一个基于妥协解的多目标线性规划分类模型

∗

1 2

1 100049

2 100190

(MCLP)

MCLP

MCLP

A compromise-based MCLP classification model

Bo Wang1Yong Shi2

(1Graduate University of Chinese Academy of Sciences,Beijing100049)

(2CAS Research Center On Ficitious Economy and Data Science,Beijing100190)

Abstract Although multiple criteria linear programming can deal with classification problem

successfully,an original MCLP model has some difficulties in choosing parameters.To overcome

the problems,compromise-based MCLP model is proposed to offer a good promotion of the

original one.In the latter model,there are also two deviations for every single point;that is,

interior deviation and exterior deviation.Similar to the original MCLP model,for each point,we

want at least one of the deviations to be zero.In addition to modeling work,this paper also gives

a proof of the existence of the parameter selection condition.

Keywords MCLP Interior deviation Exterior deviation Compromise solution

∗ ( 70921061( ),90718042( )) BHP Billiton Co.,Australia

1 (MCLP)

1.1

Ned Freed Fred Glover (Goal Programming,GP) (Multiple Criteria Linear Programming,MCLP)

MCLP

1.1.

1: MCLP

αi βi

∑

min

αi

i

∑

βi

max

i

s.t.A i X=b+αi−βi,A i∈G1,

A i X=b+αi−βi,A i∈G2,

αi,βi 0,i=1,2,...,l

1.2

(Multi-objective Programming,MOP) MOP

max g(x)=(g1(x),g2(x),...,g p(x))

over x∈X.

Pareto Pareto

g i(x) g

i

,i=1,2,...,p.

min P(α,β)

s.t.g i(x)−g

i

=αi−βi,i=1,2,...,p,

αi,βi 0,i=1,2,...,p,

x∈X.

Sawaragi

1.1. α,β R p P(α,β) α β αi βi α∗ β∗

α∗i β∗

i

=0,i=1,2,...,p.

1.

min

p

∑

i=1

a iαi−

p

∑

i=1

b iβi

s.t.g i(x)−g

i

=αi−βi,i=1,2,...,p,

αi,βi 0,i=1,2,...,p,

x∈X.

a i=

b i

min

p

∑

i=1

a i(g i(x)−g

i

) s.t.x∈X.

2. a i>b i,∀i

p

∑i=1a iαi−

p

∑

i=1

b iβi=

p

∑

i=1

b i(αi−βi)+

p

∑

i=1

(a i−b i)αi.

α β a i>b i,∀i P(α,β) α

α∗i β∗

i

=0,i=1,2,...,p.

α∗iβ∗i=0,i=1,2,...,p

1.3

1

n X∗ A i n A i X∗ X∗ l b αi βi α=(α1,α2,...,αl) β=(β1,β2,...,βl) X∗ {A1,A2,...,A l} l α·β=0.

2

2.1

MCLP Shi Yu(1989)

−∑l

i=1

αi α∗ α∗ 0 β∗ 0

∑l

i=1

βi

α∗+∑l

i=1

αi d−α d+α

d−

α

−d+α=α∗+

l

∑

i=1

αi,d−

α

+d+

α

=|α∗+

l

∑

i=1

αi|.

β∗−∑l

i=1

βi d−β d+β

d−

β

−d+

β

=β∗−

l

∑

i=1

βi,d−

β

+d+

β

=|β∗−

l

∑

i=1

βi|.

α∗,β∗ −∑l

i=1

αi

∑l

i=1

βi

|α∗+∑l

i=1

αi| |β∗−

∑l

i=1

βi|

d−

α

+d+

α

+d−

β

+d+

β

.

MCLP

min d−

α+d+

α

+d−

β

+d+

β

s.t.d−

α−d+α=α∗+

l

∑

i=1

αi,