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

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Bulgarian Academy of Sciences. 22 July, 2008

• Basic concepts• Goal Programming• Reference Point

•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

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

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

•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

Basic concepts

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

Ideal values

P ii x

)(Opt)(

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

Basic concepts

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

200300.000Biologic O2demand

600800.000Gross margin

Biologic O2demand

Gross margin

Basic concepts

• Decision space:

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

• Objective space:

pp RXRX xxfzzf ),(/)(

Basic concepts

Decision space

Basic concepts

Objective space

Basic concepts

Efficient Set

Basic concepts

Basic conceptsEfficient Set

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

Achievement function

h(nj, pj) = pj

pj + nj

Associated optimization problem

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

The decision maker gives a goal for each objective:

Associated optimization problem

Xpjtpnf

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

fj(x) ≤

≥= tj, j = 1, …, p

Goal Programming

•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

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

pnyxpnyxyx

Goal Programming

x + 2y ≤ 3001.000x + 3.000y ≥ 400.000

Satisfying solutions

Efficient and satisfying solutions

Goal Programming

f1 ≥ 400.000

f2 ≥ -300

Goal Programming

Let us now consider the following goals:

- Pollution ≤ 300

000.500000.3000.13002

000.300000.3000.1400

200300s.t.

),(min

pnyxpnyxyx

Goal Programming

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

There are no satisfying solutions

Goal Programming

f1 ≥ 500.000

f2 ≥ -300

Goal Programming

Weighted goal programming:

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

1),( pn

Goal Programming

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

- Pollution ≤ 300

000.400000.3000.13002

000.300000.3000.1400

200300s.t.

000.400300min

pnyxpnyxyx

Goal Programming

- Pollution ≤ 300

000.500000.3000.13002

000.300000.3000.1400

200300s.t.

000.500300min

pnyxpnyxyx

Goal Programming

Minmax goal programming:

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

,,1max),(

Goal Programming

- Pollution ≤ 300

000.500000.3000.13002

000.300000.3000.1400

200300s.t.

000.500,

300maxmin

pnyxpnyxyx

Goal Programming

dppnyx

pnyxyx

000.500

000.500000.3000.13002

000.300000.3000.1400

200300s.a

Goal Programming

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

4) The problem of the first level is solved

0,)()(

),(min

ii nptpnftpnf

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

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

proceed to the next level.

Goal Programming

5) The problem of the second level is solved

0,)()()(

)0(0),(

),(min

ii nptpnftpnftpnfpnh

tpnftpnf

Goal Programming

6) Problem of the third level

)0(0),(

)()()(

)0(0),(

),(min

ii nptpnf

pnphtpnftpnftpnfpnh

tpnftpnf

Goal Programming

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

- Pollution ≤ 300

3002000.300000.3000.1

400200300s.t.

300min

pnyxyx

Goal Programming

000.500000.3000.10

3002000.300000.3000.1

400200300s.t.

000.500min

pnyxyx

Goal Programming

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

- Pollution ≤ 300

000.500000.3000.1000.300000.3000.1

400200300s.t.

000.500min

pnyxyx

Goal Programming

000.500000.3000.1000.300000.3000.1

400200300s.t.

300min

pnyxyx

Goal Programming

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

Reference point

Directiondetermined by

Reference point

Non-convex problems

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

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