One dimensional cutting stock problem 1-d-csp_ with second order sustainable

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. Powar

1, 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 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 �� , ��, … , �� 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. 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).

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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 ��, ��, . . . , �� 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-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� � � 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 by

convention.

"� � 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 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 ��, ��, … , �, 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 , �- � )�# , ./�, 0 1,2,… ,$ and ./� is the sustainable trim of order one and defined as follows:

1 �!"! 2 ��"� 232 ��"�"! 2 "� 232 "� ∑ ��"��

�5!∑ "#�#5!

We next define

1- |�- � �1| (0 1, 2, … ,$ and � is an appropriate positive integer � 1, for which 1- is minimum)

where ��, ��, … , � are the stock lengths. We finally define

./� ∑ 7898:; (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 ./� and using the notations from section 2, we define

=� l!d! 232 l?d?d! 2 d� 232 d?

∑ ��"��5!∑ "#�#5!

By convention, =! 0, =� ��, =� @ABAC@;B;C@DBDBACB;CBD

, … , =� ∑ @EBEFE:A∑ BGFG:A

We next define the second order weighted means

=�� =! 2 =� 232 =�

� 2 1 . Consider

1- |�- � � =��|, 0 1,2,… ,$. � is an appropriate positive integers � 1, for which 1- 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

./� ∑ 78F8:; (3.2)

which is the sustainable trim of order two viz. ./�.

4. MATHEMATICAL FORMULATION OF THE PROBLEM

We first consider C-I and define the following ratios:

BEBG HEIJEGKGIJEG

(4.1)

(where L�# is a positive integer �, � � � ��, � M � ) Note: It is not necessary to consider always the largest common factor between "� and "#. Any other

factor L� (if exists) may be selected according to the length of stock to minimize the trim.

In view of (4.1), define the following set:

N �)�# O�� 2 P#�# Q )�# , �, O�, P# � 0, integer R� S �, �, � � � ��T� (4.2)

We are now in a position to define the sets N- U N 0 1, 2, … ,$� as follows:

N- �)�# : 0 , �- � )�# , ./W, 0 � � $�, �, � � � ��, )�# � N, X 1,2� (4.3)

where ./W is defined by (3.1) and (3.2) respectively for X 1 and λ 2.

At this stage, we may come across with the following situations:

• N- Y, Z 0 � � $�, in this case, all the order lengths have to shift in C-II.

• In view of the definition of ./W, the sets N- 0 � � $�� may or may not cover all order

lengths belonging to Category-I.

In view of above observations and the definition of the sets N-, we redefine our categories I

and II as follows:

Category-I (C-I) Let �H�, �HD , … , �H[ order lengths have been covered by the sets N- 0 1, 2, . . , $�. For convenience, we denote these order lengths by ��, ��, … , �[ 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 �# � )�#� 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 in

ascending order with respect to the length.

4.2 Remark

(i) The real numbers ./W X 1, 2� defined by 3.1� and 3.2� 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 ./W , but it has been experienced practically

in the industries that by increasing the value of ./W, the impact on the total trim loss results in a

significantly acceptable range in some particular cases. But we are strict to ./W 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 ��, ��, … , �^ with the required number of pieces

"�, "�, … , "^ respectively. For � ` �, �, � � � \�, define:

"� ��O�� 2 "�� (4.4)

"# ��P#� 2 "#� (4.5)

0 , �- � R��O�� 2 �#P#�T a- say� , ./W X 1, 2� for at least one value of k (k=1,2,…,m). The number �� has been chosen in such a way that a-

attains a minimum value lying between 0 and ./W.

Similarly, choose a number �� satisfying the following condition:

"�� O�� 2 "�� (4.6)

"#� ��P#� 2 "#� (4.7)

Proceeding this way, we finally define

"�,/c� �/O�/ 2 "�/ (4.8)

"#,/c� �/P#/ 2 "#/ (4.9)

The process would be continued till either "�/ 0 or "#/ 0 and in view of (4.4) - (4.9), we

have

"� ∑ �-O�- 2 "�//-5� O�- M O�,-C� (4.10)

"# ∑ �-P#- 2 "#//-5� P#- M P�,-C� (4.11)

"d/ �/C�ed,/C� 2 "d,/C� "d,/C� S ed,/C�� (4.12)

where f � or �, e O or P for � or � respectively. Also �- , O�- , P#- 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

� �h�#J , hd/C�, hd/C�: h�#J O�J�� 2 P#J�#, hd/C� ed,/C��d, hd/C� "d,/C��d

where h�#J , hd/C�, hd/C� , �, f � kL �, e O kL P according to

f � kL � repectively, L 1, 2, … , o. �, � 0, 1, … , \� (4.13)

Define |cp| max cps , where c a or b for fixed � and arbitrary � u�# , � (4.14)

In view of relation (4.13), we now define

�-J �h�#J , hd/C�, hd/C�: 0 S v�- � Rh�#J |hd/C�|hd/C�Tv S ./W X 1, 2�� (4.15)

L 1, 2, … , o.

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 wx from category-I

Referring relation (4.14), we consider |)[|. In view of N- [cf. relation (4.10)], there exist sets

Ny , NJ, N/, … 1 , z, L, o, … , $� containing |)[| along with some other )�#’s. Corresponding to

each set Ny , NJ, N/, … respective fixed stock lengths �y , �J , �/, … have been assigned. We select the

combination )[y corresponding to the smallest stock length �[ and focus our attention on it for the

first step of cutting.

Let v)[v )[y(say) for z � � ]�~[ where

)[y O[�[ 2 Py�y

satisfying the condition:

B|B} H|IJ|}K}IJ|}

(5.1)

In view of (5.1), it may be noted that by cutting L[y bars of stock length �[, total number of

required pieces of order lengths �[ and �y are cut.

Define

.[� L[yR�[ � v)[vT , L[y./W X 1, 2� (5.2)

• Cutting of other stock lengths from the set ~� For �, � ` ], z, we next consider the largest order length �d (say) contained in NJ and consider

|)d| for )d� � N- corresponding to the stock length �d satisfying the condition: "d"� Od I Ld�P� I L/�

for some � � � ]�~�[,y,d� Referring relation (5.1), it is clear that by cutting Ld� �bars of the stock length �d, total

number of required pieces of order lengths �d and �� have been cut. Define

.d� Ld �d � |)d|� , Ld./W X 1, 2� (5.3)

Proceeding this way, for �, � ` ], z, f, �, 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.

�� ∑ .[�[�� (5.4)

If this cutting process covers all the order lengths ��, ��, … , ��, then STOP.

• Cutting of the largest order length w� from Category-II

Referring definition of �-J [cf. relation (4.15)], we first set L 1 and consider h^#� for fixed

\ and arbitrary � and select h^#� as follows:

|h^| max#�� ~�

h^#�

Such that |h^| , �# for some � � � $�. Now, corresponding to |h^|, there exists sets �y�, �J�, �/�, … associated with the stock lengths

�y , �J, �/, … respectively containing |h^|. We select the set �y� corresponding to the smallest stock

length �y. In view of the relation (4.15), we have

h^#� O^��^ 2 P#��# for � � � \�~^.

It is clear from relations (4.10) and (4.11), that by cutting �� bars of stock length �y, we cut

��. O^� pieces of order length �^ and ��. P#� 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 "^ � ��. O^� S O^� or "# � ��P#� S P#� or both the inequalities hold together.

Case 2. Either "^ ��. O^� or "# ��P#�.



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

h^G� O^��^ 2 P#�� (� fixed as given by (4.13) )

Now, corresponding to h^G� , there exist sets �y�, �J�, �/�, … containing it. The sets �y�, �J�, �/�, …

are associated with the stock length �y , �J, �/, … respectively. We select the set �J�(say)

corresponding to the smallest stock length �J. It is clear from relations (4.10) and (4.11) that by

cutting �� bars of stock length �J, we cut ��. O^� more pieces of order length �^ and ��. P#� more

pieces of order length �#. We continue the process till either "^ ∑ �-O�-/-5� or "# ∑ �-P#-/

-5� .

Let if possible "^ ∑ �-O�-/-5� holds, then "# would be of the form

"# ∑ �-P#-/-5� 2 "#/

where we express "#/ �/C�P#,/C� 2 "#,/C� "#,/C� S P#,/C��. Referring the relation (4.13), we now consider

h#/C� P#,/C��# Now, corresponding to h#/C�, there exists sets �y/C�, �J/C�, �//C�, … containing it. The sets

�y/C�, �J/C�, �//C�, … associated with the stock lengths �y , �J, �/, … respectively. We select the set

�//C� corresponding to the smallest stock length �/(say). It is clear from relations (4.10) and (4.11)

that by cutting �/C�. "#,/C� pieces of order length �#. Now "#,/C� pieces of order length �# are left to

cut out of "#. We now consider �- � "#,/C��# for all 0 1, 2, … ,$ and select the minimum

difference corresponding to the stock length ��(say) ��, all pieces of order length �# have been cut.

5.2 Remark

At this last step of cutting �� "#,/C�. �# may exceed the sustainable trim ./. We now compute the trim loss corresponding to the order lengths �^ and �# belonging to the

Category-II.

. ��y � h^G� � �� 2 ��J � h^G� � �� 23

∑ ��@ � h^GJ ��J 2 �� "#,/C��#�/J5�

Order lengths �^ and �# belonging to category-II have been cut completely. Remaining order

lengths we again arrange in increasing order ��, ��, … , �� (say). We first consider

|h�| max#�� ~�

h�G� � � � ��~�

such that |h�| , �# for some � � � $�. Proceeding in a similar manner, we get

.JD′ ∑ ��@ � h�GJ � �J�J5� 2 R�� "#,/C��#T � � � $�

We continue the process till all order lengths are exhausted and get

�� .J;′ 2 .JD′ 23

Finally, we get total trim

� �� 2 ��.

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 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.) �� 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 I 13 0 3120 I 13 40560

2. 303498¥

42¥

2 I 8 16 2210 I 8 17680

3. 327748¥

53¥

4 I 8 32 3883 I 8 31064

4. 641733¥

12¥

3 I 23 69 2110 I 23 48530

Category-II

5. 471 9 0 I 2 0 4239 I 2 8478

6. 471801¥

33¥ 67I 1 67 3883I 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¥ 0I 13 0 3120 I 1 40560

2. 303498

¥ 42¥ 2I 8 16 2210 I 8 17680

3. 327748

¥ 53¥ 4I 8 32 3883 I 8 31064

4. 641733

¥ 12¥ 3 I 23 69 2110 I 2 48530

Category-II

5. 471 9 0 I 2 0 4239 I 2 8478 6. 471

801¥ 1

2¥ 37 I 1 37 2110 I 1 2110

7. 471801

¥ 21¥ 367 I 1 367 2210 I 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 ./� is reducing

the total trim significantly in comparison with the trim obtained by using ./�.



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.) �� ssss &'( �¨. (¨�� . &'< ©�. �((� ��.

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 ./� is exceeding the

value of trim loss obtained by ./�.

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pp. 478 - 486, ISSN Print: 0976 – 6367, ISSN Online: 0976 – 6375.