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**The Vehicle Routing Problem**

Part 2: A CP Approach

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**Outline Session 1 What is the vehicle routing problem (VRP)?**

‘Traditional’ (Non-CP) solution methods A simple CP model Playtime Session 2 A CP model for the VRP Large Neighbourhood Search revisited Propagation More Playtime TSP game after 1st CP model after 3 Carleton’s stuff after LNS

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A CP Model for the VRP

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**Model A (rich) vehicle routing problem**

n customers (fixed in this model) m routes (fixed in this model) fixed locations where things happen one for each customer + one for (each?) depot c commodities (e.g. weight, volume, pallets) Know demand from each customer for each commodity Know time between each location pair Know cost between each location pair Both obey triangle inequality

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**Vocabulary A solution is made up of routes (one for each vehicle)**

A route is made up of a sequence of visits Some visits serve a customer (customer visit) (Some tricks) Each route has a “start visit” and an “end visit” Start visit is first visit on a route – location is depot End visit is last visit on a route – location is depot Also have an additional route – the unassigned route Where visits live that cannot be assigned

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**Referencing Customers Each customer has an index in N = {1..n}**

Customers are ‘named’ in CP by their index Routes Each route has an index in M = {1..m} Unassigned route has index 0 Routes are ‘named’ in CP by their index Visits Customer visit index same as customer index Start visit for route k has index n + k ; aka startk End visit for route k has index n + m + k ; aka endk

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**Referencing Sets N = {1 .. n} – customers M = {1 .. m} – routes**

R = M {0} – includes ‘unassigned’ route V = {1 .. n + 2m} – all visits V0 = V {0} - includes a ‘spare’ S = {n+1 .. n+m} – start visits E = {n+m+1 .. n+2m} – end visits VS= N S – visits that have a sensible successor VE= N E – visits that have a sensible predecessor

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**Data We know (note uppercase) Vi The ‘value’ of customer i**

Dik Demand by customer i for commodity k Ei Earliest time to start service at i Li Latest time to start service at i Qjk Capacity of vehicle j for commodity k Tij Travel time from visit i to visit j Cij Cost (w.r.t. objective) of travel from i to j

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**Basic Variables Successor variables: si**

si gives direct successor of i, i.e. the index of the next visit on the route that visits i si VE for i in VS si = 0 for i in E Predecessor variables pi pi gives the index of the previous visit in the route pi VS for i in VE pi = 0 for i in S Redundant – but empirical evidence for its use Route variables ri ri gives the index of the route (vehicle) that visits i ri R TSP game after 1st CP model after 3 Carleton’s stuff after LNS

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**Example i si pi ri 1 4 2 7 3 8 5 9 6 1 Route 2 4 Route 1 2 End visits**

1 Route 2 4 Route 1 2 End visits 3 7 9 5 8 6 Start visits

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**Other variables Accumulation Variables**

qik Quantity of commodity k after visit i ci Objective cost getting to i For problems with time constraints ai Arrival time at i ti Start time at i (time service starts) di Departure time at i Actually, only ti is required, but others allow for expressive constraints

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**What can we model? Basic VRP VRP with time windows Multi-depot**

Heterogeneous fleet Open VRP (vehicle not required to return to base) Requires anywhere location Route end visits located at anywhere distance i anywhere = 0 Compatibility Customers on different / same vehicle Customers on/not on given vehicle Pickup and Delivery problems

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**What can we model? Variable load/unload times**

by changing departure time relative to start time Dispatch time constraints e.g. limited docks si for i in S is load-start time Depot close time Time window on end visits Fleet size and mix Add lots of vehicles Need to introduce a ‘fixed cost’ for a vehicle Cij is increased by fixed cost for all i S, all j N

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**What can’t we model Can’t handle dynamic problems**

Fixed domain for s, p, r vars Can’t introduce new visits post-hoc E.g. optional driver break must be allowed at start Can’t handle multiple visits to same customer ‘Larger than truck-load’ problems If qty is fixed, can have multiple visits / cust Heterogeneous fleet is a pain Can’t handle time- or vehicle-dependent travel times/costs Soft Constraints

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**Objective Want to minimize sum of objective (cij) over used arcs, plus**

value of unassigned visits minimize

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**Basic constraints Path ( S, E, { si | i V } )**

AllDifferent ( { pi | i VE } ) Accumulate obj. Accumulate time Time windows

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Constraints Load Consistency

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**Solving Pure CP Assign to ‘successor’ variables Form chains of visits**

Decision 1: Which visit to insert (== variable choice) Decision 2: Where to insert it (== value choice) s1= 4 1 4 s2= 1 2 ‘Rooted chain’ starts at Start ‘Free chain’ otherwise Can reason about free chains but rooted chains easier “last(k)” is last visit in chain starting at k 3 s7= 2 7 9 s5= 3 5 8 6

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**Propagation – Cycles Subtour elimination Rooted chains are fine**

For free chains: for any chain starting at j, remove j from slast(j) Some CP libraries have built-ins Comet: ‘circuit’ ILOG: Path constraint

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**Propagation – Cycles ‘Path’ constraint Propagates subtour elimination**

Also propagates cost path (S, T, succ, P, z) succ array implies path ensures path from nodes in S to nodes in T through nodes in P variable z bounds cost of path cost propagated incrementally based on shortest / longest paths

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**Propagation – Cost ‘Path’ constraint Maintains sets**

Path is consistent if it starts in S, ends in T, goes through P, and has bounds consistent with z Pos P U S U T : Possibly in the path If no consistent paths use i, then i Pos Req Pos: Required to be in the path If there is a consistent path that does not need i, then i Req Shortest path in Req lower bound on z Longest path in Pos upper bound on z == Shortest path with –ve costs Updated incrementally (and efficiently)

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**Simulated Annealing shortcut**

leverages cost estimate from Path constraint std SA: Generate sol test delta obj against uniform rand var improved SA generate random var first calc acceptable obj constrain obj Much more efficient

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**Propagation – Capacity (assume +ve qtys)**

Rooted Chain For each visit i with pi not bound For each route k in domain of ri If spare space on route k won’t allow visit i Remove k from ri Remove i from pend(k) Remove i from slast(start(k)) Free chains As above (pretty much) Before increasing chain to load L v = Count routes with free space ≥ L c = Count free chains with load ≥ L if c > v, can’t form chain Assume +ve qtys

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**Propagation – Time Time Constraints: Rooted chains: For each route k**

For each visit i in domain of slast(k) If vehicle can’t reach i before Ei Remove k from ri Remove i from slast(start(k)) Free chains: Form “implied time window” from chain Rest is as above

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**Savings Note: Clarke-Wright “Savings” method grows chains of visits**

Very successful early method Still used in many methods for an initial solution Very appropriate for CP

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**Propagation – an alternative**

Alternative relies on knowledge of “incumbent” solution Use shared data structure Can use full insertion (insert into middle of ‘chain’) CP System Var/value choice Incumbent Solution Propagators

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**Propagation Insertion ‘in the middle’**

Propagators know incumbent solution Only propagate non-binding implications e.g. don’t “set” s1 s2 here: But 1 is removed from s4 , s6 Can calculate earliest start, latest start to bound start time Can calculate feasible inserts, and update s vars appropriately Strong propagation for capacity, time constraints 1 3 4 2 7 9 8 6

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**Insertion Allows for maximum propagation**

Allows constraints to influence solution progressively e.g. Blood delivery constraints Delivery within 20 minutes of pickup As soon as one is fixed implication flows to other e.g. Mutex (mutual exclusion) constraint Either this request XOR that one Used in line-haul – deliver direct xor use a cross-dock As soon as one is used, rother = { 0 }

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** Insertion Subtours – poor man’s AllDifferent**

Can use stronger subtour elimination (Pesant) i (j) (i) j

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**Local Search Can apply std local search methods VRPs k-opt Or-opt**

exchange … Step from solution to solution, so CP is only used as rule-checker Little use of propagation But LNS is repeated insertion Makes maximum use of propagation

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**Large Neighbourhood Search revisited**

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**Large Neighbourhood Search**

Destroy & Re-create Destroy part of the solution Remove visits from the solution Re-create solution Use insert method to re-insert customers Different insert methods lead to new (better?) solutions If the solution is better, keep it Repeat

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**Large Neighbourhood Search**

Destroy part of the solution (Select method) In CP terms, this means: Relax some variable assignments In CP-VRP terms, this means Relax some successor assignments, i.e. ‘Unassign” some visits.

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**Large Neighbourhood Search**

Destroy part of the solution (Select method) Examples Remove a sequence of visits

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**Large Neighbourhood Search**

Destroy part of the solution (Select method) Examples Choose longest (worst) arc in solution Remove visits at each end Remove nearby visits Actually, choose rth worst r = n * (uniform(0,1))y y ~ 6 0.56 = 0.016 0.96 = 0.531

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**Large Neighbourhood Search**

Destroy part of the solution (Select method) Examples Dump visits from k routes (k = 1, 2, 3) Prefer routes that are close, Better yet, overlapping

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**Large Neighbourhood Search**

Destroy part of the solution (Select method) Examples Choose first visit randomly Then, remove “related” visits Based on distance, time compatibility, load

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**Large Neighbourhood Search**

Destroy part of the solution (Select method) Examples Dump visits from the smallest route Good if saving vehicles Sometimes fewer vehicles = reduced travel

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**Large Neighbourhood Search**

Destroy part of the solution (Select method) Parameter: Max to dump As a % of n? As a fixed number e.g. 100 for large problems Actual number is uniform rand (5, max)

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**Large Neighbourhood Search**

Destroy part of the solution (Select method) Examples Remove a sequence of visits Choose longest (worst) arc in solution Remove visits at each end Remove nearby visits Dump visits from k routes (k = 1, 2, 3) Choose first visit randomly Remove “related” visits Based on distance, time compatibility, load Dump visits from the smallest route (if saving vehicles) Parameter: Max to dump (as a % of n?) nxy 0.56 = 0.016 0.96 = 0.531

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**Large Neighbourhood Search**

Re-create solution Use different insert methods Insert methods differ by choice of: Which visit to insert Where to insert it Where to insert: Usually in position that increases cost by least Which to insert: Examples: Random Choose visit to insert at random Minimum insert cost Choose visit that increasess cost by the least Regret, 3-Regret, 4-Regret Choose visit with maximum difference between first and next best insert position

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Regret

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Regret

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Regret

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Regret

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Regret

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**Regret Regret = C(insert in 2nd-best route) – C(insert in best route)**

= f (2,i) – f (1,i) K-Regret = ∑k=1,K (f (k,i) – f (1,i) )

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**Large Neighbourhood Search**

If the solution is better, keep it Can use Hill-climbing Can use Simulated Annealing Can use Threshold Annealing … Repeat Can use fail limit (limit on number of infeasibilities found) Can use Restarts Can use limited number of iterations

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**Large Neighbourhood Search**

Big Advantage: We can tailor Select / Insert e.g. Routing with Equipment Some tasks need equipment Pickup /drop-off equipment at sub-depots Modelled as a PDP with special compatibility constraints Special Destroy methods Remove visits if equipment not used much Destroy routes with equip needed by unassigned visits Special Insert Methods Add multiple equips if some tasks need them Insert hardest to satisfy visits first

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**Large Neighbourhood Search**

Learning (Ropke & Pisinger) Learn which Select methods are performing well Learn which Insert methods are performing well Select according to “success” Success = New best solution New incumbent solutions Solution not previously seen

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**Advanced techniques Randomized Adaptive Decomposition**

Partition problem into subproblems Work on each subproblem in turn Decomposition ‘adapts’ Changes in response to incumbent solution 15 11 7 12 8 5 2 18 10 14 20 3 1 19 6 4 13 9 16 17

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**Advanced techniques – Re-create solution**

Complete search If subproblem is small (e.g. selected < 10 visits), use complete search Good for highly constrained problems

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**Advanced techniques – Re-create solution**

Limited Discrepancy Search Heuristic order

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Advanced techniques Limited Discrepancy Search

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Advanced techniques Limited Discrepancy Search

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Advanced techniques Limited Discrepancy Search

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Advanced techniques 2-LDS

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Advanced techniques 2-LDS

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Advanced techniques 2-LDS

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Advanced techniques 2-LDS

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Advanced techniques 2-LDS

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**Advanced techniques Limited Discrepancy Search**

Structured incomplete search Budget of discrepancies ‘Spend’ a discrepancy by going against the heuristic

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**Solving VRPs CP is “natural” for solving vehicle routing problems**

Real problems often have unique constraints Easy to change CP model to include new constraints New constraints don’t change core solve method Infrastructure for complete (completish) search in subproblems LNS is “natural” for CP Insertion leverages propagation

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**Playtime 2 Solve some VRPs using Comet**

Some differences between presentation and example: Single commodity/capacity Homogeneous fleet ‘capacity’ in model == ‘q’ here Capacity enforced by domain, not constraint Only start time modelled explicitly arrival time implicit duration fixed Successor of end visit for route k is start of route k+1 slast(k) = n + k + 1 No ‘unassigned’ route

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