A solution to a countable system of equations arising in Markovian decision processes Technical report no. 89

Optimal rules for controlling Markovian decision processes applied to solutions for problems dealing with ordering inventory supplie

Premortem light: Improving planning using a lightweight, perspective shifting technique

Teams can be overly optimistic regarding the likelihood of their plan’s success. This “planning fallacy” may result from ineffective plan-evaluation strategies. The PreMortem technique, imagining that a plan has failed and then trying to explain why, leverages a perspective-shifting strategy. However, very little research has been conducted to evaluate the validity of this technique. In this experiment, a Premortem Light was tested in a challenge course. Forty-eight members of Michigan Tech’s ROTC completed 6 novel field challenges in small teams. Each team used the Premortem Light for half of the challenges, and their typical plan-evaluation strategy (baseline) for the other half. Compared to the baseline, when teams used the Premortem Light, their plans were better, they fixated during problem solving less, and they made statistically fewer errors. Furthermore, there was no overall increase in course time. This technique can be effective in a variety of situations, including disaster response.https://digitalcommons.mtu.edu/techtalks/1023/thumbnail.jp

Representation of general and polyhedral subsemilattices and sublattices of product spaces

AbstractIt is shown that each element of the lattice of meet (resp., join) sublattices of a product S of n chains has a representation as the intersection of n subsets of S, the ith of which is decreasing (resp., increasing) for each fixed value of the ith coordinate for each i. This result is applied to show that an arbitrary element of the lattice of sublattices of S has a representation as the intersection of n2 subsets, the ijth of which is decreasing for each fixed value of the ith and increasing for each fixed value of the jth coordinate for each i, j. Irreducible representations are given in each case, providing an alternative proof of an instance of Hashimoto's (1952) representation of sublattices of a distributive lattice. Moreover, irreducible representations are given for the polyhedral members of the lattice of closed convex subsets of n-dimensional Euclidean space that are at once subsemilattices or sublattices. It is alsoshown that the polyhedral subsemilattices and sublattices can be represented as duals respectively of pre-Leontief substitution systems and generalized network-flow problems. Finally, the problems of checking whether a polyhedral set is a subsemilattice or sublattice are reduced to that of solving a system of linear inequalities, thereby showing that these recognition problems can be solved in polynomial time

Approximation algorithms for capacitated stochastic inventory systems with setup costs

We develop the first approximation algorithm with worst-case performance guarantee for capacitated stochastic periodic-review inventory systems with setup costs. The structure of the optimal control policy for such systems is extremely complicated, and indeed, only some partial characterization is available. Thus, finding provably near-optimal control policies has been an open challenge. In this article, we construct computationally efficient approximate optimal policies for these systems whose demands can be nonstationary and/or correlated over time, and show that these policies have a worst-case performance guarantee of 4. We demonstrate through extensive numerical studies that the policies empirically perform well, and they are significantly better than the theoretical worst-case guarantees. We also extend the analyses and results to the case with batch ordering constraints, where the order size has to be an integer multiple of a base load.National Science Foundation (U.S.) (CMMI-1362619)National Science Foundation (U.S.) (CMMI-1131249)National Science Foundation (U.S.) (DMS-0732175)National Science Foundation (U.S.) (CMMI-0846554)National Science Foundation (U.S.) (FA9550-08-1–0369

Subextremal functions and lattice programming

Let M and N be the set of minimizers of a function f over respective subsets K and L of a lattice, with K being lower than L. This paper characterizes the class of functions f for which M is lower (resp., weakly lower, meet lower, join lower, chain lower) than N for all K lower than L. The resulting five classes of functions, called subextremal variants, have alternate characterizations by variants of the downcrossing-differences property, i.e., their first differences change sign at most once from plus to minus along complementary chains.Comparative statics, supermodular functions

Value Iteration for Long-run Average Reward in Markov Decision Processes

Markov decision processes (MDPs) are standard models for probabilistic systems with non-deterministic behaviours. Long-run average rewards provide a mathematically elegant formalism for expressing long term performance. Value iteration (VI) is one of the simplest and most efficient algorithmic approaches to MDPs with other properties, such as reachability objectives. Unfortunately, a naive extension of VI does not work for MDPs with long-run average rewards, as there is no known stopping criterion. In this work our contributions are threefold. (1) We refute a conjecture related to stopping criteria for MDPs with long-run average rewards. (2) We present two practical algorithms for MDPs with long-run average rewards based on VI. First, we show that a combination of applying VI locally for each maximal end-component (MEC) and VI for reachability objectives can provide approximation guarantees. Second, extending the above approach with a simulation-guided on-demand variant of VI, we present an anytime algorithm that is able to deal with very large models. (3) Finally, we present experimental results showing that our methods significantly outperform the standard approaches on several benchmarks

A selective newsvendor approach to order management

Consider a supplier offering a product to several potential demand sources, each with a unique revenue, size, and probability that it will materialize. Given a long procurement lead time, the supplier must choose the orders to pursue and the total quantity to procure prior to the selling season. We model this as a selective newsvendor problem of maximizing profits where the total (random) demand is given by the set of pursued orders. Given that the dimensionality of a mixed-integer linear programming formulation of the problem increases exponentially with the number of potential orders, we develop both a tailored exact algorithm based on the L-shaped method for two-stage stochastic programming as well as a heuristic method. We also extend our solution approach to account for piecewise-linear cost and revenue functions as well as a multiperiod setting. Extensive experimentation indicates that our exact approach rapidly finds optimal solutions with three times as many orders as a state-of-the-art commercial solver. In addition, our heuristic approach provides average gaps of less than 1% for the largest problems that can be solved exactly. Observing that the gaps decrease as problem size grows, we expect the heuristic approach to work well for large problem instances. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/61330/1/20320_ftp.pd

Integrating facility location and production planning decisions

We consider a metric uncapacitated facility location problem where we must assign each customer to a facility and meet the demand of the customer in future time periods through production and inventory decisions at the facility. We show that the problem, in general, is as hard to approximate as the set cover problem. We therefore focus on developing approximation algorithms for special cases of the problem. These special cases come in two forms: (i) specialize the production and inventory cost structure and (ii) specialize the demand pattern of the customers. In the former, we offer reductions to variants of the metric uncapacitated facility location problem that have been previously studied. The latter gives rise to a class of metric uncapacitated facility location problems where the facility cost function is concave in the amount of demand assigned to the facility. We develop a modified greedy algorithm together with the idea of cost-scaling to provide an algorithm for this class of problems with an approximation guarantee of 1.52. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/64912/1/20315_ftp.pd

Approximation Algorithms for Stochastic Inventory Control Models

Approximation Algorithms for Stochastic Inventory Control Model