Modelling the requirement for supplementary nurses in an intensive care unit

The intensive care unit (ICU) of a hospital is an essential yet costly resource. Consequently, intensive care modelling has become increasingly prevalent in recent years in attempts to increase efficiency and reduce costs. Previous models have usually assumed that the numbers of beds available are restricted; when all beds are occupied, any additional patients are referred elsewhere or elective surgeries are cancelled. In this study, activities at the ICU at a large teaching hospital were modelled using data relating to all admissions to the ICU during the year 2000—a total of 1084 admissions. The unit is unusual in that the majority of patients referred for intensive care therapy are admitted. Bed numbers are increased when necessary to cope with demand. However, nurses are a restricted resource. In order to maintain the required nurse:patient ratio of at least one:one, supplementary nurses are employed during busy periods. Supplementary nurse costs are substantial and so nurse utilization must be closely monitored. The development of a model that calculates the required number of supplementary nurses per shift, and also encapsulates the time-dependent nature of elective surgery admissions and complex duration-of-stay profiles, is presented in this paper. In particular, the model is used to determine the number of rostered nurses that are required to minimize overall nursing staff costs.     

Managing the overflow of intensive care patients

Many hospitals in the Netherlands are confronted with capacity problems at their intensive care units (ICUs) resulting in cancelling operations, overloading the staff with extra patients, or rejecting emergency patients. In practice, the last option is a common choice because for legal reasons, as well as for hospital logistics, rejecting emergency patients has minimal consequences for the hospital. As a result, emergency patients occasionally have to be transported to hospitals far away. In this work, we propose a cooperative solution for the ICU capacity problem. In our model, several hospitals in a region jointly reserve a small number of beds for regional emergency patients. We present a mathematical method for computing the number of regional beds for any given acceptance rate. The analytic approach is inspired by overflow models in telecommunication systems with multiple streams of telephone calls. Simulation studies show that our model is quite accurate. We conclude that cooperation between hospitals helps to achieve a high acceptance level with a smaller number of beds resulting in improved service for all patients.     

Assigning pupils to tutor groups in a comprehensive school

Comprehensive schools in the UK have the annual task of assigning their intake of new pupils to tutor groups. This case study describes how the problem has been addressed at one such school, where several factors are taken into account in order to make the groups as similar as possible and, since pupils have an induction day in June when they assemble in their tutor groups and meet their new tutors, where the time scale is short. Manual allocation has been used in the past, needing many staff-hours to achieve a satisfactory result. Heuristics implemented on a spreadsheet have enabled substantial savings in staff time and given more evenly balanced tutor groups.     

A Simple Method of Assigning Projects to Students

Individual project work is an integral part of many degree courses in British universities. In this paper an objective method of assigning projects to students is described which is simple to administer and which takes account of student preferences.     

Assigning students to course sections using tabu search

In this paper we describe a new student registration system which has been developed at the University of Valencia, Spain. The system has two steps. First, the students make a computer-aided course selection from the courses available at the University. Thereafter, an assignment procedure allocates students to sections in order to respect two criteria: to provide the students with satisfactory schedules and to get balanced section enrollments. The assignment process has two phases. In Phase I, we obtain a set of the best solutions for each student. The algorithm is based on the construction of maximum cardinality independent sets. In Phase II, these solution sets are put together and a tabu search algorithm looks for a satisfactory balance between course sections without causing the solution obtained for each student to worsen significantly. The system was used at the beginning of the academic year 1996/97 in the Faculty of Mathematics and could be extended in the near future to the rest of the University. This revised version was published online in June 2006 with corrections to the Cover Date.     

Combining optimization with simulation to obtain credible models for intensive care units

In this paper we develop a simulation model to study bed occupancy levels in an Intensive Care Unit (ICU). The main contributions of this study are: (1) A proposal for generalized regression models to fully capture the high variability of patients’ length of stay; (2) Proof that a simulation model that does not incorporate the management decisions by clinical staff cannot be considered valid; (3) The development of a mathematical model to represent these management decisions, and (4) A proposal for a method combining optimization with simulation to estimate the model parameters. This provides a valid simulation model that includes the physician management of an ICU. Validation is accomplished by comparing distribution patterns in daily bed occupancy records against simulated bed occupancy data. The methodology is tested using data provided by the Hospital of Navarre in Spain.     

Assigning resources and targets to an organization’s activities

Each of an organization’s many activities transforms inputs into outputs. Managing these activities involves allocating input resources for some activities and assigning output targets for others. Making these decisions is especially difficult in the presence of uncertainty. In practice, many organizations address these problems by using a fairly simple “proportional allocation” heuristic (e.g., “allocate to each activity the same percentage increase (or decrease) in its resources or targets”). But proportional allocation does not consider the uncertainty inherent in the ability of each activity to make use of its resources (or meet its targets).     

Assigning arriving flights at an airport to the available gates

Unexpected changes in the flight schedules may disrupt the initial aircraft-gate assignments, and result in congestions and delays in getting aircraft onto gates. A mathematical model is developed to assign the flights with the minimum range of unutilised time periods of gates, subject to the level of service offered to passengers and other physical and managerial considerations. (The assignments are expected to be flexible enough to absorb the minor modifications in the flight schedules.) Interactive optimum and heuristic procedures, both utilising lower bounds on the ranges of future solutions, are proposed to cope with the major changes in disrupting the initial gate-assignments. Over randomly generated schedules, 74 flights can be optimally assigned to seven gates within 17 seconds when the gates are re-utilised within 30 minutes after each departure. The heuristic reaches the optimal solution after evaluating at most 20 partial solutions at one level. Over data obtained from Riyadh’s International Airport, the heuristic outperforms the existing practice: On average, 72.03% and 54.28% improvements are obtained on the number of remote served aircraft and towed aircraft, respectively.     

An Algorithm for the Fixed-Charge Assigning Users to Sources Problem

Given a set of sources with known capacities and a set of users with known demands for a product, the objective is to minimize the cost of assigning the users to the sources. The restrictions are that each user's demand is satisfied by exactly one of the sources, and that the total amount assigned to each source does not exceed its capacity. A fixed-charge version of this problem is formulated as a set partitioning problem and solved by a column-generating branch and bound procedure. Computational results are given.     

A queueing model of activities in an intensive care unit

^** Email: griffiths{at}cardiff.ac.uk Activities in an intensive care unit (ICU) at a major teachinghospital are modelled by means of a queue-theoretic approach.Using data supplied by the ICU relating to the admissions process,the bed availability and the length of stay of patients, itwas possible to fit theoretical distributions to the observed‘arrival’ and ‘service’ distributions.Queueing equations relevant to a multi-channel system havingrandom arrivals and hyper-exponential service times for eachchannel are set up, and solved iteratively. Results obtainedmatch well with observations, and the model is then utilisedto investigate several ‘what if? ’ scenarios. Referenceis made to a simulation model developed in conjunction withthe queueing model.     

Goal Programming Models for Assigning Search and Rescue Aircraft to Bases

This paper develops several variations of a goal programming model for optimally allocating a fleet of search and rescue aircraft to a fixed set of available and potentially available bases. In addition, the model determines the number of aircraft of each type from each base (at which that type has been stationed) to assign to the various search locations. The criterion for optimality is to maximize the probability of locating each distress in a specified time. These models are then modified to include fleet planning issues. Solution procedures relating to the models are discussed.     

A mathematical model for the admission process in intensive care units

A mathematical model is given for the admission process in Intensive Care Units (ICUs). It is shown that the model exhibits bistability for certain values of its parameters. In particular, it is observed that in a two-dimensional parameter space, two saddle-node bifurcation curves terminate at a single point of the cusp bifurcation, creating an enclosed region in which the model has one unstable and two stable states. It is shown that in the presence of bistability, variations in the value of parameters may lead to undesired outcomes in the admission process as the value of state variables abruptly changes. Using numerical simulations, it is also discussed how such outcomes can be avoided by appropriately adjusting the parameter values.     

Assigning cooperating UAVs to simultaneous tasks on consecutive targets using genetic algorithms

A problem of assigning multiple agents to simultaneously perform cooperative tasks on consecutive targets is posed as a new combinatorial optimization problem. The investigated scenario consists of multiple ground moving targets prosecuted by a team of unmanned aerial vehicles (UAVs). The team of agents is heterogeneous, with each UAV carrying designated sensors and all but one carry weapons as well. To successfully prosecute each target it needs to be simultaneously tracked by two UAVs and attacked by a third UAV carrying a weapon. Only for small-sized scenarios involving not more than a few vehicles and targets the problem can be solved in sufficient time using classical combinatorial optimization methods. For larger-sized scenarios the problem cannot be solved in sufficient time using these methods due to timing constraints on the simultaneous tasks and the coupling between task assignment and path planning for each UAV. A genetic algorithm (GA) is proposed for efficiently searching the space of feasible solutions. A matrix representation of the chromosomes simplifies the encoding process and the application of the genetic operators. To further simplify the encoding, the chromosome is composed of sets of multiple genes, each corresponding to the entire set of simultaneous assignments on each target. Simulation results show the viability of the proposed assignment algorithm for different sized scenarios. The sensitivity of the performance to variations in the GA tuning parameters is also investigated.     

Analysis of capacity management of the intensive care unit in a hospital

A hospital's intensive care unit (ICU) is a limited and critical resource. The efficient utilization of ICU capacity impacts on both the welfare of patients and the hospital's cost effectiveness. Decisions made in the ICU affect the operations of other departments. Yet, decision making in an ICU tends to be mainly subjective and lacking in clear criteria upon which to base any given decision. This study analyzes the admission-and-discharge processes of one particular ICU, that of a public hospital in Hong Kong, by using queuing and computer simulation models built with actual data from the ICU. The results provide insights into the operations management issues of an ICU facility to help improve both the unit's capacity utilization and the quality of care provided to its patients.     

Cardiorespiratory dynamics during transitions between mechanical and spontaneous ventilation in intensive care

Life support devices (e.g., heart pacemakers, ventilators, dialysis machines) are commonly thought to affect mainly their target organ system. This preliminary study attempts to determine the extent to which mechanical support of one organ system (lungs by mechanical ventilator) affects functions of other systems (heart and blood vessels). We studied changes in cardiorespiratory interactions and in dynamics of cardiovascular system during scheduled transitions (spontaneous breathing trials, SBT) between mechanical and spontaneous ventilation in critically ill patients. This initial study population consisted of 13 patients admitted to a surgical intensive care unit (ICU) following surgery, trauma, or complications who were judged candidates for liberation from the ventilator. We collected continuous respiratory, cardiac (ECG), and perfusion (blood pressure and pulse oximetry) signals of mechanically ventilated patients before, during, and after SBT. The data were analyzed using spectral analysis, phase dynamics, and entropy measures. We found that the mechanical ventilation not only drives lung dynamics but also affects the dynamics of cardiac and vascular systems. Spontaneous cardiovascular rhythms are entrained by the mechanical ventilator and are drawn into synchrony. Sudden interruption of mechanical ventilation typically leads to rapid desynchronization. This synchronization is restored upon reinstitution of mechanical ventilation. The initial data suggest that therapies intended to support one organ system may propagate unanticipated effects to other organ systems. Moreover, sustained therapies may disturb mechanisms that promote natural synchronization and variability and thereby adversely affect recovery of normal organ system interactions. We suggest that new measures and displays of synchronization not only could provide insight into the organ–organ coupling but also could yield information to optimize the function of devices used to support the critically ill patient. © 2008 Wiley Periodicals, Inc. Complexity, 2008