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where xi，yi and zi stand for the coordinates of the center of the envelope of the ith model; θi is the orientation of the envelope; n is the total number of parts to be packed; Xi ，Yi and Zi are the lengths of the ith envelope along x， y and z directions.
The objective function f defines a goodness value for each packing candidate to guide the search in a promising direction. In this study, the goal is to minimize the overall layout height which can be determined by the following equation:
During the search process, overlap is permitted to allow models to be moved through one another and results in a more thorough search of the solution space. Since any overlap should be avoided in a final layout, the overlap is penalized in the objective function. Thus the objective function has a second term, f2:
When two rectangular boxes overlap in parallel, the overlap is also a rectangular box. In Eq. (4), OijexT; OijeyT and Oij ezT are the lengths of the overlap between the envelopes of the ith and the jth models along x， y and z directions, respectively. Oij is the sum of the lengths of the overlap along the three orthogonal directions.
The overlap quantification in Eq. (4) and the following linearly weighted objective function are most commonly used for multi-objective optimizations [7,9]:
However, this function can seldom result in successful optimization in this packing problem. In the search process, f value of one solution was compared with that of another solution so that the search always goes in the direction of reducing the value of the objective function. With different values of the weights, the comparison results can be very different. Thus the relative value of these two weights is critical. Since the range of f2 varies dramatically according to the number of parts in a batch and the dimensions of parts, there is no single good choice of a pair of ‘weights’ for different packing instances. Even if there were a pair of weights reflecting the importance of the overlap and the height objectively, it would be difficult to decide their values mathematically. If the overlap f2 is given more weight than it deserves, namely, w2 being much larger than w1; the objective function would be more sensitive to the change of the overlap value f2: The models would be placed apart in the early stage of the search process to eliminate the overlap. Finally, the models would be stacked higher in the container. If the height is given more significance than it deserves, the objective function would be more sensitive to the change of the overall height and the overlap would be encouraged because the value of the objective function would be reduced with layouts of lower height.