Slide deck used to aid my master's thesis defense in July 2020. Program: Masters in Engineering Systems Management Affiliation: American University of Sharjah, College of Engineering, Department of Industrial Engineering

1. Design of a Theoretical
Framework for the
Operation of a Real-Time
Fire Evacuation Guidance
System
Thesis Defense Presentation
By: Arwa Abougharib

2. Outline
• Recap: Research Need, Gap and Proposed Solution
• Integration Methodology
• Map of Algorithms
• Application to a Case Study
• Conclusion
• Future Work

3. The Research
Need
• Loss of trust in fire alarms due to many false alarms in
the past [1]
• Choice of suboptimal routes: Panic, favoring known exits
over fire exits [2]
• Fire research community identified the need for a new
approach to fire alert and evacuation systems
• Provide more meaningful fire alerts that indicate [3]:
• Location
• Actions expected: Leave/Defend/ Seek refuge
floor/Wait for assistance
• Time available to egress
• Optimal egress routes
• Routes to avoid
3

4. Evacuation Research
Scenario-Independent Crowd
Management
Fire Evacuation
Let’s use Mobile
phones & IoT for real-
time dynamic path
optimization [5]
Why not solve the
network models
optimally to obtain
best paths & exit
assignment?
Your optimal
solutions do not
consider the
movement of fire!
Shortest path, maximal flow,
minimum cost, earliest arrival, game
theory [4]
4

5. The Research
Aim
Design a theoretical framework for
system that could achieve shorter
pre-movement times (and thus,
shorter RSETs) by aiding the
decision-making process of
occupants in the case of fire
5

6. Front End Components
Head
counts
Crowd sensing
•Tracking the
progression of
fire & smoke
•Fire Verification
(avoid false
alarms)
Thermal Imaging
•Tracking
Smoke
Propagation
Smoke Sensors
•Send location-
specific
directions
•Indoor
navigation
Bluetooth/Wi-Fi Beacons
•Receive
Instructions &
Directions
•Create a
database of
occupants
Mobile App
6

7. Fire Simulation
Predicted Fire & Smoke Paths
Fire + Smoke Tracking
Existing Building Layout 7

8. Route Optimization
Predicted Fire & Smoke Paths
Optimal Assignment of Evacuees to Exits +
Optimal Egress Routes
Crowd Analysis
Priority Evacuees
Network Problem is:
• Capacitated
• Dynamic
8

9. Integration
Methodology
Output Output desired action or paths (Shelter in place/Crawl/Wait for
congestion to clear/Evacuate immediately)
Optimize Optimize updated network as an Earliest Arrival
Transshipment
Compare Compare path transit times to ASET; remove high-risk arcs
Simulate Use CFAST to obtain ASET
Estimate Estimate arc capacities and arc transit times (to use later in
graph optimization)
Construct Construct currently tenable paths
Sense Use external sensing to identify and remove untenable arcs
Construct Construct building network
Eliminate Eliminate false alarms!
9

10. Algorithm 5 (Pre-Alarm Verification)
Algorithm 6: Complete
Evacuation Sequence
Tenability Checks
Extended Hydraulic
Models
Algorithm 1: Construct Paths, Estimate
capacities and travel times
Fire Thermal Model (CFAST)
Network Transformation 1
Algorithm 2: Compute the
Earliest Arrival Pattern
Network Transformation 2
Parametric search for 𝜃∗
Network Transformation 3
Algorithm 3: Compute 𝑁
and Ω
Algorithm 4: Solve for flow x
A set of strongly
polynomial algorithms to
solve the Earliest Arrival
Transshipment Problem
(Baumann, 2007) [6]
External
Sensing
Building Network Construction
10
Order
of
Execution

11. Algorithm 5 (Pre-Alarm Verification)
Order
of
Execution

12. Pre-Alarm
Verification
Sequence
Algorithm 5 (function handler: Verify()]: Fire Alarm
Verification
INPUT: Activation of smoke detectors or manual station
1 If smoke detectors are activated, do
2 Activate Infrared cameras;
Search for high-temperature zones;
3 If Fire detected OR manual station activated, do
Sound building alarm;
Evac()
Else
Notify Building Management System;
Dispatch building staff to confirm;
4 End If
5 End If
12

13. Algorithm 5 (Pre-Alarm Verification)
Algorithm 6: Complete
Evacuation Sequence
External
Sensing
Building Network Construction
13
Order
of
Execution

14. Worked Case Study
• Apartment building: Ground + 1st
Floor
• Assume all occupants are on 1st floor
• 6 Apartments  6 source nodes
• Sink node t is the safe assembly area
• All other nodes are transshipment
nodes
• Nodes connected by single directed
arcs, except AG and AB are
bidirected
1- Construct Building Network
14

15. Fire Scenario
• A fire breaks out in Apartment 𝑠5,
fills east side with smoke
• Capture measurements from
temperature and smoke sensors
• What are the safety thresholds for
temperature and smoke density?
• Temperature: 70 °C (Conservative)
• Smoke (Extinction coefficient): 0.5 /m
[7]
2- Remove Untenable Arcs
15

16. Algorithm 5 (Pre-Alarm Verification)
Algorithm 6: Complete
Evacuation Sequence
Tenability Checks
External
Sensing
Building Network Construction
16
Order
of
Execution

17. Tenability Checks
17

18. Fire Scenario
2-
Remove
Untenable
Arcs
18

19. Algorithm 5 (Pre-Alarm Verification)
Algorithm 6: Complete
Evacuation Sequence
Tenability Checks
External
Sensing
Building Network Construction
19
Order
of
Execution
Algorithm 1: Construct Paths;
Estimate capacities and travel
times

20. Construction of Egress Paths
• A tenable path is a chain of
tenable arcs
• The path must extend from a
source to a sink
• Currently tenable: Tenability for
now is confirmed, but future
tenability not yet confirmed
• For our case, 4 currently tenable
paths were found
3- Construct Currently Tenable Paths
Legend
Walking in clear conditions
Walking in thin smoke
Crawling 20

21. Construction of Egress Paths
Path 2: s1-A-G-H-I-J-K-L-N-O-t
Path 3: s3-G-H-I-J-K-L-N-O-t
Path 4: s4-A-G-H-I-J-K-L-N-O-t
21

22. Estimation of Egress Times
• Transit time on an arc can be calculated as :
• Traversal time =
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝐶𝑟𝑜𝑤𝑑 𝑆𝑝𝑒𝑒𝑑
, or, if a bottleneck
is encountered,
• Queueing time:
𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑆𝑖𝑧𝑒
𝐶𝑟𝑜𝑤𝑑 𝐹𝑙𝑜𝑤 𝑟𝑎𝑡𝑒
• We need a way to estimate a crowd’s speed and
flow rate
• The Hydraulic Model assumes streams of
crowds resemble fluid flow
• Hinges on the hydrodynamic relation
𝐹𝐶 = 𝐷𝑆𝑊
𝑒 = 𝐹𝑠 𝑊
𝑒
22

23. SFPE-Standard Hydraulic Model (hmv = 1)
• Crowd speed (S) and crowd density (D) correlation
in SFPE
𝑆 𝐷, 𝑘 ≔
𝑆𝑚𝑎𝑥, 𝐷 ≤ 0.54
0, 𝐷 ≥ 3.8
𝑘 − 0.266𝑘𝐷, Otherwise
• 𝐹𝑠 = 𝐷𝑆 = 𝑘𝐷 − 0.266 𝑘𝐷2
for D = [0.54,3.8]
• To find the speed or flowrate, the crowd density must
be found
• This is why we need crowd sensors! But if this is the
first iteration, or sensors burned due to flames,
estimate density using the quadratic formula
• Estimations using this basic model are valid only for
walking upright in clear conditions
23
SFPE Hydraulic Model Diagrams [8]

24. Algorithm 5 (Pre-Alarm Verification)
Algorithm 6: Complete
Evacuation Sequence
Tenability Checks
External
Sensing
Building Network Construction
24
Order
of
Execution
Extended Hydraulic
Models
Algorithm 1: Construct Paths;
Estimate capacities and travel
times

25. Smoke Modified Hydraulic Model (hmv = 2)
• The basic hydraulic model (BHM) accounts for
congestion effects, but not for lower visibility
in smoke, or crawling behavior
• Mobility is a speed-reduction factor dependent
on smoke density
• Applied Xie’s correlation [9]
• 𝑅𝑣𝑠𝑚𝑜𝑘𝑒 = 𝑓(𝐶𝑠)
• 𝑆𝑠 𝐷, 𝑘, 𝐶𝑠 ≔
𝑅𝑣 𝑠𝑚𝑜𝑘𝑒. 𝑆𝑚𝑎𝑥 , 𝐷 ≤ 0.54
0, 𝐷 ≥ 3.8
𝑅𝑣𝑠𝑚𝑜𝑘𝑒 . 𝑘 − 0.266𝑘𝐷 , 𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
k = 1.4
25

26. Crawl Modified Hydraulic Model (hmv = 3)
• People often crawl under thick smoke
• We recomputed the theoretical maximum
density for crawling at D = 1.6
persons/m2
• We derived a new correlation based on
empirical data [10, 11] and imposed
boundary condition
• Crawling speed further reduced by
number of 90-degree turns [12]
𝑆𝑐(𝐷, 𝑘𝑐 , 𝑛90 ) = (4 1.49 − 𝐷 𝑒−4 1.49−𝐷 + 0.69) ∙ 𝑘𝑐
𝑛90
𝑘𝑐 ≈ 0.985
26

27. Algorithm 5 (Pre-Alarm Verification)
Algorithm 6: Complete
Evacuation Sequence
Tenability Checks
External
Sensing
Building Network Construction
27
Order
of
Execution
Extended Hydraulic
Models
Algorithm 1: Construct Paths;
Estimate capacities and travel
times

28. Estimation of Flow
Capacity
• Arc Capacity = Maximum
specific flow * Effective Width
𝑢 𝑎 ∈ 𝐴 = 𝐹𝑠𝑚(ℎ𝑚𝑣𝑎) ∗ 𝑊
𝑒
• Path Capacity = Capacity of its
bottleneck
𝑢 𝑃𝑖 = min{[𝑢 𝑎 ∈ 𝑃𝑖 ]}
𝑷𝟒 𝑻𝑺𝒋𝒊 𝒌𝒋𝒊 𝒉𝒎𝒗𝒋𝒊 𝑹𝒗𝒔𝒎𝒐𝒌𝒆(𝒋𝒊) 𝑭𝒔𝒎 𝒂𝒋𝒊
𝑾𝒑(𝒂𝒋𝒊) 𝑩𝑳(𝒂𝒋𝒊) 𝑾𝒆 𝒂𝒋𝒊
𝒖 𝒂𝒋𝒊
s4A door 1.4 3 0.673 1.008 0.91 0.15 0.610 2.00
AG corridor 1.4 2 0.883 1.162 2.40 0.2 2.000 2.32
GH door 1.4 2 1.000 1.316 0.91 0.15 0.610 0.80
HI Stairway 1.08 1 1.000 1.015 0.99 0.15 0.694 0.70
IJ corridor 1.4 1 1.000 1.316 1.20 0.2 0.800 1.05
JK Stairway 1.08 1 1.000 1.015 0.99 0.15 0.694 0.70
KL corridor 1.4 1 1.000 1.316 2.40 0.2 2.000 2.63
LN corridor 1.4 1 1.000 1.316 2.40 0.2 2.000 2.63
Ot door 1.4 1 1.000 1.316 1.82 0.15 1.520 2.00
𝑢(𝑃4) 0.70
28

29. Estimation of
Travel Time
• Travel time through an arc 𝜏 𝑎 ∈ 𝑃𝑖
=
𝑀𝑃𝑖
𝑢(𝑃𝑖)
𝑖𝑓 𝑎𝑟𝑐 𝑝𝑟𝑒𝑐𝑒𝑑𝑒𝑠 𝑝𝑎𝑡ℎ′
𝑠 𝑏𝑜𝑡𝑡𝑙𝑒𝑛𝑒𝑐𝑘
𝑑𝑎
𝑆𝑎(ℎ𝑚𝑣𝑎)
𝑂𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
• Travel time through the egress path:
𝜏 𝑃𝑖 = 𝜏 𝑎 ∈ 𝑃𝑖
𝑷𝟒 𝑭𝑪 𝒂𝒋𝒊
𝑫𝒋𝒊 𝑺𝒎𝒂𝒙 𝑹𝒗𝒔𝒎𝒐𝒌𝒆(𝒋𝒊) 𝑺𝒂𝒋𝒊
𝒅𝒂𝒋𝒊 𝝉 𝒂𝒋𝒊
1 s4A 0.70 1.22 1.190 0.67 1.057 0.000 0.00
2 AG 0.70 0.27 1.190 0.88 1.051 22.500 21.42
3 GH 0.70 1.22 1.190 1.00 0.763 22.500 36.91
4 HI 0.70 1.86 0.950 1.00 0.503 3.440 6.84
5 IJ 0.70 0.80 1.190 1.00 0.984 2.400 2.44
6 JK 0.70 1.86 0.950 1.00 0.503 3.440 6.84
7 KL 0.70 0.27 1.190 1.00 1.190 23.160 19.46
8 LN 0.70 0.27 1.190 1.00 1.190 2.400 2.02
9 Ot 0.70 0.37 1.190 1.00 1.190 0.000 0.00
𝑇𝑃4
95.92
29

30. Algorithm 5 (Pre-Alarm Verification)
Algorithm 6: Complete
Evacuation Sequence
Tenability Checks
External
Sensing
Building Network Construction
30
Order
of
Execution
Algorithm 1: Construct Paths;
Estimate capacities and travel
times
Extended Hydraulic
Models
Fire Thermal Model (CFAST)

31. Run Fire Simulator
• Selected CFAST by NIST
• Deterministic, zonal model
• Surveyed 46 fire models
• Evaluated based on capabilities, availability,
documentation & support, verification and
validation history
• Used to predict the ASET along each path
• It is assumed that the input files will have been
configured offline a priori and stored in the
system before real-time operation
• If 𝑇𝑃𝑖
≥ 0.90 ASET𝑃𝑖
, eliminate arcs on path
31

32. System Outputs
32

33. Conclusion
• The framework’s value lies in:
• Frames the evacuation problem as an Earliest Arrival
Transshipment, widely regarded as the most appropriate
network problem for optimizing evacuation
• Maintaining strongly polynomial time complexity
• Using fast zonal fire simulation to ensure reliability of
recommended paths
• Flexibly ‘envelopes’ the network optimization algorithm
• As a by-product, two variants of the hydraulic model
have been proposed
• A novel approach to computing transit times was
proposed in Algorithm 1
• The two major algorithms (1 and 6) have been
mostly written in pseudocode, aiding faster future
implementation
33

34. Future Work
• Validation and verification of the proposed variants of
the hydraulic model
• Further implementation of algorithms is required
• Further study of how autonomous running of CFAST can
be achieved
• Extend the framework with
• Phased Evacuation
• Use of Fire-Safe occupant elevators
• Optimizing Ingress routing for rescue teams, while
considering counterflow
34

35. References
[1] G. Proulx, “Why building occupants ignore fire alarms,” Construction Technology Update; no. 42, Dec. 2000.
[2] F. Ozel, “Time pressure and stress as a factor during emergency egress,” Safety Science, vol. 38, no. 2, pp. 95–107, Jul. 2001.
[3] B. L. Hoskins and N. Mueller, “Evaluation of the Responsiveness of Occupants to Fire Alarms in Buildings: Phase 1,” p. 30.
[4] D. Dressler et al., “On the use of network flow techniques for assigning evacuees to exits,” Procedia Engineering, vol. 3, pp. 205–215, 2010.
[5] “eVACUATE | eVACUATE Concept.” [Online]. Available: http://www.evacuate.eu/project/evacuate-concept/. [Accessed: 11-Oct-2019].
[6] N. Baumann, “Evacuation by earliest arrival flows,” TU Dortmund University, 2007.
[7] T. Yamada and Y. Akizuki, “Visibility and Human Behavior in Fire Smoke,” in SFPE Handbook of Fire Protection Engineering, M. J. Hurley, D.
Gottuk, J. R. Hall, K. Harada, E. Kuligowski, M. Puchovsky, J. Torero, J. M. Watts, and C. Wieczorek, Eds. New York, NY: Springer, 2016, pp. 2181–2206.
[8] S. M. V. Gwynne and E. R. Rosenbaum, “Employing the Hydraulic Model in Assessing Emergency Movement,” in SFPE Handbook of Fire Protection
Engineering, M. J. Hurley, D. Gottuk, J. R. Hall, K. Harada, E. Kuligowski, M. Puchovsky, J. Torero, J. M. Watts, and C. Wieczorek, Eds. New York, NY:
Springer, 2016, pp. 2115–2151.
[9] H. Xie, “Investigation into the interaction of people with signage systems and its implementation within evacuation models,” phd, University of
Greenwich, 2011.
[10] R. Muhdi, J. Davis, and T. J. Blackburn, “Improving Occupant Characteristics in Performance-Based Evacuation Modeling,” 2006, doi:
10.1177/154193120605001118.
[11] R. A. Kady, “The development of a movement–density relationship for people going on four in evacuation,” Safety Science, vol. 50, no. 2, pp. 253–
258, Feb. 2012, doi: 10.1016/j.ssci.2011.08.058.
[12] R. A. Kady and J. Davis, “The Impact of Exit Route Designs on Evacuation Time for Crawling Occupants,” Journal of Fire Sciences, vol. 27, no. 5,
pp. 481–493, Sep. 2009, doi: 10.1177/0734904109105320.

36. Thank You!
Any Questions?

37. Introduction to Flows Over Time
• In static flows, flow units are moved
instantaneously from source(s) to sink(s)
• In flows over time/dynamic flows, flow
takes time to move through arcs
• Each arc is assigned a travel time
• Motivation? Detection of congestion
and bottlenecks
CLASSICAL FLOW DYNAMIC FLOW/ FLOW
OVER TIME
FLOW
FUNCTION
Amount pushed through
each arc
Flow per unit time at a
certain instant
CAPACITY Upper bound on total
amount through arc
Upper bound on flow rate
HOLDOVER Not allowed Allowed in principle
TRAVEL TIME N/A Arcs have travel times

38. Discrete-Time Dynamic Flow Notation
• N = (V,A) a directed graph consisting of a set of nodes, V, and a set of directed arcs A
• 𝑆+
⊆ V set of source nodes; 𝑆−
⊆ V set of sink nodes
• 𝛿−
(𝑣) , 𝛿+
(𝑣) ; sets of incoming and outgoing arcs from a node v
• b(v) ; node balance, or supply-demand function
• u(a) the capacity function
• τ(a) travel time of arc a∈ A
• 𝑥(𝑎, 𝜃) amount of flow entering arc a at time 𝜃
• 𝑥−
𝑣, 𝜃 The amount of flow that has entered a node v up to time instant 𝜃
• 𝑥+
𝑣, 𝜃 the amount of flow that has left a node v by time 𝜃

39. Chain Flows and Temporally Repeated Flows
• A simple s-t path P is a path in N; a chain of arcs from source to sink
• Can express flow as flow on arcs or flow on paths
• If we sent m units along path P, corresponding static flow is a chain flow 𝛾 =
𝑚, 𝑃
• A collection of chain flows that results in the static flow x is a chain
decomposition Γ = 𝛾1, 𝛾2, … , 𝛾𝑘
• Γ 𝑇
is a temporally repeated flow: A feasible static flow represented by Γ,
repeated at each discrete instant in time from time zero till 𝑇 − 𝜏(𝛾𝑖)
• Hence, Γ 𝑇 is a feasible dynamic flow over time horizon T
• 𝑣𝑎𝑙𝑢𝑒 Γ 𝑇 = P ∈ P 𝑇 − 𝜏 𝑃 𝑥(𝑃) = 𝑇 ∙ 𝑣𝑎𝑙𝑢𝑒 𝑥 − 𝑎 ∈ 𝐴 𝜏(𝑎) ∙ 𝑥(𝑎)

40. Computation of 𝑜𝜃
(𝑆+
)
• 𝑣𝑎𝑙𝑢𝑒 𝑥, 𝜃 is the total flow output from the
network from time zero up to time 𝜃
• 𝑜𝜃
(𝑆+
) the maximum 𝑣𝑎𝑙𝑢𝑒 𝑥, 𝜃 that can be
sent out of 𝑆+ and reach 𝑆− by time 𝜃
• 𝑜𝜃 𝑆+ = max{𝑣𝑎𝑙𝑢𝑒 Γ 𝜃 }
= max{ 𝑇 ∙ 𝑣𝑎𝑙𝑢𝑒 𝑥 − 𝑎 ∈ 𝐴 𝜏(𝑎) ∙ 𝑥(𝑎)}
• This is equivalent to minimizing the negative of
the right hand side, which is the total circulation
cost in the extended network
• 𝑜𝜃
𝑆+
or 𝑜𝜃
𝑋 (where X ⊆ 𝑆+
∪ 𝑆−
) can be
found by a single static minimum cost flow
computation in the extended network on the right

41. Network Model Constraints
• Time horizon constraints
𝑥 𝑎, 𝜃 = 0 ∀ 𝑎 ∈ 𝐴, ∀ 𝜃 ≥ 𝑇 − 𝜏(𝑎)
• Arc capacity constraints
𝑥 𝑎, 𝜃 ≤ 𝑢 𝑎 ∀ 𝑎 ∈ 𝐴 , ∀ 𝜃 ∈ ℕ0
• Node capacity constraints
𝑥−
𝑣, 𝜃 − 𝑥+
𝑣, 𝜃 + max 𝑏 𝑣 , 0 ≤ 𝑢 𝑣 ∀ 𝑣 ∈ 𝑉, ∀ 𝜃 ∈ ℕ0
• Flow conservation constraints
𝑥+ 𝑣, 𝜃 − 𝑥− 𝑣, 𝜃 ≤ max 𝑏 𝑣 , 0 ∀ 𝑣 ∈ 𝑉, ∀ 𝜃 ∈ ℕ0
• Balance constraints
𝑥+ 𝑣, 𝑇 − 𝑥− 𝑣, 𝑇 = 𝑏 𝑣 ∀ 𝑣 ∈ 𝑉

42. Objective Function
• Three evacuation objectives []:
A. Maximize 𝑣𝑎𝑙𝑢𝑒 𝑥, 𝜃 for all 𝜃 ≤ 𝑇
B. Minimize the total time needed to send the required amount of flow (or the last unit of flow)
from the source(s) to the sink(s)
1. Find minimum time horizon for feasibility 𝜃∗
2. Find a dynamic flow which is feasible within time horizon 𝜃∗
C. Minimize the average time for all flow to arrive at the sink
• A flow that fulfills any 2 also fulfills the third
• Flows with the earliest arrival property satisfy all three objectives
• An earliest arrival flow is by definition, maximum for all 𝜃
• We shall consider earliest arrival transshipment, not s-t flow. Why? (multiple
sources, multiple sinks, required amount of flow is known, supplies must not be
exceed)

43. General Solution Approaches
• Three approaches to solving dynamic flow
problems (not always mutually exclusive) []:
1. Reduce dynamic network flow problems to
static ones and use existing algorithms to
solve them e.g. Temporally repeated flows
2. Applying existing static flow algorithms to a
time-expanded network
3. Avoiding the time expansion by exploiting
some mathematical properties of the time-
dependent attributes (usually better time
complexity)
• Baumann’s thesis features a strongly
polynomial algorithm for solving the EAT
problem (3rd approach)
• However, it requires constant and integral arc
travel times and capacities.

44. Step 1: First Network Transformation
• Multiple-source multiple-sink (N) to multiple-source single-sink network (N’)

45. Step 2: Compute the EAP (Algorithm 2)
• 𝑝(𝜃) is “the maximal amount of flow that can be sent into
the sink by time 𝜃 without violating supplies at the sources,
fulfilling capacity constraints and flow conservation”
• 𝑝 𝜃 ≤ 𝑜𝜃 𝑆+ ∀ 𝜃 ≥ 0 due to bounded supplies and
demands
• 𝑝 𝜃 shall be a linear, piecewise non-decreasing function of
time.
• Each linear segment is a maximum s-t arrival pattern
resulting from a different set of sources.
• In each extended network, the supersource s is connected to
a different subset of sources
• Each linear segment is allowed to extend only up to the
time instant after which at least one of the connected
sources runs empty of supplies
𝑝 𝜃 ≔ 𝑜𝜃
𝑆𝑖 + 𝑏(𝑆+
𝑆𝑖) for 𝜃𝑖 ≤ 𝜃 < 𝜃𝑖+1

46. Construction of the EAP

47. Step 2: Compute the EAP (Algorithm 2)
Algorithm 2 (function handler: EAPComp): Computing the
Earliest Arrival Pattern [43, P. 57] 1
INPUT: (N’, 𝑆+
, t)
OUTPUT: Earliest arrival pattern 𝑝(𝜃) as a set of k breakpoints (𝜃𝑖, 𝑓𝑖) for 𝑖 =
0,1, … , 𝑘
1 𝑖 ∶= 0, 𝑆𝑖 ≔ 𝑆+
, 𝜃𝑖 ∶= 0 ;
2 While 𝑆𝑖 ≠ 𝜙 do
3 Compute 𝜃𝑖+1 ≥ 0 such that
𝜃𝑖+1 = max⁡
{𝑜𝜃𝑖+1 (𝑆′ ) ≥ 𝑜𝜃𝑖+1 (𝑆𝑖) − 𝑏(𝑆𝑖𝑆′
)} for all 𝑆′
⊆ 𝑆𝑖 ;
4 Compute an inclusion-wise minimal set2
𝑆𝑖+1 ⊊ 𝑆𝑖 such that
𝑜𝜃𝑖+1 (𝑆𝑖+1) = 𝑜𝜃𝑖+1 (𝑆𝑖) − 𝑏(𝑆𝑖𝑆𝑖+1) ;
5 Compute 𝑜𝜃 (𝑆𝑖) on the interval [𝜃𝑖 , 𝜃𝑖+1) and set
𝑝(𝜃) ≔ 𝑜𝜃 (𝑆𝑖) + 𝑏(𝑆+
𝑆𝑖) for 𝜃 𝜖 [𝜃𝑖 , 𝜃𝑖+1)
6 𝑖 = 𝑖 + 1;
End While
7 Set 𝑝(𝜃) ∶= 𝑏(𝑆+) for all 𝜃 ≥ 𝜃𝑖

48. Step 3: Second Network Transformation
• a feasible transshipment over time in N”
will induce an EAT in the original
network N, corresponding to the EAP
derived in the previously
• Add new sinks labelled 𝑡𝑖 for 𝑖 = 1, … , 𝑘
• Add k additional arcs, each of which is
directed from the supersink t’to the
additional sink 𝑡𝑖. Note that t’now
becomes an intermediate node
• Set the demand for each new sink
𝑏 𝑡𝑖 ≔ − 𝑓𝑖 − 𝑓𝑖−1 .
• Set the travel time of each arc (t’, 𝑡𝑖) to
𝜃𝑘 − 𝜃𝑖
• u((t’, 𝑡𝑖)) to the quantity
𝑓𝑖−𝑓𝑖−1
𝜃𝑖−𝜃𝑖−1
, which is
the slope of 𝑝(𝜃) over the interval
[𝜃𝑖−1, 𝜃𝑖]

49. Step 4: Find 𝜃∗
• 𝜃∗ is the minimum time horizon for which a transshipment over time is feasible
• Repeatedly checks the feasibility criterion by Klinz
• Let 𝑋 ⊆ 𝑆+ ∪ 𝑆−, 𝑏 𝑋 ≔ 𝑣 ∈𝑋 𝑏(𝑣), and let 𝑜𝜃(𝑋) be the maximum amount of flow
(ignoring supplies and demands) that can be sent from sources in 𝑆+ ∩ 𝑋 to sinks in
𝑆−
𝑋 within time 𝜃 ≥ 0.
• A feasible continuous flow over time that satisfies all supplies and demands with time
horizon 𝜃 exists if and only if
𝑜𝜃
𝑋 ≥ 𝑏 𝑋 ∀ 𝑋 ⊆ 𝑆+
∪ 𝑆−
which is a tight inequality due to the continuity of the function 𝑜𝜃
𝑋 over 𝜃 for a given 𝑋.
• 𝜃∗ is the time horizon for which 𝑜𝜃 𝑋 = 𝑏 𝑋

50. Step 5: Third Network Transformation
• Initial network transformation
in Algorithm 3
• Transformations here are
redundant so far
Illustration of the third network transformation from N'' to 𝑁0

51. Step 6: Compute 𝑁, Ω (Algorithm 3)
• 𝑁 is an instance of the LMDFP, with modified capacities of the arcs
outgoing from sources or incoming to sinks in such a way so as to
reduce the maximum flow to a level that fulfils supplies and demands.
• The LMDFP (next step) requires an ordered set of terminals to be
provided by the modeler. The chain C establishes this ordering
• C is a subset of terminals (sources and sinks) with the following
special properties:
1. It is a chain: Its subsets are nested and each two adjacent subsets differ by only
one element.
2. Each of the sets in C are tight; a subset X is tight if 𝑜𝜃∗
𝑋 = 𝑏(𝑋) holds [43,
p. 32]
3. The nested subsets are ordered by inclusion, given an ordering of terminals
𝑠0, 𝑠1, … , 𝑠𝑙
• Ω is an ordered set of terminals based on chain C, then once C is
constructed, we can construct Ω ≔{𝑠0, 𝑆1𝑆0 , 𝑆2𝑆1, … , 𝑆𝑖+1 𝑆𝑖} for
𝑖 = 0, … , 𝑙 .
Example illustration of a chain C with l =3

52. Step 6: Solve the LMDFP (Algorithm 4)
• the LMDFP seeks a feasible flow over time bounded by a time horizon (in this
case, 𝜃∗) that maximizes the amount of flow leaving sources in a given order, or
equivalently minimizes the amount of flow entering sinks in the same order.

53. Time Complexity
• A numeric algorithm has a pseudo polynomial running time if:
• Its running time is a polynomial function in the numeric value of the inputs (the largest integer present
in the inputs), rather than the length of the inputs (the number of bits used to represent it)
• It’s the other way around for polynomial time algorithms
• Cobham–Edmonds thesis states that ‘polynomial time’ means that the algorithm is
“tractable", "feasible", "efficient", or "fast“
• A problem can be solved in polynomial time is to say that there exists an algorithm that,
given an n-bit instance of the problem as input, can produce a solution in time O(nc),
where c is a constant that depends on the problem but not the particular instance of the
problem.
• Source: https://en.wikipedia.org/wiki/Cobham%27s_thesis