Interactive Fuzzy Stochastic Multi-level 0-1 Programming
through Probability Maximization
Masatoshi Sakawa and Takeshi Matsui
Faculty of Engineering, Hiroshima University, 739-8527 Higashi-Hiroshima, Japan
Keywords:
Multi-level 0-1 Programming, Random Variables, Interactive Fuzzy Programming, Probability Maximization.
Abstract:
This paper considers multi-level 0-1 programming problems involving random variable coefficients both in
objective functions and constraints. Following the probability maximization model together with the concept
of chance constraints, the formulated stochastic multi-level 0-1 programming problems are transformed into
deterministic ones. Taking into account vagueness of judgments of the decision makers, we present interactive
fuzzy programming. In the proposed interactive method, after determining the fuzzy goals of the decision
makers at all levels, a satisfactory solution is derived efficiently by updating satisfactory levels of the decision
makers with considerations of overall satisfactory balance among all levels. An illustrative numerical example
for a three-level 0-1 programming problem is provided to demonstrate the feasibility of the proposed method.
1 INTRODUCTION
The Stackelberg soluiton has been usually employed
as a solution concept to multi-level programming
problems (Sakawa and Nishizaki, 2009). To describe
the concept of the Stackelberg solution, consider a
two-level programming problem. There are two deci-
sion makers (DMs); each DM completely knows ob-
jective functions and constraints of the two DMs, and
the DM at the upper level (leader) first make a deci-
sion and then the DM at the lower level (follower)
specifies a decision so as to optimize an objective
function with full knowledge of the decision of the
leader. According to to the rule, the leader also make a
decision so as to optimize the leader’s objective func-
tion. Then a solution defined as the above-mentioned
procedure is called the Stackelberg solution.
Lai (Lai, 1996) and Shih et al. (Shih et al., 1996)
proposed solution concepts for two-level linear pro-
gramming problems or multi-level ones such that de-
cisions of DMs in all levels are sequential and all of
the DMs essentially cooperate with each other. In
their methods, the DMs identify membership func-
tions of the fuzzy goals for their objective functions,
and in particular, the DM at the upper level also spec-
ifies those of the fuzzy goals for the decision vari-
ables. The DM at the lower level solves a fuzzy pro-
gramming problem with a constraint with respect to a
satisfactory degree of the DM at the upper level. Un-
fortunately, there is a possibility that their method le-
ads a final solution to an undesirable one because
of inconsistency between the fuzzy goals of the ob-
jective function and those of the decision variables.
In order to overcome the problem in their methods,
by eliminating the fuzzy goals for the decision vari-
ables, Sakwa et al. have proposed interactive fuzzy
programming for two-level or multi-level linear pro-
gramming problems to obtain a satisfactory solution
for DMs (Sakawa et al., 1998; Sakawa and Uemura,
2000).
In actual decision making situations, however, we
must often make a decision on the basis of vague in-
formation or uncertain data. For such decision mak-
ing problems involving uncertainty, there exist two
typical approaches: probability theoretic approach
and fuzzy-theoretic one. Stochastic programming, as
an optimization method based on the probability the-
ory, have been developing in various ways (Stancu-
Minasian, 1990), including two stage problems con-
sidered by Dantzig (Dantzig, 1955) and chance con-
strained programming proposed by Charnes et al.
(Charnes and Cooper, 1959).
Under these circumstances, in this paper, we
present interactive fuzzy programming for multi-level
0-1 programming problems involving random vari-
able coefficients both in objective functions and con-
straints. Using the concept of chance constraints,
stochastic constraints are transformed into determin-
istic ones. Following the probability maximization
model, the minimization of each stochastic objec-
383
Sakawa M. and Matsui T..
Interactive Fuzzy Stochastic Multi-level 0-1 Programming through Probability Maximization.
DOI: 10.5220/0004110103830388
In Proceedings of the 4th International Joint Conference on Computational Intelligence (FCTA-2012), pages 383-388
ISBN: 978-989-8565-33-4
Copyright
c
 2012 SCITEPRESS (Science and Technology Publications, Lda.)

tive function is replaced with the maximization of the
probability that each objective function is less than or
equal to a certain value. Under some appropriate as-
sumptions for distribution functions, the formulated
stochastic multi-level 0-1 programming problems are
transformed into deterministic ones. In our interac-
tive method, after determining the fuzzy goals of the
DM at all levels, a satisfactory solution is derived ef-
ficiently by updating the satisfactory degrees of the
DMs at the upper level with considerations of overall
satisfactory balance among all levels. An illustrative
numerical example for a three-level 0-1 programming
problem is provided to demonstrate the feasibility of
the proposed method.
2 STOCHASTIC MULTI-LEVEL
0-1 PROGRAMMING
PROBLEMS
In this paper, we consider stochastic multi-level 0-
1 programming problems where each of the DMs at
all levels takes overall satisfactory balance among all
levels into consideration and tries to optimize each
objective function. Such a stochastic multi-level 0-1
programming problem is formulated as
minimize
DM1 (Level 1)
z
1
(x) = c
11
(ω)x
1
+ ··· + c
1K
(ω)x
K
.
.
.
.
.
.
minimize
DMK (Level K)
z
K
(x) = c
K1
(ω)x
1
+ ··· + c
KK
(ω)x
K
subject to A
1
x
1
+ ··· + A
K
x
K
≤ b(ω)
x
1
∈ {0, 1}
n
1
,...,x
K
∈ {0, 1}
n
K















(1)
where x
l
,l = 1,...,K, is an n
l
-dimensional 0-1 de-
cision variable column vector; ¯c
l j
,l = 1,...,K, j =
1,...,K, is an n
j
-dimensional random variable row
vectors. Here, we assume that ¯c
l j
is expressed as
¯c
l j
= c
1
l j
+
¯
t
l
c
2
l j
where
¯
t
l
,l = 1,2, ...,K are mutually
independent random variables with mean M
l
and their
distribution functionsT
l
(·),l = 1,2,...,K are continu-
ous and strictly increasing, and that
¯
α
l
,l = 1,2,...,K
are random variablesexpressed as
¯
α
l
= c
1
l
+
¯
t
l
α
2
l
. This
definition of random variables is one of the simplest
randomization modeling of coefficients using dilation
and translation of random variables, as discussed by
Stancu-Minasian (Stancu-Minasian, 1984). In addi-
tion,
¯
b
i
,i = 1,2,...,m are mutually independent ran-
dom variables whose distribution functions are also
assumed to be continuous and strictly increasing.
Since (1) contains random variable coefficients,
solution methods for ordinary mathematical program-
ming problems cannot be applied directly. Con-
sequently, we first deal with the constraints in (1)
as chance constraints (Charnes and Cooper, 1959)
which mean that the constraints need to be satisfied
with a certain probability (satisficing level) and over.
Namely, replacing constraints in (1) by chance con-
straints with a satisficing level β, the problem can be
transformed as:
minimize
DM1 (Level 1)
c
11
(ω)x
1
+ ··· + c
1K
(ω)x
K
.
.
.
.
.
.
minimize
DMK (Level K)
c
K1
(ω)x
1
+ ··· + c
KK
(ω)x
K
subject to Pr{a
i1
x
1
+ ··· + a
iK
x
K
≤ b
i
(ω)}
≥ β
i
,i = 1,...,m
x
1
∈ {0, 1}
n
1
,...,x
K
∈ {0, 1}
n
K



















.
(2)
The first constraint in (2) is rewritten as:
Pr{a
i1
x
1
+ ··· + a
iK
x
K
≤ b
i
(ω)} ≥ β
i
⇔ 1− Pr{a
i1
x
1
+ ··· + a
iK
x
K
≥ b
i
(ω)} ≥ β
i
⇔ 1− F
i
(a
i1
x
1
+ ··· + a
iK
x
K
) ≥ β
i
⇔ F
i
(a
i1
x
1
+ ··· + a
iK
x
K
) ≤ 1− β
i
⇔ a
i1
x
1
+ ··· + a
iK
x
K
≤ F
∗
i
(1− β
i
)
where F
∗
i
is a pseudo-inverse function of F
i
.
When the DM wants to maximize the probabil-
ity that the profit is greater than or equal to a certain
permissible level, probability maximization model
(Charnes and Cooper, 1959) is recommended. In this
paper, assuming that the DM wants to maximize the
probability that the profit is greater than or equal to
a certain permissible level for safe management, we
adopt the probability maximization model as a deci-
sion making model.
In the probability maximization model, the mini-
mization of each of objective function ¯z
l
(x) in (2) is
substituted with the maximization of the probability
that ¯z
l
(x) is less than or equal to a certain permissible
level h
l
under the chance constraints. Through proba-
bility maximization, problem (2) can be rewritten as:
maximize
DM1iLevel 1j
Pr{z
1
(x
1
,...,x
K
,ω) ≤ h
1
}
.
.
.
.
.
.
maximize
DMKiLevel Kj
Pr{z
K
(x
1
,...,x
K
,ω) ≤ h
K
}
subject to Pr{a
i1
x
1
+ ··· + a
iK
x
K
≤ b
i
(ω)}
≥ β
i
,i = 1,...,m
x
1
∈ {0, 1}
n
1
,...,x
K
∈ {0,1}
n
K



















.
(3)
Supposing that c for any feasible solution x to (3),
from the assumption on the distribution function T
l
(·)
of each random variable
¯
t
l
, we can rewrite objective
functions in (3) as follows.
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
384

Pr{z
l
(x
1
,... ,x
K
,ω) ≤ h
l
}
= Pr{c
l1
(ω)x
1
+ ·· · + c
lK
(ω)x
K
≤ h
l
}
= Pr



c
l1
(ω)x
1
+ ·· · + c
lK
(ω)x
K
− ( ¯c
l1
x
1
+ ·· · + ¯c
lK
x
K
)
q
(x
T
1
,. .. ,x
T
K
)V
l
(x
T
1
,. .. ,x
T
K
)
T
≤
h
l
− ( ¯c
l1
x
1
+ ·· · + ¯c
lK
x
K
)
q
(x
T
1
,. .. ,x
T
K
)V
l
(x
T
1
,. .. ,x
T
K
)
T



= φ
l


h
l
− ( ¯c
l1
x
1
+ ·· · + ¯c
lK
x
K
)
q
(x
T
1
,. .. ,x
T
K
)V
l
(x
T
1
,. .. ,x
T
K
)
T


Hence, (3) can be equivalently transformed into
the following deterministic multi-level programming
problem.
maximize
DM1iLevel 1j
φ
1


h
1
− ( ¯c
11
x
1
+ ··· + ¯c
1K
x
K
)
q
(x
T
1
,. . . , x
T
K
)V
1
(x
T
1
,. . . , x
T
K
)
T


.
.
.
.
.
.
maximize
DMKiLevel Kj
φ
K


h
K
− ( ¯c
K1
x
1
+ ··· + ¯c
KK
x
K
)
q
(x
T
1
,. . . , x
T
K
)V
K
(x
T
1
,. . . , x
T
K
)
T


subject to x ∈ X





































(4)
3 INTERACTIVE FUZZY
PROGRAMMING
In the previous section, we have dealt with random-
ness involved in the objective functions and con-
straints in the original stochastic multi-level program-
ming problem (1), and transformed into the determin-
istic multi-level programming problem (4) through
the ideas of chance constraint and probability maxi-
mization model. In this section, we take account of
fuzziness of human judgments by introducing fuzzy
goals for objectivefunction values obtained in the pre-
vious section.
To be more specific, in order to consider the im-
precise nature of the DMs’ judgments for the proba-
bilities p
l
(x),l = 1, 2, ... ,K in (4), it seems natural to
assume that the DMs have fuzzy goals such as “p
l
(x)
should be substantially greater than or equal to some
specific value.” Then, (4) can be rewritten as:
maximize
DM1iLevel 1j
µ
1
(p
1
(x
1
,. . . , x
K
))
.
.
.
.
.
.
maximize
DMKiLevel Kj
µ
K
(p
K
(x
1
,. . . , x
K
))
subject to A
1
x
1
+ ··· + A
K
x
K
≤
ˆ
b
x ∈ X















(5)
where µ
l
(·) is a membership function to quantify a
fuzzy goal for the l th objective function in (4) and it
is assumed to be nondecreasing.
Although the membership function does not al-
ways need to be linear, for the sake of simplicity, we
adopt a linear membership function. To be more spe-
cific, if the DM feels that p
l
(x) should be greater than
or equal to at least p
l,0
and p
l
(x) ≥ p
l,1
(> p
l,0
) is sat-
isfactory, the linear membership function µ
l
(p
l
(x)) is
defined as:
µ
l
(p
l
(x)) =







0 , µ
l
(p
l
(x)) < p
l,0
µ
l
(p
l
(x)) − p
l,0
p
l,1
− p
l,0
, p
l,0
≤ µ
l
(p
l
(x)) ≤ p
l,1
1 , µ
l
(p
l
(x)) > p
l,1
and it is depicted in Figure 1.
Now we are ready to propose interactive fuzzy
programming for deriving a satisfactory solution by
updating the satisfactory degree of the DM at the up-
per level with considerations of overall satisfactory
balance among all the levels.
Interactive Fuzzy Programming
Step 1: Ask the decision maker at the upper level,
DM1, to subjectively determine a satisficing level
β ∈ (0, 1) for constraints. Go to Step 2.
Step 2: In order to determine permissible levels
h
l
,l = 1,2,.. . ,K, the following problems are
solved to find the minimum values z
E
l,min
and z
E
l,M
of objectivefunctions z
E
l
(x) under the chance con-
straints with satisficing levels β
i
,i = 1,2, ..., m.
minimize ¯c
l1
x
1
+ ··· + ¯c
lK
x
K
subject to x ∈ X

, l = 1,..., K
(6)
If the set of feasible solutions to these problems is
empty, the satisficing levels β
i
,i = 1,2,.. . ,m must
be reassessed and return to step 1. Otherwise, let
z
E
l,min
be optimal objective function values to (6).
Since (6) are 0-1 programmingproblems, they can
be solved by tabu search based on strategic oscil-
lation. Ask DM1 to determine permissible levels
1.0
0
1.0
µl (pl (x))
pl (x) pl,0 pl,1
Figure 1: Linear membership function.
InteractiveFuzzyStochasticMulti-level0-1ProgrammingthroughProbabilityMaximization
385

h
l
,l = 1,2,. . .,K for objective functions in con-
sideration of z
E
l,min
and z
E
l,M
. Go to Step 3.
Step 3: Solve the following problems to find the
maximum values p
l,max
and p
l,M
of objective
functions p
l
(x) under the chance constraints with
satisficing levels β
i
,i = 1,2, ..., m.
maximize
φ
l


h
l
− ( ¯c
l1
x
1
+ ··· + ¯c
lK
x
K
)
q
(x
T
1
,. . .,x
T
K
)V
l
(x
T
1
,. . . , x
T
K
)
T


subject to x ∈ X











(7)
Then, identify the linear membership function
µ
l
(z
P
l
(x)),l = 1,2,..., K of the fuzzy goal for the
corresponding objective function. Go to step 4.
Step 4: Solve the following corresponding maxmin
problem.
maximize
x∈X
min
l=1,...,K

µ
l
(Z
P
l
(x))

(8)
Go to step 5.
Step 5: Ask DM1 to subjectively set the minimal
satisfactory level
ˆ
δ
1
. Then, solve the following
maxmin problem.
maximize
x∈X
min
l=2,...,K

µ
l
(Z
P
l
(x))

subjectto µ
1
(p
1
(x)) ≥
ˆ
δ
1
)
(9)
Set λ := 2Cλ
′
:= 1. Go to step 6.
Step 6: Ask DMλ to set the member-
ship function µ
∆
λ
(∆
λ
(x)) for the ratio
∆
λ
= (µ
λ+1
(Z
P
λ+1
(x)))/(µ
λ
(Z
P
λ
(x))) of satis-
factory degrees and the minimal satisfactory level
ˆ
δ
∆
λ
. Solve the following maxmin problem.
maximize
x∈X
min
l=λ+1,...,K

µ
l
(Z
P
l
(x))

subjectto µ
1
(Z
P
1
(x)) ≥
ˆ
δ
1
µ
∆
2
(∆
2
(x)) ≥
ˆ
δ
∆
2
.
.
.
µ
∆
λ
(∆
λ
(x)) ≥
ˆ
δ
∆
λ















(10)
Repeat this step until λ = K − 1.
Step 7: If the current solution satisfies the termina-
tion conditions, DMK−λ
′
accepts it, and K−λ
′
=
1, then the procedure stops and the current solu-
tion is determined to be a satisfactory solution.
Otherwise, ask DMK − λ
′
to update the minimal
satisfactory level
ˆ
δ
∆
K−λ
′
. If K − λ
′
= 1, ask DM1
to update the minimal satisfactory level
ˆ
δ
1
. Go to
step 8.
Step 8: Solve the following problem.
maximize v
subjectto x ∈ X
0 ≤ v ≤ 1
µ
1
(Z
P
1
(x)) ≥
ˆ
δ
1
µ
∆
2
(∆
2
(x)) ≥
ˆ
δ
∆
2
.
.
.
µ
∆
K−1
(∆
K−1
(x)) ≥
ˆ
δ
∆
K−1
Π
K−1
l=K−λ
′
+1
ˆ
∆
l
µ
K−λ
′
+1
(Z
P
K−λ
′
+1
(x)) ≥ v
·· ·
ˆ
∆
K−1
ˆ
∆
K−2
µ
K−2
(Z
P
K−2
(x)) ≥ v
ˆ
∆
K−1
µ
K−1
(Z
P
K−1
(x)) ≥ v
µ
K
(Z
P
K
(x)) ≥ v









































(11)
It should be noted that all problems (6), (7), (8),
(9), (10) and (11) in the interactive fuzzy pro-
gramming algorithm can be solved by tabu search
based on strategic oscillation (Hanafi and Freville,
1998).
4 NUMERICAL EXAMPLE
As an example for a stochastic multi-level 0-1 pro-
gramming problem, consider the following three-
level problem:
minimize
DM1 (Level 1)
c
11
(ω)x
1
+ c
12
(ω)x
2
+ c
13
(ω)x
3
minimize
DM2 (Level 2)
c
21
(ω)x
1
+ c
22
(ω)x
2
+ c
23
(ω)x
3
minimize
DM3 (Level 3)
c
31
(ω)x
1
+ c
32
(ω)x
2
+ c
33
(ω)x
3
subject to A
1
x
1
+ A
2
x
2
+ A
3
x
3
≤ b(ω)
x
1
∈ {0,1}
n
1
,. . .,x
3
∈ {0, 1}
n
3















(12)
where x
1
= (x
1
,. . . , x
15
)
T
,x
2
= (x
16
,. . . , x
30
)
T
,x
3
=
(x
31
,. . . , x
45
)
T
; each entry of 15-dimensional row
constant vectors c
ij
,i, j = 1,2,3, and each entry of
3× 15 coefficient matrices A
1
,A
2
, and A
3
are random
In step 1 of the interactive fuzzy programming,
DM1 specifies satisficing levels β
i
,i = 1,2, ..., 9 as:
(β
1
,β
2
,β
3
,β
4
,β
5
,β
6
,β
7
,β
8
,β
9
)
T
=
(0.95,0.80,0.85, 0.90,0.90, 0.85, 0.85, 0.95,0.80)
T
.
For the specified satisficing levels β
i
,i = 1,2, ..., 9,
in step 2, minimal values z
E
l,min
and maximal values
z
E
l,max
of objective functions E{¯z
l
(x
1
,x
2
,x
3
)} under
the chance constraints are calculated. By considering
these values, the DMs subjectively specify permissi-
ble levels.
In step 3, maximal values p
l,max
of p
l
(x
1
,x
2
,x
3
)
are calculated. Assume that the DMs identify the
linear membership function whose parameter values
IJCCI2012-InternationalJointConferenceonComputationalIntelligence
386

are determined by the Zimmermann method (Zim-
mermann, 1978).
In step 4, the maxmin problem is solved. The ob-
tain result is shown at the column labeled “1st” in Ta-
ble 1.
In step 5, for the obtained optimal solution, then,
the ratio of satisfactory degrees ∆
1
is equal to 0.9837.
Since DM1 is not satisfied with this solution, DM1
sets the minimal satisfactory level
ˆ
δ
1
to 0.75. (9) for
ˆ
δ
1
= 0.75 is solved. For the obtained optimal solu-
tion to (9), µ
1
(Z
P
1
(x)) = 0.7772, µ
2
(Z
P
2
(x)) = 0.6122,
and µ
3
(Z
P
3
(x)) = 0.6618, shown at the column labeled
“2nd” in Table 1.
In step 6, DM2 sets the membership function
µ
∆
2
(∆
2
(x)) for the ratio ∆
2
of satisfactory degrees and
the minimal satisfactory level as
ˆ
δ
∆
2
= 0.80. (10)
for
ˆ
δ
∆
2
= 0.80 is solved. The obtained result is
shown at the column labeled “3rd” in Table 1. For
the obtained optimal solution to (10), µ
1
(Z
P
1
(x)) =
0.7696, µ
2
(Z
P
2
(x)) = 0.6923, µ
3
(Z
P
3
(x)) = 0.6118 and
µ
∆
2
(∆
2
(x)) = 0.9767.
In step 7, since the ratio of satisfactory degrees ∆
2
is greater than
ˆ
δ
∆
2
= 0.80, the condition of termina-
tion of the interactive process is fulfilled. Then, DM1
is asked whether he is satisfied with the obtained so-
lution. Since DM1 is not satisfied, and he updates the
minimal satisfactory level
ˆ
δ
1
from 0.75 to 0.80 in or-
der to improve µ
1
(Z
P
1
(x)) and sets
ˆ
δ
∆
1
= 0.80.
In step 8, (11) for
ˆ
δ
1
= 0.80 and
ˆ
δ
∆
2
=
0.8837 is solved. The obtained result is shown
at the column labeled “4th” in Table 1. For
the obtained optimal solution to (11), µ
1
(Z
P
1
(x)) =
0.8095, µ
2
(Z
P
2
(x)) = 0.6585, µ
3
(Z
P
3
(x)) = 0.6001 and
µ
∆
1
(∆
1
(x)) = 0.9325.
In step 6, since the current solution satisfies all
termination conditions of the interactive process and
DM1 is satisfied with the current solution, the satis-
factory solution is obtained and the interaction proce-
dure is terminated.
Table 1: Interaction process.
Interaction 1st 2nd 3rd 4th
ˆ
δ
1
0.7500 0.7500 0.8000
ˆ
δ
∆
1
0.8000
ˆ
δ
∆
2
0.8000
µ
1
(Z
P
1
(x)) 0.7160 0.7772 0.7696 0.8095
µ
2
(Z
P
2
(x)) 0.7043 0.6122 0.6923 0.6585
µ
3
(Z
P
3
(x)) 0.6856 0.6618 0.6118 0.6001
∆
1
(x) 0.9837 0.7877 0.8996 0.8135
∆
2
(x) 0.9734 1.0801 0.8837 0.9123
µ
∆
1
(∆
1
(x)) 0.9325
µ
∆
2
(∆
2
(x)) 0.9767 0.8775
5 CONCLUSIONS
In this paper, we focused on stochastic multi-level
0-1 programming problems with random variable
coefficients in both objective functions and con-
straints. Through the use of the probability maximiza-
tion model in chance constrained programming, the
stochastic multi-level 0-1 programming problems are
transformed into deterministic 0-1 programming ones
under some appropriate assumptions for distribution
functions. Taking into account vagueness of judg-
ments of the DMs, interactivefuzzy programming has
been proposed. In the proposed interactive method,
after determining the fuzzy goals of the DMs at all
levels, a satisfactory solution is derived efficiently by
updating the satisfactory degree of the DM at the 1st
level with considerations of overall satisfactory bal-
ance among all levels. It is significant to note here
that the transformed deterministic problems to derive
an overall satisfactory solution can be solved through
tabu search based on strategic oscillation. An illustra-
tive numerical example for a three-level 0-1 program-
ming problem was provided to demonstrate the feasi-
bility of the proposed method. Extensions to other
stochastic programming models will be considered
elsewhere. Also extensions to multi-level 0-1 pro-
gramming problems involving fuzzy random variable
coefficients will be required in the near future.
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