Problem Solving with Ordinal Measurement

The design and optimization of a large-scale systems are the most difficalt problems. A large-scale system consists of a number of subsystems. For example, in a harvest for harvesting one can separate the following subsystems: the frame, driver's cab, platform, engine, transmission, and steering system. Different departments of the design office engaged in creating a machine optimize their ‘own’ subsystems, while ignoring others. A machine assembled from ‘autonomously optimal’ subsystems turns out to be far from perfect. A machine is a single whole. When improving one of its subsystems, we can unwittingly worsen others. This implies that it is not always possible to solve optimization problems directly even for determination of the feasible solution set. The correct determination of the feasible solution set was a major challenge in engineering optimization problems. In order to construct the feasible solution set, a method called the Parameter Space Investigation (PSI) has been created and successfully integrated into various fields of industry, science, and technology. The methods of approximation of the feasible solution and Pareto optimal sets and the regularization of the Pareto optimal set are described in our paper. These methods are applied to solving the multicriteria optimization problems of large- scale systems. For example, they were applied in an agricultural engineering to a harvester for harvesting design.

In this paper, we study general linear programs in which right hand sides are interval numbers. This model is relevant when uncertain and inaccurate factors make difficult the assignment of a single value to each right hand side. When objective function coefficients are interval numbers in a linear program, classical criteria coming from decision theory (like the worst case criterion) are usually applied to determine robust solutions. When the set of feasible solutions is uncertain, we identify a class of linear programs for which these classical approaches are no longer relevant. However, it is possible to compute the worst optimum solution. We study the complexity of this optimization problem when each right hand side is an interval number. Then, we exhibit some duality relationships between the worst optimum solution problem and the best optimum solution to the dual problem.

Optimization problems are used to model many real-life problems. Therefore, solving these problems is one of the most important goals of algorithm design. A general optimization problem can be defined by specifying a set of constraints that defines a subset in some underlying space (like the Euclidean space) called the feasible subset and an objective function that we are trying to maximize or minimize, as the case may be, over the feasible set. The difficulty of solving such problems typically depends on how ‘complex’ the feasible set and the objective function are. For example, a very important class of optimization problems is linear programming. Here the feasible subset is specified by a set of linear inequalities (in the Euclidean space); the objective function is also linear. A more general class of optimization problems is convex programming, where the feasible set is a convex subset of a Euclidean space and the objective function is also convex. Convex programs (and hence, linear programs) have a nice property that any local optimum is also a global optimum for the objective function. There are a variety of techniques for solving such problems – all of them try to approach a local optimum (which we know would be a global optimum as well). These notions are discussed in greater detail in a later section in this chapter. The more general problem, the so-called non-convex programs, where the objective function and the feasible subset could be arbitrary can be very challenging to solve. In particular, discrete optimization problems, where the feasible subset could be a (large) discrete subset of points falls under this category.

Multi-objective problems (MOPs), a class of optimization problems in the real-world, have multiple conflicting objectives. Multi-objective evolutionary algorithms (MOEAs) are known as great potential algorithms to solve difficult MOPs. With MOEAs, based on the principle of population, we have a set of optimal solutions (feasible solution set) after the search. We often use the concept of dominance relationship in population, and it is not difficult to find out set of Pareto optimal solutions during generations. However, with expensive optimization problems in the real world, it has to use a lot of fitness function evaluations during the search. To avoid expensive physical experiments, we can use computer simulations methods to solve the difficult MOPs. In fact, this way often costs expensive in computation and times for the simulation. In these cases, researchers discussed on the usage of surrogate models for evolutionary algorithms, especially for MOEAs to minimize the number of fitness callings. There are a series of proposals which were introduced with the usage of RBF, PRS, Kriging, SVM models. With the concept of machine learning, these MOEAs can solve expensive MOPS effectively. However, with our analysis, we found that using a fixed ratio of fitness functions and surrogate functions may make the unbalance of exploitation and exploration of the evolutionary process. In this paper, we suggest to use an adaptive control to determine the effective ratio during the search. The proposal is confirmed though an experiment with standard measurements on well-known benchmark sets.

Cloud storages have recently increased rapidly in many organizations as they provide many benefits and advantages for the users. These advantages include easiness in use, unlimited capacity in storage, scalability and cost effectiveness. This has motivated the concerns regarding the cloud issues including cost, performance and security. Many different schemes have been proposed to address the issues of cloud storage. Each scheme considers these issues from different perspectives without employing an optimal solution that fully satisfies the user requirements. In real world applications, the client's objectives to move to cloud storage vary from single objective to several conflicting objectives. As a result, there is a need to find a tradeoff setup for a multiple cloud storage scheme. This paper proposed Smart Cloud Optimal Resource Selection (SCORS) scheme by applying non-dominated sorting genetic algorithm (NSGA-2) to resolve the user's conflicting requirements. Our optimal solution facilitates setting up multiple cloud storages in an optimized way based on the objectives specified. Since each user has different objectives, all combinations of cloud storage client needs are collected and transformed into different multi-objective optimization problems with constraints and bounds. Experiment results show that SCORS scheme determine the optimal solution from the set of feasible solutions in an efficient way.

The traditional linear programming model is deterministic. The way that uncertainty is handled is to compute the range of optimality. After the optimal solution is obtained, typically by the simplex method, one considers the effect of varying each objective function coefficient, one at a time. This yields the range of optimality within which the decision variables remain constant. This sensitivity analysis is useful for helping the analyst get a sense for the problem. However, it is unrealistic because objective function coefficients tend not to stand still. They are typically profit contributions from products sold and are subject to randomly varying selling prices. In this paper, a realistic linear program is created for simultaneously randomizing the coefficients from any probability distribution. Furthermore, we present a novel approach for designing a copula of random objective function coefficients according to a specified rank correlation. The corresponding distribution of objective function values is created. This distribution is examined directly for central tendency, spread, skewness and extreme values for the purpose of risk analysis. This enables risk analysis and business analytics, emerging topics in education and preparation for the knowledge economy.

The results of laboratory experiments and numerical model simulations are described in which the motion of a round, negatively-buoyant, turbulent jet discharged horizontally above a slope into a rotating homogeneous fluid has been investigated. For the laboratory study, flow visualisation data are presented to show the complex three-dimensional flow fields generated by the discharge. Analysis of the experimental data indicates that the spatial and temporal developments of the flow field are controlled primarily by the lateral and vertical discharge position of the jet (with respect to the bounding surfaces of the container of width W) and the specific momentum (M 0) and buoyancy (B 0) fluxes driving the jet. The flow is seen to be characterised by the formation of (i) a primary anticyclonic eddy (PCC) close to the source, (ii) an associated secondary cyclonic eddy (SCE) and (iii) a buoyancy-driven bottom boundary current along the right side boundary wall. For the parameter ranges studied, the size L p, s and spatial location x p, s of the PCC and SCE (and the nose velocity u N of the boundary current) are shown to be only weakly-dependent upon the value of the mixed parameter M 0Ω/B 0, where Ω is the background rotation rate. Both L p and x p are shown to scale with the separation distance y*/W of the right side wall (y = 0) from the source (y = y*), both L s and x s scale satisfactorily with the length scale l M (= M 0 3/4/B 0 ½) and u N is determined by the appropriate gravity current speed [(g']0 H]½ and the separation distance y*/W. Numerical model results show good qualitative agreement with the laboratory data with regard to the generation of the PCC, SCE and boundary current as characteristic features of the flow in question. In addition, extension of the numerical model to diagnose potential vorticity and plume thickness distributions for the laboratory cases allow the differences in momentum-and buoyancy-dominated flows to be clearly delineated. Specifically, the characteristic features of the SCE are shown to be strongly dependent upon the value of M 0Ω/B 0 for the buoyant jet flow; not least, the numerical model data are able to confirm the controlling role played by the boundary walls in the laboratory experiments. Quantitative agreement between the numerical and laboratory model data is fair; most significantly, the success of the former model in simulating the dominant flow features from the latter enables the reliable extension of the numerical model to be made to cases of direct oceanic interest.

Differential evolution algorithm requires diversified population to solve multi- modalproblems. DE supports non-distributed population. Differential evolution algorithm (DE) versions include differential evolution algorithm with best selection method and species evolution (DESBS) and differential evolution algorithm with hierarchical fair competition modal (HFCDE). This article analyzed the efficiency of HFCDE and DESBS to solve the multi-modal s’ problems. HFCDE and DESBS support non-distributed population structured. HFCDE starts with set of feasible solution then it distributed them in the different hierarchy. HFCDE provides the fair competition. DESBS is another semi distributed, differential evolution algorithm.It starts with set of feasible solutions (population). Later it identifies the niches and create the sub-groups within population. Both HFCDE and DESBS have outperformed the other variant of state-of-art variants of DE. Here, the performance of DESBS and the HFCDE are rigorously tested on the multimodal problems. The success of DESBS over HFCDE in multi-modal difficulties managed to overcome the phenomena of elitism to resolve the complex problems, it has been observed that DESBS performs better than HFCDE in complex multi-modal scenarios.

Urban bus network of priority lanes: A combined multi-objective, multi-criteria and group decision-making approach

When coefficients in the objective function cannot be precisely determined, the optimal solution is fluctuated by the realisation of coefficients. Therefore, analysing the stability of an optimal solution becomes essential. Although the robustness analysis of an optimal basic solution has been developed successfully so far, it becomes complex when the solution contains degeneracy. This study is devoted to overcoming the difficulty caused by the degeneracy in a linear programming problem with interval objective coefficients. We focus on the tangent cone of a degenerate basic feasible solution since the belongingness of the objective coefficient vector to its associated normal cone assures the solution’s optimality. We decompose the normal cone by its associated tangent cone to a direct union of subspaces. Several propositions related to the proposed approach are given. To demonstrate the significance of the decomposition, we consider the case where the dimension of the subspace is one. We examine the obtained propositions by numerical examples with comparisons to the conventional techniques.KeywordsInterval linear programmingDegeneracyPolyhedral convex coneTangent coneBasic space

PurposeThis paper aims to measure the sensitivity of the structure’s deformation numerical model (NM) related to the various types of the design parameters, which is a suitable method for parameter selection to increase the time of model-updating.Design/methodology/approachIn this research, a variance-based sensitivity analysis (VBSA) approach is proposed to measure the sensitivity of NM of structures. In this way, the contribution of measurements of the structure (such as design parameter values and geometry) on the output of NM is studied using first-order and total-order sensitivity indices developed by Sobol’. In this way the generated data set of parameters by considering different distributions such as Gaussian or uniform distribution and different order as input along with, the resulted deformation variables of NM as output has been submitted to the Sobol’ indices estimation procedure. To the verification of VBSA results, a gradient-based sensitivity analysis (SA), which is developed as a global SA method has been developed to measure the global sensitivity of NM then implemented over the NM’s results of a tunnel.FindingsRegarding the estimated indices, it has been concluded that the derived deformation functions from the tunnel’s NM usually are non-additive. Also, some parameters have been determined as most effective on the deformation functions, which can be selected for model-updating to avoid a time-consuming process, so those may better to be considered in the group of updating parameters. In this procedure for SA of the model, also some interactions between the selected parameters with other parameters, which are beneficial to be considered in the model-updating procedure, have been detected. In this study, some parameters approximately (27 per cent of the total) with no effect over the all objective functions have been determined to be excluded from the parameter candidates for model-updating. Also, the resulted indices of implemented VBSA were approved during validation by the gradient-based indices.Practical implicationsThe introduced method has been implemented for a circular lined tunnel’s NM, which has been created by Fast Lagrangian Analysis of Continua software.Originality/valueThis paper plans to apply a statistical method, which is global on the results of the NM of a soil structure by a complex system for parameter selection to avoid the time-consuming model-updating process.

Depending on the nature, objectives, and constraints of the decision variables; linear programming, nonlinear programming, integer programming, mixed integer programming etc. can be classified. Extensive research has been conducted to solve all types of these problems in a parametric context. In this paper, to solve optimization problems having uncertainties represented by a single parameter on the objective function, a systematic linearization approach is developed considering the parametric expression as nonlinear. In the proposed approach, the objective function is considered as nonlinear which is converted into linear by using first order Taylor series expansion at the points making the parametric costs zero. Thus, the optimal solution is obtained from the constructed linear programming problem. In this way, by determining the intervals in which the optimal solution changes, the solution of the parametric linear programming problem is obtained. A numerical experiment is illustrated to present the effectiveness of the proposed approach.

We consider a framework for sequential bilevel linear programming in which a leader and a follower interact over multiple time periods. In each period, the follower observes the actions taken by the leader and reacts optimally, according to the follower’s own objective function, which is initially unknown to the leader. By observing various forms of information feedback from the follower’s actions, the leader is able to refine the leader’s knowledge about the follower’s objective function and, hence, adjust the leader’s actions at subsequent time periods, which ought to help in maximizing the leader’s cumulative benefit. We show that greedy and robust policies adapted from previous work in the max-min (symmetric) setting might fail to recover the optimal full-information solution to the problem (i.e., a solution implemented by an oracle with complete prior knowledge of the follower’s objective function) in the asymmetric case. In contrast, we present a family of greedy and best-case policies that are able to recover the full-information optimal solution and also provide real-time certificates of optimality. In addition, we show that the proposed policies can be computed by solving a series of linear mixed-integer programs. We test policy performance through exhaustive numerical experiments in the context of asymmetric shortest path interdiction, considering various forms of feedback and several benchmark policies.

Pump management and reservoir management have many similarities, and therefore, should ideally be analyzed in an integrated way to plan effectively the daily operation of water distribution systems. Historically, these two management activities have been evolved as separate tasks in energy-efficiency (i.e., energy optimization) studies and are often carried out in an isolated way. The latter being most often associated directly with the concepts of multimodal and multi-objective optimization problems, whereas the former is usually considered as a single optimization problem to be solved. When some single optimization problems appear at part of the solution tied to a local (i.e., regional) search-space (i.e., objective space), this artificial integration (i.e., multi-modal and multi-objective optimization) can always obtain optimal solutions. Similarly when system constraints and load conditions are considered, a set of feasible and innovative optimal solutions can be obtained in order to continue the enhancement of energy consumption that turns into a significant reduction in the overall operational cost (i.e., a potential of 6.24% cost savings) without affecting the level of services provided to the clients in a safe and protected manner.

Design engineers and decision-makers across various fields are constantly working to make optimal design decisions for multidisciplinary engineering systems in an effort to improve performance and reduce costs. The multiple disciplines that decision-makers are forced to consider can range from different physical components of a system, to competing physical phenomena influencing a component (e.g. flow forces and structural strength), to completely separate models of interest to a system (e.g. engineering performance and lifecycle cost). The common element that all these decision-making scenarios share is the presence of couplings between the considered disciplines (or subsystems). How the values for these coupling parameters are determined within a decision-making or optimization framework is the subject of countless research efforts. At present the multidisciplinary design optimization (MDO) community has settled on a few proven techniques such Collaborative Optimization (CO) and Analytic Target Cascading (ATC). However, current MDO techniques have issues that limit their effectiveness in solving various MDO problems. Many of these strategies require close coordination between subsystem optimization solvers, require significant effort by decision-makers to pose their problems in a suitable format, and/or can have large computational efficiency problems due to the fact that they involve solving nested optimization problems. In an effort to alleviate some of these issues and make MDO easier to implement and more computationally efficient, a new sequential MDO algorithm called Cooperative Design Optimization (CDO) is proposed. The CDO approach functions through a series of subsystem optimizations using a successively smaller cooperation space. The cooperation space is analogous to the design space of a traditional optimization problem, but includes only the coupling parameters that are a factor in multiple subsystems. A single iteration of the approach can be thought of as a negotiation between all of the subsystems regarding the boundaries of the cooperation space. To facilitate this cooperation, each subsystem optimization problem is restructured as a multi-objective optimization problem so that a Pareto set of optimal solutions, or agreeable design alternatives, are produced. As a result, after all subsystem optimizations are accomplished, each subsystem’s obtained Pareto optimal solution sets are compared with respect to the coupling parameters and new bounds in the cooperation space are determined for the next iteration. This process allows each subsystem to be optimized completely independently within the boundaries of the agreed upon cooperation space while coordination is achieved through an analysis of obtained optimal solutions for each subsystem. Iterations, or negotiations, are repeated until an acceptable solution or set of solutions is obtained for all included subsystems through this cooperative process.