In probabilistic approaches to engineering design, including robust design, mean and variance are commonly used as the optimization objectives. This method, however, has significant limitations. For one, some mean–variance Pareto efficient designs may be stochastically dominated and should not be considered. Stochastic dominance is a mathematically rigorous concept commonly used in risk and decision analysis, based on the cumulative distribution function (CDFs), which establishes that one uncertain prospect is superior to another, while requiring minimal assumptions about the utility function of the outcome. This property makes it applicable to a wide range of engineering problems that ordinarily do not utilize techniques from normative decision analysis. In this work, we present a method to perform optimizations consistent with stochastic dominance: the Mean–Gini method. In macroeconomics, the Gini Index is the de facto metric for economic inequality, but statisticians have also proven a variant of it can be used to establish two conditions that are necessary and sufficient for both first and second-order stochastic dominance . These conditions can be used to reduce the Pareto frontier, eliminating stochastically dominated options. Remarkably, one of the conditions combines both mean and Gini, allowing for both expected outcome and uncertainty to be expressed in a single objective which, when maximized, produces a result that is not stochastically dominated given the Pareto front meets a convexity condition. We also find that, in a multi-objective optimization, the Mean–Gini optimization converges slightly faster than the mean–variance optimization.

Introduction

In general, deterministic optimization seeks to minimize a quantity, or several quantities in the case of multi-objective optimization, given a vector of n design variables, x, subject to ng inequality constraints and nh equality constraints with constant design parameters p. The standard deterministic optimization expression is shown below. Here, gi is the ith inequality constraint; hj is the jth equality constraint; and xku and xkl are the kth upper and lower bounds, respectively, 
minf(x,p)s.t.gi(x,p)0j=1,,nghj(x,p)=0j=1,,nhxkl<xk<xkuk=1,,n
(1)
However, in reality, a plethora of uncertainties associated with both the system itself and the environment that it operates within exists. Research efforts to model and optimize with respect to uncertainties in engineering systems are referred to with different names in disciplines and contexts including but not limited to uncertainty-based multidisciplinary design optimization (MDO), nondeterministic approaches, robust design optimization (RDO), reliability-based design optimization, and stochastic programing [17]. In addition, various decision analysis techniques exist to make decisions from resulting designs. In general, there are three main steps in design optimization involving uncertainty. The first involves identifying, qualifying, and quantifying sources of uncertainty to construct mathematical models that reflect the uncertainty experienced in practice. The most popular way to model uncertainty is with probability density functions (PDFs) or, in the discrete case, probability mass functions. As a result, f(x,p) mentioned above is no longer a single quantity, but it is a distribution of possible outcomes. The second step involves numerical and analytic methods to propagate uncertainties throughout the system. Many techniques exist to perform this task, including but not limited to Monte Carlo sampling, Gaussian quadrature, and stochastic expansion [811]. Finally, objectives must be optimized to achieve the desired behavior. This final step is sometimes referred to as optimization under uncertainty (OUU) [12]. Many formulations and approaches exist, some approaches such as conditional value-at-risk and the robust counterpart approach, which is directly related to the minimax approach, take a worst-case scenario approach [1315]. Other approaches optimize with respect to higher statistical moments, but doing so increases the dimensionality of the objective space [16]. The two most common being those used in the fields known as RDO and reliability-based design optimization [4,5]. The standard practice for both involves optimizing with respect to the mean and variance of f(x,p). For example, the mathematical formulation for RDO generally used is shown in the following equation [1,3]: 
minf̃(x,p)=f(μF(x,p),σf(x,p))s.t.gi(x,p)0j=1,,nghj(x,p)=0j=1,,nhxkl<xk<xkui=1,,n
(2)

Here, two objectives, μF and σf, are combined by the function f̃, usually with a weighted sums formula. Optimizing with respect to two objectives drastically increases computational cost relative to a single objective, making it impossible for some problems and undesirable for others. The weighted sums approach creates a combined objective that accounts for both mean and uncertainty in outcome, greatly reducing computational cost. However, the weighted sums approach involves setting weights a priori without a mathematically rigorous method to determine suitable values for the weights.

In cases where a multi-objective optimization is feasible, μF and σf can be treated as competing objectives in a multi-objective optimization to achieve a set of Pareto efficient alternatives that can be plotted as a Pareto front. However, a decision must be made from within this set of Pareto efficient alternatives and such a decision must be informed by higher level information [17,18]. In reality, additional information that can accurately inform such a decision is often not available and designers often resort to variants of the weighted sum method applied a posteriori without any rigorous guidance with regard to what values should be selected for the weights. In such cases, formulating the weight, or preference vectors, is a highly subjective task, which is difficult to perform accurately and reliably [17,18].

The effectiveness of μ and σ also breaks down for non-normal distributions. They only fully parametrize the normal distribution, but in reality, uncertain outcomes very often follow non-normal distributions [19]. In such cases, using σ as the summary statistic for uncertainty can be grossly misleading. Figure 1 shows two distributions with identical mean and variance. While they are significantly different, optimizations that use μ and σ as objectives will treat the two outcomes shown identically.

Even in the case when outcomes are normally distributed, μ and σ still do not provide any information with regard to stochastic dominance. As discussed later in this paper, it is possible for a design to be stochastically dominated, even though it is Pareto efficient. Thus, when performing optimizations under uncertainty, not all designs within the Pareto efficient set are truly valid alternatives. In this work, we present a simple technique to obtain a subset of the Pareto efficient set that stochastically dominates the rest of the Pareto efficient set, which we refer to as the stochastic dominance consistent set, from which designers should make decisions. The technique centers on replacing variance and standard deviation with a variant of the Gini Index, the metric used in macroeconomics to measure income inequality. The Gini Index and its variants have many rich properties that have led to it being applied to problems in many fields including portfolio optimization, signal reconstruction, and plant population characterization [2026], among others. Replacing variance with Gini ultimately allows for an efficient optimization formulation that produces designs that stochastically dominate alternatives.

Stochastic Dominance and Gini's Mean Difference

Stochastic Dominance

An Illustration of First-Order Stochastic Dominance (FSD).

Stochastic dominance has been the subject of significant research in economics and decision analysis but has not seen significant application in the engineering design community [27]. In simplest terms, stochastic dominance is the mathematical condition where one probabilistic outcome is desirable over another, given minimal assumptions about the utility of the outcome. Here, we illustrate first-order stochastic dominance (FSD), which only requires that the utility of the outcome quantity is monotonic with respect to the quantity itself. Consider Fig. 2, which shows two example PDFs and their corresponding cumulative distribution functions (CDFs) where a higher value of the objective is desirable. Note that B provides a higher expected outcome than A, that is μB>μA, but A has less uncertainty. If we evaluate uncertainty with variance, we can clearly see that σB2>σA2. With the multi-objective approach, both A and B are potentially Pareto efficient and are valid alternatives. However, if one examines the CDFs of A and B, one sees that for every possible value q of the objective, FB(q)<FA(q). Recall that the cumulative distribution represents the probability that the outcome is at most a particular value. Therefore, the above statement is equivalent to P(Bq)<P(Aq) and therefore 
qP(B>q)>P(A>q)
(3)

In this case, B always has a higher probability of obtaining a better outcome than A. If a higher value q is always desirable, B is superior and A should not even be considered. This runs counter to conventional wisdom, in which designs within the Pareto efficient set are considered valid alternatives. However, as seen above, even designs within the Pareto efficient set may be stochastically dominated. With the multi-objective approach, stochastically dominated designs are evaluated, leading to wasted computational effort. It also allows for the pernicious possibility of a designer choosing an inferior design. This indicates a need for a means to determine stochastic dominance between design alternatives. To address this issue, Horsetail matching has been developed recently, which seeks to minimize the distance between the outcome CDF with a target [28]. As stochastic dominance is defined by the CDF, this method prevents the selection of a stochastically dominated design by basing the evaluation criterion directly off the CDF.

Relationship to Utility Theory.

In Economics and Decision Analysis, the concept of utility theory is often employed to study optimal decision-making under risk or uncertainty [2931]. These concepts have been applied to various engineering problems, including manufacturing process selection, aquifer design, and scientific payloads, among others [3234]. However, in problems studied by both economists and engineers, formulating utility functions with parameters that reliably reflect real preferences can be a challenge. With stochastic dominance, however, alternatives can be eliminated with very minimal assumptions with respect to the utility functions [27]. As a matter of fact, the only assumption necessary to establish FSD is that the utility of a quantity u(q) is monotonic and strictly increasing with respect to q; that is, (d/dq)(u(q))>0. This very minimal assumption means FSD can also be applied to physical engineering problems that do not traditionally utilize utility theory, such as maximizing lift or minimizing fuel consumption. To establish second-order stochastic dominance (SSD), the marginal utility must also be strictly decreasing with respect to q; that is (d2/dq2)(u(q))<0. When the utility function for an individual behaves with the properties necessary for SSD, such an individual is known as risk averse [31]. FSD and SSD have different definitions and implications, but in the technique demonstrated in this paper, they are established jointly.

Formal Definition of Stochastic Dominance.

In this paper, we define stochastic dominance with slight changes in notation from the standard definitions of stochastic dominance to better suit the engineering design context [27,35]. We define FSD as shown below.

Definition(First-order stochastic dominance). LetFxAandFxBbe two cumulative distribution functions corresponding to designsxAandxB, respectively. Letqbe a sample from the space of possible outcomes,Q, and the utility function,u, behaves such that(du/dq)>0. FxAFirst-order stochastically dominatesFxBif and only if: 
qϵQFxB(q)FxA(q)
(4)

Second-order stochastic dominance also assumes that there are diminishing or neutral marginal returns to utility from the quantity of interest. That is, (du/dq)>0 and (d2u/dq2)0. In the engineering context, SSD may be useful for utility-based optimization approaches or in a value-based formulation with a risk-averse decision maker. In the method presented in this paper, however, FSD and SSD are established jointly, and thus the distinction is not of importance. However, we do want to bring the existence of SSD as stochastic dominance in general remains unfamiliar to the field of engineering optimization.

Definition (Second-order stochastic dominance). LetFxAandFxBbe two cumulative distribution functions corresponding to designsxAandxB, respectively. Letqbe a sample from the space of possible outcomesQand the utility function,u, behaves such that(du/dq)>0and(d2u/dq2)0. FxASecond-order stochastically dominatesFxBif and only if 
qϵQq[FxB(t)FxA(t)]dt0
(5)

Establishing Stochastic Dominance With Gini's Mean Difference (GMD).

In this paper, we propose an alternative metric for uncertainty in probabilistic outcomes that established necessary and sufficient conditions for stochastic dominance: Gini's Mean Difference [34]. The definition for this is provided below.

Definition (Gini's mean difference). Given a probability density function,f, and an outcome space,Q, letxϵQandyϵQ. Gini's mean difference (GMD) is defined as shown in the following equation [ib3030]: 
GMD=f(x)f(y)|xy|dxdy
(6)
We are particularly interested in one half of Gini's mean difference. This quantity will be referred to as γ and is defined as shown in Eq. (7a) along with variance in Eq. (7b) 
γ=12f(x)f(y)|xy|dxdy=12GMD
(7a)
 
σ2=x2f(x)dxμ2
(7b)
In the class of Monte Carlo methods with n samples where each outcome is treated with equal weight, we can calculate γ as shown [36] 
γ=i=1nj=1n|xixj|2n2
(8)

Note that γ divided by μ is the Gini index, which is used ubiquitously in macroeconomics as a measure of income inequality [36,37]. That is, the Gini Index is a normalized variant of γ. Yitzhaki [35] proved that necessary and sufficient conditions for stochastic dominance can be established with just μ and γ. This proof was originally formulated for economic problems involving profit, so μ is considered a desirable quantity and γ is undesirable. For all optimizations performed, we minimize γ and maximize μ, that is we minimize the negative of μ. The following propositions were proven by Yitzhaki.

Proposition 1a. For two designsxAandxB, Γ1andΓ2are necessary forxA1,2xB 
Γ1:μxAμxB0
(9a)
 
Γ2:(μxAγxA)(μxBγxB)0
(9b)

Proposition 1b. IfFxAandFxBintersect at most once, thenΓ1andΓ2are sufficient forxA1,2xB.

In many cases, the CDFs of interest do not intersect more than once and sufficiency can be established. Previous research determined the normal, lognormal, exponential, and uniform distributions all have this property [38] and there are likely many others. In this study, we do not further investigate and better define the mathematical conditions under which the Γ conditions are sufficient for stochastic dominance. Future research should attempt to develop methods to determine whether sufficiency can be established for a given problem.

These two conditions provide simple but powerful tools to improve techniques for optimization under uncertainty. In the rest of the paper, we demonstrate how the conditions can be applied to robust design optimization techniques to reduce the decision space from the classic Pareto efficient set to a subset with no stochastically dominated designs. The Γ2 condition also allows for the metric for uncertainty to be combined with the mean into a single objective which, when optimized, produces a design that is not stochastically dominated. This one objective, ϒ=μγ, achieves what the weighted sums approach sought out to do in reducing the computational cost of optimization, but it has rigorous mathematical foundations without the necessity of arbitrarily setting weights a priori and produces a result consistent with stochastic dominance rules.

An Multidisciplinary Design Optimization Test Problem

Problem Overview.

As a demonstration problem, a system-level MDO simulation for a commercial satellite system was used [39,40]. This simulation captures nine design variables and 22 behavior variables divided across seven coupled subsystems, shown in the design structure matrix in Fig. 3 below. The design variables—the independent variables that are subject to optimization—and behavior variables—the dependent variables obtained through systems analysis—included in this simulation are listed in Tables 13 in the Appendix for completeness. All multi-objective optimizations are performed with MATLAB's multi-objective genetic algorithm.

An Uncertain Demand Model.

This test problem is formulated with a value-based approach, and thus, the objective to be optimized is net present profit (NPP) with an annual discount rate of 10% [39]. Uncertainty is introduced into the system objective by an uncertain transponder demand model. The model used is the Wiener process, based off geometric Brownian motion [40]. This model has been used by several aerospace researchers in the past to model demand for satellite transponders and is also very popular in quantitative finance [41,42]. We acknowledge, in reality, demand at every time-step is also a function of price, but for the purposes of this paper, the price is fixed at $2 million. The model is described by a stochastic differential equation with the closed-form solution 
ΔDD=μmΔt+σmϵΔt
(10a)
 
Dt+Δt=Dte(μm0.5σm2)Δt+σmϵΔt
(10b)

According to Intelsat's 2003–2010 financial data, revenue from leasing transponders followed a trend described by the parameters μ=0.047 and σ=0.23 [42]. These parameter values were used for the demand simulation. The stochastic variable, ϵ, is drawn from a standard normal distribution, producing 100,000 scenarios that serve as the demand model. D0 is chosen as 50, and a Δt of one month is used.

Applications of the Mean–Gini Method

Reducing the Pareto Front.

In cases where it is desirable to have a Pareto front from which to make decisions, the Γ conditions can be applied to eliminate stochastically dominated designs, arriving at a subset of the Pareto efficient set which we will refer to as the stochastic dominance efficient set, SD. Remarkably, for a certain class of Pareto efficient sets, SD can be easily visualized on the μγ Pareto front. This Pareto front reduction technique can be performed either by optimizing with respect to mean and variance and then calculating the γ values for the elements of the Pareto efficient set or by directly optimizing with respect to mean and γ in the first place. We strongly recommend performing an optimization with respect to mean and γ, as we have found the optimization converges faster than when the optimization is performed with the variance. This finding is discussed in further detail later in the paper. Either way, the Pareto front should be plotted with μ and γ, as the Γ conditions allow for a visual interpretation of the regions of the Pareto front that are stochastically dominated.

Consider some multi-objective optimization which produces the μγ Pareto efficient set P*. If the slope of P* is strictly decreasing, SD can be intuitively visualized on the Pareto front. For the purposes of this paper, we define such a set as one that can be approximated by some function, f, such that μ=f(γ), (d2/dγ2)f(γ)<0 across the entire domain of the function, and (d/dγ)f(γ)=1 for some γ. In such cases, the following proposition holds true, the proof of which is provided in the Appendix.

Proposition 2a. Suppose there exists some designx*P*such that(dμx*/dγx*)=1. Thenx*1,2x, xP*such thatμx<μx*.

Proposition 2b. LetP*be anyμγPareto efficient set with members having CDFs that overlap at most once. Suppose there exists some functionfsuch thatμ=f(γ)γ, that(d2μ/dγ2)<0and(dμ/dγ)=1exists. The setSDis defined such thatySDif and only ifyP*and(dμy/dγy)1. Then, no two designs withinSDmay stochastically dominate each other.

Thus, we have that no two designs within SD stochastically dominate each other, making them all valid alternatives in the absence of additional information. However, all other designs, x, such that xP* but xSD, are stochastically dominated by the point x* and should thus not be considered. Note that if no design exists at the exact point where (dμ/dγ)=1, one should merely choose x* as the design closest to that point. An example demonstrating this technique follows.

A multi-objective optimization of the satellite problem was performed for 100 generations with respect to μ and γ with a population size of 1000. The 426 Pareto points from the final generation of this optimization are plotted in Fig. 4 with γ on the x-axis and μ on the right-hand side y-axis. A piecewise function was found to produce the best fit for this Pareto front with a one term power function fit to the first 380 data points, corresponding to γ63,472,271 and a quadratic function fit to the remaining points. With the resulting model, the derivative is calculated numerically with the finite difference method with a step size of Δγ=1000, which is approximately five orders of magnitude smaller than the values of γ. The resulting numerical derivative is plotted. The γ value corresponding to the point where (dμ/dγ)=1 will be known as γcrit and a vertical line denotes this value on the plot. The point on the Pareto front to the right of this vertical line is SD.

The piecewise function fit to the Pareto front and the corresponding R2 values are shown below. Note that while the coefficient on the quadratic term may appear to be small, it actually considerably improves the quality of the fit considering the large values of γ being fit 
fP(γ)={1.732(107)γ0.1939+.1939,γ63,472,2711.708(108)γ2+3.859γ+3.865(108),γ>63,472,271
(11)
 
R2={0.99950.9899
(12)

By visual inspection, we noted the CDFs almost always overlap only once with each other and thus we hypothesized the sufficient condition can be established. To rigorously test this hypothesis, a stochastic dominance test, developed by researchers in mathematical ecology, was applied to the set of designs in the last generation of the optimization [43,44]. Starting from the point on the Pareto front corresponding to minimum μ and minimum γ, the test moved up the Pareto front comparing the CDFs design, xi, with that of xi1 for i=2,3m. Stochastic dominance is determined numerically by a bootstrap algorithm with 1000 samples, and the result is a p-value corresponding to the likelihood that xi does not dominate xi1. The results are shown in Fig. 5 with the Pareto front and p-values plotted on separate axes. Note that the p-values remain essentially at zero until γcrit, where designs start having higher p-values corresponding to a higher likelihood that stochastic dominance does not occur. We do acknowledge that the results are not particularly strong, which we suspect is due to both the weakness of stochastic dominance in our particular test case and the relatively poor resolution of the computationally expensive bootstrap algorithm. Nevertheless, a clear trend is visible that stochastic dominance breaks down after γcrit.

The CDF curves of the designs on the Pareto front are also calculated and plotted to visually demonstrate the stochastic dominance characteristics of the curve. In Fig. 6(a), designs are chosen from the stochastically dominated portion of the curve shown on the right-hand side of the vertical line. By visual inspection, one can see that their CDFs are nearly identical for the bottom decile, but above that, the designs further up the Pareto front clearly dominate those below. In contrast, Fig. 6(b) shows the SD designs where stochastic dominance cannot be established.

Efficiency in Optimization Convergence.

In cases where establishing a Pareto front is desirable, we propose optimizing with respect to μ and γ, instead of μ and σ as is traditionally done. We find, in our experiments, that optimizing with respect to γ is more efficient than optimizing with respect to σ. We do not provide a rigorous theoretical explanation for this improved efficiency, but we are not alone in observing it. Researchers have applied the Gini Index as a sparsity measure used in the objective of a signal reconstruction problem and found it to be more efficient than traditional sparsity measures such as the l1 and lp norms [21,25]. However, they have also not been able to determine a theoretical explanation for the improved performance. Gini is also commonly used as an objective in quantitative finance for portfolio optimization [20,24,45].

We performed optimizations with two formulations—one with the traditional μ and σ objectives, and the other with μ and γ, but with all else equal. Fifteen pairs of such optimizations were performed for 100 generations. This results in two sets of designs across n generations with m designs per generation Psn and Pgn corresponding to the optimization performed with standard deviation and Gini, respectively. For all designs in Pgn, we also find the σ associated with said design. We use the σ values associated with Pgn to obtain a fair comparison of the corresponding Pareto fronts 
[μsn,σsn]=f(Psn)
(13a)
 
[μgn,σgn]=f(Pgn)
(13b)
To evaluate the performance of the optimization, we use the S-Indicator determined through the hypervolume method, which is a standard procedure for evaluating the performance of a multi-objective optimization [46,47]. In a two-dimensional space, this is the fraction of the total area in a defined criterion space that dominates the current Pareto front. This criterion space is defined by a reference point, R, and the Nadir point, N. The Nadir point contains the worst possible outcome across all generations for both optimizations as each of its vector elements. We define the reference point as the vector of best possible quantities for each objective; this is also known as the Utopia point 
R=[max(μsn,μgn),min(σsn,σgn)]
(14a)
 
N=[min(μsn,μgn),max(σsn,σgn)]
(14b)

The size of the dominating criterion space is then determined by 106 uniform Monte Carlo samples per generation. A lower S-Indicator value corresponds to a smaller dominating criterion space, and thus, a better solution. The results from all 15 pairs of optimization runs, along with their aggregated mean, are shown in Fig. 7. We clearly see that by the standard of the S-Indicator, the optimization performed with Gini outperforms that performed with σ. After just five generations, the S-Indicator value for the Gini run is already better than the best result obtained with the σ run across all 100 generations. Figure 8 shows the evolving Pareto fronts across the optimization generations, where the faster convergence capability of the Gini objective is shown.

The improved performance in convergence of the Gini Index and its variants over variance remains unexplained both in this paper and other researchers who have reported its superior performance [21,25]. Previous work does provide a general direction to investigate. In a discussion of superiority of Gini's mean difference as a measure of variability for non-normal distributions, Yitzhaki mentions it is sensitive to whether marginal distributions are exchangeable and that this property may influence the results when comparing two marginal distributions [48]. The paper, however, does not provide more detailed results. We suspect this may be a very difficult mathematical analysis problem and would like to pose it as an open problem to the community.

A Single Objective for Expected Value and Uncertainty.

Following from Proposition 2, we obtain the following corollary, the proof of which is provided in the Appendix.

Corollary 1a. Given the conditions assumed in Proposition 2, defineϒ=μγandfϒ(x)=μxγx. We have thatargmaxxP*fϒ(x)=x*.

Corollary 1b. x*is not stochastically dominated by any design inP*.

Recall from Proposition 2a that in addition xP* s.t. μx<μx* we have x1,2x*. Since simply maximizing the value of ϒ attains x*, this allows for a single objective optimization that takes into account both expected value and variability. This optimization produces a result, which is not stochastically dominated and itself stochastically dominates a portion of the Pareto front—in our case, it dominates a very significant portion of the Pareto front. In contrast, the weighted sums method commonly used in practice does not take into account any information with regards to stochastic dominance and, thus, can lead the designer to stochastically dominated design. Thus, we define the following single objective optimization statement and perform tests with the MDO problem: 
minϒ(x,u)g1:μi1μi<0fortheithiteration
(15)
Optimizations were performed with MATLAB's interior-point algorithm starting from 100 randomly generated points between the upper and lower bounds specified in the Appendix. On average, only 893 function evaluations were needed to converge to an optimality tolerance of 106. Similarly, optimizations were performed with the following weighted sums formulation with w1=[0.1:0.2:0.9] for comparison 
minf=w1μ+(w11)γ
(16)
The results in Fig. 9 show that the ϒ optimization leads to results close to the SD designs and above γcrit with few exceptions, which are likely due to incomplete convergence. The weighted sums approach, however, often produces results that are stochastically dominated.

Recall that the result proven assumes the Pareto front takes on a shape such that (d2μ/dγ2)<0, making it not applicable to all problems in its current form. Thus, it may often still be of interest to perform the multi-objective optimization to construct the Pareto front. It may also be of interest to perform a multi-objective optimization if the designer wishes to explore the designs that neither dominate nor are dominated by x*.

Conclusion

From the results presented in this paper, one can see that stochastic dominance, which has not been the subject of much research in the engineering design community, should be considered in performing optimizations under uncertainty. The mean–variance approach to optimization under uncertainty does not provide a way to establish stochastic dominance, potentially leading to inferior design decisions being made. We propose a variant of the Gini Index which we call γ, as an alternative optimization metric to the variance that allows for stochastic dominance to be determined when the CDFs of the designs do not intersect more than once. Future research should also investigate how stochastic dominance can be established when CDFs do intersect more than once. We also present a method to reduce the Pareto efficient set to a subset that is not stochastically dominated under the assumption that the Pareto frontier can be approximated with a convex function. Further research should expand the reduction technique to greater generality. Finally, we demonstrate a single optimization objective that accounts for both expected outcome and variability that produces a design that will not be stochastically dominated. There exists an extensive literature on stochastic dominance-based decision analysis methods that may aid in developing future optimization methods for engineering design problems. The minimal assumptions with regard to the utility functions of the outcomes needed to arrive at rigorous conclusions makes stochastic dominance-based methods a promising avenue of research for improved techniques for engineering optimization and design.

Acknowledgment

The opinions, findings, and conclusions stated herein are those of the authors and do not necessarily reflect those of the sponsor.

Funding Data

  • National Science Foundation (Grant No. CMMI-1300921).

Nomenclature

     
  • D =

    market demand in number of transponders

  •  
  • fP =

    function fit to the Pareto front

  •  
  • FxA(q) =

    cumulative distribution function corresponding to design xA evaluated at q

  •  
  • P(x) =

    probability of quantity x

  •  
  • P* =

    mean Gini Pareto efficient set

  •  
  • Psn =

    designs produced with n generations of optimization with respect to standard deviation

  •  
  • Pgn =

    designs produced with n generations of optimization with respect to Gini

  •  
  • SD =

    stochastic dominance efficient set

  •  
  • Δt =

    simulation time step

  •  
  • ϵ =

    stochastic parameter

  •  
  • μm =

    mean annual demand growth

  •  
  • μgn =

    mean NPP values of designs in Pgn

  •  
  • μsn =

    mean NPP values of design in Psn

  •  
  • σm =

    market volatility

  •  
  • σsn =

    = standard deviation of NPP of designs in σsn

  •  
  • σgn =

    standard deviation of NPP of designs in σgn

  •  
  • 1,2 =

    first and second stochastically dominates

Appendix

Proof of Proposition 2

Consider any two designs xAϵP* and xBϵP*. By the definition of a Pareto efficient set, the following must be true: 
((μxA>μxB)(γxA>γxB))((μxA<μxB)(γxA<γxB))
Now consider Γ2 again. By some simple algebraic manipulation, we arrive at 
μxAμxBγxAγxB1

That is, Γ2 condition for xA1,2xB is satisfied if the inequality is true. Let P* be some Pareto efficient set that can be approximated with the function μ=fP(γ) and (d2μ/dγ2)<0. Consider some hypothetical design at the point along this function such that (dμ/dγ)=1 let us refer to this point as x*. Note for all designs xP* such that μx<μx*, we also have that γx>γx* by Pareto efficiency. Thus, we have strictly μxγx<μx*γx*. Thus both the Γ1 and Γ2 conditions for x*1,2x are met. Now consider all designs yP* such that μy>μx*. Note for y1,2x*, the Γ1 condition holds. However, at the point along the Pareto front where y lies, (dμ/dγ)<1 because (d2μ/dγ2)<0. Thus, (μyμx*/γyγx*)<1 and μyγy<μx*γx*, thus the Γ2 condition does not hold. Thus, it is not possible for y1,2x*. In fact, because the Γ conditions are necessary for all distributions, this result applies even to CDFs that intersect more than once. Thus, we denote the set SD={yP*:μy>μx*} as the stochastic dominance consistent set as no two designs within this set strictly dominate each other making all of them valid alternatives in the absence of additional information. Proof of Corollary 1. Assume all conditions established in Proposition 2. Consider the point x*. We will show x* corresponding to the maximum value of μγ for all designs in P*. From Proposition 2, we have yP* s.t. μy>μx*, μyμγ<μx*γx*. Now consider xP* s.t. μx<μx*. By the definition of the Pareto efficient set, we have γx>γx*. We also have (dμ/dγ)>1 for the portion of the Pareto front where x lies. Thus, (μx*μx/γx*γx)>1. Thus, μx*γx*>μxγx. Thus, we have shown μγ is maximized at x* because {xP*:μx<μx*}{yP*:μy>μx*}=P*. Define ϒ=μγ and fϒ(x) returns the ϒ value for the design x. Since ϒ is maximized at x*, argmaxxP*fϒ(x)=x*.

We will now show no designs within P* may stochastically dominate x*. Consider the designs xP* s.t. μx<μx*. The Γ1 necessary conditions for x1,2x* are not met. Thus, x* is not stochastically dominated by x. Now consider yP* s.t. μy>μx*. We have from Proposition 2 that μyγy<μx*γx*; thus, the Γ2 necessary conditions for y1,2x* are not met. Thus, x* is not stochastically dominated by y. Because x* is not stochastically dominated by neither x nor y and {xP*:μx<μx*}{yP*:μy>μx*}=P*, we conclude x* is not stochastically dominated by any designs in P*.

References

References
1.
Yao
,
W.
, Chen, X., Luo, W., van Tooren, M., and Guo, J.,
2011
, “
Review of Uncertainty-Based Multidisciplinary Design Optimization Methods for Aerospace Vehicles
,”
Prog. Aerosp. Sci.
,
47
(
6
), pp.
450
479
.
2.
Schwabe
,
O.
,
Shehab
,
E.
, and
Erkoyuncu
,
J.
,
2015
, “
Uncertainty Quantification Metrics for Whole Product Life Cycle Cost Estimates in Aerospace Innovation
,”
Prog. Aerosp. Sci.
,
77
, pp.
1
24
.
3.
Taguchi
,
G.
,
1986
,
Introduction to Quality Engineering: Designing Quality Into Products and Processes
, The Asian Productivity Organization, Tokyo, Japan.
4.
Park
,
G.-J.
, Lee, T.-H., Lee, K. H., and Hwang, K.-H.,
2006
, “
Robust Design: An Overview
,”
AIAA J.
,
44
(
1
), pp.
181
191
.
5.
Park
,
Y. H.
,
2003
, “
Hybrid Analysis Method for Reliability-Based Design Optimization
,”
ASME J. Mech. Des.
,
125
(
2
), pp.
221
232
.
6.
Zang
,
T. A.
, Hemsch, M. J., Hilburger, M. W., Kenny, S. P., Luckring, J. M., Maghami, P., Padula, S. L., and Stroud, W. J.,
2002
, “
Needs and Opportunities for Uncertainty-Based Multidisciplinary Design Methods for Aerospace Vehicles
,” NASA Langley Research Center, Hampton, VA, Report No.
NASA/TM-2002-211462
. https://ntrs.nasa.gov/search.jsp?R=20020063596
7.
Shapiro
,
A.
,
Dentcheva
,
D.
, and
Ruszczyński
,
A.
,
2009
,
Lectures on Stochastic Programming: Modeling and Theory
,
Society for Industrial and Applied Mathematics
,
Philadelphia, PA
.
8.
Padulo
,
M.
,
Sergio Campobasso
,
M.
, and
Guenov
,
M. D.
,
2011
, “
Novel Uncertainty Propagation Method for Robust Aerodynamic Design
,”
AIAA J.
,
49
(
3
), pp.
530
543
.
9.
Helton
,
J. C.
,
Johnson
,
J. D.
,
Sallaberry
,
C. J.
, and
Storlie
,
C. B.
,
2006
, “
Survey of Sampling-Based Methods for Uncertainty and Sensitivity Analysis
,” Sandia National Laboratories, Albuquerque, NM, Report No. SAND2006-2901.
10.
Brass
,
H.
,
Fischer
,
J. W.
, and
Petras
,
K.
,
1997
, “
The Gaussian Quadrature Method
,”
Abhandlungen Der Braunschweigischen Wissenschaftlichen Gesellschaft
,
47
, pp.
115
150
.
11.
Ghanem
,
R. G.
, and
Spanos
,
P. D.
,
1991
,
Stochastic Finite Elements: A Spectral Approach
,
Dover Publications
,
London
.
12.
Diwekar
,
U.
,
2008
, “
Optimization Under Uncertainty
,”
Introduction to Applied Optimization
, Springer, New York, pp.
1
54
.
13.
Rockafellar
,
R. T.
, and
Uryasev
,
S.
,
2000
, “
Optimization of Conditional Value-at-Risk
,”
J. Risk
,
2
(3), pp.
21
42
.
14.
Dutta
,
S. R. K.
, and
Vidyasagar
,
M.
,
1977
, “
New Algorithms for Constrained Minimax Optimization
,”
Math. Program.
,
13
(
1
), pp.
140
155
.
15.
Ryan
,
K. M.
,
Lewis
,
M. J.
, and
Yu Kenneth
,
H.
,
2015
, “
Comparison of Robust Optimization Methods Applied to Hypersonic Vehicle Design
,”
J. Aircr.
,
52
(
5
), pp.
1510
1523
.
16.
Jondeau
,
E.
, and
Rockinger
,
M.
,
2006
, “
Optimal Portfolio Allocation Under Higher Moments
,”
Eur. Financ. Manage.
,
12
(
1
), pp.
29
55
.
17.
Deb
,
K.
,
2014
, “
Multi-Objective Optimization
,”
Search Methodologies
,
Springer
,
New York
, pp.
403
449
.
18.
Deb
,
K.
,
Karthik
,
S.
, and
Jussi
,
H.
,
2016
, “
Multi-Objective Optimization
,”
Decision Sciences: Theory and Practice
,
CRC Press
, Boca Raton, FL, pp.
145
184
.
19.
Larsen
,
R. J.
, and
Marx
,
M. L.
,
1986
,
An Introduction to Mathematical Statistics and Its Applications
, Vol.
2
,
Prentice Hall
,
Englewood Cliffs, NJ
.
20.
Cheung
,
C. S.
,
Kwan
,
C. C.
, and
Miu
,
P. C. P.
,
2007
, “
Mean-Gini Portfolio Analysis: A Pedagogic Illustration
,”
Spreadsheets Educ. (eJSiE)
,
2
(
2
), p.
3
. http://epublications.bond.edu.au/ejsie/vol2/iss2/3/
21.
Hurley
,
N.
, and
Rickard
,
S.
,
2009
, “
Comparing Measures of Sparsity
,”
IEEE Trans. Inf. Theory
,
55
(
10
), pp.
4723
4741
.
22.
Rickard
,
S.
,
2006
, “
Sparse Sources Are Separated Sources
,”
IEEE
14th European Signal Processing Conference, Florence, Italy, Sept. 4–8, pp. 1–5. http://ieeexplore.ieee.org/document/7071576/
23.
Rickard
,
S.
, and
Maurice
,
F.
,
2004
, “
The Gini Index of Speech
,”
38th Conference on Information Science and Systems (CISS)
, Princeton, NJ, Mar.
24.
Shalit
,
H.
, and
Yitzhaki
,
S.
,
1984
, “
Mean‐Gini, Portfolio Theory, and the Pricing of Risky Assets
,”
J. Finance
,
39
(
5
), pp.
1449
1468
.
25.
Zonoobi
,
D.
,
Kassim
,
A. A.
, and
Venkatesh
,
Y. V.
,
2011
, “
Gini Index as Sparsity Measure for Signal Reconstruction From Compressive Samples
,”
IEEE J. Select. Top. Signal Process.
,
5
(
5
), pp.
927
932
.
26.
Weiner
,
J.
, and
Solbrig
,
O. T.
,
1984
, “
The Meaning and Measurement of Size Hierarchies in Plant Populations
,”
Oecologia
,
61
(
3
), pp.
334
336
.
27.
Levy
,
H.
,
1992
, “
Stochastic Dominance and Expected Utility: Survey and Analysis
,”
Manage. Sci.
,
38
(
4
), pp.
555
593
.
28.
Cook
,
L. W.
,
Jarrett
,
J. P.
, and
Willcox
,
K. E.
,
2017
, “
Horsetail Matching for Optimization Under Probabilistic, Interval and Mixed Uncertainties
,”
AIAA
Paper No. AIAA 2017-0590.
29.
Von Neumann
,
J.
, and
Oskar
,
M.
,
2007
,
Theory of Games and Economic Behavior
,
Princeton University Press
, Princeton, NJ.
30.
Howard
,
R. A.
,
1988
, “
Decision Analysis: Practice and Promise
,”
Manage. Sci.
,
34
(
6
), pp.
679
695
.
31.
Hazelrigg
,
G. A.
,
1998
, “
A Framework for Decision-Based Engineering Design
,”
ASME J. Mech. Des.
,
120
(4), pp.
653
658
.
32.
Gonçalves-Coelho
,
A. M.
, and
Mourao
,
A. J. F.
,
2007
, “
Axiomatic Design as Support for Decision-Making in a Design for Manufacturing Context: A Case Study
,”
Int. J. Prod. Econ.
,
109
(
1
), pp.
81
89
.
33.
Ross
,
A. M.
, Hastings, D. E., Warmkessel, J. M., and Diller, N. P.,
2004
, “
Multi-Attribute Tradespace Exploration as Front End for Effective Space System Design
,”
J. Spacecr. Rockets
,
41
(
1
), pp.
20
28
.
34.
Freeze
,
R. A.
, and
Gorelick
,
S. M.
,
1999
, “
Convergence of Stochastic Optimization and Decision Analysis in the Engineering Design of Aquifer Remediation
,”
Groundwater
,
37
(
6
), pp.
934
954
.
35.
Yitzhaki,
S.
,
1982
, “
Stochastic Dominance, Mean Variance, and Gini's Mean Difference
,”
Am. Econ. Rev.
,
72
(
1
), pp.
178
185
. http://www.jstor.org/stable/1808584
36.
Damgaard, C., and Weiner, J., 2000, “Describing Inequality in Plant Size or Fecundity,”
Ecology
, 81(4), pp. 1139–1142.
37.
Gastwirth
,
J. L.
,
1972
, “
The Estimation of the Lorenz Curve and Gini Index
,”
Rev. Econ. Stat.
,
54
(
3
), pp.
306
316
.
38.
Okunev
,
J.
,
1991
, “
The Generation of Mean Gini Efficient Sets
,”
J. Bus. Finance Account.
,
18
(
2
), pp.
209
218
.
39.
Kannan
,
H.
,
Mesmer
,
B. L.
, and
Bloebaum
,
C. L.
,
2017
, “
Increased System Consistency Through Incorporation of Coupling in Value‐Based Systems Engineering
,”
Syst. Eng.
,
20
(
1
), pp.
21
44
.
40.
Mesmer
,
B.
,
Bloebaum
,
C. L.
, and
Kannan
,
H.
,
May, 2013
, “
Incorporation of Value-Driven Design in Multidisciplinary Design Optimization
,” Tenth World Congress of Structural and Multidisciplinary Optimization (
WCSMO
), Orlando, FL, May 19–24, pp. 1–19. http://www2.mae.ufl.edu/mdo/Abstracts/5519.pdf
41.
Gardiner
,
C. W.
,
1985
,
Stochastic Methods
,
Springer-Verlag
,
Berlin
.
42.
De Weck
,
O. L.
,
Neufville
,
R. D.
, and
Chaize
,
M.
,
2004
, “
Staged Deployment of Communications Satellite Constellations in Low Earth Orbit
,”
J. Aerosp. Comput., Inf., Commun.
,
1
(
3
), pp.
119
136
.
43.
Brathwaite
,
J.
, and
Saleh
,
J. H.
,
2009
, “
Value‐Centric Framework and Pareto Optimality for Design and Acquisition of Communication Satellites
,”
Int. J. Satell. Commun. Networking
,
27
(
6
), pp.
330
348
.
44.
Moberg
,
E. A.
, and
Solow
,
A. R.
,
2015
, “
Stochastic Dominance Test (MATLAB and R Code)
,” Woods Hole Open Access Server, Woods Hole, MA.
45.
Ringuest
,
J. L.
,
Graves
,
S. B.
, and
Case
,
R. H.
,
2004
, “
Mean–Gini Analysis in R&D Portfolio Selection
,”
Eur. J. Oper. Res.
,
154
(
1
), pp.
157
169
.
46.
Cao
,
Y.
,
Smucker
,
B. J.
, and
Robinson
,
T. J.
,
2015
, “
On Using the Hypervolume Indicator to Compare Pareto Fronts: Applications to Multi-Criteria Optimal Experimental Design
,”
J. Stat. Plann. Inference
,
160
, pp.
60
74
.
47.
Zitzler
,
E.
,
Dimo
,
B.
, and
Lothar
,
T.
,
2007
, “
The Hypervolume Indicator Revisited: On the Design of Pareto-Compliant Indicators Via Weighted Integration
,”
Evolutionary Multi-Criterion Optimization
,
Springer, Berlin
.
48.
Yitzhaki
,
S.
,
2003
, “
Gini's Mean Difference: A Superior Measure of Variability for Non-Normal Distributions
,”
Metron
,
61
(
2
), pp.
285
316
.