Vietnam, March 2010
Viviane Gascon
Département des sciences de la gestion
Université du Québec à Trois-Rivières
1

The districting problem consists in partitioning a geographical region into districts in order to plan some operations while considering different criteria or constraints.
Vietnam, March 2010
2

 Contiguity
 Compactness
 Balance or equity
A district is contiguous if it is

 Socio-economic homogeneity district without having to go through any other district
Vietnam, March 2010
3

 Contiguity
 Compactness
 Balance or equity

 Socio-economic homogeneity shaped districts that is districts should be circular or square in shape rather than elongated
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4

 Contiguity
 Compactness
 Balance or equity
 Respect of natural boundaries
 Socio-economic homogeneity
Balanced in workload or in population in the districts
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5

 Contiguity
 Compactness
 Balance or equity
 Respect of natural boundaries
 Socio-economic homogeneity
Rivers, railroads, mountains, administrative boundaries, etc.
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6





Contiguity
Compactness
Having a better representation of residents who share common concerns or views (can be
Balance or equity based on income revenues, minorities, etc.)
Respect of natural boundaries
 Socio-economic homogeneity
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7

 Political districting
(Hess and Weaver (1965), Garfinkel and Nemhauser (1970), Mehotra, Johnson and
Nemhauser (1998), Bozkaya, Erkut and Laporte (2002))
 School districting
(Ferland and Guénette (1990))
 Districting for health services
(Gascon, Gorvan and Michelon (2010))
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8

The political districting problem consists in partitioning an area into electoral constituencies
(districts), each one being assigned a number of representatives.
 one representative is assigned to each district;
 each population unit is assigned to one district;
 the number of districts is usually known ( M districts);
 all districts must have approximately the same number of voters for better equity
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9

Political districting : Hess et al. (1965)
 Among the first mathematical programming approach of the political districting problem.
 The problem is modeled as an assignment problem with additional constraints where each population unit must be assigned to a district center.
Vietnam, March 2010
10

Political districting : Hess et al. (1965)
Mathematical model
 Parameters:
I : set of population units
J : set of potential district centers
M : number of district centers p i
: population of the i th population unit a : minimum population allowed for a district b : maximum population allowed for a district a and b can be considered as deviations from the average population of all population units which is given by
 i

I p i
M
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11

Political districting : Hess et al. (1965) c ij
, the cost of assigning population unit i to district center j is the
Euclidean distance between the district center i and the district center j .
d ij
: distance between the centers of population units i and j .
Minimizing the Euclidean distance between population units favours contiguous districts but do not guarantee them.
Vietnam, March 2010
12

Population unit j
Center of population unit j
District j
Population unit i
Center of population unit i
Center of district j
13
Vietnam, March 2010

Political districting : Hess et al. (1965)
Mathematical model
 Variable: x ij

1 if population unit i is assigned to district center j
0 otherwise i

I , j

J c ij
= d ij
2 p j is used in the objective function of the mathematical model by Hess et al.
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14

Political districting : Hess et al. (1965)
Mathematical model
Min
 i

I
 j

J c ij x ij
Subject to
 j

J
 j

J x x ij jj a
  i

I


1 , i
M p i x ij


I b , j

J x ij

 
, i

I , j

J
(1)
(2)
(3)
Vietnam, March 2010
15

Political districting : Hess et al. (1965)
Mathematical model
Min
 i

I
 j

J c ij x ij
Subject to
 j

J
 j

J x x ij jj a
  i

I


1 , i
M p i x ij


I b , j

J x ij

 
, i

I , j

J
(1)
(2)
(3)
Constraint (1) ensures that each population unit i is assigned to exactly one district
Vietnam, March 2010
16

Political districting : Hess et al. (1965)
Mathematical model
Min
 i

I
 j

J c ij x ij
Subject to
 j

J
 j

J x x ij jj a
  i

I


1 , i
M p i x ij


I b , j

J x ij

 
, i

I , j

J
(1)
(2)
(3)
Constraint (2) ensures that M districts are chosen.
Vietnam, March 2010
17

Political districting : Hess et al. (1965)
Mathematical model
Min
 i

I
 j

J c ij x ij
Subject to
 j

J
 j

J x x ij jj a
  i

I


1 , i
M p i x ij


I b , j

J x ij

 
, i

I , j

J
(1)
(2)
(3)
Constraint (3) ensures population equity among districts
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18

Political districting : Hess et al. (1965)
Solving method : heuristic
1.
2.
3.
4.
Define district centers
Assign population equally to the district centers at minimum costs (with a transportation algorithm)
Adjust assignment so that each population unit is entirely within one district
Compute centroids and use them as improved district centers
5.
6.
Repeat from step 2 until solution converges
Try with other initial district centers
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19

Political districting : Hess et al. (1965)
Limits of the solving method



No guaranty of convergence
Non contiguous solutions must be rejected
If many solutions, choose the most compact one and one having a good population equity by always verifying that there is no deviation form the minimum and maximum allowable population
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20

Political districting : Garfinkel and
Nemhauser (1970)
Garfinkel and Nemhauser (1970) considers predefined districts to be specified and among which the final districts are chosen.
Vietnam, March 2010
21

Political districting : Garfinkel and
Nemhauser (1970)
Mathematical model
 Parameters:
I : set of population units
J : set of potential districts
M : number of district p i
: population of the i th population unit a ij

1 if population unit i belongs to district j
0 otherwise
P(j) : population of district j where P ( j )
  i

I a ij p i
Vietnam, March 2010
22

Political districting : Garfinkel and
Nemhauser (1970)
Mathematical model
 Parameters: c j

P ( j )

 p p deviation of population of district j from the average population, p
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23

Political districting : Garfinkel and
Nemhauser (1970)
Mathematical model
 Variable x j

1 if district j is chosen
0 otherwise
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24

Political districting : Garfinkel and
Nemhauser (1970)
Mathematical programming problem
Minimise max j

J c j x j st  j

J a ij x j

1 , i

I
 j

J x j

M x j

 
, j

J
(1)
(2)
(P
1
)
Constraint (1) ensures that each population unit i is assigned to exactly one district
Vietnam, March 2010
25

Political districting : Garfinkel and
Nemhauser (1970)
Minimise max j

J st
 j

J a ij x j
 j

J x j

1 , i

M

I c j x j x j

 
, j

J
(1)
(2)
(P
1
)
Constraint (2) ensures that that M districts are chosen.
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26

Political districting : Garfinkel and
Nemhauser (1970)
The problem implies that potential districts must be defined.
 Contiguity : Let B = { b ik
}, a symmetric matrix where b ik



1
0 if units i otherwise and k have a common boundary greater th an a point
If a district is an undirected graph whose vertices are the units of the district, an arc exists between vertices i and k if and only if b ik
= 1.
A district is contiguous if and only if the graph is connected (a path exists between every pair of vertices).
A district is feasible only if it is contiguous.
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Connected graph of district j
Population unit i
Population unit j
Center of district j
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28

Political districting : Garfinkel and
Nemhauser (1970)
 A district is feasible only if
P ( j )
 p
  p , where 100

( 0
  
1 ) is the maximum allowable percentage deviation of the population of a district from the average district population.
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29

Political districting : Garfinkel and
Nemhauser (1970)
 Compactness : d(i,k) = distance between units i and k .
e(i,k) = exclusion distance between units i and k .
District j is feasible only if d(i,k) > e(i,k) implies that a ij
. a kj
= 0
( i and j can not be in the same district if the distance between them is higher than e(i,k) ) i d(i,k) e(i,k) k
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30

Political districting : Garfinkel and
Nemhauser (1970)
 Compactness : d j
= distance between the units of apart.
j for district j which are farthest d j
= max i , k

 d ( i , k ) a ij a kj

, i , k

1 ,.., N
( d j measures the “ range” of the district)
A(j) = area of district j c
' j
 d j
2
A ( j ) is a dimensionless measure of the shape compactness of district j
District j is feasible only if c
' j
 
, 0
   
.
31
Vietnam, March 2010

Political districting : Garfinkel and
Nemhauser (1970)
Solving method : two phase method
1) Phase I: Find feasible districts
Start at an arbitrary unit and adjoin contiguous units until the combined population becomes feasible.
If the district is compact, keep it.
If combined population exceeds the upper limit, backtrack on the enumeration tree.
It is verified if the district has some enclaves.
District with an enclave
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32

Political districting : Garfinkel and
Nemhauser (1970)
Solving method : two phase method
2) Phase II:
Solve the mathematical programming problem (search tree algorithm)
(see paper for more details)
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33

Political districting : Mehotra, Johnson and
Nemhauser (1998)
The problem considered by Mehotra et al.
(1998) is similar to the problem in Garfinkel and Nemhauser (1970).
But their model considers more potential districts.
They consider a graph partitioning problem where



A node is associated to every population unit (its weight is equal to the corresponding population)
An edge connects two nodes when the corresponding population units are neighbours
A solution is a connected graph (for contiguity) for which the sum of the node weights is within a population interval (for population equity).
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34

Political districting : Mehotra, Johnson and
Nemhauser (1998)
Same model as Garfinkel and Nemhauser (1970) except for c j which is the cost of district j .
The question is : how should c j be defined ?
Min
 j

J a ij x
 j

J c
 j j x j
1 , i

I
(1)
 j

J x j

M (2) x j

 
, j

J
(P
2
)
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35

Political districting : Mehotra, Johnson and
Nemhauser (1998)
The cost of district j , c j
, measures its non compactness.
V: set of population units
E: edges connecting units if they share common borders
G(V,E): graph
G’(V’,E’): connected subgraph defining a district and satisfying population limits
Non compactness of G’ will be measured by how far units in the district are from a central unit.
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36

Political districting : Mehotra, Johnson and
Nemhauser (1998) s ij
: number of edges in a shortest path from i to j in G.
Center of G’ : node u

V ' such that
 j

V ' s uj is minimized.
Cost of a district with u as the center of the district is given by
 j

V ' s uj
A district is more compact when the cost is smaller. i s ui
= 2 u j s uj
= 2 s uk
= 2 k
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37

Political districting : Mehotra, Johnson and
Nemhauser (1998)
Solving method : column generation method
1)
2)
3)
3)
Start with a subset of feasible districts, J’
Solve the linear relaxation of (P
2
) restricted to J’ where
This linear relaxation of (P
2
) is LP-P
2
(J’).
0
 x j

1
The optimal solution of the linear relaxation of (P
2
) is feasible to
LP-P
2
(J). A dual value p i is obtained for each constraint in LP-P
2
(J).
Determine if the optimal solution of LP-P
2
(J’) is optimal for
LP-P
2
(J). This is done by solving a subproblem SP.
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38

Political districting : Mehotra, Johnson and
Nemhauser (1998)
Solving method : column generation method
Parameters for SP : p i
: population of unit i p min
, p max
: lower and upper bounds on the population of a district y i
 p


1 if unit i is in the

 0 otherwise
 i

V p i district is the average population of a district
M
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39

Political districting : Mehotra, Johnson and
Nemhauser (1998)
SP problem
Min u

V where p min y i


S ( u )

S ( u )
  p n

1
 p u
 min
 i


V
 u

 s ui
 p i
 y i

0 , p u
1
  i


V
 u

V
 p i u y
 i
 p max and y
 p u satisfies contiguity constraint s
Vietnam, March 2010
40

Political districting : Mehotra, Johnson and
Nemhauser (1998)
Contiguity constraints
To ensure contiguity of districts, districts are required to be subtrees of a shortest path tree rooted at u (district center).
Constraints allowing district j to be selected only if at least one of the nodes that is adjacent to it and closer to u is also selected, are added, that is
If S j
 i
 constraint

V | s ui
 s uj

1 and ( i , j )

E
 then we add the contiguity y j
  i

S j y i ensuring that node j is selected only if all nodes along some shortest path from u to j are also selected.
Vietnam, March 2010
41

Political districting : Mehotra, Johnson and
Nemhauser (1998)
If the optimal objective value of SP is negative then a district with minimum value is added to the set J’ and LP-P
2
(J’) is solved again.
Otherwise, the current solution to LP-P
2
(J’) is also optimal to LP-P
2
(J).
In this case, if the solution is integral, then a solution to P
2 is found.
If it is not integral, a branching rule is applied, based on a depth-firstsearch strategy, to find another solution.
Vietnam, March 2010
42

Political districting : Bozkaya, Erkut and
Laporte (2003)
The political districting problem solved by Bozkaya et al.
(2003) considers the contiguity constraint as a hard constraint and all other criteria as soft constraints through a weighted objective function.
Other criteria :
 population equality
 compactness
 socio-economic homogeneity
 similar districts to the existing districts
 integrity of communities
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43

Political districting : Bozkaya, Erkut and
Laporte (2003)
Population equality :
J : set of all districts in solution x (feasible or not)
P j
(x) : population of district j in solution x
 j

J
P j
( x ) is the average population of the district
P

M
The population of a district is required to be in the interval

( 1
 
) P , ( 1
 
) P

where 0
  
1
Population equality function :

 f pop
( x )

 j

J max

P j
( x )

( 1
 
) P , ( 1
 
) P
P

P j
( x ), 0
 

It evaluates the maximum deviation of the population in the district from the maximum and the minimum allowed
Vietnam, March 2010
44

Political districting : Bozkaya, Erkut and
Laporte (2003)
Compactness : two measures
R : perimeter of the whole territory, used for scaling
R j
(x) : perimeter of district j in solution x
Compactness measure 1 : f comp 1
( x )

 j

J
R j
( x )

R
2 R
Compactness measure 2 : f comp 2
( x )

 j

J


1

2 p
M
A j
( x ) / p
R j
( x )


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45

Political districting : Bozkaya, Erkut and
Laporte (2003)
Socio-economic homogeneity : minimize the sum of the standard deviation of income
S j
(x) : standard deviation of income in district j
S : average income
Socio-economic homogeneity function: f soc
( x )

 j

J
S j
( x )
S
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46

Political districting : Bozkaya, Erkut and
Laporte (2003)
Similar districts to the existing districts :
O j
(x) : largest overlay of district j with a district contained in a solution x
A : entire area
Similarity objective function: f sim

1

 j

J
O j
( x )
A
Old and new districts
Overlaying sectors
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47

Political districting : Bozkaya, Erkut and
Laporte (2003)
Integrity of communities :
G j
(x) : largest population of a given community in district j of solution x
Integrity of communities objective function : minimize f int

1

 j

J
 j

J
G
P j j
( x )
( x )
Vietnam, March 2010
48

Political districting : Bozkaya, Erkut and
Laporte (2003)
Solving method : Tabu search
Objective function
F ( x )
  pop f pop
( x )
  comp f comp
( x )
  soc f soc
( x )
  sim f sim
( x )
  int f int
( x )
49
Vietnam, March 2010

Political districting : Bozkaya, Erkut and
Laporte (2003)
Solving method : Tabu search
Initial solution : select a seed unit for a district and add to it adjacent available.
If the number of districts created is larger than M , reduce it by merging the least populated unit with the least populated neighbour.
If the number of districts created is less than M , gradually increase it by iteratively splitting the most populated district into two while preserving contiguity.
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50

Political districting : Bozkaya, Erkut and
Laporte (2003)
Solving method : Tabu search
Type I neighbours or moves ( i,j,l ) : all solutions that can be obtained from x by moving a basic unit i from its current district j to a neighbour district l without creating a non-contiguous solution.
Type II neighbours or moves ( i,k,j,l ) : all solutions that can be obtained from x by swapping two border units i and k between their respective districts j and l without creating a non-contiguous solution.
i
District j District j i k
District l District l
Type I Type II
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51

Political districting : Bozkaya, Erkut and
Laporte (2003)
Solving method : Tabu search
Preventing cycling : for both types of moves, a move which puts unit i back into district j or unit k back into district i is said to be tabu for q iterations where q is chosen randomly in an interval.
Diversification : by adding a penalty term to the objective function value associated to the frequently performed moves.
Adaptive memory procedure : keep in a pool of solutions a set of districts belonging to some of the best solutions. Disjoint districts can be chosen form the pool and used as a basis for a new population with a higher probability.
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52

School districting problem : Ferland and
Guénette (1990)
The school districting problem consists in determining the groups of students attending each school of a school board located over a given territory.
Ferland and Guénette (1990) propose a decision support system to solve the problem.
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53

School districting problem : Ferland and
Guénette (1990)
Different constraints must be taken into account :


School capacity
Class capacity


Contiguity of school sectors
Keep students in the same school from year to year
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54

School districting problem : Ferland and
Guénette (1990)
Mathematical model
 Parameters :
G(N,A) : road network for the school board
N is the set of nodes defined as street intersections and school locations
A is the set of edges defined as the street segments.
A

A is a subset of edges with students located on it
I : number of edges in A
K : number of grades
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55

School districting problem : Ferland and
Guénette (1990)
Mathematical model
 Parameters :
 j k
 k
 number of classes of grade k (1
 k

K ) available at school j (1
 j

J )
 upper bound on the number of students in a class grade k (1
 k

K ) r i k  number of students of grade k (1
 k

K ) on edge a i
(1
 i

I )
56
Vietnam, March 2010

School districting problem : Ferland and
Guénette (1990)
Mathematical model
 Variables : x ij


1 if edge a i
( 1

 0 otherwise
 i

I ) is assigned to school j ( 1
 j

J )
Vietnam, March 2010
57

School districting problem : Ferland and
Guénette (1990)
Mathematical model
 Constraints : j
J 

1 x ij

1 , 1
 i

I (1) i
I 

1 r i k x ij
  j k
 k
, 1
 k

K , 1
 j

J (2)
Constraint (1) ensures that each edge i is assigned to exactly one school
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58

School districting problem : Ferland and
Guénette (1990)
Mathematical model
 Constraints : j
J 

1 x ij

1 , 1
 i

I (1) i
I 

1 r i k x ij
  j k
 k
, 1
 k

K , 1
 j

J (2)
Constraint (2) ensures that the capacity of each school for each grade is not exceeded
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59

School districting problem : Ferland and
Guénette (1990)
Mathematical model
 Contiguity constraints : distance is needed d ij
: distance between edge a i and school j
(distance between the node where school j is located and the endnode of a i closer to this node w : walking distance to the school
If then students on edge a i by bus. d ij
 w a i have to go to school is within walking distance of school j . j
Distance between edge a i and school j edge a i
School j
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60

School districting problem : Ferland and
Guénette (1990)
Mathematical model
 Contiguity constraints :
W j

 a i

A : d ij
 w and d il
 w , 1
 l

J , l
 j

Z

 a i

A : d ij
 w for more than one index j , 1

B

 a i

A : d ij
 w for all j , 1
 j

J
 j

J

A


 J j

1
W j

 Z  B
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61

School districting problem : Ferland and
Guénette (1990)
Mathematical model
 Walking constraints :
For any edge a i


 J j

1
W j

 Z then x ij

1 only if d ij
 w
Therefore if a i

W j then x ij

1 .
 Edges in Z should be assigned to their closest school (if capacity constraints can be satisfied) and priority should be given to edges closer to their closest school.
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School districting problem : Ferland and
Guénette (1990)
Mathematical model
 A measure to evaluate how well a solution satisfy the capacity constraints :
ECM
 j
J 

1 k
K 

1 max 0 , i
I 

1 r i k x ij
  j k
 k
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School districting problem : Ferland and
Guénette (1990)
Assignment process
 Procedure W -edges : If a i

W j then x ij

1
 Procedure Z -edges : order edges in Z in decreasing order of their distance to their closest school.
Assign each edge a i belonging to Z to the closest school j s.t.
d ij
 w and the capacity constraint is satisfied.
If it is not possible, assign a i to the closest school (even if some capacity constraints are not satisfied)
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School districting problem : Ferland and
Guénette (1990)
Assignment process
 Procedure B -edges : order edges in B in increasing order of their distance to their closest school.
Treat each edge a i belonging to B and determine S i
, the set of schools to which the edges adjacent to a i are assigned.
If S i is empty then S i
= S , set of all schools.
Assign a i satisfied.
to the closest school j in S i s.t. the capacity constraints are
If it is not possible then assign a i
ECM.
to school j in S i with smallest value
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Districting for a public medical clinic :
Gascon, Gorvan and Michelon (2010)
The territory covered by the public medical clinic is divided into districts



Each district is assigned to a given number of nurses
Each nurse is assigned to a given district
A nurse is usually assigned to a short list of patients as a follow-up nurse
The list of patients to visit varies from day to day:
 it becomes difficult to balance nurse workloads
 it becomes difficult to account for the continuity of care requirements
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Districting for a public medical clinic
The districting problem of the medical public clinic consists in determining new districts, that is, new paring of patients with nurses in such a way that nurses’ workloads do not vary much from one nurse to the other and that the same follow-up nurse is assigned to a patient, if possible.
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Mathematical model
 Variables x ik



1
0 if patient otherwise i is assigned to nurse k
 Parameters
T = Length of a working day t ij
= Traveling time from patient i to patient j t oj
= Traveling time from the public medical clinic ( o ) to patient j r i
= Time required to complete treatment to patient i
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Mathematical model f i
= Visit frequency (value between 0 and 1) f i

i
s i
= Parameter related to continuity of care s i



i
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Mathematical model p ik
= Proportion of visits made by nurse k to patient i p ik
 number of visits to patient i by nurse k during the previous month total number of visits to patient i during the previous month x ik
= Parameter related to follow-up x ik

1 if patient i was assigned to nurse k during the previous month
0 otherwise
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Nurses’ workload
 A daily nurse workload is equal to
W k

t
O k
  i

I
x ik f i
r i
 t i k
where t
O k
  i

I x ik f i t oi t i k   j j


I i x jk f i t ij
Daily mean traveling time from the medical public clinic to all patients assigned to nurse k
Daily mean traveling time from patient i to all patients assigned to nurse k
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Nurses’ workload
 Additional workload generated by assigning patient i to nurse k is defined by parameter w ik
 f i
( r i
 t i
* k
) where an estimated daily mean traveling time from the previous solution is used.
 The nurse workload W k is linearized
W k

2 t
0 k   i

I x ik w ik since w ik is a constant
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Constraints
 Each patient must be assigned to exactly one nurse (or one district).
k


K x ik

1 ,
 i

I
 A patient i can be assigned to nurse k only if he is close to her sector that is if C ik
= 1 x ik

C ik where C ik

1 if patient i can be assigned to nurse k
0 otherwise
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Constraints and objectives
 A nurse k
’s workload should be close to the average workload of all nurses.
W k

 k

K
W k
K
 q k
  q k

,
 k

K we minimise the gap between nurse k
’s workload and the average workload f equ
( x )
  k

K
 q
 k
 q k


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Constraints and objectives
 The daily working hours of a nurse k should not exceed T hours
W k
 e k
 
T ,
 k

K we minimise the excess over T in working hours f ( x ) sup
  k

K e k

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Constraints and objectives
 A patient i should be assigned to the nurse k who visited him most frequently in the previous period according to the value of p ik x ik
 p ik
  e ik
  e ik
 k

K
 i

I if following-up a patient i is essential ( s i
=1), we minimise an objective function where patient i should be assigned to the nurse k who visited him most frequently in the last period, weighted according to the frequency f i
.
f suiv
( x )
  k

K i

I f i s i e ik

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Vietnam, March 2010

Another objective f sim
( x )
 k


K i

I f i x ik

1
 x ik
 is minimised to avoid moving too many patients from one nurse to the other where x ik



1
0 if patient i otherwise
was assigned to nurse k during the previous month
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Mathematical model


Each of the four objective functions is weighted according to some

.
Global objective function
 sup f sup
( x )
  equ f equ
( x )
  sim f sim
( x )
  suiv f suiv
( x )
 The problem is an integer linear programming problem with binary variables
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Solving method
 Heuristic combined with CPLEX as a subroutine since our problem is an integer linear programming problem.
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Main steps of the algorithm
1) Update the list of patients to visit
2) Assign each new patient to nurse k whose district is the closest
3) Determine the list of patients who can be moved from one nurse’s district to another and the districts where they could be moved ( C ik or 0)
=1
4) Solve the sub problem with CPLEX
5) Repeat steps 3) and 4) while the solution changes or a given number of iterations is not reached
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




Numerical tests and results
Data generated randomly
Planning horizon for data: one year
Tests are done for one month periods
Different values of
 tested
Four types of data: A, B, C and D, and five runs for each type of data
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A – Homogenous districts of similar workloads
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B – Districts with greater density in the center; not necessarily similar workloads
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C – Greater density districts in the southwest with similar workloads
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D – Same density as c but workloads not similar
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Results
 For type A and C data , initial solutions being of good quality, computational time was low:

 few patients were moved from one district to another workloads are similar for all nurses
 For type B and D data , initial solutions being of poor quality, computational time was higher:

 overtime but limited to values acceptable different workloads among nurses but differences are acceptable
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Results
 In general, when a higher priority is given to balancing workloads , there is a greater gap between the initial and the final solution.
 When a higher priority is given to reducing movements of patients from one district to another, it produces a solution with more overtime and differences in workloads.
 When a higher priority is given to maintaining follow-up , it does not have a real impact on the solution.
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Conclusion
 Solutions generally have similar workloads which is very important to nurses
 To be tested on real data
 To be tested on daily data and to determine routes for those data
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