A D eterministic A lgorithm for G enerating O ptimal T hree-S tage layouts of H omogenous S trip P ieces

: Purpose: The time required by the algorithms for general layouts to solve the large-scale two-dimensional cutting problems may become unaffordable. So this paper presents an exact algorithm to solve above problems. Design/methodology/approach: The algorithm uses the dynamic programming algorithm to generate the optimal homogenous strips, solves the knapsack problem to determine the optimal layout of the homogenous strip in the composite strip and the composite strip in the segment, and optimally selects the enumerated segments to compose the three-stage layout. Findings: The algorithm not only meets the shearing and punching process need, but also achieves good results within reasonable time. Originality


Introduction
The unconstrained two-dimensional cutting (UTDC) problem refers to a series of small shape (or part) non-overlapping on a rectangular panel and the optimization objective of the problems is to find an arrangement for maximizing the material usage.UTDC problem is widely used in the leather, wood, metal and other manufacturing industries.Although many researchers have studied the UTDC problem, from the theory of computational complexity theory, layout problem have been proved to be a quiet difficult combinatorial optimization problem (Cui, 2013;Han, Bennell & Zhao, 2013;Thomas & Chaudhari, 2013;He & Wu, 2013;Liu & Liu, 2011;Ji, Lu & Cha, 2012;Huang & Liu, 2006;Jiang, Lv & Liu, 2008).
According to the UTDC problem, the layouts can be divided into the general layouts and the specific layouts.On the one hand, when the layouts have no any constraint, the layouts are called the general layouts (Gilmore & Gomory, 1965;Beasley, 1985;Cui, Wang & Li, 2005;Seong & Kang, 2003;Hifi & Zissimopoulos, 1996;Alvarez-Valdes, Parajon & Tamarit, 2002); on the other hand, when the layouts must meet some specific production request, the layouts are called the specific layout.Now, there are some exact algorithms for the general layouts (Gilmore & Gomory, 1965;Cui et al., 2005).But the computation results in the references indicate that the computation time of these algorithms cannot be intolerable for solving the large scale UTDC problems.So many researchers have committed to study the specific layouts.The specific layouts have three advantages: meeting the practical production technology; high computation efficiency; the results are close to the optimal results.There are many advanced specific layouts, for example, Hifi (2001) proposed the classic twostage and the three-stage layout; Fayard and Zissimopoulos (1995) presented the two-segment layout; Cui (2004a) proposed the T-shape layout.Through analysis, the T-shape layout is the superset of the two-stage layout, and is the subset of the classic three-stage layout; the twosegment is the superset of the T-shape layout, and is the subset of the classic three-stage layout.
This paper propose a new layoutthe three-stage layout based on the homogenous stripe (3HS).The 3HS layout can meet the need of the cutting technology in the practical production.
-1168-Journal of Industrial Engineering and Management -http://dx.doi.org/10.3926/jiem.1127 3HS layout is the superset of the classic three-stage, two-segment, T-shape and the classic two-stage layout, and we will introduce it in the section 2.4.
The layout decides the layout value.The sequence of the above layouts value from largest to smallest is follows: the general layout, the classic three-stage layout, the two-segment layout, the T-shape layout, and the classic two-stage layout.This paper's 3HS layout is between the general layout and the classic three-stage layout.This paper will introduce 3HS layout in part 2; the exact algorithm for generating the 3HS layout in part 3; the experiments and results analysis in part 4; conclusion in part 5.

Homogenous stripe
The homogenous stripe consists of the same size with same dimension.Figure 1(a) shows horizontal homogenous rectangular stripes, and its width is the blank width.Figure 1(b) shows vertical homogenous irregular stripes, and its width is the blank length.

Composite strip
The composite strip consists of the homogenous stripe.The composite strip can be divided into the homogenous stripe by series of cuts.When cutting, each knife cuts single horizontal or vertical homogenous stripe.In 3HS layout, if each blank takes place of the homogenous strip, the 3HS layout turns into the classic three-stage layout; if the number of the segment is 2, the 3HS layout becomes the two-segment layout; if segments are X-segment and Y-segment, the 3HS layout turns into the T-shape layout.In addition, the T-shape layout is the superset of the classic two-stage layout (Cui, 2004a).Thus, the 3HS layout is the superset of the classic three-stage, two-segment, T-shape, and the classic two-stage layout.In other words, the solution of 3HS layout is better than that of the above four layouts.

Notes and functions
Table 1 lists the various notes and functions used by the algorithms.Most of the notes and functions will be introduce again when used for the first time, this table help readers quickly finding.

L, W
Length and width of sheet l i , v i Length and value of ith blank, i = 1,...,m w i , w 0i , w 1i w i is the length of ith regular blank, w 0i , w 1i are the initial step and progressive step of ith irregular blank, i = 1,...,m Normal length and width of homogenous strip

P, Q
Normal length and width of composite strip

P ssegment
Normal size of segment n s (i) (x,y) Maximum number of ith blank in x  y homogenous strip s(x,y) Maximum value of x  y homogenous strip Maximum value of the optimal 3HSX layout v SY-3STAGE Maximum value of the optimal 3HSY layout v S-3STAGE Maximum value of the optimal 3HS layout Table 1.Notes and function

The steps of algorithm
Supposed the size of sheet and blank are integer, and the blank direction is fixed.The algorithm of 3HS layout (3HSA) includes the following steps: Step 1. Determining the optimal homogenous strip by dynamic programming algorithm; Step 2. Solving the optimal homogenous strip layout in composite strip by knapsack problem; Step 3. Solving the optimal composite strip in segment by knapsack problem; Step 4. Determining the optimal 3HSX layout by knapsack problem; Step 5. Determining the optimal 3HSY layout by knapsack problem; Step 6. Solving the optimal 3HS layout.

The normal size
The normal sizes have been used by many scholars (Ji et al., 2012;Beasley, 1985;Hifi, 2001;Fayard & Zissimopoulos, 1995;Cui, 2004a).The normal size is the length and width linear combination of blank.The layout references (Cui, 2004a) have proved that the blank maximum number of rectangle x  y is equal to the blank maximum value of rectangle x 0  y 0 , -1173-Journal of Industrial Engineering and Management -http://dx.doi.org/10.3926/jiem.1127 and x 0 is the optimal normal size that is lessen than x, and y 0 is the optimal normal size that is lessen than y.To different layout, according to normal size features, we should define it appropriately to improve the solving speed.

Definition 1. The homogenous strip normal size
According to above description, the homogenous strip consists of blanks with same shape, and the blank direction is fixed.Therefore, the homogenous strip length normal size P s (i) is the length linear combination of each blank.The equation is follows: (1) (1) The homogenous width normal size of regular blank Q s (i) is follows: (2) (2) The homogenous width normal size of irregular blank Q s (i) is follows: (3) The 0 and L are added to the normal size sequence.The P s (i) = p 1 s , p 2 s ,..., p M s represents the homogenous strip length normal size of ith blank, and M is the number of normal size; and the Q s (i) = q 1 s , q 2 s ,..., q N s represents the homogenous strip width normal size of ith blank, and N is the number of normal size.

Definition 2. The composite strip normal size
According to above description, the composite strip composes of homogenous strips.So, the composite strip length normal size P is the length linear combination of each blank: (4) (1) The composite strip width normal size of regular blank Q is follows: (5) -1174-Journal of Industrial Engineering and Management -http://dx.doi.org/10.3926/jiem.1127 (2) The composite strip width normal size of irregular blank Q is follows: (6) The 0 and L are added to the normal size sequence.The p 1 , p 2 ,...,pM represents the composite strip length normal size, and M is the number of normal size; the q 1 , q 2 ,...,qN represents the composite strip width normal size, and N is the number of normal size.

Definition 3. The segment normal size
According to above description, the segment consists of composite strip.Therefore, the segment normal width P ssegment is the collection of composite strip length normal size: (7) If both the segment width and length belong to P ssegment , then the segment is a normal segment.

The value of homogenous strip x  y
(1) Solving the maximum number that the homogenous strip x  y includes blanks Assume that n s (i) (x, y) is the maximum number of ith blank in the homogenous x  y, and there is following recursive formula, and x  P s (i) , y  Q s (i) : • The maximum number of ith regular blank in the homogenous x  y: (8) • The maximum number of ith irregular blank in the homogenous x  y: (9) Figure 7 shows the blanks number of the homogenous strip x  y. 8,326 ▲ ▲ ▲ ▲ UW5 7,780 ▲ ▲ ▲ ▲ UW6 6,615 ▲ ▲ ▲ ▲ UW7 10,464 ▲ ▲ ▲ ▲ UW8 7,692 ▲ ▲ ▲ ▲ UW9 7,038 ▲ ▲ ▲ ▲ UW10 7,507 ▲ ▲ ▲ ▲ Table 2.The computation results of different layouts From tables, we can draw conclusions: 1) The optimal results of this paper's algorithm are equal or very close to the general algorithm; 2) The optimal results of this paper's algorithm are better than the classic three-stage, two-segment, T-shape.
The optimal number of problems 39 32 32 31 Table 3.The optimal number of different layouts

Conclusions
It is very difficult to solve UTDC problem.Although there are exact algorithms, the practical computation results indicate these algorithms only solve small scale problems efficiently.These algorithm's time it takes for solving large scale problems is unaffordable.Therefore, people usually solve the problem by two types algorithms, first, the algorithms for generating specific layouts, which not only meet the practical production technology, but also solve large scale problems efficiently within reasonable time, for example, the classic three-stage layout, two-segment layout and T-shape layout; second, is general algorithm.
The paper presents an exact algorithm for generating 3HS layout.On the one hand, 3HSA is a specific layout algorithm and its optimization result is better than the classic three-stage, two-segment and T-shape layout, and 3HSA not only improves sheet utilization within reasonable time, but also meets the shearing and punching process need.On the other hand, 3HSA is the heuristic algorithm, and the computations results show that the optimization result of 3HSA is very close that of general algorithm.Therefore, 3HSA can solve a large-scale rectangular piece packing efficiency.
Figure 1.The homogenous stripe Figure 2 show the composite strip of the rectangular blanks and irregular blanks; the Figure 2(a) is the X composite strip of the irregular blanks, and the Figure 2(b) is Y composite strip of the rectangular blanks.

Figure 3
Figure 2. The composite strip

Figure 3 .Figure 5
Figure 3.The cutting process of composite strip Figure 5.The types of the 3HS layout

Figure 6 .
Figure 6.The 3HSX layout and its cutting process

Table 3
Journal of Industrial Engineering and Management -http://dx.doi.org/10.3926/jiem.1127 2SEGMENT and T-shape layout's optimal results is 32, 32 and 31 respectively.Therefore, the results of this paper algorithm are better than other layouts.
lists the optimal number of different layouts, and these statistical data come from Table2.In 43 classical benchmark problems, the number of 3HS layout's optimal results is 39, and the results ratio of the rest 4 problems and optimal is 99.9%; the number of 3STAGE, -1179-

Table 4 .
The better number problem of different layouts

Table 4
lists the optimal number of different layouts, and these statistical data come from Table2.In 43 classical benchmark problems, 1) there are 9 problems that the 3HS layout is better than 3STAGE and 2SEGMENT, and 10 problems for T-shape; 2) there are 3 problems that the 3STAGE layout is better than 2SEGMENT and 5 problems for T-shape; 3) there are 4 problems that the 2SEGMENT layout is better than T-shape.The 3HSA total time it takes for solving 43 problems from table 2 is 93.74s, and each problem's average time is 2.18s.Therefore, the time is reasonable in practical application.