Bulgarian Academy of Sciences. 22 July, 2008
Index
• Basic concepts
• Goal Programming
• Reference Point
End
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Bulgarian Academy of Sciences. 22 July, 2008
Basic concepts
•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).
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Bulgarian Academy of Sciences. 22 July, 2008
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.
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Bulgarian Academy of Sciences. 22 July, 2008
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)
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Bulgarian Academy of Sciences. 22 July, 2008
Basic concepts
•Objectives (criteria):
Maximize 1.000x1 + 3.000x2 (gross margin),
Minimize x1 + 2x2 (biologic oxygen demand).
max 1.000 x1  3.000 x2
min x  2 x
1
2

s.t.
x1  300

x2  200


x1  x2  400


1.000 x1  3.000 x2  300.000
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Bulgarian Academy of Sciences. 22 July, 2008
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).
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Bulgarian Academy of Sciences. 22 July, 2008
Basic concepts
Ideal values
Opt f i (x)
( Pi ) 
s.t. x  X
Optimal sol: x(i). Ideal value i: fi* = fi(x(i))
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Bulgarian Academy of Sciences. 22 July, 2008
Basic concepts
Payoff Matrix
f1* = f1(x(1))
f2(x(1))
…
fp(x(1))
f1(x(2))
f2* = f2(x(2))
…
fp(x(2))
…
…
…
…
f1(x(p))
f2(x(p))
…
fp* = fp(x(p))
f1*
f2*
fp*
Anti-ideals: worst value per column.
Approximation to nadir value.
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Bulgarian Academy of Sciences. 22 July, 2008
Basic concepts
Gross
margin
Biologic O2
demand
Gross
margin
800.000
600
Biologic O2
demand
300.000
200
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Bulgarian Academy of Sciences. 22 July, 2008
Basic concepts
• Decision space:


X  x  R / g j (x)  0,, j  1,, m  R
n
n
• Objective space:


f ( X )  z  R / z  f (x), x  X  R
p
p
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Bulgarian Academy of Sciences. 22 July, 2008
Basic concepts
Decision space
(3)
(1)
(2)
(4)
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Basic concepts
Objective space
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Basic concepts
Efficient Set
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Basic concepts
Efficient Set
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Goal Programming
• Building a goal
Objective function fj(x)
fj(x) ≤ tj
fj(x) ≥ tj
fj(x) = tj
Target value tj
Negative: how
short we fall from
the target value
fj(x) + nj – pj = tj
Deviation variables
Positive: how
long we fall
Undesired deviation variables
pj
nj
pj + nj
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Achievement function
h(nj, pj) =
pj
nj
pj + nj
Associated optimization problem
min

s.t.


h( n j , p j )
f j (x)  n j  p j  t j
x X
If h*(nj, pj) = 0, the
goal is satisfied.
If h*(nj, pj) > 0, the
goal is not satisfied.
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
The decision maker gives a goal for each objective:
≤
fj(x) = tj,
≥
j = 1, …, p
Associated optimization problem
min

s.t.


h(n, p)
f j ( x)  n j  p j  t j ,
j  1,, p
x X
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Bulgarian Academy of Sciences. 22 July, 2008
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.
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
In our example, let us consider the following goals:
- Pollution ≤ 300
- Gross margin ≥ 400.000
min
s.t.









h( p1 , n2 )
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
x  2 y  n1  p1  300
1.000 x  3.000 y  n2  p2  400.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Satisfying
solutions
x + 2y ≤ 300
Efficient and
satisfying
solutions
1.000x + 3.000y ≥ 400.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
f1 ≥ 400.000
f2 ≥ -300
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Let us now consider the following goals:
- Pollution ≤ 300
- Gross margin ≥ 500.000
min
s.t.









h( p1 , n2 )
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
x  2 y  n1  p1  300
1.000 x  3.000 y  n2  p2  500.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
There are no satisfying solutions
x + 2y ≤ 300
1.000x + 3.000y ≥ 500.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
f1 ≥ 500.000
f2 ≥ -300
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Weighted goal programming:
The DM gives a weight for each goal mj, j = 1, …, p
 pj
mj 
h(n, p)    n j
j 1 t j 
pj  nj
p
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Example. Goals (equal weights: m1 = m2 = 1)
- Pollution ≤ 300
- Gross margin ≥ 400.000

min
s.t.








p1
n2

300 400.000
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
x  2 y  n1  p1  300
1.000 x  3.000 y  n2  p2  400.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
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Goal Programming
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Goal Programming
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Example. Goals (equal weights: m1 = m2 = 1)
- Pollution ≤ 300
- Gross margin ≥ 500.000

min
s.t.








p1
n2

300 500.000
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
x  2 y  n1  p1  300
1.000 x  3.000 y  n2  p2  500.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
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Goal Programming
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Goal Programming
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Minmax goal programming:
The DM gives a weight for each goal mj, j = 1, …, p
 pj
mj 
h(n, p)  max
 nj
j 1,, p t
j
p  n
j
 j
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Example. Goals (equal weights: m1 = m2 = 1)
- Pollution ≤ 300
- Gross margin ≥ 500.000

min

s.t.







n2 
 p1
max 
,

 300 500.000 
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
x  2 y  n1  p1  300
1.000 x  3.000 y  n2  p2  500.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
min
s.a














d
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
x  2 y  n1  p1  300
1.000 x  3.000 y  n2  p2  500.000
p1
d
300
n2
d
500.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
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Goal Programming
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Goal Programming
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Lexicographic goal programming:
1) The goals are defined
2) The priority levels are defined
3) Each goal is assigned to the corresponding level
f1 ≤ t1
f2 ≥ t2
f3 = t3
f4 ≥ t4
f5 ≤ t5
f6 ≥ t6
Level 1
Level 2
Level 3
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
4) The problem of the first level is solved

min

s.t.





n2 p5
h1 (n, p)  
t 2 t5
x X
f 2 ( x )  n2  p 2  t 2
f 5 (x)  n5  p5  t5
pi , ni  0
•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.
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
5) The problem of the second level is solved

min

s.t.











p1 n4 n6
h2 (n, p)   
t1 t 4 t6
x X
f 2 (x)  n2  p2  t 2
f 5 (x)  n5  p5  t5
h1* (n, p)  0 (n2  p5  0)
f1 (x)  n1  p1  t1
f 4 (x)  n4  p4  t 4
f 6 (x)  n6  p6  t6
pi , ni  0
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
6) Problem of the third level

min

s.t.














p3  n3
h3 (n, p) 
t3
x X
f 2 (x)  n2  p2  t 2
f 5 (x)  n5  p5  t5
h1* (n, p)  0 (n2  p5  0)
f1 (x)  n1  p1  t1
f 4 (x)  n4  p4  t 4
f 6 (x)  n6  p6  t6
h2* (n, p)  0 ( p1  n4  p6  0)
f 3 (x)  n3  p3  t3
pi , ni  0
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Example. Goals (L1: pollution, L2: gross margin)
- Pollution ≤ 300
- Gross margin ≥ 500.000

min
s.t.


( L1) 




p2
300
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
x  2 y  n2  p2  300
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming

min
s.t.



( L 2) 






n1
500.000
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
x  2 y  n2  p2  300
p2  0
1.000 x  3.000 y  n1  p1  500.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
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Goal Programming
N2
N1
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
Example. Goals (L1: gross margin, L2: pollution)
- Pollution ≤ 300
- Gross margin ≥ 500.000

min
s.t.


( L1) 




n1
500.000
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
1.000 x  3.000 y  n1  p1  500.000
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming

min
s.t.



( L 2) 






p2
300
x  300
y  200
x  y  400
1.000 x  3.000 y  300.000
1.000 x  3.000 y  n1  p1  500.000
n1  0
x  2 y  n2  p2  300
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Bulgarian Academy of Sciences. 22 July, 2008
Goal Programming
N2
N1
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Bulgarian Academy of Sciences. 22 July, 2008
Reference point
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)
min max  j  f j (x)  q j 
xX j 1,, p
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Bulgarian Academy of Sciences. 22 July, 2008
Reference point
q
Direction
determined by 
q
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Bulgarian Academy of Sciences. 22 July, 2008
Reference point
Non-convex problems
q
54