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The aim of this study is to present an alternative approach for solving the multi-objective posynomial geometric programming problems. The proposed approach minimizes the weighted objective function comes from multi-objective geometric programming problem subject to constraints which constructed by using Kuhn-Tucker Conditions. A new nonlinear problem formed by this approach is solved iteratively. The solution of this approach gives the Pareto optimal solution for the multi-objective posynomial geometric programming problem. To demonstrate the performance of this approach, a problem which was solved with a weighted mean method by Ojha and Biswal (2010) is used. The comparison of solutions between two methods shows that similar results are obtained. In this manner, the proposed approach can be used as an alternative of weighted mean method.

Geometric Programming Problem (GPP) is a special type of nonlinear programming that often used in the applications for production planning, personal allocation, distribution, risk managements, chemical process designs and other engineer design situations. GPP is a special technique that is developed in order to find the optimum values of posynomial and signomial functions. In the classical optimization technique, a system of nonlinear equations is generally faced after taking partial derivatives for each variable and equalizing them to zero. Since the objective function and the constraints in the GPPs will be in posynomial or signomial structures, the solution of the system of nonlinear equations obtained by the classic optimization technique will be very difficult. The solution to the GPP follows the opposite method with respect to the classical optimization technique and it depends on the technique of first finding the weight values and calculating the optimum value for the objective function, then finding the values of the decision variables.

GPP has been known and used in various fields since 1960. GPP started to be modeling as part of nonlinear optimization by Zener [

When there are multiple objectives in the GPP, the problem is defined as the Multi-Objective Geometric Programming Problem (MOGPP). In general, there are two types (namely fuzzy GPP and weighted mean method) of solving approaches are exist in the literature. The studies deal with fuzzy GPP method can be given as Nasseri and Alizadeh [

Numerical approximations are widely used to solve the Multi-objective programming problems. One of the numerical approximations is the Taylor series expansion which is also given as a solution method in this study. Toksarı [

In this study, a numerical approach to solve the multi-objective posynomial geometric programming problems is proposed. This numerical approach minimizes the weighted objective function subject to Kuhn-Tucker Conditions expanded the first order Taylor series expansion about any arbitrary initial feasible solution. The same process is continued iteratively until the desired accuracy is achieved. The solution obtained at the end of the iterative processing gives the pareto optimal solution to solve the multi-objective posynomial geometric programming problem. When the results obtained are compared to the results of the weighted mean method [

In the next section of this study, MOGPP, weighted method for MOGPP and dual form of MOGPP are respectively mathematically explained. In the third section, the model that we suggest depending on the Kuhn-Tucker Conditions and first order Taylor Series expansion will be clarified. Then, the results obtained by weighted mean method and the results obtained by the approach that we suggest will be compared for a numeric example. In the last section, conclusion and comments will be included.

Let

where

where

GPP is a problem with generalized posynomial objective and inequality constraints, and monomial equality constraints. Standard form of a GPP can be written as

where

GPP in standard form is not a convex optimization problem. GP is a nonlinear, nonconvex optimization problem that can be logarithmic transformed into a nonlinear, convex problem.

Assuming for simplicity that the generalized posynomials involved are ordinary posynomials, it can express a GPP clearly, in the so-called standard form:

where

Most of these posynomial type GPP’s have zero or positive degrees of difficulty. Parameters of GPP, except for exponents, are all positive and called posynomial problems. GPP’s with some negative parameters are also called signomial problems.

The degree of difficulty is defined as the number of terms minus the number of variables minus one, and is equal to the dimension of the dual problem. If the degree of difficulty is zero, the problem can be solved analytically. If the degree of difficulty is positive, then the dual feasible region must be searched to maximize the dual objective, and if the degree of difficulty is negative, the dual constraints may be inconsistent [

GPP in standard form is not a convex optimization problem. GPP is a nonlinear, nonconvex optimization problem that can be logarithmic transformed into a nonlinear, convex problem.

General form of multi objective GPP, where p is the number of objective functions which are minimized and n is the number of positive decision variables, is defined as:

where

Definition 1

Definition 2

General form of multi objective optimization problem can be mathematically stated as:

where

A multi-objective problem is often solved by combining its multiple objectives into one single-objective scalar function. This approach is in general known as the weighted-sum or scalarization method. In more detail, the weighted-sum method minimizes a positively weighted convex sum of the objectives, that is,

that represents a new optimization problem with a single objective function. We denote the above minimization problem with

The following result by Geoffrion [

In order to the above MOGPP defined in problem (5) consider the following procedure of the weighting method, a new minimization type objective function

The basic mathematical programming problem is that of choosing values of variables so as to minimize a function of those variables subject to

This problem is a generalization of the classical optimization problem, since equality constraints are a special case of inequality constraints. By

The solution to problem (10) is then analogous to the Lagrange theorem for classical optimization problems. The Lagrange theory for a classical optimization problem can be extended to problem (10) by the following theorem.

Theorem 4.1 Assume that

The conditions (11) are necessary conditions for a local minimum of problem. The conditions (11) are called the Kuhn-Tucker conditions.

For proof of theorem, the Lagrange function can be defined as:

The necessary conditions for its local minimum are

The conditions (11) are obtained from the conditions (12)-(15) [

When there are inequalities constraints in nonlinear optimization problems, Kuhn-Tucker Conditions can be used which are based on Lagrange multipliers. The Kuhn-Tucker Conditions satisfy the necessary and sufficient conditions for a local optimum point to be a global optimum point [

The multi-objective geometric problem (5) as a single objective function using the weighting method can be rewritten as follows:

The above problem (16) may be slightly modified by introducing new variables

Assume that

ferentiable. The new function is formed by introducing

where at the point

Since the necessary conditions (17) are also the sufficient conditions for a minimum problem if the objective function of the geometric programming pro- blem (19) is convex. Therefore, optimal solution of the problem (19) gives the solution of the problem (16).

The above problem (19) is nonlinear problem since both the objective function and the constraints are nonlinear. We will use the Taylor theorem for the linearization to the problem (19). Let be both the objective function and the constraints have differentiable. Then they are expanded using the Taylor theorem about any arbitrary initial feasible solution

The linear approximation problem is solved, giving an optimal solution

The steps for the proposed solution algorithm are given below:

Step 1: Formulate the given MOGPP is as a single objective GP using the weighting method.

Step 2: Construct the constraints for the new problem from Kuhn-Tucker conditions.

Step 3: Set the nonlinear model taking the single objective function in step 1 and the constraints in step 2 to MOGPP.

Step 4:

Step 5: Expanded both the objective function and constraints of the problem obtained in step 3 using first order Taylor polynomial series about

Step 6: Solve the problem in step 5. Calculate to the approximate solution

Step 7: For eps > 0 and eps as close to 0 as possible, if

Numerical example

To illustrate the proposed model we consider the following problem which is also used in [

Find

subject to

The primal problem above can be written as below:

subject to

Using the weights

subject to

where

In this problem, the primal term number is 8, primal variable number is 4 and thus the degree of difficulty is 3.

The dual problem corresponding to the last primal problem is given below:

subject to

Problem 1 will now be solved using the proposed model. The value interval for

subject to

Then, the above problem according to the Kuhn-Tucker Conditions can be formulated as in Model 1 as follows:

From Equation (21) the problem is written as follows:

subject to

To linearize the nonlinear objective function with the nonlinear constraints in the above problem, we use the first order Taylor polynomial series at any initial feasible point

as follows:

subject to

The solution of the above problem is

When the same procedure is applied to point

By considering different values of

In this study, we proposed an alternative approach to the approximate pare to solution of MOGPP based on the weighting method. In this model, MOGPP has been reduced to a sequential linear programming problem and the Pareto optimal solution of MOGPP has been calculated approximately in an easier and more speedy way. Besides in GP problems and MOGPP the solution becomes more difficult when the degree of difficulty is a positive number whereas such a difficulty does not exist in the developed model. The solution for the problem given in the example by the weighted mean method is shown in

Variables | ||||||||
---|---|---|---|---|---|---|---|---|

0 | 5 | 3 | 7 | 6 | 2 | 10 | 88 | 0.009524 |

1 | 5.140764 | 2.470357 | 7.523979 | 5.590474 | 2.871 | 14.708 | 86.543491 | 0.010466 |

2 | 5.091219 | 2.661165 | 7.349591 | 5.737520 | 2.8738 | 14.686 | 87.849933 | 0.010043 |

3 | 5.084131 | 2.682310 | 7.332497 | 5.748260 | 2.874 | 14.65986 | 87.986130 | 0.010001 |

4 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 2.874 | 14.65962 | 87.987763 | 0.010000 |

5 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 2.874 | 14.65962 | 87.987464 | 0.010000 |

6 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 2.874 | 14.65962 | 87.987764 | 0.010000 |

Variables | ||||||||
---|---|---|---|---|---|---|---|---|

0.1 | 0.9 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987764 | 0.01000 | 5 |

0.2 | 0.8 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987764 | 0.01000 | 5 |

0.3 | 0.7 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987762 | 0.01000 | 5 |

0.4 | 0.6 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987764 | 0.01000 | 5 |

0.5 | 0.5 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987764 | 0.01000 | 6 |

0.6 | 0.4 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987764 | 0.01000 | 4 |

0.7 | 0.3 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987764 | 0.01000 | 5 |

0.8 | 0.2 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987764 | 0.01000 | 5 |

0.9 | 0.1 | 5.084056 | 2.682555 | 7.332314 | 5.748367 | 87.987764 | 0.01000 | 5 |

Variables | ||||||
---|---|---|---|---|---|---|

0.1 | 0.9 | 5.084055 | 2.682555 | 7.332315 | 5.748367 | 8.08776 |

0.2 | 0.8 | 5.084055 | 2.682555 | 7.332315 | 5.748367 | 8.08776 |

0.3 | 0.7 | 5.084055 | 2.682555 | 7.332315 | 5.748367 | 8.08776 |

0.4 | 0.6 | 5.084055 | 2.682555 | 7.332315 | 5.748367 | 8.08776 |

0.5 | 0.5 | 5.084055 | 2.682555 | 7.332315 | 5.748367 | 8.08776 |

solution by the model that we developed is shown in

Öz, E., Güzel, N. and Alp, S. (2017) An Alternative Approach to the Solution of Multi-Objective Geometric Programming Problems. Open Journal of Optimization, 6, 11-25. https://doi.org/10.4236/ojop.2017.61002