# American Institute of Mathematical Sciences

March  2022, 18(2): 873-896. 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  March 2022 Early access  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 and Management Optimization, 2022, 18 (2) : 873-896. doi: 10.3934/jimo.2021001
##### References:
 [1] H. P. Benson, An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem, Journal of Global Optimization, 13 (1998), 1-24.  doi: 10.1023/A:1008215702611. [2] I. Blöchliger, Modeling staff scheduling problems. A tutorial, European Journal of Operational Research, 158 (2004), 533-542.  doi: 10.1016/S0377-2217(03)00387-4. [3] P. Brucker, R. Qu and E. Burke, Personnel scheduling: Models and complexity, European Journal of Operational Research, 210 (2011), 467-473.  doi: 10.1016/j.ejor.2010.11.017. [4] I. Castillo, T. Joro and Y. Y. Li, Workforce scheduling with multiple objectives, European Journal of Operational Research, 196 (2009), 162-170.  doi: 10.1016/j.ejor.2008.02.038. [5] D. Creelman, Top trends in workforce management: How technology provides significant value managing your people (2014), http://audentia-gestion.fr/oracle/workforce-management-2706797.pdf, 2014. [6] A. T. Ernst, H. Jiang, M. Krishnamoorthy and D. Sier, Staff scheduling and rostering: A review of applications, methods and models, European Journal of Operational Research, 153 (2004), 3-27.  doi: 10.1016/S0377-2217(03)00095-X. [7] R. Fink and J. Gillett, Queuing theory and the Taguchi loss function: The cost of customer dissatisfaction in waiting lines, International Journal of Strategic Cost Management, 17–25. [8] D. Fluss, Workforce management: Better but not good enough, https://www.destinationcrm.com/Articles/Columns-Departments/Scouting-Report/Workforce-Management-Better-but-Not-Good-Enough-90113.aspx, 2013. [9] L. Golden, Irregular work scheduling and its consequences, Economic Policy Institute Briefing Paper, 1, No. 394, 41 pp. doi: 10.2139/ssrn.2597172. [10] Gurobi, Gurobi Optimizer 8 Reference Manual, Gurobi Optimization, Inc., 2020. [11] ICMI, The State of Workforce Management, Technical report, International Customer Management Institute, 2017. [12] S. Jütte, D. Müller and U. W. Thonemann, Optimizing railway crew schedules with fairness preferences, Journal of Scheduling, 20 (2017), 43-55.  doi: 10.1007/s10951-016-0499-4. [13] L. Kletzander and N. Musliu, Solving the general employee scheduling problem, Computers & Operations Research, 113 (2020), 104794, 13 pp. doi: 10.1016/j.cor.2019.104794. [14] G. Koole and A. Mandelbaum, Queueing models of call centers: An introduction, Annals of Operations Research, 113 (2002), 41-59.  doi: 10.1023/A:1020949626017. [15] C. K. Y. Lin, K. F. Lai and S. L. Hung, Development of a workforce management system for a customer hotline service, Computers & Operations Research, 27 (2000), 987-1004.  doi: 10.1016/S0305-0548(99)00072-6. [16] M. Liu, X. Liu, F. Chu, E. Zhang and C. Chu, Service-oriented robust worker scheduling with motivation effects, International Journal of Production Research, 1–24. doi: 10.1080/00207543.2020.1730998. [17] J. Lywood, M. Stone and Y. Ekinci, Customer experience and profitability: An application of the empathy rating index (ERIC) in UK call centres, Journal of Database Marketing & Customer Strategy Management, 16 (2009), 207-214.  doi: 10.1057/dbm.2009.24. [18] J. Manyika, S. Lund, M. Chui, J. Bughin, J. Woetzel, P. Batra, R. Ko and S. Sanghvi, Jobs lost, jobs gained: Workforce transitions in a time of automation, McKinsey Global Institute. [19] S. Mohan, Scheduling part-time personnel with availability restrictions and preferences to maximize employee satisfaction, Mathematical and Computer Modelling, 48 (2008), 1806-1813.  doi: 10.1016/j.mcm.2007.12.027. [20] E. L. Örmeci, F. S. Salman and E. Yücel, Staff rostering in call centers providing employee transportation, Omega, 43 (2014), 41-53.  doi: 10.1016/j.omega.2013.06.003. [21] R. Pastor and J. Olivella, Selecting and adapting weekly work schedules with working time accounts: A case of a retail clothing chain, European Journal of Operational Research, 184 (2008), 1-12.  doi: 10.1016/j.ejor.2006.10.028. [22] M. Rocha, J. F. Oliveira and M. A. Carravilla, Cyclic staff scheduling: optimization models for some real-life problems, Journal of Scheduling, 16 (2013), 231-242.  doi: 10.1007/s10951-012-0299-4. [23] R. K. Roy, Design of Experiments Using the Taguchi Approach: 16 Steps to Product and Process Improvement, John Wiley & Sons, 2001. [24] R. Schalk and A. Van Rijckevorsel, Factors influencing absenteeism and intention to leave in a call centre, New Technology, Work and Employment, 22 (2007), 260-274.  doi: 10.1111/j.1468-005X.2007.00198.x. [25] G. Smart, What contributes to the cost of a contact center?, https://www.niceincontact.com/blog/what-contributes-to-the-cost-of-a-contact-center-1, 2010. [26] J. Van den Bergh, J. Beliën, P. De Bruecker, E. Demeulemeester and L. De Boeck, Personnel scheduling: A literature review, European Journal of Operational Research, 226 (2013), 367-385.  doi: 10.1016/j.ejor.2012.11.029. [27] M. Van Den Eeckhout, M. Vanhoucke and B. Maenhout, A decomposed branch-and-price procedure for integrating demand planning in personnel staffing problems, European Journal of Operational Research, 280 (2020), 845-859.  doi: 10.1016/j.ejor.2019.07.069. [28] Vestel, Towards New Horizons: 2019 Annual Report, http://www.vestelinvestorrelations.com/en/financials/annual-reports.aspx, 2019. [29] WorkForceSoftware, New Survey: The 6 Most Critical Workforce Management Issues of 2017, https://www.workforcesoftware.com/blog/6-workforce-management-issues-2017/, 2017. [30] P. D. Wright and S. Mahar, Centralized nurse scheduling to simultaneously improve schedule cost and nurse satisfaction, Omega, 41 (2013), 1042-1052.  doi: 10.1016/j.omega.2012.08.004.

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##### References:
 [1] H. P. Benson, An outer approximation algorithm for generating all efficient extreme points in the outcome set of a multiple objective linear programming problem, Journal of Global Optimization, 13 (1998), 1-24.  doi: 10.1023/A:1008215702611. [2] I. Blöchliger, Modeling staff scheduling problems. A tutorial, European Journal of Operational Research, 158 (2004), 533-542.  doi: 10.1016/S0377-2217(03)00387-4. [3] P. Brucker, R. Qu and E. Burke, Personnel scheduling: Models and complexity, European Journal of Operational Research, 210 (2011), 467-473.  doi: 10.1016/j.ejor.2010.11.017. [4] I. Castillo, T. Joro and Y. Y. Li, Workforce scheduling with multiple objectives, European Journal of Operational Research, 196 (2009), 162-170.  doi: 10.1016/j.ejor.2008.02.038. [5] D. Creelman, Top trends in workforce management: How technology provides significant value managing your people (2014), http://audentia-gestion.fr/oracle/workforce-management-2706797.pdf, 2014. [6] A. T. Ernst, H. Jiang, M. Krishnamoorthy and D. Sier, Staff scheduling and rostering: A review of applications, methods and models, European Journal of Operational Research, 153 (2004), 3-27.  doi: 10.1016/S0377-2217(03)00095-X. [7] R. Fink and J. Gillett, Queuing theory and the Taguchi loss function: The cost of customer dissatisfaction in waiting lines, International Journal of Strategic Cost Management, 17–25. [8] D. Fluss, Workforce management: Better but not good enough, https://www.destinationcrm.com/Articles/Columns-Departments/Scouting-Report/Workforce-Management-Better-but-Not-Good-Enough-90113.aspx, 2013. [9] L. Golden, Irregular work scheduling and its consequences, Economic Policy Institute Briefing Paper, 1, No. 394, 41 pp. doi: 10.2139/ssrn.2597172. [10] Gurobi, Gurobi Optimizer 8 Reference Manual, Gurobi Optimization, Inc., 2020. [11] ICMI, The State of Workforce Management, Technical report, International Customer Management Institute, 2017. [12] S. Jütte, D. Müller and U. W. Thonemann, Optimizing railway crew schedules with fairness preferences, Journal of Scheduling, 20 (2017), 43-55.  doi: 10.1007/s10951-016-0499-4. [13] L. Kletzander and N. Musliu, Solving the general employee scheduling problem, Computers & Operations Research, 113 (2020), 104794, 13 pp. doi: 10.1016/j.cor.2019.104794. [14] G. Koole and A. Mandelbaum, Queueing models of call centers: An introduction, Annals of Operations Research, 113 (2002), 41-59.  doi: 10.1023/A:1020949626017. [15] C. K. Y. Lin, K. F. Lai and S. L. Hung, Development of a workforce management system for a customer hotline service, Computers & Operations Research, 27 (2000), 987-1004.  doi: 10.1016/S0305-0548(99)00072-6. [16] M. Liu, X. Liu, F. Chu, E. Zhang and C. Chu, Service-oriented robust worker scheduling with motivation effects, International Journal of Production Research, 1–24. doi: 10.1080/00207543.2020.1730998. [17] J. Lywood, M. Stone and Y. Ekinci, Customer experience and profitability: An application of the empathy rating index (ERIC) in UK call centres, Journal of Database Marketing & Customer Strategy Management, 16 (2009), 207-214.  doi: 10.1057/dbm.2009.24. [18] J. Manyika, S. Lund, M. Chui, J. Bughin, J. Woetzel, P. Batra, R. Ko and S. Sanghvi, Jobs lost, jobs gained: Workforce transitions in a time of automation, McKinsey Global Institute. [19] S. Mohan, Scheduling part-time personnel with availability restrictions and preferences to maximize employee satisfaction, Mathematical and Computer Modelling, 48 (2008), 1806-1813.  doi: 10.1016/j.mcm.2007.12.027. [20] E. L. Örmeci, F. S. Salman and E. Yücel, Staff rostering in call centers providing employee transportation, Omega, 43 (2014), 41-53.  doi: 10.1016/j.omega.2013.06.003. [21] R. Pastor and J. Olivella, Selecting and adapting weekly work schedules with working time accounts: A case of a retail clothing chain, European Journal of Operational Research, 184 (2008), 1-12.  doi: 10.1016/j.ejor.2006.10.028. [22] M. Rocha, J. F. Oliveira and M. A. Carravilla, Cyclic staff scheduling: optimization models for some real-life problems, Journal of Scheduling, 16 (2013), 231-242.  doi: 10.1007/s10951-012-0299-4. [23] R. K. Roy, Design of Experiments Using the Taguchi Approach: 16 Steps to Product and Process Improvement, John Wiley & Sons, 2001. [24] R. Schalk and A. Van Rijckevorsel, Factors influencing absenteeism and intention to leave in a call centre, New Technology, Work and Employment, 22 (2007), 260-274.  doi: 10.1111/j.1468-005X.2007.00198.x. [25] G. Smart, What contributes to the cost of a contact center?, https://www.niceincontact.com/blog/what-contributes-to-the-cost-of-a-contact-center-1, 2010. [26] J. Van den Bergh, J. Beliën, P. De Bruecker, E. Demeulemeester and L. De Boeck, Personnel scheduling: A literature review, European Journal of Operational Research, 226 (2013), 367-385.  doi: 10.1016/j.ejor.2012.11.029. [27] M. Van Den Eeckhout, M. Vanhoucke and B. Maenhout, A decomposed branch-and-price procedure for integrating demand planning in personnel staffing problems, European Journal of Operational Research, 280 (2020), 845-859.  doi: 10.1016/j.ejor.2019.07.069. [28] Vestel, Towards New Horizons: 2019 Annual Report, http://www.vestelinvestorrelations.com/en/financials/annual-reports.aspx, 2019. [29] WorkForceSoftware, New Survey: The 6 Most Critical Workforce Management Issues of 2017, https://www.workforcesoftware.com/blog/6-workforce-management-issues-2017/, 2017. [30] P. D. Wright and S. Mahar, Centralized nurse scheduling to simultaneously improve schedule cost and nurse satisfaction, Omega, 41 (2013), 1042-1052.  doi: 10.1016/j.omega.2012.08.004.
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
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