The vehicle routing problem under uncertainty via ...

The vehicle routing problem under uncertainty via robust optimization Pedro Munaria , Alfredo Morenoa , Jonathan De La Vegaa , Douglas Alemb , Jacek Gondziob , Reinaldo Morabitoa a

Federal University of Sao Carlos, Brazil

b

University of Edinburgh, Scotland, UK

The vehicle routing problem under uncertainty via robust optimization Pedro Munari ([email protected]), ISMP 2018, July 1-6, Bordeaux, France

Outline

I

Why and how to incorporate uncertainty into the VRP formulations?

I

What are the challenges of using the standard compact formulations?

I

How can we effectively solve the VRP under uncertainty via robust optimization, using a MTZ-based compact formulation?

I

The proposed modeling approach can be applied to different VRP variants and possibly to different types of combinatorial optimization problems.

1


Uncertainty in the vehicle routing problem I

The VRP is one of the most studied combinatorial optimization problems, due to its practical and theoretical importance;

I

The vast majority of papers assumes that data is perfectly known in advance: deterministic VRP;

I

What happens in practice is rather the opposite!

2


Uncertainty in the vehicle routing problem I

The VRP is one of the most studied combinatorial optimization problems, due to its practical and theoretical importance;

I

The vast majority of papers assumes that data is perfectly known in advance: deterministic VRP;

I

What happens in practice is rather the opposite!

I

Uncertainty everywhere: I I I I I

I

Travel times; Service times; Customer demands; Customer availability; and so on...

Let us face the real problem!

2


Deterministic VRP with time windows

Depot

3



Depot

3


3


Depot

i qi , si , [wia, wib]

tij , cij

j

qj , sj , [wja, wjb]



Depot

3


4

Deterministic VRP with time windows . Two-index vehicle flow formulation using MTZ min

X

cij xij

(1)

xij = 1, ∀j ∈ C

(2)

(i,j)∈E

s.t.

X (i,j)∈E

X

xij −

(i,j)∈E

X j:(0,j)∈E

X

xji = 0, ∀i ∈ C

(3)

(j,i)∈E

x0j −

X

xi,n+1 = 0,

(4)

i:(i,n+1)∈E

uj ≥ ui + qj xij − Q(1 − xij ), ∀(i, j) ∈ E,

(5)

qi ≤ ui ≤ Q, ∀i ∈ C,

(6)

wj ≥ wi + (si + tij )xij − Mij (1 − xij ), ∀(i, j) ∈ E,

(7)

wia ≤ wi ≤ wib , ∀i ∈ N ,

(8)

xij ∈ {0, 1}, ∀(i, j) ∈ E.

(9)


Uncertainty in the vehicle routing problem . Stochastic VRP

I

Incorporate uncertainty using stochastic programming; I I I

Two-stage stochastic programming (recourse models); Chance-constraints; See Oyola et al. (2016) for a comprehensive survey.

5


Uncertainty in the vehicle routing problem . Stochastic VRP

I

Incorporate uncertainty using stochastic programming; I I I

I

Two-stage stochastic programming (recourse models); Chance-constraints; See Oyola et al. (2016) for a comprehensive survey.

Recourse models: What is the minimum cost of my routes, according to different realizations of random variables and their corresponding recourses?

5


Uncertainty in the vehicle routing problem . Robust VRP

I

Incorporate uncertainty via robust optimization (RO): I

RO problems with bounded uncertainty in the sense of Ben-Tal and Nemirovski (1999), Bertsimas and Sim (2003);

6



I


I

RO problems with bounded uncertainty in the sense of Ben-Tal and Nemirovski (1999), Bertsimas and Sim (2003); Addressing the robust VRP: Sungur et al. (2008), Ordonez (2010), Lee et al. (2012), Agra et al. (2012, 2013), Gounaris et al. (2013), Jaillet et al. (2016), Hu et al. (2018), De La Vega et al. (2018);

6



I


I

I


Worst-case interval analysis: does not require a probability distribution;

6



I


I


I


I

How much do I have to pay to increase the chances of having my routes feasible in practice?

6



I


I


I


I

How much do I have to pay to increase the chances of having my routes feasible in practice? Price of robustness!

6


7

Uncertainty in the vehicle routing problem . Optimal solution of instance RC102 of the deterministic VRPTW

0

32.3

[0, 197]

[35, 65]

[59, 89]

[58, 88]

[72, 102]

[91, 121]

12

14

11

15

16

9

3

32.3

0

0

45

35.3

5.8

45.3

6

61.1

2

77.1

11.1

89.1

[65, 95]

[72, 102]

[87, 117]

[92, 122]

21

23

19

18

22

6.4

5.3

10.7

10

5

110.2

[0, 185] 2

[119, 149] [142, 172] [149, 179] 7

125.2

13

17

11.1

142.2

2

20

12.2

25

10.4

24

35

65

81.4

96.7

117.4

129.4

154

174.4

[0, 194]

[95, 125]

[91, 121]

[0, 189]

[0, 190]

[0, 191]

[141, 171]

[0, 199]

7

6

8

5

3

1

4

3

95

5.8

110.8

7

127.8

2

139.8

26 213.6

[122, 152] [154, 184] [148, 178]

45

35.3

40.3

163.3

3

152.8

7

169.8

5.3

2 185.1

26 219.4

30.8

26 225.9

I

Optimal value: 351.8 (minimum distance);

I

Risk of becoming infeasible for travel times varying by 25%: 98.25%.


8

Uncertainty in the vehicle routing problem . Optimal solution of instance RC102 of the robust VRPTW

0

32.3

[0, 197]

[35, 65]

[59, 89]

[58, 88]

[72, 102]

[91, 121]

12

14

11

15

16

9

3

32.3 40.3

0

0

45

38

5.8

45.3 53.3

6

61.1 69.1

2

77.1 85.1

11.1

89.1 97.1

[65, 95]

[72, 102]

[87, 117]

[92, 122]

21

23

19

18

22

6.4

5.3

10.7

5

110.2 118.2

[0, 185] 2

[119, 149] [142, 172] [149, 179]

10

7

125.2 133.2

13

17

11.1

142.2 150.2

2

20

12.2

25

10.4

24

35

65 68.2

81.4 84.6

96.7 99.9

117.4 120.6

129.4 132.6

154 154.8

174.4 177

[0, 191]

[0, 190]

[0, 189]

[91, 121]

[0, 194]

[95, 125]

[141, 171]

[0, 199]

1

3

5

8

7

6

4

3

51 60.5

2

63 72.5

7

91 91

5

106 107.2

26 213.6 223.6

[122, 152] [154, 184] [148, 178]

45 56.2

38 47.5

40.3

163.3 171.3

3

119 120.2

5.3

141 141

5.3

2 156.3 157.6

26 219.4 228.1

30.8

26 197.1 204.8

I

Optimal value assuming at most 1 travel time increases by 25%: 352.0;

I



9

Uncertainty in the vehicle routing problem . Optimal solution of instance RC102 of the robust VRPTW 0

[35, 65]

[59, 89]

[58, 88]

[72, 102]

[91, 121]

14

11

15

16

9

35.3

5.8

35.3 44.1

6

59 61.3

0

40

[65, 95]

19

23

6.4

38

11.1

87 89.4

[72, 102]

72 72

0

2

75 77.4

88.4 90

10

5

108.1 112.3 [0, 185]

2

21

13

7

123.1 127.3

[87, 117] 4

18

17

11.1

142 144.5

25

10.4

24

35

114.4 117

154 154

174.4 177

219.4 230.7 [0, 199]

[0, 189]

[91, 121]

[0, 194]

[95, 125]

[141, 171]

3

5

8

7

6

4

2

7

63 73.2

0

35

5

3

91 91.2

106 107.2

[92, 122]

[122, 152]

22 92 92

2

20 122 122

223.4 234.1

26

100.4 102.5

[0, 190]

51 61.2

26

32.3

181.1 185.8

[154, 184] [148, 178] 9

1

3

12

8

163.1 166.6

[0, 191]

38 47.5

[0, 197]

[119, 149] [142, 172] [149, 179]

119 120.9

35

5.3

141 141

5.3

2 156.3 157.6

30.8

26 197.1 206.1

26 167 175.7

I

Optimal value assuming at most 2 travel time increases by 25%: 401.8;

I



RVRP formulations . Sungur, Ordonez, Dessouky (2008) I

Capacitated VRP (CVRP) with demand uncertainty;

I

q is uncertain and belongs to a bounded set U (deviations around an expected demand value q 0 );

I

Three uncertainty sets: convex hull, box and ellipsoidal;

10




I


I


I

The MTZ constraints become: I

uj − ui + Q(1 − xij ) ≥ qj , ∀q ∈ U , i, j ∈ C, i 6= j;

10


10



I


I


I


I

uj − ui + Q(1 − xij ) ≥ qj , ∀q ∈ U , i, j ∈ C, i 6= j; s X uj − ui + Q(1 − xij ) ≥ qj0 + yk qjk , ∀y ∈ Y , i, j ∈ C, i 6= j; k=1


10



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I


I


I


I

Too conservative: all uncertain parameters achieve their worst case :(


10



I


I


I


I


I

Too conservative: all uncertain parameters achieve their worst case :(

I

The robust CVRP becomes one instance of the deterministic CVRP!


Robust VRP . Gounaris, Wiesemann, Floudas (2013) I

CVRP under demand uncertainty;

I

Customer demands are supported by a budget uncertainty set to avoid overly conservative solutions that hedge against the unlikely scenario where all customer demands attain their worst-case realizations simultaneously;

I

The resulting problem does not reduce to a deterministic CVRP;

11


Robust VRP . Gounaris, Wiesemann, Floudas (2013) I

CVRP under demand uncertainty;

I

Customer demands are supported by a budget uncertainty set to avoid overly conservative solutions that hedge against the unlikely scenario where all customer demands attain their worst-case realizations simultaneously;

I

The resulting problem does not reduce to a deterministic CVRP;

I

Robust counterpart of the following formulations: 1. 2. 3. 4.

Two-index vehicle flow with MTZ; Two-index vehicle flow with rounded capacity inequalities; Commodity flow formulations; Vehicle assignment formulations.

11


Robust VRP . Gounaris, Wiesemann, Floudas (2013)

I

Two-index vehicle flow with MTZ: I

n new decision variables for each demand realization q ∈ U ;

12



I

Two-index vehicle flow with MTZ: I I

n new decision variables for each demand realization q ∈ U ; Infinite-dimensional optimization problem: equivalent representation in finitely many decision variables, leading to a semi-infinite optimization problem;

12



I


I

n new decision variables for each demand realization q ∈ U ; Infinite-dimensional optimization problem: equivalent representation in finitely many decision variables, leading to a semi-infinite optimization problem; If U q is polyhedral: classical dualization techniques leads to an equivalent finite dimensional optimization problem;

12



I


I

I

n new decision variables for each demand realization q ∈ U ; Infinite-dimensional optimization problem: equivalent representation in finitely many decision variables, leading to a semi-infinite optimization problem; If U q is polyhedral: classical dualization techniques leads to an equivalent finite dimensional optimization problem; Precedence variables can also be used to obtain a finite problem.

12


Robust VRP . Agra, Christiansen, Figueiredo, Hvattum, Poss, Requejo (2012) I

Heterogeneous fleet VRPTW with uncertain travel times;

I

Robust counterpart of a layered formulation of the deterministic VRPTW;

13




I


I

n layers, to track the position of the nodes in the routes;

I

k` Additional binary variable zij : 1 when route k services customer i in position ` before servicing j;

13


13



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I

n layers, to track the position of the nodes in the routes;

I

k` Additional binary variable zij : 1 when route k services customer i in position ` before servicing j;

I

The MTZ constraints for time windows are replaced by: `2 −1

X (i,j)∈E: ˜ (i,j,`1 )∈E

k`1 + wia zij

X

X

`=`1 (i,j)∈E:

˜ (i,j,`)∈E

k` tij zij ≤

X

k`2 wib zij ,

(i,j)∈E: ˜ (i,j,`2 )∈E

∀`1 , `2 such that `1 < `2 ≤ L, ∀k ∈ K.


14


Using the classical dualization scheme leads to a robust counterpart with: a k`

X

`2 −1

wi zij 1 +

(i,j)∈E: ˜ (i,j,`1 )∈E

X

tij

(i,j)∈E

X `=`1

k`

zij +Γv

k`1 `2

+

X

(i,j)∈E

k` `2

uij 1

≤

X

b k`

wi zij 2 ,

(i,j)∈E: ˜ (i,j,`2 )∈E

∀`1 , `2 such that `1 < `2 ≤ L, ∀k ∈ K, v

k`1 `2

k` `2

+ uij 1

`2 −1

≥ tîj

X `=`1

k`

zij , ∀(i, j) ∈ E, ∀`1 , `2 such that `1 < `2 ≤ L, ∀k ∈ K.


14


Using the classical dualization scheme leads to a robust counterpart with: a k`

X

`2 −1

wi zij 1 +

(i,j)∈E: ˜ (i,j,`1 )∈E

X

tij

(i,j)∈E

X `=`1

k`

zij +Γv

k`1 `2

+

X

(i,j)∈E

k` `2

uij 1

≤

X

b k`

wi zij 2 ,

(i,j)∈E: ˜ (i,j,`2 )∈E

∀`1 , `2 such that `1 < `2 ≤ L, ∀k ∈ K, v

k`1 `2

k` `2

+ uij 1

`2 −1

≥ tîj

X

k`

zij , ∀(i, j) ∈ E, ∀`1 , `2 such that `1 < `2 ≤ L, ∀k ∈ K.

`=`1

I

The numbers of variables and constraints increase significantly!

I

Computational experiments with instances from a ship routing and scheduling problem (randomly generated based on real data), with 56 ports, 10/20 cargoes (nodes), 5 vehicles.


Robust VRP . Lee, Lee, Park (2012) I

VRP with Deadlines: travel time/demand uncertainty;

15




I

Control the degree of robustness using budgeted uncertainty sets and propose a model based on the vehicle flow formulation with MTZ;

15




I


I

The robust counterpart model is defined using: B wjB ≥ wiB +(si +tij )xij +tîj δij − Mij (1−xij ), ∀(i, j) ∈ E, ∀B ⊆ E, |B|≤ Γ.

0 ≤ wiB ≤ wib , ∀i ∈ N , ∀B ⊆ E, |B|≤ Γ.

15




I


I

The robust counterpart model is defined using: B wjB ≥ wiB +(si +tij )xij +tîj δij − Mij (1−xij ), ∀(i, j) ∈ E, ∀B ⊆ E, |B|≤ Γ.

0 ≤ wiB ≤ wib , ∀i ∈ N , ∀B ⊆ E, |B|≤ Γ. I

Large number of constraints, makes the formulation intractable: we have one set B for each subset of E with up to Γ edges;

I

The authors reported that even building the model on CPLEX for a instance with 10 customers requires too much time and memory space.

15


Novel modeling approach for the robust VRP

16



We propose a novel modeling approach with the following characteristics: I

VRP with time windows (VRPTW) and demand/travel times uncertainty; (can be used also for the CVRP and other variants!)

16





I

Based on the two-index vehicle flow formulation using MTZ constraints; (hence it is compact!)

16





I


I

Budget uncertainty sets of Bertsimas and Sim (2003);

16





I


I


I

Uses a linearization of dynamic programming recursive equations; (does not require the classical dualization scheme typically used in RO!)

16





I


I


I

Uses a linearization of dynamic programming recursive equations; (does not require the classical dualization scheme typically used in RO!)

I

Relatively few additional variables and constraints.

16



I

We assume that the demands and travel times are uncorrelated uncertain values modeled as independent random variables q˜ and t˜;

I

They fall within the symmetric and bounded ranges: I I

q˜i ∈ [qi − qî , qi + qî ]; (qi : nominal value; qî : deviation) t˜ij ∈ [tij − tîj , tij + tîj ]. (tij : nominal value; tîj : deviation)

17



I


I


I


t Normalized scale deviation: ξiq = (˜ qi − qi )/ˆ qi and ξij = (t˜ij − tij )/tîj , which are random variables in [0, 1] (without loss of generality);

17



I


I



I

t Normalized scale deviation: ξiq = (˜ qi − qi )/ˆ qi and ξij = (t˜ij − tij )/tîj , which are random variables in [0, 1] (without loss of generality);

I

The cumulative uncertainty of each random variable is bounded by its corresponding budget of uncertainty Γq or Γt (Bertsimas and Sim, 2003).

17


18


I

Using them, we define the following polyhedral uncertainty sets: U

q

=

    X q |C| q q q ξi ≤ Γ , 0 ≤ ξi ≤ 1, ∀i ∈ C ; q˜ ∈ R+ | q˜i = qi + qî ξi ,   i∈C

U

t

=

  

|E| t t˜ ∈ R+ | t˜ij = tij + tîj ξij ,

X

t ξij

t

≤Γ , 0≤

t ξij

≤ 1, ∀(i, j) ∈ E

(i,j)∈E

 

.



I

A route is robust feasible if it is (deterministic) feasible for all possible demand realizations q˜ ∈ U q and travel time realizations t˜ ∈ U t ;

I

A solution R = (r1 , . . . , rp ) is robust feasible if all routes rj , j = 1, . . . , p, are robust feasible and if they together visit all customers exactly once.


Novel modeling approach for the robust VRP . Vehicle capacity I

uvj γ : largest vehicle load at node vj of the route, j = 1, . . . , k, when up to γ ≤ Γq demand values attain their worst case;

19


19



uvj γ

   qv0 , = uvj−1 γ + qvj ,   max{uvj−1 γ + qvj , uvj−1 ,γ−1 + qvj + qˆvj },

if j = 0; if γ = 0; otherwise.


19



uvj γ

   qv0 , = uvj−1 γ + qvj ,   max{uvj−1 γ + qvj , uvj−1 ,γ−1 + qvj + qˆvj },


I

To be robust feasible, the route must satisfy uvj γ ≤ Q for all γ = 0, 1, . . . , Γq and j = 1, . . . , k.

I

Since uvj γ is nondecreasing through the nodes in the path, we can simply check uvk Γq ≤ Q.



We can extend the MTZ-based constraints using the same idea of the recursive equations!

20




I

For each i ∈ N , let uiγ be the load in the vehicle that services node i when up to γ demand values attain their worst case:

20




I

For each i ∈ N , let uiγ be the load in the vehicle that services node i when up to γ demand values attain their worst case: ujγ ≥ uiγ + qj xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 0, . . . , Γq ,

20




I

For each i ∈ N , let uiγ be the load in the vehicle that services node i when up to γ demand values attain their worst case: ujγ ≥ uiγ + qj xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 0, . . . , Γq , ujγ ≥ uiγ−1 + (qj + qˆj )xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 1, . . . , Γq ,

20




I

For each i ∈ N , let uiγ be the load in the vehicle that services node i when up to γ demand values attain their worst case: ujγ ≥ uiγ + qj xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 0, . . . , Γq , ujγ ≥ uiγ−1 + (qj + qˆj )xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 1, . . . , Γq , qi ≤ uiγ ≤ Q, ∀i ∈ C, γ = 0, . . . , Γq .

20




I

For each i ∈ N , let uiγ be the load in the vehicle that services node i when up to γ demand values attain their worst case: ujγ ≥ uiγ + qj xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 0, . . . , Γq , ujγ ≥ uiγ−1 + (qj + qˆj )xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 1, . . . , Γq , qi ≤ uiγ ≤ Q, ∀i ∈ C, γ = 0, . . . , Γq .

I

These constraints evaluate the cumulative vehicle load at customer j ∈ C if the vehicle is coming directly from customer i, according to γ demand values attaining their worst case.

20


21

Novel modeling approach for the robust VRP . Two-index vehicle flow formulation for the robust CVRP

min

X

cij xij

(10)

xij = 1, ∀j ∈ C

(11)

(i,j)∈E

s.t.

X (i,j)∈E

X

xij −

(i,j)∈E

X j:(0,j)∈E

X

xji = 0, ∀i ∈ C

(12)

(j,i)∈E

x0j −

X

xi,n+1 = 0,

(13)

i:(i,n+1)∈E

ujγ ≥ uiγ + qj xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 0, . . . , Γq

(14)

ujγ ≥ uiγ−1 + (qj + qˆj )xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 1, . . . , Γq(15) qi ≤ uiγ ≤ Q, ∀i ∈ C, γ = 0, . . . , Γq

(16)

xij ∈ {0, 1}, ∀(i, j) ∈ E.

(17)


Novel modeling approach for the robust VRP . Time windows I

wvj γ : earliest exact time from which the service can start at node vj when up to γ ≤ Γt travel times reach their worst-case values;

22


22



wvj γ

 a wv0 ,     max{wa , w vj vj−1 γ + svj−1 + tvj−1 vj }, = a  max{w , w  vj vj−1 γ + svj−1 + tvj−1 vj ,   wvj−1 ,γ−1 + svj−1 + tvj−1 vj + tˆvj−1 vj },



22



wvj γ

 a wv0 ,     max{wa , w vj vj−1 γ + svj−1 + tvj−1 vj }, = a  max{w , w  vj vj−1 γ + svj−1 + tvj−1 vj ,   wvj−1 ,γ−1 + svj−1 + tvj−1 vj + tˆvj−1 vj },

I

To be robust feasible, the route must satisfy wvj γ ≤ wvbj for all γ = 0, 1, . . . , Γt and j = 1, . . . , k;

I

It is enough to check wvj Γt ≤ wvbj , for j = 1, . . . , k.



23


Note that uvj γ can be calculated using only the Γq largest demand deviations of the customers visited up to uvj γ : uvj γ =

j X

qvi +

i=1

max

I⊆{1,...,j} |I|≤γ

X

qˆvi .

i∈I

I

Hence, the worst-case value is always achieved by taking the largest demand deviations of the visited customers;

I

It does not happen for wvj γ though!

I

wvj γ is not necessarily given by the Γt largest travel time deviations;

I

Since the service can start only after the customer time window opens, the largest deviations can be absorbed by the waiting times.


24

Novel modeling approach for the robust VRP . Two-index vehicle flow formulation for the robust VRPTW min

X

cij xij

(18)

xij = 1, ∀j ∈ C

(19)

(i,j)∈E

s.t.

X (i,j)∈E

X

X

xij −

(i,j)∈E

X j:(0,j)∈E

xji = 0, ∀i ∈ C

(20)

(j,i)∈E

X

x0j −

xi,n+1 = 0,

(21)

i:(i,n+1)∈E q

ujγ ≥ uiγ + qj xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 0, . . . , Γ

(22) q

ujγ ≥ uiγ−1 + (qj + qˆj )xij − Q(1 − xij ), ∀(i, j) ∈ E, γ = 1, . . . , Γ

(23)

q

qi ≤ uiγ ≤ Q, ∀i ∈ C, γ = 0, . . . , Γ

(24) t

wjγ ≥ wiγ + (si + tij )xij − Mij (1 − xij ), ∀(i, j) ∈ E, γ = 0, . . . , Γ

(25) t

wjγ ≥ wiγ−1 + (si + tij + tîj )xij − Mij (1 − xij ), ∀(i, j) ∈ E, γ = 1, . . . , Γ (26) a

b

t

wi ≤ wiγ ≤ wi , ∀i ∈ N , γ = 0, . . . , Γ

(27)

xij ∈ {0, 1}, ∀(i, j) ∈ E.

(28)


Computational results I

The proposed model for the RVRPTW was implemented on top of IBM CPLEX 12.7 and solved by the general purpose branch-and-cut;

25




I

We use Solomon’s instances of the deterministic VRPTW:

25




I

We use Solomon’s instances of the deterministic VRPTW: I

Demands and travel times of the instances used as nominal values;

25




I

We use Solomon’s instances of the deterministic VRPTW: I I

Demands and travel times of the instances used as nominal values; Deviations qî = trunc(αq × qi ) and tîj = 0.1 × trunc(αt × 10 × tij ), in which αq and αt belong to {0, 0.1, 0.25, 0.5};

25




I


I

Demands and travel times of the instances used as nominal values; Deviations qî = trunc(αq × qi ) and tîj = 0.1 × trunc(αt × 10 × tij ), in which αq and αt belong to {0, 0.1, 0.25, 0.5}; Budgets of uncertainty Γq and Γt assuming values 0, 1, 5, 10.

25




I


I

I


Each instance was solved 28 times:

25




I


I

I


Each instance was solved 28 times: I

using Γq = Γt = 0 and αq = αt = 0 (deterministic case);

25




I


I

I


Each instance was solved 28 times: I I

using Γq = Γt = 0 and αq = αt = 0 (deterministic case); using Γq = 1, 5, 10 and Γt = 0 for each αq = 0.1, 0.25, 0.5 and αt = 0;

25




I


I

I


Each instance was solved 28 times: I I I

using Γq = Γt = 0 and αq = αt = 0 (deterministic case); using Γq = 1, 5, 10 and Γt = 0 for each αq = 0.1, 0.25, 0.5 and αt = 0; using Γq = 0 and Γt = 1, 5, 10 for each αq = 0 αt = 0.1, 0.25, 0.5; and

25




I


I

I


Each instance was solved 28 times: I I I I

using using using using

Γq Γq Γq Γq

= Γt = 0 and αq = αt = 0 (deterministic case); = 1, 5, 10 and Γt = 0 for each αq = 0.1, 0.25, 0.5 and αt = 0; = 0 and Γt = 1, 5, 10 for each αq = 0 αt = 0.1, 0.25, 0.5; and = Γt = 1, 5, 10 for each αq = αt = 0.1, 0.25, 0.5;

25




I


I

I

Each instance was solved 28 times: I I I I

I


using using using using

Γq Γq Γq Γq

= Γt = 0 and αq = αt = 0 (deterministic case); = 1, 5, 10 and Γt = 0 for each αq = 0.1, 0.25, 0.5 and αt = 0; = 0 and Γt = 1, 5, 10 for each αq = 0 αt = 0.1, 0.25, 0.5; and = Γt = 1, 5, 10 for each αq = αt = 0.1, 0.25, 0.5;

Maximum running time limit: 3,600 seconds.

25


26

Computational results . Classes C1 and C2 (clustered), 25 customers

Γq Γt 0 0 1 0 5 0 10 0 0 1 0 5 0 10 1 1 5 5 10 10 Average

Obj 190.59 202.68 238.50 239.11 192.34 195.83 195.83 202.86 238.93 239.69 213.64

Gap (%) − − 0.011 0.020 − − − − 0.013 0.018 0.006

C1 Time 0.89 2.70 551.00 659.30 0.93 2.63 5.89 3.93 598.52 716.67 254.24

Ins 9 27 27 27 27 27 27 27 27 27

Opt 9 27 24 23 27 27 27 27 23 23

Inf 0 0 0 0 0 0 0 0 0 0

−: The gap is zero. Ins: Number of instances in the group. Opt: Number of instances solved to optimality. Inf : Number of infeasible instances. Computational times in seconds.

Obj 214.45 214.45 214.45 214.45 214.51 214.57 214.57 214.51 214.57 214.57 214.51

Gap (%) − − − − − − − − − − 0.000

C2 Time 3.50 7.46 22.29 28.75 9.04 17.83 50.38 10.88 45.00 95.83 29.10

Ins 8 24 24 24 24 24 24 24 24 24

Opt 8 24 24 24 24 24 24 24 24 24

Inf 0 0 0 0 0 0 0 0 0 0


27

Computational results . Classes R1 and R2 (random), 25 customers

Γq Γt 0 0 1 0 5 0 10 0 0 1 0 5 0 10 1 1 5 5 10 10 Average

Obj 463.37 463.37 463.37 463.37 470.33 478.16 478.49 470.33 478.19 478.48 470.75

Gap (%) − − − 0.003 − 0.002 0.005 − 0.006 0.011 0.003

R1 Time 64.17 94.67 200.61 435.06 66.81 317.28 422.63 139.97 395.44 652.59 278.92

Ins 12 36 36 36 36 36 36 36 36 36

Opt 12 36 36 33 32 31 30 32 30 28

Inf 0 0 0 0 4 4 4 4 4 4


Obj 382.15 382.15 382.15 382.15 383.86 384.80 384.89 383.86 384.80 384.89 383.57

Gap (%) − − − − − − − − − 0.001 0.000

R2 Time 17.36 27.82 72.91 147.06 26.42 77.82 160.52 39.03 176.00 430.33 117.53

Ins 11 33 33 33 33 33 33 33 33 33

Opt 11 33 33 33 33 33 33 33 33 32

Inf 0 0 0 0 0 0 0 0 0 0


28

Computational results . Classes RC1 and RC2 (mixed), 25 customers

Γq Γt 0 0 1 0 5 0 10 0 0 1 0 5 0 10 1 1 5 5 10 10 Average

Obj Gap (%) 350.24 − 360.14 0.014 432.36 0.121 438.27 0.132 356.87 − 382.91 0.023 392.57 0.037 368.53 0.014 444.25 0.136 446.52 0.150 397.26 0.063

RC1 Time Ins Opt Inf 4.75 8 8 0 619.17 24 20 0 2.457.13 24 9 0 2.687.21 24 8 0 10.13 24 23 1 526.65 24 20 1 744.22 24 19 1 583.65 24 20 1 2.492.00 24 8 1 2.650.48 24 8 1 1.277.54


Obj Gap (%) 319.28 0.110 319.28 0.116 319.28 0.128 319.28 0.141 319.60 0.115 319.78 0.129 319.90 0.141 319.58 0.119 319.77 0.140 320.08 0.152 319.58 0.129

RC2 Time Ins Opt Inf 1.464.25 8 5 0 1.700.25 24 15 0 1.838.00 24 12 0 1.859.96 24 12 0 1.595.58 24 15 0 1.823.13 24 12 0 1.835.96 24 12 0 1.736.96 24 14 0 1.856.67 24 12 0 1.900.54 24 12 0 1.761.13


Computational results . Overall results for instances with 100 customers I

21.8% of the instances were solved to optimality, considering the same variations for the budgets of uncertainty and deviations;

I

For instances not solved to optimality: I I

Average gap in C1, R1 and RC1: 11.55%, 27.03%, and 33.66%, respect.; Average gap in C2, R2 and RC2: 1.75%, 20.27%, and 24.95%, respect.;

29




I



I

Similar to the results with 25 customers, instances with higher values of the budgets of uncertainty and deviation were more challenging;

I

For example, the average gap of instances with Γq = Γt = 10 was 26.4%, whereas the average gap for instances with Γq = Γt = 0 was 14.73%;

29




I



I

Similar to the results with 25 customers, instances with higher values of the budgets of uncertainty and deviation were more challenging;

I

For example, the average gap of instances with Γq = Γt = 10 was 26.4%, whereas the average gap for instances with Γq = Γt = 0 was 14.73%;

I

Solving larger instances requires approaches based on column generation. Hence we have proposed a branch-price-and-cut method – to be presented at EURO conference next week (Monday 9th, 14:30) and described in the technical report (link at the final of this talk).

29


Computational results . Monte Carlo Simulation to check the quality of the robust solutions I

The simulation was performed by generating 10,000 random uniform realizations for demands and travel times in the half-interval [qi , qi + qî ] for all i ∈ C and [tij , tij + tîj ] for all (i, j) ∈ E, respectively;

30




I

Probability of constraint violation (Risk): empirically evaluated as the number of times a given optimal solution xα,Γ is infeasible out of the 10,000 generated random realizations given by the Monte Carlo simulation (relative frequency of infeasible solutions);

30


30



I

Probability of constraint violation (Risk): empirically evaluated as the number of times a given optimal solution xα,Γ is infeasible out of the 10,000 generated random realizations given by the Monte Carlo simulation (relative frequency of infeasible solutions);

I

Price of robustness (PoR) is defined as I z(xα,Γ ) I

z(xα,Γ )−z z

· 100%, where:

is the objective value of the solution; z is the (sub)optimal value of the equivalent deterministic (nominal) problem.


31

Computational results . Monte Carlo Simulation: Γq = 0 to 6; Γt = 0; αq = 0.25, 0.50 C1

25% deviation R1 RC1 Risk PoR Risk PoR Risk

Γq Γt

PoR

0 1 2 3 4 5 6

− 95.38 0.20 94.91 18.63 26.80 19.80 8.06 22.90 0.01 23.22 − 23.22 −

0 0 0 0 0 0 0

C1 C1 R1 R1 RC1RC1

− − − − − − −

Deviation: [0.25, 0.00] Deviation: [0.25, 0.00]

60 60 40 40

C1 C1 R1 R1 RC1RC1 100100 80 80

− − − − − − −

− 97.21 9.94 97.05 32.22 11.16 32.45 1.44 33.08 0.36 33.65 − 39.58 −


Risk Risk

Risk Risk k

60 60

Table_9

− − − − − − −

60 60 40 40

20 20 0 0 0 0 4 4 8 8 12 12PoR 16 16 20 20 24 24 28 28 32 32 36 36 PoR 80 80

PoR

50% deviation R1 RC1 Risk PoR Risk PoR Risk

− 82.91 0.00 99.98 − 82.91 18.63 99.21 9.94 42.98 23.22 40.13 21.66 0.05 24.60 29.39 21.81 0.03 33.01 0.07 32.22 − 33.57 0.20 32.22 −Table_9 34.41 −

0.25] Deviation: 0.25] C1 R1 RC1 (a)C1PoR vs R1 RiskRC1 for αq =Deviation: 0.25 and [0.00, αt [0.00, = 0.00.

20 20 0 0 0 0

5 5 10 10 15 PoR 20 20 25 25 30 30 35 35 40 40 15 PoR

C1 C1 R1 R1 RC1RC1 Deviation: 0.50] (b) PoR vs Risk for αq = 0.50 and αt[0.00, = [0.00, 0.00. Deviation: 0.50]

100100 80 80 60 60

k k

100100 80 80

− − − − − − −

C1


32

Computational results . Monte Carlo Simulation: Γt = 0; Γt = 0 to 6; αt = 0.25, 0.50

Risk Risk

6060 4040

2020 00 00

44 C1C1

− − − − − − R1R1 RC1 − RC1− − − − − − − 88 R1R1

− 71.90 − 73.74 Table_9 0.00 − 97.98 − 98.08 Table_94.53 2.61 25.45 0.74 30.45 2.76 0.10 5.40 60.60 7.81 48.97 3.11 19.98 3.02 5.31 3.34 0.10 7.51 7.19 17.09 16.61 R1 RC1 RC1 Deviation: [0.25, 0.00] Deviation: [0.50, 0.00] Deviation: [0.25, 0.00] 0.65 3.34 C1C1 Deviation: [0.50, 0.00] 3.61 0.03 5.31 0.10 R1 9.21 1.87 20.25 3.82 100 100 3.88 − 5.40 0.0980808.24 − 9.77 1.84 21.32 0.38 3.98 − 8.59 −60608.24 − 10.53 0.06 24.24 0.03 4040 3.98 − 8.59 −20208.24 − 10.78 − 29.70 −

1212 PoR 16 16 2020 2424 2828 3232 3636 PoR RC1 RC1


6060

1515 2020 PoR PoR RC1 RC1

44

88

2525

3030

3535

4040


Risk Risk

Risk Risk

2020 00 00

k k

1010 R1R1

6060 4040

4040

6060

55 C1C1

100 100 8080

8080

100 100 8080

00 00

11

22

33

4 4PoR 55 PoR

66

77

88

99

t = 0.25. R1R1 C1 vs RC1 0.25] (c) C1 PoR Risk RC1 for αq = Deviation: 0.00 and α[0.25, Deviation: [0.25, 0.25]

2020 00 00

1212 1616 PoR PoR

2020

2424

2828

3232

C1C1 vs R1 RC1 Deviation: [0.50, R1 for RC1 Deviation: [0.50, 0.50] (d) PoR Risk αq = 0.00 and αt = 0.50.0.50] 100 100 8080 6060

k k

100 100 8080

0 0 0 1 0 2 0 C1C13 0 4 0 5 0 6

Risk Risk

Γq Γt

25% deviation 50% deviation C1 R1 RC1 C1 R1 RC1 PoR Risk PoR Risk PoR Risk PoR Risk PoR Risk PoR Risk


33

Computational results . Monte Carlo Simulation: Γt = Γt = 0 to 6; αt = 0.25, 0.50 Table_9 Table_9

C1 C1 R1 R1 RC1RC1

Γ

40 40 200 20 010 0

2 3 60 4 60 40 5 40 20 6 20

Risk Risk

80 80

Γt

PoR


C1 C1 R1 R1 RC1RC1

Deviation: [0.50, 0.00]

Deviation: [0.50, 0.00] 25% deviation 50% deviation 100100 R1 RC1 80 80 C1 R1 RC1 60 Risk PoR Risk PoR Risk 60PoR Risk PoR Risk PoR Risk 40 40

0 − 95.40 − 71.90 − 99.1420 20 0.00 99.98 − 98.01 − 100.00 0 1 0.20 94.79 2.61 25.36 0.74 84.270 18.92 97.71 5.40 60.47 19.03 85.50 16PoR 0 0 5 5 10 10 15 PoR 20 20 25 25 30 30 35 35 40 40 0 4 4 8 8 12 12PoR 16 20 20 24 24 28 28 32 32 36 36 15 PoR 2 18.92 22.05 3.11 19.76 15.16 27.65 24.01 38.89 7.51 7.29 38.80 19.77 C1 C1 R1 R1 RC1RC1 Deviation: [0.00, 0.25] Deviation: [0.00, 0.25] R1 R1 RC1RC1 [0.00, 0.50] Deviation: [0.00, 0.50] 3C1 C120.40 6.59 3.61 0.03 24.64 0.05 25.77 26.66 9.21 Deviation: 1.74 40.36 2.90 100100 4 23.05 0.02 3.88 − 24.71 0.0480 8033.30 0.08 9.77 1.71 41.54 0.01 5 23.43 − 3.98 − 34.90 0.0160 6033.91 0.14 10.53 0.07 42.25 − 40 40 6 23.43 − 3.98 − 34.92 −20 2034.22 − 10.78 − 42.78 − Risk Risk

Risk Risk

60 60 q

C1

Risk Risk

100100 80 80

0 0 0 0 1 1 2 2 3 3 4 PoR 5 5 6 6 7 7 8 8 9 9 4 PoR C1 C1 R1 R1 RC1RC1 100100 80 80


0 0 0 0

4 4

8 8 12 12 16 16 20 20 24 24 28 28 32 32 PoRPoR

C1 C1 R1 R1 RC1RC1


100100 80 80 60 60

Risk Risk

Risk Risk

60 60

40 40

40 40

20 20

20 20 0 0 0 0

5 5

10 10

15 15 20 20 PoRPoR

25 25

30 30

(e) PoR vs Risk for αq = 0.25 and αt = 0.25.

35 35

0 0 0 0 5 5 10 10 15 15 20 PoR 25 25 30 30 35 35 40 40 45 45 20 PoR

(f) PoR vs Risk for αq = 0.50 and αt = 0.50.


Conclusion I

We proposed a compact formulation based on the integrating dynamic programming recursive equations into the standard two-index vehicle flow formulation with MTZ-constraints;

I

Novel strategy: robust counterpart model that does not required the commonly used dualization of constraints involving the uncertain parameters. (It can be used with other VRP variants and other problems!)

I

Practical appeal: intuitive extension of the deterministic formulation and encourages its usage by those relying on general-purpose optimization software;

I

Performance: the model is effective for solving small-scale instances (25 customers) and can be used to obtain feasible solutions with 100 customers (never reported in the RVRPTW literature).

34


Future work

I

Extend the formulation to other VRP variants and check the performance;

I

Apply the novel modeling strategy to other robust combinatorial optimization problems;

I

Explore practical applications of the RVRPTW in different logistics contexts (commercial and humanitarian logistics).

35


Merci :)

Acknowledgments

Submitted paper available at: http://www.dep.ufscar.br/docentes/munari Munari, P.; Moreno, A.; De La Vega, J.; Alem, D.; Gondzio, J.; Morabito, R. The robust vehicle routing problem with time windows: compact formulation and branch-priceand-cut method. Technical Report 002/2018, Operations Research Group, Production Engineering Department, Federal University of S˜ ao Carlos, Brazil. 2018. (Submitted)

36


Agra, A., Christiansen, M., Figueiredo, R., Hvattum, L., Poss, M., and Requejo, C. (2012). Layered formulation for the robust vehicle routing problem with time windows. In Mahjoub, A. R., Markakis, V., Milis, I., and Paschos, V. T., editors, Combinatorial Optimization: ISCO 2012, Athens, Greece, April 19-21, 2012, pages 249–260. Springer, Berlin, Heidelberg. Ben-Tal, A. and Nemirovski, A. (1999). Robust solutions of uncertain linear programming. Operations Research Letters, 25:1–13. Bertsimas, D. and Sim, M. (2003). Robust discrete optimization and network flows. Mathematical Program- ming, 98(1):49–71. De La Vega, J., Munari, P., and Morabito, R. (2018). Robust optimization for the vehicle routing problem with multiple deliverymen. Central European Journal of Operations Research. Online First. Gounaris, C. E., Wiesemann, W., and Floudas, C. A. (2013). The robust capacitated vehicle routing problem under demand uncertainty. Operations Research, 61(3):677–693. Jaillet, P., Qi, J., and Sim, M. (2016). Routing optimization under uncertainty. Operations Research, 64(1):186?200. Lee, C., Lee, K., and Park, S. (2012). Robust vehicle routing problem with deadlines and travel time/demand uncertainty. Journal of the Operational Research Society, 63:1294–1306. Oyola, J., Arntzen, H., and Woodruff, D. L. (2016a). The stochastic vehicle routing problem, a literature review, part I: models. EURO Journal on Transportation and Logistics, pages 1–29. Sungur, I., Ordonez, F., and Dessouky, M. (2008). A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty. IIE Transactions (Institute of Industrial Engineers), 40(5):509–523.

37

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