76
The difference between this problem and Zimmermann is that
p
is not initially given in this problem. Therefore, we may assume a set of
s p
, where
] ,
[ Z
b p
− ∈
. Then, the problem of each
p
given is a Zimmermann problem. The decision maker may choose a refined
p
among the solution for this given set of
s p
. Then a Zimmermann problem with the decision maker’s refined
p
is solved. This solution will be the final optimal solution for 5.22.
5.3 Interactive fuzzy linear programming
Decision processes
are better described and solved using fuzzy sets theory, rather than precise approaches. However, the decision maker himself always plays the
most important role in using fuzzy sets theory. Therefore, an interactive process between the decision maker and the decision process is necessary to solve our
problem. That is actually a user-dependent fuzzy LP technique. Furthermore, a problem-oriented concept is also a vitally important concept in
solving practical problems, as noted by Simon. By use of fuzzy sets theory, and user dependent interactive and problem,
oriented concepts, the flexibility and robustness of LP techniques are improved. An IFLP approach which is a symmetric integration of Zimmermann’s, Werner’s,
Verdegay’s and Chanas’s FLP approaches is developed and additionally it provides a decision support system for solving a specific domain of a rel-world
LP system Lai and Hwang [92]. Lai and Hwang suggested “expert decision
support system” that give an aggregate solution to all possible cases. The system determines fuzzy-efficient extreme solution and a fuzzy efficient
compromise solution. They are judged by the decision maker and he decides whether modification are necessary. In the latter case the decision maker change
membership functions assisted
by the system, Werners [202]. The application of FLP implies that the problems will be solved in an interactive
way. In the first step, the fuzzy system is modeled by using only the information
77 which the decision maker can provide without any expensive additional
information acquisition. Knowing a first “compromise solution” the decision maker can perceive which further information should be obtained and he is able to
justify the decision by comparing carefully additional advantages and arising costs. In doing so, step by step the compromise solution are improved. The
procedure obviously offer the possibility to limit the acquisition and processing information to the relevant components and therefore information costs will be
distinctly reduced, Rommefanger [154]. The most important element that affects solutions of FLP problems is
parameters which are used reflecting fuzziness of model. How these parameters define fuzzy geometry is the most sensitive point. Because the success of solution
depends on the success of reflecting the system of model. Moreover, the interactive concept provides a learning process about the
system and makes allowance for psychological convergence for the decision maker, whereby, she learns recognize good solutions, the relative importance of
factors in the system and then design a high-productivity system, instead of optimizing a given system.
This IFLP
system provides integration-oriented, adaption and learning features by considering all possibilities of specific domain of LP problems which
are integrated in logical order using an IF-THEN rule. IFLP
methods have been studied, since 1980. Typical works are Baptistella and Ollero, Fabian, Cibiobanu and Stoica, Ollero, Aracil and Camacho, Sea and
Sakawa, Slowinski, Werners and Zimmermann, Zimmermann described some general concepts and modeling methods of decision support system and expert
system in a fuzzy enviroment. Others developed interactive approaches to solve multiple criteria decision making problem, Lai and Hwang [92].
With the aim of solutions for the models like these, there are many studies on LP models. However the studies of Zimmermann, Chanas, Werners and
78 Vedegay have become quite efficient for improving LP models with decision
support to solve real-world problem.
5.3.1 Interactive fuzzy linear programming algorithm Step 1 Solve a traditional LP problem of 5.3 by use of the simplex method.
The unique optimal solution with its corresponding consumed resources is presented to the decision maker.
Step 2 Does this solution satisfy the decision maker ? Consider the following
cases. 1.If solution is satisfied the print out results an top
2.If resource i, for some i are idle then reduce available
i
b and go to Step 1.
3. If available resources are not precise and some tolerances are possible then make a parametric analysis with and go to Step 3.
Step 3 Solve a parametric LP problem of 5.4. Then results are depicted in a
table. At the same time, let us identify =
= θ
Z Z
and 1
1
= =
θ Z
Z .
Step 4 Do any of these solutions shown in table satisfy the decision maker?
Consider the following cases: 1.
If solution is satisfied then print out results ad stop. 2.
If resource i, for some i are idle then decrease
i
b and change
i
p and
then go to Step 1. 3.
If tolerance i, for same i are not acceptable then change
i
p as desired and
go to Step 3. 4.
If the objective should be considered as imprecise then to Step 5.
Step 5 After reffering to first table, the decision maker is then asked for his
subjective goal b
and its tolerance p
for solving a symmetric FLP problem. If the decision maker does not like to give his goal for the fuzzy objective, to to
Step 6. If b
is given, go to Step 8.
79
Step 5 Solve problem of 15. A unique Werners’s solution is the provided. Step 7 Is the solution of 5.16 satisfying ? Consider the following cases:
1. If the solution is satisfied then print out results and stop. 2. If the user has realized hisher goal then give the goal
b and go to Step 8.
3. If resource i, for some i are idle then decrease
i
b and change
i
p and then
go to Step 1. 4. If tolerance i, for same i are not acceptable the change
i
p as desired and go
to Step 3.
Step 8 Is p
determined by the decision maker ? If the decision maker would like to specify
p , we should provide a table to help the decision maker. Then go to
Step 9. If p
is not given, then go to Step 11.
Step 9 Solve problem of 5.21. A unique Zimmermann’s solution is obtained. Step 10 Is the solution of 5.21 satisfying ?
1. If solution is satisfied then print out results and stop. 2. If the user has realized beter hisher goal and its tolerance then give the
goal b
and p
and go to Step 8. 3. If resource i, for some i are idle then decrease
i
b and change
i
p and then
go to Step 1. 4. If tolerance i, for same i are not acceptable the change
i
p as desired and go
to Step 3.
Step 11 Solve last problem. That is, call Step 9 to solve problem of 5.21 for a set
of s
p . Then the solutions are depicted in a table.
Step 12 Are the solution satisfying ? If yes, print out the solution and then
terminate the solution procedure. Otherwise, go to Step 13.
80
Step 13 Ask the decision maker to specify the refined p
, and then go to Step 0. It is rather reasonable to ask the decision maker
p at this step, because he has a
good idea about p
now figure 5.1. For implementing the above IFLP, we need only two solution-finding techniques,
the simplex method and parametric method. Therefore, the IFLP approach proposed here can be easily programmed in a PC system for its simplicity, Lai and
Hwang [92].
5.4 Portfolio problem
