New Techniques of Weighted Sum Method for Solving Multi-Objective Geometric Programming Problems

تعد مشكلة البرمجة الهندسية متعددة الأهداف نوعًا من مشاكل التحسين التي تستخدم بشكل كبير في المشاكل الهندسية. حتى الآن لا يوجد الكثير من تقنيات التحسين التي يمكنها حساب هذا النوع من مشاكل التحسين بسهولة. في هذا البحث ، تم تقديم تقنيتين جديدتين مع خوارزميتين لتحسين مشاكل البرمجة الهندسية متعددة الأهداف. التقنية الاولى تم أنشاءها باستخدام طريقة المجموع الموزون والوسط الحسابي والتقنية الثانية باستخدام طريقة المجموع الموزون والمتوسط ​​الهندسي. تم استخدام هاتين الطريقتين لتحويل مشكلة التحسين الهندسي متعدد الأهداف إلى مشكلة التحسين الهندسي ذات الهدف الواحد. تم اخذ بعض الامثلة بالاعتبار لتوضيح النتائج. كذلك تمت مقارنة هذه النتائج مع التقنيات الشائعة الأخرى المستخدمة في حل مشاكل التحسين الهندسي متعدد الأهداف.Multi-objective geometric programming problem is a type of optimization problem that wildly used in engineering problems. Until now there are not many optimization techniques that can easily compute this type of optimization problem. In this paper, we proposed two new techniques with algorithms to optimize multi-objective geometric programming problems. We created the first technique by using the weighted sum method and Arithmetic mean, and by using the weighted sum method and geometric mean we produced the second technique. These two methods are used to convert multi-objective geometric optimization problems to single-objective geometric optimization problems Some examples are considered to illustrate the results. The results were compared with other common techniques used in solving multi-objective engineering optimization problems

Joint User-Association and Resource-Allocation in Virtualized Wireless Networks

In this paper, we consider a down-link transmission of multicell virtualized wireless networks (VWNs) where users of different service providers (slices) within a specific region are served by a set of base stations (BSs) through orthogonal frequency division multiple access (OFDMA). In particular, we develop a joint BS assignment, sub-carrier and power allocation algorithm to maximize the network throughput, while satisfying the minimum required rate of each slice. Under the assumption that each user at each transmission instance can connect to no more than one BS, we introduce the user-association factor (UAF) to represent the joint sub-carrier and BS assignment as the optimization variable vector in the mathematical problem formulation. Sub-carrier reuse is allowed in different cells, but not within one cell. As the proposed optimization problem is inherently non-convex and NP-hard, by applying the successive convex approximation (SCA) and complementary geometric programming (CGP), we develop an efficient two-step iterative approach with low computational complexity to solve the proposed problem. For a given power-allocation, Step 1 derives the optimum userassociation and subsequently, for an obtained user-association, Step 2 find the optimum power-allocation. Simulation results demonstrate that the proposed iterative algorithm outperforms the traditional approach in which each user is assigned to the BS with the largest average value of signal strength, and then, joint sub-carrier and power allocation is obtained for the assigned users of each cell. Especially, for the cell-edge users, simulation results reveal a coverage improvement up to 57% and 71% for uniform and non-uniform users distribution, respectively leading to more reliable transmission and higher spectrum efficiency for VWN

Simultaneous Optimal Uncertainty Apportionment and Robust Design Optimization of Systems Governed by Ordinary Differential Equations

The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness, suboptimal performance, and higher build costs. Treatment of small geometric uncertainty in the context of manufacturing tolerances is a well studied topic. Traditional sequential design methodologies have recently been replaced by concurrent optimal design methodologies where optimal system parameters are simultaneously determined along with optimally allocated tolerances; this allows to reduce manufacturing costs while increasing performance. However, the state of the art approaches remain limited in that they can only treat geometric related uncertainties restricted to be small in magnitude. This work proposes a novel framework to perform robust design optimization concurrently with optimal uncertainty apportionment for dynamical systems governed by ordinary differential equations. The proposed framework considerably expands the capabilities of contemporary methods by enabling the treatment of both geometric and non-geometric uncertainties in a unified manner. Additionally, uncertainties are allowed to be large in magnitude and the governing constitutive relations may be highly nonlinear. In the proposed framework, uncertainties are modeled using Generalized Polynomial Chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach allows statistical moments of the uncertain system to be explicitly included in the optimization-based design process. The framework formulates design problems as constrained multi-objective optimization problems, thus enabling the characterization of a Pareto optimal trade-off curve that is off-set from the traditional deterministic optimal trade-off curve. The Pareto off-set is shown to be a result of the additional statistical moment information formulated in the objective and constraint relations that account for the system uncertainties. Therefore, the Pareto trade-off curve from the new framework characterizes the entire family of systems within the probability space; consequently, designers are able to produce robust and optimally performing systems at an optimal manufacturing cost. A kinematic tolerance analysis case-study is presented first to illustrate how the proposed methodology can be applied to treat geometric tolerances. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design at an optimal manufacturing cost, accounting for the entire family of systems within the associated probability space. This case-study highlights the general nature of the new framework which is capable of optimally allocating uncertainties of multiple types and with large magnitudes in a single calculation