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Page 1Facility 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
Page 2Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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.
Page 3Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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
Page 4Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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)
Page 5Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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
Page 6Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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
Page 7Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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
Page 8Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Examples
Other Examples:
• School districts• Territories for social facilities• Winter services• Solid waste collection• Emergency service territories• Electrical power districting
Page 9Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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.
Page 10Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Gerrymandering
Page 11Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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.
Page 12Facility 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
Page 13Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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.
Page 14Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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:
Page 15Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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).
Page 16Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Basic Model
Example
well-balanced unbalanced
Page 17Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Basic Model
Example
contiguous not contiguous
Page 18Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Basic Model
Example
compact not compact
Page 19Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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.
Page 20Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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.
Page 21Facility 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
Page 22Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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.
Page 23Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Allocation Procedure
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
Page 24Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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
Page 25Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Allocation Procedure
Example
2 4 6
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Allocation
Location
Page 26Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Allocation Procedure
Example
2 4 6
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4
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1
2
5
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9
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Allocation
Location
Page 27Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Allocation Procedure
Example
2 4 6
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5
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Allocation
Location
Page 28Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Allocation Procedure
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
Page 29Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Location Allocation Procedure
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)
Page 30Facility 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
Page 31Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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
Page 32Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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)
Page 33Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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
2 4 6
2
4
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2
5
6
9
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7
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3
10
L1
(L3 , 1, 1)
T3 T4
L3
T3
T4
L2
T1 T2
(Bl, 2)
(Br, 2)
(B, q)
T1 T2
(L2, 1, 1)
(L1, 2, 2)
(Bl, ql ) (Br, qr )
Page 34Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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 )}
Page 35Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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).
2 4 6
2
4
61
2
5
6
9
8
7
4
3
10
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)
Page 36Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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:
Page 37Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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
2 4 6
2
4
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2
5
6
9
8
7
4
3
10
α = 0ai 7 1 10 3 5 8 6 2 9 4
wi 6 4 8 4 5 5 7 2 6 3
Page 38Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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
Page 39Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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:
Page 40Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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
Page 41Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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
Page 42Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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
Page 43Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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.
Page 44Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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.
Page 45Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
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)
Page 46Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
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)
Page 47Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
Recursive Partitioning
Example: Partitioning German Zip-Code Areas into 70 Territories
2 line directions
16 line directions
Page 48Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
GIS
Integration of heuristics for territory design into the BusinessManager®an ArcView GIS extension by geomer GmbH, Heidelberg
Page 49Facility Location and Strategic Supply Chain Management
Prof. Dr. Stefan Nickel
GISGIS Integration
data
base
GIS datastructs
GUI
maps tables legendsdialogs buttons
events
scenario manager
sc.1 sc.n
heuristics
heur. 1 heur. k
parameters
I/O
GIS (ArcView)
data
base
GIS datastructs
GUI
maps tables legendsdialogs buttons
events
scenario manager
sc.1 sc.n
heuristics
heur. 1 heur. k
parameters
I/O
data
base
data
base
GIS datastructsGIS datastructs
GUI
mapsmaps tablestables legendslegendsdialogsdialogs buttons
eventsbuttonsevents
scenario manager
sc.1 sc.n
scenario manager
sc.1sc.1 sc.nsc.n
heuristics
heur. 1 heur. k
heuristics
heur. 1heur. 1 heur. kheur. k
parametersparameters
I/O
GIS (ArcView)
optim
izatio
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optim
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GIS m
ethods
GIS m
ethods