Analytical Objective Function Research Articles

Election of rational height of steel beam structures taking into account the dynamics coefficient during episodic loading

The article develops a generalized approach to finding the rational height of a steel beam structure: a truss or a roof beam under the action of an episodic short-term pulse load. The research was carried out based on creating a generalized physical and mathematical model for finding the rational height of a steel truss of increased reliability, taking into account the action of a pulsed concentrated load acting in the middle of the span of the structure. The choice of the rational height of a steeltruss was made according to the generalized optimal design method, taking into account the constraints on strength and stability. Therefore, it can be assumed that the applied methodological approach gives a limited-optimal result given the complexity of the problem, but it is sufficient for variant design. The analytical expression of steel consumption was taken as the objective function, taking into account the dynamic coefficient for bending moment. The analytical objective function of steel consumption for a truss structure was written taking into account the unification of steel elements, and the influence of design coefficients was also taken into account. The physical and mathematical model ofthe steel truss is the design of an ideal I-beam taking into account the shear deformation of the section when determining the dynamic coefficient. Analytical expressions (rationality criteria) are obtained for determining the rational (bounded-optimal) height of the truss taking into account the influence of the dynamic coefficient. Numerical studies of the bounded-optimal height of the steel roof truss structure are carried out. It is confirmed that depending on the static and impulse load and the geometric parameters of the steel roof structure, the dynamic coefficient affects the determination of the rational height of the steel structure. The obtained criterion of the rationality of the structure also takes into account the influence of the ratio of the selfweight of the steel truss structure and the structure taking into account the enclosing load-bearing structures of the roof slabs. The analytical criterion of the rational height of the steel truss with the influence of the impulse loadalso includes the criterion of the rationality of the optimal height of the truss under static load. According to the research results, a practical method for determining the rational design of a steel roof truss has been obtained during variant design taking into account the action of impulse loads.

Data-driven method of solving computationally expensive combined economic/emission dispatch problems in large-scale power systems: an improved kriging-assisted optimization approach

Combined economic/emission dispatch (CEED) is generally studied using analytical objective functions. However, for large-scale, high-dimension power systems, CEED problems are transformed into computationally expensive CEED (CECEED) problems, for which existing approaches are time-consuming and may not obtain satisfactory solutions. To overcome this problem, a novel data-driven surrogate-assisted method is introduced firstly. The fuel cost and emission objective functions are replaced by improved Kriging-based surrogate models. A new infilling sampling strategy for updating Kriging-based surrogate models online is proposed, which improves their fitting accuracy. Through this way, the evaluation time of the objective functions is significantly reduced. Secondly, the optimization of CECEED is executed by an improved non-dominated sorting genetic algorithm-II (NSGA-II). The above infilling sampling strategy is also used to reduce the number of evaluations for original mathematic fitness functions. To improve their local convergence ability and global search abilities, the individuals that exhibit excellent performance in a single objective are cloned and mutated. Finally, information about the Pareto front is used to guide individuals to search for better solutions. The effectiveness of this optimization method is demonstrated through simulations of IEEE 118-bus test system and IEEE 300-bus test system.

An AI-based auto-design for optimizing RC frames using the ANN-based Hong–Lagrange algorithm

ABSTRACT Artificial neural networks (ANNs)-based objective functions such as costs and weights of reinforced concrete (RC) frames with four-by-four bays and four floors are optimized simultaneously based on big datasets of 330,000 designs according to ACI 318-19, whereas corresponding design parameters, which minimize objective functions, are also obtained. The Pareto frontier verified by big datasets shows reductions up to 44.983% and 33.111% in costs and weights, respectively, compared with probable designs based on averages of 688 (0.1%) best designs among 688,000 samples. Optimized designs meeting requirements imposed by codes and architects are achieved using the ANN-based Hong–Lagrange algorithm in which complex analytical objective functions are replaced by ANN-based objective functions. ANN is formulated to provide 32 forward outputs based on 18 forward inputs to minimize or maximize objective functions, such as costs and weights as a function of 18 input parameters. When good training qualities are achieved, objective functions with equality and inequality constraints are implemented in the proposed method, which determines optimal design parameters for building with accuracies and robustness equivalent to derivation-based approaches, which are hard to obtain using metaheuristic methods. The proposed AI-based auto-designs perform optimization where design variables are produced automatically while optimizing design targets.

Multi-Market Bidding Behavior Analysis of Energy Storage System Based on Inverse Reinforcement Learning

The bidding behaviors of the energy storage systems (ESS) are complicated due to time coupling and market coupling limited by their capacity states. The existing research is mainly based on optimization models and reinforcement learning (RL) models, which are idealized with analytical objective functions, rational decisions, and virtual historical data. This leads to mismatches between theoretical results and actual bidding behaviors, and the obtained results cannot be generalized to a real-world market. In recent years, the disclosure of market data has enabled data-driven analysis, while the complexity of actual data makes the analysis process challenging and calls for more suitable methods. On the basis of our investigation of ESS bidding behaviors and market data, we propose a novel inverse RL (IRL)-based framework to identify the bidding decision objective function of ESS in coupled multi-market through their historical bidding records and operation status. The identification results can be used as the model parameters to help RL-based simulation models available for a real-world market. Based on the proposed framework, we conduct an empirical analysis on the real-world data of a battery ESS in the Australian electricity market. The results show how the ESS balances its objective among different markets and maintains its state of charge (SOC) to gain more profits. This framework can help us model ESS behaviors by relying on historical data to better understand the ESS bidding decision-making mechanism, which will facilitate market equilibrium analysis and mechanism design.

Optimizing reinforced concrete beams cost based on AI-based Lagrange functions

ABSTRACT This study aims to develop artificial neural networks (ANNs)-based forward Lagrange networks optimizing ductile doubly reinforced concrete (RC) beams. Cost (CIb ) of materials and manufacture were established as an objective function, which is minimized considering constraints according to engineer’s needs. Large datasets of 100,000 were used to derive an AI-based objective function for cost (CIb , minimizing cost of an RC beams) as a function of forward input parameters, replacing complex analytical objective functions. A resilient design capable of finding concrete beams beyond human efficiency is based on equality and inequality constraints, which are implanted in AI-based Lagrange. Optimal designs are not simple especially when multiple constraining conditions are to be considered. Neither is it possible for engineers to pre-assign constraining conditions which can be sequentially calculated in an output of a conventional design. Cost of RC beams minimized by forward Lagrange networks was reduced 18%–26% compared to probable beam designs. AI-based design charts with eight forward outputs (ϕMn,Mu,Mcr,εrt_0.003,εrc_0.003,Δimme,Δlong,CIb ) based on nine forward inputs (L, h, b, fy,f’ c, ρrt,ρrc,MD,ML ) are proposed to assist engineers to design ductile doubly RC beams, presenting minimized CIb in the preliminary design stage.

An Efficient Bi-Objective Optimization Workflow Using the Distributed Quasi-Newton Method and Its Application to Well-Location Optimization

SummaryAlthough it is possible to apply traditional optimization algorithms to determine the Pareto front of a multiobjective optimization problem, the computational cost is extremely high when the objective function evaluation requires solving a complex reservoir simulation problem and optimization cannot benefit from adjoint-based gradients. This paper proposes a novel workflow to solve bi-objective optimization problems using the distributed quasi-Newton (DQN) method, which is a well-parallelized and derivative-free optimization (DFO) method. Numerical tests confirm that the DQN method performs efficiently and robustly.The efficiency of the DQN optimizer stems from a distributed computing mechanism that effectively shares the available information discovered in prior iterations. Rather than performing multiple quasi-Newton optimization tasks in isolation, simulation results are shared among distinct DQN optimization tasks or threads. In this paper, the DQN method is applied to the optimization of a weighted average of two objectives, using different weighting factors for different optimization threads. In each iteration, the DQN optimizer generates an ensemble of search points (or simulation cases) in parallel, and a set of nondominated points is updated accordingly. Different DQN optimization threads, which use the same set of simulation results but different weighting factors in their objective functions, converge to different optima of the weighted average objective function. The nondominated points found in the last iteration form a set of Pareto-optimal solutions. Robustness as well as efficiency of the DQN optimizer originates from reliance on a large, shared set of intermediate search points. On the one hand, this set of searching points is (much) smaller than the combined sets needed if all optimizations with different weighting factors would be executed separately; on the other hand, the size of this set produces a high fault tolerance, which means even if some simulations fail at a given iteration, the DQN method’s distributed-parallel information-sharing protocol is designed and implemented such that the optimization process can still proceed to the next iteration.The proposed DQN optimization method is first validated on synthetic examples with analytical objective functions. Then, it is tested on well-location optimization (WLO) problems by maximizing the oil production and minimizing the water production. Furthermore, the proposed method is benchmarked against a bi-objective implementation of the mesh adaptive direct search (MADS) method, and the numerical results reinforce the auspicious computational attributes of DQN observed for the test problems.To the best of our knowledge, this is the first time that a well-parallelized and derivative-free DQN optimization method has been developed and tested on bi-objective optimization problems. The methodology proposed can help improve efficiency and robustness in solving complicated bi-objective optimization problems by taking advantage of model-based search algorithms with an effective information-sharing mechanism.NOTE: This paper is also published as part of the 2021 SPE Reservoir Simulation Conference Special Issue.

A robust optimization method for weighted-layer-stacking proton beam therapy

The layer-stacking method can provide three-dimensional conformal dose distributions to the target based on a passive scattering method using mini-spread-out Bragg peak (SOBP). The purpose of this work is to demonstrate the effectiveness of a new weight optimization algorithm that can enhance the robustness of dose distributions against layer depth variation in layer-stacking proton beam therapy.In the robustness algorithm, the upper limit of the layer’s weight was adapted to the conventional algorithm and varied for 620 weight set evaluations. The optimal weight set was selected by using an analytical objective function based on Gaussian function with σ = 3 mm for WED variation. Then, we evaluated the stabilities of the one-dimensional depth dose distribution against WED variation generated by Gaussian samples. Three-dimensional dose distributions in the water phantom were also evaluated using the Monte-Carlo dose calculation. The variation of dose as well as dose volume histograms for the spherical target and the organ at risk (OAR) were evaluated.The robustness algorithm reduced the change of the dose distribution due to the WED variation by a factor of almost 3/4 compared to those with the conventional procedure. The rate of 91.8% in total samples was maintained within 5% change of the maximum dose, compared with the rate of 64.9% in the conventional algorithm. In the MC calculation, the high dose-volume in the OAR was reduced around the lateral penumbra and distal falloff region by the robustness algorithm.The stability of depth dose distributions was enhanced under the WED variation, compared to the conventional algorithm. This robust algorithm in layer-stacking proton therapy may be useful for treatment in which the sharpness of the distal falloff along the depth distribution needs to be maintained to spare the organ at risk and keep the dose coverage for the target tumor.

A Recommender System for Metaheuristic Algorithms for Continuous Optimization Based on Deep Recurrent Neural Networks

As revealed by the no free lunch theorem, no single algorithm can outperform any others on all classes of optimization problems. To tackle this issue, methods for recommending an existing algorithm for solving given problems have been proposed. However, existing recommendation methods for continuous optimization suffer from low practicability and transferability, mainly due to the difficulty in extracting features that can effectively describe the problem structure and lack of data for training a recommendation model. This work proposes a generic recommender system to address the above two challenges. First, a novel method is proposed to represent an analytic objective function of a continuous optimization problem as a tree, which is directly used as the features of the problem. For black-box optimization problems whose objective function is unknown, a symbolic regressor is adopted to estimate the tree structure. Second, a large number of benchmark problems are randomly created based on the proposed tree representation, providing an abundant amount of training data with various levels of difficulty. By employing a deep recurrent neural network, a recommendation model is trained to recommend a most suitable metaheuristic algorithm for white- or black-box optimization, making a significant step forward towards fully automated algorithm recommendation for continuous optimization. Experimental results on 100 000 benchmark problems show that the proposed recommendation model achieves considerably better performance than existing ones, and exhibits high transferability to real-world problems.

Language model based interactive estimation of distribution algorithm

It is very hard, if not impossible to use analytical objective functions for optimization of personalized search due to the difficulties in mathematically describing qualitative problems. To solve such optimization problems, interactive evolutionary algorithms, which can make use of human preferences, are highly desirable. However, due to the lack of effective encoding methods, interactive evolutionary algorithms have been limited to numerically encoded optimization problems. In practice, however, linguistic terms (words) are the most natural expression of human preferences, and they are also commonly used to describe items in personalized search or E-commerce; therefore, language models better suit encoding, and the optimization of personalized search is converted into a dynamic document matching problem. To optimize word-described personalized search, we propose a novel interactive estimation of distribution algorithm. This algorithm combines a language model-based encoding approach, a Dirichlet-Multinomial compound distribution-based preference expression, and a Bayesian inference mechanism. The proposed algorithm is applied to two personalized search cases to demonstrate the capability of the algorithm in ensuring a more efficient and accurate search with less user fatigue.

Решение задачи распределения ресурсов дискретного типа методами линейного программирования

Introduction. When planning projects, work packages, one often has to solve distribution-type problems associated with the optimal allocation of resources. Existing methods for solving such problems suggest the presence of an analytical dependence of a continuous type between the volumes of the distributed resource and performance indicators. However, optimization problems become inapplicable when the resource is discrete and dependencies are specified in tabular ways. Aim. To develop a mathematical model for solving the problem of the optimal distribution of resources of a discrete type with table-defined optimality criteria using linear programming methods. Describe the method of numerical solution of the problem using computer technology. Materials and methods. It is possible to solve the problem by forming an analytical dependence of a piecewise-continuous type between the volume of the distributed resource and the optimality criterion. This allows us to formulate an optimization problem solved by mathematical programming methods. It is possible to construct an analytical objective function by introducing additional parameters and restrictions. A numerical solution to the problem can be obtained both using mathematical packages of applied programs, such as, for example, “Mathcad”, and using programming systems. The paper describes the methodology for solving problems in the MS Excel environment using the add-on “Solver”. Results. A mathematical model is developed for solving a discrete distribution problem for the optimality criterion given in a table by integer linear programming methods. The technique of numerical solution in the MS Excel environment is described. Conclusion. Previously, such problems were solved by dynamic programming methods, which is more difficult in the computational plan. Conducted computational experiments showed high accuracy of model calculations and resistance to changes in the source data.

Assessing the performances of recent global search algorithms using analytic objective functions and seismic optimization problems

We have compared the performances of six recently developed global optimization algorithms: imperialist competitive algorithm, firefly algorithm (FA), water cycle algorithm (WCA), whale optimization algorithm (WOA), fireworks algorithm (FWA), and quantum particle swarm optimization (QPSO). These methods have been introduced in the past few years and have found very limited or no applications to geophysical exploration problems thus far. We benchmark the algorithms’ results against the particle swarm optimization (PSO), which is a popular and well-established global search method. In particular, we are interested in assessing the exploration and exploitation capabilities of each method as the dimension of the model space increases. First, we test the different algorithms on two multiminima and two convex analytic objective functions. Then, we compare them using the residual statics corrections and 1D elastic full-waveform inversion, which are highly nonlinear geophysical optimization problems. Our results demonstrate that FA, FWA, and WOA are characterized by optimal exploration capabilities because they outperform the other approaches in the case of optimization problems with multiminima objective functions. Differently, QPSO and PSO have good exploitation capabilities because they easily solve ill-conditioned optimizations characterized by a nearly flat valley in the objective function. QPSO, PSO, and WCA offer a good compromise between exploitation and exploration.

A method to attenuate genetic drift in genetic-algorithm optimizations: Applications to analytic objective functions and two seismic optimization problems

Genetic algorithms (GAs) usually suffer from the so-called genetic-drift effect, which reduces the genetic variability within the evolving population making the algorithm converge toward a local minimum of the objective function. We have developed an innovative method to attenuate such a genetic-drift effect that we named the drift-avoidance GA (DAGA). The implemented method combines some principles of niched GAs (NGAs), catastrophic GAs, crowding GAs, and the Monte Carlo algorithm (MCA) with the aim of maintaining an optimal genetic diversity within the evolving population, thus avoiding premature convergence. The DAGA performance is first tested on different analytic objective functions often used to test optimization algorithms. In this case, the implemented DAGA approach is compared with standard GAs, catastrophic GAs, crowding GAs, NGAs, and MCA. Then, the DAGA and the NGAs approaches are compared on two well-known nonlinear geophysical optimization problems characterized by objective functions with complex topologies: residual statics corrections and 2D acoustic full-waveform inversion. To draw general conclusions, we limit the attention to synthetic seismic optimizations. Our tests prove that the DAGA approach grants the convergence in case of objective functions with very complex topologies, where other GA implementations (such as standard GAs or NGAs) fail to converge. Differently, in case of simpler topologies, DAGA achieves similar performances with the other GA implementations considered. The DAGA approach may have a slightly higher or lower computational cost than standard GA or NGA methods, depending on its convergence speed, that is, on its ability to reduce the number of forward modelings with respect to the other methods.

A Random Forest-Assisted Evolutionary Algorithm for Data-Driven Constrained Multiobjective Combinatorial Optimization of Trauma Systems.

Many real-world optimization problems can be solved by using the data-driven approach only, simply because no analytic objective functions are available for evaluating candidate solutions. In this paper, we address a class of expensive data-driven constrained multiobjective combinatorial optimization problems, where the objectives and constraints can be calculated only on the basis of a large amount of data. To solve this class of problems, we propose using random forests (RFs) and radial basis function networks as surrogates to approximate both objective and constraint functions. In addition, logistic regression models are introduced to rectify the surrogate-assisted fitness evaluations and a stochastic ranking selection is adopted to further reduce the influences of the approximated constraint functions. Three variants of the proposed algorithm are empirically evaluated on multiobjective knapsack benchmark problems and two real-world trauma system design problems. Experimental results demonstrate that the variant using RF models as the surrogates is effective and efficient in solving data-driven constrained multiobjective combinatorial optimization problems.

On Visualizing Continuous Turbulence Scales

AbstractTurbulent flows are multi‐scale with vortices spanning a wide range of scales continuously. Due to such complexities, turbulence scales are particularly difficult to analyse and visualize. In this work, we present a novel and efficient optimization‐based method for continuous‐scale turbulence structure visualization with scale decomposition directly in the Kolmogorov energy spectrum. To achieve this, we first derive a new analytical objective function based on integration approximation. Using this new formulation, we can significantly improve the efficiency of the underlying optimization process and obtain the desired filter in the Kolmogorov energy spectrum for scale decomposition. More importantly, such a decomposition allows a ‘continuous‐scale visualization’ that enables us to efficiently explore the decomposed turbulence scales and further analyse the turbulence structures in a continuous manner. With our approach, we can present scale visualizations of direct numerical simulation data sets continuously over the scale domain for both isotropic and boundary layer turbulent flows. Compared with previous works on multi‐scale turbulence analysis and visualization, our method is highly flexible and efficient in generating scale decomposition and visualization results. The application of the proposed technique to both isotropic and boundary layer turbulence data sets verifies the capability of our technique to produce desirable scale visualization results.

Bayesian Multiobjective Optimisation With Mixed Analytical and Black-Box Functions: Application to Tissue Engineering.

Tissue engineering and regenerative medicine looks at improving or restoring biological tissue function in humans and animals. We consider optimising neotissue growth in a three-dimensional scaffold during dynamic perfusion bioreactor culture, in the context of bone tissue engineering. The goal is to choose design variables that optimise two conflicting objectives, first, maximising neotissue growth and, second, minimising operating cost. We make novel extensions to Bayesian multiobjective optimisation in the case of one analytical objective function and one black-box, i.e. simulation based and objective function. The analytical objective represents operating cost while the black-box neotissue growth objective comes from simulating a system of partial differential equations. The resulting multiobjective optimisation method determines the tradeoff between neotissue growth and operating cost. Our method exhibits better data efficiency than genetic algorithms, i.e. the most common approach in the literature, on both the tissue engineering example and standard test functions. The multiobjective optimisation method applies to real-world problems combining black-box models with easy-to-quantify objectives such as cost.

Comparing the performances of four stochastic optimisation methods using analytic objective functions, 1D elastic full‐waveform inversion, and residual static computation

ABSTRACTWe compare the performances of four stochastic optimisation methods using four analytic objective functions and two highly non‐linear geophysical optimisation problems: one‐dimensional elastic full‐waveform inversion and residual static computation. The four methods we consider, namely, adaptive simulated annealing, genetic algorithm, neighbourhood algorithm, and particle swarm optimisation, are frequently employed for solving geophysical inverse problems. Because geophysical optimisations typically involve many unknown model parameters, we are particularly interested in comparing the performances of these stochastic methods as the number of unknown parameters increases. The four analytic functions we choose simulate common types of objective functions encountered in solving geophysical optimisations: a convex function, two multi‐minima functions that differ in the distribution of minima, and a nearly flat function. Similar to the analytic tests, the two seismic optimisation problems we analyse are characterised by very different objective functions. The first problem is a one‐dimensional elastic full‐waveform inversion, which is strongly ill‐conditioned and exhibits a nearly flat objective function, with a valley of minima extended along the density direction. The second problem is the residual static computation, which is characterised by a multi‐minima objective function produced by the so‐called cycle‐skipping phenomenon. According to the tests on the analytic functions and on the seismic data, genetic algorithm generally displays the best scaling with the number of parameters. It encounters problems only in the case of irregular distribution of minima, that is, when the global minimum is at the border of the search space and a number of important local minima are distant from the global minimum. The adaptive simulated annealing method is often the best‐performing method for low‐dimensional model spaces, but its performance worsens as the number of unknowns increases. The particle swarm optimisation is effective in finding the global minimum in the case of low‐dimensional model spaces with few local minima or in the case of a narrow flat valley. Finally, the neighbourhood algorithm method is competitive with the other methods only for low‐dimensional model spaces; its performance sensibly worsens in the case of multi‐minima objective functions.

Well placement optimization using an analytical formula-based objective function and cat swarm optimization algorithm

Abstract Well placement optimization is a crucial and complex task in oil field development. Well placement is usually optimized by coupling reservoir numerical simulator with optimization algorithm. This method spends most of computing time in objective function evaluation by reservoir numerical simulator which limits its optimization efficiency. In this work, a well placement optimization method using an analytical formula-based objective function and cat swarm optimization (CSO) algorithm is established. The objective function, derived from fluid flow in porous media and material balance principle, can be calculated by the analytical formula to avoid running reservoir numerical simulator. Then the well placement optimization model is built and solved by CSO algorithm. Three examples are applied to justify the feasibility of the new objective function and the efficiency of this optimization method. Results demonstrate this method can significantly accelerate the speed of well placement optimization process. It can help to determine the optimal well placement more efficiently for actual oilfield development.

Simple algorithm for computing the communication complexity of quantum communication processes

A two-party quantum communication process with classical inputs and outcomes can be simulated by replacing the quantum channel with a classical one. The minimal amount of classical communication required to reproduce the statistics of the quantum process is called its communication complexity. In the case of many instances simulated in parallel, the minimal communication cost per instance is called the asymptotic communication complexity. Previously, we reduced the computation of the asymptotic communication complexity to a convex minimization problem. In most cases, the objective function does not have an explicit analytic form, as the function is defined as the maximum over an infinite set of convex functions. Therefore, the overall problem takes the form of a minimax problem and cannot directly be solved by standard optimization methods. In this paper, we introduce a simple algorithm to compute the asymptotic communication complexity. For some special cases with an analytic objective function one can employ available convex-optimization libraries. In the tested cases our method turned out to be notably faster. Finally, using our method we obtain 1.238 bits as a lower bound on the asymptotic communication complexity of a noiseless quantum channel with the capacity of 1 qubit. This improves the previous bound of 1.208 bits.

Revenue Sharing Based Resource Allocation for Dynamic Spectrum Access Networks

We propose a revenue sharing based resource allocation scheme for dynamic spectrum access (DSA) networks. In our scheme, based on a mutually agreed revenue sharing scheme, a primary network operator (PNO) actively shares its radio resource with a secondary network operator (SNO), which provides access service to secondary users (SUs) for its revenue maximization. To investigate the coupling effect between the revenue sharing and resource allocation, we formulate the interaction between PNO and SNO as a two-layered game, which includes a top layer game to model their revenue sharing and a bottom layer game to model their joint resource allocations. Specifically, in the top layer, based on their joint resource allocation decisions, the PNO and SNO form a Nash bargaining game to determine the revenue sharing scheme such that both of them can benefit from cooperation satisfactorily. Then, in the bottom layer, under the given revenue sharing scheme, the PNO and SNO form a Stackelberg game to determine their joint resource allocation decisions, which also influence their respective revenues. The two games work iteratively such that the PNO and SNO reach a final equilibrium state at which neither PNO nor SNO will change its decisions unilaterally in both layers. We propose efficient algorithms to solve both the top layer and bottom layer games and compute the final equilibrium of the two-layered game. Specifically, despite the non-convexity of joint resource allocation optimization problem in the bottom layer, we identify its hidden monotonic structure and propose an efficient algorithm, which is based on the polyblock approximation, to achieve the optimal solutions. Moreover, in the top layer, to tackle with the difficulty due to the lack of an analytical objective function for the revenue sharing problem, we explore its hidden unimodal property and propose a Brent's method based algorithm to achieve the optimal solution. Numerical results are presented to verify the performance of our algorithms and show that our revenue sharing based resource allocation scheme yields a win-win situation for the PNO and SNO.

MODEL AND METHOD OF MULTICRITERIA PROJECT SCOPE OPTIMIZATION WITH FUZZY INPUT DATA

The paper presents a mathematical model and method of the structural project scope optimization with fuzzy input data, which includes five objective functions. One of the functions reflects the profit for the entire project product life cycle. The other reflects the time of its realization. The third is the cost of the project. The fourth is the value of the generalized indicator of project product quality and the fifth is a risk assessment associated with the project. The model and method takes into account the restrictions on the lack of financial debt after each phase completion, the duration of the project, the quality of the separate stages products. It is assumed that the project scope is given in the form of a network model with the alternatives of the work execution. The suggested model is a multi-objective, dynamic, containing Boolean variables, algorithmic and analytical objective functions and constraints. To solve the problem the method of multi-objective structural project scope optimization with fuzzy input data with constraints, based on application of generalized criterion in conjunction with the method of implicit enumeration is proposed.