# A Robust Bayesian Approach to an Optimal Replacement Policy for Gas Pipelines

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}(i.e., per km and year). Thus, the failure probability of a single section in a planning period k is calculated as:

_{j}is the length of the j-th section (km) and t

_{k}is the k-th planning period (year). In the analyzed case, the planning period t

_{k}was taken as a year, since the planning frequency was annual. Therefore, the likelihood function of each pipeline class will be:

_{i}.

## 2. Bayesian Approach

_{i}, nor some features, like their quantiles. The experts were able to provide only qualitative judgments, which were expressed through pairwise comparison of all subnetworks about the propensity of a failure occurring in one class rather than in another. The AHP (analytic hierarchy process) methodology, proposed by [9], was used to obtain rates of occurrence of gas escapes in each subnetwork, whereas further manipulations lead to gamma conjugate priors on θ

_{i}’s. In more detail, each expert was presented with a picture of the eight possible configurations (small/large diameter, deep/not deep in the ground, walkway/traffic) and then asked to compare them pairwise. In particular, he/she was asked if, given the occurrence of a gas escape, it was more likely in a subnetwork with respect to another. Calling A and B the two subnetworks, the question was if the occurrence in A was equally/a little more/much more/clearly more/definitely more/a little less/much less/clearly less/definitely less likely than in B. A number was associated with each answer: 1, 3, 5, 7, 9, 1/3, 1/5, 1/7, 1/9, respectively. A square matrix of size eight was obtained out of the pairwise comparisons, and the eigenvector associated with its largest eigenvalue provided the probabilities P

_{1},…, P

_{8}that the gas escape had occurred in each subnetwork (mathematical details can be found in [9]). Given an estimate of the unit failure rate θ

^{∗}of the entire city network, then the guess of the j-th expert on the failure rate of the i-th subnetwork was given by ${\theta}_{i}^{(i)}={\theta}^{*}{P}_{i}$, i = 1,…, 8, using the coloring theorem (see [10]). The set of all of those guesses, for each expert and each subnetwork, is the opinions used in the current paper. In [1], they were used as a sample from a gamma prior distribution whose parameters were obtained matching mean and variance with sample mean and sample variance, respectively. The described procedure is subject to criticism, since the selected prior does not reflect the actual knowledge. Therefore, a class of priors, compatible with the prior knowledge, is entertained to solve the problem.

#### 2.1. Choice of Prior Distributions

_{i}and ${\sigma}_{i}^{2}$, ∀i, are the mean and the variance, respectively, obtained in [2]. Then, they kept the same (gamma) distribution form for priors, and they considered, for each class, the family of priors:

_{c}has mean m

_{i}and variance c.

_{i}and variance ${\sigma}_{i}^{2}$ as in [2]. They obtained the family of priors:

_{i}are given π-integrable functions and α

_{i}(i = 1,…, n) are fixed real numbers. Sometimes, the moment conditions are given by equalities. In those works, the aim is to find the range of a quantity of interest F(π) = ∫

_{Θ}F(θ)π(θ|x)dθ, when π varies over a class of prior distributions Γ.

_{i}(θ) = θ

^{i}, we have ordinary moment conditions on π; for ${H}_{i}(\theta )={I}_{{C}_{i}}(\theta )$, we have conditions on the prior probability of sets C

_{i}; and with equalities, we have the quantile class; see, e.g., [19–21].

_{i}, i=1,…,n} is a partition of the parameter space Θ. Firstly, the class is obtained considering only three sample quantiles from the distribution of experts’ assessed values. Since the range of the non-dominated sets is not “small” enough to avoid overlaps and allow us to make a clear-cut ranking of the classes, we will consider a more restricted class with seven quantiles. In that case, ranking will be more evident especially for the worst and best classes, but some overlaps will be present. Those findings could be sufficient for a decision maker (DM) about planning the replacement of the entire worst subnetwork (not just simply sections) of pipelines, whereas a precise ranking could be obtained introducing a criterion to select one representative value for each class. Here, we will consider two criteria developed in a Bayesian decision analysis framework: the posterior regret (see [22]) and, mostly, the least sensitive action (see [5]).

#### 2.2. Estimation of Failure Rate

_{θ}with density p

_{θ}(x). π(θ) denotes a prior distribution belonging to a class of distributions Γ and L a loss function belonging to a class of loss functions $\mathcal{L}$, such that for all a ∈ $\mathcal{A}$, there exists θ

_{a}, with $L(a,{\theta}_{a})=\underset{\theta \in \mathrm{\Theta}}{\mathrm{min}}L(a,\theta )=0$.

_{x}(θ) denote the posterior density when x (possibly a sample) is observed, m

_{π}(x) the marginal density, l(θ) the likelihood function and ρ(π, L, a) the posterior expected loss of a, i.e.,

**Definition 1**. a ∈ $\mathcal{A}$ is a non-dominated alternative$(a\in \mathcal{ND}(\mathcal{A}))$ if there is no other alternative b ∈ $\mathcal{A}$, such that:

_{0}, π

_{0}), such that:

**Definition 2.**For any (L, π) ∈ $\mathcal{L}$ × Γ, a Bayes action corresponding to (L, π) is an action minimizing ρ(π, L, a) in$\mathcal{A}$ and will be denoted by${a}_{(L,\pi )}^{*}$. Therefore, it holds:

_{C}(θ), the range of the posterior probability of a credible set C is obtained; see [12,26–29], among others.

**Definition 3.**Given a pair (L, π) ∈ $\mathcal{L}$ × Γ, let S(π, L, a) be the sensitivity of an alternative a with respect to the pair (L, π), defined as the difference between the posterior expected losses of a and the Bayes action, ${a}_{(L,\pi )}^{*}$, with respect to the optimal expected loss, i.e.,

**Definition 4.**a

_{ls}is the least sensitive (LS) alternative for $\mathcal{L}$ × Γ if:

**Proposition 1**. Let Γ be a class of prior distributions and $\mathcal{L}$ a class of loss functions. Then, for each a ∈ $\mathcal{A}$,

**Proof.**It is easy to see that:

_{(L,π)}, and we do not need the calculation of m

_{π}.

**Proposition 2.**Let Γ be a generalized moment class defined as in (1) and $\mathcal{L}$ a class of loss functions. Then, for each a, k ∈ $\mathcal{A}$,

_{1},…,θ

_{n+1})′,

**p**= (p

_{1},…, p

_{n+1})′ and the set T ⊂ Θ

^{n+1}× [0, 1]

^{n+1}are defined by the following conditions:

- ${\mathbf{h}}_{\mathbf{i}}^{\prime}\mathbf{p}\le {\alpha}_{i}$, i = 1,…, n, with
**h**= (H_{i}_{i}(θ_{1}),…, H_{i}(θ_{n+1}))′, - $\sum _{i=1}^{n+1}{p}_{i}=1$.

**Proof of Proposition 2**. See [8]. □

**Corollary 1.**Let Γ be a quantile class defined as in (2) and $\mathcal{L}$ a class of loss functions. Then, for each a, k ∈ $\mathcal{A}$,

**Proof of Corollary 1**. This result follows from the previous proposition and Theorem 1 in [19].

## 3. Replacement Policy of Pipelines

#### 3.1. Posterior Ranking of Pipelines

_{i}= 1/4, i = 1,…, 4. The Type 2 quantile algorithm discussed in [33] is employed to obtain the quantiles. Since we have the goal of ranking the classes, we need a unique value for the estimation of the failure rate in each class. We choose an optimal solution compatible with the “relative least sensitive” criterion (detailed in [5]). The optimal solution belongs to the set of non-dominated actions, so the first step consists of determining such a set.

_{i}. Then:

_{1}= 0.04, θ

_{2}= 0.045, θ

_{3}= 0.049 and θ

_{4}= 0.06; its posterior mean is ${\mu}_{{\pi}_{x}}$ = 0.051, and its posterior variance (or posterior expected loss) is V

_{π}= 0.000058. However, the LS action for this class is 0.0618, which has a sensitivity of 2.03. The supremum of the relative increase in posterior expected loss of this action is achieved for the distribution in Γ concentrated at θ

_{1}= 0.02, θ

_{2}= 0.04, θ

_{3}= 0.045 and θ

_{4}= 0.16; its posterior mean is 0.1325, and the posterior variance is 0.002. As was mentioned earlier, the choice of PRGM actions can be unsuitable, since these actions can provide very large relative posterior expected losses for the possible prior distributions; see [5]. Note that, in this case, the relative sensitivity measure takes in to account both the Euclidean distance between Bayes alternatives and the posterior variance (the optimal expected loss). The squared root of the relative sensitivity shows how many times the Euclidean distance is greater than the standard deviation. Then, for example, in Class 8 and in the worst case (when the supremum of the relative increase in posterior expected loss is achieved), the distance between PRGM action and the Bayes action is more than five-times larger than the standard deviation. Therefore, the PRGM action would be unlikely, if the distribution with posterior mean ${\mu}_{{\pi}_{x}}$ = 0.051 and posterior variance V

_{π}= 0.000058 were the correct one. However, and in the worst case, the distance between the least sensitive action and the corresponding Bayes action is “only” less than 1.5-times the standard deviation.

#### 3.2. Threshold Exceedance

_{x}.

## 4. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Range of posterior non-dominated set for the different pipelines classes (three quantiles).

**Figure 3.**Range of posterior non-dominated set for the different pipelines classes (seven quantiles).

**Figure 4.**Supremum and infimum of the posterior risk for any value $\tilde{\theta}$ in Classes 1, 3 and 7.

**Figure 5.**Supremum and infimum of the posterior risk for any value $\tilde{\theta}$ in Classes 2, 6 and 8.

Factors | Low Level (L) | High Level (H) |
---|---|---|

Diameter | ≤ 125 mm | > 125 mm |

Laying depth | < 0.9 m | ≥ 0.9 m |

Laying location | Sidewalk | Street |

Factors | Class 1 | Class 2 | Class 3 | Class 4 | Class 5 | Class 6 | Class 7 | Class 8 |
---|---|---|---|---|---|---|---|---|

Diameter | L | L | L | L | H | H | H | H |

Laying location | L | L | H | H | L | L | H | H |

Laying depth | L | H | L | H | L | H | L | H |

Failures x | 41 | 11 | 15 | 2 | 54 | 8 | 16 | 3 |

Length (km) × time (6 years) | 572.21 | 116.92 | 84.57 | 17.358 | 811.33 | 133.42 | 121.674 | 16.88 |

MLE | 0.072 | 0.094 | 0.177 | 0.115 | 0.067 | 0.060 | 0.131 | 0.178 |

Order | 6 | 5 | 2 | 4 | 7 | 8 | 3 | 1 |

Class 1 | Class 2 | Class 3 | Class 4 | Class 5 | Class 6 | Class 7 | Class 8 | |
---|---|---|---|---|---|---|---|---|

min | 0.03 | 0.02 | 0.11 | 0.05 | 0.02 | 0.01 | 0.08 | 0.02 |

max | 0.2 | 0.09 | 0.42 | 0.16 | 0.09 | 0.05 | 0.37 | 0.16 |

q_{0.250} | 0.090 | 0.040 | 0.290 | 0.060 | 0.040 | 0.020 | 0.120 | 0.040 |

q_{0.500} | 0.105 | 0.040 | 0.320 | 0.090 | 0.060 | 0.030 | 0.185 | 0.045 |

q_{0.750} | 0.120 | 0.090 | 0.350 | 0.130 | 0.080 | 0.040 | 0.230 | 0.060 |

pipelines | Class 1 | Class 2 | Class 3 | Class 4 |
---|---|---|---|---|

Non-dominated set | [0.0607, 0.1068] | [0.0632, 0.0857] | [0.1464, 0.3054] | [0.0883, 0.1124] |

Range | 0.0461 | 0.0225 | 0.1590 | 0.0241 |

pipelines | Class 5 | Class 6 | Class 7 | Class 8 |

Non-dominated set | [0.0600, 0.0814] | [0.0352, 0.0455] | [0.1158, 0.1666] | [0.0495, 0.1325] |

Range | 0.0214 | 0.0103 | 0.0508 | 0.0830 |

Class 1 | Class 2 | Class 3 | Class 4 | Class 5 | Class 6 | Class 7 | Class 8 | |
---|---|---|---|---|---|---|---|---|

a_{ls} | 0.0997 | 0.0721 | 0.2675 | 0.0993 | 0.0746 | 0.0399 | 0.1331 | 0.0618 |

S(a_{ls}) | 44.9602 | 0.9412 | 6.5877 | 0.1251 | 4.9896 | 0.7963 | 0.5811 | 1.9973 |

Replacement order | 3 | 6 | 1 | 4 | 5 | 8 | 2 | 7 |

a_{P}RGM | 0.0837 | 0.0801 | 0.2259 | 0.10035 | 0.0707 | 0.04035 | 0.1412 | 0.091 |

Replacement order | 5 | 6 | 1 | 3 | 7 | 8 | 2 | 4 |

Class 1 | Class 2 | Class 3 | Class 4 | Class 5 | Class 6 | Class 7 | Class 8 | |
---|---|---|---|---|---|---|---|---|

min | 0.03 | 0.02 | 0.11 | 0.05 | 0.02 | 0.01 | 0.08 | 0.02 |

max | 0.2 | 0.09 | 0.42 | 0.16 | 0.09 | 0.05 | 0.37 | 0.16 |

q_{0.125} | 0.060 | 0.030 | 0.210 | 0.050 | 0.020 | 0.020 | 0.110 | 0.030 |

q_{0.250} | 0.090 | 0.040 | 0.290 | 0.060 | 0.040 | 0.020 | 0.120 | 0.040 |

q_{0.375} | 0.090 | 0.040 | 0.310 | 0.070 | 0.050 | 0.030 | 0.150 | 0.040 |

q_{0.500} | 0.105 | 0.040 | 0.320 | 0.090 | 0.060 | 0.030 | 0.185 | 0.045 |

q_{0.625} | 0.110 | 0.050 | 0.330 | 0.110 | 0.060 | 0.030 | 0.220 | 0.050 |

q_{0.750} | 0.120 | 0.070 | 0.350 | 0.130 | 0.080 | 0.040 | 0.230 | 0.060 |

q_{0.875} | 0.130 | 0.090 | 0.390 | 0.130 | 0.090 | 0.040 | 0.260 | 0.150 |

pipelines | Class 1 | Class 2 | Class 3 | Class 4 |
---|---|---|---|---|

Non-dominated set | [0.0681, 0.0920] | [0.0735, 0.0838] | [0.1874, 0.2487] | [0.0923, 0.1039] |

Range | 0.0239 | 0.0103 | 0.0613 | 0.0116 |

pipelines | Class 5 | Class 6 | Class 7 | Class 8 |

Non-dominated set | [0.0603, 0.0692] | [0.0352, 0.0412] | [0.1272, 0.1469] | [0.0956, 0.1257] |

Range | 0.0089 | 0.0059 | 0.0196 | 0.0301 |

Class 1 | Class 2 | Class 3 | Class 4 | Class 5 | Class 6 | Class 7 | Class 8 | |
---|---|---|---|---|---|---|---|---|

a_{ls} | 0.0868 | 0.0787 | 0.2175 | 0.0978 | 0.0641 | 0.0377 | 0.1361 | 0.1106 |

S(a_{ls}) | 2.0030 | 0.1079 | 0.2802 | 0.0311 | 0.2287 | 0.2002 | 0.1053 | 0.0945 |

Order of replace. | 5 | 6 | 1 | 4 | 7 | 8 | 2 | 3 |

a_{P}RGM | 0.0801 | 0.0839 | 0.2181 | 0.0981 | 0.0648 | 0.0382 | 0.1371 | 0.11065 |

Order of replace. | 6 | 5 | 1 | 4 | 7 | 8 | 2 | 3 |

Classes | α = 0.05 | α = 0.1 | α = 0.15 | α = 0.2 | α = 0.3 | α = 0.35 |
---|---|---|---|---|---|---|

Class 1 | 0.107 | 0.105 | 0.101 | 0.098 | 0.095 | 0.093 |

Class 2 | 0.100 | 0.100 | 0.100 | 0.100 | 0.100 | 0.100 |

Class 3 | 0.350 | 0.331 | 0.330 | 0.320 | 0.320 | 0.315 |

Class 4 | 0.160 | 0.160 | 0.130 | 0.130 | 0.130 | 0.130 |

Class 5 | 0.090 | 0.089 | 0.085 | 0.083 | 0.080 | 0.080 |

Class 6 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 | 0.050 |

Class 7 | 0.225 | 0.214 | 0.194 | 0.185 | 0.185 | 0.185 |

Class 8 | 0.160 | 0.160 | 0.160 | 0.160 | 0.160 | 0.160 |

Classes | β = 0.70 | β = 0.75 | β = 0.80 | β = 0.85 | β = 0.90 | β = 0.95 |
---|---|---|---|---|---|---|

Class 1 | 0.055 | 0.054 | 0.052 | 0.051 | 0.049 | 0.046 |

Class 2 | 0.060 | 0.060 | 0.056 | 0.050 | 0.050 | 0.040 |

Class 3 | 0.100 | 0.100 | 0.100 | 0.100 | 0.100 | 0.100 |

Class 4 | 0.060 | 0.060 | 0.060 | 0.050 | 0.050 | 0.050 |

Class 5 | 0.055 | 0.053 | 0.051 | 0.050 | 0.050 | 0.048 |

Class 6 | 0.030 | 0.030 | 0.030 | 0.030 | 0.027 | 0.020 |

Class 7 | 0.110 | 0.106 | 0.098 | 0.091 | 0.083 | 0.080 |

Class 8 | 0.047 | 0.045 | 0.040 | 0.040 | 0.040 | 0.030 |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Arias-Nicolás, J.P.; Martín, J.; Ruggeri, F.; Suárez-Llorens, A. A Robust Bayesian Approach to an Optimal Replacement Policy for Gas Pipelines. *Entropy* **2015**, *17*, 3656-3678.
https://doi.org/10.3390/e17063656

**AMA Style**

Arias-Nicolás JP, Martín J, Ruggeri F, Suárez-Llorens A. A Robust Bayesian Approach to an Optimal Replacement Policy for Gas Pipelines. *Entropy*. 2015; 17(6):3656-3678.
https://doi.org/10.3390/e17063656

**Chicago/Turabian Style**

Arias-Nicolás, José Pablo, Jacinto Martín, Fabrizio Ruggeri, and Alfonso Suárez-Llorens. 2015. "A Robust Bayesian Approach to an Optimal Replacement Policy for Gas Pipelines" *Entropy* 17, no. 6: 3656-3678.
https://doi.org/10.3390/e17063656