American Institute of Mathematical Sciences

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January  2021, 17(1): 117-131. doi: 10.3934/jimo.2019102

Biobjective optimization over the efficient set of multiobjective integer programming problem

Yasmine Cherfaoui 1, and  Mustapha Moulaï 2,

1. 

USTHB, Department of Operational Research, Bp 32 El Alia, 16111 Algiers, Algeria
2. 

USTHB, LaROMaD Laboratory, Bp 32 El Alia, 16111 Algiers, Algeria

Received  May 2018 Revised  April 2019 Published  January 2021 Early access  September 2019

In this article, an exact method is proposed to optimize two preference functions over the efficient set of a multiobjective integer linear program (MOILP). This kind of problems arises whenever two associated decision-makers have to optimize their respective preference functions over many efficient solutions. For this purpose, we develop a branch-and-cut algorithm based on linear programming, for finding efficient solutions in terms of both preference functions and MOILP problem, without explicitly enumerating all efficient solutions of MOILP problem. The branch and bound process, strengthened by efficient cuts and tests, allows us to prune a large number of nodes in the tree to avoid many solutions. An illustrative example and an experimental study are reported.

Keywords: Multiobjective programming, integer programming, branch-and-cut.
Mathematics Subject Classification: Primary: 90C29, 90C10; Secondary: 90C57.
Citation: Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
References:
[1]

M. Abbas and D. Chaabane, Optimizing a linear function over an integer efficient set, European J. Oper. Res., 174 (2006), 1140-1161.  doi: 10.1016/j.ejor.2005.02.072.  Google Scholar

[2]

P. Belotti, B. Soylu and M. Wiecek, Fathoming rules for biobjective mixed integer linear programs: Review and extensions, Discrete Optim., 22 (2016), 341-363.  doi: 10.1016/j.disopt.2016.09.003.  Google Scholar

[3]

N. Boland, H. Charkhgard and and M. Savelsbergh, A new method for optimizing a linear function over the efficient set of a multiobjective integer program, European J. Oper. Res., 260 (2017), 904-919.  doi: 10.1016/j.ejor.2016.02.037.  Google Scholar

[4]

H. P. Benson, Optimization over the efficient set, J. Math. Anal. Appl., 98 (1984), 562-580.  doi: 10.1016/0022-247X(84)90269-5.  Google Scholar

[5]

M. E-A Chergui and M. Moulaï, An exact method for a discrete multiobjective linear fractional optimization, J. Appl. Math. Decis. Sci., (2008), 12pp. doi: 10.1155/2008/760191.  Google Scholar

[6]

W. Drici, F- Z Ouaïl and M. Moulaï, Optimizing a linear fractional function over the integer efficient set, Ann. Oper. Res., 267 (2018), 135-151.  doi: 10.1007/s10479-017-2691-0.  Google Scholar

[7]

J. G. Ecker, N. S. Hegner and I. A. Kouada, Generating all maximal efficient faces for multiple objective linear programs, J. Optim. Theory Appl., 30 (1980), 353-381.  doi: 10.1007/BF00935493.  Google Scholar

[8]

J. G. Ecker and I. A. Kouada, Finding all efficient extreme points for multiple objective linear programs, Math. Programming, 14 (1978), 249-261.  doi: 10.1007/BF01588968.  Google Scholar

[9]

J. G. Ecker and J. H. Song, Optimizing a linear function over an efficient set, J. Optim. Theory Appl., 83 (1994), 541-563.  doi: 10.1007/BF02207641.  Google Scholar

[10]

J. P. Evans and R. E. Steuer, A revised simplex method for linear multiple objective programs, Math. Programming, 5 (1973), 54-72.  doi: 10.1007/BF01580111.  Google Scholar

[11]

J. M. Jorge, An algorithm for optimizing a linear function over an integer efficient set, European J. Oper. Res., 195 (2009), 98-103.  doi: 10.1016/j.ejor.2008.02.005.  Google Scholar

[12]

F- Z Ouaïl, M. E-A. Chergui and M. Moulaï, An exact method for optimizing a linear function over an integer efficient set, WSEAS Transactions on Circuits and Systems, 16(3) (2017), 141-148.   Google Scholar

[13]

Y. Yamamoto, Optimization over the efficient set: Overview, J. Global Optim., 22 (2002), 285-317.  doi: 10.1023/A:1013875600711.  Google Scholar

show all references

References:
[1]

M. Abbas and D. Chaabane, Optimizing a linear function over an integer efficient set, European J. Oper. Res., 174 (2006), 1140-1161.  doi: 10.1016/j.ejor.2005.02.072.  Google Scholar

[2]

P. Belotti, B. Soylu and M. Wiecek, Fathoming rules for biobjective mixed integer linear programs: Review and extensions, Discrete Optim., 22 (2016), 341-363.  doi: 10.1016/j.disopt.2016.09.003.  Google Scholar

[3]

N. Boland, H. Charkhgard and and M. Savelsbergh, A new method for optimizing a linear function over the efficient set of a multiobjective integer program, European J. Oper. Res., 260 (2017), 904-919.  doi: 10.1016/j.ejor.2016.02.037.  Google Scholar

[4]

H. P. Benson, Optimization over the efficient set, J. Math. Anal. Appl., 98 (1984), 562-580.  doi: 10.1016/0022-247X(84)90269-5.  Google Scholar

[5]

M. E-A Chergui and M. Moulaï, An exact method for a discrete multiobjective linear fractional optimization, J. Appl. Math. Decis. Sci., (2008), 12pp. doi: 10.1155/2008/760191.  Google Scholar

[6]

W. Drici, F- Z Ouaïl and M. Moulaï, Optimizing a linear fractional function over the integer efficient set, Ann. Oper. Res., 267 (2018), 135-151.  doi: 10.1007/s10479-017-2691-0.  Google Scholar

[7]

J. G. Ecker, N. S. Hegner and I. A. Kouada, Generating all maximal efficient faces for multiple objective linear programs, J. Optim. Theory Appl., 30 (1980), 353-381.  doi: 10.1007/BF00935493.  Google Scholar

[8]

J. G. Ecker and I. A. Kouada, Finding all efficient extreme points for multiple objective linear programs, Math. Programming, 14 (1978), 249-261.  doi: 10.1007/BF01588968.  Google Scholar

[9]

J. G. Ecker and J. H. Song, Optimizing a linear function over an efficient set, J. Optim. Theory Appl., 83 (1994), 541-563.  doi: 10.1007/BF02207641.  Google Scholar

[10]

J. P. Evans and R. E. Steuer, A revised simplex method for linear multiple objective programs, Math. Programming, 5 (1973), 54-72.  doi: 10.1007/BF01580111.  Google Scholar

[11]

J. M. Jorge, An algorithm for optimizing a linear function over an integer efficient set, European J. Oper. Res., 195 (2009), 98-103.  doi: 10.1016/j.ejor.2008.02.005.  Google Scholar

[12]

F- Z Ouaïl, M. E-A. Chergui and M. Moulaï, An exact method for optimizing a linear function over an integer efficient set, WSEAS Transactions on Circuits and Systems, 16(3) (2017), 141-148.   Google Scholar

[13]

Y. Yamamoto, Optimization over the efficient set: Overview, J. Global Optim., 22 (2002), 285-317.  doi: 10.1023/A:1013875600711.  Google Scholar

Figure 1.  Search tree of the example
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Table 1.  Optimal simplex table for node 0
$ \mathcal{B}_1 $ $ x_1 $ $ x_3 $ $ x_5 $ $ RHS $
$ x_4 $ $ 5/6 $ -$ 1 $ $ 2/3 $ $ 53/6 $
$ x_2 $ $ 1/3 $ $ 0 $ $ 2/3 $ $ 16/3 $
$ x_6 $ $ 1/3 $ $ 5/2 $ -$ 2/3 $ 2/3
$ \bar{d}^1 $ -$ 7/6 $ -$ 1/2 $ -$ 10/3 $ $ 80/3 $
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$ \mathcal{B}_1 $ $ x_1 $ $ x_3 $ $ x_5 $ $ RHS $
$ x_4 $ $ 5/6 $ -$ 1 $ $ 2/3 $ $ 53/6 $
$ x_2 $ $ 1/3 $ $ 0 $ $ 2/3 $ $ 16/3 $
$ x_6 $ $ 1/3 $ $ 5/2 $ -$ 2/3 $ 2/3
$ \bar{d}^1 $ -$ 7/6 $ -$ 1/2 $ -$ 10/3 $ $ 80/3 $
Table 2.  Optimal simplex table for node 1
$ \mathcal{B}_2 $ $ x_3 $ $ x_5 $ $ x_7 $ $ RHS $
$ x_4 $ -$ 1 $ $ -1 $ $ \frac{5}{2} $ $ 8 $
$ x_1 $ $ 0 $ $ 2 $ -$ 3 $ $ 1 $
$ x_6 $ $ \frac{5}{2} $ $ 0 $ -$ 1 $ $ 1 $
$ x_2 $ $ 0 $ $ 0 $ $ 1 $ $ 5 $
$ \bar{d}^1 $ -$ \frac{1}{2} $ -$ 1 $ -$ \frac{7}{2} $ $ \frac{51}{2} $
$ \bar{d}^2 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $
$ \bar{c}^1 $ $ -1 $ $ -2 $ $ 5 $ $ -9 $
$ \bar{c}^2 $ -$ \frac{1}{2} $ $ 0 $ -$ 2 $ $ 10 $
$ \bar{c}^3 $ -$ 1 $ -$ 2 $ $ 3 $ $ 1 $
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$ \mathcal{B}_2 $ $ x_3 $ $ x_5 $ $ x_7 $ $ RHS $
$ x_4 $ -$ 1 $ $ -1 $ $ \frac{5}{2} $ $ 8 $
$ x_1 $ $ 0 $ $ 2 $ -$ 3 $ $ 1 $
$ x_6 $ $ \frac{5}{2} $ $ 0 $ -$ 1 $ $ 1 $
$ x_2 $ $ 0 $ $ 0 $ $ 1 $ $ 5 $
$ \bar{d}^1 $ -$ \frac{1}{2} $ -$ 1 $ -$ \frac{7}{2} $ $ \frac{51}{2} $
$ \bar{d}^2 $ $ 1 $ $ 0 $ $ 0 $ $ 0 $
$ \bar{c}^1 $ $ -1 $ $ -2 $ $ 5 $ $ -9 $
$ \bar{c}^2 $ -$ \frac{1}{2} $ $ 0 $ -$ 2 $ $ 10 $
$ \bar{c}^3 $ -$ 1 $ -$ 2 $ $ 3 $ $ 1 $
Table 3.  Optimal simplex table for node 3
$ \mathcal{B}_3 $ $ x_5 $ $ x_6 $ $ x_9 $ $ RHS $
$ x_4 $ 1 $ \frac{5}{2} $ $ \frac{21}{4} $ $ \frac{21}{5} $
$ x_1 $ 2 -3 $ \frac{15}{2} $ $ \frac{11}{2} $
$ x_3 $ 0 0 -1 1
$ x_2 $ 0 1 $ \frac{5}{2} $ $ \frac{7}{2} $
$ x_7 $ 0 -1 -$ \frac{5}{2} $ $ \frac{3}{2} $
$ x_8 $ 0 -1 -$ \frac{5}{2} $ $ \frac{1}{2} $
$ \bar{d}^1 $ -1 -$ \frac{7}{2} $ -$ \frac{37}{4} $ $ \frac{79}{4} $
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$ \mathcal{B}_3 $ $ x_5 $ $ x_6 $ $ x_9 $ $ RHS $
$ x_4 $ 1 $ \frac{5}{2} $ $ \frac{21}{4} $ $ \frac{21}{5} $
$ x_1 $ 2 -3 $ \frac{15}{2} $ $ \frac{11}{2} $
$ x_3 $ 0 0 -1 1
$ x_2 $ 0 1 $ \frac{5}{2} $ $ \frac{7}{2} $
$ x_7 $ 0 -1 -$ \frac{5}{2} $ $ \frac{3}{2} $
$ x_8 $ 0 -1 -$ \frac{5}{2} $ $ \frac{1}{2} $
$ \bar{d}^1 $ -1 -$ \frac{7}{2} $ -$ \frac{37}{4} $ $ \frac{79}{4} $
Table 4.  Optimal simplex table for node 4
$ \mathcal{B}_4 $ $ x_6 $ $ x_9 $ $ x_{10} $ $ RHS $
$ x_4 $ $ 1 $ $ \frac{3}{2} $ -$ \frac{1}{2} $ $ \frac{11}{2} $
$ x_2 $ $ 1 $ $ \frac{5}{2} $ 0 $ \frac{7}{2} $
$ x_3 $ $ 0 $ -$ 1 $ $ 0 $ $ 1 $
$ x_1 $ $ 0 $ $ 0 $ $ 1 $ $ 5 $
$ x_5 $ -$ \frac{3}{2} $ -$ \frac{15}{4} $ -$ \frac{1}{2} $ $ \frac{1}{4} $
$ x_7 $ -$ 1 $ -$ \frac{5}{2} $ $ 0 $ $ \frac{3}{2} $
$ x_8 $ -$ 1 $ -$ \frac{5}{2} $ 0 $ \frac{1}{2} $
$ \bar{d}^1 $ -$ 5 $ -$ 13 $ -$ \frac{1}{2} $ $ \frac{39}{2} $
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$ \mathcal{B}_4 $ $ x_6 $ $ x_9 $ $ x_{10} $ $ RHS $
$ x_4 $ $ 1 $ $ \frac{3}{2} $ -$ \frac{1}{2} $ $ \frac{11}{2} $
$ x_2 $ $ 1 $ $ \frac{5}{2} $ 0 $ \frac{7}{2} $
$ x_3 $ $ 0 $ -$ 1 $ $ 0 $ $ 1 $
$ x_1 $ $ 0 $ $ 0 $ $ 1 $ $ 5 $
$ x_5 $ -$ \frac{3}{2} $ -$ \frac{15}{4} $ -$ \frac{1}{2} $ $ \frac{1}{4} $
$ x_7 $ -$ 1 $ -$ \frac{5}{2} $ $ 0 $ $ \frac{3}{2} $
$ x_8 $ -$ 1 $ -$ \frac{5}{2} $ 0 $ \frac{1}{2} $
$ \bar{d}^1 $ -$ 5 $ -$ 13 $ -$ \frac{1}{2} $ $ \frac{39}{2} $
Table 5.  Optimal simplex table for node 5
$ \mathcal{B}_5 $ $ x_5 $ $ x_9 $ $ x_{10} $ $ RHS $
$ x_4 $ $ \frac{2}{3} $ -1 $ \frac{5}{6} $ $ \frac{29}{6} $
$ x_1 $ 0 0 -1 6
$ x_3 $ 0 -1 0 1
$ x_2 $ $ \frac{2}{3} $ 0 $ \frac{1}{3} $ $ \frac{10}{3} $
$ x_7 $ $ -\frac{2}{3} $ 0 $ -\frac{1}{3} $ $ \frac{5}{3} $
$ x_8 $ $ -\frac{2}{3} $ 0 $ -\frac{1}{3} $ $ \frac{2}{3} $
$ x_6 $ $ -\frac{2}{3} $ $ \frac{5}{2} $ $ -\frac{1}{3} $ $ \frac{1}{6} $
$ \bar{d}^1 $ $ -\frac{10}{3} $ $ -\frac{1}{2} $ $ \frac{7}{6} $ $ \frac{115}{6} $
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$ \mathcal{B}_5 $ $ x_5 $ $ x_9 $ $ x_{10} $ $ RHS $
$ x_4 $ $ \frac{2}{3} $ -1 $ \frac{5}{6} $ $ \frac{29}{6} $
$ x_1 $ 0 0 -1 6
$ x_3 $ 0 -1 0 1
$ x_2 $ $ \frac{2}{3} $ 0 $ \frac{1}{3} $ $ \frac{10}{3} $
$ x_7 $ $ -\frac{2}{3} $ 0 $ -\frac{1}{3} $ $ \frac{5}{3} $
$ x_8 $ $ -\frac{2}{3} $ 0 $ -\frac{1}{3} $ $ \frac{2}{3} $
$ x_6 $ $ -\frac{2}{3} $ $ \frac{5}{2} $ $ -\frac{1}{3} $ $ \frac{1}{6} $
$ \bar{d}^1 $ $ -\frac{10}{3} $ $ -\frac{1}{2} $ $ \frac{7}{6} $ $ \frac{115}{6} $
Table 6.  Optimal simplex table for node 6
$ \mathcal{B}_6 $ $ x_9 $ $ x_{10} $ $ x_{11} $ RHS
$ x_4 $ -1 $ -\frac{1}{2} $ 1 5
$ x_1 $ 0 1 0 5
$ x_3 $ -1 0 0 1
$ x_2 $ 0 0 0 3
$ x_5 $ 0 $ -\frac{1}{2} $ $ -\frac{3}{2} $ 1
$ x_8 $ 0 0 -1 1
$ x_{6} $ $ \frac{5}{2} $ 0 -1 $ \frac{1}{2} $
$ x_7 $ 0 0 -1 2
$ \bar{d}^1 $ $ -\frac{1}{2} $ $ -\frac{1}{2} $ -5 17
$ \bar{d}^2 $ 1 0 0 1
$ \bar{c}^1 $ -1 -1 2 -2
$ \bar{c}^2 $ $ -\frac{1}{2} $ 0 -2 $ \frac{11}{2} $
$ \bar{c}^3 $ -1 -1 0 4
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$ \mathcal{B}_6 $ $ x_9 $ $ x_{10} $ $ x_{11} $ RHS
$ x_4 $ -1 $ -\frac{1}{2} $ 1 5
$ x_1 $ 0 1 0 5
$ x_3 $ -1 0 0 1
$ x_2 $ 0 0 0 3
$ x_5 $ 0 $ -\frac{1}{2} $ $ -\frac{3}{2} $ 1
$ x_8 $ 0 0 -1 1
$ x_{6} $ $ \frac{5}{2} $ 0 -1 $ \frac{1}{2} $
$ x_7 $ 0 0 -1 2
$ \bar{d}^1 $ $ -\frac{1}{2} $ $ -\frac{1}{2} $ -5 17
$ \bar{d}^2 $ 1 0 0 1
$ \bar{c}^1 $ -1 -1 2 -2
$ \bar{c}^2 $ $ -\frac{1}{2} $ 0 -2 $ \frac{11}{2} $
$ \bar{c}^3 $ -1 -1 0 4
Table 7.  Optimal simplex table for node 8
$ \mathcal{B}_8 $ $ x_5 $ $ x_9 $ $ x_{11} $ RHS
$ x_4 $ -1 -1 $ \frac{5}{2} $ 4
$ x_2 $ 0 0 1 3
$ x_3 $ 0 -1 0 1
$ x_1 $ 2 0 -3 7
$ x_6 $ 0 $ \frac{5}{2} $ -1 $ \frac{1}{2} $
$ x_7 $ 0 0 -1 2
$ x_8 $ 0 0 -1 1
$ x_{10} $ 2 0 -3 1
$ \bar{d}^1 $ -1 $ -\frac{1}{2} $ $ -\frac{7}{2} $ 18
$ \bar{d}^2 $ 0 1 0 1
$ \bar{c}^1 $ -2 -1 5 0
$ \bar{c}^2 $ 0 $ -\frac{1}{2} $ -2 $ \frac{11}{2} $
$ \bar{c}^3 $ -2 -1 3 6
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$ \mathcal{B}_8 $ $ x_5 $ $ x_9 $ $ x_{11} $ RHS
$ x_4 $ -1 -1 $ \frac{5}{2} $ 4
$ x_2 $ 0 0 1 3
$ x_3 $ 0 -1 0 1
$ x_1 $ 2 0 -3 7
$ x_6 $ 0 $ \frac{5}{2} $ -1 $ \frac{1}{2} $
$ x_7 $ 0 0 -1 2
$ x_8 $ 0 0 -1 1
$ x_{10} $ 2 0 -3 1
$ \bar{d}^1 $ -1 $ -\frac{1}{2} $ $ -\frac{7}{2} $ 18
$ \bar{d}^2 $ 0 1 0 1
$ \bar{c}^1 $ -2 -1 5 0
$ \bar{c}^2 $ 0 $ -\frac{1}{2} $ -2 $ \frac{11}{2} $
$ \bar{c}^3 $ -2 -1 3 6
Table 8.  Optimal simplex table for the node 10
$ \mathcal{B}_{10} $ $ x_6 $ $ x_{10} $ $ x_{13} $ RHS
$ x_5 $ $ -\frac{3}{2} $ $ -\frac{1}{2} $ $ -\frac{15}{4} $ 4
$ x_4 $ 1 $ -\frac{1}{2} $ $ \frac{3}{2} $ 4
$ x_3 $ 0 0 -1 2
$ x_7 $ -1 0 $ -\frac{5}{2} $ 4
$ x_8 $ -1 1 $ -\frac{5}{2} $ 3
$ x_9 $ 0 0 -1 1
$ x_2 $ 1 0 $ \frac{5}{2} $ 1
$ x_1 $ 0 1 0 5
$ \bar{d}^1 $ -5 $ -\frac{1}{2} $ -13 $ \frac{13}{2} $
$ \bar{d}^2 $ 0 0 1 2
$ \bar{c}^1 $ 2 -1 4 1
$ \bar{c}^2 $ -2 0 $ -\frac{11}{2} $ 1
$ \bar{c}^3 $ 0 -1 -1 3
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$ \mathcal{B}_{10} $ $ x_6 $ $ x_{10} $ $ x_{13} $ RHS
$ x_5 $ $ -\frac{3}{2} $ $ -\frac{1}{2} $ $ -\frac{15}{4} $ 4
$ x_4 $ 1 $ -\frac{1}{2} $ $ \frac{3}{2} $ 4
$ x_3 $ 0 0 -1 2
$ x_7 $ -1 0 $ -\frac{5}{2} $ 4
$ x_8 $ -1 1 $ -\frac{5}{2} $ 3
$ x_9 $ 0 0 -1 1
$ x_2 $ 1 0 $ \frac{5}{2} $ 1
$ x_1 $ 0 1 0 5
$ \bar{d}^1 $ -5 $ -\frac{1}{2} $ -13 $ \frac{13}{2} $
$ \bar{d}^2 $ 0 0 1 2
$ \bar{c}^1 $ 2 -1 4 1
$ \bar{c}^2 $ -2 0 $ -\frac{11}{2} $ 1
$ \bar{c}^3 $ 0 -1 -1 3
Table 9.  Optimal simplex table for node 11
$ \mathcal{B}_{11} $ $ x_4 $ $ x_6 $ $ x_{13} $ RHS
$ x_5 $ -1 $ -\frac{5}{2} $ $ -\frac{21}{4} $ 0
$ x_3 $ 0 0 -1 2
$ x_7 $ 0 -1 $ -\frac{5}{2} $ 4
$ x_8 $ 0 -1 $ -\frac{5}{2} $ 3
$ x_{10} $ 2 2 3 7
$ x_{11} $ 0 -1 $ -\frac{5}{2} $ 2
$ x_{12} $ 0 -1 $ -\frac{5}{2} $ 1
$ x_9 $ 0 0 -1 1
$ x_2 $ 1 0 $ \frac{5}{2} $ 1
$ x_1 $ 2 2 3 13
$ \bar{d}^1 $ -1 -6 $ -\frac{29}{2} $ $ \frac{21}{2} $
$ \bar{d}^2 $ 0 0 3 6
$ \bar{c}^1 $ -2 0 1 9
$ \bar{c}^2 $ 0 -2 $ -\frac{11}{2} $ 1
$ \bar{c}^3 $ -2 -2 -4 11
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$ \mathcal{B}_{11} $ $ x_4 $ $ x_6 $ $ x_{13} $ RHS
$ x_5 $ -1 $ -\frac{5}{2} $ $ -\frac{21}{4} $ 0
$ x_3 $ 0 0 -1 2
$ x_7 $ 0 -1 $ -\frac{5}{2} $ 4
$ x_8 $ 0 -1 $ -\frac{5}{2} $ 3
$ x_{10} $ 2 2 3 7
$ x_{11} $ 0 -1 $ -\frac{5}{2} $ 2
$ x_{12} $ 0 -1 $ -\frac{5}{2} $ 1
$ x_9 $ 0 0 -1 1
$ x_2 $ 1 0 $ \frac{5}{2} $ 1
$ x_1 $ 2 2 3 13
$ \bar{d}^1 $ -1 -6 $ -\frac{29}{2} $ $ \frac{21}{2} $
$ \bar{d}^2 $ 0 0 3 6
$ \bar{c}^1 $ -2 0 1 9
$ \bar{c}^2 $ 0 -2 $ -\frac{11}{2} $ 1
$ \bar{c}^3 $ -2 -2 -4 11
Table 10.  Random instances execution times
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