Determination of robust solutions for the DARP with variations in

transportation time

Maxime Chassaing, Christophe Duhamel, Gérard Fleury and Philippe Lacomme

LIMOS, Université Blaise Pascal

France, Clermont-Ferrand

Contact : maxime.chassaing@isima.fr

Stochastic DARP Problem definition

• Dial-a-Ride Problem (DARP)

• Stochastic travel times

Solution

• Associated driver policy

• Robustness

Evaluation in the stochastic context

• Indirect robustness measure

• Simulation / Analytic evaluation

Resolution methods

• Iterative: ELS / Bi-criteria: NSGA-2

Conclusion

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Stochastic problems and vehicle routing problems

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Type 1 Variation of collected volume • Stochastic CARP : (Fleury et al., 2008)

Type 2 Variation of clients to be served • (Heilporn et al., 2011)

Type 3 Variation of travel times. • breakdowns, • route hazards, • congestion, traffic flow Review : (Gendreau et al., 2015) Stochastic Shortest Path (DSSPP) (Pattanamekar et al., 2003) VRP with Time Windows (Tas et al., 2013)

Publications

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Dial-a-ride Problem (DARP)

• Instances: (Cordeau and Laporte, 2003) (Ropke and Laporte, 2007)

• Methods VNS (Parragh and Schmid, 2013) ALNS (Masson et al., 2014) DA (Braekers et al., 2014)

Stochastic DARP (dynamic)

• On client’s demand

• (Heilporn et al., 2011)

• On travel time

• (Xiang et al., 2008)

• (Schilde et al., 2014)

event (crash) Influence area

Dial-a-Ride Problem (DARP)

Definition

• Homogeneous fleet of 𝑴 vehicles (𝑀 ≥ 1) :

• capacity : 𝐶𝑚𝑎𝑥

• 1 depot node

• 𝑵 clients (𝑁 ≥ 1) :

• 𝑅𝑖 request ( origin 𝐼+ destination 𝐼−)

Constraints

• [𝑒𝑖𝑚𝑖𝑛, 𝑙𝑖

𝑚𝑎𝑥] time windows : TW

• Total duration : 𝑻𝑹𝑻max

• Maximal riding time : 𝑻𝑫max

OBJECTIVE: find a solution to transport each client and

minimizing the total distance

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0

1+

2-

3+

1-

3-

1

1

2

2+

20,40

15,35

35,50

30,60

80,90

10,20

85 60

50

35

2515

Solution of DARP instances

• solution = set of trips

• trip is

vertex order :

depot , 1+, 1-, 2+, 3+, 2-, 3-, depot

set of variables for each vertex

• 𝑨𝒊 : arrival time on i

• 𝑩𝒊 : beginning of service at i

• 𝑾𝒊 : waiting time on i

• 𝑫𝒊 : departure time from i

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(Cordeau et Laporte, 2003)

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Di departure time from vi to vj

i

j

ej ljTime windows on vj

Aj arrival date on vj

Ai arrival Date on vi

Bi beginning of service

Wi waiting time

di service duration

Time windows on vi

ei li

time

no

de

s

Dj

Bj

tij Wj

(Cordeau et Laporte, 2003)

Deterministic case : 𝑩𝒊 (known) ⟹ 𝑫𝒊(known) ⟹ 𝑨𝒊+𝟏 (known) Evaluation : (Cordeau et al. 2003), (Firat and Woeginger, 2011)

Illustration of trip’s variables (specific to the DARP)

Di departure time from vi to vj

i

j

ej ljTime windows on vj

Aj arrival date on vj

Ai arrival Date on vi

Bi beginning of service

Wi waiting time

di service duration

Time windows on vi

ei li

time

Dj

Bj

Tij(ω’) Wj

no

de

s

Tij(ω)

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For the stochastic case : 𝑩𝒊 (known) ⟹ 𝑫𝒊(known) ⇏ 𝑨𝒊+𝟏 (unknown)

Stochastic travel times (implies choices on policy of drivers)

Illustration of trip’s variables

Main model: normal distribution

Tij~𝑁(𝑡𝑖𝑗 , 𝜎𝑖𝑗) with 𝜎𝑖𝑗= 𝑡𝑖𝑗/10

• example : 𝑡𝑖𝑗 = 10 min

• 𝑃 (9 𝑚𝑖𝑛 ≤ 𝑇𝑖𝑗 ≤ 11 𝑚𝑖𝑛) ≅ 70%

Other model: gamma distribution (Tas et al., 2013) shifted gamma distribution

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Travel time model used

expected value = 𝑡𝑖𝑗

Standard deviation = 𝑡𝑖𝑗

𝛾

Disadvantages :

- for a given 𝛾 expected value and variance are linked

𝑡𝑖𝑗 0

𝛾=4

68,2%

15,9% 15,9%

𝑡𝑖𝑗 𝑡𝑖𝑗 -𝜎𝑖𝑗 𝑡𝑖𝑗 +𝜎𝑖𝑗

0

expected value variance

𝑡𝑖𝑗 0

𝛾=1

𝑡𝑖𝑗 0

𝛾=16

• Using the expected values

• 𝑨𝒊 ; 𝑩𝒊 ; 𝑾𝒊 ; 𝑫𝒊 can be computed

But for realizations: 𝝎 and 𝝎′ travel time became 𝐓𝐢𝐣(𝝎) and 𝐓𝐢𝐣(𝝎′)

• So, become random values:

𝑨𝒊(𝝎) ; 𝑩𝒊(𝝎) ; 𝑾𝒊(𝝎) ; 𝑫𝒊(𝝎)

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si

si-1

li-1ei-1

Di-1

𝐴𝑖(𝜔)

Bi-1

𝐵𝑖

𝐴𝑖(𝜔′)

𝑇𝑖−1,𝑖(𝜔′)

Ai-1 Wi-1 di-1

𝑇𝑖−1,𝑖(𝜔)

𝐴𝑖 𝑊𝑖

theoretical value

Consequences of stochastic travel time

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Driver’s policies

Because 𝐴𝑖 𝜔 ≠ 𝐴𝑖 the driver have different choices:

• 𝑖𝑓 𝐴𝑖 𝜔 > 𝐴𝑖 then ???

• 𝑖𝑓 𝐴𝑖 𝜔 < 𝐴𝑖 then ???

Different driver’s policies can be defined: 1 « if the driver is early then he waits until 𝐵𝑖 »

2 « the driver has to respect the waiting time 𝑊𝑖 »

3 …

si

si-1

li-1ei-1

Di-1

𝐴𝑖(𝜔)

Bi-1

𝐵𝑖

Ai-1 Wi-1 di-1

𝑇𝑖−1,𝑖(𝜔)

𝐴𝑖

theoretical value

realization value

𝑊𝑖

A solution of DARP

• order of vertices

• Defined dates 𝑨𝒊, 𝑩𝒊, 𝑾𝒊, 𝑫𝒊

Robustness 𝑷(𝑺) : probability that the solution satisfies each constraint

𝑃(𝑆) = 𝑃 𝑡𝑖𝑖

with 𝑃 𝑡𝑖 probability that a trip is valid

Driver’s policy : meet 𝑩𝒊, 𝐞𝐧𝐬𝐮𝐫𝐞 𝑾𝒊, …

New criterion : robustness

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S2

S1

S3 S4

x

Vertex order ‘s space

2 – Robustness estimator

S2 ( 180 , 0.5)

S1 ( 180 , 0.1)

S3 ( 180 , 0.7)

S4 ( 180 , 0.9)

2

Solutions space (Cost known)

3

1 - Evaluation: compute Bi

Keep the solutionwith the highest

robustness

1 S -> (Cost(S), Robustness = F(S,n))

PROPOSITION

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maximizing a criterion of robustness ρ*

1'Directly used

ρ* as criterion of robustness

2'

0

1+

2-

3+

1-

3-

1

1

2

2+

20,40

15,35

35,50

30,60

80,90

10,20

85 60

50

35

2515

0

1+

2-

3+

1-

3-

1

1

2

2+

20,40

15,35

35,50

30,60

80,90

10,20

80 55

45

30

2010

1- Evaluation for the SDARP

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si

si-1

Di-1 Bi-1

𝐴𝑖 𝐴𝑖𝜔

di-1

𝑇𝑖−1,𝑖(𝜔)

si

si-1

Di-1 Bi-1

𝐴𝑖 𝐴𝑖𝜔

di-1

𝑇𝑖−1,𝑖(𝜔) 13

Evaluation for the DARP is, for a fixed order of clients, compute Bi on vertices which respect constraints

This schedule affects the robustness

Definition of valid constraint Deterministic case

• Constraints : 𝐵𝑖 ≤ 𝐶𝑠𝑡

𝐵𝑖 ≥ 𝐶𝑠𝑡

𝐵𝑖 − 𝐵𝑗 ≤ 𝐶𝑠𝑡

Stochastic case

𝐵𝑖(𝜔) depends on realizations of travel time

- for some 𝜔 the constraint is valid

- for some 𝜔 ′ the constraint is not valid

PROPOSITION : In the model a constraint is consider valid to the order ρ if:

𝑷{ 𝐵𝑖(𝜔) ≤ 𝐶𝑠𝑡 } ≥ ρ ⇔ 𝐵𝑖 +𝑐(ρ)𝜎𝑖 ≤ 𝐶𝑠𝑡

𝑷{ 𝐵𝑖 𝜔 ≥ 𝐶𝑠𝑡 } ≥ ρ ⇔ 𝐵𝑖 +𝑐(ρ)𝜎𝑖≥ 𝐶𝑠𝑡

𝑷{𝐵𝑖(𝜔) − 𝐵𝑗(𝜔) ≥ 𝐶𝑠𝑡 } ≥ ρ ⇔ 𝐵𝑖 − 𝐵𝑗 + 𝑐 ρ (𝜎𝑖 + 𝜎𝑗) ≤ 𝐶𝑠𝑡

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𝐶𝑠𝑡

𝐵𝑖 j

𝑐(𝜌)𝜎𝑖

stochastic problem :

𝐶𝑠𝑡

𝐵𝑖 j

deterministic problem

𝐵𝑖 ≤ 𝐶𝑠𝑡

14

PROPOSITION: compute 𝐵𝑖 for a fixed ρ

Using the definition of valid constraint to the order ρ

Evaluate a trip is possible if is known :

- the driver’s policy

- the ρ value

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evaluation proposed by (Firat and Woeginger , 2011) using the new constraints

A+ B+ C+0 C-0 00

B- A-00

CST CST CSTCST

CST

-CST-CST -CST -CST -CST

CST

-CST

00

-CST

-CST

-CST

i=0 i=1 i=2 i=4i=3 i=5 i=6

CST

Time Windows min constraints Time Windows max constraints Riding time max and total duration max constraints

PROPOSITION: to improve the solution robustness

Maximize a criterion of robustness

Objective : For each trip

find the maximal value of ρ possible: ρmax 𝑡

• Dichotomy to find ρmax 𝑡

using ρmax 𝑡 of each trip a criterion associated to the robustness

of a solution have been created:

ρ* = ρmax 𝑡𝑡

ρ* will by used in the metaheuristic to replace 𝑃(𝑆)

(Warning) ρ* ≠ 𝑃(𝑆)

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For a driver policy and affected dates on vertices

• The robustness of a solution can be compute :

• analytically 𝑷(𝒔)

• by simulation 𝑭 𝒔, 𝒏 an estimator of 𝑃(𝑆)

with 𝑛 number of replications

2 - Robustness estimator

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100%

Simulation n (replication)

𝐹(𝑠, 𝑛)

𝑃(𝑠)

S2

S1

S3 S4

x

Vertex order ‘s space

2 – Robustness estimator

S2 ( 180 , 0.5)

S1 ( 180 , 0.1)

S3 ( 180 , 0.7)

S4 ( 180 , 0.9)

2

Solutions space (Cost known)

3

1 - Evaluation: compute Bi

Keep the solution with the highest robustness

1 S -> (Cost(S), Robustness = F(S,n))

Evaluation: compute Bimaximizing a criterion of

robustness ρ*

1'

Directly used ρ* as

criterion of robustness

2'

On the 20 instances proposed by (Cordeau et Laporte, 2003)

BKS (Best known solution): (Braekers et al., 2014) (Parragh et Schmid, 2013)

Results for normal distribution

H1 : driver have to met Bi

𝑩𝒊 theoretical beginning of service

Comparison between ρ* and 𝐹 𝑠, 𝑛

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name m n c(BKS) ρ𝐻1∗ F𝐻1

(BKS, n)

pr01 3 24 190.02 74.8% 86.4%

pr02 5 48 301.34 76.8% 85.4%

pr03 7 72 532.00 4.8% 0.1%

pr04 9 96 570.25 14.3% 0.3%

pr05 11 120 626.93 13.1% 0.7%

pr06 13 144 785.26 3.9% 0.0%

pr07 4 36 291.71 21.9% 14.7%

pr08 6 72 487.84 8.9% 0.2%

pr09 8 108 658.31 9.4% 0.4%

pr10 10 144 851.82 1.3% 0.0%

pr11 3 24 164.46 63.2% 61.0%

pr12 5 48 295.66 32.0% 5.8%

pr13 7 72 484.83 32.5% 16.5%

pr14 9 96 529.33 59.3% 64.6%

pr15 11 120 577.29 37.9% 72.0%

pr16 13 144 730.67 16.2% 11.1%

pr17 4 36 248.21 26.8% 1.6%

pr18 6 72 458.73 62.2% 72.9%

pr19 8 108 593.49 5.8% 0.0%

pr20 10 144 785.68 3.8% 0.0%

avg. 28.4% 24.7%

value

instances

Generate robust solutions

Two methods :

1 – mono-criterion with

1𝑠𝑡 : 𝑃(𝑆) (solution robustness) ρ*(criteria associated)

2𝑛𝑑 : 𝐶(𝑆) (solution cost)

Evolutionary local search (ELS) (Wolf and Merz, 2007)

2 – multi-criteria : a solution label (ρ∗(𝑆), 𝐶(𝑆))

NSGA-II algorithm

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C(S)

ρHx(S)

optimal Pareto front

10

robustness

criterion ρ

distance

dominated solutions

Results BKS vs BFS

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𝑩𝑲𝑺 𝑩𝑭𝑺 normal distribution 𝑩𝑭𝑺 Shifted gamma

distribution 𝜸 = 𝟒

name m n c(BKS) F𝐻1(BKS, n) c(BFS) gap F 𝐻1(BFS, n) time

(min) c(BFS) gap F 𝐻1(BFS, n)

time

(min)

pr01 3 24 190.02 86.4% 200.48 5.50% 100.0% 0.25 210.94 11.01% 99.9% 0.25

pr02 5 48 301.34 85.4% 307.11 1.92% 100.0% 1.25 317.00 5.20% 100.0% 1.25

pr03 7 72 532.00 0.1% 587.96 10.52% 100.0% 2.30 628.59 18.16% 99.7% 2.30

pr04 9 96 570.25 0.3% 643.75 12.89% 100.0% 7.37 663.68 16.38% 99.3% 7.37

pr05 11 120 626.93 0.7% 744.92 18.82% 100.0% 12.07 752.20 19.98% 99.8% 12.07

pr06 13 144 785.26 0.0% 979.78 24.77% 100.0% 21.92 993.02 26.46% 99.1% 21.92

pr07 4 36 291.71 14.7% 306.69 5.13% 100.0% 0.47 320.29 9.80% 100.0% 0.47

pr08 6 72 487.84 0.2% 573.65 17.59% 100.0% 2.68 607.81 24.59% 99.3% 2.68

pr09 8 108 658.31 0.4% 774.53 17.65% 100.0% 11.25 790.74 20.12% 93.8% 11.25

pr10 10 144 851.82 0.0% 1049.23 23.18% 99.5% 21.33 1052.57 23.57% 76.6% 21.33

pr11 3 24 164.46 61.0% 168.81 2.64% 100.0% 0.28 178.97 8.82% 100.0% 0.28

pr12 5 48 295.66 5.8% 310.66 5.07% 100.0% 1.37 334.41 13.10% 100.0% 1.37

pr13 7 72 484.83 16.5% 527.10 8.72% 100.0% 3.70 605.94 24.98% 100.0% 3.70

pr14 9 96 529.33 64.6% 634.16 19.80% 100.0% 10.20 638.52 20.63% 100.0% 10.20

pr15 11 120 577.29 72.0% 634.08 9.84% 100.0% 19.93 701.94 21.59% 99.8% 19.93

pr16 13 144 730.67 11.1% 891.86 22.06% 100.0% 32.32 934.23 27.86% 99.9% 32.32

pr17 4 36 248.21 1.6% 266.83 7.50% 100.0% 0.58 294.83 18.78% 100.0% 0.58

pr18 6 72 458.73 72.9% 494.54 7.81% 100.0% 4.32 530.68 15.68% 100.0% 4.32

pr19 8 108 593.49 0.0% 699.70 17.89% 100.0% 12.43 712.33 20.02% 99.9% 12.43

pr20 10 144 785.68 0.0% 978.61 24.56% 100.0% 31.45 972.21 23.74% 99.8% 31.45

avg.

24.7% 13.19% ~100% 9.87 18.52% 98,4% 9.87

BKS: Best Known Solution for the DARP determinist (Braekers et al., 2014) (Parragh et Schmid, 2013) (Cordeau et Laporte, 2003)

BFS: Best Found Solution obtain with the ELS

Results with NSGA-2 ( Extreme solutions of the front )

column with avg. of 5 runs : 𝑛𝑏 ; 𝑔𝑎𝑝 ; 𝐹𝐻1(𝑆, 𝑛)

column with best results of 5 runs : 𝑔𝑎𝑝* ; 𝐹𝐻1(𝑆, 𝑛) *

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name Solution with minimal cost Solution with maximal robustness temps

𝒏 𝒈𝒂𝒑 𝒈𝒂𝒑∗ 𝐅𝐇𝟏(𝐬, 𝐧) 𝑭𝑯𝟏(𝑺, 𝒏) * 𝒈𝒂𝒑 𝒈𝒂𝒑∗ 𝐅𝐇𝟏(𝐬, 𝐧) 𝑭𝑯𝟏(𝑺, 𝒏)

* min

pr01 14.6 0.08% 0.00% 87.3% 85.4% 3.87% 4.31% 100.0% 100% 0.25

pr02 20.6 2.18% 0.86% 79.2% 90.3% 4.17% 5.51% 100.0% 100% 1.25

pr03 42.2 4.13% 2.84% 55.2% 86.8% 12.40% 11.03% 100.0% 100% 2.30

pr04 43.0 5.85% 3.18% 55.4% 32.9% 9.95% 12.38% 100.0% 100% 7.37

pr05 49.4 6.88% 5.16% 31.4% 38.6% 10.61% 12.21% 100.0% 100% 12.07

pr06 73.4 6.23% 4.24% 12.6% 0.3% 10.84% 10.08% 100.0% 100% 21.92

pr07 13.4 2.11% 1.21% 63.6% 81.5% 5.12% 5.79% 100.0% 100% 0.47

pr08 49.6 5.35% 3.61% 31.5% 37.3% 18.87% 20.09% 100.0% 100% 2.68

pr09 84.0 8.06% 5.90% 20.6% 5.8% 18.01% 13.18% 100.0% 100% 11.25

pr10 89.0 8.17% 7.02% 2.5% 2.2% 17.51% 20.80% 95.5% 100% 21.33

pr11 7.6 2.19% 0.25% 78.3% 75.1% 3.71% 4.52% 100.0% 100% 0.28

pr12 20.4 3.77% 2.72% 57.8% 26.3% 5.98% 5.89% 100.0% 100% 1.37

pr13 38.0 7.39% 6.22% 64.3% 43.3% 13.20% 12.23% 100.0% 100% 3.70

pr14 29.8 7.37% 6.66% 52.1% 95.3% 11.58% 12.92% 100.0% 100% 10.20

pr15 30.0 5.77% 2.09% 23.6% 48.4% 8.28% 9.33% 96.4% 100% 19.93

pr16 72.8 7.48% 6.47% 22.5% 38.7% 11.37% 9.94% 100.0% 100% 32.32

pr17 19.2 4.11% 2.04% 31.0% 5.7% 8.86% 10.80% 100.0% 100% 0.58

pr18 29.0 5.25% 3.11% 47.3% 78.1% 11.10% 12.17% 100.0% 100% 4.32

pr19 61.0 8.16% 6.80% 15.7% 1.2% 15.92% 22.34% 100.0% 100% 12.43

pr20 69.6 8.73% 7.59% 12.3% 27.1% 14.47% 19.04% 99.9% 100% 31.45

avg. 42.8 5.46% 3.90% 42.2% 45.0% 10.79% 11.73% 99.6% 100% 9.87

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Conclusion Proposition

1 - a robustness criterion (usable in a metaheuristic) 2 - an evaluation method to obtain robust schedules (according to a driver’s policy)

Methods General framework to optimize the robustness

Results obtained For classic literature DARP instances. 1 - the BKSs for the determinist DARP are not robust 2 - the solutions we computed provide a « reasonable » cost Detailed results are available online : http://fc.isima.fr/~chassain/SDARP/SDARP.php

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