Flexible linear programs with a restricted
overall flexibility level ∗
Robert Fullér
rfuller@abo.fi
Margit Kovács
margo@cs.elte.hu
György Schuster
hal@k2.jozsef.kando.hu
Abstract
It is usually supposed that tolerance levels are determined by the decision
maker a priori in flexible linear programming (FLP) problems. In this paper
we shall suppose that the decision maker does not care about the particular
values of tolerance levels, but he wishes to keep their sum below a predetermined level, which we call his overall flexibility level. We also suppose
that his overall flexibility level is soft, i.e. it is admissible to exceed it (to a
certain extent). This is a new statement of FLP problems, because here the
tolerance levels are also treated as variables, and the only restriction on them
is that their sum should not exceed very much a given level. In this setup, we
shall prove that the consistency level of FLP problems depends continuously
on the decision maker’s overall flexibility level.
Keywords: Flexible linear programming; Stability
1
FLP problems
Consider the following fuzzy linear programming problem
hc̃, xi → min; subject to Ãx . b̃, x ∈ Rn .
(1)
where hc̃, xi = c̃1 x1 + · · · + c̃n xn , denotes the weighted sum of fuzzy number
coefficients c̃j ; hãi , xi = ãi1 x1 + · · · + ãin xn denotes the weighted sum of fuzzy
number coefficients ãij ; b̃ is a vector of fuzzy numbers b̃i i = 1, . . . , m, j =
1, . . . , n; b̃0 is a fuzzy number, and . is understood in possibilistic sense.
Supposing that the decision maker has a (fuzzy) aspiration level for the objective function, represented by a fuzzy number b̃0 , problem (1) can be stated as
hc̃, xi . b̃0 ; subject to Ãx . b̃, x ∈ Rn .
∗
The final version of this paper appeared in: Fuzzy Sets and Systems, 127(2002) 177-183.
1
(2)
In the sequal we shall suppose that all of the fuzzy number coefficients in (1)
are chosen from the class of symmetric triangular fuzzy numbers. A fuzzy set of
the real line given by the membership function

|a − t|

1−
if |a − t| ≤ α, α > 0,
ã(t) =
α

0
otherwise,
is called a symmetrical triangular fuzzy number with center a ∈ R and width α,
and we shall refer to it as ã = (a, α).
Consider now (2) with fuzzy number coefficients c̃j = (cj , α), b̃i = (bi , di )
and ãij = (aij , α) of symmetric triangular form, i = 0, 1, . . . , m, j = 1, . . . , n,
and rewrite it in the form: find x ∈ Rn such that,
(hc, xi, αkxk1 ) . (b0 , d0 ); subject to {(hai , xi, αkxk1 ) . (bi , di ), i = 1, . . . , m},
(3)
where kxk1 = |x1 | + · · · + |xn |, hc, xi = c1 x1 + · · · + cn xn and hai , xi =
ai1 x1 + · · · + ain xn ; and in this case we shall call (3) a flexible linear programming
problem and interpret it as a fuzzy extension of the crisp linear inequality problem:
find x ∈ Rn such that,
hc, xi ≤ b0 ; subject to Ax ≤ b,
(4)
and di > 0 is interpreted as the level of maximal admissible violation for the i-th
crisp constraint hai , xi ≤ bi , i = 1, . . . , m; d0 is the level of maximal admissible
violation for the crisp constraint hc, xi ≤ b0 ; and b0 is interpreted as the decision
maker’s crisp aspiration level for the crisp objective function hc, xi.
Then the degree of possibility, denoted by µi (x), that x satisfies the i-th constraint in (3) is computed as in [2]

1
if hai , xi ≤ bi




hai , xi − bi
µi (x) =
1−
if bi < hai , xi ≤ bi + di

αkxk1 + di



0
if hai , xi > bi + di ,
for i = 1, 2, . . . , m, and the degree to which x satisfies the decision maker’s goal
is computed as

1
if hc, xi ≤ b0




hc, xi − b0
µ0 (x) =
1−
if bi < hc, xi ≤ b0 + d0

αkxk1 + d0



0
if hc, xi > b0 + d0 .
2
The fuzzy solution to problem (3) is defined by Bellman and Zadeh’s principle [1]
as
D(x) = min{µ0 (x), µ1 (x), . . . , µm (x)}, x ∈ Rn ,
and an optimal solution, x∗ ∈ Rn , is determined from the relationship
D(x∗ ) = κ = max{D(x)| x ∈ Rn }.
(5)
where κ is called the degree of consistency of (1). It is easy to see that problem (5)
leads to the following nonlinear mathematical programming problem,
λ → max
hc, xi − b0
≥ λ,
αkxk1 + d0
hai , xi − bi
1−
≥ λ, i = 1, . . . , m,
αkxk1 + di
λ ∈ [0, 1], x ∈ Rn
1−
Consider (2) with fuzzy number coefficients (c0j , α), (b0i , di ) and (a0ij , α), i.e.,
find x ∈ Rn such that,
(hc0 , xi, αkxk1 ) . (b00 , d0 ); subject to {(ha0i , xi, αkxk1 ) . (b0i , di ), i = 1, . . . , m}.
(6)
The fuzzy solution to problem (6) will be denoted by D0 and to find a maximizing
solution to (6) leads to the following mathematical programming problem,
λ → max
hc0 , xi − b00
≥ λ,
αkxk1 + d0
ha0 , xi − b0i
≥ λ, i = 1, . . . , m,
1− i
αkxk1 + di
λ ∈ [0, 1], x ∈ Rn .
1−
Remark 1.1 In the extremal case α = 0 (but di > 0), the problem of finding an
optimal solution to (3) from equation (5) leads to the following linear programming
3
problem,
λ → max
hc, xi − b0
≥ λ,
d0
hai , xi − bi
1−
≥ λ, i = 1, . . . , m,
di
λ ∈ [0, 1], x ∈ Rn ,
1−
which was introduced in [7]. Sensitivity analysis in FLP problems (with α = 0)
was first considered in [4], where a functional relationship between changes of
parameters of the right-hand side and those of the optimal value of the primal
objective function was derived for almost all conceivable cases.
In [6] an FLP problem (with symmetrical triangular fuzzy numbers) was formulated and the value of information was discussed via sensitivity analysis. Stable
embeddings of linear equality and inequality systems into fuzzified systems were
discussed in [5].
A stability property of the fuzzy solution (with respect to small changes of the
centers of symmetric triangular fuzzy numbers) to FLP problems (with α > 0) was
shown in [2]. Namely, it was proved that
1
1
0
|D(x) − D (x)| ≤ δ
+
, ∀x ∈ Rn ,
(7)
α min{d0 , d1 , . . . , dm }
where δ = max{|aij − a0ij |, |bi − b0i |, |cj − c0j |} denote the maximal deviation
between the centers of fuzzy parameters in (3) and (6).
2
FLP with restricted flexibility
Suppose now that the decision maker does not care about the particular values of di ,
but he wishes to reduce the overall level of violation, defined by d0 +d1 +· · ·+dm ,
as much as possible, and the membership function of his soft overall flexibility is
given by
P

1
if m

i=0 di ≤ T

P

m
P
i=0 di − T
ν(d0 + d1 + · · · + dm ) =
1−
if T < m
i=0 di ≤ T + t

t


0
otherwise,
where T is called the decision maker’s crisp overall flexibility level, and t > 0
denotes his tolerance level for exceeding T . Therefore, T is nothing else but the
cumulated violation of crisp inequalities in (4).
4
Then the fuzzy decision problem (3) under soft overall flexibility constraint can
be formulated as
hc̃, xi . b̃0
(8)
Ãx . b̃,
d0 + · · · + dm . (T, t), x ∈ Rn ,
and its fuzzy solution is then defined by
D(x, d) = min{µ0 (x), µ1 (x), . . . , µm (x), ν(d0 + d1 + · · · + dm )}, x ∈ Rn , (9)
furthermore, an optimal solution to (8) can be obtained by solving the following
nonlinear mathematical programming problem,
λ → max
hc, xi − b0
≥ λ,
αkxk1 + d0
hai , xi − bi
1−
≥ λ, i = 1, . . . , m,
αkxk1 + di
d0 + · · · + dm − T
≥λ
1−
t
λ ∈ [0, 1], d0 > 0, . . . , dm > 0, x ∈ Rn .
1−
(10)
Consider now problem (8) with (perturbed) overall flexibility degree (T 0 , t0 ).
Then we encounter the following perturbed problem
hc̃, xi . b̃0
(11)
Ãx . b̃,
d0 + · · · + dm . (T 0 , t0 ), x ∈ Rn .
The fuzzy solution to (11) is defined by
D0 (x, d) = min{µ0 (x), µ1 (x), . . . , µm (x), ν 0 (d0 +d1 +· · ·+dm )}, x ∈ Rn , (12)
where
ν 0 (d0 + d1 + · · · + dm ) =

1



1−



0
if
Pm
i=0 di
t0
− T0
Pm
i=0 di
if T 0 <
Pm
otherwise.
5
≤ T0
i=0 di
≤ T 0 + t0
It is easy to see that problem (12) leads to the following nonlinear mathematical
programming problem,
λ → max
hc, xi − b0
≥ λ,
αkxk1 + d0
hai , xi − bi
1−
≥ λ, i = 1, . . . , m,
αkxk1 + di
d0 + · · · + dm − T 0
1−
≥λ
t0
λ ∈ [0, 1], d0 > 0, . . . , dm > 0, x ∈ Rn .
1−
We will prove that the fuzzy solution and degree of consistency of FLP problem
(8) depends continuously on the degree of flexiblity, that is, small changes in the
decision maker’s overall flexibility level can cause only small deviations both in
the fuzzy solution and in the degree of consistency.
In the sequel we need the following lemma (which is a simple application of
Lemma 2.3 from [3] to fuzzy numbers ν and ν 0 ).
Lemma 2.1 Let ν = (T, t) and ν 0 = (T 0 , t0 ) be symmetric triangular fuzzy numbers. Then
1 1
0
0
0
|ν(z) − ν (z)| ≤ (|T − T | + |t − t |) max
,
, ∀z ∈ R.
t t0
The theorem in question can be stated as follows,
Theorem 2.1 Let D(x, d) and D0 (x, d) denote the fuzzy solution of FLP problems
(8) and (11) respectively. Then
1 1
0
0
n
m
kD − D k∞ = sup{|D(x, d) − D (x, d)| : x ∈ R , d ∈ R } ≤ δ max
,
t t0
(13)
0
0
where δ = (|T − T | + |t − t |).
Proof. It will be sufficient to show that
0
0
0
|D(x, d) − D (x, d)| ≤ (|T − T | + |t − t |) max
1 1
,
.
t t0
(14)
holds for any x ∈ Rn and d ∈ Rm , because from (14) folows (13). First we observe
that
|D(x, d) − D0 (x, d)| = |ν(d) − ν 0 (d)|.
6
Applying Lemma 2.1 we find
|ν(d0 + d1 + · · · + dm ) − ν 0 (d0 + d1 + · · · + dm )|
0
0
≤ (|T − T | + |t − t |) max
1 1
.
,
t t0
Which ends the proof.
Remark 2.1 From (13) it follows that
0
0
0
|κ − κ | ≤ (|T − T | + |t − t |) max
1 1
,
,
t t0
which means that small changes in the (soft) overall flexibility level can cause only
small deviations between the degrees of consistency for FLP problems (8) and (11).
If the decision maker’s overall flexibility level is enlarged, that is, T ≥ T 0 then
the consistency level of the FLP also increases, i.e., κ ≥ κ0 .
Remark 2.2 The inclusion of constraint
d0 + · · · + dm . (T, t),
into FLP problem (3) enlarges not only the dimension of the problem (by adding
(m+1) new variables), but it also effects the stability property of the fuzzy solution
under small changes in the centers of the fuzzy coefficients aij , bi and cj , because
the right-hand side of (7)
1
1
+
,
α min{d0 , d1 , . . . , dm }
could be arbitary big (there are no lower limits for the values of variables d0 , d1 ,
. . . , dm ), and therefore, the fuzzy solution may vary fortuitously even under very
small perturbations of aij , bi and cj . It is why we allowed perturbations only in
the fuzzy number representing the decision maker’s overall flexibility level, but not
in the coefficients aij , bi and cj .
3
Summary
In this paper we have considered a novel statement of flexible linear programming
problems where the decision maker does not care about the particular values of
tolerance levels, but he wishes to keep their sum below a predetermined overall soft
7
flexibility level. These types of problems may arise in portfolio selection problems
where the tolerance levels can be expressed in monetary terms and the overall
soft flexibility level denotes the amount of extra capital the investor might find
in order to improve portfolio performance. Treating tolerance levels as variables,
the dimension of the original problem (5) increases by (m + 1) new variables.
Furthermore, to find a solution to the resulting problem (10) generally requires the
use of nonlinear programming techniques, and could be tricky [8].
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