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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

Keywords—Vogel’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

j∈J

X

v∈V

djv ·cj v (1)

X

v∈V

djv = 1 ∀j∈J(2)

X

j∈J

djv ≤1∀v∈V(3)

djv ∈ {0,1} ∀j∈J, ∀v∈V(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.