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8/22/2019 One-dimensional Cutting Stock Problem _1d-Csp_ With Second Order Sustainable
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International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-
6367(Print), ISSN 0976 6375(Online) Volume 4, Issue 4, July-August (2013), IAEME
136
ONE-DIMENSIONAL CUTTING STOCK PROBLEM (1D-CSP) WITH
SECOND ORDER SUSTAINABLE TRIM: A COMPARATIVE STUDY WITH
FIRST ORDER SUSTAINABLE TRIM
P. L. Powar1, Vinit Jain
2, Manish Saraf
3, Ravi Vishwakarma
4
1Dept. of Math. & Comp. Sc., R. D. University, Jabalpur 482001, India
2KEC Int. Company, Panagar, Jabalpur, 482001, India
3HCET, Dumna Airport Road, Jabalpur, 482001, India
4Dept. of Math. & Comp. Sc., R. D. University, Jabalpur 482001, India
ABSTRACT
A method for solving one-dimensional cutting stock problem (1D-CSP) with first order
sustainable trim has been studied extensively by many researchers of Economics, Computer Science
and Mathematics. The authors have already defined the first order sustainable trim and in this paperby using the second order weighted means of order lengths and demand, a second order sustainable
trim has been defined. The cutting plan consists of cutting of at most two order lengths at a time out
of the required set of n order lengths , , , from a given set of m stock lengths , , , which resolves the problem of space constraint as well as minimization of men power significantly.
The main objective of this paper is to study the impact of two different definitions of first and
second order sustainable trims on total trim loss for the cutting of same set of data with respect to
same pattern of cutting.
Keywords: First order sustainable trim, Second order sustainable trim, 1D-CSP, Non-negative
integral valued (NIV) linear combination.
AMS (2000) subject classification: 90C90; 90C27; 90C10.
1. INTRODUCTIONThe One-Dimensional stock materials input is a very important criterion in industrial cutting
operations. Several cutting plans (cf. [1], [2]) have been designed to obtain required set of pieces
from the available stock lengths. The fundamental aim is to minimize the quantity of used stock
material or to minimize the wastage. The combination of assortment problem and the trim loss
problem is known as the cutting stock problem (CSP).
INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
TECHNOLOGY (IJCET)ISSN 0976 6367(Print)ISSN 0976 6375(Online)
Volume 4, Issue 4, July-August (2013), pp. 136-149 IAEME:www.iaeme.com/ijcet.aspJournal Impact Factor (2013): 6.1302 (Calculated by GISI)www.jifactor.com
IJCET
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The analytic method of optimization proposed by Gilmore and Gomory in 1960s has turned
non-practicable due to sufficiently large number of possible arrangement that render the solution
impossible and of no use because of its non-integral solutions (cf. [3]-[6]). Thus, instead of using the
analytic methods to obtain the ideal solution, heuristic approaches with acceptable approximation
have gained popularity (see [7], [8], [9], [10]). By using the principles introduced by Dikili [11],
Dikili et al [11] developed a method to solve 1D-CSP which completely removes the complexity ofGilmore and Gomory method.
Using genetic approaches with and without contiguity, Hinterding and khan [12] have studied
1D-CSP. Wagner [13] have studied 1D bundled CSP with contiguity in the lumber industry. In the
classical CSP, one wants to minimize the number of stock items used while satisfying the demand of
smaller sized items. However, the number of patterns/set ups to be performed on the cutting machine
is ignored. In cutting stock problem, with setup cost (CSP-S), considering cost factors for the
material and the number of set ups, the total production cost has been minimized in [14].
The main objective of present paper is to minimize the production cost by reducing the area
of working and men power. The cutting plan considered in this paper is already proposed in [8]
which consist of cutting of at most two order lengths at a time out of the required set of n order
lengths
, , . . . , from a given set of m stock lengths
, , , .This plan resolves the problem
of sorting sufficiently large number of order lengths (approximately more than one thousand) after
each stage of cutting and keeping them in the form of heaps till the entire process of cutting is over.
Our study is based mostly on the problems of transmission tower manufacturing industry.
To control the scrap or the trim loss is one of the basic factor for the sustainability of any
industry dealing with cutting of smaller lengths from the given large stock lengths. Powar et al [8]
have resolved this problem upto some extent by designing the cutting plan which works under the
pre-defined sustainable trim of order one. The mathematical model introduced in [8] involved the
classification of data and some recurrence relations. It is quite clear that the computation of total trim
loss is data dependent and the sustainable trim of order one defined in [8] is also data dependent
which works nicely for some specific set of data.
In the present paper, we have defined a sustainable trim of order two by considering the
second order weighted means of order length and demand. The impact of these two definitions viz.sustainable trim of order one and two has been explored widely on certain sets of data. It has been
noticed that the second order definition of sustainable trim is more effective in some cases to
minimize the total trim. Observations and conclusion cover the most important part of this work from
practical point of view.
2. NOTATIONS AND PRE-REQUISITESAll stock lengths and order lengths, we consider as integers throughout our analysis.
According to the requirement, the lengths can be converted into integers by multiplying them
by 10 1, integer. We use the following notations: Block of integers 0,1,, (index set), means can be any number from the set
0,1,2,, . Order lengths 0 , 1 , 2 , , arranged in ascending order with respect to length and 0 byconvention.
Required number of pieces of order length , 0. Stock length 1, 2, , arranged in ascending order with respect to length.
It has been noticed that in particular, in the transmission tower designing industry that most
of the required number of order lengths i.e. s are integral multiple of each other. In view of thisobservation, we classify the order lengths in the following two categories in accordance with their
required number of pieces:
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Category I: (C-I) We collect all those order lengths whose required number of pieces are integral
multiple of each others.
Category II: (C-II) It is the collection of all those order lengths whose required number of pieces
are prime numbers (their common multiple is 1).
3. SUSTAINABLE TRIM (ORDER ONE AND TWO)In this section, we give the definition of sustainable trim of order one described in [8] and
propose a new definition of sustainable trim of order two.
3.1Sustainable trim of order one ( )In order to cut the linear combination (say) of the two order lengths and from the
given stock lengths , , , , we have to decide upto what extent, we allow the raw material toconvert into the scrape. Throughout our cutting process (excluding the last step where it is possible
only that few piece of some order length are left to cut), we follow the restriction that 0 , 1,2,, and is the sustainable trim of order one and defined as follows:
We next define
| |( 1 , 2 , , and is an appropriate positive integer 1, for which is minimum)
where , , , are the stock lengths. We finally define
(3.1)which is the desired sustainable trim of order one.
3.1 Remark
Analytically, it has been noticed that the average value covers the acceptable over all originalvalues. Hence, we have taken the weighted mean of total required lengths.
3.2Sustainable trim of order two ( )Following the same restriction as for and using the notations from section 2, we define
ld ld
d d d
By convention, 0, , , ,
We next define the second order weighted means
1.
Consider
| |, 1,2, , .is an appropriate positive integers 1, for which is minimum. , , , are stock lengths.
3.2 Remark
is the average order length which is assumed to be cut from the given stock length 1 , , . The integer denotes the number of pieces of average stock length to be cut from
the stock length .
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We finally define
(3.2)which is the sustainable trim of order two viz. .
4. MATHEMATICAL FORMULATION OF THE PROBLEMWe first consider C-I and define the following ratios:
(4.1)(where is a positive integer , , )
Note: It is not necessary to consider always the largest common factor between and . Any otherfactor (if exists) may be selected according to the length of stock to minimize the trim.
In view of (4.1), define the following set:
, , 0, integer , , (4.2)
We are now in a position to define the sets 1, 2, , as follows:
: 0 , , , , , 1,2 (4.3)where is defined by (3.1) and (3.2) respectively for 1 and 2.
At this stage, we may come across with the following situations:
, , in this case, all the order lengths have to shift in C-II. In view of the definition of , the sets may or may not cover all order
lengths belonging to Category-I.
In view of above observations and the definition of the sets , we redefine our categories Iand II as follows:
Category-I (C-I) Let , , , order lengths have been covered by the sets 1,2 , . . , . For convenience, we denote these order lengths by , , , arranged in ascendingorder with respect to the length.
4.1RemarkThere may exist some order lengths and (say) such that and ofcourse are multiple
of each others but the length of combination exceeds the largest stock length or exceeds the sustainable trim loss. We shift all such order lengths to Category-II and finally, we
assume that the order lengths , , , have been covered by Category-I.Category-II (C-II) The remaining all order lengths denoted by , , , arranged inascending order with respect to the length.
4.2 Remark
(i) The real numbers 1, 2 defined by 3.1 and 3.2 play a crucial role in thecomputation of total trim loss. It is natural to expect that the trim loss can be minimized by
considering the minimum value lying between 0 and , but it has been experienced practicallyin the industries that by increasing the value of , the impact on the total trim loss results in asignificantly acceptable range in some particular cases. But we are strict to only.
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(ii) In order to implement the algorithm smoothly, the data of more than one tower (preferably ofsame pattern) may be clubbed.
Now consider Category-II and order lengths , , , with the required number of pieces, , , respectively. For , , , define:
(4.4)
(4.5)0 say 1, 2
for at least one value of k (k=1,2,,m). The number has been chosen in such a way that attains a minimum value lying between 0 and .
Similarly, choose a number satisfying the following condition: (4.6)
(4.7)Proceeding this way, we finally define
, (4.8)
, (4.9)The process would be continued till either 0 or 0 and in view of (4.4) - (4.9), we
have
, (4.10)
, (4.11)
, , , , (4.12)where or , or for or respectively. Also , , are positive integers, may beselected according to the length of stock in order to minimize the trim.
Referring relation (4.10)-(4.12), we now define the set , , : , , , , where , , , , according to
repectively, 1, 2, , . , 0, 1, , (4.13)
Define |c| ma xc , where c a or b for fixed and arbitrary (4.14)
In view of relation (4.13), we now define
, , : 0 | | 1, 2 (4.15)
1, 2, , .5. CUTTING PLAN
It has been noticed practically that with the preference of starting from the largest order
lengths to the smaller ones, the cutting process has been executed in general as the smaller order
lengths left behind can be adjusted easily amongst them and results in less trim loss (see Figure 7.1).
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Cutting of the largest order length from category-IReferring relation (4.14), we consider | |. In view of [cf. relation (4.10)], there exist sets
, , , 1 , , , containing | | along with some other s. Corresponding toeach set , , , respective fixed stock lengths , , , have been assigned. We select thecombination corresponding to the smallest stock length and focus our attention on it for thefirst step of cutting.
Let (say) for ~ where
satisfying the condition:
(5.1)In view of (5.1), it may be noted that by cutting bars of stock length , total number of
required pieces of order lengths and are cut.Define
1, 2 (5.2)
Cutting of other stock lengths from the setFor , , , we next consider the largest order length (say) contained in and consider| | for corresponding to the stock length satisfying the condition:
for some ~,, Referring relation (5.1), it is clear that by cutting bars of the stock length , total
number of required pieces of order lengths and have been cut. Define | | 1, 2 (5.3)
Proceeding this way, for , , , , , we consider the next largest order length out of theremaining once and applying the same technique as before, the trim loss with respect to
corresponding stock lengths
s has been computed. The process is continued till all order lengths
belonging to category I are totally exhausted.
(5.4)If this cutting process covers all the order lengths , , , , then STOP. Cutting of the largest order length from Category-II
Referring definition of [cf. relation (4.15)], we first set 1 and consider
for fixed
and arbitrary and select
as follows:
| | max ~
Such that | | for some .Now, corresponding to | |, there exists sets , , , associated with the stock lengths
, , , respectively containing
| |. We select the set
corresponding to the smallest stock
length . In view of the relation (4.15), we have
for ~ .
It is clear from relations (4.10) and (4.11), that by cutting bars of stock length , we cut
. pieces of order length and . pieces of order length . Our aim is to finish cutting ofonly two order lengths first and (fixed) at a time. Following cases may arise:Case1. Either . or or both the inequalities hold together.Case 2. Either . or .
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5.1RemarkHere two cases will not hold together because in that case and will belong to Category-I.We first deal with the case 1. In view of the relation (4.13), we next consider
( fixed as given by (4.13) )
Now, corresponding to
, there exist sets
,
,
, containing it. The sets
,
,
, are associated with the stock length , , , respectively. We select the set (say)corresponding to the smallest stock length . It is clear from relations (4.10) and (4.11) that bycutting bars of stock length , we cut . more pieces of order length and . morepieces of order length . We continue the process till either or .
Let if possible holds, then would be of the form
where we express , , , ,.Referring the relation (4.13), we now consider
,
Now, corresponding to , there exists sets , , , containing it. The sets
, , , associated with the stock lengths , , , respectively. We select the set corresponding to the smallest stock length (say). It is clear from relations (4.10) and (4.11)
that by cutting . , pieces of order length . Now , pieces of order length are left tocut out of . We now consider , for all 1 , 2 , , and select the minimumdifference corresponding to the stock length (say) , all pieces of order length have been cut.
5.2RemarkAt this last step of cutting ,. may exceed the sustainable trim .
We now compute the trim loss corresponding to the order lengths and belonging to theCategory-II.
, Order lengths and belonging to category-II have been cut completely. Remaining order
lengths we again arrange in increasing order , , , (say). We first consider| | max ~
~such that | | for some .Proceeding in a similar manner, we get
,
We continue the process till all order lengths are exhausted and get
Finally, we get total trim
.The percentages of trim lose with respect to and have been computed in accordance
with the total stock length used.
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6. DESIGN OF ALGORITHMS
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7. NUMERICAL EXAMPLES7.1Example
Consider the following data for our analysis
S.No. Order lengths
(in cm.)
Required no. of
pieces
S.No. Order lengths (in
cm.)
Required no. of
pieces
1. 801 03 6. 498 16
2. 748 24 7. 492 39
3. 733 46 8. 471 21
4. 641 23 9. 327 40
5. 548 39 10. 303 32
Table 7.1
Available stock lengthsS.No. Stock lengths (in cm.) S.No. Stock lengths (in cm.) s
. . . .
1. 2110 4. 3883
2. 2210 5. 4177
3. 3120 6. 4239
Table 7.2
Cutting Plan by using second order sustainable trim S.No. Order lengths
(in cm)
Pieces to cut Trim loss (in cm.) Used Stock lengths
(in cm.)
Category-I
1. 492548
33
0 1 3 0 31201340560
2. 303498
42
2 8 1 6 2210817680 3. 327
74853
4 8 3 2 3883831064 4. 641
73312
3 2 3 6 9 21102348530
Category-II
5. 471 9 0 2 0 423928478 6. 471801
3
3 67 1 67 3883 1 3883
Total 184 150195
Total Trim loss (%) 0.1225%Table 7.3
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Cutting Plan by using first order sustainable trim ::::
S.No. Order lengths
(in cm)
Pieces to cut Trim loss (in cm.) Used Stock lengths
(in cm.)
Category-I
1. 492548
33
0 13 0 3120 1 405602. 303
498 4
2 2 8 16 2210 8 17680
3. 327748
53
4 8 32 3883 8 31064
4. 641733
12
3 23 69 2110 2 48530
Category-II
5. 471 9 0 2 0 4239 2 84786.
471
801 1
237
1
37 2110
1
2110
7. 471801
21
367 1 367 2210 1 2110Total 521 150532
Total Trim loss (%) 0.3461%Table 7.4
7.2Figure and Screen shots of programming
Figure 7.1
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Figure 7.2
Figure 7.3
7.3ConclusionReferring tables 7.3 and 7.4, it may be noted that for this particular set of data is reducing
the total trim significantly in comparison with the trim obtained by using .
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7.4ExampleConsider the following data for our analysis
S.No. Order lengths
(in cm.)
Required no. of
pieces
S.No. Order lengths
(in cm.)
Required no. of
pieces
1. 801 03 6. 498 162. 748 24 7. 492 39
3. 733 46 8. 471 21
4. 641 23 9. 327 40
5. 548 39 10. 303 32
Table 7.5
Available stock lengths
S.No. Stock lengths (in cm.) S.No. Stock lengths (in cm.) ssss . . . .
1. 2110 4. 3883
2. 2210 5. 4170
3. 3120 --- ---Table 7.6
7.5ConclusionIt may be verified that in example 7.4, the trim loss obtained by using is exceeding the
value of trim loss obtained by .
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