Integrated pollster and vehicle routing S. Gutirrez, A. Miniguano, - PowerPoint PPT Presentation

Integrated pollster and vehicle routing S. Gutiérrez, A. Miniguano, D. Recalde, L. M. Torres, R. Torres Centro de Modelización Matemática - ModeMat Escuela Politécnica Nacional - Quito CO@Work 2020 TU Berlin - Berlin Mathematical School, September 14th - 25th, 2020

Outline Motivation and problem definition Modeling via mixed integer programming Computational results Conclusions CO@Work 2020 Integrated pollster and vehicle routing 2

The problem The National Statistics Bureau of Ecuador (INEC) is responsible for constructing the Consumer Price Index. To collect information a sample set of stores must be visited monthly. Stores are visited by a set of pollsters, transported by a fleet of hired vehicles. Pollsters also walk between stores. Tasks: schedule visits to stores within time horizon schedule daily service duties for pollsters define daily routes for vehicles CO@Work 2020 Integrated pollster and vehicle routing 3

A Mixed Linear and Integer Programming Problem 1 2 1 2 CO@Work 2020 Integrated pollster and vehicle routing 4

A Mixed Linear and Integer Programming Problem t = 0 1 2 1 2 CO@Work 2020 Integrated pollster and vehicle routing 5

A Mixed Linear and Integer Programming Problem t = 2 1 2 1 2 CO@Work 2020 Integrated pollster and vehicle routing 6

A Mixed Linear and Integer Programming Problem t = 5 2 1 1 2 CO@Work 2020 Integrated pollster and vehicle routing 7

A Mixed Linear and Integer Programming Problem t = 6 2 1 1 2 CO@Work 2020 Integrated pollster and vehicle routing 8

A Mixed Linear and Integer Programming Problem t = 15 2 2 1 1 CO@Work 2020 Integrated pollster and vehicle routing 9

A Mixed Linear and Integer Programming Problem t = 17 2 2 1 1 CO@Work 2020 Integrated pollster and vehicle routing 10

A Mixed Linear and Integer Programming Problem t = 19 2 2 CO@Work 2020 Integrated pollster and vehicle routing 11

A Mixed Linear and Integer Programming Problem t = 20 2 2 CO@Work 2020 Integrated pollster and vehicle routing 12

Definitions: sets E : pollsters K : vehicles set of stores n S : days within time horizon for planning CO@Work 2020 Integrated pollster and vehicle routing 13

Definitions: network two nodes for each store (customer) (arrival at store) C − := {1,…, n } (departure from store) C + := { n + 1,…,2 n } two nodes 0, 2 n + 1 for depot C := C − ∪ C + , V := C ∪ {0,2 n + 1} three sets of weighted arcs service arcs; service times t i , i + n A S := {( i , i + n ) : i ∈ C − } A W := {( i , j ) : i ∈ C + , j ∈ C − , j ≠ i + n } walking arcs; walk times t i , j A V := {( i , j ) : i , j ∈ C } ∪ {(0, i ) : i ∈ C } ∪ {( i ,2 n + 1) : i ∈ C } vehicle transportation arcs; travel times τ i , j CO@Work 2020 Integrated pollster and vehicle routing 14

Definitions: paths and routes A walking path for a pollster is a simple path from i ∈ C − to j ∈ C + with alternating arcs from the sets A S and A W . A vehicle path is a directed path between two nodes in using V only arcs from A V . A service route for a pollster is a dipath from to 0 2 n + 1 consisting of an alternating sequence of vehicle and walking subpaths. A vehicle route is a vehicle path from to 0 2 n + 1. The duration of a route is the sum of its arc weights. A service route is feasible if it does not exceed a maximum allowed duration, including time for a lunch break. CO@Work 2020 Integrated pollster and vehicle routing 15

Definitions: schedules Vehicles pick-up and deliver pollsters to certain stores; pollsters can share vehicles. Vehicle fleet is homogeneous, with vehicle capacity Q . Pollster “fleet” is homogeneous: any pollster can visit any store. A daily schedule consists of a set of feasible pollster routes and “compatible” vehicle routes, i.e., for any ( i , j ) ∈ A V : if ( i , j ) is contained in some service route, then it is contained in a vehicle route ( i , j ) is not contained in more than service routes Q CO@Work 2020 Integrated pollster and vehicle routing 16

The IPVRP Task: Find a set of daily schedules, at most one for each day in a given time horizon, such that: Each store is visited once. Number of working days is minimized. Number of service routes is minimized. Number of vehicle routes is minimized. CO@Work 2020 Integrated pollster and vehicle routing 17

The INEC instance CO@Work 2020 Integrated pollster and vehicle routing 18

Outline Motivation and problem definition Modeling via mixed integer programming Computational results Conclusions CO@Work 2020 Integrated pollster and vehicle routing 19

Pollster routing: variables Binary variables: x e , s x e , s i , i + n = 1 ⇔ e visits store i on day s i , i + n , ∀ ( i , i + n ) ∈ A S , e ∈ E , s ∈ S : x e , s x e , s i , j = 1 ⇔ e walks from i to j on day s i , j , ∀ ( i , j ) ∈ A W , e ∈ E , s ∈ S : z e , s z e , s i , j = 1 ⇔ e is transported from i to j on day s i , j , ∀ ( i , j ) ∈ A V , e ∈ E , s ∈ S : b e , s b e , s = 1 ⇔ i is start of a walking path for e on day s i , ∀ i ∈ C − , e ∈ E , s ∈ S : i f e , s f e , s = 1 ⇔ i is end of a walking path of e on day s i , ∀ i ∈ C + , e ∈ E , s ∈ S : i u s = 1 ⇔ s has a daily schedule assigned to it u s , s ∈ S : CO@Work 2020 Integrated pollster and vehicle routing 20

Pollster routing: constraints ∑ s ∈ S ∑ x e , s i , i + n = 1, ∀ i ∈ C − , e ∈ E ∑ x e , s j , i − x e , s i , i + n = − b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A W ∑ x e , s x e , s i , j = f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , i − n , i − ( i , j ) ∈ A W ∑ z e , s j , i ≤ 1 − x e , s i , i + n + b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A V ∑ z e , s i , j ≤ 1 − x e , s i − n , i + f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( i , j ) ∈ A V j , i − ∑ each store is visited by one pollster on one day ∑ z e , s z e , s i , j = b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A V ( i , j ) ∈ A V j , i − ∑ ∑ z e , s z e , s i , j = − f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( j , i ) ∈ A V ( i , j ) ∈ A V ∑ z e , s 0, i ≤ u s , ∀ e ∈ E , s ∈ S . i ∈ C CO@Work 2020 Integrated pollster and vehicle routing 21

Pollster routing: constraints ∑ s ∈ S ∑ x e , s i , i + n = 1, ∀ i ∈ C − , e ∈ E ∑ x e , s j , i − x e , s i , i + n = − b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A W ∑ x e , s x e , s i , j = f e , s i − n , i − i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( i , j ) ∈ A W ∑ z e , s j , i ≤ 1 − x e , s i , i + n + b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A V ∑ z e , s i , j ≤ 1 − x e , s i − n , i + f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( i , j ) ∈ A V “multicommodity flow demand” constraints for walking paths: j , i − ∑ ∑ z e , s z e , s i , j = b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , sync x variables with b and f variables ( j , i ) ∈ A V ( i , j ) ∈ A V j , i − ∑ ∑ z e , s z e , s i , j = − f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( j , i ) ∈ A V ( i , j ) ∈ A V ∑ z e , s 0, i ≤ u s , ∀ e ∈ E , s ∈ S . i ∈ C CO@Work 2020 Integrated pollster and vehicle routing 22

Pollster routing: constraints ∑ s ∈ S ∑ x e , s i , i + n = 1, ∀ i ∈ C − , e ∈ E ∑ x e , s j , i − x e , s i , i + n = − b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A W ∑ x e , s x e , s i , j = f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , i − n , i − ( i , j ) ∈ A W ∑ z e , s j , i ≤ 1 − x e , s i , i + n + b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A V ∑ z e , s i , j ≤ 1 − x e , s i − n , i + f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( i , j ) ∈ A V degree constraints for vehicle transportation: j , i − ∑ ∑ z e , s z e , s i , j = b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , forbid arcs that cannot connect properly to walking paths ( j , i ) ∈ A V ( i , j ) ∈ A V j , i − ∑ ∑ z e , s z e , s i , j = − f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( j , i ) ∈ A V ( i , j ) ∈ A V ∑ z e , s 0, i ≤ u s , ∀ e ∈ E , s ∈ S . i ∈ C CO@Work 2020 Integrated pollster and vehicle routing 23

Pollster routing: constraints ∑ s ∈ S ∑ x e , s i , i + n = 1, ∀ i ∈ C − , e ∈ E ∑ x e , s j , i − x e , s i , i + n = − b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A W ∑ x e , s x e , s i , j = f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , i − n , i − ( i , j ) ∈ A W ∑ z e , s j , i ≤ 1 − x e , s i , i + n + b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , ( j , i ) ∈ A V ∑ z e , s i , j ≤ 1 − x e , s i − n , i + f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( i , j ) ∈ A V “multicommodity flow demand” constraints for vehicle transportation j , i − ∑ (pickup and delivery of pollsters): ∑ z e , s z e , s i , j = b e , s i , ∀ i ∈ C − , e ∈ E , s ∈ S , sync z variables with b and f variables ( j , i ) ∈ A V ( i , j ) ∈ A V j , i − ∑ ∑ z e , s z e , s i , j = − f e , s i , ∀ i ∈ C + , e ∈ E , s ∈ S , ( j , i ) ∈ A V ( i , j ) ∈ A V ∑ z e , s 0, i ≤ u s , ∀ e ∈ E , s ∈ S . i ∈ C CO@Work 2020 Integrated pollster and vehicle routing 24