Resident Advisor Duty Scheduling Patrick Mannon

Transcription

1 Resident Advisor Duty Scheduling Patrick Mannon Senior, Systems Science and Engineering (217) Dr. Zachary Feinstein Assistant Professor, Electrical and Systems Engineering (314)

2 Executive Summary For this project, the problem of developing a system for scheduling Resident Advisor (RA) duty shifts on the South 40 (S40), North Side (NS) and Off Campus (OC) residential communities of Washington University in St. Louis was undertaken. The desired system distributes duty shifts based on preferences of specific dates and duty times given by the Resident Advisors who will be working the shifts. The program implements binary integer linear programs (BILP) in Matlab to minimize the total preference cost of the output schedule. The preferences given by RAs are the basis for the coefficients in the objective functions for the problems, and RA shifts pairs serve as the decision variables. Three BILPs, one for each residential area, are used to find an overall schedule for every RA. Constraints on the problems include those on the number of RAs to work each shift, the number of shifts an RA must work over the course of the entire semester, and which RAs can work which shifts. The Matlab program creates robust schedules satisfying every required constraint, and over several runs, runtime and total preference cost of the output schedules remained consistent. Schedules are produced in a very reasonable amount of time, especially given the program need only be run once per semester. The total preference cost of output schedules show no lowest preference ranked RA shift pairs being assigned when using sample data. 2

3 Problem Task The problem I wish to tackle is fair and efficient distribution of duty shifts amongst the Residential Advisors at Washington University in St. Louis (WashU). The current system is overseen by a professional employee of the Office of Residential Life (ResLife); my direct supervisor is responsible for maintaining the duty assignments this year. Thus, the idea for my project originated from hearing criticisms and limitations of the current system ResLife uses. Over 100 RAs serve ResLife this year and, ideally, duty shifts would be equitably distributed amongst them. This sounds easy in theory, but in practice, several factors exist which make this process much trickier. Background For the last two years of my undergraduate education, I have served as an RA for freshmen students at the university. RAs perform many functions, including role model, policy enforcer, and general advice-giver, among countless others. Another important task for RAs lies in RA duty. The details of when and how we serve vary based on our residential assignment but the responsibilities remain the same. We serve as the frontline in dealing with residential issues outside of the normal business hours of the Office of Residential Life (ResLife). Shift Types Three different residential areas are overseen by ResLife, the South 40 (S40), the North Side (NS), and WashU-owned Off Campus (OC) housing. The S40 is populated primarily by 3

4 freshmen and sophomores, with upperclassmen usually living on the NS or OCs. Because of the younger population, more RAs serve on the S40 to keep pace with the higher needs of students in their early years of college. Currently, 109 RAs live on the S40 compared to 14 on the NS and 12 who live OC. I live on the S40 and have only experienced RA duty from that perspective. S40 shifts consist of sitting in the Duty Office for three and a half hour shifts: 4:55 to 8:30PM and 8:25 to midnight during the week with an additional 1:25 to 5:00PM shift on the weekends. In addition to sitting in the office, one of the RAs working the 8:25 to midnight shift holds a cell phone while the office is closed. Holding the phone lasts from midnight to 8:30AM during the week and from midnight to 1:25PM over the weekend. To make things more complicated, Monday nights are an exception. Because every RA has training or meetings on Monday nights, no RAs sit in the duty office. Instead, an RA holds the phone from 5PM overnight until 8:30AM the next morning. On the other hand, NS and OC shifts are more straightforward. These areas are typically home to older residents, and as such, the RAs assigned to these communities have less intensive duties. Both areas go through week-long rotations, during which a single RA is on duty and holds a special cell phone. This rotation system leads to a significantly lower number of shifts than the S40 RAs have to fill. Constraining Factors The basic problem of assigning workers to shifts has a well-known solution which can be seen in pages 334 to 346 of Hillier and Lieberman s Introduction to Operations Research, Ninth 4

5 Edition. However, several factors exist that make RA duty assignment a more complicated process. First, each RA has a different class and exam schedule, potentially leading to conflicts with duty shifts. This is a major issue when it comes to S40 shifts. As NS and OC RAs do not sit in an office, they are expected to be on duty the entire time they hold the phone, regardless of whether or not they are in class. Another limit is the number of shifts each RA is required to work. Many shorter shifts must be assigned on the S40, whereas relatively fewer long shifts must be distributed amongst the NS and OC RAs. Subsequently, S40 RAs are scheduled for more, but much shorter, shifts than their counterparts on the NS and OC. Ideally, S40 RAs work four to six shifts per semester, while NS and OC RAs each work one or two week-long shifts. The number of RAs required to work each duty shift must also be considered when making any schedule. For the NS and OC shifts, only one RA is necessary for each week-long duty, but on the S40, two RAs must be scheduled for each office shift and one person must be assigned to each phone shift. Current Scheduling System As it stands now, three separate systems are used to distribute RA duty shifts, with each residential area utilizing an independent method. The S40 uses a computer program, while the NS and OC take the approach of simply divvying up the shifts at a staff meeting. These different techniques make sense when the number of shifts to be assigned in each area is considered. The significantly lower number of shifts on the NS and OC allows for easy manual distribution of shifts. On the other hand, (number of S40 shifts) (number of S40 RAs) possible RA shift pairs exist for the S40; this large number of pair possibilities makes manual management of S40 shift assignment unmanageable. 5

6 Strengths A decentralized structure serves as one major strength of the current duty scheduling system. By splitting up the distribution by residential area, each group can devise the solution they need instead of being tied to the requirements of other communities. Another major strength of the existing system lies in the ResLife staff s experience in using it. This system has been used for many years, and whenever a transition administrator occurs, the preceding overseer is able to rely upon experience to train their replacement. Limitations Because of the small number of shifts to assign, both the NS and OC RAs use straightforward, manual procedures to distribute their duty shifts. This simplicity leads to no major technical or procedural faults in the NS and OC solution to shift assignment. The biggest issue with the system lies in the time spent on in-person negotiation and agreement by the RAs, which takes longer than an assignment done by a computer based on their constraints and preferences. Additionally, an automatic scheduler could reduce or eliminate the meeting time dedicated to the shift allocation. Due to the large number of shifts to be assigned on the S40, the system used to distribute these shifts tends to be more complicated than the by-hand procedures from the other residential areas. This increased complication leads to more potential issues in the system. The major limitation in the current system is the need to make many manual adjustments before a final duty schedule can be assigned. The basic steps of the current S40 procedure start with the software creating a preliminary schedule. Then the duty administrator 6

7 reviews the proposed schedule and makes any necessary corrections. Unfortunately, in the case of the system used now, many adjustments must be made. First and foremost, RAs extracurricular or class conflicts are not taken into consideration until after the initial schedule has been made. The administrator cross-checks the proposed schedule against the RAs schedule constraints and makes any necessary alterations by hand. Another factor ignored by the existing system lies in the number of shifts assigned to each RA. As mentioned in the Constraining Factors section, each S40 RA should be assigned between four and six shifts. In the initial schedule, RAs could be assigned to a large range of shifts, with some scheduled to work much more than others. Again, the system overseer must examine and make changes to the proposed schedule to more evenly distribute the shifts between RAs. Proposed Scheduling System Procedure The proposed centralized system to assign RA duties will be similar to an enhanced version of the existing S40 method. As an integrated method, the RAs and shifts from the S40, NS, and OC will be handled at one time and by one process. It will rely upon a variety of information from each RA, input via a survey easily circulated via during training at the beginning of each semester. The survey will gather what community the RA serves, dates they cannot work, and their preferences on the different shift times. Once this information has been collected, the system administrator inputs this information into the software to create the schedule. 7

8 Strengths In devising the proposal of a new scheduling tool, the limitations of the existing system were reviewed. The proposed solution avoids assigning RAs to shifts conflicting with their schedules. It is not guaranteed but if possible, assignments leading to schedule conflicts will not be made. Additionally, the new system allows the user to dictate a maximum and minimum number of shifts for each RA in a given area. A centralized schedule also allows a single employee to oversee the process instead of disseminating the work to multiple people. Because of its consideration of schedule conflicts and equitability in shift distribution, this proposed system will save the administrator time in the form of fewer necessary manual corrections. The waste of valuable meeting time used to assign shifts on the NS or OC will also be eliminated. One added enhancement is the inclusion of duty shift distribution based on RA preferences in shift times. In the input survey, RAs select their preference level for the different shift times they could be assigned. This improvement predominantly affects S40 RAs, as they have a wider variety of potential times to work. RAs can select one of three levels: Conflicts with Class, Neutral, or Prefer. The system will work to create a schedule containing only assignments between RAs and shift times they ranked Prefer. If such a solution is impossible, the system will assign Neutral shifts, and finally, if a solution still does not exist, it will begin utilizing shifts denoted Conflicts with Class. Limitations The biggest foreseeable limitation of the proposed system is implementation in Matlab and lack of a clear user interface. Due to these factors, it requires very basic knowledge of computer science as well as access to Matlab software to use. Both of these issues could be 8

9 solved without much difficulty in practice, however. Defining and declaring variable values in Matlab is the extent of programming skills necessary to run the program. Such skills could be easily taught and passed on by the system administrators or contained in a step-by-step user guide. Another possible solution would be to create a friendly user interface to gather all of the parameter values for the program. Acquiring access to Matlab also has a ready solution as WashU has a Matlab license. Model The problem of scheduling RA duty shifts falls into the broad category of integer linear programming, and more specifically assignment problems. In general, assignment problems are a type of problem in the field of Operations Research in which a set of tasks is distributed amongst a group of agents with a cost associated with each agent task pair. The simplest case gives one-to-one assignments with each agent completing only one task and each task being worked on by only one agent. The optimal result will be a one-to-one solution minimizing the total cost of completing the every task. The prototypical form of an assignment problem with n agents and n tasks can be seen in Figure 1. Furthermore, Figures 2 and 3 show the problem forms for the North Side and Off Campus model and the S40 model, respectively. 9

10 Figure 1: Prototype Assignment Problem Figure 2: North Side and Off Campus Model Figure 3: South 40 Model Decision Variables To optimize an integer linear programming problem such as that of scheduling RA duties, the values assigned to a set of decision variables are considered until finding the best solution. In the prototypical problem, each agent task pair serves as a decision variable and 10

11 takes on a value of 0 if unassigned or 1 if it is assigned. Similarly, the scheduler also uses binary decision variables, each representing a different RA shift pair. Objective Functions The objective functions of Figures 2 and 3 match that seen in the prototypical example. All aim to minimize the total cost of the solution. In the case of scheduling RA duties, the goal is to devise a schedule that minimizes the total cost of all RA shift pairs. A cost coefficient, c ij, represents the cost of assigning RA i to duty shift j. These costs are set by RAs when they choose shift preferences; the lower preference a shift receives from an RA, the higher the cost associated with that RA shift pair. By summing the product of the costs and their associated decision variable, the objective function finds the sum of the cost of only the RA shift pairs that have actually been scheduled. This is the value to be minimized. Constraints Many differences also occur, especially when adding or modifying constraints. Due to their varying requirements, the different residential areas have different problem setups. As shown in Figures 2 and 3, the NS and OC models largely mirror the prototypical, but the S40 model contains many modifications. Figure 2 shows the NS and OC problem. The only change in this model lies in the first constraint; instead of requiring each RA to be assigned to only one shift, the first constraint now allows one or two duties per RA. As seen in Figure 3, the modifications to the S40 problem include constraining a range of the number of duties for each RA, setting the number of RAs assigned to work both office and phone shifts, and a constraint on which RAs can work non-monday overnight phone duties. The first constraint bounds the number of shifts an RA is designated to work between four and six. Respectively, constraints 11

12 two and three force two RAs to be scheduled to each office shift and a single RA to be assigned to each phone duty. Finally, constraint four requires that non-monday overnight phone shifts be worked by one of the two RAs working the office shift from 8:25 to midnight. Network Representation A weighted, directed graph can be used to represent the RA duty scheduling problem. Both RAs and shifts form nodes of the graph, with edges leading from RAs to their eligible shifts. RA shift pair costs serve as edge weights. Figure 4 shows an initial view of the network with all possible RA shift pairs appearing connected by edges in the graph. After solving, only pairs actually scheduled have a connecting edge. The edges colored red in Figure 4 show a possible solution and schedules the RA 1 shift 3, RA 2 shift 2, RA 3 shift 4, and RA 4 shift 1 pairs for a total cost of c 13 + c 22 + c 34 + c 41. Figure 4: Initial Network Representation of four RAs and four Shifts (Possible Solution in Red) 12

13 Implementation The RA duty scheduling program developed over the course of this project has been primarily implemented in Matlab but also utilizes Google forms and Microsoft Excel. The scheduling procedure can be split into several steps: 1. RA preference information collection 2. Input data parsing 3. Problem formation and solution a. Solver b. Decision Variables c. Objective function d. Constraints 4. Output data parsing RA Preference Collection The first step of the process utilizes a Google Form to collect information from the RAs to be scheduled. The form can be sent to every RA Figure 5: Residential Area Input and will collect their name, residential area where they work, and the required preference of shifts. To begin the form, seen in Figure 6, each RA gives their name and where they work. Based on the selected area, the RA is directed to the appropriate next screen. S40 duty shifts have much more variability than NS or OC and therefore S40 RAs have many more options when it comes to selecting their Figure 6: South 40 Preference Input Form 13

14 preferences, as can be seen in Figure 7. Each S40 RA can select up to five dates which they cannot work any shifts. RAs are also given three preference options for Tuesday through Friday evening shifts. RAs can choose to rank a day and time combination as Conflicts with Class, Neutral, or Prefer (i.e. an RA could select Conflict with Class for Tuesdays from 5 to 8:30 and Prefer for both times on Wednesday). S40 RAs were only given preference options for Tuesday through Friday shifts because no Monday night and weekend shifts should cause any regular class conflicts. It was also assumed no RAs would select prefer for the weekend shifts and thus this option was not given. NS and OC RAs have fewer options and simply choose up to three weeks for which they cannot hold the phone. Additionally, the above input system could be easily modified to accommodate more preference options for any of the three areas. Input Data Parsing After collection from the RAs, preference data must made into an easily usable form. This begins when the information is downloaded as an Excel spreadsheet and imported into Matlab. At this point the preference tables are comprised of twenty-one columns with a row for each RA. Columns contain the RA name, any dates they cannot work, and their preference rankings for shift times. However, any given RA s entry in the table contains many empty cells. This occurs because not every RA enters data into every field of the form. For example, NS and OC RAs do not weight specific day and time shifts as S40 RAs do so these columns will contain empty cells in their rows. To prevent these empty cells causing problems later in the solving process, the total preference table is split into three smaller tables based on which residential area an RA chose and labels are added to the appropriate variables. These input matrices can 14

15 be seen below in Figure 8. Breaking up the input matrices also serves to remove another step from the end user and makes the system easier to use. Figure 7: S40, NS, and OC Input Matrices Problem Formation and Solution After the preference data has been read and split into three tables, the scheduling problems seen in Figures 2 and 3 can be formed. Because of the variety of requirements on the schedules of the different areas, the software uses a specific model to create each area s schedule. Three specialized scheduling functions take the split preference tables and create a unique linear programming problem for each area. Solving these problems and combining their results will generate an overall schedule for RAs. Solver In order to solve these linear programs, the scheduler uses intlinprog, a predefined function in Matlab. The intlinprog function takes as input eight variables: f, intcon, A, b, Aeq, beq, lb, and ub. As detailed in the documentation on Matlab s website, intlinprog solves mixedinteger linear programs of the form seen in Figure 9. The input variables: 15

16 f: a vector containing the coefficients of the objective function intcon: the number of integer decision variables A: a matrix containing the left hand side coefficients of the inequality constraints b: a vector of right hand side values of the inequality constraints A eq : a matrix containing the left hand side coefficients of the equality constraints b eq : a vector of right hand side values of the equality constraints lb: a lower bound value for the integer decision variables ub: an upper bound value for the integer decision variables Decision Variables As discussed above in the Model section, each RA shift pair must be represented by a decision variable in order to find an optimal schedule. Generating these variables in Matlab requires forming a matrix of ones with a cell for each pair. Each row represents an RA and each column corresponds to a duty shift. Therefore cell ij contains the decision variable for RA i being assigned shift j. Figure 8: Linear Program Form Solved by intlinprog The duty scheduler is based upon the number of weeks in the RA contracts and a specific duty schedule for each residential area. The NS and OC have weekly phone shifts with the phone changing hands every Sunday. The total number of shifts in these areas is therefore the same as the number of weeks as the RA contract. The S40 RAs work more duties and each week twenty-one shifts must be assigned. The program sees the Sunday afternoon shift from 1:25 to 5PM on the start date as the first shift in the schedule. 16

17 Objective Functions Objective functions set a desired characteristic for creating the duty schedules. As seen in the first row of Figures 2 and 3, each of the three areas use the same objective function to schedule their RAs with a minimum total cost. The preference table determines the costs of each RA shift pair. The lower preference score an RA gives a shift, the higher that RA shift pair costs to schedule. For S40 RAs, Prefer shifts are given a cost of 1, Neutral shifts cost 5, and Conflicts with Class shifts have the highest cost of 10. Shifts on the unavailable days S40 RAs selected also cost 10. Both shifts on unavailable days and Conflicts with Class shifts possess the same cost following an assumption that days requested off by RAs are due to already scheduled events. Thus the RA cannot work a duty on these days just as they cannot work during class. In the NS and OC scheduler, weeks chosen as unavailable are given a cost of 10 and all other shifts cost 1. Selecting the values of the cost coefficients amounts to a somewhat arbitrary decision. The essential characteristic is an inverse relationship between preference and cost. Therefore the lowest preference option must have the largest cost and the highest option costs the least. The values of 1, 5, and 10 were chosen for this implementation in order to assign a high enough cost to the lowest preference option to prevent its scheduling while maintaining a constant distance between possible cost values. Forming the cost matrix in Matlab begins with a matrix of ones the same size as the decision variable matrix. The scheduler then iterates through each RA and alters the appropriate entries according to that RA s shift preferences. When crafting a solution, the scheduler will find the combination of RA shift pairs with the minimum sum of preference costs. This sum is determined by multiplying the preference 17

18 cost of the RA shift pair by the decision variable associated with that pair. Because the problems utilize binary decision variables, the sum includes only the costs of actually assigned RA shift pairs. The objective function, f, input into intlinprog is found by first reshaping the matrices of decision variables and their respective costs into column vectors. Then, element-by-element multiplication of these vectors will produce the desired objective function vector. Constraints In addition to the objective functions, constraints must be formed before solving the problems. As detailed above in the Model section, constraints on the NS and OC schedules require only small modifications from the prototypical assignment problem whereas the S40 schedule needs more changes to create. In Matlab, the linear programming solver uses matrices to contain the problem constraints. The form A x b represents the inequality constraints of the problem. In a problem with n inequality constraints and d decision variables, an nxd A matrix holds the left hand side of the constraints with an nx1 column vector b containing the right hand side of the constraint equations. Equality constraints take an analogous form, A eq x = b eq. Similar to A and b, A eq is an mxd matrix and b eq is an mx1 column vector, where m represents the number of equality constraints and d is again the amount of decision variables. The upper and lower bound vectors, ub and lb, are both dx1 columns. 18

19 Inequality Constraints A contains the coefficients for the constraints on the total number of shifts an RA must work and the constraints limiting which RA can work a non-monday phone shift. Placing upper and lower bounds on the number of shifts an RA can work must be represented by two inequality constraints, a greater-than-or-equal-to constraint for the lower bound and a less-than-or-equal-to constraint for the upper bound. Therefore 2*(number of RAs) of these constraints exist and there are two rows in A and b for each RA. Because intlinprog only works with less-than-or-equal-to constraints, both sides of the lower bound constraints must be multiplied by -1 in order to change the direction of the inequality to the necessary form. To form the rows of A representing the number-of-shift constraints, the scheduler starts with a row vector of zeros containing an entry for each decision variable. Then, for each RA, ones are placed in the entries that correspond to their RA pairs. The same row of b is set to either the upper bound or lower bound value; each RA needs one row dedicated to both the upper and lower bounds. The upper bound rows of A and b are ready to go but those of the lower bound must be multiplied by -1 before moving onto the next RA. Even more inequality constraints must be placed on the S40 scheduler. Besides constraints on the number of shifts an RA can work, additional inequality constraints also exist for the requirement forcing each non-monday phone shift being assigned to one of the RAs working the 8:25PM to midnight office shift immediately before. One such constraint must be formed for every combination of RA and non-monday phone shift, providing (number of S40 RAs)*(number of non-monday phone shifts) more rows in A. In order to add these constraints to A and b, start by iterating through each S40 RA then for each RA iterate through every non- 19

20 Monday phone shift. Each of these new rows in A contains an entry for every decision variable; the entry for the current iteration of the RA and non-monday phone shift combinations contains a value of 1, the entry for the same RA and the Monday 8:25 to midnight office shift directly before the phone shift in question holds a value of -1, and all other entries are set to 0. The column vector, b, also gains a new zero entry for each new row in A. Equality Constraints Aeq and beq contain the right- and left-hand sides of the equality constraints, respectively. Rows in the two matrices contain the restrictions on the number of RAs that must work each shift. For the NS and OC, a single RA works each shift. On the S40, two RAs must be assigned to office shifts and phone shifts are worked by a single RA. Therefore each area s problem contains the same number of equality constraints as duty shifts needing to be assigned in that community. In Aeq, rows containing entries for each decision variable represent the equality constraints. For each shift, any entry representing a possible assignment of the shift contains a 1 and all other entries hold zeros. The corresponding entry in beq is set to the appropriate number of RAs, either 1 or 2. Decision Variable Upper and Lower Bounds The vectors ub and lb hold the maximum and minimum possible values for the integer decision variables. Both have entries for every decision variable. Because the scheduler s decision variables represent binary values, ub contains only 1 s and lb contains only 0 s. 20

21 Output Data Parsing Once all solver inputs have been filled, intlinprog finds the minimum preference cost schedule, if possible. The solver s output consists of a column vector of decision variables containing a 1 if that RA shift pair was assigned and a 0 otherwise. To make the solution easier to read, this vector is reshaped into a schedule matrix with rows representing RAs and columns designating shifts. In this form, if the entry at row i and column j contains a 1, then RA i has been assigned to shift j. The output information is transformed again into an even more Figure 9: Final Output Matrix of Scheduled Duties straightforward form. The scheduler iterates through the RAs a final time and produces a list of each RA and the shifts they are assigned of the form seen in Figure 10. Implementation Observations Over the course of building the duty scheduler, many interesting linear programming and integer linear programming phenomena occurred. The development process began with the simplest possible problem, assigning n RAs to m duty shifts with no constraints and a cost matrix of all 1 s. From that point, constraints were added to the program gradually, testing with each new addition. When first programmed, the basic scheduler with no constraints used linprog, a different Matlab solver function that does not require its decision variables to be binary. On the first run of the program, using linprog produced a schedule in which each RA was assigned the 21

22 same mixed number of shifts, equivalent to the number of shifts divided by the number of RAs. After this observation, intlinprog, became the solver of choice instead. Using intlinprog in the basic problem produced a schedule in which the difference in the number of shifts assigned to RAs was at most 1. In the first test case, 12 RAs were to be assigned 26 shifts. Using intlinprog, the first two RAs were each scheduled for 3 shifts and the remaining 10 were assigned 2 each. This solution is produced by iterating through RAs and assigning duties one-at-a-time until no more remain to be scheduled. The implementation of a cost matrix with values other than 1, displayed the need for constraints upon the number of shifts an RA could work. With different cost values and no restrictions on the number of shifts, a high amount of variance existed in the distribution of shifts between RAs. RAs who gave higher preference ratings or had fewer class conflicts were assigned more shifts than their peers. Adding constraints resolved this problem and exhibited their necessity in the scheduler. Results To run and test the RA duty scheduler, specially developed functions create random RA preference matrices of the same form as the Google form input. These functions randomize the number of days requested off and the preference ratings of specific shift times. Simulating the scheduling of the current RA staff involved making random matrices of 109 S40 RAs, 14 NS RAs, and 12 OC RAs over the course of an 18 week semester. Combining these matrices provides a test input for the scheduling function. In the tests, costs for S40 RA Prefer, Neutral, and Conflicts with Class preference rankings were 1, 5, and 10, respectively. A cost of 10 was also 22

23 given to weeks requested off by NS and OC RAs. Running the scheduler with an input of this form finds a schedule satisfying every defined constraint in between 45 and 55 seconds with total preference costs of approximately For reference, the lowest possible cost, in which only Prefer S40 shifts are assigned, is 414. The highest score, in which every shift assigned conflicts with class or a requested time off, is Therefore the scheduler assigned a mixture of shift preferences. When run for a sequence of ten random RA input arrays, the program produced ten duty schedulers in 8 minutes and 41 seconds for an average runtime of 52.1 seconds per schedule. The ten schedules scored an average preference cost of 2268, analogous to the cost of a single run. The consistency in runtime and preference cost shows the reliability of the proposed scheduling system. As the program need only be run once to plan an entire semester of duties, 45 to 55 seconds is a very reasonable runtime. If, instead of 10, a value of 100 is used as the highest cost, neither the runtime nor the total preference cost of the schedule changed. This result also indicates the robustness of the duty scheduler. Increasing the highest cost value would decrease the chance of RA shift pairs with the lowest preference score being scheduled. Because the total cost of the final schedule remains the same when the highest cost value is increased, the program scheduled the same amount of lowest preference ranked shifts in both cases. 23

24 Conclusion In light of the robustness in solution schedules, both in terms of runtime and total preference cost, the goal of developing an effective RA duty scheduling system was achieved. The final system built during this project runs in an appropriately short amount of time, produces schedules with consistent preference costs, and succeeds in basing assignments on RA preferences. 24