Swing Analysis and Electoral Forecasting

This paper examines the procedure by which votes are converted into seats at U.K. General Elections. In particular it seeks to answer these questions.

1. i)
   What is the meaning of swing when more than two parties fight an election?
    
2. ii)
   How can the distribution of seats at an election be determined from a prediction of national swing?
    
3. iii)
   How can swing analysis be extended to help determine an electoral strategy for political parties?
    
4. iv)
   What explanation can be provided for the swing in Scotland?
    
5. v)
   What explanation can be provided for variations in the English swing?
    

     

Linear Programming in Air Defence Modelling

The paper describes the use of an optimizing model, together with a simulation model, to determine the most effective mix and deployment of air defence weapons to defend a given set of assets against a range of air threats. The advantages of using an optimizing model in such studies are that it:

1. a
   —Gives more consistent results by reducing player variance.
    
2. b
   —Can generate solutions that are robust to variations in threat and environment.
    
3. c
   —Ensures that good solutions are not overlooked.
    
4. d
   —Reduces the total processing time required to complete the study.
    

     

Operational Research Projects in Health and Welfare Services in the United Kingdom and Ireland

The paper presents a comparison of Health and Welfare O.R. in the U.K. and Ireland as represented by two registers of current projects, compiled in 1972 and 1977 respectively. The survey shows that

1. i)
   There appear to be very few O.R. studies of social services;
    
2. ii)
   The major focus has apparently moved away from hospital-based tactical studies, towards more broadly-based strategic studies;
    
3. iii)
   Related to this, the number of projects labelled as concerned with "planning" has increased;
    
4. iv)
   The number of projects involving "optimisation" has decreased, whilst those involving simulation have shown some increase; and
    
5. v)
   The majority of projects in both years were described as involving "applied common sense", rather than any specific technique.
    

     

An Algorithm for the Vehicle-dispatching Problem

The vehicle-scheduling problem involves the design of several vehicle tours to meet a given set of requirements for customers with known locations, subject to a capacity constraint for the vehicles and a distance (or time) constraint for vehicle tours. Three methods of solution are considered in this paper:

1. a
   A branch-and-bound approach.
    
2. b
   The "savings" approach.
    
3. c
   The 3-optimal tour method.
    

The excessive computation time and computer storage required for the first method renders it impracticable for large problems. Ten problems are examined and the results suggest that method C is superior to the other two methods.     

A Three-Dimensional Optimization Problem in Water-Resources Engineering

The problem under consideration is related to a system consisting of two storage facilities in series. Part of the water reaching the upstream reservoir may be diverted for storage in the downstream facility. However, there is a certain lag between the time when water is released from the upper reservoir and the time when it is available for utilization in the lower one. Thus the vector describing the state of the system at any one time consists of three elements:

1. a)
   amount of water stored in the upstream reservoir;
    
2. b)
   amount of water stored in the downstream reservoir;
    
3. c)
   amount of water in transit.
    

An optimum operating policy for the system had to be derived.The method of dynamic programming was applied in the analysis of this problem. The algorithm obtained was programmed for solution on an IBM 7090 digital computer. Optimal policies were obtained for each admissible state of the system.     

A Stock Control Policy for Important Spares in a Two Level Stores System

This paper describes the results of an investigation to determine correct re-order levels and order quantities. The problem had three important features:

1. i)
   a simple stock control system was required for about 15,000 items;
    
2. ii)
   run-out costs needed to be estimated;
    
3. iii)
   the solution required best stock levels and order quantities for a central store and for smaller stores supplied by it.
    

The problem arose because it was felt that production delays were being caused by the stores too frequently being unable to supply spares.Items were grouped by their importance and portability. Average run-out costs were estimated for each group. An approximate model of the two level stock holding problem was derived. An iterative process was required to obtain solutions.It was observed that the stock control rules were insensitive to run-out cost, and that items of equal run-out cost had widely differing run-out probabilities. This indicates that, in this case, it is invalid to group items by importance and to assign fixed run-out probabilities to each group.     

Operational Research and Management

The purpose of this paper is to discuss and if possible to dispel the industrialist's misunderstanding of operational research. The author lists some of the varied ideas held by industrialists of operational research including several which are contradictory, as for example (a) That operational research is new and revolutionary; and (b) That operational research proves nothing new and has always been practised without being so named.A number of suggestions are then made and developed, which should lead to a better understanding. These include, clearer definitions of operational research and a better adaptation of the specialized literature to the needs of industrial management, with particular emphasis on the publication of real rather than synthetic case histories. Headings of sections are:

1. 1
   The attitude of industrialists.
    
2. 2
   The exact nature of operational research.
    
3. 3
   Problems, criteria, models.
    
4. 4
   The practice of operational research in business.
    

     

Robust solutions and risk measures for a supply chain planning problem under uncertainty

We consider a strategic supply chain planning problem formulated as a two-stage stochastic integer programming (SIP) model. The strategic decisions include site locations, choices of production, packing and distribution lines, and the capacity increment or decrement policies. The SIP model provides a practical representation of real-world discrete resource allocation problems in the presence of future uncertainties which arise due to changes in the business and economic environment. Such models that consider the future scenarios (along with their respective probabilities) not only identify optimal plans for each scenario, but also determine a hedged strategy for all the scenarios. We

1. 1)
   exploit the natural decomposable structure of the SIP problem through Benders’ decomposition,
    
2. 2)
   approximate the probability distribution of the random variables using the generalized lambda distribution, and
    
3. 3)
   through simulations, calculate the performance statistics and the risk measures for the two models, namely the expected-value and the here-and-now.
    

     

Regularly weakly based modules over right perfect rings and Dedekind domains

A weak basis of a module is a generating set of the module minimal with respect to inclusion. A module is said to be regularly weakly based provided that each of its generating sets contains a weak basis. We study

1. (1)
   rings over which all modules are regularly weakly based, refining results of Nashier and Nichols, and
    
2. (2)
   regularly weakly based modules over Dedekind domains.
    

     

Mathematical Aspects of Manpower Planning

In this paper, concepts of linear programming-in particular the Simplex algorithm and the Dual Simplex algorithm-are used within a discrete renewal model for the development of a graded population, when the transfer rates between the various grades are given, and "wastage" is replaced by suitable recruiting.The following questions are dealt with:

1. 1)
   Which population structures (i.e. partitions into grades) can be attained from a given structure, after one or more steps?
    
2. 2)
   From which structures can a given structure be attained in one or more steps?
    
3. 3)
   If the present structure as well as a desired future structure are given, can the latter be attained from the former in one or more steps? If so, how?
    
4. 4)
   The last problem is of special interest if the starting structure is identical with that to be attained and is called re-attaining after more than one step, or strictly maintaining after one step.
    

Examples are given to illustrate the procedure, and some attention is given to the possibility of alternative routes in cases (3) and (4). It is observed that if a structure is attainable (or re-attainable) after n steps, then it is not necessarily attainable (or re-attainable) after n + 1 steps.     

Relations between the $${\mathcal {}}$$-ultrafilters

Under CH we show the following results:

1. (1)
   There is a discrete ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
2. (2)
   There is a \(\sigma \)-compact ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    
3. (3)
   There is a \({\mathcal {J}}_{\omega ^{3}}\)-ultrafilter which is not a \({\mathcal {Z}}_{0}\)-ultrafilter.
    

     

The Geographical Allocation of Community Care Resources — A Case Study

The paper presents a case study of methods of allocating a given amount of home help resource between a number of geographical areas within a County Social Services Department. Questionnaires were completed for a sample of clients covering such information as:

1. i)
   the type of client,
    
2. ii)
   the amount of home help actually received, and
    
3. iii)
   the "ideal" amount of help judged necessary.
    

Various allocations were presented to the decision makers, ranging from the simple assumption that the number of hours provided to each type of client should be the same throughout the County, to more "sophisticated" allocations which took account of such geographical differences as the level of provision of other support services and the current and forecast population structure.Although developed within the context of the home help service, the approach is general enough to be applicable to the allocation of other such services to clients, e.g. home nursing.     

Some topics in the history of harmonic analysis in the twentieth century

In December 2015 I gave a series of six lectures at the Indian Institute of Science in which I sketched the thematic development of some of the main techniques and results of 20th-century harmonic analysis. The subjects of the lectures were, briefly, as follows:

1. 1.
   Fourier series, 1900-1950.
    
2. 2.
   Singular integrals (part I).
    
3. 3.
   H ^p , BMO, and singular integrals (part II).
    
4. 4.
   Littlewood-Paley theory: the history of a technique.
    
5. 5.
   Harmonic analysis on groups.
    
6. 6.
   Wavelets.
    

I emphasized interconnections, both the way in which the material in the first lecture provided the roots out of which most of the developments in the other lectures grew, and the ways in which those developments interacted with each other. I included sketches of as many proofs as the time would permit: some very brief, but some fairly complete, especially those whose methodology is an important part of the subject. Much was omitted, of course, and there was a natural bias toward the areas where I have spent periods of my own mathematical life. Many developments, particularly those of the final quarter-century, received at most a brief mention.This paper is a written account of these lectures with a few more details fleshed out, a few topics reorganized, and a few items added. I hope that others may find it an interesting narrative and a useful reference, and that it may lead some of them to share my enjoyment of exploring the original sources. I have tried to provide the references to those sources wherever possible, and for the more recent developments I also provide references to various expository works as the occasion arises. For the pre-1950 results discussed here and their proofs, however, there is one canonical reference, which I give here once and for all: Antoni Zygmund’s treatise [96]. (The more fundamental ones can also be found in Folland [29].)     

Speed,Gradient and Workrate in Uphill Running

A series of treadmill experiments is described concerned with a runner's speed, heart-rate and the gradient. Together with the results of similar experiments, some of them carried out over 50 years ago, the results suggest that for a given heart-rate, log(speed) is linearly related to gradient, and that for a given gradient, heart-rate is linearly related to speed. The results suggest:

1. 1)
   that athletes who run p% faster on the level will run p% faster up a slope, if they maintain the same heart-rate;
    
2. 2)
   that athletes will use the same number of heart beats running up a hill of uniform slope no matter how fast or slowly they run;
    
3. 3)
   that athletes should run directly up any slope of less than about 20° and try to zigzag up slopes greater than this.
    

     

Flow equivalence of sofic shifts

We classify certain sofic shifts (the irreducible Point Extension Type, or PET, sofic shifts) up to flow equivalence, using invariants of the canonical Fischer cover. There are two main ingredients.

1. (1)
   An extension theorem, for extending flow equivalences of subshifts to flow equivalent irreducible shifts of finite type which contain them.
    
2. (2)
   The classification of certain constant to one maps from SFTs via algebraic invariants of associated G-SFTs.
    

     

The inside outside intersection problem for hexagon triple systems

We give a complete solution of the following two problems:

1. (1)
   For which (n, x) does there exist a pair of hexagon triple systems of order n having x inside triples in common?
    
2. (2)
   For which (n, x) does there exist a pair of hexagon triple systems having x outside triples in common?
    

     

An Integrated Framework for Generating and Revising Factory Schedules

Practical solutions to the production scheduling problem must provide two broad capabilities:

1. i)
   an ability to efficiently generate schedules that reflect the actual constraints and objectives of the manufacturing environment, and
    
2. ii)
   an ability to incrementally revise these schedules over time in response to unexpected executional circumstances. In this paper, we advocate a common view of predictive and reactive scheduling as an incremental problem solving process that is opportunistically focused by characteristics of the current solution constraints.
    

We describe the architecture of OPIS (opportunistic intelligent scheduler), which defines a general framework for configuring scheduling systems according to this view. We then examine the scheduling knowledge (e.g. analysis and scheduling methods, schedule generation or revision strategies) that is exploited within this architecture by the current OPIS scheduler. Experimental studies with the OPIS scheduler have demonstrated the potential of this constraint-directed scheduling methodology in both predictive and reactive scheduling contexts.     

A Decision Simulation Model for Health Services Management

This model has been designed to assist decision making in the Health Services by predicting the effects on a local population of proposed policy decisions. Policy decisions are considered to:

1. 1
   Change the amount of certain resources available, e.g. manpower, hospital beds; or
    
2. 2
   Alter the methods of using resources, e.g. changing the emphasis of health care to the community rather than the hospital; or
    
3. 3
   Change the characteristics of illnesses, e.g. by vaccination or screening.
    

The model predicts the incidence of morbidity and mortality in a specified population and the changes in resource use over a period of years. Thus it is possible to see the long term effects of changes in population size and structure alongside the effects of management decisions on the use of health resources.The model has been designed principally for research purposes but has been used in the context of a health services management game. The validity of the model's predictions have yet to be tested. It is hoped to run the model in parallel with management decision making in a single health district in order to test the model's validity and improve its predictive ability.     

Optimal Smoothing of Demand for Industrial Gas

The problem of smoothing the demand for industrial gas when there are large tariff incentives for doing so is considered. Two cases are discussed:

1. a)
   where the only control variables are the maximum hourly and daily flow rates;
    
2. b)
   where, in addition, gas storage tanks can be installed to help smooth the demand.
    

In case (a) an analytic solution to the problem of establishing the optimal hourly and daily maximum settings is derived. In case (b) the influence of the storage capacity on the demand distribution is analysed, and a method of solution is proposed. This can then be used, together with the solution for case (a), to establish the optimal number of storage tanks to install.These ideas are applied to the gas usage in a steel factory.     

Accessibility to a Road Network: Definitions and Applications

In 1953 A. Shimbel proposed definitions of integral and relative accessibility in a network. When this definition of integral accessibility is applied to a detailed route network, it does not give an exact and coherent measure of the accessibility to the territory. Instead of considering the number of edges as a basic measure to calculate the relative accessibility, we use the length of, or the travelling time of, the edges. Moreover, attempts were made to improve the definition in various other ways:

1. a)
   by limiting the reference nodes to a hexagonal grid covering the territory regularly;
    
2. b)
   by limiting the reference nodes to the principal towns or cities of economical and social significance;
    
3. c)
   by weighting the relative accessibilities in function of the distance in order to obtain local measures.
    

The accessibility is obtained from the shortest path between nodes of the network. The algorithm of DIJKSTRA was used to obtain all these shortest paths. Computer experiences will be given for the Belgian route network, built up of more than 900 nodes. Automatically plotted geographical maps will be shown for several applications (route, network, relative and integral accessibility, weighted accessibility, location of public facilities, location of distribution centres, evalution of added new routes,...,).