Basic concepts • Goal Programming • Reference Point · 1 Bulgarian Academy of Sciences.22 July,...

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Bulgarian Academy of Sciences. 22 July, 2008

Index

• Basic concepts• Goal Programming• Reference Point

End

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•A paper production firm elaborates:

- cellulose pulp obtained by mechanical means.

- cellulose pulp obtained by chemical means.

•Maximum production capacities: 300 and 200 mt/day.

•Each ton demands a working day. The firm has a staff of 400 workers.

•Gross margin per ton:

- mechanical means : 1.000 m.u.

- chemical means : 3.000 m.u.

•Cover fixed costs (300.000 m.u./day).

Basic concepts

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Objectives of the firm:

•Maximize the gross margin (economical objective)

•Minimize the hazard in the river where the factory pours the production (environmental objective).

Biologic oxygen demands in the water of the river:

- Mechanical means: 1 ut/mt,

- Chemical means: 2 ut/mt.

Basic concepts

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The multiobjective model

•Decision variables: x1 tons/day mechanical means,

x2 tons/day chemical means.

•Constraints: x1 300,

x2 200, (production capacities),

x1 + x2 400, (employment),

1.000x1 + 3.000x2 300.000 (cover costs)

Basic concepts

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•Objectives (criteria):

Maximize 1.000x1 + 3.000x2 (gross margin),

Minimize x1 + 2x2 (biologic oxygen demand).

000.300000.3000.1400

200300s.t.2min

000.3000.1max

21

21

2

1

21

21

xxxx

xx

xxxx

Basic concepts

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Efficiency

• x* is an efficient solution (Pareto optimal) of the problem, if there is not any feasible solution y such that fi(y) ≤ fi(x*) (i = 1,…, p), with some fj(y) < fj(x*).

•x* is a weakly efficient solution (weak Pareto optimal) of the problem, if there is not any feasible solution y such that fi(y) < fi(x*), (i = 1,…, p).

Basic concepts

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Ideal values

Xf

P ii x

xs.t.

)(Opt)(

Optimal sol: x(i). Ideal value i: fi* = fi(x(i))

Basic concepts

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Payoff Matrix

fp* = fp(x(p))…f2(x(p))f1(x(p))…………

fp(x(2))…f2* = f2(x(2))f1(x(2))fp(x(1))…f2(x(1))f1* = f1(x(1))

f1* f2* fp*

Anti-ideals: worst value per column.

Approximation to nadir value.

Basic concepts

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200300.000Biologic O2demand

600800.000Gross margin

Biologic O2demand

Gross margin

Basic concepts

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• Decision space:

nj

n RmjgRX ,,1,,0)(/ xx

• Objective space:

pp RXRX xxfzzf ),(/)(

Basic concepts

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(1)

(2)

(3)

(4)

Decision space

Basic concepts

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Objective space

Basic concepts

13


Efficient Set

Basic concepts

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Basic conceptsEfficient Set

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Goal Programming

• Building a goal

Objective function fj(x)

Target value tj

Deviation variables

Undesired deviation variables

fj(x) ≤ tjfj(x) ≥ tjfj(x) = tj

fj(x) + nj – pj = tj

Negative: how short we fall from the target value

Positive: how long we fall

pj nj pj + nj

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Achievement function

h(nj, pj) = pj

nj

pj + nj

Associated optimization problem

Xtpnf

pnh

jjjj

jj

xx)(s.t.

),(min If h*(nj, pj) = 0, the goal is satisfied.

If h*(nj, pj) > 0, the goal is not satisfied.

Goal Programming

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The decision maker gives a goal for each objective:

Associated optimization problem

Xpjtpnf

h

jjjj

xxpn

,,1,)(s.t.),(min

fj(x) ≤

≥= tj, j = 1, …, p

Goal Programming

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•Satisfying solution: satisfies all the goals.

•Non-satisfying solution: does not satisfy some goal

•A satisfying solution may not be efficient

•Depending on the form of the achievement function h, there are different goal programming variants:

•Weighted,

•Minmax,

•Lexicographic.

Goal Programming

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In our example, let us consider the following goals:

- Pollution ≤ 300

- Gross margin ≥ 400.000

000.400000.3000.13002

000.300000.3000.1400

200300s.t.

),(min

22

11

21

pnyxpnyxyx

yxyx

nph

Goal Programming

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x + 2y ≤ 3001.000x + 3.000y ≥ 400.000

Satisfying solutions

Efficient and satisfying solutions

Goal Programming

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f1 ≥ 400.000

f2 ≥ -300

Goal Programming

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Let us now consider the following goals:

- Pollution ≤ 300


000.500000.3000.13002

000.300000.3000.1400

200300s.t.

),(min

22

11

21

pnyxpnyxyx

yxyx

nph

Goal Programming

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x + 2y ≤ 300 1.000x + 3.000y ≥ 500.000

There are no satisfying solutions

Goal Programming

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f1 ≥ 500.000

f2 ≥ -300

Goal Programming

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Weighted goal programming:

The DM gives a weight for each goal j, j = 1, …, p

jj

j

jp

j j

j

npnp

th

1),( pn

Goal Programming

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Example. Goals (equal weights: 1 = 2 = 1)

- Pollution ≤ 300


000.400000.3000.13002

000.300000.3000.1400

200300s.t.

000.400300min

22

11

21

pnyxpnyxyx

yxyx

np

Goal Programming

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Goal Programming

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Goal Programming

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Goal Programming

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- Pollution ≤ 300


000.500000.3000.13002

000.300000.3000.1400

200300s.t.

000.500300min

22

11

21

pnyxpnyxyx

yxyx

np

Goal Programming

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Goal Programming

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Goal Programming

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Goal Programming

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Minmax goal programming:

The DM gives a weight for each goal j, j = 1, …, p

jj

j

j

j

j

pjnp

np

th

,,1max),(

pn

Goal Programming

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- Pollution ≤ 300


000.500000.3000.13002

000.300000.3000.1400

200300s.t.

000.500,

300maxmin

22

11

21

pnyxpnyxyx

yxyx

np

Goal Programming

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dn

dppnyx

pnyxyx

yxyxd

000.500

300

000.500000.3000.13002

000.300000.3000.1400

200300s.a

min

2

1

22

11

Goal Programming

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Goal Programming

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Goal Programming

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Goal Programming

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f1 ≤ t1f2 ≥ t2f3 = t3f4 ≥ t4f5 ≤ t5f6 ≥ t6

Lexicographic goal programming:1) The goals are defined2) The priority levels are defined

Level 1

Level 2

Level 3

3) Each goal is assigned to the corresponding level

Goal Programming

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4) The problem of the first level is solved

0,)()(

s.t.

),(min

5555

2222

5

5

2

21

ii nptpnftpnf

Xtp

tnh

xx

x

pn

•If h1*(n, p) > 0 (the goals of the first level are not satisfied)

stop.

•If h1*(n, p) = 0 (the goals of the first level are satisfied)

proceed to the next level.

Goal Programming

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5) The problem of the second level is solved

0,)()()(

)0(0),(

)()(

s.t.

),(min

6666

4444

1111

52*1

5555

2222

6

6

4

4

1

12

ii nptpnftpnftpnfpnh

tpnftpnf

Xtn

tn

tph

xxxpn

xx

x

pn

Goal Programming

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6) Problem of the third level

0,)(

)0(0),(

)()()(

)0(0),(

)()(

s.t.

),(min

3333

641*2

6666

4444

1111

52*1

5555

2222

3

333

ii nptpnf

pnphtpnftpnftpnfpnh

tpnftpnf

Xtnph

xpn

xxxpn

xx

x

pn

Goal Programming

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Example. Goals (L1: pollution, L2: gross margin)

- Pollution ≤ 300


3002000.300000.3000.1

400200300s.t.

300min

)1(

22

2

pnyxyx

yxyx

p

L

Goal Programming

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Goal Programming

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000.500000.3000.10

3002000.300000.3000.1

400200300s.t.

000.500min

)2(

11

2

22

1

pnyxp

pnyxyx

yxyx

n

L

Goal Programming

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Goal Programming

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N1

N2

Goal Programming

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Example. Goals (L1: gross margin, L2: pollution)

- Pollution ≤ 300


000.500000.3000.1000.300000.3000.1

400200300s.t.

000.500min

)1(

11

1

pnyxyx

yxyx

n

L

Goal Programming

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30020

000.500000.3000.1000.300000.3000.1

400200300s.t.

300min

)2(

22

1

11

2

pnyxn

pnyxyx

yxyx

p

L

Goal Programming

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N1

N2

Goal Programming

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Functions to be maximized.

Aspiration levels qj (fj ≥ qj), j = 1, …, p

A weight is assigned to each objective j, j = 1, …, p

(Ruiz et al., 2008, JORS)

jjjpjXqf

)(maxmin

,,1x

x

Reference point

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Reference point

q

Directiondetermined by

q

54


Reference point

q

Non-convex problems

Basic concepts • Goal Programming • Reference Point · 1 Bulgarian Academy of Sciences.22 July,...

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