Solving Packing Problems by a Distributed Global Optimization Algorithm

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Volume 2012, Article ID 931092,13pages doi:10.1155/2012/931092

Research Article

Solving Packing Problems by a Distributed Global

Optimization Algorithm

Nian-Ze Hu,

1

_{Han-Lin Li,}

2

_{and Jung-Fa Tsai}

3

1_{Department of Information Management, National Formosa University, Yunlin 632, Taiwan} 2_{Institute of Information Management, National Chiao Tung University, No. 1001, Ta Hsueh Road,}

Hsinchu 300, Taiwan

3_{Department of Business Management, National Taipei University of Technology,} No. 1, Sec. 3, Chung Hsiao E. Road, Taipei 10608, Taiwan

Correspondence should be addressed to Jung-Fa Tsai,jftsai@ntut.edu.tw

Received 23 February 2012; Accepted 9 May 2012 Academic Editor: Yi-Chung Hu

Copyrightq 2012 Nian-Ze Hu et al. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Packing optimization problems aim to seek the best way of placing a given set of rectangular boxes within a minimum volume rectangular box. Current packing optimization methods either find it diﬃcult to obtain an optimal solution or require too many extra 0-1 variables in the solution process. This study develops a novel method to convert the nonlinear objective function in a packing program into an increasing function with single variable and two fixed parameters. The original packing program then becomes a linear program promising to obtain a global optimum. Such a linear program is decomposed into several subproblems by specifying various parameter values, which is solvable simultaneously by a distributed computation algorithm. A reference solu-tion obtained by applying a genetic algorithm is used as an upper bound of the optimal solusolu-tion, used to reduce the entire search region.

1. Introduction

A packing optimization problem is to seek a minimal container, which can hold a given number of smaller rectangular boxes. This problem is also referred to as a container load-ing problem. Packload-ing cartons into a container is concernload-ing material handlload-ing in the manu-facturing and distribution industries. For instance, workers in the harbor have to pack more than one type of cartons into a container, and they often deal with this problem by the rule of thumb but a systematic approach. Therefore, the utilization of the container is low , which will cause additional costs.

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Similar issues can be found in fields such as knapsack 1, 2, cutting stock 3, 4,

assortment problems 5, 6, rectangular packing 7, pallet loading 8, 9, and container

loading problems10,11. In addition, researchers have dealt with various related problems.

For instance, Dowsland 12 and Egeblad and Pisinger 1, He et al. 13, Wu et al. 14,

de Almeida and Figueiredo15, Miyazawa and Wakabayashi 16, and Crainic et al. 17

proposed diﬀerent heuristic methods for solving three-dimensional packing problems, Chen et al. 10 formulated a mixed integer program for container loading problems, and Li

and Chang 6 developed a method for finding the approximate global optimum of the

assortment problem. However, Li and Chang’s method 6 requires using numerous 0-1

variables to linearize the polynomial objective function in their model, which would involve heavy computation in solving packing problems. Moreover, Chen et al.’s approach10 can

only find a local optimum of packing problems with nonlinear objective function. Recently, many global optimization methods have been developed, where Floudas’ method18 is one

of most promising methods for solving general optimization problems. Although Floudas’ method 18 can be applied to solve a packing problem to reach finite ε-convergence to

the global minimum, it requires successively decomposing the concave part of the original problem into linear functions. Besides, it adopts lower boundary techniques, which is time consuming. Li et al.19 designed a distributed optimization method to improve the

computational eﬃciency for solving packing problems. Tsai and Li 20 also presented an

enhanced model with fewer binary variables and used piecewise linearization techniques to transform the nonconvex packing problem into a mixed-integer linear problem, which is solvable to find a global solution.

Due to the complexity and hardness of three-dimensional packing problems, most results on this topic are based on heuristics21. Furthermore, parallel computing 22 was

adopted to improve the eﬃciency of combinatorial computation. Parallel Genetic Algorithm GA 23, parallel with heuristic 24, and parallel Tabu Search Algorithm TSA 25 were

proposed to solve container-packing problems under some conditions. These methods are capable of obtaining solutions with good performance relative to test examples in the litera-ture. However, the algorithms cannot guarantee to get a global optimum.

This paper proposes another method for finding the optimum of the packing problem. The major advantage of this method is that it can reformulate the nonlinear objective function of original packing problem as an increasing function with single variable and two given parameters. In addition, distributed computation and genetic algorithm are adopted to improve the eﬃciency and ensure the optimality. The proposed method then solves the reformulated programs by specifying the parameters sequentially to reach the globally opti-mal solution on a group of network-connected computers.

2. Problem Formulation

Given n rectangular boxes with fixed lengths, widths, and heights, a packing optimization problem is to allocate these n boxes within a rectangular container having minimal volume. Denote x, y, and z as the width, length, and height of the container; the packing optimization problem discussed here is stated as follows:

minimize xyz

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2 all of n boxes are within the range of x, y, and z. 3 x ≤ x ≤ x, y ≤ y ≤ y, z ≤ z ≤ z

x, y, z, x, y, z are constants.

2.1 According to Chen et al.10, the current packing model adopts the terminologies as

follows.

pi, qi, ri: Dimension of box i, pi is the length, qiis the width, and riis the height,

and pi, qi, and riare integral constants. i ∈ J, J {1, 2, 3, . . . , n} is the set of the given

boxes.

x, y, z: Variables indicating the length, width, and height of the container. xi, yi, zi: Variables indicating the coordinates of the front-left-bottom corner of box

i.

lxi, lyi, lzi: Binary variables indicating whether the length of box i is parallel to the

X-axis, Y -axis, or Z-axis. The value of lxiis equal to 1 if the length of box i is parallel

to the X-axis; otherwise, it is equal to 0. It is clear that lxi lyi lzi 1.

wxi, wyi, wzi: Binary variables indicating whether the width of box i is parallel to

the X-axis, Y -axis, or Z-axis. The value of wxi is equal to 1 if the width of box i is

parallel to the X-axis; otherwise, it is equal to 0. It is clear that wxi wyi wzi 1.

hxi, hyi, hzi: Binary variables indicating whether the width of box i is parallel to

the X-, Y -, or Z-axis. The value of hxiis equal to 1 if the height of box i is parallel to

the X-axis; otherwise, it is equal to 0. It is clear that hxi hyi hzi 1.

For a pair of boxesi, k, where i < k, there is a set of 0-1 vector aik, bik, cik, dik, eik, fik defined

as

aik 1 if box i is on the left of box k, otherwise aik 0,

bik 1 if box i is on the right of box k, otherwise bik 0,

cik 1 if box i is behind box k, otherwise cik 0,

dik 1 if box i is in front of box k, otherwise dik 0,

eik 1 if box i is below box k, otherwise eik 0,

fik 1 if boxi is above box k, otherwise fik 0.

The front-left-bottom corner of the container is fixed at the origin. The interpretation of these variables is illustrated inFigure 1.Figure 1contains two boxes i and k, where box i is located with its length along the X-axis and the width parallel to the Z-axis, and box k is located with its length along the Z-axis and the width parallel to the X-axis. We then have lxi,

wzi, hyi, lzk, wxk, and hyk equal to 1. In addition, since box i is located on the left-hand side

of and in front of box k, it is clear that aik  dik 1 and bik cik eik fik 0.

According to Chen et al.10 and Tsai and Li 20, the packing problem can be

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Problem 1.

Minimize Obj xyz 2.2

subject to xi pilxi qiwxi rihxi≤ xk 1 − aikM ∀i, k ∈ J, i < k, 2.3 xk pklxk qkwxk rkhxk≤ xi 1 − bikM ∀i, k ∈ J, i < k, 2.4 yi pilyi qiwyi rihyi≤ yk 1 − cikM, ∀i, k ∈ J, i < k, 2.5 yk pklyk qkwyk rkhyk≤ yi 1 − dikM, ∀i, k ∈ J, i < k, 2.6 zi pilzi qiwzi rihzi≤ zk 1 − eikM, ∀i, k ∈ J, i < k, 2.7 zk pklzk qkwzk rkhzk≤ zi 1− fik M, ∀i, k ∈ J, i < k, 2.8 aik bik cik dik eik fik≥ 1, ∀i, k ∈ J, i < k, 2.9 xi pilxi qiwxi rihxi≤ x, ∀i ∈ J, 2.10 yi pilyi qiwyi rihyi≤ y, ∀i ∈ J, 2.11 zi pilzi qiwzi rihzi≤ z, ∀i ∈ J, 2.12 lxi lyi lzi 1, ∀i ∈ J, 2.13 wxi wyi wzi 1, ∀i ∈ J, 2.14 hxi hyi hzi 1, ∀i ∈ J, 2.15 lxi wxi hxi 1, ∀i ∈ J, 2.16 lyi wyi hyi 1, ∀i ∈ J, 2.17 lzi wzi hzi 1, ∀i ∈ J, 2.18 where lxi, lyi, lzi, wxi, wyi, wzi, hxi, hyi, hzi, aik, bik, cik, dik, eik, fik are 0-1 variables, 2.19 M maxx, y, z, xi, yi, zi≥ 1, 1 ≤ x ≤ x ≤ x, 1 ≤ y ≤ y ≤ y, 1 ≤ z ≤ z ≤ z, x, y, z, x, y, z are constants, 2.20

x, y, z are positive variables. 2.21

The objective of this model is to minimize the volume of the container. The constraints 2.3–2.9 are nonoverlapping conditions used to ensure that none of these n boxes overlaps

with each other. Constraints 2.10–2.12 ensure that all boxes are within the enveloping

container. Constraints2.13–2.18 describe the allocation restrictions among logic variables.

For instance,2.13 implies that the length of box i is parallel to one of the axes. 2.16 implies

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Z ri rk pi pk qi qk z Y x X y (xk, yk, zk) (xi, yi yi , zi) Box i Box k (0, 0, 0)

Figure 1: Graphical illustration.

Since the objective function of Problem1is a product term, Problem1is a nonlinear mixed 0-1 program, which is diﬃcult to be solved by current optimization methods. Chen et al.10 can only solve linear objective function. Tsai and Li’s method 20 can solve Problem1

at the price of adding many extra 0-1 variables.

3. Proposed Method

Consider the objective function Obj  xyz in 2.2, where x ≥ y ≥ z, 1 ≤ x ≤ x ≤ x, 1 ≤ y ≤

y ≤ y, 1 ≤ z ≤ z ≤ z and x, y, z are positive variables. Denote r and s as two variables defined as r x − y z, s x − z. Replace y by 2x − s − r and replace z by x − s. xyz then becomes Objas follows.

Obj x2x2− r 3sx rs s2 . 3.1

We then have the following propositions.

Proposition 3.1. Suppose r and s in 3.1 are fixed values, then Objis an increasing function.

Proof. Since ∂Obj/∂x 6x2_{− 2xr 3s rs s}2_{xy yz 2xz > 0, it is clear that Obj}_{is an}

increasing function.

Proposition 3.2. The optimal solution of Problem1is integral.

Proof. Since dimensions of box i, pi, qi, ri, are integral constants for i 1, 2, . . . , n and all

of n boxes are nonoverlapping, therefore, x∗, y∗, and z∗that indicate the optimal solution of the container must be integral.

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Proposition 3.3. 3 n

i1piqiri≤ x y z/3, where piis the length, qiis the width, and riis the

height of the given box i. [19,20]

Proof. Since n_i1piqiri ≤ xyz and√3xyz ≤ x y z/3, then we can have3 ni1piqiri ≤

x y z/3.

According to the above propositions and given values of r and s denoted as r and s, consider the following program.

Problem 2. Minimize Objx 2x3− r 3sx2 rs s2x subject to 3 n i1 piqiri≤ x y z 3 , r x − y z, s x − z, 2.3 ∼ 2.21, r and s are fixed values.

3.2

Proposition 3.4. If xΔ_{, y}Δ_{, z}Δ_{is the solution of Problem} ₂ _{found by a genetic algorithm and}

x∗_{, y}∗_{, z}∗_{is the globally optimal solution of Problem}₂_{, then x}∗_{≤ x}Δ_.

Proof. Since Objis an increasing function with single variable x followingProposition 3.1

and 2x∗3_−r 3sx_∗2_{rs s}2_x∗_{≤ 2x}Δ3_−r 3sx_Δ2_{rs s}2_xΔ_{. Hence, x}∗_{≤ x}Δ_.

Adding the constraint x ≤ xΔto Problem2for reducing the search region of the opti-mal solution, we can have the following two programs.

Problem 3.

Minimize Objx 2x3− r 3sx2 rs s2x subject to x ≤ xΔ,

r x − y z, s x − z, 2.3 ∼ 2.21, 3.2, r and s are fixed values.

3.3

Problem 4.

Minimize x

subject to all the constraints in Problem 3. 3.4

Proposition 3.5. Let (x∗_{, y}∗_{, z}∗_{) be the global optimum of Problem}₃_{, then}_x∗_{, y}∗_{, z}∗_{is also the}

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Proof. Since Objx is an increasing function with single variable x following

Proposition 3.1, Problems2and3have the same global optimumx∗, y∗, z∗.

According to the above propositions, a packing optimization problem, which is a nonlinear 0-1 programming problem, can be transformed into a linear 0-1 program by intro-ducing two parameters r and s. Then we can guarantee to obtain the global optimum of a packing problem by solving the transformed linear 0-1 programs. The distributed compu-tation scheme is also proposed to enhance the compucompu-tational eﬃciency.

4. Distributed Algorithm

The solution procedure for solving Problem1to obtain a global optimum is presented in the following with a flow chart shown inFigure 2.

Step 1. Find an initial solution by GA. FromProposition 3.4, the obtained solution isxΔ, yΔ, zΔ and the constraint x ≤ xΔcan be utilized to reduce the searching space of the global solution.

Step 2. Denote sm and s1 as the upper and lower bounds of s, respectively. Find the bounds of s x − z by solving the following linear programs:

sm Maxx − z | subject to2.3 ∼ 2.21, 3.2, s1 Minx − z | subject to2.3 ∼ 2.21, 3.2.

4.1

Let x − z si, i m, and go toStep 3.

Step 3. Denote rn and r1 as the upper and lower bounds of r, respectively. Find the bounds of r x − y z by solving the following linear programs:

rn Maxx − y z | subject to2.3 ∼ 2.21, 3.2, x − z si, r1 Minx − y z | subject to2.3 ∼ 2.21, 3.2, x − z si,

4.2

Let r rj, j 1, and go toStep 4.

Step 4. Decompose main problem and perform distributed packing algorithm. According to verity of r and s, the main problem can be decomposed into several subproblems. The transformed subproblem of each iterative process is listed as follows.

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Start No No Yes No Stop Yes Yes

Find initial solution (x∆, y∆, z∆) by GA.

Let x∗= x∆

Find lower bound s(1) and upper bound s(m) of

s = x − z, let s = s(i), i = m

Find lower bound r(1) and upper bound r(n) of

x − y + z, let r = r(j), j = 1

Solve problem Pijwith x − z = s(i) and

x − y + z = r(j). Let (xij, yij, zij) be the solution

and xijyijzijbe the objective value

xij< x∗

Let (x∗, y∗, z∗) = (xij, yij, zij)

j = n

i =1

(x∗_{, y}∗_{, z}∗_{) is the optimal solution}

i = i − 1 j = j + 1

Figure 2: Solution algorithm.

Problem P

ij

.

Minimize Obj_ij x

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Obj x2x2₋_r_j_3si_{x r}_j_{si si}2 _,

1≤ i ≤ m, 1 ≤ j ≤ n.

4.3

Every sub-problem can be submitted to client computer and solved independently. Server computer controls the whole process and compares the solutionsxij, yij, zij of the

problem Pijwith the initial solutionxΔ, yΔ, zΔ. If xijis smaller than xΔ, then xΔis replaced

by xij.

The structure of distributed packing algorithm is developed based on star schema. Owing to reduce the network loading and improve the computational performance of each client computer, all results found on all clients are directly sent to host computer.

Step 5. Let j j 1. If j > n, then go toStep 6. Otherwise, go toStep 4. Step 6. Let i i − 1. If i < 1, then go toStep 3. Otherwise, go toStep 7.

Step 7. The whole process is finished and the host computer obtains the optimal solution x∗_{, y}∗_{, z}∗_{with the objective value x}∗_y∗_z∗_.

5. Numerical Examples

To validate the proposed method, several examples with diﬀerent number of boxes are solved by LINGO 11.0 26 with the distributed algorithm. Two of the test problems denoted as

Problems1and2are taken from Chen et al.10. The other examples are arbitrarily generated.

Solving these problems by the proposed method, the obtained globally optimal solutions are listed in Tables1and3. Comparison results between GA and the proposed method are shown inTable 2, and the associated graphs are presented in Figures3and4.

Packing problems often arise in logistic application. The following example Problem5 demonstrates how to apply the proposed algorithm in transportation problem

and compare the result with traditional genetic algorithm.

Problem 5. Several kinds of goods are packed into a container so as to deliver to 6 different stores on a trip. The dimensions of width and height of the container are 5 and 4. All goods are packed in cubic boxes, which have three different sizes. In order to take less time during unloading, boxes sent to the same store must be packed together. Different groups of boxes cannot overlap each other. Moreover, the packing order to each group must be ordered of the arriving time to each store. The boxes required to be sent to each store are listed inTable 4. The objective is to determine the smallest dimensions of the container.

Solution 1. The arrangement of boxes can be treated as level assortment. The boxes packed in the same level will be delivered to the same store. After performing the proposed method, list of the optimal solutions are shown in Table 5, and illustrated graph is presented in

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Table 1: Computational results.

Problem number Box number pi qi ri xi yi zi x, y, z Objective value

1 1 25 8 6 0 0 0 28, 26, 6 4368 2 20 10 5 8 0 0 3 16 7 3 8 10 2 4 15 12 6 16 11 0 2 1 25 8 6 10 20 0 35, 28, 6 5880 2 20 10 5 25 0 0 3 16 7 3 0 0 3 4 15 12 6 10 8 0 5 22 8 3 5 0 0 6 20 10 4 0 8 0 3 1 25 8 6 0 10 0 31, 16, 12 5952 2 20 10 5 0 0 0 3 16 7 3 9 0 9 4 15 12 6 25 0 0 5 22 8 3 3 7 9 6 20 10 4 0 0 5 7 10 8 4 31 0 0

Table 2: Solution comparison of the proposed algorithm and genetic algorithm GA.

Problem number GA Proposed method

x, y, z Objective value x, y, z Objective value

14 boxes 30, 30, 6 5400 28, 26, 6 4368

26 boxes 33, 26, 7 6006 35, 28, 6 5880

37 boxes 25, 25, 10 6250 31, 16, 12 5952

Table 3: Computational results for all boxes are cubic. Problem

number

Number of

cubes Side pi× qi× ri x, y, z CPU timeh:min:s Objective value

4 3 1× 1 × 1 8, 5, 5 00:00:08 200 Global optimum 3 2× 2 × 2 1 3× 3 × 3 1 5× 5 × 5 5 4 1× 1 × 1 8, 6, 5 00:01:26 240 Global optimum 2 2× 2 × 2 3 3× 3 × 3 1 5× 5 × 5 6 4 1× 1 × 1 8, 8, 5 00:10:12 320 Global optimum 3 2× 2 × 2 4 3× 3 × 3 7 4 1× 1 × 1 12, 7, 5 01:37:44 420 Global optimum 3 2× 2 × 2 3 3× 3 × 3 1 4× 4 × 4 1 5× 5 × 5

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X Y Z (0, 0, 0) (8, 0, 0) 28 26 6 (16, 11, 0) Box 1 Box 2 Box 3 Box 4 (8, 10, 2) (0, 0, 6) (8, 0, 5)

Figure 3: The graphical representation of 4 boxes.

Figure 4: The solid graphical representation of 6 boxes.

S1 S2 S3 S4 S5 S6

Figure 5: The graphical presentation of the 48 boxes for the 6 stores.

6. Conclusions

This paper proposes a new method to solve a packing optimization problem. The proposed method reformulates the nonlinear objective function of the original packing problem into a linear function with two given parameters. The proposed method then solves the reformu-lated linear 0-1 programs by specifying the parameters sequentially to reach the globally opti-mal solution. Furthermore, this study adopts a distributed genetic algorithm and distributed

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Table 4: List of stores and boxes 48 boxes. Store Goods S1 A, A, A, B, B, B, B, C S2 A, A, B, B, B, B, C, C S3 A, A, A, B, B, B, B, C S4 A, A, B, B, B, C, C, C S5 A, A, A, A, B, B, B, C S6 A, A, A, A, A, B, C, C

A: 1-inch cubic box; B: 2-inch cubic box; C: 3-inch cubic box.

Table 5: List of optimal arrangement of the boxes.

Store S1 S2 S3 S4 S5 S6 x1, y1, z1 A1, 2, 0 A0, 1, 3 A4, 0, 2 A0, 0, 3 A4, 0, 0 A0, 0, 0 x2, y2, z2 A3, 3, 3 A5, 0, 0 A0, 1, 0 A0, 2, 3 A4, 1, 3 A5, 2, 0 x3, y3, z3 A0, 4, 0 B3, 0, 0 A1, 2, 0 B3, 0, 1 A0, 2, 0 A0, 3, 0 x4, y4, z4 B4, 0, 0 B1, 0, 2 B3, 0, 0 B7, 0, 1 A4, 4, 1 A2, 4, 0 x5, y5, z5 B0, 0, 1 B3, 0, 2 B0, 3, 0 B1, 0, 2 B2, 0, 0 A5, 1, 0 x6, y6, z6 B2, 0, 0 B1, 0, 0 B1, 0, 2 C0, 2, 0 B2, 0, 2 B3, 0, 2 x7, y7, z7 C0, 2, 1 C0, 2, 0 B0, 3, 2 C6, 2, 0 B0, 0, 2 C0, 1, 1 x8, y8, z8 C3, 2, 0 C3, 2, 0 C2, 2, 1 C3, 2, 0 C1, 2, 0 C3, 2, 1 Dimension of si 6× 5 × 4 6× 5 × 4 5× 5 × 4 9× 5 × 4 5× 5 × 4 6× 5 × 4 Volume of si 120 120 100 180 100 120

The global solution is37, 5, 4, and the minimal volume of the container is 740.

packing algorithm to enhance the computational eﬃciency. Numerical examples demonstrate that the proposed method can be applied to practical problems and solve the problems to obtain the global optimum.

Acknowledgments

The authors would like to thank the anonymous referees for contributing their valuable com-ments regarding this paper and thus significantly improving its quality. The paper is partly supported by Taiwan NSC Grant NSC 99-2410-H-027-008-MY3.

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