Heterogeneous Objects are engineered with multiple materials to achieve multiple functionalities like high hardness, high toughness and low structural weight. Heterogeneous objects are increasingly used to achieve multiple and often conflicting behaviour within a single object. Developing heterogeneous objects needs computational model for design, analysis and manufacturing. The computational model should map the geometry of the object with the material composition. The most general model is the volume based model that decomposes the geometry exhaustively into simple elements and defines the material distribution over these elements. This approach can model a wide range of objects. However, defining material distribution needs manual intervention to select these elements for material continuity, and to segment or subdivide them for better material approximation. Volume based representation is quite large in size and is cumbersome to edit, query or reuse. Feature based approaches have been proposed to address some of these issues. However, current art can model only limited class of Heterogeneous Objects that includes simple material distribution over complex geometry or complex material distribution over simple geometry. The thesis presents a new method to overcome these limitations. The method, a hybrid of volume based and feature based approaches, allows the user to define the complex material distribution over complex geometries intuitively and represent the same. The complex material distribution is modelled using material reference entities that may be mixed-dimensional, inclusive of non-manifold entities. It uses Medial Axis Transform for automated segmentation of these entities into independent regions, where the material distribution can be intuitively prescribed starting from the entity and terminating at the medial axis. The spatial variation of the material is captured by a parameterized distance field from the material reference entities. It develops new constructive operators to build a complex heterogeneous object model that allows the reuse of the existing heterogeneous object models, automates handling of material continuity, and controls the gradation of the material in the interface of the constituent heterogeneous objects. Constructions using these operators can be geometry driven or material driven i.e. the geometric form controls the material distribution or the material distribution is independent of the geometric form. The proposed representation can be adaptively meshed for generation of mesh in the direction of gradation of the material for finite element analysis and process planning for additive manufacturing. An iso-material contour representation has been proposed for process planning of Heterogeneous Object Models. This avoids the stair case effect by depositing material in the direction of material gradation, and avoids over-deposition or under-deposition due to frequent start and stop of the nozzles. The proposed method has been implemented to show that it can model wide range of heterogeneous objects and can be integrated with additive manufacturing.

Allocation of objects or resources to competing agents is a ubiquitous problem in the real world. For example, a federal government may wish to allocate different types of spectrum licenses to telecom service providers; a search engine has to assign different sponsored slots to the ads of advertisers; etc. The agents involved in such situations have private preferences over the allocations. The agents, being strategic, may manipulate the allocation procedure to get a favourable allocation. If the objects to be allocated are heterogeneous (rather than homogeneous), the problem becomes quite complex. The allocation problem becomes even more formidable in the presence of a dynamic supply and/or demand. This doctoral work is motivated by such problems involving strategic agents, heterogeneous objects, and dynamic supply and/or demand. In this thesis, we model such problems in a standard game theoretic setting and use mechanism design to propose novel solutions to the problems. We extend the current state-of-the-art in a non-trivial way by solving the following problems: Optimal combinatorial auctions with single minded bidders, generalizing the existing methods to take into account multiple units of heterogeneous objects Multi-armed bandit mechanisms for sponsored search auctions with multiple slots, generalizing the current methods that only consider a single slot. Strategyproof redistribution mechanisms for heterogeneous objects, expanding the scope of the current state of practice beyond homogeneous objects Online allocation mechanisms without money for one-sided and two-sided matching markets, extending the existing methods for static settings.