Limitations of Pareto Front in Design Under Uncertainty and Their Reconciliation

2013 ◽  
Vol 135 (7) ◽  
Author(s):  
Vijitashwa Pandey ◽  
Zissimos P. Mourelatos ◽  
Efstratios Nikolaidis
Keyword(s):  
Utility Function ◽  
Engineering Design ◽  
Decision Maker ◽  
Decision Problem ◽  
Pareto Front ◽  
Optimization Problem ◽  
Reduction Gear ◽  
Design Under Uncertainty ◽  
Design Constraints ◽  
Single Attribute

Engineering design reconciles design constraints with decision maker (DM) preferences. The task of eliciting and encoding decision maker preferences is, however, extremely difficult. A Pareto front representing the locus of the nondominated designs is, therefore, often generated to help a decision maker select the best design. In this paper, we show that this method has a shortcoming when there is uncertainty in both the decision problem variables and in the model of decision maker's preferences. In this case, the Pareto front is inconsistent with multi-attribute utility (MAU) theory, unless the decision maker trades off attributes or some functions of them linearly. This is a strong restriction. To account for this, we propose a methodology that enables a decision maker to select the best design on a modified pareto front (MPF) which is acquired using envelopes of a set of certainty equivalent (CE) surfaces. The proposed method does not require separability of the multi-attribute utility function into single-attribute utilities, nor does it require the decision maker to trade the attributes (or any function of them) linearly. We demonstrate our approach on a simple optimization problem and in design of a reduction gear.

Addressing Limitations of Pareto Front in Design Under Uncertainty

2011 ◽  
Author(s):  
Vijitashwa Pandey ◽  
Zissimos P. Mourelatos ◽  
Efstratios Nikolaidis
Keyword(s):  
Decision Maker ◽  
Decision Problem ◽  
Utility Theory ◽  
Pareto Front ◽  
Optimization Problem ◽  
Reduction Gear ◽  
Design Under Uncertainty ◽  
Design Constraints ◽  
Multi Attribute Utility Theory ◽  
Single Attribute

Engineering design reconciles design constraints with decision maker preferences. The task of eliciting and encoding decision maker preferences is, however, extremely difficult. A Pareto front representing the locus of the non-dominated designs is therefore, often generated to help a decision maker select the best design. In this paper, we show that this method has a shortcoming. We show that when there is uncertainty in both the decision problem variables and in the decision maker’s preferences, this methodology is inconsistent with multi-attribute utility theory, unless the decision maker trades off attributes or some functions of them linearly. This is a strong restriction. To account for this, we propose a methodology that enables a decision maker to select the best design on a modified Pareto front which is acquired using envelopes of a set of certainty equivalent surfaces. This methodology does not require separability of the multi-attribute utility function into single attribute utilities, nor does it require the decision maker to trade the attributes (or any function of them) linearly. We demonstrate this methodology on a simple optimization problem and in design of a reduction gear.

scholarly journals A Multiobjective Genetic Algorithm for the Localization of Optimal and Nearly Optimal Solutions Which Are Potentially Useful: nevMOGA

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-22 ◽  
Author(s):  
Alberto Pajares ◽  
Xavier Blasco ◽  
Juan M. Herrero ◽  
Gilberto Reynoso-Meza
Keyword(s):  
Genetic Algorithm ◽  
Decision Making ◽  
Decision Maker ◽  
Pareto Front ◽  
Optimization Problem ◽  
Engineering Problem ◽  
Multiobjective Optimization Problem ◽  
Optimal Solutions ◽  
Multiobjective Genetic Algorithm ◽  
Pi Controllers

Traditionally, in a multiobjective optimization problem, the aim is to find the set of optimal solutions, the Pareto front, which provides the decision-maker with a better understanding of the problem. This results in a more knowledgeable decision. However, multimodal solutions and nearly optimal solutions are ignored, although their consideration may be useful for the decision-maker. In particular, there are some of these solutions which we consider specially interesting, namely, the ones that have distinct characteristics from those which dominate them (i.e., the solutions that are not dominated in their neighborhood). We call these solutions potentially useful solutions. In this work, a new genetic algorithm called nevMOGA is presented, which provides not only the optimal solutions but also the multimodal and nearly optimal solutions nondominated in their neighborhood. This means that nevMOGA is able to supply additional and potentially useful solutions for the decision-making stage. This is its main advantage. In order to assess its performance, nevMOGA is tested on two benchmarks and compared with two other optimization algorithms (random and exhaustive searches). Finally, as an example of application, nevMOGA is used in an engineering problem to optimally adjust the parameters of two PI controllers that operate a plant.

Decision Topology Assessment in Engineering Design Under Uncertainty

2014 ◽  
Author(s):  
Vijitashwa Pandey ◽  
Zissimos P. Mourelatos ◽  
Matthew P. Castanier
Keyword(s):  
Decision Analysis ◽  
Engineering Design ◽  
Decision Maker ◽  
Mathematical Optimization ◽  
Linear Functions ◽  
Design Decision ◽  
Utility Analysis ◽  
Decision Making Problem ◽  
Single Attribute ◽  
Incremental Construction

The implications of decision analysis (DA) on engineering design are well known. Recently, the authors proposed decision topologies (DT) as a visual method for making design decisions and proved that they are consistent with normative decision analysis. This paper addresses the practical issue of assessing DTs for a decision maker (DM) using their responses, particularly under uncertainty. This is critical to encoding decision maker preferences so that further analysis and mathematical optimization can be performed using the correct set of preferences. We show how multiattribute DTs can be directly assessed from DM responses. Four methods are shown to evolutionarily assess DTs among which one that requires the DM to rank alternatives and another where a utility function is first assessed. It is also shown that preferences under uncertainty can be easily incorporated. In addition, we show that topologies can be constructed using single attribute topologies similarly to multi-linear functions in utility analysis. This incremental construction simplifies the process of topology construction. The reverse problem of inferring single attribute DTs is also presented. The proposed assessment methods are used on a design decision making problem of a welded beam.

Multi-criteria shape optimization of open-spandrel concrete arch bridges: Pareto front development and decision-making

2019 ◽  
Vol 16 (5) ◽  
pp. 670-680
Author(s):  
Majid Pouraminian ◽  
Somayyeh Pourbakhshian
Keyword(s):  
Decision Making ◽  
Particle Swarm Optimization ◽  
Shape Optimization ◽  
Decision Maker ◽  
Pareto Front ◽  
Optimization Problem ◽  
Objective Functions ◽  
Swarm Optimization ◽  
Content Type ◽  
Arch Bridges

Purpose This paper aims to study the shape of the concrete arched bridge by particle swarm optimization algorithm. Design/methodology/approach Finite element model of open-spandrel concrete arch bridges was constructed using a number of parameters. Design variables of optimization problem include height of skewback abutment, height of arch crown, position of crown with respect to global axes and left and right radius of up and down arches. After parametric modeling of bridge geometry and application of multi-objective particle swarm optimization, the shape optimization of bridge arch was determined. The concrete volume used in bridge substructure construction and maximum principal tensile stress of concrete arch body was adopted as two objective functions in this study. The optimization problem aims to minimize the two objective functions. Geometric and stress constraints are also included in the problem. Findings Based on the results presented in the paper, the Pareto front is generated which helps the decision-maker or designer to pick the compromise solution from among 20 optimum designs according to their subjective preferences or engineering judgment, respectively. Moreover, to help the decision-maker, the two multiple objective decision-making methods were used for selection of the best solution from among nondominated solutions. Originality/value This research aims to solve an interesting optimization problem in structural engineering. Optimization of arch bridges structure was done for reducing construction costs and increasing safety for the first time.

scholarly journals Algebraic Solution to Constrained Bi-Criteria Decision Problem of Rating Alternatives through Pairwise Comparisons

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 303
Author(s):  
Nikolai Krivulin
Keyword(s):  
Decision Problem ◽  
Optimization Problem ◽  
Pareto Frontier ◽  
Sufficient Conditions ◽  
Approximation Error ◽  
Optimal Solution ◽  
Algebraic Solution ◽  
Pairwise Comparisons ◽  
Logarithmic Scale ◽  
Idempotent Algebra

We consider a decision-making problem to evaluate absolute ratings of alternatives from the results of their pairwise comparisons according to two criteria, subject to constraints on the ratings. We formulate the problem as a bi-objective optimization problem of constrained matrix approximation in the Chebyshev sense in logarithmic scale. The problem is to approximate the pairwise comparison matrices for each criterion simultaneously by a common consistent matrix of unit rank, which determines the vector of ratings. We represent and solve the optimization problem in the framework of tropical (idempotent) algebra, which deals with the theory and applications of idempotent semirings and semifields. The solution involves the introduction of two parameters that represent the minimum values of approximation error for each matrix and thereby describe the Pareto frontier for the bi-objective problem. The optimization problem then reduces to a parametrized vector inequality. The necessary and sufficient conditions for solutions of the inequality serve to derive the Pareto frontier for the problem. All solutions of the inequality, which correspond to the Pareto frontier, are taken as a complete Pareto-optimal solution to the problem. We apply these results to the decision problem of interest and present illustrative examples.

Parameter study and optimization design of a hat oblique tunnel portal

2021 ◽  
pp. 095440972110140
Author(s):  
Zijian Guo ◽  
Tanghong Liu ◽  
Wenhui Li ◽  
Yutao Xia
Keyword(s):  
High Speed ◽  
Optimization Design ◽  
Pareto Front ◽  
Optimization Problem ◽  
Optimization Method ◽  
Design Parameters ◽  
Parameter Study ◽  
Buffer Structure ◽  
Tunnel Entrance ◽  
The Ideal

The present work focuses on the aerodynamic problems resulting from a high-speed train (HST) passing through a tunnel. Numerical simulations were employed to obtain the numerical results, and they were verified by a moving-model test. Two responses, [Formula: see text] (coefficient of the peak-to-peak pressure of a single fluctuation) and[Formula: see text] (pressure value of micro-pressure wave), were studied with regard to the three building parameters of the portal-hat buffer structure of the tunnel entrance and exit. The MOPSO (multi-objective particle swarm optimization) method was employed to solve the optimization problem in order to find the minimum [Formula: see text] and[Formula: see text]. Results showed that the effects of the three design parameters on [Formula: see text] were not monotonous, and the influences of[Formula: see text] (the oblique angle of the portal) and [Formula: see text] (the height of the hat structure) were more significant than that of[Formula: see text] (the angle between the vertical line of the portal and the hat). Monotonically decreasing responses were found in [Formula: see text] for [Formula: see text] and[Formula: see text]. The Pareto front of [Formula: see text] and[Formula: see text]was obtained. The ideal single-objective optimums for each response located at the ends of the Pareto front had values of 1.0560 for [Formula: see text] and 101.8 Pa for[Formula: see text].

scholarly journals An Analytical Study in Multi-physics and Multi-criteria Shape Optimization

2021 ◽  
Author(s):  
Hanno Gottschalk ◽  
Marco Reese
Keyword(s):  
Shape Optimization ◽  
Physical System ◽  
Pareto Front ◽  
Optimization Problem ◽  
Analytical Study ◽  
Static Pressure ◽  
Hausdorff Metric ◽  
Objective Functions ◽  
Hölder Continuous ◽  
Turbine Vane

AbstractA simple multi-physical system for the potential flow of a fluid through a shroud, in which a mechanical component, like a turbine vane, is placed, is modeled mathematically. We then consider a multi-criteria shape optimization problem, where the shape of the component is allowed to vary under a certain set of second-order Hölder continuous differentiable transformations of a baseline shape with boundary of the same continuity class. As objective functions, we consider a simple loss model for the fluid dynamical efficiency and the probability of failure of the component due to repeated application of loads that stem from the fluid’s static pressure. For this multi-physical system, it is shown that, under certain conditions, the Pareto front is maximal in the sense that the Pareto front of the feasible set coincides with the Pareto front of its closure. We also show that the set of all optimal forms with respect to scalarization techniques deforms continuously (in the Hausdorff metric) with respect to preference parameters.

scholarly journals Two Remarks on Blackwell's Theorem

2008 ◽  
Vol 45 (02) ◽  
pp. 580-586 ◽  
Author(s):  
Ehud Lehrer ◽  
Eran Shmaya
Keyword(s):  
Decision Maker ◽  
Information Structure ◽  
Decision Problem ◽  
Partial Information ◽  
The Other ◽  
Actual State ◽  
Information Structures ◽  
Comparison Of Experiments ◽  
Classical Characterization ◽  
Blackwell's Theorem

In a decision problem with uncertainty a decision maker receives partial information about the actual state via an information structure. After receiving a signal, he is allowed to withdraw and gets zero profit. We say that one structure is better than another when a withdrawal option exists if it may never happen that one structure guarantees a positive profit while the other structure guarantees only zero profit. This order between information structures is characterized in terms that are different from those used by Blackwell's comparison of experiments. We also treat the case of a malevolent nature that chooses a state in an adverse manner. It turns out that Blackwell's classical characterization also holds in this case.

scholarly journals Moving beyond the cost–loss ratio: economic assessment of streamflow forecasts for a risk-averse decision maker

2017 ◽  
Vol 21 (6) ◽  
pp. 2967-2986 ◽  
Author(s):  
Simon Matte ◽  
Marie-Amélie Boucher ◽  
Vincent Boucher ◽  
Thomas-Charles Fortier Filion
Keyword(s):  
Risk Aversion ◽  
Utility Function ◽  
Decision Maker ◽  
Economic Value ◽  
Economic Assessment ◽  
Risk Averse ◽  
Loss Ratio ◽  
Ensemble Forecasts ◽  
Streamflow Forecasts ◽  
Cara Utility

Abstract. A large effort has been made over the past 10 years to promote the operational use of probabilistic or ensemble streamflow forecasts. Numerous studies have shown that ensemble forecasts are of higher quality than deterministic ones. Many studies also conclude that decisions based on ensemble rather than deterministic forecasts lead to better decisions in the context of flood mitigation. Hence, it is believed that ensemble forecasts possess a greater economic and social value for both decision makers and the general population. However, the vast majority of, if not all, existing hydro-economic studies rely on a cost–loss ratio framework that assumes a risk-neutral decision maker. To overcome this important flaw, this study borrows from economics and evaluates the economic value of early warning flood systems using the well-known Constant Absolute Risk Aversion (CARA) utility function, which explicitly accounts for the level of risk aversion of the decision maker. This new framework allows for the full exploitation of the information related to a forecasts' uncertainty, making it especially suited for the economic assessment of ensemble or probabilistic forecasts. Rather than comparing deterministic and ensemble forecasts, this study focuses on comparing different types of ensemble forecasts. There are multiple ways of assessing and representing forecast uncertainty. Consequently, there exist many different means of building an ensemble forecasting system for future streamflow. One such possibility is to dress deterministic forecasts using the statistics of past error forecasts. Such dressing methods are popular among operational agencies because of their simplicity and intuitiveness. Another approach is the use of ensemble meteorological forecasts for precipitation and temperature, which are then provided as inputs to one or many hydrological model(s). In this study, three concurrent ensemble streamflow forecasting systems are compared: simple statistically dressed deterministic forecasts, forecasts based on meteorological ensembles, and a variant of the latter that also includes an estimation of state variable uncertainty. This comparison takes place for the Montmorency River, a small flood-prone watershed in southern central Quebec, Canada. The assessment of forecasts is performed for lead times of 1 to 5 days, both in terms of forecasts' quality (relative to the corresponding record of observations) and in terms of economic value, using the new proposed framework based on the CARA utility function. It is found that the economic value of a forecast for a risk-averse decision maker is closely linked to the forecast reliability in predicting the upper tail of the streamflow distribution. Hence, post-processing forecasts to avoid over-forecasting could help improve both the quality and the value of forecasts.

scholarly journals Resolving Time Inconsistency of Decision Problem with Non-expectation Operator: From Internal Conflict to Internal Harmony by Strategy of Self-Coordination

2020 ◽  
Author(s):  
Xiangyu Cui ◽  
Duan Li ◽  
Yun Shi
Keyword(s):  
Decision Maker ◽  
Decision Problem ◽  
Hyperbolic Discounting ◽  
Time Inconsistency ◽  
Internal Conflict ◽  
Equilibrium Strategy ◽  
Game Model ◽  
Long Run ◽  
Short Run ◽  
Time Inconsistent

When a stochastic decision problem is time inconsistent, the decision maker would be puzzled by his conflicting decisions optimally derived from his time-varying preferences at different time instants (with different time horizons). While the long-run self (LR) of the decision maker pursues the long-term optimality, the short-run selves (SRs) of the decision maker at different time instants bow to short-term temptations. While the literature began to recognize the importance to strike a balance between LR's and SRs' interests, the existing results are not applicable to situations where the decision maker's preferences involve non-expectation operators. We propose an operable unified two-tier dual-self game model with commitment by punishment, which can cope with general time inconsistent stochastic decision problems with both expectation and non-expectation operators in the objective function. By attaching punishment terms to both the preferences of LR and SRs which quantitatively evaluate the internal conflict among different selves, our game model aligns the interests of the LR and SRs to a certain degree. The equilibrium strategy, termed strategy of self-coordination, achieves some degree of internal harmony among various selves. We successfully apply the model to the investment and consumption problem with quasi-hyperbolic discounting and the dynamic mean-variance portfolio selection problem.

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