# American Institute of Mathematical Sciences

• Previous Article
Optimal pricing and advertising decisions with suppliers' oligopoly competition: Stakelberg-Nash game structures
• JIMO Home
• This Issue
• Next Article
Analysis of $D$-$BMAP/G/1$ queueing system under $N$-policy and its cost optimization
doi: 10.3934/jimo.2021001

## Cost of fairness in agent scheduling for contact centers

 Department of Industrial Engineering, Ozyegin University, Istanbul, 34794, Turkey

* Corresponding author: erhun.kundakcioglu@ozyegin.edu.tr

Received  December 2019 Revised  October 2020 Published  December 2020

We study a workforce scheduling problem faced in contact centers with considerations on a fair distribution of shifts in compliance with agent preferences. We develop a mathematical model that aims to minimize operating costs associated with labor, transportation of agents, and lost customers. Aside from typical work hour-related constraints, we also try to conform with agents' preferences for shifts, as a measure of fairness. We plot the trade-off between agent satisfaction and total operating costs for Vestel, one of Turkey's largest consumer electronics companies. We present insights on the increased cost to have content and a fair environment on several agent availability scenarios.

Citation: Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021001
##### References:

show all references

##### References:
Required and Working Agents
Demand Volumes
Preference Scores
Distribution of Agents in Shifts
Cost and Fairness Values for P1
Total Understaffed and Working Hours for P1
Cost and Fairness Values for P2
Total Understaffed and Working Hours for P2
Model Inputs and Outputs
 Inputs Outputs Demand for a Theoretical Day Scheduling/Planning Horizon Number of Agents in Each Shift Time Intervals and Possible Shifts Total Employee Cost Break Time Distribution Rules Total Shuttle Cost Shuttle (Transportation) Costs Understaffed Hours Agent Wages and Undesirability Cost of Shifts Agent-Shift Assignments Cost of Understaffing Total Satisfaction Score Shift Preference Scores of Agents Fairness Score Distribution Fairness Bounds
 Inputs Outputs Demand for a Theoretical Day Scheduling/Planning Horizon Number of Agents in Each Shift Time Intervals and Possible Shifts Total Employee Cost Break Time Distribution Rules Total Shuttle Cost Shuttle (Transportation) Costs Understaffed Hours Agent Wages and Undesirability Cost of Shifts Agent-Shift Assignments Cost of Understaffing Total Satisfaction Score Shift Preference Scores of Agents Fairness Score Distribution Fairness Bounds
Inputs and a Sample Assignment
Break Time (Effectiveness) Factor
Preference Scoring Sample
 $\textbf{Preference Priority}$ Preference Score First 8 Second 4 Third 2 Fourth 1 Not preferred 0
 $\textbf{Preference Priority}$ Preference Score First 8 Second 4 Third 2 Fourth 1 Not preferred 0
Preference Matrix Sample
 $\textbf{agents}$ shift 1 shift 2 shift 3 shift 4 shift 5 shift 6 shift 7 shift 8 agent 1 8 4 0 0 1 0 0 2 agent 2 8 4 0 0 0 0 2 1 agent 3 4 8 0 2 0 0 0 1 agent 4 4 2 0 1 0 0 8 0 agent 5 4 2 0 1 0 8 0 0 agent 6 2 1 8 4 0 0 0 0 agent 7 1 2 0 4 8 0 0 0 agent 8 0 0 1 2 4 0 0 8 agent 9 0 8 0 4 2 0 0 1
 $\textbf{agents}$ shift 1 shift 2 shift 3 shift 4 shift 5 shift 6 shift 7 shift 8 agent 1 8 4 0 0 1 0 0 2 agent 2 8 4 0 0 0 0 2 1 agent 3 4 8 0 2 0 0 0 1 agent 4 4 2 0 1 0 0 8 0 agent 5 4 2 0 1 0 8 0 0 agent 6 2 1 8 4 0 0 0 0 agent 7 1 2 0 4 8 0 0 0 agent 8 0 0 1 2 4 0 0 8 agent 9 0 8 0 4 2 0 0 1
Model Parameters
 Description Parameter Week Index in Planning Horizon $w$ Shift Index $s$ Time Interval Index in a Day $t$ Agent Index $i$ Individual Fairness Lower Limit $h$ Overall Fairness Lower Limit $H$ Weekly Cost Per Agent $c^\text{agent}$ Cost Estimation for 1% of Understaffing $c^{\text{understaff}}$ Cost of Shift Undesirability $c^{\text{undesirable}}_s$ Average Per Person Arrival Shuttle Cost for Intervals $c^{\text{v}}_t$ Average Per Person Departure Shuttle Cost for Intervals $c'^{\text{v}}_t$ Break Time Factor (Effectiveness) of Agent in Intervals of Shift $a^s_t$ Demand in Intervals of Weeks $d^w_t$ Agents' Preference Value of Shifts $p_{is}$ Starting Interval Binary of Shifts $s_t^s$ Ending Interval Binary of Shifts $e_t^s$
 Description Parameter Week Index in Planning Horizon $w$ Shift Index $s$ Time Interval Index in a Day $t$ Agent Index $i$ Individual Fairness Lower Limit $h$ Overall Fairness Lower Limit $H$ Weekly Cost Per Agent $c^\text{agent}$ Cost Estimation for 1% of Understaffing $c^{\text{understaff}}$ Cost of Shift Undesirability $c^{\text{undesirable}}_s$ Average Per Person Arrival Shuttle Cost for Intervals $c^{\text{v}}_t$ Average Per Person Departure Shuttle Cost for Intervals $c'^{\text{v}}_t$ Break Time Factor (Effectiveness) of Agent in Intervals of Shift $a^s_t$ Demand in Intervals of Weeks $d^w_t$ Agents' Preference Value of Shifts $p_{is}$ Starting Interval Binary of Shifts $s_t^s$ Ending Interval Binary of Shifts $e_t^s$
Decision Variables
 Description Notation Binary Variable of Agents' Shift in Weeks $Y_{isw}$ Individual Average Fairness Score Auxiliary Variable of Working Weeks $A_{iw}$ Individual Average Weekly Fairness Score Variable $Z_i$ Number of Agents Variable in Shifts of Weeks $X^w_s$ Understaffed Level Variable in Intervals $U^w_t$
 Description Notation Binary Variable of Agents' Shift in Weeks $Y_{isw}$ Individual Average Fairness Score Auxiliary Variable of Working Weeks $A_{iw}$ Individual Average Weekly Fairness Score Variable $Z_i$ Number of Agents Variable in Shifts of Weeks $X^w_s$ Understaffed Level Variable in Intervals $U^w_t$
Shift Descriptions
Shuttle Costs
Parameter Values
 Description Parameter Value Number of Weeks $|W|$ 4 Number of Shifts $|S|$ 17 Number of Time Intervals $|T|$ 24 Number of Agent $|I|$ 150 Agent Cost $c^{\text{agent}}$ ＄200 Understaffing Coeffcient $c^{\text{understaff}}$ ＄10
 Description Parameter Value Number of Weeks $|W|$ 4 Number of Shifts $|S|$ 17 Number of Time Intervals $|T|$ 24 Number of Agent $|I|$ 150 Agent Cost $c^{\text{agent}}$ ＄200 Understaffing Coeffcient $c^{\text{understaff}}$ ＄10
Fairness Distribution
 $Z_i$ Range/$h$ 0 1 2 3 4 5 6 7 8 [0-1) 83 0 0 0 0 0 0 0 0 [1-2) 19 62 0 0 0 0 0 0 0 [2-3) 35 68 120 0 0 0 0 0 0 [3-4) 8 9 14 89 0 0 0 0 0 [4-5) 3 8 12 61 130 0 0 0 0 [5-6) 0 1 3 0 14 81 0 0 0 [6-7) 0 2 1 0 6 68 149 0 0 [7-8) 0 0 0 0 0 0 1 77 0 [8] 2 0 0 0 0 1 0 73 150 Total Satisfaction Score 178 289 370 519 640 824 904 1123 1200 Cost (in ＄1000) 139 139 139 139 140 143 157 522 618
 $Z_i$ Range/$h$ 0 1 2 3 4 5 6 7 8 [0-1) 83 0 0 0 0 0 0 0 0 [1-2) 19 62 0 0 0 0 0 0 0 [2-3) 35 68 120 0 0 0 0 0 0 [3-4) 8 9 14 89 0 0 0 0 0 [4-5) 3 8 12 61 130 0 0 0 0 [5-6) 0 1 3 0 14 81 0 0 0 [6-7) 0 2 1 0 6 68 149 0 0 [7-8) 0 0 0 0 0 0 1 77 0 [8] 2 0 0 0 0 1 0 73 150 Total Satisfaction Score 178 289 370 519 640 824 904 1123 1200 Cost (in ＄1000) 139 139 139 139 140 143 157 522 618
Comparison of P1 and P2
 Overall Fairness Score 640 824 904 1123 P1 Cost (＄1000) 140 143 157 522 P2 Cost (＄1000) 139 139 141 304 (P1 Cost - P2 Cost) / P2 Cost 0.7% 2.3% 10.9% 71.5%
 Overall Fairness Score 640 824 904 1123 P1 Cost (＄1000) 140 143 157 522 P2 Cost (＄1000) 139 139 141 304 (P1 Cost - P2 Cost) / P2 Cost 0.7% 2.3% 10.9% 71.5%
Fairness Distribution for P2
 $Z_i$ Range/$H$ 640 824 904 1123 [0-1) 23 16 17 0 [1-2) 10 7 6 2 [2-3) 25 9 7 3 [3-4) 5 4 2 0 [4-5) 19 17 7 11 [5-6) 11 6 3 0 [6-7) 17 20 12 1 [7-8) 2 5 12 0 [8] 38 66 84 133 Cost (in ＄1000) 139 139 141 304
 $Z_i$ Range/$H$ 640 824 904 1123 [0-1) 23 16 17 0 [1-2) 10 7 6 2 [2-3) 25 9 7 3 [3-4) 5 4 2 0 [4-5) 19 17 7 11 [5-6) 11 6 3 0 [6-7) 17 20 12 1 [7-8) 2 5 12 0 [8] 38 66 84 133 Cost (in ＄1000) 139 139 141 304
Available Shifts for Agent Groups
 Shifts Unrestricted Pregnant Disabled Student Distant 1 $\bullet$ $\bullet$ $\bullet$ $\bullet$ 2 $\bullet$ $\bullet$ $\bullet$ 3 $\bullet$ $\bullet$ $\bullet$ 4 $\bullet$ 5 $\bullet$ $\bullet$ 6 $\bullet$ 7 $\bullet$ 8 $\bullet$ 9 $\bullet$ 10 $\bullet$ 11 $\bullet$ 12 $\bullet$ 13 $\bullet$ 14 $\bullet$ 15 $\bullet$ $\bullet$ 16 $\bullet$ $\bullet$ $\bullet$ 17 $\bullet$ $\bullet$ $\bullet$
 Shifts Unrestricted Pregnant Disabled Student Distant 1 $\bullet$ $\bullet$ $\bullet$ $\bullet$ 2 $\bullet$ $\bullet$ $\bullet$ 3 $\bullet$ $\bullet$ $\bullet$ 4 $\bullet$ 5 $\bullet$ $\bullet$ 6 $\bullet$ 7 $\bullet$ 8 $\bullet$ 9 $\bullet$ 10 $\bullet$ 11 $\bullet$ 12 $\bullet$ 13 $\bullet$ 14 $\bullet$ 15 $\bullet$ $\bullet$ 16 $\bullet$ $\bullet$ $\bullet$ 17 $\bullet$ $\bullet$ $\bullet$
Number of Agents in Groups
 Scenario Unrestricted Pregnant Disabled Student Distant high restriction 30 20 20 20 60 med. restriction 90 10 10 10 30 low restriction 120 5 5 5 15 no restriction 150 0 0 0 0
 Scenario Unrestricted Pregnant Disabled Student Distant high restriction 30 20 20 20 60 med. restriction 90 10 10 10 30 low restriction 120 5 5 5 15 no restriction 150 0 0 0 0
Cost of Restriction
 no rest. low rest. medium rest. high rest. total cost (＄1000) 139 139 139 159 cost gap - 0% 0% 14%
 no rest. low rest. medium rest. high rest. total cost (＄1000) 139 139 139 159 cost gap - 0% 0% 14%
Cost of Fairness Levels with Restriction in ＄1000
 no rest. low rest. med. rest. high rest. h=4 140 140 140 193 h=5 143 144 155 224 h=6 157 160 176 243
 no rest. low rest. med. rest. high rest. h=4 140 140 140 193 h=5 143 144 155 224 h=6 157 160 176 243
Efficient Solutions for Fairness Levels with Restriction
 Cost Acceptable Solution 1 Solution 2 Tolerance Cost (＄1000) 0% 139 h=0|medium rest. scenario N/A 1% 140 h=4|medium rest. scenario 2% 141 3% 143 h=5|no rest. scenario 4% 144 5% 146 10% 153 15% 160 h=5|medium rest. scenario h=6|low rest. scenario
 Cost Acceptable Solution 1 Solution 2 Tolerance Cost (＄1000) 0% 139 h=0|medium rest. scenario N/A 1% 140 h=4|medium rest. scenario 2% 141 3% 143 h=5|no rest. scenario 4% 144 5% 146 10% 153 15% 160 h=5|medium rest. scenario h=6|low rest. scenario
Solution Times for Instances
 Restrictions Preferences Bound $h$ for P1 $H$ for P2 Time (sec) None Individual – P1 0 3 1 7 2 17 3 1257 4 117 5 126 6 49 7 14 8 5 Overall – P2 640 12 824 18 904 16 1123 10 Low Individual – P1 0 5 4 20 5 30 6 20 Medium 0 3 4 12 5 14 6 9 High 0 2 4 7 5 9 6 6
 Restrictions Preferences Bound $h$ for P1 $H$ for P2 Time (sec) None Individual – P1 0 3 1 7 2 17 3 1257 4 117 5 126 6 49 7 14 8 5 Overall – P2 640 12 824 18 904 16 1123 10 Low Individual – P1 0 5 4 20 5 30 6 20 Medium 0 3 4 12 5 14 6 9 High 0 2 4 7 5 9 6 6
 [1] Min Ji, Xinna Ye, Fangyao Qian, T.C.E. Cheng, Yiwei Jiang. Parallel-machine scheduling in shared manufacturing. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020174 [2] Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001 [3] Hongguang Ma, Xiang Li. Multi-period hazardous waste collection planning with consideration of risk stability. Journal of Industrial & Management Optimization, 2021, 17 (1) : 393-408. doi: 10.3934/jimo.2019117 [4] Hanyu Gu, Hue Chi Lam, Yakov Zinder. Planning rolling stock maintenance: Optimization of train arrival dates at a maintenance center. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020177 [5] Nguyen Huu Can, Nguyen Huy Tuan, Donal O'Regan, Vo Van Au. On a final value problem for a class of nonlinear hyperbolic equations with damping term. Evolution Equations & Control Theory, 2021, 10 (1) : 103-127. doi: 10.3934/eect.2020053 [6] Wenjun Liu, Hefeng Zhuang. Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 907-942. doi: 10.3934/dcdsb.2020147 [7] Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226 [8] Wenbin Lv, Qingyuan Wang. Global existence for a class of Keller-Segel models with signal-dependent motility and general logistic term. Evolution Equations & Control Theory, 2021, 10 (1) : 25-36. doi: 10.3934/eect.2020040 [9] Guo Zhou, Yongquan Zhou, Ruxin Zhao. Hybrid social spider optimization algorithm with differential mutation operator for the job-shop scheduling problem. Journal of Industrial & Management Optimization, 2021, 17 (2) : 533-548. doi: 10.3934/jimo.2019122 [10] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [11] Rong Wang, Yihong Du. Long-time dynamics of a diffusive epidemic model with free boundaries. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020360 [12] Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466 [13] Mingjun Zhou, Jingxue Yin. Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle. Electronic Research Archive, , () : -. doi: 10.3934/era.2020122 [14] Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403 [15] Jason Murphy, Kenji Nakanishi. Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1507-1517. doi: 10.3934/dcds.2020328 [16] Jean-Claude Saut, Yuexun Wang. Long time behavior of the fractional Korteweg-de Vries equation with cubic nonlinearity. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1133-1155. doi: 10.3934/dcds.2020312 [17] Linglong Du, Min Yang. Pointwise long time behavior for the mixed damped nonlinear wave equation in $\mathbb{R}^n_+$. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020033

2019 Impact Factor: 1.366