Problems in the classical calculus of variations

This dissertation concerns convergence analysis for nonparametric problems in the calculus of variations and sufficient conditions for weak local minimizer of a functional for both nonparametric and parametric problems. Newton's method in infinite-dimensional space is proved to be well-defined and converges quadratically to a weak local minimizer of a functional subject to certain boundary conditions. Sufficient conditions for global converges are proposed and a well-defined algorithm based on those conditions is presented and proved to converge. Finite element discretization is employed to achieve an implementable line-search-based quasi-Newton algorithm and a proof of convergence of the discretization of the algorithm is included. This work also proposes sufficient conditions for weak local minimizer without using the language of conjugate points. The form of new conditions is consistent with the ones in finite-dimensional case. It is believed that the new form of sufficient conditions will lead to simpler approaches to verify an extremal as local minimizer for well-known problems in calculus of variations.

Decentralized or centralized production : impacts to the environment, industry, and the economy

Since product take-back is mandated in Europe, and has effects for producers worldwide including the U.S., designing efficient forward and reverse supply chain networks is becoming essential for business viability. Centralizing production facilities may reduce costs but perhaps not environmental impacts. Decentralizing a supply chain may reduce transportation environmental impacts but increase capital costs. Facility location strategies of centralization or decentralization are tested for companies with supply chains that both take back and manufacture products. Decentralized and centralized production systems have different effects on the environment, industry and the economy. Decentralized production systems cluster suppliers within the geographical market region that the system serves. Centralized production systems have many suppliers spread out that meet all market demand. The point of this research is to help further the understanding of company decision-makers about impacts to the environment and costs when choosing a decentralized or centralized supply chain organizational strategy. This research explores; what degree of centralization for a supply chain makes the most financial and environmental sense for siting facilities; and which factories are in the best location to handle the financial and environmental impacts of particular processing steps needed for product manufacture. This research considered two examples of facility location for supply chains when products are taken back; the theoretical case involved shoe resoling and a real world case study considered the location of operations for a company that reclaims multiple products for use as material inputs. For the theoretical example a centralized strategy to facility location was optimal: whereas for the case study a decentralized strategy to facility location was best. In conclusion, it is not possible to say that a centralized or decentralized strategy to facility location is in general best for a company that takes back products. Each company’s specific concerns, needs, and supply chain details will determine which degree of centralization creates the optimal strategy for siting their facilities.