Optimized Border Interdiction
by
Jonathan L. Paynter
B.S., Mathematics, United States Military Academy (2006)
Submitted to the Sloan School of Management
and
Engineering Systems Division
in partial fulllment of the requirements for the degrees of
Master of Science in Operations Research and Master of Science in Technology and
Policy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
c Jonathan L. Paynter. All rights reserved.
The author hereby grants to MIT and The Charles Stark Draper Laboratory, Inc.
permission to reproduce and to distribute publicly paper and electronic copies of this
thesis document in whole or in part in any medium now known or hereafter created.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sloan School of Management and Engineering Systems Division
May 9, 2014
Certied by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stephan E. Kolitz
Technical Sta, The Charles Stark Draper Laboratory
Thesis Supervisor
Certied by. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dimitris J. Bertsimas
Boeing Professor of Operations Research
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Dava Newman
Professor of Aeronautics and Astronautics and Engineering Systems
Director, Technology and Policy Program
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Patrick Jaillet
Dugald C. Jackson Professor
Co-director, Operations Research Center
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Optimized Border Interdiction
by
Jonathan L. Paynter
Submitted to the Sloan School of Management
and
Engineering Systems Division
on May 9, 2014, in partial fulllment of the
requirements for the degrees of
Master of Science in Operations Research and Master of Science in Technology and Policy
Abstract
A feature of many conicts is the presence of a border that separates an area of on-going military
operations from an area that the enemy can use permissively. This thesis considers analytic
techniques for planning military operations designed to interdict enemy forces crossing the border.
Specically, this thesis presents optimization-based methods for scheduling patrolling units and for
positioning ground sensors in support of those patrolling units. These methods could serve as the
framework for a tactical-level decision support tool designed to assist military planners assigned to
border regions with resource allocation recommendations and trade-o comparisons. We propose
tractable mixed integer optimization formulations for these solutions based on a network model of
the routes in the region, operational constraints on the abilities of the patrolling units, and
estimates of enemy force movements. Additionally, we develop robust extensions to these
formulations that allow the model to account for a degree of enemy intelligence by incorporating
the uncertain nature of the enemy movement estimates into the formulation.
We evaluate the solutions to these formulations using simulations that account for dierent
realizations of the uncertain enemy movement. This includes cases where the realized enemy
movement closely matches the estimates made in the model and cases where the realizations are
very dierent from the model. Additionally, we provide a modied greedy heuristic to the
scheduling formulation that can serve as a tool for dynamically retasking a patrol to interdict
enemy forces in real-time after a sensor detects enemy movement. Current planning for these
operations are conducted by a sta with no decision making analytic tools. We approximate a
version of this current planning method with an algorithm and show that our method outperforms
it with both the deterministic and robust formulations. We compare the deterministic and robust
formulations and demonstrate a process for choosing between the formulations, along with an
explanation of the utility of the robust formulation.
Thesis Supervisor: Stephan E. Kolitz
Title: Technical Sta, The Charles Stark Draper Laboratory
Thesis Supervisor: Dimitris J. Bertsimas
Title: Boeing Professor of Operations Research
3
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Acknowledgments
This thesis would not have been possible without the ideas, advice, and help of many people
during the past two years.
I would like to thank my thesis and academic advisor, Dimitris Bertsimas, for his fantastic advice,
patient explanations, and condence in my ability to learn the necessary techniques to make this a
successful thesis. I would also like to thank my two thesis advisors at Draper, Stephen Kolitz and
John Irvine. Their ability to probe for the underlying ideas in the problem and emphasis on
realistic modeling was very helpful, and it ensured that the work in this thesis stayed true to the
motivating problem.
The opportunity to study at MIT would not have been possible without the Department of
Mathematical Sciences at the United States Military Academy, and I am grateful for their decision
to let me study at MIT and to bring me to West Point as a math instructor. My time spent
researching and writing was fun and enjoyable in part because of the outstanding support of MIT's
Operations Research Center and Draper Lab. I enjoyed working with some of the brightest people
in the world, and I am grateful to Dave Culver, Mark Williams, Nick Jernigan, and Louis Kim for
support during classwork and research.
The motivating experience for this thesis came from time spent deployed in southern Afghanistan,
and has its roots in operational work done by Don Nibblett, Josh Sutho, Talon Young, DJ Jones,
and the soldiers of Blackjack Troop, 2-38 CAV.
Most importantly, I owe the most to my wife - without her support, I would not have had the
opportunity to study at MIT. Her patience and dedication to our sons is what enabled me to
dedicate time to academic pursuits, and her willingness to move and support an army career is
what makes my job possible.
Jon Paynter, Captain, US Army May 9, 2014
This thesis was prepared at The Charles Stark Draper Laboratory, Inc. Publication of this thesis
does not constitute approval by Draper of the ndings herein. It is published for the exchange and
stimulation of ideas. As an active duty Army ocer, I am aware that the views, analyses, and
conclusions expressed in this document are mine and do not represent the ocial policy or position
of the United States Army, the Department of Defense, or the United States Government.
5
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Contents
1
2
3
Introduction
19
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3
Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Background
25
2.1
Tactical Planning Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2
Generalized Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4
Example Scenario - Operations in Afghanistan . . . . . . . . . . . . . . . . 29
2.5
Guards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6
Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6.1
Sensor Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6.2
Border Sensor Use . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Problem Framework
3.1
37
Functional Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.1.1
Base Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7
4
3.1.2
Base Process Plus Sensors and Data Analysis . . . . . . . . . . . . 39
3.1.3
Base Process Plus Sensors, Data Analysis, and Other ISR . . . . . . 41
3.1.4
Base Process Plus Sensors, Data Analysis, Other ISR, and Dynamic Retasking . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.1.5
Individual Functions . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2
Example As-Is Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3
Functional Focus Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4
Example Scenario - Revisited Analytically . . . . . . . . . . . . . . . . . . 47
Solution Approaches
51
4.1
Analytical Approach Overview . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2
Model Taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1
Network Interdiction Models . . . . . . . . . . . . . . . . . . . . . 54
4.3.2
Flow Capture Models . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.3
Other Border Security Models . . . . . . . . . . . . . . . . . . . . 56
4.3.4
Related Models with Military Applications . . . . . . . . . . . . . 57
4.4
Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.6
General Approach - Flow Capture . . . . . . . . . . . . . . . . . . . . . . 59
4.7
General Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.7.1
Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.7.2
Friendly Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.7.3
Enemy Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
8
4.8
4.9
Basic Patrol Scheduling (B-PS) . . . . . . . . . . . . . . . . . . . . . . . . 65
4.8.1
B-PS Model Description . . . . . . . . . . . . . . . . . . . . . . . 65
4.8.2
B-PS Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.8.3
B-PS Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.8.4
B-PS Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 70
Determine Sensor Placements (DSP) . . . . . . . . . . . . . . . . . . . . . 71
4.9.1
DSP Model Description . . . . . . . . . . . . . . . . . . . . . . . 75
4.9.2
DSP Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.9.3
DSP Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.9.4
DSP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.10 Dynamic Guard Retasking (DR) . . . . . . . . . . . . . . . . . . . . . . . 82
4.10.1 DR Model Description . . . . . . . . . . . . . . . . . . . . . . . . 83
4.10.2 DR Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.10.3 DR Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.10.4 DR Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.11 Enhanced Patrol Scheduling (E-PS) . . . . . . . . . . . . . . . . . . . . . 85
4.11.1 E-PS Model Description . . . . . . . . . . . . . . . . . . . . . . . 86
4.11.2 E-PS Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.11.3 E-PS Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.11.4 E-PS Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.12 Mission Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.13 Summary of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
9
5
6
A Robust Approach
95
5.1
Motivation and Background . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2
Robust Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3
Robust Sensor Placement (R-DSP) . . . . . . . . . . . . . . . . . . . . . . 97
5.3.1
Robust Formulation Derivation . . . . . . . . . . . . . . . . . . . . 98
5.3.2
Polyhedral Uncertainty Sets . . . . . . . . . . . . . . . . . . . . . 98
5.3.3
New Objective and Additional Constraints . . . . . . . . . . . . . . 101
5.3.4
R-DSP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 102
Results and Analysis
6.1
105
Test Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.1
Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.2
Data Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2
Tractability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3
Overall Results for the DSP Model . . . . . . . . . . . . . . . . . . . . . . 109
6.4
Robust Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4.1
Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4.2
Robust Experiment 1 . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.3
Robust Experiment 2 . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.4
Robust Experiment 3 . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4.5
Robust Experiment 4 . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4.6
Robust Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 119
10
7
Operational Evaluation
7.1
Test Scenarios - Real-world Parallels . . . . . . . . . . . . . . . . . . . . . 121
7.2
Operational Model Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 122
7.3
8
121
7.2.1
Staff Approximation Algorithm for Interdiction . . . . . . . . . . . 122
7.2.2
As-is v. Analytical Model Performance . . . . . . . . . . . . . . . 124
Guard and Sensor Placement Decisions . . . . . . . . . . . . . . . . . . . 125
7.3.1
Varying Levels of Sensors and Guards . . . . . . . . . . . . . . . . 126
7.3.2
Decisions for Different Levels of Robustness . . . . . . . . . . . . 126
Conclusion
129
8.1
Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8.2
Towards Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
8.3
8.2.1
Operational Implementation . . . . . . . . . . . . . . . . . . . . . 130
8.2.2
Adopting a Decision Support Tool . . . . . . . . . . . . . . . . . . 133
Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
A Abbreviations and Acronyms
137
11
12
List of Figures
2.4.1 An Example Scenario for Border Interdiction Operations: A combat brigade
conducting operations along a border and using limited patrolling capacity
(blue rectangles) to attempt to interdict enemy forces moving across the
border (red arrows). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 Possible Guard Configurations: Three possible elements of a guard. (From
top to bottom) A Bradley fighting vehicle in an overwatch position in northwestern Iraq (half of a guard unit); A three HMMWV platoon preparing for
an reconnaissance mission near Mosul, Iraq; A platoon of US and Afghan
soldiers moving to a new overwatch position near the Afghanistan / Pakistan border. Photos from author’s collection. . . . . . . . . . . . . . . . . 33
2.6.1 An Example of an UGS Camera - OmniSense cameras from an Unattended
Ground Sensor system made by McQ, Inc (This is an example of an available system, and not an endorsement of a particular sensor technology) . . 35
3.1.1 Functional Analysis Legend: Description of the different components used
in the Functional Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Functional Diagram - Base Process: A description of the border interdiction planning and execution cycle currently in use . . . . . . . . . . . . . . 39
3.1.3 Functional Diagram - Base Process Plus Sensors and Data Analysis: The
border interdiction planning and execution cycle augmented by the emplacement of ground sensors . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.1.4 Function Diagram - Base Process Plus Sensors, Data Analysis, and Other
ISR: The border interdiction planning and execution cycle augmented by
the emplacement of ground sensors and incorporating the need to task outside ISR assets that occasionally support border interdiction operations . . . 41
13
3.1.5 Functional Diagram - Base Process Plus Sensors, Data Analysis, Other
ISR, and Dynamic Retasking: The border interdiction planning and execution cycle augmented by the emplacement of ground sensors, ISR asset
tasking, and accounting for the ability to cue patrols in real-time based on
sensor data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.1 Operational Diagram - As-is Border Interdiction Operations: A description of the system functions as they are currently executed by army units
assigned to border interdiction tasks; the functions above the dotted line
are typically not conducted, or if they are, conducted without an analytical
component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3.1 Functional Diagram - Highlighted Focus Areas: Four functions in border
interdiction operations related to decision making that could benefit from
an analytic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1 An Improved, Example Scenario for Border Interdiction Operations: A
combat brigade conducting operations along a border and using an analytic
framework to improve limited patrolling capacity (blue rectangles) to interdict enemy elements (red arrows) by employing ground sensors (green
circles) and functional improvements to the planning process . . . . . . . . 48
4.1.1 Analytical Approach Diagram: The analytical improvements recommended
for the four critical, decision making functions in the border interdiction
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6.1 Flow Capture Scenario Example: A toy example of a situation where the
flow capture model can be used to solve for the optimal facility location
(blue rectangles) based on the flows (red arrows) in the network . . . . . . 60
4.7.1 Example Road Network: A simplified road network in a border region
where the ends of the roads in country A are modeled as origin nodes,
and the major roadway in the lower left is modeled as the destination node.
62
4.8.1 Placement of The “Patrol Scheduling” Function in the Border Interdiction
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.9.1 Placement of The “Determine Sensor Placement” Function in the Border
Interdiction System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
14
4.9.2 Sample Network 1: An illustration of employing sensors and a guard in
tandem to capture the optimal amount of enemy forces in a network with
limited connectivity. The guard (blue rectangle) can move either left or
right in response to a sensor (green circles) detection. . . . . . . . . . . . . 73
4.9.3 Sample Network 2: An illustration of employing sensors and a guard in
tandem to capture the optimal amount of enemy forces in a network with
moderate connectivity. The guard (blue rectangle) can move either left or
right in response to a sensor (green circles) detection. . . . . . . . . . . . . 74
4.9.4 Sample Network 3: Determining Sensor Placement and “Extended Reach”
- the guard (blue rectangle) has the ability to intercept enemy movement on
path 1 (through the sensors - green circles), but not path 2 or path 3 since
those flows do not travel through the sensor location and towards the guard
76
4.10.1Placement of The “Dynamic Guard Retasking” Function in the Border Interdiction System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.12.1Placement of The “Mission Feedback” Function in the Border Interdiction
System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.12.2Marginal Benefit of a Sensor: A sample graph of the utility of additional
sensors based on an updated assessment of the situation . . . . . . . . . . . 92
4.12.3Marginal Benefit of a Guard: A sample graph of the utility of additional
guards based on an updated assessment of the situation . . . . . . . . . . . 93
5.3.1 Nominal vs. Robust Flow Values: An example sketch of the difference
between solving for a single value (left) and for a range of possible values
(right), where the Ceiling parameter controls for the amount of gray area
considered in the solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.1.1 Test Scenario: A sketch of a representative border scenario with multiple
crossing points and roads connecting to a larger highway inside the country
of concern. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.2 Test Scenario as a Network: The network that we use for the initial testing
of our formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2.1 Solution Times: The solution times for different combinations of guards
(lines) and sensors (x-axis) in seconds (y-axis) for the DSP model . . . . . 108
15
6.3.1 Marginal Utility of Sensors: The percent of flow captured for a fixed quantity of guards and different numbers of sensors . . . . . . . . . . . . . . . . 110
6.3.2 The Impact of Guards and Sensors: Percent captured for different combinations of guards and sensors as a way to compare tradeoffs . . . . . . . . 110
6.3.3 The Impact of Guard Speed: Percent captured for a fast and slow guard for
different numbers of sensors . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.4.1 Simulated Flow Constructs: Four different ways of simulating the uncertain flows as compared to the uncertainty set used in the formulation. The
gray intervals represent the uncertainty sets used for certain nominal values (black lines at the mid-point of the gray intervals). These base gray
intervals are the same for all four simulation constructs. The red intervals
represent the simulation interval used with random draws uniformly from
within the red interval. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.4.2 Simulation Results for Varying Levels of Robustness: 2 guards, 4 sensors,
and a Zero-Anchored flow type under four different types of simulated flows 115
6.4.3 Simulation Results for Varying Levels of Robustness: 2 guards, 2 sensors,
and a Uniform Draw flow type under four different types of simulated flows 117
6.4.4 Simulation Results for Varying Levels of Robustness: 2 guards, 2 sensors,
and a Key Paths flow type under four different types of simulated flows . . 118
7.3.1 Optimal Guard and Sensor Locations: An illustration of the optimal locations for 2 guards and (top to bottom): 0 sensors, 1 sensor, 3 sensors and 5
sensors. The guards’ primary locations are in blue, the sensors are in green
and the guards’ base is in orange. . . . . . . . . . . . . . . . . . . . . . . 127
7.3.2 Optimal Guard and Sensor Locations for a Robust Solution: An illustration of the optimal locations for 2 guards and 4 sensors under the nominal
formulation, and the robust formulation (Γ = 2). The guards’ primary locations are in blue, the sensors are in green and the guards’ base is in orange. 128
16
List of Tables
4.2.1 Model Descriptions: A table summarizing the different models that we
present in this chapter, along with the differences in their use. . . . . . . . . 54
6.1.1 Flow Types: The different types of flows used in initially in the non-robust
simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.1 Robust Experiments: A summary of the different robust experiments conducted with flow type explanations found in table 6.4.2 . . . . . . . . . . . 112
6.4.2 Flow Types: The different types of flows used in simulations . . . . . . . . 112
6.4.3 Simulation Descriptions: The different types of realizations for the uncertain flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.4.4 The Effect of Sensor Quantity on Robust Performance: Comparing percent
of enemy traffic captured for the MIP solution, nominal solution under simulation and robust solution under uncertainty . . . . . . . . . . . . . . . . 118
7.2.1 Nodal Uncertainty (N) Comparison: A comparison of the nodal uncertainty
metric, N, for 2 guards and various numbers of sensors with both the analytic staff algorithm (optimal), and the Determine Sensor Placements function124
7.2.2 Comparison of the DSP and R-DSP Models Under Simulation to the Staff
Approximation Algorithm: The percent of enemy flow captured for various
guard and sensor combinations under the nominal, robust and staff approximation methods with the margin of improvement of the best formulation
over the staff approximation . . . . . . . . . . . . . . . . . . . . . . . . . 125
17
8.1.1 Model Comparison: A table summarizing the different models that we presented in this thesis, sorted in terms of tractability from largest to smallest.
Analogous robust versions for E-PS and B-PS follow directly from our development of the R-DSP model. . . . . . . . . . . . . . . . . . . . . . . . 131
18
Chapter 1
Introduction
1.1
Motivation
With numerous worldwide threats, the United States military continues to operate in a
wide-range of areas performing a diverse set of tasks including counterinsurgency, counterterrorism, host nation support, and general security. These deployed forces include - as a
sample from the army - a large multi-division element in Afghanistan, a joint task force in
the Horn of Africa, small special operations teams in the Central African Republic, and a
division headquarters in Jordan adjacent to the Syrian civil war. To some degree in all of
these conflicts, enemy elements are able to use the borders between countries as a means
of accessing safer locations for refit and resupply. Because of this, United States army
elements of various sizes have an interest in monitoring and securing borders.
Securing the border between two countries is a difficult operation; borders can stretch for
hundreds of miles across varied terrain, creating a diverse set of challenges for any element
trying to control the area. Military operations along a border can include active intervention (example: the Iraq-Syria border from 2003 to 2010), host-nation support (example:
the Iraq-Syria border from 2010 to 2012 and the Afghanistan-Pakistan border from 2009 to
2014), and intelligence gathering activities (example: the Libya-Algeria border in 2011).
The military must take a resource constrained approach to applying assets along a border,
usually in the context of a much broader mission inside the country. For example, in Iraq,
even the units operating along the borders with Syria and Iran had population-centric counterinsurgency tasks in addition to border security tasks. These other missions, as well as
units with missions that had no border component further inside the country (such as in
19
Baghdad and Mosul), frequently had priority for Intelligence, Surveillance and Reconnaissance (ISR) assets, which relegated the border security mission to that of an economy of
force. Even so, good border security that limits the flow of lethal material and fighters can
lead to additional security gains for many different units operating inside a country. This
means that border security operations can have a magnified effect on the rest of the friendly
forces’ operations further inside the country.
The general category of border security encompasses a wide-variety of tasks including
active searching at authorized crossing points, the bureaucratic oversight of crossings, and
a method to limit or stop traffic at unauthorized crossing points. The searching at active
crossing points is manpower intensive and nested with the bureaucratic tasks related to
trade and immigration. These tasks are broadly classified as Port of Entry operations, and
this research does not address them.
The problem of limiting traffic at unauthorized crossing points, or interdiction, is frequently
one of deploying limited resources to focused areas across vast expanses of terrain. This
is not solely a mission for soldiers and patrols; many ISR assets can assist in determining
what infiltration routes exist and how frequently they are used. As the importance of securing a border increases without an increase in the number of forces available, the efficient
employment of soldiers and ISR sensors becomes even more important. A more coordinated approach to the specific employment of different sensors and units should increase
the overall effectiveness of interdicting enemy traffic at unauthorized crossing points along
a border.
Creating this refined approach to border interdiction relies initially on analyzing relevant,
quality input data. However, collecting data on general enemy activity is difficult, not only
at the point of detection, but also in quantifying the diverse array of possible detections.
One successful example of data-driven work from recent operations is the heat mapping
of Improvised Explosive Devices (IEDs). The military adopted a number of visualization
tools that could show historical patterns of IED strikes for use in focusing future operations.
These systems were not predictive in a truly analytical sense and did not provide recommended friendly force actions, but the descriptive statistics they provided allowed many
units to better employ ambush positions, task route clearance teams, and reroute supply
convoys. Most soldiers now are very familiar with terms such “Tier One IED Hot Spot”
(although the definitions may vary) from using these techniques.
The border security problem presents an opportunity where quantifiable data can not only
enhance the process of learning about the enemy, but provide a genuine recommendation
on the employment of friendly forces. Infiltration across a border is fundamentally about
20
enemy forces moving through an area; countering these efforts hinges on the ability of the
friendly forces to learn enemy patterns and frequently used routes in order to best position
interdiction patrols and overwatch elements.
Moreover, a data-driven approach related to the frequency and types of infiltrations across
a border could greatly assist the units that are tasked to secure the border. This possibility
leads to an interest in the employment of Unattended Ground Sensors (UGS) as a means of
collecting data on various infiltration routes along a border. These sensors come in a variety
of forms, but are characterized by their ability to transmit information – acoustic, seismic,
photographic – for a long period of time after they have been emplaced in a concealed
manner. A deliberate employment of these sensors could lead to refined intelligence about
infiltration routes. Additionally, a suite a properly arrayed sensors could greatly enhance
the effective range of a security patrol by providing an early warning cue about an enemy
infiltration.
Because the friendly forces’ set of actions is more limited than a general military scenario,
the border interdiction problem presents an opportunity where a decision support tool could
be created to assist friendly planners. This tool could leverage data collection on border
crossing activity and create recommended friendly force patrols and sensor locations. Because border security operations are usually conducted as an economy of force, it is even
more critical that smaller units with less support have additional tools to assist in the planning process.
In light of this possibility for a decision support tool, this research attempts to answer the
following questions:
R ESEARCH Q UESTION 1: How can we optimally employ sensors (UGS) along a border
to improve our understanding of the enemy and our border security operations?
R ESEARCH Q UESTION 2: How can we better allocate the soldiers assigned to border
interdiction based on an improved understanding of the enemy, and in concert with the
sensors?
1.2
Contributions
In this research, we provide a holistic look at the border interdiction process that military
units must conduct when their operations are near borders that provide the enemy access
to a permissive area for refit and resupply. We provide a functional decomposition of this
21
system, and highlight functional-level analytic improvements that could provide the basis for a tactical level decision support tool. We focus on the decision-making functions,
and develop four tractable models that could be used in the process for scheduling interdiction patrols, determining where to place ground sensors, dynamically retasking interdiction
patrols based on real-time sensor cues of enemy movement, and scheduling interdiction patrols in the presence of ground sensors. We also provide a robust extension the largest of
the models (a method which extends directly to the others), which is a tractable method for
including the uncertainty present in the estimation of enemy movement. This use of robust
optimization provides a method for incorporating a degree of enemy intelligence into the
formulation without resorting to less tractable game theoretic approaches. We demonstrate
why the robust approach is valuable, and outline the details of when it is useful, which is
primarily a function of the number of sensors present. These formulation provide improved
results over an algorithm that approximates the current staff planning process. Additionally, the method that we introduce for considering the locations of patrols and sensors in
tandem could extend to mobile sensors (such as a UAV), operating in conjunction with a
ground interdiction force.
1.3
Thesis Structure
In Chapter 2 we describe the problem in more detail and present an example scenario to
help illustrate the border interdiction process. Chapter 3 provides the functional decomposition of the border interdiction system so that the we can fully evaluate where analytic
approaches could best improve the process. Chapter 4 outlines the solution approaches to
the critical decision making functions that we address. This includes the introduction of
four models, including MIP formulations for scheduling patrols and determining sensor
placements, an algorithm for dynamically retasking a patrol when a sensor detects enemy
movement, and a overview of using the formulations to evaluate operational changes. In
Chapter 5 we present the robust optimization extension to one of the MIP formulations
along with an explanation of why we need to consider uncertainty, particularly for our
estimates of the enemy movements along the border. In Chapter 6 we present computational results. This includes a tractability analysis and an understanding of how varying
parameters affects the solution. In addition, we present a comparison of the deterministic
and robust formulations, along with an explanation of when the robust formulation outperforms the deterministic formulation. In Chapter 7 we present computational results geared
towards an operational understanding of the system. This includes a comparison of our
formulations with an algorithm that approximates the current planning process, and an un22
derstanding of the implementation of the solutions found from the formulations. Chapter 8
includes our closing remarks, comments on the operational implementation of the system,
and recommendations for future work.
23
24
Chapter 2
Background
In this chapter, we present background information relevant to the border interdiction system, and the development of our models. We first describe the need for tactical planning
tools in general, followed by an explanation about how our modeling work could expand
to other domains. We then define some key terminology and outline an example scenario.
Finally, we describe in detail the guards and sensors that are involved in the interdiction
process.
2.1
Tactical Planning Tools
US military forces continue to operate in numerous locations around the world, in a variety
of conflict types. As the number of conflicts involving smaller, distributed military units
increases, the importance of developing planning tools that can assist at the tactical level
also increases. The military has a strong history of using operations research methods
for improving at the operational level, especially with logistic coordination. Examples
include the founding work of operations research in world war two with radar placement
and convoy protection, along with more recent logistics examples such as the development
of the Northern Distribution Network to support operations in Afghanistan. There is less
of a track record at the small-unit, tactical level for using decision support tools to assist at
the warfighter level.
There are acknowledged efforts in the Department of Defense about this kind of work, but
less of it is focused on tactical decision recommendations. The equivalent role of Chief
Technology Officer in the Department of Defense is the Assistant Secretary of Defense for
25
Research and Engineering. This office has a number of initiatives, and supervises many of
the research efforts in the Defense Department. One of the office’s seven technology focus
areas is Data to Decisions. This research effort includes many of the intelligence products
that assist planning staffs. A tactical planning tool of the variety discussed here would add
more of a recommendation layer to some of these existing products.
Our focus on the importance of tactical decision support tools for planning is on a missionspecific tool, and not a technology specific tool. We think it is less useful to consider
an algorithm or formulation that assists in tasking a specific type of UAV, than it is be
to consider a tool that is designed to assist in planning certain types of operations, and
incorporates the ability to task a specific type of UAV. The distinction is important, because
it indicates an emphasis on a mission-set, and not an emphasis on a proprietary technology.
With more distributed operations, decision aids at the tactical level provide two very useful
features for military units. First, they provide leverage for a smaller staff to incorporate
and take advantage of larger amounts of intelligence. For intelligence synthesis, this type
of work saw a large improvement during the conflicts in Iraq and Afghanistan. Smaller
units, down to the company-level, began employing data aggregation and pattern analysis
tools with their own intelligence sections that allowed local commanders to leverage large
amounts of information. One of the most prominent was attack visualization, in particular the ability to use density plots and heat maps to understand where areas with large
concentrations of IED attacks were occurring.
While many of these tools were fantastic enablers for intelligence staffs, they were limited
to analysis and visualization, and did not incorporate mechanisms for recommendations on
friendly force actions. We can view the decision support tool as a two layer process where
the base layer is the data analysis and intelligence fusion. The second layer is then the
recommendation layer where the intelligence is processed, and the tool provides a recommendation on possible actions. This second layer does not exist for tactical applications,
and there are places where a computer-supported recommendation layer could provide a
benefit to the normal staff process. Not every military action is easily characterized into a
system built to recommend a course of action, and so one main barrier to the development
of a tactical decision support tool is identify a problem where quantification is useful without losing too much of the situational context. Additionally, the notion of a computer-based
recommendation for the employment of tactical formations has a number of negative connotations within the military. Many officers who would agree with the notion of analytic
tools for improving a supply chain or optimizing a UAV route would hesitate to adopt a
technology that gives the impression of commanders ceding control of tactical decisions.
26
This is valid concern, and any decision support tool designed for the tactical level must be
built and implemented as an aid for planning, and not as a substitute.
These tools are even more important in settings where the US military has a smaller footprint, for example US operations in 2013 in a number of African countries, and the presence
of a large planning staff is not possible.
We will discuss additional issues associated with the implementation of this tactical decision support tools specifically for border interdiction in section §8.2.
2.2
Generalized Problem
One area where a decision support tool could assist is with security operations along a
border. The need for border interdiction operations occurs in many types of conflict, as
demonstrated by the numerous recent locations where enemy forces used a border as a
means of accessing a safe location for resupply. The questions posed in section §1 present
an opportunity for improving interdiction operations, but at a more general level, the concept of gathering data from localized sensors in order to better allocate patrols describes
most situations that include a perimeter and a need for security. The solution methods described in this paper present one approach at decomposing the overall border interdiction
system into smaller functions, each with its own solution method. Some examples of other
variants of this problem include perimeter security at a large installation, domestic border
security, the protection of a military unit’s lines of communication, smuggling routes, and
a security zone for a defensive position. In each of these cases, the ability to sense enemy
movement, cue responding forces, and collect data to better inform a security posture are
applicable.
Specifically for military applications, the possible gains from a border interdiction decision
support tool are larger in situations where the military has not had a sizable presence. For
smaller units deploying into new border regions devoid of an operational history, the ability
to have an initial recommendation for force employment based on limited ISR analysis and
the first portion of a staff intelligence assessment is of great value.
Without losing sight of these other applications, we proceed throughout this paper with
a focus on the need for resource constrained military units to execute border interdiction
operations in support of a larger military operation. Specifically we consider cases from
the American conflicts in Iraq and Afghanistan as illustrative, and leverage knowledge
27
from personal experience conducting military operations in southern Afghanistan along
the Afghanistan / Pakistan border. We develop an example scenario that characterizes these
cases in an effort to further elaborate on the problem.
2.3
Definitions
Before we describe an example scenario to detail border interdiction operations, we define
some key terms that will be used throughout the paper.
Border Region: The area adjacent to an international border where military units operate
with the task of securing the border. This region includes all of the terrain where enemy
forces can be killed or captured before getting to an area where it is easy to disperse within
the country.
Friendly forces: United States and coalition forces operating in a border region. This includes all types of forces and units, not only those dedicated to border interdiction. The
planning headquarters responsible for border security is typically the senior friendly forces
headquarters in the region. The size of this headquarters can vary substantially from scenario to scenario and could be as small as a company headquarters (responsible for 100
soldiers) or as large joint task force headquarters (responsible for over 10,000 soldiers).
Guards: The subset of the friendly forces tasked to border interdiction operations. A guard
unit has no universally defined size, but is typically a squad or platoon, and is the smallest
element that the friendly forces headquarters uses as an independent element. Guard is not
the typical term used in the military for an element patrolling the border, but the term here
carries more applicable connotations for non-military readers and for other scenarios.
Sensors: Any technology used to detect movement at a given place along the border.
Enemy: The threat group that the friendly forces are attempting to kill or capture. This
includes all re-supply that is crossing the border. Most of the enemy units do not intend to
operate full-time along the border, but are moving through in an effort to affect operations
further inland from the border.
Paths: A route that the enemy could use to move from a refit area across the border, and
into the inland area of the country.
Infiltration: Enemy movement crossing the border. This is a subset of all possible enemy
actions and relates specifically to movement.
28
Interdiction: The process of actively stopping an enemy unit that is infiltrating across the
border. Usually initiated from a covert emplacement of soldiers.
2.4
Example Scenario - Operations in Afghanistan
Consider the following scenario as an illustrative example of one application of an analytical approach to borer interdiction operations (security operations not including Port of
Entry operations) that uses as context American experiences from the last 2009-2012 in
Afghanistan. We use this as the primary example, but acknowledge that many of the need
for border interdiction operations is not restricted to major US military engagements.
United States military forces in Afghanistan include numerous combat brigades, each consisting of approximately 4000 soldiers. Each brigade has an Area of Operations (AO) for its
mission that includes training Afghan security forces and counterinsurgency tasks. Some
of these brigades are deployed along the Afghanistan-Pakistan border and consequently
conduct some border interdiction operations in an effort to prevent insurgent personnel and
materiel from crossing into Afghanistan. As an example, one of these brigades deployed
along the border consists of three battalions (800 soldiers each) and supporting assets. This
brigade assigns each battalion an Area of Operations within the brigade AO where each
battalion has a headquarters and a number of subordinate elements that can conduct security patrols or other missions. In this example, two of the brigade’s subordinate battalions
have Areas of Operation (within the brigade’s) that include the border (first and second
battalions), while the third battalion operates further inland from the border. Because their
primary missions of developing the Afghan security forces in the area and conducting counterinsurgency tasks take most of the forces, each battalion has a maximum of two platoons
(20 soldiers each) to assign to border interdiction tasks on any day as a guard force.
In this area of the country, Pakistan does not have a large security presence along the border,
and consequently, enemy forces frequently use the Pakistani side of the border to rest and
refit. Additionally, the majority of the enemy’s external supplies originates in Pakistan.
The enemy transiting the border varies in composition and could be a convoy of trucks
with supplies and fighters, a single motorcycle with bomb-making material, or a single
enemy fighter on foot.
In Figure 2.4.1, the border is denoted by the black dashed line. Enemy units are denoted
with red diamonds, and their usual infiltration routes across the border (not necessarily
29
known to the friendly forces) are marked with red arrows. Friendly forces’ bases are denoted with blue ovals, which are the locations of the headquarters elements (HQ). Friendly
forces’ security patrols serving as guards (two available platoons for each of the two borderadjacent battalions) are denoted by blue rectangles. The self-imposed boundaries between
battalions (a command and control measure) are denoted with dashed blue lines. The subordinate units of the battalions that are not conducting border interdiction operations are
not displayed. The sketch is not scaled, but the border in this region could easily extend to
a hundred kilometers.
Figure 2.4.1: An Example Scenario for Border Interdiction Operations: A combat brigade
conducting operations along a border and using limited patrolling capacity (blue rectangles) to attempt to interdict enemy forces moving across the border (red arrows).
The overriding question in the scenario is: how should the brigade / battalions employ the
patrols dedicated to border interdiction?
The enemy infiltration routes might be completely unknown or only partially known to the
friendly forces. Additionally, there could be a variety of road and trail types that cross
the border with the usual infiltration routes representing a relatively small subset of these
possible paths. The terrain might be restrictive for some of the routes - a valley, and more
open for others - a desert. These factors complicate any approach used to allocate forces.
30
A typical approach that the brigade headquarters might take would be to task the subordinate battalions along the border, first and second, to conduct operations to secure the border
region. There might be a specific requirement on the frequency of these operations or the
size of the forces involved, but specifics would be left to the battalion headquarters. This
decentralization is a necessary step for effective command and control of the forces in the
area, but also naturally complicates any centralized effort at border security for the brigade.
Additionally, any specific intelligence about an intended enemy infiltration would be sent
to the battalions and a larger, more directed operation might occur at the direction of the
brigade headquarters. Each of the two battalions with responsibility for some of the border would then conduct some security operations, and hopefully would coordinate directly
or through the brigade headquarters. Some of these operations would leverage ISR assets
and refined knowledge from previous security patrols. The battalion headquarters would
schedule these patrols based on whatever knowledge they have on hand, and would track
previous operations in an effort to create a useful log of where border interdiction patrols
have historically gone, and with what results.
Additionally, any useful information, such as aerial imagery, collected by ISR assets in the
area would be accessible through a map-based data aggregator. Some of this information
would come from ISR requests submitted by the battalions. These requests would usually
be based on which areas had not had ISR coverage recently. Intelligence analysts could
access a series of icons on a map representing available information about the border, and
this information could hopefully provide some context for future mission planning. This
information might be accessible at one time, but would probably not be aggregated in any
comprehensive estimate about overall enemy activity on the border. As a general rule,
without further intelligence, the battalion headquarters would most likely allocate patrols
on a rotational basis across the possible routes so that most possible locations are visited
with equal likelihood.
In the event that the unit had access to sensors, the battalions might choose to use them to
enhance their knowledge of the border. In that case, the unit would most likely prioritize
locations that might yield information about unknown routes or information about the types
of traffic on known routes. If the sensors were emplaced, the resulting data would be
available to the analysts. Additionally, the battalions might attempt to route the detection
alerts from the sensors into an operations center. In the event that a sensor was triggered,
the officer on duty would retask units as he saw fit based on possible enemy movement.
The use of ground sensors in this manner, in the experiences of the author, is uncommon.
However, both battalion headquarters and the brigade headquarters have a multitude of
other responsibilities, and without directed intelligence about specific enemy infiltrations,
31
border interdiction patrols frequently seem wasteful when compared with other mission
requirements. The prevailing question is still: where does a headquarters send a patrol when
there is limited information across dozens of kilometers? Even considering this and the
difficulty of gaining reliable intelligence on infiltrations, the interdiction of enemy forces
and supplies can have a significant effect on all manner of counterinsurgency operations
further inland. Therefore, any attempt at answering our two research questions - the better
incorporation of sensors and an improved allocation of soldiers - could greatly enhance the
ability of the headquarters in this scenario to secure the border.
2.5
Guards
A significant aspect of this work deals with improvements in allocating guards to interdict
enemy forces in a border region. We use the term guards as a catch-all for a variety of
possible military patrolling elements. The one requirement we place on a guard is that the
friendly forces headquarters unit allows it to operate independently.
The most common guard in the context of our example scenario would be a platoon of
soldiers, usually around 20, and if motorized, operating in three or four vehicles. This unit
size approximates the most typical US military units that operated independently in Iraq
and Afghanistan. Another possibility for a guard would be an armored vehicle section,
consisting of two tanks or Bradley fighting vehicles with 8 to 12 soldiers. This configuration was also used extensively in Iraq both for patrols and for overwatch positions of
the type considered here. This two vehicles section was allowed to operate independently
with less forces because of the increased protection afforded by a larger, armored vehicle.
Some terrain types limit vehicular elements, and the term guard could refer to a purely
dismounted element as well. Dismounted elements typically start at 9 soldiers, the basic
infantry squad in the US military, but occasional smaller scout elements could also be used.
Three variants of these guard types are depicted in Figure 2.5.1.
The most typical task associated with interdiction for a patrolling element is the emplacement of an observation post (OP) with a mobile, interdiction element from the same unit
located nearby. An OP is a small team of soldiers emplaced in an overwatch area, typically
covertly, who have the ability to conduct surveillance of a certain location. A mobile interdiction unit might be a squad of soldiers or a section (two) of vehicles that are located
32
Figure 2.5.1: Possible Guard Configurations: Three possible elements of a guard. (From
top to bottom) A Bradley fighting vehicle in an overwatch position in northwestern Iraq
(half of a guard unit); A three HMMWV platoon preparing for an reconnaissance mission
near Mosul, Iraq; A platoon of US and Afghan soldiers moving to a new overwatch position
near the Afghanistan / Pakistan border. Photos from author’s collection.
33
in a position to physically move into the path of probable enemy movement. A typical
mission for a guard unit assigned to interdict enemy forces in a certain area might look like
the following. A platoon receives the task to conduct interdiction operations of a certain
geographic area, usually specified as a Named Area of Interest (a designated location). The
platoon moves motorized towards the designated interdiction area and splits into two elements. The OP element moves on foot to a pre-planned overwatch site that gives them the
ability to surveil the NAI. The remaining portion of the platoon locates in an area near the
NAI where they can respond to any enemy forces identified by the OP.
The specifics of the platoon’s (guard unit’s) actions are generally left to the platoon leader
to plan. The higher headquarters generally plans the interdiction location (the NAI), and
leaves the execution details to the platoon.
This research could also extend guards as an aerial platform. We do not explicitly consider
it here, but this extension could easily take one of three forms. First, it could account
for a dismounted squad of soldiers that are helicopter-borne, and consequently much more
mobile. It could also account for helicopters used in a reconnaissance role. The army
frequently uses pairs of OH-58D helicopters in an aerial reconnaissance role, and will
continue the mission with AH-64D helicopters as the OH-58D phase out of service. Finally,
the research could extend to consider Unmanned Aerial Vehicles (UAVs) as guards. In this
configuration, the friendly forces would be using a UAV as a guard to interdict enemy
forces crossing a border and for possibly cueing additional forces.
2.6
Sensors
A portion of this work deals with improvements in security from friendly forces incorporating sensors in a more effective manner. The military has a wide variety of sensors available
for use, including an array of possible ground sensors.
2.6.1
Sensor Basics
The army fielded a number of UGS types during the Iraq and Afghanistan wars. Prominent
among these were the Falcon Watch system from the Harris Corporation, the Scorpion
from Northrup Grumman, and the OmniSense from McQ, Inc. These systems were used in
a number of roles, and featured the use of electro-optical and infrared cameras to capture
34
Figure 2.6.1: An Example of an UGS Camera - OmniSense cameras from an Unattended
Ground Sensor system made by McQ, Inc (This is an example of an available system, and
not an endorsement of a particular sensor technology)
images of an area after detection by a passive infrared or seismic sensor. The systems
consist of a number of components, namely a camera, some type of intrusion detection,
and a communications station that relays information back to forces, usually over a radio
frequency or satellite connection. An OmniSense camera is shown in Figure 2.6.1.
Currently, these systems are in the legacy category for procurement, as additional contracts
for development as well as Processing, Exploitation, and Dissemination (PED) have been
suspended. There are, however, a number of systems still in the army inventory. The UGS
program with current funding is called the “Expendable Unattended Ground Sensor” (EUGS) program, which is designed to provide some of the same intrusion detection capabilities but without a camera component. This program highlights their use for force protection
around a base, as well as the ability to network with other ISR sensors. The majority of
sensor use in Afghanistan has been this style of localized force protection emplaced by
units operating out of exposed outposts Shactman (2012). Even with this outpost security
focus, the E-UGS technology could be utilized in a border interdiction role. The E-UGS, if
linked to other ISR platforms or a central operating node could provide information about
enemy infiltrations along border crossing routes.
2.6.2
Border Sensor Use
Some military units deployed on a border have used UGS in an effort to gain knowledge
about different regions. However, the use has been intermittent at best, and to our knowledge, not of the scope and central purpose necessary for larger tactical gains. However,
sensors are frequently used along the United States - Mexico border as part of the Department of Homeland Security’s efforts to limit illegal crossings. These systems, because of
their dedicated use, are used in a more deliberate manner than some of the military applications. They include the goal of dynamically moving border police patrols to investigate
detections when the sensors are triggered. To our knowledge, there is not a comprehensive
35
look at leveraging sensor data and analytic techniques to improve patrol routes and sensor
locations.
Some of the UGS systems deployed on the US-Mexico border have been components of
larger systems involving mobile camera towers and fixed site sensor suites. The US government has recently been attempting to dramatically expand the number of sensors deployed
along the border. While some of the improved systems were delayed in 2013 because of
bandwidth issues, the overall concept - using UGS to trigger manned border police patrols
- remains strong Beckhusen (2013). The use of UGS by border protection services is not
limited to the US government. India recently expressed interest in installing a series of
UGS along its border with Pakistan Pandey (2012). These border security emplacements
by domestic police elements lend credence the idea of leveraging UGS in a military context
along a border.
36
Chapter 3
Problem Framework
Border interdiction operations are complex, and involve a variety of different participants,
plans, and operations. These include headquarters staff and patrolling units (guards), plans
for sensor emplacements and patrols, and mission execution. Before we develop any specific analytic techniques, we need to conduct an assessment of the different steps in the process. This will ensure that we create analytical solutions that fit into the currently executed
process without a drastic overhaul. Additionally, while we focus primarily on solutions for
the decision recommendation portion of the decision support tool, a holistic look at the entire process will better enable us to understand where the improvements fit into the overall
system. To conduct this assessment, we conduct a functional analysis of the overall border
interdiction process.
This chapter first describes are functional breakdown of the border interdiction system, and
then defines each of the functions involved in the process. We then use this functional
decomposition to show the current, as-is interdiction process. Finally, we illustrate the
areas of decision making that could benefit the most from an analytical approach, and then
describe the example scenario in terms of the analytically improved functions.
3.1
Functional Decomposition
The sub-steps involved in this problem – all the different components of border interdiction
activities – can be viewed as a system composed of various functions, with information
passing between the functions. The purpose of section §3 is to highlight this functional
structure in an effort to identify areas where improvements can enhance the effectiveness
37
of the units assigned to border interdiction missions and where we can create an analytical
process to answer our research questions. Additionally, the functional decomposition helps
present a clearer picture of the system by breaking down the tasks and connections between
those tasks that exist in border interdiction operations. The incorporation of data-driven
sensor use and dynamic retasking, outlined below, represent new steps for improving the
use of information collected along a border in an effort to enhance border security.
Figure 3.1.1: Functional Analysis Legend: Description of the different components used
in the Functional Diagrams
To present this functional analysis, we begin with the base process that describes the bulk
of military operations in a simple closed-loop fashion. We then sequentially add different
steps from the border interdiction system until we fully capture the functional processes
occurring in the system. To more accurately develop the functional analysis diagrams,
Figure 3.1.1 displays a legend that is used for the remaining functional diagrams. In the
diagrams, functions are displayed as rectangles. The green functions represent current military operations. The red function represents the current enemy operations. Blue functions
represent decision making functions designed to enhance on-going operations. The orange
function represents a guard analysis function related to enemy activities. The light blue
functions are estimation and data analysis tools. The gray circles represent exogenous systems that can have inputs into the border interdiction system. Arrows denote the products
and information that flow between functions.
3.1.1
Base Process
The initial functional decomposition of this system applies generally to most military operations. For this case, in Figure 3.1.2, the specific inputs and outputs relate to the border
interdiction mission. It is a functional representation of the normally executed patrol planning cycle that the military implements regularly, at various tactical levels.
The system begins with the initial intelligence available, and over time it can accept exogenous intelligence inputs. The intelligence staff conducts “Enemy Analysis”, which then
38
Figure 3.1.2: Functional Diagram - Base Process: A description of the border interdiction
planning and execution cycle currently in use
provides an evaluation of the enemy and terrain to the “Patrol Scheduling” function. This
function, typically executed by a planning staff, creates a patrol schedule, and units assigned to guard tasks execute missions based on that schedule. This schedule is typically
focused on identifying the time and place (NAI) for interdiction operations. These missions
impact the enemy in one of three ways: no effect, the guards capture enemy forces, or the
enemy forces observe the friendly actions and do something different from their original
plan. These results, along with debriefs from the guard units are evaluated. These evaluations and debriefs update the “Enemy Analysis” and the “Patrol Scheduling” functions to
further improve the process.
The Base Process represents operations as the currently exist. The actual implementation
of any of the above functions varies widely between units and missions, but this closed
loop process with the ability to receive new, exogenous, intelligence is the building block
for all systems that involve deliberately planned military operations. This is the process
used in the example scenario for the brigade operating along a border where the battalion
headquarters conducted the “Enemy Analysis” and “Patrol Scheduling” and any new ISR
information entered the system as new intelligence. The log of patrol activities and mission
debriefs were the main tools used in refining future missions.
3.1.2
Base Process Plus Sensors and Data Analysis
One goal of this thesis is to answer – R ESEARCH Q UESTION 1: How can we optimally
employ sensors (UGS) along a border to improve our understanding of the enemy and our
border security operations? To answer this, we can add additional functions to the base
process. In this case, in Figure 3.1.3, sensors represent the use of Unattended Ground
39
Sensors (UGS), but the framework could apply to additional types of internal ISR assets
used in a deliberate manner for gathering information about enemy border infiltrations.
Figure 3.1.3: Functional Diagram - Base Process Plus Sensors and Data Analysis: The
border interdiction planning and execution cycle augmented by the emplacement of ground
sensors
In this update, the base process continues to operate normally. However, we add functions to enhance the “Enemy Analysis”. First, the “Enemy Analysis” function provides an
infiltration specific evaluation to a function that determines the sensor placements. This
function provides recommendations to the friendly forces on the optimal sensor locations
based on the resources available. Once the sensors are emplaced, the operational sensors
detect enemy movement (on some portion of the border). These sensor detections, along
with the sensor locations, then feed the “Analysis of Sensor Data” function that provides
updated intelligence on enemy infiltration to the overall enemy analysis.
The use of ground sensors for military border security is not a new idea, but the analytic
framework here that uses the sensors deliberately to estimate enemy activity through the
“Analysis of Sensor Data” is a new construct.
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3.1.3
Base Process Plus Sensors, Data Analysis, and Other ISR
One additional improvement to this process is the ability to incorporate other types of sensor detections. These detections will primarily come from assets not organic to the friendly
forces unit responsible for border security. Consequently, we need to learn what locations
need additional collection, and then generate prioritized Intelligence, Surveillance, and Reconnaissance (ISR) requests for those locations. These requests will then be sent to the
higher headquarters in an effort to get additional ISR coverage. ISR coverage could come
from a variety of aerial platforms (manned or unmanned) that include sensors such as radar
and electro-optical. To successfully submit these requests, we need a method to prioritize
a list for where to gather additional information.
Figure 3.1.4: Function Diagram - Base Process Plus Sensors, Data Analysis, and Other
ISR: The border interdiction planning and execution cycle augmented by the emplacement
of ground sensors and incorporating the need to task outside ISR assets that occasionally
support border interdiction operations
This requires the addition of the “Determine Collection Requirements” function, as in Figure 3.1.4, which receives information gaps from the “Analysis of Sensor Data” function
and produces a prioritized list of ISR requests. This results in some additional ISR coverage, such as full motion video from an unmanned aerial vehicle (UAV) or radar coverage.
These new detections are then incorporated with the UGS detections in the “Analysis of
41
Sensor Data” function. ISR assets are widely employed in current border interdiction operations. The intent of these additional functions is to analytically refine the desired locations
for additional information and to provide a method for quantitatively including any new
ISR-provided detections.
3.1.4
Base Process Plus Sensors, Data Analysis, Other ISR, and Dynamic Retasking
The above process has the ability to provide much better intelligence for border interdiction
operations. However, the increased information about enemy actions provides updates only
to the planning process. An additional function is needed to provide real-time adjustments
to guard missions in an effort to immediately interdict enemy forces that have just been
detected. This dynamic retasking process, shown in Figure 3.1.5, and similar to crosscueing between intelligence assets, could greatly increase the utility of emplaced sensors
by integrating their detections with real-time execution.
Figure 3.1.5: Functional Diagram - Base Process Plus Sensors, Data Analysis, Other ISR,
and Dynamic Retasking: The border interdiction planning and execution cycle augmented
by the emplacement of ground sensors, ISR asset tasking, and accounting for the ability to
cue patrols in real-time based on sensor data
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We therefore add the “Dynamic Retasking” function which uses all previous inputs and
also receives real-time updates on the current statuses of various units. This function can
then provide a real-time retasking to a guard unit (or other friendly forces unit operating in
the area) in an effort to position a force for interdiction.
Once this dynamic retasking function exists, the patrol scheduling function can improve. If
guards can react in real-time to sensor detections, then this ability needs to be incorporated
into the scheduling. For example, prior to the addition of the dynamic retasking function
the optimal guard location might be at a position that also has a sensor. Because guards
and sensors have different purposes - interdict versus gather information - this could still
be optimal. However, if the guard can respond to what the sensor detects, then the new
optimal location for the guards might be further inland where they can respond to possible
detections from the sensor at the old optimal location and additionally interdict the enemy
at a different location. This extends the reach of the interdiction force.
3.1.5
Individual Functions
F UNCTION : E XECUTION OF C URRENT M ISSIONS
The “Execution of Current Missions” function represents the operations that guard units
are executing at the current time. The inputs for this function are the current guard taskings
received from the “Patrol Scheduling” function and any updated tasks from the “Dynamic
Retasking” function. Any possible number of guard units could be executing a border interdiction mission. In our example scenario, this would include a maximum of the four
platoons assigned to border interdiction. This function also accounts for non-border security missions that are happening at the current time in the border region. The outputs of
the function are the most abstract, in terms of measurement, but vital part of the system.
They are the impact of interdiction operations on enemy units attempting to infiltrate the
border. Possibilities include having no effect, capturing enemy forces, or forcing the enemy
to change future plans.
F UNCTION : O PERATIONAL S ENSORS
The “Operational Sensors” function represents active sensors that friendly forces have emplaced. The inputs to this function are recommended sensor locations, which are realized
upon emplacement, and enemy movement. This function is what allows friendly forces to
gather data about the enemy movement through whatever sensor capabilities are present:
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seismic, acoustic, or photographic. The outputs of this function are the sensor detections
that serve as the basis for data collection.
F UNCTION : OTHER O PERATIONAL ISR
The “Other Operational ISR” function represents active intelligence collection conducted
for friendly forces by assets outside of their immediate control. These include UAVs controlled by higher headquarters, and other types of airborne ISR platforms.
F UNCTION : C URRENT E NEMY ACTIONS
The “Current Enemy Actions” function encompasses all enemy movement currently occurring in the border area. The inputs to this function are unknown in our current system – they
include whatever inputs and information the enemy uses to decide on a course of action.
The outputs of this function, only some of which are visible, include any interactions with
friendly forces and detections by the sensors.
F UNCTION : E NEMY A NALYSIS
The “Enemy Analysis” function includes all steps that the friendly forces take to evaluate
and understand the enemy operating in the area. This includes the enemy composition
and activities as well as the terrain in the region. This function is the main location for
exogenous inputs, as any new intelligence gathered external to the system is incorporated in
this function. Outputs from this function include the enemy and terrain evaluations as well
as the infiltration evaluation, which is the subset of the evaluation specifically concerned
with movement across the border. The “Analysis of Sensor Data” function serves as the
quantitative counterpart to this function.
F UNCTION : PATROL S CHEDULING
The “Patrol Scheduling” function includes all steps that friendly forces take to plan for
future operations, out to some time horizon. This function receives the enemy and terrain
evaluations as an input, as well as any metrics on prior patrol performance. Any constraints
on guard operations such as the maximum patrol duration are applied in this function. The
output is the patrol schedule from the current time to the planned horizon, which consists
of tasks to guard units for all of those periods. Once the dynamic retasking function exists,
this scheduling function must also incorporate the ability for patrols to adjust in real-time
to detections.
F UNCTION : M ISSION F EEDBACK
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The “Mission Feedback” function provides feedback on the results of the operations to
provide a closed loop framework to the process. The inputs are the mission results and
patrol debriefs, which provide updates to the enemy analysis and to the patrol scheduling
functions. This function allows the friendly forces to learn from the operations they plan
and conduct.
F UNCTION : D ETERMINE S ENSOR P LACEMENTS
The “Determine Sensor Placements” function takes the current infiltration evaluation and
the available sensors and provides recommendations on the most useful places that the
sensors can be placed along the border. This is an infrequently used function because of
the high cost in time and risk for a unit to move an already emplaced sensor. It is needed
when additional sensors are provided to the system, such as when a higher headquarters
allocates more sensors to the friendly forces.
F UNCTION : A NALYSIS OF S ENSOR DATA
The “Analysis of Sensor Data” function receives the sensor locations and detections and
aggregates them for qualitative evaluation as well as applying analytics to help update the
enemy analysis function with quantitative insights about enemy traffic crossing the border.
This function is the central processor for transforming sensor detections into intelligence
on enemy infiltration patterns.
F UNCTION : D ETERMINE C OLLECTION R EQUIREMENTS
The “Determine Collection Requirements” function prioritizes the locations where additional information on enemy infiltrations is needed. This produces a list of requested ISR
locations, which if collected on, result in additional detections that can be incorporated into
the “Analysis of Sensor Data” function. This relates to assets that are controlled by a higher
headquarters, and are not organic to the friendly forces.
F UNCTION : DYNAMIC R ETASKING
The “Dynamic Retasking” function is the real-time planning update that provides the ability
to retask a currently operating unit in an effort to interdict enemy forces that have just been
detected by a sensor. This function receives all previous inputs as well as the locations
of any other friendly forces available for retasking but not executing a guard mission. It
operates concurrently with whatever Common Operating Picture the friendly forces unit
uses to maintain awareness of what is happening in the area of operations.
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3.2
Example As-Is Operations
Based on this functional decomposition, we can now revisit the example operations from
Afghanistan previously described with the example scenario and evaluate how each of the
functions are currently executed. The operational diagram in Figure 3.2.1 includes descriptions of the functions involved in the border interdiction process as they are currently
executed by most army units. Different units might execute some functions with slight
modifications, but the overall approaches remain the same.
Figure 3.2.1: Operational Diagram - As-is Border Interdiction Operations: A description
of the system functions as they are currently executed by army units assigned to border
interdiction tasks; the functions above the dotted line are typically not conducted, or if they
are, conducted without an analytical component
Of note, most of the functions above the dotted line that denote additions to the base process
are either not executed, as in the deliberate emplacement of ground sensors, or they are
executed without an analytical component. These current approaches show the opportunity
for analytical augmentation to some of the functions that would enhance the overall system.
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3.3
Functional Focus Areas
Based on the functional decomposition and operational diagram, we now frame our problem in the context of the need to improve the functions listed above. For this thesis, we
focus on the three decision support functions that involve locally controlled assets: “Patrol
Scheduling”, “Determine Sensor Placements”, and “Dynamic Retasking”. Additionally,
we outline some techniques for the “Mission Feedback” function as a way of improving
guard and sensor allocation. We leave the the “Determine Collection Requirements” for
future research since it involves assets outside the main friendly forces’ unit. Also, we
leave the data evaluation function, “Analysis of Sensor Data” for further research since it is
based more on networking information and data analysis as opposed to decision support.
Figure 3.3.1: Functional Diagram - Highlighted Focus Areas: Four functions in border interdiction operations related to decision making that could benefit from an analytic framework
3.4
Example Scenario - Revisited Analytically
We can now revisit our previous example scenario in the context of the highlighted functions above. Instead of limited security patrols based on whatever knowledge is at hand,
47
the brigade and battalion headquarters can take a more systematic approach to the problem.
First, they attempt to leverage the data collection possibilities by employing ground sensors
(green circles) in a deliberate manner using the “Determine Sensor Placements” function.
Over time, the data that these sensors collect enhances the unit’s normal enemy analysis
with outputs from the “Analysis of Sensor Data” function. These outputs are used as an analytical addition for the “Enemy Analysis” function. Additionally, intelligence collection
requests are improved with the “Determine Collection Requirements” which prioritizes the
locations for other ISR needs.
Figure 3.4.1: An Improved, Example Scenario for Border Interdiction Operations: A combat brigade conducting operations along a border and using an analytic framework to improve limited patrolling capacity (blue rectangles) to interdict enemy elements (red arrows)
by employing ground sensors (green circles) and functional improvements to the planning
process
This improved intelligence is incorporated into a more intelligent “Patrol Scheduling” function which includes any additional requirements and limitations that the commander may
have. This improved function uses previously detected infiltrations on certain routes and
the absence of detections on other routes to optimize patrolling locations. In real-time when
a sensor detects a target, the headquarters can leverage the “Dynamic Retasking” function
to develop a set of immediate courses of action. Finally, over time, the headquarters can
48
leverage the “Mission Feedback” function to improve the feedback from executed operations to enemy analysis and planning. This scenario is just one example - the terrain,
number and size of units, available assets and a host of other factors could vary between
scenarios, but the functions involved are the same.
The remainder of this thesis develops specific solution approaches to selected functions
from this process.
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50
Chapter 4
Solution Approaches
In this chapter, we develop our solution approaches to the four critical decision support
functions previously identified - “Patrol Scheduling”, “Determine Sensor Placements”,
“Dynamic Retasking” and simulation methods for “Mission Feedback”. We sequence the
chapter based on the functional buildup of the border interdiction system from base process
to a process that includes ground sensors and the ability to dynamically retask guard forces.
Additionally, we present a taxonomy of the various models that we formulate to highlight
their similar structure, and discuss their differences.
We first describe the approaches that we used to address the various functions in the context of the entire border interdiction system, and this serves as the sequence for the detailed
descriptions in the remainder of the chapter. We then conduct a literature review of the
relevant work, followed by a justification on model selection, and a discussion of the assumptions that we make. As we introduce the specific model formulations, we start by
discussing the basic construct that we adopt, followed by a detailed discussion of the necessary inputs. We then discuss a formulation for the basic “Patrol Scheduling” function
that could be implemented without sensors, followed by an expanded formulation that addressed the “Determine Sensor Placement” function. Next, we address a heuristic that
serves as the basis for the “Dynamic Retasking” function, followed by an enhanced version
of the “Patrol Scheduling” function that includes the ability to respond to sensor cues. Finally, we briefly discuss using the models to evaluate choices for the “Mission Feedback”
function, and close with a model comparison.
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4.1
Analytical Approach Overview
There are many valid solution approaches to the four functions identified in section 3.3. We
identify here the our selected approaches, with a detailed justification later in the chapter,
and highlight the linkage of the functions and their respective models in the context of the
border interdiction system.
Figure 4.1.1 outlines our analytical approaches to each of the functions using the graphic
developed in Figure 3.1.5. We proceed to outline the different functions and the specific
solution approaches using the same sequential approach used in section 3.1 to develop the
full functional decomposition of the system.
Before beginning with the base process, we need to formalize the general inputs into system - our ability to capture the necessary information about the terrain, friendly forces and
enemy forces. Initially, we describe a simplified version of the “Patrol Scheduling” function, formulated as a Mixed Integer Program (MIP), since this forms the crux of the base
process in the border interdiction system, and all security patrols and tasks tie back into the
outputs of this function (as seen in Figure 3.1.2). We then describe the inputs in the context
of the MIP formulation.
Next, we focus on using sensors for data analysis. To do this, we work to improve the
key input to the “Patrol Scheduling” formulation - enemy traffic flows - which are the
quantitative estimates of future enemy infiltrations. We describe the decision support tool
for locating the sensors with a MIP formulation for the “Determine Sensor Placements”
function. This completes the first level of improvement to the system for the decision
support tools (as seen in Figure 3.1.3).
As described previously, in this thesis we omit discussion of the “Determine Collection
Requirements” function that allows us to incorporate external collection assets into our
analysis and the “Analysis of Sensor Data” function that provides the information updating
mechanism for the system. These functions complete the next level of improvement to the
system (as seen in Figure 3.1.4).
After this, we outline the “Dynamic Retasking” function as an algorithm which allows
for an immediate attempt at interdiction after a sensor detection. Once this ability is incorporated, we can improve the “Patrol Scheduling” function MIP since the tasked guard
locations should account for the fact that the guards can adjust to real-time detections (as
seen in Figure 3.1.5).
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Finally, after revisiting and creating the enhanced “Patrol Scheduling” function, we outline
one aspect of the “Mission Feedback” function that uses simulation to evaluate the utility
of additional friendly force assets.
Figure 4.1.1: Analytical Approach Diagram: The analytical improvements recommended
for the four critical, decision making functions in the border interdiction system
4.2
Model Taxonomy
From a modeling standpoint, we present four different, but related models for implementation. These are the Basic Patrol Scheduling (B-PS) MIP, the Determine Sensor Placements
(DSP) MIP, and the Dynamic Retasking (DR) algorithm, and the Enhanced Patrol Scheduling (E-PS) MIP. In this chapter, we initially describe the flow capture construct, and then
outline a Basic Patrol Scheduling (B-PS) MIP that does not include the ability to incorporate sensor information. We then outline the DSP MIP that includes decision variables
for both scheduling and sensor placement. Then we describe the DR algorithm, which is
a greedy heuristic for the B-PS MIP. Finally, we formally describe the E-PS MIP which
incorporates knowledge of sensor location into the scheduling decisions, but works only
once the DR algorithm is implemented. We illustrate uses of the models for decisions in
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the “Mission Feedback” function. A description of the different models can be found in
table 4.2.1.
Name
B-PS
Model Description
Basic Patrol
Scheduling MIP
DSP
Determine Sensor
Placement MIP
DR
Dynamic Retasking
Algorithm
E-PS
Enhanced Patrol
Scheduling MIP
Need
“Patrol Scheduling”
in the absence of
sensors
“Determine Sensor
Placements” to
enable sensor
detections into
patrol scheduling
decisions
“Dynamic
Retasking” function
to enable a real-time
guard response to
sensor detections
“Patrol Scheduling”
function that
accounts for the
presence of sensors
Decisions
Guard locations
over time
Guard locations
over time and sensor
locations
Feasible guard
location with
highest likelihood of
intercepting
detected enemy
movement
Guard locations
over time (after
sensors are
emplaced)
Table 4.2.1: Model Descriptions: A table summarizing the different models that we present
in this chapter, along with the differences in their use.
4.3
Literature Review
This section describes some of the previous work related to border interdiction, facility
location in a network, sensor use for border security, and other applicable military models.
We consider the previous work related to two different classes of problems involving traffic
in a network.
4.3.1
Network Interdiction Models
The most frequently researched class of problems related to resource allocation with border
security is known as network interdiction. In this class of problems, the research question
is determining the most important arcs in a network on which to employ resources, and the
54
most common solution methods for this class of problems are linear and nonlinear programming. Network interdiction studies began in earnest during the Vietnam War as a means
of modeling enemy lines of communication and supply lines. Work started on identifying
the most vital arc in a network in an effort to understand the optimal location for interdiction assets or bombing (Wollmer, 1964). Two common solution methods that tie into
other network problems are minimizing the maximum flow (Wood, 1993) and maximizing
the shortest path (Israeli and Wood, 2002) where arcs are disrupted to make the use of the
network more complicated for an enemy force. This relates to supply lines as a means of
modeling the most effective ways to impede enemy supplies and logistics, with the goal
minimizing the total amount of supplies that the enemy can send through the network, or
making the supply route longer.
These network interdiction models can apply outside military contexts with drug interdiction (Wood, 1993), information security in a communications network, and the smuggling
of nuclear material (Pan et al., 2003). Most of these constructs are designed with a game
theoretic approach, with the defender and attacker (or interdictor and infiltrator) portrayed
as rationally intelligent. This game theoretic construct is selected because of the ability to
incorporate enemy intelligence, but greatly complicates the tractability of more complex
models.
Two different game theory approaches are used as part of the two player zero-sum approach to network interdiction. The first approach is sequential moves. If we consider
a situation where someone is attacking enemy supply lines, then the attacker moves first,
and then the defender (or supply line operator) then moves to operate his degraded network as efficiently as possible (Wood, 2010). The second approach is simultaneous play.
Consider a network with an infiltrator attempting to cross through and an interdictor trying to stop him. If they both attempt to complete their actions at the same time, then they
both move with knowledge of how the other can act, but without knowledge of the specific
choices made (Washburn and Wood, 1994). In the context of an infiltrator, each arc has a
known probability of capture before resources are employed on it and afterward resources
are employed. Each player then makes the optimal choice for minimizing (infiltrator) or
maximizing (interdictor) the probability of capture. This model can extend to border security with multiple types of interdictors. Sequential play can also extend to border security
but is more suited for a specific high-level smuggling attempt – the interdictors setup their
defenses and then the smuggler makes a decision about where to go (Pulat, 2005). Pulat’s
work also incorporates the possibility of placing sensors that can work in tandem with a
capturing element.
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4.3.2
Flow Capture Models
Another class of model that relates to traffic in a network is the flow capture model. Flow
capture models are a variant of the facility location problem and are based on determining
the optimal location for a certain facility based on flows in a network. This type of model
was first introduced by Fouska as work on discretionary service facilities (such as convenience stores and Automatic Teller Machines) (Berman et al., 1992) and independently by
Hodgson (Hodgson, 1990). The premise is that for some services, the traffic in a network
will stop for a service only if it is convenient while en route to another destination, and
will not make a dedicated trip. Therefore, the location of the facility must be based on the
existing flows and not on a desire by the traffic to make deliberate routing decisions based
on the discretionary services. The basic model was expanded for a number of generalizations, including traffic deviating by a small amount from their pre-planned routes for the
service (Berman et al., 1995). These models are generally used on transportation networks.
Our literature review resulted in no examples where these models were applied in a border
security context.
4.3.3
Other Border Security Models
There are a number of other models of border security measures that highlight the wide
variety of techniques possible. Bessman introduces a stochastic dynamic programming
model for an intelligent smuggler who can update his decisions to evade an interdictor in
a maritime setting (Bessman, 2010). Cfir outlines a model to show the optimal searching
patterns for cameras deployed on towers along a border (Szechtman et al., 2007). Patrascu
details heuristics for determining the best camera locations along a border based a set of
cameras with user-specified parameters and a probability of detection for different locations based on those parameters (Patrascu, 2007). Yildiz uses an agent based simulation
to compare the use of different types of UAVs and sensors to explore the utility of equipping border patrol forces with small, hand-launched UAVs Yildiz (2009). Ordonez builds a
comprehensive systems model intended to illustrate the effects of border security policies
between sectors of the US-Mexico border Ordonez (2006). All of these models capture
some specific element of border security, but have goals distinct from a military decision
support tool for sensors and patrolling guards.
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4.3.4
Related Models with Military Applications
There are a number of models with military applications that relate to this work in terms of
sensors. Haider considers the specifications needed for UGS employment to assist precision targeting using a comparative study Haider (1998). Lamm considers the use of UGS
in sensor blocks, akin to a non-lethal minefield for information collection, where forces can
employ groups of sensors to gain information about a certain region Lamm et al. (2002).
The research develops a method for understanding the necessary density of UGS in order
to inform the required number of sensors in different combat formations. Separate from
UGS, but related to the idea of linking sensor allocations with guard performance, Rozen
introduces a model for employing a single UAV and a single patrolling vessel in a maritime
corridor using dynamic programming Rozen (2009). Culver explores a model about allocating reconnaissance assets that includes the effects of synchronizing ground based units
with UAVs to enhance the information gleaned from a reconnaissance objective Culver
(2013). A framework approach that could be used in some border security contexts was
developed by Keefe and Sullivan at RAND to expand on the IED hot-spot methodology
developed for the war in Iraq. Their Actionable Hot Spot methodology uses data to create
hot spots (through clustering and other traditional means) while considering constraints on
the resources that might be used to affect the hot spots so that the final result incorporates
what is possible for friendly force actions Keefe and Sullivan (2011).
4.4
Model Selection
For our application to border interdiction in the military (with resources as sensors or
guards), we do not use the network interdiction model. Because the emphasis in network
interdiction models is on a single smuggler - or a smuggling group making single attempts
- these models are hard to generalize to a wider border context with multiple infiltrating
groups. As stated, network interdiction models work extremely well for single attempt,
high-level smuggling actions such as a group of terrorists crossing into the United States
from Mexico, where deliberate preparations are taken by both sides based on this single
attempt.
In a military context with the network interdiction class of models, there might be a number of different enemy players each attempting to get through the border region, but with
a wide variety of origins, destinations, and goals. For example, the locations of enemy
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support zones and supply caches on the sanctuary side of the border could lead to different origin nodes for different portions of enemy forces. Likewise, certain enemy elements
might have different destinations inside the country which weights their activity towards
certain destination nodes. This is further complicated by the fact that different enemy elements might be of differing levels of concern to friendly forces. These factors all lead
to a situation where the enemy may not operate in the same manner as a single, rationally
intelligent infiltrator. Consequently, this class of models does not capture the necessary
factors involved in the border interdiction system. As a consequence of not using this game
theoretic construct, we will have to use a different modeling aspect to account for a learning
enemy.
The flow capture model provides a number of benefits. First, it provides additional modeling power by providing a method for modeling the multiple enemy characteristics present
in the real problem. It also allows for a very usable set of decisions as part of the resource allocation that easily turn into a practical schedule. Additionally, it provides for
very pragmatic solving - the use of integer programming - which allow it to scale well to
realistically-sized problems. Most importantly for our system, the flow capture model provides the best method for bridging data about enemy movements into a modeling input, as
described in section 4.7.3.
4.5
Assumptions
We make a number of assumptions and approximations to successfully implement the models.
The largest assumption we make is that each guard has the same capability to interdict
an enemy force. Related to this, we also assume that any employed sensor can gather
information about any node (road intersection). These assumptions mean that the model
does not contain an exact representation of friendly forces capabilities. This means that
decision makers would need to use caution for new, unproven sensors, and would need to
only consider guards for interdiction and patrolling responsibilities if they thought that the
guard unit had the capability to interdict an enemy force.
In terms of tactical actions, we approximate the manner in which a patrol would actually
emplace an observation post or overwatch position by using the designated interdiction
location, or Named Area of Interest (NAI), as the destination for the patrol. This means
that for calculating routes and timing, we disregard the patrol-internal decisions related to
58
tactical deployment, such as internal OP locations, overwatch elements, and interdiction
team organization. We also calculate the intended route based on the ability to observe
during travel from the guard’s base, which assumes that the patrol has some limited capacity
to interdict on the move, and also that the route the patrol selects from a tactical, patrollevel standpoint will generally not deviate that greatly from the estimated route. These
assumptions are reasonable because they hold for all options the model considers, and
because they closely match the current staff planning process.
In terms of implementation, we approximate the timing needed for the schedule using the
largest granular level possible, since that allows us to solve a larger problem. For example,
if we can reasonably solve 12 time steps, then the schedule is more helpful if each time step
is 4 hours than if it is 1 hour. This is reasonable in terms of modeling enemy movement,
but puts some the smaller detailed planning within the time step back on the planning staff
and guard patrol leader.
4.6
General Approach - Flow Capture
We considered the problem through a functional decomposition in 3 to ensure that we had
an understanding of each of the main functions in the system. While the each of the critical
functions could require a unique approach for analytical improvement, we generally use the
flow capture model as the basis for starting the solutions and as the underlying formulation
for the two most critical decision support functions, “Determine Sensor Placements” and
“Patrol Scheduling”. Flow capture models are a variant of the facility location problem,
where in our adapted context the guards become the facilities to locate over time. The basic
topological input to the formulation is a network, where given the traffic flows on different
origin-destination pairs. The flow capture model uses a mixed integer linear program (MIP)
to solve for the optimal flow-capturing resource locations. Figure 4.6.1 illustrates a general
example of a flow capture scenario, as taken from its original context. In this example
there is one origin node, one destination node, and 9 additional nodes. There are 5 possible
origin-destination paths (shown in red), and each has a corresponding flow, which in this
case is the same for all paths. This network could represent traffic in an urban area, or
customers traveling through a shopping center. Given that there is one facility to locate,
such as an ATM, where should it be placed? In this simple example, the answer is at node
5 or node 8, and can be determined by inspection.
While the solution to the optimal location in Figure 4.6.1 can be determined by inspection,
in more complicated models a MIP formulation is needed. The general construct of the
59
Figure 4.6.1: Flow Capture Scenario Example: A toy example of a situation where the
flow capture model can be used to solve for the optimal facility location (blue rectangles)
based on the flows (red arrows) in the network
MIP is:
maximize: The expected amount of flow captured
subject to:
Available facilities, flows, topology
Variables:
Facility location
To properly formulate this MIP, we define the following notation.
Network
p∈P
fp
M
zj
=
yp
=
N nodes and E edges representing the trafficable routes in
the area
Paths for Origin-Destination pairs in the network
Traffic flow on path p
Number
of facilities available
(
1
if facility is located at node j
0
otherwise
(
1
if a facility is located on path p
0
otherwise
The formal flow capture model becomes:
60
max
∑
y,z
f py p
p∈P
s.t.
∑zj ≤ M
j
yp ≤
∑ zj
j∈p
y p ∈ {0, 1} ∀p
z j ∈ {0, 1} ∀ j
One key aspect of these problems, useful in our construct, is that they can account for “cannibalized flows”. If there are two facilities on the same path, no additional flow is captured
beyond what a single facility would have captured. This prevents the formulation from
placing multiple facilities in locations that would repeatedly capture the same flow, even
though we really gain no benefit from this repetitive capture. The programming formulation accounts exactly for “cannibalized flows” and it models the effects of two facilities
that impact the same flows. We will expand on this simple formulation as we develop some
of the different functions.
4.7
General Inputs
In our approaches to these different functions there are a number of common inputs. Most
of the inputs involve basic knowledge of the area of operations and the forces operating inside it – terrain, friendly forces, and enemy forces. These inputs will serve as the backbone
of the specific models we develop to implement the four functions of interest.
4.7.1
Topology
The first aspect of the border interdiction system that needs to be formalized is the terrain.
We need some capacity to capture the realities of the terrain in the border region, and as
described in section 4.6, the flow capture model is based on a directed graph.
The base input for the system is therefore a directed graph of nodes and edges that models
the trafficable routes in a border region. Edges exist along all possible routes, scaled for
61
the types of units the model is intended to plan for, and each edge would have a distance
associated with it. For example, if the traffic of concern was only motorized, then the only
routes needed in the model are those that can handle vehicular traffic. The distances are
scalable to incorporate aspects of road quality in the single distance metric. Nodes exist
at any intersection, and further additional nodes needed for methodological reasons can be
added, as needed.
The network includes an origin subset that consists of enemy source nodes on the neighboring country’s side of the border. These nodes originate the traffic in the model onto
edges that cross the border. For example, in Figure 4.7.1, the roads that start in Country
A would be modeled as beginning at an origin node. The major roadway in the lower left
could be modeled as the destination node if the enemy’s ability to reach that roadway meant
greatly increased freedom of movement and a much more difficult time interdicting them.
Additionally, the network has a subset of base nodes, which are the areas that serve as refit
points and secure areas for the guard units.
Figure 4.7.1: Example Road Network: A simplified road network in a border region where
the ends of the roads in country A are modeled as origin nodes, and the major roadway in
the lower left is modeled as the destination node.
These edges, nodes and subsets capture the simplified topology of the border region and the
terrain restrictions on friendly and enemy forces that operate in the area. Once this topology
62
is known, paths can be developed that show the possible routes between all the origindestination pairs. For example, if there were two source nodes and two sink nodes, then
there are four origin-destination pairs, and there might be multiple ways to travel the length
of each pair. This research does not address solving for these distances. The network is
known, and basically unchanging, so any shortest distance algorithm will suffice. Planners
could refine the possible enemy path choice options by pruning these possible combinations
with some common sense approaches based on knowledge of the details at the origin and
destination nodes.
4.7.2
Friendly Forces
The second major aspect of the border interdiction system that needs to be formalized is
the composition of the friendly forces.
A decision support tool for this problem would be most effective if it could incorporate a
wide variety of possible sensor types and guard units. It should not be technology-specific;
it can best serve a planner when it incorporates whatever sensors and guard types that a unit
has at the time. With this approach, the tool can assist a commander and his planners for
any ‘grab-bag’ of assets that may be available, immaterial of a specific technology.
With respect to sensor specifics, this thesis accounts only for the fact that a sensor meets
the criteria of allowing an operator to determine the presence of traffic at an intersection.
The specific capabilities of a certain sensor (range, detection type, imaging quality) are not
considered. The solution approaches discussed here require the ability for the detection to
be classified as a threat - this could be automatic, or with human-in-the-loop processing.
There is a large amount of work on sensor technology that can automatically recognize a
target and classify it as a certain type, however this technology is not a requirement.
A variety of unit types could be assigned to missions that include a border security component. In general, these units would be roughly of platoon size, and ground-based, probably
motorized. However, as long as the commander views the unit as capable of operating independently, the specific composition is not a necessary input. Each unit has four defining
characteristics that the planner inputs: speed, initial base location, patrol duration, and refit
time. This allows for mission selection based on how quickly units can travel and how long
they can operate before refitting.
Additionally, planners need to decide on a time horizon for the schedule planning. This
represents the limit of the schedule and will be decided as a function of scheduling neces63
sity, information available for future scheduling (availability of forces), and the tractability
of the formulations used.
4.7.3
Enemy Actions
The third major aspect of the border interdiction system that needs to be formalized is the
composition of the enemy forces.
Because of the methods selected for the functions, based on the flow capture MIP, we
simplify the necessary aspects of the enemy forces into flows in the network. Our main
construct for the enemy in the problem is a set of flows along origin-destination paths. If
we expect a certain number of enemy forces to traverse a path in a given time period, then
the flow on that path is the quantity of those forces. These flows are scalar values, but the
values can incorporate a number of descriptors about the enemy forces, if desired. This
allows the flow value function to serve as a weighted value function for the different types
of enemy with different destinations. For example, expected infiltration by motorized assets
might be scaled as twice as high (twice the flow value) as a dismounted enemy. Likewise,
enemy traffic headed towards a certain destination node might be weighted more heavily
than enemy traffic that is routing to an area that is less of a concern. This scaling allows
units to fine tune to whatever degree they choose the differences in enemy forces.
The more knowledge a unit has of an area, the more refined the initial enemy flow estimates
can be. However, without a lot of knowledge, the unit can still start the process. For
example, unit planners could develop the possible paths from the network and then equally
weight flows across all the different paths. With more knowledge of the network, the
planners could weight some paths, such as those with less security or better road surfaces,
higher than others, or segment the paths into clusters based on destination and equally
weight across destinations and not necessarily on individual paths. These values can be
uniform across time periods, or the planners can weight them differently in different time
periods. An example of this would be higher flows at night, or a much smaller amount of
flow on a certain day of the week.
We do not address it explicitly here, but the types of data available for analyzing enemy
movement along a border are frequently in a form that closely matches this path-based
estimate. Consequently, these enemy flows provide a relatively easy manner for capturing
the data available.
This input clearly represents the most uncertain and subjective input into the formulations.
This is partly a function of the difficulty in military planning to accurately capture detailed
64
knowledge of the enemy forces. It is, to a smaller degree, a necessary simplification to
create a workable model. Consequently, solutions to the functions in the problem should
consider a variety of different enemy flows as a means of testing the sensitivity of the
models. We introduce later the concept of a robust MIP to assist in finding good, valid
solutions for variations in the enemy flows.
These enemy flows create the set of flows for every origin-destination path considered in
the network for all time periods in the horizon. This is then the quantitative representation
of enemy movement across the border for the purposes of “Enemy Analysis”.
4.8
Basic Patrol Scheduling (B-PS)
We begin our description of the individual solution methods by describing the main decision support function in the base process, “Patrol Scheduling”, by developing the B-PS
MIP. We will add additional functionality to this formulation later in the section, and with
a robust optimization approach in the next section. “Patrol Scheduling”, as depicted in
Figure 4.8.1, is a function that takes as inputs the enemy and terrain evaluations from the
“Enemy Analysis” function and any updated performance metrics from the “Mission Feedback” function. It outputs a guard patrol schedule out to the planning horizon that serves
as the core operational planning document for the friendly forces operating in the border
region.
4.8.1
B-PS Model Description
We initially approach the patrol scheduling function as a flow capture, deterministic optimization problem as described in section 4.6. Using the inputs described in section 4.7we
develop a mixed integer linear program (MIP) that creates the best possible patrol schedule
for the available units. The concept of the formulation is:
maximize:
subject to:
Variables:
The expected amount of enemy units captured
Number of available guard units
Guard unit parameters (speed, base, patrol duration, refit time)
Network topology
Enemy flows
Guard locations
65
Figure 4.8.1: Placement of The “Patrol Scheduling” Function in the Border Interdiction
System
66
This formulation is a variant of the flow capture model that incorporates time. To fully incorporate some of the realistic limitations on guard abilities and to make a useful schedule,
we add some additional constraints to the formulation. First, we use the time parameter to
allow the model to incorporate guard speed and travel time, patrol duration limitations, and
patrol refit times. These let the model truly incorporate normal scheduling concerns that
a friendly forces headquarters would have to consider, and are reflected in subsectionsection 4.8.4, equations 4.8.4, 4.8.5, and 4.8.6. These factors are guard specific, and allow for
the model to fully account for the different factors that might impact a guard unit.
One distinct difference for this model is that we incorporate the ability to model guards
that do not capture the entire flow on a path. Multiple guards on a path would lead to complications, with subsequent guards capturing a percentage of the flow that escaped the first
guard. Adding an exact representation of this to the model would lead to a large number of
non-linear constraints, so we instead add a linear approximation of having multiple guards
on the same path to the model. Each guard has an effectiveness parameter that represents
the expected percentage of flow that the guard could capture. This parameter might be less
than 1 if the unit had limited lethality, or a limited observation range that allowed some
enemy traffic to bypass their position. This addition allows the model to consider the need
for multiple guards on the same path while incorporating a realistic limitation on patrols
and is reflected in subsection section 4.8.4, equation 4.8.10.
Additionally, we incorporate a new variable that allows us to account for the impact of
different mission configurations on the outcome. For example, we add a constraint to the
model that incorporates a penalty for a guard that patrols a location that was previously
occupied by a different guard. This is realistic since the shuffle between different guard
units at a certain location will probably degrade the overall friendly forces’ effectiveness
since some operational knowledge will be lost in the transition between units. This is
reflected in subsection section 4.8.4, equations 4.8.7, 4.8.8.
4.8.2
B-PS Terminology
To create this formulation, we first clarify additional terminology, and define our variables.
67
Network
Traffic
Guards
Sensors
Capture
Origin nodes
Destination nodes
Base nodes
p∈P
δi j
f p,t
zk, j,t
ek, j,t
y p,t
durk
re f itk
speedk
α
βk
θ
=
N nodes and E edges representing the trafficable routes in the border
region
Enemy individuals and units crossing the border
The subset of the friendly forces tasked to border security operations
Any technology used to detect traffic at a given place along the border
The generic result of a guard interdicting an enemy unit (stop and
release, capture, or kill)
The subset of the network that serves as the source for the traffic
The subset of the network that serves as the sink for the traffic
The subset of the network that serves as a base for guard units
Paths for Origin-Destination pairs in the network that
represent all the possible options for enemy travel across
the border
Shortest distance between nodes i and j in the network
Traffic
flow on path p at time t
(
1
if guard kis assigned to patrol node jat time t
(Decision
0
otherwise
Variable)
Factor between 0 and 1 for the mission-related
effectiveness of the unit based on the configuration of
guards to topology
Factor between 0 and 1 representing the percent of traffic
captured from path p at time t
The maximum patrol duration allowed for guard k
The refit time necessary for guard k
Parameter for the maximum travel speed of guard k
Parameter for controlling the value of having multiple
guards on one path
Effectiveness parameter that gives the capture value of
guard k based on the guard’s composition
Parameter that decreases the value of capture based on the
mission configuration of the node at t − 1
68
4.8.3
B-PS Inputs
The following inputs are necessary for the MIP, and are the quantified elements from section 4.7.
Topology:
Network, Origin nodes, Destination nodes, Base nodes, δi j
values
Traffic behavior:
f p,t for all paths and times, estimated enemy speed
Guard details:
Duration, refit, initial location, and speed for all guard units
Planning horizon: The number of time periods the patrol schedule is planned
for, T
69
4.8.4
B-PS Formulation
O BJECTIVE F UNCTION
max
e,y,z
∑∑
f pt y pt
t p∈P
C ONSTRAINTS
∑∑ ∑
t
zk j0 : given ∀k
(4.8.1)
zk jt = 0
(4.8.2)
k j∈O,D
∀k,t
∑ zk jt = 1
(4.8.3)
j
zk jt ≤ zk, j,t−1 +
zk,i,t−1
∑
∀k, j,t
(4.8.4)
i:δi j ≤speedk
t+durk +1
∑ ∑ zk, j,m ≥ ∑ zk, j,t − ∑ zk, j,t+1
m=t+2 j∈B
j∈B
∀k,t < T − durk
(4.8.5)
j∈B
t+re f itk
∑ ∑ zk, j,m ≤ durk ∑ zk, j,t + durk (1 − ∑ zk, j,t+1)
m=t+2 j6∈B
j∈B
ek jt ≤ 1 − θ
∀k,t < T − 1
(4.8.6)
j∈B
∑ zm, j,t−1
∀k, j,t > 0
(4.8.7)
m6=k
ek jt ≤ zk jt
∀k, j,t
y pt ≤ ∑ ∑ ek jt
(4.8.8)
∀p,t
(4.8.9)
k j∈p
y pt ≤ α + ∑ ∑ βk ek jt
∀p,t
k j∈p
y pt ∈ [0, 1] ∀p,t; zk jt ∈ {0, 1} ∀k, j,t;
ek jt ∈ [0, 1] ∀k, j,t
70
(4.8.10)
C ONSTRAINT E XPLANATIONS
The following list summarizes each of the constraints above.
(4.8.1)
(4.8.2)
(4.8.3)
(4.8.4)
(4.8.5)
(4.8.6)
(4.8.7,4.8.8)
(4.8.9)
(4.8.10)
The initial guard locations are known ahead of time, and serve as the
initializing conditions for the patrols.
No guards can patrol on the origin nodes because they are in the
neighboring country. Additionally, no guards can patrol on the sink
nodes because they represent the limiting location where no further
interdiction can happen.
Guards can only be assigned to one place at a time.
Future guard locations are restricted to locations that the guard can
reach within a time period based on the guards’ speed.
If a guard is on a base at time t, and on a patrol at time t + 1, it must
return to base prior to its maximum patrol duration in order to refit.
Upon returning from a patrol, a guard must stay on base long enough
to refit.
The effectiveness factor is below 1 for guard k at node j, when a
different guard was at node j at t − 1. This accounts for a decrease in
effectiveness due to transitioning guards on the same node.
If a guard is on path p at time t, then the capture factor, y, can take a
value greater than 0. This means that a capture can only occur for a
certain path if there is a guard on that path.
The capture factor increases linearly with the number of units on the
path. This represents an increased ability to capture traffic based on
having multiple guards.
This formulation reflects basic patrol scheduling for the guards. It completes the addition of an analytic decision support tool to the base process involved in planning border
interdiction activities.
4.9
Determine Sensor Placements (DSP)
We now proceed to the second major decision support function in the system, “Determine Sensor Placements”. The ability to effectively incorporate improved enemy estimates
hinges in large part on the initial emplacement of the sensors. Since these are difficult to
71
move once emplaced, the decision about where to place sensors has a more enduring impact than the decision on where to send a certain patrol. “Determine Sensor Placements”,
as depicted in Figure 4.9.1, is a function that takes as inputs the enemy and terrain evaluations from the “Enemy Analysis” function and any updated performance metrics from the
“Mission Feedback” function. Additionally, the availability of any new sensors is detected
by this function. “Determine Sensor Placements” outputs recommended locations for the
sensors.
Figure 4.9.1: Placement of The “Determine Sensor Placement” Function in the Border
Interdiction System
The sensors in the network provide two capabilities. First, they provide the mechanism for
refining the flow value estimates for future use in the scheduling and other functions. In
this capacity, the sensors provide the best mechanism for “learning” over time. Second, the
sensors provide an ability, through the “dynamic retasking function”, to collect information
for immediate use that allows real-time guard adjustments. Sensors at any node can provide
utility for both of these functions, but the optimal location for one may not be optimal for
the other. We focus our formulation on the ability to support a patrol in real-time, knowing
that the sensors can still provide useful information for updating the estimates of enemy
flows. To illustrate these functions we introduce a sequence of simple example scenarios.
In the following examples, we assume there are two sensors and one guard. The border
72
is denoted by the dashed black line. Consider the simple network in Figure 4.9.2, which
represents three valleys that lead across a border with a single connecting road accessible
by the guards. The guards are unable to locate at the origin because it is across the border
or at the destination since it signifies the area in the region where capture is no longer likely
because of excessive branching, population density, or another factor. If we assume equal
flows across the three different paths (in red), each with the same amount of uncertainty,
then a sensor at any node provides the same level of “learning” for future use in refining
flow values. The sensor would only provide useful information about the enemy path on
which it was located.
To assist guards in dynamically intercepting enemy forces in real-time, the optimal sensor
locations are on nodes 1 and 3 with the guard at node 8. This allows the guard to move from
the middle path to either of the other paths if the sensors detect an enemy movement. If the
guard’s speed was significantly greater than the enemy’s speed, then the guard’s location
could be either nodes 7, 8, or 9 as long as the sensors were on the two other paths. The key
is that the guard occupies the path that does not have the early detection from the sensors.
Clearly the ability to correctly position sensors relies to some degree on the relative speeds
of the guards and the enemy, the quantity of guards, and the likely location of those guards.
Figure 4.9.2: Sample Network 1: An illustration of employing sensors and a guard in tandem to capture the optimal amount of enemy forces in a network with limited connectivity.
The guard (blue rectangle) can move either left or right in response to a sensor (green
circles) detection.
Consider now the same network, but with connecting roads from the outer routes to the
73
central route, as depicted in Figure 4.9.3. There are now five different paths in the network
from origin to destination. In this example, a sensor at any node without branching yields
no additional information for “learning”. A sensor at a node with branching could conceivably provide more information - by narrowing down the uncertainty in the flow estimate
for the combined paths. However, it could also provide less information if the future detections from the sensor produced no updates or useful distinctions about the traffic on the
subsequent paths that go through that node. For example, a sensor at node 5 would provide
detections on the majority of the paths, but without any distinction about which path (2, 4,
or 5) the enemy traffic was traveling on.
Figure 4.9.3: Sample Network 2: An illustration of employing sensors and a guard in
tandem to capture the optimal amount of enemy forces in a network with moderate connectivity. The guard (blue rectangle) can move either left or right in response to a sensor
(green circles) detection.
Additionally, in this case, the sensors are optimally placed on nodes 4 and 6 since placement
on nodes 1 and 3 would not allow for information on if the traffic was staying on the outer
route (paths 1 or 3) or taking the connector road to the central route (paths 4 or 5). This
indicates that the utility of a sensor for dynamically intercepting enemy threats is not simply
a function of providing more space between the sensor and guard or a function of how close
the sensor is to the border. The topology of the network distinctly matters.
We can quantify this feature using an approach we call “extended reach”. In the above
example, the guard receives value by capturing the flow along paths 2, 4, and 5 by locating
at node 8. If however, there are sensors at nodes 4 and 6 and the enemy’s speed is assumed
74
to be roughly equal (or less) than the guard’s speed, then a sensor detection at node 4 or
node 6 provides the guard with enough warning time to move to nodes 7 or 9 respectively
to capture that enemy as well. In effect, the guard’s reach is extended beyond node 8 to
include both nodes 7 and 9 because of the sensors located at nodes 4 and 6. This “extended reach” assumes some initial knowledge of the enemy’s speed, and also assumes that
multiple detections will not happen simultaneously for the same guard. Both of these are
operationally reasonable.
Based on the difficulty of quantifying the ability to “learn” in a network, we focus on the
“extended reach” reach concept in formulating the “Determine Sensor Placement” function. Because this concept applies directly to the utility of the guard’s placement in the
scheduling function, ideally the sensor placement would be determined at the same time
as the guard schedule, thereby allowing for a fully synchronized plan. In essence, if the
two functions occurred on the same cycle (for example, every three days), then those two
functions could be solved simultaneously in a coupled manner However, for operational
purposes, this is not feasible. Over time, different guard quantities may be available along
with adjustments in flow values, but the sensors will generally remain in place. For example, over the course of a month the scheduling function may provide updates ten times
with a changing configuration of available guard units and with new flow value estimates
halfway through the month. The sensors however, remain in place. Consequently, even if
we can solve a coupled problem, we need some method to decouple the sensor placement
from the scheduling function for operational implementation. .
One additional network example is instructive for understanding the model that follows. In
the subset of a network, depicted in Figure 4.9.4, there exists a sensor at node 1 and a guard
at node 4 (for reasons not reflected in the subset that is depicted). In this case, the guard
should get “extended reach” credit for node 3 because the guard’s speed is equal to the
enemy’s speed. However, the guard will only know to move to node 3 if the sensor at node
1 detects a threat. Therefore, the “extended reach” credit does not include path 3 since that
does not travel through a sensor. Additionally, path 2 traffic will be detected by sensor 1,
but will not travel through node 3. So, the “extended reach” of the guard at node 4 includes
node 3, but only for traffic on path 1. This example highlights some of the complications
in formulating an “extended reach” concept.
4.9.1
DSP Model Description
We approach this problem as an optimization problem similar to the “Patrol Scheduling”
function from section 4.8.4. We take the same formulation, but add in additional decision
75
Figure 4.9.4: Sample Network 3: Determining Sensor Placement and “Extended Reach” the guard (blue rectangle) has the ability to intercept enemy movement on path 1 (through
the sensors - green circles), but not path 2 or path 3 since those flows do not travel through
the sensor location and towards the guard
variables for sensors. Then, using the estimated enemy speed and a reasonable assumption for the number of guards that will generally be available, we optimize the coupled
problem of sensor placement and guard placement. Of note is the use of the estimated
guard configuration. Because the sensors will be emplaced in the chosen locations well
beyond the friendly forces’ initial knowledge of guard availability, we use an estimate of
likely guard configuration as opposed to exactly what is available at that time. The result
is a good idea about the sensor locations that provide the best in-tandem results when employed with guards in the future. The model we develop, because it solves the coupled
problem of sensor emplacement and guard scheduling could extend to other applications
where an operational implementation requires deciding on the two results at the same time.
The concept of the formulation is:
maximize:
subject to:
Variables:
The expected amount of enemy captured
Number of available guard units
Guard unit parameters (speed, base, patrol duration, refit time)
Network topology
Enemy flows
Sensor locations, Guard locations
76
To accomplish the goal of placing sensors, we need to quantify the concept of “extended
reach”. We do this by adding variables that account for extended reach and for capture
along a path because of extended reach. These two variables are analogous to the basic flow
capture model: guard location is denoted by z, and extended reach capability is denoted by
r. Flow capture on a path from a guard’s presence is denoted by y, and the extended reach
capture on a path is denoted by d. We then combine the two types of flow capture on paths
as seen in section section 4.9.4, equation 4.9.16.
To solve for which nodes a guard can reach based on certain sensor locations, we create a
composite variable w, based on sensor and guard locations at a certain time. We then restrict
the extended reach variable r based on this composite variable, the network topology and
guard and enemy speeds as reflected in section section 4.9.4, equation 4.9.13.
4.9.2
DSP Terminology
To create this formulation, we first clarify additional terminology, and define our variables.
Additions to the formulation from section 4.8.4 are outlined.
Network
Traffic
Guards
Sensors
Capture
Origin nodes
Destination nodes
Base nodes
N nodes and E edges representing the trafficable routes in the border
region
Enemy individuals and units crossing the border
The subset of the friendly forces tasked to border security operations
Any technology used to detect traffic at a given place along the border
The generic result of a guard interdicting an enemy unit (stop and
release, capture, or kill)
The subset of the network that serves as the source for the traffic
The subset of the network that serves as the sink for the traffic
The subset of the network that serves as a base for guard units
77
p∈P
δi j
f p,t
zk, j,t
=
ek, j,t
y p,t
d p,t
rk,l,t
=
si
=
wi,k, j,t
=
Reachk
durk
re f itk
speedk
α
βk
θ
Paths for Origin-Destination pairs in the network that represent all the
possible options for enemy travel across the border
Shortest distance between nodes i and j in the network
Traffic
flow on path p at time t
(
1
if guard kis assigned to patrol node jat time t
0
otherwise
Factor between 0 and 1 for the mission-related effectiveness of the unit
based on the configuration of guards to topology
Factor between 0 and 1 representing the percent of traffic captured
from path p at time t
Factor between 0 and 1 representing the percent of traffic captured from path
p at time t from the extended reach of guards
(
1
if guard k can reach node l, within certain criteria, at time t
0
otherwise
(
1
if a sensor is assigned to node i
0
otherwise
(
1
if a sensor is at node i, guard k is assigned to patrol node j at time t
0
otherwise
An n x n matrix where each cell {i, l}is the set of nodes such that a guard at
one of the nodes in the set can capture the enemy at node l after notification
from a sensor at node i, j 6= l
Parameter for the maximum patrol duration allowed for guard k
The refit time necessary for guard k
Parameter for the maximum travel speed of guard k
Parameter for controlling the value of having multiple guards on one
path
Effectiveness parameter that gives the capture value of guard k based
on the guard’s composition
Parameter that decreases the value of capture based on the mission
configuration of the node at t − 1
78
4.9.3
DSP Inputs
The following inputs are necessary for the MIP, and are the quantified elements from section 4.7.
Topology:
Network, Origin nodes, Destination nodes, Base nodes, δi j
values
Traffic behavior:
f p,t for all paths and times, estimated enemy speed
Guard details:
Duration, refit, initial location, and speed for all guard units
Planning horizon: The number of time periods the patrol schedule is planned
for, T
4.9.4
DSP Formulation
O BJECTIVE F UNCTION
max
d,e,r,s,w,y,z
∑∑
t p∈P
79
f pt y pt
C ONSTRAINTS
zk j0 : given ∀k
∑∑ ∑
t
zk jt = 0
(4.9.1)
∑
si = 0
(4.9.2)
∑ zk jt = 1
∀k,t
(4.9.3)
k j∈O,D
i∈O,D
j
zk jt ≤ zk, j,t−1 +
∀k, j,t
zk,i,t−1
∑
(4.9.4)
i:δi j ≤speedk
t+durk +1
∑ ∑ zk, j,m ≥ ∑ zk, j,t − ∑ zk, j,t+1
m=t+2 j∈B
j∈B
∀k,t < T − durk
(4.9.5)
j∈B
t+re f itk
∑ ∑ zk, j,m ≤ durk ∑ zk, j,t + durk (1 − ∑ zk, j,t+1)
m=t+2 j6∈B
j∈B
ek jt ≤ 1 − θ
∀k,t < T − 1
(4.9.6)
j∈B
∀k, j,t > 0
∑ zm, j,t−1
(4.9.7)
m6=k
ek jt ≤ zk jt
∀k, j,t
(4.9.8)
∑ si ≤ p
(4.9.9)
si = 0
(4.9.10)
i
∑
i∈O,D
wik jt ≤ si
∀i, k, j,t
wik jt ≤ zk jt
rklt ≤
(4.9.11)
∀i, k, j,t
∑
∑
(4.9.12)
wi,k, j,t
∀k, l,t
(4.9.13)
i∈l + j∈Reach(i,l)
d pt ≤ ∑ ∑ rk, j,t
∀p,t
(4.9.14)
k j∈p
d pt ≤ ∑ si
∀p,t
(4.9.15)
i∈p
y pt ∑ ∑ (ek, j,t ) + d p,t
∀p,t
(4.9.16)
k j∈p
y pt ≤ α + ∑ ∑ βk (ek jt ) + d p,t
∀p,t
(4.9.17)
k j∈p
y pt ∈ [0, 1] ∀p,t;
zk jt ∈ {0, 1} ∀k, j,t;
si ∈ {0, 1} ∀i;
d pt ∈ [0, 1] ∀p,t;
(4.9.18)
ek jt ∈ [0, 1] ∀k, j,t;
rklt ∈ {0, 1} ∀k, l,t;
wik jt ∈ {0, 1} ∀i, k, j,t
80
(4.9.19)
(4.9.20)
C ONSTRAINT EXPLANATIONS
The following list summarizes each of the constraints above.
(4.9.1)
The initial guard locations are known ahead of time, and serve as the
initializing conditions for the patrol.
(4.9.2)
No guards can patrol on the origin nodes because they are in the
neighboring country. Additionally, no guards can patrol on the sink
nodes because they represent the limiting location where no further
interdiction can happen. Likewise for sensors.
(4.9.3)
Guards can only be assigned to one place at a time.
(4.9.4)
Future guard locations are restricted to locations that the guard can
reach within a time period based on the guards’ speed.
(4.9.5)
If a guard is on a base at time t, and on a patrol at time t + 1, it must
return to base prior to its maximum patrol duration in order to refit.
(4.9.6)
Upon returning from a patrol, a guard must stay on base long enough
to refit.
(4.9.7,4.9.8)
The effectiveness factor is below 1 for guard k at node j, when a
different guard was at node j at t − 1. This accounts for a decrease in
effectiveness due to transitioning guards on the same node.
(4.9.9)
The sensors are limited to the quantity on hand.
(4.9.11,4.9.12) The construction of a composite variable, w, based on sensor and guard
locations
(4.9.13)
Indicator for which nodes a guard and sensor combination provide
extended reach
(4.9.14,4.9.15) If there is a node with extended reach coverage is on path p at time t,
then the extended reach capture factor, d, can take a value greater than
0.
(4.9.16)
If a guard is on path p at time t, or has extended reach to path p, then
the capture factor, y, can take a value greater than 0.
(4.9.17)
The capture factor increases linearly with the number of units on the
path. This represents an increased ability to capture traffic based on
having multiple guards.
This formulation adds the capacity to analytically decide on where to place sensors. It
provides the initial work for adding sensor data and cueing to planning border interdiction
activities.
81
4.10
Dynamic Guard Retasking (DR)
This section outlines the analytical approach to dynamic guard retasking. The intent is to
create a tool for employment in the friendly forces headquarters that allows for guards on
mission to respond to sensor detections, thereby taking advantage of the utility created in
the “extended reach” emplacement of the sensors.
To make full use of the extended reach sensor capabilities, we need to ensure that forces
have a fast, reliable method for determining when, and which units can respond to a sensor
detection. To do that, we develop the “Dynamic Guard Retasking” function. “Dynamic
Guard Retasking”, as depicted in Figure 4.10.1, is a function that takes as inputs the enemy and terrain evaluations from the “Enemy Analysis” function, the sensor locations and
any detections, the guard taskings and the current status of forces operating in the area. It
outputs new taskings to the guard units currently operating based on their ability to immediately respond to the sensor detections.
Figure 4.10.1: Placement of The “Dynamic Guard Retasking” Function in the Border
Interdiction System
82
4.10.1
DR Model Description
We approach the dynamic retasking function as an algorithm needed to quickly determine
the best possible retaskings for the guards after a sensor cue. Because this function operates
in real time, speed is more crucial here than in other functions. To facilitate fast computation, we limit our approach to possible retasking options that lead to the possibility of
capturing the detected enemy forces within one hour. This assumption is reasonable in real
world conditions since the likelihood of capturing the enemy after an hour based is very
low in most cases because of the number of possible paths the enemy can travel on.
The algorithm we develop is a modified version of a greedy heuristic for the flow capture
model. In the truncated network that we consider, the node with the sensor detection serves
as the origin, and the destination nodes are the outermost nodes that the enemy could travel
to within one hour. A key distinction though is that not every node within the network is
feasible, since the guard would need to get there prior to the enemy. We consider the probability of the enemy traveling to a certain node, and use the quantity of possible flows that
travel through both the detection node and the considered node as the means of developing
the probability.
4.10.2
DR Terminology
We first clarify some new terminology.
s
δi j
N js ∀ j, s
ts ji
tkli
Sk
B
pi
The speed classification for an enemy element, where s ∈ {1, 2, 3}for
slow, medium or fast with corresponding, pre-determined, estimated
speeds. This allows for refining the previously estimated speed based
on information from the sensor detection. This is important since the
estimated enemy speed initially my not match the true enemy speed as
estimated from the sensor detection.
The distance from node i to node j
The set of all nodes accessible within 1 hour from node j for an enemy
with a speed classification s
The time for an enemy with speed s to travel from node j to node i
The time for guard k to travel from node l to node i
The set of nodes that are possible retasking options for guard k
The retasking vector with non-empty components bk if guard k is
retasked to node bk
The probability of the detected enemy traveling to node i
83
4.10.3
DR Inputs
Basic Topology: Network, Origin nodes, Destination nodes, Base nodes, δi j values
N js ∀ j, s
The set of all nodes accessible within 1 hour from node j for an enemy
with a speed classification s where s ∈ {1, 2, 3}for slow, medium or fast
Guard details:
Speed for all guard units, and the current locations
4.10.4
DR Algorithm
Trigger: Enemy is detected at node j by a sensor
1. Classify the enemy speed (could be human in the loop or based on simple criteria
about the mode of travel)
(a) Determine s, where s ∈ {1, 2, 3}
2. Determine possible enemy locations within one hour
(a) Access N js
3. Determine if any guards can intercept at a possible location within one hour
(a) For all k guards:
i. Sk ∈ {}
ii. Access guard’s current location, l:
iii. For all i ∈ N js :
A. If tkli < ts ji , add i to Sk
4. Determine the best possible guard retaskings:
(a) Iter = 0
(b) retasking vector:B = { {}∀k }
(c) If there exists a guard k | Sk 6= ∅:
i. Issue retasking preparatory order to all guards k | Sk 6= ∅ with all information about node j
84
ii. Iter + +
iii. total jIter =
fp
∑
p| j∈p b∈p
/
iv. For i ∈ N js , i ∈
/ B:
∑
A.
pIter
i
=
fp
p|i, j∈p b∈p
/
total jIter
v. For all guards k | Sk 6= ∅:
A. Determine c(k): argmax pi ∀i ∈ Sk
vi. Select argmax pc(k) ∀k
vii. Set bk = c(k) f or k ∈ argmax
viii. Issue retasking order to guard k for location bk
ix. Repeat step (c)
This algorithm develops the set of retasking orders for all guards that have the opportunity
to intercept the enemy. The friendly forces headquarters can use the probability pi of the
enemy traveling to a certain node to refine the yes/no decision to send an individual guard.
The algorithm, in step 4(c), removes from consideration the flows that already have a guard
assigned to a node that might intercept them. As such, this algorithm is a version of a
greedy heuristic for a flow capture problem, but with flows limited to only those that travel
through the node with the enemy detection.
The implementation of this algorithm in a friendly forces’ headquarters would allow the
unit to capitalize on emplaced sensors by retasking guards to react to sensor detections.
Importantly, because the emphasis is on calculation speed, the friendly forces staff could
have time to evaluate the possible nodes in N js and select an intercept node based on additional information or intelligence assessments not captured in the initial flow values that
are used to calculate pi .
4.11
Enhanced Patrol Scheduling (E-PS)
We can now revisit the “Patrol Scheduling” function to incorporate the ability of the guards
to dynamically respond to sensor detections. Now that the friendly forces have the capacity
to dynamically retask guards, the patrol schedule that they develop should position guards
with the knowledge that they can respond dynamically to a detection. For example, in
general it now does not make sense to place a guard at the same location as a sensor since
85
the guard could probably be re-positioned in a different location with more effectiveness.
This incorporates the “extended reach” capability developed in section 4.9.4.
With the sensor locations fixed, the formulation is less complex than an exact implementation of the “Determine Sensor Placements” function. In fact, the formulation is exactly the
same, except the sensor locations, s, are no longer decision variables, but input data to the
formulation. Because the sensor locations are no longer decisions, the formulation also no
longer requires the composite variable for sensor and guard locations, w, and the extended
reach path capture indicator, d (since the effects can be added directly into the normal path
capture indicator). So, after the sensors are emplaced, the friendly forces headquarters
can now implement a slightly simplified version of the “Determine Sensor Placements”
function, but can use the exact details of the guard units on hand.
4.11.1
E-PS Model Description
We take the “Determine Sensor Placements” function and change the sensor decision variable into input data.
4.11.2
E-PS Terminology
To create this formulation, we first clarify additional terminology, and define our variables.
Network
Traffic
Guards
Sensors
Capture
Origin nodes
Destination nodes
Base nodes
N nodes and E edges representing the trafficable routes in the border
region
Enemy individuals and units crossing the border
The subset of the friendly forces tasked to border security operations
Any technology used to detect traffic at a given place along the border
The generic result of a guard interdicting an enemy unit (stop and
release, capture, or kill)
The subset of the network that serves as the source for the traffic
The subset of the network that serves as the sink for the traffic
The subset of the network that serves as a base for guard units
86
p∈P
δi j
f p,t
zk, j,t
=
ek, j,t
y p,t
Paths for Origin-Destination pairs in the network that represent all the
possible options for enemy travel across the border
Shortest distance between nodes i and j in the network
Traffic
flow on path p at time t
(
1
if guard kis assigned to patrol node jat time t
0
otherwise
Factor between 0 and 1 for the mission-related effectiveness of the unit
based on the configuration of guards to topology
Factor between 0 and 1 representing the percent of traffic captured
from path p at time t
(
rk,l,t
Reachk
durk
re f itk
speedk
α
βk
θ
4.11.3
=
1
if guard kcan reach node l, within certain criteria, at time t
0
otherwise
An n x n matrix where each cell {i, l}is the set of nodes such that a
guard at one of the nodes in the set can capture the enemy at node l
after notification from a sensor at node i, j 6= l
Parameter for the maximum patrol duration allowed for guard k
The refit time necessary for guard k
Parameter for the maximum travel speed of guard k
Parameter for controlling the value of having multiple guards on one
path
Effectiveness parameter that gives the capture value of guard k based
on the guard’s composition
Parameter that decreases the value of capture based on the mission
configuration of the node at t − 1
E-PS Inputs
The following inputs are necessary for the MIP, and are the quantified elements from section 4.7.
87
Topology:
Network, Origin nodes, Destination nodes, Base nodes, δi j
values
Traffic behavior:
f p,t for all paths and times, estimated enemy speed
Guard details:
Duration, refit, initial location, and speed for all guard units
Planning horizon: The number of time periods the patrol schedule is planned
for, T
4.11.4
E-PS Formulation
O BJECTIVE F UNCTION
max
e,r,y,z
min
f ∈U
∑∑
t p∈P
88
f pt y pt
C ONSTRAINTS
zk j0 , si : given ∀k, i
∑∑ ∑
t
(4.11.1)
zk jt = 0
(4.11.2)
k j∈O,D
∀k,t
∑ zk jt = 1
(4.11.3)
j
zk jt ≤ zk, j,t−1 +
∀k, j,t
zk,i,t−1
∑
(4.11.4)
i:δi j ≤speedk
t+durk +1
∑ ∑ zk, j,m ≥ ∑ zk, j,t − ∑ zk, j,t+1
m=t+2 j∈B
j∈B
∀k,t < T − durk
(4.11.5)
j∈B
t+re f itk
∑ ∑ zk, j,m ≤ durk ∑ zk, j,t + durk (1 − ∑ zk, j,t+1)
m=t+2 j6∈B
j∈B
ek jt ≤ 1 − θ
∀k,t < T − 1
(4.11.6)
j∈B
∀k, j,t > 0
∑ zm, j,t−1
(4.11.7)
m6=k
ek jt ≤ zk jt
∀k, j,t
rklt ≤
(4.11.8)
∀k, l,t
(4.11.9)
y pt ≤ ∑ ∑ (ek, j,t + rk, j,t ) ∀p : i ∈ p,t
(4.11.10)
y pt ≤ ∑ ∑ ek, j,t
(4.11.11)
zk, j,t
∑
j| j∈Reachk (i,l)∀(i=1)∈l +
k j∈p
∀p : i ∈
/ p,t
k j∈p
y pt ≤ α + ∑ ∑ βk (ek jt + rk jt ) ∀p : i ∈ p,t
(4.11.12)
y pt ≤ α + ∑ ∑ βk ek jt
(4.11.13)
k j∈p
∀p : i ∈
/ p,t
k j∈p
y pt ∈ [0, 1] ∀p,t;
ek jt ∈ [0, 1] ∀k, j,t
zk jt ∈ {0, 1} ∀k, j,t;
rklt ∈ {0, 1} ∀k, l,t
89
(4.11.14)
C ONSTRAINT E XPLANATIONS
(4.11.1)
The initial guard and sensor locations are known ahead of time, and
serve as the initializing conditions for the patrol.
(4.11.2)
No guards can patrol on the origin nodes because they are in the
neighboring country. Additionally, no guards can patrol on the sink
nodes because they represent the limiting location where no further
interdiction can happen.
(4.11.3)
Guards can only be assigned to one place at a time.
(4.11.4)
Future guard locations are restricted to locations that the guard can
reach within a time period based on the guards’ speed.
(4.11.5)
If a guard is on a base at time t, and on a patrol at time t + 1, it must
return to base prior to its maximum patrol duration in order to refit.
(4.11.6)
Upon returning from a patrol, a guard must stay on base long enough
to refit.
(4.11.7,4.11.8)
The effectiveness factor is below 1 for guard k at node j, when a
different guard was at node j at t − 1. This accounts for a decrease in
effectiveness due to transitioning guards on the same node.
(4.11.9)
Indicator for which nodes a guard and sensor combination provide
extended reach
(4.11.10,4.11.11) If a guard is on path p at time t, or has extended reach to path p, then
the capture factor, y, can take a value greater than 0.
(4.11.12,4.11.13) The capture factor increases linearly with the number of units on the
path. This represents an increased ability to capture traffic based on
having multiple guards.
The enhancement to the “Patrol Scheduling” function now allows the unit to fully incorporate the impact of the sensors on the guard taskings.
4.12
Mission Feedback
To truly implement a closed-loop system where the results of the missions are evaluated
and incorporated into the planning process, we need to develop a “Mission Feedback” function. Many aspects of this - for example: assessments of certain guard units or qualitative
90
analyses of population centers - are not easily quantified, and formulaic approaches probably harm a true evaluation of the on-the-ground reality. However, there are some aspects
of border interdiction operations that benefit from further analytic inquiry.
Figure 4.12.1: Placement of The “Mission Feedback” Function in the Border Interdiction
System
To evaluate operations, we can run simulations to compare other mission plans, or the
missions actually executed (after the fact), to the optimal solutions. For example, if some
sensors have already been emplaced, we can compare the optimal location of sensors with
the current locations. We can then weigh the costs of changing the placements in terms
of mission risk with the possible gain in security improvements. We can also analyze the
impact of changes in force allocation. For example, we can compare the change in effectiveness for if the friendly forces have additional sensors, or a different quantity of available
guards. With this information, the friendly forces headquarters can weigh resource allocation decisions between border security missions and other missions. This also provides
evidence to a unit for friendly force requests.
We focus our “Mission Feedback” work on using analytic methods for decision support to
assist the commander and staff in determining the effects of additional forces. The first aspect of this is establishing the marginal utility of a sensor. We need to develop an approach
that gives us the incremental value of having an additional sensor in terms of interdicting
91
Figure 4.12.2: Marginal Benefit of a Sensor: A sample graph of the utility of additional
sensors based on an updated assessment of the situation
enemy forces for the given situation at hand. Specifically, for a certain topology, usual
guard force, and updated enemy flows based current data we want to present the commander with the ability to weigh the utility of additional sensor emplacement missions.
Figure 4.12.2 illustrates the basic graphical approach. We iterate the “Determine Sensor
Placements” function for different levels of sensors to determine an upper bound on the
effectiveness of border interdiction efforts based on the guard forces on hand. Then, based
on the commander’s internal evaluation of mission risk, he can decide about the utility of
future sensor emplacements.
The second aspect is presenting the utility of additional guard forces. In general, guard
units would be harder to add into an area than sensors based on general availability of
forces. Figure 4.12.3 illustrates the basic graphical approach. We iterate the “Determine
Sensor Placements” function for different levels of sensors to determine an upper bound
on the effectiveness of border interdiction efforts for a certain quantity of emplaced or
on hand sensors. The commander can then decide if he wants to allocate forces from a
different tasking to border interdiction, request and try to justify additional forces, continue
with the current amount of forces, or re-assign forces from border interdiction to different
taskings.
Other aspects of “Mission Feedback” relate predominately to determining if the actual outcomes are matching the expected outcomes. Based on discrepancies between the functions
provided and real results (in terms of capturing enemy forces), friendly forces could update
enemy flows are refine the estimation of the enemy situation.
92
Figure 4.12.3: Marginal Benefit of a Guard: A sample graph of the utility of additional
guards based on an updated assessment of the situation
4.13
Summary of Models
In this chapter, we presented analytic solutions to our four decision-focused, critical functions from the border interdiction system. We created four models, with two options for the
“Patrol Scheduling” function, based on the presence of sensors, and highlighted the use of
the models for evaluating decisions in the “Mission Feedback” function.
93
94
Chapter 5
A Robust Approach
In this chapter, we introduce a robust optimization approach to solving the main flow capture formulation, R-DSP, that takes the solution to the “Determine Sensor Placements”
function DSP), and includes some consideration of uncertainty. This approach extends
similarly for the other models. We explain the the need for a robust approach, and conduct
a literature review of robust linear programming. Then we introduce a robust formulation
for the “Determine Sensor Placements” function. A comparable formulation for “Patrol
Scheduling” could be created analogously.
5.1
Motivation and Background
So far we have developed a series a strictly deterministic mixed integer programming formulations. We assume in each of these formulations that we know the exact true values for
each of the different parameters and coefficients. Clearly we must make some assumptions
in order to make a tractable model of a complex event. However, some of these assumptions
are more onerous than others. If we make a slight error in estimating the maximum patrol
duration for a guard, durK , or a slight error in estimating the effectiveness parameter of a
guard, βk , (which represents the proportion of enemy flow intersecting the guard’s location
is captured), it probably will not have drastic effects on the solutions for the sensor and
guard locations. We can also conduct a sensitivity analysis on these parameters to further
understand their effects.
While perturbations in most of the parameters have little overall affect on our formulations,
an error in estimating the enemy flows could have drastic effects. The objective function
95
hinges on the ability of positioning guards on enemy paths, based on these flows, f pt . This
is a general problem for many military applications of operations research because of the
very stochastic environment where estimates of enemy performance and actions are highly
uncertain. Consequently, we need some method of accounting for this uncertainty. Using
the most likely value for a coefficient, as we did previously in the deterministic formulations, is a good starting point, but the solutions to deterministic formulations are sometimes
brittle. Because of the number of variables and the size of the feasible space, an optimal
solution can be very specific to the parameters and coefficients used in the formulation.
Small perturbations in these coefficients and parameters can have large effects on the decision variables and the resulting real-world actions. These changes typically impact the
formulation in one of two, not mutually exclusive, ways. First, when the realizations of the
coefficients and parameters are considered, the solution to the formulation may turn out to
be a sub-optimal solution to the problem. Second, the solution to the formulation may turn
out to be completely infeasible.
One approach to protecting against this uncertainty when using a MIP, is robust optimization, which protects against possible changes in coefficients by solving for the optimal
value that remains feasible for a set of coefficient values instead of a single, nominal value.
In this method, the final answer is optimal for all possible values of the coefficients inside
the uncertainty set, which usually means a loss of optimality when compared with the nominal values, but a large increase in the amount of time that the solution remains feasible.
This method does not require (or use) a probability distribution on the set of considered
values, but rather considers any value in the range given. This approach remains tractable
for large problems, and with some techniques the problem remains in its original form - for
example, a MIP becomes a robust MIP.
5.2
Robust Literature Review
Robust optimization generally attempts to capture the details of possible perturbations in
an uncertainty set, which contains the possible values. Work in this type of optimization
started with Soyster, who introduced a method that captured the worst case outcome based
on a collection of uncertainty sets within a problem, Soyster (1973). Additional work expanded this concept to account for techniques where the modeler could control some level
of robustness in the model, and not rely exclusively on checking the worst case outcome.
Ben-Tal and Nemirovski, along with El Ghaoui, developed a technique that uses ellipsoidal
uncertainty sets which allows for the formulation to account for an overall limit within
96
an uncertainty set based on an accumulated distance from the nominal value to the ellipsoidal boundary. See Ben-Tal and Nemirovski (1998, 2000); Ghaoui et al. (1998). This
method alters the complexity of the underlying formulation, and a linear program when
transformed with robust ellipsoidal uncertainty becomes a second order cone problem. Another method of controlling for uncertainty was developed by Bertsimas and Sim, and can
take many forms, including limiting the total uncertain deviations in the coefficients, or by
limiting the number of coefficients within an uncertainty set that can vary. See Bertsimas
and Sim (2004); Bertsimas et al. (2011). This method allows for control over the amount of
uncertainty considered, but allows the formulation to retain its original complexity, where
a linear program with robust polyhedral uncertainty is still a linear program.
Some military applications have been studied using robust optimization with polyhedral
uncertainty. These applications include a mission planning tool for Unmanned Aerial Vehicles, Sakamoto (2006), operational planning Bryant (2006), and reconnaissance asset
allocation Culver (2013).
5.3
Robust Sensor Placement (R-DSP)
We outline here a robust sensor placement formulation since a robust approach to patrol
scheduling is analogous, and just a simplification of this formulation. The intent is to get
solutions for the sensor locations and guard locations over time that are close to optimal for
a range of possible enemy flows.
We previously discussed the two primary benefits of a robust approach: ensuring that the
solution remains feasible for perturbed coefficients, and that the feasible solution is the
best possible solution for the set of coefficients considered. Much of the robust literature
highlights the importance of a robust approach when dealing with hard constraints - the
ones where infeasibility is catastrophic. For example, in a manufacturing context with a
budgetary constraint where the optimal solution results in an over-budget process once certain cost parameters are realized, the result is catastrophic to the manufacturing company.
This type of robust approach is equally important in a military context. The importance of
addressing uncertainty for these hard constraints was addressed in well in Culver (2013),
where the amount of risk that a unit is exposed to must be managed, but the measurements
are uncertain.
We could evaluate similar concerns by testing the importance of robustness when estimating friendly and enemy speeds (which could be a substitute for varying road conditions)
97
or patrol durations. We expect the results to yield similar results, with an increase in the
feasibility coming in tandem with a slight decrease in optimality. But because these constraints in our problem are not as fixed, we focus instead on the most uncertain coefficients
in the formulation. We apply robustness to the enemy flows since that constitutes the crux
of the formulation, and also represents the coefficients with the most uncertain measurements. If the robust optimization approach assists us in creating a decision support tool that
addresses uncertainty in the enemy evaluation, then it is much more realistic that it could
be implemented in an operational context. We proceed focusing on the uncertainty in the
enemy flows.
5.3.1
Robust Formulation Derivation
In general, there are two solution methods for robust problems, either cutting planes or
a reformulation. Because we are searching for a formulation that solves in a reasonable
amount of time, but not conducting an exact comparison of the two techniques, we focus
here in the reformulation. We can use linear programming duality to transform a MIP
with an uncertainty set into a new MIP that accounts for all realizations of our uncertain
parameters in the uncertainty set. For details on this procedures, see Bertsimas and Sim
(2004); Bertsimas et al. (2011). The key notion of this construct is that for any possible
solution, the solver will have to consider the worst possible realization of the uncertainty
for that solution. In essence, “nature plays second”, and the formulation considers the worst
outcome that could realize in nature for a fixed solution. For our problem, we are therefore
solving:
max
min
y
f p ∈Uncertainty
∑∑
f pt y pt
(5.3.1)
t p∈P
Where we have an inner minimization problem that we reformulate into the robust counterpart formulation below.
5.3.2
Polyhedral Uncertainty Sets
For this formulation, we focus on implementing a polyhedral uncertainty set, as introduced
in (Bertsimas and Sim, 2004). The intent is to replace the fixed value enemy flows in
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the existing formulation with a set of flows that reasonably captures the likely uncertainty
inherent in the evaluation of enemy forces. We do not bound the set with the absolute
extremes - the goal is to capture reasonable outcomes as opposed to all possible outcomes.
This technique is not probabilistic. We consider all elements within the uncertainty set as
possible, and do not assign a probability distribution to the uncertainty set. This reinforces
the notion of forming the set based on reasonable uncertainty, and not absolute extremes.
This uncertainty set is motivated by the Central Limit Theorem, where we bound the total
amount of variability allowed in the formulation since we expect an increase in the flows on
one path might be generally offset by decreases in a different path. Consequently, we can
define a somewhat ’smaller’ uncertainty set than one that accounts for every path deviating
in one direction at the same time.
To create the uncertainty set, we define two equations. First, we now want to treat the
enemy flows, f p,t , as uncertain, and to define lower and upper bounds on those flows.
Lower Bound ≤ f pt ≤ U pper Bound
∀p,t
(5.3.2)
Additionally, we know that is is extremely unlikely to be wrong about all of the enemy
flows at the same time. We therefore want a parameter in the formulation that can put a
ceiling on the aggregate amount of deviations for the flows on all the paths for any time
period. This controls limits the total amount of error in the flow estimations considered by
the formulation. Of particular use is that we do not specify which flows might vary, but
create the robust formulation to account for the worst possible combination of deviations
for any given solution.
∑ (Flow Deviation pt ) ≤ Ceiling
∀t
(5.3.3)
p∈P
As an example, Figure 5.3.1, illustrates the difference in what flow values we consider between a nominal problem and a robust problem. The robust solution, while it may have
a slightly lower objective function because it considers many possible flow values, gives
us a solution that is good for any realization of flow values within the set depicted in the
gray area. If we cap the total amount of deviation considered as in equation (5.3.3), then
the we can control the amount of “gray area” in the set that we include in our solution.
99
Because the robust formulation accounts for the worst possible outcome within the uncertainty set, we know that our solution will be protected against the worst case realization of
flow deviations.
Figure 5.3.1: Nominal vs. Robust Flow Values: An example sketch of the difference between solving for a single value (left) and for a range of possible values (right), where the
Ceiling parameter controls for the amount of gray area considered in the solution
Before detailing the specifics of the uncertainty set, we first define some additional terminology.
p∈P
f p,t
f¯pt
fˆpt
Paths for Origin-Destination pairs in the network that
represent all the possible options for enemy travel across
the border
The true traffic flow on path p at time t
Our estimate of the mean traffic flow; the nominal value
used in the deterministic formulation
The estimated half-length of the uncertainty for f p,t . We
consider twice this length as the range for the possible flow
incorporated in the formulation
We can now formally write equation 5.3.2 as:
| f pt − f¯pt |
≤1:
fˆpt
We can now formally write equation 5.3.3 as:
100
∀p,t
(5.3.4)
| f pt − f¯pt |
≤Γ
fˆpt
p∈P
(5.3.5)
∑∑
t
Our uncertainty set, U, is now:
(
U:
5.3.3
| f pt − f¯pt |
∑ ∑ fˆpt ≤ Γ;
t p∈P
| f pt − f¯pt |
≤1
fˆpt
)
∀p,t
(5.3.6)
New Objective and Additional Constraints
To create the new formulation, we add the following variables to the previously explained
set in section 4.9.2.
q pt
v pt
h
Decision variable associated with the first robust constraint in U, 5.3.4
Decision variable associated with the first robust constraint in U, 5.3.4
Decision variable associated with the robust accumulation constraint in
U, 5.3.5
Additionally, we reformulate the same objective in terms of these new variables and U.
max
d,e,r,s,w,y,z,h,q,v
∑∑
t p∈P
f¯pt y pt − ∑ ∑ q pt − hΓ
(5.3.7)
t p∈P
The robust formulation also includes the addition of three constraints, and bounds on the
robust variables.
h + q pt ≥ fˆpt v pt
y pt ≤ v pt
y pt ≥ −v pt
q pt , h ≥ 0 ∀p,t;
∀p,t
∀p,t
(5.3.8)
(5.3.9)
∀p,t
(5.3.10)
v p,t : f ree ∀p,t
(5.3.11)
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5.3.4
R-DSP Formulation
Combining all of the above new information, we can now reformulate our solution to the
“Determine Sensor Placements” function as follows.
O BJECTIVE F UNCTION
max
d,e,r,s,w,y,z,h,q,v
∑∑
t p∈P
f¯pt y pt − ∑ ∑ q pt − hΓ
C ONSTRAINTS
102
t p∈P
(5.3.12)
zk j0 : given ∀k
∑∑ ∑
t
zk jt = 0
(5.3.13)
∑
si = 0
(5.3.14)
∑ zk jt = 1
∀k,t
(5.3.15)
k j∈O,D
i∈O,D
j
zk jt ≤ zk, j,t−1 +
zk,i,t−1
∑
∀k, j,t
(5.3.16)
i:δi j ≤speedk
t+durk +1
∑ ∑ zk, j,m ≥ ∑ zk, j,t − ∑ zk, j,t+1
m=t+2 j∈B
j∈B
∀k,t < T − durk
(5.3.17)
j∈B
t+re f itk
∑ ∑ zk, j,m ≤ durk ∑ zk, j,t + durk (1 − ∑ zk, j,t+1)
m=t+2 j6∈B
j∈B
∀k,t < T − 1
j∈B
(5.3.18)
ek jt ≤ 1 − θ
∑ zm, j,t−1
∀k, j,t > 0
(5.3.19)
m6=k
ek jt ≤ zk jt
∀k, j,t
(5.3.20)
∑ si ≤ p
(5.3.21)
i
wik jt ≤ si
∀i, k, j,t
wik jt ≤ zk jt
rklt ≤
(5.3.22)
∀i, k, j,t
∑
∑
(5.3.23)
wi,k, j,t
∀k, l,t
(5.3.24)
i∈l + j∈Reach(i,l)
d pt ≤ ∑ ∑ rk, j,t
∀p,t
(5.3.25)
k j∈p
d pt ≤ ∑ si
∀p,t
(5.3.26)
i∈p
y pt ∑ ∑ (ek, j,t ) + d p,t
∀p,t
(5.3.27)
k j∈p
y pt ≤ α + ∑ ∑ βk (ek jt ) + d p,t
∀p,t
(5.3.28)
k j∈p
h + q pt ≥ fˆpt v pt
y pt ≤ v pt
y pt ≥ −v pt
y pt ∈ [0, 1] ∀p,t;
zk jt ∈ {0, 1} ∀k, j,t;
si ∈ {0, 1} ∀i;
q, h ≥ 0 ∀p,t;
∀p,t
(5.3.29)
∀p,t
(5.3.30)
∀p,t
(5.3.31)
d pt ∈ [0, 1] ∀p,t;
ek jt ∈ [0, 1] ∀k, j,t;
(5.3.32)
rklt ∈ {0, 1} ∀k, l,t;
wik jt ∈ {0, 1} ∀i, k, j,t;
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v : f ree ∀p,t
(5.3.33)
(5.3.34)
C ONSTRAINT EXPLANATIONS
(5.3.13)
(5.3.14)
(5.3.15)
(5.3.16)
(5.3.17)
(5.3.18)
(5.3.19,5.3.20)
(5.3.21)
(5.3.22,5.3.23)
(5.3.24)
(5.3.25,5.3.26)
(5.3.27)
(5.3.28)
(5.3.29,5.3.30,5.3.31)
The initial guard locations are known ahead of time, and serve as the
initializing conditions for the patrol.
No guards can patrol on the origin nodes because they are in the
neighboring country. Additionally, no guards can patrol on the sink
nodes because they represent the limiting location where no further
interdiction can happen. Likewise for sensors.
Guards can only be assigned to one place at a time.
Future guard locations are restricted to locations that the guard can
reach within a time period based on the guards’ speed.
If a guard is on a base at time t, and on a patrol at time t + 1, it must
return to base prior to its maximum patrol duration in order to refit.
Upon returning from a patrol, a guard must stay on base long enough
to refit.
The effectiveness factor is below 1 for guard k at node j, when a
different guard was at node j at t − 1. This accounts for a decrease in
effectiveness due to transitioning guards on the same node.
The sensors are limited to the quantity on hand.
The construction of a composite variable, w, based on sensor and guard
locations
Indicator for which nodes a guard and sensor combination provide
extended reach
If there is a node with extended reach coverage is on path p at time t,
then the extended reach capture factor, d, can take a value greater than
0.
If a guard is on path p at time t, or has extended reach to path p, then
the capture factor, y, can take a value greater than 0.
The capture factor increases linearly with the number of units on the
path. This represents an increased ability to capture traffic based on
having multiple guards.
Additional constraints related to the robust reformulation of the MIP
The robust approach to the “Patrol Scheduling” function is formulated in a similar manner
and includes the addition of the same variables and constraints, and the same new objective
function. Next, we consider the computational results for the different models.
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Chapter 6
Results and Analysis
We now proceed with numerical tests of our formulations. In this chapter, we focus our
analysis on the “Determine Sensor Placements” function, since that is the most computationally complex of the formulations, and specifically consider the robust version of the
model, R-DSP. Tractability and solution quality for this mode, R-DSP, imply similar results
for the other, smaller models. We will expand on more of the results in the Operational
Evaluation section.
This chapter begins with an outline of the test structure, including the scenario that we
use and the details of our data generation. We then describe the tractability of the models,
followed by a description of the types of results that we get with the DSP model. Finally,
we discuss in detail the results of the R-DSP formulation, and describe the utility of the
robust approach.
6.1
6.1.1
Test Structure
Scenario
To test the different models, we create a network graph that is representative of a possible border scenario. Building on operational experience, as opposed to a specific pair of
countries, we construct a network that represents a border scenario in a relatively open area
where roads and trails crossing a border connect Country B (where enemy forces refit) to
Country A (which friendly forces are securing) into a larger highway running parallel to
105
the border inside the country of interest. This scenario is sketched in Figure 6.1.1. This
sketch represents a region is approximately 10 kilometers by 40 kilometers. Because of the
vast differences in the number of forces employed in different types of areas, the size of an
area could represent the border security responsibilities for a company, battalion or brigade
size element depending on a host of operational factors. We proceed by considering it in a
similar manner to the example outlined in Figure 2.4.1, where a brigade allocates only two
patrolling elements as border guards on a regular basis, with only occasional changes.
Figure 6.1.1: Test Scenario: A sketch of a representative border scenario with multiple
crossing points and roads connecting to a larger highway inside the country of concern.
We then construct this representative scenario as a network of nodes and edges to use in our
formulation. The resulting graph, shown in Figure 6.1.2, includes 91 nodes with 143 edges,
where we use 4 nodes in Country B as Origin nodes where the enemy traffic originates, and
4 nodes in Country A as Destination nodes that represent when enemy forces have started
traveling down the main highway and have the possibility of distinguishing enemy forces
from the local populace and capturing them becomes much more unlikely.
To test this formulation, we set parameters values within a range that are representative of
actual forces in terms of patrol duration, refit ability, and speed. We initially experiment
with scheduling patrols for one day at a time while determining the optimal sensor locations. This basic level of testing includes guards that have a patrol duration of 24 hours,
it increments the guard taskings in two hour blocks. We test the scenario with two guard
speeds, one representing a motorized force on degraded roads (approximately 15 kilometers
per hour), and one representing a primarily dismounted force (approximately 7 kilometers
per hour).
106
Figure 6.1.2: Test Scenario as a Network: The network that we use for the initial testing
of our formulation
6.1.2
Data Generation
To test the formulations, we also create likely scenarios for enemy flows. To do this, we
establish two types of enemy traffic flows. Of note, the emphasis for the numerical values
for the flows is on their relative values, and not the absolute values since the formulation
solves for the maximum flow capture given a certain set of parameters.
Flow Name
Uniform
Key Paths
Flow Description
Flows are randomly drawn from the same range for all paths in the
network.
Flows are randomly drawn from a range of small values for most paths,
with a couple of key paths with larger flows representing the enemy’s
primary paths.
Table 6.1.1: Flow Types: The different types of flows used in initially in the non-robust
simulations
6.2
Tractability Analysis
We presented four non-robust models - a mixed integer formulation for basic patrol scheduling (B-PS), a mixed integer formulation that included sensor placements as a decision
(DSP), a mixed integer formulation that included the sensors locations a priori (E-PS), and
a greedy heuristic for solving a condensed version of the basic scheduling function (DR).
We also introduced a robust extension to the DSP model (R-DSP). The DSP model is the
largest, and the following versions are effectively subset problems of it. Consequently, we
107
demonstrate tractability results only for DSP, since acceptable run times for it will indicate
acceptable run times for the other models.
We view tractability not from a computational complexity standpoint, but from a practical solvability standpoint. For example, the DR algorithm requires an almost immediate
response, hence it is not an integer programming formulation, and we cap the considered
region of the network for our algorithm. The “Patrol Scheduling” models, B-PS and EPS, require relatively quick solution times so that a staff could iterate through a variety
of options, but solution times less than ten minutes would be completely reasonable. The
“Determine Sensor Placements” model occurs at more infrequent intervals than the other
formulations and less rapid solving is acceptable. Solutions on the order of two hours are
still reasonable for a staff conducting planning for the sensor emplacements that will serve
them for the next couple of months.
To test the solutions, we used the commercial solver Gurobi called from an interface in
Python. The code ran on a department computer that had eight cores (2.8 GHz), 8 GB
cache, and 16 GB of RAM.
Figure 6.2.1: Solution Times: The solution times for different combinations of guards
(lines) and sensors (x-axis) in seconds (y-axis) for the DSP model
After many different iterations of the “Determine Sensor Placement” function (the largest
of the models involved), we determined that the main factor affecting tractability was the
number of sensors involved. More sensors results in many additional variables, both the indicator variables for the sensors and the composite variables that are needed to capture the
108
extended reach that the sensors provide. For the largest combination tested here, 3 guards
and 5 sensors, the calculation takes less than an hour. The results for various combinations
are displayed in Figure 6.2.1. Of note, the solution time with no sensors is almost negligible, which is generally the result that we get for the B-PS and E-PS models, since neither
includes sensors. Because the robust version retains the model power of the original, the
R-DSP model takes similar times to the DSP model.
This means that the different models are definitely practical for implementation in a decision support tool. A staff could run the DSP model for the “Determine Sensor Placements”
function whenever they needed to consider emplacing new sensors or moving existing sensors with results in a reasonable amount of time (less than an hour). This would provide
plenty of time for a staff that is considering the moves infrequently, and in a very deliberate,
methodical manner. A staff could run the E-PS model for the “Patrol Scheduling” formulation iteratively during a planning process with almost real-time results (less than a minute).
This provides the flexibility for a staff to easily consider many different options. The DR
algorithm for the “Dynamic Retasking” function also runs in real-time, which means that
it could be successfully used to generate immediate retasking orders for patrols conducting
interdiction operations.
6.3
Overall Results for the DSP Model
Generally, the DSP model outputs a patrol schedule, along with a list of the nodes that
need sensors for optimal performance. As we compare the performance of the formulation
for different numbers of sensors, we see solutions that have the basic characteristics that
we expect - including monotonicity and a decreasing marginal improvement from each
additional sensor given a fixed allocation of guards. Both of these features can be seen in
Figure 6.3.1, which illustrates the percent of the enemy flow captured with one guard for
different sensor allocations on the test network.
One quality of this formulation is that it allows us to compare tradeoffs between the number
of guards and the number of sensors involved. This is useful for the planning staff when
understanding how many guards to allocate to the border interdiction problem compared
with other missions, and for understanding when it is useful to attempt to acquire additional
sensors from their higher headquarters. An illustration of the different levels of effectiveness for different combinations of guards and sensors can be seen using a heatmap, where
the color transitions from blue to red as the percent of enemy flow captured increases.
The effects from additional guards can be seen by moving up the y-axis, and the effects
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Figure 6.3.1: Marginal Utility of Sensors: The percent of flow captured for a fixed quantity
of guards and different numbers of sensors
from additional sensors can be seen by moving right along the x-axis. This type of comparison also allows a planning staff to consider tradeoffs between allocating guards, and
intelligently emplacing sensors, where additional sensors might mean fewer guards were
required. An example of this type of comparison can be seen in .
Figure 6.3.2: The Impact of Guards and Sensors: Percent captured for different combinations of guards and sensors as a way to compare tradeoffs
110
For a given topology, one of the key parameters that affects the importance of sensors is
guard speed. With no sensors, a slow guard and a fast guard capture roughly the same
amount of enemy traffic, as seen in Figure 6.3.3. However, as more sensors are allocated
(moving right across the x-axis), the fast guard has the larger improvement. The fast guard
benefits more from additional sensors because the area over which he can respond to a sensor detection is larger, because of his increased speed, and consequently there are additional
useful places for sensors. Similar benefits can be seen with multiple employed guards.
Figure 6.3.3: The Impact of Guard Speed: Percent captured for a fast and slow guard for
different numbers of sensors
6.4
Robust Results
To demonstrate the effects of the robust optimization, we describe our methodology for
testing the robust formulation, R-DSP, and then illustrate four different experiments. Our
methodology includes how we select the flow values and the uncertainty set half-lengths,
along with how we simulate flows during the test. Our four experiments highlight different
aspects of the robust performance.
111
Experiment
1
2
3
4
# of Guards
2
2
2
2
# of Sensors
4
2
2
Vary
Flow Type
Zero-Anchored
Uniform Draw
Key Paths
Uniform
Table 6.4.1: Robust Experiments: A summary of the different robust experiments conducted with flow type explanations found in table 6.4.2
6.4.1
Methodology
To determine the utility of the robust problem, we consider the same medium network
as before, and solve for the sensor placement and approximated guard schedule from the
“Determine Sensor Placements” formulation for a variety of robust levels. In particular, we
consider various Γ values across a range that considers a completely deterministic solution
(Γ = 0) to one that considers uncertainty on every path. Our intuition was that for certain
types of flows, the deterministic solution would be the optimal choice only for the exact
nominal flow values, and that any deviation from nominal for those flow values would lead
to the robust solution being the better choice.
To account for simulating uncertain half-lengths for each of the flows, we update our descriptions of the two types of flows considered above, and add a third to capture the utility
of the uncertainty set.
Flow Name
Uniform Draw
Key Paths
Zero-Anchored
Flow and Uncertainty Description
The flows and uncertain half-lengths for every path are drawn
uniformly from the same range of values.
The flows and uncertain half-lengths for most paths are drawn
uniformly from the same, small range of values. The flows and
uncertain half-lengths for a small subset of paths are drawn uniformly
from a larger range of values. This mimics a situation where certain
paths are expected to have more enemy traffic at times, but that
estimation also has greater uncertainty.
The flows for every path are drawn uniformly from the same range of
values, but a larger range than the Uniform Draw. This induces some
aspects of a small number of key paths. The uncertain half-lengths are
equal to the flows to create a situation where there is always the
possibility that no flow will travel on a certain path. This captures the
realism of not having a certain, finite floor on the possible enemy
traffic on a certain path.
Table 6.4.2: Flow Types: The different types of flows used in simulations
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Of these three types, the “Zero-Anchored” method of generating flows is the most realistic
because it allows for the possibility of any path to have zero flow during a certain time
window, which mirrors the uncertainty present in understanding enemy infiltrations.
Figure 6.4.1: Simulated Flow Constructs: Four different ways of simulating the uncertain
flows as compared to the uncertainty set used in the formulation. The gray intervals represent the uncertainty sets used for certain nominal values (black lines at the mid-point of the
gray intervals). These base gray intervals are the same for all four simulation constructs.
The red intervals represent the simulation interval used with random draws uniformly from
within the red interval.
To evaluate the effects of a robust solution, we simulate uncertain flow values in a variety
of ways. We construct the uncertainty set as a range of values for each path, with the nominal value at the mid-point. This set is a reasonable approximation, but not all-inclusive
of possible values. Consequently, we simulate uncertain flows from a variety of different
constructs so that we can see the comparison in performance between solutions with different levels of robustness. For each of the flow types that are used in the formulation and
113
solver, we can vary the simulation that we consider. Specifically, we consider four simulation constructs that can each be applied to any of the flow types. These simulation types
are described in table 6.4.3, and depicted in Figure 6.4.1.
Simulation Name
Baseline
Random Half High
Large Uncertainty - High
Large Uncertainty - Low
Simulation Description
Flow values drawn randomly for the full uncertainty set
considered in the formulation
Half the flow values skew high. For half of the flow values,
the simulated value is drawn randomly from the the lower
half-length. For the other half, it is drawn randomly from
the upper half length and an additional half length above
the high-end of the uncertainty set for that path. This
allows for some flow values that the formulation did not
consider.
The flow values with large uncertainty skew high. For the
smaller half-lengths (less uncertainty), the simulated flow
values is drawn randomly from the lower half-length. For
the larger half-lengths (more uncertainty), the simulated
flow value is drawn randomly for the upper half-length and
an additional half length above the high end of the
uncertainty set for that path. This allows for some flow
values that the formulation did not consider.
The flow values with large uncertainty skew low. For the
larger half-lengths (more uncertainty), the simulated flow
values is drawn randomly from the lower half-length. For
the smaller half-lengths (less uncertainty), the simulated
flow value is drawn randomly for the upper half-length and
an additional half length above the high end of the
uncertainty set for that path. This allows for some flow
values that the formulation did not consider.
Table 6.4.3: Simulation Descriptions: The different types of realizations for the uncertain
flows
6.4.2
Robust Experiment 1
We can now use these different types of simulations to test the utility of the robust solutions
found using R-DSP. The most realistic flow type that we consider is a Zero-Anchored since
it accounts for the possibility of capturing nothing on any of the given paths and has a wide
114
enough range to allow for significant differences in nominal values between paths, which
has some aspects of the Key Paths flow type. We solved for the solution under various Γ
values for 2 guards and 4 sensors in the test network. The solution for these formulations
was exposed to the four types of simulations above.
To compare the results of the different levels of robustness, we first determined which of
the Γ settings yielded unique solutions. In this case, we consider the nominal solution and
five different robust solutions, each with a different level of protection. To compare them,
we plot the histograms for the percent of enemy flow captured (our objective) for each of
the simulation types considered. Our results for the 2 guard, 4 sensor, Zero-Anchored flows
are shown in Figure 6.4.2.
Figure 6.4.2: Simulation Results for Varying Levels of Robustness: 2 guards, 4 sensors,
and a Zero-Anchored flow type under four different types of simulated flows
Of note in these results is that a robust solution under Γ = 2 is better than the deterministic
in almost every realization. Even under Baseline simulation (with a range from just the
uncertainty set considered in the formulation), the robust solution outperforms the nominal
solution. When random realizations occur on the high end of the flow sets, the robust solution is preferable. Of particular note, when the flow values with large uncertainties realize
on the high end and beyond the uncertainty set considered (Large Uncertainty - High), the
robust solution is better. This means that the robust solution is better than the deterministic
when our model of the uncertainty set was incorrect and larger flows in absolute terms are
crossing the border. The only solution where the nominal outperforms the robust is when
the larger uncertainties realize on the lower end. This is also when the model of the uncertainty set was incorrect, but happens when the absolute number of enemy forces crossing
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the border is lower. Considering all of these results, we would select the robust solution
with Γ = 2.
In terms of robust modeling, it is particularly important to consider different simulations
and robust levels to determine the best level of protection. A Γ = 10 value, while robust,
has so much additional robust protection that it ends up being too conservative to be worth
selecting.
While these results are the most promising in terms of robust considerations, we include
two other examples that show cases where the robust considerations do not prove worthwhile in most cases. Critically here though, is the notion that the Zero-Anchored flow type
is the most realistic consideration, and the quantity of guards and sensors is realistic for a
task force sized element.
6.4.3
Robust Experiment 2
We show the results here for a scenario with 2 guards and 2 sensors with a Uniform Draw
flow type. This means that for almost every path, there is always the possibility of capturing
some enemy traffic. This formulation results in three unique solutions under the range
of protection levels that we considered. The results for this simulation are depicted in
Figure 6.4.3. In this scenario, the robust solution only outperforms the nominal solution
when the flows larger uncertainties realize on the low end of their uncertain ranges. This
is a worthwhile consideration for a decision maker, but probably not enough to make the
use of a robust formulation useful. We obtained similar results for 2 guards and 2 sensors
under the Zero-Anchored flow type.
6.4.4
Robust Experiment 3
Next, we consider a scenario with 2 guards and 2 sensors with a Key Paths flow type. In this
case, we considered three paths as having much high nominal flow values along with much
larger uncertain ranges. Again in this case, the uncertain ranges are not zero-anchored, so
for almost every path, there is always the possibility of capturing some enemy traffic. This
formulation results in three unique solutions under the range of protection levels that we
considered. The results for this simulation are depicted in Figure 6.4.4. In this scenario, the
robust solution only outperforms the nominal solution when the flows larger uncertainties
realize on the low end of their uncertain ranges. This is perhaps more useful in this case
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Figure 6.4.3: Simulation Results for Varying Levels of Robustness: 2 guards, 2 sensors,
and a Uniform Draw flow type under four different types of simulated flows
than in the Uniform Draw flow type. In this case, if the decision makers want to consider
the possibility of large flow values on the key paths, but do not find it likely, they might
consider the tradeoff that comes from using robust protection in Large Uncertainty - Low
versus the case where large amounts of traffic cross on the key paths.
6.4.5
Robust Experiment 4
Clearly there is some distinction as to when the robust formulation proves valuable. To
determine this, we conducted a series of simulations on a formulation with 2 guards and
various sensor quantities. As described in table 6.4.4, the quantity of sensors involved in
the formulation has a large impact on the utility of the robust solution. The table shows, for
different numbers of sensors, the expected percent captured from solving the MIP deterministically, from simulating the deterministic answer under the Baseline simulation type,
and for simulating the best robust solution under the Baseline simulation type. These results hold to varying degrees for other simulation types as well. Of note is the difference
in the values on each row between the nominal solution under simulation and the robust
solution under simulation. For very low quantities of sensors, the deterministic solution
outperforms the robust solution most likely because the deterministic guard and sensor
configuration is relatively stable, and not fragile. As the number of sensors increases, the
deterministic solution becomes more fragile, and the robust solution outperforms the deter117
Figure 6.4.4: Simulation Results for Varying Levels of Robustness: 2 guards, 2 sensors,
and a Key Paths flow type under four different types of simulated flows
ministic solution even under very basic simulation. Then, as the number of sensors reaches
a saturation point, where based on guard speeds and the topology there are more sensors
than can be effectively employed to distinctly improved the “extended reach” of a patrol,
the deterministic solution begins to perform better. This is because there are enough sensors
to make the deterministic solution less fragile once again.
# of Sensors
0
1
2
3
4
5
6
7
MIP Solutions
27.9
35.1
40.5
47.5
52
56.2
60.6
62.4
Nominal (Simulation)
27.9
33
38.1
38.5
38
39.8
39.9
45.2
Robust (Simulation)
27.7
30.6
31.8
44.3
47.6
43.1
45.7
43.8
Table 6.4.4: The Effect of Sensor Quantity on Robust Performance: Comparing percent of
enemy traffic captured for the MIP solution, nominal solution under simulation and robust
solution under uncertainty
This indicates that for a certain number of employed guards, there is a range of utilized
sensors where the robust solution provides better results. Additionally, we can see that
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some of the marginal gain in percent captured from placing a sensor, that we expected from
the MIP solution, is not realized under the Baseline simulation, but the specifics vary based
on the simulation type considered.
6.4.6
Robust Conclusions
We conclude based on these experiments that the robust formulation has distinct value for
our different functions. In general, the flow type considered does not have a large effect
on when the robust solution is useful, which is promising, since it demonstrates that the
robust formulation is valuable in a number of different operational situations. Additionally,
we conclude that the value of the robust formulation varies with the number of sensors
available. For realistic numbers of sensors (resource constrained, but non-zero), the robust
formulation provides value. However, for very small or very large numbers of sensors, the
robust formulation does not provide value over the deterministic formulation.
We present additional considerations for the robust formulation in the next section, where
we consider the differences in the decisions made under the robust formulation versus the
nominal.
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Chapter 7
Operational Evaluation
In addition to evaluating general characteristics of the modeling results as well as a simulated comparison of the robust performance, we evaluate the operational aspects of these
results. This evaluation has two parts. First, we compare the method with the methods
currently in use. This is a difficult comparison, as the current process is not founded on
an algorithmic decision tool, and so the comparison will necessitate some type of approximation of current military planning. Second, we show what the solutions mean in terms
of implementing the critical decision variables, along with an understanding of how the
implementable solutions differ between the nominal and robust solutions.
We proceed in this chapter with a description of a staff approximation algorithm and the
comparison of the algorithm to our analytic approach for both the placement of sensors and
the scheduling of patrols. We then illustrate what the operational solutions created by the
formulations would be during implementation. Finally, we discuss the factors that would
impact the operational employment of the model.
7.1
Test Scenarios - Real-world Parallels
As described in section 6.1, the scenario that we test is a realistic approximation of a battalion level border security situation measuring about 10 kilometers by 40 kilometers. The
scenario represents a generally connected border region with many roads and trails that
connect through hubs between the border and the main thoroughfare which represents that
in-country boundary of the border region. These hubs can generally be viewed as small
villages or main traveling points that include numerous intersecting roads. We deliberately
avoid a specific international border region to retain a more general focus.
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7.2
Operational Model Evaluation
In this section we focus on comparing our decision support tool the current military staff
method of determining the sensor placement and patrol schedule. Specifically, we compare
the DSP and R-DSP models to an approximation of the current staff process. The current
process is generally not as structured as our method and it is dependent on the individual
planners conducting the analysis. Our main method for conducting this comparison is to
explore the differences in the objective value, the percent of enemy captured, for the current
process and our method.
Before we present this comparison, we first compare the methods in terms of using the
sensors to reduce uncertainty in the network. The sensor emplacement process currently
employed is very individualized, and a planner might have his or her own view of how best
to place sensors. The purpose could be gaining information about an area, as described in
section 4.9, and less about extending the reach of patrolling units. As described previously,
this learning process is very difficult to quantify. We consider reducing uncertainty in the
system as a proxy for learning about an area since no set definition of learning exists in this
context.
Because of this, before we show the overall comparison, we first compare our method to
a analytical method of reducing uncertainty that a staff could employ if they had access to
the synthesized data from an “Analysis of Sensor Data” function. A staff might choose to
do this if they lacked a dynamic retasking tool and did not have a plan for an overall sensor
emplacement strategy that combined with other interdiction efforts. This difference necessitates two different comparison metrics - one based on learning and reducing uncertainty
and one based on magnifying the reach of active patrols.
7.2.1
Staff Approximation Algorithm for Interdiction
A approximation of a how a staff could conduct sensor placement with a learning objective
is to assign sensors to the locations with the highest level of uncertainty. In this sense, the
long-term goal of the headquarters is to reduce the uncertainty in the system. We therefore
create a greedy heuristic that places sensors in the locations that have the highest nodal
uncertainty based on the formulation that we use. This method uses the analytic framework
that we create (in order to quantify uncertainty), even though in practice, no such methods
exist. This comparison is therefore best viewed as an ideal version of what a staff could
do for reducing quantitative uncertainty if they had access to the refined data that a fully
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functioning border interdiction system could provide. In terms of a comparison, this means
that this method is an upper bound on what a staff could produce.
We define the nodal uncertainty metric as follows, where S in the set of paths that travel
through node j. This metric is the percent of the total uncertainty that travels through a
node that has a sensor. With no sensors present, the metric equals 0, and with sensors on
every path, the metric would equal 100. fˆpt is the half-length of the uncertainty set, which
is the measure of uncertainty in the system.
∑
N=
∑ ∑ fˆpt
j∈Sensors p∈S t
∑
∑ ∑ fˆpt
∗ 100
(7.2.1)
j∈Sensors p∈Paths t
The staff approximation algorithm then selects the nodes that produce the highest value of
N for a certain number of sensors using a sorted list of all the total uncertainty at every
node in the network.
The scheduling process is also less structured. However, a characteristic of current scheduling is that staffs will combine the knowledge of the most likely place that might have a
crossing with considerations on patrolling in a variety of different places. To create this
approximation we therefore consider a scheduling approximation that is a greedy heuristic
for selecting the nodes with the largest expected flow as the destinations for the patrols. We
define the total flow G similarly to the Nodal Uncertainty, N.
∑
G=
∑ ∑ f pt
j∈Sensors p∈S t
∑
∑ ∑ f pt
∗ 100
(7.2.2)
j∈Sensors p∈Paths t
After selecting the goal destination node for the patrol, the algorithm tasks the patrol to
get there on the shortest path available. The algorithm includes a feature to approximate
the problem of cannibalized flows, so that guards do not attempt to capture the same traffic
from nearby nodes. To do this, the algorithm prevents the goal nodes from the sorting
algorithm on G from being closer than 10 distances units. As a function of the current
topology, this ensures that guards are “reasonably” spaced and not on the same path. This
is a good approximation of how a planning staff would space out guards that are employed
at the same time.
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7.2.2
As-is v. Analytical Model Performance
To compare these results, we consider two aspects. First, and of secondary importance,
we compare the results of sensor placement for our formulations against the staff approximation method to see if there are differences in how the sensors could reduce uncertainty
over time. Placing sensors at nodes as currently done could lead to a reduced amount of
uncertainty about a specific node, but will not reduce the uncertainty about other nodes, or
more specifically, the associated paths. This was the primary reason that without a formal
“Analysis of Sensor Data” function we implemented the “Determine Sensor Placement”
function based on the concept of extended reach. Consequently, we use a metric of nodal
uncertainty to compare the differences between the two methods for reducing long term
uncertainty. The staff approximation algorithm can be seen as the optimal solution for this
metric, since it uses a sorting algorithm to select the nodes with the highest uncertainty.
Therefore, what we are comparing here is how our method, which does not incorporate this
aspect as a factor in sensor placement, compares.
As can be seen in table 7.2.1, the Determine Sensor Placements function is outperformed
by the staff algorithm, but not by an extremely large margin. Of note, the staff algorithm
is designed to produce the optimal answer under this metric, which gives a rough idea
for a method of reducing uncertainty over time. This means that the “Determine Sensor
Placements” function gives reasonable results even when compared using a metric it is not
designed to consider.
# of Sensors
0
1
2
3
4
5
6
7
Staff Algorithm
0
3.4
6.5
9.2
11.8
14.4
16.9
19.5
DSP Model
0
2.0
3.5
5.9
7.7
6.4
9.7
10.7
Table 7.2.1: Nodal Uncertainty (N) Comparison: A comparison of the nodal uncertainty
metric, N, for 2 guards and various numbers of sensors with both the analytic staff algorithm (optimal), and the Determine Sensor Placements function
The main comparison metric that we consider is how the staff approximation algorithm performs for interdicting enemy forces. This is the crux of the task for the border interdiction
task, and should serve as the primary comparison metric. Even though the “Dynamic Retasking” function does not exist in practice, we include in the staff approximation method
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the ability to retask patrols and leverage the extended reach of the sensors. A comparison of
the staff approximation method to the our method when zero sensors are present provides
a benchmark of what happens without any incorporation of extended reach.
Some results for this comparison can be seen in table 7.2.2. With no sensors present (so
with no additional ability for extended reach), the DSP (nominal) and R-DSP (robust) models outperform the staff approximation algorithm by a total of 4 percent of the enemy flow,
which is an improvement of almost 20 percent. As the number of sensors is increased (in
this example with 2 guards present), the difference becomes even larger, which is a function of our formulation including a method for retasking guards, and including an explicit
ability to incorporate sensor location based on extended reach.
# of Guards
1
2
3
2
2
2
2
2
2
2
# of Sensors
0
0
0
1
2
3
4
5
6
7
DSP
16.1
27.9
38.9
33.0
38.1
38.5
38.0
39.8
39.9
45.2
R-DSP
16.1
27.7
38.7
30.6
31.8
44.3
47.6
43.1
45.7
43.8
Staff Approx
12.2
23.3
34.6
23.3
24.6
24.6
24.6
24.6
24.6
24.6
Margin
3.9
4.6
4.3
9.7
13.5
19.7
23.0
18.5
21.1
20.6
Table 7.2.2: Comparison of the DSP and R-DSP Models Under Simulation to the Staff
Approximation Algorithm: The percent of enemy flow captured for various guard and
sensor combinations under the nominal, robust and staff approximation methods with the
margin of improvement of the best formulation over the staff approximation
7.3
Guard and Sensor Placement Decisions
In this section we focus on the operational decisions about where sensors are guards are
located based on the analytical methods created. The intent is to show the operational
aspects of the decisions as opposed to a synthetic value of the percent of flow captured. We
focus on visualizing the implementation of the key decision variables.
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7.3.1
Varying Levels of Sensors and Guards
To develop a better understanding of the utility of additional sensors, and how the guards
can be optimally positioned, we illustrate the locations of the guards and sensors in Figure 7.3.1 with four combinations: 2 guards and 0 sensors, 1 sensor, 3 sensors and 5 sensors.
We omit the travel locations for the guards, and illustrate the primary guard locations in
blue, the sensors in green and the base in orange.
Generally, the solutions focus in the same area (the left half of the network). There is little
overlap in sensor locations for different quantities of sensors. For example, the solution for
3 sensors is not the solution for 2 sensors plus an additional location. However, the sensor
locations focus emplacements in the same regions. This type of visualization is useful for
understanding the optimal locations for emplacing a certain number of sensors knowing
that the possibility exists for others. If the additional sensors are ready for emplacement
after many patrols (a month later for example), then the estimates of enemy actions may
have significantly changed, and the locations for the new sensors may be in a completely
different area. If however, the additional sensors are available very soon, then the an understanding of the different emplacement options between the current quantity of sensors
and the updated quantity after more arrive could influence the decision on where to place
sensors.
As the numbers of sensors increases, the guards’ tasked locations are farther from the border, which allows for more reaction time for the guards in response to sensor detections.
This matches our original intuition of how sensor placement could aid guard forces. Additionally, the resulting guard locations provides the benefit of a probable decrease in operational risk since the tasked locations are not as exposed to the border, and generally in
more supportable positions from the base.
7.3.2
Decisions for Different Levels of Robustness
We illustrate in Figure 7.3.2 a comparison of the nominal solution and the robust solution
for 2 guards and 4 sensors. For this example, the robust solution outperforms the nominal
solution. Additionally, we can see from the illustration that we get some benefits in terms
of operational employment, since the guard locations under the robust solution are closer
to the base (in this example), which has presumably less operational risk. The sensor
emplacement under the robust solution also has a slightly different configuration of sensors
with the sensors focused on areas slightly farther (laterally) from the base.
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Figure 7.3.1: Optimal Guard and Sensor Locations: An illustration of the optimal locations for 2 guards and (top to bottom): 0 sensors, 1 sensor, 3 sensors and 5 sensors. The
guards’ primary locations are in blue, the sensors are in green and the guards’ base is in
orange.
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Figure 7.3.2: Optimal Guard and Sensor Locations for a Robust Solution: An illustration
of the optimal locations for 2 guards and 4 sensors under the nominal formulation, and the
robust formulation (Γ = 2). The guards’ primary locations are in blue, the sensors are in
green and the guards’ base is in orange.
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Chapter 8
Conclusion
8.1
Summary of Results
In this thesis, we provide an analytical approach to military border interdiction problems.
In chapter 2 we provide the background for the problem, with an overview of the needs for
tactical decision support tools in general, along with the other uses for the specific models
that we set out to create. We outline the basics of the problem, and provide details on the
sensors and guards that we consider. In chapter 3 we conduct a functional decomposition of
the border interdiction system, define the specifics of each function in the process, outline
the current, as-is process, and then detail the four decision-making functions that could
benefit the most from an analytical approach.
In chapter 4 we present four different models three of the functions considered, along with
an explanation of how we could use these models to evaluate different options as part of
evaluating past missions. In chapter 5 we introduce the robust extension to the model used
for the “Determine Sensor Placements”, and the robust approach could extend similarly for
the “Patrol Scheduling” function. This robust approach allows us to incorporate a degree
of enemy intelligence into the otherwise deterministic formulation. In chapter 6 we present
our computational results, with a description of the tractability involved, along with an
explanation of the basic outputs from the models. We also include a detailed description
of the results we get for the robust formulation. In chapter 7 we compare our models to
an algorithm that approximates the current staff planning process, and then describe the
results in terms of the resulting decision variables.
Overall, this thesis includes five different models which are summarized in table 8.1.1.
These provide tractable solutions to some of the decision making functions in the border
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interdiction system, and our functional decomposition of that system provides avenues for
further research. Our chosen model structure provides the power to consider many different aspects of friendly and enemy forces, along with the ability to tailor details about the
mission-specific considerations. The robust extension that we provide allows for a tractable
way of dealing with the uncertainty inherent in the estimation of the enemy movement. Will
illustrate the method for determining when the robust approach will yield better results,
which is generally when the number of sensors is in a range between very few (0 or 1) and
a saturation amount (so many that the marginal benefit of any one sensor is minimal). Our
method for accounting for sensors, “extended reach”, also provides a workable method for
optimizing models that consider moving guards and moving sensors. This could be used
for UAVs employed in tandem with patrols, or for other search and interdict pairings.
8.2
Towards Implementation
The emphasis of this work was the decision support portion of a border interdiction tool.
A decision support tool is a system that uses mathematical models to provide assistance
to the people conducting planning. The intent is to create a framework that can provide
recommended decisions along with decision analysis tools to help a planning staff make
decisions and understand the implications of the decisions that they make. The goal is not
to automate the process and remove the human element from the planning staff. Rather, it
is to augment the planning staff’s own knowledge and understanding of the situation with
additional, useful information.
8.2.1
Operational Implementation
The initial results from our models are promising, and indicate that a border interdiction
support tool can be created that is useful for operational units. However, for it to be worthwhile to invest more resources into developing this tool, a number of additional steps need
to be taken. To ensure that the conceptual model of the formulations used here are valid,
further research should explore different geographic settings by expanding this work to
different network types of network topologies. For example, this conceptual validation
process could include topologies reflecting open desert with sparse, hard-packed roads and
more isolated mountain valleys. Once this analysis was conducted, and the results confirmed for other types of terrain, then the concept of the system would be confirmed as
widely applicable, and additional development steps could be taken.
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Name
Model Description
Need
R-DSP
Robust Determine
Sensor Placement MIP
“Determine Sensor
Placements” to enable sensor
detections into patrol
scheduling decisions while
considering uncertainty in
the enemy flows.
DSP
Determine Sensor
Placement MIP
“Determine Sensor
Placements” to enable sensor
detections into patrol
scheduling decisions.
E-PS
Enhanced Patrol
Scheduling MIP
“Patrol Scheduling” function
that accounts for the presence
of sensors
B-PS
Basic Patrol
Scheduling MIP
DR
Dynamic Retasking
Algorithm
“Patrol Scheduling” in the
absence of sensors. Used for
border interdiction
scheduling when there are no
ground sensors.
“Dynamic Retasking”
function to enable a real-time
guard response to sensor
detections
Frequency of
Implementation
Would be used on
an infrequent basis
with an
approximation of
the guard forces
available over the
next month in order
to get results for the
best sensor
locations.
Would be used on
an infrequent basis
with an
approximation of
the guard forces
available over the
next month in order
to get results for the
best sensor
locations.
Guard locations
over time (after
sensors are
emplaced)
Would be used on
almost daily basis
for scheduling
patrols 2-3 days in
the future.
Would be available
for use all the time
with the algorithm
triggered by a
sensor detection.
Table 8.1.1: Model Comparison: A table summarizing the different models that we presented in this thesis, sorted in terms of tractability from largest to smallest. Analogous
robust versions for E-PS and B-PS follow directly from our development of the R-DSP
model.
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Once the concept is validated for a wide variety of terrains, the border interdiction tool
needs an interface with existing intelligence systems that enables it to get input data and
to provide useful data outputs. The inputs to the model could come from a some of the
existing GIS software in use by the army. The modifications needed would be the ability
to output a network structure based on roads and trails, along with the associated distances.
Most of the software already includes a layer feature where users can build a road overlay,
along with distances from the coordinate systems in the software. For other inputs, the
interface could take direct inputs from the user , with friendly forces input information
could come directly from the planning staff based on the capabilities of the units involved.
The most difficult of the inputs to model would be the initial estimates of the enemy flows.
We believe that a combination of ISR data and intelligence reports could yield good starting points that could then be adjusted by the planning staff based on any additional understanding of the situation. This adjustment could also include the weighting of various
enemy units, origin sources, and destinations as more or less important based on the mission at hand. These adjustments represent a quantification of judgments already made by
the planning staff.
In addition to the inputs, a successful implementation of the model would require a system
for implementing the “Analysis of Sensor Data” function. Intelligence software exists that
can combine reports and create a single visualization with all of the known intelligence
on an area: human, electronic, imagery, and others. This system would need to take the
combined data and synthesize it into a single estimate of enemy traffic. This system would
also allow for a more structured process in gathering initial intelligence since part of the
unit’s initial ISR requests could be based on providing information to this system.
The system implementation for the functions described here would not occur on the same
frequency. A realistic example could be that a set of ground sensors are emplaced every
month, but the patrol schedule is generated every two days based on updated friendly forces
in the area. This frequent implementation of the patrol scheduling function allows for the
incorporation of learning about the enemy without a drastically more complex formulation. Since any updates from the sensor data can be incorporated into the revised flows
for the next patrol scheduling, a realistic implementation includes the ability to account
for learning about the enemy. Additionally, the frequency of scheduling also allows for a
simple adjustment to account for reasonably randomizing where the patrols go. If a unit is
concerned about not repeating security patrols in the same area, they can add an additional
constraint to the scheduling formulation to account for it, or add a feature that decreases
the estimated enemy flows for any paths that have been recently patrolled. This ensures
that the friendly forces do not always go to the same locations.
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8.2.2
Adopting a Decision Support Tool
The adoption of a tactical-level decision support tool in the operational army would involve
a number of hurdles. Most immediately, many commanders could have the impression that
any support tool that provides tactical options is something that limits their independence
and autonomy. The point of a decision support tool is to enable a planning staff, but without good examples and and an education initiative on how it is best implemented, many
commanders will have a first impression of something that is encroaching on their decision
making. The goal of educating people on its use is to convince them that a decision support
tool is not designed to automate the planning process. It is designed to assist, and it can
do that by synthesizing data, presenting options, and providing a means to compare staffgenerated plans. In this tool, we specifically do not attempt to include a measure of risk
since a commander ultimately must decide on when to execute a mission. Additionally, the
tool does not provide a complete allocation of a commander’s forces. It provides options
within a specific mission set; it can provide a measure of the marginal impact of additional
resources applied to border interdiction, but does not measure the impact of those forces
leaving a different mission.
Related to this, an automated tool of this type presents opportunities where a friendly
force’s higher headquarters has access to the same decision support recommendations. This
could create a scenario where the opportunity for micro-management from a physically removed and less involved headquarters is higher. These scenarios occur even without automated recommended actions, but the possibilities are greater once a decision support tool
is utilized. The flip side of this concern is that planning staffs might become too dependent and perhaps even lazy, and not consider their own understanding of a situation and
the additional factors involved when tasking tactical units. An active education initiative
on the best use for a decision support tool along with commanders who stay attuned to the
delegation of responsibilities will be needed to limit this influence.
One of the most difficult aspects to influence when adopting the use of a decision support
tool is the user’s risk aversion and willingness to deviate from the recommendations. It is
crucial that staffs who come to different conclusions than the decision support tool feel free
to implement them. A model cannot capture every aspect of a situation, and if an involved
staff brings other factors into consideration that change their understanding of the situation,
then they should not feel bound to recommendation.
To increase the likelihood of widespread adoption for this type of tool, the formal model
building process would require additional legitimacy from the involvement of operating
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forces. The involvement of a major combatant command (such as Africa Command or
Central Command) with real-world data would add legitimacy to the model development
process for future commanders, and increase the credibility of the model because of its
development with training data from a real-world mission. This data would also provide
the only reasonable avenue for model validation.
8.3
Future Work
In addition to the further development of this work into an implementable decision support
tool, there are a number of related applications that could be avenues for expanding this
work. The models developed here could be extended for use in a perimeter security setting
for a large installation or area. In a military context, this extension could serve as a framework for understanding how to allocate security patrols for logistics areas, resupply lines
or large formations mobilizing for further operations. For large logistic support areas, there
are generally only a few security units available, and these formulations could provide an
approach for scheduling security patrols. For resupply lines leading from a logistics hub to
a larger military formation or base, the formulation could be adapted for understanding how
to schedule security up and down this critical corridor. Additionally, the same techniques
for a logistics support area could apply to securing and prevent enemy reconnaissance of a
large formation mobilizing for follow-on operations.
The work synchronizing the emplacement of ground sensors with mobile patrols could be
easily extended to incorporate sensors that move each time step, similar to the employment
of the guards. This could give insights into the best ways of employing mobile sensors
(UAVs) with patrols to enhance overall interdiction. Further work in this area could also
give insights into when it is useful to task small UAVs to directly support ground patrols
for security.
This aerial extension could also lend insight into developing a formulation for determine
intelligence collection priorities for external ISR assets. This extension could prioritize
subsets of nodes for further collection as a Named Area of Interest for additional ISR
platforms, and the objective function could hinge on the ability to reduce uncertainty in the
system or assist in analyzing the already provided sensor data.
Other applications from network interdiction models could also be studied under this framework. These include a different version of interdicting supply lines based on varying the
representation of enemy movements and destinations. The functional decomposition and
134
combinations of analyzing sensor data with allocating scarce patrols could also apply to
interdicting drug trafficking or human smuggling across a region similarly modeled as a
network, which is one application are for network interdiction models.
135
136
Appendix A
Abbreviations and Acronyms
AO
B-PS
DR
DSP
E-PS
E-UGS
HMMWV
HQ
IED
ISR
MIP
PED
R-DSP
UAV
UGS
US
Area of Operations
Basic Patrol Scheduling Model
Dynamic Retasking Model
Determine Sensor Placement Model
Enhanced Patrol Scheduling Model
Expendable Unattended Ground Sensors
High Mobility Multipurpose Vehicle
Headquarters
Improvised Explosive Device
Intelligence, Surveillance, Reconnaissance
Mixed Integer Program
Processing, Exploitation and Dissemination
Robust Determine Sensor Placement Model
Unmanned Aerial Vehicle
Unattended Ground Sensors
United States
137
138
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