An optimization algorithm for cutting stock problems in the TFT-LCD industry

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An optimization algorithm for cutting stock problems in the TFT-LCD industry

Jung-Fa Tsai

a,*

, Ping-Lun Hsieh

a

, Yao-Huei Huang

b

a

Institute of Commerce Automation and Management, National Taipei University of Technology, No. 1, Sec. 3, Chung-Hsiao E. Road, Taipei 10608, Taiwan b

Institute of Information Management, National Chiao Tung University, No. 1001, Ta-Hsueh Road, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Article history:

Received 27 August 2008

Received in revised form 31 December 2008 Accepted 13 March 2009

Available online 24 March 2009 Keywords:

Cutting stock problem Optimization Production

a b s t r a c t

Due to lack of efﬁcient approaches of mixed production, the present production approach of the TFT-LCD industry is batch production that each glass substrate is cut into LCD plates of one size only. This study proposes an optimization algorithm for cutting stock problems of the TFT-LCD industry. The proposed algorithm minimizes the number of glass substrates required to satisfy the orders, therefore reducing the production costs. Additionally, the solution of the proposed algorithm is a global optimum which is different from a local optimum or a feasible solution that is found by the heuristic algorithm. Numerical examples are also presented to illustrate the usefulness of the proposed algorithm.

1. Introduction

This study considers the cutting stock problem (CSP) of TFT-LCD (thin-ﬁlm transistor liquid–crystal display) industry which aims to seek an optimal production schedule of cutting several LCD panels with different sizes from a given glass substrate to meet the orders. The glass substrate is one of important raw materials in the whole manufacturing process of LCD. If an enterprise designs a produc-tion scheme using the minimum number of glass substrates based on the orders received, it can reduce the manufacturing costs and increase the product’s competitiveness in the market.

The CSP attempts to plan the optimal production schedule for minimizing the production costs, i.e. minimum trim loss. Different variants of CSP are available. An important variant of the CSP is the one-dimensional CSP. Many approaches for this problem have been proposed. For instance, Holthaus (2002) considered the integer one-dimensional cutting stock problem with different types of standard lengths and the objective of cost minimization.Umetani, Yagiura, and Ibaraki (2003)designed an approach which is based on meta-heuristics, and incorporates an adaptive pattern genera-tion technique.Gradisar and Trkman (2005)also proposed a com-bined method for the solution to the general one-dimensional cutting stock problem (G1D-CSP).

Another variant of CSP is the two-dimensional CSP. In this var-iant, a set of order pieces is cut from a large supply of rectangular stock sheets of ﬁxed size in a way that minimizes the total cost. Based on this goal, we are interested in ﬁnding ‘cutting patterns’ that minimize the unused area (trim loss). The problem is called

two-dimensional cutting problem. Cutting and Packing problems belong to an old and very well-known family, called CP inDyckhoff (1990)andSweeney and Paternoster (1992). This is a family of nat-ural combinatorial optimization problems.

The two-dimensional cutting and packing problem is widely ap-plied in optimally cutting raw materials such as glass, steel and pa-per, in two-dimensional bin packing, and in layout designing problems. Many scholars have devoted themselves to developing many methods one after another to solve the problem; these meth-ods can be grouped into two major types.

(i) Deterministic: Deterministic methods take advantage of ana-lytical properties of the problem to generate a sequence of points that converge to a global solution. For example,Chen, Sarin, and Balasubramanian (1993)presented a mixed inte-ger programming model for a class of assortment problems.

Li and Tsai (2001)proposed a new method which ﬁnds the optimum of cutting problems by solving few linear mixed 0–1 problems. Li, Chang, and Tsai (2002) developed an approach using the piecewise linearization technique of the quadratic objective function to improve an approximate model for two-dimensional cutting problems.

(ii) Heuristic: Heuristic algorithms can obtain a solution quickly, but the quality of the solution cannot be guaranteed.G and Kang (2001)developed a heuristic that ﬁnds efﬁcient layouts with low complexity for two-dimensional pallet loading problems of large size.Wu, Huang, Lau, Wong, and Young (2002)introduced an effective deterministic heuristic, Less Flexibility First, for solving the classical NP-complete rectangle-packing problem.Leung, Chan, and Troutt (2003)

proposed an application of a mixed simulated annealing–

* Corresponding author. Tel.: +886 2 27712171x3420; fax: +886 2 27763964. E-mail address:[email protected](J.-F. Tsai).

Contents lists available atScienceDirect

Computers & Industrial Engineering

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genetic algorithm heuristic for the two-dimensional orthog-onal packing problem. Beasley (2004) also presented a heuristic algorithm for the constrained two-dimensional non-guillotine cutting problem. The main defect of these heuristic algorithms is that they fail to claim the solution obtained is a global optimum unless the whole solution space is completely searched. Toward TFT-LCD industry involving mass production, the costs can be further reduced substan-tially if a global optimum can be derived instead of a local optimum or a feasible solution. For more detailed articles about the cutting optimization problem, readers can consult

Lodi, Martello, and Monaci (2002) and Valério de Carvalho (2002).

Many approaches for two-dimensional cutting stock problem have also been proposed.Hifi (1997)discussed one of the best-known exact algorithms, due toViswanathan and Bagchi (1993), for solving constrained two-dimensional cutting stock problem optimally. They proposed a modification of this algorithm in order to improve the computational performance of the standard ver-sion.Cung, Hifi, and Cun (2000)developed a new version of the algorithm inHifi (1997)for solving exactly some variants of (un)-weighted constrained two-dimensional cutting stock problems.

Leung, Yung, and Troutt (2001)applied a genetic algorithm and a simulated annealing approach to the two-dimensional non-guillo-tine cutting stock problem and carried out experimentation on

sev-eral test cases. Vanderbeck (2001) developed a nested

decomposition approach for two-dimensional cutting stock prob-lem.Burke, Kendall, and Whitwell (2004)presented a new best-ﬁt heuristic for the two-dimensional rectangular stock-cutting problem and demonstrated its effectiveness.

The two-dimensional cutting stock problem considered in this study is to derive the minimum number of glass substrates based on the available cutting combinations to meet the order demands. Due to lack of efﬁcient approaches of mixed production, the pres-ent production approach is batch production that each glass sub-strate is cut into LCD plates of one size only. However, the computational result shows that the mixed production has a high-er utilization of a glass substrate. Moreovhigh-er, the proposed optimi-zation algorithm can signiﬁcantly reduce production costs to enhance the competitiveness of products.

The main advantages of the proposed method are listed as follows:

1. In comparison with the batch production, the proposed method presents a mixed production approach that generates LCD plates of various sizes in a glass substrate to increase the utili-zation of glass substrates (i.e. total area of produced products/ total area of glass substrates).

2. The proposed method provides the optimal production scheme according to the quantities of the orders.

3. The proposed method is able to ﬁnd out all the alternative solu-tions with the same optimal objective value (i.e., different cutting combinations under the same utilization of a material substrate). The rest of this paper is organized as follows: in Section2, the mathematical models of a cutting optimization problem are formu-lated. In Section3, a cutting stock optimization algorithm is pro-posed. Section 4 presents numerical examples to illustrate the proposed method and concluding remarks are included in Section5.

2. Mathematical models

Since the mixed production has a higher utilization of a glass substrate than batch production, herein we construct some models

to ﬁnd out all cutting combinations in a glass substrate. To facili-tate the discussion, the following notations are introduced ﬁrst:

(l0, w0): The length and width of the glass substrate.

Z: Number of LCD products with different sizes which have to be produced.

(lz, wz): The length and width of the zth LCD product, lzwz lzþ1wzþ1, z = 1, 2, . . ., Z.

J: Index of multiple solutions with the same objective value. T: Number of possible cutting combinations with different objective values.

ct

zj: The cutting quantity of the zth product of the jth alternative solution with the same objective value in the tth iteration, z = 1, 2, . . ., Z, t = 1, 2, . . ., T, j = 1, 2, . . ., J.

First, let Obj(0) = l0w0, consider the following model: 2.1. Model 1.1 Max ObjðtÞ ¼X Z z¼1 ct z1ðlzwzÞ ð1Þ subject to ObjðtÞ Objðt 1Þ: ð2Þ

This model aims to ﬁnd the possible cutting combinations with different objective values. For t = 1, the constraint Obj(1) 6 Obj(0) represents to use the maximum portion of a glass substrate. For t = 2, 3, . . . T, the constraint Obj(t) 6 Obj(t 1) represents that the utilization of a glass substrate decreases as t increases. After each t iteration, we can obtain a solution and an objective value. Because distinct solutions may exist under the same objective value, we de-velop the following model for ﬁnding multiple solutions:

2.2. Model 1.2 MaxX Z z¼1 ct zjðlzwzÞ ð3Þ subject to XZ z¼1 ct zj c t z;j1 P 1; for all j: ð4Þ

If the objective value obtained of Model 1.2 is equal to Obj(t) deriving from Model 1.1 in the tth iteration, then alternative solu-tions exist. The purpose of constraint(4)is to search for an alterna-tive solution. For example, let Obj(0) = 200, (l1, w1) = (5, 5), (l2, w2) = (5, 10) and (l3, w3) = (10, 10). The possible cutting combi-nations can be obtained by using Models 1.1 and 1.2 as follows:

Step 1: Let t = 1, j = 1, and Obj(0) = 200. Using Model 1.1 to ﬁnd the possible cutting combination, we have the following model:

Max Objð1Þ ¼ c1 11ð5 5Þ þ c121ð5 10Þ þ c131ð10 10Þ s:t: Objð1Þ Objð0Þ; c1 11;c 1 21;c 1 312 integer:

Solving the above model, we can obtain a possible cutting com-bination ðc1

11;c121;c131Þ ¼ ð8; 0; 0Þ; and the objective value is Obj(1) = 200.

Step 2: Using Model 1.2 to ﬁnd the alternative solutions, we have the following model:

Max c1 12ð5 5Þ þ c 1 22ð5 10Þ þ c 1 32ð10 10Þ s:t: jc1 12 8j þ jc122 0j þ jc132 0j 1; c121;c122;c1322 integer:

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Solving the above model, we can ﬁnd an alternative solution ðc1

12;c122;c132Þ ¼ ð6; 1; 0Þ with the same objective value 200. By performing this process iteratively, all the alternative solutions can be found. However, the absolute terms in constraint (4)

must be linearized so that Model 1.2 can be transferred into

a linear programming problem. Consider the following

propositions for linearizing an absolute term contained in a constraint:

Proposition 1. Let

a

2 f0; 1g; b 0 then:

8 > < > : Proof 1.

(i) If x a P 0, then

a

= 0, b = 0 based on (i) and (iii); which results in x a + 2a

a

2b = x a.

(ii) If x a 6 0, then

a

= 1, b = x based on (i) and (ii); which results in x a + 2a

a

2b = a x. h

By Proposition 1, Model 1.2 is equivalently transformed into an-other linear program formulated as below.

2.3. Model 1.3 MaxX Z z¼1 ct zjðlzwzÞ subject to XZ z¼1 ðct zj ctz;j1þ 2ctz;j1

az

2bzÞ 1; for all j; ð5Þ M

az

ct

zj ctz;j1 Mð1

az

Þ; for all j and z; ð6Þ

0 bz M

az

; for all j and z; ð7Þ

Mð

az

1Þ þ ct

zj bz Mð1

az

Þ þ ctzj; for all j and z; ð8Þ where

a

zare 0–1 variables, M is a large constant and the other vari-ables are deﬁned as before.

Model 1.3 is a linear programming problem solvable to obtain a global optimum and capable of ﬁnding out all possible cutting combinations even there are multiple solutions.

According to the above discussions, for the iteration of t (t = 1, 2, . . ., T), Model 1.1 is applied to solve the possible cutting combinations with different utilizations of a glass substrate. For the iteration of j, Model 1.3 is applied to solve the multiple solu-tions under a certain utilization derived from Model 1.1. Therefore some possible cutting combinations can be acquired by Model 1.1 and Model 1.3. However Models 1.1 and 1.3 only check the total measure of the LCD plates is less than that of a glass substrate. Next this study aspires to verify all rectangles of each cutting combina-tion can be allocated into a glass substrate. Suppose the lengths and widths of n rectangles are given. A two-dimensional cutting optimization problem is to allocate all of these rectangles within an enveloping rectangle on x-axis and y-axis which occupies min-imum area. The concept of the problem is stated as follows: 2.4. Model 2.1

Min xy

subject to

1. All of the n rectangles are non-overlapping.

2. All of the n rectangles are within the range of x and y. 3. 0 < x 6 l0and 0 < y 6 w0.

The related terminologies used in the model, referring toLi and Tsai (2001)are stated below:

(x, y): The top right corner coordinates of the enveloping rectangle.

(pi, qi): The dimension of rectangle i, piis the long side and qiis the short side, piand qiare constants, i 2 K, K is the set of given rectangles.

x0

i: Distance between center of rectangle i and original point along the x-axis.

y0

i: Distance between center of rectangle i and original point along the y-axis.

si: An orientation indicator for rectangle i, i 2 K. si= 1 if piis par-allel to the x-axis; si= 0 if piis parallel to the y-axis.

The conditions of non-overlapping between rectangles i and k can be reformulated by introducing two binary variables uik,

v

By Model 2.2, we know whether we are able to use the least square measure of the lager rectangle to produce the needed smal-ler rectangles or not. In addition, we try our best to make the sur-plus area centralized for cutting even smaller products once again.

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By using Model 2.2 to examine all possible cutting combina-tions, we can get the feasible cutting combinations for designing the optimal production plan. Assume that cm_{¼ ðc}m

1;cm2; . . . ;cmZÞ de-notes the mth feasible cutting combination, ozrepresents the quan-tity ordered of the zth product and gmdenotes the number of glass substrates that have to be cut by the mth feasible cutting combina-tion. In order to minimize the total quantity of glass substrates re-quired for fulﬁlling the demand of orders, the optimal production model is formulated as follows:

2.6. Model 3 Min X M m¼1 gm ð18Þ subject to XM m¼1 cm zgmPoz; for z ¼ 1; 2; . . . ; Z; ð19Þ where gm2 integer.

According to the solution of Model 3, we can know the min-imal number of the required glass substrates and how many glass substrates are cut by each feasible cutting combination, respectively.

3. Solution algorithm

The algorithm of cutting stock problems is described as follows:

Input:{(l0, w0), (lz, wz), oz, t = 1, and m = 1} Processes:

{

Step 1: Let j = 1 and do Model 1.1 if ðct

zj¼ 0Þ for all z then go to Step 5 Step 2: Compare cm

z with ctzj if ðcm

z >ctzjfor some m or ctzj¼ 0Þ then go to Step 5 if ðcm

z ctzjfor some m) then go to Step 4 Step 3: Verify the feasibility of combination ct

zj do Model 2.2

if (feasible) then ðcm

1;cm2; . . . ;cmZÞ ¼ ðct1j;ct2j; . . . ;ctZjÞ and m = m + 1 Step 4: Find possible cutting combination in the same utilization

j = j + 1 and do Model 1.3

if (Objective of Model 1.3 = Obj(t)) then go to Step 2 else t = t + 1 and go to Step 1

Step 5: Find the optimal production combination if (m P 2) do Model 3

} Output: {

if (m P 2) output (the optimal production combination (g1, g2, . . ., gM)) else output (no feasible production combination)

}

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According to the above algorithm, we can obtain the optimal pro-duction scheme to produce plates most efﬁciently. The process of the algorithm is depicted inFig. 1.

4. Example

In a TFT-LCD plant, assume the dimension of the glass substrate is 150 cm 180 cm. This plant wants to produce three kinds of products (40 in., 42 in. and 46 in.). The ratio of length to width of the LCD plates is 16:10. The information of these products is listed inTable 1.

To enhance the understanding of the proposed algorithm, the following illustrates the solution process of the example problem step by step.

Initial: Let t = 1 and m = 1.

Step 1-1: Let j = 1. By using Model 1.1, we ﬁnd the solution (2, 1, 2) with the objective value 26,620.

Step 1-2: Because current feasible cutting combination set is empty, proceed straight to Step 3.

Step 1-3: Verify the possible cutting combination (2, 1, 2) by model 2.2. The result reveals that (2, 1, 2) is infeasible. There-fore, remove (2, 1, 2).

Step 1-4: Let j = 2 and add the constraint

jc1

12 2j þ jc122 1j þ jc132 2j 1 to Model 1.2 for ﬁnding alter-native solutions. By solving Model 1.3, we ﬁnd that the objec-tive obtained is no equal to 26,620. Then let t = 2 and go to Step 1.

Step 2-1: Let j = 1 and solve Model 1.1. We ﬁnd another possible cutting combination (0, 4, 1) with the objective value 26,360. Step 2-2: The feasible cutting combination set is empty, then proceed to Step 3.

Step 2-3: Verify the feasibility of (0, 4, 1) by Model 2.2. The result shows that it is infeasible. Therefore, remove (0, 4, 1).

jc2

12 0j þ jc222 4j þ jc232 1j 1 to Model 1.2 for ﬁnding alter-native solutions. By solving Model 1.3, the objective value found is not equal to 26,360. Then let t = 3 and go to Step 1.

Step 3-1: Let j = 1 and solve Model 1.1. We ﬁnd another possible cutting combination (0, 5, 0) with the objective value 25,200. Step 3-2: The feasible cutting combination set is empty, then proceed to Step 3.

Step 3-3: Verify the feasibility of (0, 5, 0) by Model 2.2. The result indicates that (0, 5, 0) is feasible. Then record the feasible cut-ting combination ðc1

1;c12;c13Þ ¼ ð0; 5; 0Þ and let m = 2.

jc3

12 0j þ jc322 5j þ jc332 0j 1 to Model 1.2 for finding alter-native solutions. By solving Model 1.3, the objective value obtained is not equal to 25,200. Then let t = 4 and go to Step 1. Step 4-1: Let j = 1 and solve Model 1.1. We find another possible cutting combination (3, 1, 1) with the objective value 25,010. Step 4-2: (3, 1, 1) does not meet the stop condition and there is no feasible cutting combination satisfies the condition, cm

1 3; cm2 1; cm3 1 for some m. Then proceed to Step 3. Step 4-3: Verify the feasibility of (3, 1, 1) by Model 2.2. The result shows that (3, 1, 1) is infeasible. Therefore, remove (3, 1, 1).

jc4

12 3j þ jc422 1j þ jc432 1j 1 to Model 1.2 for ﬁnding alter-native solutions. By solving Model 1.3, the objective value acquired is not equal to 25,010. Then let t = 5 and go to Step 1. Continuing the solution process, we ﬁnd that the possible cut-ting combination (0, 0, 3) is also a feasible cutcut-ting combination. Then add (0, 0, 3) into the feasible cutting combination set. After several iterations, we get the cutting combination (0, 0, 2). This

cut-ting combination satisﬁes the termination condition

cm

z ctzj; z ¼ 1; 2 and cm3 >ct3j (i.e., 3 > 2) for some m. Therefore, stop the solution process.

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According to the orders of products and the obtained set of fea-sible cutting combination as shown inTable 2, we solve Model 3 with LINGO (2004)to ﬁnd the best objective value 700 and the optimal solution g1= 200, g7= 500 and the other variables are zero. The solution process of this problem takes 546 s by using a Pentium 4 CPU 3.2 G PC. The graphic solutions of the optimal solutions are shown inFigs. 2 and 3.

If the plant adopts the production way suggested by the pro-posed algorithm, it can use g1and g7to achieve the highest utiliza-tion of the glass substrate and only consume 700 glass substrates to accomplish the orders. By the batch production approach, the number of glass substrates needed to fulﬁll the orders is 734. Addi-tionally, the difference of the order quantities does not change the result that the proposed method is superior to the batch

produc-tion.Table 3lists results compared between the proposed method and the batch production of four examples with different order quantities. Since the computational time is not affected by the or-der quantities very much, the computation times of these four problems are all about 550 s. The results demonstrate that the pro-posed algorithm can solve the cutting stock problems of big size products in the TFT-LCD industry effectively.

In the proposed algorithm, the process of finding all feasible cutting combinations is the most time-consuming according to our testing. The computational time mainly depends on how many various LCD plates can be cut from a glass substrate. Therefore, the proposed method is suitable for the cutting stock problem of big size LCD products. The problems with more than 30 various small LCD plates that have to be cut from a glass substrate may integrate some heuristic or distributed algorithms to reduce the computa-tional time for finding the feasible cutting combinations and that is an interesting issue for further research. The extended problems with the same size of glass substrate 180 cm 150 cm and eight products listed inTable 4are solved within 10 min. The results shown inTable 5reveal that the proposed method utilizes fewer glass substrates than the batch method and has significant saving ratios.

Fig. 3. The graphic solution of the optimal cutting combination (2, 0, 2).

Table 1

Dimensions and orders of the products.

Product (in.) Length (cm) Width (cm) Area Order

40 85 54 4590 1000

42 90 56 5040 1000

46 100 62 6200 1000

Table 2

Feasible cutting combinations. Feasible cutting combinations ðcm

1;cm2;cm3Þ m = 1: (0, 5, 0) m = 2: (1, 4, 0) m = 3: (2, 3, 0) m = 4: (3, 2, 0) m = 5: (4, 1, 0) m = 6: (5, 0, 0) m = 7: (2, 0, 2) m = 8: (0, 3, 1) m = 9: (1, 2, 1) m = 10: (2, 1, 1) m = 11: (3, 0, 1) m = 12: (0, 0, 3) m = 13: (0, 1, 2) Table 3

Differences between the mixed production and the batch production. Order quantities

(40 in., 42 in., 46 in.)

Number of glass substrates Saving Proposed method (m) Batch method (b) Quantities Ratio (%) (1000, 2000, 3000) 1567 1600 33 2.06 (2000, 1000, 3000) 1534 1600 66 4.13 (3000, 2000, 1000) 1300 1334 34 2.55 (1000, 1000, 1000) 1400 1467 34 4.57 Saving ratio = 1 (m/b).

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5. Conclusions

This study proposes a cutting stock optimization method for TFT-LCD industry, which can find the optimal cutting way accord-ing to the quantities of orders. The optimization techniques for finding alternative solutions and the approach for linearizing absolute terms are also presented. The results of numerical exam-ples illustrate the usefulness of the proposed method. Especially to the TFT-LCD industry which needs mass production, an effec-tive method can reduce production costs and promote the com-petitiveness of products. The directions for further research are to take more production situations and factors into consideration, such as the costs, the defect rate and the time of delivery and to integrate heuristic or distributed algorithms to enhance the com-putational efficiency. Using column generation techniques to solve this problem is also a very interesting topic for further investigation.

References

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Table 4

Dimensions and orders of the products.

LCD panels Order #

Product # (Length, width) 1 2 3 4

1 (90, 56) 5500 8500 9000 9900 2 (93, 60) 7700 8500 9000 10,000 3 (99, 63) 8100 8500 8500 8950 4 (95, 66) 8500 8700 8750 9550 5 (98, 64) 9000 9900 9950 10,500 6 (100, 62) 9500 9650 10,500 11,550 7 (108, 72) 13,000 13,500 15,550 17,000 8 (126, 82) 15,000 15,500 20,500 21,550 Table 5

Comparison results of the proposed method and batch method. Order # Number of glass substrates Saving

Proposed method (m) Batch method (b) Quantities Ratio (%)

1 24,715 34,702 9987 28.78

2 26,350 36,835 10,485 28.46

3 30,042 43,052 13,010 30.22

4 32,268 46,049 13,781 29.93