American Institute of Mathematical Sciences

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January  2022, 18(1): 593-611. doi: 10.3934/jimo.2020170

Cooperation in traffic network problems via evolutionary split variational inequalities

Shipra Singh 1, , Aviv Gibali 2,3, and  Xiaolong Qin 4,,

1. 

Department of Mathematics, Technion - Israel Institute of Technology, Haifa, 3200003, Israel
2. 

Department of Mathematics, ORT Braude College, Karmiel, 2161002, Israel
3. 

The Center for Mathematics and Scientific Computation, University of Haifa, Mt. Carmel, Haifa, 3498838, Israel
4. 

Department of Mathematics, Zhejiang Normal University, Zhejiang, China

* Corresponding author: Xiaolong Qin

Received  August 2020 Revised  September 2020 Published  January 2022 Early access  November 2020

In this paper, we construct an evolutionary (time-dependent) split variational inequality problem and show how to reformulate equilibria of the dynamic traffic network models of two cities as such problem. We also establish existence result for the proposed model. Primary numerical results of equilibria illustrate the validity and applicability of our results.

Keywords: Evolutionary split variational inequality problem, Wardrop condition, traffic equilibrium problems, equilibrium solutions, projected dynamical system.
Mathematics Subject Classification: Primary:90C33, 90C39, 49J40.
Citation: Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2022, 18 (1) : 593-611. doi: 10.3934/jimo.2020170
References:
[1]

Q. H. Ansari and M. Rezaei, Existence results for Stampacchia and Minty type vector variational inequalities, Optimization, 59 (2010), 1053-1065.  doi: 10.1080/02331930903395725.  Google Scholar

[2]

D. Aussel and J. Cotrina, Existence of time-dependent traffic equilibria, Appl. Anal., 91 (2012), 1775-1791.  doi: 10.1080/00036811.2012.692364.  Google Scholar

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D. Aussel, R. Gupta and A. Mehra, Evolutionary variational inequality formulation of the generalized Nash equilibrium problem, J. Optim. Theory Appl., 169 (2016), 74-90.  doi: 10.1007/s10957-015-0859-9.  Google Scholar

[4]

A. Barbagallo, Degenerate time-dependent variational inequalities with applications to traffic equilibrium problems, Comput. Methods Appl. Math., 6 (2006), 117-133.  doi: 10.2478/cmam-2006-0006.  Google Scholar

[5]

A. Barbagallo and M.-G. Cojocaru, Dynamic equilibrium formulation of the oligopolistic market problem, Math. Comput. Modelling, 49 (2009), 966-976.  doi: 10.1016/j.mcm.2008.02.003.  Google Scholar

[6]

A. Barbagallo, P. Daniele and A. Maugeri, Variational formulation for a general dynamic financial equilibrium problem: Balance law and liability formula, Nonlinear Anal., 75 (2012), 1104-1123.  doi: 10.1016/j.na.2010.10.013.  Google Scholar

[7]

H. Brezis, Inéquations d'évolution abstraites, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A732-A735.  Google Scholar

[8]

C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.   Google Scholar

[9]

L.-C. Ceng and S. Huang, Existence theorems for generalized vector variational inequalities with a variable ordering relation, J. Global Optim., 46 (2010), 521-535.  doi: 10.1007/s10898-009-9436-9.  Google Scholar

[10]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[11]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[12]

Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[13]

C. Ciarcià and P. Daniele, New existence theorems for quasi-variational inequalities and applications to financial models, European J. Oper. Res., 251 (2016), 288-299.  doi: 10.1016/j.ejor.2015.11.013.  Google Scholar

[14]

M. Chen and C. Huang, A power penalty method for a class of linearly constrained variational inequality, J. Ind. Manag. Optim., 14 (2018), 1381-1396.  doi: 10.3934/jimo.2018012.  Google Scholar

[15]

M. G. Cojocaru, P. Daniele and A. Nagurney, Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications, J. Optim. Theory Appl., 127 (2005), 549-563.  doi: 10.1007/s10957-005-7502-0.  Google Scholar

[16]

M.-G. Cojocaru and L. B. Jonker, Existence of solutions to projected differential equations in Hilbert spaces, Proc. Amer. Math. Soc., 132 (2004), 183-193.  doi: 10.1090/S0002-9939-03-07015-1.  Google Scholar

[17]

S. Dafermos, Traffic equilibrium and variational inequalities, Transportation Sci., 14 (1980), 42-54.  doi: 10.1287/trsc.14.1.42.  Google Scholar

[18]

P. Daniele, Time-dependent spatial price equilibrium problem: Existence and stability results for the quantity formulation model, J. Global Optim., 28 (2004), 283-295.  doi: 10.1023/B:JOGO.0000026449.29735.3c.  Google Scholar

[19]

P. Daniele, A. Maugeri and W. Oettli, Time-dependent traffic equilibria, J. Optim. Theory Appl., 103 (1999), 543-555.  doi: 10.1023/A:1021779823196.  Google Scholar

[20]

P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., 35 (1991), 31-62.  doi: 10.1080/17442509108833688.  Google Scholar

[21]

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res., 44 (1993), 9-42.  doi: 10.1007/BF02073589.  Google Scholar

[22]

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545.  Google Scholar

[23]

G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambique condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8), 7 (1963/64), 91-140.   Google Scholar

[24]

G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 34 (1963), 138-142.   Google Scholar

[25]

S. Giuffrè, G. Idone and S. Pia, Some classes of projected dynamical systems in Banach spaces and variational inequalities, J. Global Optim., 40 (2008), 119-128.  doi: 10.1007/s10898-007-9173-x.  Google Scholar

[26]

S. Lawphongpanich and D. W. Hearn, Simplical decomposition of the asymmetric traffic assignment problem, Transportation Res. Part B, 18 (1984), 123-133.  doi: 10.1016/0191-2615(84)90026-2.  Google Scholar

[27]

J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar

[28]

S.-Y. Matsushita and L. Xu, On finite convergence of iterative methods for variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 161 (2014), 701-715.  doi: 10.1007/s10957-013-0460-z.  Google Scholar

[29]

A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26 (2010), 6pp. doi: 10.1088/0266-5611/26/5/055007.  Google Scholar

[30]

A. Nagurney, D. Parkes and P. Daniele, The Internet, evolutionary variational inequalities, and the time-dependent Braess paradox, Comput. Manag. Sci., 4 (2007), 355-375.  doi: 10.1007/s10287-006-0027-7.  Google Scholar

[31]

B. Panicucci, M. Pappalardo and M. Passacantando, A path-based double projection method for solving the asymmetric traffic network equilibrium problem, Optim. Lett., 1 (2007), 171-185.  doi: 10.1007/s11590-006-0002-9.  Google Scholar

[32]

F. Raciti, Equilibrium conditions and vector variational inequalities: A complex relation, J. Global Optim., 40 (2008), 353-360.  doi: 10.1007/s10898-007-9202-9.  Google Scholar

[33]

L. Scrimali and C. Mirabella, Cooperation in pollution control problems via evolutionary variational inequalities, J. Global Optim., 70 (2018), 455-476.  doi: 10.1007/s10898-017-0580-3.  Google Scholar

[34]

Y. Shehu and O. Iyiola, On a modified extragradient method for variational inequality problem with application to industrial electricity production, J. Ind. Manag. Optim., 15 (2019), 319-342.  doi: 10.3934/jimo.2018045.  Google Scholar

[35]

M. J. Smith, The existence, uniqueness and stability of traffic equilibria, Transportation Res., 13 (1979), 295-304.  doi: 10.1016/0191-2615(79)90022-5.  Google Scholar

[36]

G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416.   Google Scholar

[37]

X. K. Sun and S. J. Li, Duality and gap function for generalized multivalued $\epsilon$-vector variational inequality, Appl. Anal., 92 (2013), 482-492.  doi: 10.1080/00036811.2011.628940.  Google Scholar

[38]

J. Yang, Dynamic power price problem: An inverse variational inequality approach, J. Ind. Manag. Optim., 4 (2008), 673-684.  doi: 10.3934/jimo.2008.4.673.  Google Scholar

show all references

References:
[1]

Q. H. Ansari and M. Rezaei, Existence results for Stampacchia and Minty type vector variational inequalities, Optimization, 59 (2010), 1053-1065.  doi: 10.1080/02331930903395725.  Google Scholar

[2]

D. Aussel and J. Cotrina, Existence of time-dependent traffic equilibria, Appl. Anal., 91 (2012), 1775-1791.  doi: 10.1080/00036811.2012.692364.  Google Scholar

[3]

D. Aussel, R. Gupta and A. Mehra, Evolutionary variational inequality formulation of the generalized Nash equilibrium problem, J. Optim. Theory Appl., 169 (2016), 74-90.  doi: 10.1007/s10957-015-0859-9.  Google Scholar

[4]

A. Barbagallo, Degenerate time-dependent variational inequalities with applications to traffic equilibrium problems, Comput. Methods Appl. Math., 6 (2006), 117-133.  doi: 10.2478/cmam-2006-0006.  Google Scholar

[5]

A. Barbagallo and M.-G. Cojocaru, Dynamic equilibrium formulation of the oligopolistic market problem, Math. Comput. Modelling, 49 (2009), 966-976.  doi: 10.1016/j.mcm.2008.02.003.  Google Scholar

[6]

A. Barbagallo, P. Daniele and A. Maugeri, Variational formulation for a general dynamic financial equilibrium problem: Balance law and liability formula, Nonlinear Anal., 75 (2012), 1104-1123.  doi: 10.1016/j.na.2010.10.013.  Google Scholar

[7]

H. Brezis, Inéquations d'évolution abstraites, C. R. Acad. Sci. Paris Sér. A-B, 264 (1967), A732-A735.  Google Scholar

[8]

C. Byrne, Y. Censor, A. Gibali and S. Reich, The split common null point problem, J. Nonlinear Convex Anal., 13 (2012), 759-775.   Google Scholar

[9]

L.-C. Ceng and S. Huang, Existence theorems for generalized vector variational inequalities with a variable ordering relation, J. Global Optim., 46 (2010), 521-535.  doi: 10.1007/s10898-009-9436-9.  Google Scholar

[10]

Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numer. Algorithms, 8 (1994), 221-239.  doi: 10.1007/BF02142692.  Google Scholar

[11]

Y. Censor, T. Elfving, N. Kopf and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, 21 (2005), 2071-2084.  doi: 10.1088/0266-5611/21/6/017.  Google Scholar

[12]

Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numer. Algorithms, 59 (2012), 301-323.  doi: 10.1007/s11075-011-9490-5.  Google Scholar

[13]

C. Ciarcià and P. Daniele, New existence theorems for quasi-variational inequalities and applications to financial models, European J. Oper. Res., 251 (2016), 288-299.  doi: 10.1016/j.ejor.2015.11.013.  Google Scholar

[14]

M. Chen and C. Huang, A power penalty method for a class of linearly constrained variational inequality, J. Ind. Manag. Optim., 14 (2018), 1381-1396.  doi: 10.3934/jimo.2018012.  Google Scholar

[15]

M. G. Cojocaru, P. Daniele and A. Nagurney, Projected dynamical systems and evolutionary variational inequalities via Hilbert spaces with applications, J. Optim. Theory Appl., 127 (2005), 549-563.  doi: 10.1007/s10957-005-7502-0.  Google Scholar

[16]

M.-G. Cojocaru and L. B. Jonker, Existence of solutions to projected differential equations in Hilbert spaces, Proc. Amer. Math. Soc., 132 (2004), 183-193.  doi: 10.1090/S0002-9939-03-07015-1.  Google Scholar

[17]

S. Dafermos, Traffic equilibrium and variational inequalities, Transportation Sci., 14 (1980), 42-54.  doi: 10.1287/trsc.14.1.42.  Google Scholar

[18]

P. Daniele, Time-dependent spatial price equilibrium problem: Existence and stability results for the quantity formulation model, J. Global Optim., 28 (2004), 283-295.  doi: 10.1023/B:JOGO.0000026449.29735.3c.  Google Scholar

[19]

P. Daniele, A. Maugeri and W. Oettli, Time-dependent traffic equilibria, J. Optim. Theory Appl., 103 (1999), 543-555.  doi: 10.1023/A:1021779823196.  Google Scholar

[20]

P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stochastics Stochastics Rep., 35 (1991), 31-62.  doi: 10.1080/17442509108833688.  Google Scholar

[21]

P. Dupuis and A. Nagurney, Dynamical systems and variational inequalities, Ann. Oper. Res., 44 (1993), 9-42.  doi: 10.1007/BF02073589.  Google Scholar

[22]

K. Fan, Some properties of convex sets related to fixed point theorems, Math. Ann., 266 (1984), 519-537.  doi: 10.1007/BF01458545.  Google Scholar

[23]

G. Fichera, Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambique condizioni al contorno, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8), 7 (1963/64), 91-140.   Google Scholar

[24]

G. Fichera, Sul problema elastostatico di Signorini con ambigue condizioni al contorno, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8), 34 (1963), 138-142.   Google Scholar

[25]

S. Giuffrè, G. Idone and S. Pia, Some classes of projected dynamical systems in Banach spaces and variational inequalities, J. Global Optim., 40 (2008), 119-128.  doi: 10.1007/s10898-007-9173-x.  Google Scholar

[26]

S. Lawphongpanich and D. W. Hearn, Simplical decomposition of the asymmetric traffic assignment problem, Transportation Res. Part B, 18 (1984), 123-133.  doi: 10.1016/0191-2615(84)90026-2.  Google Scholar

[27]

J.-L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar

[28]

S.-Y. Matsushita and L. Xu, On finite convergence of iterative methods for variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 161 (2014), 701-715.  doi: 10.1007/s10957-013-0460-z.  Google Scholar

[29]

A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Problems, 26 (2010), 6pp. doi: 10.1088/0266-5611/26/5/055007.  Google Scholar

[30]

A. Nagurney, D. Parkes and P. Daniele, The Internet, evolutionary variational inequalities, and the time-dependent Braess paradox, Comput. Manag. Sci., 4 (2007), 355-375.  doi: 10.1007/s10287-006-0027-7.  Google Scholar

[31]

B. Panicucci, M. Pappalardo and M. Passacantando, A path-based double projection method for solving the asymmetric traffic network equilibrium problem, Optim. Lett., 1 (2007), 171-185.  doi: 10.1007/s11590-006-0002-9.  Google Scholar

[32]

F. Raciti, Equilibrium conditions and vector variational inequalities: A complex relation, J. Global Optim., 40 (2008), 353-360.  doi: 10.1007/s10898-007-9202-9.  Google Scholar

[33]

L. Scrimali and C. Mirabella, Cooperation in pollution control problems via evolutionary variational inequalities, J. Global Optim., 70 (2018), 455-476.  doi: 10.1007/s10898-017-0580-3.  Google Scholar

[34]

Y. Shehu and O. Iyiola, On a modified extragradient method for variational inequality problem with application to industrial electricity production, J. Ind. Manag. Optim., 15 (2019), 319-342.  doi: 10.3934/jimo.2018045.  Google Scholar

[35]

M. J. Smith, The existence, uniqueness and stability of traffic equilibria, Transportation Res., 13 (1979), 295-304.  doi: 10.1016/0191-2615(79)90022-5.  Google Scholar

[36]

G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416.   Google Scholar

[37]

X. K. Sun and S. J. Li, Duality and gap function for generalized multivalued $\epsilon$-vector variational inequality, Appl. Anal., 92 (2013), 482-492.  doi: 10.1080/00036811.2011.628940.  Google Scholar

[38]

J. Yang, Dynamic power price problem: An inverse variational inequality approach, J. Ind. Manag. Optim., 4 (2008), 673-684.  doi: 10.3934/jimo.2008.4.673.  Google Scholar

Figure 1.  Transportation network patterns of the City $ X\; \text{and}\; Y $
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Figure 2.  Traffic network pattern of two cities $ X $ and $ Y $
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Figure 3.  Traffic network pattern of City $ Y $
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Table 1.  Numerical Results
$ t_{i} $ $ x_{1}^{*}(t_{i}) $ $ x_{2}^{*}(t_{i}) $ $ x_{3}^{*}(t_{i}) $ $ x_{4}^{*}(t_{i}) $
$ 0 $ 1 1 1.5 1.5
$ \frac{1}{8} $ 1.125 1.125 1.6875 1.6875
$ \frac{1}{4} $ 1.25 1.25 1.875 1.875
$ \frac{3}{8} $ 1.375 1.375 2.0625 2.0625
$ \frac{1}{2} $ 1.5 1.5 2.25 2.25
$ \frac{5}{8} $ 1.625 1.625 2.4375 2.4375
$ \frac{3}{4} $ 1.75 1.75 2.625 2.625
$ \frac{7}{8} $ 1.875 1.875 2.8125 2.8125
$ 1 $ 2 2 3 3
$ \frac{9}{8} $ 2.125 2.125 3.1875 3.1875
$ \frac{5}{4} $ 2.25 2.25 3.375 3.375
$ \frac{11}{8} $ 2.375 2.375 3.5625 3.5625
$ \frac{3}{2} $ 2.5 2.5 3.75 3.75
$ \frac{13}{8} $ 2.625 2.625 3.9375 3.9375
$ \frac{7}{4} $ 2.75 2.75 4.125 4.125
$ \frac{15}{8} $ 2.875 2.875 4.3125 4.3125
$ 2 $ 3 3 4.5 4.5
Table Options

Download as excel

$ t_{i} $ $ x_{1}^{*}(t_{i}) $ $ x_{2}^{*}(t_{i}) $ $ x_{3}^{*}(t_{i}) $ $ x_{4}^{*}(t_{i}) $
$ 0 $ 1 1 1.5 1.5
$ \frac{1}{8} $ 1.125 1.125 1.6875 1.6875
$ \frac{1}{4} $ 1.25 1.25 1.875 1.875
$ \frac{3}{8} $ 1.375 1.375 2.0625 2.0625
$ \frac{1}{2} $ 1.5 1.5 2.25 2.25
$ \frac{5}{8} $ 1.625 1.625 2.4375 2.4375
$ \frac{3}{4} $ 1.75 1.75 2.625 2.625
$ \frac{7}{8} $ 1.875 1.875 2.8125 2.8125
$ 1 $ 2 2 3 3
$ \frac{9}{8} $ 2.125 2.125 3.1875 3.1875
$ \frac{5}{4} $ 2.25 2.25 3.375 3.375
$ \frac{11}{8} $ 2.375 2.375 3.5625 3.5625
$ \frac{3}{2} $ 2.5 2.5 3.75 3.75
$ \frac{13}{8} $ 2.625 2.625 3.9375 3.9375
$ \frac{7}{4} $ 2.75 2.75 4.125 4.125
$ \frac{15}{8} $ 2.875 2.875 4.3125 4.3125
$ 2 $ 3 3 4.5 4.5
Table 2.  Numerical Results
$ t_{i} $ $ y_{1}^{*}(t_{i}) $ $ y_{2}^{*}(t_{i}) $ $ y_{3}^{*}(t_{i}) $ $ y_{4}^{*}(t_{i}) $ $ y_{5}^{*}(t_{i}) $
$ 0 $ 2.5 2.5 2 2 2
$ \frac{1}{8} $ 2.8125 2.8125 2.25 2.25 2.25
$ \frac{1}{4} $ 3.125 3.125 2.5 2.5 2.5
$ \frac{3}{8} $ 3.4375 3.4375 2.75 2.75 2.75
$ \frac{1}{2} $ 3.75 3.75 3 3 3
$ \frac{5}{8} $ 4.0625 4.0625 3.25 3.25 3.25
$ \frac{3}{4} $ 4.375 4.375 3.5 3.5 3.5
$ \frac{7}{8} $ 4.6875 4.6875 3.75 3.75 3.75
$ 1 $ 5 5 4 4 4
$ \frac{9}{8} $ 5.3125 5.3125 4.25 4.25 4.25
$ \frac{5}{4} $ 5.625 5.625 4.5 4.5 4.5
$ \frac{11}{8} $ 5.9375 5.9375 4.75 4.75 4.75
$ \frac{3}{2} $ 6.25 6.25 5 5 5
$ \frac{13}{8} $ 6.5625 6.5625 5.25 5.25 5.25
$ \frac{7}{4} $ 6.875 6.875 5.5 5.5 5.5
$ \frac{15}{8} $ 7.1875 7.1875 5.75 5.75 5.75
$ 2 $ 7.5 7.5 6 6 6
Table Options

Download as excel

$ t_{i} $ $ y_{1}^{*}(t_{i}) $ $ y_{2}^{*}(t_{i}) $ $ y_{3}^{*}(t_{i}) $ $ y_{4}^{*}(t_{i}) $ $ y_{5}^{*}(t_{i}) $
$ 0 $ 2.5 2.5 2 2 2
$ \frac{1}{8} $ 2.8125 2.8125 2.25 2.25 2.25
$ \frac{1}{4} $ 3.125 3.125 2.5 2.5 2.5
$ \frac{3}{8} $ 3.4375 3.4375 2.75 2.75 2.75
$ \frac{1}{2} $ 3.75 3.75 3 3 3
$ \frac{5}{8} $ 4.0625 4.0625 3.25 3.25 3.25
$ \frac{3}{4} $ 4.375 4.375 3.5 3.5 3.5
$ \frac{7}{8} $ 4.6875 4.6875 3.75 3.75 3.75
$ 1 $ 5 5 4 4 4
$ \frac{9}{8} $ 5.3125 5.3125 4.25 4.25 4.25
$ \frac{5}{4} $ 5.625 5.625 4.5 4.5 4.5
$ \frac{11}{8} $ 5.9375 5.9375 4.75 4.75 4.75
$ \frac{3}{2} $ 6.25 6.25 5 5 5
$ \frac{13}{8} $ 6.5625 6.5625 5.25 5.25 5.25
$ \frac{7}{4} $ 6.875 6.875 5.5 5.5 5.5
$ \frac{15}{8} $ 7.1875 7.1875 5.75 5.75 5.75
$ 2 $ 7.5 7.5 6 6 6
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