Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 521386, 9 pages

http://dx.doi.org/10.1155/2014/521386

Research Article

Flexible Stock Allocation and Trim Loss Control for Cutting Problem in the Industrial-Use Paper Production

Fu-Kwun Wang and Feng-Tai Liu

Department of Industrial Management, National Taiwan University of Science and Technology, Taipei 106, Taiwan

Received 29 November 2013; Accepted 25 May 2014; Published 17 June 2014

Academic Editor: Hsiao-Fan Wang

Copyright © 2014 Fu-Kwun Wang and Feng-Tai Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider a one-dimensional cutting stock problem (CSP) in which the stock widths are not used to fulfill the order but kept for use in the future for the industrial-use paper production. We present a new model based on the flexible stock allocation and trim loss control to determine the production quantity. We evaluate our approach using a real data and show that we are able to solve industrial-size problems, while also addressing common cutting considerations such as aggregation of orders, multiple stock widths, and cutting different patterns on the same machine. In addition, we compare our model with others, including trim loss minimization problem (TLMP) and cutting stock problem (CSP). The results show that the proposed model outperforms the other two models regarding total flexibility and trim loss ratio.

1. Introduction

A one-dimensional cutting stock problem (CSP) is one of the famous combinatorial optimization problems, which has many applications in industries, such as paper, wood, textiles, steel, space, ship construction, and logistic transportation [1–6]. Most studies focus on minimizing the trim loss that is the amount of residual pieces of processed stock lengths. A standard one-dimensional cutting stock problem (S1D-CSP) as a kind of the above problems is known as an NP-complete one [7]. Numerous studies have examined how to fulfill orders and optimize production planning. Gilmore and Gomory [8] presented a delayed pattern generation technique for solving a one-dimensional cutting problem using linear programming. Other methods, including pattern-oriented approach, item-oriented approach, mixed approach and exact approach, can be found in [9–27].

In the industrial-use paper industry, the production quantity is usually greater than the customers’ order. Using the traditional CSP, the trim loss can be significant. Thus, we need to consider usable leftovers to prevent the trim loss generated after optimization. This issue becomes a one-dimensional CSP with usable leftovers. Yanasse [28] reported that the literature on usable leftovers is scarce and the problem still lacks clear and appropriate definitions. Kos and Duhovnik [29] proposed usable leftover material used in the next cutting plan to reduce trim loss. Related studies can be found in [6, 29–35]. Cherri et al. [31] presented several modifications in some well-known heuristics to solve a one-dimensional CSP with usable leftovers. Poldi and Arenales [32] presented a study with the classical one-dimensional integer CSP, which consists of cutting a set of available stock lengths in order to produce smaller ordered items. Cui and Yang [33] considered a one-dimensional CSP with useful leftover in the cutting plan. Cherri et al. [35] proposed a priority-in-use heuristic approach to solve a one-dimensional CSP with usable leftovers. However, these models cannot be directly used for solving the CSP in the industrial-use production that each reel can only be produced a certain number of rolls depending on its cutting machine. Wang and Liu [36] presented a new decision model for reducing trim loss and inventory in the paper industry.

In this study, we present a new model based on the flexible stock allocation and trim loss control to determine the production quantity. Our proposed model is a flexibility maximization problem (FAP). Under a certain condition of trim loss control, FAP can be confined to cutting stock problem (CSP) or trim loss minimization problem (TLMP). The remainder of this paper is organized as follows. In Section 2, the definition of problem in the paper industry is presented. A new model is developed in Section 3. In Section 4, some examples illustrate the application of the proposed model. Finally, conclusions are drawn in Section 5.

2. Problem Definition

The production of industrial-use paper starts from raw material to reels and then from reels to the production of rolls as finished goods. The entire operation mode is cyclical production, which is the only method for achieving production efficiency. Therefore, the leftover material is not used in a follow-up production. For the production planning (see Figure 1), the customer’s paper requirements are obtained and the marketing demand is predicted. Then, during the combined production-marketing meeting, the number of production days and the production quantity of paper types are determined. The production quantity indicates the number of reels, and each reel can produce the number of rolls that depends on the paper type. It should be noted that the unit of the paper width is millimeter (mm).

Figure 1: Production planning in the paper industry.

To formulate the models of CSP, TLMP and FAP, see the notations section are used.

The main research question is how to improve the stock allocation and trim loss of a CSP with useful leftovers in the paper industry. This problem can be studied for either one- or multidimensional CSPs. In this study, one-dimensional CSP with useful leftovers was used. We first provide two examples to illustrate the differences between CSP, TLMP, and manual adjustment (MA). In practice, MA is used in the industrial-use paper industry in which the CSP and TLMP solutions are candidates as manually selecting as MA solution. The CSP and TLMP are usually solved through column generation [8, 9]. To obtain the solutions of CSP and TLMP, the computer program was written in Lingo 11 Software [37].

The formulation of CSP is defined as follows:

where and are decision variables and integer variables. Minimizing the total number of patterns is the objective function (1) of the model. Constraint (2) guarantees the cutting stocks regarding the reel width. Constraint (3) guarantees the cutting stocks regarding the demand. In the industrial-use paper production, there exists the maximum trim loss for each cutting, and then constraint (4) guarantees the waste of each roll during the cutting process.

In order to reduce the trim loss, a modified model called TLMP is given as follows:

where and are decision variables and integer variables. Minimizing the total trim loss is the objective function (5) of the model. Constraint (6) guarantees the cutting stocks regarding the reel width. Constraint (7) guarantees the cutting stocks regarding the demand. In industry paper production, the maximum trim loss for each cutting and the limit production volume are considered; then constraint (8) guarantees the number of rolls for each reel, and constraint (9) guarantees the waste of each roll during the cutting process.

For example 1, we assume that the reel width is 10 units, is 3, is 3, the demand of order widths is , and the stock widths are . In Figure 2, we provide CSP, TLMP, and MA solutions. The trim loss using CSP is 5 units. And the trim loss using TLMP is zero. We found that the stock width using CSP is obtained as and the stock widths using TLMP are . In order to obtain flexible stock widths, using MA based on CSP and TMLP solutions, the extending stock width is determined as , and the stock width is obtained as . Thus, MA can provide more flexible stock width .

(a)

(b)

(c)

Figure 2: (a) CSP solution, (b) TLMP solution, and (c) MA solution for example 1.

For example 2, we assume that the reel width is 10 units, is 3, is 3, the demand of order widths is , and the stock widths are . In Figure 3, we provide CSP, TLMP, and MA solutions. The trim loss using CSP is 2 units. And the trim loss using TLMP is zero. We found that the stock width using CSP is zero, the unused rolls are 2 and 3, and the stock widths using TLMP are , , and . Using MA based on CSP and TMLP solutions, the extending stock widths are determined as or , the stock width is obtained as zero or , and the trim loss is obtained as 2 units or zero. Thus, MA can provide more flexible stock widths or .

(a)

(b)

(c)

Figure 3: (a) CSP solution, (b) TLMP solution, and (c) MA solution for example 2.

Based on the above discussions, we conclude that the MA approach can provide more flexible stock widths in a one-dimensional CSP with useful leftovers. This motivates the development of a mathematical formulation for a one-dimensional CSP with useful leftovers in the industrial-use industry.

3. The Proposed Model

To provide more flexible leftovers, we propose the flexible stock allocation approach. It should be noted that the concept of extending stock widths is similar to the usable leftovers [31]. In FAP, we have stock items of width , , the reel width is , each paper width is based on a fixed number called GAP to differentia, and the extending stock width can be obtained from to . Thus, the extending stock item of width is established as

where , , is integer, and is the number of stock widths. Using (10), we can combine and as production width . This new set of paper widths can provide more flexible stock allocation for cutting plan.

In order to evaluate the benefit of the flexible stock allocation, we define a flexible coefficient for each production width that is obtained as follows:

where , , p is the number of patterns, is the number of widths in pattern , and is a positive integer. For instance, we set . Since has only one combination, the value is assigned to one. Since can be divided into two combinations , the value is assigned to two. Furthermore, can be divided into fourteen combinations ,; the value is assigned to fourteen. In addition, a flexible coefficient for each extending width is obtained as follows:

where and .

We introduce a coefficient for controlling the trim loss. Thus, the proposed formulation is as follows:

where and are decision variables and integer variables. Maximizing the total flexibility is the objective function (13) of the model that is the summation of total production coefficient and total trim loss coefficient. The composition of each production coefficient includes nonorder quantity, production width, and flexible coefficient, and the trim loss coefficient includes the coefficient and the total trim loss. Constraint (14) guarantees the cutting stocks regarding the reel width. Constraint (15) guarantees the cutting stocks regarding the demand. Constraint (16) guarantees the number of rolls for each reel. Constraint (17) guarantees the waste of each roll during the cutting process.

When , the objective function (13) becomes to be a maximize function of; that is, it does not consider trim loss. If the production capacity fails to satisfy (15) during the problem-solving process, a full roll is generated. Subsequently, because the flexibility of is greater than any of the leniency and flexibility coefficient combinations, the full roll is substituted by . In addition, the optimal CSP solution also generates a full roll and the full roll is substituted by ; thus, the FAP results approximate the CSP target function; that is, Minimize . Therefore, the difference between FAP() and the of CSP can be compared.

When R = , can be neglected and the objective function (13) approaches Maximize . In addition, approximates the TLMP target function; that is, Minimize . Therefore, the difference between FAP() and the of TLMP can be compared.

In summary, when , flexible stock becomes the optimal condition and trim loss is maximized. Conversely, when , flexible stock becomes the least favorable condition and trim loss is minimized. Therefore, the control of variable is a flexible stock and trim loss strategy that decision makers adopt during the production process.

4. Illustrative Examples

We consider a real case from an industrial-use paper production and five simulated datasets to illustrate the application of our proposed method. We set the current scheduling quantity as reels, and each reel can produce number of rolls. The cutting machine width limit is , and the maximum trim loss is . These parameters are defined as ,  mm, reels = 135 rolls, and  mm.

To obtain the solutions of CSP, TLMP, and FAP, the computer program is divided into the engine and the user interface. The engine interface was written in Lingo 11 Software [37]. The user interface in Visual Basic 5 enables the navigation of data flow from various input sources, such as a common company database and a random number dataset.

4.1. A Real Case from an Industrial-Use Paper Production

According to the FAP model in Section 3, the details are as follows.

Step 1. Define , , and , for (see Table 1).

Table 1: Product information and data for , , and .

1
X
0
1000

2
X
0
1100

3
1200
7
1200

4
1300
22
1300

5
X
0
1400

6
1500
28
1500

7
1600
58
1600

8
1700
47
1700

9
1800
43
1800

10
1900
7
1900

11
2000
20
2000

12
2100
9
2100

13
2200
30
2200

14
2300
7
2300

15
2400
12
2400

16
2500
20
2500

Note: X = not available.

Step 2. Using (10) to obtain the extending widths, since and , we can obtain that and .

Step 3. Aggregate to , to , and to , for and .

Step 4. Use (11)-(12) to compute the flexible coefficient for (see Table 2).

Table 2: Product information and data for , , and .

1
1000
0
1
17
#2600
0
23

2
1100
0
2
18
#2700
0
24

3
1200
7
3
19
#2800
0
25

4
1300
22
4
20
#2900
0
26

5
1400
0
5
21
#3000
0
27

6
1500
28
6
22
#3100
0
28

7
1600
58
7
23
#3200
0
29

8
1700
47
8
24
#3300
0
30

9
1800
43
9
25
#3400
0
31

10
1900
7
10
26
#3500
0
32

11
2000
20
11
27
#3600
0
33

12
2100
9
12
28
#3700
0
34

13
2200
30
14
29
#3800
0
35

14
2300
7
16
30
#3900
0
36

15
2400
12
19
31
#4000
0
37

16
2500
20
22
32
#4100
0
38

33
#4200
0
39

34
#4300
0
40

35
#4400
0
41

36
#4500
0
42

37
#4600
0
43

Note: # = extending stock width.

Using FAP to perform optimization, must be set to 0, thereby allowing FAP results to approximate those of CSP. In this case, we obtained the production capacities of FAP and CSP, stock, trim loss ratio (TLR), and total flexible coefficient (TF), where and .

The primary reason for comparing CSP was to determine whether FAP effectively reduced TLR and whether the flexible stock of FAP is superior to that of CSP (see Table 3). The TLRs for CSP and FAP were 3.3 and 1.4, respectively; the flexible stocks for CSP, FAP, and extending stock were , and , respectively. Thus, the results suggest that FAP outperforms CSP in reducing TLR and that the flexible stock and TF of FAP are superior to that of CSP.

Table 3: Optimal solutions using FAP and CSP methods.

CSP
FAP

1
1000
0
1
31
31
21
21

2
1100
0
2
0
0
0
0

3
1200
7
3
7
0
12
5

4
1300
22
4
22
0
23
1

5
1400
0
5
0
0
10
10

6
1500
28
6
30
2
28
0

7
1600
58
7
58
0
58
0

8
1700
47
8
47
0
47
0

9
1800
43
9
43
0
43
0

10
1900
7
10
7
0
7
0

11
2000
20
11
20
0
20
0

12
2100
9
12
9
0
9
0

13
2200
30
14
30
0
30
0

14
2300
7
16
7
0
7
0

15
2400
12
19
12
0
12
0

16
2500
20
22
20
0
20
0

21
#3000
0
27
0
0
1
1

37
#4600
0
43
1
1
1
1

TLR (%)
3.3

1.4

TF

86

160

Note: rolls = 135, R = 0, and = − .

To compare FAP to TLMP, the flexible variable of FAP was set to −1000, which denotes minimal TLR. In this case, we obtained the production capacities of FAP and TLMP, stock, TLR, and TF (see Table 4).

Table 4: Optimal solutions using FAP and TLMP method.

TLMP
FAP

1
1000
0
1
39
39
37
37

2
1100
0
2
2
2
1
1

3
1200
7
3
9
2
8
1

4
1300
22
4
22
0
23
1

5
1400
0
5
9
9
6
6

6
1500
28
6
28
0
28
0

7
1600
58
7
58
0
58
0

8
1700
47
8
47
0
47
0

9
1800
43
9
43
0
43
0

10
1900
7
10
7
0
7
0

11
2000
20
11
20
0
20
0

12
2100
9
12
9
0
9
0

13
2200
30
14
30
0
30
0

14
2300
7
16
7
0
7
0

15
2400
12
19
12
0
15
3

16
2500
20
22
20
0
20
0

TLR (%)
0.42

0.42

TF

94

133

Note: rolls = 135, R = −1000, and = −.

The primary reason for comparing TLMP was to determine whether the TLR of FAP is similar to that of TLMP or not and whether the flexible stock of FAP is superior to that of TLMP or not (see Table 4). The TLRs for TLMP and FAP were 0.42 and 0.42, respectively; the flexible stocks for TLMP and FAP were , and , respectively. Notably, the flexible stock of FAP was considerably more lenient. Therefore, based on the results, the TLR of FAP was identical to that of TLMP, and the flexible stock and TF of FAP were superior to those of TLMP.

Moreover, we employed sensitivity analysis to observe the influence that has on TLR and TF. When , and is an integer, the results as shown in Table 5 are obtained.

Table 5: The sensitivity analysis of coefficient R for using FAP.

1
0

1
1000
1
11
21
22
28
36
37
37

2
1100
2
0
0
1
1
1
1
1

3
1200
3
5
5
4
1
1
1
1

4
1300
4
1
1
1
1
1
1
1

5
1400
5
10
10
10
10
6
6
6

6
1500
6
0
0
0
0
0
0
0

7
1600
7
0
0
0
0
0
0
0

8
1700
8
0
0
0
0
0
0
0

9
1800
9
0
0
0
0
0
0
0

10
1900
10
0
0
0
0
0
0
0

11
2000
11
0
0
0
0
0
0
0

12
2100
12
0
0
0
0
0
0
0

13
2200
14
0
0
0
0
0
0
0

14
2300
16
0
0
0
0
0
0
0

15
2400
19
0
0
0
0
0
1
3

16
2500
22
0
0
0
0
0
0
0

21
#3000
27
1
1
1
1
1
0
0

37
#4600
43
1
1
1
1
1
1
0

TLR (%)
4.2
1.4
1.3
0.9
0.52
0.45
0.42

TF
150
160
160
157
145
138
133

Note: the number of rolls = 135.

According to Table 5, when , TLR increased and changed to . This was primarily because trim loss was equivalent to the flexible coefficient of , causing the stock capacity of to decrease. Thus, when is defined, we can directly use for solution identification.

When R < 0, we observed that the TF gradually reduced from 160 to 133 and the TLR reduced from 1.4 to 0.42. These results suggest that, when has a value less than 0, the TF decreases and the TLR declines. Regarding flexible stock, we found that, when ranged between −1 and −2, the production capacity of was 22; subsequently, as decreased to between −3 and −8 and −534 and , the production capacity of increased to 28 and to 37, respectively. These results suggest that, as decreases, the allocation of stock gradually coagulates at a lower leniency, negating the effects of extended stock. The decrease in TF from 160 to 133 implies that the degree of permitted flexibility for adjusting stock had already diminished. Therefore, we suggest that be maintained within a range between and 0.

Because the trim loss value at each interval of is a fixed value, we selected the medians of each interval and tabulated them into Table 6, which enabled us to select the desired results. Consequently, the number of medians can be defined by decision makers based on actual conditions.

Table 6: The range and midpoint of R.

Range of R
0

Midpoint
0
−1
−5
−47
−310
−1000

4.2. Simulated Examples

To verify the superiority of the flexible stock and trim loss produced by using FAP over those produced using CSP and TLMP, we selected 5 Cases for comparison, and randomly obtained the (where ), which was achieved by using the RANDBETWEEN function in Microsoft Office Excel 2007. The range of this function was set between (see Table 7). The optimization calculations were then performed for FAP, CSP, and TLMP.

Table 7: Information of for simulated examples.

Case

1
2
3
4
5

1
1000
1
46
2
6
37
9

2
1100
2
3
15
2
47
41

3
1200
3
33
11
31
5
33

4
1300
4
22
21
22
42
27

5
1400
5
39
12
20
36
1

6
1500
6
46
25
6
31
2

7
1600
7
22
26
21
38
12

8
1700
8
34
21
34
6
41

9
1800
9
30
25
31
32
29

10
1900
10
38
14
44
26
48

11
2000
11
1
20
12
48
22

12
2100
12
23
42
37
42
16

13
2200
14
12
20
48
11
48

14
2300
16
6
31
33
2
19

15
2400
19
27
12
41
28
30

16
2500
22
14
38
41
13
27

We compared FAP(), FAP(), CSP, and TLMP, and the results were tabulated in Table 8. Because using FAP necessitates the consideration of the flexible coefficients, FAP() should effectively reduce TLR when an excessively large CSP’s TLR value is produced. Cases 2, 4, and 5 verified that FAP reduced CSP’s TLR. FAP() and TLMP were then examined to determine whether FAP’s TLR presented similarities with TLMP’s TLR. Consequently, the TLR values observed in all the 5 Cases were consistent.

Table 8: The results of TF and TLR for CSP, TLMP, and FAP.

Method
Measure
Case

1
2
3
4
5

CSP
TF
94
98
10
43
0

TLR (%)
0
1.1
1.6
0.5
1.8

TLMP
TF
49
19
6
10
9

TLR (%)
0
0.15
1.2
0.13
1.2

FAP 

(R = 0)
TF
109
115
25
65
26

TLR (%)
0
1
1.6
0.16
1.4

FAP 

(R = −1000)
TF
109
19
6
20
9

TLR (%)
0
0.15
1.2
0.13
1.2

Rolls
144
141
183
162
162

Subsequently, we endeavored to determine whether FAP could effectively increase the flexibility of stock adjustment (see Table 9). The FAP() for Cases 1 and 3 indicated that the stock leniency demonstrated a merging action. In addition, the extended stock was used in all of the case samples. Furthermore, uncut rolls were presented in Cases 1, 2, 4, and 5. Because is the lowest production capacity model, this model is equivalent to CSP. The FAP() for Cases 2, 3, and 5 was similar. However, FAP() presented increased stock adjustment flexibility and extended stock usage in Cases 1 and 4. Thus, FAP() can effectively reduce CSP’s TLR and increase stock adjustment flexibility when TLR is at a minimum level. The TLR in FAP() was equivalent to that of TLMP, which increased stock adjustment flexibility.

Table 9: Stock information for CSP, TLMP, and FAP.

Case
Method
Solution

1
CSP

TLMP

FAP (R = 0)

FAP (R = −1000)

2
CSP

TLMP

FAP (R = 0)

FAP (R = −1000)

3
CSP

TLMP

FAP (R = 0)

FAP (R = −1000)

4
CSP

TLMP

FAP (R = 0)

FAP (R = −1000)

5
CSP
NA

TLMP

FAP (R = 0)

FAP (R = −1000)

Note: {paper width, stock quantity}.

A sensitivity analysis was employed to determine the performance of FAP in the 5 Cases and the influence of on TLR and TF. Consequently, was set at where was an integer. The results are tabulated in Table 10.

Table 10: The sensitivity analysis of coefficient R for all cases using FAP.

Case
Measure
R

1
TLR (%)
0

TF
109

2
TLR (%)
1
0.94
0.76
0.29
0.22
0.15

TF
115
112
91
46
39
19

3
TLR (%)
1.6
1.2

TF
25
6

4
TLR (%)
0.16
0.13

TF
65
20

5
TLR (%)
1.4
1.3
1.2

TF
26
25
9

The medians tabulated in Table 6 were used for data reconstruction and the results are presented in Table 11. Subsequently, we collected the values at each interval for Cases 1, 3, 4, and 5. For Case 2, we were unable to collect the values at intervals of −55–−79, −80–−124, and −125–−156. Decision makers can determine whether they wish to incorporate the medians at these intervals or not; however, this method of incorporating medians can be used to control the majority of TLR changes.

Table 11: The results of fixed values of R for all cases using FAP.

Case
Measure
R

0
−1
−5
−47
−310
−1000

1
TLR (%)
0
0
0
0
0
0

TF
109
109
109
109
109
109

2
TLR (%)
1
1
1
0.94
0.15
0.15

TF
115
115
115
112
19
19

3
TLR (%)
1.6
1.6
1.6
1.2
1.2
1.2

TF
25
25
25
6
6
6

4
TLR (%)
0.16
0.16
0.16
0.16
0.16
0.13

TF
65
65
65
65
65
20

5
TLR (%)
1.4
1.4
1.4
1.3
1.2
1.2

TF
26
26
26
25
9
9

5. Conclusion

The results of the case study analysis indicate that FAP() was similar to CSP in that both methods could be used to determine the minimal production capacity and the maximal, flexible adjusted stock. Because of the unique production characteristics of industrial-use paper, using the CSP method may produce full rolls and, thus, cannot obtain optimized trim loss problems. Similar to the CSP method, FAP() generates stock that cannot be flexibly adjusted, despite possessing minimal trim loss. Furthermore, CSP and TLMP failed to control the changes of TLR; therefore, FAP can utilize to control and maintain TLR in a range betweenCSP and TLMP’TLR. This approach eliminates the trim loss problem exhibited in CSP and the adjustability problem exhibited in TLMP and allows decision makers to effectively control stock and trim loss according to actual situations.

Future research may consider solving extending stock in stock allocation. In addition, the cost effects during the production process should be addressed.

Notations

i: | The index number and is the number of stock/order widths
: | A stock width with
: | An order width with
: | Demand for with
L: | Reel width
NR: | The number of rolls for a reel
GAP: | The difference between two paper widths
: | An extending production width, where is the index number and
pwk: | A production width, where is the index number
: | Quantity for the production width,
: | Flexible coefficient for the production width , with
R: | Flexible coefficient for trim loss
SRQ: | Production scheduling of reel quantity
UB: | Upper bound for trim loss
: | The number of widths in pattern
: | The number of patterns , where is the index number and is the number of patterns.

Conflict of Interests

The authors declare that they have no conflict of interests.

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Lingo Software, Version 11, Lindo Systems, Inc, Chicago, Ill, USA, 2009