Modeling and Management of a Hospital Department via Petri Nets

Transcription

1 Modeling and Management of a Hospital Department via Petri Nets M. Dotoli, M.P. Fanti, G. Iacobellis, L. Martino Dept. of Electrical and Electronic Engineering, Polytechnic of Bari, Bari, Italy A.M. Moretti Pulmonology Department, General Hospital, Bari, Italy. W. Ukovich Dept. of Electronic, Electrical Engineering and Computer Science, University of Trieste, Trieste, Italy Abstract This paper addresses the management and performance analysis of the pulmonology department of the general hospital of Bari, Italy, focusing on the department workflow and drug distribution system. To this aim, we present a discrete event system model which can be employed as a support for taking decisions regarding the information and automation integration in order to improve the drug procurement logistics and management. The model employs a timed Petri net framework to describe in a concise and detailed way the workflow in the pulmonology department, while considering various ordering and management policies of the department drug storehouse. Keywords-Hospital management, drug distribution system, pharmacy storehouse inventory, Petri nets, performance evaluation, simulation. I. INTRODUCTION During the last decades, cost pressures on hospitals have forced hospital executives to run their organizations in an ever more business-like manner [1]: the constant challenge is to provide high-quality service at ever reduced costs. In order to achieve these purposes, inefficient use of resources should be identified and actions should be taken to eliminate sources of waste. Consequently, researchers and practitioners investigate new approaches that can enable the application to healthcare systems of methodologies and tools coming from different industrial fields. In particular, techniques belonging to the manufacturing and logistics fields and suitable to manage operations, inventories, and resources are increasingly explored for application to the healthcare sector. Formal models and simulation are both useful and effective tools for capacity planning and efficiency improvement of healthcare systems. A hospital system may be effectively described as a Discrete Event System (DES) [2] whose dynamics depends on the interaction of discrete events exhibiting a high degree of concurrency and parallelism. The DES model is the starting point to study the system dynamics and to perform discrete event simulations [3], [4]. Among the DES models, Petri Nets (PN) [5] offer significant advantages because of their twofold representation: graphical and mathematical. Hence, PN may be employed to effectively model emergency medical services and hospitals. In the related literature, high-level PN are employed for modeling workflows in hospital operating theatres [6] or in emergency cardiology departments [7]. In [8] PN are used to verify and simulate an ubiquitous RFID (Radio Frequency IDentification) healthcare system. Moreover, emergency departments are modeled and simulated by PN in [9], [10] and [11]. This paper presents a model which describes in detail the structure and the dynamics of the Pulmonology Department (PD) of the general hospital of Bari (Italy). The model describes in a Timed Petri Net (TPN) framework the complete workflow of patients and the management of the PD pharmacy storehouse. Indeed, a crucial problem of the PD workflow management is identified in the storehouse inventory system. Hence, the paper focuses on the application of different inventory management policies to the PD pharmacy storehouse. Some simulation studies show that the model is able to efficiently describe the PD as well as to foresee the impact of the inventory management policies on the selected performance indices. Accordingly, in order to improve the system management, we propose some inventory control strategies based on the integration of informative tools into the system. The remainder of the paper is organized as follows. Section 2 describes in detail the considered hospital department and Section 3, after reporting some basic definitions about the structure and dynamics of TPN, presents the department TPN model. Moreover, Section 4 reports the simulation data and discusses the results obtained by the numerical simulation. Finally, Section 5 reports the conclusions. II. THE PULMONOGY DEPARTMENT DESCRIPTION This section describes two crucial management issues of the considered case study, represented by the PD of the general hospital of Bari (Italy): the department workflow and the department pharmacy inventory. A. The department workflow description Reception of patients is taken care of by administrative staff of the department. In particular, patients that are admitted in the PD can come from different origins: patients may come from the hospital emergency department; patients may be transferred from a different department of the general hospital; patients are admitted on the basis of the family doctor s prescription /10/$ IEEE

2 Waiting for next day Pulmonology Department Nurse Control of the inventory Counting of drug depletions Drawing up the order [urgent] [Friday] [Monday] Emergency order [Wednesday] Ordinary order Integration order Load of drugs into the department Pharmacy Employee Fulfilling the order within 8h Fulfilling the order within 16h Delivery of the drugs to the department Figure 1. The UML activity diagram of the pharmacy storehouse management. Doctors examine patients with the help of nurses daily between 8 and 9 o clock. All the clinical data and therapies are written on paper forms. The PD includes C 1 =20 available beds to hospitalize patients. In case of lack of beds, patients are either transferred to a different hospital department or placed on a stretcher in the hallway for a maximum of 24 hours. Typically, patients are discharged after an average time period T=120 hours=5 days. B. Management of the department pharmacy storehouse The PD has a dedicated pharmacy and a suitable drug storehouse that requires drugs from the central pharmacy. The ordinary orders to the central pharmacy are performed in a specific week day and are dispatched within 8 hours. Moreover, integration orders to the central pharmacy are performed in two additional specific days; finally, emergency orders may be placed every day in order to integrate drugs or face unpredictable lacks of specific medications. Integration orders are dispatched within 16 hours and emergency orders are dispatched as soon as possible. However, for a good work organization of the central pharmacy, it is convenient to both minimize the overall number of orders and, in particular, the integration and emergency orders by an efficient management policy of the storehouse inventory. This paper focuses on the crucial issue of the management of the PD pharmacy. Inventory management addresses two fundamental issues: when a stock should replenish its inventory (order timing choice) and how much it should order from suppliers for each replenishment (order size choice) [12]. In this paper, the so-called (T,R,Q) inventory control rule is applied [13]. In this policy, at every time step T, if the stock level drops below the reorder point R, quantity Q is ordered. In this case, we assume that R=Q. Moreover, the determination of the quantity Q is performed by taking into account the patients that are currently hospitalized, assuming that the drugs prescribed by the doctors for the hospitalized patients are already administered and not present in the pharmacy. This paper compares three policies in which the quantity Q is determined in three different ways (respectively denoted P1, P2, and P3) described in the following. P1: This is the currently used inventory control policy of the PD. The order quantity Q is constant and determined by the nurses on the basis of a qualitative evaluation of the necessary drugs and of the hospitalized patients. P2: In this technique the order quantity Q is fixed and determined on the basis of an information system that monitors the drug storehouse. In particular, in this strategy the number of hospitalized patients is considered constant in each day. P3: By this policy the order quantity Q is variable and determined based on a data base that monitors the state of the drug storehouse and the number of hospitalized patients. More precisely, in the P2 and P3 strategies the amount Q of drugs to be ordered is chosen by the following rule: Q=D med * S med * N rep, (1) where D med is the number of days that have to elapse till the next ordinary order day, S med is the average number of drugs that are prescribed for each patient per day and N rep is the number of patients hospitalized in the department. In particular, in case P2 N rep is assumed constant and is an average number evaluated on the basis of historical data. Moreover, in case P3 N rep is the monitored real number of hospitalized patients. To concisely describe the system dynamics, we employ UML activity diagrams [14]: UML is a graphic and textual modeling language intended to understand and describe systems from various viewpoints, and activity diagrams aim at describing the logic of the involved processes and the workflow of the system. They are similar to flowcharts, but they allow representations of parallel elaborations in order to explain the critical points in the processes and workflow of the whole system by pointing out all the possible paths, parallel activities, and their subdivisions. The main elements of UML activity diagrams are: initial activities (denoted by solid circles); final activities (denoted by bull s eye symbols); activities, represented by a rectangle with rounded edges; arcs, representing flows, connecting activities; forks and joins, depicted by a horizontal split, used for representing concurrent activities and actions; decisions, representing alternative flows and depicted by a diamond, with options written on either sides of the arrows emerging from the diamond;

3 swim lanes, highlighting responsibilities; signals representing activities sending or receiving a message which can be of two types: input signals (message receiving activities), shown by a concave polygon, and output signals (message sending activities), shown by a convex polygon. Figure 1 shows the UML activity diagram of the PD describing the management of the drug storehouse, enlightening when the different order types are performed. Moreover, the diagram points out the role of nurses and of the pharmacy employees. III. THE TIMED PETRI NET MODEL OF THE PULMONOLOGY DEPARTMENT A. Overview of Timed Petri Nets A TPN (Peterson 1981) is a bipartite digraph described by the six-tuple TPN=(P, T, Pre, Post, F, RS), where P is a set of places, T is a set of transitions partitioned into the set T I of immediate transitions (represented by bars), the set T E of stochastic transitions (represented by boxes), and the set T D of deterministic timed transitions (represented by black boxes). Matrices Pre and Post are the pre-incidence and the postincidence matrices, respectively, of dimension P T. Note that we use symbol A to denote the cardinality of the generic set A. Moreover, F is a firing time vector. The firing time of transition t j T E is an exponentially distributed random variable with mean F j =1/λ j (i.e., the j-th element of vector F), where λ j is the average firing rate of the exponential transition. Each t j T I has zero firing time, i.e., F j =0 and the generic transition t j T D is associated with the constant firing delay F j =δ j. The state of a TPN is given by its current marking which is a mapping M: P N, where N is the set of non-negative integers. M is described by a P -vector and the i-th component of M, indicated with M(p i ), represents the number of tokens in the i-th place p i P. A TPN system <PN,M 0 > is a TPN with initial marking M 0. Given a TPN and a transition t T, the following sets of places may be defined: t={p P: Pre(p,t)>0}, named pre-set of t; t ={p P: Post(p,t)>0}, named post-set of t. A transition t j T is enabled at a marking M if and only if for each p i t j, M(p i ) Pre(p i,t j ). When fired, t j produces a new marking M, denoted as M[tj>M, where for each p i P it holds: M (p i )=M(p i )+Post(p i,t j )-Pre(p i,t j ). (2) Finally, we define a set RS of elements called random switches which associate uniform probability distributions to subsets of conflicting transitions. B. The TPN model of the PD workflow This section describes the TPN model of the PD workflow that is shown in Fig. 2. The TPN can be divided into three parts: the TPN modeling the patients input, the TPN modeling the PD workflow, and the TPN modeling the storehouse management system (see Fig. 2). The TPN modeling the patients input. Places p i with i=1,, 8 and transitions t j with j=1,,8 in Fig. 2 model the input of patients into the PD. In particular, places p i for i=1,, 7 model the day of the week and tokens in place p 8 represent the number of patients waiting for a bed in the department. Transitions t j for j=1,,7 are deterministic transitions modeling the elapsing of days and they fire every 24 hours. Moreover, transitions t j for j=8,,14 are stochastic transitions and model patient inputs into the PD each day of the week. The TPN modeling the PD workflow. Transition t 15 T E in Fig. 2 models the delay for patients registration that is performed by the administrative staff of the PD. Tokens in p 9 are the available beds, tokens in p 66 are hospitalized patients, and tokens in p 50 model patients waiting for the discharge. Moreover, each hospitalized patient is modeled by two places and two transitions: for instance, place p 10 (p 30 ), when marked, is an available (occupied) bed and, when transition t 16 (t 36 ) fires, a patient occupies the bed (is ready for discharge). Hence, when a new patient enters the department a token is added to p 51 and the daily visit phase can start. In particular, a token in p 51 models a doctor visiting patients and place p 52 models a nurse registering the doctor s prescription. When t 58 fires, a token reaches p 63 and another one reaches p 53 : the token in p 63 represents a patient that waits for the next visit, the token in p 53 models the patient receiving drugs. Furthermore, if p 64 (p 65 ) is marked, then the visit process is currently performed (is inhibited until the next day). Moreover, when a drug is administered to a patient, the token representing it is taken from the storehouse (place p 56 ) and one token is simultaneously added to the complementary place p 55 modeling the storehouse capacity. When M(p 55 ) reaches a fixed value, t 60 can fire and an emergency order occurs (token in p 57 ), at the same time the token in p 54 is removed and the drugs dispensing is stopped until the requested drugs get to storehouse (t 61 ). Q3 t67 Q2 t65 C1 t63 p1 t7 t1 p2 t2 p3 t3 p4 t4 p5 t5 t8 t9 t10 t11 t12 p9 p59 p60 p61 Q1 C2-2Q11 C 2-2Q 2 C2-2Q3 W 1 t62 p10 t66 t16 t36 t64 Q3 Q2 C2-Q1 C2-Q2 Q 1 p58 p30 C2-Q3 p50 p8 t15 p66 C2 p29 C2-Q2 p62 C2-Q3 t55 t35 p49 W1 C2-Q1 p6 t13 t56 p56 t61 t6 t14 p7 C3-2Q4 t60 p57 Figure 2. The TPN modeling the PD workflow. W2 W2 C3 t57 p52 t58 t59 C3-Q4 p51 p53 p55 p64 t70 p54 p65 t68 p63 t69

4 TABLE I. TRANSITION DESCRIPTION OF FIG. 2. Transition Firing time in minutes Description Distribution t 1 - t One day elapse Deterministic t 8 - t Patient arrival on working days Exponential t 13 - t Patient arrival on holidays Exponential t 15 5 Patient registration Triangular t 16 - t 35 5 Bed assignment Triangular t 36 - t Average patient stay in hospital Exponential t 56 1 Patient discharge Triangular t Doctor s visit Triangular t Prescription registration Triangular t Drug administering Triangular t 60 1 Emergency occurrence Immediate t Emergency visit Triangular t 62 1 Ordinary order occurrence Immediate t Ordinary order fulfillment time Triangular t 64 1 Monday integration order occurrence Immediate t Monday integration order fulfillment time Triangular t 66 1 Wednesday integration order occurrence Immediate t Wednesday integration order fulfillment time Triangular t 68 1 Daily visits Immediate t 69 5 Daily visits stop until next day Triangular t Daily visits restart Triangular TABLE II. PLACE DESCRIPTION OF FIG. 2. Place p 1 - p 7 p 8 p 9 p 10 - p 29 p 30 - p 49 p 50 p 51 p 52 p 53 p 54 p 55 p 56 p 57 p 58 p 59 p 60 p 61 p 62 p 63 p 64 p 65 p 66 Description Week days Number of patient waiting for entering the PD PD capacity Available beds Occupied beds Number of discharged patients Number of patients to visit Number of prescriptions Number of patients waiting for drugs Enable/inhibit drugs administering Storehouse capacity Storehouse Emergency order Virtual storehouse Ordinary order Monday integration order Wednesday integration order Virtual storehouse capacity Patients visited Daily visits take place Daily visits are inhibited Patients waiting for a bed The TPN modeling the storehouse management system. The storehouse management system is depicted in the lower part of Fig. 2. Place p 56 models the storehouse with its marking M(p 56 ) representing the available drugs and p 55 is the storehouse capacity (with C2=3000 medication units). Place p 58 and its complementary place p 62 represent the virtual storehouse and its capacity (with C2=3000 medication units) In particular, M(p 58 ) represents the number of available drugs, taking into account the number of patients in the PD. More precisely, when a new patient enters the department a fixed quantity of drugs is taken from the virtual storehouse, modeling a booking of drugs for the patient. Hence, M(p 58 ) provides an estimate of the quantity of drugs available in the storehouse for new patients and new orders are given on the basis of this estimate: when M(p 62 ) reaches the reorder point R, the corresponding order is performed. Order types in Fig.2 are differentiated as follows: a token in p 5 enables an ordinary order, while a token in p 3 or p 1 enables an integration order. Tables I and II summarize the meaning of the transitions and the places of the TPN in Fig. 2. Note that in Fig. 2 the various order quantities Q described in section II.B are attached a pedex i with i=1,,4 since they are specified for the different order days. In particular, they are computed according to (1) with the following parameter values: D med1 = 7 days (ordinary orders), D med2 = 4 days (integration orders on Mondays), D med3 = 2 days (integration orders on Wednesdays), D med4 = 1 days (emergency orders); S med = 8 drugs on average administered per patient per day; N repi (i=1,,4) depends on the considered policy and is the evaluated number of patients currently hospitalized in the department. In particular, in case P2 we set N repi=10 with i=1, 4. Conversely, in case P3 N repi =20-M(p 9 ) with i=1,2,3, where 20 is the PD capacity and M(p 9 ) represents the number of free beds. Hence, under P3 the ordered quantity with ordinary and integration orders is variable. In addition, for emergency orders we set N rep4=10 as under strategy P2. As a result, weights Q i with i=1,,4 in Fig. 2 are determined by (1). Finally, in Fig. 2 weights W 1 and W 2 respectively represent prescriptions and administerings associated to each patient. Namely, upon hospitalization of a patient W 1 = S med * T medication units are reserved with W 2 = S med doses per day. IV. SIMULATION SPECIFICATION AND RESULTS A. Data collection and model distributions In this section we discuss the data collection procedures and the related identification of the firing time distributions of transitions t j for j=8,,14 in Fig. 2 modeling the patients input. In order to identify the distribution of the firing times, historical data are collected referring to a period of about 644 days. The firing times identification is performed in the Matlab software [15] that detects the probability distribution that better fits the real data. Figures 3 and 4 compare in the two cases of week days and holidays three probability distributions with the real data: normal, Weibull, and Poisson. From a careful analysis it turns out that the Poisson distribution with mean λ w =2.64 patients/day (working days) and λ h =1.90 patients/day (holidays), better approaches the real data, in comparison with the others that we have estimated. B. Simulation specification The system is analyzed in three different scenarios corresponding to the application of the storehouse management policies P1, P2, and P3 described in section IIB. The management policy P1 corresponds to the real data and current workflow organization. On the contrary, cases P2 and P3 are the results obtained in a simulation framework in order to evaluate information based solutions. Indeed, the storehouse

5 management policies aim at reducing the total number orders, while trying to handle mainly ordinary orders (limiting the socalled integration and emergency orders). The system dynamics is analyzed via numerical simulation using the data reported in Table I. In particular, Table I shows the average values of the exponential transitions and the constant firing delay of deterministic transitions. Furthermore, Table I shows the modal values δ of the processing times. Accordingly, the maximum and minimum values of the range in which the firing delay can vary, are respectively indicated by D δ =1,2δ and d δ =0,8δ. In order to analyze the system behavior, the following performance indices are selected: N O : average number of ordinary orders per year; N I : average number of integration orders per year; N E : average number of emergency orders per year. Number of orders Estimated error (week days) Estimated error (holidays) Maximum Average Weibull Normal Poisson Figure 3. Probability distribution error of input transition firing times (week days). Maximum Average Weibull Normal Poisson Figure 4. Probability distribution error of input transition firing times (holidays) P1 P2 P3 Figure 5. Average number of orders per year. The TPN model of the case study is simulated in the MATLAB environment [15]: such a matrix-based engineering software appears particularly appropriate for simulating the dynamics of TPN based on the matrix formulation of the marking update (2), as well as to describe and simulate PN systems with a large number of places and transitions, such as the system at hand. In each case the system is simulated by a long simulation run of 4 years. The performance indices estimates are deduced by 10 independent replications with a 95% confidence interval. C. Simulation results Figure 5 reports the obtained simulation results, depicting the average number of different orders per year. In particular, the figure shows that the total number of orders, equaling on average N O +N I +N E =102 orders per year in case P1, dramatically diminishes under the P2 and P3 policies, being respectively about 62 (-40%) and 42 (-59%) orders per year in the two cases. As a consequence, under the latter two inventory control strategies, the orders congestion is greatly reduced and the storehouse management is hence more agile and efficient. In particular, under the P2 strategy the ordinary orders number N O is only slightly reduced (-8%) with respect to the current policy P1, but the integration orders number N I is considerably limited (-80%), at the price of only about 3 additional emergency orders per year. On the other hand, with the P3 strategy the ordinary orders number N O is much more reduced with respect to P1 (-55%), but the integration orders number N I is less limited (-70%), with about the same value of performance index N E than P2. Further, comparing policies P2 and P3 it is interesting to remark that under the former strategy each order requires the maximum amount of drugs that can be ordered by the PD storehouse, eventually leading to warehouse dimensioning problems as well as medication stock time issues. Conversely, with the P3 policy orders are managed in a more dynamic way, requesting from the central pharmacy only the necessary amounts of drugs at the right time. Accordingly, under P3 the PD storehouse can be smaller and issues on drugs best before dates are less probable to arise than with P2. Additionally, the simulations show that, under both the suggested drug inventory control policies P2 and P3, the minimum average inventory level equals about 500 drug doses, with virtually no shortage risks. Moreover, the simulation test show that lower inventory levels, corresponding to savings in fixed costs, characterize the variable order quantity policy P3, as opposed to strategy P2. Summing up, the simulations clearly show that information integration into the department inventory management can deeply increase the efficiency of the storehouse and distribution system. V. CONCLUSIONS We propose a timed Petri net model for analyzing and simulating the workflow of a hospital department -starting from the arrival of patients to their discharge- while considering the drug distribution system management, a key process in the department workflow. In particular, several inventory control policies are considered and compared by

6 means of some discrete event simulations carried out on a real case study, showing that information integration can deeply increase the efficiency of the storehouse and distribution system. The proposed model can be used as a starting point for setting up a decision tool for the design and dimensioning of hospital departments. Further research includes enhancing the model of drug prescription and administering taking into account the personalized amount of medication units prescribed to patients (rather than average dosages) and the actual implementation of the presented inventory strategies in the real case study. ACKNOWLEDGEMENT This work was supported by Fondazione Cassa di Risparmio di Puglia under the research project Modeling and control of logistic systems characterized by high information integration. REFERENCES [1] J. Belien and E. Demeulemeester, A branch-and price approach for integrating nurse and surgery scheduling, European Journal of Operational research 189, 2008, pp [2] C.G. Cassandras and S. Lafortune, Introduction to Discrete Event Systems. Second Edition, New York, NY, USA: Springer, [3] M.M. Gunal and M. Pidd, Interconnected DES Models of Emergency, Outpatient and Inpatient Departments of a Hospital, Proceedings of the 2007 Winter Simulation Conference, [4] A. Kumar and S.J. Shim, Eliminating Emergency Department Wait by BPR Implementation, proceedings of the 2007 IEEE IEEM, [5] J.L. Peterson, Petri Net Theory and the Modeling of Systems. In: Prentice Hall, Englewood Cliffs, NJ, USA, [6] Y.T. Kotb and A.S. Baumgart, An extended Petri net for modelling workflow with Critical sections, proceedings of the 3th INFORMS Workshop on Data Mining and Health Informatics, 2005, pp [7] M. Dotoli, M.P. Fanti, Mangini A.M., and W. Ukovich, A Continuous Petri Net Model for the Management and Design of Emergency Cardiology Departments, 3rd IFAC Conference on Analysis and Design of Hybrid Systems, Zaragoza, Spain, September 16-18, [8] S.S. Choi, M.K. Choi, W.J. Song, and S.H. Son, Ubiquitous RFID Healthcare Systems Analysis on PhysioNet Grid Portal Services Using Petri Nets, proceedings of IEEE ICICS, 2005, [9] H.H. Xiong, M.C. Zhou, and C.N. Manikopoulos, Modeling and Performance Analysis of Medical Services Systems Using Petri Nets, proceedings of IEEE International Conference on Systems, Man and Cybernetics, 1994, [10] M.Criswell, I. Hasan, R. Kopach, S. Lambert, M. Lawley, D. Mc.Williams, G. Trupiano, and N. Varadarajan, Emergency Department divert avoidance using Petri nets, proceedings of the Systems Enginenering conference, San Antonio, Texas, 2007, USA. [11] G. Amodio, M.P. Fanti, A.M. Mangini, and W. Ukovich, Modelling and Performance Evaluation of Cardiology Emergency Departments by Petri Nets, ORHAS 2009, July 12-17, Leuven, Belgium.. [12] H. Chen, L. Amodeo, F. Chu, and K. Labadi, Modeling and performance evaluation of supply chains using batch deterministic and stochastic Petri nets, IEEE Trans. Automat. Sci. Eng., vol. 2, pp , [13] R. Furcas, A. Giua, A. Piccaluga, and C. Seatzu, Hybrid Petri net modelling of inventory management systems, European Journal of Automation APII-JESA, vol. 35, no. 4, pp , [14] R. Miles and K. Hamilton, Learning UML 2.0. O Reilly Media, Sabastopol CA USA, [15] The Mathworks, MATLAB Release Notes for Release 14. Natick, MA: The Mathworks, 2006.