Modelling nationwide hospital length of stay: opening the black box

Hospital length of stay is considered to be a reliable and valid proxy for measuring the consumption of hospital resources. Average length of stay, however, albeit easy to quantify and calculate, does not suitably reflect the nature of such underlying distributions and may therefore mask the effects that the different streams of patients have on the system. This paper uses routinely collected and readily available nationwide data on stroke-related patients, aged 65 years and over, who were discharged from English hospitals over a 1-year period. This will be the basis for a running example illustrating the alternative methods of analysis and models of patients' length of stay. The methods include statistical methods: survival analysis, mixed exponential and phase-type distributions; and decision modelling techniques: compartmental and simulation models. The paper concludes by summarizing these various modelling techniques and by highlighting the similarity of the estimated parameters of patient flow as calculated by the phase-type distribution and compartmental modelling techniques.     

A mixed integer programming approach for allocating operating room capacity

We have developed a methodology for allocating operating room capacity to specialties. Our methodology consists of a finite-horizon mixed integer programming (MIP) model which determines a weekly operating room allocation template that minimizes inpatients' cost measured as their length of stay. A number of patient type priority (eg emergency over non-emergency patient) and clinical constraints (eg maximum number of hours allocated to each specialty, surgeon, and staff availability) are included in the formulation. The optimal solution from the analytical model is inputted into a simulation model that captures some of the randomness of the processes (eg surgery time, demand, arrival time, and no-show rate of the outpatients) and non-linearities (eg the MIP assumes proportional allocation of demand satisfaction (output) with room allocation (input)). The simulation model outputs the average length of stay for each specialty and the room utilization. On a case example of a Los Angeles County Hospital, we show how the hospital length of stay pertaining to surgery can be reduced.     

Modelling activities at a neurological rehabilitation unit

A queuing model of a specialist neurological rehabilitation unit is studied. The application is to the Neurological Rehabilitation Centre at Rookwood Hospital (Cardiff, UK), the national rehabilitation unit for Wales. Due to high demand this 21-bed inpatient facility is nearly always at maximum occupancy, and with a significant bed-cost per day this makes it a prime candidate for mathematical modelling. Central to this study is the concept that treatment intensity has an effect on patient length of stay. The model is constructed in four stages. First, appropriate patient groups are determined based on a number of patient-related attributes. Second, a purpose-built scheduling program is used to deduce typical levels of treatment to patients of each group. These are then used to estimate the mean length of stay for each patient group. Finally, the queuing model is constructed. This consists of a number of disconnected homogeneous server queuing systems; one for each patient group. A Coxian phase-type distribution is fitted to the length of time from admission until discharge readiness and an exponential distribution models the remainder of time until discharge. Some hypothetical scenarios suggested by senior management are then considered and compared on the grounds of a number of performance measures and cost implications.     

Modelling the flow of congestive heart failure patients through a hospital system

The number of hospital admissions in England due to heart failure is projected to increase by over 50% during the next 25 years. This will incur greater pressures on hospital managers to allocate resources in an effective manner. A reliable indicator for measuring the quantity of resources consumed by hospital patients is their length of stay (LOS) in care. This paper proposes modelling the length of time heart failure patients spend in hospital using a special type of Markov model, where the flow of patients through hospital can be thought of as consisting of three stages of care—short-, medium- and longer-term care. If it is assumed that new admissions into the ward are replacements for discharges, such a model may be used to investigate the case-mix of patients in hospital and the expected patient turnover during some specified period of time. An example is illustrated by considering hospital admissions to a Belfast hospital in Northern Ireland, between 2000 and 2004.     

A pre-assignment heuristic algorithm for the Master Surgical Schedule Problem (MSSP)

In this paper a 0–1 linear programming model and a solution heuristic algorithm are developed in order to solve the so-called Master Surgical Schedule Problem (MSSP). Given a hospital department made up of different surgical units (i.e. wards) sharing a given number of Operating Rooms (ORs), the problem herein addressed is determining the assignment among wards and ORs during a given planning horizon, together with the subset of patients to be operated on during each day. Different resource constraints related to operating block time length, maximum OR overtime allowable by collective labour agreement and legislation, patient length of stay (LOS), available OR equipment, number of surgeons, number of stay and ICU beds, are considered. Firstly, a 0–1 linear programming model intended to minimise a cost function based upon a priority score, that takes into proper account both the waiting time and the urgency status of each patient, is developed. Successively, an heuristic algorithm that enables us to embody some pre-assignment rules to solve this NP-hard combinatorial optimisation problem, is presented. In particular, we force the assignment of each patient to a subset of days depending on his/her expected length of stay in order to allow closing some stay areas during the weekend and hence reducing overall hospitalisation cost of the department. The results of an extensive computational experimentation aimed at showing the algorithm efficiency in terms of computational time and solution effectiveness are given and analysed.     

A Length of Stay Index to Monitor Efficiency of Service to General Surgery In-Patients

An index, based on patient length of stay, has been designed for use in the regional monitoring system of Health Authorities. The index, reflecting efficiency of service to general surgery patients, is based upon homogeneous patient groups defined in terms of Hospital Activity Analysis data. The groups and percentile norms of stay under full efficiency have been set up in consultation with members of the medical profession. The index' asymptotic distribution is given, and its use is illustrated for a sample of twelve hospitals and 70 groups of general surgery patients.     

On the determination of natural frequencies and mode shapes of cable-stayed bridges

An accurate analysis of the natural frequencies and mode shapes of a cable-stayed bridge is fundamental to the solution of its dynamic responses due to seismic, wind and traffic loads. In most previous studies, the stay cables have been modelled as single truss elements in conventional finite element analysis. This method is simple but it is inadequate for the accurate dynamic analysis of a cable-stayed bridge because it essentially precludes the transverse cable vibrations. This paper presents a comprehensive study of various modelling schemes for the dynamic analysis of cable-stayed bridges. The modelling schemes studied include the finite element method and the dynamic stiffness method. Both the mesh options of modelling each stay cable as a single truss element with an equivalent modulus and modelling each stay cable by a number of cable elements with the original modulus are studied. Their capability to account for transverse cable vibrations in the overall dynamic analysis as well as their accuracy and efficiency are investigated.     

OR and energy: Problems of modelling

Attention is focused on three problem areas in energy modelling: (1) identifying the essential elements of the system, (2) coping with multiple criteria, (3) incorporating learning in the system. These aspects are illustrated through examples in current energy systems research, involving the oil market, power systems planning, and the role of nuclear energy.In modelling the word oil market, too heavy emphasis is placed on economic forces, but practically non on the political forces. However, certain economics-oriented studies indicate that financial gains of OPEC may actually be intensitive to the oil pricing policy followed. If that is indeed the case, the significant (political) elements and their motives have to be captured in the model in order to arrive at consistent results.Modelling the different objectives in power systems planning is an area where advances are urgently needed. A method is proposed, where decision alternatives are generated a posteriori, in contrast to recent approaches involving a priori articulation of preferences, or interactive methods. A multicriterion dynamic linear programming model and a fast algorithm are used in generating efficient solutions, which are then grouped, based on the clustering in objective values.The problem associated with changing system objectives is discussed and the nuclear programme is given as an example of how the system objectives move from costs to perceived risks. In line with the real system that ‘learns’ from its experience, we need models that change their objective functions as a result of their own outputs at prior times.     

Building cyclic master surgery schedules with leveled resulting bed occupancy

This paper proposes and evaluates a number of models for building surgery schedules with leveled resulting bed occupancy. The developed models involve two types of constraints. Demand constraints ensure that each surgeon (or surgical group) obtains a specific number of operating room blocks. Capacity constraints limit the available blocks on each day. Furthermore, the number of operated patients per block and the length of stay of each operated patient are dependent on the type of surgery. Both are considered stochastic, following a multinomial distribution. We develop a number of mixed integer programming based heuristics and a metaheuristic to minimize the expected total bed shortage and present computational results.     

Eventual convexity of chance constrained feasible sets

In decision-making problems where uncertainty plays a key role and decisions have to be taken prior to observing uncertainty, chance constraints are a strong modelling tool for defining safety of decisions. These constraints request that a random inequality system depending on a decision vector has to be satisfied with a high probability. The characteristics of the feasible set of such chance constraints depend on the constraint mapping of the random inequality system, the underlying law of uncertainty and the probability level. One characteristic of particular interest is convexity. Convexity can be shown under fairly general conditions on the underlying law of uncertainty and on the constraint mapping, regardless of the probability-level. In some situations, convexity can only be shown when the probability-level is high enough. This is defined as eventual convexity. In this paper, we will investigate further how eventual convexity can be assured for specially structured chance constraints involving Copulae. The Copulae have to exhibit generalized concavity properties. In particular, we will extend recent results and exhibit a clear link between the generalized concavity properties of the various mappings involved in the chance constraint for the result to hold. Various examples show the strength of the provided generalization.     

Experimenting with the Coxian Phase-Type Distribution to Uncover Suitable Fits

In the past few decades, Coxian phase-type distributions have become increasingly more popular as a means of representing survival times. In healthcare, they are considered suitable for modelling the length of stay of patients in hospital and more recently for modelling the patient waiting times in Accident and Emergency Departments. The Coxian phase-type distribution has not only been shown to provide a good representation of real survival data, but its interpretation seems reasonably initiative to the medical experts. The drawback, however, is fitting the distribution to the data. There have been many attempts at accurately estimating the Coxian phase-type parameters. This paper wishes to examine the most promising of the approaches reported in the literature to determine the most accurate. Three performance measures are introduced to assess the fitting process of the algorithms along with the likelihood values and AIC to examine the goodness of fit and complexity of the model. Previous research suggests that the fitting process is strongly influenced by the initial parameter estimates and the data itself being quite variable. To overcome this, one experiment in this research paper will use the same initial parameter values for each estimation and perform the fits on the data simulated from a Coxian phase-type distribution with known parameters.     

Using phase-type models to cost stroke patient care across health,social and community services

Stroke disease places a heavy burden on society, incurring long periods of time in hospital and community care, and associated costs. Also stroke is a highly complex disease with diverse outcomes and multiple strategies for therapy and care. Previously a modeling framework has been developed which clusters patients into classes with respect to their length of stay (LOS) in hospital. Phase-type models were then used to describe patient flows for each cluster. Also multiple outcomes, such as discharge to normal residence, nursing home, or death can be permitted. We here add costs to this model and obtain the Moment Generating Function for the total cost of a system consisting of multiple transient phase-type classes with multiple absorbing states. This system represents different classes of patients in different hospital and community services states. Based on stroke patients’ data from the Belfast City Hospital, various scenarios are explored with a focus on comparing the cost of thrombolysis treatment under different regimes. The overall modeling framework characterizes the behavior of stroke patient populations, with a focus on integrated system-wide costing and planning, encompassing hospital and community services. Within this general framework we have developed models which take account of patient heterogeneity and multiple care options. Such complex strategies depend crucially on developing a deep engagement with the health care professionals and underpinning the models with detailed patient-specific data.     

Matching patient and physician preferences in designing a primary care facility network

This paper introduces an integer programming model for planning primary care facility networks, which accounts for the interests of different stakeholders while maximizing access to health care. Physician allocation to health-care facilities is explicitly modelled, which allows consideration of physician incentives in the planning phase. An illustrative case study in the Turkish primary care system is presented to show the implications of focusing on patient or physician preferences in the planning phase. A discussion of trade-offs between the different stakeholder preferences and some recommendations for modelling choices to match these preferences are provided. In the context of this case, we found that using an access measure that decays with distance, and incorporating nearest allocation constraints improves performance for all stakeholders. We also show that increasing the number of physicians may have adverse affects on access measures when physician preferences are addressed.     

Goodwill,inventory penalty,and adaptive supply chain management

Information visibility is generally useful for decision makers distributed across supply chains. Availability of information on inventory levels, price, lead times, demand, etc. can help reduce uncertainties as well as alleviate problems associated with bullwhip effect. A majority of extant literature in this area assume a static supply chain network configuration. While this was sufficient a few decades ago, advances in e-commerce and the ease with which order processing can be performed over the Internet necessitates appropriate dynamic (re)configuration of supply chains over time. Each node in the supply chain is modeled as an actor who makes independent decisions based on information gathered from the next level upstream. A knowledge-based framework is used for dynamic supply chain configuration and to consider the effects of inventory constraints and ‘goodwill,’ as well as their effects on the performance dynamics of supply chains. Preliminary results indicate that neither static nor dynamic configurations are consistently dominant. Scenarios where static configurations perform better than the modeled system are identified.     

A system dynamics-based simulation study for managing clinical governance and pathways in a hospital

This paper examines the development of clinical pathways (CP) in a hospital in Australia based on empirical clinical data of patient episodes. A system dynamics (SD)-based decision support system is developed and analysed for this purpose. The study highlights the scenarios that will help hospital administrators to redistribute caseloads among admitting clinicians with a focus on multiple diagnostic-related groups (DRGs) as the means to improve the patient turnaround and hospital throughput without compromising quality patient care. DRGs are the best known classification system used in a casemix funding model. Casemix is a DRG-based government funding model for hospitals with a mix of performance measures aiming to reward initiatives that increase efficiencies in hospitals. The classification system groups inpatient stays into clinically meaningful categories of similar levels of complexity that consume similar amounts of resources. Policy explorations reveal various combinations of the dominant policies that hospital management can adopt. With the use of visual interfaces, executives can manipulate the DSS to test various scenarios. Experimental evidence based on focus groups demonstrated that it can enhance group learning processes and improve decision making. The findings are supported by other recent studies of CP implementation on various DRGs. These showed substantial reduction in length of stay, costs and resource utilization.     

Cost minimization models: Applications in a teaching hospital

The primary objective of this article is to explore the feasibility of the application of cost minimization analysis in a teaching hospital environment. The investigation is concerned with the development of cost per admission and cost per patient day models. These models are further used for determining the value of the length of stay that would minimize cost per patient day (projected length of stay) and for estimating the costs. This study is based on total of 109,060 observations (2002), obtained from a teaching hospital in South Florida. The top 10 diagnosis-related groups (DRGs) with the highest volume are selected for the study. The cost models are fitted to the data for an average R² value of 87.3%, and a mean absolute percentage error (MAPE) value of 16.1%. The result demonstrates that if a hospital can control the length of stay at the projected level, on average, the cost per admission and the cost per patient day will decrease. Based on 8703 admissions for the selected DRGs in 2002, the total cost per year and the cost per patient day are decreased by approximately 8.56% ($15,453,841) and 4.02% ($66.30), respectively. Overall, these results confirm that the concept of cost minimization analysis in economic theory can be applied to healthcare industries for the purpose of reducing of costs. Cost minimization and cost variation analyses offer useful information to hospital management for better decision-making. It would be an important aid in making management decisions, particularly for cost reduction.     

Exploring communication networks to understand organizational crisis using exponential random graph models

In recent social network studies, exponential random graph (ERG) models have been used comprehensively to model global social network structure as a function of their local features. In this study, we describe the ERG models and demonstrate its use in modelling the changing communication network structure at Enron Corporation during the period of its disintegration. We illustrate the modelling on communication networks, and provide a new way of classifying networks and their performance based on the occurrence of their local features. Among several micro-level structures of ERG models, we find significant variation in the appearance of A2P (Alternating k-two-paths) network structure in the communication network during crisis period and non-crisis period. We also notice that the attribute of hierarchical positions of actors (i.e., high rank versus low rank staff) have impact on the evolution process of networks during crisis. These findings could be used in analyzing communication networks of dynamic project groups and their adaptation process during crisis which could lead to an improved understanding how communications network evolve and adapt during crisis.     

Equations of Motion for Constrained Multibody Systems and their Control

This paper presents some recent advances in the dynamics and control of constrained multibody systems. The constraints considered need not satisfy the D’Alembert principle and therefore the results are of general applicability. They show that, in the presence of constraints, the constraint force acting on the multibody system can always be viewed as made up of the sum of two components whose explicit form is provided. The first of these components consists of the constraint force that would have existed were all the constraints ideal; the second is caused by the nonideal nature of the constraints, and though it needs specification by the mechanician who is modeling the specific system at hand, it has a specific form. The general equations of motion obtained herein provide new insights into the simplicity with which Nature seems to operate. They point toward the development of new and novel approaches for the exact control of complex multibody nonlinear systems. In honor of Bob kalaba, friend, colleague, and mentor.     

Stabilization of the motions of mechanical systems in prescribed phase-space manifolds

A method for constructing a mathematical model of the dynamics of a mechanical system is proposed. An algorithm is constructed for determining the expressions for the control forces and the components of the constraint reactions. A modification is made to the dynamic equations which enables one to solve the problem of stabilizing the constraints and which ensures the required accuracy in the numerical solution of the corresponding system of differential-algebraic equations describing the constraints imposed on the system, its kinematics and dynamics. By virtue of well-known dynamic analogies, the proposed method can be used to investigate the dynamics of different physical systems. The problem of modelling the dynamics of an adaptive optical system with two degrees of freedom is considered.