# Combining an Extended SMAA-2 Method with Integer Linear Programming for Task Assignment of Multi-UCAV under Multiple Uncertainties

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

- We propose a novel method to aggregate multiple criteria value matrices from different information sources (ISs) under the condition that the criteria weight information is unavailable and the weight information about ISs is partially available.
- We integrate the aggregation method into the original SMAA-2 method. Thus, we propose an extended SMAA-2 method.
- We propose to convert the task assignment model for multi-UCAV with a variety of uncertain parameters into an integer linear programming model to maximize the holistic acceptability index of the task assignment scheme according to the meaning of holistic acceptability indices, and then apply integer linear programming to achieve task assignment.

## 3. Problem Formulation

## 4. The Stochastic Multicriteria Acceptability Analysis-2 (SMAA-2) Method

## 5. The Extended SMAA-2 Method

#### 5.1. Normal Distribution Interval Number and Weighted Arithmetic Averaging Operator

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1.**

**Proof.**

#### 5.2. Aggregation Method of Criterion Value Matrices from Different Information Sources

#### 5.2.1. WAA Operator-Based Criteria Value Matrices Aggregation

**Definition**

**5.**

**Definition**

**6.**

#### 5.2.2. Iterative Algorithm for Computing Objective Weights

**Definition**

**7.**

^{−8}), and $\Delta \rho $ is computed as follows:

Algorithm 1: Pseudo-code of the aggregation method | |

Input: | number of alternatives, criteria, and information sources: $N$, $n$, $K$; |

criteria value matrices: ${S}^{\left(1\right)},{S}^{\left(2\right)},\cdots ,{S}^{\left(t\right)}$; | |

subjective weight vector $\mathsf{\eta}=\left[{\eta}_{1},{\eta}_{2},\cdots ,{\eta}_{K}\right]$ and initial objective weight vector, ${\mathsf{\rho}}^{\left(1\right)}=\left[1/K,1/K,\cdots ,1/K\right]$; | |

criteria weight vector $\mathsf{\omega}=\left[{\omega}_{1},{\omega}_{2},\cdots ,{\omega}_{n}\right]$; | |

error margin used in computing the objective weight vector: $\zeta $. | |

Output: | final criteria value matrix ${S}^{\left(0\right)}$. |

Procedures: | $l\leftarrow 1$ |

$\Delta {\rho}^{\left(l\right)}\leftarrow \infty $ | |

while$\Delta {\rho}^{\left(l\right)}\ge \zeta $do | |

compute $\mathsf{\lambda}$ by (33) | |

$l\leftarrow l+1$ | |

for $k\leftarrow 1$ to $K$ do | |

compute $Dev\left({S}^{\left(k\right)},{S}^{\left(0\right)}\right)$ by (22), (23), (31), (32) and (34) | |

end for | |

update ${\mathsf{\rho}}^{\left(l\right)}$ by (35) | |

update $\Delta {\rho}^{\left(l\right)}$ by (36) | |

end while | |

return${S}^{\left(0\right)}$ |

#### 5.3. Integrating the Aggregation Method into the Original SMAA-2 Method

Algorithm 2: Pseudo-code of the extended SMAA-2 method | ||

Symbols: | ${h}_{i}^{j}$, number of times alternative ${A}_{i}$ is evaluated into rank j in the Monte Carlo simulation; | |

${N}_{ite}$, number of iterations, set as 10000; | ||

$r=\left[{r}_{1},{r}_{2},\cdots ,{r}_{N}\right]$, vector of ranks of alternatives; | ||

$U=\left[{U}_{1},{U}_{2},\cdots ,{U}_{N}\right]$, utility value vector of alternatives; | ||

$Agg\left(\mathsf{\omega}\right)$, function returning the final criteria value matrix ${S}^{\left(0\right)}$ based on the criteria weight vector $\mathsf{\omega}$ (details are presented in Algorithm 1) | ||

$RAN{D}_{\mathsf{\Omega}}\left(S\right)$, function returning a random criteria value matrix from criteria distribution $f\left(\mathsf{\Omega}\right)$; | ||

$RAN{D}_{W}$, function returning a random weight vector from weight distribution $g\left(W\right)$; | ||

$RANK\left(U\right)$, function returning a vector of ranks corresponding to $U$. | ||

Input: | number of alternatives, criteria, and decision-makers: $N$, $n$, $K$; | |

criteria value matrices: ${S}^{\left(1\right)},{S}^{\left(2\right)},\cdots ,{S}^{\left(K\right)}$; | ||

subjective weight vector and initial objective weight vector $\mathsf{\eta}=\left[{\eta}_{1},{\eta}_{2},\cdots ,{\eta}_{K}\right]$, ${\mathsf{\rho}}^{\left(1\right)}=\left[1/K,1/K,\cdots ,1/K\right]$; | ||

error margin used in computing the objective weight vector: $\zeta $; | ||

number of iterations of the Monte Carlo simulation: ${N}_{ite}$. | ||

Output: | rank acceptability index ${b}_{i}^{r}$; | |

central weight vector ${\mathsf{\omega}}_{i}^{c}$; | ||

confidence factor ${p}_{i}^{c}$; | ||

holistic acceptability index ${e}_{i}$. | ||

Procedures: | // Initialize ${\mathsf{\omega}}_{i}^{c}$, ${h}_{i}^{j}$ and ${e}_{i}$for $i\leftarrow 1$ to $N$ do${\mathsf{\omega}}_{i}^{c}\leftarrow 0$ ${e}_{i}\leftarrow 0$ for $j\leftarrow 1$ to $N$ do${h}_{i}^{j}\leftarrow 0$ end forend for// Phase 1, compute ${b}_{i}^{r}$, ${\mathsf{\omega}}_{i}^{c}$ and ${e}_{i}$// Main loop for $k\leftarrow 1$ to ${N}_{ite}$ do$\mathsf{\omega}\leftarrow RAN{D}_{W}$ //produce ${S}^{\left(0\right)}$ by $Agg(\cdot )$ ${S}^{\left(0\right)}\leftarrow Agg\left(\mathsf{\omega}\right)$ //$S$ is the generated random criteria value matrix $S\leftarrow RAN{D}_{\mathsf{\Omega}}\left({S}^{\left(0\right)}\right)$ for $i\leftarrow 1$ to $N$ do${U}_{i}\leftarrow U\left({\mathsf{\xi}}_{i},\mathsf{\omega}\right)$ end for$r\leftarrow RANK\left(U\right)$ for $i\leftarrow 1$ to $N$ do${h}_{i}^{{r}_{i}}\leftarrow {h}_{i}^{{r}_{i}}+1$ if ${r}_{i}=1$ then${\mathsf{\omega}}_{i}^{c}\leftarrow {\mathsf{\omega}}_{i}^{c}+\mathsf{\omega}$ end ifend forend for// Compute ${\mathsf{\omega}}_{i}^{c}$, ${b}_{i}^{r}$ and ${e}_{i}$ for $i\leftarrow 1$ to $N$ doif ${h}_{i}^{1}>0$ then${\mathsf{\omega}}_{i}^{c}\leftarrow {\mathsf{\omega}}_{i}^{c}/{h}_{i}^{1}$ end iffor $j\leftarrow 1$ to $N$ do${b}_{i}^{j}\leftarrow {h}_{i}^{j}/{N}_{ite}$ ${e}_{i}\leftarrow {e}_{i}+{\alpha}_{j}{b}_{i}^{j}$ end forend for | // Phase 2, compute ${p}_{i}^{c}$// Initialize ${p}_{i}^{c}$, and generate the final criteria value matrix ${S}_{i}^{\left(0\right)}$ corresponding to ${\mathsf{\omega}}_{i}^{c}$ for $i\leftarrow 1$ to $N$ do${p}_{i}^{c}\leftarrow 0$ ${S}_{i}^{\left(0\right)}\leftarrow Agg\left({\mathsf{\omega}}_{i}^{c}\right)$ end for// Main loop for $j\leftarrow 1$ to ${N}_{ite}$ dofor $i\leftarrow 1$ to $N$ do$S\leftarrow RAN{D}_{\mathsf{\Omega}}\left({S}_{i}^{\left(0\right)}\right)$ ${U}_{i}=U\left({\mathsf{\xi}}_{i},{\mathsf{\omega}}_{i}^{c}\right)$ for $k\leftarrow 1$ to $N$ and $k\ne i$ doif $U\left({\mathsf{\xi}}_{k},{\mathsf{\omega}}_{i}^{c}\right)>{U}_{i}$ thengo to worseend ifend for${p}_{i}^{c}\leftarrow {p}_{i}^{c}+1$ worse: end forend forfor $i\leftarrow 1$ to $N$ do${p}_{i}^{c}\leftarrow {p}_{i}^{c}/{N}_{ite}$ end for |

## 6. Converting the Task Assignment Model Based on Holistic Acceptability Indices

## 7. Case Study

## 8. Conclusions

- Due to the integration of the WAA operator-based aggregation method, the extended method is able to perform stochastic multicriteria acceptability analysis under circumstances of multiple criteria value matrices. This is also the most prominent advantage of the extended method over the original SMAA-2 method.
- The extended method employs the iterative algorithm to compute the objective weight of a sensor based on the accuracy and credibility of the target information it provides so as to minimize the subjective judgment errors of experts on the sensor performance. By this means, the actual importance of a sensor can be reflected to the greatest extent so that the final criteria value matrix obtained could be as accurate as possible. Through these measures, the credibility and accuracy of task assignment schemes can be further increased.
- The foundation of the extended method is the stochastic multicriteria acceptability analysis, so it inherits the advantage of the SMAA technique in that it does not require decision-makers to express their preferences explicitly [30]. This property is particularly useful in operational planning because the commander could only express implicit preferences for multiple operational objectives most of the time.
- The extended method is based on the assumption that the uncertain target information can be represented by normal distribution interval numbers. This assumption may limit the application of the method. When the uncertain target information is represented by fuzzy sets or rough sets, the extended method cannot be applied.
- The extended method employs a linear utility value function to map criteria values to utility values. Although Lahdelma et al. have proved it may often be safe to apply a linear utility value function, they also mentioned that when the convexity or concavity of the utility value function is severe, the results would be slightly different [45]. This means that in some special cases, the linear utility value function may not be the best choice, and we would need to construct a more appropriate one.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviation

$UCA{V}_{i}$ | The ith UCAV |

${T}_{j}$ | The jth target |

${O}_{B}$, ${\omega}_{B}$ | Operational benefit and corresponding weight |

${O}_{C}$, ${\omega}_{C}$ | Operational cost and corresponding weight |

${x}_{ij}$ | Decision variables. If ${T}_{j}$ is assigned to $UCA{V}_{i}$, ${x}_{ij}=1$ and otherwise ${x}_{ij}=0$ |

$o\_{b}_{ij}^{\left(k\right)}$ | Operational benefit of $UCA{V}_{i}$ attacking ${T}_{j}$ |

$o\_{c}_{ij}^{\left(k\right)}$ | Operational cost of $UCA{V}_{i}$ attacking ${T}_{j}$ |

${w}_{k}$ | Weight of the kth sensor |

${v}_{j}^{\left(k\right)}$ | Value of ${T}_{j}$, provided by the kth senor |

${p}_{ij}$ | Lethality probability of missiles loaded on $UCA{V}_{i}$ against ${T}_{j}$ |

${q}_{i}$ | The number of missiles loaded on $UCA{V}_{i}$ |

${d}_{ij}^{\left(k\right)}$ | Distance between $UCA{V}_{i}$ and ${T}_{j}$, provided by the kth senor |

${c}_{ij}^{fly}$ | Flight cost per unit distance from $UCA{V}_{i}$ and ${T}_{j}$ |

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**Figure 2.**The operational process (four unmanned combat aerial vehicles (UCAVs) attack four targets).

**Figure 5.**The holistic acceptability index matrices when (

**a**) ${\omega}_{B}\ge {\omega}_{C}$ and (

**b**) ${\omega}_{B}<{\omega}_{C}$.

**Figure 6.**Task assignment diagrams when (

**a**) ${\omega}_{B}\ge {\omega}_{C}$ and (

**b**) ${\omega}_{B}<{\omega}_{C}$.

Sensors | Target Value $({\mathit{v}}_{\mathit{j}}=[{\mathit{v}}_{\mathit{j}}^{-},{\mathit{v}}_{\mathit{j}}^{+}])$ | $\mathbf{Flight}\text{}\mathbf{Distance}\text{}\mathbf{between}\text{}\mathit{U}\mathit{C}\mathit{A}{\mathit{V}}_{\mathit{i}}$$\text{}\mathbf{and}\text{}{\mathit{T}}_{\mathit{j}}$$({\mathit{d}}_{\mathit{i}\mathit{j}}=[{\mathit{d}}_{\mathit{i}\mathit{j}}^{-},{\mathit{d}}_{\mathit{i}\mathit{j}}^{+}]$$,\text{}\mathit{i}=1,2,3,4;\mathit{j}=1,2,3,4)$ |
---|---|---|

Sensor 1 | $\begin{array}{c}\left[7.0,9.0\right]\\ \left[6.0,8.0\right]\\ \left[3.0,7.0\right]\\ \left[2.0,5.0\right]\end{array}$ | $\begin{array}{cccc}\left[725,762\right]& \left[721,755\right]& \left[595,637\right]& \left[612,651\right]\\ \left[751,792\right]& \left[720,759\right]& \left[672,715\right]& \left[661,704\right]\\ \left[642,684\right]& \left[567,609\right]& \left[666,703\right]& \left[608,647\right]\\ \left[630,672\right]& \left[578,620\right]& \left[602,642\right]& \left[565,607\right]\end{array}$ |

Sensor 2 | $\begin{array}{c}\left[6.5,8.5\right]\\ \left[4.0,7.5\right]\\ \left[5.0,8.0\right]\\ \left[4.0,7.0\right]\end{array}$ | $\begin{array}{cccc}\left[708,745\right]& \left[738,771\right]& \left[614,655\right]& \left[602,641\right]\\ \left[731,772\right]& \left[732,771\right]& \left[690,732\right]& \left[650,692\right]\\ \left[619,660\right]& \left[570,613\right]& \left[679,716\right]& \left[597,636\right]\\ \left[608,650\right]& \left[587,628\right]& \left[617,658\right]& \left[554,596\right]\end{array}$ |

Sensor 3 | $\begin{array}{c}\left[5.0,9.0\right]\\ \left[5.0,8.5\right]\\ \left[4.0,9.0\right]\\ \left[5.0,8.0\right]\end{array}$ | $\begin{array}{cccc}\left[715,752\right]& \left[709,742\right]& \left[597,638\right]& \left[611,650\right]\\ \left[739,780\right]& \left[703,742\right]& \left[671,713\right]& \left[661,703\right]\\ \left[627,669\right]& \left[545,588\right]& \left[658,695\right]& \left[609,648\right]\\ \left[616,658\right]& \left[559,601\right]& \left[596,637\right]& \left[565,607\right]\end{array}$ |

Sensor 4 | $\begin{array}{c}\left[6.0,8.0\right]\\ \left[4.0,6.5\right]\\ \left[3.0,8.5\right]\\ \left[4.5,7.5\right]\end{array}$ | $\begin{array}{cccc}\left[728,765\right]& \left[733,767\right]& \left[610,651\right]& \left[621,660\right]\\ \left[749,790\right]& \left[733,772\right]& \left[683,726\right]& \left[665,707\right]\\ \left[630,671\right]& \left[581,623\right]& \left[667,705\right]& \left[602,642\right]\\ \left[623,665\right]& \left[592,634\right]& \left[607,648\right]& \left[564,606\right]\end{array}$ |

UCAVs | $\mathbf{No}.\text{}\mathbf{of}\text{}\mathbf{Missiles}\text{}\mathbf{on}\text{}\mathit{U}\mathit{C}\mathit{A}{\mathit{V}}_{\mathit{i}}$$\left({\mathit{q}}_{\mathit{i}}\right)$ | $\mathbf{Flight}\text{}\mathbf{Cost}\text{}\mathbf{Per}\text{}\mathbf{Unit}\text{}\mathbf{Distance}\text{}\mathbf{of}\text{}\mathit{U}\mathit{C}\mathit{A}{\mathit{V}}_{\mathit{i}}$$\left({\mathit{c}}_{\mathit{i}\mathit{j}}^{\mathit{f}\mathit{l}\mathit{y}}\right)$ | $\mathbf{Lethality}\text{}\mathbf{Probability}\text{}\mathbf{of}\text{}\mathbf{Missiles}\text{}\mathbf{on}\text{}\mathit{U}\mathit{C}\mathit{A}{\mathit{V}}_{\mathit{i}}$$\text{}\mathbf{against}\text{}{\mathit{T}}_{\mathit{j}}$$\left({\mathit{p}}_{\mathit{i}\mathit{j}}\right)$ |
---|---|---|---|

$UCA{V}_{1}$ | 6 | $\left[\begin{array}{cccc}5& 10& 5& 6\\ 4& 6& 6& 8\\ 4& 7& 10& 8\\ 9& 9& 8& 9\end{array}\right]$ | $\left[\begin{array}{cccc}0.82& 0.63& 0.86& 0.68\\ 0.91& 0.47& 0.49& 0.72\\ 0.46& 0.78& 0.59& 0.55\\ 0.58& 0.57& 0.71& 0.49\end{array}\right]$ |

$UCA{V}_{2}$ | 8 | ||

$UCA{V}_{3}$ | 6 | ||

$UCA{V}_{4}$ | 6 |

Parameters | Subjective Weight Vector of Sensors $\left(\mathsf{\eta}\right)$ | Initial Objective Weight Vector of Sensors $({\mathsf{\rho}}^{\left(1\right)})$ | Error Margin $\left(\mathit{\zeta}\right)$ | Adjustment Factor $\left(\mathit{\tau}\right)$ |
---|---|---|---|---|

Value | [0.20, 0.10, 0.35, 0.35] | [0.25, 0.25, 0.25, 0.25] | 10^{−8} | 0.5 |

Sensors | Target Value $({\mathit{v}}_{\mathit{j}}=\left({\mathit{\mu}}_{\mathit{j}},{\mathit{\sigma}}_{\mathit{j}}\right))$ | $\mathbf{Flight}\text{}\mathbf{Distance}\text{}\mathbf{between}\text{}\mathit{U}\mathit{C}\mathit{A}{\mathit{V}}_{\mathit{i}}$$\text{}\mathbf{and}\text{}{\mathit{T}}_{\mathit{j}}$$({\mathit{d}}_{\mathit{i}\mathit{j}}=\left({\mathit{\mu}}_{\mathit{i}\mathit{j}},{\mathit{\sigma}}_{\mathit{i}\mathit{j}}\right)$$,\text{}\mathit{i}=1,2,3,4;\mathit{j}=1,2,3,4)$ |
---|---|---|

Sensor 1 | $\begin{array}{c}\left(8.00,0.33\right)\\ \left(7.00,0.33\right)\\ \left(5.00,0.67\right)\\ \left(3.50,0.50\right)\end{array}$ | $\begin{array}{cccc}\left(743.4,6.26\right)& \left(738.3,5.69\right)& \left(615.9,6.90\right)& \left(631.6,6.57\right)\\ \left(771.8,6.87\right)& \left(739.3,6.56\right)& \left(693.5,7.07\right)& \left(682.5,7.03\right)\\ \left(662.9,6.94\right)& \left(588.0,7.06\right)& \left(684.2,6.14\right)& \left(627.7,6.53\right)\\ \left(651.2,7.06\right)& \left(599.1,6.93\right)& \left(621.8,6.75\right)& \left(586.0,7.00\right)\end{array}$ |

Sensor 2 | $\begin{array}{c}\left(7.50,0.33\right)\\ \left(5.75,0.53\right)\\ \left(6.50,0.50\right)\\ \left(5.50,0.50\right)\end{array}$ | $\begin{array}{cccc}\left(726.2,6.18\right)& \left(754.5,5.61\right)& \left(634.2,6.70\right)& \left(621.7,6.55\right)\\ \left(751.2,6.84\right)& \left(751.4,6.49\right)& \left(711.0,7.07\right)& \left(671.4,7.03\right)\\ \left(639.6,6.94\right)& \left(591.3,7.07\right)& \left(697.2,6.20\right)& \left(616.8,6.52\right)\\ \left(628.8,7.06\right)& \left(607.8,6.88\right)& \left(637.3,6.79\right)& \left(574.6,7.00\right)\end{array}$ |

Sensor 3 | $\begin{array}{c}\left(7.00,0.67\right)\\ \left(6.75,0.58\right)\\ \left(6.50,0.83\right)\\ \left(6.50,0.50\right)\end{array}$ | $\begin{array}{cccc}\left(733.6,6.20\right)& \left(725.7,5.58\right)& \left(617.4,6.87\right)& \left(630.6,6.58\right)\\ \left(759.4,6.85\right)& \left(722.4,6.51\right)& \left(691.7,7.07\right)& \left(682.2,7.04\right)\\ \left(647.9,6.95\right)& \left(566.4,7.06\right)& \left(676.7,6.19\right)& \left(628.8,6.52\right)\\ \left(637.3,7.06\right)& \left(579.8,6.90\right)& \left(616.8,6.79\right)& \left(586.4,6.99\right)\end{array}$ |

Sensor 4 | $\begin{array}{c}\left(7.00,0.33\right)\\ \left(5.25,0.42\right)\\ \left(5.75,0.92\right)\\ \left(6.00,0.50\right)\end{array}$ | $\begin{array}{cccc}\left(746.8,6.15\right)& \left(749.8,5.74\right)& \left(630.8,6.86\right)& \left(640.5,6.49\right)\\ \left(769.3,6.81\right)& \left(752.4,6.58\right)& \left(704.4,7.07\right)& \left(686.2,7.01\right)\\ \left(650.5,6.98\right)& \left(602.1,7.06\right)& \left(685.9,6.24\right)& \left(621.7,6.61\right)\\ \left(644.1,7.05\right)& \left(613.0,6.93\right)& \left(627.9,6.82\right)& \left(584.9,7.02\right)\end{array}$ |

${\overline{\mathbf{S}}}^{\left(\mathit{i},\mathit{k}\right)}$ | Sensor 1 | Sensor 2 | Sensor 3 | Sensor 4 |
---|---|---|---|---|

UCAV1 | $\begin{array}{ll}(1.000,0.042)\hfill & (0.829,0.004)\hfill \\ (0.873,0.042)\hfill & (0.417,0.008)\hfill \\ (0.625,0.083)\hfill & (1.000,0.005)\hfill \\ (0.637,0.062)\hfill & (0.813,0.005)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.044)\hfill & (0.873,0.004)\hfill \\ (0.765,0.078)\hfill & (0.420,0.007)\hfill \\ (0.867,0.067)\hfill & (1.000,0.005)\hfill \\ (0.733,0.067)\hfill & (0.850,0.005)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.095)\hfill & (0.842,0.004)\hfill \\ (0.962,0.083)\hfill & (0.425,0.008)\hfill \\ (0.929,0.119)\hfill & (1.000,0.005)\hfill \\ (0.928,0.071)\hfill & (0.816,0.005)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.048)\hfill & (0.845,0.004)\hfill \\ (0.748,0.059)\hfill & (0.421,0.008)\hfill \\ (0.822,0.131)\hfill & (1.000,0.005)\hfill \\ (0.856,0.071)\hfill & (0.821,0.005)\hfill \end{array}$ |

UCAV2 | $\begin{array}{ll}(1.000,0.042)\hfill & (1.000,0.005)\hfill \\ (0.870,0.041)\hfill & (0.696,0.007)\hfill \\ (0.622,0.083)\hfill & (0.742,0.008)\hfill \\ (0.438,0.063)\hfill & (0.565,0.010)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.044)\hfill & (1.000,0.005)\hfill \\ (0.762,0.077)\hfill & (0.667,0.007)\hfill \\ (0.863,0.066)\hfill & (0.704,0.008)\hfill \\ (0.733,0.067)\hfill & (0.559,0.011)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.095)\hfill & (1.000,0.005)\hfill \\ (0.958,0.083)\hfill & (0.701,0.007)\hfill \\ (0.924,0.119)\hfill & (0.732,0.008)\hfill \\ (0.929,0.071)\hfill & (0.557,0.010)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.048)\hfill & (1.000,0.005)\hfill \\ (0.745,0.059)\hfill & (0.682,0.007)\hfill \\ (0.818,0.130)\hfill & (0.728,0.008)\hfill \\ (0.857,0.071)\hfill & (0.561,0.010)\hfill \end{array}$ |

UCAV3 | $\begin{array}{ll}(1.000,0.042)\hfill & (1.000,0.004)\hfill \\ (0.897,0.043)\hfill & (0.644,0.007)\hfill \\ (0.638,0.085)\hfill & (0.388,0.009)\hfill \\ (0.445,0.094)\hfill & (0.528,0.008)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.044)\hfill & (1.000,0.004)\hfill \\ (0.786,0.080)\hfill & (0.618,0.007)\hfill \\ (0.885,0.068)\hfill & (0.367,0.009)\hfill \\ (0.746,0.068)\hfill & (1.000,0.008)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.095)\hfill & (1.000,0.004)\hfill \\ (0.989,0.085)\hfill & (0.654,0.007)\hfill \\ (0.948,0.122)\hfill & (0.383,0.009)\hfill \\ (0.944,0.073)\hfill & (1.000,0.008)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.048)\hfill & (1.000,0.004)\hfill \\ (0.769,0.061)\hfill & (0.617,0.007)\hfill \\ (0.838,0.134)\hfill & (0.379,0.009)\hfill \\ (0.872,0.073)\hfill & (0.523,0.008)\hfill \end{array}$ |

UCAV4 | $\begin{array}{ll}(1.000,0.042)\hfill & (0.849,0.011)\hfill \\ (0.874,0.042)\hfill & (0.923,0.011)\hfill \\ (0.628,0.084)\hfill & (1.000,0.009)\hfill \\ (0.432,0.062)\hfill & (0.943,0.011)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.044)\hfill & (0.901,0.011)\hfill \\ (0.766,0.078)\hfill & (0.932,0.011)\hfill \\ (0.871,0.067)\hfill & (1.000,0.010)\hfill \\ (0.724,0.066)\hfill & (0.986,0.011)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.095)\hfill & (0.860,0.011)\hfill \\ (0.964,0.083)\hfill & (0.946,0.011)\hfill \\ (0.933,0.120)\hfill & (1.000,0.010)\hfill \\ (0.917,0.071)\hfill & (0.935,0.011)\hfill \end{array}$ | $\begin{array}{ll}(1.000,0.048)\hfill & (0.867,0.011)\hfill \\ (0.745,0.060)\hfill & (0.911,0.011)\hfill \\ (0.826,0.132)\hfill & (1.000,0.009)\hfill \\ (0.847,0.071)\hfill & (0.954,0.011)\hfill \end{array}$ |

**Table 6.**The holistic acceptability indices when ${\omega}_{B}\ge {\omega}_{C}$ and ${\omega}_{B}<{\omega}_{C}$.

${\mathit{e}}_{\mathit{i}\mathit{j}}$ | ${\mathit{\omega}}_{\mathit{B}}\ge {\mathit{\omega}}_{\mathit{C}}$ | ${\mathit{e}}_{\mathit{i}\mathit{j}}$ | ${\mathit{\omega}}_{\mathit{B}}<{\mathit{\omega}}_{\mathit{C}}$ | ||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ | ||

$UCA{V}_{1}$ | 0.9664 | 0.1899 | 0.5141 | 0.2496 | $UCA{V}_{1}$ | 0.5689 | 0.1200 | 0.9511 | 0.2800 |

$UCA{V}_{2}$ | 0.9999 | 0.4003 | 0.3808 | 0.1390 | $UCA{V}_{2}$ | 1.0000 | 0.3094 | 0.4906 | 0.1200 |

$UCA{V}_{3}$ | 0.9996 | 0.4901 | 0.2272 | 0.2031 | $UCA{V}_{3}$ | 1.0000 | 0.5200 | 0.1216 | 0.2784 |

$UCA{V}_{4}$ | 0.9724 | 0.3728 | 0.4219 | 0.1530 | $UCA{V}_{4}$ | 0.3646 | 0.3100 | 0.9176 | 0.3278 |

**Table 7.**Task assignment schemes when ${\omega}_{B}\ge {\omega}_{C}$ and ${\omega}_{B}<{\omega}_{C}$.${x}_{ij}$.

${\mathit{\omega}}_{\mathit{B}}\ge {\mathit{\omega}}_{\mathit{C}}$ | ${\mathit{x}}_{\mathit{i}\mathit{j}}$ | ${\mathit{\omega}}_{\mathit{B}}<{\mathit{\omega}}_{\mathit{C}}$ | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ | ${\mathit{T}}_{1}$ | ${\mathit{T}}_{2}$ | ${\mathit{T}}_{3}$ | ${\mathit{T}}_{4}$ | ||

$UCA{V}_{1}$ | 0 | 0 | 0 | 1 | $UCA{V}_{1}$ | 0 | 0 | 1 | 0 |

$UCA{V}_{2}$ | 1 | 0 | 0 | 0 | $UCA{V}_{2}$ | 1 | 0 | 0 | 0 |

$UCA{V}_{3}$ | 0 | 1 | 0 | 0 | $UCA{V}_{3}$ | 0 | 1 | 0 | 0 |

$UCA{V}_{4}$ | 0 | 0 | 1 | 0 | $UCA{V}_{4}$ | 0 | 0 | 0 | 1 |

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## Share and Cite

**MDPI and ACS Style**

Wang, J.; Luo, P.; Hu, X.; Zhang, X.
Combining an Extended SMAA-2 Method with Integer Linear Programming for Task Assignment of Multi-UCAV under Multiple Uncertainties. *Symmetry* **2018**, *10*, 587.
https://doi.org/10.3390/sym10110587

**AMA Style**

Wang J, Luo P, Hu X, Zhang X.
Combining an Extended SMAA-2 Method with Integer Linear Programming for Task Assignment of Multi-UCAV under Multiple Uncertainties. *Symmetry*. 2018; 10(11):587.
https://doi.org/10.3390/sym10110587

**Chicago/Turabian Style**

Wang, Jun, Pengcheng Luo, Xinwu Hu, and Xiaonan Zhang.
2018. "Combining an Extended SMAA-2 Method with Integer Linear Programming for Task Assignment of Multi-UCAV under Multiple Uncertainties" *Symmetry* 10, no. 11: 587.
https://doi.org/10.3390/sym10110587