# An Approach for Integrating Uncertainty When Selecting an Anti-Torpedo Decoy in Brand New Warships

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

- Logistics issues. This allows evaluating different aspects: storage of decoys, volume they occupy, existence of available means to train personnel in the use of the type of decoy.
- Operational capacities or effectiveness issues. This is an essential criterion to make an adequate decision to the tactical and strategic demands of a surface ship in the field of ASW.

_{ij}denote the intensity of importance of the criteria being compared, that is, logistics, and operational capabilities. Let y

_{ij}denote the intensity of importance of the sub-criteria belonging logistics criterion. Finally, z

_{ij}denotes the intensity of importance given to the sub-criteria related to operational capabilities criterion. Pairwise comparison matrices for criteria and sub-criteria were given to experts of the Navy in order to fill in the x

_{ij}, y

_{ij}, and z

_{ij}values. The numerical values to denote the intensity of the importance of criteria and sub-criteria are the integers from 1 to 9. The results were treated statistically to obtain the weights of the criteria, which were used in the AHP method.

_{i}, for i from 1 to n. A score is assigned to each alternative (w

_{i}is the score of alternative x

_{i}), providing a weight vector. A square matrix of pairwise comparison is used to solve the multi-criteria decision-making (MCDM). Let A be the pairwise comparison matrix:

_{ij})

_{ij}= 1/a

_{ji}and a

_{ii}= 1 for all i and j from 1 to n [17]. Saaty proposed a consistency index (CI) to evaluate the consistency of the pairwise comparison matrix:

_{max}− n)/(n − 1)

_{max}· w

_{i}), sub-criteria (sc

_{ij}), and alternatives (a

_{i}), as shown in Table 1.

_{SV}= C

_{1}· x

_{SV}+ C

_{2}

_{SV}= 5 m

^{3}if y

_{SV}= 0 and x

_{SV}= 1 m

^{3}if y

_{SV}= 1

_{RT}= C

_{3}· x

_{RT}+ C

_{4}

_{RT}= 30 s if y

_{RT}= 0 and x

_{RT}= 0 s if y

_{RT}= 1

^{3}the minimum utility y

_{SV}is assigned, i.e., a value of 0%, given that this volume is too large because it reduces space to other logistic needs of the vessel. Then, for a volume of 1 m

^{3}the maximum utility y

_{SV}is assigned, i.e., a value of 100%, since it is not possible to store decoys in a smaller space.

_{RT}(0%) happens when x

_{RT}is 30 s. Half a minute is considered as the maximum time established to make effective use of decoy capabilities. That is, once that time has passed, it is considered that the vessel is no longer able to make effective use of any of the decoys. Likewise, the maximum utility y

_{RT}(100%) happens when the reaction time is 0 s. That is to say, the optimal value of the reaction time is a null value, which would represent the ideal situation. However, it is only a reference, because in practice reaction time can never be zero. Hence, the utility functions are:

_{SV}= −25 · x

_{SV}+ 125 and y

_{RT}= −(10/3) · x

_{RT}+ 100

_{ij})

_{A}the set of alternatives and S

_{S}the set of scenarios. Then:

_{A}= {a

_{i}} for i = 1 to m

_{S}= {s

_{j}} for j = 1 to n

_{ij}is the penalty obtained after choosing the alternative a

_{i}when the given scenario is s

_{j}.

_{i}

^{0}= (c

_{ik}− c

_{im})/(1 · c

_{im}+ Σµ

_{ij}· c

_{ij}+ 0 · c

_{ik})

_{ik}= max c

_{ij}and c

_{im}= min c

_{ij}with µ

_{ij}ϵ [0,1]

_{i}= λ

_{i}

_{1}· p

_{i}

_{1}+ … + λ

_{nn}· p

_{nn}

_{i}+ (max g

_{i}

^{0}− g

_{i}

^{0}) + 1/Σc

_{ij}

## 3. Results and Discussion

_{0}. Therefore, to apply GMUBO, three additional scenarios are going to be assumed in order to show how this new approach works.

_{1}, S

_{2}, and S

_{3}are considered, taking into account that these scenarios present different conditions that will modify the valuations given by the experts of the Navy. In the following points the characteristics and implications of each scenario are explained.

- Scenario S
_{1}. In this new scenario, the implementation of important logistical improvements in the Navy is assumed. This has produced a change in the valuations of the criteria and sub-criteria, which is presented in Table 3. The values given for “possibility of training” and “resupply needs” have changed. - Scenario S
_{2}. The existence of this scenario is based on the subjectivity of the values given to two of the sub-criteria of the operational capabilities criterion. This means that the sub-criteria “radius of action” and “constraints” are very variable. Therefore, the assessments given to them have been reviewed. The values are presented in Table 4. - Scenario S
_{3}. In this case, it is simply assumed that the weight of the logistics criterion is reduced by up to 15% and the operational capabilities criterion increases up to 85% (Table 5).

_{ij}represents the penalty incurred when choosing an alternative a

_{i}when a scenario s

_{j}has been given. Therefore, P becomes:

_{1}

^{0}= 1.4598

_{2}

^{0}= 1.6920

_{ij}in each column and 1 to the smallest. The other values are assigned from the linear interpolation between 0 and 1. Since it has the smallest uncertainty, alternative 1 should be chosen. However, according to the methodology exposed in Section 2, it is necessary to calculate the weighted sum to avoid high penalties. Then, calculating W

_{i}for each alternative:

_{1}= 0.0630

_{2}= 0.1070

_{i}, which would be alternative 2. Because both uncertainty (g

_{i}

^{0}) and weighted sum (W

_{i}) led to more than one alternative, then the alternative with the highest final sum FS must be selected:

_{1}= W

_{1}+ (max g

_{i}

^{0}− g

_{1}

^{0}) + 1/Σc

_{1j}= 3.2364

_{2}= W

_{2}+ (max g

_{i}

^{0}− g

_{2}

^{0}) + 1/Σc

_{2j}= 2.3342

## 4. Managerial Implications

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Difference between towed and expendable devices: (

**a**) a towed device is dragged by the warship; and (

**b**) an expendable device is launched from the warship.

**Figure 2.**Initial methodology versus the GMUBO methodology: (

**a**) Initially, neither the uncertainty nor different scenarios were considered; (

**b**) GMUBO considers the uncertainty since it foresees different scenarios that could occur.

**Figure 6.**Sensitivity analysis by considering different criteria: (

**a**) Sensitivity of the weight assigned to the logistics criterion; and (

**b**) sensitivity of the weight assigned to the operational capabilities criterion.

**Figure 7.**Graphical interface of GMUBO [17].

Goal | |||
---|---|---|---|

c_{1} | c_{2} | … | c_{n} |

sc_{11}… | sc_{21}… | … | sc_{n}_{1}… |

a_{1} | a_{2} | … | a_{m} |

Alternatives | Logistics (35%) | Operational Capabilities (65%) |
---|---|---|

N (49.5%) | 84% | 41% |

L (50.5%) | 16% | 59% |

Alternatives | Logistics (40%) | Operational Capabilities (60%) |
---|---|---|

N (55.4%) | 73.9% | 43.1% |

L (44.6%) | 26.1% | 56.9% |

Alternatives | Logistics (30%) | Operational Capabilities (70%) |
---|---|---|

N (59.6%) | 83.5% | 49.3% |

L (40.4%) | 16.5% | 50.7% |

Alternatives | Logistics (15%) | Operational Capabilities (85%) |
---|---|---|

N (42.0%) | 68.6% | 37.9% |

L (58.0%) | 31.4% | 62.1% |

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**MDPI and ACS Style**

Carreño, R.M.; Martínez, J.; Bouza, J.B. An Approach for Integrating Uncertainty When Selecting an Anti-Torpedo Decoy in Brand New Warships. *Math. Comput. Appl.* **2019**, *24*, 5.
https://doi.org/10.3390/mca24010005

**AMA Style**

Carreño RM, Martínez J, Bouza JB. An Approach for Integrating Uncertainty When Selecting an Anti-Torpedo Decoy in Brand New Warships. *Mathematical and Computational Applications*. 2019; 24(1):5.
https://doi.org/10.3390/mca24010005

**Chicago/Turabian Style**

Carreño, Rafael M., Javier Martínez, and José Benito Bouza. 2019. "An Approach for Integrating Uncertainty When Selecting an Anti-Torpedo Decoy in Brand New Warships" *Mathematical and Computational Applications* 24, no. 1: 5.
https://doi.org/10.3390/mca24010005