one-dimensional cutting stock problem _1d-csp_ a study on data dependent tri

7/29/2019 One-dimensional Cutting Stock Problem _1d-Csp_ a Study on Data Dependent Tri

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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

524

ONE-DIMENSIONAL CUTTING STOCK PROBLEM (1D-CSP): A STUDY ON

DATA DEPENDENT TRIM LOSS

P. L. Powar1, Vinit Jain

2, Manish Saraf

3, Ravi Vishwakarma

4

1Deptt. of Math. & Comp. Sc., R. D. University, Jabalpur 482001, India

2KEC Int. Company, Panagar, Jabalpur, 482001, India

3HCET, Dumna Airport Road, Jabalpur, 482001, India

4Deptt. of Math. & Comp. Sc., R. D. University, Jabalpur 482001, India

ABSTRACT

A generalized approach is introduced for getting minimal trim in one- dimensional cutting

stock problem (1D-CSP) which occurs extensively while manufacturing of different engineering

objects. Our study is especially focused on transmission tower manufacturing industry. The concept

of computing total trim by using the idea of pre-defined sustainable trim has been explored inprevious work of authors. Given m stock lengths , , , , the cutting plan consists of cutting ofat most two order lengths at a time out of the demanded set of n order lengths , , , .

Considering the given data, the total trim has been computed corresponding to two different

sustainable trims of order one (viz. ) and order two (viz.). Introducing various values ofsustainable trims as knots between and , the total trim has been computed corresponding toeach knot, the linear approximation has been constructed which predict the total trim loss at any

arbitrary point lying between and .KEYWORDS: First order sustainable trim, Second order sustainable trim, 1D-CSP, Non-negative

integral valued (NIV) linear combination, Hat function, Linear approximation.

AMS (2000) SUBJECT CLASSIFICATION: 90C90; 90C27; 90C10.

1. INTRODUCTIONVarious sizes of items are ordered in industries dealing with processing of materials such as

steel, film, textiles etc. The ordered items are cut from bars, plates or sheets of raw materials that

have a fixed size (cf. [10], [14], [15], [16]). During the process of cutting of these items, the problem

for trying to minimize the waste of raw material is referred to as a cutting stock problem (CSP). The

CSP plays an important role in the economic status of the processing industries and it has been

INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &

TECHNOLOGY (IJCET)ISSN 0976 6367(Print)ISSN 0976 6375(Online)

Volume 4, Issue 4, July-August (2013), pp. 524-538 IAEME:www.iaeme.com/ijcet.aspJournal Impact Factor (2013): 6.1302 (Calculated by GISI)

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I A E M E


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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 TRIMS3.1 Order One

In order to cut the linear combination (say) of the two order lengths and from thegiven 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

original values. Hence, we have taken the weighted mean of total required lengths.

3.2 Order TwoFollowing the same restriction as for and using the notations from section 2, we define

ld ldd 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.


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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 fromthe stock length . We finally define

(3.2)which is the sustainable trim of order two viz. .4. MATHEMATICAL MODELING

We 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.


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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 a

significantly acceptable range in some particular cases. But we are strict to only.(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: (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 as repectively, 1 , 2 , , . , 0 , 1 , , (4.13)


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Define |c| m a x c , where c a or b for fixed and arbitrary (4.14)

In view of relation (4.13), we now define

, , : 0 | | (4.15) 1, 2, 1 , 2 , ,

5. CUTTING PLANIt 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 8.1).

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 to

each 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 set

For , , , 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 , totalnumber of required pieces of order lengths and have been cut. Define

| | 1, 2 (5.3)


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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 lengthsbelonging 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 fixedand 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 stocklength . 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 of

only 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 .5.1Remark

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

( 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 , , , .,


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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 of stock length , we cut . , pieces of order length . Now , pieces oforder length are left to cut out of . We now consider , for all 1 , 2 , , andselect the minimum difference 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 orderlengths 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. ALGORITHM


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7. HAT FUNCTIONS AND THEIR APPLICATIONS IN 1D-CSPPiecewise linear approximation may not have the practical significance in comparison to

cubic spline or even higher order approximation. But it shows most of the essential features of

piecewise polynomial approximation in a simple and easily understandable setting.

Consider the closed interval [a, b] on the real line and define a partition P:

: , Let $denote the collection or linear space of all continous broken lines on [a, b] with breaks at, , (see figure 7.1). We define the basis for $(cf. [3])Let : , : and set

,

/ ,

0 , otherwise (7.1)

Figure 7.1 Hat Functions

It may be checked easily that the set S= is linearly independent. If we denote thebroken line interpolation to g by , then it is given by

(7.2)which satisfy the interpolatory conditions

1 , . . , 7.1 Applications

For the given data, the average order lengths

have to be computed say

is the shortest and

is the longest average order length. Consider the closed interval , on and define thepartition as follows:: . . (7.3)

Corresponding to each 0 , . , the total trim loss by using thealgorithm described in section 6 has to be computed.


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Similarly, corresponding to the sustainable trims of order one and two , we define apartition and compute the broken line interpolation , .

Since are known, the basis functions (Hat functions) as described in (7.1) can becomputed for the partition P in (7.3). Referring relation (7.2), we can write the broken line

interpolant

for the approximate trim loss as follows:

which gives the information of approximate trim loss at any arbitrary point , .7.1 Remark

Broken lines are neither very smooth nor very efficient approximation. Both for a smoother

and more efficient approximation one has to go to piecewise polynomial approximation with higher

order pieces. The most popular choice continues to be a piecewise cubic approximating function.

Since, it is the first step of splines to enter into a field of cutting-stock problem (CSP), authors have

started with the simplest formulation of approximate function for the trim loss. Higher order

approximation would be constructed in accordance with the demand of this method in the practical

field.

8. ILLUSTRATIVE EXAMPLETables of data

Order lengths and required number of pieces

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 8.1

Available stock lengths

S.No.Stock lengths (in cm.) S.No. Stock lengths (in cm.)

1. 2110 4. 3883

2.2210 5. 4177

3.3120 6. 4239

Table 8.2


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Figure 8.1 Cutting Plan

Figure 8.2 Screen shot of the programming

Figure 8.3 Screen shot of the programming


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Computations as discussed in sections 3 and 5

First order sustainable trim = 66.3591,Second order sustainable trim = 68.1928

S. No. Sustainable Trims Total Trim in %

1. 66.3591 0.3461

2. 66.8176 0.3461

3. 67.2756 0.1225

4. 67.7342 0.1225

5. 68.1928 0.1225

Table 8.15

Partition = { 66.3591 66.8176 67.2756 67.7342 68.1928}

Figure 8.4 Linear splines for total trim loss corresponding to various sustainable trims

9. CONCLUSIONIt is interesting to note that the impact of sustainable trim on the total trim plays key role in

1D-CSP. Referring Figure 8.4, it is apparent that corresponding to the second order sustainable trim which is greater than the sustainable trim of order one , the total trim loss is minimalcorresponding to in the interval , . It is suggested that for a given data, and given stocklengths, after computing and , the minimal trim loss may be predicated locally as per theproposed cutting plan.

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one-dimensional cutting stock problem _1d-csp_ a study on data dependent tri

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