Improving hospital layout planning through clinical pathway mining
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Abstract
Clinical pathways (CPs) are standardized, typically evidencebased health care processes. They define the set and sequence of procedures such as diagnostics, surgical and therapy activities applied to patients. This study examines the value of datadriven CP mining for strategic healthcare management. When assigning specialties to locations within hospitals—for new hospital buildings or reconstruction works—the future CPs should be known to effectively minimize distances traveled by patients. The challenge is to dovetail the prediction of uncertain CPs with hospital layout planning. We approach this problem in three stages: In the first stage, we extend a machine learning algorithm based on probabilistic finite state automata (PFSA) to learn significant CPs from data captured in hospital information systems. In that stage, each significant CP is associated with a transition probability. A unique feature of our approach is that we can generalize the data and include those CPs which have not been observed in the data but which are likely to be followed by future patients according to the pathway probabilities obtained from the PFSA. At the same time, rare and nonsignificant CPs are filtered out. In the second stage, we present a mathematical model that allows us to perform hospital layout planning decisions based on the CPs, their probabilities and expert knowledge. In the third stage, we evaluate our approach based on different performance measures. Our case study results based on realworld hospital data reveal that using our CP mining approach, distances traveled by patients can be reduced substantially as compared to using a baseline method. In a second case study, when using our approach for reconstructing a hospital and incorporating expert knowledge into the planning, existing layouts can be improved.
Keywords
Machine learning Sequential pattern mining Clinical pathways Hospital layout planning Healthcare operations management1 Introduction
Planning the layout of a new hospital or reconfiguring existing ones is a complex task and the development of quantitative planning approaches has gained attention since the late 1970s (Elshafei 1977). Confusing layouts can add to patients’ anxiety (Landro 2014) and uncertainty in patient flows challenges strategic decision making in healthcare (Blumenthal 2009). Also, new treatment methods, length of stay reduction and shifting from inpatient to outpatient care can lead to variation and uncertainty in hospitalwide patient flows. Therefore, learning significant clinical pathways (CPs) from data and dovetailing them with strategic hospital decision making in the context of hospital layout planning is the focus of this study.
We approach the problem in three stages: In the first stage, we choose an algorithm to learn significant CPs from large transactional data. In that stage, we address the problem to determine CP probabilities including those which have not been observed in the data but are likely to occur in the future. In the second stage, we present a mathematical model that allows us to perform hospital layout planning decisions based on the CPs and their probabilities as learned in the first stage. In the third stage, we evaluate our approach based on a realworld setting using different performance measures.
The remainder of this paper is structured as follows. In the next section, we review literature on CP mining and the use of CPs for hospital operations management. In this section, we also differentiate our work from other hospital layout planning approaches and highlight similarities and differences with the estimation of rare event probabilities. In Sect. 3, we provide a description of the sequential pattern mining approach employed in our study. In this section, we also present a mathematical model for hospital layout planning and define performance metrics that will be evaluated in our study. In Sect. 4, a brief computational study is provided in order to demonstrate the effectiveness of our approach based on hypothetical data. In Sect. 5, we give a presentation of our results using real data. We finally provide a conclusion and outline streams for further research.
2 Related work
We break down related work into the following four streams: In the first stream, we delimit our work from evidencebased use of CPs since our work follows paradigms from datadriven CP and process mining. In the second stream, we review the use of CPs driven by healthcare operations management. We then highlight similarities and differences to related work on hospital layout planning problems and delimit our work from the estimation of rare event probabilities and sequential pattern mining approaches. We finally provide a summary of similarities and differences with existing work.
2.1 Medical, datadriven clinical pathway and process mining
CPs support a consistent application of evidencebased medicine for the best patient outcomes. Often this has the effect of placing an emphasis on the reduction of unwarranted variation in clinical practice (Wennberg et al. 1977). Similarly, van de Klundert et al. (2010) define CPs as standardized, typically evidencebased health care processes. Instead of building our research on the paradigm of CPs as a subdiscipline of evidencebased medicine, we follow a datadriven approach to infer CPs from data which has been studied by Zhang et al. (2015a), among others. The novelty of our approach is, however, that expert opinions can be incorporated into the learning process as Sect. 3 will reveal and that we bridge the gap between CP mining and operational decision making.
Mining healthcare processes has been the focus of previous literature and a review of approaches is provided by Rojas et al. (2016). Mans et al. (2015) conclude that data mining techniques cannot be used for process discovery, conformance checking, and other forms of process analysis. Some research exists that analyzes process variation in healthcare based on similarity measures between CPs (e.g. Huang et al. 2013; Combi et al. 2009). This is different to our work since we will allow for variations in CPs while filtering out nonsignificant ones. Adherence in CPs is investigated by van de Klundert et al. (2010). Zhang et al. (2015b) and Zhang et al. (2014) apply a hierarchical clustering approach to determine the most likely CP. The authors study the patients’ health conditions and treatment approaches. Iwata et al. (2013) use a clustering and temporal analysis approach in order to identify missing information in existing CPs.
2.2 Healthcare operationsdriven use of clinical pathways
Using CPs as an input of their model, Cardoen and Demeulemeester (2008) provide a strategic instrument for evaluating future changes to the hospital setting. Their simulation can be used to evaluate extreme or unrealistic conditions which can provide insights in the system. On an operational decision level, CPs are used in a variety of patient scheduling applications (Gartner and Padman 2017). For example, Gartner and Kolisch (2014) use CPs to schedule elective patients hospitalwide on scarce resources. Their work uses elective patients’ CPs as an input parameter in their models and assume that once an elective patient contacts the hospital, the pathways are fixed. However, our work bridges strategic decisions with sequential pattern mining for both, elective and nonelective patients.
2.3 Hospital layout planning
In general, layout planning aims at arranging organizational units inside a building such that the available area is used optimally and total distances are minimized. Most layout planning applications arise in industrial environments. When optimizing manufacturing facilities, the most common objective is to minimize traffic areas and traveled distances for produced goods. Thus, reliable information about movements of goods during the production process is needed. Hospital layout planning is typically located on a strategic decision level. Problems are reviewed, among others, in Arnolds and Nickel (2015) where Elshafei (1977) is most relevant for our work because of their travel distance minimization objective and model formulation as a quadratic assignment problem. However, we not only consider the planning of new hospitals but also the reconstruction and improvement of existing ones. We allow for an existing hospital to fix facilities at their location by fixing variables. Furthermore, we add constraints which bound the maximum travel distances between specialties.
2.4 Estimation of rare event probabilities and sequential pattern mining
Estimating the probability of rare CPs has similarities to estimating rare event probabilities. Bachoc et al. (2015) approached the latter problem by adapting the Hastings–Metropolis algorithm on Markov chains. Guyader et al. (2011) estimate the tail probability given quantiles or the other way around to predict quantiles using a tail probability. Similarly to Guyader et al. (2011), they extend the Hastings–Metropolis algorithm. Both approaches are different to our study because, based on the similarity of patient flows, we merge states and rare events that occurred in our data can be filtered out. Also, rare events which have not been observed yet but might occur in the future can be assigned a probability.
Sequential pattern mining discovers frequent subsequences as patterns in a sequence database and a taxonomy of different algorithms is provided by Mabroukeh and Ezeife (2010). Carrasco and Oncina (1994) described an algorithm which is based on generating a prefix tree and then statemerging is carried out using a similarity measure. Jacquemont et al. (2009) extended this algorithm. Similarly, Herbst and Karagiannis (1998) use a Bayesian model merging approach for the induction of Hidden Markov Models. The difference between (Herbst and Karagiannis 1998; Carrasco and Oncina 1994; Jacquemont et al. 2009) lies in particular in the state merging process where Jacquemont et al. (2009) employ a statistical view of it which we follow in our work. Moreover, we incorporate blacklisting into the merging which is not addressed in any of the discussed works.
In conclusion, the approaches proposed in this paper can be categorized into and differentiated from the literature as follows: First, we select and implement a machine learning approach in which clinical activities and their relations between each other are learned from data. Second, we provide a mathematical model that incorporates this information. Finally, we present evaluation criteria based on crossvalidation in order to evaluate the machine learning approach in combination with layout planning. The sequential pattern mining can cope with forbidden state merges so that e.g. expert knowledge can be taken into account by introducing a blacklist. Thus, we are able to incorporate a decision maker’s opinion into the layout planning procedure which is especially important when reconfiguring a layout as our study will reveal in Sect. 5.
3 Methods
In order to detect significant CPs, we evaluate a sequential pattern mining algorithm devised by Carrasco and Oncina (1994) which is extended by Jacquemont et al. (2009). The rationale to select this algorithm from the literature on sequential pattern mining is because the learning approach has only a weak representational bias (van der Aalst 2011) on the process model (Weber 2014). One explanation is that we focus on the hospitalwide specialty flow and therefore, we can neglect parallel bookings on different specialties. The algorithm learns a probabilistic deterministic finite state automaton (PDFA) which is, under certain conditions, equivalent to learning a hidden Markov model (see Dupont et al. 2005). For standard textbooks covering automata theory we refer to Hopcroft et al. (2007).
3.1 Learning a probabilistic deterministic finite state automaton (PDFA)
The algorithm first learns a probabilistic prefixtree acceptor and afterwards, states are merged recursively by using a similarity measure.
Probabilistic Prefix Tree Acceptor (PPTA)
In a PPTA which can be drawn as a graph, states are represented by a circle. In each circle, we write the index of the state and, after a colon, the probability to be final. States which have a probability greater than zero to be final are doublecircled. The initial state is labeled by a “start” arc pointing it. Each transition between the predecessor and the successor state is represented by an arc. The label on each transition consists of the transition symbol and the corresponding transition probability in parentheses. The transition symbol is, in our healthcare application, the clinical procedure which leads from one state to another. The proportion of sequences coming from the previous state to the following state is the probability of this transition. Using training sequences e.g. from Table 1a in Sect. 4, a PPTA can be constructed which is shown in Fig. 1 in Sect. 4.
Probabilistic Deterministic Finite State Automaton(PDFA) PDFAs are a generalization of PPTAs and we can formally describe them as follows: Let Q denote a finite set of states with state \(q_i \in Q \quad \forall i\in \mathbb {N}_{\ge 0}\). Let \(q_0\in Q\) denote the initial state. Let \(\varSigma \) be an alphabet in which letters are denoted by \(z\in \varSigma \) and \(z=\#\) denotes the termination letter of a sequence. In our study, a letter corresponds to a ‘clinical activity’. The term is used as a synonym with ‘clinical procedure’. A clinical activity is performed on a facility (such as an operating room, imaging device or ward). Let \(q(q_i,z)\rightarrow q_j\) be an injective transition function leading to state \(q_j\) from state \(q_i\) with letter z. Specifically, \(q_i\) is equal to \(q_j\) when we have a loop on state \(q_i\) and letter z. Let \(\pi (q_i,z) \in [0,1]\) be a probability function on the transitions and let \(\pi _F(q_i)\in [0,1]\) be a function that assigns to each state a probability to be final. Then, \(A {:}{=} \left( Q, \varSigma , q\left( q_i, z\right) , q_0, \pi \left( q_i, z\right) , \pi _F\left( q_i\right) \right) \) is a tuple that defines our PDFA.
3.2 State merging
In order to avoid sequences to be generated that are from a medical point of view irrelevant, we check each time when we merge two states whether the incoming letter to and the outgoing letter from that new state are reasonable. To avoid that two letters forbiddingly follow each other, we introduce a set \(\mathcal {B}\) (blacklist) of forbidden tuples of letters, denoted by \((z_i,z_j)\in \mathcal {B}\).
3.3 Improving hospital layout planning through clinical pathway mining
In addition to material transportation costs that arise in industrial applications the most significant characteristic in a service environment of a hospital are the distances traveled by patients: Long travel distances do neither support the healing process nor patient satisfaction. To minimize total distances, accurate movement probabilities have to be determined which we will evaluate by combining the PDFA with a layout planning problem which will be introduced next.
3.3.1 Problem description
We extend the wellknown quadratic assignment problem by the possibility to fix specialties to locations as well as taking into account maximum distances between specialties. The formal description of our hospital layout problem reads as follows: Let \(\mathcal {S}\) denote the set of specialties and let \(\mathcal {L}\) denote possible locations in a hospital in which specialties can be located. Let \(f_{i,k}\) be the transition frequency between specialty \(i\in \mathcal {S}\) and specialty \(k\in \mathcal {S}\). Here, a specialty does not only refer to specialty departments such as the radiology department but also to facilities such as the operating theater. Furthermore, let \(d_{j,l}\) denote the distance between location \(j\in \mathcal {L}\) and location \(l\in \mathcal {L}\). Let \(\mathcal {C}\) denote a set of specialty tuples and let \(\overline{D}_{i,k}\) be the maximum distance allowed between two specialties \((i,k) \in \mathcal {C}\), for example, between the surgery room and the ICU. Let \(\mathcal {W}\) denote a whitelist which is a set of tuples \((i,j)\in \mathcal {W}\) representing specialty \(i\in \mathcal {S}\) and location \(j\in \mathcal {L}\). Especially, when specialties such as the emergency department must be located in a defined area as for example near the entrance or when a hospital evaluates reorganization this is an important feature as we will learn in our experimental study.
3.3.2 Model formulation
3.4 Evaluation methods and metrics
The clinical pathways and the layouts can be evaluated using different methods and metrics which are introduced in the following.
3.4.1 Mean absolute deviation (MAD) between clinical pathway probabilities
A trivial baseline approach is to split the transactional data into a training and testing set. For each pathway, the relative frequency is computed. The absolute differences in probabilities between the pathway from the training set and the testing set are then calculated and averaged. Appendix refapp:corssval provides implementation details of this method in Java.
Another approach to calculate the MAD between pathway probabilities is to calculate the pathway probabilities by using the PDFA obtained from the learning set. The absolute differences in probabilities between the PDFA approach and the testing set are then calculated and averaged.
3.4.2 Significant clinical pathways
3.4.3 Error of the layout planning problem (ELPP)
In order to demonstrate the effectiveness of the automaton approach for layout planning, we can incorporate significant clinical pathways into the layout planning problem as formulated in models (2)–(7). Based on the assignment of specialties to locations determined by the mathematical program, we calculate the walking distances using the test data. We finally compare them with the distances obtained using perfect information. In doing so, we assume that both the training and test data are known. We denote this measure as ELPP.
3.4.4 Crossvalidation
Once we have learned the probability of each significant pathway \(p\in \mathcal {P}\), we split them into a set of tuples \((i,j)\in \mathcal {E}\) which we denote as edges where i and j represent letters i.e. clinical activities in the pathways or facilities to be visited. Based on the layout \(\mathcal {X}_f\) learned in fold \(f\in \mathcal {F}\), we can now compute the walking distances based on the distribution of pathways in the test set of pathways.
Based on the MAD and ELPP measures, we can determine confidence intervals. In our experimental study, we use the paired corrected ttest (Nadeau and Bengio 2001) implemented in the Javabased WEKA machine learning library from Witten and Frank (2011).
4 Hospitalwide layout planning under uncertain clinical pathways: an example
The following example illustrates the approach for learning significant CPs. We will demonstrate the effectiveness of the automaton approach by using two different values for the generalization parameter \(\alpha ^\text {aut}\). Furthermore, we show how we feed the result into the layout planning problem.
Sample of 20 sequences broken down by a training (a) and a testing (b) subsample
(a)  
AB  ABA  ABB  ABCA  AC 
ACC  BA  BAA  BC  BCA 
(b)  
BCA  BCA  ABCA  ABCA  AAC 
BAAC  CBAA  CB  CBA  BCAA 
4.1 Learning a probabilistic deterministic finite state automaton (PDFA)
Now, if we evaluate the probability of, for example, pathway ABB, we start at state 0 and reach state 1 by the first letter A. The state is reached by 6 of 10 sequences and as a consequence the transition probability is \(\pi (0,A)=\frac{6}{10}\). Using the second letter B as transition symbol, we reach state 3 with transition probability \(\pi (1,B)=\frac{4}{6}\). Now, the third letter B of the sample pathway leads to state 6 with transition probability \(\pi (3,B)=\frac{1}{4}\). Since our pathway has no more letters, we terminate at this state by probability \(\pi _{F}=1\). We can now compute the probability of the pathway ABB by the product of the transition probabilities times the acceptance probability of the state reached after the last transition. \(\pi ^{ABB}=\frac{6}{10}\cdot \frac{4}{6}\cdot \frac{1}{4}\cdot \frac{1}{1}=\frac{24}{240}=\frac{1}{10}\).
4.2 State merging with \(\alpha ^\text {aut}\) = 0.2
 \(\pi (0,A)\)=

\(\frac{n(0,A)\,+\,n(1,A)}{\sum \nolimits _{z^{\prime }\in \varSigma \cup \{\#\}}n(0,z^{\prime })+\sum \nolimits _{z^{\prime }\in \varSigma \cup \{\#\}}n(1,z^{\prime })} = \frac{6\,+\,0}{10\,+\,6} = \frac{6}{16}\),
 \(\pi (0,B)\)=

\(\frac{n(0,B)\,+\,n(1,B)}{\sum \nolimits _{z^{\prime }\in \varSigma \cup \{\#\}}n(0,z^{\prime })+\sum \nolimits _{z^{\prime }\in \varSigma \cup \{\#\}}n(1,z^{\prime })} = \frac{4\,+\,4}{10\,+\,6} = \frac{8}{16}\) and
 \(\pi (0,C)\)=

\(\frac{n(0,C)\,+\,n(1,C)}{\sum \nolimits _{z^{\prime }\in \varSigma \cup \{\#\}}n(0,z^{\prime })+\sum \nolimits _{z^{\prime }\in \varSigma \cup \{\#\}}n(1,z^{\prime })} = \frac{0\,+\,2}{10\,+\,6} = \frac{2}{16}\).
 \(\pi _F(0)=\)

\(\pi (q(0,\#)) =\frac{n(0,\#)\,+\,n(1,\#)}{\sum \nolimits _{z^{\prime }\in \varSigma \cup \{\#\}}n(0,z^{\prime })+\sum \nolimits _{z^{\prime }\in \varSigma \cup \{\#\}}n(1,z^{\prime })} = \frac{0\,+\,0}{10\,+\,6}= 0.\)
4.3 Improving layout planning through clinical pathway mining
Now, assume we have the following distance matrix between the locations \(d_{i,j} = \left( (0, 50, 200), (50, 0, 50), (200, 50, 0)\right) \). We solve the hospital layout planning problem with the probability distributions from the trivial and the automaton approach using the training sample of pathways. The results are shown in Sect. 4.4.3.
4.4 Evaluation results
Comparing the two PDFAs that were generated above underlines that \(\alpha ^\text {aut}\) has a substantial influence on the merging process and thus on the generalization of the original data. Using \(\alpha ^\text {aut}=0.2\), a very generalized PDFA with only node 0 is left while using \(\alpha ^\text {aut}=0.9\) leads to a less generalized automaton with 3 from the original set of 14 states. Another observation is that all sequences from Table 1(a) are represented by both PDFAs. In addition, the automaton for \(\alpha ^\text {aut}=0.2\) can represent the sequence CBAA which is not included in the learning data, see Table 1(a). However, that sequence is not represented by the automaton’s language for \(\alpha ^\text {aut}=0.9\), see Fig. 4b. The reason for this difference in the generalization of the original data is that \(\alpha ^\text {aut}\) appears in the denominator of the Hoeffding’s bound calculation. The bigger the value of \(\alpha ^\text {aut}\) the lower the threshold and, as a consequence, the lower the generalization. On the contrary, a small value of \(\alpha ^\text {aut}\) leads to a more general automaton.
4.4.1 Mean absolute deviation (MAD) between clinical pathway probabilities
Pathway probability distribution for the three approaches for the training and test data
i  AB  ABA  ABB  ABCA  AC  ACC  BA  BAA 

(a) Pathways 1 to 8  
\(\pi _i^{{ trivial}}\)  0.1  0.1  0.1  0.1  0.1  0.1  0.1  0.1 
\(\pi _{i,0.2}^{{ PDFA}}\)  0.021\(*\)  0.007  0.005  0.001  0.014  0.002  0.021\(*\)  0.007 
\(\pi _{i,0.9}^{{ PDFA}}\)  0.055\(*\)  0.027\(*\)  0.016  0.005  0.005  0.006  0.044\(*\)  0.022\(*\) 
\(\pi _i^{{ test}}\)  0.0  0.0  0.0  0.2  0.0  0.0  0.0  0.0 
BC  BCA  AAC  BAAC  CBAA  CB  CBA  BCAA  MAD 

(b) Pathways 9 to 16 and mean absolute deviations of probabilities  
0.1  0.1  0.0  0.0  0.0  0.0  0.0  0.0  0.1 
0.011  0.003  0.005  0.001  0.001  0.011  0.003  0.001  0.066 
0.015  0.007  0.000  0.000  0.000  0.018  0.009  0.000  0.072 
0.0  0.2  0.1  0.1  0.1  0.1  0.1  0.1 
The figures reveal that the trivial approach fails to estimate probabilities of pathways which are not in the training set. Using the automaton approach, however, the pathway CB which is not in the training set receives a probability of 0.011 and 0.018 for \(\alpha ^\text {aut}=0.2\) and \(\alpha ^\text {aut}=0.9\), respectively. The pathway CBAA which is failed to be discovered by the automaton approach with \(\alpha ^\text {aut}=0.9\), receives a probability of 0.001 for \(\alpha ^\text {aut}=0.2\), similar to the ones of BAAC and AAC.
4.4.2 Significant clinical pathways
To compute significant pathways, we decide to use a large \(\alpha ^\text {sig}=0.33\) because using a small one would give us no significant CP at all. We get \(z_\alpha =0.440\) while we observed \(N=10\) pathways from the training data (see Table 1(a)). For example in the case of \(w=ABA\) and the automaton with \(\alpha ^\text {aut}=0.9\), \(p(ABA)=0.027\), the threshold comes up to \(k=0.440\cdot \sqrt{\frac{0.027\cdot (10.027)}{10}}=0.0226\). Since this is smaller than the probability of the pathway (which was 0.027), the pathway is significant. Significant CPs are flagged by an asterisk in Table 2.
4.4.3 Error of the layout planning problem (ELPP)
Layouts obtained with the trivial, the automata and the perfect information approaches
Approach  Location  ELPP  

1  2  3  
(a)  
Trivial  B  C  A  450 
PDFA \(\alpha ^\text {aut}=0.9\)  B  C  A  450 
PDFA \(\alpha ^\text {aut}=0.2\)  A  B  C  300 
Perfect information  C  A  B  0 
5 Experimental investigation
In the following, we provide an experimental investigation of the presented methods. We first give a description how we generated the sequences (CPs) followed by an overview of the hospital and its distances between different locations. Afterwards, our evaluation metrics are introduced followed by a presentation of the results.
5.1 Data and sequence generation
We tested the sequential pattern mining and layout planning approach experimentally on data from a 350bed sized hospital in Germany. Similarities between the U.S. healthcare system and other developedworld countries are that the data was collected for the billing of diagnosisrelated groups (DRGs). As a consequence, we expect a similar data quality in other DRG systems such as U.S. and developedworld countries that employ DRG systems. We extracted 15,858 CPs from the hospital information system which corresponds to the same number of patients observed in the year 2011.
Overview of the alphabet that corresponds to the different specialties, functional units as well as entrance and exit
A  Internal medicine 
B  Surgery department 
C  Urology department 
D  Gynecology department 
E  ENT department 
F  Orthopedics department 
G  Intensivecare 
H  Ophthalmology department 
I  Radiology department 
J  Operating theater 
N  Functional diagnostics 
X  Entrance and exit 
5.2 Layout of the collaborating hospital
5.3 Distance matrix generation
To run our experiments, we set up transfer time matrix (9). The order of the columns/rows represent the order of the letters, see Table 4. For example, the first column/row represents the current position of the internal medicine department while the second column/row represents the current position of the surgery department and so on. The fifth floor has three positions in which the urology, ear, nose, throat (ENT) and ophthalmology deparment are located. As a consequence, distances are zero between these locations, see column/row 3.
5.4 Evaluation metrics
The sequential pattern mining approach is assessed using two evaluation metrics: Mean absolute deviation (MAD) and the error based on the layout planning problem (ELPP). The MAD is assessed by calculating the mean absolute difference between the probability distribution of significant CPs using the automaton approach and the actual probability distribution. We also assess a trivial baseline approach which uses the prior probability distribution of CPs.
5.5 Results
All computations were performed on a 2.4 GHz PC (Intel Core i7 4700MQ) with 32 GB RAM running a Windows 7 operating system. The mathematical model was coded in Java in an ILOG Concert environment. The solver used was ILOG CPLEX 12.6 (64 bit). We implemented the sequential pattern mining approach in Java, too.
We now compare the performance of the approaches broken down by MAD and ELPP and provide a comparison of the layout of our collaborating hospital with the layout that minimizes the ELPP based on the optimal parameter combination found by varying \(\alpha ^\text {sig}\) and \(\alpha ^\text {aut}\). Finally, we show the results of our discussion of the solution with the hospital.
5.5.1 Crossvalidation results
Overview of the results of the crossvalidation with different \(\alpha \)
\(\alpha ^\text {aut}\)  MAD  ELPP 

0.001  0.083\(*\)  147.222\(*\) 
0.005  0.083\(*\)  147.222\(*\) 
0.01  0.083\(*\)  147.222\(*\) 
0.05  0.083\(*\)  147.222\(*\) 
0.1  0.083\(*\)  147.222\(*\) 
0.2  0.083\(*\)  147.222\(*\) 
0.3  0.083\(*\)  147.222\(*\) 
0.4  0.083\(*\)  147.222\(*\) 
0.5  0.088\(*\)  235.722 
0.6  0.088\(*\)  235.722 
0.7  0.091\(*\)  203.833 
0.8  0.097\(*\)  183.722\(*\) 
0.9  0.086\(*\)  148.722\(*\) 
1.0  0.111\(*\)  229.833 
Trivial approach  0.120  200.970 
The table reveals that for the crossvalidation experiments the lowest MAD is 0.083 and the lowest ELPP is 147.222. The figures also show that the MAD increases with increasing \(\alpha ^{aut}\) which leads to the hypothesis that an overgeneralized automaton overestimates some pathway probabilities.
5.5.2 Results of the MAD and ELPP metrics
Figure 7a, b show the MAD and ELPP results, respectively. We varied \(\alpha ^\text {aut}\) while we fixed \(\alpha ^\text {sig}=0.001\). In addition, we restricted the maximum length of each CP to 5.
The results using MAD as metric show that using a small \(\alpha ^\text {aut}\) value outperforms the trivial approach. More precisely, at a level of \(\alpha ^\text {aut}=0.001\) the MAD becomes approximately 1.65E−4 which is lower than the MAD of the trivial approach which is 1.69E−4. Another observation is that the slope remains negative but it becomes more and more flat until \(\alpha ^\text {aut}=0.8\) is reached.
5.5.3 Evaluation of the hospital layouts
Comparison of the allocation results for the collaborating hospital
Department  Location  

Original  Trivial  Automaton  
AInternal medicine  6.2  2  2 
BSurgery  6.3  6.3  6.3 
CUrology  6.4  6.4  6.4 
DGynecology  6.5  3  3 
EENT department  6.4  6.5  6.5 
FOrthopedics  6.3  6.4  6.2 
GIntensivecare  4  6.4  6.4 
HOphthalmology  6.4  5  5 
IRadiology  2  6.2  6.4 
JOperating theater  5  6.3  6.3 
NFunctional diagnostics  3  4  4 
XEntrance and exit  1  1  1 
5.5.4 Fixing hospital specialties based on recommendations from the hospital
Once again, the entrance and exit have been fixed on their current location. The only specialties that remain on their original position according to both approaches are the surgery department and urology. Both approaches locate the operating theater next to the surgery department which is reasonable. In most cases the trivial and the automaton approach give the same recommendations. Only the radiology department and orthopedic department are interchanged.
Optimal allocation of specialties at the collaborating hospital with some fixed assignments
Department  Location  

Original  Trivial  Automaton  
AInternal medicine  6.2  6.3  6.2 
BSurgery  6.3  6.2  6.3 
CUrology  6.4  6.4  6.4 
DGynecology  6.5  6.3  6.3 
EENT department  6.4  6.4  6.4 
FOrthopedics  6.3  6.4  6.4 
GIntensivecare  4  4  4 
HOphthalmology  6.4  6.5  6.5 
IRadiology  2  2  2 
JOperating theater  5  5  5 
NFunctional diagnostics  3  3  3 
XEntrance and exit  1  1  1 
Having fixed some specialties, both approaches deliver the same result except for the internal medicine and the surgery department which are interchanged. They stay on their current position with the automaton approach. The urology department and ENT specialty are recommended to stay at floor 4. Departments that should be located in upper floors than before are the ophthalmology department from floor 4 to 5, the orthopedics department from floor 3 to 4 and internalmedicine, following the trivial approach, from floor 2 to 3. It is recommended that the gynecology department moves down from floor 5 to 3 and the surgery department regarding to the trivial approach from floor 3 to 2. A possible explanation for this setup may be that the gynecology department and surgery department should be located closer to the functional areas in the ground and first floor.
5.6 Discussion and generalizability of the results
5.6.1 Limitation of using transfer times
When setting up the transfer time matrix, we argue that patients use elevators to get from one specialty to another. This is true for patients that have to be transported, for example, from a ward to the operating theater. However, some patients may simply use the stairs for getting from one floor to another. Also, waiting times for an elevator may vary considerably during a working day. For example, there may be high traffic during breakfast, lunch and dinner times when food has to be transported to the specialties and back. Furthermore, times are depending on walking speeds which might be very different for different patient types or their transportation mode (walking, wheelchair and bed).
5.6.2 Generalizability of the results
The approaches presented in this paper enhance the current state of the art in literature on the strategic decision level in healthcare operations management. It links the work of Cardoen and Demeulemeester (2008) on the strategic decision level with clinical medical work on CP mining. From an operations management point of view, the sequential pattern mining approach presented could be used in a patient scheduling problem which is located on an operational decision level, see Gartner and Kolisch (2014).
5.6.3 Calibration of \(\alpha ^\text {aut}\)
As could be seen in the example and the computational results, the automatonbased approach is sensitive to the \(\alpha ^\text {aut}\) parameter which controls the state merging process. A parameter optimization of \(\alpha ^\text {aut}\) should be carried out before applying the approach in practice. Otherwise, it can happen that a trivial approach outperforms the automatonbased pathway mining approach.
5.6.4 Applicability for existing and new hospitals
Our approach can be used for both applications: reorganization of existing and building of new hospitals. The difference is that for reorganizing a hospital, specialties would be fixed to locations (see Constraints (5)). This might also be necessary when planning a new hospital where for example the emergency room should be near to the entrance area on the ground floor. Fixing any specialty in advance leaves enough room for any other improvement (total travel distances or transfer times) that might be achieved by changing the location of the remaining specialties. However, the improvement potential may be reduced if some facilities are fixed to their locations rather than reorganizing or building the hospital without fixing variables.
6 Conclusion
In this paper, we have dovetailed clinical pathway (CP) mining with hospitalwide layout planning: First, we have selected and extended a machine learning approach to learn CPs from data. Then, we have presented a mathematical model for hospital layout planning which takes into account clinical pathways. It features not only the planning of new hospitals but also the reconfiguration of existing ones by partially fixing specialties to locations. We evaluated the approach in a crossvalidation setting and have shown results based on different evaluation measures and level of detail. Depending on its generalization parameter, the chosen automaton approach outperformed a baseline approach significantly.
Future work will focus on parameter optimization. For example, a full factorial test design will be run to determine a (near optimal) generalization parameter \(\alpha ^{aut}\), paired with \(\alpha ^{sig}\) which can filter out nonsignificant CPs. Alternatively, a heuristic search approach may be beneficial to determine both parameters.
Further extensions will be to evaluate patient types and transportation modes and trading off walking distances and transportation costs. Finally, we will test the applicability of our approach towards operational decisions such as hospitalwide patient scheduling.
Notes
Acknowledgements
The authors sincerely thank the editor and the two anonymous referees for their careful review and excellent suggestions for improvement of this paper.
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