Vehicle routing and demand forecasting in recyclable waste collection

Transcription

1 Vehicle routing and demand forecasting in recyclable waste collection Iliya Markov Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering École Polytechnique Fédérale de Lausanne Joint work with Yousef Maknoon, Matthieu de Lapparent, Jean-François Cordeau, Sacha Varone, Michel Bierlaire CIRRELT November 26, 2015 Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

2 Source: EPFL Facebook page Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

3 Source: EPFL Facebook page Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

4 Source: EPFL website Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

5 EPFL facts and figures One of two Federal Institutes of Technology in Switzerland Located in Lausanne Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

6 EPFL facts and figures One of two Federal Institutes of Technology in Switzerland Located in Lausanne Comprises 5 schools, 2 colleges, 26 institutes, and 350 laboratories Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

7 EPFL facts and figures One of two Federal Institutes of Technology in Switzerland Located in Lausanne Comprises 5 schools, 2 colleges, 26 institutes, and 350 laboratories 10,124 students of over 115 nationalities - 5,205 bachelor students (40% international) - 2,646 master students (50% international) - 2,077 PhD students (80% international) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

8 EPFL facts and figures One of two Federal Institutes of Technology in Switzerland Located in Lausanne Comprises 5 schools, 2 colleges, 26 institutes, and 350 laboratories 10,124 students of over 115 nationalities - 5,205 bachelor students (40% international) - 2,646 master students (50% international) - 2,077 PhD students (80% international) 5,630 staff faculty - 1,228 postdocs Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

9 TRANSP-OR Michel full professor Mila secretary Anne IT Discrete Choice Matthieu postdoc Eva PhD student Anna PhD student Meritxell PhD student Operations Research Yousef postdoc Shadi postdoc Iliya PhD student Tomáš PhD student Stefan PhD student Riccardo postdoc Marija PhD student Pedestrians Flurin PhD student Antonin PhD student Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

10 Contents 1 Introduction 2 Vehicle Routing 3 Demand Forecasting 4 Integration 5 Conclusion Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

11 Introduction Contents 1 Introduction 2 Vehicle Routing 3 Demand Forecasting 4 Integration 5 Conclusion Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

12 Introduction Ecological waste management (CTI project) Ecopoint in Rue de Neucha tel, Geneva; Source: self Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

13 Introduction Ecological waste management (CTI project) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

14 Introduction Ecological waste management (CTI project) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

15 Introduction Introduction Sensorized containers for recyclables periodically send waste level data to a central database. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

16 Introduction Introduction Sensorized containers for recyclables periodically send waste level data to a central database. Level data is used for container selection and vehicle routing, with tours often planned several days in advance. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

17 Introduction Introduction Sensorized containers for recyclables periodically send waste level data to a central database. Level data is used for container selection and vehicle routing, with tours often planned several days in advance. Vehicles are dispatched to carry out the daily schedules produced by the routing algorithm. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

18 Introduction Introduction Sensorized containers for recyclables periodically send waste level data to a central database. Level data is used for container selection and vehicle routing, with tours often planned several days in advance. Vehicles are dispatched to carry out the daily schedules produced by the routing algorithm. Efficient waste collection thus depends on the ability to - make good forecasts of the container levels at the time of collection, - and optimally route the vehicles to serve the selected containers. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

19 Vehicle Routing Contents 1 Introduction 2 Vehicle Routing 3 Demand Forecasting 4 Integration 5 Conclusion Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

20 Vehicle Routing Problem description Multiple depots, containers, and dumps (recycling plants) with TW Maximum tour duration, interrupted by a break Site dependencies (accessibility restrictions) Tours are sequences of collections and disposals at the available dumps, with a mandatory disposal before the end. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

21 Vehicle Routing Problem description Multiple depots, containers, and dumps (recycling plants) with TW Maximum tour duration, interrupted by a break Site dependencies (accessibility restrictions) Tours are sequences of collections and disposals at the available dumps, with a mandatory disposal before the end. Tours need not finish at the depot they started from. - flexible assignment of destination depots - practiced in sparsely populated rural areas Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

22 Vehicle Routing Problem description Multiple depots, containers, and dumps (recycling plants) with TW Maximum tour duration, interrupted by a break Site dependencies (accessibility restrictions) Tours are sequences of collections and disposals at the available dumps, with a mandatory disposal before the end. Tours need not finish at the depot they started from. - flexible assignment of destination depots - practiced in sparsely populated rural areas There is a heterogeneous fixed fleet. - different volume and weight capacities, speeds, costs, etc... Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

23 Vehicle Routing Problem description Figure 1: Tour illustration c c c c c c c c depots dump dump c c c c c c c c c c = container Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

24 Vehicle Routing State of the art VRP with intermediate facilities (VRP-IF) - Bard et al. (1998), Kim et al. (2006), Crevier et al. (2007) Electric and alternative fuel VRP - Conrad and Figliozzi (2011), Erdoğan and Miller-Hooks (2012), Schneider et al. (2014), Schneider et al. (2015) Heterogeneous fixed fleet VRP - Taillard (1996), Baldacci and Mingozzi (2009), Subramanian et al. (2012), Penna et al. (2013) - Hiermann et al. (2014) and Goeke and Schneider (2014) use some form of heterogeneity in the electric VRP Flexible assignment of depots - Kek et al. (2008) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

25 Vehicle Routing Contributions Integration of dynamic destination depot assignment into the VRP-IF - consideration of relocation costs Integration of heterogeneous fixed fleet into the VRP-IF - challenges posed by intermediate facility visits Benchmarking to several classes of simpler problems from the literature and state of practice - E-VRPTW (modified from Schneider et al., 2014) - MDVRPI (Crevier et al., 2007) - optimal solutions, state of practice, etc... Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

26 Vehicle Routing Formulation Sets O = set of origins O = set of destinations D = set of dumps P = set of containers N = O O D P K = set of vehicles Parameters π ij = length of arc (i, j) τ ijk = travel time of vehicle k on arc (i, j) [λ i, µ i ] = time window lower and upper bound at point i ε i = service duration at point i ρ i = pickup quantity at point i Ω k = capacity of vehicle k φ k = fixed cost of vehicle k β k = unit-distance running cost of vehicle k θ k = unit-time running cost of vehicle k α ik = 1 if point i is accessible for vehicle k, 0 otherwise H = maximum tour duration η = maximum continuous work limit after which a break is due δ = break duration Ψ = weight of relocation cost term Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

27 Vehicle Routing Formulation Decision variables: binary { 1 if vehicle k traverses arc (i, j) x ijk = 0 otherwise { 1 if i and j are, respectively, the origin and destination of vehicle k z ijk = 0 otherwise { 1 if vehicle k takes a break on arc (i, j) b ijk = 0 otherwise { 1 if vehicle k is used y k = 0 otherwise Decision variables: continuous S ik = start-of-service time of vehicle k at point i Q ik = cumulative quantity on vehicle k at point i Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

28 Vehicle Routing Formulation min r = φ k y k + β k π ij x ijk + θ k S jk S ik k K i N j N j O k i O k + Ψ (1) ( ) βk π ji + θ k τ jik zijk k K s.t. A i O k j O k x ijk = 1, i P (2) k K j D P A x ijk = y k, k K (3) i O k j N x ijk = y k, k K (4) i D j O k A x ijk = 0, k K, j O (O \ O k (5) i N x ijk = 0, k K, i O (O \ O k ) (6) j N Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

29 Vehicle Routing Formulation min r = φ k y k + β k π ij x ijk + θ k S jk S ik k K i N j N j O k i O k + Ψ (1) ( ) βk π ji + θ k τ jik zijk k K s.t. A i O k j O k x ijk = 1, i P (2) k K j D P A x ijk = y k, k K (3) i O k j N x ijk = y k, k K (4) i D j O k A x ijk = 0, k K, j O (O \ O k (5) i N x ijk = 0, k K, i O (O \ O k ) (6) j N Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

30 Vehicle Routing Formulation min r = φ k y k + β k π ij x ijk + θ k S jk S ik k K i N j N j O k i O k + Ψ (1) ( ) βk π ji + θ k τ jik zijk k K s.t. A i O k j O k x ijk = 1, i P (2) k K j D P A x ijk = y k, k K (3) i O k j N x ijk = y k, k K (4) i D j O k A x ijk = 0, k K, j O (O \ O k (5) i N x ijk = 0, k K, i O (O \ O k ) (6) j N Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

31 Vehicle Routing Formulation min r = φ k y k + β k π ij x ijk + θ k S jk S ik k K i N j N j O k i O k + Ψ (1) ( ) βk π ji + θ k τ jik zijk k K s.t. A i O k j O k x ijk = 1, i P (2) k K j D P A x ijk = y k, k K (3) i O k j N x ijk = y k, k K (4) i D j O k A x ijk = 0, k K, j O (O \ O k (5) i N x ijk = 0, k K, i O (O \ O k ) (6) j N Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

32 Vehicle Routing Formulation s.t. i N : i j x A ijk = i N : i j x jik, k K, j P D (7) A x imk + x mjk 1 z ijk, m N m D k K, i O k, j O k (8) A x ijk α ik, k K, i P D (9) j N ρ i Q ik Ω k, k K, i P (10) Q ik = 0, k K, i N \ P (11) Q ik + ρ j Q jk + Ω k ( 1 xijk ), k K, i O k P D, j P (12) S ik + ε i + δb ijk + τ ijk S jk + M ( ) A 1 x ijk, k K, i O k P D, j P D O k (13) λ i x ijk S ik, k K, i O k P D (14) j N S jk µ j x ijk, k K, j P D O k (15) i N 0 S jk S ik H, k K (16) j O k i O k Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

33 Vehicle Routing Formulation s.t. i N : i j x A ijk = i N : i j x jik, k K, j P D (7) A x imk + x mjk 1 z ijk, m N m D k K, i O k, j O k (8) A x ijk α ik, k K, i P D (9) j N ρ i Q ik Ω k, k K, i P (10) Q ik = 0, k K, i N \ P (11) Q ik + ρ j Q jk + Ω k ( 1 xijk ), k K, i O k P D, j P (12) S ik + ε i + δb ijk + τ ijk S jk + M ( ) A 1 x ijk, k K, i O k P D, j P D O k (13) λ i x ijk S ik, k K, i O k P D (14) j N S jk µ j x ijk, k K, j P D O k (15) i N 0 S jk S ik H, k K (16) j O k i O k Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

34 Vehicle Routing Formulation s.t. i N : i j x A ijk = i N : i j x jik, k K, j P D (7) A x imk + x mjk 1 z ijk, m N m D k K, i O k, j O k (8) A x ijk α ik, k K, i P D (9) j N ρ i Q ik Ω k, k K, i P (10) Q ik = 0, k K, i N \ P (11) Q ik + ρ j Q jk + Ω k ( 1 xijk ), k K, i O k P D, j P (12) S ik + ε i + δb ijk + τ ijk S jk + M ( ) A 1 x ijk, k K, i O k P D, j P D O k (13) λ i x ijk S ik, k K, i O k P D (14) j N S jk µ j x ijk, k K, j P D O k (15) i N 0 S jk S ik H, k K (16) j O k i O k Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

35 Vehicle Routing Formulation s.t. i N : i j x A ijk = i N : i j x jik, k K, j P D (7) A x imk + x mjk 1 z ijk, m N m D k K, i O k, j O k (8) A x ijk α ik, k K, i P D (9) j N ρ i Q ik Ω k, k K, i P (10) Q ik = 0, k K, i N \ P (11) Q ik + ρ j Q jk + Ω k ( 1 xijk ), k K, i O k P D, j P (12) S ik + ε i + δb ijk + τ ijk S jk + M ( ) A 1 x ijk, k K, i O k P D, j P D O k (13) λ i x ijk S ik, k K, i O k P D (14) j N S jk µ j x ijk, k K, j P D O k (15) i N 0 S jk S ik H, k K (16) j O k i O k Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

36 Vehicle Routing Formulation s.t. i N : i j x A ijk = i N : i j x jik, k K, j P D (7) A x imk + x mjk 1 z ijk, m N m D k K, i O k, j O k (8) A x ijk α ik, k K, i P D (9) j N ρ i Q ik Ω k, k K, i P (10) Q ik = 0, k K, i N \ P (11) Q ik + ρ j Q jk + Ω k ( 1 xijk ), k K, i O k P D, j P (12) S ik + ε i + δb ijk + τ ijk S jk + M ( ) A 1 x ijk, k K, i O k P D, j P D O k (13) λ i x ijk S ik, k K, i O k P D (14) j N S jk µ j x ijk, k K, j P D O k (15) i N 0 S jk S ik H, k K (16) j O k i O k Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

37 Vehicle Routing Formulation s.t. S ik S mk + ε i η M ( ) 1 b ijk, k K, i O k P D, j P D O k (17) m O k η S jk S mk M ( ) 1 b ijk, k K, i O k P D, j P D O k (18) m O k b ijk x ijk, k K, i, j N (19) S jk S ik η (H η) b ijk, k K (20) i N j N j O k i O k x ijk, z ijk, b ijk, y k {0, 1}, k K, i, j N (21) S ik Q ik 0, k K, i N (22) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

38 Vehicle Routing Formulation s.t. S ik S mk + ε i η M ( ) 1 b ijk, k K, i O k P D, j P D O k (17) m O k η S jk S mk M ( ) 1 b ijk, k K, i O k P D, j P D O k (18) m O k b ijk x ijk, k K, i, j N (19) S jk S ik η (H η) b ijk, k K (20) i N j N j O k i O k x ijk, z ijk, b ijk, y k {0, 1}, k K, i, j N (21) S ik Q ik 0, k K, i N (22) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

39 Vehicle Routing Solution methodology: Exact approach We apply variable fixing and several additional inequalities Impossible traversals x iik = 0, k K, i N (23) x ijk = 0, k K, i O k, j D O k (24) x ijk = 0, k K, i P, j O k (25) x ijk = 0, k K, i D, j D : i j (26) Time-window infeasible traversals x ijk = 0, Bounds on time k K, i O k P D, j P D O k : λ i + ε i + τ ijk > µ j (27) S jk S ik x ijk (ε i + τ ijk ), k K (28) j O k i O k i N j N S ik max m P (µm τ imk) y k, k K, i O k (29) ( ) S jk min λm + ε m + τ mjk x mjk, m D m D k K, j O k (30) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

40 Vehicle Routing Solution methodology: Exact approach Symmetry breaking for subsets K of identical vehicles ρ v i x ijk ρ v g i x ijk, g 1,..., ( K 1 ) (31) g+1 i P j P D i P j P D Symmetry breaking for replications of the same dump D ix ij g k ix ij g+1 k, k K, g 1,..., ( D 1 ) (32) i P i P Bounds on dump visits x ijk 1, k K, j D (33) i P x ijk min ( D 1, P 1), k K (34) i D j P Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

41 Vehicle Routing Solution methodology: Heuristic approach To solve instances of realistic size, we developed a heuristic algorithm. It constructs a feasible initial solution using an insertion procedure. It improves the initial solution through a multiple neighborhood search procedure admitting intermediate infeasibility with a dynamically evolving penalty. Periodically, we restart from the best feasible solution because feasibility may be hard to restore. Periodically, we also reassign dump visits and evaluate vehicle reassignments because the fleet is heterogeneous and fixed. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

42 Vehicle Routing Results We test the heuristic against the mathematical model on modifications of the Schneider et al. (2014) E-VRPTW instances. - recharging facilities are regarded as dumps - additional features relevant to our problem are added Additionally, we test the heuristic on - the Crevier et al. (2007) instances for the purpose of evaluating the benefit of flexible depot assignment, - and on state-of-practice data For each instance, the heuristic is run 10 times. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

43 Vehicle Routing Results: Modified Schneider et al. (2014) instances 36 instances derived from the Solomon (1987) VRPTW instances 3 groups of 12 instances with 5, 10, and 15 customers Number of recharging stations: 2 to 8 Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

44 Vehicle Routing Results: Modified Schneider et al. (2014) instances 36 instances derived from the Solomon (1987) VRPTW instances 3 groups of 12 instances with 5, 10, and 15 customers Number of recharging stations: 2 to 8 Modifications - regard recharging stations as dumps - use 2 vehicle classes with different capacities, costs, and site dependencies - apply a maximum tour duration, a maximum working time limit, and a break duration Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

45 Vehicle Routing Results: Modified Schneider et al. (2014) instances 36 instances derived from the Solomon (1987) VRPTW instances 3 groups of 12 instances with 5, 10, and 15 customers Number of recharging stations: 2 to 8 Modifications - regard recharging stations as dumps - use 2 vehicle classes with different capacities, costs, and site dependencies - apply a maximum tour duration, a maximum working time limit, and a break duration We compare the heuristic against the mathematical model. For each instance, the heuristic is run 10 times. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

46 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 5 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c101c , c103c , c206c c208c , r104c r105c r202c r203c , rc105c , rc108c , rc204c , rc208c , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

47 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 5 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c101c , c103c , c206c c208c , r104c r105c r202c r203c , rc105c , rc108c , rc204c , rc208c , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

48 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 5 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c101c , c103c , c206c c208c , r104c r105c r202c r203c , rc105c , rc108c , rc204c , rc208c , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

49 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 5 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c101c , c103c , c206c c208c , r104c r105c r202c r203c , rc105c , rc108c , rc204c , rc208c , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

50 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 10 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c101c , , c104c , c202c , , c205c , , r102c , r103c , r201c , r203c , , rc102c , rc108c , rc201c , rc205c , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

51 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 10 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c101c , , c104c , c202c , , c205c , , r102c , r103c , r201c , r203c , , rc102c , rc108c , rc201c , rc205c , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

52 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 10 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c101c , , c104c , c202c , , c205c , , r102c , r103c , r201c , r203c , , rc102c , rc108c , rc201c , rc205c , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

53 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 10 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c101c , , c104c , c202c , , c205c , , r102c , r103c , r201c , r203c , , rc102c , rc108c , rc201c , rc205c , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

54 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 15 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c103c , , c106c , , c202c , , c208c , , r102c , , r105c , r202c , , r209c , , rc103c , , rc108c , , rc202c , , rc204c , , Average , , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

55 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 15 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c103c , , c106c , , c202c , , c208c , , r102c , , r105c , r202c , , r209c , , rc103c , , rc108c , , rc202c , , rc204c , , Average , , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

56 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 15 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c103c , , c106c , , c202c , , c208c , , r102c , , r105c , r202c , , r209c , , rc103c , , rc108c , , rc202c , , rc204c , , Average , , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

57 Vehicle Routing Results: Modified Schneider et al. (2014) instances Table 1: Heuristic vs solver on modified Schneider et al. (2014) instances with 15 customers Heuristic Solver on model with adtl inequalities Solver on model without adtl inequalities Runtime MIP Runtime Improve- MIP Runtime Improve- Instance Vehicles Best Average avg(s.) Objective Gap(%) (s.) ment(%) Objective Gap(%) (s.) ment(%) c103c , , c106c , , c202c , , c208c , , r102c , , r105c , r202c , , r209c , , rc103c , , rc108c , , rc202c , , rc204c , , Average , , Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

58 Vehicle Routing Results: Crevier et al. (2007) instances 22 instances, with a limited homogeneous fleet stationed at one depot All depots can act as intermediate facilities. BKS by Hemmelmayr et al. (2013) We apply the MNS heuristic to evaluate the benefits from flexible destination depot assignments. Keeping the home depot and optimizing the destination depot obtains % average savings over 10 runs % savings in the best case Optimizing the home depot and the destination depot obtains % average savings over 10 runs % savings in the best case Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

59 Vehicle Routing Results: Comparison to the state of practice 35 tours planned by specialized software for the canton of Geneva 7 to 38 containers per tour, up to 4 dump visits per tour MNS heuristic improves tours by 1.73% to 34.91%, on avg 14.75% Extrapolating annually, cost reductions of at least 300,000 USD Figure 2: Comparison to the state of practice (average of 10 runs per tour) State of Practice Multiple Neighborhood Search Distance (km) Tours Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

60 Demand Forecasting Contents 1 Introduction 2 Vehicle Routing 3 Demand Forecasting 4 Integration 5 Conclusion Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

61 Demand Forecasting State of the art Much of it is focused on city and regional level. And a fairly small amount on the container (micro) level, e.g. - inventory levels in pharmacies (Nolz et al., 2011, 2014) - recyclable materials from old cars (Krikke et al., 2008) - charity donation banks (McLeod et al., 2013) - waste container levels (Johansson, 2006; Faccio et al., 2011; Mes, 2012; Mes et al., 2014) Contributions - operational level forecasting rather than critical levels - estimated and validated on real data, compared to most of the literature which uses simulated data Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

62 Demand Forecasting Methodology Let n itg denote the number of deposits in container i on day t of size q g. The data generating process of the daily demands is ρ it = m g=1 q g n itg (35) Let n itg iid Pois (λ itg ) and have a probability π itg. Then we obtain E (ρ it ) = m g=1 q g λ itg π itg (36) We minimize the sum of squared errors between observed ρ o it and expected E (ρ it ) for p containers and h days p h min (ρ o λ,π i=1 t=1 it m ) 2 q g λ itg π itg (37) g=1 assuming strict exogeneity and errors modeled as white noise. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

63 Demand Forecasting Methodology Given vectors of covariates c it and vectors of parameters β λ g and β π g, we define Poisson rates and logit-type probabilities ( ) λ itg = exp c T itβ λ g (38) exp ( c T ) it π itg = βπ g m j=1 exp ( ) c T (39) it βπ j Then, in compact form, the minimization problem writes as p h m min ρ o exp ( c T it it βλ g + c T it βπ g + ln (q g ) ) m B j=1 exp ( ) c T it βπ j i=1 t=1 g=1 2 (40) B = (β λ g, β π g : g), and β π g = 0 for one arbitrarily chosen g. We will refer to this minimization problem as the mixture model. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

64 Demand Forecasting Methodology In case of only one deposit quantity, it degenerates to a pseudo-count data process min B p h i=1 t=1 ( ( 2 ρ o it exp c T itβ + ln(q))) (41) We will refer to this minimization problem as the simple model. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

65 Demand Forecasting Methodology Using new sets of covariates ċ it, we can generate a forecast as E(ρ it ) = m g=1 exp ( ċ T it βλ g + ċ T it βπ g + ln (q g ) ) m j=1 exp ( ) ċ T (42) it βπ j Given the operational nature of the problem, the covariates should be quick and easy to obtain. Examples include days of the week, months, weather data, holidays, etc... Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

66 Demand Forecasting Data 36 containers for PET in the canton of Geneva with capacity of 3,040 or 3,100 liters Balanced panel covering March to June, 2014 (122 days), which brings the total number of observations to 4,392 The final sample excludes unreliable level data (removed after visual inspection). Missing data is linearly interpolated for the values of ρ o it. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

67 Demand Forecasting Residual plots Figure 3: Residual plot of the mixture model All residuals Value over standard deviation Figure 4: Residual plot of the simple model All residuals Value over standard deviation Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

68 Demand Forecasting Seasonality pattern Waste generation exhibits strong weekly seasonality. Peaks are observed during the weekends. There also appear to be longer-term effects for months. Figure 5: Mean daily volume deposited in the containers 600 Amount (liters) (Mar) 10 (Mar) 11 (Mar) 12 (Mar) 13 (Mar) 14 (Apr) 15 (Apr) 16 (Apr) 17 (Apr) 18 (May) Weeks in (May) 20 (May) 21 (May) 22 (Jun) 23 (Jun) 24 (Jun) 25 (Jun) 26 (Jun) Mean daily volume Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

69 Demand Forecasting Covariates Table 2: Table of covariates Variable Container fixed effect Day of the week Month Minimum temperature in Celsius Precipitation in mm Pressure in hpa Wind speed in kmph Type dummy dummy dummy continuous continuous continuous continuous Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

70 Demand Forecasting Evaluating the fits Coefficient of determination R 2 = 1 SS res SS tot (43) with higher values for a better model. Akaike information criterion (AIC): ( ) ( ) SSres 2 #param AIC = exp #obs #obs (44) with lower values for a better model. The exponential penalizes model complexity. SS res is the residual sum of squares. SS tot is the total sum of squares. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

71 Demand Forecasting Estimation on full sample Mixture model: R 2 of (AIC 52,900) with 5L and 15L Simple model: R 2 of (AIC 53,700) with 10L Table 3: Estimated coefficients of mixture model ˆβ λ 1 (5L)*** ˆβ λ 2 (15L)*** ˆβ π 2 (15L)*** Minimum temperature in Celsius 1, Precipitation in mm Pressure in hpa Wind speed in kmph Monday Tuesday 1, Wednesday Thursday 1, Friday 1, Saturday -6, Sunday 1, March April 1, May 1, June -2, Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

72 Demand Forecasting Validation 50 experiments The mixture and the simple model are estimated on a random sample of 90% of the panel, and validated on the remaining 10%. In both cases the values are significantly different at 90% confidence level. Table 4: Mean R 2 for estimation and validation sets Mixture model mean R 2 Simple model mean R 2 Estimation (AIC 51,400) (AIC 53,600) Validation Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

73 Demand Forecasting Validation Figure 6: Histograms for estimation and validation samples Estimation 6 Validation 10 4 Count 5 Count R R 2 Mixture Simple Mixture Simple Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

74 Demand Forecasting Work so far Markov, I., Varone, S., and Bierlaire, M. (under review). Integrating a heterogeneous fixed fleet and a flexible assignment of destination depots in the waste collection VRP with intermediate facilities. Markov, I., Lapparent, M. (de), Bierlaire, M., and Varone, S. (2015). Modeling a waste disposal process via a discrete mixture of count data models. Proceedings of the 15 th Swiss Transport Research Conference (STRC), April 15-17, 2015, Ascona, Switzerland. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

75 Integration Contents 1 Introduction 2 Vehicle Routing 3 Demand Forecasting 4 Integration 5 Conclusion Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

76 Integration Inventory routing: Setup Integration of forecasting, container selection and routing over a planning horizon adds up to solving an IRP. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

77 Integration Inventory routing: Setup Integration of forecasting, container selection and routing over a planning horizon adds up to solving an IRP. We do not allow overflows in expected terms. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

78 Integration Inventory routing: Setup Integration of forecasting, container selection and routing over a planning horizon adds up to solving an IRP. We do not allow overflows in expected terms. But container demands are stochastic and we cannot ignore the probability of overflow. lower routing cost higher overflow probability higher routing cost lower overflow probability Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

79 Integration Inventory routing: Setup Integration of forecasting, container selection and routing over a planning horizon adds up to solving an IRP. We do not allow overflows in expected terms. But container demands are stochastic and we cannot ignore the probability of overflow. lower routing cost higher overflow probability higher routing cost lower overflow probability To achieve this balance, we need to penalize a state of overflow weighted by its probability. Notation - I it inventory of container i on day t - s it0 container i on day t is in a state of no overflow - s it1 container i on day t is in a state of overflow Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

80 Integration Figure 7: Container index i omitted for readability. P(I 0 +ρ 0 <C) s 10 State probability tree P(I 0 +ρ 0 +ρ 1 <C I 0 +ρ 0 <C) P(I 0 +ρ 0 +ρ 1 C I 0 +ρ 0 <C) s 20 s 21 P(I 0 +ρ 0 +ρ 1 +ρ 2 <C I 0 +ρ 0 +ρ 1 <C) P(I 0 +ρ 0 +ρ 1 +ρ 2 C I 0 +ρ 0 +ρ 1 <C) P(0+ρ 2 <C) P(0+ρ 2 C) s 30 s 31 s 30 s 31 P(I 0 +ρ 0 +ρ 1 +ρ 2 +ρ3<c I 0 +ρ 0 +ρ 1 +ρ 2 <C) P(I 0 +ρ 0 +ρ 1 +ρ 2 +ρ3 C I 0 +ρ 0 +ρ 1 +ρ 2 <C) P(0+ρ 3 <C) P(0+ρ 3 C) P(0+ρ 2 +ρ 3 <C 0+ρ 2 <C) P(0+ρ 2 +ρ 3 C 0+ρ 2 <C) P(0+ρ 3 <C) P(0+ρ 3 C) s 40 s 41 s 40 s 41 s 40 s 41 s 40 s 41 s 00 P(I 0 +ρ 0 C) P(0+ρ 1 <C) s 20 P(0+ρ 1 +ρ 2 <C 0+ρ 1 <C) P(0+ρ 1 +ρ 2 C 0+ρ 1 <C) s 30 s 31 P(0+ρ 1 +ρ 2 +ρ 3 <C 0+ρ 1 +ρ 2 <C) P(0+ρ 1 +ρ 2 +ρ 3 C 0+ρ 1 +ρ 2 <C) P(0+ρ 3 <C) P(0+ρ 3 C) s 40 s 41 s 40 s 41 s 11 P(0+ρ 1 C) P(0+ρ 2 <C) s 30 P(0+ρ 2 +ρ 3 <C 0+ρ 2 <C) P(0+ρ 2 +ρ 3 C 0+ρ 2 <C) s 40 s 41 s 21 P(0+ρ 2 C) s 31 P(0+ρ 3 <C) P(0+ρ 3 C) s 40 s 41 Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

81 Integration Calculating the probabilities The forecasting errors are assumed to be iid normal, therefore ρ o it = E(ρ it ) + ε it, ε it iid Normal(0, σ 2 ) (45) An unbiased and consistent estimate of the variance, with p containers and h days of historical data, is given by p h σ 2 i=1 t=1 = (ρo it E(ρ it)) 2 ph #params (46) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

82 Integration Calculating the probabilities The forecasting errors are assumed to be iid normal, therefore ρ o it = E(ρ it ) + ε it, ε it iid Normal(0, σ 2 ) (45) An unbiased and consistent estimate of the variance, with p containers and h days of historical data, is given by p h σ 2 i=1 t=1 = (ρo it E(ρ it)) 2 ph #params An unconditional probability can be calculated simply as P(I i0 + ρ i0 C i ) = P (ε it C i I i0 E(ρ i0 )) ( ) Ci I i0 E(ρ i0 ) = 1 Φ σ (46) (47) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

83 Integration Calculating the probabilities To calculate a conditional probability, we need to evaluate P ( I i0 + h ) h 1 ρ it C i I i0 + ρ it < C i = t=0 t=0 ( h ) (48) h h 1 h 1 = P ε it C i I i0 E(ρ it ) ε it < C i I i0 E(ρ it ) t=0 t=0 t=0 t=0 Substitute a = C i I i0 h 1 t=0 E(ρ it) and X = h 1 t=0 ε it, where X N (0, hσ 2 ) and X is independent of ε ih. After a standardization, expression (48) rewrites as P (X + ε ih a E(ρ ih ) X < a) = P(ε ih a E(ρ ih ) X, X < a) = P(X < a) 1 a/σ h = 2πΦ(a/σ a E(ρ h) ih ) xσ exp( x 2 /2) exp( y 2 /2)dxdy = h σ ( 1 a/σ h = 2 2πΦ(a/σ exp( x 2 a E(ρ ih ) xσ ) h /2) erfc h) σ dx 2 (49) Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

84 Integration Formulation outline All unconditional and conditional probabilities in the tree can be precomputed. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

85 Integration Formulation outline All unconditional and conditional probabilities in the tree can be precomputed. We are now in a position to formulate a new objective function that captures the probabilistic information. Expected overflow and emergency visit cost + Routing cost + Expected route failure cost Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55

86 Integration Formulation outline All unconditional and conditional probabilities in the tree can be precomputed. We are now in a position to formulate a new objective function that captures the probabilistic information. Expected overflow and emergency visit cost + Routing cost + Expected route failure cost Constraints use expected values only. We add a day-index and inventory constraints to the VRP model. Iliya Markov (TRANSP-OR, EPFL) Recyclable waste collection November 26, / 55