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A technique is developed for finding a closed form expression for the cumulative distribution function of the maximum value of the objective function in a stochastic linear programming problem, where either the objective function coefficients or the right hand side coefficients are continuous random vectors with known probability distributions. This is the “wait and see” problem of stochastic linear programming. Explicit results for the distribution problem are extremely difficult to obtain; indeed, previous results are known only if the right hand side coefficients have an exponential distribution [1]. To date, no explicit results have been obtained for stochastic c, and no new results of any form have appeared since the 1970’s. In this paper, we obtain the first results for stochastic c, and new explicit results if b an c are stochastic vectors with an exponential, gamma, uniform, or triangle distribution. A transformation is utilized that greatly reduces computational time.

Consider the linear programming problem,

where _{i},

Early work on the distribution problem can be found in Babbar [

Following [

For all

and is feasible if

For the case in which the b vector is random, let the probability space be defined by the m-tuple

where

Thus,

Now, let

Since

Then

Thus,

Now, consider the case in which only the c vector is random. Let the probability space C be defined by the n-tuple

where

Thus,

where

To evaluate the right-hand side of equation Equation (15) let

Since

Then

where

Thus, if only the c vector is random the distribution function of

In the case of stochastic b, Let

By substituting for b we have:

The probability that a basis G remains feasible is

where

where

Because

Note that since is the basis matrix, its determinant is nonzero; thus

The problems were run using the Mathematica software version 8.0.1.0 utilizing the supercomputer at the University of Oklahoma.

CPUs: All compute nodes have dual Intel Xeon E5-2650 “Sandy Bridge” oct core 2.0 GHz CPUs; there is also one “fat node” with quad Intel Xeon E7-4830 “Westmere” oct core 2.13 GHz CPUs.

RAM: Most of the compute nodes have 32 GB of 1333 MHz RAM and 23 with 64 GB of 1333 MHz RAM; the one “fat node” has 1 TB of 1066 MHz RAM, which is called large memory.

Accelerators: There are 18 NVIDIA Tesla M2075 cards, for an aggregate of an additional approximately 9 TFLOPs double precision.

In order to compare the run times, four types of distributions were considered as shown in

Distribution | Defined Equation | Parameters | |
---|---|---|---|

Exponential | |||

Uniform | |||

Gamma | |||

Triangular |

The different distributions were solved using both Bereanu’s method and the Ewbank, Foote and Kumin transformation method to compare the two.

Size | Sample of number in result | Difference between run times | |
---|---|---|---|

Exponential | 2 × 2 | 2851 | 3.06 |

3 × 3 | 10,071 | 7.71 | |

6 × 6 | 187,191,798,507,739 | 4.62 | |

9 × 9 | 264,776,529,949,169,000,000 | 11.00 | |

Uniform | 2 × 2 | 2,929,968 | 2.05 |

3 × 3 | 46,970,460,160 | 2.06 | |

6 × 6 | 8,538,555,554,355,150,000 | 5.41 | |

9 × 9 | 844,697,996,409,499,233,632,305,152 | 86.61 | |

Gamma | 2 × 2 | 549,615,780 | 2.60 |

3 × 3 | 15,629,133,492 | 1.41 | |

6 × 6 | 243,545,558,927,209,970,255,163,031,323,401,871,559 | 4.70 |

Dimention | Bereanu’s Method | EFK Method | |
---|---|---|---|

Exponential | 2 × 2 | 2.386 | 1.747 |

3 × 3 | 78.68 | 17.97 | |

6 × 6 | No Result | 9176.28 | |

Uniform | 2 × 2 | 3.12 | 2.606 |

3 × 3 | 210.4 | 105.144 | |

Gamma | 2 × 2 | No Result | 11.544 |

3 × 3 | No Result | 115.004 | |

Triangular | 2 × 2 | No Result | 13.292 |

3 × 3 | No Result | 575.846 |

the EFK method substantially reduces the computational time. In addition, Bereanu’s method is not able to solve some larger sizes of the problem. All times are measured in seconds.

Ansaripour, A., Ma- ta, A., Nourazari, S. and Kumin, H. (2016) Some Explicit Results for the Distribution Problem of Stochastic Linear Programming. Open Journal of Optimization, 5, 140-162. http://dx.doi.org/10.4236/ojop.2016.54014