Retail Distribution Inside a City or small Region

The Traveling Salesman Problem Given a set of cities and distances among them, find a closed tour, not visiting any city more than once, and passing through all cities Given a set of cities and distances among them, find a closed tour, not visiting any city more than once, and passing through all cities

Exemple

Heuristic n An aproximate procedure n Does not garantie optimality n Is not bad n Easy to perform

Triangle Heuristic n Start with any partial solution, e.g the convex envelop of the cities n Por each remaining point, find the triangle with a vertex on it and two other points in the partial solution, whose angle at the point is larger n Insert the new point between the oposite existing points n Use the new solution as the new partial solution n Keep going

Initial Tour

Insertion

Solution

Can you improve it?

Vehicle routing

Euclidean Distances Y calculables por una simple fórmula

Examples Straight line p = 2 Manhattan p = 1

Ejemplos Intermedias p = 1.5 Superiores p = 3

Problem Data

Angle Heuristic Assume that each vehicle has a capacity of 8 customers plus euclidean 2-distance

Angle Heuristic

Improvement Algorithm

Initial

Improvement? ?

? a b d(0,a) + d(b,0) - d(a,b)

Procedimiento (más usado) n Compute all improvements n Order in decreasing order n Keep adding n Fullfilling constraints

¿Optimization? n Too complicated! n However use aproximations n The most usual one u Assume a route for each point u The route passes through the point, ¿Shall we use it? u Is it better to assign the point to other routes? u Its an assignment problem u Let y i = 1 if you use route i 0 otherwise u Let x ji = 1 if node j is assigned to the route containing i

VILP ? Savins?d(0,j) -d(0,i)+d(i,j) i j Cost of opening route at i fixed = d(i,0)+d(0,i)

Formulation

Resumen n All construct routes n No closed tours n Need TSP n But disturbs the solution n Heuristics n Available software with many variations