The document discusses using process network synthesis and the P-graph framework to plan optimal evacuation routes. It presents an illustrative example of evacuating people from three rooms of a building. The problem is modeled as a PNS problem by defining resources, targets, and movement activities. The P-graph software generates multiple alternative evacuation plans and identifies the optimal plan in a matter of seconds or minutes. This approach allows flexible evacuation planning to ensure safety during emergency situations.

1. Planning Evacuation Routes with the P-
graph Framework
Juan C. Garcia-Ojeda*, Botond Bertok, Ferenc
Friedler, and Mate Hegyhati
University of Pannonia, Veszprém, Hungary
* Autonomous University of Bucaramanga, Colombia
PRES 2012
August 25-29, 2012
Prague, Czech Republic

2. OutlineOutline
 Evacuation route planning problem
 Illustrative example
 Process-network synthesis (PNS)
by P-graph framework
 Evacuation route planning
by process-network synthesis
 Results and discusion
2www.p-graph.com

3. Evacuation Route PlanningEvacuation Route Planning
ProblemProblem
 Aim is to ensure
safest and fastest movement of individuals
away from any threat
(e.g., bomb threat, taking of hostages)
or the actual occurrence of a hazard
(e.g., traffic, industrial, or nuclear accidents;
natural disasters, fire, viral outbreak)
/Stringfield, 1996/
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4. Evacuation Route PlanningEvacuation Route Planning
Problem (Cont’d)Problem (Cont’d)
 Evacuation plans lack flexibility. This may lead individuals
into dangerous situations (e.g., blocked exits, or spaces
with gas leakage)
/NFPA, 1996; Pu and Zlatanova, 2005/
 Optimization software for supporting human decisions is
essential
/Cova and Johnson, 2003; Dimakis et al., 2010; Pu and
Zlatanova, 2005/
 Optimal or near optimal evacuation plan may imply the
evaluation of a myriad of evacuation routes because of the
combinatorial nature of the problem
/Cova and Johnson, 2003; Hamacher and Tjandra, 2002;
Kim et al., 2008/
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5. Evacuation Route Planning:Evacuation Route Planning:
Illustrative ExampleIllustrative Example
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Room A: 3/5
Room B:
4/8
Room C:
3/20
Number of evacuees / Room capacity
Door AC: 2/1
Door BC: 2/1
Door BO: 3/2
Outside: safe area
Door AB: 2/1
Door CO: 2/1
Door capacity:
evacuees / time unit
Question:
Evacuation plan
Given:
Floor map

6. Process-Network Synthesis:Process-Network Synthesis:
Problem DefinitionProblem Definition
 Inputs:
 A set P of demands to be satisfied
 A set R of resources (available)
 A set O of potential activities
(manufacturing, transportation, etc.)
 Cost data and capacity constraints
 Output:
 A ranked list of the n-best networks
6www.p-graph.com

7.  Optimization Methodology
 „P-graph framework”
(F. Friedler & L.T. Fan, 1990)
 Algorithmic model
generation
(MIP)
Process-Network SynthesisProcess-Network Synthesis
7
Problem
Generation of the
Mathematical Model
Solver
Mathematical model
Solution
7www.p-graph.com

8. P-graph FrameworkP-graph Framework
(F. Friedler & L. T. Fan, 1990)(F. Friedler & L. T. Fan, 1990)
 Three cornerstones of the P-graph Framework
 Structural representation: P-graph
 Axioms of the combinatorially feasible process
networks
 Algorithms for
- Generating the rigorous superstructure
- Generating the combinatorially feasible
process structures
- Generating optimal or n-best networks
8www.p-graph.com

9. P-graph Framework:P-graph Framework:
Structural RepresentationStructural Representation
 Unambiguous representation of structural properties
Process element P-graph representation
Resource or precursor
Final target
Intermediate entity
Activity
9www.p-graph.com

10. P-graph Framework:P-graph Framework:
Structural Representation (Cont’d)Structural Representation (Cont’d)
 Bipartite graph
Activities
Inputs and outputs (entities)
Input: Resources or precondition
mj
Output: Effect or target
Activity Oi
10www.p-graph.com

11. P-graph Framework:P-graph Framework:
Structural Representation (Cont’d)Structural Representation (Cont’d)
 Clear logical interpretation
Each precondition
(E AND F) has to be
satisfied to operate an
activity (O2)
Any of the activities
(O2 OR O3) having the
same effect can
potentially be sufficient
to initiate the
consecutive step (O1)
OR
AND
11www.p-graph.com

12. Axioms of Combinatorialy Feasible ProcessAxioms of Combinatorialy Feasible Process
Structures (Friedler et al., 1992)Structures (Friedler et al., 1992)
(S1) Every final target is represented in the structure.
(S2) An entity represented in the structure has no
input to if and only if it represents a precursor.
(S3) Every activity represented in the structure is
defined in the synthesis problem.
(S4) Any activity represented in the structure has at
least one path leading to a final target.
(S5) If an entity belongs to the structure, it must be
an activating entity to or resulting entity from at
least one activity represented in the structure.
12www.p-graph.com

13. P-graph Framework:P-graph Framework:
Reduction of the Search SpaceReduction of the Search Space
Feasible structures
Search Space
Combinatorially feasible
structures
Optimal structure
2nd
best
structure 3rd
best
structure
13www.p-graph.com

14. P-graph Framework: AlgorithmsP-graph Framework: Algorithms
(Friedler et al., 1992, 1993, 1995)(Friedler et al., 1992, 1993, 1995)
 Algorithm MSG (Maximal Structure Generation)
generates the rigorous superstructure: the union
of the combinatorially feasible structures.
 Algorithm SSG (Solution-Structure Generator)
generates each combinatorially feasible structure
exactly once.
 Algorithm ABB (Accelerated Branch and Bound)
generates the n-best solutions of the problem
while the search space is reduced to the set of
combinatorially feasible structures.
14www.p-graph.com

15. Problem Definition:Problem Definition:
Software PNS-DrawSoftware PNS-Draw
15www.p-graph.com

16. Problem Definition:Problem Definition:
Software PNS-StudioSoftware PNS-Studio
16www.p-graph.com

17. MINLP vs. P-graph FrameworkMINLP vs. P-graph Framework
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MINLP
P-graph
framework
Problem given by
Variables,
Constraints
Resources, Targets,
Potential activities
Generation of the
math. model
Manual Automatic
Structural properties
of process-networks
Hidden in the
math. model
Exploited
Number of solutions Single Multiple
Handling special
constraints
Can be
incorporated to
the math.
model
May require
modifications of the
model generator and
solver

18. Evacuation Route Planning byEvacuation Route Planning by
Process-Network SynthesisProcess-Network Synthesis
18
PNS Problem
P-graph Framework
Solution
18www.p-graph.com
Evacuation Route Planning Problem
Model
Transformation

19. Process-Network Synthesis ProblemProcess-Network Synthesis Problem
of the Building Evacuationof the Building Evacuation
 Resources:
Evacuees at their initial locations
 Activities:
Movement of evacuees
 Targets:
Evacuees at safe locations
 Objective:
As soon as possible
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20. Steps of Model TransformationSteps of Model Transformation
1. Transform floor map to P-graph representation
2. Estimate the time horizon (hazard spreading)
3. Determine the time unit
4. Multiplication of the graph of floor map for
each time unit
5. Formulate PNS problem based on the time
extended graph
6. Simplify the problem by algorithm MSG
7. Assign penalty more on later evacuations
8. Generate alternative evacuation plans by
algorithm ABB
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21. Process-Network Synthesis ProblemProcess-Network Synthesis Problem
of the Illustrative Exampleof the Illustrative Example
 Resources:
 Evacuees at their initial locations: A, B, C
 Targets:
 Evacuees outside the building at time 1, 2, 3…:
- D1, D2, D3…
 Activities:
 Movement of evacuees at time 1, 2, 3…:
- Leaving Room A at time 1 and
arriving in Room B at time 2: A_B_1_2
…
 Stay of evacuees at time 1, 2, 3…:
- Staying in Room A from time 1 to time 2: A_A_1_2
…
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22. Process-Network Synthesis ProblemProcess-Network Synthesis Problem
of the Illustrative Example (Cont’d)of the Illustrative Example (Cont’d)
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Activity Precondition Result Upper bound
A_A_0_1 A A_1 9
A_B_0_1 A B_1 2
A_C_0_1 A C_1 2
B_B_0_1 B B_1 8
B_C_0_1 B C_1 2
B_D_0_2 B D_2 2
C_C_0_1 C C_1 20
C_D_0_1 C D_1 2
A_A_1_2 A_1 A_2 5
A_C_1_2 A_1 C_2 2
B_C_1_2 B_1 C_2 2
B_D_1_3 B_1 D_3 3
C_C_1_2 C_2 C_3 20
C_D_1_2 C_1 D_2 2

23. Maximal Structure for the IllustrativeMaximal Structure for the Illustrative
Example Considering 3 Units of TimeExample Considering 3 Units of Time
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Time 1
Time 2
D_1
Time 3
Room A Room B Room C
Outside
Stay
Move

24. Solution #1 provided by Algorithm ABBSolution #1 provided by Algorithm ABB
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Time 0

Time 1

  
Time 2
    
Time 3
  

25. Solution #2 provided by Algorithm ABBSolution #2 provided by Algorithm ABB
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Time 0

Time 1

  
Time 2
    
Time 3
   

26. Solution #3 provided by Algorithm ABBSolution #3 provided by Algorithm ABB
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Time 0

Time 1

  
Time 2
    
Time 3
   

27. Solution #4 provided by Algorithm ABBSolution #4 provided by Algorithm ABB
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Time 0

Time 1

 
Time 2
    
Time 3
  

28. Concluding RemarksConcluding Remarks
 Evacuation Route Planning Problem can be
converted to PNS problem and solved by the P-
graph framework
 P-graph software generates optimal and n-best
alternative evacuation plan
 Alternative evacuation plans can be evaluated by
in-depth analysis
 P-graph software is available from p-graph.com
 The computational time required is typically
a few seconds or minutes
28www.p-graph.com

29. AcknowledgementAcknowledgement
Authors acknowledge the support of the Hungarian
Research Fund under project OTKA 81493 K.
29www.p-graph.com

30. Call for PartitipationCall for Partitipation
Veszprém Optimization Conference:
Advanced Algorithms
11-14 December, 2012
University of Pannonia, Veszprém, Hungary
Deadline for Abstract Submission: September 17, 2012
http://vocal.dcs.vein.hu
30www.p-graph.com

31. Thank you for your attention!Thank you for your attention!