In muti-objective geometric programming problem there are more than one objective function. There is no single optimal solution which simultaneously optimizes all the objective functions. Under these conditions the decision makers always search for the most ‘‘preferred’’ solutionو in contrast to the optimal solution. A few mathematical programming methods namely fuzzy programmingو goal programming and weighting methods have been applied in the recent past to find the compromise solution. In this paper _-constraint method has been applied to find the non-inferior solution. A brief solution procedure of constraint method has been presented to find the non-inferior solution of the multi-objective programming problems. Furtherو the multi-objective programming problems is solved by the fuzzy programming technique to find the optimal compromise solution. Finallyو two numerical examples are solved by both the methods and compared with their obtained solutions.

 Multi objective Programming (MOP) has become famous among many researchers due to more practical and realistic applications. 
‎A lot of methods have been proposed especially during the past four decades.‎
‎In this thesis‎, ‎we develop a new algorithm based on a new approach to solve MOP by starting from a utopian point‎,
‎which is usually infeasible‎, ‎and moving towards the feasible region via stepwise movements and a simple continuous interaction with decision maker‎.
‎We consider the case where all objective functions and constraints are linear‎.‎
‎The implementation of the proposed algorithm is demonstrated by two numerical examples‎.

This thesis investigates decentralized bi-level  multi-objective linear programming (DBL-MOLP) problems with a single decision-maker (DM) at the higher level and more than one DM at the lower level. Each DM can have more than one objective function, which is formulated  as a fuzzy goal. To characterize the decision decentralization in a DBL-MOLP- problem, this thesis proposes an assignment scheme of relative satisfaction for the higher-level DM to ensure his leadership and therefore prevent the paradox problem reported in the literature, where lower-level DMs have higher satisfaction degrees than that of the higher-level DM. Through the assignment scheme, if the higher-level DM is not satisfied with the resulting solutions of objective functions, the re-solving process is easily conducted by adjusting the level of relative satisfaction for the associated lower-level DMs. A linearization transformation approach is also presented to facilitate the solution process. To emphasize some important fuzzy goals, a weighting scheme is considered in this thesis. A numerical example  is used for illustration, and comparisons with existing approaches are conducted to demonstrate the feasibility of the proposed method.

It is usually difficult to determine the coefficients of an objective function as a real number since, most often, these possess inherent uncertainty and/or inaccuracy.
Given that this is the usual state, we consider fuzzy-valued objective function as one approach to tackle uncertainty and inaccuracies in the objective function coefficients of mathematical programming models.‎
In this dissertation we consider optimization problems where the objectives are fuzzy functions. For this class of fuzzy optimization problems we discuss the Newton method to find a non-dominated solution. For this purpose, we use the generalized Hukuhara differentiability notion, which is the most general concept of existing differentiability for fuzzy functions. This work improves and correct the Newton Method previously proposed in the literature.

In this thesis ,we present a new modified method for solving multi-level multi objective linear fractional programming problems (ML-MOLFPPs) based on fuzzy goal programming (FGP) approach with some modifications in the algorithm suggested by Baky (2010) which dealt with multi-level multi objective linear programming problem (ML-MOLPP). In proposed modified approach, numerator and denominator function of each objective at each level are individually transformed in to fuzzy goals and their aspiration levels are determined using individual best solutions. Different linear membership functions are defined for numerator and denominator function of each objective function. Then highest degree of each of these membership goals is achieved by minimizing the sum of negative deviational variables. The proposed algorithm simplifies the ML-MOLFPP by eliminating solution preferences by the decision makers at each level, thereby avoiding difficulties associate with multi-level programming problems and decision deadlock situations. The aim of this thesis is to present simple and efficient technique to obtain compromise optimal solution of ML-MOLFP problems .Numerical examples are illustrated in order to support the proposed modified FGP technique.

This thesis is concerned with the study of necessary and sufficient optimality conditions for convex-concave fractional programming problems for which the decision set is the union of convex sets. The lagrangian function for such problems is defined and the Kuhn-tucker saddle and stationary points are characterized. In addition, some important theorems related to the Kuhn-tucker problem for saddle and stationary points are established. Moreover, a general duall problem is formulated, and weak, strong and convers duality theorems are proved. Throughout the presented thesis illustrated examples are given to clarify and implement the developed theory

 Generalized geometric programming (GGP) problems occur frequently in engineering design and
management. Some exponential-based decomposition methods have been developed for solving
global optimization of GGP problems. However, the use of logarithmic/exponential transformations
restricts these methods to handle the problems with strictly positive variables. This paper proposes
a technique for treating non-positive variables with integer powers in GGP problems. By means of
variable transformation, the GGP problem with non-positive variables can be equivalently solved
with another one having positive variables. In addition, we present some computationally efficient
convexification rules for signomial terms to enhance the efficiency of the optimization approach.
Numerical examples are presented to demonstrate the usefulness of the proposed method in GGP
problems with non-positive variables.

‎It is well known that in the standard traffic network equilibrium model with a single value of time (VOT) ‎for all users‎, ‎a so-called marginal-cost toll can drive a user equilibrium flow pattern to a system optimum‎.
‎This result holds when either cost (money) or time units are used in expressing the objective function of the‎system optimum and the criterion for user equilibrium‎. ‎This paper examines the multi-criteria or the costversus-time network equilibrium and system optimum problem in a network with a discrete set of VOTs for‎several user classes‎. ‎Specifically‎, ‎the following questions are investigated‎: ‎Are the user-optimal flows dependent upon the unit (time or money) used in measuring the travel disutility in the presence of road pricing?‎
‎Are there any uniform link tolls across all individuals (link tolls that are identical for all user classes) that can‎support a multi-class user equilibrium flow pattern as a system optimum when the system objective function‎is measured by either money or time units? What are the general properties of the valid toll set?
 

Geometric programming provides a powerful tool for solving a variety of nonlinear optimization problems‎. ‎Many applications of geometric programming are engineering design problems in which some of the problem parameters are imprecise. ‎When the parameters in the problem are interval, ‎the calculated objective value should be interval as well‎. ‎This dissertation develops a procedure to derive the interval objective value of the interval posynomial geometric programming problem when the exponents of decision variables in the objective function‎, ‎the cost and the constraint coefficients‎, ‎and the right-hand sides are interval numbers‎. First we transform the interval geometric programming problem into a pair of two-level of mathematical programs‎. ‎Based on duality theorem, ‎the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs‎. ‎The upper bound and lower bound of the objective value are obtained by solving the pair of geometric programs‎. ‎

In this thesis, we introduce a fuzzy mathematical programming with generalized fuzzy number as objective coefficients.
We also examine a transportation problem with additional restriction. There is an additional entropy objective
function in the transportation problem besides transportation cost objective function. Using new fuzzy mathematical
programming, this multi-objective entropy transportation problem with generalized trapezoidal fuzzy number costs
has been reduced to a primal geometric programming problem. In the end‎, ‎the ‎‎pareto optimal solution of the transportation model
is found and numerical examples are provided to illustrate the effectiveness and methodology of the problem. 

This dissertation presents an iterative parametric approach for solving multiobjective linear fractional programming problems which solves one linear programming problem in each iteration to obtain efficient solutions. This approach parameterizes the objective function and adds constraints to the feasible region in order to guarantee convergency . The number of constraints in each iteration is fixed and feasible region of each iteration is a subset of the feasible region of the previous iteration. An interesting property of this approach is that the number of iterations is independent of the size of problems. In fact it could solve MOLFP problems after at most 2 iterations with admissible tolerance ϵ≥〖10〗^(-8). Two numerical examples are investigated to demonstrate the performance of the present approach.

In this dissertation, a method is suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Based on an order relation of interval number, the uncertain objective function is transformed into two deterministic objective functions, in which the robustness of design is considered. Through a modified possibility degree, the uncertain inequality and equality constraints are changed to deterministic inequality constraints. The two objective functions are converted into a single-objective problem through the linear combination method, and the deterministic inequality constraints are treated with the penalty function method. The intergeneration projection genetic algorithm is employed to solve the finally obtained deterministic and non-constraint optimization problem. Two numerical examples are investigated to demonstrate the effectiveness of the present method

In this thesis, a global optimization algorithm, which relies on the exponential variable transformation of the signomial geometric programming (SGP) and the Lagrangian duality of the transformed programming, is roposed for solving the signomial geometric programming (SGP). The difficulty in utilizing Lagrangian duality within a global optimization context is that the restricted Lagrangian function for a given estimate of the Lagrangian multipliers is often nonconvex. Minimizing a linear underestimation of the restricted Lagrangian overcomes this difficulty and facilitates the use of Lagrangian duality within a global optimization framework. In the new algorithm the lower bounds are obtained by minimizing the linear relaxation of restricted Lagrangian function for a given estimate of the Lagrange multipliers. A branch-and-bound algorithm is presented that relies on these Lagrangian relaxations to rovide lower bounds and on the interval Newton method to facilitate convergence in the neighborhood of the global solution. Computational results show that the algorithm is efficient.

Geometric programming provides a powerful tool for solving a variety of engineering optimization problems‎. ‎Many applications of geometric programming are engineering design problems in which some of the problem parametersare estimates of actual values‎. ‎When the parameters in the problem are fuzzy‎, ‎the calculated objective value should be fuzzy as well‎. ‎This dissertation develops a procedure to derive the fuzzy objective value of the fuzzy posynomial geometric programming problem when the exponents of decision variables in the objective function‎, ‎the cost and the constraint coefficients‎, ‎and the right-hand sides are fuzzy numbers‎. ‎The idea is based on Zadeh's extension principle to transform the fuzzy geometric programming problem into a pair of two-level of mathematical programs‎. ‎Based on duality algorithm‎, ‎the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs‎. ‎The upper bound and lower bound of the objective value are obtained by solving the pair of geometric programs‎. ‎From different values of α‎, ‎the membership function of the objective value is constructed.