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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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;

11. 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

12. 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

13. 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

14. 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

15. 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

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

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

18. 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