Chapter 7

1. Chapter 7 Part B: Locational analysis www.spatialanalysisonline.com

2. Locational analysis • Types of problem: • Planar – demand and facilities can be located anywhere in the plane • Discrete – nodes are fixed, so a discrete set of locations for demand and possibly facilities, but no network defined • Network – demand and facilities assumed to occur at network nodes (vertices); travel restricted to network – more common now www.spatialanalysisonline.com

3. Locational analysis • Location problems – key issues: • How many facilities required? • What size/type of facilities? • What objectives? e.g. cost minimisation (as in distribution from warehouses to stores) or service maximisation (ensuring every potential customer can be served within a given time/distance) • Do capacity constraints apply? – facilities, networks, delivery/collection vehicles etc www.spatialanalysisonline.com

4. Locational analysis • Location problems – further key issues: • Distance metric • Static/dynamic • Private/public • Single/multi-objective • Unique/diverse service • Elastic/inelastic demand • Deterministic/adaptive/stochastic • Hierarchical/single level • Desirable/undesirable facilities www.spatialanalysisonline.com

5. Locational analysis • P-median • Broadly, locate p facilities to service demand at n>p locations, at minimal cost • If p=1, demand locations are fixed, and the facility can be located anywhere in the plane, the solution is the MAT point, sometimes referred to as the spatial median. Find exactly by iterative algorithm • If p>1 requires more specialised methods – can be solved exactly by ‘branch and bound’ methods, but simple heuristics may be better • If demand is defined to lie at the vertices of a network, and travel must be via the network, facilities ‘will’ be located at network nodes www.spatialanalysisonline.com

6. Locational analysis • P-median – Cooper’s heuristic (Planar problem) • Randomly select p points in the MBR or the convex hull of the customer point set, V, as the initial locations for the median points • Allocate every point in V to its (Euclidean) closest median point. This partitions V into p subsets, Vp • For each of the p subsets of V, compute the MAT point using standard iterative equation • Iterate steps 2 and 3 until the change in the objective function falls below a preset tolerance level • Optionally repeat process from step 1 www.spatialanalysisonline.com

7. Locational analysis • P-median – 1 and 2 facility planar solutions www.spatialanalysisonline.com

8. Locational analysis • P-median – 1 facility planar solution, weighted demand www.spatialanalysisonline.com

9. Locational analysis • P-median – T&B heuristic (Network problems) Let V be the set of m candidate vertices, then: • Randomly select p vertices from V and call this set Q • For each vertex i in Q and each j not in Q (i.e. in the set V but not in Q) swap i and j and see if the value of the objective function is improved; if so keep this new solution as the new set Q • Iterate step 2 until no further improvements are found • Optionally repeat from step 1 www.spatialanalysisonline.com

10. Locational analysis • P-median – Other heuristics (Network problems) May commence with T&B or greedy algorithm as a starting solution, then apply improved procedure • Greedy add – very fast • Candidate list search (CLS) – very fast • Variable neighbourhood search (VNS) - fast • Lagrangian relaxation – slow, but provides upper and lower bounds on optimality www.spatialanalysisonline.com

11. Locational analysis • P-median: p=5, network problem, CLS solution Tripolis, Greece: 1358 vertices (variable demand per vertex; 2256 edges. 5 facility p-median solution. Colours show demand allocations (serviced vertices). Darker lines show network routes employed. Note – all facilities located fairly centrally www.spatialanalysisonline.com

12. Locational analysis • P-centre • Broadly, locate p facilities to service demand at n>p locations, such that the maximum distance travelled by customers is minimised • This is a form of ‘coverage’ problem – attempting to provide facilities for all customers in a manner which ensures that (i) every customer can been serviced and (ii) travel distances/times/costs are not excessive • Max travel time is often computed from (network) distance. If a maximum acceptable value is specified then may require a lot of facilities (i.e. p becomes a variable) www.spatialanalysisonline.com

13. Locational analysis • P-centre – 1 and 2 facility planar solutions www.spatialanalysisonline.com

14. Locational analysis • P-centre – 1 facility planar solution, weighted demand www.spatialanalysisonline.com

15. Locational analysis • P-centre: p=5, network problem, CLS solution Tripolis, Greece: 1358 vertices (variable demand per vertex; 2256 edges. 5 facility p-centre solution. Colours show demand allocations (serviced vertices). Darker lines show network routes employed. Note - much wider spread of facilities vs the p-median solution www.spatialanalysisonline.com

16. Locational analysis • Arc Routing • Visit all links in a network (exactly once if possible) • Applications: rubbish collection; snow clearance; door-to-door deliveries/meter reading • Variants: subset of links to be covered; variable link costs/directional constraints; capacity constraints on vehicles; preferential links www.spatialanalysisonline.com

17. Locational analysis • Arc routing • Check network for Eulerian circuit condition (degree=even if undirected) – ‘repair’ where necessary • Solve the ECP – e.g. using Fleury’s algorithm Example: solution (as map) for snow clearance (TransCAD demo dataset) www.spatialanalysisonline.com