This research is aimed to address the optimization of a product portfolio problem under uncertainty using the principles of , nancial portfolios theory. Since the success of a product portfolio is dependent on strategic decision making as well as on future changes of return, the return is best considered when it is deemed an uncertain parameter. The speci, c innovation of this research is the use of a robust optimization approach and providing an Exact Solution Algorithm based on the model of Bertsimas and Sim. Given the uncertainty of the returns, the product portfolio model was developed based on the robust counterpart formulation of Bertsimas and Sim. An Exact Solution Algorithm was also formulated to reduce the Solution time. The results obtained by applying the model to a real case study of the dairy industry in Iran showed that increasing the con, dence level would decrease total returns of the portfolio and increase its total risk. A comparison between the proposed Algorithm and similar methods showed that, on average, it would make 3% improvement in the Solution time.

‎In this paper‎, ‎we present an Exact analytical Solution for five interacting quarks‎. ‎We solve the Schr\"{o}dinger equation for pentaquarks in the framework of five-body and two-body problems‎. ‎For this purpose‎, ‎we utilize Yukawa potential in Jacobi coordinates‎. ‎Also finding the relation between the reduced masses and coupling constants of pentaquarks‎, ‎we obtain the coupling constant of Yukawa potential for pentaquark systems‎. ‎We calculate the energy of these systems in their ground state‎. ‎The results are well consistent with the theoretical results‎. ‎Our procedure to obtain these results is appropriate for other potentials and $n$-body systems‎.

In this work, we extend the existing local fractional Sumudu decomposition method to solve the nonlinear local fractional partial differential equations. Then, we apply this new Algorithm to resolve the nonlinear local fractional gas dynamics equation and nonlinear local fractional Klein-Gordon equation, so we get the desired non-differentiable Exact Solutions. The steps to solve the examples and the results obtained, showed the flexibility of applying this Algorithm, and therefore, it can be applied to similar examples.

In this work, we investigate the the problem of an unsteady tank drainage while considering an isothermal and incompressible Ellis fluid. Exact Solution is gotten for a resulting non-linear PDE (partial differential equation)-subject to proper boundary conditions-. The special cases such as Newtonian, Power law, and as well as Bingham Solution are retrieved from this suggested model of Ellis fluid. Expressions for velocity profile, shear stress on the pipe, volume flux, average velocity, and the relationship between the time vary with the depth of a tank and the time required for complete drainage are obtained. Impacts of different developing parameters on velocity profile vz and depth H(t) are illustrated graphically. The analogy of the Ellis, power law, Newtonian, and Bingham Plastic fluids for the relation of depth with respect to time, unfold that the tank can be empty faster for Ellis fluid as compared to its special cases.

In the present paper, an Exact mathematical Solution is obtained for the nonlinear free transverse vibration of beams for the  rst time. The governing nonlinear partial di erential equation in un-deformed coordinates system is converted in two coupled partial di erential equations in deformed coordinates system. Then, a mathematical explanation is obtained for the nonlinear mode shapes as well as natural frequencies versus geometrical and material properties of the beam. It is shown that as the sth mode of transverse vibration is excited, the 2sth mode of the in-plane vibration will be developed. The results of the present work is compared with those obtained by the Galerkin method and the observed agreement will con rm the Exact mathematical Solution. It is shown that the governing equation is linear in the time domain. As a parameter, amplitude to length ratio ( =l) is proposed to show when the nonlinear terms become dominant in the behavior of structure.

In the present paper, the analytical Solution for an Exact second order fuzzy initial value problem under generalized Hukuhara differentiability is obtained. First the Solution of first order linear fuzzy differential equation under generalized Hukuhara differentiability is investigated using integration factor methods in two cases. The second based on the type of generalized Hukuhara differentiability, the analytical Solution of an Exact second order fuzzy initial value problem is derived in two senses of differentiability. Also, several properties for generalized Hukuhara differentiability are obtained on the topics, such as, the univariate fuzzy chain rules, fuzzy integration by parts, among others. At the end, some illustrative examples are presented and these examples show the behavior of the ‎Solutions.

In this paper, the classic coupled poro-thermoelasticity model of hollow and solid cylinders under radial symmetric loading condition is considered. A full analytical method is used and an Exact unique Solution of the classic coupled equations is presented. The thermal and pressure boundary conditions, the body force, the heat source and the injected volume rate per unit volume of a distribute water source are considered in the most general forms and no limiting assumption is used. This generality allows simulation of several of the applicable problems.

In this paper the classic coupled thermoporoelasticity model of hollow and solid cylinders under radial symmetric loading condition (r, t) is considered. A full analytical method is used and an Exact unique Solution of the classic coupled equations is presented. The thermal and pressure boundary conditions, the body force, the heat source, and the injected volume rate per unit volume of a distribute water source are considered in the most general forms, and no limiting assumption is used. This generality allows simulation of various applicable problems.