The Clustered Dial-a-Ride Problem

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Public Transportation in Rural Areas:

The Clustered Dial-a-Ride Problem

Fabian Feitsch November 16th, 2018

Attributions of third party images can be found on slide 12.

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Public Transportation

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Public Transportation

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Public Transportation

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Public Transportation

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Public Transportation

So far, so good?

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Public Transportation

So far, so good?

loooooooomoooooooon walking distance?

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Public Transportation

So far, so good?

wheather?

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Public Transportation

So far, so good?

wheather?

waiting time?

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Public Transportation

So far, so good?

wheather?

waiting time?

Ñ Probably in the city, but not in villages!

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Public Transportation

So far, so good? Ñ Probably in the city, but not in villages!

Ñ Doorstep Service in Rural Areas

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The Dial-a-Ride Problem

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The Dial-a-Ride Problem

A Dial-a-Ride instance is a triple I “ pn, rd_i_,js, Sq.

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The Dial-a-Ride Problem

A Dial-a-Ride instance is a triple I “ pn, rd_i_,js, Sq. n :“ number of riders

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The Dial-a-Ride Problem

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The Dial-a-Ride Problem

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The Dial-a-Ride Problem

Number of persons m “ n ` 1

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The Dial-a-Ride Problem

Number of persons m “ n ` 1

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The Dial-a-Ride Problem

Number of persons m “ n ` 1 rd_i_,js :“ distance matrix

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The Dial-a-Ride Problem

start with 0 (driver’s pickup) 0

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The Dial-a-Ride Problem

start with 0 (driver’s pickup) 0 enumerate pickups

1

2 3

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The Dial-a-Ride Problem

enumerate dest‘s in same order

1

2 3

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The Dial-a-Ride Problem

4

1

2 3

5

6

7

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The Dial-a-Ride Problem

4

1

2 3

5

6

7

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The Dial-a-Ride Problem

4

1

2 3

5

6

7 S :“number of seats in the vehicle

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The Dial-a-Ride Problem

4

1

2 3

5

6

for this presentation: S “ 8

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The Dial-a-Ride Problem

4

1

2 3

5

6

Objective: Feasible tour minimizing the sum of total distances.

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The Dial-a-Ride Problem

4

1

2 3

5

6

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The Dial-a-Ride Problem

4

1

2 3

5

6

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An Exact Algorithm

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An Exact Algorithm

There is an exact algorithm by Psaraftis, 1980.

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An Exact Algorithm

It works similar to the Held-Karp-algorithm.

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An Exact Algorithm

Running Time: O^˚p3^n´1q.

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An Exact Algorithm

Can be generalized to solve partial instances:

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An Exact Algorithm

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An Exact Algorithm

Find best tour such that a) girl is delivered

b) waiting customer is fetched c) boy is still on board.

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Back to Rural Areas . . .

Hogsmeade

Springfield Minas Tirith

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Back to Rural Areas . . .

A rural Dial-a-Ride instance typically looks like this:

Hogsmeade

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Back to Rural Areas . . .

Hogsmeade

Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Ñ Locations are inside clusters.

Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Ñ Bypasses do not exist.

Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Ñ All riders head in the same direction.

Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Seems to be simpler than the Dial-a-Ride Problem . . . Ñ All riders head in the same direction.

Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Ñ ÝÑ

T ^˚-algorithm Assumptions:

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Back to Rural Areas . . .

Hogsmeade

Goal:

Ñ ÝÑ

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Back to Rural Areas . . .

Hogsmeade

Goal:

Classify instances whose optimal tour is unidirectional.

Ñ ÝÑ

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Back to Rural Areas . . .

Hogsmeade

Goal:

Classify instances whose optimal tour is unidirectional.

(without computing it)

Ñ ÝÑ

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A Classifier

Hogsmeade

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A Classifier

Hogsmeade

Idea: Distribute the costs of a tour to the clusters.

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A Classifier

Hogsmeade

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A Classifier

Hogsmeade

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A Classifier

Hogsmeade

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A Classifier

Hogsmeade

Let C₁, . . . C_q be the clusters.

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A Classifier

Hogsmeade

Let ΥpT , C_i q P R^` such that @T : ř_q

i“1 ΥpT , C_iq “ cpT q

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A Classifier

Hogsmeade

i“1 ΥpT , C_iq “ cpT q

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A Classifier

Hogsmeade

i“1 ΥpT , C_iq “ cpT q Let ΦpC_i q be a lower bound on ΥpT ^˚, C_iq.

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A Classifier

Hogsmeade

Theorem (= Classifier):

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A Classifier

Hogsmeade

Theorem (= Classifier): @C_i : ΦpC_iq “ ΥpÝÑ

T ^˚, C_iq ñ T ^˚ “ ÝÑ T ^˚

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A Classifier

Hogsmeade

T ^˚, C_iq ñ T ^˚ “ ÝÑ T ^˚ Proof. Via exchange argument.

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A Classifier

Hogsmeade

T ^˚, C_iq ñ T ^˚ “ ÝÑ T ^˚ Proof. Via exchange argument.

TODO!

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

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Distribute Costs to Clusters

Hogsmeade

Assign the parts of a tour to clusters.

7

2 8

3 4 9

10 5

m “ 6

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

Obs.: Edges of a tour are weighted.

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

weight: 2

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

weight: 3

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

weight: 3

Ñ Count atomic journeys!

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

Obs.: Edges of a tour are weighted. Ñ Count atomic journeys!

Every cluster C_i has four counters:

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

α :“ #rightbound persons with p_r ď i.

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

α :“ #rightbound persons with p_r ď i.

α “ 3

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

α :“ #rightbound persons with p_r ď i. β :“ #leftbound persons with d_r ě i.

pickup cluster of r

dropoff cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

α :“ #rightbound persons with p_r ď i. β :“ #leftbound persons with d_r ě i.

β “ 1

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

α :“ #rightbound persons with p_r ď i. β :“ #leftbound persons with d_r ě i. γ :“ #left-entering persons with p_r ě i.

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

α :“ #rightbound persons with p_r ď i. β :“ #leftbound persons with d_r ě i. γ :“ #left-entering persons with p_r ě i. δ :“ #right-entering persons with d_r ď i.

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

ΥpT , C_iq “ inpC_iq ` αC_iC_i_`1 ` βC_iC_i_´1 ` γC_i_´1C_i ` δC_i_`1C_i

See thesis

for proof of cpTq “ ř

ΥpT, C_i q.

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

See thesis

ΥpT, C_i q.

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

See thesis

ΥpT, C_i q.

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

See thesis

ΥpT, C_i q.

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

See thesis

ΥpT, C_i q.

pickup cluster of r

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Distribute Costs to Clusters

Hogsmeade

7

2 8

3 4 9

10 5

m “ 6

See thesis

ΥpT, C_i q.

Todo: ΦpC_iq ď ΥpT ^˚, C_iq

pickup cluster of r

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6 ³

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Idea: Any T induces an ordered partition on every cluster.

Hogsmeade

7

2 8

4 9

10 5

m “ 6 ³

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6 ³ “

t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

Other Possibilities?

3 “

t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

Other Possibilities? S “ rt4u, t8us S “ rt4, 8us

3 “

t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

Other Possibilities? S “ rt4u, t8us S “ rt4, 8us Additionally: List of Portals P.

3 “

t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

“rl, ls, rl, rs‰

3 “

t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

Given S and P the lower bound can be estimated.

3 “

t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

Solve internal tours.

3 “

t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

3

Compute lower bounds for α, β, γ and δ.

“t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

3

Add costs up and obtain lower bound Φ_S_,PpC_iq.

“t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

3

Add costs up and obtain lower bound Φ_S_,PpC_iq. ñ min ΦpC_i q_S,P “ ΦpC_iq ď ΥpT ^˚, C_iq

“t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

3

|C_i | “ 6:

5 227 236 choices

kk kik hk

kk kj

“t8u, t4u‰

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Lower Bound on Υ p T

^˚

, C

_i

q (Sketch)

Hogsmeade

7

2 8

4 9

10 5

m “ 6

3

|C_i | “ 6:

5 227 236 choices

kk kik hk

kk kj

Practical

Limit!

“t8u, t4u‰

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Evaluation

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Evaluation

Ñ First artificial instances, then realistic instances.

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Evaluation

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Evaluation

for n “ 12

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Evaluation

Runtimes:

for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

T ^˚-Algorithm: 3 ms Classifier: 4 s for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

T ^˚-Algorithm: 3 ms Classifier: 4 s Classifier’s Accuracy:

for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

T ^˚-Algorithm: 3 ms Classifier: 4 s

Ratio T ^˚ “ ÝÑ T ^˚ Classifier’s Accuracy:

for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

Ratio T ^˚ “ ÝÑ T ^˚ Clusters close together („ 6km):

Classifier’s Accuracy:

59 % for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

far apart (ě 16 km):

Classifier’s Accuracy:

59 % 100 % for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

Recall Classifier’s Accuracy:

59 % 100 % for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

59 % 0.4 100 % for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

0.4 0.9 59 %

100 % for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

0.4 0.9 ÝÑ

T ^˚-Algorithm as Heuristic:

59 % 100 % for n “ 12

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Evaluation

Runtimes:

Exact: 120 s ÝÑ

0.4 0.9 ÝÑ

T ^˚-Algorithm as Heuristic:

Approximation Quality (empiric): ď 1.1

59 % 100 % for n “ 12

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Topology of Street Networks

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Topology of Street Networks

Street Networks often do not meet the assumptions.

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Topology of Street Networks

Example #1:

Rural Instance

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Topology of Street Networks

Example #1:

Rural Instance

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Topology of Street Networks

Example #1:

Rural Instance

T ^˚ bypasses a cluster!

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Topology of Street Networks

Example #1:

Rural Instance

Yet, no false positive.

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Topology of Street Networks

Example #1:

Rural Instance

Yet, no false positive.

ñ Classifier is robust to some extent.

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Topology of Street Networks

Example #2:

Regional Instance

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Topology of Street Networks

Example #2:

Regional Instance

Really hard scenario . . .

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Topology of Street Networks

Example #2:

Regional Instance

Really hard scenario . . .

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Topology of Street Networks

Example #2:

Regional Instance

Really hard scenario . . . False positives are to be expected in this case.

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Conclusion

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Conclusion

The Exact Algorithm considers unsensible tours.

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Conclusion

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Conclusion

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Conclusion

Intuition yields the ÝÑ

T ^˚-algorithm.

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Conclusion

T ^˚-algorithm.

A classifier decides if the ÝÑ

T ^˚-algorithm can be used.

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Conclusion

T ^˚-algorithm.

If yes, only a fraction of time is needed to get T ^˚.

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Conclusion

T ^˚-algorithm.

If no, virtually no time is wasted.

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Conclusion

T ^˚-algorithm.

No false-positives: Optimal route is guaranteed.

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Conclusion

T ^˚-algorithm.

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Conclusion

T ^˚-algorithm.

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Attributions

The above icons are made by Freepik from flaticon.com

Ð CC 3.0 BY by SimpleIcon from flaticon.com (c) Map Images from OpenStreetMap (osm.org)

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13/11 The following slides were abandoned at some point

and not officially shown at the presentation. They may contain errors or are incomplete. Maybe they help you nonetheless.

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The Objective Function

0

m 1

2 3

5

6

7

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The Objective Function

A tour T is a permutation of r0, 2m ´ 1s.

0

m 1

2 3

5

6

7

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The Objective Function

0

m 1

2 3

5

6

7

T “ r0, 3, 1, 5, 7, 2, 6, 4s

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The Objective Function

0

m 1

2 3

5

6

7 T feasible ô

T “ r0, 3, 1, 5, 7, 2, 6, 4s

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The Objective Function

0

m 1

2 3

5

6

7 T feasible ô T r1s “ 0 & T r2ms “ m

T “ r0, 3, 1, 5, 7, 2, 6, 4s

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The Objective Function

0

m 1

2 3

5

6

& precedences obeyed

T “ r0, 3, 1, 5, 7, 2, 6, 4s

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The Objective Function

0

m 1

2 3

5

6

& S not violated

T “ r0, 3, 1, 5, 7, 2, 6, 4s

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The Objective Function

0

m 1

2 3

5

6

& S not violated

S ě 3 T “ r0, 3, 1, 5, 7, 2, 6, 4s

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The Objective Function

0

m 1

2 3

5

6

& S not violated

S ě 3

T minfeasible

ř_2m

i“2 kpi ´ 1q ¨ d

”

T ri ´ 1s, T ris ı

Objective: T “ r0, 3, 1, 5, 7, 2, 6, 4s

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The Objective Function

0

m 1

2 3

5

6

& S not violated

S ě 3

kpjq is the number of persons after step j of T .

T minfeasible

ř_2m

i“2 kpi ´ 1q ¨ d

”

T ri ´ 1s, T ris ı

Objective: T “ r0, 3, 1, 5, 7, 2, 6, 4s

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An Exact Algorithm

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An Exact Algorithm

1

2 3

4

5

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An Exact Algorithm

1

2 3

4 Find a tour with 6 steps: 5

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An Exact Algorithm

1

2 3

1 2 3 4 5

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An Exact Algorithm

1

2 3

1 2 3 4 5

0

cost: 0

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An Exact Algorithm

1

2 3

1 2 3 4 5

0

cost: 0

1

cost: 9

2

cost: 20

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An Exact Algorithm

1

2 3

1 2 3 4 5

0

cost: 0

1

cost: 9

2

cost: 20

4

cost: 41

2

cost: 49

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An Exact Algorithm

1

2 3

1 2 3 4 5

0

cost: 0

1

cost: 9

2

cost: 20

4

cost: 41

2

cost: 49

1

cost: 60

5

cost: 49

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An Exact Algorithm

1

2 3

1 2 3 4 5

0

cost: 0

1

cost: 9

2

cost: 20

4

cost: 41

2

cost: 49

1

cost: 60

5

cost: 49

2

cost: 57

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An Exact Algorithm

1

2 3

1 2 3 4 5

0

cost: 0

1

cost: 9

2

cost: 20

4

cost: 41

2

cost: 49

1

cost: 60

5

cost: 49

2

cost: 57

4

cost: 98

5

cost: 92