A new heuristic algorithm for the circular packing problem with

SCIENCE CHINA Information Sciences

. RESEARCH PAPERS .

August 2011 Vol. 54 No. 8: 1572–1584 doi: 10.1007/s11432-011-4351-3

A new heuristic algorithm for the circular packing problem with equilibrium constraints LIU JingFa1,2 ∗ , LI Gang3 & GENG HuanTong1 1School

of Computer and Software, Nanjing University of Information Science & Technology, Nanjing 210044, China; 2Network Information Center, Nanjing University of Information Science & Technology, Nanjing 210044, China; 3School of Mathematics and Physics, Nanjing University of Information Science & Technology, Nanjing 210044, China Received October 14, 2010; accepted April 8, 2011

Abstract The circular packing problem with equilibrium constraints is an optimization problem about simplified satellite module layout design. A heuristic algorithm based on tabu search is put forward for solving this problem. The algorithm begins from a random initial configuration and applies the gradient method with an adaptive step length to search for the minimum energy configuration. To jump out of the local minima and avoid the search doing repeated work, the algorithm adopts the strategy of tabu search. In the process of tabu search, we improve the traditional neighboring solutions, tabu objects and the acceptance criteria of the current solution effectively. We test two sets of benchmarks consisting of 11 representative instances from the current literature. The numerical results show that the proposed algorithm breaks the records in seven out of 11 instances, and obtains the optimal solutions for the other four instances. Keywords

equilibrium constraints, packing problem, heuristic algorithm, tabu search, layout optimization

Citation Liu J F, Li G, Geng H T. A new heuristic algorithm for the circular packing problem with equilibrium constraints. Sci China Inf Sci, 2011, 54: 1572–1584, doi: 10.1007/s11432-011-4351-3

1

Introduction

The packing problem is concerned with how to pack some objects into a limited spacing container without overlapping. Its objective is to increase the space utilization ratio of the container as much as possible. It is widely applied to the freightage, cutting, telecommunication, aerospace, etc. [1–3]. Solving this problem can economize on resources, and reduce the cost of produce and the fee of transportation. The packing problem is an NP-hard problem [4]. Interiorly there are only few publications discussing it. Most published research [5–9] on the packing problem mainly focused on the circular and rectangular packing problem, and did not study additional behavioral constraints (for instance, equilibrium, inertia, stability, etc.). This paper studies the circular packing problem with equilibrium constraints. This kind of packing problem requires the packing system satisfying with constraints of the static non-equilibrium, in addition to the requirement of non-overlapping and high space utility as the general circular packing problem [10]. ∗ Corresponding

author (email: jfl[email protected])

c Science China Press and Springer-Verlag Berlin Heidelberg 2011

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In recent years, some authors have studied the circular packing problem with equilibrium constraints and proposed some algorithms. Teng et al. [10] proposed heuristic algorithms including the method of model-changing iteration and the method of main objects topo-models. Tang and Teng [11] gave a modified genetic algorithm called decimal coded adaptive genetic algorithm. Qian et al. [12] developed a human-computer interactive genetic algorithm, which makes the artificial individual a part of the chromosome population. Yu et al. [13] developed a learning-based genetic algorithm that adds a learning operation to the traditional genetic algorithm and lets every individual be able to search for a local optimum efficiently in the quasi-Newton directions. Li et al. [14] proposed a so-called mutation particle swarm optimization (PSO) algorithm, which adds a mutation operator to the PSO algorithm. By proposing a constraint handing strategy suitable for PSO, and combining direct local search and the PSO algorithm, Zhou et al. [15] gave a hybrid algorithm. Lei and Qiu [16] gave a novel adaptive particle swarm optimizer by modifying on the traditional PSO algorithm. Huang and Chen [17] proposed an improved quasi-physical algorithm that introduces an efficient strategy of accelerating the search process into the quasi-physical algorithm [18]. Wang et al. [19] developed an improved scatter search algorithm that incorporates two types of local search methods, the gradient decent method and Nelder-Mead simplex algorithm, into the scatter search. These algorithms consist roughly of two categories: stochastic algorithms and heuristic algorithms. Although all these approaches work relatively well for the equilibrium constraint circular packing problem, their efficiency still needs to be improved. Different search methods have their respective limitations. Relative to stochastic algorithms, heuristic algorithms (for example, the method of model-changing iteration [10] and the improved quasi-physical algorithm [17]) generally adopt more aimed heuristic strategies, but they lack broad adaptability. Stochastic algorithms (for example, the genetic algorithm [11–13], the particle swarm optimizer [14–16], and the scatter search algorithm [19]) have global search ability and broad adaptability, but they lack mechanism of effective local search and speed of convergence. In addition, the deterministic local search algorithms, such as the gradient method, the conjugate gradient method, and the quasi-Newton method, can also execute global search. These methods have high speed and high accuracy, but they tend to be trapped into local minima. By [19–21], a reasonable combination of the stochastic algorithm with global search ability, the local accurate search method and some heuristic strategies may be an efficient mechanism to construct high-performance algorithm in a certain field. In [21], Liu and Li improved the energy landscape paving (ELP) method [22], and put forward a new global search algorithm, basin filling algorithm, which combines the improved ELP method, the gradient method based on local search, and some heuristic configuration update mechanisms. The effectiveness of the basin filling algorithm is verified by solving a set of well-known instances for the circular packing problem with equilibrium constraints. However, because the ELP method in the basin filling algorithm is essentially a stochastic method based on Metropolis sampling, it is easy to miss some promising configurations in the process of search. Therefore, aiming to the concrete problem, how to construct an effective global stochastic search method (which can execute intensification search and diversification search, what is more, has the ability of traversing the whole solution space within a reasonable time), and how to comprise the advantages of the global stochastic search method and the local search method may be the breakthrough of devising a high-performance algorithm. Tabu search (TS) method [23–25] is a class of global intelligent optimization algorithm with local search ability. To avoid the search being trapped into local minima, and prevent cycling, a structure called tabu list is introduced to make the search explore different approaches. In addition, TS adopts the so-called aspiration criteria to avoid the search missing promising configurations. By executing local search based on the promising configurations, TS is apt to find the global optimal solution. Therefore, TS is a kind of global optimization algorithm with very powerful search ability. In this paper, by combining the local minimizing method and the TS method, a heuristic algorithm based on tabu search (HTS) is proposed. In HTS, we improve the traditional TS effectively, including putting forward the aimed neighboring solutions of the problem, the new tabu objects, and the acceptance criteria of the current solution. The numerical results show that HTS is an effective algorithm for solving the circular packing problem with equilibrium constraints. We test two sets of 11 representative instances. HTS breaks the

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Figure 1

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The schematic diagram of packing of the circular objects in Cartesian coordinate system.

records in seven instances, and obtains the optimal solutions for the other four instances.

2

Description and conversion of the problem

The circular packing problem with equilibrium constraints, based on the background of the man-made satellite module layout design [21], is in fact a layout optimization for the dishes installed on a rotating table [10]. There is a rotating circular table with radius R0 and angle velocity ω (see Figure 1) and n circular objects Ci (i ∈ N ={1, 2,. . . , n}), which are installed on the circular table such that all n circular objects tend to the center of the table as much as possible and satisfy the following constraints: 1) There is no interference (i.e., overlap) between any two different circular objects; 2) Each circular object does not extend outside the table; 3) The static non-equilibrium value of the packing system should not exceed a permissible value δ J (> 0). Cartesian coordinate system is built as Figure 1. Given n circular objects Ci with radii ri and masses mi (i ∈ N ={1, 2,. . . , n}). Let the coordinates of the center of Ci be (xi , yi ). We call X=(x1 , y1 , x2 , y2 ,. . . , xn , yn ) a solution of layout, i.e., a configuration. The circular packing problem with equilibrium constraints is in fact how to pack all n circular objects into the circular container C0 without overlapping so that the radius of C0 is as small as possible. It is also the following constrained optimization problem: ⎧ 2 + y2 + r ⎪ Minimize R , = max x 0 i ⎪ i i ⎪ i∈N ⎪ ⎪ ⎪ ⎪ (xi − xj )2 + (yi − yj )2 ri + rj , i = j, i, j ∈ N, ⎨ Subject to (1) x2i + yi2 R0 − ri , i ∈ N, ⎪

⎪ ⎪

2 2 ⎪ n n ⎪ ⎪ ⎪ mi xi + mi y i δJ , J= ⎩ i=1

i=1

where J denotes the static non-equilibrium value of the packing system. The physical implication of the static non-equilibrium constraints, i.e., J δJ , is that the static non-equilibrium value J (or magnitude of non-equilibrium centrifugal force) induced by the masses of all circular objects that deviate from the center of the circular container (i.e., the spinning clapboard of satellite module) is within a permissible value. The smaller J is, the better the packing system satisfies the static equilibrium constraint. When J=0, the whole system is at a static balance. We imagine that all n circular objects and the circular container as smooth elastic solids. By the quasiphysics strategy and the penalty function method [21], to solve the feasible solution to the constrained

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optimization problem (1) can be converted into solving the following unconstrained optimization problem: ⎡ 2 n 2 ⎤ n−1 n n

2 Minimize U (X) = (2) mi xi + mi y i ⎦ , dij +l ⎣ i=0 j=i+1

i=1

i=1

2 where dij = dji = max(0, ri +rj − (xi − xj )2 + (y i − yj ) is the overlapping depth between the circular 2 objects Ci and Cj , and di0 = d0i = max(0, ri + xi + yi2 − R0 ) is the overlapping depth between the circular object Ci and the container C0 , and l is a penalty coefficient and a very small positive number. If we can find an efficient algorithm to solve the optimization problem “minimize U (X)”, we can obtain the optimal or approximate optimal solution of the original circular packing problem with equilibrium constraints (1) by using the dichotomous search. Consequently, in the following section, we will focus our discussion on the optimization problem “minimize U (X)”.

3

Tabu search algorithm

Tabu search (TS) proposed by Glover [23–25] in 1986 is a generalization of local neighboring search algorithm, and a successful application of artificial intelligence in combinatorial optimization problems. The so-called “tabu” in TS indicates the algorithm tries to avoid the search doing repeated work. To avoid the search cycling and explore different effective search approaches, and prevent the algorithm from being trapped into the local minima, TS adopts a so-called tabu list to record recently visited local minima, which in the next search the algorithm will no longer or selectively visit according to the information in the tabu list. The basic frame of TS is as follows. Give an initial solution, and construct a neighborhood of the current solution, then some candidate solutions are chosen from the neighborhood. If the objective function value of the best candidate solution is better than that of the current optimal solution, the tabu restriction of the best candidate solution will be canceled. Replace the current optimal solution and the current solution by it, and add it into the tabu list, and at the same time modify the term of each tabu object in tabu list and release the object whose term is 0. If there is no such a candidate solution, the best solution being not tabooed in the neighborhood is chosen as the new current solution and added into the tabu list. Modify the term of each tabu object in the tabu list and release the object whose term is 0. This process is repeated until the algorithm satisfies stopping criterion. About more complete descriptions of TS, interested readers may refer to [25].

4 4.1

Heuristic algorithm based on tabu search Generation of initial configuration

TS is a global optimization method based on local search. Starting from good initial solution, the algorithm is generally prone to find good solutions, and from bad initial solution, the convergence speed of the algorithm will reduce. So most algorithms based on tabu search select good initial solutions to enhance the efficiency of algorithms. However, in this paper we do not lay emphasis on the generation of initial solution. The algorithm may start from a random initial configuration and even from a configuration where all circular objects are put at the center of the circular container. It is because even if the algorithm starts from a random configuration, its efficiency can be ensured by updating configuration according to the neighborhood structure and subsequently executing the gradient method, where all overlapping circular objects will carry on moving under the interaction of elastic force, and at last the whole system will attain a balance, which may be a good configuration with low energy. 4.2

Generation of neighboring configuration

Definition 1. Action: For the current configuration X=(x1 , y1 ,. . . ,xi , yi ,. . . ,xn , yn ), we call an action as a placement of the circular object Ci which has maximum Ei /ri on a vacant point of the

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circular container, where Ei = nj=0,j=i d2ij denotes the sum of the extrusive elastic potential energy which other n − 1 circular objects and the container C0 exert on the circular object Ci , and the so-called vacant point denotes a point of the vacant region of the circular container. The method of determining a vacant point of the circular container is as follows. Randomly produce a point in the circular container. If the distance between this point and the center of every circular object in the container is larger than the radius of the corresponding circular object, this point is a vacant point in the circular container; otherwise, we produce randomly a new point in the circular container. This process repeats until the new produced point is a vacant point in the circular container. Definition 2. Neighboring configuration and neighborhood: For the current configuration X=(x1 , y1 , . . . , xi , yi , . . . , xn , yn ), we call X =(x1 , y1 , . . . , xi , yi , . . . , xn , yn ) as a neighboring configuration of X, where (xi ,yi ) is the new coordinates of the center of the circular object Ci , which are produced by doing an action. The set of all neighboring configurations of X is called the neighborhood of X, denoted by N (X). To enhance the search efficiency of TS and make TS jump out of local minima efficiently, we apply the heuristic neighborhood structure to update configuration. At first, we pick out the circular object Ci with maximum Ei /ri from the current configuration. Then we “pseudo-do” randomly Ci 10 actions within the circular container, where “pseudo-do” means the circular object Ci is packed temporarily and will be removed from the container after computing the corresponding extrusive elastic potential energy of the circular object Ci in this position. With the positions of other circular objects unchanged, we put the center of Ci at the point, where the extrusive elastic potential energy of Ci is smallest, and gain a new configuration X . Definition 3. Candidate neighborhood: For the current configuration X=(x1 , y1 ,. . . , xi , yi ,. . . , xn , yn ), suppose Ci is the circular object with maximum Ei /ri , we call 10 neighboring configurations produced by pseudo-do randomly Ci 10 actions within the circular container as Ci -based candidate neighborhood of X, denoted by N (X, Ci ). Obviously, N (X, Ci ) ⊆ N (X). 4.3

Local search strategy

After producing new configuration X by neighborhood structure, to search further lower-energy configurations near X and avoid the search far from the global optimal solution, we start from X and execute accurate local search algorithm, the gradient method (GM) based on an adaptive step length [21], to speed up searching for the global optimal configuration. 4.4

Tabu list

Tabu list contains two important elements: tabu object and tabu length. A tabu object denotes the object in the tabu list. When applying neighborhood structure to update configuration, cycling is possible, namely some circular objects may be selected repeatedly and at each time, they cannot be determined at appropriate positions. To avoid this, we forbid some circular objects to be picked. However, we do not set tabu object just as general tabu search method, which selects the best solution from the candidate solutions, and if its objective function value is better than that of the current found optimal solution, TS ignores the tabu restriction of this circular object and replaces the current optimal solution and the current solution by it, and adds it into the tabu list; otherwise, the best solution being not tabooed within the candidate neighborhood is chosen as the new current solution and the corresponding circular object is added into tabu list as a tabu object. Now, we do not consider whether the circular object Ci is a tabu object. If only Ci has been picked out and done five actions, and at every time the updated configuration by doing an action and subsequently running local search (see subsection 4.3) is not accepted (the acceptance criteria are shown in subsection 4.5), we set Ei /ri =0, and add Ci into the tabu list as a tabu object. Thus, when we once again pick the circular object with the maximum Ei /ri from the current configuration by the neighborhood structure to do an action, we will pick Ci no longer, which avoid efficiently the search revisiting the local minima that have been reached. In addition, this improved strategy can give the current minimum-energy configuration (which includes promising configurations

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next to the global optimal solution) several chances to search further lower-energy configurations, that is, at each time the algorithm starts from the current minimum-energy configuration, after updating the configuration by the neighborhood structure and subsequently executing local search, the improved strategy can search the global optimal solution efficiently. Tabu length denotes the maximum number of times which tabu objects are not picked out of tabu list on the condition without considering the aspiration criteria. In this paper, we set tabu length at 3, and adopt unidirectional chained list to record the objects tabooed in the search process and a so-called first-in first-out queue structure to update tabu list. 4.5

Aspiration criterion and the acceptance criteria of the current configuration

When the objective function value of the configuration corresponding to a certain tabu object is better than that of the current optimal configuration, we adopt a so-called aspiration criterion to ignore the tabu restriction of this tabu object and accept this configuration as the current configuration. The aspiration criterion is a key ingredient of TS in avoiding the search missing new promising configurations and obtaining the global optimal solution. In the traditional TS, if the objective function values of all tabu objects in tabu list are not superior to the objective value of the current optimal configuration, the best solution being not tabooed within the candidate neighborhood is chosen as the current configuration. This strategy has a technical flaw. It does not provide sufficient chances for the preceding minimumenergy configuration, including promising configurations next to the global optimal solution, to search further lower-energy configurations. What is more, by this strategy, TS is apt to accept higher-energy configuration as the current configuration. On the premise of retaining the aspiration criterion, in this paper we improve the acceptance criteria of the current configuration as follows: 1) If the circular object Ci which is picked from the preceding configuration X and will do an action is a tabu object, and the energy U (X ) of the configuration X obtained by doing an action on Ci and subsequently executing local search is lower than the energy Uopt of the current optimal configuration, we release compulsively Ci , and modify the terms of all objects in the tabu list and release the object whose term is 0. Replace the current optimal configuration and the current configuration by X . 2) If Ci is a tabu object and the energy U (X ) of the configuration X obtained by doing an action on Ci and subsequently executing local search is not lower than Uopt , we reject X and restore X as the current configuration. 3) If Ci is not a tabu object and the energy U (X ) of the configuration X obtained by doing an action on Ci and subsequently executing local search is lower than the energy U (X) of the preceding configuration X, we accept still X as the current configuration and modify the terms of all objects in tabu list and release the object whose term is 0. Moreover, if the energy U (X ) of the configuration X is lower than that of the current optimal configuration, we replace further the current optimal configuration by X . 4) If Ci is not a tabu object and the energy U (X ) of the configuration X obtained by doing an action on Ci and subsequently executing local search is not lower than the energy U (X) of the preceding configuration X, we do not accept X and restore X as the current configuration. It is obviously that this improved strategy is advantageous for HTS to start from the current promising configuration to search further the global optimal configuration. 4.6

Stopping criterion

HTS is stopped when one of the following conditions is reached: 1) The optimal solution is obtained, namely U (X) 10−20, which means the algorithm stops with success; 2) the iterative step number t > 105 , which denotes the algorithm stops with failure. 4.7

Description of the algorithm

By integrating the gradient method with the adaptive step length into the improved tabu search process, a new global search algorithm, the heuristic algorithm based on tabu search (HTS), for the circular

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packing problem with equilibrium constraints is developed. The calculating procedure is described as follows: 1) Randomly give the coordinates (x1 , y1 ), (x2 , y2 ),. . . ,(xn , yn ) of the centers of n circular objects Ci (i=1, 2,. . . ,n) in the circular container C0 with R0 as the radius, whereby the initial configuration X= (x1 , y1 , x2 , y2 ,. . . , xn , yn ) is determined. Let t=1, ε=10−20, and Uopt =1010 . 2) Compute U (X, t). 3) Pick out the circular object Ci with maximum Ei /ri in the current configuration X, and copy X. Let j=1. 4) Pseudo-do randomly Ci 10 actions within the circular container and gain Ci -based candidate neighborhood N (X, Ci ) of X, then pick the optimal candidate configuration from N (X, Ci ) and denote it by X . 5) Call the gradient method (GM) based on the adaptive step length, and let X =GM(X ). 6) If U (X , t) < ε, then output X and stop with success; Else if Ci is a tabu object in the tabu list, then If U (X , t) < Uopt , then Delete Ci from the tabu list and let Uopt = U (X , t), go to 7); Else do not accept X , and restore X as the current configuration. Let j = j+1, and go to 8); Else if U (X , t) < Uopt , then let Uopt = U (X , t); If U (X , t) < U (X, t), then go to 7); Else do not accept X , and restore X as the current configuration. Let j = j+1, and go to 8). 7) Accept X as the current configuration, i.e., let X = X , U (X, t) = U (X , t), and the term of each tabu object in tabu list subtracts 1. Release the object whose term is 0, and go to 9). 8) If j > 5, then let Ei /ri =0 for the circular object Ci , and add Ci into the tabu list, and go to 3); Else go to 4). 9) If t < 105 , then let t = t+1, and go to 3); Else stop with failure.

5

Computational results and analysis

We implement the heuristic algorithm based on tabu search (HTS) in Java language and run it on a Notebook PC with Intel Core 2 Duo, 1.6 GHz processor and 1.0 GB of RAM. We test two sets of instances from the literature. 5.1

Test instances

The first test set consists of five instances. Instances 1.1 and 1.5 are from [11]; instance 1.2 is from [12]; instances 1.3 and 1.4 are from [10]. These five instances are usually used as benchmark for the circular packing problem with equilibrium constraints by the current literatures [10–19, 21]. The optimal solutions of instances 1.1 and 1.2 are 120.7106782 and 1.0000000, respectively. For each instance, the number n of the circular objects to be packed, and the permissible value of the static non-equilibrium term δJ (g·mm), and the radius ri (mm) and mass mi (g) of each circular object are reported in Table 1, respectively. The second test set which consists of six instances is from [17]. The optimal solutions of instances 2.1 and 2.2 can be deduced easily to be 60.0000000 and 215.4700539, respectively. This set of instances is usually used for the pure public packing without equilibrium constraints [26–28]. However, these instances that vary from regular to non-regular, from identical to non-identical, from small to large problems are a set of good benchmarks that are usually used to test the efficiency of the proposed algorithm. Presently we apply this set of instances to test further the performance of HTS for the circular packing problem

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Table 1 Instance

n

δJ (g·mm)


1579

The first set of test instances

ri (mm) and mi (g), i=1, 2, . . . , n

Instance 1.1 n=5

0.1

Instance 1.2 n =9

0.1

r1 = m1 =20.71, ri = mi =50, i=2, 3, 4, 5 √ √ r1 = r2 = r3 = r4 = ( 2−1), r5 = r6 = r7 = r8 = r9 = (3 − 2 2), mi = ri , i=1, 2, . . . , 9

Instance 1.3 n=5

1.0

r1 =6, r2 =7,r3 =8,r4 =9, r5 =11, mi = ri2 ,i=1, 2, . . . , 5

Instance 1.4 n=7

3.4

r1 =10, r2 =11,r3 =12,r4 =11.5, r5 =9.5, r6 =8.5, r7 =10.5, mi = ri2 ,i=1, 2, . . . , 7

Instance 1.5 n=40

20

r1 =106, m1 =11, r2 =112, m2 =12, r3 =98, m3 =9, r4 =105, m4 =11, r5 =93, m5 =8, r6 =103, m6 =10, r7 =82, m7 =6, r8 =93, m8 =8, r9 =117, m9 =13, r10 =81, m10 =6, r11 =89, m11 =7, r12 =92, m12 =8, r13 =109, m13 =11, r14 =104, m14 =10, r15 =115, m15 =13, r16 =110, m16 =12, r17 =114, m17 =12, r18 =89, m18 =7, r19 =82, m19 =6, r20 =120, m20 =14, r21 =108, m21 =11, r22 =86, m22 =7, r23 =93, m23 =8, r24 =100, m24 =10, r25 =102, m25 =10, r26 =106, m26 =11, r27 =111, m27 =12, r28 =107, m28 =11, r29 =109, m29 =11, r30 =91, m30 =8, r31 =111, m31 =12, r32 =91, m32 =8, r33 =101, m33 =10, r34 =91, m34 =8, r35 =108, m35 =11, r36 =114, m36 =12, r37 =118, m37 =13, r38 =85, m38 =7, r39 =87, m39 =7, r40 =98, m40 =9

Table 2 δJ (g·mm)

ri (mm) and mi (g), i=1, 2, . . . , n

Instance 2.1 n=7

0.01

ri =20, mi = ri2 ,i=1, 2, . . . ,7

Instance 2.2 n=12

0.01

Instance 2.3 n=15

0.01

Instance 2.4 n=17

0.01

Instance 2.5 n=37

0.01

Instance 2.6 n=50

0.01

Instance

n

The second set of test instances

r1 = r2 = r3 =100, r4 = r5 = r6 =48.26, r7 =· · · =r12 =23.72, mi = ri2 , i=1, 2, . . . , 12 r1 =1, ri+1 = ri +1, i=1, 2, . . . , 14, mi = ri2 , i=1, 2, . . . , 15

r1 =25, r2 =20, r3 = r4 =15, r5 = r6 = r7 =10, r8 = r9 =· · · =r17 =5, mi = ri2 , i = 1, 2, . . . , 17 ri =20, mi = ri2 , i=1, 2, . . . , 37 ri =20, mi = ri2 , i=1, 2, . . . , 50

with equilibrium constraints. For each of the second set of instances, the number n of the circular objects to be packed, and the permissible value of the static non-equilibrium term δJ (g·mm), and the radius ri (mm) and mass mi (g) of each circular object are listed in Table 2, respectively. 5.2

Experimental results and comparison

For the first set of instances, we choose the penalty coefficient of the static non-equilibrium l=10−4, 10−4 , 10−5 , 10−7 and 10−4 for instances 1, 2,. . . , 5, respectively. For each instance, HTS is run five times independently. The minimal radius R0 (mm) of the circular container, the average static nonequilibrium value J (g·mm) and average running time T (s) by HTS over five independent runs are listed in Table 3, in comparison with those by the mutation particle swarm optimization (MPSO) algorithm [14], the adaptive particle swarm optimizer (APSO) [16], the improved quasi-physical (IQP) algorithm [17], the improved scatter search (ISS) [19], and the basin filling (BF) algorithm [21]. Table 3 shows that we find the optimal solutions for instances 1.1 and 1.2. It is noteworthy that [17], [19], [21] and this paper use the same target energy function obtained by quasi-physical method by Huang et al. [18, 29], but the objective function values of the resulting configurations obtained by IQP in [17] and by ISS in [19] for (2) satisfy U (X)