DOI 10.1007/s11590-012-0598-x
O R I G I NA L PA P E R
A note on “Reducing the number of binary variables
in cutting stock problems”
Hao-Chun Lu · Yu-Chien Ko · Yao-Huei Huang
Received: 17 February 2011 / Accepted: 30 November 2012 / Published online: 22 December 2012 © Springer-Verlag Berlin Heidelberg 2012
Abstract This study proposes a deterministic model to solve the two-dimensional
cutting stock problem (2DCSP) using a much smaller number of binary variables and thereby reducing the complexity of 2DCSP. Expressing a 2DCSP with m stocks and n cutting rectangles requires 2n2+ n(m + 1) binary variables in the traditional model.
In contrast, the proposed model uses n2+ nlog2m binary variables to express the
2DCSP. Experimental results showed that the proposed model is more efficient than the existing model.
Keywords Deterministic model· Cutting stock problem · Binary variables
1 Introduction
This study considers the two-dimensional cutting stock problems (2DCSPs) in real-world applications, such as cutting steel tubes, paper tubes, carpet, and glass. The 2DCSP seeks optimal cutting patterns to minimize the total number of stocks required to fulfill orders and reduce the total amount of scrap for each stock in a schedule. For example, in paper mills, paper tubes (i.e., raw materials) are cut into different
H.-C. Lu
Department of Information Management, College of Management, Fu Jen Catholic University, No.510, Jhongjheng Road, Sinjhuang City, Taipei County 242, Taiwan
Y.-C. Ko
Department of Information Management, Chung Hua University, 707, Sec.2, WuFu Road, Hsinchu 300, Taiwan
Y.-H. Huang (
B
)Institute of Information Management, National Chiao Tung University, Management Building 2, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan, ROC e-mail: yaohuei.huang@gmail.com
products with different sizes [6,13] according to customer requirements, and the trim loss of the tubes needs to be minimized. Examples of the cutting stock problems in other fields include placing all devices into a system-on-a-chip circuit [24], container loading and shipping problems in the transport industry [28], and cutting Thin Film Transistor-Liquid Crystal Display (TFT-LCD) plates from glass substrate [31]. An optimal production scheme minimizes the number of stock sheets required to com-plete customer orders, thereby reducing manufacturing costs and increasing company competitiveness.
Research on optimal solutions for trim-loss problems dates back to several decades, as shown by the classic papers of Gilmore and Gomory [11], Chambers and Dyson [3], and Hinxman [14]. Their methods attempted to minimize the stock wastage subject to customer demands, setup costs, processing times, and characteristics of cutting patterns. More recently, Holthaus [15] has proposed an integer decomposition method using different types of patterns of standard length, whereas Umetani et al. [32] utilized meta-heuristics and adaptive pattern generation techniques to minimize the number of patterns. Gradisar and Trkman [12] developed a mixed hybrid approach that combined sequential heuristic procedures to improve the performance of the branch-and-bound algorithm. In some works, the trim-loss problem is called the “(strip) packing problem” or the “loading problem” [8,9,17,26,27,30].
The 2DCSP concept was first proposed by Brooks et al. [2]. Since then, various methods based on different algorithms have been developed. Generally, these algo-rithms can be divided into two classes.
(i) Deterministic algorithms: deterministic algorithms are based on mathematical programs, which utilize the branch-and-bound algorithm to derive an optimal solution. For example, Chen et al. [4] proposed a mixed integer programming model for a class of assortment problems, and their model minimized trim-loss in only one rectangular area. Li and Chang [20] and Li et al. [21,22] reformu-lated the mathematical model to improve the approximate solution and speed up the computation time. However, the above models were not suitable for treating 2DCSP. If the Li’s original model for solving the 2DCSP is directly extended, a large number of binary variables may be required, resulting in high computational complexity [10].
(ii) Heuristic algorithms: numerous heuristic approaches are available in the liter-atures, and their main advantage lies in the ease in solving 2DCSP within an acceptable time. For example, Jakobs [17] developed an application of genetic algorithms to solve the two-dimensional packing problems. Leung et al. [18,19] also proposed a mixed simulated annealing-genetic algorithm for the two-dimensional packing problems, and Lin [25] designed a genetic algorithm that incorporated a novel random packing process and an encoding scheme to solve a special 2DCSP within one stock. In real-world cases, minimum trim loss in the 2DCSP is an important issue within an acceptable solving time, such as cut-ting rectangular steel bars in manufacturing, and guillotine-cutcut-ting problem in paper industry [1,12,27,29]. Column generation algorithms are also widely used to solve the 2DCSPs [5,33]. Cui et al. [7] developed a recursive version of the branch-and-bound algorithm to obtain an approximate solution, whereas Tsai
et al. [31] proposed a cutting stock algorithm for the TFT-LCD industry. The latter algorithm sought a feasible fixed-size cutting pattern for raw materials to minimize the number of stocks required to satisfy customer requirements. All the above algorithms can find easily feasible solutions; however, the solution quality cannot be guaranteed.
Based on comparisons of the above works, we propose a novel deterministic model to solve the 2DCSP. The advantages of the proposed model are as follows:
(i) It solves 2DCSPs effectively using a much smaller number of binary variables. (ii) It guarantees that an optimal solution is achievable.
The rest of this paper is organized as follows. In Sect.2, we discuss existing reference models for solving 2DCSPs, and in Sect.3, we introduce the proposed model, which uses a much smaller number of 0–1 variables. The results of the numerical tests on practical examples are presented in Sect.4. Section5presents some concluding remarks.
2 Reference model
The parameters and decision variables used in this paper are listed below [4,20]:
Parameters
m The number of the stock sheets.
S The set of cutting rectangles, S= {1, 2, . . . , n}. (pi, qi) The length and width of the cutting rectangle i, i ∈ S,
( pi and qi are constants).
(Width, Length) The length and width of the stock sheet.
Decision variables
(X, Y ) The top right-hand corner coordinates of the enveloping
rectangle.
(xi, yi) The bottom-left coordinates of the cutting rectangle i, i ∈ S(xi
and yi are variables).
si An orientation indicator for the given cutting rectangle
i, i ∈ S. si = 1 if pi is parallel to the x-axis; otherwise,
si = 0 if piis parallel to the y-axis (si denotes a binary
variable).
(ai, j, bi, j, ci, j, di, j) The non-overlapping condition for a pair of cutting rectangles(i, j).
Chen et al. [4] proposed mixed-integer program as a basic model for 2DCSP using only one stock sheet, aimed at minimizing the size of the stock sheet, also called an assortment problem. The concept of the basic model proposed by Chen et al. [4] is introduced as follows:
P1 (basic model)
Min X Y
s.t.(i) all the rectangles are non-overlapping,
(ii) all the rectangles are within the range of X and Y.
Li and Chang [20] proposed a method that employed a much smaller number of binary variables to reformulate the non-overlapping constraints, and Li et al. [21] tried to linearize approximately the cross term (i.e., X Y) in the objective function by using a piecewise linearization technique. The accuracy of the approximative solution depends on the number of break points in piecewise linearization [23]. As the cross term is a nonlinear programming problem that is difficult to solve when searching for an optimal solution and the original model is only suitable for assortment problem, reformulating P1 (basic model) is necessary to solve 2DCSP. We modify the objective function in P1 as Min Y , fix the width of the stock sheets as a given value, and give the number of stocks as m. P1 can be extended to another model for general 2DCSPs. The specific 2DCSP linear program is reformulated as follows:
P2 (modified 2DCSP model) Min Y s.t. (xi − xj) + M(1 − ai, j) ≥ pjsj + qj(1 − sj), i, j ∈ S, i < j, (1) (xj− xi) + M(1 − bi, j) ≥ pisi + qi(1 − si), i, j ∈ S, i < j, (2) (yi − yj) + M(1 − ci, j) ≥ qjsj+ pj(1 − sj), i, j ∈ S, i < j, (3) (yj− yi) + M(1 − di, j) ≥ qisi+ pi(1 − si), i, j ∈ S, i < j, (4) ai, j+ bi, j+ ci, j+ di, j = 1, i, j ∈ S, i < j, (5) xi+ pisi+ qi(1 − si) ≤ Width, i ∈ S, (6) yi+ qisi+ pi(1 − si) ≤ Length · m k=1 (Qi,k· k), i ∈ S, (7) yi ≥ Length · m k=1 (Qi,k· (k − 1)), i ∈ S, (8) Y ≥ yi+ qisi+ pi(1 − si), i ∈ S, (9)
where ai, j, bi, j, ci, j, di, j, Qi,k, si ∈ {0, 1}, and M is a sufficiently large constant.
Here, Constraints (1)–(5) ensure that the rectangles are non-overlapping. Constraints (6)–(8) indicate that each rectangle is fitly packed into only one of the stock sheets. The decision variable Y in Constraint (9) denotes the length of the accumulated stock sheets.
Remark 1 In the P2 model, the numbers of binary variables and constraints are 2n2+
To reduce the complexity of 2DCSP (i.e., 2n2+ n(m + 1) binary variables), we propose a novel model that uses a much smaller number of binary variables in 2DCSP (i.e., fixed-width stocks). The model is described in detail in the next section.
3 Proposed model
We first introduce the binary vector wi = (wi,1, wi,2, . . . , wi,θ), where i denotes a
small rectangle cut from the kth stock sheet for k= 1, . . . , m, and θ = log2m. We
then have the following expressions:
k= 1 +
θ
r=1
2r−1wi,r, θ = log2m, wi,r ∈ {0, 1}. (10)
Let S(k) ⊆ {1, . . . , θ} be a subset of indexes such that
k= 1 +
r∈S(k)
2r−1. (11)
In addition, let|S(k)| be the number of elements in S(k); for example, S(1) = φ and |S(1)| = 0, S(2) = {1} and |S(2)| = 1, S(4) = {1, 2} and |S(4)| = 2, and so on. We then introduce the following propositions:
Proposition 1 Define m equations to represent a binary vector wi based on
(wi,1, . . . , wi,θ) as follows: Fk(wi) = |S(k)| − r∈S(k) wi,r+ r/∈S(k) wi,r for k= 1, . . . , m. (12) Proof (i) If k= 1+rθ=12r−1wi,r, then|S(k)| =
r∈S(k)wi,rand
r/∈S(k)wi,r =
0, which ensures that Fk(wi) = 0.
(ii) If k= 1 +θr=12r−1wi,r, then|S(k)| > r∈S(k)wi,rand r/∈S(k)wi,r ≥ 0 or |S(k)| =r∈S(k)wi,r and
r/∈S(k)wi,r ≥ 1, which ensures that Fk(wi) ≥ 1.
(iii) We then prove that Fk(wi) = 0 if and only if k = 1+
θ
r=12r−1wi,r; otherwise,
Fk(wi) ≥ 1.
Remark 2 Onlylog2m binary variables are used in the Proposition1.
We utilize the Proposition1 to express m equations using onlylog2m binary
variables. As it is straightforward to introduce Proposition1, we can derive the Propo-sition2for 2DCSP afterward.
Proposition 2 n small rectangles need to be cut from m stock sheets. Referring to the Proposition1, we introduceθ = log2m binary variables (i.e., wi,r f or r =
1, . . . , θ) to express m functions (i.e., Fk(wi)) for each cutting rectangle i (i ∈ S) cut exactly from one of the stock sheets as follows:
1+ θ r=1 2r−1wi,r ≤ m, i ∈ S, (13) yi+ (k − 1) · Length · Fk(wi) ≥ (k − 1) · Length, i ∈ S, k = 1, . . . , m, (14) yi+ qisi+ pi(1 − si) − (m − k) · Length · Fk(wi) ≤ k · Length, i ∈ S, k = 1, . . . , m, (15)
where the Lengt h means the limited length of a stock sheet and the function Fk(wi) is the same as Eq. (12).
Proof Based on the Proposition1, if Fk∗(wi) = 0 (k∗is an arbitrary integer, and k∗∈
{1, . . . , m}), then the other Fk(wi) ≥ 1 (k = 1, . . . , m, and k = k∗). Constraints
(14) and (15) will only be active if (i) yi ≥ (k∗− 1) · Length,
(ii) yi + qisi+ pi(1 − si) ≤ k∗· Length.
On the other hand, Constraints (14) and (15) will be inactive, and Fk(wi) ≥ 1. The
conditions will ensure that rectangle i is cut from the stock sheet k∗. The Proposition2
is therefore proven.
The P2 model refers to Chen’s method (1994), whose model used four binary vari-ables(ai, j, bi, j, ci, j, di, j) to handle the non-overlapping issue of the pair of rectangular
items(i, j). By referring to Li’s model, the non-overlapping issue can be expressed using only two binary variables(ui, j, vi, j) to reduce the complexity of 2DCSP.
Based on Li’s model, P2 , and the Proposition2, a novel model of 2DCSP can be formulated using a much smaller number of binary variables as follows:
P3 (proposed 2DCSP model)
Min Y
s.t. (13)−(15),
(xi−xj)+Width(ui, j+vi, j) ≥ pjsj+qj(1−sj), i, j ∈ S, i < j, (16)
(xj−xi)+Width(1−ui, j+vi, j) ≥ pisi+qi(1−si), i, j ∈ S, i < j, (17)
(yi−yj)+m · Length · (1+ui, j−vi, j)≥qjsj+ pj(1−sj), i, j ∈ S, i < j,(18)
(yj−yi)+m · Length · (2−ui, j−vi, j)≥qisi+ pi(1−si), i, j ∈ S, i < j, (19)
xi+ pisi+ qi(1 − si) ≤ Width, i ∈ S, (20)
yi+ qisi+ pi(1 − si) ≤ Y, i ∈ S, (21)
xi ≥ 0, yi ≥ 0, i ∈ S, (22)
where ui, j, vi, j, wi,r, si ∈ {0, 1}.
Remark 3 P3 requires n2+nlog2m binary variables and 2n2+n(2m+1) constraints.
By comparing Remark3with Remark1, the complexity of the proposed 2DCSP model is much less than that of the original model. The numerical experiments
conducted to evaluate the performance of the proposed model are discussed in the next section.
4 Numerical experiments
Two numerical examples are presented to demonstrate the effectiveness of the pro-posed model and to compare its performance with that of the original model. The first example is the sound box design assembly problem, and the second is the TFT-LCD cutting stock problem. In both cases, the objective is to minimize the number of stocks required to satisfy customer’s requirements. The numerical examples were coded in IBM ILOG CPLEX [16] environment, and run on a PC with an Intel Pentium(D) 2.8 GHz CPU and 2 GB RAM.
Example 1 This problem arises in the sound box design, which requires cutting of
rectangular items from a standard size piece of wood (60 cm× 110 cm), i.e., Width = 60 and Lengt h = 110 for each stock. The parameters of the required plates are shown in Table1. The given number of stock sheets is m = 4. The proposed model utilizes two binary variables,wi,1andwi,2, to construct Fk(wi) for each plate because
log24 = 2. The problem is formulated as follows:
Min Y s.t. (16)−(22), 1+ wi,1+ 2wi,2 ≤ 4, i = 1, . . . , 7, yi+ 440(wi,1+ wi,2) ≥ 0, i = 1, . . . , 7, yi+ qisi + pi(1 − si) + 440(wi,1+ wi,2) ≤ 110, i = 1, . . . , 7, yi+ 440(1 − wi,1+ wi,2) ≥ 110, i = 1, . . . , 7, yi+ qisi + pi(1 − si) + 440(1 − wi,1+ wi,2) ≤ 220, i = 1, . . . , 7, yi+ 440(1 + wi,1− wi,2) ≥ 220, i = 1, . . . , 7, yi+ qisi + pi(1 − si) + 440(1 + wi,1− wi,2) ≤ 330, i = 1, . . . , 7, yi+ 440(2 − wi,1− wi,2) ≥ 330, i = 1, . . . , 7, yi+ qisi + pi(1 − si) + 440(2 − wi,1− wi,2) ≤ 440, i = 1, . . . , 7.
This problem is solved using CPLEX. The maximal number of binary variables and linear constraints required under the proposed model is 63 and 161, respectively. Moreover, the optimal solution Y = 277 is obtained in a feasible number of iterations and at a reasonable time (i.e., iterations = 1,066,902 and CPU time = 274.36 seconds), corresponding to three pieces of stock. Table2lists the results of the original and the proposed models, and Fig.1shows the solution in graphical form.
Table 1 The parameters of the required plates in Example1
#1 (82,60) #2 (90,30) #3 (85,27) #4 (60,25) #5 (60,20) #6 (55,29) #7 (57,30)
Table 2 Experiment results of Example1
Items 0–1 variables # of constraints Iterations CPU Time (s)
P2 (reference model) 196 448 n/a n/a
P3 (proposed model) 63 105 1,066,902 274.36
m= 4. The CPU time in P1 is outside the limit (time >2,000)
100 40 30 20 10 110 50 60 70 80 90 0 20 40 60 0 20 40 60 100 40 30 20 10 110 50 60 70 80 90 1 m= m=2 m=3 #5 #2 #3 #1 #4 (33,110) (60,110) (3,110) (33,25) (33,20) (3,20) (6,20) (60,107) (60,82) 0 100 40 30 20 10 110 50 60 70 80 90 20 40 60 #6 #7 (29,55) (30,57) (60,57) (29,0) (30,0)
Fig. 1 Graphic solution of Example1
Example 2 In the TFT-LCD industry example [30], the dimensions of each glass substrate (i.e., stock) are fixed at (150 cm× 180 cm), where the number of stocks is four. Assuming that a certain production line needs 18 different-sized products, as shown in Table3, the problem is formulated as follows:
Min Y
s.t. (16)−(22),
1+ wi,1+ 2wi,2 ≤ 4, i = 1, . . . , 18,
yi+ 720Fk(wi) ≥ 180(k − 1), i = 1, . . . , 18, k = 1, . . . , 4,
yi+ qisi+ pi(1 − si) − 720Fk(wi) ≤ 180k, i = 1, . . . , 18, k = 1, . . . , 4.
This problem is also solved using CPLEX. The maximal number of binary variables and linear constraints required under the proposed model is 360 and 810, respectively. Moreover, the optimal solution Y = 525 is obtained in a feasible number of iterations and at a reasonable time (i.e., approximately 24 million iterations and 2,000 s), corre-sponding to three pieces of stocks. The result of Example2under P2 is not available due to the limited solution time (i.e., insufficient memory). Table2lists the results of
Table 3 Eighteen kinds of products in Example2 #1 (130,30) #2 (130,10) #3 (120,25) #4 (100,100) #5 (95,95) #6 (90,90) #7 (95,85) #8 (80,80) #9 (80,75) #10 (70,70) #11 (60,60) #12 (55,50) #13 (40,40) #14 (50,40) #15 (100,30) #16 (45,20) #17 (20,15) #18 (25,10) Plate #( pi, qi)
Fig. 2 Graphic result of the TFT-LCD example
Table 4 Experiment results of Example2
Items 0–1 variables # of constraints Iterations CPU time (s)
P2 (reference model) 1,296 2,736 n/a n/a
P3 (proposed model) 360 655 24,879,402 2,023.94
m= 4. The CPU time in P1 is outside the limit (time >5,000)
the original and the proposed models, and Fig.2shows the solution in graphical form (Table4).
The results of Examples1and2demonstrate that, compared with the reference model, the proposed model requires a logarithmic number of binary variables to for-mulate a model of 2DCSP, and the binary variables are used to ensure that each assigned rectangles is exactly cut from one of the stocks. Thus, it is computationally more efficient due to the reduction of the complexity of binary variables. From this point of view, Example2is considered as an illustration which randomly generates different-sized products with various numbers of 0–1 variables, and the stock size is also fixed at (150 cm× 180 cm). After solving ten tests, we investigate the tendency of P2 and P3 using different m and n with various 0–1 variables, as shown in Fig.3. Here, we mark the running time for each test. Examining the results of these tests, we have the following observations:
(i) Both models were able to find the same optimal solution for each of the first five tests (tests 1–5).
222 362 562 699 775 843 1043 1288 1987 5023 Out of limitation Out of limitation Out of limitation Out of limitation Out of limitation 401 980 1311 2974 9921 (n/m) # of bin a ry va ri a bles (Test #) #1 #2 #3 #4 #5 #6 #7 #8 #9 #10
Fig. 3 Trend of CPU time in the ten tests
(ii) Owing to limitation problem (CPLEX default setting) caused by the large number of 0–1 variables and constraints, P2 failed to provide solution in 3 h in our experiments using n≥ 12 and m ≥ 6.
(iii) P3 successfully solved all ten tests within the default limitation of the CPLEX software.
5 Conclusions
We have proposed a deterministic model that only requires logarithmic binary vari-ables and additional constraints to solve 2DCSPs. Compared with the current model, the proposed model can solve the same problem with larger scale size. On the other hand, to obtain a feasible solution within a reasonable time, merging the column gener-ation techniques, distributed algorithms, or heuristic methods (i.e., genetic algorithms, simulated annealing, and tabu-search) is a sensible practice direction to enhance the computational efficiency in future research.
Acknowledgments The authors would like to thank the editor and anonymous referee for providing most valuable comments for us to improve the quality of this manuscript. This research was supported by the project granted by ROC NSC 99-2221-E-030-005- and NSC 100-2221-E-030 -009-.
References
1. Beasley, J.E.: An algorithm for the two-dimensional assortment problem. Eur. J. Oper. Res. 19, 253–261 (1985)
2. Brooks, R.L., Smith, C.A.B., Stone, A.H., Tutte, W.T.: The dissection of rectangles into squares. Duke Math. J. 7(1), 312–340 (1940)
3. Chambers, M.L., Dyson, R.G.: The cutting stock problem in the flat glass industry selection of stock sizes. Oper. Res. Q. 27(4), 949–957 (1976)
4. Chen, C.S., Sarin, S., Balasubramanian, R.: A mixed-integer programming model for a class of assort-ment problems. Eur. J. Oper. Res. 63, 362–367 (1993)
5. Cintra, G.F., Miyazawa, F.K., Wakabayashi, Y., Xavier, E.C.: Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation. Eur. J. Oper. Res. 191(1), 61–85 (2008)
6. Correia, M.H., Oliveira, J.F., Ferreira, J.S.: Reel and sheet cutting at a paper mill. Comput. Oper. Res. 31(8), 1223–1243 (2004)
7. Cui, Y., Yang, Y., Cheng, X., Song, P.: A recursive branch-and-bound algorithm for the rectangular guillotine strip packing problem. Comput. Oper. Res. 35(4), 1281–1291 (2008)
8. Demir, M.C.: A pattern generation-integer programming based formulation for the carpet loading problem. Comput. Ind. Eng. 54(1), 110–117 (2008)
9. Dyckhoff, H.: A typology of cutting and packing problems. Eur. J. Oper. Res. 44(2), 145–159 (1990) 10. Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization, 2nd edn. Springer, Berlin (2009) 11. Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting stock problem. Oper. Res.
9, 849–859 (1961)
12. Gradisar, M., Trkman, P.: A combined approach to the solution to the general one-dimensional cutting stock problem. Comput. Oper. Res. 32(7), 1793–1807 (2005)
13. Hajizadeh, I., Lee, C.G.: Alternative configurations for cutting machines in a tube cutting mill. Eur. J. Oper. Res. 183(3), 1385–1396 (2007)
14. Hinxman, A.I.: The trim-loss and assortment problem: a survey. Eur. J. Oper. Res. 5(1), 8–18 (1980) 15. Holthaus, O.: Decomposition approaches for solving the integer one dimensional cutting stock problem
with different types of standard lengths. Eur. J. Oper. Res. 141(2), 295–312 (2002) 16. IBM/ILOG. 2009. Cplex 12.0 reference manual.http://www.ilog.com/products/cplex/
17. Jakobs, S.: On genetic algorithms for the packing of polygons. Eur. J. Oper. Res. 88(1), 165–181 (1996) 18. Leung, T.W., Chan, C.K., Troutt, M.D.: Application of a mixed simulated annealing-genetic algorithm
for a 2D packing problem. Eur. J. Oper. Res. 145(3), 530–542 (2003)
19. Leung, T.W., Yung, C.H., Troutt, M.D.: Applications of genetic search and simulated. annealing to the two-dimensional non-guillotine cutting stock problem. Comput. Ind. Eng. 40(3), 201–214 (2001) 20. Li, H.L., Chang, C.T.: An approximately global optimization method for assortment problems. Eur. J.
Oper. Res. 105(3), 604–612 (1998)
21. Li, H.L., Chang, C.T., Tsai, J.F.: Approximately global optimization for assortment problems using piecewise linearization techniques. Eur. J. Oper. Res. 140(3), 584–589 (2002)
22. Li, H.L., Tsai, J.F., Hu, N.Z.: A distributed global optimization method for packing problems. J. Oper. Res. Soc. 54, 419–425 (2003)
23. Li, H.L., Lu, H.C., Huang, C.H., Hu, N.Z.: A superior representation method for piecewise linear functions. INFORMS J. Comput. 21(2), 314–321 (2009)
24. Lim, S.K.: Physical design for 3D system on package. Des. Test Comput. 22(6), 532–539 (2005) 25. Lin, C.C.: A genetic algorithm for solving the two-dimensional assortment problem. Comput. Ind.
Eng. 50(1–2), 175–184 (2006)
26. Martello, S., Monaci, M., Vigo, D.: An exact approach to the strip packing problem. INFORMS J. Comput. 15(3), 310–319 (2003)
27. Page, E.: A note on a two-dimensional dynamic problem. Oper. Res. Q. 26, 321–324 (1975) 28. Pisinger, D.: Heuristics for the container loading problem. Eur. J. Oper. Res. 141(2), 382–392 (2002) 29. Suliman, S.M.A.: A sequential heuristic procedure for the two-dimensional cutting-stock problem. Int.
J. Prod. Econ. 99(1), 177–185 (2006)
30. Sweeney, P., Paternoster, E.R.: Cutting and packing problems: a categorized application-oriented research bibliography. J. Oper. Res. Soc. 43, 691–706 (1992)
31. Tsai, J.F., Hsieh, P.L., Huang, Y.H.: An optimization algorithm for cutting stock problems in the TFT-LCD industry. Comput. Ind. Eng. 57(3), 913–919 (2009)
32. Umetani, S., Yagiura, M., Ibaraki, T.: One-dimensional cutting stock problem to minimize the number of different patterns. Eur. J. Oper. Res. 146(2), 388–402 (2003)
33. Vanderbeck, F.: A nested decomposition approach to a three-stage, two-dimensional cutting-stock problem. Manag. Sci. 47(6), 864–879 (2001)
