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An Introduction to Stochastic Vehicle Routing Michel Gendreau CIRRELT and MAGI École Polytechnique de Montréal PhD course on Local Distribution Planning Molde University College − March 12-16, 2012

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Introduction to Stochastic Vehicle Routing Outline 1. Introduction 2. Basic Concepts in Stochastic Optimization 3. Modeling Paradigms 4. Problems with Stochastic Demands 5. Problems with Stochastic Customers 6. Problems with Stochastic Service or Travel Times 7. Conclusion and perspectives

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Acknowledgements Walter Rei CIRRELT and ESG UQÀM Ola Jabali CIRRELT and HEC Montréal Tom van Woensel and Ton de Kok School of Industrial Engineering Eindhoven University of Technology Introduction to Stochastic Vehicle Routing

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Introduction

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Introduction to Stochastic Vehicle Routing Vehicle Routing Problems Introduced by Dantzig and Ramser in 1959 One of the most studied problem in the area of logistics The basic problem involves delivering given quantities of some product to a given set of customers using a fleet of vehicles with limited capacities. The objective is to determine a set of minimum- cost routes to satisfy customer demands.

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Introduction to Stochastic Vehicle Routing Vehicle Routing Problems Many variants involving different constraints or parameters: Introduction of travel and service times with route duration or time window constraints Multiple depots Multiple types of vehicles...

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Introduction to Stochastic Vehicle Routing What is Stochastic Vehicle Routing? Basically, any vehicle routing problem in which one or several of the parameters are not deterministic: Demands Travel or service times Presence of customers …

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Basic Concepts in Stochastic Optimization

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Dealing with uncertainty in optimization Very early in the development of operations research, some top contributors realized that : In many problems there is very significant uncertainty in key parameters; This uncertainty must be dealt with explicitly. This led to the development of : Chance-constrained programming (1951) Stochastic programming with recourse (1955) Dynamic programming (1958) Robust optimization (more recently) Introduction to Stochastic Vehicle Routing

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Information and decision-making In any stochastic optimization problem, a key issue is: How do the revelation of information on the uncertain parameters and decision-making (optimization) interact? When do the values taken by the uncertain parameters become known? What changes can I (must I) make in my plans on the basis of new information that I obtain? Introduction to Stochastic Vehicle Routing

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Chance-constrained programming Proposed by Charnes and Cooper in 1951. The key idea is to allow some constraints to be satisfied only with some probability. E.g., in VRP with stochastic demands, Pr{total demand assigned to route r ≤ capacity } ≥ 1-α Introduction to Stochastic Vehicle Routing

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Stochastic programming with recourse Proposed separately by Dantzig and by Beale in 1955. The key idea is to divide problems in different stages, between which information is revealed. The simplest case is with only two stages. The second stage deals with recourse actions, which are undertaken to adapt plans to the realization of uncertainty. Basic reference: J.R. Birge and F. Louveaux, Introduction to Stochastic Programming, 2 nd edition, Springer, 2011. Introduction to Stochastic Vehicle Routing

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Dynamic programming Proposed by Bellman in 1958. A method developed to tackle effectively sequential decision problems. The solution method relies on a time decomposition of the problem according to stages. It exploits the so-called Principle of Optimality. Good for problems with limited number of possible states and actions. Basic reference: D.P. Bertsekas, Dynamic Programming and Optimal Control, 3rd edition, Athena Scientific, 2005. Introduction to Stochastic Vehicle Routing

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Here, uncertainty is represented by the fact that the uncertain parameter vector must belong to a given polyhedral set (without any probability defined) E.g., in VRP with stochastic demands, having set upper and lower bounds for each demand, together with an upper bound on total demand. Robust optimization looks in a minimax fashion for the solution that provides the best “worst case”. Robust optimization Introduction to Stochastic Vehicle Routing

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Modelling paradigms

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Also called re-optimization Based on the implicit assumption that information is revealed over time as the vehicles perform their assigned routes. Relies on Dynamic programming and related approaches (Secomandi et al.) Routes are created piece by piece on the basis on the information currently available. Not always practical (e.g., recurrent situations) Real-time optimization Introduction to Stochastic Vehicle Routing

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A priori optimization A solution must be determined beforehand; this solution is “confronted” to the realization of the stochastic parameters in a second step. Approaches: Chance-constrained programming (Two-stage) stochastic programming with recourse Robust optimization [“Ad hoc” approaches] Introduction to Stochastic Vehicle Routing

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Probabilistic constraints can sometimes be transfor- med into deterministic ones (e.g., in the case above if customer demands are independent and Poisson). This model completely ignores what happens when things do not “turn out correctly”. Chance-constrained programming Introduction to Stochastic Vehicle Routing

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Not used very much in stochastic VRP up to now. Model may be overly pessimistic. Robust optimization Introduction to Stochastic Vehicle Routing

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Recourse is a key concept in a priori optimization What must be done to “adjust” the a priori solution to the values observed for the stochastic parameters! Another key issue is deciding when information on the uncertain parameters is provided to decision-makers. Solution methods: Integer L-shaped (Laporte and Louveaux) Heuristics (including metaheuristics) Probably closer to actual industrial practices, if recourse actions are correctly defined! Stochastic programming with recourse Introduction to Stochastic Vehicle Routing

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VRP with stochastic demands

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A probability distribution is specified for the demand of each customer. One usually assumes that demands are independent (this may not always be very realistic...). Probably, the most extensively studied SVRP: Under the reoptimization approach (Secomandi) Under the a priori approach (several authors) using both the chance-constrained and the recourse models. VRP with stochastic demands (VRPSD) Introduction to Stochastic Vehicle Routing

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Probably, the most extensively studied SVRP: Under the reoptimization approach (Secomandi et al.) Under the a priori approach (several authors) using both the chance-constrained and the recourse models. Classical recourse strategy: Return to depot to restore vehicle capacity Does not always seem very appropriate or “intelligent” However, recently many authors have started proposing more creative recourse schemes: Pairing routes (Erera et al.) Preventive restocking (Yang, Ballou, and Mathur) VRP with stochastic demands Introduction to Stochastic Vehicle Routing

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Additional material: M. Gendreau, W. Rei and P. Soriano, “A Hybrid Monte Carlo Local Branching Algorithm for the Single Vehicle Routing Problem with Stochastic Demands”, presented at GOM 2008, August 2008. W. Rei and M. Gendreau, “An Exact Algorithm for the Multi-Vehicle Routing Problem with Stochastic Demands”, presented at the TU Eindhoven, November 2009. VRP with stochastic demands Introduction to Stochastic Vehicle Routing

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VRP with stochastic customers

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Each customer has a given probability of requiring a visit. Problem grounded in the pioneering work of Jaillet (1985) on the Probabilistic Traveling Salesman Problem (PTSP). At first sight, the VRPSC is of no interest under the reoptimization approach. VRP with stochastic customers (VPRSC) Introduction to Stochastic Vehicle Routing

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Recourse action: “Skip” absent customers Has been extensively studied by Gendreau, Laporte and Séguin in the 1990’s: Exact and heuristic solution approaches Has also been used to model the Consistent VRP (following slides). VRP with stochastic customers (VPRSC) Introduction to Stochastic Vehicle Routing

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The Consistent VRP with Stochastic Customers The consistent vehicle routing problem First introduced by Groër, Golden, and Wasil (2009) Have the same driver visiting the same customers at roughly the same time each day that these customers need service Focus is on the customer Planning is done for D periods, known demand, m vehicles Arrival time variation is no more than L Minimize travel time over D periods Introduction to Stochastic Vehicle Routing day 1day 2 Depot

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Problem definition The consistent vehicle routing problem with stochastic customers Each customer has a probability of occurring Same driver visits the same customers A delivery time window is quoted to the customer → (Self-imposed TW) Cost structure Penalties for early and late arrivals Travel times Introduction to Stochastic Vehicle Routing Depot a priori approach Stage 1 Plan routes and set targets Stage 2 Compute travel times and penalties

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Problem data An undirected graph G =( V,A ) V ={ v 1,..,v n } is a set of vertices E ={( v i, v j ): v i, v j V, i

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Model First-stage decision variables : x ij = 1, if client j is visited immediately after client i for 2 ≤ i

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Model Introduction to Stochastic Vehicle Routing

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Model Introduction to Stochastic Vehicle Routing Reformulation the objective function: is a lower bound on the expected travel time Gendreau, Laporte and Séguin (1995) And

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Model Assumption: early arrivals do not wait for the time window Evaluation of the second stage cost Q r, δ : expected recourse cost corresponding to route r if orientation δ is chosen Q P r, δ : total average penalties associated with time window deviations for route r if orientation δ is chosen Q T r : total average travel time for route for route r Introduction to Stochastic Vehicle Routing

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Model Given a route r, we relabel the vertices on the route according to a given orientation δ as follows: Introduction to Stochastic Vehicle Routing : the minimum expected penalty associated with customer

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Model Introduction to Stochastic Vehicle Routing v0v0 v5v5 v4v4 v7v7 v0v0 v5v5 v4v4 v0v0 v4v4 v7v7 v0v0 v4v4 Setting of and evaluation of Parameters: – the collection of random events where customer requires a visit w – half length of the time window p ω – probability of – arrival time at considering β – late arrival penalty Variables: – target arrival time at customer – early arrival at customer by – late arrival at customer by

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Solution procedure Based on the Integer 0-1 L-Shaped Method proposed by Laporte and Louveaux (1993) Variant of branch-and-cut Assumption 1: Q(x) is computable Assumption 2: There exists a finite value L = general lower bound for the recourse function. Operates on the current problem (CP) on each node of the search tree In the VRP context, CP is relaxed: I. Integrality constraints II. Subtour elimination and route duration constraints III. Introduction to Stochastic Vehicle Routing

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0-1 Integer L-Shaped Algorithm List contains initial relaxed CP Choose next pendent node Integer Introduce violated feasibility constraints and LBF Introduce optimality cutsUpdate Apply branching Fathom node yes no yes Introduction to Stochastic Vehicle Routing

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General lower bound We create an auxiliary graph with all distances equal to l and all probabilities are set to p and q → a lower bound on average travel time is ( n-1 ) pl → a lower bound on the penalties associated with time window deviations can be determined also Introduction to Stochastic Vehicle Routing

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v3v3 v2v2 v6v6 Lower bounding functionals Introduced by Hjorring and Holt (1999) bound Q(x) using partial routes Partial route h consist of : Introduction to Stochastic Vehicle Routing v0v0 v5v5 v4v4 v7v7 v8v8 v9v9 v1v1 v0v0

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Lower bounding functionals We look for a lower bound on the recourse associated with route h, P h Bounds on S → compute exactly Bounds on U : assume each node, separately, directly succeeds v S → → compute for each node in U Bounds on T : assume general sequence with U as in L → compute for a subset of scenarios where at most one customer is absent Introduction to Stochastic Vehicle Routing v0v0 v5v5 v4v4 v7v7 v0v0 v5v5 v4v4 v7v7 v6v6 v0v0 v5v5 v4v4 v7v7 v8v8 v9v9 v1v1 v0v0 vgvg vgvg vgvg

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Lower bounding functionals Let Let or if and are consecutive in and For r partial routes the following is a valid inequality: Introduction to Stochastic Vehicle Routing

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Preliminary results Experimental sets: Vertices were generated similar to Laporte, Louveaux and van Hamme (2002) p values are randomly generated within 0.6 and 0.9 20 customers with 4 vehicles or 15 with 3 vehicles Introduction to Stochastic Vehicle Routing

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Preliminary results Introduction to Stochastic Vehicle Routing SetN Initial best integer Initial best node Initial GAPFinal solutionFinal GAPRun time 115356.7303.115.0%332.8<1%64 215352.5264.325.0%285.2<1%55 315397.3360.39.3%388.1<1%263 415363.3303.016.6%312.8<1%69 515407.7357.012.4%393.3<197 620616.2554.110.1%597.2<1%879 720486.4 6.2%461.0<1%340 920476.5405.414.9%451.3<1%27564 1020520.0414.020.4%455.1<164638 1120449.4368.518.0%397.8<167501 1220526.9475.39.8%478.6<12242 1320474.0436.57.9%448.0<11244 1520571.9472.117.5%486.7 9.35%25200 1620444.3397.710.5%416.2<125200 1720442.5390.911.7%414.2 3.13%25200 820494.4401.018.9%433.13.41%86400 1420522.8449.514.0%503.51.53%86400

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Future research A subset of customers that occur with probability 1 Multiple partial routes Improving the LBF Improving the bound on the objective function Sampling approach for larger sets Introduction to Stochastic Vehicle Routing

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VRP with stochastic service or travel times

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The travel times required to move between vertices and/or service times are random variables. The least studied, but possibly the most interesting of all SVRP variants. Reason: it is much more difficult than others, because delays “propagate” along a route. Usual recourse: Pay penalties for soft time windows or overtime. All solution approaches seem relevant, but present significant implementation challenges. Introduction to Stochastic Vehicle Routing

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Conclusions and perspectives

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Introduction to Stochastic Vehicle Routing Conclusion and perspectives Stochastic vehicle routing is a rich and promising research area. Much work remains to be done in the area of recourse definition. SVRP models and solution techniques may also be useful for tackling problems that are not really stochastic, but which exhibit similar structures Up to now, very little work on problems with stochastic travel and service times, while one may argue that travel or service times are uncertain in most routing problems! Correlation between uncertain parameters is possibly a major stumbling block in many application areas, but no one seems to work on ways to deal with it.

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