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** Case study background and problem formulations**

PROBLEM1: problem_linear

PROBLEM1: problem_linear

maximizing Linear

subject to

Ax ≤0 (multiple linear constraints ensuring that for all companies efficiency is not higher than 1)

d’x=1 (constraint maintaining denominator of ratio for company k = 0 equal to 1)

Box constraints (lower and upper bounds on variables)

——————————————————————–——————

Linear = Linear function

Box constraints = constraints on individual decision variables

——————————————————————–——————

Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 2.83GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem statement | Data | Solution | 6 | 5 | 0.58433 | <1 |

Dataset2 | Problem statement | Data | Solution | 6 | 5 | 0.81256 | <1 |

Dataset3 | Problem statement | Data | Solution | 6 | 5 | 1.00000 | <1 |

Dataset4 | Problem statement | Data | Solution | 6 | 5 | 0.76799 | <1 |

Dataset5 | Problem statement | Data | Solution | 6 | 5 | 0.85086 | <1 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 6 | 5 | 0.58433 | <1 |

Dataset2 | Matlab code | Data | Solution | 6 | 5 | 0.81256 | <1 |

Dataset3 | Matlab code | Data | Solution | 6 | 5 | 1.00000 | <1 |

Dataset4 | Matlab code | Data | Solution | 6 | 5 | 0.76799 | <1 |

Dataset5 | Matlab code | Data | Solution | 6 | 5 | 0.85086 | <1 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 3.50GHz (sec) | ||
---|---|---|---|---|---|---|

Dataset1 | R code | Data | 6 | 5 | 0.58433 | <1 |

Dataset2 | R code | Data | 6 | 5 | 0.81256 | <1 |

Dataset3 | R code | Data | 6 | 5 | 1.00000 | <1 |

Dataset4 | R code | Data | 6 | 5 | 0.76799 | <1 |

Dataset5 | R code | Data | 6 | 5 | 0.85086 | <1 |

**PROBLEM2: problem_max_risk **

maximizing Linear

subject to

max_risk≤0 (constraint ensures that for all companies efficiency is not higher than 1)

d’x=1 (constraint maintaining denominator of ratio for company k = 0 equal to 1)

Box constraints ( lower and upper bounds on variables)

——————————————————————–——————

max_risk = maximum risk function

Box constraints = constraints on individual decision variables

——————————————————————–——————

Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 2.83GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem statement | Data | Solution | 6 | 5 | 0.58433 | <1 |

Dataset2 | Problem statement | Data | Solution | 6 | 5 | 0.81256 | <1 |

Dataset3 | Problem statement | Data | Solution | 6 | 5 | 1.00000 | <1 |

Dataset4 | Problem statement | Data | Solution | 6 | 5 | 0.76799 | <1 |

Dataset5 | Problem statement | Data | Solution | 6 | 5 | 0.85086 | <1 |

Data and solution in MATLAB Environment

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 3.50GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Matlab code | Data | Solution | 6 | 5 | 0.58433 | <1 |

Dataset2 | Matlab code | Data | Solution | 6 | 5 | 0.81256 | <1 |

Dataset3 | Matlab code | Data | Solution | 6 | 5 | 1.00000 | <1 |

Dataset4 | Matlab code | Data | Solution | 6 | 5 | 0.76799 | <1 |

Dataset5 | Matlab code | Data | Solution | 6 | 5 | 0.85086 | <1 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 3.50GHz (sec) | ||
---|---|---|---|---|---|---|

Dataset1 | R code | Data | 6 | 5 | 0.58433 | <1 |

Dataset2 | R code | Data | 6 | 5 | 0.81256 | <1 |

Dataset3 | R code | Data | 6 | 5 | 1.00000 | <1 |

Dataset4 | R code | Data | 6 | 5 | 0.76799 | <1 |

Dataset5 | R code | Data | 6 | 5 | 0.85086 | <1 |

**PROBLEM3: problem_linear _in_one_shot**

For 5 problems

maximizing Linear

subject to

Ax ≤0 (multiple linear constraints ensuring that for all companies efficiency is not higher than 1)

d’x=1 (constraint maintaining denominator of ratio for company k = 0 equal to 1)

Box constraints (lower and upper bounds on variables)

end for

——————————————————————–——————

Linear = Linear function

Box constraints = constraints on individual decision variables

——————————————————————–——————

Download Problem Data

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 2.83GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Cycle statement | Data | Solution | 6 | 5 | 0.58433 | <1 |

Dataset2 | 6 | 5 | 0.81256 | <1 | |||

Dataset3 | 6 | 5 | 1.00000 | <1 | |||

Dataset4 | 6 | 5 | 0.76799 | <1 | |||

Dataset5 | 6 | 5 | 0.85086 | <1 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 3.50GHz (sec) | ||
---|---|---|---|---|---|---|

Dataset1 | R code | Data | 6 | 5 | 0.58433 | <1 |

**PROBLEM4: problem_max_risk_in_one_shot **

For 5 problems

maximizing Linear

subject to

max_risk≤0 (constraint ensures that for all companies efficiency is not higher than 1)

d’x=1 (constraint maintaining denominator of ratio for company k = 0 equal to 1)

Box constraints ( lower and upper bounds on variables)

end for——————————————————————–——————

max_risk = maximum risk function

Box constraints = constraints on individual decision variables

——————————————————————–——————Download Problem Data< 650.81256<1

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 2.83GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Cycle statement | Data | Solution | 6 | 5 | 0.58433 | <1 |

Dataset2 | 6 | 5 | 0.81256 | <1 | |||

Dataset3 | 6 | 5 | 1.00000 | <1 | |||

Dataset4 | 6 | 5 | 0.76799 | <1 | |||

Dataset5 | 6 | 5 | 0.85086 | <1 |

Data and solution in R Environment

Problem Datasets | # of Variables | # of Rows | Objective Value | Solving Time, PC 3.50GHz (sec) | ||
---|---|---|---|---|---|---|

Dataset1 | R code | Data | 6 | 5 | 0.58433 | <1 |

CASE STUDY SUMMARY

This case study compares the relative managerial efficiency of five companies by applying the CCR Model from Data Envelopment Analysis (DEA) (Charnes et al, 1978). The comparison is made for the companies, Sempra Energy, American Electric Power Co., Inc., AGL Resources Inc., CenterPoint Energy, Inc. and Duke Energy Corporation from Utilities industrial sector. We used Earnings per Share, Debt to Equity Ratio, Leverage Ratio, Solvency Ratio as inputs and Net Profit Margin and Price to Earnings Ratio as outputs (collected from the financial statements of these companies). The categorization of inputs and outputs was made according to Zhang, 2007. The model maximize the ratio of the weighted outputs and the weighted inputs of the company for every company, subject to constraints prohibiting that the ratio of the other companies to be higher than 1.

For further readings about DEA, consult “Handbook on data envelopment analysis” from Cooper and “Service Productivity Management: Improving Service Performance Using Data Envelopment Analysis (DEA)” from Sherman.

The optimization problem is solved with two equivalent formulations utilizing the Multilinear and Max_Risk PSG functions.

**References**

• Charnes, A., Cooper, W. W. , and E. Rhodes (1978): “Measuring the efficiency of decision making units.” European Journal of Operational Research.

• Cooper, W.W, Lawrence M. S, and Z. Joe. (2011): “Handbook on data envelopment analysis”. Springer.

• Sherman, H. D. (2006): “Service Productivity Management: Improving Service Performance Using Data Envelopment Analysis (DEA). Includes DEAFrontier Software”. Springer.

• Zhang, X. (2007): “The Generalized DEA Model of Fundamental Analysis of Public Firms, with Application to Portfolio Selection.”. Doctoral Dissertation, University of Tennessee – Knoxville