Some Models for a Hierarchical Manpower System

This paper is an extension of previous work which described a model for a hierarchical manpower system with two grades. The model is first extended to a system with several grades, and then to a system with several types of entrant or in which the type of entrant has changed over time. Finally, a calculation is described which can be used to determine the best mix of qualifications amongst the entrants.     

A Markov Chain Application to Manpower Supply Planning

Markov chains have been suggested since the early sixties for modeling manpower supply, mostly in military, governmental and public agencies. An actual industrial application is presented. This will be of interest to managers as well as practitioners and students of operational research. The approach, use and validation of the Markov model are discussed, along with some limitations and extensions.     

Note on the Convergence of a Manpower Distribution

An estimate of the rate of convergence to its limit is given of the stochastic matrix describing the structure at time t of a stratified population in which the proportional wastage is compensated by recruitment to the lowest rank. An example is given.     

Entropy as a Measure of Stability in a Manpower System

It is important when studying a manpower system to be able to measure stability of the experience distribution and thus evade possible trauma. This paper develops the use of entropy as such a measure, both for the comparison of different companies in steady state, and as a means of monitoring the changes in tenure distribution in time for a particular firm. It is shown for several companies that the entropy measure does in fact reflect the changes in length of service structure taking place within the firm.     

A Markovian Goal Programming Approach to Aggregate Manpower Planning

The combination of a manpower supply model (a Markov type model based on historical probabilities of losses, promotions and gains) and goal programming with preemptive priorities provides a useful tool for developing a future year manpower plan under conflicting socio-econo-organizational objectives. Successful utilization requires a close management involvement in adjusting probabilities and specifying goals, priorities and impending policy changes. Such an approach is presented in this paper and illustrated by means of an industrial case study example. The presentation is kept simple, yet detailed and unified, so that is is easily understood by practitioners and students of operational research/management science.     

Note on a Stochastic Manpower Smoothing and Production Model

It is shown that the optimal policy for a stochastic manpower smoothing and production model, with regular and overtime hours can be characterized by a set of four critical numbers for each state and each period, resulting in substantial computational simplifications.     

An Extension to McClean's Steady State Model of Manpower Systems

McClean's model focusses on length of service or stability, making no mention of the size of the system. However, it is shown that size can also be extracted from the same formulation.     

A Dynamic Programming Approach to Determine Optimal Manpower Recruitment Policies

The manpower planning models available in the literature have dealt with how changes take place in a manpower planning system, under various operating and policy constraints. However, none of these models has identified the manpower system costs. In this paper we have identified various manpower system costs. Further, we have developed a manpower planning model with the objective of minimizing the manpower system costs. The model has been found to be analogous to the Wagner-Whitin model in production/inventory management. A numerical example has been given to illustrate the model.     

Helping students build a path of understanding from ratio and proportion to decimal notation

Various studies have shown that students of all levels struggle to understand decimal numbers. This paper discusses a novel approach to increasing students’ conceptual understanding of decimal numbers. Rather than approach decimal notation as a discrete and separate mathematical topic, this approach enables students to work with contextual problems to gain a solid understanding of ratio and proportion. Using their understanding of ratio and proportion as a foundation, students can then build connected and related understandings of fractions, decimals and percents. The study discussed in this paper illustrates that grounding decimal instruction in the broader context of ratio can help students gain deeper conceptual understandings of decimal notation as well as fractions and percents.     

The Steady-State Behaviour of a Manpower Planning Model in Which Class Corresponds to Length of Service

We consider a discrete time Markov population model in which class corresponds to length of service. For this particular system it is shown that the steady state distributions of class sizes are independently Poisson. The model is applied to eight firms for which the steady state tenure distribution is evaluated.     

Entropy as a Measure of the Experience Distribution in a Manpower System

It is important when studying a manpower system to be able to measure stability of the experience distribution and thus evade possible trauma. This paper develops the use of entropy as such a measure, both for the comparison of different companies in steady state, and as a means of monitoring the changes in tenure distribution in time for a particular firm. This is achieved by finding the probability of a person to be in a specific length of service class as t→∞ under the more general assumptions possible.     

合作联盟资源集成计划一种新方法

合作联盟里，资源集成计划往往是联盟成员群体谈判博弈的结果。本以两人博弈为例，对联盟的资源集成计划给出一个谈判博弈模型，能够较好地模仿和反映合作联盟资源整合计划的制订过程。     

Helping Students Come to Grips With the Meaning of Division

Many years ago, Arons pointed out the incomprehension science students exhibit of the basic mathematical operations multiplication and division and the need to address the problem in physics classes to assure student understanding of the physical world. McDermott et al.'s Physics by Inquiry program does address this need directly and in detail (by defining two meanings for division). However, in the author's classes many students had relatively low scores (ranging from 60–80%) when trying to explain simple operations. Reported in this paper are ways to supplement the text that force students to address the actual meaning of division by stressing the relation between a “whole” and a “package,” and connect that meaning with previously learned operational definitions for area and volume.     

The Optimal Allocation of Resources to a Variable Timetable

In this note we describe a programming method for allocating resources to a variable timetable. It is illustrated in terms of a restricted version of a general programming model for resource allocation to a CPA network, but it can be used for more general versions of the CPA model, and for the scheduling of ships, aeroplanes, buses and so on. It makes use of the standard trans-shipment transportation problem framework, although in practice the assignment matrix need not be constructed. A highly efficient initial solution is obtained first, then optimized via a primal method of dealing with the "either-or" constraints which arise because of the variability in the timetable.