American Institute of Mathematical Sciences

Advanced
March  2022, 18(2): 873-896. doi: 10.3934/jimo.2021001

Cost of fairness in agent scheduling for contact centers

Onur Şimşek and  O. Erhun Kundakcioglu ,

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.

Keywords: Workforce scheduling, fairness, contact centers, employee satisfaction, long term planning.
Mathematics Subject Classification: Primary: 90B90; Secondary: 90-10.
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.

show all references

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.

Figure 1.  Forecasted Intraday Call Volumes
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Figure 2.  Required and Working Agents
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Figure 3.  Demand Volumes
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Figure 4.  Preference Scores
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Figure 5.  Distribution of Agents in Shifts
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Figure 6.  Cost and Fairness Values for P1
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Figure 7.  Total Understaffed and Working Hours for P1
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Figure 8.  Cost and Fairness Values for P2
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Figure 9.  Total Understaffed and Working Hours for P2
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Table 1.  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
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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
Table 2.  Inputs and a Sample Assignment
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Table 3.  Break Time (Effectiveness) Factor
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Table 4.  Preference Scoring Sample
$ \textbf{Preference Priority} $ Preference Score
First 8
Second 4
Third 2
Fourth 1
Not preferred 0
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$ \textbf{Preference Priority} $ Preference Score
First 8
Second 4
Third 2
Fourth 1
Not preferred 0
Table 5.  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
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$ \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
Table 6.  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 $
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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 $
Table 7.  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 $
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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 $
Table 8.  Shift Descriptions
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Table 9.  Shuttle Costs
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Table 10.  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
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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
Table 11.  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
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$ 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
Table 12.  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%
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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%
Table 13.  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
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$ 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
Table 14.  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 $
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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 $
Table 15.  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
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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
Table 16.  Cost of Restriction
no rest. low rest. medium rest. high rest.
total cost (＄1000) 139 139 139 159
cost gap - 0% 0% 14%
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no rest. low rest. medium rest. high rest.
total cost (＄1000) 139 139 139 159
cost gap - 0% 0% 14%
Table 17.  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
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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
Table 18.  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
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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
Table 19.  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
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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]

Semih Yalçındağ, Ettore Lanzarone. Merging short-term and long-term planning problems in home health care under continuity of care and patterns for visits. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1487-1504. doi: 10.3934/jimo.2021029
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Omer Faruk Yilmaz, Mehmet Bulent Durmusoglu. A performance comparison and evaluation of metaheuristics for a batch scheduling problem in a multi-hybrid cell manufacturing system with skilled workforce assignment. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1219-1249. doi: 10.3934/jimo.2018007
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Qiong Liu, Jialiang Liu, Zhaorui Dong, Mengmeng Zhan, Zhen Mei, Baosheng Ying, Xinyu Shao. Integrated optimization of process planning and scheduling for reducing carbon emissions. Journal of Industrial and Management Optimization, 2021, 17 (3) : 1025-1055. doi: 10.3934/jimo.2020010
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Masataka Kato, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. Effect of energy-saving server scheduling on power consumption for large-scale data centers. Journal of Industrial and Management Optimization, 2016, 12 (2) : 667-685. doi: 10.3934/jimo.2016.12.667
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Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discrete and Continuous Dynamical Systems - S, 2011, 4 (2) : 273-309. doi: 10.3934/dcdss.2011.4.273
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Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 687-705. doi: 10.3934/dcdsb.2011.15.687
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Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205
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Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1521-1543. doi: 10.3934/cpaa.2021031
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Dariush Mohamadi Zanjirani, Majid Esmaelian. An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1235-1259. doi: 10.3934/jimo.2018202
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Xiaoying Han, Peter E. Kloeden, Basiru Usman. Long term behavior of a random Hopfield neural lattice model. Communications on Pure and Applied Analysis, 2019, 18 (2) : 809-824. doi: 10.3934/cpaa.2019039
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Tyson Loudon, Stephen Pankavich. Mathematical analysis and dynamic active subspaces for a long term model of HIV. Mathematical Biosciences & Engineering, 2017, 14 (3) : 709-733. doi: 10.3934/mbe.2017040
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Mouhamadou Aliou M. T. Baldé, Diaraf Seck. Coupling the shallow water equation with a long term dynamics of sand dunes. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1521-1551. doi: 10.3934/dcdss.2016061
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M. Hadjiandreou, Raul Conejeros, Vassilis S. Vassiliadis. Towards a long-term model construction for the dynamic simulation of HIV infection. Mathematical Biosciences & Engineering, 2007, 4 (3) : 489-504. doi: 10.3934/mbe.2007.4.489
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Brian Pigott, Sarah Raynor. Long-term stability for kdv solitons in weighted Hs spaces. Communications on Pure and Applied Analysis, 2017, 16 (2) : 393-416. doi: 10.3934/cpaa.2017020
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Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
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Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110
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Igor Chueshov, Peter E. Kloeden, Meihua Yang. Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 991-1009. doi: 10.3934/dcdsb.2018139
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Raffaele D'Ambrosio, Martina Moccaldi, Beatrice Paternoster. Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2763-2773. doi: 10.3934/dcdsb.2018105
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Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
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Ian Johnson, Evelyn Sander, Thomas Wanner. Branch interactions and long-term dynamics for the diblock copolymer model in one dimension. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3671-3705. doi: 10.3934/dcds.2013.33.3671

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