Exact Algorithms for the Chance-Constrained Vehicle Routing Problem - - PowerPoint PPT Presentation

Exact Algorithms for the Chance-Constrained Vehicle Routing Problem

Ricardo Fukasawa

Department of Combinatorics & Optimization University of Waterloo

January 7, 2016 Aussois 2016 joint work with Thai Dinh and James Luedtke (University of Wisconsin)

Dinh, Fukasawa, Luedtke Chance-constrained VRP 1 / 28

The deterministic vehicle routing problem

depot

G = (V , E) V = {0} ∪ V+ Edge lengths ℓe, e ∈ E K vehicles, capacity b Find a set of K routes with minimum total length Client demands di, ∀i ∈ V+

Dinh, Fukasawa, Luedtke Chance-constrained VRP 2 / 28

The deterministic vehicle routing problem

depot

G = (V , E) V = {0} ∪ V+ Edge lengths ℓe, e ∈ E K vehicles, capacity b Find a set of K routes with minimum total length Client demands di, ∀i ∈ V+ Let Sj be the set of clients served by route j. Then d(Sj) ≤ b

Dinh, Fukasawa, Luedtke Chance-constrained VRP 2 / 28

The stochastic vehicle routing problem

depot

G = (V , E) V = {0} ∪ V+ Edge lengths ℓe, e ∈ E K vehicles, capacity b Find a set of K routes with minimum total length Client demands di, ∀i ∈ V+ Demands Di, ∀i ∈ V+: random variables that only get realized after routes have been decided Let Sj be the set of clients served by route j. Then d(Sj) ≤ b

Dinh, Fukasawa, Luedtke Chance-constrained VRP 2 / 28

The stochastic vehicle routing problem

depot

G = (V , E) V = {0} ∪ V+ Edge lengths ℓe, e ∈ E K vehicles, capacity b Find a set of K routes with minimum total length Client demands di, ∀i ∈ V+ Demands Di, ∀i ∈ V+: random variables that only get realized after routes have been decided Let Sj be the set of clients served by route j. Then d(Sj) ≤ b

Question

What to do when sum of loads exceed b?

Dinh, Fukasawa, Luedtke Chance-constrained VRP 2 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Recourse-based model (two-stage) Follow planned routes

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan Common assumption: Make round-trip to depot and back (simpler) Minimize expected cost (including cost

• f recourse)

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan Common assumption: Make round-trip to depot and back (simpler) Minimize expected cost (including cost

• f recourse)

Other recourse options Alternate recourse actions may be better (e.g., Novoa et al., 2006)

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan Common assumption: Make round-trip to depot and back (simpler) Minimize expected cost (including cost

• f recourse)

Other recourse options Alternate recourse actions may be better (e.g., Novoa et al., 2006)

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Recourse-based model (two-stage) Follow planned routes If capacity is exceeded at a customer, determine alternative service plan Common assumption: Make round-trip to depot and back (simpler) Minimize expected cost (including cost

• f recourse)

Other recourse options Alternate recourse actions may be better (e.g., Novoa et al., 2006) but more expensive computationally

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded!

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! Let Sj be the set of clients served by route j. Then P {D(Sj) ≤ b} ≥ 1 − ǫ

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! Let Sj be the set of clients served by route j. Then P {D(Sj) ≤ b} ≥ 1 − ǫ Advantages No need to model complicated recourse actions Customers more likely to receive “regular” service Possible driver efficiency

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! Let Sj be the set of clients served by route j. Then P {D(Sj) ≤ b} ≥ 1 − ǫ Advantages No need to model complicated recourse actions Customers more likely to receive “regular” service Possible driver efficiency Disadvantage: When you are the unlucky customer that is left stranded with probability ǫ

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

depot

Chance-constrained model Limit the likelihood that capacity in each truck is exceeded to a specified threshold ǫ And ignore cases when capacity is exceeded! Let Sj be the set of clients served by route j. Then P {D(Sj) ≤ b} ≥ 1 − ǫ Advantages No need to model complicated recourse actions Customers more likely to receive “regular” service Possible driver efficiency Disadvantage: When you are the unlucky customer that is left stranded with probability ǫ e.g. Dey and F. (Sunday at train station)

Dinh, Fukasawa, Luedtke Chance-constrained VRP 3 / 28

What to do when capacity is violated? (2-stage vs. chance-constrained)

Experiment: For an instance, take routing decisions based on chance-constrained (ǫ = 5%) and recourse (2-stage) model Evaluate each solution by probability that a truck’s capacity is violated and by its expected 2-stage cost (2-stage solution is optimal) Four instances, size up to 22 nodes, all independent normal (low and high variance) Max Violation Prob. % % Increase Var CC Sol 2-stage Sol Expected Cost Low 1.7 50.0 2.3% 5.0 7.8 0.9% 2.4 2.4 3.1 6.4 0.6% High 4.0 8.3 3.4% 3.6 23.7 2.9% 1.0 1.0 0.7 16.9 0.3%

Dinh, Fukasawa, Luedtke Chance-constrained VRP 4 / 28

Literature review

Deterministic VRP State-of-the-art methods use branch-and-cut-and-price E.g., F. et al. (2006), Baldacci et al. (2008), Baldacci et al. (2011), Pecin et al. (2014) Stochastic VRP (2-stage)

Heuristics: Stewart & Golden (1983), Dror & Trudeau (1986), Savelsbergh & Goetschalckx (1995), Novoa et al. (2006), Secomandi and Margot (2009), . . . Integer L-Shaped: Gendreau et al. (1994), Laporte et al. (2002), . . . Branch-and-cut: Laporte et al. (1989), . . . Branch-and-price: Christiansen et al. (2007) Branch-and-cut-and-price: Gauvin et al. (2014) Stochastic VRP (chance-constrained) Reduction to deterministic case: Stewart & Golden (1983) Branch-and-cut: Laporte et al. (1989)

Dinh, Fukasawa, Luedtke Chance-constrained VRP 5 / 28

Literature review

Deterministic VRP State-of-the-art methods use branch-and-cut-and-price E.g., F. et al. (2006), Baldacci et al. (2008), Baldacci et al. (2011), Pecin et al. (2014) Stochastic VRP (2-stage)

Heuristics: Stewart & Golden (1983), Dror & Trudeau (1986), Savelsbergh & Goetschalckx (1995), Novoa et al. (2006), Secomandi and Margot (2009), . . . Integer L-Shaped: Gendreau et al. (1994), Laporte et al. (2002), . . . Branch-and-cut: Laporte et al. (1989), . . . Branch-and-price: Christiansen et al. (2007) Branch-and-cut-and-price: Gauvin et al. (2014) Stochastic VRP (chance-constrained) Reduction to deterministic case: Stewart & Golden (1983) Branch-and-cut: Laporte et al. (1989) Not aware of any exact SVRP method that has been tested for problems with correlation (most assume independent normal)

Dinh, Fukasawa, Luedtke Chance-constrained VRP 5 / 28

Literature review

Deterministic VRP State-of-the-art methods use branch-and-cut-and-price E.g., F. et al. (2006), Baldacci et al. (2008), Baldacci et al. (2011), Pecin et al. (2014) Stochastic VRP (2-stage)

Heuristics: Stewart & Golden (1983), Dror & Trudeau (1986), Savelsbergh & Goetschalckx (1995), Novoa et al. (2006), Secomandi and Margot (2009), . . . Integer L-Shaped: Gendreau et al. (1994), Laporte et al. (2002), . . . Branch-and-cut: Laporte et al. (1989), . . . Branch-and-price: Christiansen et al. (2007) Branch-and-cut-and-price: Gauvin et al. (2014) Stochastic VRP (chance-constrained) Reduction to deterministic case: Stewart & Golden (1983) Branch-and-cut: Laporte et al. (1989) Not aware of any exact SVRP method that has been tested for problems with correlation (most assume independent normal)

Goal

Develop exact methods for chance-constrained SVRP with very few assumptions on the demand uncertainty.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 5 / 28

Edge formulation for deterministic VRP

di: deterministic demand at customer i ∈ V+ r(S): number of trucks required to serve S ⊆ V+ xe: number of times a vehicle traverses edge e ∈ E min

x

• e∈E

ℓexe s.t.

• e∈δ({i})

xe = 2, ∀i ∈ V+

• e∈δ({0})

xe = 2K

• e∈δ(S)

xe ≥ 2r(S), ∀S ⊆ V+ xe ≤ 1, ∀e ∈ E \ δ({0}) xe ∈ Z+, ∀e ∈ E.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 6 / 28

Edge formulation for deterministic VRP (2)

Capacity inequalities

• e∈δ(S)

xe ≥ 2r(S), ∀S ⊆ V+ For determinstic VRP, r(S) requires solving bin-packing. Valid formulation by substituting r(S) by the lower bound

• d(S)

b

• .

Exact separation is possible for x ∈ Z|E|

+ .

Heuristic separation for x ∈ R|E|

+ .

Many other classes of valid inequalities

Dinh, Fukasawa, Luedtke Chance-constrained VRP 7 / 28

Edge formulation for chance-constrained VRP

Modified capacity inequalities

• e∈δ(S)

xe ≥ 2rǫ(S), ∀S ⊆ V+ rǫ(S): Minimum number of trucks required to serve customer set S, where probability of capacity violation is at most ǫ for each truck Requires solving stochastic bin-packing

Dinh, Fukasawa, Luedtke Chance-constrained VRP 8 / 28

Edge formulation for chance-constrained VRP

Modified capacity inequalities

• e∈δ(S)

xe ≥ 2rǫ(S), ∀S ⊆ V+ rǫ(S): Minimum number of trucks required to serve customer set S, where probability of capacity violation is at most ǫ for each truck Requires solving stochastic bin-packing

Challenge

How to obtain valid lower bounds on rǫ(S)? Laporte et al. (1989): If demands are independent normal, can use Q1−ǫ(S) b

• where Qp(S) be pth quantile of the random variable

i∈S Di, i.e.

Qp(S) := inf

• b′ : P{

i∈S Di ≤ b′} ≥ p

• .

Dinh, Fukasawa, Luedtke Chance-constrained VRP 8 / 28

Edge formulation for chance-constrained VRP

Modified capacity inequalities

• e∈δ(S)

xe ≥ 2rǫ(S), ∀S ⊆ V+ rǫ(S): Minimum number of trucks required to serve customer set S, where probability of capacity violation is at most ǫ for each truck Requires solving stochastic bin-packing

Challenge

How to obtain valid lower bounds on rǫ(S)? Laporte et al. (1989): If demands are independent normal, can use Q1−ǫ(S) b

• where Qp(S) be pth quantile of the random variable

i∈S Di, i.e.

Qp(S) := inf

• b′ : P{

i∈S Di ≤ b′} ≥ p

• .

Not valid in general.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 8 / 28

Bad example for Laporte et al. bound

1 2 3 4 Scenarios 1 2 3 Clients 1 1 2 1 2 1 1 1 3 1 1 2 4 1 1 1 Probability 0.8 0.1 0.1

Table: Demands in each scenario

b = 2 ǫ = 0.1

Dinh, Fukasawa, Luedtke Chance-constrained VRP 9 / 28

Bad example for Laporte et al. bound

1 2 3 4 Scenarios 1 2 3 Clients 1 1 2 1 2 1 1 1 3 1 1 2 4 1 1 1 Probability 0.8 0.1 0.1

Table: Demands in each scenario

b = 2 ǫ = 0.1 Solution depicted is feasible

Dinh, Fukasawa, Luedtke Chance-constrained VRP 9 / 28

Bad example for Laporte et al. bound

1 2 3 4 Scenarios 1 2 3 Clients 1 1 2 1 2 1 1 1 3 1 1 2 4 1 1 1 Probability 0.8 0.1 0.1

Table: Demands in each scenario

b = 2 ǫ = 0.1 Solution depicted is feasible However, for S = {1, 2, 3, 4}, Q0.9(S) = 5

Dinh, Fukasawa, Luedtke Chance-constrained VRP 9 / 28

Bad example for Laporte et al. bound

1 2 3 4 Scenarios 1 2 3 Clients 1 1 2 1 2 1 1 1 3 1 1 2 4 1 1 1 Probability 0.8 0.1 0.1

Table: Demands in each scenario

b = 2 ǫ = 0.1 Solution depicted is feasible However, for S = {1, 2, 3, 4}, Q0.9(S) = 5 Thus using

• Q1−ǫ(S)

b

• requires 3

vehicles to enter S = {1, 2, 3, 4}

Dinh, Fukasawa, Luedtke Chance-constrained VRP 9 / 28

Bounds on required trucks more generally

Simple general bound

kǫ(S) =    1 P

• i∈S

Di ≤ b

• ≥ 1 − ǫ

2

• therwise

kǫ(S) ≤ rǫ(S) but sufficient to define a valid formulation Cheap to compute for a given set S Cuts may be weak

Dinh, Fukasawa, Luedtke Chance-constrained VRP 10 / 28

Improved general bound

Qp(S): p-th quantile of the random variable

i∈S Di

Qp(S) := inf

• b′ : P{

i∈S Di ≤ b′} ≥ p

• .

Lemma

rǫ(S) ≥ Q1−ǫrǫ(S)(S) b

• Dinh, Fukasawa, Luedtke

Chance-constrained VRP 11 / 28

Improved general bound

Qp(S): p-th quantile of the random variable

i∈S Di

Qp(S) := inf

• b′ : P{

i∈S Di ≤ b′} ≥ p

• .

Lemma

rǫ(S) ≥ Q1−ǫrǫ(S)(S) b

• But we don’t know rǫ(S)!

Dinh, Fukasawa, Luedtke Chance-constrained VRP 11 / 28

Improved general bound

Qp(S): p-th quantile of the random variable

i∈S Di

Qp(S) := inf

• b′ : P{

i∈S Di ≤ b′} ≥ p

• .

Lemma

rǫ(S) ≥ Q1−ǫrǫ(S)(S) b

• But we don’t know rǫ(S)!

Lemma

For any k ≥ 2, rǫ(S) ≥ min

• k,

Q1−ǫ(k−1)(S) b

• Proof: Either rǫ(S) ≥ k or rǫ(S) ≤ k − 1.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 11 / 28

Improved general bound (2)

Use best k: k∗

ǫ (S) = max

• min
• k,

Q1−ǫ(k−1)(S) b

• : k = 2, . . . , K
• .

Always at least as good as first simple bound

Dinh, Fukasawa, Luedtke Chance-constrained VRP 12 / 28

Improved general bound (2)

Use best k: k∗

ǫ (S) = max

• min
• k,

Q1−ǫ(k−1)(S) b

• : k = 2, . . . , K
• .

Always at least as good as first simple bound Improvements are possible for special cases: Independent normal: Can use bound of Laporte et al. (1989) Correlated normal: Can obtain bound based on smallest eigenvalue of covariance matrix Σ

Dinh, Fukasawa, Luedtke Chance-constrained VRP 12 / 28

Joint normal demands

{Di}i∈V+ are joint normal with mean µ and covariance matrix Σ 0. For S ⊆ V+:

◮ Let ΣS be the submatrix of Σ corresponding to S ◮ Let λS be the smallest eigenvalue of ΣS ◮ Let ¯

di(S) = µi + Φ−1(1 − ǫ)

• λS/|S|

Define: kJ

ǫ (S) =

• 1,

if P{D(S) ≤ b} ≥ 1 − ǫ max

• i∈S ¯

di(S)/b

• , 2
• ,
• therwise.

Theorem

If the demands follow a joint normal distribution with mean vector µ and covariance matrix Σ and ǫ ≤ 0.5, then kǫ(S) ≤ kJ

ǫ (S) ≤ rǫ(S), ∀S ⊆ V+.

Note: For the independent normal case, we have kJ

ǫ (S) = kI ǫ(S)

Note 2: Neither kJ

ǫ (S) nor k∗ ǫ (S) dominate each other.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 13 / 28

Joint normal demands (2)

¯ di(S) = µi + Φ−1(1 − ǫ)

• λS/|S|

Proof.

Model rǫ(S) as stochastic bin-packing. zt if use bin t, t = 1, . . . , K; yit if item i goes in bin t Items fit in bin t if P{

i∈S

Diyit ≤ Czt} ≥ 1 − ǫ P{

• i∈S

Diyit ≤ Czt} ≥ 1 − ǫ ⇐ ⇒

• i∈S

µiyit + Φ−1(1 − ǫ)

• y T

StΣSySt ≤ Czt

(1)

• y T

StΣSySt ≥

• λSySt2 =
• λSySt ≥
• λS/|S|

i∈S

yit.

• i∈S

¯ di(S)yit ≤ Czt. (2)

• i∈S

¯ di(S)/b

• is a lower bound for the (deterministic) bin-packing using (2), thus,

rǫ(S) ≥

• i∈S

¯ di(S)/b

• Dinh, Fukasawa, Luedtke

Chance-constrained VRP 14 / 28

Branch-and-cut-and-price for DETERMINISTIC VRP

Sets : Ω: set of feasible routes Parameters : qe

j : number of times edge e

appears in route j Variables : λj: (binary) whether to choose route j

Relation of x and λ

• j∈Ω

qe

j λj = xe, ∀e ∈ E

min

λ

• e∈E

ℓexe s.t.

• e∈δ({i})

xe = 2, ∀i ∈ V+

• e∈δ({0})

xe = 2K

• e∈δ(S)

xe ≥ 2r(S), ∀S ⊆ V+ xe ≤ 1, ∀e ∈ E \ δ({0}) x ∈ ZE

+

Dinh, Fukasawa, Luedtke Chance-constrained VRP 15 / 28

Branch-and-cut-and-price for DETERMINISTIC VRP

Sets : Ω: set of feasible routes Parameters : qe

j : number of times edge e

appears in route j Variables : λj: (binary) whether to choose route j

Relation of x and λ

• j∈Ω

qe

j λj = xe, ∀e ∈ E

min

λ

• e∈E

ℓe

• j∈Ω

qe

j λj

s.t.

• e∈δ({i})
• j∈Ω

qe

j λj = 2, ∀i ∈ V+

• e∈δ({0})
• j∈Ω

qe

j λj = 2K

• e∈δ(S)
• j∈Ω

qe

j λj ≥ 2r(S), ∀S ⊆ V+

• j∈Ω

qe

j λj ≤ 1, ∀e ∈ E \ δ({0})

λj ∈ {0, 1}, ∀j ∈ Ω

Dinh, Fukasawa, Luedtke Chance-constrained VRP 15 / 28

Branch-and-cut-and-price for DETERMINISTIC VRP

Sets : Ω: set of feasible routes Parameters : qe

j : number of times edge e

appears in route j Variables : λj: (binary) whether to choose route j

Relation of x and λ

• j∈Ω

qe

j λj = xe, ∀e ∈ E

min

λ

• e∈E

ℓe

• j∈Ω

qe

j λj

s.t.

• e∈δ({i})
• j∈Ω

qe

j λj = 2, ∀i ∈ V+

• e∈δ({0})
• j∈Ω

qe

j λj = 2K

• e∈δ(S)
• j∈Ω

qe

j λj ≥ 2r(S), ∀S ⊆ V+

• j∈Ω

qe

j λj ≤ 1, ∀e ∈ E \ δ({0})

λj ∈ {0, 1}, ∀j ∈ Ω Can be solved within a branch-cut-and-price framework.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 15 / 28

Branch-and-cut-and-price for DETERMINISTIC VRP

Sets : Ω: set of feasible routes Parameters : qe

j : number of times edge e

appears in route j Variables : λj: (binary) whether to choose route j

Relation of x and λ

• j∈Ω

qe

j λj = xe, ∀e ∈ E

min

λ

• e∈E

ℓe

• j∈Ω

qe

j λj

s.t.

• e∈δ({i})
• j∈Ω

qe

j λj = 2, ∀i ∈ V+

• e∈δ({0})
• j∈Ω

qe

j λj = 2K

• e∈δ(S)
• j∈Ω

qe

j λj ≥ 2r(S), ∀S ⊆ V+

• j∈Ω

qe

j λj ≤ 1, ∀e ∈ E \ δ({0})

λj ∈ {0, 1}, ∀j ∈ Ω Can be solved within a branch-cut-and-price framework. Need to solve the pricing problem ⇒ Finding a minimum cost route - strongly NP-hard

Dinh, Fukasawa, Luedtke Chance-constrained VRP 15 / 28

Dealing with deterministic pricing

Column generation for deterministic VRP: Route 0, v1, . . . , vk, 0

k

• i=1

dvi ≤ b Customers visited at most once

Dinh, Fukasawa, Luedtke Chance-constrained VRP 16 / 28

Dealing with deterministic pricing

Column generation for deterministic VRP: Route 0, v1, . . . , vk, 0

k

• i=1

dvi ≤ b Customers visited at most once

Dinh, Fukasawa, Luedtke Chance-constrained VRP 16 / 28

Dealing with deterministic pricing

Column generation for deterministic VRP: Route 0, v1, . . . , vk, 0

k

• i=1

dvi ≤ b Customers visited at most once Walk (or q-route) relaxation: allow cycles

q-route example

Dinh, Fukasawa, Luedtke Chance-constrained VRP 16 / 28

Dantzig-Wolfe formulation for DETERMINISTIC

Sets : Ω: set of feasible routes Parameters : qe

j : number of times edge e

appears in route j Variables : λj: (binary) whether to choose route j

Relation of x and λ

• j∈Ω

qe

j λj = xe, ∀e ∈ E

min

λ

• j∈Ω
• e∈E

ℓeqe

j λj

s.t.

• j∈Ω
• e∈δ({i})

qe

j λj = 2, ∀i ∈ V+

• j∈Ω
• e∈δ({0})

qe

j λj = 2K

• j∈Ω
• e∈δ(S)

qe

j λj ≥ 2r(S), ∀S ⊆ V+

• j∈Ω

qe

j λj ≤ 1, ∀e ∈ E \ δ({0})

λj ∈ {0, 1}, ∀j ∈ Ω

Dinh, Fukasawa, Luedtke Chance-constrained VRP 17 / 28

Dantzig-Wolfe formulation for DETERMINISTIC

Sets : Ω: set of feasible routes Ω′: set of walks j such that

• i∈j di ≤ b. And Ω ⊆ Ω′

Parameters : qe

j : number of times edge e

appears in route j Variables : λj: (binary) whether to choose route j

Relation of x and λ

• j∈Ω

qe

j λj = xe, ∀e ∈ E

min

λ

• j∈Ω′
• e∈E

ℓeqe

j λj

s.t.

• j∈Ω′
• e∈δ({i})

qe

j λj = 2, ∀i ∈ V+

• j∈Ω′
• e∈δ({0})

qe

j λj = 2K

• j∈Ω′
• e∈δ(S)

qe

j λj ≥ 2r(S), ∀S ⊆ V+

• j∈Ω′

qe

j λj ≤ 1, ∀e ∈ E \ δ({0})

λj ∈ {0, 1}, ∀j ∈ Ω Remains a valid formulation since in any integer solution, only variable corresponding to routes will be 1. Pseudo-polynomial time pricing using dynamic programming Bounds become weaker Improvements based on eliminating some cycles

Dinh, Fukasawa, Luedtke Chance-constrained VRP 17 / 28

Dantzig-Wolfe formulation for STOCHASTIC

Sets : Ωs: set of routes satisfying chance-constraint Parameters : qe

j : number of times edge e

appears in route j Variables : λj: (binary) whether to choose route j

Relation of x and λ

• j∈Ω

qe

j λj = xe, ∀e ∈ E

min

λ

• j∈Ωs
• e∈E

ℓeqe

j λj

s.t.

• j∈Ωs
• e∈δ({i})

qe

j λj = 2, ∀i ∈ V+

• j∈Ωs
• e∈δ({0})

qe

j λj = 2K

• j∈Ωs
• e∈δ(S)

qe

j λj ≥ 2rǫ(S), ∀S ⊆ V+

• j∈Ωs

qe

j λj ≤ 1, ∀e ∈ E \ δ({0})

λj ∈ {0, 1}, ∀j ∈ Ω

Dinh, Fukasawa, Luedtke Chance-constrained VRP 17 / 28

Dantzig-Wolfe formulation for STOCHASTIC

Sets : Ωs: set of routes satisfying chance-constraint Ω′

s: set of walks satsifying

chance-constriant. And Ωs ⊆ Ω′

s

Parameters : qe

j : number of times edge e

appears in route j Variables : λj: (binary) whether to choose route j

Relation of x and λ

• j∈Ω

qe

j λj = xe, ∀e ∈ E

min

λ

• j∈Ω′

s

• e∈E

ℓeqe

j λj

s.t.

• j∈Ω′

s

• e∈δ({i})

qe

j λj = 2, ∀i ∈ V+

• j∈Ω′

s

• e∈δ({0})

qe

j λj = 2K

• j∈Ω′

s

• e∈δ(S)

qe

j λj ≥ 2rǫ(S), ∀S ⊆ V+

• j∈Ω′

s

qe

j λj ≤ 1, ∀e ∈ E \ δ({0})

λj ∈ {0, 1}, ∀j ∈ Ω

Dinh, Fukasawa, Luedtke Chance-constrained VRP 17 / 28

Column generation for stochastic VRP: Find a walk 0, v1, . . . , vk, 0 such that P k

• i=1

Dvi ≤ b

• ≥ 1 − ǫ.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 18 / 28

Column generation for stochastic VRP: Find a walk 0, v1, . . . , vk, 0 such that P k

• i=1

Dvi ≤ b

• ≥ 1 − ǫ.

Theorem

Finding the least cost walk (q-route) in a graph that respects the capacity chance constraint is strongly NP-hard.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 18 / 28

Column generation for stochastic VRP: Find a walk 0, v1, . . . , vk, 0 such that P k

• i=1

Dvi ≤ b

• ≥ 1 − ǫ.

Theorem

Finding the least cost walk (q-route) in a graph that respects the capacity chance constraint is strongly NP-hard. What to do? Trivial special cases: independent, exponential or poisson demands Exact pricing also possible for independent normal In general: Relax further

Dinh, Fukasawa, Luedtke Chance-constrained VRP 18 / 28

Relaxed pricing scheme

Exact capacity chance constraint

yi: binary indicator of whether or not node i is visited F ǫ =

• y ∈ {0, 1}V+ : P
• DTy ≤ b
• ≥ 1 − ǫ
• Idea

Find w ∈ ZV+

+

and τ ∈ Z+ such that: F ǫ ⊆ R(w, τ) :=

• y ∈ {0, 1}V+ : w Ty ≤ τ
• Use R(w, τ) instead of F ǫ:

No feasible routes are excluded Pricing problem reduces to deterministic case using weights w and capacity τ Complexity of pricing problem controlled by magnitude of w

Dinh, Fukasawa, Luedtke Chance-constrained VRP 19 / 28

Generic relaxed pricing scheme (cont’d)

How to choose coefficients? Natural choice: wi = E[Di] Given w, optimize τ in preprocessing phase: τ = max

• w T y : P
• DTy ≤ b
• ≥ 1 − ǫ, y ∈ {0, 1}V+

Stochastic binary knapsack problem Joint normal random demands ⇒ Binary second-order cone program Scenario model of random demands ⇒ Structured binary integer program (Song et al., 2014) Any easily computable upper bound on the above maximum can be used.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 20 / 28

Relaxed pricing with joint normal demands

With joint normal random demands, binary second-order cone program can be replaced with a semidefinite program With mean vector µ and covariance matrix Σ: P

• DTy ≤ b
• ≥ 1 − ǫ ⇐

⇒ µT y + Φ−1(1 − ǫ)

• y TΣy ≤ b

Idea

Get a lower bound on y TΣy in terms of µTy Find η such that ηµT y ≤ y TΣy for all y ∈ {0, 1}V+

Dinh, Fukasawa, Luedtke Chance-constrained VRP 21 / 28

Pricing with joint normal demands (cont.)

Key idea

Decompose Σ = D + Q, where D = diag(p1, ..., pn), Q 0 Constraint on D: pi ≥ ηµi y T Σy = y T Dy + y TQy ≥ y TDy = pTy ≥ ηµTy Semidefinite program: η∗ = max

η,p,Q η

(5a) s.t. µiη ≤ pi i ∈ V+ (5b) Σ = diag(p1, ..., pn) + Q (5c) Q 0, (5d)

Dinh, Fukasawa, Luedtke Chance-constrained VRP 22 / 28

Pricing with joint normal demands (cont.)

Key idea

Decompose Σ = D + Q, where D = diag(p1, ..., pn), Q 0 Constraint on D: pi ≥ ηµi y T Σy = y T Dy + y TQy ≥ y TDy = pTy ≥ ηµTy Semidefinite program: η∗ = max

η,p,Q η

(5a) s.t. µiη ≤ pi i ∈ V+ (5b) Σ = diag(p1, ..., pn) + Q (5c) Q 0, (5d)

The relaxation

µT y + Φ−1(1 − ǫ)

• η∗µTy ≤ b

Dynamic programming still works because only need to keep track of µT y

Dinh, Fukasawa, Luedtke Chance-constrained VRP 22 / 28

Pricing with independent normal demands

Exact pricing for independent normal with mean vector µ and variance vector σ2 P

• DTy ≤ b
• ≥ 1 − ǫ ⇐

⇒ µT y + Φ−1(1 − ǫ)

• y Tσ2 ≤ b

Dynamic program for exact pricing: maintain states for µT y and for y T σ2

Dinh, Fukasawa, Luedtke Chance-constrained VRP 23 / 28

Computational tests overview

Test instances Based on deterministic VRP instances 32 to 55 customers, 5 to 8 vehicles 7 Small: 32-45 customers 3 Large: 50-55 customers Two variance settings: “low” (≈ 10% of mean) and “high” (≈ 20% of mean) Three distribution assumptions: independent normal, joint normal, scenario Implementation details Cplex 12.4.0 Implemented in BCP code based from F. et al. (2006) 7200 second time limit BC BC ∗ BC J BCPr BCPe kǫ(S) k∗

ǫ (S)

max{k∗

ǫ (S), kJ ǫ (S)}

• Rel. pricing

Exact pricing (for ind. normal)

Table: Strategies used

Dinh, Fukasawa, Luedtke Chance-constrained VRP 24 / 28

Results: Independent normal

BC BC ∗ BC J BCPr BCPe Low variance Small AvT (s) 2646 71 40 153 125 NumSolv 3 7 7 6 7 AvGap (%) 4.73 0.76 Large AvT (s) 7200 7200 7200 3311 257 NumSolv 3 3 AvGap (%) 14.52 8.59 7.27 High variance Small AvT (s) 3436 354 144 939 1722 NumSolv 1 4 6 4 4 AvGap (%) 10.40 3.17 1.34 1.70 1.44 Large AvT (s) 7200 7200 7200 7200 2379 NumSolv 2 AvGap (%) 14.74 11.12 11.45 5.72 1.10 Table: Summary of computational results for independent normal random demands.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 25 / 28

Results: Joint normal

BC BC ∗ BC J BCPr Low variance Small AvT (s) 3238 107 109 585 NumSolv 3 6 6 5 AvGap (%) 4.67 1.10 1.12 1.29 Large AvT (s) 7200 7200 7200 5743 NumSolv 1 AvGap (%) 14.04 8.73 8.73 2.4 High variance Small AvT (s) 4628 414 393 2335 NumSolv 1 4 4 4 AvGap (%) 11.72 3.41 3.41 2.76 Large AvT (s) 7200 7200 7200 891 NumSolv 2 AvGap (%) 14.58 10.74 10.74 3.06 Table: Summary of computational results for joint normal random demands.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 26 / 28

Results: Scenarios

BC BC ∗ BCPr Low variance Small AvT (s) 1947 217 803 NumSolv 3 6 7 AvGap (%) 5.54 0.79 Large AvT (s) 7200 7200 7200 NumSolv AvGap (%) 13.68 9.07 4.89 High variance Small AvT (s) 4328 499 4207 NumSolv 1 5 2 AvGap (%) 11.76 2.64 5.09 Large AvT (s) 7200 7200 7200 NumSolv AvGap (%) 19.26 15.78 11.16 Table: Summary of computational results for scenario model of random demands.

Dinh, Fukasawa, Luedtke Chance-constrained VRP 27 / 28

Concluding remarks

Summary

Chance-constrained formulation avoids difficulties in modeling recourse actions Proposed method can solve stochastic VRP with correlations Builds on successful approaches for solving deterministic VRP

Future work

Incorporate more “advanced features” of deterministic VRP into solution approach (e.g. ng-routes, other cuts) Seek improved “pricing friendly” relaxation of chance-constrained capacity constraint

Dinh, Fukasawa, Luedtke Chance-constrained VRP 28 / 28