This document presents a solution algorithm for solving a multiobjective cutting stock problem in the aluminum industry where scrap is considered a fuzzy parameter. The problem involves casting molten aluminum into rods and cutting them into logs to meet customer demands while minimizing costs from inventory and scrap. The algorithm formulates the problem using fuzzy set concepts and models scrap as a fuzzy number. It then finds α-Pareto optimal solutions for different α-levels using a weighted objective function and nonlinear programming solved with branch-and-bound methods. An example demonstrates implementing the method.

1. ON THE SOLUTION OF MULTIOBJECTIVE CUTTING
STOCK PROBLEM IN THE ALUMINUM INDUSTRY
UNDER FUZZY ENVIRONMENT
OMAR M. SAAD
Department of Mathematics, Faculty of Science,
Helwan University, Ain Helwan, P.O.Box 11795,
Cairo, Egypt.
e-mail: omarsd55@hotmail.com

2. ABSTRACT
 In this paper a solution algorithm for solving
multiobjective cutting stock problem in the
aluminum industry under fuzzy environment
is proposed. It is considered that the scrap is
the fuzzy parameter. The concept of level set
together with the definition of this fuzzy
parameter and its membership function are
introduced. A practical example of the
method implementation for the solution
algorithm is presented.

3. INTRODUCTION
 In this paper, a factory production rate S in tons/
(day, week, month or year) of molten aluminum is
considered which casts as long cylindrical rods
and swan into logs according to customer's desire
to be used for extrusion purposes. These logs
differ in the dimensions and the amount of alloying
elements added to the aluminum. A continuous
flow of molten aluminum passes to holding
furnaces then the metal is cast into long cylindrical
rods and this casting produces T rods / unit time.

4.  The molten metal release into a number of
circular moulds of the same diameter, lying
on a casting table that is lowered to allow
more metal to enter until required depth is
reached. Simultaneous casting for many
rods is defined as "drop" and the required
depth is defined as the "drop length". The
butt ends of the rods are removed (treated
as scrap) and the remainder swan is cut into
logs.

5.  Meeting customer's demand leads to high
production rate that requires inventory space and
the sawing process leads to scrap. Also, the price
of aluminum and the competitiveness of the
industry require that the costs of inventory and
scrap have to be minimized. Throughout this
paper, it is assumed that the scrap is a fuzzy
parameter. All of these lead to a multiobjective
mixed-integer nonlinear programming cutting stock
problem which will be formulated in the following
section.

6.  According to our experiences, it is believed
that this problem has not been treated in
literature before.

7. PROBLEM FORMULATIN AND
THE SOLUTION CONCEPT
),
)
,
(
),...,
,
(
),
,
(
(
:
)
(
2
1 P
Z
F
P
Z
F
P
Z
F
F
Minimize
FMCS
n

inte ge r,
and
,
0
,
,
,...,
2
,
1
,
,...,
2
,
1










ij
i
i
j
j
ij
ij
j
i
ij
Z
X
m
j
d
P
Z
n
n
i
X
M
Z

8.  


ij j
j
j
ij
ij
i P
k
Z
r
P
Z
F
~
)
,
(
where
= the costs of the scrap and the inventory.
Zij : The number of rods of length swan into logs
of length
Pj: The number of logs of length produced in
excess of the demand for those logs,

9. :
~
ij
r i
L
,
j
l
The length left from cutting rods of length
into logs called scrap and it is assumed
to be fuzzy parameter,
:
j
k j
l
A measure of the value of a log of length
that goes to inventory,
:
M The number of rods produced per drop (fixed).

10. :
i
X The number of additional drops of length i
L
:
ij
n The number of logs, where ]
/
[ j
i
ij l
L
n 
and ij
j
ij
i c
l
n
L 
 , where:
:
ij
c is the value of the scrap aluminum when a rod of length
i
L is swan into logs of length ,
j
l ,
j
ij l
c 
:
j
d j
l
The reduced demand for logs
.

11. FUZZY CONCEPTS
 Fuzzy set theory has been developed for
solving problems in which descriptions of
activities and observations are imprecise,
vague and uncertain.
 The term "fuzzy" refers to the situation in
which there are no well-defined boundaries
of the set of activities or observations to
which the descriptions apply. A fuzzy set is a
class of objects with membership grades.

12.  A membership function, which assigns to each
object a grade of membership, is associated
with each fuzzy set. Usually the membership
grades are in [0, 1].
 When the grade of membership for an object in
a set is one, this object is absolutely in that
set; when the grade of membership is zero, the
object is absolutely not in that set. Borderline
cases are assigned numbers between zero and
one.

13. :
~
ij
r
i
L
j
l
* In the following, it is assumed that
the lengths left from cutting rods of length
into logs are fuzzy scrap and those parameters
are characterized by fuzzy numbers.
A fuzzy number is defined differently by many authors
and the most frequently used definition is the following
one.

14. )
(
~ ij
r
r
ij

Definition 1. (Fuzzy number) [3]
A real fuzzy number is a convex continuous fuzzy
, and is defined as:
~
ij
r
subset of the real line R whose membership function,
(1) A continuous mapping from R to the closed interval [0, 1],
(2) ],
,
(
0
)
( 1
~ q
r
r ij
ij
rij






(3) Strictly increasing on ],
,
[ 2
1 q
q
denoted by

15. (4) ],
,
[
1
)
( 3
2
~ q
q
r
r ij
ij
r ij




(5) Strictly decreasing on ],
,
[ 4
3 q
q
(6) ).
,
[
0
)
( 4
~ 



 q
r
r ij
ij
rij


16. Definition 2. (-Level set) [11]
 The - level set of the fuzzy numbers
is defined as the ordinary set for
which the degree of their membership
function exceeds the level
~
ij
r
)
(
~
ij
r
L
:
]
1
,
0
[



17. (

L )
~
ij
r  :
ij
r
 )
(
~ ij
r
r
ij
 ,



m
j
n
i ,...,
2
,
1
;
,...,
2
,
1 

* For a certain degree ,
]
1
,
0
[


),
)
,
(
),..,
,
(
),
,
(
( 2
1 P
Z
F
P
Z
F
P
Z
F
F
Minimize n

)
( MMINLCS


subject to

18. integer,
and
,
0
,
,
,...,
2
,
1
,
,...,
2
,
1

 



 


ij
i
i
j
j
ij
ij
j
i
ij
Z
X
m
j
d
P
Z
n
n
i
X
M
Z
where
,
)
,
(  


ij j
j
j
ij
ij
i P
k
Z
r
P
Z
F
and
).
(
~
ij
ij r
L
r 


19. * Problem (-MMINLCS) can be rewritten in
the following equivalent form as:
)
( MMINLCS


),
)
,
(
),..,
,
(
),
,
(
( 2
1 P
Z
F
P
Z
F
P
Z
F
F
Minimize n

subject to
integer,
and
,
0
,
,
,...,
2
,
1
,
,...,
2
,
1

 



 


ij
i
i
j
j
ij
ij
j
i
ij
Z
X
m
j
d
P
Z
n
n
i
X
M
Z

20. where
,
)
,
(  


ij j
j
j
ij
ij
i P
k
Z
r
P
Z
F
and
,
ij
ij
ij U
r
u 

such that ij
ij U
u ,
on the variables
are lower and upper bounds
ij
r , respectively.

21. Definition 3. (-Pareto-optimal solution) [11]
)
,
( *
*
j
ij P
Z 



),
,
( j
ij P
Z
)
(
~
ij
ij r
L
r 

A point is said to be an Pareto optimal
MMINLCS), if and only if there
such that:
Solution to problem (
Does not exist another
.
,...,
2
,
1
),
,
(
)
,
( *
*
n
s
P
Z
F
P
Z
F j
ij
s
j
ij
s 


22.  with strictly inequality holding for at least
one s, where the corresponding values of
parameters are called level
optimal parameters.
*
ij
r 


23.  To find an - Pareto optimal solution
to problem (-MMINLCS), a weighted
objective function is minimized by
multiplying each objective function in
problem (- MMINLCS) by a weight,
then adding them together, see [2].

24. * This leads to find a solution of the
following problem C (w):
to
subjet
P
Z
F
w
Minimize
w
C
n
s
s
s

1
),
,
(
:
)
(
integer,
and
,
0
,
,
,...,
2
,
1
,
,...,
2
,
1

 



 


ij
i
i
j
j
ij
ij
j
i
ij
Z
X
m
j
d
P
Z
n
n
i
X
M
Z

25. where
,
)
,
(  


ij j
j
j
ij
ij
i P
k
Z
r
P
Z
F
and
,
ij
ij
ij U
r
u 

provided that .
1
and
)
,...,
2
,
1
(
,
0
n
1
s
s 




w
n
s
ws
* It should be noted that problem C(w) above is a
mixed-integer nonlinear programming problem with
a single-objective function that can be solved using
LINGO software along with the branch-and-bound
method [14].

26. SOLUTION ALGORITHM
Step0.
Start with a degree .
0
*



Step1.
Determine the points (q1,q2,q3,q4) for the fuzzy parameters
~
ij
r in problem (FMCS) with the corresponding
)
(
~ ij
r
r
ij

assumptions (1)-(6) in Definition 1.
membership function satisfying

27. Step2.
Convert problem (FMCS) into the non-fuzzy
version of problem (-MMINLCS).
Step3.
Use the nonnegative weighted sum approach [2]
to formulate problem )
( *
w
C at certain
 


n
s
s
s w
w
w
1
*
*
.
1
,

28. Step4.
Find the -optimal solution of the problem
using the LINGO software along with the branch-and-
bound method [14].
)
( *
w
C
)
step
( *

 
 ]
1
,
0
[

Step5.
Set and go to step 1.
Step6.
Repeat again the above procedure until the interval [0, 1]
is fully exhausted. Then, stop.

29. PRACTICAL EXAMPLE
* Suppose a factory has an order ),
pieces
400
( 
j
d
where the rods are of the length 1,
i
for
)
5
( 
 m
Li
and swan into logs of length
).
70
,
55
,
50
( 3
2
1 cm
l
cm
l
cm
l 


where j
i
ij l
L
n /
 then .
7
,
9
,
10 13
12
11 

 n
n
n
Also, the inventory values are given as
.
600
,
500 2
1 
 k
k

30.  The number of rods produced from one drop
is (M = 20 rods). There is an additional drop
determined by (xi = 30) for i =1. It is assumed
that the constraint of the over production is
500 Pj 600.
 In order to minimize the scarp and the
inventory, the following multiobjective
mixed-integer nonlinear cutting stock
problem can be formulated as:

31. to
subject
P
Z
F
P
Z
F
P
Z
F
F
Minimize ),
)
,
(
),
,
(
),
,
(
( 3
2
1

integer,
and
,
0
,
,
,...,
2
,
1
,
,...,
2
,
1

 



 


ij
i
i
j
j
ij
ij
j
i
ij
Z
X
m
j
d
P
Z
n
n
i
X
M
Z
Where
.
)
,
(
,
)
,
(
,
)
,
(
3
3
13
13
~
3
2
2
12
12
~
2
1
1
11
11
~
1
P
k
Z
r
P
Z
f
P
k
Z
r
P
Z
f
P
k
Z
r
P
Z
f







32. * Assume that the membership function has
the following trapezoidal form:




























.
,
0
,
)
(
1
,
,
1
,
)
(
1
,
,
0
)
(
4
4
3
2
3
4
3
3
2
2
1
2
2
1
2
1
~
q
r
q
r
q
q
q
q
r
q
r
q
q
r
q
q
q
q
r
q
r
r
ij
ij
ij
ij
ij
ij
ij
ij
r ij


33. * Assume also that the fuzzy parameters are
given by the following fuzzy numbers shown below:
~
i j
r
q4
q3
q2
q1
20
15
10
5
35
30
20
10
10
7
5
2
11
~
r
12
~
r
13
~
r

34. • For a certain degree
(say), it is easy to find:
,
36
.
0
*



.
4
.
9
6
.
2
,
34
12
,
19
6 13
12
11 




 r
r
r
•Therefore, the non-fuzzy multiobjective cutting
stock problem can be written in the following form:

35. ,
600
),
;
;
(
13
12
11
3
3
13
13
2
2
12
12
1
1
11
11







Z
Z
Z
to
subject
P
k
Z
r
P
k
Z
r
P
k
Z
r
F
Minimize
,
100
10
,
200
20
,
300
15
2
12
11
12
12
11





Z
Z
Z
Z
Z
.
4
.
9
6
.
2
,
34
12
,
19
6
13
12
11






r
r
r

36. * Using the weighting method [2] and setting
,
3
/
1
13
12
11 

 w
w
w
then the cutting stock problem with a single-objective
function will take the form:
),
600
(
3
/
1
)
550
(
3
/
1
)
500
(
3
/
1 3
13
13
2
12
12
1
11
11 P
Z
r
P
Z
r
P
Z
r
F
Minimize 





subject to
,
600
13
12
11 

 Z
Z
Z

37. ,
100
10
,
200
20
,
300
15
2
12
11
12
12
11





Z
Z
Z
Z
Z
,
400
)
7
(
)
9
(
)
10
( 3
13
2
12
1
11 




 P
Z
P
Z
P
Z
.
4
.
9
6
.
2
,
34
12
,
19
6
13
12
11






r
r
r
.
600
500
,
600
500
,
600
500
3
2
1






P
P
P

38.  The above mixed-integer nonlinear
programming problem can be solved using
the LINGO software along with the branch-
and-bound method [14] to obtain the
following -Pareto mixed-integer optimal
solution:
5
.
276075
with
,
500
,
500
,
502
,
246
,
10
,
9
,
4
.
7
,
28
,
14
3
2
1
13
12
11
13
12
11










F
P
P
P
Z
Z
Z
r
r
r

39.  It should be noted that a systematic
variation of the degree will yield
another -Pareto optimal solution.
]
1
,
0
[



40. CONCLUSIONS
 In our opinion, many aspects and general
questions remain to be studied and explored
in the area of multiobjective cutting stock
problem in the aluminum industry. There
are, however, several unsolved problems
should be discussed in the future. Some of
these problems are:

41.  An algorithm is required for treating
multiobjective cutting stock problem in
the aluminum industry with fuzzy
parameters in the resources (the right-
hand side of the constraints).
 An algorithm is needed for dealing with
multiobjective cutting stock problem in
the aluminum industry with fuzzy
parameters in the objective functions
and in the resources.

42.  It is required to continue research work
in the area of large-scale multiobjective
cutting stock problem in the aluminum
industry under fuzzy environment.
 A parametric study on multiobjective
cutting stock problem in the aluminum
industry should be carried out for
different values of level sets of the fuzzy
parameters.

43. ACKNOWLEDGMENT
 The author is deeply grateful to Prof. H.
A. El-Hofy, Production Engineering
Department, Faculty of Engineering,
Alexandria University, Egypt for
reviewing the paper, useful discussions,
and valuable comments

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Packing", European Journal of Operational Research
84 (1995) 503-505.
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Holland Series in Systems Science and Engineering
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(1980).
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46.  [9] Haessler, R. W. and Vonderembse, M. A.," A
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