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We consider the problem of ranking the top

Discrete event system (DES) simulation has been widely used for analyzing and evaluating complex systems since the assumptions for deriving an analytical solution are rarely satisfied in real situation. While DES simulation has been successful in solving many practical problems in a variety of areas such as supply chain systems, healthcare systems, and manufacturing systems, the concerns on the efficiency have never ended [

Ordinal optimization (OO) which aims to obtain a good estimate through ordinal comparisons although the estimated value is still poor emerges as a way to improve the simulation efficiency [

To the best of our knowledge, no previous research has considered the simulation budget allocation for ranking the top

In this paper, we consider the problem of ranking the top

Consider the problem of ranking the top

Denote the mean performances of each design by

Given that

Hence, an optimal computing budget allocation problem can be formulated by maximizing the probability of correctly ranking the top

Maximizing the probability of correctly ranking the top

The performance of every design is independently simulated.

The independence of each design ensures that the samples

Define the cumulant generating function of sample mean

For each

the limit

the origin belongs to

Assumption

We now derive the probability of false ranking other than the correct ranking probability, followed by the corresponding derivation of its large deviation principle. Recall that the probability of correctly ranking the top

Theorem

The rate function of

If there exists a function

Then, it can be concluded that

Now we are in the position to derive the assumed function

Under Assumption

By the Gärtner-Ellis Theorem [

Hence, from large deviation principle,

Similarly,

Therefore, the convergence rate function of the false ranking probability can be expressed as follows:

The objective is to maximize the probability of correctly ranking the top

By [

The result can similarly be applied to

Since model (

From the KKT conditions on problem (

Under Assumptions

We assume that a point satisfying the KKT condition of (

Suppose

Suppose the optimal solution to (

We are now in the position to prove that a point satisfying the KKT conditions of (

Since the problem (

If

Based on the results that

So we have proved the assertion that a point satisfying the KKT conditions of (

Therefore, we could conclude that the optimal allocation rule to rank the top

Suppose that the performance of each design follows the normal distribution; that is,

Based on the results from Theorem

INPUT:

INITIALIZE: Perform

WHILE

(1) Increase the computing budget by Δ

(2) Use (

The population distribution parameters are estimated by the sample statistics.

(3) Simulate additional

END OF WHILE

Define

As the simulation continues, design

Furthermore, we need to take note of

It is also worthy to note that there will be significant variation in the estimate of the performance value. Denote the empirical cumulant generating function and rate function of system

In this section, we test the proposed simulation budget allocation rule for ranking the top

To compare the performance of the procedures, we carried out numerical experiments for the different allocation procedures discussed above. In comparing the procedures, the effectiveness of the procedures is measured by the probability of correctly ranking the top

Each of the procedures simulates each of the

Parameters for the numerical experiments.

Design | Equal spacing | Equal variance | Increasing spacing decreasing variance | |||
---|---|---|---|---|---|---|

Mean | Variance | Mean | Variance | Mean | Variance | |

I | 2 | 400 | 1 | 100 | 1 | 400 |

II | 4 | 361 | 2 | 100 | 2 | 361 |

III | 6 | 324 | 4 | 100 | 4 | 324 |

IV | 8 | 289 | 7 | 100 | 7 | 289 |

V | 10 | 256 | 11 | 100 | 11 | 256 |

VI | 12 | 225 | 16 | 100 | 16 | 225 |

VII | 14 | 196 | 22 | 100 | 22 | 196 |

VIII | 16 | 169 | 29 | 100 | 29 | 169 |

IX | 18 | 144 | 37 | 100 | 37 | 144 |

X | 20 | 121 | 46 | 100 | 46 | 121 |

XI | 22 | 100 | 56 | 100 | 56 | 100 |

XII | 24 | 81 | 67 | 100 | 67 | 81 |

XIII | 26 | 64 | 79 | 100 | 79 | 64 |

XIV | 28 | 49 | 92 | 100 | 92 | 49 |

XV | 30 | 36 | 106 | 100 | 106 | 36 |

XVI | 32 | 25 | 121 | 100 | 121 | 25 |

XVII | 34 | 16 | 137 | 100 | 137 | 16 |

XVIII | 36 | 9 | 154 | 100 | 154 | 9 |

XIX | 38 | 4 | 172 | 100 | 172 | 4 |

XX | 40 | 1 | 191 | 100 | 191 | 1 |

Figure

Probability of correctly ranking the top

The experiment results show that the proposed simulation budget allocation rule OCBA-Rm performs the best in all three experiments. It is also interesting to note that OCBA-

In this paper, we study the problem of simulation budget allocation of ranking the top

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the editor and the anonymous reviewers for the constructive comments which have improved the quality of this paper.