results demonstrated
that both VAM and VAM-nq are five to ten times faster in
computing results than HM and ILP across all tested matrix
sizes. However, we proved that VAM is not able to generate
optimal solutions in large quadratic matrices constantly (starting
at approx. 15 ×15) or small non-quadratic matrices (starting at
approx. 5×6). In fact, we show that VAM produces insufficient
results especially for non-quadratic matrices. The result deviate
further from the optimum if the matrix size increases. Our
proposed VAM-nq is able to provide similar results as the original
VAM for quadratic matrices, but delivers much better results
in non-quadratic instances often reaching an optimum solution.
This is especially important for practical use cases since quadratic
matrices are rather rare.
LIS T OF ABBREVIATIONS
GAP Generalised Assignment Problem
HM Hungarian Method
ILP Integer Linear Programming
KPI Key Performance Indicator
SDV Self-driving Vehicle
VAM Vogel’s Approximation Method
VAM-nq Vogel’s Approximation Method for non-quadratic
Matrices
I. INTRODUCTION
The transportation problem is an extensively studied topic
in operational research (D´
ıaz-Parra et al. 2014). The methods
for solving the mentioned problem aim to minimise the total
transportation cost while bringing goods from several supply
points (e. g. warehouses) to demand locations (e. g. customers).
In general, each transport origin features a fixed amount of
goods that can be distributed. Correspondingly, every point
of transport destination requires a certain amount of units
(Shore 1970). The underlying use case, where tasks have to
be assigned to self-driving vehicles (SDVs), differs in some
regards from the classical transportation problem. In our case,
each vehicle has a capacity restriction of one, i. e. a maximum
of one load carrier can be transported at a time. Furthermore,
each task corresponds to a demand of one. This basically
means that every task can only be allocated to one single
vehicle. Additionally, the amount of available vehicles does
rarely match the number of unassigned tasks in practice. Since
the size of the matrices depends on those two factors, non-
quadratic matrices (e. g. 10 ×50) are common. There are
different approaches that can be applied to solve this kind of
problem, e. g. ILP, HM and VAM. While ILP and HM manage
to always generate an optimal solution, VAM often fails to do
so. Furthermore, those methods vary greatly in the computa-
tional demand necessary to solve assignment problems. There
are two major reasons involved in the motivation for improving
the original VAM. For one, the authors wanted to keep the
great performance (computational time) of the original VAM.
Secondly, the insufficient results for non-quadratic matrices
should be improved significantly, i. e. reaching the optimum.
Following these considerations, an improved VAM version, as
proposed in this paper, was developed and compared with the
three established methods. The goal was to find a solution
that provides optimal or near-optimal results while at the same
requiring a small amount of resources (computing power).
Despite its age, the approximation method proposed by
those authors is still in use nowadays and is subject to recent
operations research as the contributions by Banik and Hasan
(2018), Ezekiel and Edeki (2018), Hlatk´
a et al. (2017), Ahmed
et al. (2016), and Gani and Baskaran (2014) show.
Already Shimshak et al. (1981) extended the original VAM
with rules that apply in case of ties, e. g. the same maximum
cost differences occur. Out of the four cases using either
individual rules or a combination of them, only one case
manages to generate slightly better solutions than the original
VAM in regards to costs. Furthermore, they used only small
matrices (5×5,5×10 and 10×10) which does not provide any
information on the results achieved in large-scale applications.
Goyal (1984) further tried to improve Shimshak’s approach
in case of unbalanced transportation problems, i. e. the total
supply does not correspond to the total demand. Again, only a
small 3×3matrix was used, thus, lacking any real informative
value. Balakrishnan (1990) realised this drawback and tested
Goyal’s approach with other not specified examples concluding
that it is not always better than Shimshak’s approach. He
in turn proposed an extended approach which was tested
in different scenarios and compared with those of the other
authors mentioned above leading to even better solutions.
In a more recent contribution, Koruko˘
glu and Ballı (2011)
improved the original VAM by constructing a total opportunity
cost matrix which they obtained through the addition of the
row and column opportunity cost matrix. A row opportunity
matrix is for example generated by subtracting the smallest
cost value in each row from all other values in the same row.
The column opportunity matrix is obtained in the same way.
Koruko˘
glu and Ballı (2011) further deviate from the classical
approach by selecting the three rows or columns with the
highest penalty costs instead of choosing only the highest one.
Out of those three, the cell with the lowest transportation cost
is consequently selected and used for resource allocation.
This paper is structured as follows: in the second chapter a
detailed description of VAM as well as a brief explanation of
the HM and the ILP are given. The third chapter features the
description of the proposed VAM-nq. Chapter four will provide
an overview of the experiments as well as the discussion of
the corresponding results. The last chapter contains a brief
conclusion to this paper.
II. ES TABLI SH ED SOLUTION ME TH OD S FOR THE
GENERALISED ASS IG NM EN T PRO BL EM
This chapter is intended to provide a description on es-
tablished solution methods for the Generalised Assignment
Problem. These are in particular the basic VAM as well as
the HM and the ILP. Since the use case at hand differs in
some areas from conventional examples (e.g. the vehicles can
only transport one load carrier at a time and have thus a supply
of one), these variations will be considered in the description
of VAM and the HM. The ILP approach will be adapted to
the underlying use case as well, i. e. an appropriate objective
function as well as necessary constraints will be formulated.
A. Vogel’s Approximation Method
The following description of VAM is based on the original
proposal by Reinfeld and Vogel (1958). VAM solves transport
matrices by repeating the steps as seen below until a feasible
solution is found. The cells of the matrices are filled with costs
cij associated with allocating a task to a vehicle. Those costs
occur when a vehicle brings goods from a point of origin ito
a destination j. Each source (origin) features a specific amount
of goods that can be allocated (supply). Correspondingly, each
sink (destination) usually requires a certain number of units
(demand). In order to carry out the allocation under these
circumstances, the following steps are necessary:
1) Calculate the difference between the smallest and the
second-smallest cell value for each row and each column.
2) Select the row or column which features the biggest
difference. If there is a tie, choose the row or column
containing the smallest cell value.
3) Choose the smallest cell value of the selected row or
column and allocate the corresponding task to a vehicle.
4) Eliminate the row and column that has been used for the
allocation.
5) Check if there are still vehicles and tasks left to allocate,
and repeat steps 1 - 4 in case that there are.
Apart from the later proposed adoption of VAM in this
paper, there are different authors that tried to improve or
change the classic VAM in order to achieve better results and
move closer to an optimal solution which can be achieved for
example by ILP or HM (Paul 2018; Dinagar and Keerthivasan
2018; Nahar et al. 2018; Ahmed et al. 2016; Koruko˘
glu and
Ballı 2011; Balakrishnan 1990; Goyal 1984; Shimshak et al.
1981).
An example for VAM can be found in Table I through Ta-
ble III. Here, the rows are represented by vehicles (Vi) and the
columns by tasks (Tj). The costs (cij ) are the corresponding
cell values. The row differences can be found in iwhile the
column differences are saved in j. Starting with the initial
matrix (Table I), it is evident that the biggest difference can be
found in the third row featuring the lowest value in the third
column (Table II). Accordingly, task 3 is assigned to vehicle 3.
After the allocation, the third row and column are eliminated
(Table III).
TABLE I. INITIAL MATR IX TO B E SO LVED B Y VAM
cij T1 T2 T3 T4 Δi
V1 200 100 400 50 50
V2 60 80 30 350 30
V3 210 300 70 150 80
V4 120 510 340 80 40
V5 70 80 40 400 30
Δj 10 0 10 30
TABLE II. MATR IX FE ATUR IN G TH E IDE NT IFI ED BI GG ES T DIFF ER EN CE
(80)
cij T1 T2 T3 T4 Δi
V1 200 100 400 50 50
V2 60 80 30 350 30
V3 210 300 70 150 80
V4 120 510 340 80 40
V5 70 80 40 400 30
Δj 10 0 10 30
TABLE III. MATR IX AF TE R EL IM INATI NG A SSI GN ED ROW A ND
COLUMN
cij T1 T2 T3 T4 Δi
V1 200 100 400 50 50
V2 60 80 30 350 30
V3 210 300 70 150 80
V4 120 510 340 80 40
V5 70 80 40 400 10
Δj 10 0 10 30
B. Integer Linear Programming
As already stated previously, ILP is able to find an optimal
solution for different scenarios, even large-scale problems.
Initially, one has to formulate an objective function as well
as applicable restrictions in order to receive correct results.
According to Osman (1995) and following the adoption of the
ILP to fit the use case at hand, the objective function reads as
follows:
min X
jJ
X
vV
djv ·cj v (1)
X
vV
djv = 1 jJ(2)
X
jJ
djv 1vV(3)
djv ∈ {0,1} ∀jJ, vV(4)
The goal of the objective function (1) is to minimise the
sum of all costs (cjv ) for all jobs J= 1, . . . , m and for all
vehicles V= 1, . . . , n which is the result of multiplying the
decision variable (djv ) with the corresponding costs which
arise when a job jis assigned to a vehicle v. The first constraint
(2) ensures that every job is assigned to a vehicle while the
second constraint (3) makes sure that each vehicle’s capacity
of 1 is not exceeded, i. e. each vehicle can execute a maximum
of one job at a time. The last constraint (4), which applies for
both jobs and vehicles, restricts the decision variable djv to
binary values.
C. Hungarian Method
The Hungarian Method was initially proposed by Kuhn
(1955) to solve the Generalised Assignment Problem (GAP).
Similar to the ILP, the HM is able to find an optimal solution
to said problem. The algorithm solves n×nmatrices (e. g.
10 ×10) by carrying out the following steps until an optimum
solution is found:
1) Find the minimum value in each column and subtract this
value from all other values in the corresponding column.
2) Find the minimum value in each row and subtract this
value from all other values in the corresponding row.
3) Draw lines through the columns and rows so that all
zero values of the matrix are covered by as few lines
as possible.
4) Check if the number of lines equals n. If it does, an
optimal allocation of the zero values is possible. If the
number of lines is smaller than n, an optimal allocation
is not yet feasible and step 5 has to be carried out.
5) Find the smallest value which is not covered by a line
and a) subtract this value from each not covered row and
b) add it to each covered column.
6) Return to step 3.
It has to be noted that in case of n×mmatrices (e. g.
10 ×40), an extension takes places to generate n×nmatrices
(e. g. 40 ×40) since the method only works with quadratic
matrices. The additional cells are filled with values that are of
the same size as the highest value of the original matrix. This
extension requires additional computing power since instead
of 400 cells (10 ×40), the algorithm has to consider 1600
cells (40 ×40). It is evident that this is a drawback when non-
quadratic matrices are to be solved. This is always the case
when more tasks than vehicles have to be considered or vice
versa.
D. Comparison ILP / VAM / HM
In order to compare the three methods, experiments have
been carried out with different quadratic and non-quadratic
matrices using an Intel Core i7-6820HQ 2.70 GHz featuring
32 GB RAM. Figure 1 shows clearly that ILP requires the
most computational time for quadratic matrices. Especially,
in large matrices the time it takes to finish the calculations
rises significantly. HM and VAM on the other hand do not
require a lot of time to finish calculating the matrices. In fact,
there is almost no difference between them up until 80 ×80
matrices where the HM starts to take longer than VAM. From
this point forward, the difference between HM and VAM grows
continuously with increasing matrix size. This might lead to
the conclusion that it is more sensible to use the HM since
it is able to produce optimal solutions while maintaining a
relatively low computation time. However, looking at Figure 2
shows that the computation time for HM increases significantly
if non-quadratic matrices are involved. This is due to the
fact that HM has to generate additional rows or columns to
produce quadratic matrices since it is not able to deal with
non-quadratic problem instances (see section II). VAM on the
other hand can deal with quadratic and non-quadratic matrices
regardless of their size in a relatively small amount of time
which shows VAMs great scalability.
III. IMP ROVED VAM FO R NO N-Q UADR ATIC M ATRICES
(VAM-NQ)
Prior experiments have shown, that the original VAM is
not able to produce optimal or at least near-optimal results for
non-quadratic matrices (see Figure 3). In fact, the results are
in some cases more than 100 % worse than the optimum. It
was determined that choosing the row or column featuring the
maximum difference from the smaller dimension leads to those
insufficient results. This means for example that if the matrix
contains more columns than rows, choosing a column with
the maximum difference (which is achieved by subtracting
cell values in the smaller dimension) might result in worse
outcomes. The same obviously applies vice versa if there are
more rows than columns. This can be explained with the fact
that the bigger dimension obviously features more values and
10 ×10
20 ×20
30 ×30
40 ×40
50 ×50
60 ×60
70 ×70
80 ×80
90 ×90
100 ×100
110 ×110
120 ×120
130 ×130
140 ×140
150 ×150
160 ×160
170 ×170
180 ×180
190 ×190
200 ×200
0
2,000
4,000
6,000
matrix size
mean computation time in µs
Origin ILP HM VAM
Fig. 1. Mean computational time for ILP (CPLEX-solver), HM and VAM
for quadratic matrices in microseconds (5.000 samples each)
50 ×50
50 ×75
50 ×100
50 ×125
50 ×150
50 ×175
50 ×200
50 ×225
50 ×250
50 ×275
50 ×300
50 ×325
50 ×350
50 ×375
50 ×400
50 ×425
50 ×450
50 ×475
50 ×500
50 ×525
0
500
1,000
1,500
2,000
matrix size
mean computation time in µs
Origin ILP HM VAM
Fig. 2. Mean computational time for ILP (CPLEX-solver), HM and VAM
for non-quadratic matrices in microseconds (5.000 samples each)
the chance is therefore higher to find a smaller cell value within
those. In order to mitigate the above stated disadvantage of
VAM, an improved version of VAM was developed.
Figure 3 shows that the results produced by VAM start to
deteriorate immediately if the matrix size is increased in only
one dimension, i. e. a non-quadratic matrix is created. It is
evident that while VAM is able to generate optimal solutions
in some cases, the cases where it fails are up to 200 % worse
than the optimum (see Figure 3). The deviations increase
continuously with increasing matrix size, even in rather small
instances. In case of 5×10 matrices for example, the results
can be twice as bad as the optimum value.
5×5
5×6
5×7
5×8
5×9
5×10
5×11
5×12
5×13
5×14
5×15
5×16
5×17
5×18
5×19
5×20
0 %
25 %
50 %
75 %
100 %
125 %
150 %
175 %
200 %
matrix size
deviation from optimum
Fig. 3. Deviation of VAM from the optimal solution with increasing matrix
size (5.000 samples each)
In general, there are two possible versions of non-quadratic
matrices. Either there are more columns than rows or more
rows than columns. The description below is based on the
first case when a matrix contains more columns than rows.
Accordingly, the rows and columns in the description have
to be switched when the second case occurs. VAM-nq solves
allocation matrices featuring more columns than rows by
carrying out the following steps:
1) Calculate the difference between the smallest and the
second-smallest cell value for each row.
2) Select the row featuring the biggest difference. If there is
a tie among rows, choose the row containing the smallest
cell value.
3) Determine the smallest cell value for the selected row and
allocate the corresponding task to a vehicle.
4) Eliminate the corresponding row and column that have
been used for the allocation.
5) Check if there are still vehicles and tasks left to allocate,
and repeat steps 1 - 4 in case that there are.
Upon comparison of the original and the adapted VAM,
it becomes evident that there are some variations and simpli-
fications. For one, VAM-nq considers only the rows in case
that there are more columns than rows (step 1). Accordingly,
only the biggest differences in the rows and the corresponding
smallest cell values are considered (step 2 and 3). Applying
those variations to the second case (more rows than columns)
would mean that only columns, their biggest differences and
smallest cell values are considered in steps 1 through 3. With
Table IV and Table V, the example of subsection II-A is
solved with both versions showing that the proposed VAM-nq
provides significant better results even in small non-quadratic
cases.
TABLE IV. SOLUTION OF THE ORIGINAL METHOD (VAM) WITH
OBJECTIVE OF 320
cij T1 T2 T3 T4
V1 200 100 400 50
V2 60 80 30 350
V3 210 300 70 150
V4 120 510 340 80
V5 70 80 40 400
TABLE V. SOLUTION OF VAM- NQ WITH OBJECTIVE OF 290
cij T1 T2 T3 T4
V1 200 100 400 50
V2 60 80 30 350
V3 210 300 70 150
V4 120 510 340 80
V5 70 80 40 400
IV. EXP ERIME NT S
In order to evaluate the performance of VAM-nq as well as
its ability to reach optimal solutions, experiments have been
carried out by using AnyLogic to generate matrices of different
sizes.
A. Design
The matrices have been randomly generated and randomly
filled with uniformly distributed costs ranging from 0 to 1.400.
Each matrix has been solved 5.000 times to provide meaningful
results. The following overview shows which matrices have
been used to generate and evaluate the corresponding key
performance indicators (KPIs):
Mean Deviation of VAM from the optimal solution as
seen in Figure 3: 15 non-quadratic 5×nmatrices with
n={6,...,20}
Mean Computation Times for VAM, HM and ILP as seen
in Figure 1 and Figure 2:
a) 20 quadratic matrices starting with 10 ×10 and rising
to 200 ×200 in steps of 10
b) 20 non-quadratic matrices starting with 50 ×50 and
rising to 50 ×525 in steps of 25
Mean Deviation of VAM and VAM-nq from the optimal
solution:
a) 50 non-quadratic 50 ×nmatrices with n=
{51,...,100}
b) 17 different mixed matrices (5×5,5×50,10 ×10,
10 ×20,10 ×30,10 ×40,20 ×20,10 ×60,20 ×60,
30×30,10 ×100,40 ×40,50×50,50×100,100 ×100,
100 ×200,100 ×300)
B. Results of the Experiments
As can be seen from Figure 4, both the original VAM and
VAM-nq are not always able to produce an optimal solution,
but are instead on average deviating from it. It is also evident
50 ×50
50 ×55
50 ×60
50 ×65
50 ×70
50 ×75
50 ×80
50 ×85
50 ×90
50 ×95
50 ×100
0 %
10 %
20 %
30 %
40 %
50 %
60 %
70 %
matrix size
mean deviation from optimum
Origin VAM VAM-nq
Fig. 4. Mean deviation of the original VAM and VAM-nq from the optimal
solution for non-quadratic matrices in percent (5.000 samples each)
that in non-quadratic instances (as seen in Figure 4) the
deviation gap between the original VAM and VAM-nq rises
continuously when the size of the non-quadratic matrix is
increased. While the deviation of the original VAM continuous
to grow, the deviation of VAM-nq approaches 0 %, i. e. an
optimal solution is generated more often. This shows clearly
that the proposed method is more suitable to deal with non-
quadratic instances than the original method. Figure 5 shows
the results of experiments performed by using the original
VAM and VAM-nq for different problem instances. In this
case, it is also evident that in non-quadratic instances the
original VAM produces results that are up to 300% worse
than the corresponding optimal solution. VAM-nq on the other
hand displays almost no deviation and manages on average
to generate an optimal solution in all non-quadratic cases.
However, it is also recognisable that in quadratic matrices the
original VAM is always slightly better than VAM-nq, but the
differences in those cases are negligible.
V. CONCLUSION
Experiments have shown that VAM is substantially faster in
calculating results than HM and CPLEX-solver (ILP) across all
matrix sizes. However, VAM is not able to generate optimum
solutions in large quadratic matrices (starting with approx.
15×15) or small non-quadratic matrices (starting with approx.
5×6). In fact, VAM produces insufficient results in those
cases and deviates greatly from the optimum. The proposed
adapted version of VAM, introduced as VAM-nq, is able
to provide slightly worse results than the original VAM for
quadratic instances, but delivers much better results in non-
quadratic instances reaching an optimum solution in most of
the cases. Based on those findings, the authors propose to
use an algorithm that includes both the original VAM and
the improved VAM-nq and which is able to switch between
those two according to the underlying situation. In case that
the underlying matrix is quadratic, the original VAM method
5×5
5×50
10 ×10
10 ×20
10 ×30
10 ×40
20 ×20
10 ×60
20 ×60
30 ×30
10 ×100
40 ×40
50 ×50
50 ×100
100 ×100
100 ×200
100 ×300
0 %
50 %
100 %
150 %
200 %
250 %
300 %
350 %
matrix size
deviation from optimum
Origin VAM VAM-nq
Fig. 5. Deviation of the original VAM and VAM-nq from the optimal solution
for different matrix sizes (5.000 samples each)
should be used by the algorithm. For non-quadratic problem
instances however, the algorithm should switch to the improved
VAM-nq.
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AUTHOR BIOGRAPHIES
Maximilian Selmair is doctoral student at the Univer-
sity of Plymouth. Recently employed at the SimPlan AG,
he was in charge of projects in the area of material flow
simulation. Currently he is working on his doctoral thesis
with a fellowship of the BMW GROUP. His email address is:
maximilian.selmair@bmw.de and his website can be found at
maximilian.selmair.de.
Alexander Swinarew is a masters graduate in logistics at
OTH Regensburg who worked at BMW on his masters thesis
during which the proposed method was developed. His email
address is: alexander.swinarew@gmail.com.
Prof. Dr. Klaus-J¨
urgen Meier holds the professor-ship for
production planning and logistic systems in the Department
of Engineering and Management at the University of Applied
Sciences Munich and he is the head of the Institute for Pro-
duction Management and Logistics (IPL). His e-mail address
is: klaus-juergen.meier@hm.edu.
Dr. Yi Wang is a lecturer in business decision making
in the Faculty of Business, University of Plymouth, UK. He
has special research interests in supply chain management,
logistics, operation management, culture management, infor-
mation systems, game theory, data analysis, semantics and
ontology analysis, and neuromarketing. His e-mail address is:
yi.wang@plymouth.ac.uk.
... It follows that internal logistics have to adapt efficiently to the increasing dynamics and complexity of production processes. The visionary concept of Logistics 4.0 is proposed to offer a wide range of solutions for this purpose [2,3]. The use of Autonomous Mobile Robots (AMRs), in particular, is gaining attention and has become the focal point of researchers across different industries [4][5][6][7]. ...
... For this purpose, all driving efforts are transferred to a matrix that constitutes the associated Assignment Problem (AP) (for more details see Section 4.1.4). The solution to the AP, referred to as Vogels Approximation Method for non-quadratic Matrices (VAM-nq), was developed by Selmair et al. [3], and is an extension of the original Vogels Approximation Method (VAM) introduced by Reinfeld et al. [19]. Unlike exact methods like the Hungarian Method, proposed by Kuhn [20], and Integer Linear Programming, VAM-nq approximates a solution for the AP. ...
... As a result, the quality of the solution is sufficient for most cases, but the algorithm computes solutions substantially faster than exact methods. For a detailed review, we refer the reader to Selmair et al. [3]. ...
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The allocation of tasks to Autonomous Mobile Robots in a production setting in combination with the most efficient parking and charging processes are the focus of this paper. This study presents a simulative evaluation of the theoretical allocation methods developed in Selmair and Maurer (2020) combined with either hard or dynamic availability rules to ascertain the most efficient parameters of an Autonomous Mobile Robot System. In order to quantify this efficiency, the following Key Performance Indicator (KPI) were considered: number of delayed orders, driven fleet metres and the percentage of available Autonomous Mobile Robot as determined by their state of charge. Additionally, as an alternative energy source, a fast-charging battery developed by Battery Streak Inc. was included in this study. The results show that, in comparison to a conventional and commonly used trivial strategy, our developed strategies provide superior results in terms of the relevant KPI.
... It follows that internal logistics have to adapt efficiently to the increasing dynamics and complexity of production processes. The visionary concept of Logistics 4.0 is proposed to offer a wide range of solutions for this [2] [3]. The use of AGVs, in particular, is gaining attention and is becoming the focal point of researchers across different industries [4]- [7]. ...
... For this purpose, all driving efforts are transferred to a matrix that constitutes the associated GAP (for more details see Section IV-A4). The solution to the GAP, referred to as Vogel's Approximation Method for non-quadratic Matrices (VAM-nq), was developed by Selmair et al. [3], and is an extension of the original Vogel's Approximation Method (VAM) introduced by Reinfeld et al. [19]. Unlike exact methods like the Hungarian Method, proposed by Kuhn [20], and Integer Linear Programming, VAM-nq approximates a solution for the GAP. ...
... As a result, the quality of the solution is sufficient for most cases, but the algorithm computes solutions substantially faster than exact methods. For a detailed review, we refer the reader to Selmair et al. [3]. ...
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This paper presents our work in progress for the development of an efficient charging & parking strategy. Our research aim is to develop a strategy that not only provides an efficient approach to charging AGV batteries, but also reduces traffic density in a highly utilised large-scale AGV system. Alongside the current state-of-the-art solution, three new allocation methods are introduced: Trivial+, Pearl Chain and a method based on the Generalised Assignment Problem (GAP). These four methods vary in their scope, in terms of number of vehicles considered, when calculating a decision for a specific vehicle. Furthermore, two types of availability rules for vehicles are introduced and evaluated. Their combination with the allocation methods lay the foundation for future research. All allocation methods and availability rules are explained in detail and this is followed by a summary of the expected outcomes.
... The first attempt at solving the AP dates back to 1955 with Kuhn's Hungarian Method. In the subsequent decades, numerous researchers proposed a multitude of methods and solutions as listed hereafter in their chronological order: (Kuhn, 1955;Munkres, 1957;Reinfeld and Vogel, 1958;Witzgall and Zahn, 1965;Edmonds, Johnson and Lockhart, 1969;Dinic and Kronrod, 1969;Hopcroft and Karp, 1973;Pulleyblank, 1973;Gabow, 1976;Lawler, 1976;Karzanov, 1976;Cunningham and Marsh, 1978;Micali and Vazirani, 1980;Burkard and Derigs, 1980;Kazakidis, 1980;Derigs, 1981;Shimshak, Kaslik and Barclay, 1981;Galil, Micali and Gabow, 1982;Minoux, 1982;Havel, Kuntz and Crippen, 1983;Gabow, 1985;Grötschel and Holland, 1985;Derigs, 1986;Derigs and Metz, 1986;Trick, 1987;Jonker and Volgenant, 1987;Derigs, 1988;Lessard, Rousseau and Minoux, 1989;Gabow, Galil and Spencer, 1989;Gabow, 1990;Gabow and Tarjan, 1991;Derigs and Metz, 1991;Gerngross, 1991;Applegate and Cook, 1993;Atamtürk, 1993;Miller and Pekny, 1995;Feder and Motwani, 1995;Cheriyan, Hagerup and Mehlhorn, 1996;Goldberg and Kennedy, 1997;Goldberg and Karzanov, 2004;Mucha and Sankowski, 2004;Harvey, 2006;Duan and Pettie, 2014;Cygan, Gabow and Sankowski, 2015;Gabow, 2017;Selmair, Meier and Wang, 2019) The analysis of the relevant literature has yielded a total of four methods that are frequently applied in research (Paul, 2018;Dinagar and Keerthivasan, 2018;Nahar, Rusyaman and Putri, 2018;Ahmed et al., 2016;Korukoglu and Ballı, 2011;Freling, Wagelmans and Paixão, 2001;Li, Mirchandani and Borenstein, 2007;Balakrishnan, 1990). These are Integer Linear Programming (ILP), the Hungarian Method (HM), Vogel's Approximation Method (VAM) and the Jonker Volgenant Castanon (JVC) algorithm. ...
... After the allocation, the third row and column are eliminated (Table I c). Selmair, Meier and Wang (2019) have shown that the original VAM is not always suitable when it comes to determining optimal or near-optimal results for non-quadratic matrices (see Figure 5 a). In fact, the calculated objective values are in some cases more than 100 % higher than the optimal objective value. ...
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Full-text available
The optimal allocation of transportation tasks to a fleet of vehicles, especially for large-scale systems of more than 20 Autonomous Mobile Robots (AMRs), remains a major challenge in logistics. Optimal in this context refers to two criteria: how close a result is to the best achievable objective value and the shortest possible computational time. Operations research has provided different methods that can be applied to solve this assignment problem. Our literature review has revealed six commonly applied methods to solve this problem. In this paper, we compared three optimal methods (Integer Linear Programming, Hungarian Method and the Jonker Volgenant Castanon algorithm) to three three heuristic methods (Greedy Search algorithm, Vogel's Approximation Method and Vogel's Approximation Method for non-quadratic Matrices). The latter group generally yield results faster, but were not developed to provide optimal results in terms of the optimal objective value. Every method was applied to 20.000 randomised samples of matrices, which differed in scale and configuration, in simulation experiments in order to determine the results' proximity to the optimal solution as well as their computational time. The simulation results demonstrate that all methods vary in their time needed to solve the assignment problem scenarios as well as in the respective quality of the solution. Based on these results yielded by computing quadratic and non-quadratic matrices of different scales, we have concluded that the Jonker Volgenant Castanon algorithm is deemed to be the best method for solving quadratic and non-quadratic assignment problems with optimal precision. However, if performance in terms of computational time is prioritised for large non-quadratic matrices (50 × 300 and larger), the Vogel's Approximation Method for non-quadratic Matrices generates faster approximated solutions.
... The first attempt at solving the AP dates back to 1955 with Kuhn's Hungarian Method. In the subsequent decades, numerous researchers proposed a multitude of methods and solutions as listed hereafter in their chronological order: (Kuhn, 1955;Munkres, 1957;Reinfeld and Vogel, 1958;Witzgall and Zahn, 1965;Edmonds, Johnson and Lockhart, 1969;Dinic and Kronrod, 1969;Hopcroft and Karp, 1973;Pulleyblank, 1973;Gabow, 1976;Lawler, 1976;Karzanov, 1976;Cunningham and Marsh, 1978;Micali and Vazirani, 1980;Burkard and Derigs, 1980;Kazakidis, 1980;Derigs, 1981;Shimshak, Kaslik and Barclay, 1981;Galil, Micali and Gabow, 1982;Minoux, 1982;Havel, Kuntz and Crippen, 1983;Gabow, 1985;Grötschel and Holland, 1985;Derigs, 1986;Derigs and Metz, 1986;Trick, 1987;Jonker and Volgenant, 1987;Derigs, 1988;Lessard, Rousseau and Minoux, 1989;Gabow, Galil and Spencer, 1989;Gabow, 1990;Gabow and Tarjan, 1991;Derigs and Metz, 1991;Gerngross, 1991;Applegate and Cook, 1993;Atamtürk, 1993;Miller and Pekny, 1995;Feder and Motwani, 1995;Cheriyan, Hagerup and Mehlhorn, 1996;Goldberg and Kennedy, 1997;Goldberg and Karzanov, 2004;Mucha and Sankowski, 2004;Harvey, 2006;Duan and Pettie, 2014;Cygan, Gabow and Sankowski, 2015;Gabow, 2017;Selmair, Meier and Wang, 2019) The analysis of the relevant literature has yielded a total of four methods that are frequently applied in research (Paul, 2018;Dinagar and Keerthivasan, 2018;Nahar, Rusyaman and Putri, 2018;Ahmed et al., 2016;Korukoglu and Ballı, 2011;Freling, Wagelmans and Paixão, 2001;Li, Mirchandani and Borenstein, 2007;Balakrishnan, 1990). These are Integer Linear Programming (ILP), the Hungarian Method (HM), Vogel's Approximation Method (VAM) and the Jonker Volgenant Castanon (JVC) algorithm. ...
... After the allocation, the third row and column are eliminated (Table I c). Selmair, Meier and Wang (2019) have shown that the original VAM is not always suitable when it comes to determining optimal or near-optimal results for non-quadratic matrices (see Figure 5 a). In fact, the calculated objective values are in some cases more than 100 % higher than the optimal objective value. ...
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Full-text available
The optimal allocation of transportation tasks to a fleet of vehicles, especially for large-scale systems of more than 20 Autonomous Mobile Robots (AMRs), remains a major challenge in logistics. Optimal in this context refers to two criteria: how close a result is to the best achievable objective value and the shortest possible computational time. Operations research has provided different methods that can be applied to solve this assignment problem. Our literature review has revealed six commonly applied methods to solve this problem. In this paper, we compared three optimal methods (Integer Linear Programming, Hungarian Method and the Jonker Volgenant Castanon algorithm) to three three heuristic methods (Greedy Search algorithm, Vogel's Approximation Method and Vogel's Approximation Method for non-quadratic Matrices). The latter group generally yield results faster, but were not developed to provide optimal results in terms of the optimal objective value. Every method was applied to 20.000 randomised samples of matrices, which differed in scale and configuration, in simulation experiments in order to determine the results' proximity to the optimal solution as well as their computational time. The simulation results demonstrate that all methods vary in their time needed to solve the assignment problem scenarios as well as in the respective quality of the solution. Based on these results yielded by computing quadratic and non-quadratic matrices of different scales, we have concluded that the Jonker Volgenant Castanon algorithm is deemed to be the best method for solving quadratic and non-quadratic assignment problems with optimal precision. However, if performance in terms of computational time is prioritised for large non-quadratic matrices (50 × 300 and larger), the Vogel's Approximation Method for non-quadratic Matrices generates faster approximated solutions.
... There are several methods that solve the GAP. Beside optimisation methods like Integer Linear Programming (ILP), there are, for example, algorithms like the Hungarian Method (HM) proposed by Kuhn (1955), Vogel's Approximation Method (VAM) proposed by Reinfeld and Vogel (1958) or Vogel's Approximation Method for non-quadratic Matrices (VAM-nq) proposed by Selmair et al. (2019). The last method was used in the simulation study of this paper due to its superior calculation performance and satisfactory results in terms of non-quadratic matrices. ...
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The efficient distribution of scarce resources has been a challenge in many different fields of research. This paper focuses on the area of operations research, more specifically, Automated Guided Vehicles intended for pick-up and delivery tasks. In time delivery in general and flexibility in particular are important KPIs for such systems. In order to meet in time requirements and maximising flexibility, three prioritisation methods embedded in a task allocation system for autonomous transport vehicles are introduced. A case study within the BMW Group aims to evaluate all three methods by means of simulation. The simulation results have revealed differences between the three methods regarding the quality of their solutions as well as their calculation performance. Here, the Flexible Prioritisation Window was found to be superior.
... There are several methods that solve the GAP. Beside optimisation methods like Integer Linear Programming (ILP), there are, for example, algorithms like the Hungarian Method (HM) proposed by Kuhn (1955), Vogel's Approximation Method (VAM) proposed by Reinfeld and Vogel (1958) or Vogel's Approximation Method for non-quadratic Matrices (VAM-nq) proposed by Selmair et al. (2019). The last method was used in the simulation study of this paper due to its superior calculation performance and satisfactory results in terms of non-quadratic matrices. ...
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The efficient distribution of scarce resources has been a challenge in many different fields of research. This paper focuses on the area of operations research, more specifically, Automated Guided Vehicles intended for pickup and delivery tasks. In time delivery in general and flexibility in particular are important KPIs for such systems. In order to meet in time requirements and maximising flexibility, three prioritisation methods embedded in a task allocation system for autonomous transport vehicles are introduced. A case study within the BMW Group aims to evaluate all three methods by means of simulation. The simulation results have revealed differences between the three methods regarding the quality of their solutions as well as their calculation performance. Here, the Flexible Prioritisation Window was found to be superior.
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