hospital bed assignment process

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Flexible bed allocations for hospital wards

René bekker.

1 Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

Dennis Roubos

2 HOTflo Company, Schoutlaan 26, 6002 EA Weert, The Netherlands

Flexibility in the usage of clinical beds is considered to be a key element to efficiently organize critical capacity. However, full flexibility can have some major drawbacks as large systems are more difficult to manage, lack effective care delivery due to absence of focus and require multi-skilled medical teams. In this paper, we identify practical guidelines on how beds should be allocated to provide both flexibility and utilize specialization. Specifically, small scale systems can often benefit from full flexibility. Threshold type of control is then effective to prioritize patient types and to cope with patients having diverse lengths of stay. For large scale systems, we assert that a little flexibility is generally sufficient to take advantage of most of the economies of scale. Bed reservation (earmarking) or, equivalently, organizing a shared ward of overflow, then performs well. The theoretical models and guidelines are illustrated with numerical examples. Moreover, we address a key question stemming from practice: how to distribute a fixed number of hospital beds over the different units?

Introduction

Inpatient beds are a critical capacity in the patient care process within a hospital. Traditionally, the clinical organization is according to medical disciplines, resulting in separate nursing units for, e.g., medicine, surgery, cardiology, obstetrics, neurology, gynaelogy. Over the years other classifications have been introduced, such as length of stay (e.g., short and long stay, see for example [ 29 ]), level of care (intensive, medium, special or normal care), or urgency (elective, urgent and emergent), each having organizational advantages. A disadvantage of a strict classification of inpatient beds is that this may result in small scale hospital units. Such small scale units suffer severely from the variability of health care processes [ 7 ]. More generally, it is well known that the efficiency of service systems often increases as the system becomes larger [ 32 ]. This is referred to as ‘economies of scale’ (abbreviated as EOS). Flexibility in bed usage is thus a key concept for an efficient management of beds, as has been recognized in, e.g., [ 4 , 9 , 12 , 18 ], and is of fundamental importance for the increasing pressure to reduce costs.

On the opposite, in manufacturing it has long been recognized that focus on a limited range of tasks improves efficiency. This principle of specialization advocates to divide capacity to patient groups with similar medical conditions, see, e.g., [ 13 , 27 , 30 ] and references therein for some health care related studies. The increasing focus on more complex cases further advocates to organize specialized hospital units, which is evidently necessary to some extent. A further disadvantage of flexibility is that this requires the medical staff, such as nurses, to be able to treat multiple patient types. This may require costly additional training efforts. Moreover, small wards guarantee personalized patient care and may improve work satisfaction and efficiency of nurses.

Apart from medical specializations and the potential improvements from economies of focus, there are some other issues with full flexibility. First, the overall performance may improve, but that may be at the expense of one type of patients. This may be unwanted in case that a particular patient type should be prioritized (e.g., receive specialized care). Related is the example in [ 9 ] of cardiac and thoracic surgery, where cardiac patients have priority over thoracic patients. Under their average delay constraints and taking the priority for cardiology into account, it follows that a combined unit would actually need more beds than two separate units. Second, the overall performance may even decrease in case of non-identical average service times (also referred to as average length of stay, abbreviated as ALOS). This observation goes back to [ 24 ]. In that case, patients with prolonged hospital stay block access for patients with high turnovers.

In this paper, we propose an intermediate organizational bed assignment that utilizes the efficiency gains of large systems and avoids the drawbacks mentioned above. More specifically, we consider the following bed allocation policies:

• Separate wards : Each patient type has dedicated beds.
• Simple merging : All patient types share all beds.
• Earmarking : Each patient type has dedicated (earmarked) beds, whereas all patient types share a joint ward of overflow with fully flexible beds.
• Threshold policy : All beds are fully flexible, but there is a hierarchy in admission of patient types. The most important (e.g., most urgent) patients are always admitted when beds are available, but other patient types are only admitted when the number of available beds exceeds some (prespecified) threshold.

The advantages and disadvantages of the different bed allocation policies are indicated in Table  1 . These findings are further supported in the rest of the paper. Specialization refers to all benefits of having small scale units, such as specialized medical teams, single-skilled nurses and efficiency in task performance due to routine operations. Flexibility and EOS refer to all benefits of large systems, such as the ability to handle peaks in demand, flexibility in allocation of beds and flexibility in nurse rostering (see, e.g., Burke et al. [ 5 ]). Bed guarantees means that different patient types have allocated beds, making bed management significantly easier. Prioritization and the efficiency in accommodating patients with severely different LOS are further addressed in Section  4.2 .

Pros and cons of different bed allocation policies

For large scale systems specialization often is a major requirement, leading to the distribution of beds over different medical units. The earmarking policy is then effective (see Section  4.3 ). At a smaller scale, i.e., within a single unit, further specialization might be unnecessary and the focus is rather on efficient bed usage and accommodation of different patient types (see Section  4.2 ).

Goals and contribution

The issue of how to allocate partially flexible capacity for clinical wards has not yet been addressed in the literature. Therefore, our contribution is two-fold. First, we identify which structure of the bed allocation policy is appropriate for balancing between flexibility and the issues of large scale systems. For this structure we distinguish two cases that differ in system size, as they require a different approach. Bed allocation for small scale systems : at the unit level (like an ICU), the number of beds is shared by different patient groups. For instance, a patient group may represent a medical discipline, patients with a similar diagnosis, or similar level of urgency. As the sizes of the patient groups are small, specialization is inefficient whereas an earmarking policy often is less effective. In this setting, threshold policies are effective when there is a difference in priority for patient types, or patient types have an entirely different ALOS. Bed distribution for large scale systems : at the hospital level, the total number of staffed beds should be distributed over the different (often medical) units. To allow for flexible bed usage and avoid large-system size issues at the same time, earmarking is an effective policy. We see that some flexibility is sufficient to accommodate most of the peaks in bed demand. The beds at each ward are dedicated (earmarked) that can be handled by specialized medical teams, whereas the beds at the joint ward are flexible.

In the literature the commonly addressed question is ‘how many hospital beds?’ [ 4 , 8 ]. In practice, the overall number of beds is limited due to the building construction and obtained licenses [ 12 ]. The typical question for hospital managers therefore is ‘how to distribute hospital beds?’. We provide rules of thumb based on square-root staffing for the distribution of the fixed number of total beds across units.The second contribution is that we provide models to support strategic and tactical decision making regarding ward sizes and the level of flexibility. Specifically, using these models, the exact number of beds and its allocation for the corresponding policy can be determined. For large scale systems, the performance of an earmarking policy can easily be calculated due to the product-form solution. To enhance application of threshold policies, the models are suitable for a form of decision support as well. We like to emphasize that well-founded hospital management of bed capacity requires quantitative models to visualize the impact of strategic management decisions and policies.

Queueing literature

We now briefly review some of the basic queueing literature related to pooling. The term bed pooling is also often encountered in the literature when different units fully share their capacity. As mentioned, Smith and Whitt [ 24 ] seem to be the first to give counterexamples to show that full flexibility or resource sharing is not always beneficial. Another early paper supporting this from a qualitative perspective is Rothkopf and Rech [ 23 ]. In Mandelbaum and Reiman [ 20 ], the authors consider queueing networks in which both servers (beds) and queues can be pooled. They quantify the effect of pooling in terms of an efficiency index and show that pooling always helps in light traffic, but that pooling effects can go either way in heavy traffic. We refer to the references in [ 20 ] for the application of pooling in different application areas.

In the context of call centers, van Dijk and van der Sluis [ 25 ] gave some instructive examples where pooling is not beneficial and they proposed overflow pooling as an alternative. In overflow pooling the servers are dedicated to a queue, but they can serve customers from the other queue in case the server becomes idle. The concept of pooling is also related to skill-based routing in call centers. For instance, Wallace and Whitt [ 28 ] showed that “a little flexibility goes a long way”, meaning that only a few generalists are required to approach near optimal performance. In Chevalier et al. [ 6 ], the authors find that a 80/20 rule works well for a remarkably wide range of parameters. Here, the 80/20 rule means that 20 % of the staffing budget should be spent on flexible (multi-skilled) servers while 80 % should be spent on dedicated (single-skilled) servers. This already hints that flexibility and specialization can go hand in hand in hospital systems.

From a different angle, van Essen et al. [ 26 ] consider how departments should be clustered to benefit from scale effects. The authors take into account that not all departments can be clustered and that patients should not be spread over the hospital. Clustering is formulated as an optimization problem where blocking probabilities impose constraints. As the optimization problem is strongly NP-hard, the authors provide two heuristic approaches in addition to the exact formulation.

Organization

The paper is organized as follows. We introduce the general model and assumptions in Section  2 . The bed allocation policies and its performance analysis are discussed in Section  3 . In Section  4 we show numerical results. The allocation of beds over different patient groups within a unit is studied in Section  4.2 . In Section  4.3 we consider the distribution of beds over different units at the hospital level. Section  5 concludes.

We analyze the patient flow through the clinical wards in the spirit of the Erlang loss model. The aim of this model is to support managerial decision making at the strategic and tactical level. We first introduce the main assumptions in Section  2.1 and then formally define the model in Section  2.2 .

Basic assumptions

The assumptions of the model are based on the data analysis in [ 4 ] of 24 hospital wards of the VU medical center in addition to our experience with other Dutch hospitals.

Arrival process

The model assumes that patients arrive according to a Poisson process. This has been widely accepted for urgent patients, see for example [ 34 ]. Surprisingly, the number of elective admissions varies significantly as well. This variation can even be larger than the variation in urgent admissions [ 4 , 21 ]. The Poissonian assumption therefore seems a reasonable approximation for the elective admission process (see also [ 31 ]).

Length of stay

The model assumes that the lengths of stay (abbreviated as LOS) are independent and identically distributed for each patient type. This seems an appropriate assumption as long as the patient mix and medical practice do not change. In practice, deviations from this assumption can occur, as the LOS may be affected by the level of congestion and delays in the care chain. In some cases we further assume, for mathematical convenience, exponentially distributed LOS. This often slightly underestimates the amount of variability present, but the impact on the results is typically very small (see Section  4.1 ).

The capacity of a unit is based on the number of operational beds. The number of operational beds is important for the distribution of budgets and is generally constant and evaluated on a yearly basis. The actual number of staffed beds may fluctuate slightly, but this rather is at an operational or tactical level.

Bed blocking

The model assumes that patients are blocked and lost from the system in case all appropriate beds are occupied. For urgent patients this means ambulance diversions and reallocation of patients at the Accident & Emergency department (A&E). For elective patients, unavailability of beds implies canceled admissions or surgeries. Such patients are often rescheduled, but this may affect the admissions of patients from the waiting list. As a rough approximation, we consider the rescheduled patients as new admissions.In Dutch hospitals, the waiting time at A&E departments for inpatient beds is usually short, whereas the fraction of transfers to other hospitals due to unavailability of beds is significant (estimated at about 10 %). In addition to our experience with Dutch hospitals, where excessive waiting for beds is uncommon, we chose to incorporate blocking. In the literature, delay models for bed capacity have also been proposed [ 7 , 8 ]. Note that for the classical models, there is a direct relation between the probability of waiting (delay model) and the blocking probability (loss model). The delay models typically do not take flexible bed allocations into account. We refer to [ 19 ] where routing policies from emergency departments to internal wards are addressed in an asymptotic queueing framework.

Model and notation

We consider the allocation of beds for J types of patients. A patient type typically refers to a medical discipline or to a specific diagnosis group. Patients of type j are assumed to arrive according to a Poisson process with rate λ j , j = 1,…, J . Denote the overall arrival rate by λ = ∑ j λ j . Let the LOS of type j be denoted by S j with mean 𝔼 S j , j = 1,…, J . The traffic intensities are then ρ j : =  λ j 𝔼 S j . In case the LOS of type j is exponentially distributed, we let μ j denote the corresponding rate.

The total number of beds available is N . There is no waiting room for patients. This means that when a patient arrives and all beds are occupied, the arriving patient is refused, see Section  2.1 . However, patients can also be refused in other situations. For instance, when each ward has its own number of beds (say N j , with ∑ j N j =  N ), patients are also refused when the preferred ward is fully occupied.

The Lagrange relaxation of this problem is

which is again a linear combination of b j ’s; take c j + γ j , j = 1,…, J , as coefficients in Eq.  1 .Our main performance measure is the loss fraction, reflecting the quality of the care process. Due to PASTA 1 , the loss fraction is equivalent to the fraction of time during which no bed is available for a certain patient type (bed blocking).

Another important performance measure focusing on efficiency is the occupancy rate. In case of only dedicated beds, by Little’s law, the occupancy (in %) for type j is given by

Since the number of shared beds can differ for different patient types, it is not always clear how the occupancy should be determined (i.e., what the appropriate value for the denominator of Eq.  2 is). However, as the arrival process is assumed to be exogenous, a decrease in the loss fraction directly implies an increase in the average number of occupied beds of the particular type (numerator of Eq.  2 ). For conciseness and ease of presentation, we only give the loss fraction throughout the paper.

Bed allocations and analysis

In this section, we describe the bed allocation policies (Section  3.1 ) and consider their performance analysis (Section  3.2 ). Denote the number of type j patients present at an arbitrary arrival epoch by x j , j = 1,…, J , with x = ( x 1 ,…, x J ) the corresponding vector.

Bed allocations

The bed allocation strategies differ by the rule used for accepting newly arriving patients.

Separate wards

Simple merging, earmarking beds.

where ( x ) + = max( x ,0). Here, ( x i − M i ) + represents the number of beds of the joint ward occupied by patients of type i .

The earmarking policy may be considered as an intermediate option between separate wards and simple merging. In case ∑ j M j =  N the policy of earmarking reduces to the situation of J separate wards, whereas in case M j ≡0 this bed allocation policy corresponds to simple merging.

Threshold policies

There can be a hierarchy in the admission of patients. To reserve a number of beds for patients with high priority we employ a threshold policy. For type j there is a threshold value T j that represents a maximum on the number of occupied beds for which patients of type j are admitted. More specifically, an arriving patient of type j is admitted in case ∑ i x i  <  T j . Note that the patients of highest priority have T j = N . The threshold policy with thresholds T 1 ,…, T J is denoted by ( T 1 ,…, T J ).

Optimal policy

The main aim of the optimal policy is to compare the performance of the other proposed policies to best achievable values in case of fully dynamic admission control. Hence, it provides a benchmark for what is ideally possible and allows to evaluate the relative performance of the corresponding policy. Specifically, the optimal admission policy minimizes the objective function b ( c ). This implies that upon arrival of each type of patient, given the number of patients of each type present x , it is decided whether the patient is admitted or refused. Such a policy might be difficult to implement in a hospital, unless bed occupancy is digitally registered in real time.

Performance analysis

Roughly speaking, the performance models can be classified in three categories, as addressed below. Some structural properties are discussed in Section  3.3 .

Separate wards and simple merging

For the cases of separate wards or simple merging, the performance can be immediately obtained using the Erlang loss model. The blocking probability or loss fraction for separate ward i reads

The total traffic load for the J type of patients equals ρ = ∑ j ρ j . Using the Erlang loss formula again yields b i = B ( ρ , N ) for all i ∈{1,…, J } in case of simple merging.

Let π ( x ) denote the stationary distribution of x ( t ), which has the following product form:

To obtain the fraction of refused admissions, define the sets 𝒮 j = { x  ∈ 𝒮: x j =  M j +  M joint  − ∑ i ≠ j ( x i − M i ) + } for j = 1,…, J . Using PASTA,we have b j = ∑ x ∈𝒮 j π ( x ) .

Finally, we note that the product-form result is insensitive to the LOS distribution, see Bonald [ 2 ] and references therein. Hence, we only require the average length of stay to determine the performance of the earmarking policy without assuming exponential LOS.

Optimal and threshold policies

Contrary to the allocation policies of separate wards, simple merging and earmarking, there are no closed-form results for the performance of the optimal and threshold policies. For the optimal policy, we use dynamic programming to find the policy that minimizes the relative costs b ( c ). A similar iterative procedure, based on dynamic programming, can be used to determine the performance of threshold policies. For these policies, we require that the LOS follows an exponential distribution.

The dynamic programming value function V n at the n th epoch can then be determined by

where we use the convention that V n ( x ) = ∞ for x  ∉ 𝒮 . Here, the first term represents an arrival, the second a departure, and the third term is due to uniformization. We note that at an arrival there is a decision to make. Either the patient of type j is accepted and the system moves to state x + e j , or the patient is refused and the systems stays in state x and incurs costs α j / λ . The minimal long-run average costs and the optimal policy can be found using value iteration.

We determine the value function and long-run average costs using value iteration again.

Structural properties

In this part, we discuss a number of structural properties of the bed allocation policies.

• (i) The optimal and threshold policies coincide in case the ALOS of the different patient types are identical (and the LOS is exponentially distributed), as can also be observed from the first example in Section  4.1 . This is easy to explain by noting that the bed occupancy can then be modeled as a one-dimensional birth-and-death process. Since the ALOS are identical, the decision to accept or refuse an arriving patient now only depends on the available number of beds, and is independent of the type of patients present. This results can already be found in Lippman [ 17 ].
• (ii) In the setting of call centers, Gurvich et al. [ 10 ] and Koçağa and Ward [ 16 ] have considered (partly) comparable control problems for Erlang C and Erlang A models, respectively. The authors show that threshold policies are asymptotically optimal, i.e. the limiting control scheme is of a threshold type for a sequence of systems with increasing arrival rates. Although the models are slightly different and the analysis involves an asymptotic framework, this supports the idea that threshold policies should work well in many practical situations.
• (iii) In some cases, the patient groups can be indexed according to a priority list based on the values of α j μ j . In case of two patient classes and μ 1 ≥ μ 2 and α 1 ≥ α 2 (and thus α 1 μ 1 ≥ α 2 μ 2 ) it holds that if it is optimal to accept patient type 2 in some state, then it is also optimal to accept patient type 1, see Altman et al. [ 1 ]. A formal proof of a stronger result seems rather involved (see also [ 1 ]), and the structure of the optimal policy may differ, see [ 22 , Example 3].
• (iv) The priority list discussed above can be directly used to determine parameter values for threshold and earmarking policies. Again, without loss of generality, let α 1 μ 1 ≥⋯≥ α J μ J . For threshold policies, it can then be argued that N = T 1 ≥⋯≥ T J , also see [ 1 ]. For earmarking, we can directly conclude that M J = 0, as class J needs no protection from other classes.

Results on bed allocations

For determining suitable bed allocation policies, we need to consider two different cases that are related by the size of the system under consideration. Small scale systems tend not to suffer that severely from multi-skilled staffing issues, and are treated in Section  4.2 . Multi-skilled staffing is only partly possible in large scale systems, thereby limiting the type of control. Large scale systems and the distribution of beds over the different units is discussed in Section  4.3 . To clarify drawbacks related to full flexibility (holding for both small and large scale systems), we start with two instructive examples in Section  4.1 .

Instructive examples

We consider two cases in which differentiating between patient types might be desirable.

Example I: Specialized care

Consider two types of patients in which one type is of specific interest, e.g., it receives specialized care. Assume that the ALOS of both patient types is 4 days, i.e., μ 1 = μ 2 = 0.25, which roughly equals the ALOS at an Intensive Care (see also [ 4 ]). Let λ 1 = 5 and λ 2 = 2, yielding ρ 1 = 20 and ρ 2 = 8.

Now, assume that N 1 = 20 and N 2 = 12, such that N = 32. In that case, the loss fraction for type 1 and 2 patients are 15.9 % and 5.1 %, respectively, with a weighted average loss fraction of b tot = 12.8 % . The difference in loss fractions may be a deliberate choice due to, for instance, the specialized care of type 2. Motivated by economies of scale, the bed allocation policy may be changed into simple merging. In that case, the loss fraction for both type of patients becomes 6.6 %. Hence, the average performance improves, but type 2 (specialized care) is negatively affected.

It is possible to prioritize type 2 patients using one of the three alternative bed allocation policies. The relative importance of type 2 is then quantified by c j (or α j ), j = 1,2. In general, it is not directly clear how to value this relative importance, unless the weights are identical. Using different weight combinations, the hospital manager obtains valuable insights to make this trade-off.

The case in which both patient types are equally important ( α 1 = α 2 ) is trivial, since the ALOS of both types are also identical and the optimal policy is then simple merging. Consider now the situation that the hospital manager decides that the loss fraction of type 2 should be well below 5.1 % such that type 2 also benefits from the reallocation of beds. In case the value of type 2 patients is twice the value of type 1, i.e. α 2 = 2 α 1 , the optimal values for the three policies can be found in the first part of Table  2 , where the parameters of the earmarking and threshold policy are chosen such that b ( c ) is minimized under the corresponding policy. An earmarking policy ( M 1 , M 2 ) denotes that M i beds are dedicated to class i , i = 1,2, whereas the remaining beds, N − M 1 − M 2 are fully flexible. Note that b 2 is still above 5.1 % for the optimal earmarking policy. To further decrease the value of b 2 at least 8 or 9 beds should be earmarked for type 2, see the second part of Table  2 . The optimal and threshold policies thus outperform earmarking.

Loss fractions in % for case I; first part corresponds to optimal values in case α 2 = 2 α 1 , the second part to some earmarking policies

Consider the case that (0,9) is a preferable earmarking allocation (e.g., it is optimal for α 2 = 4 α 1 ). In that case, only 9 beds require single-skilled staff and 23 beds require multi-skilled staff. The limited number of multi-skilled staff (one of the main advantages) can then be exploited by choosing a much larger value for M 1 with only a minor loss in performance. For instance, in the second part of Table  2 can be found that the difference in performance of (0, 8) and (16, 8) and (0, 9) and (16, 9) is negligible.

Alternatively, the possible optimal combinations of loss fractions for the three policies can be depicted by the efficiency frontier, see Fig.  1 . The values on this line give combinations of b 1 and b 2 that are optimal for the considered policy class. From Fig.  1 , it follows that the threshold and optimal policy coincide (see also Section  3.3 ) and that they (slightly) outperform earmarking especially for highly unbalanced loss fractions. In turn, earmarking outperforms separate wards, in particular for non-extreme blocking probabilities. Note that given the practical disadvantages of the threshold and optimal policy, it may be preferable to apply earmarking in some practical scenario’s.

Efficiency frontier in case of specialized care at one ward

Example II: Patient types with different ALOS

Consider two types of patients with a large difference in ALOS. As an illustration, assume that the ALOS of type 2 patients is 10 times as large as the ALOS of type 1 patients; we take μ 1 = 1 and μ 2 = 0.1. Let λ 1 = 20 and λ 2 = 2 such that the traffic loads are identical, i.e., ρ 1 = ρ 2 = 20.

The current bed allocation is often determined based on historically acquired privileges. For instance, assume that N 1 = 27 and N 2 = 17, such that N = 44. In that case, the loss fraction for type 1 and 2 patients are 2.7 % and 25.6 %, respectively. This yields an average loss fraction of b tot = 4.8 % . Motivated by economies of scale, the bed allocation policy may be changed into simple merging. However, using the Erlang loss model, the loss fraction then turns out to increase to 6.5 %. Similar results in a different setting can be found in [ 25 ], indicating that simple merging does not necessarily work well in case of patient groups with a large difference in ALOS.

Since the load is identical for both types of patients it could be suggested to equally divide the number of beds over the two wards, that is N 1 = N 2 = 22. In that case, the loss fraction is 10.7 % for both patient types, which is much higher than the average of 4.8 % in case of allocation policy (27, 17). We note that the optimal bed allocation for separate wards in terms of minimal weighted average loss fraction is (30, 14) yielding an average loss fraction of 4.1 %.

For the moment, let us assume that both patient types are of equal importance, i.e., α 1 = α 2 . The loss fractions (in %) for the different bed allocation policies can be found in Table  3 . Note that the loss fraction of type 2 is well above 25 % for all optimal policies (except simple merging). This evidently follows from the large ALOS of type 2 and the fact that type 1 and 2 are of equal relative importance.

Loss fractions in % for various policies in case II

The impact of changing the relative importance (i.e. c j , j = 1,2) can be seen using the efficiency frontier, see Fig.  2 . In this example, both threshold policies and earmarking perform nearly as well as the optimal policy. It can also be observed that both type 1 and 2 may benefit from a different bed allocation compared to separate wards if the blocking probability of type 2 is not too large.

Efficiency frontier for two wards with different ALOS

LOS distribution

For the analysis of the threshold and optimal policy, we assumed that the LOS is exponentially distributed. In practice we sometimes observe that the lognormal distribution gives a better fit for the length of stay. To investigate the sensitivity of our approach to the lognormal LOS distribution we have run several simulation experiments. The average blocking probability for lognormally distributed LOS is obtained using 100M events divided among 25 sub runs so that a confidence interval for the average blocking probability can be obtained using the student’s t distribution. The confidence was found to be such that the obtained blocking probabilities are accurate up to two decimal places.

The parameters of the lognormal distribution, denoted by μ and σ 2 , are chosen such that the ALOS remains the same, whereas we varied the coefficient of variation. Specifically, the expectation and the variance of a lognormal random variable X are

Loss fractions of types 1 and 2 in % for the threshold and optimal policy for lognormal LOS

We conclude from the simulation experiments that there is no significant difference in results between exponential and lognormal LOS for Example I. This can be explained by the structure of the threshold and optimal policy. Both admit arriving patients if beds are available, except for a type 1 patient if there is only 1 bed available, and is therefore similar to an Erlang loss model. Example II shows some difference between the exponential LOS and lognormal LOS. However, the difference is very small and only becomes apparent to some extent for very small or relatively large values of σ 2 .

Small scale systems: bed allocation

At the level of a single unit, the patient population is often diverse. This diversity may be related to medical diagnosis, clinical pathway, urgency, or medical discipline for combined units (such as at an IC that is used by different disciplines). Since diseconomies of scale are large for small unit sizes, organizing dedicated beds for small patient groups should be avoided. Moreover, the medical staff in general can treat all patient types visiting the unit so disadvantages related to multi-skill workers are of minor concern. In terms of our bed allocation policies, a unit usually acts in practice as ‘simple merging’.

From the examples in Section  4.1 , it follows that such a policy may not always deal well with different patient types in terms of prioritization and ALOS. To determine effective allocation policies, we now study the performance of bed allocations for a set of different problem instances. Since the optimal policy is hard to implement in practice, we compare it with the performance of simple merging, earmarking and the threshold policy. To this end, we generate 50 problem instances at random with the following specifications

• 2 types of patient classes;
• the number of beds N i is uniformly distributed on [6,36];
• the average length of stay β i is uniformly distributed on [1,14];
• the importance of a patient class α i is uniformly distributed on [1,10]; thus, α i / α j is the relative importance of class i compared to class j ;
• the arrival rate of patients λ i is such that the relative offered load (i.e., λ i β i / N i ) is uniformly distributed on [0.5,1.3].

The performance is measured by comparing the costs of each policy to the optimal policy. Denote by c ∗ the optimal costs, and let c ( s ) , c ( e ) and c ( t ) denote the costs associated with simple merging, earmarking, and the threshold policy, respectively. The performance is then calculated as ( c (⋅) − c ∗ )/ c ∗ . Figure  3 shows boxplots of the 50 problem instances for the three policies. The boxes in the plots are bounded by the 25th and 75th percentiles, while the central mark is the median. The whiskers are the lower and upper adjacent values, respectively, that are within 1.5 times the interquartile range.

Relative difference in average costs for simple merging ( a ), earmarking policy ( b ) and threshold policy ( c ) compared to the average costs for the optimal policy

The results show that the threshold policy performs almost optimally. The maximum relative difference for the threshold policy is below 3.5 % compared to the optimal policy, and the average relative difference for all 50 problem instances is approximately 0.3 %. Simple merging is the worst among all studied policies with an average relative difference that equals 27 %. A huge benefit is obtained when we switch from simple merging to earmarking, with an average of the relative difference that is approximately 9 %. We note that the difference between simple merging and earmarking becomes larger when the load is larger. For instance, in case we take λ i such that the relative offered load is uniformly distributed on [0.8, 1.3] the average relative differences are 0.4 %, 9 %, and 49 % for threshold policies, earmarking, and simple merging, respectively.

It is hard to say something about the situations in which a certain kind of policy performs well. From our numerical results, we have seen that the simple merging policy deviates more from the optimal policy as the difference between ρ 1 and ρ 2 increases. The same holds for the threshold policy. For the earmarking policy it turns out that the higher the difference between α 1 and α 2 , the bigger the difference compared with the optimal policy.

Conclusion for practice

Threshold policies turn out to be effective for distinguishing between patient types. Moreover, the rules for admitting patients is relatively simple as it is based only on the number of available beds present at the unit. We therefore advocate to use policies of the threshold type. Our experience in practice is that doctors find it hard to reject patients when beds are still available. An exception might be the distinction between urgency classes, which is supported by medical staff.

Large scale systems: bed distribution

The distribution of beds among different medical disciplines usually involves tens or hundreds of beds. The scenario of simple merging will then be infeasible in practice, as this would require all medical staff to be trained to treat all patient types. The threshold and optimal policy suffer from the same multi-skill problem. So, on a large scale separate beds for each patient class or earmarking allocations are the only feasible alternatives.

For the earmarking policy, the shared or flexible beds may provide sufficient flexibility to utilize scale effects to a large extent. The lower part of Table  2 already suggested that some flexibility is sufficient for an efficient bed usage. As an illustrative example, consider the case of five symmetrical wards, each having a load of 20. Note that for the performance of any earmarking policy, only the load is required and not the specific arrival rate and ALOS. The total number of beds available for the five wards is 115. In Fig.  4 the blocking probability is displayed against the number of flexible beds (on the horizontal axis). If all beds are dedicated, then each ward gets 23 beds and the blocking probability is 8.49 %. When each ward allows only one bed to be flexible, resulting in 5 flexible beds overall, the blocking probability decreases to 4.89 %. Full flexibility, i.e. letting all 115 beds be flexible, results in 1.36 % blocking probability. As can be seen from Fig.  4 , blocking probabilities below 2 % are already attained with 20 flexible beds. This illustration shows that a little flexibility is often sufficient to benefit from economies of scale. In particular, the blocking probability decreases as the number of flexible beds increases, but this happens in a convex way.

Blocking probability as a function of the number of flexible beds

The illustration above indicates how many beds should be flexible. Another prime practical question is how to distribute beds across all units. This often is the most relevant issue as the total number of beds in the hospital is fixed, or changes in bed allocation should be such that the total number of beds remain fixed. Such questions can be explored by trying all combinations of bed allocations, but this number increases exponentially fast with the number of units in the hospital. Below, we identify guidelines for how many beds should be allocated to each unit. This allocation generally is a good starting point, but it may be tuned as there often are issues that are specific to local conditions. Examples of such conditions are construction of the building, nurse-to-patient ratios making it effective to be the number of beds being a multiple of some integer, historically obtained rights, policy considerations, etc.

The principle we propose for the distribution of beds is based on square-root staffing. Recalling that ρ i is the offered load for unit i , the capacity should roughly be

Bed allocation without flexible beds

In the current situation, hospitals generally organize the clinic using separate wards. Admission of patients at other wards do occur, but this is commonly not organized on a structural basis. In Dutch hospitals it is common that admissions at non-preferred wards happen after rather exhaustive personal communications between medical supervisors of different medical units. As such, distribution of beds without organized flexibility is a prominent practical issue at the moment.

Using staffing rule ( 3 ), the blocking probability for unit i may be approximated by [ 14 ]

where ϕ (⋅) and Φ(⋅) are the density and cumulative distribution function of the standard normal distribution. The beds should now be allocated such that the sum of the capacities is N (see Eq.  4 ) and that the blocking probabilities satisfy the relative priorities (see Eq.  5 ). This yields the following system of J non-linear equations with as many unknowns β i , i = 1,…, J :

In extreme cases, this system of equations may be infeasible, for instance when blocking probabilities above 1 are required to satisfy relative patient values. In that case, it is recommended to carefully consider the specifications as such situations reflect unusual behavior in hospital operations. Otherwise, we opt to minimize the squared difference between the lhs and rhs of ( 5 ) constrained by ( 4 ).

As an illustration, we apply the concept above to a specific example. Consider 5 units representing, for instance, the different surgical disciplines. Let the load ρ i and relative value α i for each discipline be as given in Table  5 . Hence, unit 5 is large, whereas units 2 and 4 have some preference over the other units. We note that blocking probabilities b i are calculated using the continuous extension of the Erlang loss model, such that non-integral values of s i can be taken into account.

Bed allocation without flexibility

From Table  5 can be observed that the solution to the system of Eqs.  4 and  5 provides satisfying results and yields a good starting point to determine the final allocation. For the latter, we need at least rounding of s i .

Bed allocation with flexible beds

We assume that the number of flexible beds M joint is given, and is not part of the allocation (otherwise, it could be beneficial to make almost all beds flexible as we did not consider costs for multi-skilled staff explicitly). This seems reasonable, as the decision on M joint is typically influenced by many factors that are difficult to quantify. We note that the example above (Figure  4 ) provides a good intuition for appropriate choices of M joint .

Since there are now closed-form approximations for the blocking probability, we propose to use the following approximation scheme. Suppose that the flexible capacity is infinite. The number of type i patients in the system then has a Poisson distribution with mean ρ i , which is approximately normally distributed for ρ i not too small. The probability that an arriving patient needs a flexible bed is then ℙ ( X i  ≥  s i ) ≈ 1 − Φ( β i ) , with X i the number of type i patients at an arbitrary arrival instant. The fraction of time that type i needs flexible beds should respect the relative value α i between the different patient types. This is not precisely the same as the ratio between blocking probabilities, but the relative difference is typically small (unless the blocking probabilities are large).

The reasoning above leads to another set of J non-linear equations with as many unknowns β i , i = 1,…, J (see above in case this system of equations is infeasible). Again, the beds should be allocated such that the sum of the capacities is N (see Eq.  6 ) and that the fraction of time flexible beds are needed satisfy the relative priorities (see Eq.  7 ):

Consider the example above from Table  5 , but now assume that it has been decided that 15 beds are flexible. The allocation of beds and the corresponding blocking probabilities b i can be found in Table  6 . Note that the blocking probabilities have decreased significantly compared to the situation without flexible beds. Units 1, 3, and 5 now have slightly less beds than their offered load.

Bed allocation with 15 flexible beds

Having some flexibility in bed usage is generally sufficient to cope with peaks in demand. As such, earmarking allocations are effective. Moreover, appropriate bed allocations can easily be supported with quantitative models. Our experience is that having fully flexible beds that are shared by all disciplines in hospitals are scarce (except for ICs or acute admission units). The same concept can also be carried out on a slightly smaller scale: related medical disciplines can partly share their beds according to an earmarking allocation. When the scale is large enough, such a cooperation is expected to perform well.

Conclusion and discussion

In this paper we considered different practical alternatives to full flexibility of clinical beds or simple merging. The benefit of full flexibility can be easily explained by the economies of scale. However, full flexibility can be difficult to manage and may suffer from limited options of specialization in addition to issues in training many multi-skilled medical teams.

Our first contribution is that we propose structural and practically achievable bed allocation policies that perform well. For small scale systems, e.g., different patient groups at a medical unit, the benefits of a larger scale outweighs the drawbacks. To accommodate priorities of patient types and differences in lengths of stay, a threshold type of control is effective. In our numerical experiments we have seen that the threshold policy is nearly optimal, and in special cases coincides with the optimal policy.

For large scale systems, e.g., different medical disciplines, full flexibility is usually not desirable. However, a little flexibility is generally sufficient to benefit from most of the scale advantages. This can be implemented using an earmarking policy. Only a few members of the medical team need to be multi-skilled for little flexibility and yet the advantages are significant. In addition, we have addressed a prominent practical question of ‘how to distribute a fixed number of beds over different units?’. Using a square-root staffing principle, this can be efficiently determined by solving a set of equations.

The second contribution is that we provide models to support strategic and tactical decision making about the number of hospital beds. The performance analysis for earmarking is being implemented in a decision support system, exploiting the product-form solution, to facilitate hospital management in well-founded decisions about bed management.

From a practical point of view, we envisage that implementation of fair and flexible allocations is involved due to historically obtained privileges. Moreover, some specific characteristics of patient flows may be further explored to improve the accuracy of the model. For instance, in some situations, a delay model could be more appropriate than a loss model. Nonetheless, with the current models we display some key organizational concepts that are valid in a broader setting.

From a scientific standpoint, it is of future interest to find optimal bed allocations when costs are involved for single-skilled and multi-skilled medical teams (although it is not straightforward to quantify this in practice). Asymptotic regimes may give further theoretical support for the different bed allocations. Finally, extending the assumptions of the model could strengthen the conclusion.

1 PASTA is the acronym for Poisson Arrivals See Time Averages, stating that the distribution of the number of customer seen by an arriving customers equals the time average distribution, see [ 33 ].

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CANTON, Ohio, July 1, 2024 – US Acute Care Solutions (USACS), the nation’s largest physician-owned provider of hospital-based emergency and inpatient medicine, announced today that it has named Tanveer Gaibi, MD, MBA, FACEP, a Regional Vice President of its East Division. Most recently, Dr. Gaibi served as Chairman of Emergency Medicine at Inova Fairfax and Vice President of Strategic Development for the USACS East Division. With a diverse leadership background spanning rural, urban, and tertiary care academic centers, Dr. Gaibi has been a key contributor in USACS' business development endeavors across the East Division. His extensive clinical experience and proven operational success have been instrumental in shaping business strategies and driving organic growth initiatives. President of the USACS East Division, Martin Brown, MD, FACEP, said, “Dr. Gaibi has been a tremendous asset for both our division and our practice. He continually embraces an enterprise-wide perspective and utilizes his certified coaching skills to invest in the development of his colleagues. We are thrilled to have him join our leadership team.” Dr. Gaibi said, “I am eager to serve our practice in this capacity and make an impact. Our talented divisional leadership team has generated an outstanding amount of momentum that I look forward to perpetuating. I am grateful for their confidence in my leadership abilities to fill this critical role.” Dr. Gaibi is a board-certified emergency physician who completed a seven-year BS/MD program at SUNY Upstate Medical University. He went on to complete his emergency medicine residency at the University of Maryland. Later, he earned his MBA from the University of Virginia Darden School of Business. About USACS Founded by emergency medicine and inpatient physicians across the country, USACS is solely owned by its physicians and hospital system partners. The group is a national leader in integrated acute care, including emergency medicine, hospitalist, and critical care services. USACS provides high-quality care to approximately ten million patients annually across more than 400 programs and is aligned with many of the leading health systems in the country. Visit usacs.com for more. ### Media Contact Marty Richmond Corporate Communications Department US Acute Care Solutions 330.493.4443 x1406 [email protected]

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Four Rules for Bed Assignment In An Efficient Hospital

Editor’s Note: the following is an excerpt from Dr. Robbin Dick’s forthcoming book on Hospital Capacity Management. Dr. Dick is MEP’s Director of Observation Services. He will be speaking on hospital capacity management and other subjects at MEP’s third annual observation medicine conference,  Observation Care ’15 .

Bed assignment often sets the pulse for the entire hospital, affecting every patient and every department from minute to minute, yet is often poorly managed. 

I have been amazed over the years at how many people desire the position of bed assigner. Nurses, Surgeons, Private internists, Hospitalists, ED providers and even Hospital Administrators at some point in time want to assign beds to patients. They often have no idea, however, how their decisions will affect ED wait times, operating room and cath lab needs, and the flow of non-ED patients throughout the hospital, to name just a few factors that need to be considered. Yet bed assignment has far reaching effects.

Many think that bed assignment is a simple task. In theory it should be. A bed is available and the patient gets assigned. There are certain constraints—sex, semi-private versus private, isolation issues, acuity, telemetry and specialty needs. All need to be taken into account to ensure that each patient goes to the right place and receives the proper care. But good capacity management demands that bed assignment be carefully considered and executed.

Centralize bed assignment authority. All beds need to be assigned by a centralized authority – and no one else. Patients cannot be moved without a reason. Patients cannot be assigned by others. All discharges must quickly and accurately be handed over to environmental services for bed cleaning then immediately handed back to bed assignment.

I can recall an analysis of environmental services staffing on the evening shift. Beds vacated by discharged patients at 5 p.m. weren’t being cleaned until 1 or 2 a.m.  All the data said that there was sufficient staffing. Drilling down we discovered that nursing staff had decided to transfer patients from bed to bed, for multiple reasons, without contacting bed assignment. This was being done so frequently that it consumed 1/3 of the environmental staff that was allocated to clean beds for discharged patients. Thus the need for rule 1.

Use Non-clinical staff to do bed assignment. Set the rules and parameters on when and where patients can go and let them do their job. Nurses tend to look through clinical content to assign just the right bed. This wastes time and energy while producing no better results than non-clinical staff. The main reason clinical staff are involved at all is that there is an idea that bed assignment provides a report to the accepting unit. This is not only inappropriate but dangerous since the true clinical status of a patient can rarely be gleaned from reviewing the chart not to mention that care handoffs need to be done by the current care provider (where they are) and the receiving care provider. Bed assignment simply provides the proper location based on specific patient attributes like sex, isolation, telemetry, acuity and specialty needs.

Provide a mechanism of escalation. When problems arise the bed assignment staff need help. Assistance with inadequate telemetry availability, limits on private room use, prevention of patient movement within the hospital all can have a significant impact on bed assignment and timely patient placement.

Provide priorities and establish mechanisms to maintain them. Priorities for bed assignment need to be linked to the needs of the sickest patients in the organization wherever they may be. Those patients requiring ICU level of care whether in the ED, PACU, Floor or another facility needing to transfer the patient would have the highest priority. It has been demonstrated that patients requiring this level of care have a lower mortality rate and suffer the fewest complications the quicker they are placed in the intensive care unit.

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• Bed Management
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Access Bed Control

Support for efficient patient placement & bed management  , save time and increase efficiency with automated bed management.

Effective hospital bed management requires real-time patient and bed visibility, as well as coordination among staff groups. If you’re running these processes manually — as many medium sized hospitals do — you could be losing dozens of available bed days per year. Access Bed Control (ABC) provides a single, up-to-date portal for hospital staff to manage all bed needs.

On average, a 150-bed hospital can add 38 available bed days per year by using Access Bed Control. What would that mean for your revenue and patient satisfaction?  

Patient Placement

Access Bed Control eliminates the need to make calls between several different departments. It provides all information the patient access staff and admitting department need to make the correct placement decisions. With the ABC patient placement module, you can:

1. Identify Patient Characteristics and Needs

Patient clinical characteristics and bed needs are collected from your hospital information system or entered manually. These can include things such as diagnosis, medical service, level of care, isolation, telemetry, fall precaution, near nursing station, observation, bariatric, and behavioral issues. This system works for all admits, pre-admits, and transfers.

2. Match a Patient with Ideal Bed

Patient access staff can focus on beds throughout the entire hospital or a specific unit that matches the incoming patient’s needs. The interface provides information about which beds are clean, room types available (private, semi-private), bed status (clean, dirty, in process, occupied), and other details.

3. Manage Bed Cleaning

If the target bed is dirty, the system will automatically assign the cleaning task and give an estimate of when the bed will be ready. It will also notify you upon completion of the assignment.

4. Enter Patient Transporation Requests

When a bed is selected and the patient is ready, the admitting staff member can enter a transport request. The system checks to avoid conflicts in transport, dispatches the most appropriate transporter, and updates the progress of the transport from pickup to completion.

This module integrates with our patient transportation system and discharge bed cleaning module to address all aspects of patient flow.  

Discharge Cleaning

ABC collects and communicates patient discharges and transfers, then selects the appropriate employee to clean the bed and room for the next patient. The system also provides information about potentially infectious organisms for the safety of all involved in the cleaning process.

ABC follows the assignment through to the end, as it is programmed to monitor the bed cleaning process until it is available for the next patient. Once an assignment is completed, ABC stores and reports the results for future analysis by hospital directors and managers.

Contact us today to schedule a free demo of Access Bed Control, or view a complete benefit analysis.

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• Research article
• Open access
• Published: 07 January 2013

Decision support for hospital bed management using adaptable individual length of stay estimations and shared resources

• Robert Schmidt 1 ,
• Sandra Geisler 2 &
• Cord Spreckelsen 1  

BMC Medical Informatics and Decision Making volume  13 , Article number:  3 ( 2013 ) Cite this article

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Elective patient admission and assignment planning is an important task of the strategic and operational management of a hospital and early on became a central topic of clinical operations research. The management of hospital beds is an important subtask. Various approaches have been proposed, involving the computation of efficient assignments with regard to the patients’ condition, the necessity of the treatment, and the patients’ preferences. However, these approaches are mostly based on static, unadaptable estimates of the length of stay and, thus, do not take into account the uncertainty of the patient’s recovery. Furthermore, the effect of aggregated bed capacities have not been investigated in this context. Computer supported bed management, combining an adaptable length of stay estimation with the treatment of shared resources (aggregated bed capacities) has not yet been sufficiently investigated. The aim of our work is: 1) to define a cost function for patient admission taking into account adaptable length of stay estimations and aggregated resources, 2) to define a mathematical program formally modeling the assignment problem and an architecture for decision support, 3) to investigate four algorithmic methodologies addressing the assignment problem and one base-line approach, and 4) to evaluate these methodologies w.r.t. cost outcome, performance, and dismissal ratio.

The expected free ward capacity is calculated based on individual length of stay estimates, introducing Bernoulli distributed random variables for the ward occupation states and approximating the probability densities. The assignment problem is represented as a binary integer program. Four strategies for solving the problem are applied and compared: an exact approach, using the mixed integer programming solver SCIP; and three heuristic strategies, namely the longest expected processing time, the shortest expected processing time, and random choice. A baseline approach serves to compare these optimization strategies with a simple model of the status quo. All the approaches are evaluated by a realistic discrete event simulation: the outcomes are the ratio of successful assignments and dismissals, the computation time, and the model’s cost factors.

A discrete event simulation of 226,000 cases shows a reduction of the dismissal rate compared to the baseline by more than 30 percentage points (from a mean dismissal ratio of 74.7% to 40.06% comparing the status quo with the optimization strategies). Each of the optimization strategies leads to an improved assignment. The exact approach has only a marginal advantage over the heuristic strategies in the model’s cost factors (≤3 % ). Moreover,this marginal advantage was only achieved at the price of a computational time fifty times that of the heuristic models (an average computing time of 141 s using the exact method, vs. 2.6 s for the heuristic strategy).

Conclusions

In terms of its performance and the quality of its solution, the heuristic strategy RAND is the preferred method for bed assignment in the case of shared resources. Future research is needed to investigate whether an equally marked improvement can be achieved in a large scale clinical application study, ideally one comprising all the departments involved in admission and assignment planning.

Peer Review reports

Bed capacity is a crucial but limited hospital resource. Therefore, professional bed management aims at an optimal allocation of beds, one involving short waiting periods for the patients and a low rate of canceled admissions, yet with a high occupancy rate. Optimal allocation is hampered by the inherent uncertainty of the patients’ actual length of stay. Furthermore, the bed capacity of a ward tends to be increasingly shared by different clinical units. Bed management is often part of the more general effort at improving patient treatment and maintaining a constant throughput of patients. Currently, many hospitals have formed teams of case and bed managers dedicated to these tasks[ 1 , 2 ].

Bed management—the current situation in German hospitals

Since 2003, drastic organizational changes have taken place in German hospitals. These were triggered by the ending of payments to the hospital for individual treatments and instead a lump sum compensation based on the internationally established classification of Diagnosis Related Groups (DRG)[ 3 ] has been introduced. The current compensation scheme calculates the costs of treatment based on the average length of stay (LoS) of a patient according to the assigned DRG. Therefore, a marked economic loss for the hospital could be the result if the actual LoS exceeds the average. Furthermore, only the number of actually treated cases matters, solely providing the clinical infrastructure is no longer rewarded. Due to these changes, the traditional one-to-one link between a ward and a specific clinic was abandoned. Today, patients treated by different clinical units may be assigned to the same ward, provided that there are no medical reasons not to do so. The rate of elective patients varies between 30% and 80%, depending on the clinical unit[ 4 ]. Up to 80% of a surgical unit’s patients are elective, and their admission could thus be planned in advance. Patient admission planning and assignment in German hospitals are often performed by the Case Management and Standard Care departments. The patient admission planning decision process of case managers has been reported to be a complex process, heavily influenced by a multitude of factors[ 5 ].

In the course of this work, a comprehensive requirements analysis based on five semi-structured interviews with several representatives of the Standard Care (SC), Case Management (CM), and IT departments of the local hospital RWTH University Hospital Aachen, Germany (UK Aachen) has been performed. The CM department has been established at the hospital to improve all processes concerning the treatment, nursing, and after-treatment of the patients. Preliminary investigations revealed an incremental shift of the tasks of patient admission planning, from clinics to case managers. Thus, the CM department could be regarded as a centralized unit, responsible for admission planning. The SC department was established as an interdisciplinary department responsible for nursing the patients who are treated in different clinics. The SC department is currently facing manifold challenges concerning bed management, e.g., assigning patients so that the spatial distance to the treating clinic is minimized. The IT department has to provide all necessary data for resource planning and an IT infrastructure for decision support.

The interviewees of the SC and CM departments are primarily involved in bed and case management at UK Aachen. All interviews were prepared in advance, recorded, and analyzed afterwards. The interviews lasted approximately 30 to 90 minutes each. The interview protocols were transcribed immediately after the interviews. The interviews aimed at:

The elicitation of information about the departments, the responsibilities, and the daily challenges of the persons in charge concerning admission planning and assignment.

The identification of those aspects that must be respected in the development of the Decision Support System (DSS).

Results of the interviews

According to the interviews, bed management has gained increasing attention in Germany since the establishment of the new financial compensation model for hospitals (based on Diagnosis Related Groups). There is a great need for computer-based decision support in this context, due to the high complexity of the planning problem.

Bed management aims at finding a suitable admission date with respect to the preferences of the patient and the bed capacities available. The increasing responsibility of the hospital’s case managers for the patient admission planning tasks has been reported to have resulted in a more centralized planning process and in reducing the inherent organizational complexity of a decentralized planning scenario. Current approaches to bed management using multi-agent systems[ 6 – 8 ] involve decentralized planning processes and thus were not further considered. Futhermore, the interviewees reported an inherent uncertainty in the LoS and in the duration of the specific treatment steps, which results in uncertainty of the bed capacity available at a given time. Only a fraction of the admissions are planned in advance. Acute patients must be assigned immediately. The possible planning time frame of a patient depends on the individual case of the patient. It was suggested by the interviewees that patients may be categorized into priority classes, taking their individual treatment needs into account. The grouping of patients with respect to their treatment needs has recently been considered by Wang et al.[ 9 ] as well, who categorize patients into different priority classes with respect to their state of illness. The interviewees reported that the planning time frame can be between a day and several months. In general, patients are scheduled within two months. In practice, patients are categorized into priority groups with respect to their planning time frame. Three priority groups have been depicted by the interviewees: priority 1 patients must be scheduled within 24 hours, priority 2 patients within one week, and priority 3 patients may be scheduled fairly long-term. Restrictions exist even for admissions which can be planned in advance: e.g., treatments may have to be started before or after a given date, for medical or personal reasons. Additional restrictions involve the preferred, allowed, or excluded combinations of patients in the same room or a patient’s demand for a bed in a single room. Furthermore, an increasing relaxation of the former tight linkage between clinics and wards has been reported. Interdisciplinary units, such as the Standard Care department, were established with the main objective of nursing patients treated by different clinics. However, clinics may prefer special wards for their patients for medical or organizational reasons. Lists are used for structuring the planning task. The overall planning collective on a list does not exceed 100 patients in general. Further aspects and statements from the interviews are summarized in Table 1 .

The new challenges concerning bed management in German hospitals can be summarized as follows:

Wards must be considered to be partially shared and central resources.

Patients treated by different clinical units may be assigned to the same ward.

There are special limitations due to medical, insurance, or social reasons, which restrict the sharing of the same room or even ward by different patients (e.g., the exclusion of mixed gender rooms).

The planning process has to cope with the inherent uncertainty of the outcome of the single treatment steps and the overall duration of the patient’s stay.

Patients may be planned in advanced and categorized into priority groups reflecting the urgency of the treatment.

Related work

The challenges of bed management without computer assistance leading to special training programs for the persons in charge have been analyzed intensively by Proudlove et al.[ 10 , 11 ]. Bed management has to solve optimization problems in a context with a high level of uncertainty: the outcome of a treatment cannot be predicted fully, and emergency patients need to be treated immediately.

Case Management as described above has been reported to potentially improve the treatment and care trajectory of cancer patients and is increasingly being implemented in the organizational structure of hospitals[ 12 ]. The challenges for case managers are more complex than those for bed managers in terms of resource allocation planning, since case managers need to consider the entire clinical pathway. Due to interdependencies between resources, the decision making process concerning resource allocation was reported to be highly sophisticated[ 5 ]. Besides the complex interdependencies, the variability in the usage of resources plays an important role[ 13 ]. Gallivan et al. investigate the variability of the patients’ LoS in intensive care after cardiac surgery[ 13 ]. Their results indicate that this variability has a considerable impact on the intensive care capacity requirements. They conclude that a booking admission system should be treated with caution regarding inpatient admissions if there is a high variability in the LoS.

Mathematical approaches and computer-based assistance designed to solve the optimization tasks described above were early on proposed and developed. Queuing Theory and Compartmental Flow Models have been successfully applied in a clinical context: McClean reports on a Decision Support System (DSS) based on a compartmental flow model[ 14 ]. Fomundam and Herrmann give an overview of the application of queuing theory approaches in the health care sector[ 15 ]. A recent review of queuing models applied to clinical problems especially addresses patient planning approaches[ 16 ]. Approaches based on queuing theory have also been frequently combined with simulation models. For example, Cochran and Bharti describe a multistage methodology to balance inpatient bed utilization in a hospital[ 17 ]. The approach combines a queuing model and a discrete event simulation in order to achieve flow-maximization in the system. However, queuing theory approaches are unsuitable for providing decision support on the operational level in the presence of frequent revisions of LoS estimates, since the inherently highly variable dynamics resulting from the revisions would not be sufficiently taken into account. In the related field of operation room planning, Discrete Event Simulation has been applied to situations where queuing models cannot be used accurately[ 18 ].

Stochastic scheduling approaches have already been proposed for admission planning tasks as well. Connors describes a stochastic scheduling algorithm employing deterministic and stochastic constraints[ 19 ]. The patient’s characteristics and requirements as well as the hospital’s status are considered by the algorithm. The patient’s LoS is modeled stochastically, reflecting the probability of the LoS. A gamma density function is used to model the patient’s LoS. The aggregation of similar resources with predefined characteristics is not considered. The LoS estimates cannot be modified after the patient is admitted and are thus static.

Stochastic scheduling has been applied to the task of maximizing operating room utilization, which bears considerable similarity to the bed assignment problem: Arnaout developed and evaluated an approach[ 20 ] based on the Longest Expected Processing Time first (LEPT) strategy[ 21 ]. The LEPT strategy is a stochastic scheduling strategy that processes jobs in decreasing order with respect to the expected processing time. Similarly, the Shortest Expected Processing Time (SEPT) strategy processes jobs in increasing order. Both strategies are stochastic extensions of the deterministic strategies, Longest Processing Time (LPT) and Shortest Processing Time (SPT). Variants of the LEPT and LPT strategies have been applied in the context of surgery planning as well[ 22 – 24 ]. Hans et al. consider a robust surgery loading problem with the subject of surgery assignment to operating room day schedules[ 22 ]. A part of their algorithm contains a dispatching rule which is based on the LEPT strategy. The LPT and SPT strategies are considered by Lamiri et al. as a constructive heuristic in order to calculate a basic solution of the planning problem[ 23 ]. The LEPT and SEPT strategies are used to compute an approximate solution of the mathematical model within this work as well.

Mathematical Programming approaches have also been successfully applied to health care related problems: Zhang et al. used a Mixed Integer Program (MIP) to optimize the capacity allocation of operating rooms to specialties[ 25 ]. MIPs and Binary Integer Programs (BIPs) belong to the complexity class of NP-hard problems and are thus in general challenging[ 26 ]. Several approximation and heuristic strategies have been developed to compute a solution in reasonable time, which, however, cannot be guaranteed to be optimal[ 27 – 30 ]. Heuristic approaches differ from approximate ones in that they include an unforeseeable error in the approximate solution. Hans et al. describe optimization models for surgery planning: their approach tries to maximize the capacity utilization, minimize the risk of overtime, and minimize the number of canceled admissions[ 22 , 31 ]. That approach allows operating room capacity to be freed up for additional surgeries. Lamiri et al. propose several approaches to improve the scheduling of surgeries[ 23 , 32 , 33 ]. Surgery times as well as operating room capacities were modeled by random variables in order to represent their inherent stochastic variability. Belien et al. proposed and evaluated models for generating surgery schedules with leveled resulting bed occupancy[ 34 ]. However, the existing approaches in general focus on the rooms or clinical staff. Hospital beds and ward capacities are usually regarded as an auxiliary condition—if considered at all.

Chow et al. describe a combination of a mixed integer optimization model and a Monte Carlo simulation method in order to improve scheduling practices of operating room use and the resulting downstream bed utilization[ 35 ]. Surgical schedules are simulated with reference to historical case records and the resulting bed requirements are predicted accordingly. The mixed integer optimization model allows scheduling both surgeon blocks and patient types. The simulation allows predicting the impact of the scheduling rules.

Demeester describes a patient admission scheduling algorithm considering hospital beds and supporting operational decisions concerning hospital bed assignments[ 36 ]. That approach assigns patients to individual rooms and hospital beds for exact dates. The hospital beds and hospital rooms are considered a single, individual resource and it does not contemplate a suitable variety of hospital beds and rooms in a ward.

This paper addresses the feasibility of a computer-supported bed management taking into account aggregated bed capacities (shared resources) and LoS estimates that can be individually updated during the patients’ treatment. This study aims at answering the question of how to choose and combine modeling and optimization approaches in order to fulfill the following requirements (which were identified by the preliminary investigation):

The treatment priority of the patients must be the major concern and must, thus, be considered with rights to appeal.

The approach should address the bed capacity of a ward by aggregating beds into groups, instead of focusing on the assignment of patients to individual beds. In these groups, beds are equivalent with respect to the limitations mentioned above.

The planning process should be based on LoS estimates and representational means of uncertainty, namely appropriate probability distributions for the LoS adjusted to the variance of the empirical data.

It should be possible to dynamically adapt a plan to changing estimates.

The approach to be presented in this paper aims at improving the efficiency and effectiveness of the bed and case management process. This approach, fulfilling the above criteria, should enable a decision process that respects the patients’ treatment priorities and individual preferences (e.g., gender, private or double room, diagnosis), relies on ward capacities, deals with the inherent uncertainty of the patients’ recoveries, and allows a dynamic adjustment to changing LoS estimates. Due to the complexity of this optimization task, different types of algorithmic approaches (e.g., exact solution vs. heuristic strategies) needed to be evaluated with respect to their performance and the quality of their results.

As far as we know, none of the published approaches fulfills all these requirements.

Rationale of the approach

As found by the requirements analysis, the LoS is an important factor in estimating the capacity utilization. Therefore, an adequate decision support should rely on a system architecture which supports an interactive adjustment of the estimated LoS by the users. The adjustment of the LoS estimates yields a more precise estimate of the ward’s usage rate, since the usage rate can be calculated using the up to date LoS estimates. Our approach focuses on the assignment of patients to bed contingents respecting the given patient preferences, in contrast to an assignment of patients to individual beds as carried out, for instance, by Demeester et al.[ 36 ]. The LoS is treated as a stochastic variable, hence, optimization has to be based on expected values. Different optimization strategies will be compared, including the exact solution and three heuristic approaches.

Software architecture

Figure 1 shows the software architecture of the DSS developed. The ward staff in charge of a patient may submit a patient LoS estimate at any time. An initial LoS estimate may be ascertained using data from the clinical information system (gray). A special user interface, Figure 2 , supports the LoS estimate submission with an interactive diagram.

Architecture of the DSS. The architecture of the DSS consists of two interfaces: one for LoS submission and another one for query submission. The LoS estimate can be submitted at any time and is immediately considered in the decision support. An initial LoS estimate may be derived from the clinical information system (KIS). An admission planning query can be submitted at any time as well. The DSS calculates an optimal admission date and assignment for the patient group.

Length of stay submission interface. The expected LoS as well as an uncertainty factor may be entered. Visual feedback is provided by presenting the resulting cumulative distribution function (CDF) and highlighting the assignment probability in red.

The persons in charge of admission planning and assignments may submit queries to the system. Such queries specify a set of patients, whose admission dates and assignments have to be planned. The decision support module calculates an optimal admission date and assignment for each patient. The implemented prototype user interface is shown in Figure 3 .

Query interface for admission and assignment planning. Each Row contains a patient dataset. The columns labeled with green contain the data that has to be entered in advance. The red labeled columns contain the result set of the DSS. The dropdown menu allows choosing the optimization strategy (EXACT, RAND, LEPT, SEPT) to be used.

Optimization model

Length of stay estimation.

The LoS of patient i is modeled by a log-normally distributed random variable D i , representing the LoS estimate. The density P ( D i  =  t ) represents the probability that patient i is assigned to a ward at time t . The cumulative distribution P ( D i ≥ t ) = 1− P ( D i < t ) =  p i t represents the probability that patient i is assigned to a ward at least until time t . This event can be regarded as a Bernoulli trial with success probability p i t . In general, the distribution may be chosen arbitrarily. Nonetheless, Marazzi et al.[ 37 ] and Ruffieux et al.[ 38 ] showed that the log-normal distribution is superior to other closed-form distributions in the context of LoS modeling.

The persons in charge submit their estimates of the expected date of discharge and specify their uncertainty by defining a time interval for the patient’s discharge. The system interprets the submitted parameters as the expectation value μ and the variance σ of the log-normal distribution.

Expected ward capacity

The expected free capacity of a ward at a given time is calculated based on the LoS estimates of the patients assigned to the ward in question within the time frame being considered. The expected free capacity can be updated dynamically based on revised LoS estimates.

Let j be the ward in question with a total capacity of n j beds, and let m j be the number of patients assigned to the ward. In order to derive the probability distribution of the ward’s usage rate, a Bernoulli distributed random variable X i t is introduced. The event X i t  = 1 represents the event that patient i is still assigned to the ward in question at time t with the corresponding probability P ( X i t  = 1) =  p i t . The probability distribution of the use of ward j at time t can now be calculated with the random variable S jt = ∑ i = 1 m j X it . The densities P ( S j t  =  k ), k  = 0,…, p j may be approximated through a reduction to the normal distribution, according to the central limit theorem. However, the expected use E [ S j t ] of the ward j at time t can be calculated directly, since E [ S jt ] = E [ ∑ i = 1 m j X it ] = ∑ i = 1 m j E [ X it ] = ∑ i = 1 m j p it . The expected free capacity is, thus, n j − E [ S j t ]. The expected usage rate is E [ S jt ] n j = : c jt .

Affinity between clinics and wards

A patient cannot be assigned to an arbitrary ward in the hospital. In general, only a subset of the wards is suitable for a given patient. This subset depends on the clinic responsible for the treatment of the patient. Furthermore, usually there exists a ranking between suitable wards. The suitability (and ranking) of wards with respect to a clinic is modeled by a corresponding mapping (affinity), assigning a value α ∈ [0,1] to each pair of a ward and a clinic.

Cost factors

Aspects of an assignment benefit are represented by cost factors: affinity costs , ward occupancy , change of ward occupancy , and assignment delay . Affinity costs: The affinity α ∈ [0,1] introduced above represents a cost factor which has to be defined by the administrator of the DSS for each pair of ward and clinic in advance. A high affinity (close to 1) represents a high preference of the ward for nursing a patient treated in the corresponding clinic. If α =0, patients treated in the corresponding clinic cannot be assigned to the ward in question. This cost factor is referred to as the α cost factor. Ward occupancy: As solely providing a clinical infrastructure is no longer rewarded by the DRG-based compensation scheme, the use of the ward’s capacity is an important factor in the hospital’s economic status. Wards should have a high average rate of use. Therefore, the optimization approach needs to consider the estimated ward usage as a cost factor (which will be referred to as the β cost factor). The β cost factor is defined for each patient i , ward j , and admission date t : ∑ m = t E [ D i ] c jm .

Change of ward occupancy: Frequent changes of the ward during a clinical stay exposes the patients and the nursing staff to additional stress and should therefore be avoided. The corresponding measure is referred to as the γ cost factor, and is defined for patient i , ward j , and admission date t by ∑ m = t E [ D i ] | c j ( m + 1 ) − c jm | .

Assignment delay: The δ cost factor weights the delay until the patient’s assignment by their treatment priority in order to reduce the waiting time and to provide a timely start for urgent treatments.

Mathematical program

The formal description of the admission planning and assignment problem is given by a Binary Integer Program (BIP)[ 27 ].

b = 1 , … , x = | ℬ | index of patient preferences i = 1 , … , n = | P | index of patients j = 1 , … , m = | S | index of wards t = 1 , … , k = | T | index of allowed days of admissions U j t b random variable representing the number of used beds of ward j , fulfilling the patient preference b at date t V i random variable representing the LoS of patient i E ( U j t b ) expected number of used beds on ward j at date t , fulfilling the patient preference b E ( V i ) expected LoS of patient i K j b overall bed capacity of ward j , fulfilling the patient preference b c jtb = E ( U jtb ) K jb cost resulting from an assignment to ward j at date t by a patient with preferences b C i treating clinic of patient i m α affinity weight factor m β ward usage weight factor m γ ward usage change weight factor m δ admission delay weight factor

AFF : S × K → [ 0 , 1 ] , Mapping of the affinities between wards and clinics

Cons : P × ℬ → P , Mapping of patients who demand the preference b ∈ ℬ

Cons : S × ℬ → S , Mapping of wards satisfying the patient preference b ∈ ℬ

L max : P → T Mapping of the maximal LoS of a patient within the set P

Prio : P → { 0 , 1 , 2 , 3 } , Mapping of patients to their treatment priority

Decision variable

• x ijt = 1 , iff patient i is admitted to ward j at time t 0 , else

Objective function

Constraints.

• ∀ b ∈ ℬ : ( ∑ i ∈ Cons ( P , b ) ∑ j ∈ S ∑ t ∈ T x ijt = | Cons ( P , b ) | )

The above restriction represents the requirement that an admission date and a ward shall be derived for every patient. The set of patients is partitioned according to the patient preferences.

In order to ensure the solvability of the BIP, a dummy ward with high capacity has been modeled. Patient’s assignment to the dummy ward is accompanied by an enormous penalty cost and is interpreted as a dismissal.

The above restriction implements the requirement that no more than one admission date and one ward is calculated for a given patient, thus preventing multiple assignments.

The capacity of a ward shall never be exceeded. To prevent exceeding the capacity of the ward, all assignments of previous admission dates must be considered. The expression K j b  −  E ( U j t b ) represents the expected number of free beds that fulfill the patient’s preference b at date t of ward j . Exceeding the ward’s bed capacity can find its cause in the following scenarios:

Earlier patient assignments to ward j have led to the excess the ward’s capacity.

The current patient assignment exceeds the ward’s capacity.

To prevent both cases of incorrect assignment, a time-window is defined. The time-window is defined by the expression of the inner sum: ∑ t 1 = max ( 0 , t − L max ) t .

The variable L m a x represents the maximum LoS of the considered set of patients.

The BIP is tackled by the software tool SCIP[ 39 ] within the EXACT approach.

Optimization methods

The admission planning and assignment problem has been described as a BIP in the previous section.

Exact solution

The exact solution represents the mathematically exact and optimal solution of the BIP. Several software tools have been developed to provide methods to solve such problems.

The software tool SCIP a [ 39 ], currently the most powerful non-commercial software tool to solve BIPs b , is used in this project. The exact approach to solve the program is referred to as the EXACT approach. Notably, dismissals are modeled as assignments of patients to a residual category of beds (at a maximal cost) in order to guarantee the solvability of the assignment problem. Due to the complexity of the exact algorithm, a timeout has to be defined. If the algorithm fails to reach a solution in time, it is stopped without creating an assignment. Thus, a timeout results in dismissals.

Heuristic strategies

Several heuristic strategies have been proposed to compute solutions in reasonable time. Pinedo[ 21 ] provides an overview of the common heuristic approaches. In heuristic strategies, the assignment problem is usually simplified by a reduction to an online problem instance. An online algorithm tackles a given problem instance in a sequential piece-by-piece manner, whereas an offline algorithm approaches the problem at hand as a whole, considering the relevant interdependencies of its parts[ 40 ].

Heuristic strategies involve two parameters: the assignment order of the patients and a cost criterion. The cost criterion represents the individual assignment cost of the current patient, date, and ward.

The assignment order of the patients is determined by the following strategies: Longest Expected Processing Time (LEPT): The LEPT approach sorts the patients in descending order solely according to their expected LoS. Afterwards, it assigns them to the wards sequentially following a greedy minimal cost strategy. Shortest Expected Processing Time (SEPT): The SEPT approach is similar to the LEPT strategy. The only difference is that patients are sorted in ascending order with respect to their expected LoS. Random choice (RAND): The patients to be assigned are randomly chosen. The RAND strategy has been implemented as a baseline in order to analyze the effect of sorting.

All heuristic strategies use a minimal cost strategy for the final assignment by checking all possible assignments for the patient under consideration and greedily selecting the one with minimal cost.

The worst case complexity of the heuristic approaches is O ( | P | · log ( | P | ) + | S | · | T | · L max ) for LEPT and SEPT (due to the sorting procedure) and O ( | S | · | T | · L max ) for RAND, where | P | is the number of patients to be assigned, | S | is the number of wards to be considered, | T | is the number of days in the planning time frame, and L m a x is the maximal LoS.

Status Quo approach

In order to allow a comparison between the aforementioned optimization methods (heuristic strategies and optimal solution) and the status quo proceeding as carried out by the human bed managers, the STATQUO approach was defined. STATQUO neglects those cost factors that are usually neglected by the human decision making as well. Therefore, only the affinity factor ( α factor) and the ward occupancy ( β cost factor) are considered. The assignment order of patients corresponds to the RAND strategy.

All of the above specified planning strategies were evaluated and compared in a Discrete Event Simulation (DES)[ 41 ] study. A virtual hospital environment containing wards and clinics was designed to represent the system. An event represents a new admission and assignment planning task for a group of patients, referred to as the planning collective, of random size. Each event occurs at a point in time and affects the state of the system: the occupancy rate of the wards.

In order to allow a detailed analysis, the states before and after the event are logged as well as the computed assignment of the patients and the computation time. The general proceeding of the simulation is depicted in detail in Figure 4 .

Activity diagram of the simulation flow.

Simulation environment

A highly configurable simulation environment was developed, allowing a close to reality simulation, based on a realistic hospital model. In each run of the simulation, a group of patients with random configurations is generated, and then at the point of each admission, an assignment strategy was applied to this group. Characteristics such as the detailed assignment costs, the wards’ states before and after the assignment, and the computation time, were logged for further analysis. A simulation is specified in general by the following four constants: 1) PPY: the total number of patients to be considered over one year (the basic time interval of a simulation run), 2) MXP: the maximal number of patients to be assigned to beds in one single assignment cycle (the actual number is randomly chosen from the interval between zero and this maximum), 3) CPD: The number of assignment cycles per day 4) SPF: the shift of the planning frame, (in days), when all CPD assignment cycles have been accomplished.

Hospital simulation model

The modeling framework allows to specify a hospital by defining the available clinical units and wards. Different bed capacities can be set up for each ward by giving the number of available beds with specific features (e.g., single vs. double room bed). For each pair of ward and clinical unit, the affinity quantifies whether (i.e., to what degree) it makes sense to assign a patient treated in the clinical unit to a bed of the respective ward.

Patient simulation model

Furthermore, the modeling environment is able to generate data sets of patient groups with individual patient characteristics: treatment priority, gender, treating clinic, initial LoS estimate, and individual preferences (single vs. double room, etc.). Valid admission dates are derived from the treatment priority. An exemplary data set for one patient would be: (TreatmentPriority=2, Gender=male, TreatingClinic=Urology, LoS=5, Preference=SingleRoom).

The individual characteristics of the patients are randomly generated based on given statistical distributions: the LoS distribution is based on publicly available, official data of the German DRG statistics c (LoS mean values, upper and lower LoS bounds for the different groups according to the DRG classification system, and prevalence data). See Figure 5 . The statistical distributions of treatment priority, gender, treating clinic, and patient preferences must be defined accordingly in order to perform the simulation.

Mean and variance of the LoS frequency distribution. Three-dimensional representation of the relative frequency distribution of the mean and variance LoS attributes according to the DRG browser. The relative frequencies are colored to highlight their position.

Statistical analysis

The simulation data as well as the simulation results are analyzed by descriptive statistics. The mean, median, standard deviation, first and third quartile are calculated and presented in boxplot diagrams.

Ethics approval and informed consent

Neither institutional ethics approval nor written consent from participants were required to perform this study, since all patients were virtually generated according to publicly available statistical data.

Informed consent was provided by the case management and bed management department.

Hard- and software platform

The DSS was developed and evaluated on a Lenovo Thinkpad R61, Intel Core 2 Duo CPU 2.0 GHz and 2048 MB RAM.

The operating system was Ubuntu 10.04 GNU/Linux i686 running a 2.6.28 Kernel. The major part of the DSS was developed in Java version 6. The software tool SCIP[ 39 ] was used to solve the BIP.

Simulation data

The general parameters were set as follows: PPY:= 45,000, MXP:= 110, CPD:= 5, SPF:=1.

Hospital environment

The model configuration for this study was derived from the actual situation at UK Aachen using datasets provided by our local medical controllers. Additional information was elicited by interviews with the relevant representatives of the UK Aachen. The model contained 27 clinics, 43 wards of 15 different types, and 72 affinities, quantifying the associative strength between a clinic and a ward based on the results of the interviews within the requirements analysis. The data characterizing the clinics is shown in Table 2 and the data of the wards in Table 3 respectively.

Patient planning collectives

The LoS distribution was derived from the DRG-Browser d and is portrayed in Figure 5 . The distribution of patients treated by specific clinics over the year (treating clinic) was inferred from the annual report of UK Aachen[ 42 ] and is summarized in Table 2 . The sizes of the planning collective were uniformly chosen from the set {1,…,110} according to the interviews of the requirements analysis. The treatment priority was chosen based on the statistics portrayed in Table 4 . The gender distribution was estimated to be 56% male and 44% female patients. The distribution of patient preferences reflected the characteristics of the ward types portrayed in Figure 3 , e.g., for “Ward_G”: the probability for a single bed is 8/30 and for a female patient in a double bed room 10/30.

Results of the simulation

Different assignments were calculated and recorded, for each of the assignment strategies. The results were subsequently analyzed considering the following aspects:

The ratio of successful assignments and dismissals

The performance of the assignment calculation

The distribution of the cost factors

Successful assignments vs. dismissals

The boxplot in Figure 6 depicts the statistical aspects of the patient dismissal ratio for each planning strategy. The red dot indicates the mean value. The boxplot shows that there is a great similarity in their statistical aspects between the heuristic strategies LEPT, SEPT, and RAND (mean ca. 0.43, median ca. 0.44, first quartile ca. 0.37, third quartile ca. 0.51). Furthermore, the boxplot shows that the EXACT strategy has the lowest mean (0.4060) and median (0.4123) dismissal ratio. However, the range between the lower (0.2571) and upper quartile (0.5696) of about 0.3125 is greater than those of heuristic strategies (ca. 0.14). Thus, the lower mean and median outcome is accompanied by a higher variance (0.0455 for the EXACT and ca. 0.027 for the heuristic strategies). The simple status quo proceeding has a high dismissal ratio, almost 0.7471, and thus discloses the advantage of using one of the proposed admission and assignment strategies.

Boxplot of dismissal ratios refined by strategy. The Statistical characteristics of the dismissal ratios refined by planning strategy are depicted by a boxplot.

Figure 7 portrays in a boxplot the statistical aspects of the EXACT strategy with respect to the size of the planning contingent. The boxplot shows an adjusted mean value indicated as a blue dot, besides the mean value highlighted in red. The adjusted mean value represents the mean dismissal ratio with respect to the planning contingent size disregarding dismissals resulting from a penalty timeout of the mathematical program. The computational time limit of the EXACT approach was set at 300 seconds. The boxplot reveals an increase in the proportion of dismissals resulting from the penalty timeout with a growing size of the planning contingent. For groups of sizes between 81 and 110 patients, 11.63% of the dismissals were due to the penalty timeout. This phenomenon may be explained by considering the complexity of the mathematical program:

An admission and assignment is computed for a maximum of 110 patients at once.

The virtual hospital environment contains 43 wards in total.

The maximal planning time frame is limited to 40 days.

Thus, the size of the mathematical program is up to:

110 · 43 · 40 = 189,200 decision variables,

1 + 110 + 43 · 40 = 1831 constraints.

Boxplot of the EXACT strategy’s dismissal ratios refined by planning contingent size. The Statistical characteristics of the EXACT strategy’s dismissal ratios refined by planning contingent size are depicted by a boxplot.

Although the number of variables and constraints does not by itself allow of reasoning about the computational complexity in general, MIPs with these characteristics are usually considered hard to solve. The boxplot portrayed in Figure 7 shows that the dismissal ratio realized by the EXACT strategy, neglecting dismissals from the penalty timeout, has a maximum which is 0.4119 lower than those of the heuristic strategies with ca. 0.43 as portrayed in Figure 8 .

Boxplot of the RAND strategy’s dismissal ratios refined by planning contingent size. The statistical characteristics of the RAND strategy’s dismissal ratios refined by planning contingent size are depicted by a boxplot.

Figure 8 shows the dismissal ratio with respect to the planning contingent size of the RAND strategy in a boxplot. The statistical characteristics (mean ca. 0.43, median ca. 0.44, first quartile ca. 0.38, third quartile ca. 0.51) of this aspect are almost the same as those of the heuristic strategies and are thus only portrayed for the RAND strategy and not for the LEPT and SEPT strategies. The boxplot reveals that the planning contingent size has no influence on the dismissal ratio. Interestingly, the interquartile range is greater for planning contingent classes of up to 20 patients (ca. 0.27) compared to all other classes (ca. 0.14) in both the EXACT and the heuristic strategies.

Performance

Figure 9 depicts the statistical aspects of the computation time for the different strategies in a boxplot (the y-axis is base two logarithmically scaled). The boxplot reveals a drastically longer computation time needed by the EXACT strategy compared to the heuristic ones. The EXACT strategy is characterized by a mean value of ca. 141 seconds and a median of ca. 106 seconds, whereas the heuristic strategies are characterized by a mean value of ca. 2.6 seconds and median value of ca. 2 seconds, thus the EXACT strategy takes almost fifty times longer than the heuristic ones.

Boxplot of computation time with respect to strategy. The statistical characteristics of the computation time of an assignment, differentiated by strategy, are highlighted by a boxplot. The y-axis is logarithmically scaled (base two).

Furthermore, the boxplot shows a strong similarity in the statistical aspects (first quartile ca. 842, mean ca. 2589, median ca. 2006, third quartile ca. 3739) between the heuristic strategies, which indicates that the sorting overhead can be neglected.

Figure 10 shows statistical aspects of the total cost outcome of the planning strategies in a boxplot. Dismissals were not considered in the statistical analysis of the cost factors, since the dismissal penalties would bias the cost outcome drastically. The boxplot indicates a slightly lower mean and median realization for the EXACT strategy (mean ca. 3.4338, median ca. 2.9049) compared to the heuristic ones (mean ca. 3.58, median ca. 3.12). Furthermore, the boxplot reveals similar statistical aspects between the heuristic strategies (mean ca. 3.58, median ca. 3.12, first quartile ca. 1.72, third quartile ca. 4.80).

Boxplot of total cost comparison of strategies. The boxplot shows the statistical characteristics of the total cost outcome refined by strategy.

Figure 11 shows statistical aspects of the cost factor contribution of the EXACT strategy in a boxplot. The boxplot shows that the β cost factor (mean ca. 2.369, median ca. 1.92, first quartile ca. 1.019, third quartile ca. 3.093) is the most influential cost factor on the total outcome. The second most influential factor is the δ cost factor (mean ca. 0.56, median ca. 0.456, first quartile ca. 0.305, third quartile ca. 0.869).

Boxplot of the EXACT strategy’s cost factors. The boxplot shows the statistical characteristics of the EXACT strategy refined by cost factor.

Figure 12 shows statistical aspects of the cost factor contribution of the RAND strategy in a boxplot. The statistical characteristics with respect to the partial cost factors were very similar between the heuristic strategies. Hence, only the statistical characteristics of the RAND strategy are analyzed further. The boxplot shows that the β cost factor (mean ca. 2.645, median ca. 2.247, first quartile ca. 1.027, third quartile ca. 3.662) is the most influential cost factor on the total outcome as well.

Boxplot of the RAND strategy’s cost factors. The boxplot shows the statistical characteristics of the RAND strategy refined by cost factor.

Regarding the mean value, the contribution of the β cost factor to the overall costs was approximately 74% when using heuristic strategies and 69% in the case of the EXACT approach. Accordingly, the contribution of the δ cost factor was 12% for the heuristic strategies and 16% for the EXACT approach. Thus, considering the EXACT strategy, the δ cost factor contributes slightly more to the total cost, while the β cost factor contributes less, in contrast to the heuristic strategies. The overall lower cost of the EXACT approach was thus accompanied by a reduction of the β and an increase of the δ cost factor, compared to the heuristic strategies. A higher contribution of the δ cost factor is attributed to a wider use of the planning time frame. Hence, the planning time frame is used more broadly by the EXACT strategy compared to the heuristic one and results in an overall lower cost outcome and a lower dismissal ratio.

Prior work, and the interviews conducted for the elicitation of the requirements, showed the great potential of computer aided decision support (CDS) applied to patient admission planning and assignment. CDS has been shown to improve the hospital’s resource utilization. In general, all the resources necessary for the treatment of each patient must be considered during the planning process. In contrast, our work focuses on a single resource: the bed capacity. Other important and critical resources, such as the hospital staff or the operating theater, and their interdependencies, were neglected. Nonetheless, the case managers must consider these resources during their planning task as well. Therefore, the improvements achieved by the optimization approaches may not be reproduced at the same level in real life. However, bed capacity is one of the most important resources and considerably affects decisions in patient admission planning. The availability of intensive care beds has been reported to be a major bottleneck at UK Aachen, having a direct influence on the cancellation of surgeries. Our decision support approach, based on up to date LoS estimates, is likely to be more accurate than the static approach which uses fixed LoS estimates, since changes of resource use are taken into account by the algorithmic planning methodology immediately. Our approach closely follows – often unpredictable – changes of the real situation and can thus be assumed to trigger more accurate decisions

Using individual estimates of LoS has specific advantages and drawbacks: on the one hand, changes in the LoS can be considered immediately in planning, and may reflect the complex individual conditions of patients. On the other hand, the individual estimates may be inaccurate and depend on the staffs’ experience and training. Chow et al. proposed an alternative, by deriving scheduling guidelines from recurring patterns of optimized schedules generated by their simulations: the guidelines are designed to improve the scheduling decisions without implementing optimization algorithms[ 35 ]. Demeester describes a patient admission scheduling algorithm considering hospital beds and providing operational decision support concerning the hospital bed assignment task as well[ 36 ]. In contrast to our approach, Demeester assumed that the patient’s admission date and expected LoS are known in advance[ 36 ]. Furthermore, adaptations of the patients’ LoS cannot be carried out, and hence are not considered by the decision support method. The consolidated approach proposed by our work may yield an improved robustness in the reliability of the capacity estimates. Furthermore, our approach can be generalized by considering universal resource capacities by future investigations. The evaluation of our approach reveals its principal qualification. Further investigations should study the potential of this developed DSS in clinical practice.

The limitation to a single resource, i.e., hospital beds, could be relaxed by considering the patients’ clinical pathways. Following this approach might further improve this method of decision support[ 43 , 44 ]. However, the consideration of individual clinical pathways will probably increase the planning complexity and uncertainty. Furthermore, clinical pathways are neither comprehensively standardized nor clinically well established.

In addition, the existence of emergency patients which have to be treated immediately heavily influences the planning problem. Emergency patients arrive at random and cannot be scheduled in advance. A common approach to considering emergency patients in advance is to provide a contingent of extra beds. However, the provision of extra beds causes further expenses for the hospital. The fixing of an extra bed contingent size is a strategic decision to be made by the hospital’s management. Decisions on the choice of resource capacity can be supported by queuing theory approaches[ 15 ], in combination with a simulation and flow analysis[ 45 , 46 ], or by stochastic processes[ 47 ] in general. All these approaches require accurate and detailed annual statistics of all the relevant planning aspects of emergency patients. The proposed DSS does not take into account the occurrence of emergency patients, but assumes a sufficient amount of extra beds reserved for emergency patients. However, an explicit consideration of emergency patients by the DSS might further improve this decision support method as well.

The transfer of patients to other beds is considered only implicitly. Due to ward abstraction, i.e., bed allocation irrespective of individual beds, transfers within the same wards can be ignored. Transfers to different wards may be interpreted as a discharge from the prior ward and an assignment to the subsequent ward. Following this interpretation, transfers are implicitly regarded by the DSS. However, taking transfers explicitly into account in advance might also further improve the DSS.

Overall, the DSS might be improved by considering further critical resources, the clinical pathway of patients, emergency patients, algorithmic improvements, and a weight adaptation of the cost factors.

Requirements analysis

The interviewees reported a classification of patients into priority groups reflecting their treatment urgency. The characteristics of the patients in the different priority groups were not further analyzed. However, an analysis of the priority groups might reveal interesting patterns which could be used to improve patient admission planning as well. For instance, it might happen that certain clusters of patients appear in the groups, perhaps leading to the exhaustion of certain resources. Besides, further analysis of the characteristics of the priority groups might be interesting in order to adapt the resource availability to reflect upcoming demand and to avoid congestion.

Comparison between the heuristic approaches and the exact solution

Given the similarity of all the heuristic approaches in their performance, we argue that it is sufficient to compare the performance of only one heuristic approach with the exact solution. The complexity of the heuristic algorithms (reported in the methods section) justifies this argument: considering a maximum of 110 patients to be scheduled, the sorting complexity may well be neglected. The comparison of the dismissal ratios is complicated by the fact that there are two genuinely different types of dismissals in the case of the EXACT approach: 1) dismissals due to exhausted bed capacity 2) dismissals due to exceeding the time limit of the algorithm. The first kind of dismissal should clearly be minimized by the exact solution of the optimization problem. Considering the effect of the timeout, the strong positive correlation between the size of the group to be scheduled and the dismissal ratio in the EXACT case is not surprising.

Comparison with the baseline STATQUO approach

The baseline approach used in our study only pays regard to a subset of the aspects already considered by the optimization approach. In a real life scenario, further aspects, such as the operating room capacities and other critical resource capacities, are usually taken into account by the persons in charge. Therefore, the STATQUO criteria might not always guide real life decisions.

Weight adaptation of the cost factors

All weight factors of the partial cost factors were fixed to 1.0. Hence, each cost factor is regarded as having equal weight. The weight factors can serve to adjust the impact of the partial cost factors, depending on the aim of the optimization strategy. A suitable weight combination for a given strategy needs to be investigated by an additional study.

Algorithm improvements

Although our approach allows of adaptable LoS estimates and ward utilization estimates, stochastic variability is neglected in the final BIP. To enable an exact solution within an acceptable time, the stochastic mathematical program was reduced to a deterministic BIP. A stochastic mathematical program might lead to an improved assignment, due to a more accurate representation of the modeling reality. Discretization of a stochastic program is usually performed by substituting the random variables by constant values. The constant values are usually obtained by adding some slack value, e.g. the standard deviation, to a constant factor, e.g. the mean outcome of the substituted random variable. Thus, the resulting program does not represent the reality as accurate as the stochastic program, due to neglecting the stochastic variability. However, an exact stochastic mathematical program of the necessary complexity will most likely fail to solve the problem within an acceptable time[ 48 ].

The bed assignments calculated by the heuristic strategies might be improved by applying meta-heuristic strategies, such as Tabu Search[ 36 ], Evolutionary Approaches[ 49 ], or Simulated Annealing[ 29 , 50 ].

Simulation procedure

The simulation conducted to evaluate the model used data taken from a real hospital setting. The main characteristics and quantities of the model, namely the capacity and configuration of the wards, the linkage between the wards and the clinics, and the distribution of the treating clinics, correspond to the situation at UK Aachen. Besides, the patients’ characteristics were based on the real distributions as well. The distributions of the planning collective size, gender, treatment priority, and individual patient preferences were derived from the interviews. The statistics concerning the number and type of treatments refer to the annual report of UK Aachen[ 42 ]. The LoS distribution was given by the German national DRG statistics as provided by the DRG-Browser. In this way, errors resulting from unrealistic or biased simulation data could be minimized. However, the simulation probably still possesses unrealistic patient datasets, for instance, male gynecology patients.

The simulation revealed a quite high dismissal ratio, of approximately 40% realized by the EXACT and heuristic strategies, and approximately 74% realized by the STATQUO approach. This high ratio may be attributed to the fact that each patient is considered exactly once for admission planning. Patients who cannot be planned for at the planning time point are dismissed and will not be planned for in the future. In reality, a patient that cannot be planned for at the planning time point will usually be considered for planning again during the patient’s planning period, and is thus not considered as dismissed at that point in time. A dismissal ratio of 40% may be interpreted as meaning that 60% of the patients on the list are planned successfully, whereas 40% could not be assigned.

The effect of LoS revision was not analyzed within the simulation study due to the complexity of such an analysis. In contrast to the LoS estimations (which are based on the DRG statistics), a reliable data source for a realistic simulation of LoS adaptions was not available. Thus, we had to postpone a simulation based evaluation of the effects of LoS adaptations. In the future, the necessary data can be systematically acquired during the routine use of the module proposed in this paper; the organizational prerequisites for the routine use have yet to be provided. Nonetheless, we could show the feasibility of using a cost-based patient admission assignment methodology, which takes into account adaptable length of stay estimations and aggregated resources. Precisely analyzing LoS adaption effects clearly is an important and valuable topic for ongoing research.

Our requirements analysis revealed a strong need for computer-based decision support in the context of case and bed management.

Our work focused on the implementation of a decision support system for admission planning and bed assignment, taking into account the availability of suitable hospital beds. Decision support relies on an algorithmic core, providing the calculation of an optimal admission and assignment plan for a given group of patients and its implementation within a software system.

Patient admission and assignment is based on up to date and adaptable LoS estimates, taking into account the aggregated contingents of hospital beds, treatment priorities, patient preferences, and a linkage between clinics and wards.

The admission planning and assignment problem was formally described by a BIP. In order to solve the BIP, two classes of strategies were developed and analyzed: the first class consisted of an exact approach, and the other class contained three heuristic strategies. Four partial cost factors have been introduced in order to represent the advantage of an assignment: affinity costs, ward occupancy, change of ward occupancy, and assignment delay. The weighted sum of the partial cost factors results in the assignment costs. The objective of the BIP is to minimize the total assignment costs and is, thus, following a min-cost strategy.

Discrete event simulation revealed the following facts: the application of optimization strategies following a min-cost approach yields a marked reduction in patient dismissals compared to a status-quo approach. In theory, this results in an increase in ward utilization. In addition, calculating the exact solution with a MIP solver resulted in only minor advantages with respect to costs and the dismissal ratio compared to the heuristic strategies. Moreover, calculating an adequate exact solution requires almost fifty times more computational time than the heuristic strategies. Our study analyzed and compared three heuristic strategies: the LEPT, SEPT, and RAND strategies. LEPT and SEPT did not reveal considerable advantages over the RAND strategy, and should thus be omitted because of the computational load resulting from the sorting procedure. An adequate solution to the bed assignment problem is calculated much faster by the heuristic strategies than by the exact optimization. Thus, the RAND strategy utilizing a min-cost optimization approach can be considered as the preferred method for bed assignment in the case of shared resources.

The simulation revealed a promising reduction in the patient dismissal rate by applying the proposed DSS strategies and, hence, may present a way to increase the hospital’s throughput in the future.

a http://scip.zib.de/ b http://plato.asu.edu/ftp/milpc.html c http://www.g-drg.de d http://www.g-drg.de

Abbreviations

• Binary Integer Program

Case Management

Diagnosis Related Group

Decision Support System

Exact strategy of solving a BIP

Longest Expected Processing Time

Length of Stay

Mixed Integer Program

Random assignment strategy

Standard Care

Shortest Expected Processing Time

Status quo assignment strategy

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Acknowledgements

This work was supported by the Research Cluster on Ultra High-Speed Mobile Information and Communication UMIC ( http://www.umic.rwth-aachen.de ).

We would like to thank the Standard Care Team, the Case Management Team, the IT/Medico Team, and the Care Management Team of UK Aachen for their extensive contributions to the interviews of the requirements analysis and for interesting and inspiring discussions.

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All authors contributed to the interviews of the requirements elicitation, the design of the mathematical model, and the conceptual design of the decision support system. RS prepared, conducted, and evaluated the interviews, developed and implemented the mathematical model resulting in the decision support system, and designed and performed the evaluation. SG and CS were involved in the design and the evaluation of the interviews, the design of the mathematical model, of the decision support system, and its evaluation. All three authors have read and approved the final manuscript.

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Schmidt, R., Geisler, S. & Spreckelsen, C. Decision support for hospital bed management using adaptable individual length of stay estimations and shared resources. BMC Med Inform Decis Mak 13 , 3 (2013). https://doi.org/10.1186/1472-6947-13-3

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Using patient waiting-time data to improve the hospital bed-assignment process

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An improvement activity involving use of real-time waiting-time data resulted in reductions in bed-assignment times and overall diversion hours.

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Early bed assignments of elective surgical patients can be a useful planning tool for hospital staff; they provide certainty in patient placement and allow nursing staff to prepare for patients’ arrivals to the unit. However, given the variability in the surgical schedule, they can also result in timing mismatches—beds remain empty while their assigned patients are still in surgery, while other ready-to-move patients are waiting for their beds to become available. In this study, we used data from four surgical units in a large academic medical center to build a discrete-event simulation with which we show how a Just-In-Time (JIT) bed assignment, in which ready-to-move patients are assigned to ready-beds, would decrease bed idle time and increase access to general care beds for all surgical patients. Additionally, our simulation demonstrates the potential synergistic effects of combining the JIT assignment policy with a strategy that co-locates short-stay surgical patients out of inpatient beds, increasing the bed supply. The simulation results motivated hospital leadership to implement both strategies across these four surgical inpatient units in early 2017. In the several months post-implementation, the average patient wait time decreased 25.0% overall, driven by decreases of 32.9% for ED-to-floor transfers (from 3.66 to 2.45 hours on average) and 37.4% for PACU-to-floor transfers (from 2.36 to 1.48 hours), the two major sources of admissions to the surgical floors, without adding additional capacity.

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Acknowledgements

The authors thank Ann L. Prestipino and Peter L. Slavin, MD (Senior Vice President of MGH and President of MGH, respectively, at the time of this work) for their continuous support throughout this process. We are grateful to the nursing directors from the units who participated and contributed to the implementation of the strategy, and to Gianna Wilkins from Process Improvement for helping to map and document the new processes during implementation. Finally, we thank the editor and two anonymous reviewers for detailed feedback which has improved the completeness of the work herein.

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Braaksma, A., Copenhaver, M.S., Zenteno, A.C. et al. Evaluation and implementation of a Just-In-Time bed-assignment strategy to reduce wait times for surgical inpatients. Health Care Manag Sci 26 , 501–515 (2023). https://doi.org/10.1007/s10729-023-09638-3

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Optimizing inpatient bed management in a rural community-based hospital: a quality improvement initiative

• Brian N. Bartlett 1 ,
• Nadine N. Vanhoudt 2 ,
• Hanyin Wang 3 ,
• Ashley A. Anderson 4 ,
• Danielle L. Juliar 5 ,
• Jennifer M. Bartelt 6 ,
• April D. Lanz 7 ,
• Pawan Bhandari 8 &
• Gokhan Anil 9  

BMC Health Services Research volume  23 , Article number:  1000 ( 2023 ) Cite this article

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Appropriate use of available inpatient beds is an ongoing challenge for US hospitals. Historical capacity goals of 80% to 85% may no longer serve the intended purpose of maximizing the resources of space, staff, and equipment. Numerous variables affect the input, throughput, and output of a hospital. Some of these variables include patient demand, regulatory requirements, coordination of patient flow between various systems, coordination of processes such as bed management and patient transfers, and the diversity of departments (both inpatient and outpatient) in an organization.

Mayo Clinic Health System in the Southwest Minnesota region of the US, a community-based hospital system primarily serving patients in rural southwestern Minnesota and part of Iowa, consists of 2 postacute care and 3 critical access hospitals. Our inpatient bed usage rates had exceeded 85%, and patient transfers from the region to other hospitals in the state (including Mayo Clinic in Rochester, Minnesota) had increased. To address these quality gaps, we used a blend of Agile project management methodology, rapid Plan-Do-Study-Act cycles, and a proactive approach to patient placement in the medical-surgical units as a quality improvement initiative.

During 2 trial periods of the initiative, the main hub hospital (Mayo Clinic Health System hospital in Mankato) and other hospitals in the region increased inpatient bed usage while reducing total out-of-region transfers.

Our novel approach to proactively managing bed capacity in the hospital allowed the region’s only tertiary medical center to increase capacity for more complex and acute cases by optimizing the use of historically underused partner hospital beds.

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Introduction

Establishing an ideal number of staffed inpatient beds is a complex issue for health care organizations. Numerous factors, such as planned and predicted inflows into the hospital system, contribute to this challenge. Historically, bed demand for elective and other surgical procedures requiring an inpatient bed has been easier to predict than that for admissions from the emergency department, urgent care, and clinic visits [ 1 ]. Other variables, such as labor markets, space constraints, and equipment use, add to the complexity of predicting bed demand. Inadequate inpatient bed availability has led to emergency department crowding and, in turn, concern for the quality of services delivered [ 2 , 3 ]. These factors combine to create a challenging operational workflow when managing inpatient bed usage [ 4 , 5 ].

Our hypothesis centered on the strategic allocation of beds among various hospital sites. Specifically, we hypothesized that proactive bed assignment at tertiary centers for patients with presenting signs of severe illness who require specialized care unavailable at smaller sites will enhance regional health care provision. Concurrently, we hypothesized that optimizing the capacity at smaller hospitals by admitting patients with non-severe illness will improve patient distribution and thereby augment the ability of hospital systems to deliver comprehensive care throughout a health care region. To test this hypothesis, we developed a novel approach (via a combination of various methods) to bed hold to ensure that hospital beds were available for patients with severe illness (those requiring specialized services such as Cardiology, Neurosurgery, Orthopedic Surgery, Trauma Care, Interventional Radiology or Gastroenterology procedures) at the Mayo Clinic Health System (MCHS) hospital in Mankato, Minnesota, USA (MCHS-Mankato). This dual-pronged approach anticipates both unplanned, acute admission requirements and the ongoing need for beds for less critical cases. MCHS-Mankato is the only hospital that provides acute care, primary care, continuous emergency care, a level II nursery, critical care, advanced trauma care, and specialized medicine in southwestern Minnesota. Therefore, reducing or preventing transfers to outside of the region and providing care to patients at the right place and at the right time were important aims of our study. However, certain emergency situations required transfer to another tertiary hospital in which advanced care is provided.

The Admission Transfer Center (ATC) at MCHS acts as a critical node for patient distribution, operating like an air traffic control center to manage patient placement. The ATC, staffed by triage nurses at a centralized Southern Minnesota location, connects to all MCHS sites via an advanced information technology network. It uses data-driven strategies to facilitate timely transfers and improve patient flow, which contributes to operational efficiency. Additionally, its unique model provides a rich data source for investigating the dynamics of patient transfers, which thereby fosters various research opportunities.

The MCHS Southwest Minnesota region (SWMN) is based on a hub-and-spoke model that includes 2 acute care hospitals and 3 critical access hospitals. MCHS-Mankato is the only tertiary medical center and is the hub in this region. Four spoke hospitals in the region (Fairmont, St James, New Prague, and Waseca, Minnesota) serve patients with less severe illness. MCHS-Mankato consistently had greater than 95% occupancy and remained in diversion status on most days since the COVID-19 pandemic began because of a combination of increased demand, bed closures, staffing shortages, and other challenges. During the same period, tertiary hospitals throughout Minnesota also had high occupancy rates, which limited their capacities to accept patient transfers. These limitations resulted in previously unseen delays in hospital care throughout southwest Minnesota.

In SWMN, patients requiring care for severe illness and subspecialty care could often be served only at MCHS-Mankato. Therefore, these patients would be transferred out of the region when no inpatient beds were available at MCHS-Mankato. At the start of this project, the spoke hospitals in SWMN were not consistently reaching full occupancy. The historical method of admitting patients to MCHS-Mankato until full capacity was reached failed to make use of open spoke hospital beds for appropriate patients.

Study process

In the third quarter of 2021, MCHS-Mankato started the design and implementation of a project to optimize hospital bed management in SWMN. This project aimed to ensure the availability of beds for local patients with severe illness, maximize capacity opportunities in the surrounding spoke hospitals, and reduce out-of-region transfers. The spoke hospitals, all under the purview of MCHS leadership and operationally integrated via the ATC, were pivotal as key stakeholders in the formulation of our study design.

The initiative for optimizing hospital bed management was divided into 2 phases: phase 1, which comprised discovery (opportunity identification), planning, and proposal steps, and phase 2, which consisted of implementation. Through direct engagement with key stakeholders, phase 1 of the project followed the project engagement approach outlined in Table 1 . This approach was closely aligned with Agile methodology and principles and our work breakdown structure (Fig.  1 ) to yield our overarching deliverables in the form of tasks or work buckets. In phase 2 of the project, 4 work streams were proposed after brainstorming sessions with subject matter experts and key stakeholders. The 4 work streams included the pre-admission process, transfer plan, updated admission guidelines, and updated diversion plan (Table 2 ). These work streams were led by respective operations, nursing, and executive partners. Each work stream was critical to ensuring that all aspects of hospital input were addressed in pursuit of optimizing bed management.

Work Breakdown Structure. CMI indicates case mix index; FA, Fairmont, Minnesota; GI, gastrointestinal; MA, Mankato, Minnesota; NP, New Prague, Minnesota; SJ, St James, Minnesota; SWMN, Mayo Clinic Health System – Southwest Minnesota region; WA, Waseca, Minnesota

Deliverables and procedures

Deliverables were designated for each of the 4 work streams, with a project leader and key stakeholders assigned from various work areas to assess the current operational state and identify opportunities for improvement.

The pre-admission process work stream was identified as an essential component because it is a source of planned admissions, with an aspect of volume control structured by the hospital. The pre-admission process includes surgical and procedural patients requiring hospital care. The transfer plan work stream investigated the systems and processes used by MCHS-Mankato to leverage its partnership with the internal patient transfer center, external hospital transfer centers, and the emergency medical system to direct and triage patients awaiting admission into hospitals that have the capability and available bed capacity to admit patients.

The admission guidelines work stream was heavily focused on the admission criteria that MCHS-Mankato and the spoke hospitals have established to admit patients according to their respective capabilities. This exercise provided an opportunity to distinguish between patients with various levels of illness severity and quantify the volume or demand for inpatient care. These criteria were used to prioritize a selection of beds to ensure safe and timely placement of patients with critical illness. The diversion plan work stream leveraged what we learned from each of the previous deliverables to identify a diversion service area and activation/deactivation criteria for hospital and/or service line diversions. Each work stream deliverable had an essential component for structuring hospital bed optimization, with interdependencies between each component.

Pre-admission process

Throughout the COVID-19 pandemic, access to critical care beds was considerably reduced because of unprecedented demand. As surgical case volume began to quickly increase, implementing a process that allowed local teams to adequately plan for surgical admissions that require postoperative critical care placement became essential. Historically, the routine ebb and flow of hospital discharges provided adequate capacity for surgical admissions. However, the substantial effect of the COVID-19 pandemic on capacity necessitated a change in practice. To support the timely placement of postoperative critical care patients, a workflow was established that provided advanced notification of all planned cases by the level of care to patient placement teams. The night before a surgical procedure, a systems-based software program generated a bed request that was visible to the bed planning team for all scheduled cases. As part of the enhanced pre-admission process, bed planning staff pre-assigned critical care surgical patients to an inpatient bed for cases completed before 4 pm . This was a fluid process, allowing for analysis of emergent situations with an immediate need for a critical care bed in the hospital. Teams would reevaluate the pre-admitted bed to balance the emergent need and anticipated surgical patient. Pre-assigned patients were factored into hospital occupancy, which was therefore directly associated with diversion status. This practice allowed SWMN hospitals to mitigate capacity constraints through advanced planning.

Transfer plan

A vital patient-based perspective was incorporated into our research design through the engagement of both our MCHS-Mankato and SWMN regional boards. These entities not only granted approval for our initiative but also actively encouraged its execution, which effectively served as a patient advisory board. Furthermore, our Emergency Medical Treatment and Labor Act compliance team conducted a thorough review of the transfer policy. The team’s endorsement stemmed from the optimized regional capacity to provide an advanced triaging system, which thereby ensured that the most critically ill patients were less likely to require transfer out of the SWMN region.

SWMN established a plan to determine the best location for a patient to be admitted to an inpatient setting throughout the region according to clinical needs and capacity. Because of the development of regional admission guidelines, physicians and practitioners understood the capabilities of all spoke hospitals. Each of the critical access spoke hospitals also reinforced staffing plans to accommodate patients who did not require a high level of care, which permitted admissions during capacity restraints. Emergency department physicians and practitioners were made aware of bed capacity for the day through secure chat messaging. This allowed practitioners to identify the level of care a patient required and to determine which location could best meet the needs of the patient. If a current location had no capacity or ability to treat the patient, the patient was then parallel transferred to another location in our region for inpatient admission. This transfer plan improved care for patients by allowing them to remain in the region for treatment while also maximizing the capacity of all hospitals in the region.

Updated admission guidelines

SWMN admission guidelines were established to support safe patient placement according to site, unit, and nurse capability, which maximized bed usage for each hospital in the region. The guidelines delineated the placement of patients on the basis of their specific clinical needs and identified which units could provide care, as well as those that could not provide care. The guidelines were analyzed and benchmarked against other hospital facilities for best practices.

Admission guidelines served as the foundation for optimizing hospital bed management, which established a precedent for ensuring that each patient was placed in the right bed after admission. Historically, patients without severe illness may have been placed in beds designated for patients with severe illness when no other internal capacity options were available. This practice reduced critical care bed capacity, contributed to the underuse of regional beds, and negatively affected hospital patient flow. Work stream reevaluation of admission guidelines indicated that restricting the critical care units for patients with severe illness who met the identified admission criteria was a foundational component to optimizing SWMN hospital bed management.

Updated diversion plan

The diversion plan was the most complex work stream. Admission patterns throughout SWMN were studied. An analysis of admission guidelines and patient flow patterns throughout each of the 5 SWMN hospitals identified that underuse of beds at regional spoke hospitals occurred in conjunction with overcapacity at MCHS-Mankato. Patients requiring high-level and specialty care had barriers to bed access at MCHS-Mankato because of a lack of capacity. Specific specialty practices (ie, service lines), including orthopedics, cardiology, neurosurgery, and gastroenterology, were disproportionately affected by these capacity constraints. Because MCHS-Mankato is the only multispecialty tertiary care center in an approximately 80-mile radius, patients unable to access inpatient care were transferred distantly outside of the SWMN region. This placed a considerable strain on local emergency medical service resources, which resulted in a systemwide patient flow deficit. Admission pattern analysis indicated that an average of 23 patients per day without severe illness who would have met the admission criteria of regional spoke hospitals occupied a bed in MCHS-Mankato. These beds were identified as potential capacity for patients with severe illness at MCHS-Mankato.

To maximize spoke hospital bed usage, it was essential to shift from the traditional approach of admitting patients to MCHS-Mankato on a first-come, first-served basis until full capacity was reached. The foundational practices implemented by the other 3 work streams readied the teams for the next step (ie, enhancing the diversion procedure) in optimizing hospital capacity throughout the region. To reduce out-of-region transfers of patients with serious illness, specific thresholds were established that activated diversion procedures according to each level of care, including intensive, progressive, and medical/surgical care. When a diversion procedure was activated, specific criteria were used to prioritize admission for select service area hospitals, levels of care, and tertiary services to maximize use of the limited volume of beds. This action preserved local bed capacity for more critical care patients, which improved quality, reduced the distance of transfers, and minimized the overall strain on the regional health care system.

Diversion thresholds were established after an in-depth analysis of patient demand, which included historical admission, discharge, and transfer rates for each level of care and primary admission services. Service line diversion was activated when the facilitywide medical-surgical bed volume was reduced to 4 beds. These beds were then reserved for patients in the diversion service area who required inpatient care from orthopedics, neurosurgery, cardiology, or gastroenterology because these service lines had the highest frequency of out-of-region transfers. When a request for admission was made, bed management would perform a real-time bed capacity assessment and connect the hospital medicine physician or admitting specialist with the referring clinician for admission screening. If the patient did not require care through the service lines, transfer options were pursued to maximize regional bed capacity. Occasionally, exceptions were made at the discretion of the hospitalist and emergency department physician. Diversion thresholds for high-level care areas were activated when bed volume was reduced to 3 beds in progressive care and to 2 beds in intensive care. Admission to either progressive or intensive care was limited to the regional service area during the diversion but not to specific service lines.

Implementation plan summary

The project was implemented with a series of Plan-Do-Study-Act (PDSA) cycles in 2 trials performed during the second quarter of 2022. Trial 1 was conducted from April 27, 2022, through May 3, 2022, in which the new diversion procedure was implemented. Trial 2 was performed from June 1, 2022, through June 7, 2022. Because of the success of trial 1, we continued use of the new diversion procedure during the period between trials 1 and 2. A key change from trial 1 to trial 2 was allowing a thorough review of exceptions to the established admission criteria during diversion activation. Patients requiring interventional radiology or expert trauma care were not excluded in trial 2 but were excluded in trial 1. The hospitalist team was allowed to make 2 exceptions per day to the admission criteria, when appropriate.

Project outcomes and analysis

Measures of success were identified in 2 primary categories (transfer and census). Transfer rates were calculated by dividing transfer volume by all admissions (transfer volume plus total admissions) of all adult patients (excluding behavioral health patients) during the period. Transfer rates were determined for total transfers from all SWMN hospitals (MCHS-Mankato alone and the 4 spoke hospitals combined); transfers from SWMN hospitals (MCHS-Mankato alone and the 4 spoke hospitals combined) to Mayo Clinic in Rochester, Minnesota; and parallel transfers from MCHS-Mankato to SWMN spoke hospitals or to hospitals in the MCHS Southeast Minnesota region (SEMN). Transfer rates after each implementation trial/PDSA cycle were compared with the average 2021 transfer rate (ie, baseline).

Trial 1 outcomes

During implementation trial 1, the reduction from baseline in the total, all-cause transfer rate of patients out of the region from the 4 spoke hospitals (3 critical access hospitals and MCHS-Fairmont) was 71%, which changed from 35% at baseline to only 10% after trial 1 (Fig.  2 ) In addition, the average daily census of patients with severe illness who were admitted through the prioritized service lines (orthopedics, neurosurgery, cardiology, and gastroenterology) to MCHS-Mankato remained the same ( n  = 37) during both trial 1 and trial 2, despite the reduced number of available beds due to bed blocking. The rate of transfers from MCHS-Mankato to Mayo Clinic in Rochester, Minnesota, a destination medical center, decreased by 43%, from 7 to 4% (Fig.  3 ). Additionally, the rate of parallel transfers from MCHS-Mankato to SWMN and SEMN increased by 20% from a baseline of 5% to 6% (Fig.  4 ).

Transfer Rates for Mayo Clinic Health System (MCHS) Southwest Minnesota (SWMN) Regional Hospitals. Adult patients (excluding behavioral health patients) were transferred from the MCHS hospital in Mankato, Minnesota, or 1 of the 4 SWMN spoke hospitals to an external hospital, a hospital in another MCHS region, or Mayo Clinic in Rochester, Minnesota. Total transfer rates for all causes were determined for the 2021 calendar year (baseline) and during 2 project trial periods of 7 days

Transfer Rates of Patients Transferred to Mayo Clinic in Rochester, Minnesota. Adult patients with acute illness (excluding behavioral health patients) were transferred from the Mayo Clinic Health System (MCHS) hospital in Mankato, Minnesota, or 1 of the 4 MCHS Southwest Minnesota spoke hospitals to Mayo Clinic in Rochester, Minnesota. Transfer rates were determined for the 2021 calendar year (baseline) and during 2 project trial periods of 7 days

Parallel Transfer Rates Among Mayo Clinic Health System (MCHS) Hospitals. Adult patients (excluding behavioral health patients) were transferred from the MCHS hospital in Mankato, Minnesota, to hospitals in the MCHS Southwest Minnesota (SWMN) and Southeast Minnesota regions. Parallel transfer rates were determined for the 2021 calendar year (baseline) and during 2 project trial periods of 7 days

Trial 2 outcomes

Considerable inpatient nurse staffing shortages occurred during trial 2, which did not occur during trial 1. This resulted in closure of an average of 22 inpatient beds per day (20% reduction) during trial 2, which further reduced hospital capacity. However, MCHS-Mankato continued to admit 97% of our average historical admission volume. During trial 2, the rate of all-cause transfers from MCHS-Mankato increased by 22% from baseline (Fig.  2 ), but the mean transfer rate of patients from MCHS-Mankato to Mayo Clinic in Rochester, Minnesota, decreased from a baseline of 7% to 4% (Fig.  3 ). Although service line admission volume did not increase during trial 2, the mean daily census for all 4 service lines increased from 0 beds at baseline to 30 beds after trial 2, despite inpatient capacity reduction. Parallel transfers from MCHS-Mankato to the SWMN spoke hospitals and SEMN hospitals increased from 5% at baseline to 10% during trial 2 (Fig.  4 ).

The complexity of treating patients requiring hospital care has been increasing, while the challenges of providing continuous staffing in hospitals have been mounting for health care organizations. Many recommendations on how hospitals can improve their performance despite these challenges have been proposed [ 6 ]. Improvements in patient flow and movement pathways, as well as care coordination through optimization of admission and discharge processes, are methods that can positively affect quality and patient care outcomes [ 7 ]. Intentionally delineating roles among hospital frontline staff and management teams has also shown positive results [ 8 ]. Our study used a novel strategy of combining some of these well-documented recommendations with new strategies to optimize our regional inpatient service, which routinely exceeds 85% capacity.

The PDSA cycle outcomes of the 2 implementation trials yielded results that supported our hypothesis. These findings suggest that hospital systems may benefit from proactively assigning beds for patients with severe illness at their tertiary centers and optimizing capacity at their smaller hospital sites by admitting patients without severe illness. Often, hospital systems will fill available capacity on a first-come, first-served basis. This approach prioritizes speed of admission and discounts matching patient needs with medical and surgical resource availability at the right location and time.

Research regarding the strategy of proactively transferring patients between acute care hospitals and non-acute care hospitals in a health system appears to be limited. However, some studies have reported that patients with complex conditions and who have been transferred to another hospital may have poor outcomes [ 9 , 10 ]. Traditionally, patient transfers occur only after a tertiary hospital activates diversion procedures. Our findings suggest that identifying bed demand for patients with severe illness should not be limited to intensive and progressive care units only. Alternatively, a clearly delineated projection of bed demand for patients who require complex services must be considered when creating operational capacity designations.

Although our preliminary findings indicated a positive outcome of our quality improvement initiative, these findings were limited by an increase in bed shortages because of staffing constraints and fluctuating patient volumes in clinics, urgent care centers, and emergency departments. The bed shortages that occurred during the second implementation trial could have affected the integrity of our optimization project data. Increased barriers to inpatient bed access also challenged the enhancement of the exception criteria process, which remains an opportunity for improvement. Emergency medicine practitioners felt particularly challenged in finding recipient hospitals for patients in a timely manner. However, MCHS leadership helped reinforce the value of this project, which prioritized bed access for patients with severe illness and thereby reduced the risk of boarding or transferring critical care patients. Our analysis also lacked sufficient statistical power to establish significant differences. Despite these limitations, our results suggest that our project to optimize hospital bed management in a regional health care system had a generally positive effect.

Operational inefficiencies in the ATC and transfer delays posed some challenges. Future ATC flow enhancements and more robustly staffed ambulance services may help mitigate these issues. Our spoke hospitals were crucial for developing this process and were engaged partners in the trials. The primary challenge was ensuring timely patient transfers from the emergency department and managing the ATC operational flow. Envisaging further collaboration between the ATC and emergency medical service, we aim to use their expertise for process streamlining, enhancing the patient experience, and creating a more coordinated system in SWMN.

Future iterations of regional capacity optimization studies will most likely include analytical forecasts, such as inpatient service line demand beyond the current projection of 6 to 8 weeks [ 11 , 12 , 13 , 14 , 15 , 16 , 17 ]. Bed designations can remain fluid and responsive to analytical projections for months and even years and as hospital data analytics develop. A better understanding of population health needs and advanced at-home care options, including advanced digital health and telemedicine offerings, will affect future hospitalization needs and forecasting of inpatient bed usage.

Some hospital systems may benefit from proactively transferring patients to hospitals in their system that primarily serve patients without severe illness at an early time in the admission cycle and reserving beds at a larger tertiary medical center for patients with severe illness. The traditional approach of filling a hospital, activating diversion procedures, and then seeking alternative placement fails to maximize resource capabilities in a large regional system. As capacity constraints continue to strain hospital systems, innovative approaches to the admission process must be explored. Historical operational models that are relevant to only a single hospital fail to meet the demands resulting from contemporary resource constraints.

Availability of data and materials

The data supporting the conclusions of this article are included within the article.

Abbreviations

Admission Transfer Center

Mayo Clinic Health System

Mayo Clinic Health System hospital in Mankato, Minnesota, USA

Plan-Do-Study-Act

MCHS Southeast Minnesota region

MCHS Southwest Minnesota region

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Brian N. Bartlett

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P.B. served as project manager; N.N.V. provided statistical analysis and created the tables; B.N.B., N.N.V., H.W., D.L.J., J.M.B., A.D.L., P.B., and G.A. directly contributed to writing and editing the manuscript. All authors read and approved the final manuscript.

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Bartlett, B.N., Vanhoudt, N.N., Wang, H. et al. Optimizing inpatient bed management in a rural community-based hospital: a quality improvement initiative. BMC Health Serv Res 23 , 1000 (2023). https://doi.org/10.1186/s12913-023-10008-6

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Received : 13 February 2023

Accepted : 07 September 2023

Published : 18 September 2023

DOI : https://doi.org/10.1186/s12913-023-10008-6

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