A Quasiphysical and Dynamic Adjustment Approach for Packing the Orthogonal Unequal Rectangles in a Circle with a Mass Balance : Satellite Payload Packing

Packing orthogonal unequal rectangles in a circle with a mass balance (BCOURP) is a typical combinational optimization problem with the NP-hard nature. This paper proposes an effective quasiphysical and dynamic adjustment approach (QPDAA). Two embedded degree functions between two orthogonal rectangles and between an orthogonal rectangle and the container are defined, respectively, and the extruded potential energy function and extruded resultant force formula are constructed based on them. By an elimination of the extruded resultant force, the dynamic rectangle adjustment, and an iteration of the translation, the potential energy and static imbalance of the system can be quickly decreased to minima. The continuity and monotony of two embedded degree functions are proved to ensure the compactness of the optimal solution. Numerical experiments show that the proposed QPDAA is superior to existing approaches in performance.


Introduction
2D rectangle packing problems are derived from the industry and antiaircraft field [1][2][3].They occur in logistics packing, plate cutting, the layout design of the very large scale integration (VLSI), and satellite modules.They can be divided into unconstrained rectangle packing problems [1] and constrained ones [3].Both are NP-hard problems and are difficult to be solved.However they have attracted much attention and some packing approaches for different containers have been reported in literatures.
The layout design problem of the satellite module described in [32] is an important packing problem, which can be transformed into the problem of packing 2D orthogonal unequal rectangles within a circular container with the mass balance (BCOURP).In 1999, Feng et al. [33] built a mathematical model of this problem and analyzed the isomorphism and equivalent intrinsic properties among its layout schemes by using the graph theory and group theory and proposed a theoretical global optimization algorithm.In 2007, Xu et al. [34] defined embedded degree functions between two rectangles and between the rectangle and circular container and presented a compaction algorithm with the particle swarm local search (CA-PSLS).Their idea is that a feasible solution with a smaller envelope radius obtained through the gradient method is taken as an elite individual and the optimal solution is obtained by the PSO iteration.In 2010, Xu 2 Mathematical Problems in Engineering et al. [35] suggested a heuristic algorithm ordered by GA (GA-HA, see its algorithm steps in Appendix A and Figure 1).Its key technology is the positioning strategy of constructing the feasible solution.By combining it with GA, the computational efficiency and solution quality are improved.
Generally, there exist strong points and deficiencies for each type of approaches.
(i) For approaches based on the graph theory, there exists combinatorial explosion when the adjacent topological relation is transformed into the layout diagraph without a size limit for the large-scale layout problem.This is because only two limited relations called the vicinity and distance can be used in pruning branch.(ii) The heuristic method can be used to quickly construct a feasible solution.But it is generally not easy to devise a good heuristic strategy, unless the designer makes a long time painstaking effort and has good luck.(iii) Stochastic algorithms have the global search ability, but there exists the bottleneck of time-costing overlapping area calculation for them [33].By combining the heuristic method with the stochastic algorithm, their respective advantages can be exerted to the utmost.Based on this mechanism, CA-PSLS and GA-HA are consecutively proposed to solve this problem.
According to No Free Lunch Theory [36], how to obtain the knowledge from the problem itself and its area and fuse it into the heuristic and stochastic search mechanism is a way of designing a high performance approach for this problem.
Huang et al. [37][38][39][40] presented a quasiphysical and quasihuman heuristic algorithm and its variants for the circle packing problem.They obtained excellent results.For BCOURP, Xu et al. [34] proposed CA-PSLS based on embedded degrees between circumcircles of two rectangles and between the container and rectangle's circumcircle.But due to the discontinuity of the two embedded degree functions, it is difficult to obtain a high quality solution by using CA-PSLS.That is, constructing the continuous rectangular embedded function and exploring a better optimized mechanism are necessary for this problem.Therefore, in this paper, we consider two definitions of monotonous and continuous embedded degrees between two orthogonal rectangles and between an orthogonal rectangle and the container and suggest a dynamic adjustment strategy.We merge them into the proposed QPDAA to improve the solution quality of BCOURP.Numerical experiments will test effectiveness of the considered QPDAA.
The remainder of this paper is organized as follows.The problem statement and mathematical model are in Section 2. The compact and feasible solution strategy and dynamic adjustment strategy are given in Sections 3 and 4, respectively.This algorithm is presented in Section 5. Section 6 is experiments and analysis.The conclusion is shown in Section 7. The final part is acknowledgment.

Problem Statement and Mathematical Model
Consider the following two related definitions where I  = {1, 2, . . ., } and  is the number of rectangles.
Definition 1.As shown in Figure 2, let the origin of the Cartesian coordinate system be the center of the container.
Definition 2. For a layout scheme X  , if   = 0 ∘ or 90 ∘ , then X  is an orthogonal rectangle packing scheme,   ( ∈ I  ) is an orthogonal rectangle, and the packing is the orthogonal rectangle packing (see Figure 3).
Herein this paper considers only orthogonal rectangle packing schemes.
In Formula (1), (X) denotes the radius of the enveloping circle of the scheme X whose circular center is at (0, 0).Formula (2) indicates that there is no overlap region between two rectangles   and   .Formula (3) indicates that all rectangles are contained in the container.In Formulas (2) and (3), int(  ) denotes the interior region of the rectangle   .Formula (4) means that the static imbalance (X) of the solution X is less than its threshold , where  > 0.

Compact and Feasible Solution Strategy
Based on the potential energy function of the embedded degree between two circles, Huang et al. [38][39][40] proposed the quasiphysical strategy and its variants for the circle packing problem.Inspired by the quasiphysical idea, we suggest a compact and feasible strategy for BCOURP.

Embedded Degree Function and Related
Properties.Xu et al. [34] defined the embedded degrees between two rectangles and between the rectangle and container by Definitions 3 and 4, respectively.Definition 3. Let    and    (,  = 1, 2, . . .,  and  ̸ = ) denote the radii of the circumscribed circles of two rectangles   and   , respectively, and   (see Figure 4(a)) the embedded degree between them (see Figure 4(a)); then   can be calculated by The embedded degree between the rectangle and container Figure 4: Two embedded degree definitions in [34].
Figure 5: The embedded degree between two rectangles in the critical state.
degree between the rectangle and container; then  0 can be calculated by Both of the above two embedded degree functions are discontinuous in the critical state from overlapping to separating, as has been discussed by Stoyan and Yaskov [41].For example, as shown in Figure 5, by moving one rectangle along a direction, two rectangles with the overlap area of 10 −5 (state 1) are changed into   (2.5, 5, 6, 5, 30, 0 ∘ ) and   (0, 0, 5, 4, 20, 90 ∘ ) (state 2).By Formula (5), we know that   =   +   −     ≈ 1.52 for state 1, but   = 0 for state 2. So,   in Definition 3 is discontinuous where   = 0 (,  = 1, 2, . . .,  and  ̸ = ).Similarly,  0 ( = 1, 2, . . ., ) in Definition 4 is also discontinuous (see Figure 6).Owing to their discontinuity, it is difficult to select an appropriate step length to obtain the feasible and compact layout scheme for the gradient iteration of CA-PSLS.Inspired by [41], Definitions 5 and 6 are given for the considered QPDAA.Definition 5.For two rectangles   and   (  ,   = 0 ∘ or 90 ∘ , ,  = 1, 2, . . .,  and  ̸ = ) (shown in Figure 7), let   and   be the radii of their circumscribed circles, respectively, and let   denote their embedded degree; then   can be calculated by Otherwise. (7) Figure 6: The embedded degree between the rectangle and container in the critical state.In Formula (7) In Formula (7), the embedded degree between two rectangles is the moving distance of the rectangle   from an overlap state with the stationary   to the separation state along the direction from the center (  ,   ) to the center (  ,   ).If   and   are two squares, and the center of   is on the diagonal line of   and enough close to its center (i.e.,   →   and   →   ), the moving distance of   from the initial state (shown in Figure 8(a)) to the separate state (shown in Figure 8(b)) along the direction of their diagonal lines is about   +   .Thus in the initial state, their embedded degree   is close to the maximal value   +   .In addition, when 2|  −   | →   +   and/or 2|  −   | →   +   ,  → 0 and/or V → 0. That is,   → 0. Definition 6.For the rectangle   (  = 0 ∘ or 90 ∘ ,  = 1, 2, . . ., ) and the circle container (0, 0,   ) as shown in Figure 7, let  0 denote the embedded degree between the rectangle   and container; then  0 can be calculated by The geometric interpretation of Definition 6 is that when the farthest vertex of the rectangle   ( = 1, 2, . . ., ) from the coordinate origin is within the container, their embedded degree  0 = 0; otherwise it is the length of the straight line segment pointed by  0 in Figure 9.
For embedded degree functions in Definitions 3 and 5, their geometric figures are two curved semi-cone surfaces shown in Figures 10(a) and 10(b), respectively, where it is obvious that there is a gap between the semi-cone surface and  plane in Figure 10(a) but there is no gap between them in Figure 10(b).It is not difficult to assert that the difference of geometric figures of two embedded degree functions in Definitions 4 and 6 is the same as the above one.After describing Lemma 7, we propose properties of two embedded degree functions in Definitions 5 and 6, respectively.Proof.∀,  ∈ I  and  ̸ = , set D 1 = {(, V) | 0 <  ≤ 1 and 0 < V ≤ 1}.From Definition 5, we know that the function   (, V) is continuous on both D 1 and D − D 1 .Here, we prove that it is continuous on the domain Simultaneously, lim This is because According to Lemma 7, the binary function   (, V) is continuous on the domain D 2 .Therefore,   (, V) is continuous on the domain D. Property 2. ∀,  ∈ I n and  ̸ = , for the container with the radius   and rectangle   (  = 0 ∘ or 90 ∘ ,  = 1, 2, . . ., ), set  =   and V =   ; then the binary function  0 in Definition 6 is continuous on the domain

Extruded Force and Energy Function.
In order to quickly decrease the overlapping area of the rectangle packing system, we define the extruded forces between two rectangles and between the rectangle and container.is the direction from the center of the container to the furthermost rectangular vertex (see Figure 6): By experiments, we can know that  > .
So, the extruded resultant force  →   of   ( = 1, 2, . . ., ) in the rectangle packing scheme can be calculated by Definition 10.Let   and  0 (,  = 1, 2, . . .,  and  ̸ = ) denote extruded potential energies of   with respect to   and the container, respectively.Then   can be calculated by Formula (15), where   and  0 denote two embedded degrees between two rectangles   and   and between the rectangles   and container: Definition 11.The total extruded potential energy   of   can be calculated by Let   be the area of the rectangle   ; then   and   /  ( = 1, 2, . . ., ) are its absolute and relative extruded potential energies, respectively.

Dynamic Adjustment Strategy
Considering the problem of the low efficiency and possible local optimum (e.g., large static imbalance) of the iteration of two steps in Section 3.3, we propose Property 3 and dynamic adjustment strategy for QPDAA.

Related Property.
For optimizing the static imbalance of the layout scheme, we introduce Property 3.

Rotation and Off-Trap. (i)
A rectangle with a larger pain degree is found out from X and is rotated 90 ∘ round its center counterclockwise direction to relieve its pain.(ii) A rectangle with a larger pain degree is found out from X and is moved to such a place in the container where its pain degree is smaller.We know that the role of (ii) is similar to a construction of the nonisomorphic layout pattern [42,43].

The Proposed Algorithm
Through an organic combination of the compact and feasible solution strategy and dynamic adjustment strategy, we present QPDAA for BCOURP.
Let  0 and  * be the predetermined value and an allowable maximum of the envelope radius of the solution, respectively;  is the number of rectangles.  (  ,   ), (X), and (X) denote the mass center, envelope radius, and extruded potential energy of the packing scheme X, respectively.  (X) ( = 1, 2, . . ., ) is the extruded potential energy of the rectangle   ; ℎ is the step length;  is the maximum translation times.Then steps of the proposed QPDAA are shown in Algorithm 1.

Experiments and Analysis
6.1.Experiments.The proposed QPDAA is coded in VC++ 6.0 and carried on a Pentium 3 GHZ PC with 512 MB memory.CA-PSLS [34], GA-HA [35] are coded in VC++ 6.0 and two algorithms are carried on a Pentium 1.83 GHZ with 512 MB memory; IGA [44] is carried on an IBM 586 166 MHz.
Experiment 2. For testing the effects of  0 and  * on the minimal radius and running time with the proposed QPDAA, we take another set of  0 and  * (five examples and other parameters are the same as those of Experiment 1) and run the proposed QPDAA procedure 30 times for each example.
The minimal radii and running times are given in Table 7.
Their layout schemes and layout diagraphs are shown in Tables 8, 9, 10, and 11 and Figure 13.It can be found from  Table 7 that changing values of  0 and  * we can obtain the packing scheme with a smaller envelop radius, but it costs more time for each example.So, values  0 and  * in Experiment 1 can be applied to make a tradeoff between the computational effectiveness and solution quality.
In order to further test the effectiveness of the proposed QPDAA, we consider Experiment 3. for Examples 1-3, respectively.For GA + HA, the population size, mutation probability, and max number of the iteration are 30, 0.125, and 50, respectively.By running HA + GA and the proposed QPDAA 30 times for three examples, respectively, their optimal envelop radii and average times are shown in Table 12, respectively.The optimal packing scheme diagrams of GA + HA and the proposed QPDAA are shown in Figures 14(a)-14(c) and Figures 14(d)-14(f).We can know from Table 12 that both the solution quality and computational quality of the proposed QPDAA are obviously higher than those of GA + HA.
Note that, in Experiment 3, the procedure of GA + HA is coded by author and is carried on a Pentium 3 GHZ with 512 MB memory.

Analysis.
From data of Tables 2, 7, and 12, we know that the solution quality of the proposed QPDAA algorithm is higher than those of CA-PSLS and GA-HA.Compared with those of CA-PSLS, embedded degree functions of the proposed QPDAA can make the layout scheme more compact.Due to the fixed candidate positions of GA + HA, it is difficult to find the best position for some rectangles close to the marginal region of the container.The deficiency of a mechanism of decreasing the static balance in the process of packing rectangles also limits the solution quality of GA + HA.The experimental results illustrate the effectiveness of the proposed QPDAA.The computational efficiency of the proposed QPDAA is one magnitude higher than those of CA-PSLS.There are two reasons.(i) Owing to orthogonal packing, searching the optimal solution in the 2D solution of the proposed QPDAA is easier than that in the 3D solution space of CA-PSLS.(ii) In order to improve the solution quality of CA-PSLS, PSO is used to optimize the feasible solution obtained through the gradient method based on two discontinuous embedded degree functions, but the proposed QPDAA need not do the PSO optimization without reducing its solution quality.Except for Example 1, the computational efficiency of the proposed QPDAA is higher than that of GA + HA.And with the increase of the size of the packing problem, the advantage of the proposed QPDAA is more obvious.This is because computational complexities of the extruded resultant force and potential energy are () in this paper, but the computational complexity of noninterference judgment is ( 2 ).In addition, for GA + HA, with the increasing of the number of rectangles, the number of candidate positions of each rectangle increases dramatically.These reasons lead to the computational efficiency of the proposed QPDAA higher than that of GA + HA for the BCOURP with a large size.

Conclusions
Taking the layout design of a satellite module as the application background, we have proposed the QPDAA for the BCOURP problem in this paper.Two continuous embedded functions between orthogonal rectangles and between the rectangle and container are constructed to overcome the weakness of embedded functions in [34].And the suggestion of the extruded resultant force formula and the potential energy function of the rectangle packing system based on the proposed embedded functions make solving the BCOURP problem simple and effective as solving the circle packing problem [37][38][39][40].The proposed dynamic adjustment strategy can quickly decrease the static imbalance of the packing scheme and make the iteration skip the local optimum.The experiment results show that the proposed QPDAA is superior to existing algorithms in performance for the BCOURP problem, especially for the BCOURP problem with the large size.The next work is to extend the above algorithm into solving the 3D satellite module payload packing problem.
Step 3.For  = 1, 2, . . .,  − 1, calculate centers and direction angles of 16 candidate positions (see Figure 1) of the rectangle  () with respect to the rectangle  () .From its candidate positions eliminate unfeasible ones and calculate the optimal one (i.e., compared with other feasible candidate positions, it makes the packing scheme of the first  rectangles have less envelop radius).
Step 4. If  <  then update the current packing scheme,  + +, and go to Step 3; otherwise, go to Step 5.
Step 5.If  <  then update the optimal packing scheme and use GA to generate a new placing sequence set {(),  = 1, 2, . . ., } and go to Step 2; otherwise, go to Step 6.
Step 6.Output the optimal packing scheme and envelop radius; algorithm ends.

Figure 1 :
Figure 1: The sketch map for available positions of the rectangle.

Figure 2 :
Figure 2: The definition of a rectangle.

Figure 7 :
Figure 7: Two cases of the embedded degree definition between two rectangles.

Figure 8 :
Figure 8: The geometric interpretation of definition of the embedded degree between two rectangles.

Lemma 7 .
If a binary function (, ) is continuous for each variable in a domain, respectively, and is monotonous for the variable  or , then the function (, ) is continuous in the domain.Property 1. ∀,  ∈ I  , let   and   ( ̸ =  and   ,   = 0 ∘ or 90 ∘ ) be two rectangles, and the domain D = { ≤ 1 and V ≤ 1}.Then   (, V) in Definition 5 is a continuous binary function in the domain D.

Figure 9 :Figure 10 :
Figure 9: The schematic diagram overlapped between the orthogonal rectangle and container.

Figure 12 :Figure 13 :
Figure 12: The layout diagraphs of five examples for the proposed QPDAA.

Figure 14 :
Figure 14: Packing scheme diagrams of GA + HA and the proposed QPDAA for Experiment 3.

Table 1 :
Parameters of rectangles for five layout examples.

Table 2 :
Performance comparisons of four algorithms.

Table 3 :
The layout schemes of the proposed QPDAA for Examples 1 and 2.

Table 4 :
The layout schemes of the proposed QPDAA for Example 3.

Table 5 :
The layout scheme of the proposed QPDAA for Example 4.

Table 6 :
The layout scheme of the proposed QPDAA for Example 5.

Table 7 :
The effect of parameters  0 and  * on the optimal radius and running time for the proposed QPDAA.

Table 8 :
The optimal layout schemes of Examples 1 and 2 for the proposed QPDAA.

Table 9 :
The optimal layout schemes of Example 3 for the proposed QPDAA.

Table 10 :
The optimal layout scheme of Example 4 for the proposed QPDAA.

Table 11 :
The optimal layout scheme of Example 5 for the proposed QPDAA.

Table 12 :
The effect of  0 and  * on the optimal radius and running time for the proposed QPDAA.