University of Göttingen

:Introduction :The Proposed Algorithm :Bicriteria Location Problems :Conclusions 

On the Solution of Multicriteria Continuous Location 

Problems using the Big Square Small Square Method 

Daniel Scholz 

Institute for Numerical and Applied Mathematics, University of Göttingen 

joint work with Anita Schöbel 

Institute for Numerical and Applied Mathematics, University of Göttingen 

September 18, 2008 

EWGLA XVII, Elche (Alicante), Spain 

Daniel Scholz University of Göttingen September 18, 2008 1 / 26


Outline 

1. Introduction 

1.1 Motivation 

1.2 Definitions and Notations 

1.3 Epsilon Efficency 

2. The proposed Algorithm 

2.1 The BSSS Method 

2.2 Our Algorithm 

2.3 The Output Set 

3. Bicriteria Location Problems 

3.1 Semi-Obnoxious Problem 

3.2 Minimax and Maximin 

3.3 Numerical Results 

4. Conclusions 

4.1 Extensions 

4.2 References 



:Motivation :Definitions and Notations :Epsilon Efficency 

A Semi-obnoxious Location Problem 

Figure: Where to locate a new waste dump? 














Notations 

Notation 1 

A multicriteria optimization problem can be formulated as 

vec min 

x∈X f (x) = (f1(x), . . . , fp(x)), 

where X ⊂ R n and fi(x) : R n → R. Define Y := f (X ) ⊂ R p . 

Notation 2 

Let x = (x1, . . . , xp) and y = (y1, . . . , yp) be two vectors in R p . 

1. We write x ≦ y if and only if xi ≤ yi for i = 1, . . . , p. 

2. We write x ≤ y if and only if xi ≤ yi for i = 1, . . . , p and x �= y. 

3. We write x < y if and only if xi < yi for i = 1, . . . , p. 

Define R p 

≥ := {x ∈ Rp : x ≥ 0}. 




Efficient Solutions 

Definition 1 (Ehrgott 2000) 

A solution y ∈ X is called efficient if there is no x ∈ X satisfying 

f (x) ≤ f (y). 

Denote by XE ⊂ X the set of all efficient solutions and YN = f (XE ). 




ε-efficient Solutions 

Definition 2 (Loridan 1984; White 1986) 

A solution y ∈ X is called ε-efficient, ε ∈ R p >, if there is no x ∈ X 

satisfying 

f (x) + ε ≤ f (y). 

Denote by X ε E the set of all ε-efficient solutions and Y ε N = f (X ε E ). 




The Definition of ε-efficient Solutions 

Figure: For the definition of ε-efficient solutions. 



:The BSSS Method :The Multicriteria BSSS Method :The Output Set 

The General Idea of the BSSS Method 

Figure: The general idea of the BSSS algorithm. 





























The General Idea Multicriteria BSSS Method 

Figure: The general idea of the multicriteria BSSS algorithm. 
























The Big Square Small Square Method for Multicriteria Optimization 




Properties of the Output Set (1) 

Figure: All efficient solutions are contained in the output set. 




Properties of the Output Set (2) 

Figure: All points in the output set are ε-efficient solutions. 



:Semi-Obnoxious Location Problem :Maximin and Minimax Problem :Numerical Results 

A Semi-Obnoxious Location Problem 

Consider n existing locations ak = (ak,1, ak,2) on the plane with positive 

weights wk, vk > 0 for k = 1, . . . , n. We want to find one new facility 

x = (x1, x2) with respect to the following criteria. 

Minimize the sum of service costs from the locations to the new facility: 

min 

x∈X f1(x) = 

n� 

wk · d(ak, x). 

k=1 

Minimize the sum of the reciprocal squared distance from the locations to 

the new facility: 

min 

x∈X f2(x) 

n� vk 

= 

. 

d(ak, x) 2 

k=1 

See Brimberg and Juel (1998) and Skriver and Anderson (2003). 




Example of a Semi-Obnoxius Location Problem (1) 

We adopted the input data from Brimberg and Juel (1998). 

For ε = (ε1, ε2) we chose 

k a 1 k a 2 k wk vk 

1 0.20 0.80 5.0 1.0 

2 0.72 0.32 7.0 1.0 

3 0.88 0.64 2.0 1.0 

4 0.56 0.68 3.0 1.0 

5 0.28 0.08 6.0 1.0 

6 0.20 0.60 1.0 1.0 

7 0.48 0.16 5.0 1.0 

Table: The input data for the example instance. 

ε1 = 0.02 · (f1(x ∗ 2 ) − f1(x ∗ 1 )) and ε2 = 0.02 · (f2(x ∗ 1 ) − f2(x ∗ 2 )). 




Example of a Semi-Obnoxius Location Problem (2) 




The Bicriteria Maximin and Minimax Problem 

Minimize the distance from the new facility to the furthest location: 

min 

x∈X f1(x) = max 

k=1,...,n wk · d(ak, x). 

Maximize the distance between the new facility and the nearest resident: 

Therefore, the bicriteria problem is 

max 

x∈X f2(x) = min 

k=1,...,n vk · d(ak, x). 

vec min 

x∈X (f1(x), −f2(x)). 

See Ohsawa (2000) and Ohsawa and Tamura (2003). 




Example for the Bicriteria Maximin and Minimax Problem 




Computational Results for the Semi-Obnoxious Location Problem 

Technical Information 

Implemented in JAVA, using double precision arithmetic, run on a 1.3 GHz 

computer. 

Input Data 

We generated 10 ≤ n ≤ 10, 000 existing locations randomly in [0, 1] 2 and 

weights randomly in [0, 1]. 

Ten problems were run for various values of n for every problem. 

For ε = (ε1, ε2) we chose 

ε1 = 0.05 · (f1(x ∗ 2 ) − f1(x ∗ 1 )) and ε2 = 0.05 · (f2(x ∗ 1 ) − f2(x ∗ 2 )) 

throughout all problems. 




Computational Results for the Semi-Obnoxious Location Problem 

Run time (sec.) Iterations 

n Min Max Ave. Min Max Ave. 

10 0.02 0.85 0.33 442 2,238 1,374.8 

20 0.10 3.64 1.02 870 3,494 1,925.7 

50 0.32 17.82 3.92 1,371 7,798 3,345.6 

100 0.63 22.30 6.07 1,914 7,880 4,246.8 

200 0.77 29.53 9.59 1,289 10,166 5,370.7 

500 1.61 51.23 14.74 1,818 11,850 5,741.3 

1, 000 8.96 41.26 22.26 3,954 10,804 7,409.9 

2, 000 31.47 54.77 42.91 7,703 11,591 9,450.3 

5, 000 98.23 198.13 155.86 10,654 17,593 15,054.1 

10, 000 252.34 456.46 339.15 14,280 20,198 17,680.5 

Table: Results for the semi-obnoxious location problem for 10 runs using random 

input. 




Computational Results for the Maximin and Minimax Problem 

Run time (sec.) Iterations 

n Min Max Ave. Min Max Ave. 

10 0.03 0.22 0.10 450 1,082 797.6 

20 0.07 1.52 0.61 517 2,665 1,428.6 

50 0.07 3.94 1.48 511 4,271 2,365.9 

100 0.11 2.59 1.43 663 4,090 2,705.7 

200 0.49 8.66 2.52 1,459 4,976 3,037.3 

500 2.76 37.60 11.91 2,956 11,674 6,049.5 

1, 000 9.43 36.93 18.39 4,450 10,493 6,769.3 

2, 000 12.48 36.97 27.00 3,374 8,822 6,702.0 

5, 000 83.09 134.84 105.41 8,938 13,445 11,094.2 

10, 000 189.68 373.32 266.95 12,196 20,818 15,839.5 

Table: Results for the maximin and minimax location problem for 10 runs using 

random input. 



:Extensions :References 

Further Ideas 

Some Extensions and Further Ideas 

1. The method is also applicable for objectives with more than two 

variables with just slightly changes. 

2. Approximating all ε-efficient solutions. 

3. Combining the BSSS approach with other multicriteria scalarization 

method, e.g. the ε-constrained method. 




Thank you! 




References 




References (1) 

J. Brimberg, H. Juel. (1998): 

A bicriteria model for locating a semi-desirable facility in the plane. 

European Journal of Operational Research, 106: 144–151. 

Z. Drezner, A. Suzuki (2004): 

The Big Trinangle Small Triangle Method for the Solution of Nonconvex 

Facility Location Problems. 

Operations Research, 52: 128–135. 

M. Ehrgott (2000): 

Multicriteria optimization. 

Springer Verlag, Berlin, 1st edition. 

H.W. Hamacher, S. Nickel (1996): 

Multicriteria planar location problems. 






P. Hansen, D. Peeters, D. Richard, J.F. Thisse (1985): 

The Minisum and Minimax Location Problems Revisited. 

Operations Research, 33: 1251–1265. 

K. Ichida, Y. Fujii (1990): 

Multicriterion Optimization Using Interval Analysis. 

Computing, 44: 47–57. 

P. Loridan (1984): 

ε-Solutions in Vector Minimizatin Problems. 

Journal of Optimization Theory and Applications, 43: 265–276. 

Y. Ohsawa (2000): 

Bicriteria Euclidean Location Associated with Maximin and Minimax Criteria. 

Naval Research Logistics, 47: 581–592. 





Y. Ohsawa, K. Tamura (2003): 

Efficient Location for a Semi-Obnoxious Facility. 

Annals of Operations Research, 123: 173–188. 

F. Plastria (1992): 

GBSSS: The generalized big square small square method for planar 

single-facility location. 


A.J.V. Skriver and K.A. Anderson (2003): 

The bicriterion semi-obnoxious location (BSL) problem solved by an 

ε-approximation. 


D.J. White (1986): 

Epsilon Efficiency. 

Journal of Optimization Theory and Applications, 49: 319–337.

University of Göttingen

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