This dissertation covered distinct flavors of Multidimensional Packing Problems, considering instances in two and three dimensions, with different objective functions. In all instances considered here, both items and bins have rectangular shapes.
In Chapter I, besides introducing the notation and the typology of MD-PP, and
reviewing previous work, we analyze the effect of restricting rotation and certain types of pattern restrictions on the number of bins used on 2D-BPP. It is shown that restricting the orientation of items can lead to solutions using twice as many bins as when 90º rotations are allowed. The restrictions of guillotine cut patterns and 1st-order non-guillotine patterns are also considered, with asymptotic results presented. For the case when items have to be packed together, or connected, up to a 33% increase in the number of bins can be expected.
Chapters III and IV are dedicated to Pallet Loading Problems (PLP).
In Chapter III, we define the idea of the Minimum Size Instance (MSI) of an
equivalence class of PLP, and show that every class has one and only one MSI. This makes the MSI extremely helpful in distinguishing equivalence classes. We also develop bounds on the dimensions of item and pallet in the MSI of a class, given the set of efficient partitions of an instance of that class. We also show that a bound used for almost 15 years [Dowsland 1987] is incorrect.
Applying the newly developed bounds to the MSI, we enumerate all equivalence classes with area ratio bound as large as 100. Previous work in this area considered at most a subset of these classes.
The first of these new bounds is the Single Homogeneous Perfect Partition (SHPP) bound. This bound, together with simple heuristics, proves optimality of almost 50% of all classes that have no proven optimal solution after applying other simple bounds.
The second bound is the Single Perfect Partition (SPP) bound. It can be useful when the MSI of a class presents only one perfect X-partition or Y-partition.
The third bound is the Relaxed Class (RC) bound, requiring the definition of new relations among distinct equivalence classes of PLP – restricted and relaxed classes. Every feasible packing pattern in a restricted class is shown to be adoptable by a relaxed class.
The fourth bound, the Combined Perfect Partition and Relaxed Class (CPPRC) bound, combines ideas of the SHPP and the RC bounds.
Bounds based on similarity of classes and an improved LP bound are presented, but not completely explored. These two bounds are applied to only a few instances of PLP.
Among the algorithms, the Hollow Block Heuristic is the simplest one proposed in
Chapter IV. It is related to the diagonal heuristic, described in Nelissen [1993], but has a
simpler structure. The G5-heuristic generates 1st-order non-guillotine packing patterns, and
finds optimal solutions to all instances of PLP with an area ratio (AR) bound of up to 51 boxes, 99.999% of all instances with an AR bound of up to 100 boxes, and differs at most by one box in the 0.001% remaining instances.
Three other heuristics are proposes based on non-guillotine cut patterns of higher order, with at most eight blocks. These heuristics solve some instances not solved with the G5-heuristic, nor by other heuristics from the literature.
After the application of the proposed bounding procedures and heuristics, 6,952 equivalence classes of PLP with an AR bound of up to 100 boxes remain without a proven optimal solution. An exact algorithm is developed to solve these last remaining instances. The HVZ algorithm is based on the idea of representing a packing pattern with a ternary string, with the characters H, V, and Z. This new exact algorithm, together with the application of the RC bound, is able to identify the optimal solution to the remaining instances, obtaining in only three equivalence classes a solution not generated with the heuristics proposed in this dissertation.
Chapters V and VI are dedicated to knapsack problems, in two and three
dimensions (2D-KP and 3D-KP).
In Chapter V we implement a mixed integer program (MIP) model of 2D-KP and
show that it can be used, with a commercial optimization software to solve some instances from the literature.
In Chapter VI, we extend the HVZ algorithm for PLP to develop the Diagonal Fill
Algorithm (DFA) for 2D-KP and 3D-KP, and show that these new algorithms are able to solve orthogonal versions of instances from the literature only solved before considering fixed orientation or guillotine cut patterns. The two proposed algorithms, 2D-DFA and 3D- DFA, are the first algorithms in the literature to solve orthogonal non-guillotine instances of 2D-KP and 3D-KP.
Multidimensional Bin Packing Problems (MD-BPP) are addressed in Chapter VII.
New branch-and-price algorithms for 2D-BPP and 3D-BPP, based on the 2D-DFA and 3D- DFA are developed. These are also the first algorithms in the literature to solve the orthogonal non-guillotine instances.
For cases where a good solution soon is better than an optimal solution later, we substitute the exact algorithms for MD-KP in the branch-and-price procedure for MD-BPP. This new heuristic finds better solutions to instances of MD-BPP from the literature.