SOLVING GEOMETRIC PROGRAMMING PROBLEMS WITH TRIANGULAR AND TRAPEZOIDAL UNCERTAINTY DISTRIBUTIONS

. The geometric programming problem is an important optimization technique that is often used to solve different nonlinear optimization problems and engineering problems. The geometric programming models that are commonly used are generally based on deterministic and accurate parameters. However, it is observed that in real-world geometric programming problems, the parameters are frequently inaccurate and ambiguous. In this paper, we consider chance-constrained geometric programming problems with uncertain coefficients and with geometric programming techniques in the uncertain-based framework. We show that the associated chance-constrained uncertain geometric programming problem can be converted into a crisp geometric programming problem by using triangular and trapezoidal uncertainty distributions for the uncertain variables. The main aim of this paper is to provide the solution procedures for geometric programming problems under triangular and trape-zoidal uncertainty distributions. To show how well the procedures and algorithms work, two numerical examples and an application in the inventory model are given.


Introduction
Geometric programming (GP) is a powerful optimization tool to solve various nonlinear optimization problems under posynomial constraints.In 1967, Duffin et al. [8] developed the fundamental theories of the GP technique.Subsequently, this technique is used by many researchers to solve various engineering design problems such as circuit design, inventory modelling, production planning, risk management, chemical processing, information theory, and structural design [2-6, 10-12, 15, 24, 25, 33].
For the traditional GP problems, the coefficients of cost and constraint functions are precisely known.However, it is observed that the values of the parameters in real-world GP problems may be imprecise and ambiguous.As a result, numerous strategies in connection with inaccurate and inconclusive data in GP models are proposed.
Stochastic GP was established by Avriel and Wilde [1], with deterministic exponents along with the coefficients, which are nonnegative random variables.Peykani et al. [20] introduced a novel Fuzzy Data Envelopment Analysis (FDEA) model by the use of possibility, necessity, and credibility approaches.They displayed comparative research work between the FDEA model and the Robust Fuzzy Data Envelopment Analysis (RFDEA) model.Shiraz et al. [29] studied a chance-constrained Data Envelopment Analysis (DEA) model with randomrough data inputs and outputs and investigated the stability and robustness through sensitivity analysis.The authors considered a chance-constrained DEA model with random and rough input and output data.They proposed a deterministic equivalent model with quadratic constraints to solve the model.
Liu [16] developed a strategy to find out the range of the values of the posynomial objective function for the GP problems with interval coefficients and exponents.Subsequently, the same author [17] solved the GP problems with fuzzy coefficients and exponents.
Liu [18] worked on uncertainty theory, which is a new area of mathematics.Solano-Charris et al. [32] solved the bi-objective robust vehicle routing problem with uncertain costs and demands by nontraditional methods such as the Non-dominated Sorting Genetic Algorithm (NSGAII) and the Multiobjective Evolutionary Algorithm (MOEA).The authors considered it a robust optimization problem with uncertain demands and travel times.They used the min-max optimization criterion to address the uncertainties in the objective function.Shiraz et al. [27] constructed the GP problems based on an uncertainty frame model with the cost and constraint coefficients which follow normal, linear, and zigzag uncertainty distributions.Subsequently, Shiraz et al. [28] solved the fuzzy chance-constrained GP problems with possibility, necessity, and credibility approaches.Shiraz and Fukuyama [26] later developed the GP problems using the rough set theory.Recently, Shiraz et al. [30] used the copula theory approach to solve the stochastic GP problems.The authors considered a joint chanceconstrained GP problem with copula and converted it into deterministic convex programming.They used sequential convex approximation and piecewise tangent approximation to find the lower and upper limits of the objective function.
From the literature survey, we see that much research has been done on the traditional GP problems with precise coefficients of posynomial objective and constraint functions.However, in real-world GP problems, the coefficients may be imprecise.The GP problems with uncertainty have been solved under normal, linear, zigzag uncertainty distributions, and L-R fuzzy and rough fuzzy environments.However, the GP problems may have triangular and trapezoidal uncertainty in the coefficients of posynomial objective and constraint functions.There is no previous research work on the GP problems when the coefficients of cost and constraint functions follow triangular and trapezoidal uncertainty distributions.Thus, we make an attempt to study the GP problems with triangular and trapezoidal uncertainty distributions.The main contributions to this paper are the followings.
-We introduce the uncertain GP problems along with the coefficients of cost and constraint functions, which are independent uncertain variables.-The equivalent chance-constrained GP problem is developed under triangular and trapezoidal uncertainty distributions.-We show how the uncertain chance-constrained GP models can be converted into crisp GP models and, consequently, the crisp GP models can be solved in a conventional way.-Two numerical examples and one application in an inventory model are incorporated to validate our proposed method.
The organization of the rest of the paper is as follows.
The preliminaries are discussed in Section 2. In Section 3, we consider the chance-constrained GP problems with triangular and trapezoidal uncertain coefficients and show how the chance-constrained uncertain GP models can be converted into crisp GP models.In Section 4, two numerical examples are given to demonstrate the efficiency of the procedures.Furthermore, an application based problem in an inventory model is solved under triangular and trapezoidal uncertainty distributions in Section 5. Finally, a conclusion on the work is incorporated in Section 6.

Preliminaries
In this section, we discuss some basic concepts of uncertainty theory and credibility theory referred to [18].
Definition 2.14.A UV  is said to be trapezoidal if and only if it has UD of the form where , , ,  ∈ R,  <  <  < .In notation, it can be written as  ∼  (, , , ).
The graphical representation of trapezoidal UD is given in Figure 2.
Definition 2.15.If  is a UV, then the mean value of  is provided at least one of the two integral is finite.

Chance-constrained geometric programming model with uncertain coefficients
A traditional GP problem permits solving a minimization problem where the objective function as well as the constraints are posynomials.GP problems whose coefficients and decision variables are all positive except the exponents, are called posynomial problems.A classical primal GP model and its dual form are defined as follows.
where   > 0,   > 0,   ∈ R, ∀, , .Here,  0 and   are the total number of terms present in the objective function and  ℎ constraints respectively.
To define the dual problem of the above primal problem (3.1), let  =  ∑︀ =0   be the total numbers of terms presents in the primal GP problem (3.1) and   be the dual variables such that Here is the relationship between primal and dual problems.
Primal-dual relationship.The relationship between primal and dual GP problems due to strong duality theorem [7,8] is as follows.
Now, depending on  , we have the following two cases.
Case I.If  ≥  + 1, then the dual problem given in equation (3.2) is feasible as the number of equations is less than or equal to the number of dual variables, which guarantees the existence of solution for dual variables (Beightler and Philips [3]).Case II.If  <  + 1, then the dual problem given in equation (3.2) is inconsistent as the number of equations is greater than the number of dual variables.It guarantees that there is no analytical solution for dual variables.However, an approximate solution can be found by least square or linear programming method (Sinha et al. [31]).Now, we expand a chance-constrained GP model with uncertain coefficients.First, we convert the standard GP model given in equation (3.1) in an uncertain GP model as where ̃︁   are UVs,   > 0,   ∈ R, ∀, , .The chance-constrained uncertain GP model can be formulated based on equation (3.4) where  ∈ (0, 1) is predefined uncertainty level,   > 0,  = 1, 2, . . ., .In the following two subsections, we develop the equivalent crisp primal-dual GP model of the above uncertain model (3.5) under triangular and trapezoidal UDs.

Uncertain chance-constrained geometric programming problem with triangular uncertainty distribution
We propose a solving technique for the chance-constrained uncertain GP problem given in equation (3.5) considering the coefficients to be UVs with triangular UDs.Let the coefficients ̃︁   given in equation (3.5) be independent triangular UVs i.e., ̃︁   ∼ ∇(  ,   ,   ).The following lemmas are helpful to transform the chance-constrained uncertain GP model given in equation (3.5) to a crisp or deterministic model.Accordingly, we solve it by converting into the dual form.

3
. Subsequently, by Lemma (3.2), the constraints given in equation (3.5) can be written as where Equations (3.6) and (3.7) are the objective function and the constraints of the transformed crisp primal GP model, respectively.Therefore, the equivalent crisp GP model of (3.5)

Uncertain chance-constrained geometric programming problem with trapezoidal uncertainty distribution
We propose a solving technique for the chance-constrained uncertain GP problem given in equation (3.5).We consider the coefficients to be UVs with trapezoidal UDs.Let the coefficients ̃︁   given in equation (3.5) be independent trapezoidal UVs i.e., ̃︁   ∼  (  ,   ,   ,   ).Similar to the previous discussion, we present the following two lemmas that transform the chance-constrained uncertain GP model given in equation (3.5) to a crisp or deterministic model and, hence, we solve it by converting into the dual form as earlier.

Numerical examples
Here, we present two numerical examples to demonstrate the efficiency of the triangular and trapezoidal uncertain GP models.
where all the coefficients are treated as UVs and  1 , Consequently, we solve the dual problem (4.5) for different uncertainty levels  ∈ (0, 1) and provide the primaldual solutions corresponding to different uncertainty levels in Table 2.
The expected values of the objective function under triangular and trapezoidal cases for different uncertainty levels  ∈ (0, 1) are shown in Figure 3.
where all the coefficients are treated as UVs and  1 ,  2 ,  3 > 0.
Triangular case.

Let
The equivalent dual problem is max We determine the values of  11 ,  21 by (3.7) as , if 1 2 ≤  < 1. Subsequently, solve the dual problem (4.8) for different uncertainty levels  ∈ (0, 1) and provide the primal-dual solutions corresponding to different uncertainty levels in Table 3.
The expected values of the objective function under triangular and trapezoidal cases for different uncertainty levels  ∈ (0, 1) are shown in Figure 4.

Application in inventory model
We solve a real application based problem under the uncertainty based framework using GP technique applied in an inventory model.Shiraz et al. [27] solved an uncertain inventory model under normal, linear, and zigzag UDs.We consider similar kind of inventory model under triangular and trapezoidal UDs.In this model, our assumptions are the followings.
-Demand is affected by selling price.
-No shortage is allowed.
-, price per unit in Rs.
-, order quantity in units. (5.1) So, the uncertain inventory model is formulated as max (5.2) In standard form, the model (5.2) is of the form min  −1 s.t. (5.3) Let us assume that  1 ,  2 ,  3 ,  4 be independent UVs.We choose the price elasticity of demand  1 = 1.5 and the cost elasticity of product  3 = 0.02.Therefore, the equivalent deterministic primal GP model is (5.4) Consequently, the dual problem of model (5.4) is max Hence, we solve the dual problem (5.5) for different uncertainty levels  ∈ (0, 1) for triangular UDs.In particular, for the uncertainty levels  = 0.25,  = 0.50, and  = 0.75, we find the optimal price per unit, optimal order quantity in units and maximum profit which are given in Table 5.

Trapezoidal case.
Let ≤  < 1.Hence, as earlier, we solve the dual problem (5.5) for different uncertainty levels  ∈ (0, 1) for trapezoidal UDs.In particular, for the uncertainty levels  = 0.25,  = 0.50, and  = 0.75, we find the optimal price per unit, optimal order quantity in units and maximum profit which are given in the Table 6.
One obvious question from the preceding two cases is what happens if we change the uncertainty level .In that situation, we get a different solution due to the changes in feasible space.Therefore, the decision maker has to be very careful about uncertainty levels and what kind of uncertainty is present in that model.

Conclusion
Geometric programming (GP) is a powerful optimization tool for solving a wide range of optimization and engineering design problems.Traditional GP problems assume that the parameters of GP problems are deterministic and specific.However, in real-world GP problems, the parameters may be uncertain and imprecise.Recently, uncertainty theory has been developed to deal with such kinds of uncertain problems.In this article, we solve an uncertain GP problem under triangular and trapezoidal uncertainty distributions.Here, we develop the equivalent chance-constrained GP models.We solve it conventionally by transforming the uncertain GP models into crisp GP models.Two numerical examples are given to demonstrate the effectiveness of the techniques and algorithms.We solve two numerical GP problems under triangular and trapezoidal uncertainty distributions for different uncertainty levels  and find the corresponding expected values of the cost function.Following that, we plot the expected cost function in relation to the uncertainty level .Lastly, we add an example of how our proposed method could be used in an inventory model to make the most profit.

Example 4 . 1 .
Consider the following GP problem with uncertain coefficients min  0

Figure 3 .
Figure 3. Expected values of objective function under triangular and trapezoidal cases.

Example 4 . 2 .
Let us consider the GP problem with uncertain coefficients as

Figure 4 . 2 𝛽 4
Figure 4. Expected values of objective function under triangular and trapezoidal cases.

Table 1 .
Optimal solutions under triangular UD.Consequently, we solve the dual problem (4.3) for different uncertainty levels  ∈ (0, 1) and provide the primaldual solutions corresponding to different uncertainty levels in

Table 3 .
Optimal solutions under triangular UD.

Table 5 .
Optimal solutions under triangular UD.