Solvability in Discrete, Nonstationary, Infinite Horizon Optimization

For several time-staged operations management problems, the optimal immediate decision is dependent on the choice of problem horizon. When that horizon is very long or indefinite, an appropriate modeling technique is infinite horizon optimization. For problems that have stationary data over time, optimizing system performance over an infinite horizon is generally no more difficult than optimizing over a finite horizon. However, restricting problem data to be stationary can render the models unrealistic, failing to include nonstationary aspects of the real world. The primary difficulty in nonstationary, infinite horizon optimization is that the problem to solve can never be known in its entirety. Thus, solution techniques must rely upon increasingly longer finite horizon problems. Ideally, the optimal immediate decisions to these finite horizon problems converge to an infinite horizon optimum. When finite detection of that optimal decision is possible, we call the underlying infinite horizon problem well-posed. The literature on nonstationary, infinite horizon optimization has generally relied upon either uniqueness of the optimal immediate decision or monotonicity of that decision as a function of horizon length. In this thesis, we require neither of these, instead developing a more general structural condition called coalescence that is equivalent to well-posedness. Chapters 2-4 study infinite horizon variants of three deterministic optimization applications: concave cost production planning, single machine replacement, and capacitated inventory planning. For each problem, we show that coalescence is equivalent to well-posedness. We also give a solution procedure for each application that will uncover an infinite horizon optimal immediate decision for any well-posed problem. In Chapter 5, we generalize the results of these applications to a generic classes of optimization problems expressible as dynamic programs. Under two different sets of assumptions concerning the finiteness of and reachability between states, we show that coalescence and well-posedness are equivalent. We also give solution procedures that solve any well-posed problem under each set of assumptions. Finally, in Chapter 6, we introduce a stochastic application: the infinite horizon asset selling problem, and again show that coalescence and well-posedness are equivalent and give a solution procedure to solve any such well-posed problem.Ph.D.Industrial & Operations EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/60810/1/tlortz_1.pd