decision support scheduling for maritime search and rescue ...xs3d.kaist.ac.kr/lab activity/2014...

2014-15 Winter Seminar

Decision Support Scheduling for Maritime Search and Rescue Planning

with a System of UAVs and Fuel Service Stations

XS3D in Industrial and System Engineering

Friday March 6th, 2015

- Seunghyeon Lee -

2014-15 Winter Seminar -2-

Contents

• Introduction– Background

– Goal

• Problem Description– Search and Rescue(SAR)

– Real time rolling horizon

– Assumption

• Mathematical Formulation

• Numerical Example

• Concluding Remarks


Introduction - Background

• When a maritime accident occurs, search and rescue planning for survivors is an essential task.

• Recently, UAVs have been used to conduct Search and Rescue(SAR).

• Limitation to use multiple UAVs for SAR– Fuel limitation of commercial UAV

– Change of priority of search areas by current and wind

– Orchestration of many UAVs and fuel service stations


Introduction – Goal

• Goal for this research– Development of a method to efficiently plan the tasks for an

unmanned aircraft system(UAS) for maritime SAR that overcomes limitations via mixed integer linear programming(MILP)


Problem Description - Search

• Purpose– Maximize the probability of finding survivors

Figure 1. Possibility area

Possibility area: The smallest area containing all possible survivor locations that are consistent with the facts and assumptions of a situationPossibility sub area: The possibility area is often assumed to be an m by n grid of square cells referred to as possibility sub-areas.

0.16POC(probability of containment): probability that a search objective is in that possibility sub-area



• Purpose– Maximize the probability of finding survivors

– It means that maximize the sum of probability of success(POS)

• POS– POSi = POCi∙PODi

– Probability of detection of cell i (PODi) is the probability that a survivor is detected during the search duration Pi

– POCi is the probability that a survivor is located in cell i.

– In this research, I assume that POSi = POCi



0.12 0.08 0.14

0.07 0.04 0.18

0.15 0.13 0.9

1. Generates UAV task sequence2. UAV starts from initial location3. UAV moves to cell along schedule4. Conducts search task5. After search task in a cell, moves to

another cell or station to conduct SAR or recharging


Problem Description - Rescue

• Purpose– Deliver relief goods to target

– Transmit the location and information the status of the survivor to SAR control center

– Follows the survivor until rescue team arrives at the location of target

• Survivor– Relationship between UAV and survivor is cooperative

– Survivor is moving along a known deterministic path


0.12 0.08 0.14

0.07 0.04 0.18

0.15 0.13 0.9

1. When a UAV detects a survivor during search task, it starts rescue task(deliveries the relief goods and follows the survivor)

2. If UAV following the survivor become weary, another UAV replace the rescue task(following the customer)

3. Rescue task is terminated when rescue team arrives

Survivor !!

Problem Description - Rescue


Problem Description – Real time rolling horizon

• When a UAV detects a survivor, planning for the new rescue tasks should be incorporated into existing SAR schedule

• To incorporate new rescue task, MILP should be reset using current information such as UAVs’ fuel levels and locations

• Thus, new schedule is generated to conduct incorporated SAR


Problem Description – Assumption

• Initial location of UAV and station are known

• UAVs are initially located at station with full fuel level

• POS in all cells is given

• Survivor’s moving path is already determined


Mathematical Formulation

𝑖, 𝑗 : Indices for jobs

𝑆 : Index for stations

𝐾 : Index for UAV

𝑅 : Index for 𝑟𝑡ℎ flight of UAV

𝑁𝐽 : Total number of search and rescue jobs

𝑁𝑆𝐽 : Number of search jobs

𝑁𝑅𝐽 : Number of rescue jobs

𝑆𝑜𝐾 : Initial location of UAV 𝑘

𝐸𝑖 : Start time of job 𝑖

𝛺𝑆𝐽 : = {1,… ,𝑁𝑆𝐽}, set of search jobs

𝛺𝑅𝐽 : = {𝑁𝑆𝐽 + 1,… ,𝑁𝑆𝐽 + 𝑁𝑅𝐽}, set of rescue jobs

𝛺𝐽 : = {1,… ,𝑁𝑆𝐽 , 𝑁𝑆𝐽 + 1,… ,𝑁𝑆𝐽 + 𝑁𝑅𝐽}, set of total jobs

𝑁𝑆𝑇𝐴 : Number of recharge stations

𝑁𝑅 : Maximum number of UAV’s flight during whole periods

𝑀 : Large positive number

𝐷𝑖𝑗 : Distance from job or station 𝑖 to job or station 𝑗

𝑃𝑖 : Processing time of job 𝑖

𝑃𝑂𝑆𝑖 : Probability of success on search job 𝑖

• Notation



𝐻 : Required time for fully recharge the empty fuel tank

𝑈 : Setup time for recharge process

𝑞𝑘 : Maximum traveling time of UAV 𝑘

𝑞𝑘,𝑖𝑛𝑖 : Initial fuel level of UAV 𝑘

𝑇𝑆𝑘 : Travel speed of UAV 𝑘

𝑁𝑇 : Number of periods

𝑇 : = 1,… ,𝑁𝑇 , set of periods

𝑆𝑇𝑡 : Start time of period 𝑡

𝐸𝑇𝑡 : End time of period 𝑡

𝑆𝑡 : = {𝑆1, … , 𝑆𝑁𝑇}, set of jobs during period 𝑡

𝛺𝐼𝑁𝐼 : = {1𝐼𝑁𝐼, … , 𝐾𝐼𝑁𝐼}, set of initial locations of UAVs

𝛺𝑆𝑆 : = {𝑁𝐽 + 1,𝑁𝐽 + 3,… ,𝑁𝐽 + 2 ∙ 𝑁𝑆𝑇𝐴 − 1}, set of start stations of UAV

𝛺𝑆𝐸 : = {𝑁𝐽 + 2,𝑁𝐽 + 4,… ,𝑁𝐽 + 2 ∙ 𝑁𝑆𝑇𝐴}, set of end stations of UAV

𝛺𝐴 : = {𝛺𝐽 ∪ 𝛺𝑆𝑆 ∪ 𝛺𝑆𝐸 ∪ 𝛺𝐼𝑁𝐼}, set of jobs, stations, and initial locations of UAV

• Notation

• Decision Variables𝑋𝑖𝑗𝑘𝑟 :

Binary decision variable, 1 if UAV 𝑘 processes job 𝑗 or recharges at station 𝑗 after processing job 𝑖 or recharging at station 𝑖 during the 𝑟𝑡ℎ flight of UAV 𝑘; 0, otherwise

𝐶𝑖𝑘𝑟 :Real number decision variable, job 𝑖′s start time by UAV 𝑘 or UAV 𝑘′s recharge start time atstation 𝑖 during its 𝑟𝑡ℎflight

𝑞𝑘𝑟 : Real number decision variable, total fuel consumption for UAV 𝑘 during 𝑟𝑡ℎ flight



• Recharge/refuel time function– RTf : Fuel replenish time required after the first flight

• 𝑅𝑇𝑓 =𝐻

𝑞𝑘∙ 𝑞𝑘𝑟−1 + 𝑞𝑘 − 𝑞𝑘,𝑖𝑛𝑖 + 𝑈

– RTr : Fuel replenish time required after the r-1th flight

• 𝑅𝑇𝑟 =𝐻

𝑞𝑘∙ 𝑞𝑘𝑟−1 + 𝑈

𝐻 : Required time for fully recharge the empty fuel tank

𝑈 : Setup time for recharge process

𝑞𝑘 : Maximum traveling time of UAV 𝑘

𝑞𝑘,𝑖𝑛𝑖 : Initial fuel level of UAV 𝑘



• Mixed Integer Linear Program(MILP)

Maxmize

𝑖∈Ω𝑆𝐽

𝑗∈Ω𝐽∪Ω𝑆𝐸

𝑘∈𝐾

𝑟∈𝑅

𝑃𝑂𝑆𝑖 ∙ 𝑋𝑖𝑗𝑘𝑟

𝑆𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜


𝑋𝑆𝑜𝑘𝑗𝑘𝑟 = 1 𝑘 ∈ 𝐾, 𝑟 = 1

𝑠∈Ω𝑆𝑆∪Ω𝐼𝑁𝐼


𝑋𝑠𝑗𝑘𝑟 = 1 𝑘 ∈ 𝐾, 𝑟 > 1

𝑠∈Ω𝑆𝐸

𝑖∈Ω𝐽∪Ω𝑆𝑆∪Ω𝐼𝑁𝐼

𝑋𝑖𝑠𝑘𝑟 = 1 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅


𝑋𝑖𝑠𝑘𝑟 =

𝑖∈Ω𝐽∪Ω𝑆𝐸

𝑋𝑠−1,𝑖𝑘𝑟+1 𝑘 ∈ 𝐾, 𝑟 = 1,… , 𝑁𝑅 − 1, 𝑠 ∈ Ω𝑆𝐸


𝑋𝑖𝑠𝑘𝑟 = 0 𝑠 ∈ Ω𝑆𝑆 ∪ Ω𝐼𝑁𝐼 , 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅

𝑖∈Ω𝐽∪Ω𝑆𝐸

𝑋𝑠𝑖𝑘𝑟 = 0 𝑠 ∈ Ω𝑆𝐸 , 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅

𝑖∈Ω𝐴

𝑘∈𝐾

𝑟∈𝑅

𝑋𝑖𝑗𝑘𝑟 ≤ 1 𝑗 ∈ Ω𝑆𝐽

𝑖∈Ω𝐴

𝑘∈𝐾

𝑟∈𝑅

𝑋𝑖𝑗𝑘𝑟 = 1 𝑗 ∈ Ω𝑅𝐽

Maximize the sum of POSi

Basic properties of UAV flight

Search and rescue job



• Mixed Integer Linear Program(MILP)

𝑗∈Ω𝐴

𝑋𝑖𝑗𝑘𝑟 −

𝑗∈Ω𝐴

𝑋𝑗𝑖𝑘𝑟 = 0 𝑖 ∈ Ω𝐽 , 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅

𝐶𝑠𝑘𝑟 = 𝐶𝑠−1,𝑘𝑟+1 𝑘 ∈ 𝐾, 𝑟 = 1, . . . , 𝑁𝑅 − 1, 𝑠 ∈ Ω𝑆𝐸

𝑘∈𝐾

𝑟∈𝑅

𝐶𝑖𝑘𝑟 =𝐸𝑖 𝑖 ∈ Ω𝑅𝐽 ,

M ∙


𝑋𝑖𝑗𝑘𝑟 ≥ 𝐶𝑖𝑘𝑟 𝑖 ∈ Ω𝐽 ∪ Ω𝑆𝑆 ∪ Ω𝐼𝑁𝐼 , 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅

𝐶𝑖𝑘𝑟 + 𝑃𝑖 +𝐷𝑖𝑗

𝑇𝑆𝑘− 𝐶𝑗𝑘𝑟 ≤ 𝑀 1 − 𝑋𝑖𝑗𝑘𝑟 𝑖 ∈ Ω𝐽 , 𝑗 ∈ Ω𝐽 ∪ Ω𝑆𝐸 , 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅 ,

𝐶𝑆𝑜𝑘,𝑘𝑟 + 𝐷𝑖𝑗/𝑇𝑆𝑘 − 𝐶𝑗𝑘𝑟 ≤ 𝑀 1 − 𝑋𝑖𝑗𝑘𝑟 𝑗 ∈ Ω𝐽 ∪ Ω𝑆𝐸 , 𝑘 ∈ 𝐾, 𝑟 = 1 ,

𝐶𝑖𝑘𝑟 + 𝑅𝑇𝑓 + 𝐷𝑖𝑗/𝑇𝑆𝑘 − 𝐶𝑗𝑘𝑟 ≤ 𝑀(1 − 𝑋𝑖𝑗𝑘𝑟) 𝑖 ∈ Ω𝑆𝑆 , 𝑗 ∈ Ω𝐽 ∪ Ω𝑆𝐸 , 𝑘 ∈ 𝐾, 𝑟 = 2 ,

𝐶𝑖𝑘𝑟 + 𝑅𝑇𝑟 + 𝐷𝑖𝑗/𝑇𝑆𝑘 − 𝐶𝑗𝑘𝑟 ≤ 𝑀(1 − 𝑋𝑖𝑗𝑘𝑟) 𝑖 ∈ Ω𝑆𝑆 , 𝑗 ∈ Ω𝐽 ∪ Ω𝑆𝐸 , 𝑘 ∈ 𝐾, 𝑟 > 2 ,

Basic properties of UAV flight

Relationship between 𝑋𝑖𝑗𝑘𝑟 and 𝐶𝑖𝑘𝑟



• Mixed Integer Linear Program(MILP)𝐶𝑠𝑘𝑟 ≥


𝑋𝑠𝑗𝑘𝑟 − 1 ∙ 𝑆𝑇1 + 𝑆𝑇1 𝑠 ∈ Ω𝑆𝑆 ∪ Ω𝐼𝑁𝐼 , 𝑘 ∈ 𝐾, 𝑟 ∈ R

𝐶𝑠𝑘𝑟 + 𝑅𝑇𝑓𝐼{𝑟=2} + 𝑅𝑇𝑟𝐼{𝑟>2} +

𝑖∈Ω𝐴

𝑗∈Ω𝐴

(𝐷𝑖𝑗/𝑇𝑆𝑘) ∙ 𝑋𝑖𝑗𝑘𝑟 +

𝑖∈Ω𝑆𝐽

𝑗∈Ω𝐴

(𝑃𝑖) ∙ 𝑋𝑖𝑗𝑘𝑟 ≤ 𝐸𝑇𝑁𝑇

𝑠 ∈ Ω𝑆𝑆 ∪ Ω𝐼𝑁𝐼 , 𝑘 ∈ 𝐾, 𝑟 ∈ R

𝐶𝑖𝑘𝑟 ≥ (


𝑋𝑖𝑗𝑘𝑟 − 1) ∙ 𝑆𝑇1 + 𝑆𝑇1 i ∈ 𝑆1, k ∈ K, r ∈ R

𝐶𝑖𝑘𝑟 ≥


𝑋𝑖𝑗𝑘𝑟 − 1 ∙ 𝐸𝑇𝑡−1 + 𝐸𝑇𝑡−1 i ∈ 𝑆𝑡 , k ∈ K, r ∈ R, t = 2. . 𝑁𝑇

𝐶𝑖𝑘𝑟 + 𝑃𝑖 ≤ 𝐸𝑇𝑡 i ∈ 𝑆𝑡 , k ∈ K, r ∈ R, t = 1. . 𝑁𝑇

𝑠∈Ω𝑆𝐸

𝐶𝑠𝑘𝑟 − 𝐶𝑆𝑜𝑘,𝑘𝑟 = 𝑞𝑘𝑟 k ∈ K, r = 1

𝑠∈Ω𝑆𝐸

𝐶𝑠𝑘𝑟 − (

𝑖∈Ω𝑆𝑆

𝐶𝑖𝑘𝑟 + 𝑅𝑇𝑓) = 𝑞𝑘𝑟 k ∈ K, r = 2

𝑠∈Ω𝑆𝐸

𝐶𝑠𝑘𝑟 − (

𝑖∈Ω𝑆𝑆

𝐶𝑖𝑘𝑟 + 𝑅𝑇𝑟) = 𝑞𝑘𝑟 k ∈ K, r > 2

𝑖∈Ω𝐴

𝑗∈Ω𝐴

(𝐷𝑖𝑗/𝑇𝑆𝑘) ∙ 𝑋𝑖𝑗𝑘𝑟 +

𝑖∈Ω𝐽

𝑗∈Ω𝐴

𝑃𝑖 ∙ 𝑋𝑖𝑗𝑘𝑟 ≤ 𝑞𝑘 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅

𝑞𝑘𝑟 ≤ 𝑞𝑘,𝑖𝑛𝑖 (k ∈ K, r = 1)

𝑞𝑘𝑟 ≤ 𝑞𝑘 (k ∈ K, r ≠ 1)

𝐶𝑖𝑘𝑟 ≥ 0 𝑖 ∈ Ω𝐴, 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅

𝑞𝑘𝑟 ≥ 0 (k ∈ K, 𝑟 ∈ 𝑅)

𝑋𝑖𝑗𝑘𝑟 ∈ {0,1} 𝑖 ∈ Ω𝐴, 𝑗 ∈ Ω𝐴, 𝑘 ∈ 𝐾, 𝑟 ∈ 𝑅

Start and end constraint for search

Time periodfor searchable job

Fuel consumption

Maximum flight duration

Decision variable constraints


Numerical Example - Search

• Ex1) Situation at Time = 0

0 200 400 600

200

400

UAV #

Station

SearchJob #

1,7

1 2

3

#

#

4,10

5,11

2,8 3,9

6,12



• Ex1) Task sequence for search during time period 15

UAV Task Start Time

1 𝐼𝑁𝐼1,0 01 SJ5 4.071 STA 7.071 SJ10 141 STA 19.482 𝐼𝑁𝐼2,0 12 SJ4 6.482 STA 11.963 𝐼𝑁𝐼3,0 03 SJ2 23 STA 53 SJ8 93 SJ11 143 STA 17

During conducting search task, UAV 1 found out a survivor at time = 5


Numerical Example - Rescue

• Ex2) Situation at Time = 5

5,11

2,8

0 200 400 600

200

400

2

3

1 Survivor

Rescue Job #

#

1,7

4,10

3,9

6,12

UAV #

Station

SearchJob #

#

#

213

45



• Ex2) Task sequence for search and rescue from T = 5

UAV Task Start Time

1 𝐼𝑁𝐼1,5 5

1 RJ1 51 RJ2 5.11 RJ3 71 RJ4 8.21 STA 10

2 𝐼𝑁𝐼2,5 5

2 SJ4 6.482 STA 11.96

3 𝐼𝑁𝐼3,5 5

3 STA 73 RJ5 93 SJ11 143 STA 23

Rescue tasks are conducted by UAV 1 and 3


Numerical Example – Result Comparison

• In Example 1, the sum total POS obtained is 1.29

• In Example 2, the sum total POS is 0.95– The SAR plan changed when UAV1 detected a survivor at T = 5

– From T = 5, search and rescue were conducted simultaneously


Concluding Remarks

• SAR is the most important task when a maritime accident takes place

• Goal of my research is to develop a decision support scheduling for maritime SAR with UAVs and station.– UAVs’ Fuel limitation, priority change, and orchestration

• MILP was introduced to generate a schedule for SAR– Objective is to maximize the probability of finding a survivor

• Sum of total probability can be changed by rescue

decision support scheduling for maritime search and rescue ...xs3d.kaist.ac.kr/lab activity/2014...

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