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Assignment Problems with Weighted andNonweighted 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 PhysicsUniversity of the Philippines Los Baños

International Conference in Mathematics and Applications, 2011

Bosaing et al. (IMSP, UPLB) Assignment Problems... ICMA 2011 1 / 42

Introduction The Assignment Problem

Outline1 Introduction

The Assignment ProblemWeighted Neighborhood ConstraintNonweighted Neighborhood ConstraintParameters and Decision Variables

2 Weighted Neighborhood Constraint36 Tiling44 Tiling63 Tiling

3 Nonweighted Neighborhood Constraint36 Tiling44 Tiling63 Tiling

4 Illustrative ExampleWeighted and Nonweighted Neighborhood Constraint

Assignment Problem

ASSIGNMENT: elements of given finite sets should be assignedto 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 differentsets are minimized

two compartments are adjacent if they share a common edgewe assign weights ωg and ωg to sets g and g, respectivelycost of adjacency=

∣∣ωg − ωg∣∣

Assignment Problem

∣∣ωg − ωg∣∣

Assignment Problem

∣∣ωg − ωg∣∣

Assignment Problem

∣∣ωg − ωg∣∣

Assignment Problem

∣∣ωg − ωg∣∣

Assignment Problem

∣∣ωg − ωg∣∣

Assignment Problem

Figure: Assignment Problem as Weighted Bipartite Graph

Introduction Weighted Neighborhood Constraint

Weighted Neighborhood Constraint

Figure: Weighted Neighborhood Constraint

Introduction Nonweighted Neighborhood Constraint

Nonweighted Neighborhood Constraint

Figure: Weighted VS Nonweighted Neighborhood Constraint

Introduction Parameters and Decision Variables

Weighted Neighborhood Constraint36 Tiling

Let the binary-valued decision variables be

xgij =

0, if an element from set g is not assigned to the

compartment at the i−th row and j−th column1, otherwise

for i = 1,2, . . . , r and j = 1,2, . . . , cr is the number of rowsc is the number of columns

Let Ng be the number of elements in set g for g = 1,2, . . . , k where kis the number of sets.

Weighted Neighborhood Constraint 36 Tiling

Figure: Starting with adjacent (column) compartment and starting withnon-adjacent (column) compartment

Figure: Graph representation

Figure: Adjacencies

Weighted Neighborhood Constraint36 Tiling: Starting with adjacent (column) compartment

The integer program is

Minimize

r∑i=1

c−1∑j=1

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxgi(j+1)

∣∣∣∣∣∣ (O1)

dr/2e−1∑i=1

dc/2e∑j=1

∣∣∣∣∣∣k∑

ωgxg(2i)(2j) −k∑

ωgxg(2i+1)(2j)

∣∣∣∣∣∣ (O2)

br/2c∑i=1

bc/2c∑j=1

∣∣∣∣∣∣k∑

ωgxg(2i−1)(2j−1) −k∑

ωgxg(2i)(2j−1)

∣∣∣∣∣∣ (O3)

subject to

Constraint 1: For i = 1,2, . . . , r and j = 1,2, . . . , c,

k∑g=1

{= 0, if ij−th compartment is a dummy compartment≤ 1, otherwise

Constraint 2: For g = 1,2, . . . , k ,

r∑i=1

c∑j=1

xgij = Ng

Constraint 3: For i = 1,2, . . . , r , j = 1,2, . . . , c and g = 1,2, . . . , k ,

xgij ∈ {0,1}

The linearized objective function is

Minimize

r∑i=1

c−1∑j=1

αij +

dr/2e−1∑i=1

dc/2e∑j=1

β(2i)(2j) +

br/2c∑i=1

bc/2c∑j=1

γ(2i−1)(2j−1)

subject to

Constraint 1: For i = 1,2, . . . , r and j = 1,2, . . . , c − 1,

k∑g=1

ωgxgij −k∑

ωgxgi(j+1) − αij ≤ 0

continuation...

Constraint 2: For i = 1,2, . . . , r and j = 1,2, . . . , c − 1,

−k∑

ωgxgij +k∑

ωgxgi(j+1) − αij ≤ 0

Constraint 3: For i = 1,2, . . . , dr/2e − 1 and j = 1,2, . . . , dc/2e,k∑

ωgxg(2i+1)(2j) − β(2i)(2j) ≤ 0

Constraint 4: For i = 1,2, . . . , dr/2e − 1 and j = 1,2, . . . , dc/2e,

−k∑

ωgxg(2i)(2j) +k∑

ωgxg(2i+1)(2j) − β(2i)(2j) ≤ 0

continuation...

Constraint 5: For i = 1,2, . . . , br/2c and j = 1,2, . . . , bc/2c,

k∑g=1

ωgxg(2i−1)(2j−1) −k∑

ωgxg(2i)(2j−1) − γ(2i−1)(2j−1) ≤ 0

Constraint 6: For i = 1,2, . . . , br/2c and j = 1,2, . . . , bc/2c,

−k∑

ωgxg(2i−1)(2j−1) +k∑

ωgxg(2i)(2j−1) − γ(2i−1)(2j−1) ≤ 0

Weighted Neighborhood Constraint36 Tiling: Starting with non-adjacent (column) compartment

The objective function of the integer program is

Minimize

r∑i=1

c−1∑j=1

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxgi(j+1)

∣∣∣∣∣∣ (O1)

dr/2e−1∑i=1

dc/2e∑j=1

∣∣∣∣∣∣k∑

ωgxg(2i)(2j−1) −k∑

ωgxg(2i+1)(2j−1)

∣∣∣∣∣∣ (O2)

br/2c∑i=1

bc/2c∑j=1

∣∣∣∣∣∣k∑

ωgxg(2i−1)(2j) −k∑

ωgxg(2i)(2j)

∣∣∣∣∣∣ (O3)

Minimize

r∑i=1

c−1∑j=1

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxgi(j+1)

∣∣∣∣∣∣+r−1∑i=1

c∑j=1

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxg(i+1)j

∣∣∣∣∣∣

Figure: Adjacencies in square tiling

Figure: Adjacencies in hexagonal tiling

Minimize

r∑i=1

c−1∑j=1

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxgi(j+1)

∣∣∣∣∣∣ (O1)

+r−1∑i=1

c∑j=1

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxg(i+1)j

∣∣∣∣∣∣ (O2)

+r−1∑i=1

c∑j=2

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxg(i+1)(j−1)

∣∣∣∣∣∣ (O3)

Nonweighted Neighborhood Constraint 36 Tiling

Nonweighted Neighborhood ConstraintSuppose a1, a2, . . . , ak , b1, b2, . . . , bk be the dummy weights associated to thedecision variables y1, y2, . . . , yk , z1, z2, . . . , zk , respectively, then define the relationρ(O?) as

ρ(O?)(|(a1y1 + a2y2 + · · ·+ ak yk )− (b1z1 + b2z2 + · · ·+ bk zk )|)=

∣∣κ1(O?)

∣∣+ ∣∣κ2(O?)

∣∣+ · · ·+ ∣∣κk(O?)

∣∣where

κ1(O?) = (a1y1 + a2y2 + · · ·+ ak−1yk−1 + ak yk )−(b1z1 + b2z2 + · · ·+ bk−1zk−1 + bk zk )

κ2(O?) = (a2y1 + a3y2 + · · ·+ ak yk−1 + a1yk )−(b2z1 + b3z2 + · · ·+ bk zk−1 + b1zk )

κ3(O?) = (a3y1 + a4y2 + · · ·+ a1yk−1 + a2yk )−(b3z1 + b4z2 + · · ·+ b1zk−1 + b2zk )

...κ(k−1)(O?) = (ak−1y1 + ak y2 + · · ·+ ak−3yk−1 + ak−2yk )

−(bk−1z1 + bk z2 + · · ·+ bk−3zk−1 + bk−2zk )κk(O?) = (ak y1 + a1y2 + · · ·+ ak−2yk−1 + ak−1yk )

−(bk z1 + b1z2 + · · ·+ bk−2zk−1 + bk−1zk ).

For simplicity, we let ωg = g.

Figure: Circular Shift Permutation of the dummy weights

Nonweighted Neighborhood Constraint36 Tiling: Starting with adjacent (column) compartment

Minimize

r∑i=1

c−1∑j=1

ρ(O1)

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxgi(j+1)

∣∣∣∣∣∣ (O1)

dr/2e−1∑i=1

dc/2e∑j=1

ρ(O2)

∣∣∣∣∣∣k∑

ωgxg(2i+1)(2j)

∣∣∣∣∣∣ (O2)

br/2c∑i=1

bc/2c∑j=1

ρ(O3)

∣∣∣∣∣∣k∑

ωgxg(2i−1)(2j−1) −k∑

ωgxg(2i)(2j−1)

∣∣∣∣∣∣ (O3)

The linearized objective function is

Minimize

r∑i=1

c−1∑j=1

k∑h=1

αhij +

dr/2e−1∑i=1

dc/2e∑j=1

k∑h=1

βh(2i)(2j) +

br/2c∑i=1

bc/2c∑j=1

k∑h=1

γh(2i−1)(2j−1)

subject to

Constraint 1: For h = 1,2, . . . , k , i = 1,2, . . . , r and j = 1,2, . . . , c − 1,

κh(O1) − αhij ≤ 0

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 andj = 1,2, . . . , dc/2e,

κh(O2) − βh(2i)(2j) ≤ 0

Constraint 4: For h = 1,2, . . . , k , i = 1,2, . . . , dr/2e − 1 andj = 1,2, . . . , dc/2e,

−κh(O2) − βh(2i)(2j) ≤ 0

continuation...

Constraint 5: For h = 1,2, . . . , k , i = 1,2, . . . , br/2c andj = 1,2, . . . , bc/2c,

κh(O3) − γh(2i−1)(2j−1) ≤ 0

Constraint 6: For h = 1,2, . . . , k , i = 1,2, . . . , br/2c andj = 1,2, . . . , bc/2c,

−κh(O3) − γh(2i−1)(2j−1) ≤ 0

Nonweighted Neighborhood Constraint36 Tiling: Starting with non-adjacent (column) compartment

Minimize

r∑i=1

c−1∑j=1

ρ(O1)

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxgi(j+1)

∣∣∣∣∣∣ (O1)

dr/2e−1∑i=1

dc/2e∑j=1

ρ(O2)

∣∣∣∣∣∣k∑

ωgxg(2i)(2j−1) −k∑

ωgxg(2i+1)(2j−1)

∣∣∣∣∣∣ (O2)

br/2c∑i=1

bc/2c∑j=1

ρ(O3)

∣∣∣∣∣∣k∑

ωgxg(2i−1)(2j) −k∑

ωgxg(2i)(2j)

∣∣∣∣∣∣ (O3)

Nonweighted Neighborhood Constraint44 Tiling

Minimize

r∑i=1

c−1∑j=1

ρ(O1)

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxgi(j+1)

∣∣∣∣∣∣ (O1)

+r−1∑i=1

c∑j=1

ρ(O2)

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxg(i+1)j

∣∣∣∣∣∣ (O2)

Nonweighted Neighborhood Constraint63 Tiling

Minimize

r∑i=1

c−1∑j=1

ρ(O1)

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxgi(j+1)

∣∣∣∣∣∣ (O1)

+r−1∑i=1

c∑j=1

ρ(O2)

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxg(i+1)j

∣∣∣∣∣∣ (O2)

+r−1∑i=1

c∑j=2

ρ(O3)

∣∣∣∣∣∣k∑

ωgxgij −k∑

ωgxg(i+1)(j−1)

∣∣∣∣∣∣ (O3)

Illustrative Example Weighted and Nonweighted Neighborhood Constraint

Illustrative Example

Table: Distribution of elements per group.Group Number of elements

Group 1 3Group 2 4Group 3 5

Assume ωg = g

Weighted Neighborhood Constraint

Figure: Optimal Solutions

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 inOperational 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 OverpopulationUsing 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 IndustrialEngineering Conference: Research, Applications and Best Practices August 2010, Cebu,Philippines.

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