Possibility and evidence theory based design optimization: A survey

To obtain reliable designs, aleatory and epistemic uncertainties are considered recently in the structural analysis and design optimization. The reliability based de-sign optimization (RBDO) method [1] is used when the amount of input data is sufficient enough to create accurate statistical distribution. On the other hand, when the sufficient input data are not available due to limitations in time, human, and facility resources, the optimum design may not be reliable if RBDO method is used. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method can be used for structural analysis and possibility based design optimization (PBDO) [2]. However, in many industry design problems, we may have to deal with design problems that involve with the mixed input statistical random and fuzzy variables simultaneously. For these problems, RBDO may yield unreliable optimum designs because of insufficient data. On the other hand, treating the random variables as fuzzy variables and invoking PBDO to solve the mixed design variable problem may yield too conservative designs with higher optimum costs. This paper proposes a new mixed variable design optimization (MVDO) problem based on the performance measure approach (PMA) [1]. To evaluate the possibilistic constraint in MVDO, a sub-optimization problem for inverse analysis is carried out using a hyper-cylinder domain. To solve this sub-problem efficiently and effectively, a new numerical algorithm, maximum failure search (MFS) method, is proposed in this paper by combining the enhanced hybrid mean value (HMV+) method [3] for the inverse reliability analysis in RBDO and the maximal possibility search (MPS) method [2] for the inverse possibility analysis in PBDO. Some mathematical examples are used to demonstrate the efficiency and effectiveness of the proposed numerical MFS method. Some physical design examples are used to compare the proposed MVDO results with RBDO and PBDO results.

Aircraft manufacturing companies have to consider multiple derivatives to satisfy various market requirements. They modify or extend an existing aircraft to meet new market demands while keeping the development time and cost to a minimum. Many researchers have studied the derivative design process, but these research efforts consider baseline and derivative designs together, while using the whole set of design variables. Therefore, an efficient process that can reduce cost and time for aircraft derivative design is needed. In this research, a more efficient design process is proposed which obtains global changes from local changes in aircraft design in order to develop aircraft derivatives efficiently. Sensitivity analysis was introduced to remove unnecessary design variables that have a low impact on the objective function. This prevented wasting computational effort and time on low priority variables for design requirements and objectives. Additionally, uncertainty from the fidelity of analysis tools was considered in design optimization to increase the probability of optimization results. The Reliability Based Design Optimization (RBDO) and Possibility Based Design Optimization (PBDO) methods were proposed to handle the uncertainty in aircraft conceptual design optimization. In this paper, Collaborative Optimization (CO) based framework with RBDO and PBDO was implemented to consider uncertainty. The proposed method was applied for civil jet aircraft derivative design that increases cruise range and the number of passengers. The proposed process provided deterministic design optimization, RBDO, and PBDO results for given requirements.

Structural analysis and design optimization have recently been extended to stochastic approach to take various uncertainties into account. However in areas whwew it is not possible to produce accurate statistical information, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, since improper modeling of uncertainty could cause greater degree of statistical uncertainty than those of physical uncertainty. For uncertainty with insufficient information, possibility-based (or fuzzy set) methods have recently been introduced in stochastic structural analysis and design optimization. The main advantage of the fuzzy analysis is that it preserves the intrinsic random nature of physical variables through their membership functions and, when used for evaluation of designs, yields more conservative design than those from the probabilistic methods. There are two computational aspects in the fuzzy analysis compared to the probability analysis. First, the input fuzzy variables can be defined easier than the input random variables when no or very limited statistical data are available. Secondly, extended fuzzy operations are much simpler than those required to use probability, especially when a number of variables are involved. For possibility-based design optimization (PBDO), like for reliability-based design optimization (RBDO), the performance measure approach (PMA) with an inverse fuzzy analysis is more appropriate than other approaches, such as a possibility index approach. This paper proposes a new formulation of PBDO using PMA. It is also found that PBDO is more desirable than RBDO, since PBDO can inherently handle a system level possibility unlike RBDO. For the inverse fuzzy analysis, the maximal possibility search (MPS) method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the

Purpose – The purpose of this paper is to study the consideration of uncertainty from analysis modules for aircraft conceptual design by implementing uncertainty-based design optimization methods. Reliability-Based Design Optimization (RBDO), Possibility-Based Design Optimization (PBDO) and Robust Design Optimization (RDO) methods were developed to handle uncertainties of design optimization. The RBDO method is found suitable for uncertain parameters when sufficient information is available. On the other hand, the PBDO method is proposed when uncertain parameters have insufficient information. The RDO method can apply to both cases. The RBDO, PBDO and RDO methods were considered with the Multidisciplinary Design Optimization (MDO) method to generate conservative design results when low fidelity analysis tools are used. Design/methodology/approach – Methods combining MDO with RBDO, PBDO and RDO were developed and have been applied to a numerical analysis and an aircraft conceptual design. This research eva...

Aircraft manufacturing companies have to consider multiple derivatives to satisfy various market requirements. They modify or extend an existing aircraft to meet the new market demands while keeping the development time and the cost to a minimum. Many researchers have studied the derivative design process, but these research considered the baseline and the derivatives together, while using the whole set of design variables. Therefore, an efficient process that can reduce the cost and the time for the aircraft derivative design is needed. In this dissertation, Aircraft Derivative Design Optimization process (ADDOPT) was developed which obtains the global changes from the local changes in the aircraft design to develop the aircraft derivatives efficiently. The sensitivity analysis was implemented to ignore design variables that have low impact on the objective function. This avoids wasting computational effort and time on low priority variables for design requirements and objectives. Additionally, the classification of uncertainty from its characteristics and sources of uncertainty involved in the aircraft design process were suggested to consider with design optimization. Uncertainty from the fidelity of analysis tools was applied in design optimization to increase the probability of optimization results. To handle uncertainty in low fidelity analysis tools on aircraft conceptual design optimization, Reliability Based Design Optimization (RBDO) and Possibility Based Design Optimization (PBDO) methods were performed. In this research, Extended Fourier Amplitude Sensitivity Test (eFAST) method was implemented in ADDOPT for Global Sensitivity Analysis (GSA) method and Collaborative Optimization (CO) based framework with RBDO and PBDO were also used. These methods were evaluated using numerical examples. ADDOPT was carried through on the civil jet aircraft derivative design. The objective of the optimization problem was to increase cruise range while satisfying the requirement such as the number of passengers. The proposed process reduced computation effort by reducing the number of design variables and achieved the target probability of failure when considering uncertainty from low fidelity analysis tools.

For problems involving uncertainties in design variables and parameters, a bi-objective evolutionary algorithm (EA) based approach to design optimization using evidence theory is proposed and implemented in this paper. In addition to a functional objective, a plausibility measure of failure of constraint satisfaction is minimized. Despite some interests in classical optimization literature, this is the first attempt to use evidence theory with an EA. Due to EA's flexibility in modifying its operators, nonrequirement of any gradient, its ability to handle multiple conflicting objectives, and ease of parallelization, evidence-based design optimization using an EA is promising. Results on a test problem and two engineering design problems show that the modified evolutionary multi-objective optimization algorithm is capable of finding a widely distributed trade-off frontier showing different optimal solutions corresponding to different levels of plausibility failure limits. Furthermore, a single-objective evidence-based EA is found to produce better optimal solutions than a previously reported classical optimization algorithm. Furthermore, the use of a graphical processing unit (GPU) based parallel computing platform demonstrates EA's performance enhancement around 160–700 times in implementing plausibility computations. Handling uncertainties of different types are getting increasingly popular in applied optimization studies and this EA based study is promising to be applied in real-world design optimization problems.

This article is devoted to applications of fuzzy set theory, possibility theory and evidence theory in civil engineering, presenting current work of a group or researchers at the University of Innsbruck. We argue that these methods are well suited for analyzing and processing the parameter uncertainties arising in soil mechanics and construction management. We address two specific applications here: finite element computations in foundation engineering and a queueing model in earth work.

In most industrial applications, only limited statistical information is available to describe the input uncertainty model due to expensive experimental testing costs. It would be unreliable to use the estimated input uncertainty model obtained from insufficient data for the design optimization. Furthermore, when input variables are correlated, we would obtain non-optimum design if we assume that they are independent. In this paper, two methods for problems with a lack of input statistical information--possibility-based design optimization (PBDO) and reliability-based design optimization (RBDO) with confidence level on the input model--are compared using mathematical examples and an Abrams M1A1 tank roadarm example. The comparison study shows that PBDO could provide an unreliable optimum design when the number of samples is very small. In addition, PBDO provides an optimum design that is too conservative when the number of samples is relatively large. Furthermore, the obtained PBDO designs do not converge to the optimum design obtained using the true input distribution as the number of samples increases. On the other hand, RBDO with confidence level on the input model provides a conservative and reliable optimum design in a stable manner. The obtained RBDO designs converge to the optimum design obtained using the true input distribution as the number of samples increases.

<div class="htmlview paragraph">Early in the engineering design cycle, it is difficult to quantify product reliability or compliance to performance targets due to insufficient data or information for modeling the uncertainties. Design decisions are therefore, based on fuzzy information that is vague, imprecise qualitative, linguistic or incomplete. The uncertain information is usually available as intervals with lower and upper limits. In this work, the possibility theory is used to assess design reliability with incomplete information. The possibility theory can be viewed as a variant of fuzzy set theory. A possibility-based design optimization method is proposed where all design constraints are expressed possibilistically. It is shown that the method gives a conservative solution compared with all conventional reliability-based designs obtained with different probability distributions. A general possibility-based design optimization method is also presented which handles a combination of random and possibilistic design variables. Numerical examples demonstrate the application of possibility theory in design.</div>

Mathematical theories of uncertainty in which measures of uncertainty are now well established are fuzzy set theory, evidence theory, and possibility theory, in addition to classical set theory and probability theory. Some of these theories are connected. Classical set theory is subsumed under fuzzy set theory. Probability theory and possibility theory are branches of evidence theory, while evidence theory is, in turn, a branch of fuzzy measure theory. Although information aspects of fuzzy measure theory have not been investigated as yet, the theory is briefly introduced in this chapter because it represents a broad framework for future research.

An evidence-based framework for optimization of structural components under material field uncertainty is developed. The approach is applied to the evidence-based design optimization (EBDO) of a stiffened tube under axial impact load using an isotropic-elastic-plastic constitutive model to simulate dynamic material behavior. Uncertainty modeling considers the changes in material parameters that are caused by variability in material properties as well as incertitude in the experimental data and the procedure for determination of these parameters. Spatial variability is modeled using a joint field belief structure that includes the effects of both incertitude and randomness on the estimated material properties across the structural component. With propagation of field uncertainties, the evidence-based objective and design constraint functions are modeled. Surrogate models are used in different steps to facilitate both the field uncertainty propagation and subsequent solution of the EBDO problem. The results of the application of the developed methodology to EBDO of an energy-absorbing component are presented and discussed.

The paper proposes a parallel survey of revision and updating operations available in the probability theory and in the possibility theory frameworks. In these two formalisms the current state of knowledge is generally represented by a [0,1]-valued function whose domain is an exhaustive set of mutually exclusive possible states of the world. However, in possibility theory, the unit-interval can be viewed as a purely ordinal scale. Two general kinds of operations can be defined on this assignment function: conditioning, and imaging (or “projection”). the difference between these two operations is analogous to the one made between belief revision à la Gärdenfors and updating à la Katsuno and Mendelzon in the logical framework. In the probabilistic framework these two operations are respectively Bayesian conditioning and Lewis' imaging. Counterparts to these operations are presented for the possibilistic framework including the case of conditioning upon uncertain observations, and justifications are given which parallel the ones existing for the probabilistic operations. More particularly, it is recalled that possibilistic conditioning satisfies all the postulates proposed by Alchourrón, Gärdenfors and Makinson for belief revision (stated in possibilistic terms), and it is proved that possibilistic imaging satisfies all the postulates proposed by Katsuno and Mendelzon. the situation where our current knowledge is stated in terms of weighted logical propositions is discussed in connection to possibility theory. Revision in other more complex numerical formalisms, namely belief and plausibility functions, and upper and lower probabilities is also surveyed. Recent results on the revision of conditional knowledge bases are also reviewed. the frameworks of belief functions, upper and lower probabilities and conditional bases are more sophisticated than the previous ones because they enable to distinguish between factual evidence and generic knowledge in a cognitive state. This, framework leads to two forms of belief revision respectively taking care of the revision of evidence and the revision of knowledge. © 1994 John Wiley & Sons, Inc.

Artificial Intelligence has emerged and been considerably developed in the last fifty years together with the advent of the computer age, as a new scientific area aiming at processing information in agreement with the way humans do. Inevitably, such a range of efforts, both theoretical and application-oriented, and motivated by new concerns, has led to modify and enlarge the way basic notions such as uncertainty, belief, knowledge, or evidence could be thought, represented and handled in practice. In particular, there has been a major trend in uncertainty (more specifically, partial belief) modelling, emphasizing the idea that the degree of confidence in an event is not totally determined by the confidence in the opposite event, as assumed in probability theory. Possibility theory (Dubois and Prade, 1988) belongs to this trend that describes partial belief in terms of certainty and plausibility, viewed as distinct concepts. Belief and plausibility functions (Shafer, 1976), or lower and upper probabilities (Walley, 1991) are other important representatives of this trend. The distinctive features of possibility theory are its computational simplicity, and its position as a bridge between numerical and symbolic theories of partial belief for practical reasoning. Possibility theory, based on a non-additive setting that contrasts with probability theory, provides a potentially more qualitative treatment of partial belief, since the operations “max” and “min” play a role somewhat analogous to the sum and the product in probability calculus. There are two different kinds of possibility theory: one is qualitative and the other is quantitative. They share the same kind of set-functions, but they differ when it comes to conditioning and combination

In this paper a robust optimization problem with uncertain objective function is considered. The uncertainty is modeled by specifying a scenario set, containing a finite number of objective function coefficients, called scenarios. Additional knowledge in scenario set can be represented by using a mass function defined on the power set of scenarios. This mass function defines a belief function, which in turn induces a family of probability distributions in scenario set. One can then use a generalized Hurwicz criterion, i.e. a convex combination of the upper and lower expectations, to solve the uncertain problem. Recently, possibility theory has been applied to extend the model of uncertainty based on belief functions. Namely, belief function can be induced by a random fuzzy set. In this paper we show how this generalized model can be applied to robust optimization.

Contribution of the mathematical modelling of knowledge to the evaluation of uncertainty margins of a LBLOCA transient (LOFT-L2-5)