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Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization
Cost of fairness in agent scheduling for contact centers
Department of Industrial Engineering, Ozyegin University, Istanbul, 34794, Turkey |
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.
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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
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. 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. |
[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. |
[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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |









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 |
Preference Score | |
First | 8 |
Second | 4 |
Third | 2 |
Fourth | 1 |
Not preferred | 0 |
Preference Score | |
First | 8 |
Second | 4 |
Third | 2 |
Fourth | 1 |
Not preferred | 0 |
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 |
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 | |
Shift Index | |
Time Interval Index in a Day | |
Agent Index | |
Individual Fairness Lower Limit | |
Overall Fairness Lower Limit | |
Weekly Cost Per Agent | |
Cost Estimation for 1% of Understaffing | |
Cost of Shift Undesirability | |
Average Per Person Arrival Shuttle Cost for Intervals | |
Average Per Person Departure Shuttle Cost for Intervals | |
Break Time Factor (Effectiveness) of Agent in Intervals of Shift | |
Demand in Intervals of Weeks | |
Agents' Preference Value of Shifts | |
Starting Interval Binary of Shifts | |
Ending Interval Binary of Shifts |
Description | Parameter |
Week Index in Planning Horizon | |
Shift Index | |
Time Interval Index in a Day | |
Agent Index | |
Individual Fairness Lower Limit | |
Overall Fairness Lower Limit | |
Weekly Cost Per Agent | |
Cost Estimation for 1% of Understaffing | |
Cost of Shift Undesirability | |
Average Per Person Arrival Shuttle Cost for Intervals | |
Average Per Person Departure Shuttle Cost for Intervals | |
Break Time Factor (Effectiveness) of Agent in Intervals of Shift | |
Demand in Intervals of Weeks | |
Agents' Preference Value of Shifts | |
Starting Interval Binary of Shifts | |
Ending Interval Binary of Shifts |
Description | Notation |
Binary Variable of Agents' Shift in Weeks | |
Individual Average Fairness Score Auxiliary Variable of Working Weeks | |
Individual Average Weekly Fairness Score Variable | |
Number of Agents Variable in Shifts of Weeks | |
Understaffed Level Variable in Intervals |
Description | Notation |
Binary Variable of Agents' Shift in Weeks | |
Individual Average Fairness Score Auxiliary Variable of Working Weeks | |
Individual Average Weekly Fairness Score Variable | |
Number of Agents Variable in Shifts of Weeks | |
Understaffed Level Variable in Intervals |
Description | Parameter | Value |
Number of Weeks | 4 | |
Number of Shifts | 17 | |
Number of Time Intervals | 24 | |
Number of Agent | 150 | |
Agent Cost | $200 | |
Understaffing Coeffcient | $10 |
Description | Parameter | Value |
Number of Weeks | 4 | |
Number of Shifts | 17 | |
Number of Time Intervals | 24 | |
Number of Agent | 150 | |
Agent Cost | $200 | |
Understaffing Coeffcient | $10 |
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 |
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% |
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% |
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 |
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 |
Shifts | Unrestricted | Pregnant | Disabled | Student | Distant |
1 | |||||
2 | |||||
3 | |||||
4 | |||||
5 | |||||
6 | |||||
7 | |||||
8 | |||||
9 | |||||
10 | |||||
11 | |||||
12 | |||||
13 | |||||
14 | |||||
15 | |||||
16 | |||||
17 |
Shifts | Unrestricted | Pregnant | Disabled | Student | Distant |
1 | |||||
2 | |||||
3 | |||||
4 | |||||
5 | |||||
6 | |||||
7 | |||||
8 | |||||
9 | |||||
10 | |||||
11 | |||||
12 | |||||
13 | |||||
14 | |||||
15 | |||||
16 | |||||
17 |
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 |
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% |
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 |
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 |
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 |
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] |
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