Developing the Pathologists’ Monthly Assignment Schedule: A Case Study at the Division of Anatomical Pathology of The Ottawa Hospital

Abstract

In the Division of Anatomical Pathology of a teaching hospital at the beginning of each month, clinical managers assign expected daily pathology requests to the pathologists on duty. Since the number of these requests is usually large and a division employs a number of pathologists with different sub-specialties, the size of the problem is significant and finding a feasible assignment schedule manually is time-consuming. Moreover, every time there is a need to change, a new assignment schedule needs to be developed taking into account all the pre-defined constraints including pathologists’ availability, sub-specialty mix, teaching/research releases, etc. In this paper we describe an analytics optimization model embedded in a decision support tool that helps the clinical managers of the division determine the optimal monthly assignment schedule. The decision support tool has been validated using data from the Division of Anatomical Pathology at The Ottawa Hospital in Ottawa, Ontario, Canada.

Introduction and Problem Statement

Developing an assignment schedule for the pathology laboratory at a teaching hospital is a process that assigns the individual pathologists to pathology requests taking into account pathology sub-specialties, the volume of requests, and any teaching/research functions that impacts each pathologist’s availability. As the complexity of clinical pathology grows, the time spent on creating the pathologists’ assignment schedule increases creating a need for a computer-aided scheduling support tool¹. In this paper we describe a support tool that was developed in collaboration with the clinical managers and pathologists from the Division of Anatomical Pathology (DAP) at The Ottawa Hospital (TOH). TOH is a teaching hospital affiliated with the University of Ottawa that provides inpatient and outpatient services on three campuses located in different parts of the city. The DAP is located on one of these campuses (General Hospital campus) and serves the entire TOH. It also receives pathology requests from the community hospitals and from private pathology laboratories serving the City of Ottawa and surrounding areas. Each day about 200 pathology requests of differing complexity arrive from the operating rooms of TOH, the clinics affiliated with the hospital, the community hospitals, and from small laboratories. These requests are recoded as the “cases”, with each case involving the analysis of one or more specimens that are further subdivided into slides. In this paper we are concerned with the assignment schedule at the level of the specimen. We do not describe the process of creating the specimens from a tissue but want to stress that each specimen may include (depending on its type) anywhere from 3 to over 200 slides that need to be examined by a pathologist to whom a particular specimen is assigned. The 36 full time pathologists working at the DAP are salaried and their responsibilities, apart from clinical work include teaching and research that impact their availability in a given day, week, and month. These pathologists cover services spanning 26 sub-specialties (i.e. liver, breast, neurology, etc.).

In order to process all pathology requests, at the beginning of each month clinical managers develop the assignment schedule for the following month. This schedule assigns each type of specimen to a pathologist for each working day of the month. It must take into account the following requirements:

• Pathologist’s sub-specialty: Each pathologist can assess only those specimens that are within his/her expertise such as breast, head and neck, musculoskeletal, to name a few.

• Pathologist’s availability: Due to a number of clinical and non-clinical responsibilities, each pathologist’s availability needs to be assessed on daily and weekly basis. We measure availability in a given week using “full time equivalent (FTE)” fraction (i.e. for a pathologist working full time in a given week FTE = 1).

• The service weight: The amount of time it will take to assess all slides of a given specimen type in a given day is clearly a stochastic variable. However, these times are estimated using pre-determined service weights that are different for each specimen (i.e. a service weight of 2.0 for neuropathology indicates that the expected workload on a given day for neuropathology should require the services of two pathologists). There is no agreement in the literature (and practice) as to how the service weights ought to be determined but this discussion is beyond the scope of our research. Clinical managers at the DAP rely on weights that are based on the L4E indicator used in a number of North American laboratories to determine pathologists’ workload.

The above requirements define what we call “hard requirements” as they need to be satisfied by any assignment schedule. However, there is a number of “soft requirements” that should be considered when a reasonable assignment schedule is created and these requirements include:

• Consistent assignment in a week: It is desirable that within a given week a pathologist is assigned to work on specimens belonging to a single sub-specialty.

• Rotation of the specimens: In order to maintain the clinical skills required for the analysis of a given sub-specialty, each pathologist should regularly rotate through his/her areas of expertise.

• Prioritization of the specimens: Due to vacations and other absences, it is not always possible to cover every sub-specialty on every day. Faced with this dilemma, the DAP prioritizes between subspecialties in order to ensure that the crucial ones are covered. For instance, full coverage requires three pathologists to be assigned to a given sub-specialty. Faced with insufficient resources, the DAP will choose to reduce that coverage to two pathologists as a first step. This ranking of coverage priorities needs to be respected.

Currently the clinical managers at the DAP use a manual approach to scheduling that relies on data recorded in a spreadsheet. Since the size of the pathologists’ assignment schedule is large, finding a satisfactory assignment manually is time-consuming and can take a number of iterations over a number of days to complete. Moreover, every time there is a need to revise the schedule the process must start over again.

In this paper we describe a computer-based decision support tool for developing an assignment schedule. This tool has two main components – an analytics optimization model that is used to develop the optimal assignment schedule taking into account both the hard and soft requirements and a spreadsheet-driven interface that is similar to what is being used now in the division and allows the managers to manipulate and revise the assignment schedule in order to assess a number of scheduling scenarios or consider special cases.

The next section presents a brief review of the research on using the assignment problem in healthcare. This is followed by a description of the analytics model that we have developed. The decision support tool is described in the Implementation section. The comparison of the assignment schedules developed automatically with those used by the division is presented in the Results section. The paper ends with some concluding thoughts.

Related Work

The development of an assignment schedule is an important analytics problem that aims to find an optimal allocation of n available staff members to m positions or tasks in a system. During the last few decades, researchers have developed a number of assignment models for various settings such as industrial systems, educational institutions and healthcare organizations all designed to help managers allocate tasks to resources (including human resources and equipment)¹¹.

A number of assignment models, specific to healthcare, have been developed⁶. The type of model developed depends on the type of organization and the specific characteristics of the problem to be addressed². Assigning nurses and physicians (mostly in the Emergency Department) to shifts so that staffing requirements are met represents one of the most common problems and the analytics models include stochastic programming³, goal programming⁴ and genetic programming models⁵. Home care staff assignments require the development of an assignment schedule connecting health providers with patients based on factors such as workload restrictions, provider qualifications and preferences, acuity levels of patients, an overtime penalty and staff satisfaction. The majority of the home care assignment problems use optimization models^6,7,8,9,10.

To the best of our knowledge, our work is the first attempt to apply analytics to develop the optimal assignment schedule in a pathology department. The proposed model is similar to those used elsewhere but has a number of distinct features: it deals with a situation where each pathologist has multiple sub-specialties and explicitly considers the need for pathologists to rotate through the types of specimens in order to maintain the required skills in a given clinical sub-specialty. It is worth mentioning that there are commercial scheduling systems designed to develop an assignment schedule for the pathologists. However they are mostly concerned with implementing the existing scheduling practice in terms of rules rather than developing the kind of optimal assignment schedule provided by the system described in this paper. A good example of such assignment is Q-Genda’s pathologist scheduling software (Q-Genda corporation) that allows the managers to develop different scheduling scenarios according to a set of decision rules. However, because of the way that these scenarios are created, it is not possible to determine if proposed assignment is truly the best one or if it just meets basic requirements.

Methods

The development of the pathologists’ assignment schedule is a decision-making problem that involves the analysis and assessment of a number of alternative schedules each with different characteristics. This type of problem is solved using analytics methods that allow the user not only to assess possible alternatives very quickly but also to identify the optimal one based on the measurable criteria. In this paper we describe the use of an analytics approach that involves the development of a mixed-integer optimization model that incorporates both hard and soft requirements. This model has the following four components: decision variables, model parameters, objective function, and constraints. Each component is described in greater detail below and the full model is provided in the Appendix.

Decision variables

The decision variables represent the unknown (to be determined by the model) daily assignments of the pathologists to specimens on each day. For the DAP, assuming that a monthly assignment schedule needs to be developed for 20 business days on average (DAP does not work on the weekends and statutory holidays), this implies that there are 18720 possible daily assignment combinations. However, the incorporation of the soft constraints requires the creation of auxiliary decision variables bringing the total to 43649.

Model parameters

Every model requires known and measurable inputs that are called “parameters”. We use five types of parameters:

• The FTE for each pathologist working full time has a value of 1 while for the part timer it can be any number between 0 and 1 (i.e. for a pathologist working two out of five days a week the value of his/her FTE is 0.4). Values for the FTE fraction parameters were obtained from the DAP.

• The service weights for each type of specimen were determined by the DAP clinical managers as explained earlier.

• The availability of a pathologist on a given day is captured by a binary parameter with value 1 if the pathologist is available and 0 if the pathologist is unavailable. These values change from month to month and reflect each pathologist’s workload taking into account teaching/research obligations, holidays, etc.

• Another binary parameter captures which specimens are within each pathologist’s area of expertise. (When a pathologist is able to diagnose a given specimen type then the value of this parameter is 1. It is 0 otherwise). This information was provided by the DAP.

Objective function

The purpose of the objective function is to establish a performance measure for assignment schedules in order to derive the optimal solution. In our model the objective function is created by amalgamating the soft requirements outlined earlier. Thus, the objective function has three parts, each with a specific weight of importance that can be assigned by a clinical manager. In the case study described later in the paper the values of the weights were determined experimentally so the resulting assignment schedule best matches the expectations of the clinical managers.

Constraints

The constraints describe in a formal way the constraining factors that limit the development of an assignment schedule. Model constraints include hard requirements as well as a number of additional constraints related to the need to produce feasible daily assignments over 20 days. The model we developed for the DAP has 52057 constraints excluding binary and non-negativity constraints imposed on the decision variables.

The model was solved using variant of Simplex algorithm implemented in IBM ILOG CPLEX Optimization Studio running on a Dell T7600 desktop computer with Windows7. On average it took seven to ten minutes to obtain the optimal assignment schedule for a given month.

Implementation

Responding to the clinical managers request that the user interface to the model should have a “spreadsheet – like” feel, we embedded the model within a customized Microsoft Excel platform that provides a user-friendly interface. The resulting scheduling decision support tool was named the Automatic Pathologists’ Scheduler (APS) system. It integrates the analytics model and the Microsoft Excel spreadsheet using a number of macros. The APS system has a hierarchical structure with a main menu (see Figure 1) that allows the clinical manager to access the different functions of the system.

Figure 1.

Main menu of the APS system.

In order to use the APS system, the clinical manager needs to enter the values of the parameters required by the analytics model. These parameters can be categorized into:

• Dynamic parameters: the values of these parameters have to be updated every month before solving the model. They include information about the holidays in a given month, pathologists’ availability and specific “hard-wired” assignments that task a given pathologist to a specific specimen type on a given day.

• Static parameters: the values of these parameters are updated infrequently (or do not need to be updated at all). They include the roster of pathologists working in a division, the list of sub-specialties, the FTE fractions and the service weights.

The APS system has the built-in functionality that allows for extended editing and control of the development of assignment schedules. This functionality includes:

• Editing: revising values of all parameters, including anticipated daily pathology requests. This function also allows for the manual development of a partial assignment schedule that is later used by the analytics model as a starting point, as needed.

• Optimization control: The user interface allows the manager to experiment with different values for the weights assigned to the components of the objective function. The purpose of this functionality is to provide the clinical manager with the ability to develop different assignment schedule scenarios depending on the changing importance of each of the soft requirements.

• Schedule analysis: A separate spreadsheet interface for analysing the schedule allows the clinical managers to evaluate and revise the schedule, if needed.

The output produced by the APS system is illustrated on Figure 2 (due to space limitations only a fragment of the spreadsheet is presented). The assignment schedule is organized by the days in a month (columns), types of the specimens (coded with the abbreviations used in the DAP and represented as the rows), and labels associated with individual pathologists (cell values). In the original schedule the cell values include pathologists’ initials, but for privacy reasons they are replaced here with P# labels. The dash (-) on the schedule indicates a non-business day. As an example pathologist P14 is assigned all cardiac specimens from April 7 to April 10. Starting on April 13 this type of specimen is re-assigned to pathologist P5 (until April 17, inclusive). In the schedule, two separate rows have been allocated to breast specimens (indicated by “Breast1” and “Breast2” labels) as historically two pathologists are required to analyze all daily breast specimen requests.

Figure 2.

Sample output from the APS system.

Currently the APS system generates combined schedule for all pathologists (as requested by DAP managers), however we will be also implementing the disaggregation of the schedule so individualized assignments for each pathologist are created automatically.

Results

The system’s performance has been validated by the clinical managers at the DAP and also by comparing manually developed assignment schedules with those created by the model. The results show that the APS system is effective in developing a schedule that meets most of the pre-defined scheduling criteria and as such represents a significant step forward in managing the pathologists’ scheduling problem. To illustrate, below we present and compare the assignment schedules generated by the APS system for the months of October and November 2014 against the schedules that were developed manually for those same months and implemented in the DAP. When conducting these comparisons, following the advice of the clinical managers, we looked at:

• Percentage of unassigned specimen types: in the schedule developed by the APS system, higher priority sub-specialties are assigned to the pathologists first. This may result in some lower priority sub-specialties being left unassigned due to an insufficient number of pathologists available. This measure is calculated the percentage of all unassigned specimen types within a month in relation to all the assignments in this month.

• Percentage of inconsistent assignments: this measures the consistency of assignments within a given week. It is calculated as the percentage of all assignments that are not consistent within a week in relation to all the assignments in this week. An assignment schedule that has a lower value for this measure is preferred.

• Percentage of missed rotations: each pathologist ought to cycle through all sub-specialties within his/her area of expertise. This is accomplished by rotating what specimen type a pathologist is covering each week. Missed rotations (in %) are the number of sub-specialties within each pathologist’s area of expertise that is not covered in a given month divided by the total number of sub-specialties for each pathologist. This metric is aggregated over all pathologists to derive a single measure. It is important to note that it may not be possible to cover all sub-specialties in a given pathologist’s area of expertise while still maintaining consistency with the week as some pathologists have 5 or more sub-specialties. However, an assignment schedule that has less missed rotations is clearly better.

Looking at the results presented in Table 1, we note that the APS system consistently outperforms the manually developed assignment schedule when considering the percentage of inconsistent assignments. The differences between the APS and the manual schedule on the other measures are negligible indicating that the model quite accurately reflects current practice.

Table 1.

Comparison of automatic and manual assignment schedules for October and November 2014.

Period	Unassigned specimen types (%)		Inconsistent assignments (%)		Missed rotation (%)
	APS system	Manual assignment schedule	APS system	Manual assignment schedule	APS system	Manual assignment schedule
October 2014	1.5%	0.8%	14%	50%	17%	14%
November 2014	0.6%	0.1%	12%	44%	15%	16%

Discussion

In this paper we describe the APS system that helps clinical managers of the DAP to develop a monthly assignment schedule. The system creates the schedule by solving an analytics model that takes into account all hard and soft scheduling requirements considered currently by the managers. Providing the clinical managers with such an easy to use support tool has value for a number of reasons. First, the APS system helps to create the optimal assignment schedule very quickly. Secondly, the proposed schedule can be used by the managers to either make additional adjustments taking into account intrinsic requirements or to create a number of scenarios considering different staffing possibilities, specimens’ volume levels, etc. Finally, the APS system gives flexibility in establishing and revising all model parameters to reflect for example a sudden change in the pathologists’ availability.

Currently we are using the system to develop assignment schedules for the next couple months of 2015 and at the same time we are working on improving and customizing the user interface. While the APS system was created for DAP and using data from that division, it can be easily customized and ported to different pathology departments provided that there is data to revise the model’s parameters. A possible extension of the model would be to consider non-deterministic character of daily requests for pathology services. This extension requires additional data of different granularity and type that, at present, is not easily available at the DAP.

Appendix. Mixed Integer Linear Programming Model

Decision Variables

$$X_{ijt}=\{\begin{array}{c}1,\text{if specimens of type}j\text{is assigned to pathologist}i\text{on day}t\\ 0,\text{otherwise}.\end{array}$$

i=1,2,3,…36 j=1,2,3…,26 t=1,2,3,..25

$$Y_{ijk}=\{\begin{array}{c}1,\text{if pathologist}i\text{is given specimens of type}j\text{in week}k\\ 0,\text{otherwise}.\end{array}$$

i=1,2,3,…36 j=1,2,3…,26 k=1,2,3,4

β_ij: Auxiliary variable for “Rotation of the specimens” that keeps track of the number of different types of specimens that have been assigned to each pathologist.

Model Parameters

• b_i: the FTE fraction of pathologist i availability per week

• a_j: the proportion of a full day’s work required for daily pathology requests of type j

• α_ij: equals 1 if specimen type j belongs to one of the sub-specialties that pathologist i posses.

• T_it: equals 1 if pathologist i is available on day t, otherwise 0

• S_ij: equals 1 if pathologist i is able to review specimen type j, otherwise 0

• C₁, C₂,C₃: weights for each part of the objective function

Objective Function

$$\min z=C_1{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{k=1}^K{Y}_{ijk}-C_2}}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^m{\beta }_{ij}+C_3}}{\displaystyle \sum _{j=1}^m{\displaystyle \sum _{t=1}^T{d}_{jt}}}$$

The objective function consists of three parts. The first part penalizes assignment schedules each time a pathologist is given a different specimen type in the same week. The second part rewards assignment schedules every time they cover another specimen type that lies within their expertise. The final part penalizes assignment schedules for every uncovered assignment. The weights C₁, C₂ and C₃ allow the user to adjust the importance of the three components to the performance of an assignment schedule. Typically the third component is deemed the most important.

Constraints

1. Consistent assignment in a week (4 weeks within a month, 5 business days within a week):

   $$\begin{array}{c}\forall (i,j)\\ {\displaystyle \sum _{t=1}^5{X}_{ijt}\le M{Y}_{ij1}}\\ {\displaystyle \sum _{t=6}^{10}{X}_{ijt}\le M{Y}_{ij2}}\\ {\displaystyle \sum _{t=11}^{15}{X}_{ijt}\le M{Y}_{ij3}}\\ {\displaystyle \sum _{t=16}^{20}{X}_{ijt}\le M{Y}_{ij4}}\end{array}$$

   where M is a very large number. Each time a new specimen type is assigned to a given week, the above constraints for a new Y_ijk to be positive. Ideally, a pathologist is given only one specimen type in a week so that, for each week, there is only one Y_ijk that is positive.
2. Rotation of the specimens

   $$\begin{array}{c}\forall (i,j)\\ {\displaystyle \sum _t{\alpha }_{ij}{X}_{ijt}\ge {\beta }_{ij}}\\ {\beta }_{ij}\le 5\end{array}$$

   The ideal schedule would give a week’s worth of one specimen type to a pathologist and then switch to a different specimen type for the subsequent week in order to rotate through his/her areas of expertise. This set of constraints rewards schedules that assign a specimen type with a pathologist area of expertise up to a maximum of a reward of 5 for any particular specimen type. This encourages the model to give full weeks of a given specimen and then switch.

3. Weekly assignment

   The FTE fractions are per week, but in a scale of 0 to 1, so their values (b_i) have to be multiplied by 5 in order to derive the number of days that each pathologist works per week (5 business days in a week).

   $$\begin{array}{c}\forall i\\ {\displaystyle \sum _{t=1}^5{\displaystyle \sum _{j=1}^m{a}_j{X}_{ijt}\le 5b_i}}\\ {\displaystyle \sum _{t=6}^{10}{\displaystyle \sum _{j=1}^m{a}_j{X}_{ijt}\le 5b_i}}\\ {\displaystyle \sum _{t=11}^{15}{\displaystyle \sum _{j=1}^m{a}_j{X}_{ijt}\le 5b_i}}\\ {\displaystyle \sum _{t=16}^{20}{\displaystyle \sum _{j=1}^m{a}_j{X}_{ijt}\le 5b_i}}\end{array}$$

   These constraint force the number of assignments (weighted by the service weight) to respect the number of days that a given pathologist is available based on their FTE fraction.

4. Daily assignments

   $$\begin{array}{c}\forall (i,t)\\ {\displaystyle \sum _{j=1}^m{a}_j{X}_{ijt}\le 1}\end{array}$$

   This set of constraints ensures that no pathologist is given more than a full day’s work on any given day.

5. Demand coverage

   $$\begin{array}{c}\forall (j,t)\\ {\displaystyle \sum _{i=1}^n{X}_{ijt}+d_{jt}=1}\end{array}$$

   This set of constraint ensures that if a given specimen on a given day (j,t) is not assigned to a particular pathologist then the value of d_jt is forced to be one and thus the objective function captures the number of unassigned slots.

6. Availability

   $$\forall (j,t)X_{ijt}\le T_{it}$$

   These constraints allow the manager to explicitly state what days a given pathologist may not be available.

7. Skill set

   $$\forall (i,j)X_{ijt}\le S_{ij}$$

   These constraints again allow the manager to explicitly state which type of specimens a given pathologist is unable to analyse.