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 2012August 25-29, 2012

Prague, Czech Republic

OutlineOutline

Evacuation route planning problem Illustrative exampleProcess-network synthesis (PNS)

by P-graph frameworkEvacuation route planning

by process-network synthesisResults and discusion

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Evacuation Route Planning Evacuation Route Planning ProblemProblem

Aim is to ensuresafest 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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Evacuation Route Planning Evacuation 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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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

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

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Optimization Methodology „P-graph framework”

(F. Friedler & L.T. Fan, 1990) Algorithmic model

generation (MIP)

Process-Network SynthesisProcess-Network Synthesis

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Problem

Generation of the Mathematical Model

Solver

Mathematical model

Solution

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P-graph Framework P-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

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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

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P-graph Framework: P-graph Framework: Structural Representation (Cont’d)Structural Representation (Cont’d)

Bipartite graphActivities Inputs and outputs (entities)

Input: Resources or precondition mj

Output: Effect or target

Activity Oi

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P-graph Framework: P-graph Framework: Structural Representation (Cont’d)Structural Representation (Cont’d)

Clear logical interpretationEach 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

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Axioms of Combinatorialy Feasible Process Axioms 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.

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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

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P-graph Framework: Algorithms P-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.

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Problem Definition: Problem Definition: Software PNS-DrawSoftware PNS-Draw

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Problem Definition: Problem Definition: Software PNS-StudioSoftware PNS-Studio

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MINLP vs. P-graph FrameworkMINLP vs. P-graph Framework

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MINLP P-graph framework

Problem given byVariables, 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

Evacuation Route Planning by Evacuation Route Planning by Process-Network SynthesisProcess-Network Synthesis

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PNS Problem

P-graph Framework

Solution

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Evacuation Route Planning Problem

Model Transformation

Process-Network Synthesis Problem Process-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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Steps of Model TransformationSteps of Model Transformation

1. Transform floor map to P-graph representation2. Estimate the time horizon (hazard spreading)3. Determine the time unit4. Multiplication of the graph of floor map for

each time unit5. Formulate PNS problem based on the time

extended graph6. Simplify the problem by algorithm MSG7. Assign penalty more on later evacuations8. Generate alternative evacuation plans by

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Process-Network Synthesis Problem Process-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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Process-Network Synthesis Problem Process-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 9A_B_0_1 A B_1 2A_C_0_1 A C_1 2B_B_0_1 B B_1 8B_C_0_1 B C_1 2B_D_0_2 B D_2 2C_C_0_1 C C_1 20C_D_0_1 C D_1 2A_A_1_2 A_1 A_2 5A_C_1_2 A_1 C_2 2B_C_1_2 B_1 C_2 2B_D_1_3 B_1 D_3 3C_C_1_2 C_2 C_3 20C_D_1_2 C_1 D_2 2

Maximal Structure for the Illustrative Maximal 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

Solution #1 provided by Algorithm ABB Solution #1 provided by Algorithm ABB

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Time 0

Time 1

Time 2

Time 3

Solution #2 provided by Algorithm ABB Solution #2 provided by Algorithm ABB

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Time 0

Time 1

Time 2

Time 3

Solution #3 provided by Algorithm ABB Solution #3 provided by Algorithm ABB

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Time 0

Time 1

Time 2

Time 3

Solution #4 provided by Algorithm ABB Solution #4 provided by Algorithm ABB

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Time 0

Time 1

Time 2

Time 3

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.comThe computational time required is typically

a few seconds or minutes

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AcknowledgementAcknowledgement

Authors acknowledge the support of the Hungarian Research Fund under project OTKA 81493 K.

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Call for PartitipationCall for Partitipation

Veszprém Optimization Conference:Advanced Algorithms

11-14 December, 2012University of Pannonia, Veszprém, Hungary

Deadline for Abstract Submission: September 17, 2012

http://vocal.dcs.vein.hu

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Thank you for your attention!Thank you for your attention!