Michael Z. Spivey

The Dynamic Assignment Problem

There has been considerable recent interest in the dynamic vehicle routing problem, but the complexities of this problem class have generally restricted research to myopic models. In this paper, we address the simpler dynamic assignment problem, where a resource (container, vehicle, or driver) can serve only one task at a time. We propose a very general class of dynamic assignment models, and propose an adaptive, nonmyopic algorithm that involves iteratively solving sequences of assignment problems no larger than what would be required of a myopic model. We consider problems where the attribute space of future resources and tasks is small enough to be enumerated, and propose a hierarchical aggregation strategy for problems where the attribute spaces are too large to be enumerated. Finally, we use the formulation to also test the value of advance information, which offers a more realistic estimate over studies that use purely myopic models.

The Lah Numbers and the nth Derivative of e1/x

Summary We give five proofs that the coefficients in the nth derivative of exp(1/x) are the Lah numbers, a triangle of integers whose best-known applications are in combinatorics and finite difference calculus. Our proofs use tools from several areas of mathematics, including binomial coefficients, Faà di Brunos formula, set partitions, Maclaurin series, factorial powers, the Poisson probability distribution, and hypergeometric functions.

Fibonacci Identities via the Determinant Sum Property

Mike Spivey (mspivey@ups.edu) received his B.S. from Samford University and his Ph.D. from Princeton, and he currently teaches at the University of Puget Sound (Tacoma, WA 98416). His research background is in optimization, but he also enjoys dabbling in combinatorics and elementary number theory. Lately, he hasn’t been able to stop himself from playing around with matrices made up of various mathematical entities.

The Euler-Maclaurin Formula and Sums of Powers

Mathematicians have long been intrigued by the sum 1 + 2 + · · ·+ n of the first n integers, where m is a nonnegative integer. The study of this sum of powers led Jakob Bernoulli to the discovery of Bernoulli numbers and Bernoulli polynomials. There are expressions for sums of powers in terms of Eulerian numbers and Stirling numbers [5, p. 199]. In addition, past articles in this MAGAZINE contain algorithms for producing a formula for the sum involving powers of m + 1 from that involving powers of m [1, 4]. (The algorithm in Bloom is actually Bernoulli’s method.) This note involves a curious property concerning sums of integer powers, namely,

A Combinatorial Proof for the Alternating Convolution of the Central Binomial Coefficients

Abstract We give a combinatorial proof of the identity for the alternating convolution of the central binomial coefficients. Our proof entails applying an involution to certain colored permutations and showing that only permutations containing cycles of even length remain.

Some Fixed-Point Results for the Dynamic Assignment Problem

In previous work the authors consider the dynamic assignment problem, which involves solving sequences of assignment problems over time in the presence of uncertain information about the future. The algorithm proposed by the authors provides generally high-quality but non-optimal solutions. In this work, though, the authors prove that if the optimal solution to a dynamic assignment problem in one of two problem classes is unique, then the optimal solution is a fixed point under the algorithm.

Probabilistic Proofs of a Binomial Identity, Its Inverse, and Generalizations

Abstract We give elementary probabilistic proofs of a binomial identity, its binomial inverse, and generalizations of both of these. The proofs are obtained by interpreting the sides of each identity as the probability of an event in two different ways. Each proof uses a classic balls-and-jars scenario.

Optimal Discounts for the Online Assignment Problem

Abstract We prove that, for two simple functions d r l t , solving the online assignment problem with c r l − d r l t as the contribution for assigning resource r to task l at time t gives the optimal solution to the corresponding offline assignment problem (provided the optimal offline solution is unique). We call such functions d r l t optimal discount functions.

The Poisson process and associated probability distributions on time scales

Duals of probability distributions on continuous (R) domains exist on discrete (Z) domains. The Poisson distribution on R, for example, manifests itself as a binomial distribution on Z. Time scales are a domain generalization in which R and Z are special cases. We formulate a generalized Poisson process on an arbitrary time scale and show that the conventional Poisson distribution on R and binomial distribution on Z are special cases. The waiting times of the generalized Poisson process are used to derive the Erlang distribution on a time scale and, in particular, the exponential distribution on a time scale. The memoryless property of the exponential distribution on R is well known. We find conditions on the time scale which preserve the memorylessness property in the generalized case.

Asymptotic Moments of the Bottleneck Assignment Problem

One of the most important variants of the standard linear assignment problem is the bottleneck assignment problem. In this paper we give a method by which one can find all of the asymptotic moments of a random bottleneck assignment problem in which costs (independent and identically distributed) are chosen from a wide variety of continuous distributions. Our method is obtained by determining the asymptotic moments of the time to first complete matching in a random bipartite graph process and then transforming those, via a Maclaurin series expansion for the inverse cumulative distribution function, into the desired moments for the bottleneck assignment problem. Our results improve on the previous best-known expression for the expected value of a random bottleneck assignment problem, yield the first results on moments other than the expected value, and produce the first results on the moments for the time to first complete matching in a random bipartite graph process.