Location Theory - UNI-SB.DE WS 0809... · Sales territories • one territory ... • The activity...

Facility Location and Strategic Supply Chain Management

Prof. Dr. Stefan Nickel

Location Theory

Chapter 6 – Territory Design

Contents

• Introduction

• Basic Model

• Location Allocation Procedure

• Recursive Partitioning



Introduction

Territory Design

Territory design is the problem of grouping small geographic areas, called basic areas, into larger geographic clusters, called territories, in a way that the latter are acceptable according to relevant planning criteria.

These criteria can either be economically motivated or have a demographicbackground. Moreover, spatial restrictions, like compactness and contiguityare often demanded.

In addition, for each of the territories, often a center is sought.



Introduction

Components and Criteria for Territory Design

• atomic building blocks, the basic areas(points, lines, geographical areas)

• one- or more-dimensional activity measure associated with the basic areas(e.g. number of population, sales potential,...)

• a territory is a subset of the set of basic areasget activity measure of a territory by additive aggregation

• territory centers associated with each territory(e.g. salesman residence or office, the geographical center of the territory)

• balance: activity measure of all territories has to be approximately equal• compactness of territories: round-shape, undistorted, good accessibility• connectedness: each territory consists of one piece of land• contiguity: territories should be geographically connected



Examples

Example 1: Electoral DistrictsElectoral districts

• balanced population(“one man – one vote“)

• minority representation• compactness & contiguity

(prevent gerrymandering)• respect existing boundaries

and community integrity• political data

(e.g. to achieve socio-economichomogeneity)



Examples

Example 2: Sales Force TerritoriesSales territories

• one territory per salesman• group together small sales

coverage units• balance workload, sales

potential, ...• design compact, connected

and easily accessible territories• find locations for salesmen



promising catchment areas

Examples

Example 3: Entry into a New MarketDetermine areas to locate new branches

• a given number• restriction on radius• capture maximal market

potential



Examples

Example 4: Street Areas for Leaflet DeliveryGroup street segmentsfor leaflet delivery intoservice territories

• equal workload for delivery boys• upper bounds on number of

house-holds and time• good accessibility• connected areas



Examples

Other Examples:

• School districts• Territories for social facilities• Winter services• Solid waste collection• Emergency service territories• Electrical power districting



Gerrymandering

GerrymanderingThe word "gerrymander" is named for the Governor of Massachusetts Elbridge Gerry and is a blend of his name with the word "salamander", used to describe the shape of a tortuous electoral districtpressed through the Massachusetts legislature in 1812.Gerrymandering is a form of redistribution in which electoral district or constituency boundaries are manipulated for electoral advantage. It may be used to help or hinder particular constituents, such as members of a political, racial, linguistic, religious or class group.



Gerrymandering



Gerrymandering

How Gerrymandering can influence electoral results ona non-proportional systemThere is a state with two parties (green and red) and it has to be subdivided in 4 districts. At the beginning the green party has a 36:28 majority in the whole state :

The green wins the 3 rural/suburban districts.The result expresses and enhances the fact that G is the state-wide majority party.

The green party wins 4:0. That‘s a disproportional result considering the state-wide reality.

With classical Gerrymandering techniques it is even possible to ensure a 1:3 win to the state-wide minority, red party.

There is a 2:2 tie. This solution is closer to proportionality, but masking the fact that green is the state-wide majority party.



Location Theory


Contents

• Introduction

• Basic Model





Basic Model

A Basic Model for Territory Design• A territory design problem comprises a set (e.g.,

points, lines, or polygons) of basic areas.• Basic area represented by its center .• For each basic area , a single quantifiable attribute, called

activity measure (e.g., s are workload for serving or visiting the customers within the area, estimated sales potential, number of inhabitants) is given.

• Territories are disjoint subsets of the basic set , so that every basic area is contained in exactly one territory:

and• The activity measure or size of a territory is the sum of the activity

measures of its basic areas:• denotes the center of territory . In general, this center is

identical to one of the basic areas of the territory.



Basic Model

Criteria• Balance:

All territories should be balanced, i.e., of equal size, with respect to the activity measure of the territory.If a territory was perfectly balanced, its activity measure would be equal to the average territory size :

Because of the discrete structure of the problem, perfectly balanced territories can generally not be accomplished.Therefore, the relative percentage deviation of the territory sizes from their average size is considered:



Basic Model

• Contiguity:Two basic areas are called neighbored, if their geographical representations have a nonempty intersection.A territory is called contiguous, if the convex hull of the basic areas comprising the territory does not intersect the convex hull of the basic areas of another territory:

• Compactness:A territory is said to be geographically compact if it is somewhat round-shaped and undistorted.One way to measure the compactness is to compute the dispersion of the district area about its center ( the sum of the squared Euclidean distances from the center of the district to a finite set of „selected“ points).



Basic Model

Example

well-balanced unbalanced



Basic Model

Example

contiguous not contiguous



Basic Model

Example

compact not compact



Basic Model

ObjectiveThe objective can be informally described as follows:Partition all basic areas into a number of territories that satisfy the planning criteria of balance, compactness, contiguity, and disjointness, and locate a center within each territory.



Optimization Approaches

Optimization Approaches for Territory Design

• Location allocation approachesFormulate the territory design problem as a discrete capacitated location problem; iteratively locate territory centers and then allocate basic areas.

• Set-partitioning approachesGenerate (many) candidate territories and determine an optimal subset.

• Divisional methodsIteratively subdivide the considered region.

• ClusteringTreat each basic area initially as a single territory and iteratively merge pairs of territories together forming new and bigger territories

• Local search based heuristicsImprove an existing territory plan by successively shifting basic areas between neighboring territories with the aim of minimizing a weighted additive function of different planning criteria. Examples: Simulated Annealing, Tabu Search

• Genetic algorithmsImprove existing territory solutions by combining/shifting the fittest solutions.



Location Theory


Contents

• Introduction

• Basic Model





Location Allocation Procedure

Essentially the Location Allocation Procedure is to:

1. Guess initial centers of the territories ( Location).2. Use a tree partitioning problem to assign the basic areas to these centers

( Allocation).3. If necessary, adjust the territories such that each basic area is entirely

within one of them (split resolution).4. Compute a centroid for each territory. These are the new centers of the

territories ( Location).5. Go back to Step 2 until the solution converges.




Location Allocation ProcedureStep 1: Location

Find (new) centers of the territories:• Randomly• Based on preceding iteration• Based on a local search method• Based on the solution of a

Lagrangian subproblemStep 2: Allocation

Assign basic areas to the centers:• Allocation to closest• Based on the solution of a

capacitated network flow problem

Step 3: Stopping criterionContinue procedure until a stopping criterion is fulfilled



Location Allocation ProcedureFrom a Basic Solution to an Allocation: Tree Partitioning

basic edge

Subset of basic edges is a spanning tree.

To yield an allocation: solve a tree partitioningproblem

- partition into subtrees- each subtree contains exactly one center- balance activity measure of subtrees




Example

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Allocation

Location




Allocation - A classical IP for assigning basic areas to centers (Hess et. al., 1965)

set τ = 0

relax integrality... this yields a classical transportation problem




Why isn’t it useful to solve IP directly?

• The running time of solver like CPLEX depends very much on theproblem parameters

• It is not applicable to problems with many thousands of basic areas

Therefore:• Solve the transportation problem and round the fractional variables

optimally (split resolution)



Location Theory


Contents

• Introduction

• Basic Model





Recursive Partitioning

Recursive Partitioning Approach

Recursively subdivide the problem geometrically using lines into smaller and smaller subproblems, until an elemental level is reached where we can efficiently solve the territory design problem.

V Vl Vr Vll

Vlr

Vrl

Vrr




Line PartitionsDefinition

PP = (B, q) is called partition problem if B ⊆ V and 1 ≤ q ≤ p.

PP is trivial, if q = 1 ⇒ B is a territory

DefinitionLP = (Bl , Br , ql , qr ) is called line partition of PP = (B, q), if1. Bl ∪ Br = B and Bl ∩Br = ∅ and

∃ line L: Bl = B∩H≤(L) and Br = B∩H>(L)2. 1 ≤ ql, qr ≤ q and ql + qr = q

PPl = (Bl , ql ) and PPr = (Br , qr ): left and right sub-problem of PP = (B, q)PP: father problem of PPl and PPr

Alternatively: LP = (L, ql , qr ), with L being a line.

L

(B, q)Bl

Br

(Bl, ql)

(Br, qr)




ExampleB = {1, …, 10} and q = 4.

Line partition LP = (Bl , Br , ql , qr ) of PP = (B, q):ql = qr = 2 and L1 = { y = 3.5 } ⇒ Bl = {1, 3, 5, 7, 10}, Br = {2, 4, 6, 8, 9}

Recursive partitioning resembles a binary tree⇒ Partition tree

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(L3 , 1, 1)

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(Bl, 2)

(Br, 2)

(B, q)

T1 T2

(L2, 1, 1)

(L1, 2, 2)

(Bl, ql ) (Br, qr )




Balance of a Line PartitionBalance of a

• partition problem PP = (B, q): bal(PP) = bal(B, q) = | w(B) / q – μ | / μ

• line partition LP = (Bl , Br , ql , qr ): bal(LP) = max { bal(Bl, ql ), bal(Br, qr ) }

• territory layout TL = {T1, …, Tp }: bal(TL) = maxj=1,…,p bal(Tj, 1)

Note: bal(V, p) = 0.

PropositionLet LP = (Bl , Br , ql , qr ) be a line partition of PP = (B, q).Then:

bal(PP) ≤ bal(LP) = max {bal(Bl, ql ), bal(Br, qr )}




ExampleWeights:

For LP = (L, 2, 2):

Determine all line partitions for a given line L = (•, α), 0 ≤ α < 2πFor α = π / 2: L = vertical line⇒ sort basic areas in B by non-decreasing x-coordinate xi.

For α ≠ π / 2: rotate coordinate system such that line is vertical⇒ sort basic areas in B by non-decreasing value xi sin(α) – yi cos(α).

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L

i 1 2 3 4 5 6 7 8 9 10

wi 4 2 4 3 5 7 6 5 6 8(Bl, 2)

(Br, 2)



Recursive PartitioningDenote sorted sequence of basic areas: a1, …, aM

All non-trivial line partitions for L: where 1≤ k ≤ M and Bl

k = {a1, …, ak }, Brk = B \ Bl

k

Well balanced line partition for L:

Determine index k’ such that

and let index k* such that

Then choose line partition:




ExampleFor ql = qr = 2 and α = 0

Sorted sequence:

Then:

⇒ k’ = 4 and k* = 5.

PropositionGiven a partition problem (B, q), angle α, and ql, qrThen:

where wmax = maxi∈B wi

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α = 0ai 7 1 10 3 5 8 6 2 9 4

wi 6 4 8 4 5 5 7 2 6 3




ExampleFor ql = qr = 2:

For • α = 0: and• α = π / 4: ⇒ the bound is tight

How to choose ql and qr ?To minimize the right hand side, choose ql and qr as

•

•

CorollaryLet PP and PP' be two partition problems, and PP be the father problem of PP'. If PP' = (B’, q’) is generated by a line partition LP(k*) of PP, then




TheoremGiven the territory design problem (V, p), where p = 2s, s ≥ 1. Then:

where wmax = maxi∈V wi

Sketch of Proof:Let PPi = (B, q) be a partition problem at level 0 ≤ i ≤ s of the binary partition tree.Then, q = p/2i = 2s-i.If PPi-1 is the father problem of PPi, then

For i = s, the height of the partition tree:



Recursive PartitioningExample

We obtain:

For L1, L2, and L3 as before:

RemarkWe even have: This bound is tight.

TheoremGiven the territory design problem (V, p), where 2s < p < 2s+1, s ≥ 1.

Then:

Sketch of Proof:Using q ≥ 2s-i we obtain:

Problem: the height of the partition tree is now s+1.

j 1 2 3 4

bal(Tj, 1) 0.04 0.12 0.04 0.12




Contiguity of a Line PartitionRecall: Territory Tj is connected if ch( Tj ) ∩ V \ Tj = ∅.

PropositionLet (Bl, Br, ql , qr ) be a line partition of (B, q). Then: ch(Bl ) ∩ ch(Br ) = ∅.

Compactness of a Line PartitionEvaluate compactness indirectly by means of the line partitions.

Idea:

orV




Possible measures for the compactness of a line partition:

length of intersection length of stripewith convex hull

Measure: cp(LP) = l2(c1, c2 ).

Vl

Vr

length ofintersection

convex hull

εc2

c1

c2c1




The Successive Dichotomies HeuristicTwo questions:

• How do we perform the partitioning of a problem into subproblems?• How do we explore the partition tree?

Before:Upper U and lower bound L on the size of the territories:

where τ is the maximal allowed deviation.Partition problem PP = (B, q) is called feasible, if: L ≤ w(B) / q ≤ U.




Subdividing a Partition Problem Two decisions for a partition problem PP = (B, q)

1. Select numbers ql, qr with ql + qr = q and2. Select a line L partitioning the point-set B.

Concerning 1.Set •

•

Concerning 2.Consider a fixed number K of line directions: αi := π i / K, i = 1, …, K.Compute LP(k*) for all αi, i = 1, …, K, and all possible values of ql and qr .Insert all feasible line partitions in a set FLP.



Recursive Partitioning Rank all line partitions in LP ∈ FLP in terms of balance and compactness:

where • β is a weighting factor, 0 ≤ β ≤ 1, and • balmax and cpmax is the max. balance and compactness of a line partition in FLP

Sort the partitions in FLP in nondecreasing order of their ranking value.Implement the highest ranked partition

Complexity of these two steps: O(K |B| log |B| + K log K)




LP1 , LP2 , …, LPt

Exploring the Partition Tree Greedily choosing the highest ranked partition for each partition problem may still result in infeasible sub-problems further down in the partition tree.

⇒ Integrate backtracking mechanism to revise decisions:if one of the sub-problems of a partition problem PP is infeasible, choose a different line partition for PP.

LP1 , LP2 , …, LPt

LP1 , LP2 , …, LPt

LP1 , LP2 , …, LPt

LP1 , LP2 , …, LPtLP1 , LP2 , …, LPt

LP1 , LP2 , …, LPt

(Bll, qll ) (Blr, qlr )

(Br, qr )(Bl, ql )

FLP = ∅

(B, q)




Example: Partitioning German Zip-Code Areas into 70 Territories

2 line directions

16 line directions



GIS

Integration of heuristics for territory design into the BusinessManager®an ArcView GIS extension by geomer GmbH, Heidelberg



GISGIS Integration

data

base

GIS datastructs

GUI

maps tables legendsdialogs buttons

events

scenario manager

sc.1 sc.n

heuristics

heur. 1 heur. k

parameters

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maps tables legendsdialogs buttons

events

scenario manager

sc.1 sc.n

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heur. 1 heur. k

parameters

I/O

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data

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GIS datastructsGIS datastructs

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mapsmaps tablestables legendslegendsdialogsdialogs buttons

eventsbuttonsevents

scenario manager

sc.1 sc.n

scenario manager

sc.1sc.1 sc.nsc.n

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heur. 1heur. 1 heur. kheur. k

parametersparameters

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Location Theory - UNI-SB.DE WS 0809... · Sales territories • one territory ... • The activity...

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Transcript of Location Theory - UNI-SB.DE WS 0809... · Sales territories • one territory ... • The activity...