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** Case study background and formulation of problems**

**PROBLEM 1: problem_best_ranking**

Minimize pr_pen (minimizing probability that Loss exceeds zero)

subject to

Linearmulti =1 (non-zero requirement)

Box constraints (constraints on decision variables)

————————————————————————-

Pr_pen = Probability of Exceedance

Box constraints = constraints on individual decision variables

————————————————————————-

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset |
8 | 248 | 0 | 0.03 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 2: problem_worst_ranking**

Minimize pr_pen_g (minimizing probability that gain exceeds zero)

subject to

Linearmulti =1 (non-zero requirement)

Box constraints (constraints on decision variables)

————————————————————————-

Pr_pen_g = Probability of Exceedance Penalty for Gain

Box constraints = constraints on individual decision variables

————————————————————————-

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset |
8 | 248 | 0.16532258 | 4.22 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 3: problem_best_buffered-ranking**

Minimize bPOE ( minimizing buffered probability of exceedance)

Box constraints (constraints on decision variables)

————————————————————————-

bPOE = Buffered Probability of Exceedance

Box constraints = constraints on individual decision variables

————————————————————————-

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset |
8 | 248 | 4.03258 | 0.01 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

**PROBLEM 4: problem_worst_buffered-ranking**

Minimize bPOE_g ( minimizing buffered probability of exceedance for gain)

Box constraints (constraints on decision variables)

————————————————————————-

bPOE_g = Buffered Probability of Exceedance for Gain

Box constraints = constraints on individual decision variables

————————————————————————-

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset |
8 | 248 | 0.49441096 | 0.01 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

Run-File | Problem Statement | Data | Solution | ||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

**CASE STUDY SUMMARY**

This case study investigates the best buffered-ranking characteristic for the Decision Making Units (DMUs) which is defined as the minimum number k that the efficiency measure of this DMU exceeds the average of the top k efficient DMUs. The best buffered-ranking is similar to the best ranking (see, Salo, A., & Punkka, A.). The best buffered-ranking assesses not only the number of top efficient DMUs, but also their overall magnitude of efficiency. Computing the best ranking is equivalent to minimizing the probability of exceedance, while computing the best buffered-ranking is equivalent to minimizing the buffered-probability of exceedance. This case study considers also performance of the worst buffered-ranking that is a characteristic opposite to the best buffered-ranking. Experimental results obtained using two data sets clearly demonstrate that buffered-rankings have a great computational advantage compared to rankings.

References

• Mafusalov, A., & Uryasev, S. (2018). Buffered probability of exceedance: mathematical properties and optimization. SIAM Journal on Optimization, 28, 1077–1103.

• Norton, M., Mafusalov, A., & Uryasev, S. (2018). Cardinality of upper average and its application to network optimization.SIAM Journal on Optimization, 28, 1726–1750.

• Norton, M., & Uryasev, S. (2019). Maximization of AUC and buffered AUC in binary classification. Mathematical Programming,174, 575–612.

• Salo, A., & Punkka, A. (2011). Ranking intervals and dominance relations for ratio-based efficiency analysis. ManagementScience, 57, 200–214.

• Wang, Y., & Uryasev, S. (2019). Buffered ranking in efficiency analysis, Working paper.