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On a modified extragradient method for variational inequality problem with application to industrial electricity production
Pricing decisions for complementary products in a fuzzy dual-channel supply chain
1. | School of Economic and Management, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, China |
2. | School of Business, State University of New York at Oswego, New York 13126, USA |
The paper considers the pricing problem of complementary products in a fuzzy dual-channel supply chain environment where there are two manufacturers and one retailer. Four different decision models are established to study this problem: the centralized decision model, MS-Bertrand model, RS-Bertrand model and Nash game model, where the consumer demand and manufacturing cost for each product are characterized as fuzzy variables. A closed form solution has been obtained for each model by using game theory and fuzzy theory. Numerical examples are presented to compare the maximal expected profits and optimal pricing decisions, and to provide additional managerial insights. The finding shows that the decision makers are more likely to choose industries with higher self-price elastic coefficient and lower complementarity in the retail channel to cooperate. We can obtain that consumers can benefit from the cooperation of the two manufacturers because of lower prices. We can also find that it might not be bad for retailer because it can expand demand and obtain more maximal expected profits.
References:
[1] |
P. Berger, J. Lee and B. Weinberg, Optimal cooperative advertising integration strategy for organizations adding a direct online channel, Journal of the Operational Research Society, 57 (2006), 920-927. Google Scholar |
[2] |
G. Cai,
Channel selection and coordination in dual-channel supply chains, Journal of Retailing, 86 (2010), 22-36.
doi: 10.1016/j.jretai.2009.11.002. |
[3] |
E. Cao, Y. Ma and C. Wan,
Contracting with asymmetric cost information in a dual-channel supply chain, Operation Research Letters, 41 (2013), 410-414.
doi: 10.1016/j.orl.2013.04.013. |
[4] |
C. Cheng,
Group opinion aggregation based on a grading process: A method for constructing triangular fuzzy numbers, Computers and Mathematics with Applications, 48 (2004), 1619-1632.
doi: 10.1016/j.camwa.2004.03.008. |
[5] |
W. Chiang, D. Chhajed and J. Hess,
Direct marketing, indirect profits: A strategic analysis of dual-channel supply chain design, Management Science, 49 (2003), 1-20.
doi: 10.1287/mnsc.49.1.1.12749. |
[6] |
S. Choi,
Price competition in a channel structure with a common retailer, Marketing Science, 10 (1991), 271-296.
doi: 10.1287/mksc.10.4.271. |
[7] |
B. Dan, J. Xiao and X. Zhang, A collaboration strategy complementary products in dual-channel supply chains, Journal of Industrial Engineering and Engineering Management, 25 (2011), 162-166. Google Scholar |
[8] |
Z. Ding and Y. Liu, Revenue sharing contract in dual channel supply chain in case of free riding, Journal of Systems Engineering, 28 (2013), 370-376. Google Scholar |
[9] |
H. Kurata, D. Yao and J. Liu, Pricing policies under direct vs. indirect channel competition and nation vs. store brand competition, European Journal of Operational Research, 180 (2007), 262-281. Google Scholar |
[10] |
N. Leila and S. Mehdi, Modeling and solving a pricing problem considering substitutable products in a dual supply chain with internet and traditional distribution channels, Cumhuriyet University Faculty of Science Science Journal, 36 (2015), 956-970. Google Scholar |
[11] |
T. Li, X. Zhao and J. Xie, Inventory management for dual sales channels with inventory-level-dependent demand, Journal of the Operational Research Society, 66 (2015), 488-499. Google Scholar |
[12] |
Q. Li and B. Li,
Dual-channel supply chain equilibrium problems regarding retail services and fairness concerns, Applied Mathematical Modelling, 40 (2016), 7349-7367.
doi: 10.1016/j.apm.2016.03.010. |
[13] |
H. Lin and W. Chang,
Order selection and pricing methods using flexible quantity and fuzzy approach for buyer evaluation, European Journal of Operational Research, 187 (2008), 415-428.
doi: 10.1016/j.ejor.2007.03.003. |
[14] |
B. Liu, Theory and Practice of Uncertain Programming, Physica-Verlag: Heidelberg, 2002. Google Scholar |
[15] |
B. Liu, Uncertainty Theory: An Introduction to Its Axiomatic Foundations, Springer-Verlag: Berlin, 2004. |
[16] |
S. Liu and Z. Xu,
Stackelberg game models between two competitive retailers in fuzzy decision environment, Fuzzy Optimization and Decision Making, 13 (2014), 33-48.
doi: 10.1007/s10700-013-9165-x. |
[17] |
S. Nahmias,
Fuzzy variables, Fuzzy Sets and Systems, 1 (1978), 97-110.
doi: 10.1016/0165-0114(78)90011-8. |
[18] |
J. Raju, R. Sethuraman and S. Dhar,
The introduction and performance of store brands, Management Science, 41 (1995), 957-978.
doi: 10.1287/mnsc.41.6.957. |
[19] |
S. Sang,
Price competition of manufacturers in supply chain under a fuzzy decision environment, Fuzzy Optimization and Decision Making, 14 (2015), 335-363.
doi: 10.1007/s10700-014-9202-4. |
[20] |
S. Sayman, S. Hoch and J. Raju,
Positioning of store brands, Marketing Science, 21 (2002), 378-397.
doi: 10.1287/mksc.21.4.378.134. |
[21] |
H. Shin and W. Benton,
Quantity discount-based inventory coordination: Effectiveness and critical environmental factors, Production and Operations Management, 13 (2004), 63-76.
doi: 10.1111/j.1937-5956.2004.tb00145.x. |
[22] |
W. Soon,
A review of multi-product pricing models, Applied Mathematics and Computation, 217 (2011), 8149-8165.
doi: 10.1016/j.amc.2011.03.042. |
[23] |
L. Wang, H. Song and Y. Wang,
Pricing and service decisions of complementary products in a dual-channel supply chain, Computers and Industrial Engineering, 105 (2017), 223-233.
doi: 10.1016/j.cie.2016.12.034. |
[24] |
J. Wei and J. Zhao,
Reverse channel decisions for a fuzzy closed-loop supply chain, Applied Mathematical Modeling, 37 (2013), 1502-1513.
doi: 10.1016/j.apm.2012.04.003. |
[25] |
E. Wolfstetter, Topics in Microeconomics: Industrial Organization, Auction, and Incentives, Cambridge, UK: Cambridge University Press, 1999.
doi: 10.1017/CBO9780511625787. |
[26] |
C. Wu, C. Chen and C. Hsieh,
Competitive pricing decisions in a two-echelon supply chain with horizontal and vertical competition, International Journal of Production Economics, 135 (2011), 265-274.
doi: 10.1016/j.ijpe.2011.07.020. |
[27] |
J. Xiao, B. Dan and X. Zhang, Study on cooperation strategy between electronic channels and retailers in dual-channel supply chain, Journal of Systems Engineering, 24 (2009), 673-679. Google Scholar |
[28] |
D. Yang and T. Xiao,
Pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties, Journal of Cleaner Production, 149 (2017), 1174-1187.
doi: 10.1016/j.jclepro.2017.02.138. |
[29] |
X. Yue and J. Liu,
Demand forecast sharing in a dual-channel supply chain, European Journal of Operational Research, 174 (2006), 646-667.
doi: 10.1016/j.ejor.2004.12.020. |
[30] |
X. Yue, S. Mukhopadhyay and X. Zhu,
A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192.
doi: 10.1016/j.jbusres.2005.06.005. |
[31] |
J. Zhao, W. Liu and J. Wei,
Competition under manufacturer service and price in fuzzy environments, Knowledge-Based Systems, 50 (2013), 121-133.
doi: 10.1016/j.knosys.2013.06.003. |
[32] |
J. Zhao, W. Tang and J. Wei,
Pricing decision for substitutable products with retail competition in a fuzzy environment, International Journal of Production Economics, 135 (2012), 144-153.
doi: 10.1016/j.ijpe.2010.12.024. |
[33] |
J. Zhao and L. Wang,
Pricing and retail service decisions in fuzzy uncertainty environments, Applied Mathematics and Computation, 250 (2015), 580-592.
doi: 10.1016/j.amc.2014.11.005. |
[34] |
C. Zhou, R. Zhao and W. Tang,
Two-echelon supply chain games in a fuzzy environment, Computers and Industrial Engineering, 55 (2008), 390-405.
doi: 10.1016/j.cie.2008.01.014. |
[35] |
H. Zimmermann,
Application-oriented view of modeling uncertainty, European Journal of Operational Research, 122 (2000), 190-198.
doi: 10.1016/S0377-2217(99)00228-3. |
show all references
References:
[1] |
P. Berger, J. Lee and B. Weinberg, Optimal cooperative advertising integration strategy for organizations adding a direct online channel, Journal of the Operational Research Society, 57 (2006), 920-927. Google Scholar |
[2] |
G. Cai,
Channel selection and coordination in dual-channel supply chains, Journal of Retailing, 86 (2010), 22-36.
doi: 10.1016/j.jretai.2009.11.002. |
[3] |
E. Cao, Y. Ma and C. Wan,
Contracting with asymmetric cost information in a dual-channel supply chain, Operation Research Letters, 41 (2013), 410-414.
doi: 10.1016/j.orl.2013.04.013. |
[4] |
C. Cheng,
Group opinion aggregation based on a grading process: A method for constructing triangular fuzzy numbers, Computers and Mathematics with Applications, 48 (2004), 1619-1632.
doi: 10.1016/j.camwa.2004.03.008. |
[5] |
W. Chiang, D. Chhajed and J. Hess,
Direct marketing, indirect profits: A strategic analysis of dual-channel supply chain design, Management Science, 49 (2003), 1-20.
doi: 10.1287/mnsc.49.1.1.12749. |
[6] |
S. Choi,
Price competition in a channel structure with a common retailer, Marketing Science, 10 (1991), 271-296.
doi: 10.1287/mksc.10.4.271. |
[7] |
B. Dan, J. Xiao and X. Zhang, A collaboration strategy complementary products in dual-channel supply chains, Journal of Industrial Engineering and Engineering Management, 25 (2011), 162-166. Google Scholar |
[8] |
Z. Ding and Y. Liu, Revenue sharing contract in dual channel supply chain in case of free riding, Journal of Systems Engineering, 28 (2013), 370-376. Google Scholar |
[9] |
H. Kurata, D. Yao and J. Liu, Pricing policies under direct vs. indirect channel competition and nation vs. store brand competition, European Journal of Operational Research, 180 (2007), 262-281. Google Scholar |
[10] |
N. Leila and S. Mehdi, Modeling and solving a pricing problem considering substitutable products in a dual supply chain with internet and traditional distribution channels, Cumhuriyet University Faculty of Science Science Journal, 36 (2015), 956-970. Google Scholar |
[11] |
T. Li, X. Zhao and J. Xie, Inventory management for dual sales channels with inventory-level-dependent demand, Journal of the Operational Research Society, 66 (2015), 488-499. Google Scholar |
[12] |
Q. Li and B. Li,
Dual-channel supply chain equilibrium problems regarding retail services and fairness concerns, Applied Mathematical Modelling, 40 (2016), 7349-7367.
doi: 10.1016/j.apm.2016.03.010. |
[13] |
H. Lin and W. Chang,
Order selection and pricing methods using flexible quantity and fuzzy approach for buyer evaluation, European Journal of Operational Research, 187 (2008), 415-428.
doi: 10.1016/j.ejor.2007.03.003. |
[14] |
B. Liu, Theory and Practice of Uncertain Programming, Physica-Verlag: Heidelberg, 2002. Google Scholar |
[15] |
B. Liu, Uncertainty Theory: An Introduction to Its Axiomatic Foundations, Springer-Verlag: Berlin, 2004. |
[16] |
S. Liu and Z. Xu,
Stackelberg game models between two competitive retailers in fuzzy decision environment, Fuzzy Optimization and Decision Making, 13 (2014), 33-48.
doi: 10.1007/s10700-013-9165-x. |
[17] |
S. Nahmias,
Fuzzy variables, Fuzzy Sets and Systems, 1 (1978), 97-110.
doi: 10.1016/0165-0114(78)90011-8. |
[18] |
J. Raju, R. Sethuraman and S. Dhar,
The introduction and performance of store brands, Management Science, 41 (1995), 957-978.
doi: 10.1287/mnsc.41.6.957. |
[19] |
S. Sang,
Price competition of manufacturers in supply chain under a fuzzy decision environment, Fuzzy Optimization and Decision Making, 14 (2015), 335-363.
doi: 10.1007/s10700-014-9202-4. |
[20] |
S. Sayman, S. Hoch and J. Raju,
Positioning of store brands, Marketing Science, 21 (2002), 378-397.
doi: 10.1287/mksc.21.4.378.134. |
[21] |
H. Shin and W. Benton,
Quantity discount-based inventory coordination: Effectiveness and critical environmental factors, Production and Operations Management, 13 (2004), 63-76.
doi: 10.1111/j.1937-5956.2004.tb00145.x. |
[22] |
W. Soon,
A review of multi-product pricing models, Applied Mathematics and Computation, 217 (2011), 8149-8165.
doi: 10.1016/j.amc.2011.03.042. |
[23] |
L. Wang, H. Song and Y. Wang,
Pricing and service decisions of complementary products in a dual-channel supply chain, Computers and Industrial Engineering, 105 (2017), 223-233.
doi: 10.1016/j.cie.2016.12.034. |
[24] |
J. Wei and J. Zhao,
Reverse channel decisions for a fuzzy closed-loop supply chain, Applied Mathematical Modeling, 37 (2013), 1502-1513.
doi: 10.1016/j.apm.2012.04.003. |
[25] |
E. Wolfstetter, Topics in Microeconomics: Industrial Organization, Auction, and Incentives, Cambridge, UK: Cambridge University Press, 1999.
doi: 10.1017/CBO9780511625787. |
[26] |
C. Wu, C. Chen and C. Hsieh,
Competitive pricing decisions in a two-echelon supply chain with horizontal and vertical competition, International Journal of Production Economics, 135 (2011), 265-274.
doi: 10.1016/j.ijpe.2011.07.020. |
[27] |
J. Xiao, B. Dan and X. Zhang, Study on cooperation strategy between electronic channels and retailers in dual-channel supply chain, Journal of Systems Engineering, 24 (2009), 673-679. Google Scholar |
[28] |
D. Yang and T. Xiao,
Pricing and green level decisions of a green supply chain with governmental interventions under fuzzy uncertainties, Journal of Cleaner Production, 149 (2017), 1174-1187.
doi: 10.1016/j.jclepro.2017.02.138. |
[29] |
X. Yue and J. Liu,
Demand forecast sharing in a dual-channel supply chain, European Journal of Operational Research, 174 (2006), 646-667.
doi: 10.1016/j.ejor.2004.12.020. |
[30] |
X. Yue, S. Mukhopadhyay and X. Zhu,
A Bertrand model of pricing of complementary goods under information asymmetry, Journal of Business Research, 59 (2006), 1182-1192.
doi: 10.1016/j.jbusres.2005.06.005. |
[31] |
J. Zhao, W. Liu and J. Wei,
Competition under manufacturer service and price in fuzzy environments, Knowledge-Based Systems, 50 (2013), 121-133.
doi: 10.1016/j.knosys.2013.06.003. |
[32] |
J. Zhao, W. Tang and J. Wei,
Pricing decision for substitutable products with retail competition in a fuzzy environment, International Journal of Production Economics, 135 (2012), 144-153.
doi: 10.1016/j.ijpe.2010.12.024. |
[33] |
J. Zhao and L. Wang,
Pricing and retail service decisions in fuzzy uncertainty environments, Applied Mathematics and Computation, 250 (2015), 580-592.
doi: 10.1016/j.amc.2014.11.005. |
[34] |
C. Zhou, R. Zhao and W. Tang,
Two-echelon supply chain games in a fuzzy environment, Computers and Industrial Engineering, 55 (2008), 390-405.
doi: 10.1016/j.cie.2008.01.014. |
[35] |
H. Zimmermann,
Application-oriented view of modeling uncertainty, European Journal of Operational Research, 122 (2000), 190-198.
doi: 10.1016/S0377-2217(99)00228-3. |


Paper | Supply chain structure | Product status | Dual channel | Fuzzy or not | Decisions |
Chiang et al.[5] | one M | single | Yes | No | pricing policy |
one R | ordering policy | ||||
Berger and Weinberg[1] | one firm | single | Yes | No | pricing policy |
advertising | |||||
Dan et al.[7] | one M | single | Yes | No | pricing policy |
one R | service decision | ||||
Zhao et al.[32] | one M | substitutable | No | Yes | pricing policy |
two Rs | |||||
Zhao and Wang[33] | one M | single | No | Yes | pricing policy |
two Rs | service decision | ||||
Leila and Mehdi[10] | two Ms | substitutable | Yes | No | pricing policy |
one R | |||||
Sang[19] | two Ms | substitutable | No | Yes | pricing policy |
one R | |||||
Wang et al.[23] | two Ms | complementary | Yes | No | pricing policy |
one R | service decision | ||||
Paper | Supply chain structure | Product status | Dual channel | Fuzzy or not | Decisions |
Chiang et al.[5] | one M | single | Yes | No | pricing policy |
one R | ordering policy | ||||
Berger and Weinberg[1] | one firm | single | Yes | No | pricing policy |
advertising | |||||
Dan et al.[7] | one M | single | Yes | No | pricing policy |
one R | service decision | ||||
Zhao et al.[32] | one M | substitutable | No | Yes | pricing policy |
two Rs | |||||
Zhao and Wang[33] | one M | single | No | Yes | pricing policy |
two Rs | service decision | ||||
Leila and Mehdi[10] | two Ms | substitutable | Yes | No | pricing policy |
one R | |||||
Sang[19] | two Ms | substitutable | No | Yes | pricing policy |
one R | |||||
Wang et al.[23] | two Ms | complementary | Yes | No | pricing policy |
one R | service decision | ||||
Linguistic expression | Triangular fuzzy variable | |
Large (about 500) | (350,500,700) | |
Small (about 200) | (150,200,300) | |
Large (about 800) | (600,800,900) | |
Small (about 300) | (200,300,350) | |
Large (about 1200) | (800,1200,1500) | |
Small (about 600) | (500,600,800) | |
Very sensitive(about 5) | (4, 5, 6) | |
Sensitive (about 2) | (1, 2, 2.5) | |
Very sensitive(about 8) | (5, 8, 10) | |
Sensitive (about 4) | (3, 4, 5) | |
Very sensitive(about 30) | (25, 30, 35) | |
Sensitive (about 20) | (10, 20, 25) | |
Very sensitive(about 0.5) | (0.3, 0.5, 0.8) | |
Sensitive(about 0.2) | (0.1, 0.2, 0.3) | |
Very sensitive(about 0.4) | (0.3, 0.4, 0.6) | |
Sensitive (about 0.2) | (0.1, 0.2, 0.3) | |
Very sensitive(about 0.3) | (0.1, 0.3, 0.5) | |
Sensitive (about 0.1) | (0.05, 0.1, 0.15) | |
High(about 30) | (25, 30, 40) | |
Low(about 15) | (10, 15, 20) | |
High (about 5) | (4, 5, 6) | |
Low (about 2) | (1, 2, 4) |
Linguistic expression | Triangular fuzzy variable | |
Large (about 500) | (350,500,700) | |
Small (about 200) | (150,200,300) | |
Large (about 800) | (600,800,900) | |
Small (about 300) | (200,300,350) | |
Large (about 1200) | (800,1200,1500) | |
Small (about 600) | (500,600,800) | |
Very sensitive(about 5) | (4, 5, 6) | |
Sensitive (about 2) | (1, 2, 2.5) | |
Very sensitive(about 8) | (5, 8, 10) | |
Sensitive (about 4) | (3, 4, 5) | |
Very sensitive(about 30) | (25, 30, 35) | |
Sensitive (about 20) | (10, 20, 25) | |
Very sensitive(about 0.5) | (0.3, 0.5, 0.8) | |
Sensitive(about 0.2) | (0.1, 0.2, 0.3) | |
Very sensitive(about 0.4) | (0.3, 0.4, 0.6) | |
Sensitive (about 0.2) | (0.1, 0.2, 0.3) | |
Very sensitive(about 0.3) | (0.1, 0.3, 0.5) | |
Sensitive (about 0.1) | (0.05, 0.1, 0.15) | |
High(about 30) | (25, 30, 40) | |
Low(about 15) | (10, 15, 20) | |
High (about 5) | (4, 5, 6) | |
Low (about 2) | (1, 2, 4) |
63.0097 | 60.8519 | 20.8009 | |||||
61.17 | 21.36 | 63.3476 | 82.0076 | 29.2987 | 20.8376 | 7.9387 | |
40.02 | 13.20 | 63.3317 | 82.0065 | 29.3002 | 41.9865 | 16.0902 | |
47.14 | 15.97 | 63.4526 | 74.9997 | 26.6042 | 27.8597 | 10.6342 |
63.0097 | 60.8519 | 20.8009 | |||||
61.17 | 21.36 | 63.3476 | 82.0076 | 29.2987 | 20.8376 | 7.9387 | |
40.02 | 13.20 | 63.3317 | 82.0065 | 29.3002 | 41.9865 | 16.0902 | |
47.14 | 15.97 | 63.4526 | 74.9997 | 26.6042 | 27.8597 | 10.6342 |
220.56 | 330.2375 | 505.94 | 37076 | ||||
226.36 | 163.9087 | 244.52 | 19578 | 4173 | 5358 | 29109 | |
226.44 | 163.9089 | 244.48 | 16111 | 2180 | 10818 | 29109 | |
223.31 | 219.0836 | 327.41 | 18895 | 3761 | 9585 | 32240 |
220.56 | 330.2375 | 505.94 | 37076 | ||||
226.36 | 163.9087 | 244.52 | 19578 | 4173 | 5358 | 29109 | |
226.44 | 163.9089 | 244.48 | 16111 | 2180 | 10818 | 29109 | |
223.31 | 219.0836 | 327.41 | 18895 | 3761 | 9585 | 32240 |
(1800,2000,2200) | (3, 4, 5) | (0.4, 0.5, 0.6) | |
(3201,3302,3403) | (6, 8, 10) | (0.1, 0.3, 0.5) | |
(800,1200,1500) | (15, 20, 25) | (0.1, 0.2, 0.3) | |
(500,600,700) | (25, 30, 35) | (0.3, 0.4, 0.6) | |
(1000,1156,1312) | (15, 20, 25) | (0.4, 0.5, 0.6) |
(1800,2000,2200) | (3, 4, 5) | (0.4, 0.5, 0.6) | |
(3201,3302,3403) | (6, 8, 10) | (0.1, 0.3, 0.5) | |
(800,1200,1500) | (15, 20, 25) | (0.1, 0.2, 0.3) | |
(500,600,700) | (25, 30, 35) | (0.3, 0.4, 0.6) | |
(1000,1156,1312) | (15, 20, 25) | (0.4, 0.5, 0.6) |
-36.3415 | -28.0346 | -19.6351 | -20.7345 | -19.284 | |
-22.3776 | -18.0985 | -2.4605 | -2.5192 | -0.827 | |
322.6005 | 271.3165 | 38.6035 | 37.2342 | 13.1502 | |
12.2558 | 14.0035 | 18.0239 | 17.8967 | 18.3916 | |
8.6757 | 49.2929 | 89.1796 | 83.7494 | 90.7422 | |
-26.3475 | -23.7178 | -12.0458 | -11.9774 | -10.7595 | |
640.6542 | 1070.9 | 342.9444 | 319.2612 | 129.3525 | |
-3141 | 0 | 5177 | 4690 | 5528 | |
102830 | 143660 | 6082 | 5501 | 611.5339 | |
92082 | 133632 | 417 | 0 | -4373 | |
88941 | 133632 | 5594 | 4690 | 155 |
-36.3415 | -28.0346 | -19.6351 | -20.7345 | -19.284 | |
-22.3776 | -18.0985 | -2.4605 | -2.5192 | -0.827 | |
322.6005 | 271.3165 | 38.6035 | 37.2342 | 13.1502 | |
12.2558 | 14.0035 | 18.0239 | 17.8967 | 18.3916 | |
8.6757 | 49.2929 | 89.1796 | 83.7494 | 90.7422 | |
-26.3475 | -23.7178 | -12.0458 | -11.9774 | -10.7595 | |
640.6542 | 1070.9 | 342.9444 | 319.2612 | 129.3525 | |
-3141 | 0 | 5177 | 4690 | 5528 | |
102830 | 143660 | 6082 | 5501 | 611.5339 | |
92082 | 133632 | 417 | 0 | -4373 | |
88941 | 133632 | 5594 | 4690 | 155 |
(0.2, 0.5, 0.8) | 225.6026 | 163.7939 | 244.5593 | 19452 | 4175.0 | 5354.1 | 28981 | |
(0.3, 0.5, 0.7) | 225.6886 | 163.8355 | 244.5621 | 19427 | 4175.1 | 5355.9 | 28958 | |
(0.4, 0.5, 0.6) | 225.7746 | 163.8772 | 244.5649 | 19403 | 4175.2 | 5357.6 | 28936 | |
(0.45, 0.5, 0.55) | 225.8176 | 163.8980 | 244.5663 | 19391 | 4175.3 | 5358.5 | 28925 | |
(0.2, 0.5, 0.8) | 225.6799 | 163.7941 | 244.5216 | 15990 | 2180.7 | 10810 | 28980 | |
(0.3, 0.5, 0.7) | 225.7658 | 163.8357 | 244.5244 | 15964 | 2180.8 | 10813 | 28958 | |
(0.4, 0.5, 0.6) | 225.8518 | 163.8774 | 244.5272 | 15938 | 2180.9 | 10817 | 28936 | |
(0.45, 0.5, 0.55) | 225.8948 | 163.8982 | 244.5286 | 15925 | 2181.0 | 10818 | 28924 |
(0.2, 0.5, 0.8) | 225.6026 | 163.7939 | 244.5593 | 19452 | 4175.0 | 5354.1 | 28981 | |
(0.3, 0.5, 0.7) | 225.6886 | 163.8355 | 244.5621 | 19427 | 4175.1 | 5355.9 | 28958 | |
(0.4, 0.5, 0.6) | 225.7746 | 163.8772 | 244.5649 | 19403 | 4175.2 | 5357.6 | 28936 | |
(0.45, 0.5, 0.55) | 225.8176 | 163.8980 | 244.5663 | 19391 | 4175.3 | 5358.5 | 28925 | |
(0.2, 0.5, 0.8) | 225.6799 | 163.7941 | 244.5216 | 15990 | 2180.7 | 10810 | 28980 | |
(0.3, 0.5, 0.7) | 225.7658 | 163.8357 | 244.5244 | 15964 | 2180.8 | 10813 | 28958 | |
(0.4, 0.5, 0.6) | 225.8518 | 163.8774 | 244.5272 | 15938 | 2180.9 | 10817 | 28936 | |
(0.45, 0.5, 0.55) | 225.8948 | 163.8982 | 244.5286 | 15925 | 2181.0 | 10818 | 28924 |
(0.2, 0.5, 0.8) | 60.9727 | 21.3595 | 63.0687 | 81.7999 | 29.3032 | 20.8272 | 7.9437 | |
(0.3, 0.5, 0.7) | 60.9608 | 21.3597 | 63.0508 | 81.7934 | 29.3034 | 20.8326 | 7.9437 | |
(0.4, 0.5, 0.6) | 60.9489 | 21.3599 | 63.0329 | 81.7869 | 29.3037 | 20.8380 | 7.9438 | |
(0.45, 0.5, 0.55) | 60.9430 | 21.3600 | 63.0240 | 81.7836 | 29.3038 | 20.8406 | 7.9438 | |
(0.2, 0.5, 0.8) | 39.8371 | 13.2063 | 63.0530 | 81.7988 | 29.3047 | 41.9617 | 16.0984 | |
(0.3, 0.5, 0.7) | 39.8198 | 13.2064 | 63.0351 | 81.7923 | 29.3049 | 41.9725 | 16.0985 | |
(0.4, 0.5, 0.6) | 39.8025 | 13.2065 | 63.0173 | 81.7858 | 29.3052 | 41.9833 | 16.0987 | |
(0.45, 0.5, 0.55) | 39.7939 | 13.2066 | 63.0083 | 81.7825 | 29.3053 | 41.9886 | 16.0987 |
(0.2, 0.5, 0.8) | 60.9727 | 21.3595 | 63.0687 | 81.7999 | 29.3032 | 20.8272 | 7.9437 | |
(0.3, 0.5, 0.7) | 60.9608 | 21.3597 | 63.0508 | 81.7934 | 29.3034 | 20.8326 | 7.9437 | |
(0.4, 0.5, 0.6) | 60.9489 | 21.3599 | 63.0329 | 81.7869 | 29.3037 | 20.8380 | 7.9438 | |
(0.45, 0.5, 0.55) | 60.9430 | 21.3600 | 63.0240 | 81.7836 | 29.3038 | 20.8406 | 7.9438 | |
(0.2, 0.5, 0.8) | 39.8371 | 13.2063 | 63.0530 | 81.7988 | 29.3047 | 41.9617 | 16.0984 | |
(0.3, 0.5, 0.7) | 39.8198 | 13.2064 | 63.0351 | 81.7923 | 29.3049 | 41.9725 | 16.0985 | |
(0.4, 0.5, 0.6) | 39.8025 | 13.2065 | 63.0173 | 81.7858 | 29.3052 | 41.9833 | 16.0987 | |
(0.45, 0.5, 0.55) | 39.7939 | 13.2066 | 63.0083 | 81.7825 | 29.3053 | 41.9886 | 16.0987 |
(10, 30, 50) | 226.3554 | 163.9024 | 243.2656 | 19576 | 4320.2 | 5338.1 | 29234 | |
(15, 30, 45) | 226.3583 | 163.9045 | 243.6821 | 19577 | 4271.3 | 5344.8 | 29192 | |
(20, 30, 40) | 226.3613 | 163.9066 | 244.0987 | 19577 | 4222.4 | 5351.5 | 29151 | |
(25, 30, 35) | 226.3642 | 163.9087 | 244.5152 | 19578 | 4173.6 | 5358.2 | 29109 | |
(10, 30, 50) | 226.4335 | 163.9026 | 243.2279 | 16110 | 2347.0 | 10777 | 29234 | |
(15, 30, 45) | 226.4364 | 163.9047 | 243.6443 | 16110 | 2291.3 | 10791 | 29192 | |
(20, 30, 40) | 226.4394 | 163.9068 | 244.0608 | 16111 | 2235.7 | 10804 | 29151 | |
(25, 30, 35) | 226.4423 | 163.9089 | 244.4773 | 16111 | 2180.0 | 10818 | 29109 |
(10, 30, 50) | 226.3554 | 163.9024 | 243.2656 | 19576 | 4320.2 | 5338.1 | 29234 | |
(15, 30, 45) | 226.3583 | 163.9045 | 243.6821 | 19577 | 4271.3 | 5344.8 | 29192 | |
(20, 30, 40) | 226.3613 | 163.9066 | 244.0987 | 19577 | 4222.4 | 5351.5 | 29151 | |
(25, 30, 35) | 226.3642 | 163.9087 | 244.5152 | 19578 | 4173.6 | 5358.2 | 29109 | |
(10, 30, 50) | 226.4335 | 163.9026 | 243.2279 | 16110 | 2347.0 | 10777 | 29234 | |
(15, 30, 45) | 226.4364 | 163.9047 | 243.6443 | 16110 | 2291.3 | 10791 | 29192 | |
(20, 30, 40) | 226.4394 | 163.9068 | 244.0608 | 16111 | 2235.7 | 10804 | 29151 | |
(25, 30, 35) | 226.4423 | 163.9089 | 244.4773 | 16111 | 2180.0 | 10818 | 29109 |
(10, 30, 50) | 61.1638 | 21.4399 | 63.3457 | 82.0067 | 29.3404 | 20.8429 | 7.9005 | |
(15, 30, 45) | 61.1644 | 21.4121 | 63.3463 | 82.0070 | 29.3265 | 20.8426 | 7.9144 | |
(20, 30, 40) | 61.1650 | 21.3844 | 63.3469 | 82.0073 | 29.3126 | 20.8423 | 7.9282 | |
(25, 30, 35) | 61.1656 | 21.3566 | 63.3476 | 82.0076 | 29.2987 | 20.8420 | 7.9421 | |
(10, 30, 50) | 40.0141 | 13.3298 | 63.3298 | 82.0055 | 29.3418 | 41.9914 | 16.0120 | |
(15, 30, 45) | 40.0144 | 13.2881 | 63.3305 | 82.0059 | 29.3279 | 41.9915 | 16.0398 | |
(20, 30, 40) | 40.0147 | 13.2465 | 63.3311 | 82.0062 | 29.3141 | 41.9915 | 16.0676 | |
(25, 30, 35) | 40.0150 | 13.2048 | 63.3317 | 82.0065 | 29.3002 | 41.9915 | 16.0954 |
(10, 30, 50) | 61.1638 | 21.4399 | 63.3457 | 82.0067 | 29.3404 | 20.8429 | 7.9005 | |
(15, 30, 45) | 61.1644 | 21.4121 | 63.3463 | 82.0070 | 29.3265 | 20.8426 | 7.9144 | |
(20, 30, 40) | 61.1650 | 21.3844 | 63.3469 | 82.0073 | 29.3126 | 20.8423 | 7.9282 | |
(25, 30, 35) | 61.1656 | 21.3566 | 63.3476 | 82.0076 | 29.2987 | 20.8420 | 7.9421 | |
(10, 30, 50) | 40.0141 | 13.3298 | 63.3298 | 82.0055 | 29.3418 | 41.9914 | 16.0120 | |
(15, 30, 45) | 40.0144 | 13.2881 | 63.3305 | 82.0059 | 29.3279 | 41.9915 | 16.0398 | |
(20, 30, 40) | 40.0147 | 13.2465 | 63.3311 | 82.0062 | 29.3141 | 41.9915 | 16.0676 | |
(25, 30, 35) | 40.0150 | 13.2048 | 63.3317 | 82.0065 | 29.3002 | 41.9915 | 16.0954 |
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