# A Two-Stage Approach for Routing Multiple Unmanned Aerial Vehicles with Stochastic Fuel Consumption

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Contributions

## 2. Notation

## 3. Mathematical Formulation

#### 3.1. Objective Function

#### 3.2. First-Stage Routing Constraints

#### 3.3. Second-Stage Constraints

#### 3.4. Tightening the Two-Stage Stochastic Formulation

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 4. Solution Approach: Sample Average Approximation (SAA)

#### 4.1. Statistical Estimate of a Lower Bound for ${C}^{\ast}$

#### 4.2. Statistical Estimate of an Upper Bound for ${C}^{\ast}$

## 5. Heuristic Solution Technique

#### 5.1. Construction Heuristic

Algorithm 1 Construction heuristic | |

Input: the instance data, $\Delta \subset \Omega $ | |

Output: a heuristic solution ${\mathit{x}}^{h}$ | |

1: for $\omega \in \Delta $ do | ▹ Parallelizable for each $\omega \in \Delta $ |

2: solve the deterministic FCMURP with fuel consumption values set to $\mathit{f}(\omega )$ | |

3: ${\mathit{x}}^{\omega}\leftarrow $ optimal solution of the deterministic problem | |

4: end for | |

5: for $(i,j)\in E$ do | |

6: ${d}_{ij}\leftarrow 1-{\sum}_{\omega \in \Delta}p(\omega ){x}_{ij}^{\omega}$ | ▹ weighting function |

7: ${\overline{c}}_{ij}\leftarrow {c}_{ij}{d}_{ij}$ | ▹ travel-cost transformation |

8: ${\overline{f}}_{ij}\leftarrow {\sum}_{\omega \in \Delta}p(\omega ){f}_{ij}(\omega )$ | ▹ fuel consumption update |

9: end for | |

10: ${\mathit{x}}^{h}\leftarrow $ optimal solution of deterministic FCMURP solved with ${\overline{c}}_{ij}$ and ${\overline{f}}_{ij}$ |

#### 5.2. Improvement Heuristic: Tabu Search Method

Algorithm 2 Tabu search heuristic. | |

Input: initial feasible solution
$\overline{\mathit{x}}$ | |

Output: a heuristic solution ${\mathit{x}}^{\ast}$ | |

1: set ${\mathit{x}}^{\ast}:=\overline{\mathit{x}}$, $C({\mathit{x}}^{\ast}):=C(\overline{\mathit{x}})$, $k:=1$, $\Xi :=\varnothing $, and stop := 0 | ▹$\Xi $ - tabu list |

2: while $k\u2a7d\theta $ and stop = 0 do | |

3: generate $g(\overline{\mathit{x}})$ | ▹ neighborhood generation |

4: choose ${\mathit{x}}^{\prime}\in g(\overline{\mathit{x}})$ that satisfies $C({\mathit{x}}^{\prime})<C(\overline{\mathit{x}})$ and is not in tabu list $\Xi $ or satisfies aspiration criteria | |

5: if ${\mathit{x}}^{\prime}=\varnothing $ then | |

6: choose a solution ${\mathit{x}}^{\prime}=argmin\{C(\mathit{x}):\mathit{x}\in g(\overline{\mathit{x}}),{x}_{i,j}\notin \Xi \}$ | ▹ best non-tabu solution |

7: end if | |

8: if ${\mathit{x}}^{\prime}$ is feasible and $C({\mathit{x}}^{\prime})<C({\mathit{x}}^{\ast})$ then | ▹ feasibility check |

9: set ${\mathit{x}}^{\ast}:={\mathit{x}}^{\prime}$ and $C({\mathit{x}}^{\ast}):=C({\mathit{x}}^{\prime})$ | |

10: end if | |

11: if ${\mathit{x}}^{\ast}$ is not updated for $\sqrt{k}$ iterations then | |

12: set $\overline{\mathit{x}}={\mathit{x}}^{\ast}$, $C(\overline{\mathit{x}}):=C({\mathit{x}}^{\ast})$ | ▹ reset neighborhood search |

13: else | |

14: set $\overline{\mathit{x}}:={\mathit{x}}^{\prime}$ and $C(\overline{\mathit{x}}):=C({\mathit{x}}^{\prime})$ | |

15: end if | |

16: update $\Xi $ based on $\varrho $ | ▹ tabu list update |

17: if ${\mathit{x}}^{\ast}$ is not updated for $\tau $ iterations then | ▹ secondary termination criterion |

18: set stop := 1 | |

19: end if | |

20: set $k:=k+1$ | |

21: end while |

## 6. Computational Results

#### 6.1. Instance Generation

#### 6.2. Parameters for SAA and the Heuristic

#### 6.3. Performance Evaluation Criteria

#### 6.4. Results for Small-Scale Test Instances

#### 6.5. Results for Large-Scale Instances

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of a feasible first-stage solution and recourse action for a particular realization of the uncertainty. (

**a**) A feasible first-stage solution to the two-stage FCMURP with two UAVs; the first-stage routes for the UAVs correspond to the ‘here-and-now’ decisions. (

**b**) A feasible recourse action for the two-stage FCMURP with two UAVs; for a given realization of the fuel consumed by any vehicle to traverse an edge, the refuel trips (colored in brown) are added to the first-stage solution.

**Figure 2.**Histogram of run times to compute the optimal solution of each SAA instance for the small-scale problems.

**Figure 4.**Box plot of the heuristic run times for the second set of instances. The mean run times, in seconds, for m = 2, m = 3, and m = 4 for $\left|T\right|\in \{20,30\}$ are given by $(243,310)$, $(275,354)$, and $(327,360)$, respectively.

Symbol | Explanation |
---|---|

$T=\{{t}_{1},\cdots ,{t}_{n}\}$ | set of n targets |

${d}_{0}$ | depot where m UAVs are initially stationed |

$\phantom{\rule{0.83328pt}{0ex}}\overline{\phantom{\rule{-0.83328pt}{0ex}}D\phantom{\rule{-0.83328pt}{0ex}}}\phantom{\rule{0.83328pt}{0ex}}=\{{d}_{1},\cdots ,{d}_{k}\}$ | additional depots/refueling sites |

$D=\phantom{\rule{0.83328pt}{0ex}}\overline{\phantom{\rule{-0.83328pt}{0ex}}D\phantom{\rule{-0.83328pt}{0ex}}}\phantom{\rule{0.83328pt}{0ex}}\bigcup \left\{{d}_{0}\right\}$ | set of depots |

F | fuel capacity of each UAV |

$G=(V,E)$ | directed graph with $V=T\cup D$ on which FCMURP is formulated |

${c}_{ij}$ | travel cost for the edge $(i,j)\in E$ |

${\widehat{f}}_{ij}$ | nominal fuel consumed by the UAV to traverse the edge $(i,j)\in E$ |

$\mathit{f}$ | random variable vector representing the fuel consumed by any UAV |

$\Omega $ | set of scenarios for $\mathit{f}$ |

$\omega \in \Omega $ | realization of random variable $\mathit{f}$ |

$p(\omega )$ | probability of occurrence of $\omega $ |

$\mathbb{E}$ | expectation operator |

**Table 2.**Computational results for the small-scale instances. The values in brackets for the columns EEV, LB-SAA, UB-SAA, and H denote the standard deviation of these estimates.

# | EV | EEV | LB-SAA | UB-SAA | H |
---|---|---|---|---|---|

1 | 391.00 | 513.20 (3.36) | 451.70 (1.70) | 455.24 (1.74) | 477.68 (2.24) |

2 | 432.00 | 559.25 (1.79) | 503.34 (2.04) | 504.65 (4.01) | 526.44 (4.59) |

3 | 429.00 | 578.77 (2.11) | 553.53 (1.62) | 564.70 (2.74) | 558.95 (2.97) |

4 | 396.00 | 483.50 (2.69) | 443.50 (2.56) | 446.31 (2.74) | 449.05 (3.58) |

5 | 405.00 | 541.33 (2.07) | 487.00 (1.62) | 490.60 (2.38) | 489.38 (2.64) |

F | $\left|\mathit{T}\right|=20$ | $\left|\mathit{T}\right|=30$ | ||||||
---|---|---|---|---|---|---|---|---|

# | EV | EEV | H | # | EV | EEV | H | |

$2.25\lambda $ | 1 | 442.00 | 510.12 (1.87) | 479.63 (1.58) | 1 | 575.00 | 642.95 (2.69) | 596.00 (3.06) |

$2.25\lambda $ | 2 | 467.00 | 564.83 (1.28) | 492.50 (1.04) | 2 | 549.00 | 603.40 (2.60) | 585.00 (3.43) |

$2.25\lambda $ | 3 | 442.00 | 511.65 (1.72) | 483.75 (2.24) | 3 | 549.00 | 576.00 (2.51) | 566.00 (2.51) |

$2.25\lambda $ | 4 | 482.00 | 506.00 (1.47) | 494.00 (0.53) | 4 | 530.00 | 586.00 (2.42) | 561.00 (2.71) |

$2.25\lambda $ | 5 | 471.00 | 578.57 (1.40) | 521.15 (0.71) | 5 | 541.00 | 610.00 (2.33) | 588.00 (3.22) |

$2.50\lambda $ | 1 | 467.00 | 565.45 (0.72) | 489.00 (0.96) | 1 | 524.00 | 615.30 (2.24) | 547.00 (2.69) |

$2.50\lambda $ | 2 | 442.00 | 512.15 (1.16) | 482.25 (0.25) | 2 | 516.00 | 600.00 (2.15) | 560.00 (2.22) |

$2.50\lambda $ | 3 | 442.00 | 502.73 (2.68) | 484.45 (1.81) | 3 | 505.00 | 584.30 (2.06) | 535.00 (2.78) |

$2.50\lambda $ | 4 | 471.00 | 588.17 (3.31) | 529.63 (3.12) | 4 | 539.00 | 611.00 (1.97) | 604.00 (2.66) |

$2.50\lambda $ | 5 | 467.00 | 564.18 (0.83) | 491.25 (0.43) | 5 | 534.00 | 606.00 (1.88) | 585.00 (1.95) |

$2.75\lambda $ | 1 | 442.00 | 512.77 (2.47) | 481.63 (2.83) | 1 | 523.00 | 584.00 (1.79) | 540.00 (2.18) |

$2.75\lambda $ | 2 | 442.00 | 493.58 (3.04) | 485.68 (2.93) | 2 | 518.00 | 573.00 (1.70) | 539.90 (2.55) |

$2.75\lambda $ | 3 | 487.00 | 587.85 (2.01) | 544.00 (3.70) | 3 | 546.00 | 580.00 (1.61) | 576.00 (2.18) |

$2.75\lambda $ | 4 | 485.00 | 584.98 (0.90) | 516.25 (1.25) | 4 | 542.00 | 587.00 (1.52) | 571.00 (1.97) |

$2.75\lambda $ | 5 | 460.00 | 523.65 (1.02) | 512.20 (4.13) | 5 | 509.00 | 555.00 (1.43) | 535.00 (2.37) |

$3.00\lambda $ | 1 | 505.00 | 584.85 (7.45) | 545.40 (1.59) | 1 | 509.00 | 543.00 (1.34) | 539.00 (1.87) |

$3.00\lambda $ | 2 | 438.00 | 534.05 (5.3) | 476.28 (2.95) | 2 | 562.00 | 650.00 (1.25) | 629.55 (1.80) |

$3.00\lambda $ | 3 | 433.00 | 509.52 (7.34) | 473.73 (4.59) | 3 | 540.00 | 618.30 (1.16) | 579.00 (1.17) |

$3.00\lambda $ | 4 | 442.00 | 513.62 (1.83) | 477.12 (2.02) | 4 | 525.00 | 598.30 (1.07) | 540.00 (1.12) |

$3.00\lambda $ | 5 | 451.00 | 534.68 (4.86) | 505.18 (8.74) | 5 | 505.00 | 586.00 (0.98) | 580.00 (1.50) |

F | $\left|\mathit{T}\right|=20$ | $\left|\mathit{T}\right|=30$ | ||||||
---|---|---|---|---|---|---|---|---|

# | EV | EEV | H | # | EV | EEV | H | |

$2.25\lambda $ | 1 | 460.00 | 525.10 (3.33) | 513.48 (4.33) | 1 | 502.00 | 597.03 (2.63) | 538.23 (3.53) |

$2.25\lambda $ | 2 | 485.00 | 587.18 (1.20) | 528.75 (1.64) | 2 | 477.00 | 550.23 (2.89) | 524.85 (3.87) |

$2.25\lambda $ | 3 | 460.00 | 526.27 (1.50) | 505.53 (1.01) | 3 | 561.00 | 706.97 (3.71) | 673.95 (3.17) |

$2.25\lambda $ | 4 | 460.00 | 517.65 (2.51) | 495.00 (3.05) | 4 | 542.00 | 598.00 (1.85) | 574.00 (2.23) |

$2.25\lambda $ | 5 | 552.00 | 775.90 (1.69) | 637.38 (0.89) | 5 | 552.00 | 729.65 (2.39) | 634.43 (2.03) |

$2.50\lambda $ | 1 | 497.00 | 628.82 (1.50) | 598.43 (0.92) | 1 | 539.00 | 688.65 (2.24) | 602.00 (2.64) |

$2.50\lambda $ | 2 | 460.00 | 524.38 (1.05) | 507.60 (1.88) | 2 | 527.00 | 619.00 (1.21) | 595.40 (1.31) |

$2.50\lambda $ | 3 | 492.00 | 605.67 (2.91) | 549.73 (2.07) | 3 | 524.00 | 604.92 (1.29) | 577.27 (1.59) |

$2.50\lambda $ | 4 | 486.00 | 620.45 (1.98) | 563.45 (2.52) | 4 | 575.00 | 713.32 (3.88) | 641.48 (3.35) |

$2.50\lambda $ | 5 | 486.00 | 608.18 (2.00) | 587.20 (1.99) | 5 | 568.00 | 669.62 (3.39) | 644.20 (2.67) |

$2.75\lambda $ | 1 | 460.00 | 526.30 (0.75) | 504.95 (1.20) | 1 | 566.00 | 651.95 (2.23) | 638.00 (2.38) |

$2.75\lambda $ | 2 | 460.00 | 515.05 (2.24) | 499.23 (2.23) | 2 | 556.00 | 604.00 (1.52) | 589.00 (2.26) |

$2.75\lambda $ | 3 | 487.00 | 587.85 (2.01) | 544.00 (2.43) | 3 | 588.00 | 702.30 (4.77) | 660.80 (4.76) |

$2.75\lambda $ | 4 | 485.00 | 584.98 (0.90) | 516.25 (1.46) | 4 | 567.00 | 657.38 (4.29) | 627.53 (3.31) |

$2.75\lambda $ | 5 | 460.00 | 523.65 (1.02) | 512.20 (1.57) | 5 | 549.00 | 660.47 (1.71) | 622.05 (1.69) |

$3.00\lambda $ | 1 | 460.00 | 515.15 (1.16) | 490.12 (0.75) | 1 | 549.00 | 605.00 (3.10) | 597.00 (2.38) |

$3.00\lambda $ | 2 | 438.00 | 534.05 (1.29) | 476.28 (1.90) | 2 | 574.00 | 781.55 (3.11) | 675.17 (3.73) |

$3.00\lambda $ | 3 | 433.00 | 509.52 (1.42) | 473.73 (0.80) | 3 | 558.00 | 674.20 (2.05) | 642.08 (1.24) |

$3.00\lambda $ | 4 | 460.00 | 526.55 (0.49) | 506.75 (0.42) | 4 | 543.00 | 654.70 (3.37) | 632.23 (2.48) |

$3.00\lambda $ | 5 | 460.00 | 514.40 (1.49) | 499.75 (0.77) | 5 | 511.00 | 586.32 (0.87) | 545.30 (0.05) |

F | $\left|\mathit{T}\right|=20$ | $\left|\mathit{T}\right|=30$ | ||||||
---|---|---|---|---|---|---|---|---|

# | EV | EEV | H | # | EV | EEV | H | |

$2.25\lambda $ | 1 | 480.00 | 551.45 (3.65) | 546.25 (2.71) | 1 | 618.00 | 784.63 (5.31) | 720.20 (5.59) |

$2.25\lambda $ | 2 | 505.00 | 568.72 (2.91) | 544.92 (2.21) | 2 | 583.00 | 698.80 (3.51) | 670.95 (2.69) |

$2.25\lambda $ | 3 | 480.00 | 549.62 (0.89) | 543.00 (0.50) | 3 | 583.00 | 657.60 (2.02) | 645.95 (2.98) |

$2.25\lambda $ | 4 | 520.00 | 544.00 (2.09) | 534.00 (1.83) | 4 | 566.00 | 633.15 (3.32) | 605.00 (2.44) |

$2.25\lambda $ | 5 | 574.00 | 836.52 (1.17) | 645.00 (1.36) | 5 | 573.00 | 740.77 (4.20) | 677.45 (3.28) |

$2.50\lambda $ | 1 | 519.00 | 672.40 (4.96) | 609.55 (4.18) | 1 | 561.00 | 683.90 (2.37) | 642.07 (1.73) |

$2.50\lambda $ | 2 | 519.00 | 628.45 (3.52) | 613.75 (3.27) | 2 | 549.00 | 643.75 (2.12) | 618.77 (1.39) |

$2.50\lambda $ | 3 | 514.00 | 617.72 (1.26) | 582.35 (0.45) | 3 | 520.00 | 584.32 (3.58) | 568.70 (4.57) |

$2.50\lambda $ | 4 | 523.00 | 683.30 (3.62) | 587.52 (3.85) | 4 | 593.00 | 721.37 (4.79) | 672.08 (5.64) |

$2.50\lambda $ | 5 | 505.00 | 588.75 (2.75) | 540.28 (1.88) | 5 | 586.00 | 695.88 (2.84) | 656.03 (3.72) |

$2.75\lambda $ | 1 | 493.00 | 541.20 (2.04) | 534.55 (1.21) | 1 | 582.00 | 668.90 (3.69) | 650.83 (3.41) |

$2.75\lambda $ | 2 | 480.00 | 534.80 (1.93) | 529.40 (2.49) | 2 | 572.00 | 642.05 (4.54) | 618.00 (4.12) |

$2.75\lambda $ | 3 | 505.00 | 625.50 (4.16) | 562.70 (3.72) | 3 | 603.00 | 722.95 (5.76) | 630.48 (5.91) |

$2.75\lambda $ | 4 | 505.00 | 584.85 (3.73) | 545.40 (3.29) | 4 | 585.00 | 668.70 (1.59) | 656.40 (2.58) |

$2.75\lambda $ | 5 | 480.00 | 550.57 (1.62) | 541.33 (0.98) | 5 | 567.00 | 650.95 (0.97) | 643.10 (1.28) |

$3.00\lambda $ | 1 | 480.00 | 506.00 (3.50) | 497.00 (4.14) | 1 | 567.00 | 623.00 (2.02) | 590.00 (2.82) |

$3.00\lambda $ | 2 | 505.00 | 629.20 (4.06) | 558.50 (4.30) | 2 | 583.00 | 760.67 (5.70) | 695.58 (6.15) |

$3.00\lambda $ | 3 | 451.00 | 534.68 (4.86) | 505.18 (4.66) | 3 | 576.00 | 687.48 (3.19) | 655.98 (3.04) |

$3.00\lambda $ | 4 | 480.00 | 547.93 (2.42) | 537.50 (2.33) | 4 | 565.00 | 682.97 (3.34) | 658.93 (3.00) |

$3.00\lambda $ | 5 | 480.00 | 538.73 (2.21) | 529.20 (3.04) | 5 | 532.00 | 593.25 (1.89) | 586.38 (2.24) |

© 2018 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Venkatachalam, S.; Sundar, K.; Rathinam, S.
A Two-Stage Approach for Routing Multiple Unmanned Aerial Vehicles with Stochastic Fuel Consumption. *Sensors* **2018**, *18*, 3756.
https://doi.org/10.3390/s18113756

**AMA Style**

Venkatachalam S, Sundar K, Rathinam S.
A Two-Stage Approach for Routing Multiple Unmanned Aerial Vehicles with Stochastic Fuel Consumption. *Sensors*. 2018; 18(11):3756.
https://doi.org/10.3390/s18113756

**Chicago/Turabian Style**

Venkatachalam, Saravanan, Kaarthik Sundar, and Sivakumar Rathinam.
2018. "A Two-Stage Approach for Routing Multiple Unmanned Aerial Vehicles with Stochastic Fuel Consumption" *Sensors* 18, no. 11: 3756.
https://doi.org/10.3390/s18113756