CONSTRAINT: costs of having adjacent elements from different sets are minimized two compartments are adjacent if they share a common edge we assign ...
Assignment Problems with Weighted and Nonweighted Neighborhood Constraints in 36 , 44 and 63 Tilings A.A.D. Bosaing
J.F. Rabajante
M.L.D. De Lara
Institute of Mathematical Sciences and Physics University of the Philippines Los Baños
International Conference in Mathematics and Applications, 2011
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
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Introduction
The Assignment Problem
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
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Introduction
The Assignment Problem
Assignment Problem
ASSIGNMENT: elements of given finite sets should be assigned to the compartments of a finite tiling regular tilings of regular polygons in Euclidean plane (36 , 44 and 63 )
CONSTRAINT: costs of having adjacent elements from different sets are minimized two compartments are adjacent if they share a common edge we assign weights ω g and ωg to sets g and g, respectively cost of adjacency= ωg − ωg
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
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Introduction
The Assignment Problem
Assignment Problem
ASSIGNMENT: elements of given finite sets should be assigned to the compartments of a finite tiling regular tilings of regular polygons in Euclidean plane (36 , 44 and 63 )
CONSTRAINT: costs of having adjacent elements from different sets are minimized two compartments are adjacent if they share a common edge we assign weights ω g and ωg to sets g and g, respectively cost of adjacency= ωg − ωg
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
3 / 42
Introduction
The Assignment Problem
Assignment Problem
ASSIGNMENT: elements of given finite sets should be assigned to the compartments of a finite tiling regular tilings of regular polygons in Euclidean plane (36 , 44 and 63 )
CONSTRAINT: costs of having adjacent elements from different sets are minimized two compartments are adjacent if they share a common edge we assign weights ω g and ωg to sets g and g, respectively cost of adjacency= ωg − ωg
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
3 / 42
Introduction
The Assignment Problem
Assignment Problem
ASSIGNMENT: elements of given finite sets should be assigned to the compartments of a finite tiling regular tilings of regular polygons in Euclidean plane (36 , 44 and 63 )
CONSTRAINT: costs of having adjacent elements from different sets are minimized two compartments are adjacent if they share a common edge we assign weights ω g and ωg to sets g and g, respectively cost of adjacency= ωg − ωg
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
3 / 42
Introduction
The Assignment Problem
Assignment Problem
ASSIGNMENT: elements of given finite sets should be assigned to the compartments of a finite tiling regular tilings of regular polygons in Euclidean plane (36 , 44 and 63 )
CONSTRAINT: costs of having adjacent elements from different sets are minimized two compartments are adjacent if they share a common edge we assign weights ω g and ωg to sets g and g, respectively cost of adjacency= ωg − ωg
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
3 / 42
Introduction
The Assignment Problem
Assignment Problem
ASSIGNMENT: elements of given finite sets should be assigned to the compartments of a finite tiling regular tilings of regular polygons in Euclidean plane (36 , 44 and 63 )
CONSTRAINT: costs of having adjacent elements from different sets are minimized two compartments are adjacent if they share a common edge we assign weights ω g and ωg to sets g and g, respectively cost of adjacency= ωg − ωg
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
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Introduction
The Assignment Problem
Assignment Problem
Figure: Assignment Problem as Weighted Bipartite Graph Bosaing et al. (IMSP, UPLB)
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Introduction
Weighted Neighborhood Constraint
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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Introduction
Weighted Neighborhood Constraint
Weighted Neighborhood Constraint
Figure: Weighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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Introduction
Nonweighted Neighborhood Constraint
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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Introduction
Nonweighted Neighborhood Constraint
Nonweighted Neighborhood Constraint
Figure: Weighted VS Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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Introduction
Nonweighted Neighborhood Constraint
Nonweighted Neighborhood Constraint
Figure: Weighted VS Nonweighted Neighborhood Constraint
Bosaing et al. (IMSP, UPLB)
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Introduction
Parameters and Decision Variables
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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ICMA 2011
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Introduction
Parameters and Decision Variables
Weighted Neighborhood Constraint 36 Tiling
Let the binary-valued decision variables be 0, if an element from set g is not assigned to the compartment at the i−th row and j−th column xgij = 1, otherwise for i = 1, 2, . . . , r and j = 1, 2, . . . , c r is the number of rows c is the number of columns Let Ng be the number of elements in set g for g = 1, 2, . . . , k where k is the number of sets.
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
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Weighted Neighborhood Constraint
36 Tiling
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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ICMA 2011
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Weighted Neighborhood Constraint
36 Tiling
Weighted Neighborhood Constraint 36 Tiling
Figure: Starting with adjacent (column) compartment and starting with non-adjacent (column) compartment
Bosaing et al. (IMSP, UPLB)
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Weighted Neighborhood Constraint
36 Tiling
Weighted Neighborhood Constraint 36 Tiling
Figure: Graph representation
Figure: Adjacencies Bosaing et al. (IMSP, UPLB)
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Weighted Neighborhood Constraint
36 Tiling
Weighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
The integer program is Minimize k r X c−1 X X X k ωg xgij − ωg xgi(j+1) (O1) g=1 i=1 j=1 g=1
+
i=1
+
k X ωg xg(2i)(2j) − ωg xg(2i+1)(2j) (O2) g=1 g=1
dr /2e−1 dc/2e k X X X j=1
k X (O3) ω x − ω x g g g(2i−1)(2j−1) g(2i)(2j−1) g=1 g=1
br /2c bc/2c k X X X i=1
j=1
Bosaing et al. (IMSP, UPLB)
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Weighted Neighborhood Constraint
36 Tiling
Weighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
subject to Constraint 1: For i = 1, 2, . . . , r and j = 1, 2, . . . , c, k X g=1
xgij
= 0, if ij−th compartment is a dummy compartment ≤ 1, otherwise
Constraint 2: For g = 1, 2, . . . , k , r X c X
xgij = Ng
i=1 j=1
Constraint 3: For i = 1, 2, . . . , r , j = 1, 2, . . . , c and g = 1, 2, . . . , k , xgij ∈ {0, 1} Bosaing et al. (IMSP, UPLB)
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Weighted Neighborhood Constraint
36 Tiling
Weighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
The linearized objective function is Minimize c−1 r X X
br /2c bc/2c
dr /2e−1 dc/2e
αij +
i=1 j=1
X
X
i=1
j=1
β(2i)(2j) +
X X i=1
γ(2i−1)(2j−1)
j=1
subject to Constraint 1: For i = 1, 2, . . . , r and j = 1, 2, . . . , c − 1, k X g=1
Bosaing et al. (IMSP, UPLB)
ωg xgij −
k X
ωg xgi(j+1) − αij ≤ 0
g=1
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Weighted Neighborhood Constraint
36 Tiling
Weighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
continuation... Constraint 2: For i = 1, 2, . . . , r and j = 1, 2, . . . , c − 1, −
k X
ωg xgij +
k X
ωg xgi(j+1) − αij ≤ 0
g=1
g=1
Constraint 3: For i = 1, 2, . . . , dr /2e − 1 and j = 1, 2, . . . , dc/2e, k X
ωg xg(2i)(2j) −
g=1
k X
ωg xg(2i+1)(2j) − β(2i)(2j) ≤ 0
g=1
Constraint 4: For i = 1, 2, . . . , dr /2e − 1 and j = 1, 2, . . . , dc/2e, −
k X
ωg xg(2i)(2j) +
g=1 Bosaing et al. (IMSP, UPLB)
k X
ωg xg(2i+1)(2j) − β(2i)(2j) ≤ 0
g=1 Assignment Problems...
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Weighted Neighborhood Constraint
36 Tiling
Weighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
continuation... Constraint 5: For i = 1, 2, . . . , br /2c and j = 1, 2, . . . , bc/2c, k X
ωg xg(2i−1)(2j−1) −
g=1
k X
ωg xg(2i)(2j−1) − γ(2i−1)(2j−1) ≤ 0
g=1
Constraint 6: For i = 1, 2, . . . , br /2c and j = 1, 2, . . . , bc/2c, −
k X
ωg xg(2i−1)(2j−1) +
g=1
Bosaing et al. (IMSP, UPLB)
k X
ωg xg(2i)(2j−1) − γ(2i−1)(2j−1) ≤ 0
g=1
Assignment Problems...
ICMA 2011
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Weighted Neighborhood Constraint
36 Tiling
Weighted Neighborhood Constraint 36 Tiling: Starting with non-adjacent (column) compartment
The objective function of the integer program is Minimize k r X c−1 X X X k ωg xgij − ωg xgi(j+1) (O1) g=1 i=1 j=1 g=1
+
k X ωg xg(2i)(2j−1) − ωg xg(2i+1)(2j−1) (O2) g=1 g=1
dr /2e−1 dc/2e k X X X i=1
+
j=1
k X (O3) ω x − ω x g g g(2i−1)(2j) g(2i)(2j) g=1 g=1
br /2c bc/2c k X X X i=1
j=1
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
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Weighted Neighborhood Constraint
44 Tiling
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
Assignment Problems...
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Weighted Neighborhood Constraint
44 Tiling
Weighted Neighborhood Constraint 44 Tiling
The objective function of the integer program is Minimize r −1 c k k k c−1 X r X X X X X X X k ωg xgij − ωg xg(i+1)j ωg xgij − ωg xgi(j+1) + i=1 j=1 g=1 g=1 g=1 i=1 j=1 g=1
Figure: Adjacencies in square tiling Bosaing et al. (IMSP, UPLB)
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Weighted Neighborhood Constraint
63 Tiling
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
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Weighted Neighborhood Constraint
63 Tiling
Weighted Neighborhood Constraint 63 Tiling
Figure: Adjacencies in hexagonal tiling
Bosaing et al. (IMSP, UPLB)
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Weighted Neighborhood Constraint
63 Tiling
Weighted Neighborhood Constraint 63 Tiling
The objective function of the integer program is Minimize k r X c−1 X X X k ωg xgij − ωg xgi(j+1) (O1) g=1 i=1 j=1 g=1 r −1 X c X k X X k + ωg xgij − ωg xg(i+1)j (O2) i=1 j=1 g=1 g=1 r −1 X c X k X X k (O3) + ω x − ω x g g gij g(i+1)(j−1) i=1 j=2 g=1 g=1 Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
36 Tiling
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
36 Tiling
Nonweighted Neighborhood Constraint Suppose a1 , a2 , . . . , ak , b1 , b2 , . . . , bk be the dummy weights associated to the decision variables y1 , y2 , . . . , yk , z1 , z2 , . . . , zk , respectively, then define the relation ρ(O?) as ρ(O?) (|(a1 y1 + a2 y2 + · · · + ak yk ) − (b1 z1 + b2 z2 + · · · + bk zk )|) = κ1(O?) + κ2(O?) + · · · + κk (O?) where κ1(O?) = (a1 y1 + a2 y2 + · · · + ak −1 yk −1 + ak yk ) −(b1 z1 + b2 z2 + · · · + bk −1 zk −1 + bk zk ) κ2(O?) = (a2 y1 + a3 y2 + · · · + ak yk −1 + a1 yk ) −(b2 z1 + b3 z2 + · · · + bk zk −1 + b1 zk ) κ3(O?) = (a3 y1 + a4 y2 + · · · + a1 yk −1 + a2 yk ) −(b3 z1 + b4 z2 + · · · + b1 zk −1 + b2 zk ) .. . κ(k −1)(O?) = (ak −1 y1 + ak y2 + · · · + ak −3 yk −1 + ak −2 yk ) −(bk −1 z1 + bk z2 + · · · + bk −3 zk −1 + bk −2 zk ) κk (O?) = (ak y1 + a1 y2 + · · · + ak −2 yk −1 + ak −1 yk ) −(bk z1 + b1 z2 + · · · + bk −2 zk −1 + bk −1 zk ). Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
36 Tiling
Nonweighted Neighborhood Constraint For simplicity, we let ωg = g.
Figure: Circular Shift Permutation of the dummy weights
Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
36 Tiling
Nonweighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
The objective function of the integer program is Minimize X k X k ρ(O1) ωg xgij − ωg xgi(j+1) g=1 j=1 g=1
r X c−1 X i=1
dr /2e−1 dc/2e
+
X
X
i=1
j=1
br /2c bc/2c
+
X X i=1
j=1
(O1)
k k X X ρ(O2) ωg xg(2i)(2j) − ωg xg(2i+1)(2j) g=1 g=1
X k X k ρ(O3) ωg xg(2i−1)(2j−1) − ωg xg(2i)(2j−1) g=1 g=1
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
(O2)
(O3)
ICMA 2011
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Nonweighted Neighborhood Constraint
36 Tiling
Nonweighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
The linearized objective function is Minimize r X c−1 X k X
αhij +
i=1 j=1 h=1
dr /2e−1 dc/2e k X XX i=1
βh(2i)(2j) +
j=1 h=1
br /2c bc/2c k X XX i=1
γh(2i−1)(2j−1)
j=1 h=1
subject to Constraint 1: For h = 1, 2, . . . , k , i = 1, 2, . . . , r and j = 1, 2, . . . , c − 1, κh(O1) − αhij ≤ 0
Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
36 Tiling
Nonweighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
continuation... Constraint 2: For h = 1, 2, . . . , k , i = 1, 2, . . . , r and j = 1, 2, . . . , c − 1, −κh(O1) − αhij ≤ 0 Constraint 3: For h = 1, 2, . . . , k , i = 1, 2, . . . , dr /2e − 1 and j = 1, 2, . . . , dc/2e, κh(O2) − βh(2i)(2j) ≤ 0 Constraint 4: For h = 1, 2, . . . , k , i = 1, 2, . . . , dr /2e − 1 and j = 1, 2, . . . , dc/2e, −κh(O2) − βh(2i)(2j) ≤ 0
Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
36 Tiling
Nonweighted Neighborhood Constraint 36 Tiling: Starting with adjacent (column) compartment
continuation... Constraint 5: For h = 1, 2, . . . , k , i = 1, 2, . . . , br /2c and j = 1, 2, . . . , bc/2c, κh(O3) − γh(2i−1)(2j−1) ≤ 0 Constraint 6: For h = 1, 2, . . . , k , i = 1, 2, . . . , br /2c and j = 1, 2, . . . , bc/2c, −κh(O3) − γh(2i−1)(2j−1) ≤ 0
Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
36 Tiling
Nonweighted Neighborhood Constraint 36 Tiling: Starting with non-adjacent (column) compartment
The objective function of the integer program is Minimize X k X k ρ(O1) ωg xgij − ωg xgi(j+1) g=1 j=1 g=1
r X c−1 X i=1
dr /2e−1 dc/2e
+
X
X
i=1
j=1
k k X X ρ(O2) ωg xg(2i)(2j−1) − ωg xg(2i+1)(2j−1) g=1 g=1
br /2c bc/2c
+
X X i=1
j=1
(O1)
X k X k ρ(O3) ωg xg(2i−1)(2j) − ωg xg(2i)(2j) g=1 g=1
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
(O2)
(O3)
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Nonweighted Neighborhood Constraint
44 Tiling
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
44 Tiling
Nonweighted Neighborhood Constraint 44 Tiling
The objective function of the integer program is Minimize k k X X ωg xgij − ωg xgi(j+1) (O1) ρ(O1) g=1 g=1 j=1
r X c−1 X i=1
+
k k X X ρ(O2) ωg xgij − ωg xg(i+1)j (O2) g=1 j=1 g=1
r −1 X c X i=1
Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
63 Tiling
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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Nonweighted Neighborhood Constraint
63 Tiling
Nonweighted Neighborhood Constraint 63 Tiling
The objective function of the integer program is Minimize k k X X ωg xgij − ωg xgi(j+1) (O1) ρ(O1) g=1 g=1 j=1
r X c−1 X i=1
+
i=1
+
k k X X ρ(O2) ωg xgij − ωg xg(i+1)j (O2) g=1 j=1 g=1
r −1 X c X
k k X X ρ(O3) ωg xgij − ωg xg(i+1)(j−1) (O3) g=1 j=2 g=1
r −1 X c X i=1
Bosaing et al. (IMSP, UPLB)
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Illustrative Example
Weighted and Nonweighted Neighborhood Constraint
Outline 1
2
3
4
Introduction The Assignment Problem Weighted Neighborhood Constraint Nonweighted Neighborhood Constraint Parameters and Decision Variables Weighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Nonweighted Neighborhood Constraint 36 Tiling 44 Tiling 63 Tiling Illustrative Example Weighted and Nonweighted Neighborhood Constraint Bosaing et al. (IMSP, UPLB)
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Illustrative Example
Weighted and Nonweighted Neighborhood Constraint
Illustrative Example
Table: Distribution of elements per group. Group Number of elements Group 1 3 Group 2 4 Group 3 5 Assume ωg = g
Bosaing et al. (IMSP, UPLB)
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Illustrative Example
Weighted and Nonweighted Neighborhood Constraint
Weighted Neighborhood Constraint
Figure: Optimal Solutions
Bosaing et al. (IMSP, UPLB)
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Illustrative Example
Weighted and Nonweighted Neighborhood Constraint
Nonweighted Neighborhood Constraint
Figure: Optimal Solutions
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
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Appendix
References
References Taha, H.A. Operations Research: An Introduction. Prentice Hall, 2006. Diaby, M. Linear programming formulation of the vertex colouring problem. Int. J. Mathematics in Operational Research, 2(3):259–289, 2010. Esteves, R.J.P., Villadelrey, M.C. and Rabajante, J.F. Determining the Optimal Distribution of Bee Colony Locations To Avoid Overpopulation Using Mixed Integer Programming. Journal of Nature Studies, 9(1):79–82, 2010. Kaatz, F.H., Bultheel, A. and Egami, T. Order in mathematically ideal porous arrays: the regular tilings. http://nalag.cs.kuleuven.be/papers/ade/regulartiles/index.html De Lara, M.L.D. and Rabajante, J.F. Population assignments in grids with neighborhood constraint. International Industrial Engineering Conference: Research, Applications and Best Practices August 2010, Cebu, Philippines.
Bosaing et al. (IMSP, UPLB)
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Appendix
References
References Taha, H.A. Operations Research: An Introduction. Prentice Hall, 2006. Diaby, M. Linear programming formulation of the vertex colouring problem. Int. J. Mathematics in Operational Research, 2(3):259–289, 2010. Esteves, R.J.P., Villadelrey, M.C. and Rabajante, J.F. Determining the Optimal Distribution of Bee Colony Locations To Avoid Overpopulation Using Mixed Integer Programming. Journal of Nature Studies, 9(1):79–82, 2010. Kaatz, F.H., Bultheel, A. and Egami, T. Order in mathematically ideal porous arrays: the regular tilings. http://nalag.cs.kuleuven.be/papers/ade/regulartiles/index.html De Lara, M.L.D. and Rabajante, J.F. Population assignments in grids with neighborhood constraint. International Industrial Engineering Conference: Research, Applications and Best Practices August 2010, Cebu, Philippines.
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
42 / 42
Appendix
References
References Taha, H.A. Operations Research: An Introduction. Prentice Hall, 2006. Diaby, M. Linear programming formulation of the vertex colouring problem. Int. J. Mathematics in Operational Research, 2(3):259–289, 2010. Esteves, R.J.P., Villadelrey, M.C. and Rabajante, J.F. Determining the Optimal Distribution of Bee Colony Locations To Avoid Overpopulation Using Mixed Integer Programming. Journal of Nature Studies, 9(1):79–82, 2010. Kaatz, F.H., Bultheel, A. and Egami, T. Order in mathematically ideal porous arrays: the regular tilings. http://nalag.cs.kuleuven.be/papers/ade/regulartiles/index.html De Lara, M.L.D. and Rabajante, J.F. Population assignments in grids with neighborhood constraint. International Industrial Engineering Conference: Research, Applications and Best Practices August 2010, Cebu, Philippines.
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
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Appendix
References
References Taha, H.A. Operations Research: An Introduction. Prentice Hall, 2006. Diaby, M. Linear programming formulation of the vertex colouring problem. Int. J. Mathematics in Operational Research, 2(3):259–289, 2010. Esteves, R.J.P., Villadelrey, M.C. and Rabajante, J.F. Determining the Optimal Distribution of Bee Colony Locations To Avoid Overpopulation Using Mixed Integer Programming. Journal of Nature Studies, 9(1):79–82, 2010. Kaatz, F.H., Bultheel, A. and Egami, T. Order in mathematically ideal porous arrays: the regular tilings. http://nalag.cs.kuleuven.be/papers/ade/regulartiles/index.html De Lara, M.L.D. and Rabajante, J.F. Population assignments in grids with neighborhood constraint. International Industrial Engineering Conference: Research, Applications and Best Practices August 2010, Cebu, Philippines.
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
42 / 42
Appendix
References
References Taha, H.A. Operations Research: An Introduction. Prentice Hall, 2006. Diaby, M. Linear programming formulation of the vertex colouring problem. Int. J. Mathematics in Operational Research, 2(3):259–289, 2010. Esteves, R.J.P., Villadelrey, M.C. and Rabajante, J.F. Determining the Optimal Distribution of Bee Colony Locations To Avoid Overpopulation Using Mixed Integer Programming. Journal of Nature Studies, 9(1):79–82, 2010. Kaatz, F.H., Bultheel, A. and Egami, T. Order in mathematically ideal porous arrays: the regular tilings. http://nalag.cs.kuleuven.be/papers/ade/regulartiles/index.html De Lara, M.L.D. and Rabajante, J.F. Population assignments in grids with neighborhood constraint. International Industrial Engineering Conference: Research, Applications and Best Practices August 2010, Cebu, Philippines.
Bosaing et al. (IMSP, UPLB)
Assignment Problems...
ICMA 2011
42 / 42
