International Journal of Computer Science Engineering and Information Technology Research (IJCSEITR) ISSN 2249-6831 Vol. 3, Issue 3, Aug 2013, 227-240 ÂŠ TJPRC Pvt. Ltd.

ONE-DIMENSIONAL CUTTING STOCK PROBLEM (1D-CSP) WITH FIRST ORDER SUSTAINABLE TRIM: A PRACTICAL APPROACH P. L. POWAR1, VINIT JAIN2, MANISH SARAF3 & RAVI VISHWAKARMA4 1,4

Department of Mathematics & Computer Science, R. D. University, Jabalpur, Madhya Pradesh, India 2

KEC International Company, Panagar, Jabalpur, Madhya Pradesh, India 3

HCET, Dumna Airport Road, Jabalpur, Madhya Pradesh, India

ABSTRACT Given a set of

order lengths

, our aim in this paper is to consider the space constraint and propose a

cutting plan by cutting at most two order lengths at a time from given stock lengths. The first order sustainable trim has been defined by considering first order weighted mean of order lengths. We have defined a sustainable trim in such a way that our cutting plan will work within the limit of pre defined sustainable trim except at the end of cutting where few order lengths are left to cut. This problem has been carried over by the authors in accordance with the practical working in the transmission tower industry. The proposed cutting plan works smoothly with minimum working space and limited men power.

KEYWORDS: First Order Sustainable Trim, 1D-CSP, Non-Negative Integral Valued (NIV) Linear Combination AMS (2000) Subject Classification: 90C90; 90C27; 90C10.

1. INTRODUCTION A Cutting Stock Problem (CSP) consists of cutting a set of available stock lengths into smaller items by optimizing trim loss (Objective function). These problems mostly occur in the production planning of various industries such as paper, steel, plastic, aluminium, iron and wood (cf. [9], [11], [12], [13]). Initially, a Cutting Stock Problem (CSP) was formulated as an integer linear optimization problems (cf. [3]) but the large number of cutting patterns (columns) due to the large possibilities of combinations of items made the problem complex. In 1961, Gilmore and Gomory ([3], see also [4], [8]) proposed the first solution methodology for the CSP in which they have relaxed the integrity constraints and applied the Simplex method with column generation technique to solve the continuous relaxation of CSP. For one-dimensional case, the new patterns are introduced by solving an auxiliary optimization problem called the Knapsack problem using dual variable information from the linear programming. The Knapsack problem has well-known methods to solve it, such as branch-and-bound, dynamic programming, delayed column generation method which can be more efficient than the original approach, particularly as the size of problem grows. The column generation approach applied to the CSP was pioneered by Gilmore and Gomory in a series of papers [2] (cf. [4], [5]). Gilmore and Gomory show that this approach is guaranteed to converge to the (fractional) optimal solution without needing to enumerate all the possible patterns in advance. The problem of evaluating and computing different one-dimensional stock cutting algorithms regarding trim loss has been discussed in [7]. Considering the leftover after fulfilling current order sets and unutilized stock lengths, Trkman et

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P. L. Powar, Vinit Jain, Manish Saraf & Ravi Vishwakarma

al [14] have discussed the problem of reduction of trim loss and costs over a broader time span (cf. [14]). With the constraint of several stock lengths in limited quantities, Poldi et al [10] dealt with the classical one-dimensional integer cutting stock problem, which consists of cutting a set of available stock lengths in order to produce smaller ordered items. A linear programming procedure and a sequential heuristic procedure for one-dimensional CSP with usable leftover (residual length) are presented in [1]. Recently, a method for solving one-dimensional cutting stock problem with usable leftover (CSPUL) in cases where the ratio between the average stock and average order length is less than 3; has been initiated by Gradisar et al in [6]. The authors of this paper visited one of the leading industries dealing with design and manufacturing of transmission towers continuously for a month to observe different stages of tower manufacturing; starting from procurement of stock, working of CNC machines for cutting of bars, marking, etc. till the erection of tower at required place. The present work has been carried over by considering the solutions of some of the problems which occur during the period of entire manufacturing process. The design of our algorithm is based on the following facts:

Recently, the working space for the industries has been squeezed remarkably as the industrial development is growing very fast globally. In almost all industries where the cutting of smaller lengths from the longer stock lengths of different dimensions is desired, maintenance of stock with different lengths is a major problem. Due to this space constraint only some of the transmission tower industries in European countries like Brazeel and

Maxico were not in a position to sustain even. For all types of tower designing the production departments were using a fixed stock length which resulted in a uncontrolled trim loss; ultimately turned into non-sustainability of the industry and finally took over by KEC, a multinational company in India.

Sorting of sufficient 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 cutting process is over, is another tedious job done manually which is also space consuming.

Looking into the limitations of supplier, it has been also studied that in view of the convenience of transportation, the supplier can supply only the stock lengths varying from minimum step of

meters to maximum

meters with the

meter.

Lastly, the following observation plays a key role for designing the algorithm in the present paper. Let

be the number of pieces required corresponding to order lengths

respectively. It is

interesting to note that in almost all cases of different types of tower designing the following relation holds:

where

and

is a positive integer.

So far, the above constraints and situations have been studied partially in a generalized form (cf. [7], [8]). In the present paper, we have focused our attention at only one industry viz. transmission tower manufacturing and tried to resolve almost all problems stated above. We shall first discuss the algorithm and then give the mathematical formulation of the problem along with the definition of trim loss. On the basis of the order lengths, the required number of pieces of order lengths and stock lengths, we have defined mathematically a sustainable trim of order one i.e. the trim which can be sustained by the Industries

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One-Dimensional Cutting Stock Problem (1D-CSP) with First Order Sustainable Trim: A Practical Approach

resulting without loss. The mathematical modeling has been designed by using integral valued linear combinations of order lengths for the first category and recurrence relations involving required number of pieces of order lengths for the second category. In view of the modeling, the algorithm for the cutting plan has been proposed which will predict the trim before the actual cutting. Considering the actual data extracted from the industry where the authors carried out their study, the implementation part has been covered widely. Finally, we ended the paper with some conclusions and remarks.

NOTATIONS AND PRE-REQUISITES Throughout our discussion, we consider lengths as integer. If they are not integers, we assume that it is always possible to multiply them with Block of integers Order lengths

. The following notations are used: (index set),

means can be any number from the set

arranged in ascending order with respect to length and

Required number of pieces of order length Stock lengths

by convention.

.

arranged in ascending order with respect to length.

It has been noticed particularly that in the transmission tower designing industry most of the required number of order lengths i.e.

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: Category I: (C-I) We collect all those order lengths whose required number of pieces are integral multiples 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).

2. FIRST ORDER SUSTAINABLE TRIM In order to cut the linear combination

(say) of the two order lengths

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

and

is the sustainable trim and would be defined later.

Define

We next define

( where

and is an appropriate positive integer

, for which

are the stock lengths. We finally define

is minimum)

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P. L. Powar, Vinit Jain, Manish Saraf & Ravi Vishwakarma

(3.1) which is the desired sustainable trim. 2.1 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. MATHEMATICAL FORMULATION OF THE PROBLEM 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 other factor

(if exists)

may be selected according to the length of stock to minimize the trim. In view of (4.1), define the following set: (4.2) We are now in a position to define the sets

where

as follows:

is defined by (3.1)

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

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 Category-I (C-I) Let

we redefine our categories I and II as follows:

order lengths have been covered by the sets

convenience, we denote these order lengths by

For

arranged in ascending order with respect to the length.

3.1 Remark There may exist some order lengths exceeds the largest stock length

or

(say) such that

and

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) C-II consists of the remaining all order lengths ascending order with respect to the length.

ofcourse are multiple of each others but

denoted by

arranged in

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One-Dimensional Cutting Stock Problem (1D-CSP) with First Order Sustainable Trim: A Practical Approach

3.2 Remark ď‚ˇ

The real number

defined by

plays 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 been experienced practically in the industries that by increasing the value of

, but it has

, the impact on the total trim loss

results in a significantly acceptable range in some particular cases. But we are strict to ď‚ˇ

and

only.

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

respectively. For

,

with the required number of pieces

, define: (4.4) (4.5) (for at least one value of k (k=1,2,â€Ś,m))

The number

has been chosen in such a way that

Similarly, choose a number

attains a minimum value lying between 0 and

.

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

or

and in view of (4.4) - (4.9), we have (4.10) (4.11) (4.12)

where

for

or

respectively. Also

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

(4.13) Define In view of relation (4.13), we now define

for fixed and arbitrary

(4.14)

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P. L. Powar, Vinit Jain, Manish Saraf & Ravi Vishwakarma

(4.15)

4. 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. Cutting of the Largest Order Length

from Category-I

Referring relation (4.14), we consider containing respective fixed stock lengths smallest stock length Let

. In view of

along with some other

(cf. relation (4.10)), there exist sets â€™s. Corresponding to each set

have been assigned. We select the combination

corresponding to the

and focus our attention on it for the first step of cutting. (say) for

where

satisfying the condition: (5.1) In view of (5.1), it may be noted that by cutting order lengths

and

bars of stock length

, total number of required pieces of

are cut.

Define (5.2) Cutting of Other Stock Lengths from the Set For

, we next consider the largest order length

corresponding to the stock length

(say) contained in

and consider

for

satisfying the condition:

for some Referring relation (5.1), it is clear that by cutting of order lengths

and

bars of the stock length

, total number of required pieces

have been cut. Define (5.3)

Proceeding this way, for

, 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

â€™s has been computed.

The process is continued till all order lengths belonging to category I are totally exhausted.

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One-Dimensional Cutting Stock Problem (1D-CSP) with First Order Sustainable Trim: A Practical Approach

(5.4) If this cutting process covers all the order lengths Cutting of the Largest Order Length Referring definition of select

, then STOP.

from Category-II

(cf. relation (4.15)), we first set

and consider

for fixed

and arbitrary and

as follows:

Such that

for some

Now, corresponding to respectively containing We select the set

. , there exists sets

associated with the stock lengths

. 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 order length

and

bars of stock length

, we cut

pieces of

pieces of order length .

Our aim is to finish cutting of only two order lengths first Case 1: Either

or

Case 2: Either

or

and

(fixed) at a time. Following cases may arise:

or both the inequalities hold together. .

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 ( fixed as given by (4.13) ) Now, corresponding to the stock length

, there exist sets

containing it. The sets

respectively. We select the set

(say) corresponding to the smallest stock length

It is clear from relations (4.10) and (4.11) that by cutting of order length

and

more pieces of order length

bars of stock length

where we express

holds, then

, we cut

. We continue the process till either

. Let if possible

are associated with

would be of the form

.

.

more pieces or

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P. L. Powar, Vinit Jain, Manish Saraf & Ravi Vishwakarma

Referring the relation (4.13), we now consider

Now, corresponding to

, there exists sets

associated with the stock lengths We select the set

respectively. corresponding to the smallest stock length

(4.11) that by cutting bars of stock length length

are left to cut out of

, we cut

.

(say). It is clear from relations (4.10) and

pieces of order length . Now

pieces of order

.

We now consider stock length

containing it. The sets

for all

and select the minimum difference corresponding to the

(say). By cutting only one bar of the stock length

, all pieces of order length

have been cut.

4.1 Remark At this last step of cutting

may exceed the sustainable trim

We now compute the trim loss corresponding to the order lengths

Order lengths

and

(say). We first consider

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 .

belonging to the Category-II.

belonging to category-II have been cut completely. Remaining order lengths we again

arrange in increasing order

such that

and

.

235

5. FLOW-CHART FOR CUTTING PLAN

Figure 1

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P. L. Powar, Vinit Jain, Manish Saraf & Ravi Vishwakarma

Figure 2

237

6. EXAMPLE We have considered the following data for our analysis. Table 1 S. No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Order Lengths (in cm.) 310 660 800 890 550 640 750 400 230 460

Required No. of Pieces 40 16 13 36 39 23 24 47 32 21

Available Stock Lengths Table 2 S. No. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

Stock Lengths (in cm.) 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300

Cutting Plan Table 3 S. No.

Order Lengths (in cm)

Pieces Trim Loss (in to Cut cm.) Category-I

1. Category-II 2. 3. 4. 5. 6. 7. 8.

Used Stock Lengths (in cm.)

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P. L. Powar, Vinit Jain, Manish Saraf & Ravi Vishwakarma

Table 3: Contd., 9. Total Total Trim Loss (%)

Figure 3: Cutting Pattern

Figure 4: Screen Shot of the Program

7. CONCLUSIONS It has been noticed practically that any profitable industry dealing with tower manufacturing is permitted to yield 2%-3% trim. In view of example given in section 7, it is very interesting to note that, the proposed cutting plan requires minimum space, less manpower and in particular the acceptable trim within the range of the industry.

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Cui, Y., Yang, Y. (2010), A heuristic for the one-dimensional cutting stock problem with usable leftover. European Journal of Operational Research, 204 (2), 245-250

2.

Gilmore, P. C. (1979), Cutting stock, linear programming, knapsacking, dynamic programming and integer programming; some interconnections, Annals of Discrete Mathematics, 4, 217-235.

3.

Gilmore, P., Gomory, R. (1961), A linear programming approach to the cutting stock problem, Operations Research 9(6), 848-859.

4.

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Gilmore, P., Gomory, R. (1963), A linear programming approach to the cutting stock problem, part-II, Operations Research 11, 863-888.

5.

Gilmore, P.C., Gomory, R.E. (1965), Multistage cutting stock problems of two and more dimensions. INFORMS Operations Research 13(1), 94-120.

6.

Gradisar, M., Erjavec, J., Tomat L. (2011), One dimensional cutting stock optimization with usable leftover: A case of low stockâ€“to-order ratio, International journal of Decision support system technology, 3(1), 54-66.

7.

Gradisar, M., Resinovic G., Kljajic, M. (2002), Evaluation of algorithms of one-dimensional cutting, Computer & Operations Research 29, 1207-1220.

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Gradisar, M., Trkman, Peter (2005), A combined approach for solution of general one dimensional cutting stock problem, Computer & Operation research 32 (7) 1793-1807.

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Haessler, R. W. and Vonderembse, M. A. (1979), A procedure for solving the master slab cutting stock problem in the steel industry, AIIE Transactions, 11, 160-165.

10. Poldi, K. C., Arenales, M.N. (2009), Heuristics for the one-dimensional cutting stock problem with limited multiple stock lengths, Computers and Operations Research 36, 2074-2081. 11. Stadtler, H. (1990), A one-dimensional cutting problem in the aluminum industry and its solution, European Journal of Operational Research, 44, 209-223. 12. Tilanus, C., Gerhardt, C. (1976), An application of cutting stock in the steel industry, Operational Research, 669675. 13. Tokuyama, H., and Ueno, N. (1981), The cutting stock problems in the iron and steel industries in J.P. Brans (ed.), Operational Research, North- Holland, 809-823. 14. Trkman, Peter, Gradisar, M. (2007), One-dimensional cutting stock optimization in consecutive time periods. Eur. J. Oper. Res., 179(2), 291-301.