1. Introduction
Nowadays, mathematical modeling is the most considered tool in simulation and has inevitable applications in many fields, especially in financial management, economics, wireless networking, and engineering in general [
1,
2,
3]. In this respect, several techniques have been recommended to minimize the cost and maximize the profits. Mathematical models are classified into two general categories: (I) linear programming (LP) problems and (II) geometric programming (GP) problems that can be defined on crisp and fuzzy numbers. Usually, we use crisp numbers in academic cases, in order to learn the concept of LP and GP problems as well as obtain the optimal solution to them in different forms [
4,
5]. Realistically, most things are not crisp. Non-crisp things have different definitions in various cases, thus fuzzy sets were introduced into mathematical modeling to obtain more accurate results.
At first, Zadeh [
6] presented fuzzy sets theory and the concept of fuzzy numbers. Subsequently, the first formulation of fuzzy mathematical programming was proposed by Tanaka et al. [
7] in 1974. Afterward, Zimmermann [
8] introduced the fundamental mathematical formulation of fuzzy linear programming problems. Additionally, the authors in [
9] introduced an approach for consensus modeling in collaborative and distributed design, in which the design cluster was defined as a fuzzy evaluation relationship between a group of functional requirements and a group of conjecture regions. In [
10], a new algorithm was proposed for solving the fully fuzzy linear programming problem. This was based on a new lexicographic ordering on fuzzy triangular numbers, by converting it to its equivalent multi-objective linear programming problem. This was explored by several researchers in diverse fields [
11,
12].
In order to solve decision-making problems, Garg [
13] developed a nonlinear programming model based on the technique of order preference by similarity to an ideal solution and in which the criterion values and their importance are given in the form of interval neutrosophic numbers. The comprehension of decision making on fuzzy sets was proposed by Bellman and Zadeh [
14]. However, the meaning of decision making does not necessarily imply finding the optimum solution for the programming problem. On the contrary, the decision maker wants to get to its desired level of satisfaction, which might not precisely maximize or minimize the objective function. Therefore, the effect of the constraint is not the same as the classical form [
15,
16]. Initial efforts in multilevel decision making are mainly aimed on determining the optimal conditions and solution algorithms to solve linear, nonlinear, and discrete fundamental problems, wherein for each decision level there is only one decision that is assigned for optimizing the unique objective. Faísca et al. [
17] studied a multi-parametric programming approach for solving trilevel hierarchical problems. Ranking function on fuzzy numbers was taken into great consideration due to the fact that fuzzy numbers are not comparable to each other. So, many researchers are utilizing the ranking function to compare fuzzy numbers, especially in network systems, decision making, and measuring the fuzzy paths [
18,
19]. In [
20], multi-parametric programming is suggested to find an optimal solution in a linear programming problem.
However, mathematical modeling in the case of linear programming was not adequate in a realistic application to formulate the situation and reach the expected goal. Geometric programming, which is an extension type of linear programming, was created to make an accurate formulation of the problem. In 1987, the author in [
21] developed the geometric programming problem for the fuzzy mathematical problem, which consists of more information compared to the classical models. After that, many authors worked on fuzzy non-linear programming and fuzzy GP problems to obtain an optimal solution with fewer errors and shortages, in line with the different types of fuzzy numbers [
22,
23,
24]. Jafarian et al. [
25] proposed a novel method to support the process of solving multi-objective nonlinear programming problems, which are subjected to strict or flexible constraints. The concept of intuitionistic fuzzy sets is integrated into the solving procedure and continuously interacts with the decision maker. Ruan et al. [
26] discussed the optimal conditions and related geometric properties of a linear trilevel decision problem with dominated objective functions. Lai [
27] proposed a fuzzy approach for finding a satisfactory solution for the linear multilevel decision problem and used the concepts of membership functions as well as the satisfaction degree of individual decision power. Shih et al. [
28] extended Lai’s concepts and adopted tolerance membership functions and multiple objective optimizations in order to find a better solution for the above problem. Sakawa et al. [
29] presented an interactive fuzzy programming approach for linear multilevel decision problems. This is accomplished by updating the satisfaction degrees of the decision maker by considering the overall satisfaction balance at all levels. Their interactive fuzzy programming approach is dominated with inconsistency between the fuzzy goals of objectives and decision variables that existed in the research developed by Lai [
27] and Shih et al. [
28].
In this paper, in order to improve the satisfaction degree in the solution of a fuzzy geometric programming problem that has various applications in the real world, a novel multi-parametric vector is proposed. In addition, the ranking function is also applied to the fuzzy numbers to make a correlation between coefficients and the exponents and find the maximum satisfaction degree under the optimal value. The main properties of this method are: (i) giving more flexibility to decision maker, (ii) designing a new satisfaction degree in a fuzzy geometric programming problem, (iii) turning the objective function into a constraint, which makes one more restriction on a feasible solution, (iv) creating a precise fuzzy geometric programming problem under a decision making tolerance value, (v) satisfying the maximum of the decision maker’s desire, (vi) increasing the decision-making power of the decision maker, (vii) the suggested approach in this study is applicable and utilized.
As a whole society and industry are collectively seeking newly optimized methods for staying competitive, complex problems in this direction can be solved through GP. Therefore, in mathematical programming, fuzzy GP is mainly more applicable in modern industries for carrying out the decision making. The moderator and decision makers tend to provide results through GP that was not easy in beforehand. Decision maker adds a limitation on the region of the feasible solution, which has the role of a condition. In this way, we need to formulate the programming problem toward the decision maker’s desire.
The rest of this paper is organized as follows:
Section 2 reviews some definitions and theorems that are concerned with the geometric programming problem, fuzzy sets, and some related remarks. In
Section 3, we introduce the concept of confidence level and satisfaction degree. For the sake of this definition, we propose a multi-parametric vector
to compute the optimal solution and optimal value for the fuzzy geometric programming problem, and suggest its membership function and survey the theorem. In
Section 4, the two-phase method is proposed for solving the fuzzy GP problem under the proposed
. In
Section 5, previous works are investigated and compared with the proposed method and the advantages of the proposed method are considered. In
Section 6, with the help of a numerical example, a technical method is illustrated and compared with the non-multi-parametric vector
while being scrutinized. Finally, we draw our conclusions in
Section 7.
3. New Concepts of Feasibility and Efficiency
In this section, a multi-parametric operator
is presented that indicates the confidence level extracted from the feasibility and efficiency of the optimal solution. The related properties and theorems are considered and the membership function is presented. There is a discussion about the tolerance level. This tolerance value is exerted in the programming problem, which is formulated as a novel membership function. This circumscription is effected as a condition and can play an important role in achieving the feasible solution. Also, the feasible solution will be closer to the decision maker’s satisfaction. Assuming problem (
2), and supposing that the
denotes all
of fuzzy constraints that is related to the fuzzy inequalities constraint
, then the membership function is defined as follows:
where
is the maximum tolerance value of the
i-th fuzzy function
that is determined by the decision maker. This tolerance value is the decision maker’s command, which makes the GP programming more complicated. In this approach, different tolerance values make several categories of feasible solutions. Thus, selecting one tolerance value, which is decided during the decision making, trying to satisfy the decision maker, and subsequently increasing his satisfaction degree, also increases the efficiency level. Actually, the decision maker can attain groups of the optimal solution under the various tolerance values
, so with the help of the multi-parametric vector
, the difficulty of obtaining the satisfactory solution can be controlled towards arriving at an optimal solution in line with the desire of the decision maker.
By contemplating the problem (
3), a confidence level
that is denoted as a multi-parametric vector
is applied to the programming problem, where
stands for the satisfaction degree relevant to the objective function, which will be changed into the constraint, and
for
denotes the satisfaction degree relevant to each constraint. Here, a multi-parametric
is proposed to increase the satisfaction degree of the decision maker under his tolerance value. The point is that the confidence value is applied to any constraint, specifically to arrive at an optimal satisfaction degree. Moreover, the tolerance value is enforced on the objective function that increases the efficiency of the optimal solution in a fuzzy geometric programming problem. Suppose that
is a continuous and monotone fuzzy function. Then the fuzzy reversed PGP problem is equivalent to
where every
is defined on its own corresponding constraints and objective function. For the purpose of managing the membership function in any distinct fuzzy constraints and objective function, if
and
, then the
i-th constraint is satisfied. But if we choose
as the maximum tolerance determined by the decision maker and
, then the
i-th constraint has been breached. In this case, if we consider PGP as
and by turning the objective function into a constraint, a membership function will be defined for it exclusively. Therefore, the membership function can be defined as follows:
and
Here, the multi-parameter vector
is applied to a constraint and objective function. The promising framework of this method is to add the objective function as a constraint, which is capable of being influenced by the restriction of the feasible solution directly. On the other hand, by employing this technique, the solution area is going to be limited and the optimal solution will be more accurate. The parameter
is relevant to the
i-th fuzzy constraint and
is the satisfaction degree, which is defined on the objective function. In order to find the optimal solution with respect to the decision maker’s satisfaction with the maximum tolerance level under problem (
3), the following model is proposed:
Definition 15. If is an m-dimensional vector andthen a vector will be an α-feasible solution for problem (2). Let
be an
-feasible solution of the fuzzy geometric programming problem (
2), i.e., the satisfaction degree
is the same for all constraints and objective functions. So that for problem (
9), we have
, or on the other hand,
. Then
is an
-feasibility to problem (
2). Whereas we can say that for the GP problem (
2), if it is feasible, then
is non-empty.
Theorem 5. Suppose that , and for , is an α-feasible solution to problem (2). Then can be an α-efficient optimal solution iff is an optimal solution of following programming problem:where is the maximum tolerance amount. Proof. Suppose that
, and for
,
is an
-feasible solution to problem (
2). With the help of Definition 15 and programming problem (
11), we have
, so
is a feasible solution. On the other hand, as
is an
-efficient solution, there will be no other
that satisfies
. Therefore, this means that
is an optimal solution.
Again, suppose that
is an optimal solution for problem (
14), and obviously, it is an
-feasible solution. Thus, it implies that
is an
-efficient solution.
Now, assume that
is an optimal solution to problem (
14) in Theorem 5. Just the following programming problem needs to be solved:
where
is the maximum tolerance amount. □
Note that the mentioned tolerance,
, for every constraint and objective function will be determined by an expert in the real-life problems that are adapted for practical situations.
Remark 4. It is worth mentioning that any real (crisp) number could be written as a fuzzy number with the right and left zero endpoints of the intervals.
4. The Main Process of the Two-Phase Method
The process of the method which was proposed will be discussed in this section. In this case, the method is described in two steps as follows.
The first phase is concerned with making a suitable GP problem that can be solved. In phase I, Theorems 3 and 4 are applied to the GP problem in order to build a correct linear programming problem that is able to set the tolerance value on it. Here, the decision maker applies his demand, so that it is likely to satisfy him. The decision maker can select different levels of tolerance value, which gives different groups of feasible solutions, therefore, among these feasible solutions, we need to create a technique to find the optimal solution.
Phase II starts with a given feasible solution from phase I. The target of phase II is to improve the satisfaction degree with an optimal solution. In the current method, the decision maker adds the general order that sometimes is not useful or does not coincide for every constraint. Definitely, finding an optimal solution is preceded over the decision maker’s desire in solving a fuzzy geometric programming problem. In the cases where the satisfaction level and the tolerance value are the same for all the contrarians, the decision maker will not be satisfied with the optimal solution because this is a kind of push for the decision maker to embrace the optimal solution. If one is more adventurous, the preference is to reduce and restrict the interval of the feasible solution to gain a better optimal value with the maximum satisfaction degree of the decision maker. The order and limitation can be chosen and assigned separately for every constraint, and even for the objective function. The multi-parametric confidence vector is applied to match the satisfaction degree with its corresponding condition.
In this step, the objective function is turned into a constraint with its corresponding initial solution and satisfaction degree . By the genesis of the tolerance degree, , it can be excelled for every constraint and objective function separately so that the satisfaction degree can be maximized in each constraint. In the end, the optimal solution is achieved for the original problem with the best satisfaction degree.
In the first step, suppose that the GP is the fuzzy posynomial geometric programming problem in (
10). By applying Theorems 3 and 4 and setting
, where
,
, then we have
In the next step, we should compose problem (
14) according to Theorem 5 so that, the multi-parametric vector
is used in the objective function and constraints. In addition, we need to predict the decision maker’s satisfaction. As for these two criteria, the mentioned geometric programming problem can be turned into
Now it is easy to find the optimal solution with the current software (MATLAB).
The method is depicted as a flowchart in
Figure 1.
6. Illustration of the Superiority of the Proposed Method
In this section, an instance of the fuzzy geometric programming problem to illustrate the proposed method is given. First, we solve the example with the multi-parametric vector
and then express the superiority of the multi-parametric vector
over the one-dimensional vector
.
Example 1. A septic productive factory produces concrete pipes for its projects. This factory needs to produce utmost , and (kg) of three types of concrete pipes , , and , respectively. The pipes have to be produced by using four different kinds of concrete , , , and . The percentage of each type of raw concrete needed in each pipe (kg) and its unit price ($/kg) are listed in Table 1. Find the maximum raw concrete needed under the owner’s tolerance level. Solution. This problem can be induced in the fuzzy form of a PGP problem as follows:
Here, we perform the steps of the method as expressed above.
Due to the definition of the membership function, for constraints and the objective function, we have the following membership functions, respectively:
Utilize
,
. Problem (
18) is changed into the following fuzzy problem under Theorems 3 and 4:
The first step is to apply the ranking function on trapezoidal fuzzy numbers under the Remark 3.
The initial optimal solution is taken from the basic variables as , , , and the optimal value equals .
From the first step, by applying , we find .
According to the two-phase method, we can change problem (
22) into the following programming problem:
Suppose that
, and
are the maximum toleration values for
, which is given by decision maker, then substitute
and
into problem (
23). We have:
Table 2 shows the decision maker’s satisfaction at different
-efficiency confidence levels. If
and
denote the optimal value of the objective function at each step under different
s, we can have the table as follows, which is procured by the MATLAB software:
The values observed in
Table 2 demonstrates that
and
are the most inefficient components. From columns 2, 3, and 5 in
Table 2, it can be deduced that by increasing
and
, the optimal solution will be strayed away. Reducing
and
achieves a better result. Again from columns 1 and 4 in
Table 2, one can get closer to the optimal solution by increasing
and
. Now from here, we will try to minimize
and
by choosing the
-efficient solution with optimal value
as an initial solution.
Figure out the LP problem below affected by
:
By maximizing the satisfaction degree, the optimal solution of problem (
25) will be obtained as
under the confidence level
and the optimal value computed with respect to
as
. Subsequently, the optimal solution for GP programming problem (
18) will be
,
,
, and the optimal value will be equal to
.
To compare this GP problem when the vector
is not multi-parametric, see
Table 3.
A comparison of
Table 2 and
Table 3 reveals that in
Table 3, by increasing the confidence level
, the optimal solution is minimized, whereas we expect to maximize the optimal solution when the confidence level rises. Indeed, the satisfaction degree for each constraint should be increased or decreased by the same amount, while in the proposed method, there is a chance to decrease or increase the satisfaction level for each constraint separately. In addition, an opportunity lies in dedicating a satisfaction level to the objective function. In
Table 3, it is obvious that by increasing the confidence level, we get opposite results for the optimization, and the optimal value goes to its normal optimal solution, while in
Table 2 the optimal value is kept optimal, as long as the satisfaction degree components are changing. The reason is that the vector
is not appropriate for all constraints, and particularly with turning the objective function to the constraint. In
Table 3,
can be noted as
for
, whereas in
Table 2, the satisfaction degree is
. This implies that the one-dimensional satisfaction degree is meaningless in this case. It is clear that the flexibility of the
dimensional confidence level
can help to reach a better optimal solution that implements decision maker’s goal.