Time decomposition of multi-period supply chain models

Many supply chain problems involve discrete decisions in a dynamic environment. The inventory routing problem is an example that combines the dynamic control of inventory at various facilities in a supply chain with the discrete routing decisions of a fleet of vehicles that moves product between the facilities. We study these problems modeled as mixed-integer programs and propose a time decomposition based on approximate inventory valuation. We generate the approximate value function with an algorithm that combines data fitting, discrete optimization and dynamic programming methodology. Our framework allows the user to specify a class of piecewise linear, concave functions from which the algorithm chooses the value function. The use of piecewise linear concave functions is motivated by intuition, theory and practice. Intuitively, concavity reflects the notion that inventory is marginally more valuable the closer one is to a stock-out. Theoretically, piecewise linear concave functions have certain structural properties that also hold for finite mixed-integer program value functions. (Whether the same properties hold in the infinite case is an open question, to our knowledge.) Practically, piecewise linear concave functions are easily embedded in the objective function of a maximization mixed-integer or linear program, with only a few additional auxiliary continuous variables. We evaluate the solutions generated by our value functions in a case study using maritime inventory routing instances inspired by the petrochemical industry. The thesis also includes two other contributions. First, we review various data fitting optimization models related to piecewise linear concave functions, and introduce new mixed-integer programming formulations for some cases. The formulations may be of independent interest, with applications in engineering, mixed-integer non-linear programming, and other areas. Second, we study a discounted, infinite-horizon version of the canonical single-item lot-sizing problem and characterize its value function, proving that it inherits all properties of interest from its finite counterpart. We then compare its optimal policies to our algorithm's solutions as a proof of concept.PhDCommittee Chair: George Nemhauser; Committee Member: Ahmet Keha; Committee Member: Martin Savelsbergh; Committee Member: Santanu Dey; Committee Member: Shabbir Ahme

Approximately Bisimilar Discrete Abstractions of Nonlinear Systems Using Variable-resolution Quantizers

Abstract-This paper presents a method for the design of discrete abstract models of nonlinear continuous-state systems under the framework of approximate bisimulation. First, the notion of quantizer embedding, which transforms a continuousstate system into a finite-state system, is extended to a variableresolution setting. Next, it is shown that the series of conditions for approximate bisimulation can be converted into a set of linear inequalities, which can be verified by a linear programming solver. From this result, we obtain an algorithm that repeatedly refines a variable-resolution mesh until approximate bisimulation with a prescribed error specification is achieved

Aerodynamic shape optimization via discrete adjoint formulation using Euler equations on unstructured grids

A methodology for aerodynamic shape optimization on two-dimensional unstructured grids using Euler equations is presented. The sensitivity derivatives are obtained using the discrete adjoint formulation. The Euler equations are solved using a fully implicit, upwind, cell-vertex, median-dual finite volume scheme. Roe\u27s upwind flux-difference-splitting scheme is used to determine the inviscid fluxes. To enable discontinuities to be captured without oscillations, limiters are used at the reconstruction stage;The derivation of the accurate discretization of the flux Jacobians due to the conserved variables and the entire mesh required for the costate equation is developed and its efficient accumulation algorithm on an edge-based loop is implemented and documented. Exact linearization of Roe\u27s approximate Riemann solver is incorporated into the aerodynamic analysis as well as the sensitivity analysis. Higher-order discretization is achieved by including all distance-one and -two terms due to the reconstruction and the limiter, although the limiter is not linearized. Two-dimensional body conforming grid movement strategy and grid sensitivity are obtained by considering the grid to be a system of interconnected springs. Arbitrary airfoil geometries are obtained using an algorithm for generalized von Mises airfoils with finite trailing edges. An incremental iterative formulation is used to solve the large sparse linear systems of equations obtained from the sensitivity analysis. The discrete linear systems obtained from the equations governing the flow and those from the sensitivity analysis are solved iteratively using the preconditioned GMRES (Generalized Minimum Residual) algorithm. For the optimization process, a constrained nonlinear programming package which uses a sequential quadratic programming algorithm is used;This study presents the process of analytically obtaining the exact discrete sensitivity derivatives and computationally cost-effective algorithms to efficiently use them in a design environment. Storage issues are circumvented by developing algorithms which perform matrix-vector operations during the construction itself. To validate the unstructured shape optimization procedure, an arbitrary symmetrical airfoil is optimized in an inviscid transonic flow regime