Applying cost minimization techniques to hospitals: A comment

In a challenging paper, [Suthummanon S., Omachonu V.K., 2007. Cost minimization models: Applications in a teaching hospital. European Journal of Operational Research 186, 1175–1183] (hereafter SO), present an interesting cost minimization model to be applied to improve hospital efficiency. Though the technical aspects of the statistical analysis SO present cannot be criticized, it is argued that the application to hospital costs the authors present is fraught with conceptual and practical problems. These need to be resolved before a length of stay approach to minimizing hospital costs can have any practical relevance.     

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.     

Nonparametric analysis of educational costs

Traditional economic analyses of the public sector that assume cost minimization are not consistent with political models of bureaucracy. If costs are not minimized then estimated cost functions will be biased. The purpose of this paper is to provide a flexible nonparametric technique based on Farrell-type efficiency measures to estimate public sector costs. Standard indices need to be modified to fit the special nature of public sector service provision which is characterized by an influence of exogenous variables on cost. A useful by-product is an index of environmental harshness faced by local governments. For illustrative purposes, this technique is applied to a sample of New York State school districts. It is found that nearly 64% of districts are cost inefficient, spending on average $1200 per pupil above the cost minimizing level. In addition, it is estimated that the average school district faces environmental cost of over $1700 per pupil.     

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.     

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.     

Where to treat the older patient? Can Markov models help us better understand the relationship between hospital and community care?

We have previously used Markov models to describe movements of patients between hospital states; these may be actual or virtual and described by a phase-type distribution. Here we extend this approach to a Markov reward model for a healthcare system with Poisson admissions and an absorbing state, typically death. The distribution of costs is evaluated for any time and expressions derived for the mean and variances of costs. The average cost at any time is then determined for two scenarios: the Therapeutic and Prosthetic models, respectively. This example is used to illustrate the idea that keeping acute patients longer in hospital to ensure fitness for discharge, may reduce costs by decreasing the number of patients that become long-stay. In addition we develop a Markov Reward Model for a healthcare system including states, where the patient is in hospital, and states, where the patient is in the community. In each case, the length of stay is described by a phase-type distribution, thus enabling the representation of durations and costs in each phase within a Markov framework. The model can be used to determine costs for the entire system thus facilitating a systems approach to the planning of healthcare and a holistic approach to costing. Such models help us to assess the complex relationship between hospital and community care.     

Optimal spare ordering policy under a rebate warranty

This paper develops, from the customer’s perspective, the optimal spare ordering policy for a non-repairable product with a limited-duration lifetime and under a rebate warranty. The spare unit for replacement is available only by order and the lead time for delivery follows a specified probability distribution. Through evaluation of gains due to the rebate and the costs due to ordering, shortage, and holding, we derive the expected cost per unit time and cost effectiveness in the long run and examine the optimal ordering time by minimizing or maximizing these cost expressions. We show that there exists a unique optimum solution under mild assumptions. We provide a numerical example and illustrate sensitivity analysis.     

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.     

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.     

An approach for analysing transportation costs and a case study

One of the important parameters in the determination of optimal transportation system is economy. Therefore, a realistic method based on the technical, economical and operational parameters of various transportation modes, namely, road, railway, and sea routes is required in the analysis of costs. This method will take into consideration the probable price escalations during the lifetime of a certain transportation system. The cost of a unit of cargo or passenger per route length should be considered since it is an indicator of economics. In this paper, an approach for transportation cost analysis based on the economic analysis of the alternative modes of cargo or passenger transportation, is presented.     

Optimal Paths And Costs Of Adjustment In Dynamic DEA Models: With Application To Chilean Department Stores

In this paper we propose a range of dynamic data envelopment analysis (DEA) models which allow information on costs of adjustment to be incorporated into the DEA framework. We first specify a basic dynamic DEA model predicated on a number of simplifying assumptions. We then outline a number of extensions to this model to accommodate asymmetric adjustment costs, non-static output quantities, non-static input prices, and non-static costs of adjustment, technological change, quasi-fixed inputs and investment budget constraints. The new dynamic DEA models provide valuable extra information relative to the standard static DEA models—they identify an optimal path of adjustment for the input quantities, and provide a measure of the potential cost savings that result from recognising the costs of adjusting input quantities towards the optimal point. The new models are illustrated using data relating to a chain of 35 retail department stores in Chile. The empirical results illustrate the wealth of information that can be derived from these models, and clearly show that static models overstate potential cost savings when adjustment costs are non-zero. This paper arises out the senior author's PhD thesis at the University of New England, Australia. The authors gratefully acknowledge Dr. George E. Battese for his comments on earlier drafts of this work.     

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.     

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.     

Cost allocation and convex data envelopment

This paper considers allocation rules. First, we demonstrate that costs allocated by the Aumann–Shapley and the Friedman–Moulin cost allocation rules are easy to determine in practice using convex envelopment of registered cost data and parametric programming. Second, from the linear programming problems involved it becomes clear that the allocation rules, technically speaking, allocate the non-zero value of the dual variable for a convexity constraint on to the output vector. Hence, the allocation rules can also be used to allocate inefficiencies in non-parametric efficiency measurement models such as Data Envelopment Analysis (DEA). The convexity constraint of the BCC model introduces a non-zero slack in the objective function of the multiplier problem and we show that the cost allocation rules discussed in this paper can be used as candidates to allocate this slack value on to the input (or output) variables and hence enable a full allocation of the inefficiency on to the input (or output) variables as in the CCR model.     

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.     

三种类型汽车的能耗和使用成本的建模研究

本文对电动,传统,混合动力三种类型汽车的能耗和使用成本问题进行了建模、求解及分析．首先定义了电动汽车的能量利用率ηE。根据能量守恒将所有电量转换成汽油消耗量,而传统车型则直接反映到汽油的消耗上,混合动力汽车则综合电动汽车和传统汽车的能耗模型,从而统一能耗水平评价标准．这样,我们就得到了三种不同类型车的能耗模型再进行模型求解以此来分析比较三种类型车哪一种节能效果好．对电动汽车来说,使用成本主要包括驾驶维护成本、报废处理成本,而对传统汽车来说,使用成本主要包括驾驶维护成本、报废处理成本、环境成本和其他成本,混合动力汽车则结合二者使用成本．因此,根据不同车型使用成本类型建立相应的数学模型,按照建立的公式,可以得到电动汽车的使用成本LCC1、传统汽车的使用成本LCC2和混合动力汽车的使用成本LCC3,从而进行模型求解．     

The optimization of repair decision using life-cycle cost Parameters

Life-cycle cost models typically minimize system repair costsas a function of various cost coefficients associated with agiven repair option such as discarding upon failure or repairingthe components. Fixed costs such as set-up costs pose specialproblems for the minimization of life-cycle costs: first ofall, fixed costs are characterized by step functions linkedto capacity constraints, while variable costs are representedby continuous functions. Secondly, both categories of cost shouldbe evaluated simultaneously for all components of a physicalsystem and for all repair options. Thirdly, there are operational–researchtools such as mixed integer programming which can solve largeproblems of this type under a set of acceptable simplifyingassumptions. Heuristic methods can be used to minimize the searchtime for a global optimum solution. This paper describes anapproach which has been successfully applied to maintenanceplanning, vibration analysis, and expert fault diagnostics.One of these applications is discussed in detail as an illustrationof the method proposed.     

The management of intrusion detection: Configuration,inspection, and investment

This paper analyzes intrusion detection decisions in the presence of multiple alarm types, which differ in occurrence probabilities, damage and investigation costs. Specifically, multi-period optimization models are used to study three critical decisions associated with intrusion detection: (i) Allocation of the investigation budget to different periods and to different alarm types; (ii) Configuration of an intrusion detection system (IDS), i.e. choosing a false alarm rate for a given IDS; and (iii) Allocation of an appropriate amount of the investigation budget in the presence of alternative investment opportunities. Three models that cascade onto each other are presented. We minimize the sum of security costs including damages, due to ignored alarms, the investigation cost and the undetected intrusion cost. We show that it can be optimal to ignore non-critical alarms in order to allocate more of the investigation budget to critical alarms that may occur in the future. We establish that the security costs decrease as the investigation budget increases. Our last model deals with security investments—in the form of an investigation budget. The investigation budget must be increased until the rate of increase in savings in security costs due to the additional budget are equal to the internal rate of return of an organization. These analyses are done with explicit (derived) cost functions, as opposed to implicit (assumed) cost functions. We conclude by providing additional managerial insights and numerical examples.     

Analysis of a two-echelon inventory system with two supply modes

In this paper, we consider a serial two-echelon periodic review inventory system with two supply modes at the most upstream stock point. As control policy for this system, we propose a natural extension of the dual-index policy, which has three base-stock levels. We consider the minimization of long run average inventory holding, backlogging, and both per unit and fixed emergency ordering costs. We provide nested newsboy characterizations for two of the three base-stock levels involved and show a separability result for the difference with the remaining base-stock level. We extend results for the single-echelon system to efficiently approximate the distributions of random variables involved in the newsboy equations and find an asymptotically correct approximation for both the per unit and fixed emergency ordering costs. Based on these results, we provide an algorithm for setting base-stock levels in a computationally efficient manner. In a numerical study, we investigate the value of dual-sourcing in supply chains and illustrate that dual-sourcing can lead to significant cost savings in cases with high demand uncertainty, high backlogging cost or long lead times.     

A queueing model for bed-occupancy management and planning of hospitals

The aim of this paper is, on the one hand, to describe the movement of patients through a hospital department by using classical queueing theory and, on the other hand, to present a way of optimising the use of hospital resources in order to improve hospital care. A queueing model is used to determine the main characteristics of the access of patients to hospital, such as mean bed occupancy and the probability that a demand for hospital care is lost because all beds are occupied. Moreover, we present a technique for optimising the number of beds in order to maintain an acceptable delay probability at a sufficiently low level and, finally, a way of optimising the average cost per day by balancing costs of empty beds against costs of delayed patients.