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 01 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 solution obtained by applying a genetic algorithm is used as an upper bound of the optimal solution, used to reduce the entire search region.
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 loading problem. Packing cartons into a container is concerning material handling in the manufacturing 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.
Similar issues can be found in fields such as knapsack [
Due to the complexity and hardness of threedimensional packing problems, most results on this topic are based on heuristics [
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 optimal solution on a group of networkconnected computers.
Given
According to Chen et al. [
: Dimension of box
For a pair of boxes
The frontleftbottom corner of the container is fixed at the origin. The interpretation of these variables is illustrated in Figure
Graphical illustration.
According to Chen et al. [
The objective of this model is to minimize the volume of the container. The constraints (
Since the objective function of Problem
Consider the objective function
Suppose
Since
The optimal solution of Problem
Since dimensions of box
Since
According to the above propositions and given values of
If
Since
Adding the constraint
Let (
Since
According to the above propositions, a packing optimization problem, which is a nonlinear 01 programming problem, can be transformed into a linear 01 program by introducing two parameters
The solution procedure for solving Problem
Solution algorithm.
(Find an initial solution by GA). From Proposition
Denote
Denote
Decompose main problem and perform distributed packing algorithm. According to verity of
Every subproblem can be submitted to client computer and solved independently. Server computer controls the whole process and compares the solutions (
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.
Let
Let
The whole process is finished and the host computer obtains the optimal solution
To validate the proposed method, several examples with different number of boxes are solved by LINGO 11.0 [
Computational results.
Problem number  Box number 






( 
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 
Solution comparison of the proposed algorithm and genetic algorithm (GA).
Problem number  GA  Proposed method  

( 
Objective value  ( 
Objective value  
1 (4 boxes)  (30, 30, 6)  5400  (28, 26, 6)  4368 
2 (6 boxes)  (33, 26, 7)  6006  (35, 28, 6)  5880 
3 (7 boxes)  (25, 25, 10)  6250  (31, 16, 12)  5952 
Computational results for all boxes are cubic.
Problem number  Number of cubes  Side 
( 
CPU time (h:min:s)  Objective value 

4  3 

(8, 5, 5)  00:00:08  200 Global optimum 
3 
 
1 
 
1 


 
5  4 

(8, 6, 5)  00:01:26  240 Global optimum 
2 
 
3 
 
1 


 
6  4 

(8, 8, 5)  00:10:12  320 Global optimum 
3 
 
4 


 
7  4 

(12, 7, 5)  01:37:44  420 Global optimum 
3 
 
3 
 
1 
 
1 

The graphical representation of 4 boxes.
The solid graphical representation of 6 boxes.
Packing problems often arise in logistic application. The following example (Problem
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 in Table
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: 1inch cubic box; B: 2inch cubic box; C: 3inch cubic box.
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
List of optimal arrangement of the boxes.
Store  S1  S2  S3  S4  S5  S6 


A (1, 2, 0)  A (0, 1, 3)  A (4, 0, 2)  A (0, 0, 3)  A (4, 0, 0)  A (0, 0, 0) 
( 
A (3, 3, 3)  A (5, 0, 0)  A (0, 1, 0)  A (0, 2, 3)  A (4, 1, 3)  A (5, 2, 0) 
( 
A (0, 4, 0)  B (3, 0, 0)  A (1, 2, 0)  B (3, 0, 1)  A (0, 2, 0)  A (0, 3, 0) 
( 
B (4, 0, 0)  B (1, 0, 2)  B (3, 0, 0)  B (7, 0, 1)  A (4, 4, 1)  A (2, 4, 0) 
( 
B (0, 0, 1)  B (3, 0, 2)  B (0, 3, 0)  B (1, 0, 2)  B (2, 0, 0)  A (5, 1, 0) 
( 
B (2, 0, 0)  B (1, 0, 0)  B (1, 0, 2)  C (0, 2, 0)  B (2, 0, 2)  B (3, 0, 2) 
( 
C (0, 2, 1)  C (0, 2, 0)  B (0, 3, 2)  C (6, 2, 0)  B (0, 0, 2)  C (0, 1, 1) 
( 
C (3, 2, 0)  C (3, 2, 0)  C (2, 2, 1)  C (3, 2, 0)  C (1, 2, 0)  C (3, 2, 1) 
 
Dimension of 






 
Volume of 
120  120  100  180  100  120 
 
The global solution is (37, 5, 4), and the minimal volume of the container is 740. 
The graphical presentation of the 48 boxes for the 6 stores.
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 reformulated linear 01 programs by specifying the parameters sequentially to reach the globally optimal solution. Furthermore, this study adopts a distributed genetic algorithm and distributed 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.
The authors would like to thank the anonymous referees for contributing their valuable comments regarding this paper and thus significantly improving its quality. The paper is partly supported by Taiwan NSC Grant NSC 992410H027008MY3.