IMA Journal of Management Mathematics - recent issues

Editorial

Utley, M., Chaussalet, T., Baker, R. — Thu, 27 Aug 2009 06:37:44 PDT

Non-homogeneous Markov models for sequential pattern mining of healthcare data

Garg, L., McClean, S., Meenan, B., Millard, P. — Thu, 27 Aug 2009 06:37:44 PDT

Sequential pattern mining has been a popular data mining technique for extracting useful information from large databases and has successfully been used for numerous industrial and commercial problems. This paper presents a new mathematical modelling application to healthcare, providing important information to health service managers and policy makers to help them identify sequential patterns which require attention for efficiently managing scarce healthcare resources and developing effective healthcare management policies. In healthcare, these sequential patterns are analogous to the patient pathways. We present a non-homogeneous Markov model for identifying not only patient pathways which have high probability but also for identifying pathways which incur high cost or time. In order to have a more realistic model, we also consider time-dependent covariates and their impact on the pathways. An algorithm based on branch and bound global optimization is presented which can efficiently extract a required number of such patient pathways of interest. The approach is illustrated using historical data on geriatric patients from an administrative database of a London hospital.

Analytical methods for calculating the distribution of the occupancy of each state within a multi-state flow system

Utley, M., Gallivan, S., Pagel, C., Richards, D. — Thu, 27 Aug 2009 06:37:44 PDT

We present analytical techniques for estimating the time-varying occupancy of each state within any multi-state flow system that can be represented as a particular type of directed graph called a rooted directed tree. Such systems have a single point of entry from which each other state within the system can be reached by exactly one directed path. The discrete-time model presented incorporates the use of time-varying and general distributions for the number of individuals entering the system and of general sojourn time distributions for each state. We illustrate the use of such analysis in the context of the delivery of mental health services in the community for people with common mental health problems and then discuss the possibility of adapting these methods with relation to systems that have a structure more complex than that of a rooted directed tree.

Modelling risk of readmission with phase-type distribution and transition models

Demir, E., Chaussalet, T., Xie, H., Millard, P. H. — Thu, 27 Aug 2009 06:37:44 PDT

A patient with frequent past readmissions may have an increased risk of future readmission. The principal objective of this paper was to determine the risk of readmission, given individual patient's history of readmissions. First, we develop a modelling approach to systematically tackle the issue surrounding the appropriate choice of a time window which defines readmission. Discharged patients can be divided into two groups: a group at high risk of readmission and a group at low risk. Using national data (England), the estimated time window for chronic obstructive pulmonary disease is 38 days. Using this time window, we classify ‘high’ and ‘low’ risk of readmission groups. We use transition models to incorporate patients’ history of readmissions along with additional covariates. Solely using patients’ history of readmissions, the model has a receiver operating characteristic c statistic of 0.71, illustrating that such a simple model with no covariates has the potential of estimating risk of readmission.

A genetic algorithm approach to the nurse scheduling problem with fuzzy preferences

Duenas, A., Tutuncu, G. Y., Chilcott, J. B. — Thu, 27 Aug 2009 06:37:44 PDT

The nurse scheduling problem (NSP) consists of generating a work schedule for nursing staff in a hospital. The approach presented in this paper considers a multi-objective NSP involving the nurses’ preferences. These preferences are modelled by fuzzy sets and aggregated to determine an overall preference cost function. The schedules are generated by a hybrid approach based on an interactive sequential multi-objective problem solving method combined with a genetic algorithm (GA). The head nurse is identified as the decision maker. Different versions of the GA are developed to test the efficiency of the approach. The results reveal that the proposed approach generates good quality solutions and due to its flexibility may be applicable in real-life NSPs.

Exploring potential consequences on mortality estimates of errors in clinical databases

Pagel, C., Gallivan, S. — Thu, 27 Aug 2009 06:37:44 PDT

Much time and energy are spent collecting and recording clinical data relating to outcomes, either within a single hospital or at a national level, in order to assess performance. Analysis of these data, particularly with respect to mortality rates, is a key part of clinical governance. However, most analyses are based on the assumption that data are accurate whereas, given the complexity of the data-gathering procedures, errors are not uncommon. Thus, it is useful to have insight into potential problems that might arise from analyses based on incomplete or erroneous data. We have developed a mathematical model that can be used to assess the effects of errors on estimates of mortality rates derived from clinical databases. Using simple assumptions about the nature of such errors, we have conducted thought experiments to investigate their potential impact. We show that there are plausible circumstances in which errors could cause systematic biases potentially leading to serious misinterpretation of clinical performance.

Equilibrium replenishment in a supply chain with a single distributor and multiple retailers

Kogan, K., Perlman, Y., Hovav, S. — Thu, 27 Aug 2009 06:37:44 PDT

This paper addresses a problem encountered by a large-scale health service supply chain operating in a periodic review mode. Due to the vital nature of the products it provides, the number and timing of urgent orders are not limited. As a result, increasingly high transportation costs are incurred and the problem is to select an inventory replenishment (review) period that minimizes the transportation cost. Moreover, the supply chain involves multiple retailers which inevitably and independently respond to any change in replenishment policy since it may affect their inventory costs. Such a relationship results in a game between a distribution centre and retailers. Since the problem is intractable due to its scale and stochastic nature, we combine a game theoretic approach with an empirical analysis. We show that this system is predictable using equilibria and that the current replenishment equilibrium of the health service supply chain is close to the Nash solution. Numerical analysis shows that the transportation costs are cut if the distribution centre implements in reality its formal (Stackelberg) leadership by reducing the replenishment period. However, this does not coordinate the supply chain and greater system-wide savings are possible by increasing the replenishment period if the supply chain is vertically integrated or the parties cooperate.

Introduction to max-linear programming

Butkovic, P., Aminu, A. — Tue, 09 Jun 2009 20:17:48 PDT

Let a b = max(a, b) and a b = a + b for a, b R. Extend this pair of operations to matrices and vectors in the same way as in linear algebra. Being motivated by scheduling of multiprocessor interactive systems, we introduce max-linear programs of the form f^T x -> min (or max) subject to A x c = B x d and develop solution methods for both of them. We prove that these methods are pseudo-polynomial if all entries are integers. This result is based on an existing pseudo-polynomial algorithm for solving the systems of the form A x = B y.

An efficient method for calculating the minimum distance from an operating point to a specific (hyperbolic) efficient frontier

Silver, E. A., Bischak, D. P., Da Silveira, G. J. C. — Tue, 09 Jun 2009 20:17:48 PDT

This paper is concerned with movement from a current operating point so as to reach a 2D efficient frontier. After a discussion of different criteria for deciding on which point on the frontier to target, we focus, as an illustration, on a particular inventory management context and the use of the criterion of minimum distance from the current point to the frontier. Specifically, the efficient frontier turns out to be a hyperbola in a 2D representation of total (across a population of items) average stock (in monetary units) versus total fixed costs of replenishments per year. Any current (or proposed) operating strategy, differing from the class along the frontier, is located above the frontier. Finding the minimum distance from the current point to the frontier requires determining the smallest root of a quartic equation within a restricted range.

Energy consumption and environmental pollution: a stochastic model

Tapiero, C. S. — Tue, 09 Jun 2009 20:17:48 PDT

This paper considers an energy–pollution economic problem which uses a stochastic queue framework. This approach is shown to establish a link between a firm's energy-consuming economic activity and its effects on the pollution that it generates as a function of its activity. An economic model is then used to draw some inferences regarding the effects of selected economic parameters on the propensity to pollute and prevent pollution. In particular, a number of theories pertaining to the effects of pollution and its regulation on the output and competitiveness of firms are elaborated. For example, the Porter hypothesis presuming that a stringent environmental policy improves a firm's competitiveness is analysed in terms of our model. For demonstration purposes, a simple example is used.

Mathematical modelling of maintained systems using point processes

Syamsundar, A., Naikan, V. N. A. — Tue, 09 Jun 2009 20:17:48 PDT

In this paper, the analytical needs of industrial maintained systems are articulated and the mathematical modelling of maintained systems using the theory of point (counting) processes is presented. The adaptation of point processes to the complex nature of maintained systems is brought out. The models with their estimation, inference, validation and prediction based on the data generated by the maintained systems are depicted. Applications of these models to the data from maintained systems are shown and how they have been used to meet the objectives of maintenance is brought out. The issues resolved and the further work to be done is indicated.

Analysing alternatives in financial services for wealth management banks: the analytic network process and the balanced scorecard approach

Wu, C.-R., Lin, C.-T., Tsai, P.-H. — Tue, 09 Jun 2009 20:17:48 PDT

The main purpose of this study is to evaluate business performance of wealth management (WM) banks by applying the balanced scorecard (BSC) and the analytic network process (ANP). In the proposed model, the dimensions of financial services for WM banks have been taken from four perspectives derived from the BSC approach, viz. finance, customer, internal business and learning and growth. The proposed approach, therefore, links the financial and non-financial, tangible and intangible and internal and external factors, thus providing a holistic framework for the evaluation of an alternative for WM banks. The ability of ANP to consider interdependencies among and between levels of decision makes it a multi-criteria decision-making tool. Thus, a combination of BSC- and ANP-based approach proposed in this paper can provide decision makers with a more balanced, realistic and accurate representation of the challenge of conducting business performance for WM banks.

Editorial

Percy, D. F. — Tue, 10 Mar 2009 08:34:32 PDT

Uses and limitations of mathematics in sport

Haigh, J. — Tue, 10 Mar 2009 08:34:32 PDT

A number of topics are reviewed under the theme of the interaction between mathematics and sport, mainly based on probabilistic or statistical considerations. Some differences and similarities that arise from different scoring systems, exemplified by motor racing and table tennis, are described. Ways of resolving ties in league tables, or drawn matches in individual contests, are explored, with examples from chess and soccer: a suggestion is made for modifying the usual penalty shoot-out in soccer so as to minimize its expected length. Two different ways of assessing the ‘importance’ of single incidents within a larger contest are offered: both suggest that such incidents are equally important to both contestants. Sample calculations indicate that the importance of incidents may be fairly constant across a wide range of values for the parameters that govern the outcome of a contest. Two statistical phenomena—Simpson's paradox and regression to the mean—are illustrated by basketball and golf. Applications of dynamic programming, to American football and soccer, are outlined. Different ways of computing batting averages in cricket, perhaps more meaningful than those generally used, are noted.

A new modelling approach demonstrating the inability to make up for lost time in endurance running events

Morton, R. H. — Tue, 10 Mar 2009 08:34:32 PDT

The tolerable duration of high-intensity exercise can be described by a simple hyperbolic function of power or velocity, with an asymptote referred to as the ‘critical power/velocity’ and a curvature constant referred to as the ‘anaerobic work/distance capacity’. More recently, this hyperbola has been generalized by permitting a non-zero temporal asymptote. Using this three-parameter model, we consider the consequences of running the initial part of a race at a speed different from the constant rate proscribed by the hyperbola. We show that for any distance split, an improved time is achievable and that the least time occurs when both parts of the race are run at speeds determined by applying the hyperbola to each part. Further improvement is possible by an appropriate selection of initial distance, with the first part being run at a higher speed than the second. Still further improvement is possible if the athlete follows an all-out running strategy, and we prove that for this model an all-out strategy is uniquely optimal. Significant performance gains appear possible for events of less than 10-min duration. Thus, under this model, an athlete, who at any time during a race drops below their all-out pace, can never make up for lost time. This result is contrary to conventional wisdom. Accordingly, we examine some recent empirical evidence which confirms the predicted nature of all-out power development over short time periods and suggests that pace variation, at least to some degree, may not be as suboptimal as previously assumed.

A random point process model for the score in sport matches

Volf, P. — Tue, 10 Mar 2009 08:34:32 PDT

A sequence of goals scored during sport match is modelled as a realization of two dependent random point processes. It is assumed that the scoring intensity of each team has several components depending on time or on factors describing the teams and other conditions of the match. This dependence is modelled with the aid of a semi-parametric multiplicative regression model of intensity. A method of model evaluation is presented and demonstrated on a real data set. Prediction obtained from the model via the Monte Carlo simulation is compared with real results.

Bayesian modelling of football outcomes: using the Skellam's distribution for the goal difference

Karlis, D., Ntzoufras, I. — Tue, 10 Mar 2009 08:34:33 PDT

Modelling football match outcomes is becoming increasingly popular nowadays for both team managers and betting funs. Most of the existing literature deals with modelling the number of goals scored by each team. In this paper, we work in a different direction. Instead of modelling the number of goals directly, we focus on the difference of the number of goals, i.e. the margin of victory. Modelling the differences instead of the scores themselves has some major advantages. Firstly, we eliminate correlation imposed by the fact that the two opponent teams compete each other, and secondly, we do not assume that the scored goals by each team are marginally Poisson distributed. Application of the Bayesian methodology for the Skellam's distribution using covariates is discussed. Illustrations using real data from the English Premiership for the season 2006–2007 are provided. The advantages of the proposed approach are also discussed.

Predicting score difference versus score total in rugby and soccer

Stefani, R. T. — Tue, 10 Mar 2009 08:34:33 PDT

A common paradigm is used to show that, in rugby and soccer, score difference is more accurately predicted based on past data than is score total. Offensive and defensive ratings evolve by applying least-squares or exponential smoothing to past home scores and visiting scores, corrected for home advantage. Smoothed ratings provide a first-order prediction of score difference and score total for future matches. The average absolute error of such predictions may be minimized by employing a ‘shrinking factor’ L. If L is one, predictions are based solely on smoothed past performances. If L is zero, all predicted score differences are zero and all predicted score totals equal the average total score of past matches. L between one and zero shrinks predicted score difference towards zero and predicted score total towards average total score. Based on more than 3000 games contested in rugby union (Zurich/Guinness Premiership and Super 12/14) and soccer (English Premier League and Italian Serie A), L for score difference was more than that for score total. Thus, past performance was a better predictor of score difference than score total. It is suggested that a team focuses strategy on score difference (to win or draw) while score total is of no strategic interest, thus affecting prediction based on past data. Based on an evaluation of more than 2000 games, it was concluded that betting on score total is undesirable in contrast to score difference because of a relatively small number of possible bets and an unreliable profit.

Evaluating the performance of an ice hockey team using interactive phases of play

Bedford, A., Baglin, J. — Tue, 10 Mar 2009 08:34:33 PDT

This study measures the interaction between two opposing teams in ice hockey by regressing a number of performance measures to a single measure which enables assessment of a team's performance during the course of a game. The notion of ‘phases of play’, whereby players and teams fluctuate through periods of ‘high phase’ and ‘low phase’ during a game, is used as a theoretical underpinning for the research. We also consider ‘relative phase’ that describes the overall interaction between the two teams. Team game data from the 2005/06 National Hockey League season were collected along with data for the games played up to time of research in the 2006/07 season. An optimized binary logistic regression model for both home and away teams was naturally found to model match outcome better than other methods when a team's score was included as a performance variable. This model correctly classified 91% of games as either a win or a loss. Using live match data, these logistic regression models were then used to create phases of play plots of a home team's and away team's performance throughout the progress of a number of games in the 2006/07 season. These scores were smoothed using a Tukey's T4253H smoother to eliminate excess noise in the phases of play plots. It was concluded that the results of this analysis gave an objective, simple and all-round measure of a team's performance which would be a valuable evaluative asset to coaches, the media and spectators.

Markowitz portfolio theory for soccer spread betting

Fitt, A. D. — Tue, 10 Mar 2009 08:34:33 PDT

Soccer spread betting is analysed using standard probabilistic methods assuming that goals are scored in a match according to Poisson distributions with constant means. A number of different possible forms of ‘edge’ (betting advantage) is identified. It is shown how the centre spreads of the more common bets in the ‘bet universe’ may be calculated. A more general question is then addressed, namely, how a punter should invest if they take a view that the online bookmakers have fixed the goal means incorrectly or some other edge is in their favour. It is shown that a Markowitz portfolio theory framework may be set up in such cases. This leads to the definitions of an ‘efficient betting frontier’ and an ‘optimal bet portfolio’. Examples are used throughout to illustrate the theory that is developed.

Evaluating relative performances in disabled sports competitions

Percy, D. F., Warner, D. B. — Tue, 10 Mar 2009 08:34:33 PDT

Handicapping systems play important roles in many sports. This paper focuses on disabled sports competitions, which primarily aim to encourage participation. Nevertheless, prizes are awarded for relative performances and should be allocated fairly. We review handicapping procedures specifically for disabled Alpine skiing and identify imperfections with this system, particularly how historical and new results are combined to determine scaling factors. Significant problems also arise due to different class sizes and variations in courses and conditions. We investigate the difficulties in establishing a suitable handicapping system and propose modifications to the existing procedures that improve their fairness. We develop simple mathematical models and statistical analyses that might be suitable, based on results data relating to several years of competition, and we evaluate the performance of our proposed method in comparison with that of the existing approach.

A ranking for the Olympic Games with unitary input DEA models

de Mello, J. C. C. B. S., Angulo-Meza, L., Branco da Silva, B. P. — Tue, 10 Mar 2009 08:34:33 PDT

There is no official method to establish a final ranking for the Olympic Games. The usual ranking is based on the lexicographic multicriteria method, the main drawback of which is to overvalue gold medals. Furthermore, it does not take into account that the various sports may be of different importance. This work proposes a ranking model to eliminate those drawbacks. We use firstly a modified cross-evaluation data envelopment analysis model with weighted restrictions for each cluster. The outputs are the number of gold, silver and bronze medals and the input is a unitary constant for all countries. After obtaining a rank for each and every cluster, we build a general ranking by aggregation of the partial ones. Our model uses the results of the Athens Olympic Games.

Organization of a college baseball tournament

Smith, J. C. — Tue, 10 Mar 2009 08:34:34 PDT

The National Collegiate Athletic Association baseball tournament involves 64 teams representing American universities and colleges in a series of win-and-advance weekend tournaments. In the first weekend, a four-team ‘regional’ tournament is played at a host institution. The 16 regionals are also paired a priori, with the winners of paired regionals playing in the second weekend at one of the two institutions’ home sites. The eight teams remaining after the second weekend play a final tournament at a neutral location. Given a selection of 64 tournament teams and their seeding classifications, prohibited four-team groupings during the first weekend and prohibited regional pairings in the second weekend, we examine the problem of creating regionals and regional pairings in order to minimize expected team travel costs. The problem is modelled as a non-linear mixed-integer program and solved by a combinatorial cutting plane approach. We examine the performance and output of the proposed algorithm on 2006 and 2007 tournament data.