Optimizing investments in interdependent infrastructures
Garcia Llinas, Guisselle Adriana
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The design of infrastructure networks challenges managers and researchers due to the size of current networks, their interconnectedness and the disruptions they are exposed to. Many studies address the topic of the design of these infrastructures ignoring the interdependencies to which they are subject. Others acknowledge the relevance of interdependencies when disruptions occur but limit their work to restore the infrastructures to their original design. Our main concern is to develop an integrated model for the robust design of interdependent infrastructures that minimizes the total expected cost of operation of the system. Making decisions about all the infrastructures at the same time allows the distribution of investment budgets to the most critical portions of the systems to optimize the resilience of the infrastructures. Our work is divided in three chapters. We start from a single but complex infrastructure (i.e., transportation system,) moving to the development of the integrated mathematical model for multiple infrastructures and concluding with optimization methods to solve the integrated model. Chapter 1 is a review of the ``Discrete Network Design Problem in Transportation" and the available \ methods to solve it. A traditional method by Leblanc (1975) is proven to fail if a certain parameter is not adequately fixed. Due to the uncertainty to do this selection, we propose linearization methods to modify the method and avoid pitfalls. Results are presented to measure the negative effects of obtaining wrong solutions to a problem and to demonstrate the efficiency of the modified method. Chapter 2 proposes the integration of multiple infrastructures in a single model for their robust design. Failure scenarios are used to provide robustness to the optimal solution, and dependency relationships of various types are modeled to integrate the infrastructures. These dependencies play an important role especially when the systems are under failure and the flows of material have to be re-accommodated. This is the reason why failures are spread through different infrastructures (i.e., cascade effects.) The optimal design provides the minimum total expected cost of operation of the system of systems subject to a fixed budget for investments. The model is a mixed-integer linear program whose complexity challenges the operations research community. A semi-real case is solved using a commercial solver to illustrate the usefulness of the model. Chapter 3 develops a decomposition method to solve the integrated model of Chapter 2. A Benders' decomposition (BD) scheme is used to separate the integer decisions from the continuous ones. New cuts to the master problem for the selection of values for the integer variables are added after each iteration of the BD algorithm. The algorithm is proven to find the optimal solution of a very small instance. For larger instances the repetitive solution of the linear subproblem causes efficiency issues. The Dantzig-Wolfe (DW) decomposition method is implemented to solve this linear program. The convenient block diagonal structure of the matrix allows the reduction to linear programs for independent infrastructures, after separating the dependency constraints. The results show that numerical issues should be solved to make the double-decomposed method usable for large instances. The integrated model has the capacity to distribute among the infrastructures the resources for the improvement of the networks. Failure scenarios are integrated to the model to generate robust designs. The dependency relationships are relevant to find realistic optimal solutions for the design of the current highly interdependent networks.