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IMPROVING THE EFFICIENCY OF A CLINICAL LABORATORY: A MATHEMATICAL APPROACH
IMPROVING THE EFFICIENCY OF A CLINICAL LABORATORY: A MATHEMATICAL APPROACH Bruni Maria Elena, Patrizia Beraldi Domenico Conforti
Dipartimento di Elettronica, Informatica, Sistemistica, Universit´ a della Calabria, Via P. Bucci 41C, 87030 Rende (Cosenza), Italy Tel. +39-0984-494733 E-mail: [email protected]
Abstract: The increasing automation of laboratory equipment has produced strong impacts on the organizational structure and technical requirements of clinical laboratories. To achieve an appropriate level of service quality, it is important to involve well-orchestrated multidisciplinary teams in the design effort. A good program optimizes facilities use, fully accomodating user needs without being wasteful. This complex task involves the study and the definition of critical spatial and organizational relationships to ensure adequate capacity across the entire system. This study provides an optimization model to address decisions regarding the organization of clinical laboratories. Easily quantifiable outcomes are used including those with intrinsic financial benefits. The model was validated on the basis of real data collected from two clinical laboratories. The computational experiments carried out have shown the validity of the proposed model. c Copyright 2006 IFAC Keywords: Operations research; Work organization; Time delay.
1. INTRODUCTION Clinical laboratory is an important part of medical practice because laboratory information is essential for diagnosis and management of patients. Health care organizations are reevaluating the role of these services and attempting to drive down costs and improving patient outcomes. Outcome in this context has more than one meaning depending on the subject involved in the definition process. When evaluating processes, managers often use a very limited set of performance measures and indicators. This set typically includes financial measures as profit or the percentage of cost reduction. From a different point of view, non-financial
measures are often more important. For the patient, for example, receiving prompt and effective treatment represents an important goal. The challenge for a clinical laboratory in managed care is to optimize outcomes for both patient and health care system. Approaching this goal will require a clear understanding of the important performance measures to consider. Laboratory management faces various types of planning problems, being responsible for information management, human resource management, cost/profit analysis, technology assessment, patient management to mention a few. The different functions of the laboratory can not be merged efficiently without the help of quantitative methods to assist the management in handling such variegate planning
problems. The purpose of our study is to develop a mathematical model to represent the laboratory organization and to validate it on real case studies. In order to capture the specific aspects considered in our analysis we briefly describe the functioning and the general structure of a clinical laboratory. The laboratory is logically divided into different working areas as accessioning/specimen processing, urine/stool analysis, hematology, clinical chemistry, immunology etc. The laboratory section for endocrinology and metabolic studies is not considered here, because of its predominant research orientation. The primary architectural determinants of the optimal organization of laboratory services include administrative costs, labor costs, reliability of results and turnaround times defined as the time from the initial order to the receipt of the result. As we will explain in the next section, we address the problem of optimal laboratory organization by using a multiperiod programming model, in which the number of requests to process is defined dynamically in each period. Once the sample (also called test or specimen in this work) has been collected and delivered, the laboratory operations can be performed. The operations can be separated into three phases: preanalytical, analytical and postanalytical. The preanalytical operations in general involve only specimen handling; a few samples need to be prepared before the analysis. We will focus on the analytical phase, i.e. the analysis of the specimen by an appropriate instrument. The postanalytical operations are devoted to the collection and transmission of results. Not all the samples are equal. Critical-care medicine requires rapidity of treatment decisions and clinical management. To meet the objectives of critical-care medicine, the critical-care laboratory (referred also as a stat laboratory) must consider as primary objective the immediate data transmission for an efficient management of medical emergencies. Usually, the stat laboratory is physically part of the routine laboratory, and personnel and instrumentation are common for both the stat and the routine laboratories. Under this plan, specimens submitted for stat analysis must have priority over all other works in the routine laboratory. It is evident that the stat samples represent a critical factor impacting the laboratory performances because they require interruption of workflow in the main laboratory routine work. Our model explicitly considers the importance of the stat analysis accounting for the urgency in a straightforward way. Laboratory behavior depends on how the laboratory is staffed, equipped and organized. All these aspects are intimately related. To give an example, the laboratory automation is changing the role of the clinical technicians. In fact, automated analytical systems decrease the need for skilled technicians in the various practical
manual steps involved in laboratory work, but also strengthen the consultative and managerial roles of the clinical staff. Due to these changes the number of senior technicians (who usually are biologists or clinical chemists) is rapidly decreasing and the working role of these specialists is eminently devoted to postanalytical tasks as recording and transmitting test results, performing administrative work, making telephone calls, and so on. Further we assume that technicians are interchangeable due to the nature of the working tasks and to cross-training. In such complex system the resources involved have to be allocated efficiently and the relevant outcomes optimized. In the next section we propose a mathematical model able to face the complex designing problem of a clinical laboratory. Given the variability and the complexity of the problem we divide the modeling process into two parts. First we provide a detailed description of a general laboratory layout and organization, second we define a set of performance measures and indicators relevant for the purpose of our study. Finally, we build a mathematical model and examine results-related validation.
2. MATHEMATICAL MODELING OF CLINICAL LABORATORY STRUCTURE AND OPTIMIZATION MANAGEMENT In the last years the techniques of operations research have been applied to various laboratory planning problems. In order to place our work in the correct perspective we review the existing literature in the field. Simulation models have been proposed for determining the performance of different planning rules given the equipment and staffing of the clinical laboratory (Van Merode, 1996), describing the dynamics of a workstation that receives a mixture of routine and emergency specimens (Winkel, 1984), describing the functioning of an entire laboratory (Vogt and Klose, 1994), investigating which analyzer should be purchased to fulfill existing needs (Groothuis and van Merode, 2002). It is worth noting that, while simulation techniques have been used to describe laboratory functioning, the use of optimization techniques to support laboratory management is rather original. In (Van Merode and Hasman, 1998) a mathematical model is presented to face the problem of the assignment of technicians to organizational unit. Our model has similar motivations of (Van Merode and Hasman, 1998), but is different for many aspects. In (Van Merode and Hasman, 1998) the only critical resource is the number of technicians. This choice restricts the applicability of the model to existing laboratories where the technology equipment is already given.
Fig. 1. Example of a lay-out of a laboratory. We have N analytical areas, M workstations and P methods. We would mention that one of the major cost component of a laboratory in the design phase is the cost of instruments (Travers, 1997). The set of instruments to select and purchase has a dramatic influence on the overall performance of the laboratory from both a financial and quality perspective (Burtis, 1987). Furthermore, the technology equipment of a laboratory should be dimensioned on the basis of the type and level of the demand. A second difference between the two works is the assumption made on the due time of orders. In (Van Merode and Hasman, 1998) all production process that started in a certain period is assumed to be completed within that period. In other words the release time and the due time of each test are forced to coincide with the starting time and the ending time of a period. As the authors acknowledged, for orders with a large processing time the optimality of the solution may be negatively influenced. To overcome this difficulty our work proposes a multiperiod model that, in addition, allow us to give an upper bound on the turnaround time of the order. Here we report our findings on the use of modeling techniques of operations research to represent the complex management of clinical laboratory.
2.1 Planning problems and performance measures. As mentioned in section 1, a laboratory can be distinguished in different working areas. These areas (also called analytical areas) are logically separated. In each analytical area different groups of tests (methods) can be performed. Each method is a combination of tests requested on samples that have been prepared in the preanalitycal area. After preparation, samples are processed in an analytical area equipped with a set of workstations of the same or different type. A workstation is a machine that can perform only certain methods. In figure 1 an example of a laboratory layout is reported. As we can see, each analytical area may have a number of workstations able to perform a number of methods. In general the strategic problem of choosing the appropriate technological equipment of a laboratory has a significant
impact on how the clinical laboratory is staffed, organized, and managed. The greatest cost attributable to the laboratory is certainly due to the technological equipment. How best to cope with a rising clinical demand while responding to mandated reductions in operating costs strongly depends on the workstations available and on the assignment of the tests to be performed to these workstations. In our model we address the problem of the identification of an optimal set of methods in each analytical area. This information can be used as a valid recommendation for the laboratory director who is concerned about the ordering and the the purchasing of the workstations. Because todays decisions will have consequences for some years to come, selection of an instrument or structural changes of laboratory organization requires a detailed understanding of the field and the support of quantitative methods for making better decisions. In order to choose the best equipment, it is important to have an insight into the expected number of test requests that have to be processed in the laboratory. Different request forms may contain different method requests; each method has different frequency and importance. The demand for each period within a day can be derived in terms of the volume of tests distributed over the method types and priority degrees. Priority indicates the length of the interval between the actual time and the due-time. For example for stat tests the due-time can be less than one hour. The flow of requests can vary substantially over the day and in general has a peak during working hours. We take into account the variability of the demand flow during the day dividing the day into periods. In each period we may have a different arrival rate and a different request mix. Because technicians are requested to be at the workstation when a request has to be processed, in our model the staffing problem is addressed and solved. It is worth recalling that we assume that all technicians are equally skilled. As far as the performance measures are concerned, we report in table 2.1 1 examples of performance measures and the corresponding indicators. The set of performance measures to optimize should relate to all aspects which management considers to be important. Table 1. Performance measures and indicators Performance measures Financial performance Internal processes Learning and growth
Client perspective
1
Example Cost reduction Profit realization Quality control Throughput times Research and development Staff development Investments Turnaround times Test diversity Value for money
Source (Van Merode and Goldschmidt, 1998)
Indicator Money Money Score Time N/A Nmb of Fte Money Time Nmb of tests N/A
In order to formulate an optimization model aimed at determining the optimal structure of the clinical laboratory, we have selected from this list some relevant performance measures. We formulate the selected performance measures using mathematical expressions to include in our model. In particular, we detect one financial performance (the overall costs), and two performances reflecting the client perspective, the turnaround time and the test diversity. Using this set of performance measures, the best laboratory organization should achieve the client satisfaction, minimizing costs. Now we are ready to present a formal description of the proposed optimization model (LABOPT).
min
X X X X ( cml Nmle + hIe,l ) + e∈E l∈L m∈M
X X
l∈L
(1)
aml yml
l∈L m∈M
where
X
Iel = Ule De − X
(2) Nmle topm ∀l, ∀e
(3)
m∈M
yml ≥ 2 ∀m ∈ Mu
(4)
yml ≥ 1 ∀m ∈ (M \ Mu )
(5)
l∈L
X l∈L
Nmle ≤ yml
e X
δmef
f =1
∀m ∈ M, ∀l ∈ Lm , ∀e ∈ E (6) g g X XX X Nmlg ≥ δmef − Nml(g−1) e=1 f =e
l∈L
2.2 Mathematical programming formulation. Let L be the set of the analytical areas present in the laboratory, M the set of all possible methods and E the set of periods in which a working day is divided. We can define two subsets Mu and Lm ; Lm is the set of analytical areas in which the method m can be processed and Mu ⊏ M is the set of methods usually requested for emergencies. We would like to observe that Lm can coincide with L or, if the model is used to explore the possibility of improving an existing laboratory, be a proper subset of L. We will use indices l to indicate analytical areas, m to indicate methods and e, f, g to indicate periods. We define the following list of model parameters; let UeT OT be the total number of technicians available in a period, aml the activation cost of the method m in the area l, h the cost per hour of the technicians, cml the processing cost of one sample in the area l with the method m, δmef the number of samples arrived in period e that have to be processed within the period f with the method m, topm the time spent by the technician to process one sample with the method m, wm the processing time of one sample with the method m, De the duration of the period e, P the fraction of the period devoted to manual operations. The decision variables are : • Nmle the number of samples to process with method m in the area l during the period e • Ule the number of technicians to assign to the area l in the period e • yml a binary decision that takes value one if the method m is activated in the area l and zero otherwise The model can be stated as follows:
l∈Lm
∀m ∈ M, ∀g = 1 . . . |E| − 1, X topm Nmle ≤ Ule De P ∀e, ∀l
(7) (8)
m∈M
X
Nmle (wm + topm ) ≤ De ∀e ∈ E
(9)
m∈M
X
Ule ≤ UeT OT ∀e ∈ E
l∈L
X X X
Nmle =
l∈L m∈M e∈E
X
X XX
(10) δmef (11)
m∈M e∈E f ≥e
Ule ≥ 2 ∀e ∈ E
(12)
l∈L
yml ∈ {0, 1}, Nmle , Ule integer ∀m ∈ M , ∀l ∈ L, ∀e ∈ E
(13)
Constraints (4),(5) impose that each method has to be assigned at least to one area, that is the laboratory must be equipped with one or more workstations able to process that type of method. In particular constraint (4) imposes the presence of backup instruments so that the stat tests can be processed even when the primary instrument is out of service. Constraints (6) assure that the number of samples to process with a particular method in a period does not exceed the number of samples arrived until that period. Constraints (7) express the multiperiod linking constraints and state that the number of samples to be processed at the current period must be equal to the number of samples to be processed within the current period minus the number of samples processed at the previous period. Constraints (8) limit the time devoted to manual operations to a fraction of the total duration of the period considered, while constraints (9) assure that the time assigned to scheduled samples is less than the time available in the period. The number of technicians assigned in a period can not exceed the maximum staff available (10). Constraint (11) assures that all the
requests are assigned and processed according to the relative due-time. Finally, the reliability of the operations is guaranteed imposing in each period at least two technicians in the laboratory. The objective function optimizes three measures for the P efficiency P Pof the laboratory. The first term ( e∈E l∈L m∈M cml Nmle ) minimizes the total cost of containers required to collect the liquid to be examined with relative gain in quality (less invasivity) P from P patient’s point of view. The second term l∈L m∈M (aml yml ) reflects the activation cost of the methods in the laboratory working areas. Imposing the activation cost equal to one the objective function achieves the minimization of the methods to be activated. These two terms would appear to reduce the laboratory costs both by curbing the number of tests performed and by reducing the costs P of individual tests. Last, but not least the term l∈L h(Ule De − P N top ) involves the cost reduction due mle m m∈M to the idle time of the technicians. In fact the first component is the total time of assigned technicians in a period, whereas the second one is the effective working time. It is worth noting that all the performance measures that we considered relevant are taken into account in our model. As a final remark, the three terms are expressed in the same unit leading to a standard optimization problem.
3. DISCUSSION OF RESULTS AND CONCLUSIONS The validity of the model LABOPT was demonstrated by comparison of the optimal solution obtained by LABOPT with the actual situation, and the optimization output was considered plausible by laboratory experts. However, none of the two laboratories has implemented the proposed solution. Achieving a genuine partnership between operational researchers and health institutions requires the partners to be equally open to new methods of analysis and to enter into new modes of discourse, and this may requires long time. The model has been tested on data from two italian laboratories, differing greatly in layout and workload. To instruct the optimization model, specific data have been collected by studying written specifications and through interviews with technicians. The data collection has concerned the technical description of each method, the processing time per sample, the arrival time of samples throughout the day at the laboratory and the total number of arrivals within each period. Demand data were estimated on the basis of historical data collected form the laboratory over the past years. As we mentioned before, the two laboratories under study are very different. In fact one of them (a) is a private research laboratory, as opposite
Fig. 2. Idle time per period lab (b).
Fig. 3. Total idle time for lab (a). the second one (b) is a laboratory embedded in a hospital. The difference is also evident in the arrival request flow which is concentrated only in the morning in the laboratory (a), while the laboratory (b) works all the day without any interruption. The width of each period as well as the number of periods considered is different in the two cases. Furthermore the laboratory (b) has also a flow of emergency requests to be processed within the arrival period. In table 3 the distribution and the length of periods in the day is reported for both the laboratories. We set respectively the number of analytical areas and methods equal to the actual configurations of the laboratories (99 methods and 14 analytical areas for both the laboratories). The number and the mix of methods activated in each analytical area is different between the laboratories. The full problem data are reported in (Bruni M. E., 2005) For each laboratory the solution obtained by our model LABOPT is compared with the real existing laboratory organizations. In figure 2 the results concerning the technicians idle time are presented for the laboratory (b). As we can observe the idle time of our solution is less than the actual idle time of operators. A reduction of the total idle time can be observed also for the laboratory (a) in figure 3. In figure 4 is reported the number of technicians assigned by our model compared with the number of technicians working in the laboratory. For the sake of clarity we note that this comparison is
Table 2. Periods Period Lab (a) Lab (b)
Slot time Length (min) Slot time 7:30-8:30 Length (min)
I 7:30-8:30 60 8:30-10:30 60
II 8:30-10:30 120 10:30-12:30 120
III 10:30-14:30 240 12:30-14:30 120
IV
V
VI
14:30-18:30 120
18:30-7:30 240
780
REFERENCES
Fig. 4. Number of technicians; laboratory (b). made without further refinements without considering the delicate issue of the crew resource management in a clinical context. Nonetheless we observe that the maximum number of technicians assigned by our model is ever less then the actual number of technicians working at the laboratory. Similar considerations can be drawn for the laboratory (a). Our model provides useful information for assessing the performance of the laboratory and for choosing the appropriate laboratory structure. However, further attention should be paid to the estimation of the parameters used within the model. We do not believe that description and analysis of the existing situation will simply suggest alternative solutions in a deterministic manner, but we are strongly convinced that the proposed model could be used to gain insights into the existing situation, as well as possibilities for redesign. In this context the first item to be examined is how the laboratory is divided into areas and how the staff is assigned to these areas. Finally it is worth noting that the turn around time is in any case less then the difference between the arrival time and the due time. This allowed us to tackle efficiently the emergency request arrivals and to limit the turnaround time to be less then a given upper bound. We think that operations research methods can be a valuable tool in supporting laboratory directors to make better management decisions, based on rationally understandable experiments rather than on subjective feelings and beliefs, even if these are grounded on a long personal experience in the field.
Bruni M. E., Beraldi P., Conforti D. (2005). A mathematical approach for the management of a clinical laboratory. Technical report. DeHealthLab-Unical. Burtis, C.A. (1987). advanced technologyand its impact on the clinical laboratory. Clinical chemistry 33(3), 352–357. Groothuis, S., H. M.J. Goldschmidt E. J. Drupsteen J. C.M. de Vries A. Hasman and G.G. van Merode (2002). Turn-around time for chemical and endocrinology analyzers studied using simulation. Clin. Chem. Lab. Med. 40(2), 174–181. Travers, E. (1997). Clinical Laboratory Management.. number ISBN 0-683-08376-7.. Williams and Wilkins. Van Merode, G. G., S. Groothuis and H. J.M. Goldschmidt (1998). Workflow management : changing your organization through simulation. 4 th Conference on Quality Revolution in Clinical Laboratories, Antwerp, Belgium. Van Merode, G.G, A. Hasmanb J. Derks B. Schoenmaker H.M. J. Goldschmidt (1996). Advanced management facilities for clinical laboratories. Computer Methods and Programs in Biomedicine (50), 195–205. Van Merode, G.G., M. Oosten O.J. Vrieze J. Derks and A. Hasman (1998). Optimisation of the structure of the clinical laboratory. European Journal of Operational Research (105), 308–316. Vogt, W., S. L. Braun F. Hanssmann F. Liebl G. Berchtotd H. Blaschke M. Eckert G. E. Hoffmann and S. Klose (1994). Realistic Modeling of Clinical Laboratory Operation by Computer Simulation. Clinical Chemistry 40(6), 922–928. Winkel, P. (1984). Operational research and cost containment: A general mathematical model of a workstation. Clinical Chemistry 30(11), 1758–1764.
Dipartimento di Elettronica, Informatica, Sistemistica, Universit´ a della Calabria, Via P. Bucci 41C, 87030 Rende (Cosenza), Italy Tel. +39-0984-494733 E-mail: [email protected]
Abstract: The increasing automation of laboratory equipment has produced strong impacts on the organizational structure and technical requirements of clinical laboratories. To achieve an appropriate level of service quality, it is important to involve well-orchestrated multidisciplinary teams in the design effort. A good program optimizes facilities use, fully accomodating user needs without being wasteful. This complex task involves the study and the definition of critical spatial and organizational relationships to ensure adequate capacity across the entire system. This study provides an optimization model to address decisions regarding the organization of clinical laboratories. Easily quantifiable outcomes are used including those with intrinsic financial benefits. The model was validated on the basis of real data collected from two clinical laboratories. The computational experiments carried out have shown the validity of the proposed model. c Copyright 2006 IFAC Keywords: Operations research; Work organization; Time delay.
1. INTRODUCTION Clinical laboratory is an important part of medical practice because laboratory information is essential for diagnosis and management of patients. Health care organizations are reevaluating the role of these services and attempting to drive down costs and improving patient outcomes. Outcome in this context has more than one meaning depending on the subject involved in the definition process. When evaluating processes, managers often use a very limited set of performance measures and indicators. This set typically includes financial measures as profit or the percentage of cost reduction. From a different point of view, non-financial
measures are often more important. For the patient, for example, receiving prompt and effective treatment represents an important goal. The challenge for a clinical laboratory in managed care is to optimize outcomes for both patient and health care system. Approaching this goal will require a clear understanding of the important performance measures to consider. Laboratory management faces various types of planning problems, being responsible for information management, human resource management, cost/profit analysis, technology assessment, patient management to mention a few. The different functions of the laboratory can not be merged efficiently without the help of quantitative methods to assist the management in handling such variegate planning
problems. The purpose of our study is to develop a mathematical model to represent the laboratory organization and to validate it on real case studies. In order to capture the specific aspects considered in our analysis we briefly describe the functioning and the general structure of a clinical laboratory. The laboratory is logically divided into different working areas as accessioning/specimen processing, urine/stool analysis, hematology, clinical chemistry, immunology etc. The laboratory section for endocrinology and metabolic studies is not considered here, because of its predominant research orientation. The primary architectural determinants of the optimal organization of laboratory services include administrative costs, labor costs, reliability of results and turnaround times defined as the time from the initial order to the receipt of the result. As we will explain in the next section, we address the problem of optimal laboratory organization by using a multiperiod programming model, in which the number of requests to process is defined dynamically in each period. Once the sample (also called test or specimen in this work) has been collected and delivered, the laboratory operations can be performed. The operations can be separated into three phases: preanalytical, analytical and postanalytical. The preanalytical operations in general involve only specimen handling; a few samples need to be prepared before the analysis. We will focus on the analytical phase, i.e. the analysis of the specimen by an appropriate instrument. The postanalytical operations are devoted to the collection and transmission of results. Not all the samples are equal. Critical-care medicine requires rapidity of treatment decisions and clinical management. To meet the objectives of critical-care medicine, the critical-care laboratory (referred also as a stat laboratory) must consider as primary objective the immediate data transmission for an efficient management of medical emergencies. Usually, the stat laboratory is physically part of the routine laboratory, and personnel and instrumentation are common for both the stat and the routine laboratories. Under this plan, specimens submitted for stat analysis must have priority over all other works in the routine laboratory. It is evident that the stat samples represent a critical factor impacting the laboratory performances because they require interruption of workflow in the main laboratory routine work. Our model explicitly considers the importance of the stat analysis accounting for the urgency in a straightforward way. Laboratory behavior depends on how the laboratory is staffed, equipped and organized. All these aspects are intimately related. To give an example, the laboratory automation is changing the role of the clinical technicians. In fact, automated analytical systems decrease the need for skilled technicians in the various practical
manual steps involved in laboratory work, but also strengthen the consultative and managerial roles of the clinical staff. Due to these changes the number of senior technicians (who usually are biologists or clinical chemists) is rapidly decreasing and the working role of these specialists is eminently devoted to postanalytical tasks as recording and transmitting test results, performing administrative work, making telephone calls, and so on. Further we assume that technicians are interchangeable due to the nature of the working tasks and to cross-training. In such complex system the resources involved have to be allocated efficiently and the relevant outcomes optimized. In the next section we propose a mathematical model able to face the complex designing problem of a clinical laboratory. Given the variability and the complexity of the problem we divide the modeling process into two parts. First we provide a detailed description of a general laboratory layout and organization, second we define a set of performance measures and indicators relevant for the purpose of our study. Finally, we build a mathematical model and examine results-related validation.
2. MATHEMATICAL MODELING OF CLINICAL LABORATORY STRUCTURE AND OPTIMIZATION MANAGEMENT In the last years the techniques of operations research have been applied to various laboratory planning problems. In order to place our work in the correct perspective we review the existing literature in the field. Simulation models have been proposed for determining the performance of different planning rules given the equipment and staffing of the clinical laboratory (Van Merode, 1996), describing the dynamics of a workstation that receives a mixture of routine and emergency specimens (Winkel, 1984), describing the functioning of an entire laboratory (Vogt and Klose, 1994), investigating which analyzer should be purchased to fulfill existing needs (Groothuis and van Merode, 2002). It is worth noting that, while simulation techniques have been used to describe laboratory functioning, the use of optimization techniques to support laboratory management is rather original. In (Van Merode and Hasman, 1998) a mathematical model is presented to face the problem of the assignment of technicians to organizational unit. Our model has similar motivations of (Van Merode and Hasman, 1998), but is different for many aspects. In (Van Merode and Hasman, 1998) the only critical resource is the number of technicians. This choice restricts the applicability of the model to existing laboratories where the technology equipment is already given.
Fig. 1. Example of a lay-out of a laboratory. We have N analytical areas, M workstations and P methods. We would mention that one of the major cost component of a laboratory in the design phase is the cost of instruments (Travers, 1997). The set of instruments to select and purchase has a dramatic influence on the overall performance of the laboratory from both a financial and quality perspective (Burtis, 1987). Furthermore, the technology equipment of a laboratory should be dimensioned on the basis of the type and level of the demand. A second difference between the two works is the assumption made on the due time of orders. In (Van Merode and Hasman, 1998) all production process that started in a certain period is assumed to be completed within that period. In other words the release time and the due time of each test are forced to coincide with the starting time and the ending time of a period. As the authors acknowledged, for orders with a large processing time the optimality of the solution may be negatively influenced. To overcome this difficulty our work proposes a multiperiod model that, in addition, allow us to give an upper bound on the turnaround time of the order. Here we report our findings on the use of modeling techniques of operations research to represent the complex management of clinical laboratory.
2.1 Planning problems and performance measures. As mentioned in section 1, a laboratory can be distinguished in different working areas. These areas (also called analytical areas) are logically separated. In each analytical area different groups of tests (methods) can be performed. Each method is a combination of tests requested on samples that have been prepared in the preanalitycal area. After preparation, samples are processed in an analytical area equipped with a set of workstations of the same or different type. A workstation is a machine that can perform only certain methods. In figure 1 an example of a laboratory layout is reported. As we can see, each analytical area may have a number of workstations able to perform a number of methods. In general the strategic problem of choosing the appropriate technological equipment of a laboratory has a significant
impact on how the clinical laboratory is staffed, organized, and managed. The greatest cost attributable to the laboratory is certainly due to the technological equipment. How best to cope with a rising clinical demand while responding to mandated reductions in operating costs strongly depends on the workstations available and on the assignment of the tests to be performed to these workstations. In our model we address the problem of the identification of an optimal set of methods in each analytical area. This information can be used as a valid recommendation for the laboratory director who is concerned about the ordering and the the purchasing of the workstations. Because todays decisions will have consequences for some years to come, selection of an instrument or structural changes of laboratory organization requires a detailed understanding of the field and the support of quantitative methods for making better decisions. In order to choose the best equipment, it is important to have an insight into the expected number of test requests that have to be processed in the laboratory. Different request forms may contain different method requests; each method has different frequency and importance. The demand for each period within a day can be derived in terms of the volume of tests distributed over the method types and priority degrees. Priority indicates the length of the interval between the actual time and the due-time. For example for stat tests the due-time can be less than one hour. The flow of requests can vary substantially over the day and in general has a peak during working hours. We take into account the variability of the demand flow during the day dividing the day into periods. In each period we may have a different arrival rate and a different request mix. Because technicians are requested to be at the workstation when a request has to be processed, in our model the staffing problem is addressed and solved. It is worth recalling that we assume that all technicians are equally skilled. As far as the performance measures are concerned, we report in table 2.1 1 examples of performance measures and the corresponding indicators. The set of performance measures to optimize should relate to all aspects which management considers to be important. Table 1. Performance measures and indicators Performance measures Financial performance Internal processes Learning and growth
Client perspective
1
Example Cost reduction Profit realization Quality control Throughput times Research and development Staff development Investments Turnaround times Test diversity Value for money
Source (Van Merode and Goldschmidt, 1998)
Indicator Money Money Score Time N/A Nmb of Fte Money Time Nmb of tests N/A
In order to formulate an optimization model aimed at determining the optimal structure of the clinical laboratory, we have selected from this list some relevant performance measures. We formulate the selected performance measures using mathematical expressions to include in our model. In particular, we detect one financial performance (the overall costs), and two performances reflecting the client perspective, the turnaround time and the test diversity. Using this set of performance measures, the best laboratory organization should achieve the client satisfaction, minimizing costs. Now we are ready to present a formal description of the proposed optimization model (LABOPT).
min
X X X X ( cml Nmle + hIe,l ) + e∈E l∈L m∈M
X X
l∈L
(1)
aml yml
l∈L m∈M
where
X
Iel = Ule De − X
(2) Nmle topm ∀l, ∀e
(3)
m∈M
yml ≥ 2 ∀m ∈ Mu
(4)
yml ≥ 1 ∀m ∈ (M \ Mu )
(5)
l∈L
X l∈L
Nmle ≤ yml
e X
δmef
f =1
∀m ∈ M, ∀l ∈ Lm , ∀e ∈ E (6) g g X XX X Nmlg ≥ δmef − Nml(g−1) e=1 f =e
l∈L
2.2 Mathematical programming formulation. Let L be the set of the analytical areas present in the laboratory, M the set of all possible methods and E the set of periods in which a working day is divided. We can define two subsets Mu and Lm ; Lm is the set of analytical areas in which the method m can be processed and Mu ⊏ M is the set of methods usually requested for emergencies. We would like to observe that Lm can coincide with L or, if the model is used to explore the possibility of improving an existing laboratory, be a proper subset of L. We will use indices l to indicate analytical areas, m to indicate methods and e, f, g to indicate periods. We define the following list of model parameters; let UeT OT be the total number of technicians available in a period, aml the activation cost of the method m in the area l, h the cost per hour of the technicians, cml the processing cost of one sample in the area l with the method m, δmef the number of samples arrived in period e that have to be processed within the period f with the method m, topm the time spent by the technician to process one sample with the method m, wm the processing time of one sample with the method m, De the duration of the period e, P the fraction of the period devoted to manual operations. The decision variables are : • Nmle the number of samples to process with method m in the area l during the period e • Ule the number of technicians to assign to the area l in the period e • yml a binary decision that takes value one if the method m is activated in the area l and zero otherwise The model can be stated as follows:
l∈Lm
∀m ∈ M, ∀g = 1 . . . |E| − 1, X topm Nmle ≤ Ule De P ∀e, ∀l
(7) (8)
m∈M
X
Nmle (wm + topm ) ≤ De ∀e ∈ E
(9)
m∈M
X
Ule ≤ UeT OT ∀e ∈ E
l∈L
X X X
Nmle =
l∈L m∈M e∈E
X
X XX
(10) δmef (11)
m∈M e∈E f ≥e
Ule ≥ 2 ∀e ∈ E
(12)
l∈L
yml ∈ {0, 1}, Nmle , Ule integer ∀m ∈ M , ∀l ∈ L, ∀e ∈ E
(13)
Constraints (4),(5) impose that each method has to be assigned at least to one area, that is the laboratory must be equipped with one or more workstations able to process that type of method. In particular constraint (4) imposes the presence of backup instruments so that the stat tests can be processed even when the primary instrument is out of service. Constraints (6) assure that the number of samples to process with a particular method in a period does not exceed the number of samples arrived until that period. Constraints (7) express the multiperiod linking constraints and state that the number of samples to be processed at the current period must be equal to the number of samples to be processed within the current period minus the number of samples processed at the previous period. Constraints (8) limit the time devoted to manual operations to a fraction of the total duration of the period considered, while constraints (9) assure that the time assigned to scheduled samples is less than the time available in the period. The number of technicians assigned in a period can not exceed the maximum staff available (10). Constraint (11) assures that all the
requests are assigned and processed according to the relative due-time. Finally, the reliability of the operations is guaranteed imposing in each period at least two technicians in the laboratory. The objective function optimizes three measures for the P efficiency P Pof the laboratory. The first term ( e∈E l∈L m∈M cml Nmle ) minimizes the total cost of containers required to collect the liquid to be examined with relative gain in quality (less invasivity) P from P patient’s point of view. The second term l∈L m∈M (aml yml ) reflects the activation cost of the methods in the laboratory working areas. Imposing the activation cost equal to one the objective function achieves the minimization of the methods to be activated. These two terms would appear to reduce the laboratory costs both by curbing the number of tests performed and by reducing the costs P of individual tests. Last, but not least the term l∈L h(Ule De − P N top ) involves the cost reduction due mle m m∈M to the idle time of the technicians. In fact the first component is the total time of assigned technicians in a period, whereas the second one is the effective working time. It is worth noting that all the performance measures that we considered relevant are taken into account in our model. As a final remark, the three terms are expressed in the same unit leading to a standard optimization problem.
3. DISCUSSION OF RESULTS AND CONCLUSIONS The validity of the model LABOPT was demonstrated by comparison of the optimal solution obtained by LABOPT with the actual situation, and the optimization output was considered plausible by laboratory experts. However, none of the two laboratories has implemented the proposed solution. Achieving a genuine partnership between operational researchers and health institutions requires the partners to be equally open to new methods of analysis and to enter into new modes of discourse, and this may requires long time. The model has been tested on data from two italian laboratories, differing greatly in layout and workload. To instruct the optimization model, specific data have been collected by studying written specifications and through interviews with technicians. The data collection has concerned the technical description of each method, the processing time per sample, the arrival time of samples throughout the day at the laboratory and the total number of arrivals within each period. Demand data were estimated on the basis of historical data collected form the laboratory over the past years. As we mentioned before, the two laboratories under study are very different. In fact one of them (a) is a private research laboratory, as opposite
Fig. 2. Idle time per period lab (b).
Fig. 3. Total idle time for lab (a). the second one (b) is a laboratory embedded in a hospital. The difference is also evident in the arrival request flow which is concentrated only in the morning in the laboratory (a), while the laboratory (b) works all the day without any interruption. The width of each period as well as the number of periods considered is different in the two cases. Furthermore the laboratory (b) has also a flow of emergency requests to be processed within the arrival period. In table 3 the distribution and the length of periods in the day is reported for both the laboratories. We set respectively the number of analytical areas and methods equal to the actual configurations of the laboratories (99 methods and 14 analytical areas for both the laboratories). The number and the mix of methods activated in each analytical area is different between the laboratories. The full problem data are reported in (Bruni M. E., 2005) For each laboratory the solution obtained by our model LABOPT is compared with the real existing laboratory organizations. In figure 2 the results concerning the technicians idle time are presented for the laboratory (b). As we can observe the idle time of our solution is less than the actual idle time of operators. A reduction of the total idle time can be observed also for the laboratory (a) in figure 3. In figure 4 is reported the number of technicians assigned by our model compared with the number of technicians working in the laboratory. For the sake of clarity we note that this comparison is
Table 2. Periods Period Lab (a) Lab (b)
Slot time Length (min) Slot time 7:30-8:30 Length (min)
I 7:30-8:30 60 8:30-10:30 60
II 8:30-10:30 120 10:30-12:30 120
III 10:30-14:30 240 12:30-14:30 120
IV
V
VI
14:30-18:30 120
18:30-7:30 240
780
REFERENCES
Fig. 4. Number of technicians; laboratory (b). made without further refinements without considering the delicate issue of the crew resource management in a clinical context. Nonetheless we observe that the maximum number of technicians assigned by our model is ever less then the actual number of technicians working at the laboratory. Similar considerations can be drawn for the laboratory (a). Our model provides useful information for assessing the performance of the laboratory and for choosing the appropriate laboratory structure. However, further attention should be paid to the estimation of the parameters used within the model. We do not believe that description and analysis of the existing situation will simply suggest alternative solutions in a deterministic manner, but we are strongly convinced that the proposed model could be used to gain insights into the existing situation, as well as possibilities for redesign. In this context the first item to be examined is how the laboratory is divided into areas and how the staff is assigned to these areas. Finally it is worth noting that the turn around time is in any case less then the difference between the arrival time and the due time. This allowed us to tackle efficiently the emergency request arrivals and to limit the turnaround time to be less then a given upper bound. We think that operations research methods can be a valuable tool in supporting laboratory directors to make better management decisions, based on rationally understandable experiments rather than on subjective feelings and beliefs, even if these are grounded on a long personal experience in the field.
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