American Institute of Mathematical Sciences

Advanced
January  2023, 19(1): 549-572. doi: 10.3934/jimo.2021196

The inventory replenishment policy in an uncertain production-inventory-routing system

Chenyin Wang 1, , Yaodong Ni 2,, and  Xiangfeng Yang 2,

1. 

School of Information Science, Guangzhou Xinhua University, Guangzhou 510520, China
2. 

School of Information Technology and Management, University of International Business and Economics, Beijing 100029, China

* Corresponding author: Yaodong Ni (ni@uibe.edu.cn)

Received  June 2021 Revised  August 2021 Published  January 2023 Early access  November 2021

Fund Project: This work was supported in part by the National Natural Science Foundation of China under Grant 71471038, in part by the Program for Huiyuan Distinguished Young Scholars, University of International Business and Economics (UIBE) under Grant 17JQ09, in part by the Fundamental Research Funds for the Central Universities in UIBE under Grant CXTD10-05, in part by the Youth Innovative Talent Projects of Guangdong's Universities Provincial Key Platforms and Significant Scientific Research Projects under Grant 2018KQNCX362, in part by the Educational Science Research Projects for 13th Five-Year Plan in Guangdong under Grant 2020GXJK163, and in part by the Key Discipline Project of Guangzhou Xinhua University under Grant 2020XZD02. The authors are grateful to the editor and the referees for their valuable comments and suggestions

This study introduces an uncertain programming model for the integrated production routing problem (PRP) in an uncertain production-inventory-routing system. Based on uncertainty theory, an uncertain programming model is proposed firstly and then transformed into a deterministic and equivalent model. The study further probes into different types of replenishment policies under the condition of uncertain demands, mainly the uncertain maximum level (UML) policy and the uncertain order-up to level (UOU) policy. Some inequalities are put forward to define the UML policy and the UOU policy under the uncertain environments, and the influences brought by uncertain demands are highlighted. The overall costs with optimal solution of the uncertain decision model grow with the increase of the confidence levels. And they are simultaneously affected by the variances of uncertain variables but rely on the value of confidence levels. Results show that when the confidence levels are not less than 0.5, the cost difference between the two policies begins to narrow along with the increase of the confidence levels and the variances of uncertain variables, eventually being trending to zero. When there are higher confidence levels and relatively large uncertainty in realistic applications, in which the solution scale is escalated, being conducive to its efficiency advantage, the comprehensive advantages of the UOU policy is obvious.

Keywords: Integrated production routing problem, uncertain variable, uncertain programming, replenishment policy.
Mathematics Subject Classification: Primary: 90B05, 90C70; Secondary: 90B99.
Citation: Chenyin Wang, Yaodong Ni, Xiangfeng Yang. The inventory replenishment policy in an uncertain production-inventory-routing system. Journal of Industrial and Management Optimization, 2023, 19 (1) : 549-572. doi: 10.3934/jimo.2021196
References:
[1]

N. Absi, C. Archetti, S. Dauzère-Pérès and D. Feillet, A two phase iterative heuristic approach for the production routing problem, Transportation Science, 49 (2015), 784-795.  doi: 10.1287/trsc.2014.0523.

[2]

N. Absi, C. Archetti, S. Dauzère-Pérès, D. Feillet and M. G. Speranza, Comparing sequential and integrated approaches for the production routing problem, European Journal of Operational Research, 269 (2018), 633-646.  doi: 10.1016/j.ejor.2018.01.052.

[3]

Y. Adulyasak, J.-F. Cordeau and R. Jans, Benders decomposition for production routing under demand uncertainty, Operational Research, 63 (2015), 851-868.  doi: 10.1287/opre.2015.1401.

[4]

A. Agra, M. Christiansen, K. S. Ivarsøy, I. E. Solhaug and A. Tomasgard, Combined ship routing and inventory management in the salmon farming industry, Annals of Operations Research, 253 (2017), 799-823.  doi: 10.1007/s10479-015-2088-x.

[5]

A. Agra, C. Requejo and F. Rodrigues, A hybrid heuristic for a stochastic production-inventory-routing problem, Electronic Notes in Discrete Mathematics, 64 (2018), 345-354.  doi: 10.1016/j.endm.2018.02.009.

[6]

C. Archetti, L. Bertazzi, G. Paletta and M. G. Speranza, Analysis of the maximum level policy in a production distribution system, Computers and Operations Research, 38 (2011), 1731-1746.  doi: 10.1016/j.cor.2011.03.002.

[7]

C. Archetti and M. G. Speranza, The inventory routing problem: The value of integration, International Transactions in Operational Research, 23 (2016), 393-407.  doi: 10.1111/itor.12226.

[8]

J. F. Bard and N. Nananukul, Heuristics for a multiperiod inventory routing problem with production decisions, Computers and Industrial Engineering, 57 (2009), 713-723.  doi: 10.1016/j.cie.2009.01.020.

[9]

J. F. Bard and N. Nananukul, The integrated production inventor distribution routing problem, Journal of Scheduling, 12 (2009), 257-280.  doi: 10.1007/s10951-008-0081-9.

[10]

T. Bayley, H. Süral and J. H. Bookbinder, A hybrid benders approach for coordinated capacitated lot-sizing of multiple product families with set-up times, International Journal of Production Research, 56 (2018), 1326-1344.  doi: 10.1080/00207543.2017.1338778.

[11]

T. Bektas, G. Laporte and D. Vigo, Integrated vehicle routing problems, Computers and Operations Research, 55 (2015), 126.  doi: 10.1016/j.cor.2014.08.008.

[12]

L. Bertazzi and M. G. Speranza, Inventory routing problems: An introduction, EURO J. Transp Logist, 1 (2012), 307-326.  doi: 10.1007/s13676-012-0016-7.

[13]

Y. Boutarfa, A. Senoussi, N. K. Mouss and N. Brahimi, A tabu search heuristic for an integrated production distribution problem with clustered retailers, IFAC-PapersOnLine, 49 (2016), 1514-1519.  doi: 10.1016/j.ifacol.2016.07.794.

[14]

N. Brahimi and T. Aouam, Multi-item production routing problem with backordering: A MILP approach, International Journal of Production Research, 54 (2015), 1076-1093.  doi: 10.1080/00207543.2015.1047971.

[15]

P. Chandra, A dynamic distribution model with warehouse and customer replenishment requirements, Journal of the Operational Research Society, 44 (1993), 681-692.  doi: 10.1057/jors.1993.117.

[16]

Z. Chen, Y. Lan, R. Zhao and C. Shang, Deadline-based incentive contracts in project management with cost salience, Fuzzy Optimization and Decision Making, 18 (2019), 451-473.  doi: 10.1007/s10700-019-09302-y.

[17]

J. F. Côté, G. Guastaroba and M. G. Speranza, The value of integrating loading and routing, European Journal of Operational Research, 257 (2016), 89-105.  doi: 10.1016/j.ejor.2016.06.072.

[18]

M. Darvish and L. C. Coelho, Sequential versus integrated optimization: Production, location, inventory control, and distribution, European Journal of Operational Research, 268 (2018), 203-214.  doi: 10.1016/j.ejor.2018.01.028.

[19]

X. Fang, Y. Du and Y. Qiu, Reducing carbon emissions in a closed-loop production routing problem with simultaneous pickups and deliveries under carbon cap-and-trade, Sustainability, 9 (2017), 2198.  doi: 10.3390/su9122198.

[20]

L.-L. Fu, M. A. Aloulou and C. Triki, Integrated production scheduling and vehicle routing problem with job splitting and delivery time windows, International Journal of Production Research, 55 (2017), 5942-5957.  doi: 10.1080/00207543.2017.1308572.

[21]

A. Ghasemkhani, R. Tavakkoli-Moghaddam, Y. Rahimi, S. Shahnejat-Bushehri and H. Tavakkoli-Moghaddam, Integrated production-inventory-routing problem for multi-perishable products under uncertainty by meta-heuristic algorithms, International Journal of Production Research, (2021). doi: 10.1080/00207543.2021.1902013.

[22]

A. H. Golsefidi and M. R. A. Jokar, A robust optimization approach for the production-inventory-routing problem with simultaneous pickup and delivery, Computers and Industrial Engineering, 143 (2020), 106388.  doi: 10.1016/j.cie.2020.106388.

[23]

F. Hein and C. Almeder, Quantitative insights into the integrated supply vehicle routing and production planning problem, International Journal of Production Economics, 177 (2016), 66-76.  doi: 10.1016/j.ijpe.2016.04.014.

[24]

H. Ke, T. Su and Y. Ni, Uncertain random multilevel programming with application to production control problem, Soft Computing, 19 (2015), 1739-1746.  doi: 10.1007/s00500-014-1361-2.

[25]

R. S. Kumar, K. Kondapaneni, V. Dixit, A. Goswami, L. S. Thakur and M. K. Tiwari, Multi-objective modeling of production and pollution routing problem with time window: A self-learning particle swarm optimization approach, Computers and Industrial Engineering, 99 (2016), 29-40.  doi: 10.1016/j.cie.2015.07.003.

[26]

M. Lai, X. Cai and X. Li, Mechanism design for collaborative production-distribution planning with shipment consolidation, Transportation Research Part E: Logistics and Transportation Review, 106 (2017), 137-159.  doi: 10.1016/j.tre.2017.07.014.

[27]

L. Lei, S. Liu, A. Ruszczynski and S. Park, On the integrated production, inventory, and distribution routing problem, IIE Transactions, 38 (2006), 955-970.  doi: 10.1080/07408170600862688.

[28]

Y. Li, F. Chu and K. Chen, Coordinated production inventory routing planning for perishable food, IFAC-PapersOnLine, 50 (2017), 4246-4251.  doi: 10.1016/j.ifacol.2017.08.829.

[29]

Y. Li, F. Chu, C. Feng, C. Chu and M. Zhou, Integrated production inventory routing planning for intelligent food logistics systems, IEEE Transactions on Intelligent Transportation Systems, 3 (2019), 867-878.  doi: 10.1109/TITS.2018.2835145.

[30]

Y. Li, X. Li and S. Zhang, Optimal pricing of customized bus services and ride-sharing based on a competitive game model, Omega, 103 (2021), 102413.  doi: 10.1016/j.omega.2021.102413.

[31]

B. Liu, Uncertainty Theory, 2$^nd$ edition, Springer, Berlin, 2007.

[32]

B. Liu, Some research problems in uncertainty theory, Journal of Uncertain systems, 3 (2009), 3-10. 

[33]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer, Berlin, 2010.

[34]

P. Liu, A. Hendalianpour, J. Razmi and M. S. Sangari, A solution algorithm for integrated production-inventory-routing of perishable goods with transshipment and uncertain demand, Complex and Intelligent Systems, 7 (2021), 1349-1365.  doi: 10.1007/s40747-020-00264-y.

[35]

W. Ma, Y. Che, H. Huang and H. Ke, Resource-constrained project scheduling problem with uncertain durations and renewable resources, International Journal of Machine Learning and Cybernetics, 7 (2016), 613-621.  doi: 10.1007/s13042-015-0444-4.

[36]

P. L. Miranda, R. Morabito and D. Ferreira, Optimization model for a production, inventory, distribution and routing problem in small furniture companies, Top, 26 (2018), 30-67.  doi: 10.1007/s11750-017-0448-1.

[37]

I. Moon, Y. J. Jeong and S. Saha, Fuzzy bi-objective production distribution planning problem under the carbon emission constraint, Sustainability, 8 (2016), 798.  doi: 10.3390/su8080798.

[38]

A. Nananukul, Lot-Sizing and Inventory Routing for a Production-Distribution Supply Chain, phdthesis, Ph.D thesis, The University of Texas at Austin, 2008.

[39]

Y. Qiu, M. Ni, L. Wang, Q. Li, X. Fang and P. M. Pardalos, Production routing problems with reverse logistics and remanufacturing, Transportation Research Part E: Logistics and Transportation, 111 (2018), 87-100.  doi: 10.1016/j.tre.2018.01.009.

[40]

Y. Qiu, L. Wang, X. Xu, X. Fang and P. M. Pardalos, A variable neighborhood search heuristic algorithm for production routing problems, Applied Soft Computing, 66 (2018), 311-318.  doi: 10.1016/j.asoc.2018.02.032.

[41]

Y. Qiu, L. Wang, X. Xu, X. Fang and P. M. Pardalos, Formulations and branch-and-cut algorithms for multi-product multi-vehicle production routing problems with startup cost, Expert Systems with Applications, 98 (2018), 1-10.  doi: 10.1016/j.eswa.2018.01.006.

[42]

V. Schmid, K. F. Doerner and G. Laporte, Rich routing problems arising in supply chain management, European Journal of Operational Research, 224 (2013), 435-448.  doi: 10.1016/j.ejor.2012.08.014.

[43]

A. Senoussi, S. Dauzère-Pérès, N. Brahimi, B. Penz and N. K. Mouss, Heuristics based genetic algorithm for capacitated multi vehicle production inventory distribution problem, Computers and Operations Research, 96 (2018), 108-119.  doi: 10.1016/j.cor.2018.04.010.

[44]

J. Shen and K. Zhu, Uncertain supply chain problem with price and effort, International Journal of Fuzzy Systems, 20 (2018), 1145-1158.  doi: 10.1007/s40815-017-0407-x.

[45]

O. Solyali, J. F. Cordeau and G. Laporte, Robust inventory routing under demand uncertainty, Transportation Science, 46 (2012), 327-340.  doi: 10.1287/trsc.1110.0387.

[46]

M. Stalhane, H. Andersson, M. Christiansen and K. Fagerholt, Vendor managed inventory in tramp shipping, Omega, 47 (2014), 60-72.  doi: 10.1016/j.omega.2014.03.004.

[47]

B. Vahdani, S. T. A. Niaki and S. Aslanzade, Production-inventory-routing coordination with capacity and time window constraints for perishable products: Heuristic and meta-heuristic algorithms, Journal of Cleaner Production, 161 (2017), 598-618.  doi: 10.1016/j.jclepro.2017.05.113.

[48]

Q. Zhang, A. Sundaramoorthy, I. E. Grossmann and J. M. Pinto, Multiscale production routing in multicommodity supply chains with complex production facilities, Computers and Operations Research, 79 (2017), 207-222.  doi: 10.1016/j.cor.2016.11.001.

[49]

J. Zhou, F. Yang and K. Wang, Multi-objective optimization in uncertain random environments, Fuzzy Optimization and Decision Making, 13 (2014), 397-413.  doi: 10.1007/s10700-014-9183-3.

show all references

References:
[1]

N. Absi, C. Archetti, S. Dauzère-Pérès and D. Feillet, A two phase iterative heuristic approach for the production routing problem, Transportation Science, 49 (2015), 784-795.  doi: 10.1287/trsc.2014.0523.

[2]

N. Absi, C. Archetti, S. Dauzère-Pérès, D. Feillet and M. G. Speranza, Comparing sequential and integrated approaches for the production routing problem, European Journal of Operational Research, 269 (2018), 633-646.  doi: 10.1016/j.ejor.2018.01.052.

[3]

Y. Adulyasak, J.-F. Cordeau and R. Jans, Benders decomposition for production routing under demand uncertainty, Operational Research, 63 (2015), 851-868.  doi: 10.1287/opre.2015.1401.

[4]

A. Agra, M. Christiansen, K. S. Ivarsøy, I. E. Solhaug and A. Tomasgard, Combined ship routing and inventory management in the salmon farming industry, Annals of Operations Research, 253 (2017), 799-823.  doi: 10.1007/s10479-015-2088-x.

[5]

A. Agra, C. Requejo and F. Rodrigues, A hybrid heuristic for a stochastic production-inventory-routing problem, Electronic Notes in Discrete Mathematics, 64 (2018), 345-354.  doi: 10.1016/j.endm.2018.02.009.

[6]

C. Archetti, L. Bertazzi, G. Paletta and M. G. Speranza, Analysis of the maximum level policy in a production distribution system, Computers and Operations Research, 38 (2011), 1731-1746.  doi: 10.1016/j.cor.2011.03.002.

[7]

C. Archetti and M. G. Speranza, The inventory routing problem: The value of integration, International Transactions in Operational Research, 23 (2016), 393-407.  doi: 10.1111/itor.12226.

[8]

J. F. Bard and N. Nananukul, Heuristics for a multiperiod inventory routing problem with production decisions, Computers and Industrial Engineering, 57 (2009), 713-723.  doi: 10.1016/j.cie.2009.01.020.

[9]

J. F. Bard and N. Nananukul, The integrated production inventor distribution routing problem, Journal of Scheduling, 12 (2009), 257-280.  doi: 10.1007/s10951-008-0081-9.

[10]

T. Bayley, H. Süral and J. H. Bookbinder, A hybrid benders approach for coordinated capacitated lot-sizing of multiple product families with set-up times, International Journal of Production Research, 56 (2018), 1326-1344.  doi: 10.1080/00207543.2017.1338778.

[11]

T. Bektas, G. Laporte and D. Vigo, Integrated vehicle routing problems, Computers and Operations Research, 55 (2015), 126.  doi: 10.1016/j.cor.2014.08.008.

[12]

L. Bertazzi and M. G. Speranza, Inventory routing problems: An introduction, EURO J. Transp Logist, 1 (2012), 307-326.  doi: 10.1007/s13676-012-0016-7.

[13]

Y. Boutarfa, A. Senoussi, N. K. Mouss and N. Brahimi, A tabu search heuristic for an integrated production distribution problem with clustered retailers, IFAC-PapersOnLine, 49 (2016), 1514-1519.  doi: 10.1016/j.ifacol.2016.07.794.

[14]

N. Brahimi and T. Aouam, Multi-item production routing problem with backordering: A MILP approach, International Journal of Production Research, 54 (2015), 1076-1093.  doi: 10.1080/00207543.2015.1047971.

[15]

P. Chandra, A dynamic distribution model with warehouse and customer replenishment requirements, Journal of the Operational Research Society, 44 (1993), 681-692.  doi: 10.1057/jors.1993.117.

[16]

Z. Chen, Y. Lan, R. Zhao and C. Shang, Deadline-based incentive contracts in project management with cost salience, Fuzzy Optimization and Decision Making, 18 (2019), 451-473.  doi: 10.1007/s10700-019-09302-y.

[17]

J. F. Côté, G. Guastaroba and M. G. Speranza, The value of integrating loading and routing, European Journal of Operational Research, 257 (2016), 89-105.  doi: 10.1016/j.ejor.2016.06.072.

[18]

M. Darvish and L. C. Coelho, Sequential versus integrated optimization: Production, location, inventory control, and distribution, European Journal of Operational Research, 268 (2018), 203-214.  doi: 10.1016/j.ejor.2018.01.028.

[19]

X. Fang, Y. Du and Y. Qiu, Reducing carbon emissions in a closed-loop production routing problem with simultaneous pickups and deliveries under carbon cap-and-trade, Sustainability, 9 (2017), 2198.  doi: 10.3390/su9122198.

[20]

L.-L. Fu, M. A. Aloulou and C. Triki, Integrated production scheduling and vehicle routing problem with job splitting and delivery time windows, International Journal of Production Research, 55 (2017), 5942-5957.  doi: 10.1080/00207543.2017.1308572.

[21]

A. Ghasemkhani, R. Tavakkoli-Moghaddam, Y. Rahimi, S. Shahnejat-Bushehri and H. Tavakkoli-Moghaddam, Integrated production-inventory-routing problem for multi-perishable products under uncertainty by meta-heuristic algorithms, International Journal of Production Research, (2021). doi: 10.1080/00207543.2021.1902013.

[22]

A. H. Golsefidi and M. R. A. Jokar, A robust optimization approach for the production-inventory-routing problem with simultaneous pickup and delivery, Computers and Industrial Engineering, 143 (2020), 106388.  doi: 10.1016/j.cie.2020.106388.

[23]

F. Hein and C. Almeder, Quantitative insights into the integrated supply vehicle routing and production planning problem, International Journal of Production Economics, 177 (2016), 66-76.  doi: 10.1016/j.ijpe.2016.04.014.

[24]

H. Ke, T. Su and Y. Ni, Uncertain random multilevel programming with application to production control problem, Soft Computing, 19 (2015), 1739-1746.  doi: 10.1007/s00500-014-1361-2.

[25]

R. S. Kumar, K. Kondapaneni, V. Dixit, A. Goswami, L. S. Thakur and M. K. Tiwari, Multi-objective modeling of production and pollution routing problem with time window: A self-learning particle swarm optimization approach, Computers and Industrial Engineering, 99 (2016), 29-40.  doi: 10.1016/j.cie.2015.07.003.

[26]

M. Lai, X. Cai and X. Li, Mechanism design for collaborative production-distribution planning with shipment consolidation, Transportation Research Part E: Logistics and Transportation Review, 106 (2017), 137-159.  doi: 10.1016/j.tre.2017.07.014.

[27]

L. Lei, S. Liu, A. Ruszczynski and S. Park, On the integrated production, inventory, and distribution routing problem, IIE Transactions, 38 (2006), 955-970.  doi: 10.1080/07408170600862688.

[28]

Y. Li, F. Chu and K. Chen, Coordinated production inventory routing planning for perishable food, IFAC-PapersOnLine, 50 (2017), 4246-4251.  doi: 10.1016/j.ifacol.2017.08.829.

[29]

Y. Li, F. Chu, C. Feng, C. Chu and M. Zhou, Integrated production inventory routing planning for intelligent food logistics systems, IEEE Transactions on Intelligent Transportation Systems, 3 (2019), 867-878.  doi: 10.1109/TITS.2018.2835145.

[30]

Y. Li, X. Li and S. Zhang, Optimal pricing of customized bus services and ride-sharing based on a competitive game model, Omega, 103 (2021), 102413.  doi: 10.1016/j.omega.2021.102413.

[31]

B. Liu, Uncertainty Theory, 2$^nd$ edition, Springer, Berlin, 2007.

[32]

B. Liu, Some research problems in uncertainty theory, Journal of Uncertain systems, 3 (2009), 3-10. 

[33]

B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer, Berlin, 2010.

[34]

P. Liu, A. Hendalianpour, J. Razmi and M. S. Sangari, A solution algorithm for integrated production-inventory-routing of perishable goods with transshipment and uncertain demand, Complex and Intelligent Systems, 7 (2021), 1349-1365.  doi: 10.1007/s40747-020-00264-y.

[35]

W. Ma, Y. Che, H. Huang and H. Ke, Resource-constrained project scheduling problem with uncertain durations and renewable resources, International Journal of Machine Learning and Cybernetics, 7 (2016), 613-621.  doi: 10.1007/s13042-015-0444-4.

[36]

P. L. Miranda, R. Morabito and D. Ferreira, Optimization model for a production, inventory, distribution and routing problem in small furniture companies, Top, 26 (2018), 30-67.  doi: 10.1007/s11750-017-0448-1.

[37]

I. Moon, Y. J. Jeong and S. Saha, Fuzzy bi-objective production distribution planning problem under the carbon emission constraint, Sustainability, 8 (2016), 798.  doi: 10.3390/su8080798.

[38]

A. Nananukul, Lot-Sizing and Inventory Routing for a Production-Distribution Supply Chain, phdthesis, Ph.D thesis, The University of Texas at Austin, 2008.

[39]

Y. Qiu, M. Ni, L. Wang, Q. Li, X. Fang and P. M. Pardalos, Production routing problems with reverse logistics and remanufacturing, Transportation Research Part E: Logistics and Transportation, 111 (2018), 87-100.  doi: 10.1016/j.tre.2018.01.009.

[40]

Y. Qiu, L. Wang, X. Xu, X. Fang and P. M. Pardalos, A variable neighborhood search heuristic algorithm for production routing problems, Applied Soft Computing, 66 (2018), 311-318.  doi: 10.1016/j.asoc.2018.02.032.

[41]

Y. Qiu, L. Wang, X. Xu, X. Fang and P. M. Pardalos, Formulations and branch-and-cut algorithms for multi-product multi-vehicle production routing problems with startup cost, Expert Systems with Applications, 98 (2018), 1-10.  doi: 10.1016/j.eswa.2018.01.006.

[42]

V. Schmid, K. F. Doerner and G. Laporte, Rich routing problems arising in supply chain management, European Journal of Operational Research, 224 (2013), 435-448.  doi: 10.1016/j.ejor.2012.08.014.

[43]

A. Senoussi, S. Dauzère-Pérès, N. Brahimi, B. Penz and N. K. Mouss, Heuristics based genetic algorithm for capacitated multi vehicle production inventory distribution problem, Computers and Operations Research, 96 (2018), 108-119.  doi: 10.1016/j.cor.2018.04.010.

[44]

J. Shen and K. Zhu, Uncertain supply chain problem with price and effort, International Journal of Fuzzy Systems, 20 (2018), 1145-1158.  doi: 10.1007/s40815-017-0407-x.

[45]

O. Solyali, J. F. Cordeau and G. Laporte, Robust inventory routing under demand uncertainty, Transportation Science, 46 (2012), 327-340.  doi: 10.1287/trsc.1110.0387.

[46]

M. Stalhane, H. Andersson, M. Christiansen and K. Fagerholt, Vendor managed inventory in tramp shipping, Omega, 47 (2014), 60-72.  doi: 10.1016/j.omega.2014.03.004.

[47]

B. Vahdani, S. T. A. Niaki and S. Aslanzade, Production-inventory-routing coordination with capacity and time window constraints for perishable products: Heuristic and meta-heuristic algorithms, Journal of Cleaner Production, 161 (2017), 598-618.  doi: 10.1016/j.jclepro.2017.05.113.

[48]

Q. Zhang, A. Sundaramoorthy, I. E. Grossmann and J. M. Pinto, Multiscale production routing in multicommodity supply chains with complex production facilities, Computers and Operations Research, 79 (2017), 207-222.  doi: 10.1016/j.cor.2016.11.001.

[49]

J. Zhou, F. Yang and K. Wang, Multi-objective optimization in uncertain random environments, Fuzzy Optimization and Decision Making, 13 (2014), 397-413.  doi: 10.1007/s10700-014-9183-3.

Figure 1.  The changes of cost in the UMLI policy
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Figure 2.  The changes of cost in the UML policy
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Figure 3.  The changes of cost in the UOU policy
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Figure 4.  The change of the total cost with cost uncertainty
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Table 1.  Indices and sets
$ i,j $: Indices for retailers, where $ 0 $ corresponds to the plant.
$ t $: Index for periods or days, $ |T|=\tau $.
$ N $: Set of retailers, $ N_{0}=N\bigcup\{0\} $.
$ K $: Set of vehicles, $ K=\{1,2,...,m\} $.
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$ i,j $: Indices for retailers, where $ 0 $ corresponds to the plant.
$ t $: Index for periods or days, $ |T|=\tau $.
$ N $: Set of retailers, $ N_{0}=N\bigcup\{0\} $.
$ K $: Set of vehicles, $ K=\{1,2,...,m\} $.
Table 2.  Parameters
$ \tilde{d}_{it} $: Uncertain demand at retailer $ i $ in period $ t $.
$ \tilde{f} $: Uncertain fixed production setup cost.
$ \tilde{u} $: Uncertain unit production cost.
$ \tilde{h}_{i} $: Uncertain unit inventory holding cost at the plant or retailer.
$ \tilde{c}_{ij} $: Uncertain transportation cost from node $ i $ to node $ j $.
$ C $: Production capacity of the plant.
$ m $: The number of vehicles.
$ Q_{k} $: Capacity of vehicle $ k $.
$ B_{i} $: Initial inventory at retailer $ i $, where $ 0 $ corresponds to the plant.
$ L_{i} $: Maximum inventory level at the plant and retailers.
$ \alpha $: Confidence level about uncertain costs.
$ \beta _{i} $: Confidence level of node $ i $ (satisfaction degree of uncertain demands).
$ \gamma_{i} $: Confidence level of node $ i $ (satisfaction degree of uncertain demands).
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$ \tilde{d}_{it} $: Uncertain demand at retailer $ i $ in period $ t $.
$ \tilde{f} $: Uncertain fixed production setup cost.
$ \tilde{u} $: Uncertain unit production cost.
$ \tilde{h}_{i} $: Uncertain unit inventory holding cost at the plant or retailer.
$ \tilde{c}_{ij} $: Uncertain transportation cost from node $ i $ to node $ j $.
$ C $: Production capacity of the plant.
$ m $: The number of vehicles.
$ Q_{k} $: Capacity of vehicle $ k $.
$ B_{i} $: Initial inventory at retailer $ i $, where $ 0 $ corresponds to the plant.
$ L_{i} $: Maximum inventory level at the plant and retailers.
$ \alpha $: Confidence level about uncertain costs.
$ \beta _{i} $: Confidence level of node $ i $ (satisfaction degree of uncertain demands).
$ \gamma_{i} $: Confidence level of node $ i $ (satisfaction degree of uncertain demands).
Table 3.  Decision variables
$ z_{t} $: Equal to 1 if there is production at the plant in period $ t $, 0 otherwise.
$ p_{t} $: Production quantity in period $ t $.
$ x_{ijkt} $: Equal to 1 if vehicle $ k $ travels directly from node $ i $ to node $ j $ in period $ t $, 0 otherwise.
$ w_{ikt} $: Load of vehicle $ k $ immediately before making a delivery to retailer $ i $ in period $ t $.
$ q_{ikt} $: Quantity delivered to retailer $ i $ by vehicle $ k $ in period $ t $.
$ y_{ikt} $: Equal to 1 if node $ i $ is visited by vehicle $ k $ in period $ t $, 0 otherwise.
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$ z_{t} $: Equal to 1 if there is production at the plant in period $ t $, 0 otherwise.
$ p_{t} $: Production quantity in period $ t $.
$ x_{ijkt} $: Equal to 1 if vehicle $ k $ travels directly from node $ i $ to node $ j $ in period $ t $, 0 otherwise.
$ w_{ikt} $: Load of vehicle $ k $ immediately before making a delivery to retailer $ i $ in period $ t $.
$ q_{ikt} $: Quantity delivered to retailer $ i $ by vehicle $ k $ in period $ t $.
$ y_{ikt} $: Equal to 1 if node $ i $ is visited by vehicle $ k $ in period $ t $, 0 otherwise.
Table 4.  The lower and upper limit, and the interval in the MLI and UMLI policy
Situation Policy Lower Limit Upper Limit Interval
Deterministic MLI 0 +$ \infty $ [0, $ +\infty $]
Linear UMLI $ R_{L}(\beta_{i}) $ +$ \infty $ [$ R_{L}(\beta_{i}) $, $ +\infty $]
Normal UMLI $ R_{N}(\beta_{i}) $ +$ \infty $ [$ R_{N}(\beta_{i}) $, $ +\infty $]
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Situation Policy Lower Limit Upper Limit Interval
Deterministic MLI 0 +$ \infty $ [0, $ +\infty $]
Linear UMLI $ R_{L}(\beta_{i}) $ +$ \infty $ [$ R_{L}(\beta_{i}) $, $ +\infty $]
Normal UMLI $ R_{N}(\beta_{i}) $ +$ \infty $ [$ R_{N}(\beta_{i}) $, $ +\infty $]
Table 5.  The lower and upper limit, and the interval in the ML and UML policy
Situation Policy Lower Limit Upper Limit Interval
Deterministic ML 0 $ L[i] $ [0, $ L[i] $]
Linear UML $ R_{L}(\beta_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $ [$ R_{L}(\beta_{i}) $, $ L_{i}-R_{L}(\gamma_{i}) $]
Normal UML $ R_{N}(\beta_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $ [$ R_{N}(\beta_{i}) $, $ L_{i}-R_{N}(\gamma_{i}) $]
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Situation Policy Lower Limit Upper Limit Interval
Deterministic ML 0 $ L[i] $ [0, $ L[i] $]
Linear UML $ R_{L}(\beta_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $ [$ R_{L}(\beta_{i}) $, $ L_{i}-R_{L}(\gamma_{i}) $]
Normal UML $ R_{N}(\beta_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $ [$ R_{N}(\beta_{i}) $, $ L_{i}-R_{N}(\gamma_{i}) $]
Table 6.  The lower and upper limit, and the interval in the OU and UOU policies
Situation Policy Lower Limit Upper Limit Interval
Deterministic OU 0 $ L[i] $ $ L[i] $
Linear UOU $ R_{L}(\beta_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $
Normal UOU $ R_{N}(\beta_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $
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Situation Policy Lower Limit Upper Limit Interval
Deterministic OU 0 $ L[i] $ $ L[i] $
Linear UOU $ R_{L}(\beta_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $ $ L_{i}-R_{L}(\gamma_{i}) $
Normal UOU $ R_{N}(\beta_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $ $ L_{i}-R_{N}(\gamma_{i}) $
Table 7.  The uncertain variables of the PRP in linear uncertain environment
Parameters Values
$ \tilde{d_{it}} $ $ \mathcal{L}(d_{it}(1-\epsilon^{ld}),d_{it}(1+\epsilon^{ld})) $
$ \tilde{f} $ $ \mathcal{L}(f(1-\epsilon^{lf}),f(1+\epsilon^{lf})) $
$ \tilde{p} $ $ \mathcal{L}(p(1-\epsilon^{lp}),p(1+\epsilon^{lp})) $
$ \tilde{h_{0}} $ $ \mathcal{L}(h_{0}(1-\epsilon^{lh}),h_{0}(1+\epsilon^{lh})) $
$ \tilde{h_{i}} $ $ \mathcal{L}(h_{i}(1-\epsilon^{lh}),h_{i}(1+\epsilon^{lh})) $
$ \tilde{c_{ij}} $ $ \mathcal{L}(c_{ij}(1-\epsilon^{lc}),c_{ij}(1+\epsilon^{lc})) $
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Parameters Values
$ \tilde{d_{it}} $ $ \mathcal{L}(d_{it}(1-\epsilon^{ld}),d_{it}(1+\epsilon^{ld})) $
$ \tilde{f} $ $ \mathcal{L}(f(1-\epsilon^{lf}),f(1+\epsilon^{lf})) $
$ \tilde{p} $ $ \mathcal{L}(p(1-\epsilon^{lp}),p(1+\epsilon^{lp})) $
$ \tilde{h_{0}} $ $ \mathcal{L}(h_{0}(1-\epsilon^{lh}),h_{0}(1+\epsilon^{lh})) $
$ \tilde{h_{i}} $ $ \mathcal{L}(h_{i}(1-\epsilon^{lh}),h_{i}(1+\epsilon^{lh})) $
$ \tilde{c_{ij}} $ $ \mathcal{L}(c_{ij}(1-\epsilon^{lc}),c_{ij}(1+\epsilon^{lc})) $
Table 8.  The change trend of lower and upper limit, the interval, and the cost in the UML policy
($ \beta_{i}, \gamma_{i}) $ Lower Limit Upper limit Gap $ Cost_{M} $
($<0.5, <0.5) $ $ \downarrow $ $ \uparrow $ $ \uparrow $ $ \downarrow $
($<0.5, =0.5) $ $ \downarrow $ = $ \downarrow $ $ \downarrow $
($<0.5,>0.5) $ $ \downarrow $ $ \downarrow $ $ \ast $ $ \ast $
($ =0.5,<0.5) $ = $ \uparrow $ $ \uparrow $ $ \downarrow $
($ =0.5, =0.5) $ = = = =
($ =0.5,>0.5) $ = $ \downarrow $ $ \downarrow $ $ \uparrow $
($>0.5,<0.5) $ $ \uparrow $ $ \uparrow $ $ \ast $ $ \ast $
($>0.5, =0.5) $ $ \uparrow $ = $ \downarrow $ $ \uparrow $
($>0.5,>0.5) $ $ \uparrow $ $ \downarrow $ $ \downarrow $ $ \uparrow $
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($ \beta_{i}, \gamma_{i}) $ Lower Limit Upper limit Gap $ Cost_{M} $
($<0.5, <0.5) $ $ \downarrow $ $ \uparrow $ $ \uparrow $ $ \downarrow $
($<0.5, =0.5) $ $ \downarrow $ = $ \downarrow $ $ \downarrow $
($<0.5,>0.5) $ $ \downarrow $ $ \downarrow $ $ \ast $ $ \ast $
($ =0.5,<0.5) $ = $ \uparrow $ $ \uparrow $ $ \downarrow $
($ =0.5, =0.5) $ = = = =
($ =0.5,>0.5) $ = $ \downarrow $ $ \downarrow $ $ \uparrow $
($>0.5,<0.5) $ $ \uparrow $ $ \uparrow $ $ \ast $ $ \ast $
($>0.5, =0.5) $ $ \uparrow $ = $ \downarrow $ $ \uparrow $
($>0.5,>0.5) $ $ \uparrow $ $ \downarrow $ $ \downarrow $ $ \uparrow $
Table 9.  The change trend of lower and upper limit, the interval, and the cost in the UOU policies
($ \beta_{i}, \gamma_{i}) $ Lower Limit Upper limit Gap $ Cost_{U} $
($<0.5,<0.5) $ $ \downarrow $ $ \uparrow $ $ \uparrow $ $ \uparrow $
($<0.5, =0.5) $ $ \downarrow $ = $ \downarrow $ =
($<0.5,>0.5) $ $ \downarrow $ $ \downarrow $ $ \ast $ $ \downarrow $
($ =0.5,<0.5) $ = $ \uparrow $ $ \uparrow $ $ \uparrow $
($ =0.5, =0.5) $ = = = =
($ =0.5,>0.5) $ = $ \downarrow $ $ \downarrow $ $ \downarrow $
($>0.5,<0.5) $ $ \uparrow $ $ \uparrow $ $ \ast $ $ \uparrow $
($>0.5, =0.5) $ $ \uparrow $ = $ \downarrow $ =
($>0.5,>0.5) $ $ \uparrow $ $ \downarrow $ $ \downarrow $ $ \downarrow $
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($ \beta_{i}, \gamma_{i}) $ Lower Limit Upper limit Gap $ Cost_{U} $
($<0.5,<0.5) $ $ \downarrow $ $ \uparrow $ $ \uparrow $ $ \uparrow $
($<0.5, =0.5) $ $ \downarrow $ = $ \downarrow $ =
($<0.5,>0.5) $ $ \downarrow $ $ \downarrow $ $ \ast $ $ \downarrow $
($ =0.5,<0.5) $ = $ \uparrow $ $ \uparrow $ $ \uparrow $
($ =0.5, =0.5) $ = = = =
($ =0.5,>0.5) $ = $ \downarrow $ $ \downarrow $ $ \downarrow $
($>0.5,<0.5) $ $ \uparrow $ $ \uparrow $ $ \ast $ $ \uparrow $
($>0.5, =0.5) $ $ \uparrow $ = $ \downarrow $ =
($>0.5,>0.5) $ $ \uparrow $ $ \downarrow $ $ \downarrow $ $ \downarrow $
Table 10.  The cost relative difference value between the UML and UOU policies with $ Lx = 1 $
$ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
[0.0, 0.3) 0.1 0.936 0.858 0.780 0.702 0.624
[0.0, 0.3) 0.3 0.809 0.736 0.663 0.590 0.457
[0.0, 0.3) 0.5 0.698 0.630 0.561 0.438 0.372
[0.0, 0.3) 0.7 0.560 0.472 0.409 0.347 0.264
[0.0, 0.3) 0.9 0.443 0.384 0.325 0.347 0.190
[0.3, 0.6) 0.1 2.028 1.726 1.424 1.122 0.803
[0.3, 0.6) 0.3 1.364 1.284 0.893 0.651 0.354
[0.3, 0.6) 0.5 0.948 0.754 0.561 0.294 0.000
[0.3, 0.6) 0.7 0.593 0.421 0.233 0.000 0.000
[0.3, 0.6) 0.9 0.361 0.193 0.000 0.000 0.000
[0.6, 0.9) 0.1 4.432 3.646 2.860 2.055 1.205
[0.6, 0.9) 0.3 2.118 1.664 1.211 0.736 0.218
[0.6, 0.9) 0.5 1.197 0.879 0.561 0.154 $ \circ $
[0.6, 0.9) 0.7 0.630 0.380 0.106 $ \circ $ $ \circ $
[0.6, 0.9) 0.9 0.304 0.080 $ \circ $ $ \circ $ $ \circ $
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$ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
[0.0, 0.3) 0.1 0.936 0.858 0.780 0.702 0.624
[0.0, 0.3) 0.3 0.809 0.736 0.663 0.590 0.457
[0.0, 0.3) 0.5 0.698 0.630 0.561 0.438 0.372
[0.0, 0.3) 0.7 0.560 0.472 0.409 0.347 0.264
[0.0, 0.3) 0.9 0.443 0.384 0.325 0.347 0.190
[0.3, 0.6) 0.1 2.028 1.726 1.424 1.122 0.803
[0.3, 0.6) 0.3 1.364 1.284 0.893 0.651 0.354
[0.3, 0.6) 0.5 0.948 0.754 0.561 0.294 0.000
[0.3, 0.6) 0.7 0.593 0.421 0.233 0.000 0.000
[0.3, 0.6) 0.9 0.361 0.193 0.000 0.000 0.000
[0.6, 0.9) 0.1 4.432 3.646 2.860 2.055 1.205
[0.6, 0.9) 0.3 2.118 1.664 1.211 0.736 0.218
[0.6, 0.9) 0.5 1.197 0.879 0.561 0.154 $ \circ $
[0.6, 0.9) 0.7 0.630 0.380 0.106 $ \circ $ $ \circ $
[0.6, 0.9) 0.9 0.304 0.080 $ \circ $ $ \circ $ $ \circ $
Table 11.  The cost relative difference value between the UML and UOU policies with $ Lx = 2 $
$ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
[0.0, 0.3) 0.1 1.448 1.369 1.278 1.199 1.120
[0.0, 0.3) 0.3 1.268 1.195 1.215 1.048 0.975
[0.0, 0.3) 0.5 1.128 1.060 0.991 0.923 0.854
[0.0, 0.3) 0.7 0.991 0.927 0.862 0.798 0.734
[0.0, 0.3) 0.9 0.870 0.809 0.749 0.689 0.629
[0.3, 0.6) 0.1 2.843 2.531 2.219 1.906 1.570
[0.3, 0.6) 0.3 1.959 1.718 1.478 1.194 0.938
[0.3, 0.6) 0.5 1.378 1.184 0.991 0.798 0.590
[0.3, 0.6) 0.7 0.989 0.828 0.666 0.487 0.299
[0.3, 0.6) 0.9 0.710 0.571 0.405 0.267 0.000
[0.6, 0.9) 0.1 5.495 4.709 3.923 3.118 2.253
[0.6, 0.9) 0.3 2.850 2.384 1.918 1.441 0.928
[0.6, 0.9) 0.5 1.627 1.309 0.991 0.666 0.278
[0.6, 0.9) 0.7 0.989 0.748 0.507 0.234 $ \circ $
[0.6, 0.9) 0.9 0.599 0.385 0.178 $ \circ $ $ \circ $
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$ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
[0.0, 0.3) 0.1 1.448 1.369 1.278 1.199 1.120
[0.0, 0.3) 0.3 1.268 1.195 1.215 1.048 0.975
[0.0, 0.3) 0.5 1.128 1.060 0.991 0.923 0.854
[0.0, 0.3) 0.7 0.991 0.927 0.862 0.798 0.734
[0.0, 0.3) 0.9 0.870 0.809 0.749 0.689 0.629
[0.3, 0.6) 0.1 2.843 2.531 2.219 1.906 1.570
[0.3, 0.6) 0.3 1.959 1.718 1.478 1.194 0.938
[0.3, 0.6) 0.5 1.378 1.184 0.991 0.798 0.590
[0.3, 0.6) 0.7 0.989 0.828 0.666 0.487 0.299
[0.3, 0.6) 0.9 0.710 0.571 0.405 0.267 0.000
[0.6, 0.9) 0.1 5.495 4.709 3.923 3.118 2.253
[0.6, 0.9) 0.3 2.850 2.384 1.918 1.441 0.928
[0.6, 0.9) 0.5 1.627 1.309 0.991 0.666 0.278
[0.6, 0.9) 0.7 0.989 0.748 0.507 0.234 $ \circ $
[0.6, 0.9) 0.9 0.599 0.385 0.178 $ \circ $ $ \circ $
Table 12.  The cost relative difference value between the UML and UOU policies with $ Lx = 4 $
$ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
[0.0, 0.3) 0.1 2.482 2.402 2.322 2.243 2.163
[0.0, 0.3) 0.3 2.224 2.150 2.076 2.002 1.928
[0.0, 0.3) 0.5 2.000 1.932 1.863 1.794 1.725
[0.0, 0.3) 0.7 1.799 1.735 1.671 1.606 1.542
[0.0, 0.3) 0.9 1.628 1.568 1.507 1.447 1.387
[0.3, 0.6) 0.1 4.233 3.921 3.608 3.296 2.960
[0.3, 0.6) 0.3 3.029 2.789 2.548 2.308 2.049
[0.3, 0.6) 0.5 2.251 2.057 1.863 1.669 1.460
[0.3, 0.6) 0.7 1.711 1.550 1.388 1.226 1.052
[0.3, 0.6) 0.9 1.329 1.190 1.051 0.912 0.762
[0.6, 0.9) 0.1 7.621 6.835 6.049 5.244 4.379
[0.6, 0.9) 0.3 4.110 3.644 3.178 2.701 2.188
[0.6, 0.9) 0.5 2.501 2.182 1.863 1.536 1.185
[0.6, 0.9) 0.7 1.641 1.400 1.159 0.913 0.648
[0.6, 0.9) 0.9 1.123 0.930 0.736 0.538 0.313
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$ \epsilon^{ld} $ $ \beta_{i} $ $ \gamma_{i}=0.1 $ $ \gamma_{i}=0.3 $ $ \gamma_{i}=0.5 $ $ \gamma_{i}=0.7 $ $ \gamma_{i}=0.9 $
[0.0, 0.3) 0.1 2.482 2.402 2.322 2.243 2.163
[0.0, 0.3) 0.3 2.224 2.150 2.076 2.002 1.928
[0.0, 0.3) 0.5 2.000 1.932 1.863 1.794 1.725
[0.0, 0.3) 0.7 1.799 1.735 1.671 1.606 1.542
[0.0, 0.3) 0.9 1.628 1.568 1.507 1.447 1.387
[0.3, 0.6) 0.1 4.233 3.921 3.608 3.296 2.960
[0.3, 0.6) 0.3 3.029 2.789 2.548 2.308 2.049
[0.3, 0.6) 0.5 2.251 2.057 1.863 1.669 1.460
[0.3, 0.6) 0.7 1.711 1.550 1.388 1.226 1.052
[0.3, 0.6) 0.9 1.329 1.190 1.051 0.912 0.762
[0.6, 0.9) 0.1 7.621 6.835 6.049 5.244 4.379
[0.6, 0.9) 0.3 4.110 3.644 3.178 2.701 2.188
[0.6, 0.9) 0.5 2.501 2.182 1.863 1.536 1.185
[0.6, 0.9) 0.7 1.641 1.400 1.159 0.913 0.648
[0.6, 0.9) 0.9 1.123 0.930 0.736 0.538 0.313
Table 13.  The change trend of the cost relative difference between the UML and UOU policies
$ \beta_{i} $}{$ \gamma_{i} $ $<0.5 $ $ =0.5 $ $>0.5 $
$<0.5 $ $ \uparrow(\uparrow, \downarrow) $ $ \uparrow(=, \downarrow) $ $ \ast(\downarrow, \ast) $
$ =0.5 $ $ \uparrow(\uparrow, \downarrow) $ $ =(=, =) $ $ \downarrow(\downarrow, \uparrow) $
$>0.5 $ $ \ast(\uparrow, \ast) $ $ \downarrow(=, \uparrow) $ $ \downarrow(\downarrow, \uparrow) $
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$ \beta_{i} $}{$ \gamma_{i} $ $<0.5 $ $ =0.5 $ $>0.5 $
$<0.5 $ $ \uparrow(\uparrow, \downarrow) $ $ \uparrow(=, \downarrow) $ $ \ast(\downarrow, \ast) $
$ =0.5 $ $ \uparrow(\uparrow, \downarrow) $ $ =(=, =) $ $ \downarrow(\downarrow, \uparrow) $
$>0.5 $ $ \ast(\uparrow, \ast) $ $ \downarrow(=, \uparrow) $ $ \downarrow(\downarrow, \uparrow) $
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