The districting problem: applications and solving methods

Transcription

1 The districting problem: applications and solving methods Viviane Gascon Département des sciences de la gestion Université du Québec à Trois-Rivi Rivières 1

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

3 Main criteria Contiguity Compactness Balance or equity Respect of natural boundaries Socio-economic homogeneity A district is contiguous if it is possible to travel from any point in the district to any other in the district without having to go through any other district 3

4 Main criteria Contiguity Compactness Balance or equity Respect of natural boundaries Socio-economic homogeneity Compactness is a criterion used to prevent the formation of odd-shaped districts that is districts should be circular or square in shape rather than elongated 4

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

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

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

8 Applications 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)) 8

9 Political districting 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 9

10 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. 10

11 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 p i I i M 11

12 Political districting : Hess et al. (1965) c ij 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. 12

13 District j Population unit i Population unit j Center of population unit i Center of population unit j Center of district j 13

14 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 ij2 p j is used in the objective function of the mathematical model by Hess et al. 14

15 Political districting : Hess et al. (1965) Mathematical model Min i I j J c ij x ij Subject to j J j J a x x ij = 1, i I jj = i I p M i x ij b, j J (1) (2) (3) x ij { 0,1 }, i I j J, 15

16 Political districting : Hess et al. (1965) Mathematical model Min i I j J c ij x ij Subject to j J j J a x x ij = 1, i I jj = i I p M i x ij b, j J (1) (2) (3) Constraint (1) ensures that each population unit i is assigned to exactly one district x ij { 0,1 }, i I j J, 16

17 Political districting : Hess et al. (1965) Mathematical model Min i I j J c ij x ij Subject to j J j J a x x ij = 1, i I jj = i I p M i x ij b, j J (1) (2) (3) Constraint (2) ensures that M districts are chosen. x ij { 0,1 }, i I j J, 17

18 Political districting : Hess et al. (1965) Mathematical model Min i I j J c ij x ij Subject to j J j J a x ij x x ij = 1, i I jj = i I p M i x ij b, j J { 0,1 }, i I j J, (1) (2) (3) Constraint (3) ensures population equity among districts 18

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

20 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 20

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

22 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 22

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

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

25 Political districting : Garfinkel and Nemhauser (1970) Mathematical programming problem Minimise max st j J a ij j J x j x = 1, j = M j J i I c j x j (1) (2) (P 1 ) Constraint (1) ensures that each population unit i is assigned to exactly one district x j { 0,1 }, j J 25

26 Political districting : Garfinkel and Nemhauser (1970) Minimise max st j J a ij j J x j x = 1, j = M j J i I c j x j (1) (2) (P 1 ) Constraint (2) ensures that that M districts are chosen. x j { 0,1 }, j J 26

27 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 if units i and k have a common boundary greater than a = 0 otherwise point If a district is an undirected graph whose vertices are the units s 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. 27

28 Connected graph of district j Population unit i Population unit j Center of district j 28

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

30 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 30

31 Political districting : Garfinkel and Nemhauser (1970) Compactness : d j = distance between the units of j for district j which are farthest apart. d j = max, = d( i, k) a a, i, k = 1 { } N i k ij kj,.., (d j measures the range of the district) A(j) = area of district j c ' j = d 2 j A( j) is a dimensionless measure of the shape compactness of district j District j is feasible only if ' c j β, 0 β. 31

32 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 32

33 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) 33

34 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). 34

35 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 j J j J x j x c j x j = 1, j = M i I (1) (2) (P 2 ) x j { 0,1 }, j J 35

36 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 G will be measured by how far units in the district are from a central unit. 36

37 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 G : node u V ' such that s uj j V ' is minimized. Cost of a district with u as the center of the district is given by s uj j V ' A district is more compact when the cost is smaller. i s ui = 2 s uk = 2 k u j s uj = 2 37

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

39 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 1 if unit i is in the district y i = 0 otherwise pi i V p = is the average population of a district M 39

40 Political districting : Mehotra,, Johnson and Nemhauser (1998) SP problem Min u V p y i { S( u) } where S( u) = π min p u i p { V u} i y i p p u { V u} ( s ) n+ 1 π u + min ui π i i max { 0,1 }, i { V u} and y satisfies contiguity constraints y i 40

41 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 V sui = suj 1 and ( i, j) E} then we add the contiguity constraint y j y i i S j ensuring that node j is selected only if all nodes along some shortest path from u to j are also selected. 41

42 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 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 depthd epth-first- search strategy, to find another solution. 42

43 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 43

44 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 P = j J P ( x) j M is the average population of the district The population of a district is required to be in the interval [( 1 α ) P,(1 + α) P ] where 0 α < 1 Population equality function : j f x = ( ) pop J max { P ( x) (1 + α) P,(1 α) P P ( x),0} It evaluates the maximum deviation of the population in the district from the maximum and the minimum allowed j P j 44

45 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 : R j ( x) j J f ( comp1 x) = 2R R Compactness measure 2 : f ( comp2 x) = j J 1 2π M R A j j ( x) / π ( x) 45

46 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 S j ( x) 46

47 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 A ( x) Old and new districts Overlaying sectors 47

48 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) 48

49 Political districting : Bozkaya, Erkut and Laporte (2003) Solving method : Tabu search Objective function F( x) = α pop f pop ( x) + α comp f comp ( x) + α soc f soc ( x) + α sim f sim ( x) + α int f int ( x) 49

50 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 units until the district population attains P or when no adjacent units are 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. 50

51 Political districting : Bozkaya, Erkut and Laporte (2003) Solving method : Tabu search Type I neighbours or moves (i,j,l( 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( 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 l i k District j District l Type I Type II 51

52 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 θ iterations where θ 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. 52

53 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. 53

54 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 54

55 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 55

56 School districting problem : Ferland and Guénette (1990) Mathematical model Parameters : α φ r k j k k i = 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 = number of students of grade k (1 k K) on edge a i (1 i I ) k K) 56

57 School districting problem : Ferland and Guénette (1990) Mathematical model Variables : x ij = 1 if edge ai (1 i I) is assigned to school j (1 j J ) 0 otherwise 57

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

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

60 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 end- node of a i closer to this node w : walking distance to the school If d ij > w and xij = 1 then students on edge a i have to go to school j by bus. If d ij w then a i is within walking distance of school j. edge a i Distance between edge a i and school j School j 60

61 School districting problem : Ferland and Guénette (1990) Mathematical model Contiguity constraints : W Z B j { a A : d w and d > w, 1 l J l j} =, i ij il { a A : d w for more than one index j, j J} = 1 i ij { a A : d > w for all j, j J} = 1 i ij A = ( J W ) Z B j = 1 j 61

62 School districting problem : Ferland and Guénette (1990) Mathematical model Walking constraints : J ( W ) Z then x = only if d w For any edge a 1 Therefore if a i i j= 1 j W j then x ij = 1. ij ij 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. 62

63 School districting problem : Ferland and Guénette (1990) Mathematical model A measure to evaluate how well a solution satisfy the capacity constraints : ECM = J K I k k max 0, ri xij α jφk = = = j 1 k 1 i 1 63

64 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) 64

65 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 to the closest school j in S i s.t.. the capacity constraints are satisfied. If it is not possible then assign a i to school j in S i with smallest value ECM. 65

66 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 66

67 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. 67

68 Mathematical model Variables x ik = 1 if patient i is assigned to nurse k 0 otherwise 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 68

69 Mathematical model f i = Visit frequency (value between 0 and 1) f i = number of visits planned to patient i during a number of days during a month month s i = Parameter related to continuity of care s i = 1 if continuity of care is important to patient i 0 otherwise 69

70 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 x ik = Parameter related to follow-up 1 if patient i was assigned to nurse k = 0 otherwise during the previous month 70

71 Nurses workload A daily nurse workload is equal to where k k W = 2t + [ x f ( r + t )] k O i I ik i i i t = x k O i I ik f i t oi Daily mean traveling time from the medical public clinic to all patients assigned to nurse k t = x k i j I j i jk f i t ij Daily mean traveling time from patient i to all patients assigned to nurse k 71

72 Nurses workload Additional workload generated by assigning patient i to nurse k is defined by parameter w = f ( r + t ik i where an estimated daily mean traveling time from the previous solution s is used. i *k i ) The nurse workload W k is linearized W since w ik is a constant k k = 2t 0 + i I x ik w ik 72

73 Constraints Each patient must be assigned to exactly one nurse (or one district). ict). 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 xik C ik where C ik = 1 if patient i can be assigned to nurse k 0 otherwise 73

74 Constraints and objectives A nurse k s 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 s workload and the average workload f ( x) equ = ( q + + ) k qk k K 74

75 Constraints and objectives The daily working hours of a nurse k should not exceed T hours W k e + T, k k K we minimise the excess over T in working hours f ( x) sup = k K + e k 75

76 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 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 ( x) suiv + = f isieik k K i I I 76

77 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 1 x ik = 0 if patient i otherwise was assigned to nurse k during the previous month 77

78 Mathematical model Each of the four objective functions is weighted according to some α. Global objective function α f ( x) + α f ( x) + α f ( x) + α sup sup equ equ ( x) The problem is an integer linear programming problem with binary variables sim sim suiv f suiv 78

79 Solving method Heuristic combined with CPLEX as a subroutine since our problem is an integer linear programming problem. 79

80 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 =1 or 0) 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 80

81 Numerical tests and results Data generated randomly Planning horizon for data: one year Tests are done for one month periods Different values of α tested Four types of data: A, B, C and D, and five runs for each type of o data 81

82 A Homogenous districts of similar workloads 82

83 B Districts with greater density in the center; not necessarily similar workloads 83

84 C Greater density districts in the southwest with similar workloads 84

85 D Same density as c but workloads not similar 85

86 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 86

87 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 up,, it does not have a real impact on the solution. 87

88 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 88