Cost of fairness in agent scheduling for contact centers

Onur Şimşek; O. Erhun Kundakcioglu

doi:10.3934/jimo.2021001

Previous Article
The finite-time ruin probability of a risk model with a general counting process and stochastic return
JIMO Home
This Issue
Next Article
Robust optimal strategies of DC pension plans with stochastic volatility and stochastic income under mean-variance criteria

doi: 10.3934/jimo.2021001

Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Management Optimization, 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	L. Golden, Irregular work scheduling and its consequences, Economic Policy Institute Briefing Paper, 1, No. 394, 41 pp. doi: 10.2139/ssrn.2597172. Google Scholar
[10]	Gurobi, Gurobi Optimizer 8 Reference Manual, Gurobi Optimization, Inc., 2020. Google Scholar
[11]	ICMI, The State of Workforce Management, Technical report, International Customer Management Institute, 2017. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	R. K. Roy, Design of Experiments Using the Taguchi Approach: 16 Steps to Product and Process Improvement, John Wiley & Sons, 2001. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	Vestel, Towards New Horizons: 2019 Annual Report, http://www.vestelinvestorrelations.com/en/financials/annual-reports.aspx, 2019. Google Scholar
[29]	WorkForceSoftware, New Survey: The 6 Most Critical Workforce Management Issues of 2017, https://www.workforcesoftware.com/blog/6-workforce-management-issues-2017/, 2017. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[9]	L. Golden, Irregular work scheduling and its consequences, Economic Policy Institute Briefing Paper, 1, No. 394, 41 pp. doi: 10.2139/ssrn.2597172. Google Scholar
[10]	Gurobi, Gurobi Optimizer 8 Reference Manual, Gurobi Optimization, Inc., 2020. Google Scholar
[11]	ICMI, The State of Workforce Management, Technical report, International Customer Management Institute, 2017. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[23]	R. K. Roy, Design of Experiments Using the Taguchi Approach: 16 Steps to Product and Process Improvement, John Wiley & Sons, 2001. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[28]	Vestel, Towards New Horizons: 2019 Annual Report, http://www.vestelinvestorrelations.com/en/financials/annual-reports.aspx, 2019. Google Scholar
[29]	WorkForceSoftware, New Survey: The 6 Most Critical Workforce Management Issues of 2017, https://www.workforcesoftware.com/blog/6-workforce-management-issues-2017/, 2017. Google Scholar
[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. Google Scholar

Figure 1. Forecasted Intraday Call Volumes

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. Required and Working Agents

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3. Demand Volumes

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4. Preference Scores

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5. Distribution of Agents in Shifts

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6. Cost and Fairness Values for P1

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7. Total Understaffed and Working Hours for P1

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8. Cost and Fairness Values for P2

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9. Total Understaffed and Working Hours for P2

Figure Options

Download full-size image

Download as PowerPoint slide

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

$ \textbf{Preference Priority} $	Preference Score
First	8
Second	4
Third	2
Fourth	1
Not preferred	0

$ \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

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	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	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

$ 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

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%

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 $

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

	no rest.	low rest.	medium rest.	high rest.
total cost (＄1000)	139	139	139	159
cost gap	-	0%	0%	14%

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

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

	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

[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 & Management Optimization, 2021 doi: 10.3934/jimo.2021029
[2]	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 & Management Optimization, 2018, 14 (3) : 1219-1249. doi: 10.3934/jimo.2018007
[3]	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 & Management Optimization, 2021, 17 (3) : 1025-1055. doi: 10.3934/jimo.2020010
[4]	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 & Management Optimization, 2016, 12 (2) : 667-685. doi: 10.3934/jimo.2016.12.667
[5]	Elena Bonetti, Giovanna Bonfanti, Riccarda Rossi. Long-time behaviour of a thermomechanical model for adhesive contact. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 273-309. doi: 10.3934/dcdss.2011.4.273
[6]	Stanisław Migórski, Anna Ochal, Mircea Sofonea. Analysis of a frictional contact problem for viscoelastic materials with long memory. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 687-705. doi: 10.3934/dcdsb.2011.15.687
[7]	Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205
[8]	Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1521-1543. doi: 10.3934/cpaa.2021031
[9]	Dariush Mohamadi Zanjirani, Majid Esmaelian. An integrated approach based on Fuzzy Inference System for scheduling and process planning through multiple objectives. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1235-1259. doi: 10.3934/jimo.2018202
[10]	Xiaoying Han, Peter E. Kloeden, Basiru Usman. Long term behavior of a random Hopfield neural lattice model. Communications on Pure & Applied Analysis, 2019, 18 (2) : 809-824. doi: 10.3934/cpaa.2019039
[11]	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
[12]	Mouhamadou Aliou M. T. Baldé, Diaraf Seck. Coupling the shallow water equation with a long term dynamics of sand dunes. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1521-1551. doi: 10.3934/dcdss.2016061
[13]	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
[14]	Brian Pigott, Sarah Raynor. Long-term stability for kdv solitons in weighted H^s spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 393-416. doi: 10.3934/cpaa.2017020
[15]	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 & Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
[16]	Chao Yang, Yanbing Yang. Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4643-4658. doi: 10.3934/dcdss.2021110
[17]	Igor Chueshov, Peter E. Kloeden, Meihua Yang. Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 991-1009. doi: 10.3934/dcdsb.2018139
[18]	Raffaele D'Ambrosio, Martina Moccaldi, Beatrice Paternoster. Numerical preservation of long-term dynamics by stochastic two-step methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2763-2773. doi: 10.3934/dcdsb.2018105
[19]	Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934/dcdsb.2019046
[20]	Ian Johnson, Evelyn Sander, Thomas Wanner. Branch interactions and long-term dynamics for the diblock copolymer model in one dimension. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3671-3705. doi: 10.3934/dcds.2013.33.3671

American Institute of Mathematical Sciences

Cost of fairness in agent scheduling for contact centers

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors