IJCET Volume 4 Issue 4 explores 1D cutting stock problem

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 Jain2
, Manish Saraf3
, Ravi Vishwakarma4
1
Dept. of Math. & Comp. Sc., R. D. University, Jabalpur 482001, India
2
KEC Int. Company, Panagar, Jabalpur, 482001, India
3
HCET, Dumna Airport Road, Jabalpur, 482001, India
4
Dept. 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 paper
by 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 ݈ଵ , ݈ଶ, … , ݈௡ from a given set of m stock lengths ܷଵ, ܷଶ, … , ܷ௠
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. INTRODUCTION
The 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.asp
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IJCET
© I A E M E

137
The analytic method of optimization proposed by Gilmore and Gomory in 1960’s 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 of
Gilmore 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 ݈ଵ, ݈ଶ, . . . , ݈௡ from a given set of m stock lengths ܷଵ, ܷଶ, … , ܷ௠. 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-REQUISITES
All 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௡
ሺ݊ ൒ 1, integerሻ. We use the following notations:
‫ܤ‬ሺ݊ሻ െ Block of integers 0,1, … , ݊ (index set), ݆ ‫א‬ ‫ܤ‬ሺ݊ሻ means ݆ can be any number from the set
ሼ0, 1, 2, … , ݊ሽ.
݈௜ െ Order lengths ݅ ൌ 0, 1, 2, … , ݊ arranged in ascending order with respect to length and ݈଴ ൌ 0 by
convention.
݀௜ െ Required number of pieces of order length ݈௜, ݀଴ ൌ 0.
ܷ௝ െ Stock length ሺ݆ ൌ 1, 2, … , ݉ሻ 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 this
observation, 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.
3.1 Sustainable trim of order one (࢚࢙
૚
)
In order to cut the linear combination ܽ௜௝ (say) of the two order lengths ݈௜ and ݈௝ from the
given stock lengths ܷଵ, ܷଶ, … , ܷ௠, we have to decide upto what extent, we allow the raw material to
convert 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 ൑ ܷ௞ െ
ܽ௜௝ ൑ ‫ݐ‬௦
ଵ
, ݇ ൌ 1,2, … , ݉ and ‫ݐ‬௦
ଵ
is the sustainable trim of order one and defined as follows:
‫ܮ‬ ൌ
݈଴݀଴ ൅ ݈ଵ݀ଵ ൅ ‫ڮ‬ ൅ ݈௡݀௡
݀଴ ൅ ݀ଵ ൅ ‫ڮ‬ ൅ ݀௡
ൌ
∑ ݈௜݀௜
௡
௜ୀ଴
∑ ݀௝
௡
௝ୀ଴
We next define
‫ܮ‬௞ ൌ |ܷ௞ െ ݅‫|ܮ‬
(݇ ൌ 1, 2, … , ݉ and ݅ is an appropriate positive integer ൒ 1, for which ‫ܮ‬௞ is minimum)
where ܷଵ, ܷଶ, … , ܷ௠ are the stock lengths. We finally define
‫ݐ‬௦
ଵ
ൌ
∑ ௅ೖ
೘
ೖసభ
௠
(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 original
values. Hence, we have taken the weighted mean of total required lengths.
3.2 Sustainable trim of order two (࢚࢙
૛
)
Following the same restriction as for ‫ݐ‬௦
ଵ
and using the notations from section 2, we define
ܵ௡ ൌ
l଴d଴ ൅ ‫ڮ‬ ൅ l୬d୬
d଴ ൅ dଵ ൅ ‫ڮ‬ ൅ d୬
ൌ
∑ ݈௜݀௜
௡
௜ୀ଴
∑ ݀௝
௡
௝ୀ଴
By convention, ܵ଴ ൌ 0, ܵଵ ൌ ݈ଵ, ܵଶ ൌ
௟బௗబା௟భௗభା௟మௗమ
ௗబାௗభାௗమ
, … , ܵ௡ ൌ
∑ ௟೔ௗ೔
೙
೔సబ
∑ ௗೕ
೙
ೕసబ
We next define the second order weighted means
ܵ௡
ଵ
ൌ
ܵ଴ ൅ ܵଵ ൅ ‫ڮ‬ ൅ ܵ௡
݊ ൅ 1
.
Consider
‫ܮ‬௞ ൌ |ܷ௞ െ ݅ ܵ௡
ଵ|, ݇ ൌ 1,2, … , ݉.
݅ is an appropriate positive integers ൒ 1, for which ‫ܮ‬௞ is minimum. ܷଵ, ܷଶ , … , ܷ௠ are stock lengths.
3.2 Remark
ܵ௡
ଵ
is the average order length which is assumed to be cut from the given stock length
ܷ௝ሺ݆ ൌ 1, … , ݉ሻ. The integer ݅ denotes the number of pieces of average stock length to be cut from
the stock length ܷ௞.

139
We finally define
‫ݐ‬௦
ଶ
ൌ
∑ ௅ೖ
೙
ೖసభ
௠
(3.2)
which is the sustainable trim of order two viz. ‫ݐ‬௦
ଶ
.
4. MATHEMATICAL FORMULATION OF THE PROBLEM
We first consider C-I and define the following ratios:
ௗ೔
ௗೕ
ൌ
ఈ೔ൈ௥೔ೕ
ఉೕൈ௥೔ೕ
(4.1)
(where ‫ݎ‬௜௝ is a positive integer ݅, ݆ ‫א‬ ‫ܤ‬ሺ݊ሻ, ݅ ൐ ݆ )
Note: It is not necessary to consider always the largest common factor between ݀௜ and ݀௝. Any other
factor ‫ݎ‬௜ (if exists) may be selected according to the length of stock to minimize the trim.
In view of (4.1), define the following set:
‫ܣ‬ ൌ ሼܽ௜௝ ൌ ߙ௜݈௜ ൅ ߚ௝݈௝ ‫׷‬ ܽ௜௝ ൑ ܷ௠, ߙ௜, ߚ௝ ൒ 0, integer ൫݅ ൏ ݆, ݅, ݆ ‫א‬ ‫ܤ‬ሺ݊ሻ൯ሽ (4.2)
We are now in a position to define the sets ‫ܣ‬௞ ‫ك‬ ‫ܣ‬ ሺ݇ ൌ 1, 2, … , ݉ሻ as follows:
‫ܣ‬௞ ൌ ሼܽ௜௝: 0 ൑ ܷ௞ െ ܽ௜௝ ൑ ‫ݐ‬௦
ఒ
, ݇ ‫א‬ ‫ܤ‬ሺ݉ሻ, ݅, ݆ ‫א‬ ‫ܤ‬ሺ݊ሻ, ܽ௜௝ ‫א‬ ‫,ܣ‬ ߣ ൌ 1,2ሽ (4.3)
where ‫ݐ‬௦
ఒ
is defined by (3.1) and (3.2) respectively for ߣ ൌ 1 and λ ൌ 2.
At this stage, we may come across with the following situations:
• ‫ܣ‬௞ ൌ ߶, ‫׊‬ ݇ ‫א‬ ‫ܤ‬ሺ݉ሻ, in this case, all the order lengths have to shift in C-II.
• In view of the definition of ‫ݐ‬௦
ఒ
, the sets ‫ܣ‬௞ ሺ݇ ‫א‬ ‫ܤ‬ሺ݉ሻሻ may or may not cover all order
lengths belonging to Category-I.
In view of above observations and the definition of the sets ‫ܣ‬௞, we redefine our categories I
and II as follows:
Category-I (C-I) Let ݈ఈଵ
, ݈ఈమ
, … , ݈ఈ௣
order lengths have been covered by the sets ‫ܣ‬௞ ሺ݇ ൌ
1, 2, . . , ݉ሻ. For convenience, we denote these order lengths by ݈ଵ, ݈ଶ, … , ݈௣ arranged in ascending
order with respect to the length.
4.1 Remark
There 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 ሺܷ௝ െ ܽ௜௝ሻ
exceeds the sustainable trim loss. We shift all such order lengths to Category-II and finally, we
assume that the order lengths ݈ଵ, ݈ଶ, … , ݈௣ have been covered by Category-I.
Category-II (C-II) The remaining all order lengths ܰ ൌ ݊ െ ‫݌‬ denoted by ݈ଵ, ݈ଶ, … , ݈ே arranged in
ascending order with respect to the length.
4.2 Remark
(i) The real numbers ‫ݐ‬௦
ఒ
ሺߣ ൌ 1, 2ሻ defined by ሺ3.1ሻ and ሺ3.2ሻ play a crucial role in the
computation 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 ‫ݐ‬௦
ఒ
, but it has been experienced practically
in the industries that by increasing the value of ‫ݐ‬௦
ఒ
, the impact on the total trim loss results in a
significantly acceptable range in some particular cases. But we are strict to ‫ݐ‬௦
ఒ
only.

140
(ii) In order to implement the algorithm smoothly, the data of more than one tower (preferably of
same pattern) may be clubbed.
Now consider Category-II and order lengths ݈ଵ, ݈ଶ, … , ݈ே with the required number of pieces
݀ଵ, ݀ଶ, … , ݀ே respectively. For ݅ ് ݆, ݅, ݆ ‫א‬ ‫ܤ‬ሺܰሻ, define:
݀௜ ൌ ݊ଵߙ௜ଵ ൅ ݀௜ଵ (4.4)
݀௝ ൌ ݊ଵߚ௝ଵ ൅ ݀௝ଵ (4.5)
0 ൑ ܷ௞ െ ൫݈௜ߙ௜ଵ ൅ ݈௝ߚ௝ଵ൯ ൌ ሺ‫ݓ‬௞ sayሻ ൑ ‫ݐ‬௦
ఒ
ሺߣ ൌ 1, 2ሻ
for at least one value of k (k=1,2,…,m). The number ݊ଵ has been chosen in such a way that ‫ݓ‬௞
attains a minimum value lying between 0 and ‫ݐ‬௦
ఒ
.
Similarly, choose a number ݊ଶ satisfying the following condition:
݀௜ଵ ൌ ݊ଶߙ௜ଶ ൅ ݀௜ଶ (4.6)
݀௝ଵ ൌ ݊ଶߚ௝ଶ ൅ ݀௝ଶ (4.7)
Proceeding this way, we finally define
݀௜,௦ିଵ ൌ ݊௦ߙ௜௦ ൅ ݀௜௦ (4.8)
݀௝,௦ିଵ ൌ ݊௦ߚ௝௦ ൅ ݀௝௦ (4.9)
The process would be continued till either ݀௜௦ ൌ 0 or ݀௝௦ ൌ 0 and in view of (4.4) - (4.9), we
have
݀௜ ൌ ∑ ݊௞ߙ௜௞ ൅ ݀௜௦
௦
௞ୀଵ ߙ௜௞ ൐ ߙ௜,௞ାଵ (4.10)
݀௝ ൌ ∑ ݊௞ߚ௝௞ ൅ ݀௝௦
௦
௞ୀଵ ߚ௝௞ ൐ ߚ௜,௞ାଵ (4.11)
݀௨௦ ൌ ݊௦ାଵߜ௨,௦ାଵ ൅ ݀௨,௦ାଵ ሺ݀௨,௦ାଵ ൏ ߜ௨,௦ାଵሻ (4.12)
where ‫ݑ‬ ൌ ݅ or ݆, ߜ ൌ ߙ or ߚ for ݅ or ݆ respectively. Also ݊௞, ߙ௜௞, ߚ௝௞ are positive integers, may be
selected according to the length of stock in order to minimize the trim.
Referring relation (4.10)-(4.12), we now define the set
‫ܤ‬ ൌ ሼܾ௜௝
௥
, ܾ௨
௦ାଵ
, ܾ௨
௦ାଶ
: ܾ௜௝
௥
ൌ ߙ௜௥݈௜ ൅ ߚ௝௥݈௝, ܾ௨
௦ାଵ
ൌ ߜ௨,௦ାଵ݈௨, ܾ௨
௦ାଶ
ൌ ݀௨,௦ାଵ݈௨
where ܾ௜௝
௥
, ܾ௨
௦ାଵ
, ܾ௨
௦ାଶ
൑ ܷ௠, ‫ݑ‬ ൌ ݅ ‫ݎ݋‬ ݆, ߜ ൌ ߙ ‫ݎ݋‬ ߚ according to
‫ݑ‬ ൌ ݅ ‫ݎ݋‬ ݆ repectively, ‫ݎ‬ ൌ 1, 2, … , ‫.ݏ‬ ݅, ݆ ൌ 0, 1, … , ܰሽ (4.13)
Define |c୧| ൌ max c୧୨ , where c ൌ a or b for fixed ݅ and arbitrary ݆
ܿ௜௝ ൑ ܷ௠ (4.14)
In view of relation (4.13), we now define
‫ܤ‬௞
௥
ൌ ሼܾ௜௝
௥
, ܾ௨
௦ାଵ
, ܾ௨
௦ାଶ
: 0 ൏ หܷ௞ െ ൫ܾ௜௝
௥
|ܾ௨
௦ାଵ|ܾ௨
௦ାଶ
൯ห ൏ ‫ݐ‬௦
ఒ
ሺߣ ൌ 1, 2ሻሽ (4.15)
‫ݎ‬ ൌ 1, 2, … , ‫.ݏ‬
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).

141
• Cutting of the largest order length ࢒࢖ from category-I
Referring relation (4.14), we consider |ܽ௣|. In view of ‫ܣ‬௞ [cf. relation (4.10)], there exist sets
‫ܣ‬௤, ‫ܣ‬௥, ‫ܣ‬௦, … ሺ1 ൑ ‫,ݍ‬ ‫,ݎ‬ ‫,ݏ‬ … ൑ ݉ሻ containing |ܽ௣| along with some other ܽ௜௝’s. Corresponding to
each set ‫ܣ‬௤, ‫ܣ‬௥, ‫ܣ‬௦, … respective fixed stock lengths ܷ௤, ܷ௥, ܷ௦, … have been assigned. We select the
combination ܽ௣௤ corresponding to the smallest stock length ܷ௣ and focus our attention on it for the
first step of cutting.
Let หܽ௣ห ൌ ܽ௣௤(say) for ‫ݍ‬ ‫א‬ ‫ܤ‬ሺ‫݌‬ሻ~௣ where
ܽ௣௤ ൌ ߙ௣݈௣ ൅ ߚ௤݈௤
satisfying the condition:
ௗ೛
ௗ೜
ൌ
ఈ೛ൈ௥೛೜
ఉ೜ൈ௥೛೜
(5.1)
In view of (5.1), it may be noted that by cutting ‫ݎ‬௣௤ bars of stock length ܷ௣, total number of
required pieces of order lengths ݈௣ and ݈௤ are cut.
Define
‫ݐ‬௣
ଵ
ൌ ‫ݎ‬௣௤൫ܷ௣ െ หܽ௣ห൯ ൑ ‫ݎ‬௣௤‫ݐ‬௦
ఒ
ሺߣ ൌ 1, 2ሻ (5.2)
• Cutting of other stock lengths from the set ࡭࢘
For ݅, ݆ ് ‫,݌‬ ‫,ݍ‬ we next consider the largest order length ݈௨ (say) contained in ‫ܣ‬௥ and consider
|ܽ௨| for ܽ௨௩ ‫א‬ ‫ܣ‬௞ corresponding to the stock length ܷ௨ satisfying the condition:
݀௨
݀௩
ൌ
ߙ௨ ൈ ‫ݎ‬௨௩
ߚ௩ ൈ ‫ݎ‬௦௩
for some ‫ݒ‬ ‫א‬ ‫ܤ‬ሺ‫݌‬ሻ~ሼ௣,௤,௨ሽ
Referring relation (5.1), it is clear that by cutting ‫ݎ‬௨௩ െbars of the stock length ܷ௨, total
number of required pieces of order lengths ݈௨ and ݈௩ have been cut. Define
‫ݐ‬௨
ଵ
ൌ ‫ݎ‬௨ ሺܷ௨ െ |ܽ௨|ሻ ൑ ‫ݎ‬௨‫ݐ‬௦
ఒ
ሺߣ ൌ 1, 2ሻ (5.3)
Proceeding this way, for ݅, ݆ ് ‫,݌‬ ‫,ݍ‬ ‫,ݑ‬ ‫,ݒ‬ we consider the next largest order length out of the
remaining 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.
ܶଵ ൌ ∑ ‫ݐ‬௣
ଵ
௣‫א‬஻ሺ௡ሻ (5.4)
If this cutting process covers all the order lengths ݈ଵ, ݈ଶ, … , ݈௡, then STOP.
• Cutting of the largest order length ࢒ࡺ from Category-II
Referring definition of ‫ܤ‬௞
௥
[cf. relation (4.15)], we first set ‫ݎ‬ ൌ 1 and consider ܾே௝
ଵ
for fixed
ܰ and arbitrary ݆ and select ܾே௝
ଵ
as follows:
|ܾே| ൌ max
௝‫א‬஻ሺ௡ሻ~ಿ
ܾே௝
ଵ
Such that |ܾே| ൑ ܷ௝ for some ݆ ‫א‬ ‫ܤ‬ሺ݉ሻ.
Now, corresponding to |ܾே|, there exists sets ‫ܤ‬௤
ଵ
, ‫ܤ‬௥
ଵ
, ‫ܤ‬௦
ଵ
, … associated with the stock lengths
ܷ௤, ܷ௥, ܷ௦, … respectively containing |ܾே|. We select the set ‫ܤ‬௤
ଵ
corresponding to the smallest stock
length ܷ௤. In view of the relation (4.15), we have
ܾே௝
ଵ
ൌ ߙேଵ݈ே ൅ ߚ௝ଵ݈௝ for ݆ ‫א‬ ‫ܤ‬ሺܰሻ~ே.
It is clear from relations (4.10) and (4.11), that by cutting ݊ଵ bars of stock length ܷ௤, we cut
݊ଵ. ߙேଵ pieces of order length ݈ே and ݊ଵ. ߚ௝ଵ pieces of order length ݈௝. Our aim is to finish cutting of
only two order lengths first ݈ே and ݈௝ (fixed) at a time. Following cases may arise:
Case1. Either ݀ே െ ݊ଵ. ߙேଵ ൏ ߙேଵ or ݀௝ െ ݊ଵߚ௝ଵ ൏ ߚ௝ଵ or both the inequalities hold together.
Case 2. Either ݀ே ൌ ݊ଵ. ߙேଵ or ݀௝ ൌ ݊ଵߚ௝ଵ.

142
5.1 Remark
Here 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
ܾேೕ
ଶ
ൌ ߙேଶ݈ே ൅ ߚ௝ଶ݈ଶ (݆ fixed as given by (4.13) )
Now, corresponding to ܾேೕ
ଶ
, there exist sets ‫ܤ‬௤
ଶ
, ‫ܤ‬௥
ଶ
, ‫ܤ‬௦
ଶ
, … containing it. The sets ‫ܤ‬௤
ଶ
, ‫ܤ‬௥
ଶ
, ‫ܤ‬௦
ଶ
, …
are associated with the stock length ܷ௤, ܷ௥, ܷ௦, … respectively. We select the set ‫ܤ‬௥
ଶ
(say)
corresponding to the smallest stock length ܷ௥. It is clear from relations (4.10) and (4.11) that by
cutting ݊ଶ bars of stock length ܷ௥, we cut ݊ଶ. ߙேଶ more pieces of order length ݈ே and ݊ଶ. ߚ௝ଶ more
pieces of order length ݈௝. We continue the process till either ݀ே ൌ ∑ ݊௞ߙ௜௞
௦
௞ୀଵ or ݀௝ ൌ ∑ ݊௞ߚ௝௞
௦
௞ୀଵ .
Let if possible ݀ே ൌ ∑ ݊௞ߙ௜௞
௦
௞ୀଵ holds, then ݀௝ would be of the form
݀௝ ൌ ∑ ݊௞ߚ௝௞
௦
௞ୀଵ ൅ ݀௝௦
where we express ݀௝௦ ൌ ݊௦ାଵߚ௝,௦ାଵ ൅ ݀௝,௦ାଵ ሺ݀௝,௦ାଵ ൏ ߚ௝,௦ାଵሻ.
Referring the relation (4.13), we now consider
ܾ௝
௦ାଵ
ൌ ߚ௝,௦ାଵ݈௝
Now, corresponding to ܾ௝
௦ାଵ
, there exists sets ‫ܤ‬௤
௦ାଵ
, ‫ܤ‬௥
௦ାଵ
, ‫ܤ‬௦
௦ାଵ
, … containing it. The sets
‫ܤ‬௤
௦ାଵ
, ‫ܤ‬௥
௦ାଵ
, ‫ܤ‬௦
௦ାଵ
, … associated with the stock lengths ܷ௤, ܷ௥, ܷ௦, … respectively. We select the set
‫ܤ‬௦
௦ାଵ
corresponding to the smallest stock length ܷ௦(say). It is clear from relations (4.10) and (4.11)
that by cutting ݊௦ାଵ. ݀௝,௦ାଵ pieces of order length ݈௝. Now ݀௝,௦ାଵ pieces of order length ݈௝ are left to
cut out of ݀௝. We now consider ܷ௞ െ ݀௝,௦ାଵ݈௝ for all ݇ ൌ 1, 2, … , ݉ and select the minimum
difference corresponding to the stock length ܷ௧(say) ܷ௧, all pieces of order length ݈௝ have been cut.
5.2 Remark
At this last step of cutting ܷ௧ െ ݀௝,௦ାଵ. ݈௝ may exceed the sustainable trim ‫ݐ‬௦.
We now compute the trim loss corresponding to the order lengths ݈ே and ݈௝ belonging to the
Category-II.
‫ݐ‬ ൌ ቀܷ௤ െ ܾேೕ
ଵ
ቁ ݊ଵ ൅ ቀܷ௥ െ ܾேೕ
ଶ
ቁ ݊ଶ ൅ ‫ڮ‬
ൌ ∑ ቀܷ௟ െ ܾேೕ
௥
ቁ݊௥ ൅ ሺܷ௧ െ ݀௝,௦ାଵ݈௝ሻ௦
௥ୀଵ
Order lengths ݈ே and ݈௝ belonging to category-II have been cut completely. Remaining order
lengths we again arrange in increasing order ݈ଵ, ݈ଶ, … , ݈ெ (say). We first consider
|ܾெ| ൌ max
௝‫א‬஻ሺெሻ~ಾ
ܾெೕ
ଵ
݆ ‫א‬ ‫ܤ‬ሺ‫ܯ‬ሻ~ெ
such that |ܾெ| ൑ ܷ௝ for some ݆ ‫א‬ ‫ܤ‬ሺ݉ሻ.
Proceeding in a similar manner, we get
‫ݐ‬௥మ
′
ൌ ∑ ቀܷ௟ െ ܾெೕ
௥
ቁ ݊௥
௧
௥ୀଵ ൅ ൫ܷ௧ െ ݀௝,௦ାଵ݈௝൯ ݈ ‫א‬ ‫ܤ‬ሺ݉ሻ
We continue the process till all order lengths are exhausted and get
ܶଶ ൌ ‫ݐ‬௥భ
′
൅ ‫ݐ‬௥మ
′
൅ ‫ڮ‬
Finally, we get total trim
ܶ ൌ ܶଵ ൅ ܶଶ.
The percentages of trim lose with respect to ‫ݐ‬௦
ଵ
and ‫ݐ‬௦
ଶ
have been computed in accordance
with the total stock length used.

143
6. DESIGN OF ALGORITHMS

International Journal of Computer Engineering and Technology (IJCET), ISSN 0976
6367(Print), ISSN 0976 – 6375(Online) Volu
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976
6375(Online) Volume 4, Issue 4, July-August (2013), © IAEME
144
August (2013), © IAEME

145
7. NUMERICAL EXAMPLES
7.1 Example
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 lengths
S.No. Stock lengths (in cm.) S.No. Stock lengths (in cm.) ‫܍ܔ܊܉ܖܑ܉ܜܛܝ܁‬ ‫ܕܑܚ܂‬s
࢚࢙
૚
ൌ ૟૟.૜૞૛૚ ࢉ࢓.
࢚࢙
૛
ൌ ૟ૡ.૚ૢ૛ૡ ࢉ࢓.
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. 492
548
ቅ
3
3
ቅ 0 ൈ 13 ൌ 0 3120 ൈ 13 ൌ 40560
2. 303
498
ቅ
4
2
ቅ 2 ൈ 8 ൌ 16 2210 ൈ 8 ൌ 17680
3. 327
748
ቅ
5
3
ቅ 4 ൈ 8 ൌ 32 3883 ൈ 8 ൌ 31064
4. 641
733
ቅ
1
2
ቅ 3 ൈ 23 ൌ 69 2110 ൈ 23 ൌ 48530
Category-II
5. 471 9 0 ൈ 2 ൌ 0 4239 ൈ 2 ൌ 8478
6. 471
801
ቅ
3
3
ቅ 67 ൈ 1 ൌ 67 3883 ൈ 1 ൌ 3883
Total 184 150195
Total Trim loss (%) 0.1225%
Table 7.3

146
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. 492
548
ቅ
3
3
ቅ 0 ൈ 13 ൌ 0 3120 ൈ 1 ൌ 40560
2. 303
498
ቅ
4
2
ቅ 2 ൈ 8 ൌ 16 2210 ൈ 8 ൌ 17680
3. 327
748
ቅ
5
3
ቅ 4 ൈ 8 ൌ 32 3883 ൈ 8 ൌ 31064
4. 641
733
ቅ
1
2
ቅ 3 ൈ 23 ൌ 69 2110 ൈ 2 ൌ 48530
Category-II
5. 471 9 0 ൈ 2 ൌ 0 4239 ൈ 2 ൌ 8478
6. 471
801
ቅ
1
2
ቅ 37 ൈ 1 ൌ 37 2110 ൈ 1 ൌ 2110
7. 471
801
ቅ
2
1
ቅ 367 ൈ 1 ൌ 367 2210 ൈ 1 ൌ 2110
Total 521 150532
Total Trim loss (%) 0.3461%
Table 7.4
7.2 Figure and Screen shots of programming
Figure 7.1

147
Figure 7.2
Figure 7.3
7.3 Conclusion
Referring tables 7.3 and 7.4, it may be noted that for this particular set of data ‫ݐ‬௦
ଶ
is reducing
the total trim significantly in comparison with the trim obtained by using ‫ݐ‬௦
ଵ
.

148
7.4 Example
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 7.5
Available stock lengths
S.No. Stock lengths (in cm.) S.No. Stock lengths (in cm.) ‫܍ܔ܊܉ܖܑ܉ܜܛܝ܁‬ ‫ܕܑܚ܂‬ssss
࢚࢙
૚
ൌ ૡ૙. ૚૙૜ૡ ࢉ࢓.
࢚࢙
૛
ൌ ૠૢ. ૜૚૚ૡ ࢉ࢓.
1. 2110 4. 3883
2. 2210 5. 4170
3. 3120 --- ---
Table 7.6
7.5 Conclusion
It may be verified that in example 7.4, the trim loss obtained by using ‫ݐ‬௦
ଶ
is exceeding the
value of trim loss obtained by ‫ݐ‬௦
ଵ
.
REFERENCE
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3. Gilmore P. C., Cutting stock, linear programming, knapsacking, dynamic programming and
integer programming; some interconnections, Annals of Discrete Mathematics, 4, 217-235,
1979.
4. Gilmore P., Gomory R., A linear programming approach to the cutting stock problem,
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