A non-homogeneous discrete time Markov model for admission scheduling and resource planning in a cost or capacity constrained healthcare system
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DOI: 10.1007/s10729-009-9120-0
- Cite this article as:
- Garg, L., McClean, S., Meenan, B. et al. Health Care Manag Sci (2010) 13: 155. doi:10.1007/s10729-009-9120-0
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Abstract
Healthcare resource planners need to develop policies that ensure optimal allocation of scarce healthcare resources. This goal can be achieved by forecasting daily resource requirements for a given admission policy. If resources are limited, admission should be scheduled according to the resource availability. Such resource availability or demand can change with time. We here model patient flow through the care system as a discrete time Markov chain. In order to have a more realistic representation, a non-homogeneous model is developed which incorporates time-dependent covariates, namely a patient’s present age and the present calendar year. The model presented in this paper can be used for admission scheduling, resource requirement forecasting and resource allocation, so as to satisfy the demand or resource constraints or to meet the expansion or contraction plans in a hospital and community based integrated care system. Such a model can be used with both fixed and variable numbers of admissions per day and should prove to be a useful tool for care managers and policy makers who require to make strategic management decisions. We also describe an application of the model to an elderly care system, using a historical dataset from the geriatric department of a London hospital.
Keywords
Resource managementAdmission schedulingNon-homogeneous Markov modelStochastic optimal control1 Introduction
Admission scheduling [1, 2] and resource planning [3] are fundamental problems which require complex strategies to effectively manage care services ensuring optimum utilization of scarce resources and efficient quality of service delivery. Long waiting lists are considered to be symptomatic of an inefficient care system [4–6]. To avoid long waiting lists, care professionals and policy makers are required to estimate the resource requirements for a given time in the future and to allocate the budget/ resources to satisfy this demand [7]. Also if the resource availability is limited and cannot meet an ever increasing demand, it is necessary to estimate the number of admissions that can be satisfied each day and schedule accordingly, from the waiting list [8]. This will help the health care managers to optimally exploit capacity and limit the size of the waiting list [7].
Neither admission scheduling nor resource requirements forecasting are new problems and numerous models have been proposed. Some of these [5, 9–13] are queueing models, others [14–19] utilise simulation or population/ratio based approaches which use average lengths of stay to quantify the number of patients requiring care resources, for example [20–24]. However, some of these methods, which are mostly stochastic or simulation based, are computationally very complex and require substantial data and logistics [25]. Other simpler methods, on the other hand, do not properly represent case mix [3] or local specificities [25] and thus require more sophisticated models [26]. In addition, as suggested by [3, 27], other approaches are inaccurate or misleading. Also previous models mainly model patient flow in a department or hospital and do not incorporate readmissions and care in the community. Therefore they are not very suitable for admission scheduling and capacity planning in an integrated care system, which includes both hospital and community care [28].
In our previous work [29] we have developed a continuous time non-homogeneous Markov model for admission scheduling and resource forecasting by enumerating patient pathways. The limitation of this model is that it assumes a fixed number of admissions per day, so it cannot be used in many practical scenarios (such as a care system with expansion or contraction plans). A variable number of admissions each day can therefore prove to be a more realistic assumption with correspondingly better solutions that ensure optimal resource utilization. Also we might need to schedule a variable number of admissions each day so as to allocate resources to satisfy the fluctuating demand for care services [12, 30]. In this paper, we present a discrete time Markov model for admission scheduling and capacity planning. Based on our initial homogeneous model, a novel, more realistic non-homogeneous model is developed, incorporating time dependent covariates (in this case the patient’s present age and the present calendar year). This approach can effectively be used for resource requirement forecasting and resource allocation to satisfy the demand or resource constraints or to meet the expansion or contraction plans. We can also use this new model to compare different admission scheduling strategies for a care system. Finally we illustrate use of the model by applying it to a historical dataset of all male patients from a geriatric department of a London hospital admitted during a 16 year period [31]. Throughout the paper we use mathematical notation which is described in Appendix 2 (Table 5).
2 Background
2.1 Patient flow through the care system
A patient pathway for such a system is defined as the way a patient sequentially moves from one hospital phase to the next and similarly one community phase to the next. Patients can be discharged from any hospital phase to the first community phase or the patient can die in any hospital or community phase. Re-admission into the first hospital phase is possible from any community phase. We can therefore represent this care system by an n + m + 1 state discrete time Markov chain with death as an absorbing state. The transition matrix Q for the absorbing Markov chain can then be represented as follows [33, 35]:
Here λ_{i} represents the rate of transition from hospital phase i to hospital phase i + 1. ν_{i} represents the rate of transition from hospital phase i to the first community phase and μ_{i} is the rate of transition from hospital phase i to the absorbing phase death. Similarly α_{i} is the rate of transition from community phase i to community phase i + 1, γ_{i} is the rate of transition from community phase i to the first hospital phase and β_{i} is the rate of transition from community phase i to death.
3 Model description
3.1 The homogeneous model
3.2 The non-homogeneous model
In the model presented here we consider two time dependent covariates: the patient’s present age and the present calendar year. In order to have a more realistic model, we update the covariates each day for each patient currently in the system, recalculate the parameter values for each patient separately and recompute the discrete time transition probability matrix P.
Once the expected numbers of patients in different phases of the system are known, we use these expressions to calculate the optimum rate of admissions to satisfy the given constraints and to estimate the resource requirements based on the given admission schedule.
3.3 Incorporating admissions
For a more realistic model it is necessary to incorporate new admissions into the system. We therefore develop our model to include, in the first instance, a fixed number of admissions each day and subsequently a variable number of admissions (increasing or decreasing) each day are assumed.
3.3.1 Fixed number of admissions
3.3.2 Variable number of admissions
The results obtained in this sub-section can be used for admission scheduling as discussed in the next sub-section.
3.4 Admission scheduling
Admission scheduling can be defined as designing an admission policy to ensure optimum utilization of the future available resources satisfying given constraints or meeting a given target. Possible constraints can be availability based on budgets, beds or other resources (such as nursing staff, physicians, or specialists) at a given time in the future or the total daily, monthly or yearly expenditure. We will consider two scenarios of interest: first a new care system and second a pre-existing care system.
3.4.1 A new care system
We assume that the constraint is to satisfy the available number of beds, so there will be only B(t_{given}) beds available at time t_{given}.
From (20) it is clear that the value of r will be negative for a hospital with a contraction plan.
3.4.2 A pre-existing care system
We can use (28) and (29) to estimate the expected admission rate to satisfy the given bed availability constraints and the expected cost of care constraints respectively.
4 An illustration
4.1 Daily resource requirement
We can estimate the daily resource requirements in a care unit or a care system if we know the expected number of patients in each phase in the care unit at a given time in future. To calculate the expected cost we have used the indicative cost for the 8-phase care system estimated by McClean & Millard [38] using relative weightings for each phase based on the then current data for geriatric patients in the UK. These relative weightings are 15, 10, 8 and 6 for hospital phases acute, treatment, rehabilitative and long-stay respectively and 5, 4 and 0.5 for community phases dependent, convalescent and recovered respectively [38]. Also McClean & Millard [38] estimated average daily cost of the care in acute phase as £150. Here we will consider 2 cases, one with a fixed number of admissions each day and the other with a variable number of admissions each day.
4.1.1 A fixed number of admissions
Estimated number of patients and cost (new system with 1 admission/ day)
Days | Phase1 | Phase2 | Phase3 | Phase4 | Phase5 | Phase6 | Phase7 | Phase8 | Total Daily Cost(In £) |
---|---|---|---|---|---|---|---|---|---|
100 | 11.07 | 8.95 | 10.16 | 0.27 | 11.16 | 31.65 | 2.85 | 23.89 | 5,223.0 |
200 | 11.97 | 9.67 | 14.29 | 0.88 | 12.50 | 70.13 | 16.28 | 64.27 | 7,471.3 |
300 | 12.60 | 10.16 | 15.94 | 1.50 | 12.95 | 95.47 | 38.94 | 112.44 | 8,931.8 |
400 | 13.04 | 10.50 | 16.94 | 2.06 | 13.10 | 110.38 | 68.06 | 165.92 | 9,895.4 |
500 | 13.36 | 10.74 | 17.70 | 2.55 | 13.09 | 118.00 | 101.21 | 223.35 | 10,527.1 |
600 | 13.60 | 10.93 | 18.32 | 2.97 | 12.99 | 120.78 | 136.51 | 283.90 | 10,940.3 |
700 | 13.81 | 11.08 | 18.87 | 3.34 | 12.84 | 120.51 | 172.51 | 347.04 | 11,214.3 |
800 | 13.99 | 11.22 | 19.39 | 3.66 | 12.66 | 118.41 | 208.18 | 412.47 | 11,403.4 |
900 | 14.18 | 11.36 | 19.91 | 3.94 | 12.49 | 115.29 | 242.78 | 480.04 | 11,543.0 |
1,000 | 14.38 | 11.51 | 20.43 | 4.19 | 12.32 | 111.66 | 275.83 | 549.68 | 11,655.6 |
1,100 | 14.59 | 11.66 | 20.98 | 4.42 | 12.15 | 107.82 | 307.00 | 621.37 | 11,754.1 |
1,200 | 14.82 | 11.83 | 21.55 | 4.62 | 12.00 | 103.93 | 336.12 | 695.12 | 11,846.0 |
1,300 | 15.07 | 12.01 | 22.16 | 4.80 | 11.86 | 100.08 | 363.08 | 770.93 | 11,934.9 |
1,400 | 15.33 | 12.21 | 22.80 | 4.99 | 11.74 | 96.32 | 387.82 | 848.80 | 12,022.4 |
1,500 | 15.61 | 12.41 | 23.47 | 5.16 | 11.62 | 92.64 | 410.34 | 928.74 | 12,109.1 |
1,600 | 15.91 | 12.63 | 24.18 | 5.32 | 11.51 | 89.07 | 430.66 | 1,010.71 | 12,195.2 |
1,700 | 16.22 | 12.86 | 24.92 | 5.49 | 11.41 | 85.59 | 448.82 | 1,094.68 | 12,280.8 |
1,800 | 16.55 | 13.10 | 25.70 | 5.66 | 11.31 | 82.21 | 464.88 | 1,180.59 | 12,365.8 |
1,900 | 16.89 | 13.35 | 26.51 | 5.82 | 11.22 | 78.92 | 478.91 | 1,268.38 | 12,450.3 |
2,000 | 17.24 | 13.61 | 27.36 | 5.99 | 11.14 | 75.72 | 490.99 | 1,357.96 | 12,534.5 |
4.1.2 A variable number of admissions
Estimated values of change multipliers for different phases (new system with 1 admission/ day)
Days | Phase1 | Phase2 | Phase3 | phase4 | phase5 | Phase6 | phase7 | Phase8 | Total daily cost (in £) |
---|---|---|---|---|---|---|---|---|---|
100 | 167.3 | 210.4 | 516.6 | 18.4 | 417.1 | 1,975.9 | 210.2 | 1,434.1 | 189,509 |
200 | 310.4 | 321.8 | 1,084.7 | 110.1 | 636.1 | 7,582.6 | 2,262.6 | 7,591.6 | 518,548 |
300 | 484.2 | 455.4 | 1,463.7 | 263.2 | 817.0 | 13,819.4 | 7,770.0 | 19,777.1 | 883,538 |
400 | 665.4 | 594.2 | 1,777.9 | 455.7 | 978.6 | 19,163.7 | 17,421.6 | 38,742.9 | 1,231,388 |
500 | 845.8 | 731.4 | 2,069.0 | 671.6 | 1,123.0 | 23,077.0 | 31,232.7 | 64,999.4 | 1,541,227 |
600 | 1,026.9 | 868.2 | 2,351.0 | 899.2 | 1,254.4 | 25,572.0 | 48,793.5 | 98,934.8 | 1,812,459 |
700 | 1,215.0 | 1,009.1 | 2,634.8 | 1,130.1 | 1,379.7 | 26,921.5 | 69,466.6 | 140,893.1 | 2,054,930 |
800 | 1,417.2 | 1,159.9 | 2,932.3 | 1,359.2 | 1,505.8 | 27,477.1 | 92,528.9 | 191,219.5 | 2,281,731 |
900 | 1,640.1 | 1,325.7 | 3,255.0 | 1,583.3 | 1,638.4 | 27,564.7 | 117,264.4 | 250,278.3 | 2,504,815 |
1,000 | 1,888.2 | 1,510.2 | 3,612.6 | 1,801.3 | 1,780.8 | 27,436.4 | 143,015.9 | 318,454.7 | 2,732,909 |
1,100 | 2,164.2 | 1,715.4 | 4,011.9 | 2,013.0 | 1,934.6 | 27,259.1 | 169,206.2 | 396,145.7 | 2,971,016 |
1,200 | 2,468.7 | 1,941.9 | 4,457.2 | 2,219.2 | 2,099.1 | 27,124.0 | 195,340.7 | 483,749.1 | 3,220,849 |
1,300 | 2,800.9 | 2,189.4 | 4,950.4 | 2,421.2 | 2,272.9 | 27,066.0 | 220,998.7 | 581,650.5 | 3,481,661 |
1,400 | 3,159.0 | 2,456.4 | 5,491.5 | 2,620.1 | 2,453.7 | 27,083.2 | 245,822.4 | 690,213.8 | 3,751,133 |
1,500 | 3,540.3 | 2,741.0 | 6,079.0 | 2,817.4 | 2,638.7 | 27,153.9 | 269,506.6 | 809,773.2 | 4,026,123 |
1,600 | 3,941.8 | 3,041.0 | 6,710.4 | 3,014.2 | 2,825.2 | 27,248.3 | 291,791.6 | 940,627.5 | 4,303,206 |
1,700 | 4,360.3 | 3,353.9 | 7,382.8 | 3,211.8 | 3,010.6 | 27,336.0 | 312,458.2 | 1,083,036.5 | 4,579,022 |
1,800 | 4,792.2 | 3,677.1 | 8,092.5 | 3,411.0 | 3,192.4 | 27,390.4 | 331,326.2 | 1,237,218.1 | 4,850,474 |
1,900 | 5,234.3 | 4,008.0 | 8,835.7 | 3,612.7 | 3,368.4 | 27,389.5 | 348,253.7 | 1,403,347.7 | 5,114,825 |
2,000 | 5,682.8 | 4,343.9 | 9,608.4 | 3,817.4 | 3,536.7 | 27,317.5 | 363,136.8 | 1,581,556.5 | 5,369,748 |
4.2 Resource constraints
4.2.1 A new care system
Estimated number of admissions per day according to budget or beds availability
Day | Estimating the number of admissions per day if the constraint is the daily cost | Estimating the number of admissions per day if the constraint is the beds availability | ||||
---|---|---|---|---|---|---|
Daily cost limit (in £) | Number of beds available in the hospital | Average admissions per day | Number of beds available in the hospital | Daily cost (in £) | Average admissions per day | |
500 | 800,000 | 3,370 | 76 | 6,000 | 628,690 | 135 |
1,000 | 400,000 | 1,734 | 34 | 8,000 | 822,597 | 158 |
1,500 | 500,000 | 2,340 | 41 | 4,500 | 458,330 | 79 |
2,000 | 750,000 | 3,841 | 60 | 1,700 | 171,991 | 26 |
Estimated rate of change in admission rate (r) and estimated number of admissions per day (a) to satisfy given budget constraints at two time points in the future
t_{1} in days | C(t_{1}) in £ | t_{2} in days | C(t_{2}) in £ | A_{req} = a (Estimated initial admission rate) | r (estimated rate of change in admission rate) |
---|---|---|---|---|---|
500 | 800,000 | 1,000 | 800,000 | 88.23 | −0.00095 |
500 | 800,000 | 1,000 | 120,240 | 30.83 | 0.01 |
1,000 | 400,000 | 1,500 | 500,000 | 17.64 | 0.04 |
1,000 | 500,000 | 1,200 | 400,000 | 100 | −0.0024354 |
1,500 | 500,000 | 2,000 | 610,000 | 15.73 | 0.049 |
1,600 | 200,000 | 2,000 | 200,000 | 18.47 | −0.0032 |
4.2.2 A pre-existing care system
We can estimate the expected resource requirements for such a care system by using (33) and (34). For example if there are no new admissions to a care system with initially only 500 patients in the acute phase (and no patient in any other phase) then the expected number of patients left in care system after 2,000 days will be 0.65, 0.45, 0.9, 0.3, 0.35, 0.25 and 41.45 patients respectively in the acute phase, treatment phase, rehabilitative phase, long-stay phase, dependent phase, convalescent phase and recovered phase.
From Table 1, \( {\Omega_{{t_{\text{given}}}}}_{ = 1000} = \pounds11655.60 \)
Therefore, A_{req} = 8.0753 patients/day.
From Table 1, \( \left( {{\mathbf{x}}*{\mathbf{c}}} \right)\,{\Omega_{{t_{\text{given}}} = 1000}} = \pounds11655.60 \)
Therefore, A_{req} = 1.8336 patients/day.
If the bed availability is a constraint, then we can use (28) to estimate the mean rate of admissions allowed for a care system with a fixed number of admissions each day and (30) to estimate the rate of change in admission rate (r) for a care system with variable number of admissions each day.
5 Discussion
In our model, we assume that there is always a waiting list of patients who can be admitted to the first phase of the care system whenever there is a bed available. This assumption is realistic in many practical situations. However, as the transition probabilities are estimated separately for each patient in the care system, the model can be modified for resource requirements planning in other scenarios where patients arrive arbitrarily and admissions can be modeled as a Poisson stream. Another limitation of our model is that it is based largely on expected values. The model can therefore be used to predict the expected number of patients in each phase and expected daily cost of care. However, these expected numbers represent long term averages, but do not reflect variability. We are currently working on enhancing our model to estimate the variability of these averages and to examine its effects. Such variability and its effects can also be estimated using a stochastic simulation models (such as a discrete event simulation).
An obvious application of our model is to facilitate decision making of health and social care planners, managers and policy makers for resource planning and budget allocation, especially for the elderly care. However, this model can be customized for other care systems as well. Our model equips care planners, managers and policy makers with a tool for resource planning, budget allocation and strategic decision making for care management and service improvement. In a community based setting, they can use our model with demographic data and regional health statistics data, to allocate budget for health and social care for the population of their region. Bed availability, optimum utilization of the allocated budget, minimization of waiting time and the size of patient queues can thus be assured Individual resource requirements for hospital care and community care can also be estimated. Another application of the tool is in health insurance where an insurance company might use the model to forecast resource requirements and then decide the optimum premium for the patient’s health insurance policy.
Our model can also be used for better informed budget allocation and resource planning decisions evaluating the combined cost measure of the dead-alive trade off, quality of life and economic costs [39]. This can be achieved by using our model coupled with a partially observable Markov decision process based on, for example, the patient management model proposed by [39] to select the care pathway which optimizes the combined cost for the integrated care system. This will also facilitate cost effectiveness analysis (CEA) [40–42] and cost-utility analysis (CUA) [40, 43] of different admission scheduling policies and resource management strategies. For example we might decide if it is cost-effective to discharge (from a hospital phase) a patient to a care home or nursing home or his/her own home, with a care package such as is described in [44]. We can estimate the resource availability and schedule the admissions accordingly. Quality of life can be measured in terms of quality adjusted life years gained (QALY) [45], disability adjusted life years (DALYs) [45], life years gained (LYs gained) [45] or Sen’s capability approach [46, 47]. It can also be measured for individual patients in the care system facilitating combined cost effectiveness analysis for the whole the system as in the model the transition probabilities are estimated separately for each patient.
The mean number of patients in each phase on a given day in the future increases exponentially with the change in a transition rate (a parameter of transition matrix Q) (see Eqs. 35 and 36, Appendix 1).
The mean number of patients in each phase on a given day in the future increases linearly with the change in the initial (or fixed) admission rate (see Eq. 37, Appendix 1).
The mean number of patients in each phase on a given day in the future increases linearly with the change in the rate of change in the admission rate (see Eq. 38, Appendix 1).
Changes in the expected total daily cost on a given day in the future increases linearly with changes in daily cost of care in different phases (and similarly with the change in the relative weighting for a phase for indicative cost) (see Eq. 39, Appendix 1).
If the beds availability on a given day in the future is a constraint, then the expected number of admissions allowed each day decreases exponentially with the change in a transition rate (see Eqs. 40 and 41, Appendix 1).
If the daily cost of care on a given day in the future is a constraint, then the expected number of admissions allowed each day decreases approximately linearly with changes in daily cost of care in different phases (and similarly with the change in the relative weighting for a phase for indicative cost) (see Eqs. 42 and 43, Appendix 1).
Our model is very efficient in terms of algorithm complexity. Implementation of the algorithm discussed in the last section for resource requirements forecasting and/or admission scheduling would require matrix multiplications for each new admission and for each day throughout the duration we wish to forecast the resource requirement. Also the number of admissions is equal to the time horizon of the resource requirement. Therefore the time complexity of the algorithm is O(k^{2}) where k is the number of days ahead for which the resource requirement requires to be forecasted. Also, we require to store the previous value of matrix s_{k} for each iteration. Therefore the space complexity of the algorithm is O(n).
6 Conclusion
We have demonstrated how a discrete time non-homogeneous Markov models can be effectively used in more sophisticated admission scheduling and resource requirement forecasting and allocation. We allow both fixed and variable rate of admissions to satisfy the demand or resource constraints. This can be a very useful tool for care managers and policy makers so as to facilitate strategic decision making for care management and service improvement. We are currently working to develop our approach as a decision (what-if analysis) tool.
Acknowledgements
The authors acknowledge support for this work from the Engineering and Physical Sciences Research Council funded RIGHT and MATCH projects (Grant References EP/E019900/1 and GR/S29874/01). Any views or opinions presented herein are those of the authors and do not necessarily represent those of RIGHT or MATCH, their associates or their sponsors.