www.spatialanalysisonline.com chapter 7 part b: locational analysis

www.spatialanalysisonline.com

Chapter 7

Part B: Locational analysis

3rd edition 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


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


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


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


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


Locational analysis

P-median – 1 and 2 facility planar solutions


Locational analysis

P-median – 1 facility planar solution, weighted demand


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


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


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


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)


Locational analysis

P-centre – 1 and 2 facility planar solutions


Locational analysis

P-centre – 1 facility planar solution, weighted demand


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


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


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 chapter 7 part b: locational analysis

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p facilities

locational analysis

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p vertices

p subsets of v

p points

network defined network

weighted demand slide