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Call Center Scheduling Problem Federal Reserve Bank United States Treasury May 6, 2009 Morgan Brunz Brittany McCluskey

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Background of Problem Situation Every month the U.S. Treasury sends out 11 million checks for Social Security and other government payments Cost = $1 per check every month To eliminate cost –> Switch over to direct deposit Two methods of contacting the FED –Call Center - 70% Usage Rate –Website - 30% Usage Rate

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Background of Problem Situation Call center can receive between 9,000 to 1,000 calls per day Call center has 23 permanent employees and hires between 100 to 5 temporary agents per day FED must determine how many temporary employees should be hired per day Call Center Hours –7am - 7pm, Monday - Friday 4 Shifts – 7am, 8am, 9am, 10:15am –Lunch - 45 minutes –Two Breaks– 15 minutes

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Problem Objective Optimization functionalities –Minimize cost by not over hiring temporary workers –Maximize customer service by eliminating dropped calls Our analysis focused on 2 nd, 3 rd, and 4 th weeks –Most checks delivered in the 1 st week of the month Highest call volume demands

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Two Model Approach 1. Waiting Line System –Arrival rate of calls per 15 minutes –Service rate per 15 minutes –Use number of agents needed for each 15-minute time interval for second model 2. Integer Programming Model –Variables – Possible employee schedules –Equation – Each 15-minute period Number of Agents working >= Number of Agents needed

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Integer Programming Model Example 7am7:304:457:157:455:004:30 # agents ≥ # needed for response time this interval 1 st shift: on at 7, off at 4:30 2 nd shift: on at 7:15, off at 4:45 3rd shift: on at 7:30, off at 5 Waiting Line System

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Technical Description – Waiting Line System Service Rate –Average Call = 4 minutes 20 seconds –15 min / 4.33 min = 3.46 calls per 15-minute interval Arrival Rate –Given past 3 months of call volume by 30-minute intervals –Added call volume received across all days for each 30-minute interval –Averaged and divided by 2 to get 15-minute intervals Obtained employees needed per 15-minute interval by dividing average arrival rates by service rate of 3.46

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Technical Description – Integer Programming Model Used programming language AMPL/CPLEX Created 2 text files for model –FedAmpl –FedFile 24 variables = possible temporary agents schedules currently used by FED –A Matrix Row T = 15-minute interval Column S = Possible schedules x1x2x3x4 7-7151111 715-7301111 730-7451111 745-81111 8-815 0 011 815-83011 01 830-845111 0 845-91111 9-9151111 915-9301111 930-9451111 945-101111 10-10151111 1015-10301111

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Technical Description – Integer Programming Model Constraints –One constraint generated 48 = one per 15-minute time period –Agents Working >= Agents Needed (RHS) RightHandSide –Represents the employees needed per 15-minute time period –48 RHS values –Total Employees – Permanent Employees Working = Temporary Agents Needed per 15-minute Interval 15 Minute Interval Number of Agents Needed Permanent Employees Agents Needed - Permanents = RHS 7-715312 715-730312 730-745514

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Analysis and Interpretation – Solution Model 1 Found number of temporary employees and their schedules Kept permanent employees’ schedules as they are currently Objective Function = 9 Temporary Employees For Example: Schedule 4: need 1 temporary employee to come in at 7:00 to 3:45, breaks 8:30 and 1:45, and lunch at 10:30 CPLEX Output CPLEX 11.2.0: optimal integer solution; objective 9 6 MIP simplex iterations x [*] := 1 0 4 1 7 0 10 0 13 0 16 0 19 0 22 0 2 0 5 0 8 3 11 0 14 0 17 0 20 0 23 0 3 0 6 3 9 0 12 0 15 0 18 1 21 0 24 1;

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Analysis and Interpretation – Solution Model 2 Changed the RHS by taking out the permanent employees Models temporary and permanent employees Objective Function = 21 Total Employees Does not take into account that permanent employees answer the phones only 80% of their day CPLEX Output CPLEX 11.2.0: optimal integer solution; objective 21 16 MIP simplex iterations x [*] := 1 0 4 0 7 2 10 0 13 0 16 4 19 0 22 0 2 2 5 0 8 0 11 0 14 0 17 0 20 2 23 1 3 2 6 1 9 3 12 0 15 0 18 0 21 0 24 4

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Analysis and Interpretation – Solution Model 3 Factored in the fact that the permanent employees only answer the phone 80% of their day Increased the RHS by multiplying by 1.2 and rounding up Objective Function = 25 Total Employees Next model we will add additional schedules CPLEX Output CPLEX 11.2.0: optimal integer solution; objective 25 18 MIP simplex iterations x [*] := 1 0 4 0 7 2 10 1 13 0 16 4 19 0 22 0 2 3 5 0 8 0 11 0 14 0 17 0 20 3 23 1 3 2 6 2 9 3 12 0 15 0 18 0 21 0 24 4

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Analysis and Interpretation – Solution Model 4 Added 10 more possible schedules with start times staggered every 15 minutes, and breaks and lunches systematically staggered 9 employees used new schedules Objective Function = 24 Total Employees –Objective improved by 1 CPLEX Output CPLEX 11.2.0: optimal integer solution; objective 24 34 MIP simplex iterations x [*] := 1 0 5 1 9 2 13 0 17 0 21 3 25 0 29 0 33 0 2 0 6 2 10 1 14 2 18 0 22 0 26 2 30 2 34 2 3 1 7 0 11 0 15 0 19 0 23 2 27 0 31 0 4 1 8 0 12 0 16 0 20 0 24 0 28 2 32 1

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Analysis and Interpretation – Solution Model 5 Adjusted RHS by subtracting permanent employees Determined temporary employees needed and their schedules using all possible schedules, including 10 new schedules Objective Function = 9 Temporary Employees Additional schedules did not reduce temporary employees needed CPLEX Output CPLEX 11.2.0: optimal integer solution; objective 9 7 MIP simplex iterations x [*] := 1 0 5 0 9 0 13 0 17 0 21 0 25 0 29 0 33 0 2 0 6 3 10 0 14 0 18 1 22 0 26 1 30 0 34 0 3 0 7 0 11 0 15 0 19 0 23 1 27 3 31 0 4 0 8 0 12 0 16 0 20 0 24 0 28 0 32 0

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15 Minute Interval Agents Needed - Permanents 15 Minute Interval Agents Needed - Permanents 7-71521-1156 715-7302115-1304 730-7454130-1451 745-84145-22 8-81542-2151 815-8304215-2301 830-8454230-2453 845-93245-32 9-91533-3150 915-9305315-3300 930-9454330-3450 945-104345-4 10-101544-415-4 1015-10301415-430-4 1030-10451430-445-5 1045-110445-5 11-111515-5150 1115-11301515-530 1130-11453530-5450 1145-123545-61 12-121526-6150 1215-12304615-6300 1230-12456630-645 1245-19645-7 Conclusions Analysis of employees needed per 15-minute interval –Too many permanent employees are starting late in the day –More employees need to begin earlier in the day

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Conclusions Best Solution = Model 4 –Models permanent and temporary employee schedules –Total Employees = 24

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Conclusions Alternative Solution = Model 1 –Models temporary employee schedules –Permanent employee schedules are kept as they are currently –Temporary Employees = 9

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Conclusions Realistic Recommendation – Add two to three employees to CPLEX output Break and Lunch Schedule Recommendation –Limit lunches between 12:15 to 1:15 due to high call volume Additional Suggestions –Part-Time Employee Schedules Additional shifts from 10:00 – 2:00 –4 Day Work Week Go Green

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