Journal of Applied Mathematics and Physics, 2014, 2, 14-20
Published Online January 2014 (http://www.scirp.org/journal/jamp)
http://dx.doi.org/10.4236/jamp.2014.21003
OPEN ACCESS JAMP
Mathematical Model for the Homogenization
of Unit Load Formation
Béla Illés, Gabriella Bognár
Faculty of Mechanical Engineering and Informatics, University of Miskolc, Miskolc, Hungary
Email: a ltilles@uni-miskolc.hu, v.bognar.gabriella@uni-miskolc.hu
Received September 2013
ABSTRACT
One of the most important issues in storage and transport processes is the formation of unit loads. Our main goal
is to investigate the homogenization of unit load formation cases. We provide a model involving the major factors
and parameters for the optimal selection of the unit load formations. Objective functions and constraints related
to the basic tasks are formulated. We give a method for the selection of the optimal unit load formation equip-
ment for a given number of products under given constraints.
KEYWORDS
Unit Load; Unit Load Formation Device; Homogeneous Unit Load; Branch and Bound Method
1. Introduction
In the development of transport, storage and distribution design within supply chain, great emphasis is put on the
planning of low task consuming material flows. The fast and efficient way of handling and storing components,
raw materials, semi-finished and finished products plays a significant role in this process. Therefore, one of the
most important issues in storage and transport processes is the formation of unit loads. It is designed for simpli-
fying the cargo and storage operations as well as for reducing their frequency.
The most common areas applicable on any area of materials handling are the materials handling inside the
plant, the materials handling between plants, the in-plant storage, the outside transportation, the commercial
storage and the distribution systems [1-6,8].
According to the design of the unit load formation devices the most important devices are
standardized pallets,
columnar pallets,
box pallets,
platform pallets (wood, metal, wire mesh),
roller pallets,
shock absorbent pallets,
disposable pallets,
skids,
storage baskets,
tote pans,
containers
The two main tasks during the unit load formation are [7]
1. to choose the proper unit load formation equipment to the goods,
2. to determine the way of loading goods into the unit load formation equipment
The following conditions must be satisfied during the selection of the equipment
1. the goods must fit into the unit load formation equipment,
2. the weight of the goods cannot exceed the carrying capacity of the unit load formation equipment,
3. the position of the goods must be fixed inside the equipment,
4. the unit load formation equipment must fit into the storage areas, loading devices and transporting vehicles
B. ILLÉS, G. BOGNÁR
OPEN ACCESS JAMP
15
The unit load types can be classified as
1) Homogeneous (only one type of goods placed in the unit load formation equipment)
2) Non-homogeneous (or mixed when several kinds of goods placed in the equipment)
We review the main factors which appear in the homogenization method of the unit load formation. An opti-
mization process is introduced for the selection of the most appropriate equipments under different constraints.
The method is exhibited through the solution to a given example.
2. Homogenization Process
Definition: The method of choosing optimal number of unit load formation equipment types subject to given
number of goods and maximized number of unit load formation equipments is called the homogenization of the
unit load formation equipments. We review four different cases depending on the constraints.
In our investigations the following indices will be applied:
product identifier: i = 1, ..., m
equipment model number: ν = 1, ..., r
2.1. No Constraint on the Type of Unit Load Formation Equipments
Let us define the loading matrix A by
A
1
1
i
r
a
i
m
ν
ν



=





, (3 )
where the number of goods
0
l
is not greater than the type number of unit load formation equipments
r
, i.e.,
0
lr
.
In this case, we search for the maximal item of each row in the loading matrix and we choose the unit load
formation equipment belonging to the column of goods with maximal item.
2.2. Constraint on the Unit Load Formation Equipment Types
Let us suppose that
0
lr<
.
1) If
0
ml
, i.e., the product type number is not greater than the equipment type number, then the optimal
unit load formation equipment type can be given by choosing a maximal per line item.
2) If
0
ml>
, i.e., the product type number is greater than the equipment type number, then
11
rm
iii
i
Kk a xmax!
ννν
ν
= =
= =
∑∑
,
where
1
i
x
ν
=
, if the
ν
-th equipment is chosen for the i-th product, otherwise
0
i
x
ν
=
, and
is the weight
factor of the i-th product for the
ν
-th equipment. A further condition of the model is that for each product ex-
actly one equipment has to be chosen;
1
1
r
ii
xx
ν
ν
=
= =
. Let
1
m
i
sgn x
νν
ν
β
=

=

and
1
ν
β
=
if the
ν
-th equipment is chosen. Therefore the upper bound condition for the unit load formation
equipment is
00
1
r
l
ν
ν
ββ
=
= ≤
.
B. ILLÉS, G. BOGNÁR
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16
Summarizing the problem is formulated as:
{ }
01
i
x,
1
1
r
ii
xx
ν
ν
=
= =
0
11
rm
i
sgn xl
ν
νν
= =



∑∑
11
rm
iii
i
Kk a xmax!
ννν
ν
= =
= =
∑∑
This problem can be solved by Branch and Bound Method or dynamic programming.
2.3. Constraint on the Number of Unit Load Formation Equipments by Type
If more type of equipments can be applied for a product then let us denote by
i
M
the quantity of
i
-th product
placed into the equipment. The following conditions has to be met
1) each product has to be placed such that no space is left on either equipment
1
r
ii i
ax M
νν
ν
=
=
,
2) the number of equipment is the bound:
1
m
i
xc
νν
ν
=
,
3) the product numbers are integers:
i
x Integer
ν
=
The objective function can be formulated as
11
rm
iii
i
Kk a xmax!
ννν
ν
= =
= =
∑∑
If there is no solution, the first strong condition can be substituted by a less rigorous one
1
r
ii i
ax M
νν
ν
=
.
If only one type of equipment can be applied for a product then let us define matrix B
B
1
1
i
r
b
i
m
ν
ν



=





,
which gives the equipment demand for the average amount of the product. Hence
05
i
ii
M
b Entier.
z
ν
ν

= +


,
where
i
z
ν
denotes the quantity of goods placed on the equipment. The conditions are the following
1) for each product only one equipment can be chosen,
2)
1
i
x
ν
=
if the
ν
-th equipment is chosen for the
i
-th product, otherwise
0
i
x
ν
=
,
1
1
r
ii
xx
ν
ν
=
= =
,
{ }
01
i
x,
.
B. ILLÉS, G. BOGNÁR
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17
3) The equipment number is the bound
1
m
ii
yxb c
ννν ν
ν
=
= ≤
.
The objective function can be given by
11
rm
ii i
i
Kk bxmax!
νν
ν
= =
= =
∑∑
,
where the weight factor
i
k
can be chosen as an average quantity:
ii
kM=
.
The model can be solved by linear programming.
2.4. Lower Bound on the Number of Equipment Purchases
Only one equipment can be selected for a product:
1)
1
i
x
ν
=
if the
ν
-th equipment is chosen otherwise
0
i
x
ν
=
;
1
1
r
ii
xx
ν
ν
=
= =
,
{ }
01
i
x,
2) the bound on the number of equipment
1
m
ii
yxb d
ννν ν
ν
=
= ≥
The objective function is
11
rm
ii
i
Kkbxmin!
ννν
ν
= =
= =
∑∑
where
k
ν
denotes the cost of the equipment. The model is solvable by linear programming.
3. The Branch and Bound Method
The selection of the unit load equipment is made by product [3].
1) If the number of product types
m
is less than the number of the unit load equipments
p
then max
lm
=.
2) If
mp>
then max
lm
=,
3) If the types of the allowed unit load equipments is
0
l
, then
0max
ll=
:
a) when
0
ll
, there is no need to homogenization,
b) when
0
ll>
then the asset diversity must be reduced. The weight factor applied for the selection of the
device for the
ν
-th equipment and i-th product is
.
In the weighting factor
0
ii
ii
ux
tux
ν
ν
=
,
denotes the relative frequency of the i-th product, where
01
m
i
i
uu
=
=
and
1
ii
uu
ν
δ
δ
=
=
is the relative fre-
quency of the
ν
-th quantity. Let us form the matrices
X
1
1
ik
k
x
i
m
ν



=





,
KO
1
1
ii
r
kb
i
m
νν
ν



=





.
B. ILLÉS, G. BOGNÁR
OPEN ACCESS JAMP
18
The objective function is
11
pm
ii i
i
KMk bxmax!
νν
ν
= =
= =
∑∑
and
1
i
x
ν
=
if the i-th product is selected for the
ν
-th equipment, otherwise
0
i
x
ν
=
,
{ }
0, 1
i
x
1
1
r
ii
xx
ν
ν
=
= =
.
For the selection of the initial value of
0
l
we form the sums of the matrix column KO
1
m
ii
kb
ν νν
ν
β
=
=
,
then we form the equipments in descending order according to
ν
β
.
We introduce a column order investigation. Let
mp>
. It is advisable to take more objective functions into
account. We form the efficiency matrix
A
1
1
ij
jm
a
i
n



=





,
where
{ }
ij ij
amax f
µ
µ
=
denotes the quantity of the i-th product applicable to the j-th equipment with the
µ
-th
loading method. The optimal selection is achieved if we find the maximal element in the efficiency matrix
{ }
i ij
j
smax a=
. If we chose the most efficient equipment to each product then the upper bound for the efficiency
is
01
m
i
i
ss
=
=
. In order to review the optimal variants we form the inefficiency matrix
B
1
1
ij
jm
b
i
n



=





,
with elements
iji ij
b sa= −
. Let us reduce matrix B. We take the least element in each column
{ }
i ij
j
dmin b=
. If
0
j
d>
, then the column can be omitted (as these equipments are not optimal for none of the products). If
0
j
d=
then the j-th column is kept (i.e., it is optimal for at least one product)
B’
1
1
ij
jk
b
i
n



=





,
km
,
and
k
denotes the number of column left. Moreover, let
p
is the number of applicable equipments. If
kp
then the obtained equipment number is allowed, and the applied number is
k
. If
kp<
, then the equipment
numbers have to be reduced. Let us form the sum of the columns:
1
n
j ij
i
cb
=
=
. We sort in order the obtained
B. ILLÉS, G. BOGNÁR
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19
values and take the
p
δ
+
column, where
( )
p'
p'
cc
δ
ε
+−<
for
δ
and
ε
is an appropriate small value;
( )()
12 1
'' p'p '
cc cc
δ
++
≤ ≤≤≤≤
. On the base of this order we form a possible
p
column combination from
the
p
δ
+
-th column of A. Let the
r
-th combination of A is
A’(r)
1
1
ij
j pp
a'(r )
i
n



=





,
1rw≤≤
,
1, ,rw=
. We form the maximal value of the rows:
( )( )
{ }
i ij
j
s'r maxa'r=
and determine the resultant effi-
ciency:
( )( )
01
m
i
i
s'rs'r
=
=
. The optimum version of all the possible variants corresponds to the greatest effi-
ciency:
()() () ()
{ }
00000
1,2,,,,
r
s''maxs's's' rs' w=
.
4. Example
Let us select the most appropriate 4 unit load formation equipment from 5 different equipments for 5 different
products.
Then
5,5, 4nmp== =
. The efficiency matrix and row maximums can be calculated as
A =
12345
1 592149
2 424399
3 358578
4 266256
5 734117








and
5
01
9986739
i
i
ss
=
= =++++=
. Let us form the inefficiency matrix by
iji ij
b sa= −
:
B =
12345
1 59214
2 42439
3 35857
4 26625
5 73411








00030
That gives the values of
{ }
i ij
j
dmin b=
:
[ ]
00030
, i.e., the third column can be neglected as it is posi-
tive. The reduced B matrix is
B’ =
12345
1 4075
2 5750
3 5301
4 4001
5 0436








.
B. ILLÉS, G. BOGNÁR
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20
As
kp=
, we found the optimal solution. The optimal selection of the equipments for the given problem can
be summarized as follows:
Product Equipment
1
2
2
5
3
3
4
2-3
5
1
Acknowledgements
The research work presented in this paper is based on the results achieved within the TÁMOP-4.2.1.B-
10/2/KONV-2010-0001 project and carried out as part of the TÁMOP-4.1.1.C-12/1/KON V -2012-0002 “Coop-
eration between higher education, research institutes and automotive industryproject in the framework of the
New Széchenyi Plan. The realization of this project is supported by the Hungarian Government, by the Euro-
pean Union, and co-financed by the European Social Fund.
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