Mitsushi Tamaki

A secretary problem with uncertain employment and best choice of available candidates

A finite number of candidates appear one-by-one in random order with all permutations equally likely. We are able, at any time, to rank the candidates that have so far appeared according to some order of preference. Each candidate may be classified into one of two types independent of the other candidates: available or unavailable. An unavailable candidate does not accept an offer of employment. The goal is to find a strategy that maximizes the probability of employing the best among the available candidates based on both the relative ranks and the availabilities observed so far. According to when the availability of a candidate can be ascertained, two models are considered. The availability is ascertained only by giving an offer of employment MODEL 1, while the availability is ascertained just after the arrival of the candidate MODEL 2.

The Secretary Problem with Optimal Assignment

We consider the secretary problem with assignment of the applicants to m positions to be staffed in order to maximize the probability that we select only the m best, and assign the kth best to the kth position. A simple strategy is proposed. Although the optimal strategies for m = 1, 2 are known to be simple, we show that the optimal strategy for m = 3 is not necessarily simple. However, a simple strategy turns out to be a fairly good suboptimal strategy.

Choosing either the best or the second best when the number of applicants is random

Abstract Gusein-Zade considered a version of the secretary problem in which we are allowed to make one choice and regard the choice as successful when the chosen applicant is either the best or the second best among all N applicants, where N is a given constant. Here we attempt to generalize his problem to the one in which N is a random variable whose distribution is known. The optimality equations are derived for any given distribution of N, but our main concerns are to investigate, in detail, the case where N is uniformly distributed on [1, m]. It can be shown that, in this case, the optimal policy has the form similar to that of the Gusein-Zade problem: pass s1 - 1 applicants, then choose the relatively best applicant thereafter (if any), but beginning with the s2(≥ s1)th stage, choose also the relatively second best applicant (if any). Some asymptotics concerning the critical numbers s1 and s2, and the success probability, are also obtained.

Optimal dispatching control of an AGV in a JIT production system

This paper deals with an FMS (Flexible Manufacturing System) in a JIT (Just-In-Time) production system. The FMS consists of m workstations, one dispatching station and a single AGV (Automated Guided Vehicle). Each workstation has an input buffer of limited capacity and its processing times are distributed stochastically. When the processing of a new component starts at the workstation, a withdrawal Kanban attached to it is sent to the dispatching station. The AGV chooses one from workstations whose withdrawal Kanbans are accumulated at the dispatching station, and conveys a component with a withdrawal Kanban from the dispatching station to the workstation. The main purpose of this paper is to find an optimal dispatching policy of the AGV that maximizes the long-run expected average reward per unit time. The problem is formulated as a semi-Markov decision process and an optimal dispatching policy is computed. Numerical experiments are performed to make several comparisons.

Shelf life of candidates in the generalized secretary problem

The study presents a version of the secretary problem called the duration problem in which the objective is to maximize the time of possession of the relatively best or the second best objects. It is shown that in this duration problem there are threshold numbers such that the optimal strategy is determined by them. When the number of objects tends to infinity the thresholds values are ź 0.120381 N ź and ź 0.417188 N ź , respectively, and the asymptotic mean time of shelf life is 0.403827 N .

Optimal stopping rule for the full-information duration problem with random horizon

As a version of the secretary problem, Ferguson, Hardwick and Tamaki (1992) considered an optimal stopping problem called the duration problem. The basic duration problem is the classical duration problem, in which the objective is to maximize the time of possession of a relatively best object when a known number of rankable objects appear in random order. In this paper we generalize this classical problem in two directions by allowing the number N (of available objects) to be a random variable with a known upper bound n and also allowing the objects to appear in accordance with Bernoulli trials. Two models can be considered for our random horizon duration problem according to whether the planning horizon is N or n. Since the form of the optimal rule is in general complicated, our main concern is to give to each model a sufficient condition for the optimal rule to be simple. For N having a uniform, generalized uniform, or curtailed geometric distribution, the optimal rule is shown to be simple in the so-called secretary case. The asymptotic results, as n → ∞, will also be given for these priors.

Duration problem with multiple exchanges

We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the

Maximizing the probability of stopping on any of the last m successes in independent Bernoulli trials with random horizon

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Duration problem on trajectories

--choice duration problem, there exists a sequence (s1,s2,...,sm) of critical numbers such that, whenever there remain k choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time

A Bayesian Approach to the Best-Choice Problem

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