Multi-Objective Optimization and Decision Aid for
Spread Footing Design in Uncertain Environment
Nicolas PIEGAY, Denys BREYSSE
University of Bordeaux, I2M, Civil and Environmental Engineering Department GCE, Bat. B18, Avenue des facultés, 33405 Talence, France.
Abstract. This paper aims to present an evaluation, optimization and comparison approach for the design of isolated spread foundations in an uncertain environment. The choice of the optimal solution is made according to the decision-maker's preferences to find the best compromise between various conflicting objectives. Furthermore, this approach takes into account a set of uncertain parameters (soil properties, loading, critical values…) modeled using Monte-Carlo simulations.
Keywords. Foundation design, multi-objective optimization, multi-criteria decision aid, Monte-Carlo simulations
1. Introduction
Foundation design should satisfy a set of safety objectives based on the ultimate limit states (ULS) and the serviceability limit states (SLS). In large industrial plants the number of footings can be very large (several hundred) and the economic and production stakes have to be considered in balance with safety objectives. In this work, risk is viewed as the inability to get performance requirements considering all uncertainties which may be due to the inherent soil variability, to the economic and production data but also to the expression of design requirements.
This paper aims to present a global approach for the evaluation, the multi-objective optimization and the choice of the optimal foundation in a uncertain context. At each step of the design process, and considering risk-aversion behavior, the decision-maker's preferences can be formalized through: (a) a satisfaction level regarding each performance, (b) a robustness degree associated with each objective function, (c) the weighting degree expressed on each design objective.
In a first part we will describe the multi-objective evaluation approach of spread foundations, then an optimization methodology will be presented.
2. Evaluation of Objectives 2.1. Input Data
Input parameters include the "decision parameters" and "environmental parameters" used to describe the design configurations of a foundation.
The "decision parameters" of the foundation consist of "design variables" and "project variables". The "design variables" for the spread footing are depth D, width B, and length L. Assigning values to all "design variables" leads to a particular configuration, called an alternative. The "project variables", like "environmental variables", are set for all the alternatives. They represent the design load applied by the superstructure, the number of work crews, or the number of footings. The boundary between "design variables" and "project variables", defined from specifications, can move according to the ability of the team in charge of geotechnics to influence the project at a higher level (for instance, to review the design parameters of the superstructure).
The "environmental variables" represent the features of the natural environment and the socio-economic context. They are considered as uncertain and modeled through probability distributions. The "socio-economic variables" are associated with the unit price and unit time of spread foundation construction. The "natural © 2015 The authors and IOS Press.
This article is published online with Open Access by IOS Press and distributed under the terms of the Creative Commons Attribution Non-Commercial License.
environment variables" are related to the soil properties and climatic loading.
2.2. Description of Soil Variability
The soil physical and mechanical parameters are subject to uncertainty and are modeled as random variables. The Monte Carlo method is used to analyze the statistical distribution of results by generating pseudo-random values in agreement with the probability distribution of input parameters.
As mentioned by Cherubini (2000), the main statistical parameters characterizing the soil variability are the expected mean values, the standard deviations, the kinds of probability distribution and the cross-correlations between physic-chemical properties of soil. In this paper, the spatial correlation of soil properties is not considered. In order to assign the probability distribution and the coefficient of variation to each soil parameter, we refer to the literature, using values from former foundation reliability studies.
However, as pointed out by Phoon and Kulhawy (1999a, 1999b), such data have to be used carefully because the observed variability through standard deviation is actually a result of the combination of random and epistemic uncertainties. We select lognormal distributions to model the variability of soil properties. This kind of probability distribution is very well suited for strictly positive parameters and commonly used in technical literature (Breysse 2011; Fenton and Griffiths 2003; Popescu et al. 2005). Table 1 shows the mean and the coefficient of variation of some soil properties that are considered in this work.
Table 1. Statistical parameters of "natural environment variables"
Soil properties Mean COV
c' 14 kPa 25%
߮′ 26° 10%
Unit weight 18 kN/m3 5%
Young's modulus 40 MPa 30%
Poisson's ratio 0.4 7%
Wind load Mean COV
Vertical (downward)
12 kN 15%
Horizontal 16 kN 15%
Various values of correlation between effective cohesion (c') and friction angle (ᇱ) can be considered, depending on the soil. If this correlation is generally assumed to be negative, its appropriate value is still a matter of discussion, as well because it is uncertain as its effects on reliability are not well known. A negative cross-correlation ( = −0.61 [Cherubini 2000]) between c' and ᇱ of the soil will be considered in a first stage.
Wind load is also viewed as uncertain (Table 1) and represented by a lognormal distribution with a coefficient of variation of 15% (Orr and Breysse 2008).
2.3. The Design Model
The parameters identified in the previous sections are the input data of a design model describing the system requisites and how it works. This model is based on appropriate construction rules and regulations. The outputs of the model are called "performance indicator" (PI) and they represent the set of values of the design objectives. PI and objectives are equivalent in this paper but an objective may be comprised of a set of performance indicators in other design problems. They will contribute to the evaluation of the alternatives. PI are defined in order to quantify the reliability of the foundation, the execution cost and the construction duration. To describe the failure modes of the footings, the model is analytical and in agreement with the recommendations of Eurocode 7 (CEN 2005) regarding ULS and SLS. Safety objectives are evaluated through a set of four performance indicators (FSBC, FSSR, R, s) respectively corresponding to three ULSs (bearing failure, sliding failure, excessive load eccentricity) and one SLS (excessive settlement). Strictly excessive load eccentricity is not a ULS but it can cause a ULS of bearing failure or possibly toppling if the foundation is on a very strong ground.
Construction cost (CC) and construction duration (CD) are based on the five work tasks identified by Wang and Kulhawy (2008) and related to excavation, formwork, reinforcement, concrete, and compacted backfill. The quantities of these work tasks can be expressed as functions of the design variables B, L and D. Costs and
productivity can vary depending on the location, the worker qualification, and the socio-economic context. Thus, the unit prices and unit times are uncertain parameters, that are modeled using triangular distributions (+/- 10% around the mean).
Finally, for a given footing configuration (i.e. a set of input data), six "performance indicators" are calculated, namely FSBC, R, FSSR, s, CC, and CD. The four first PI correspond to safety requirements. Each PI is represented by a statistical distribution, which results from the stochastic nature of the input variables ("environmental variables" precisely).
2.4. Desirability Function
For each PI, a desirability function is introduced. The benefits of such a function is twofold: (a) it improves the concept of "acceptable threshold" in order to reflect its non-deterministic nature due to the vagueness of the decision-maker's preferences; (b) it makes possible more or less risk-aversion to be taken into account with each PI.
Desirability values range from 0 to 1 and reflect the satisfaction degree of the decision-maker regarding the values of PI. A value of 0 means a full non-compliance of the PI while 1 expresses a complete satisfaction. When the measurement units of the various PI and their range of values are very different, the use of desirability functions standardizes the overall performances on a dimensionless single scale of desirability.
The one-sided desirability functions of Derringer (Derringer and Suich 1980) drawn in Figure 1 are used. Monotonically increasing or decreasing desirability functions correspond to
an PI that must be maximized or minimized, respectively. For each value Yi,j taken by a PI i during the jth Monte-Carlo simulation, the associated desirability di,j is calculated. For an increasing function, Ymin is the minimum acceptable value of Yi,j and Ymax is the minimum value above which di,j reaches 1 (and conversely for a decreasing function). The parameter r adjusts the curvature according to the decision-maker's preferences. For the same value of Yi,j, the larger the r value, the quicker di,j tends to 0, thus the decision-maker is more risk-averse. For r=1, the user is risk-neutral.
Figure 1. Increasing one-sided desirability function
In practice, a series of interviews have been planned with experts involved in the industrial project in order to identify their preferences regarding the various objectives and the desirability functions have been shaped accordingly. Table 2 describes their preferences regarding the desirability function for the six PI in the case study detailed here.
The desirability function being known for each performance indicator, the next step consists in computing the cumulative distribution functions of desirability ( , ).
Table 2. Parameters of the desirability functions of each "performance indicator"
PI Units r One-sided function Ymin Ymax
FS_BC - 1 increasing 1.1 1.5 R - 1 decreasing 0.2 0.33 FS_SR - 1 increasing 1.1 1.5 s mm 1 decreasing 1 10 CC k€ 1 decreasing 0 600 CD days 1 decreasing 0 84
3. Optimal Solution
Each alternative can be evaluated, in a probabilistic way, regarding the set of PI accounting for the desirability level reached on each performance value. This section explains how two alternatives can be compared and how an optimization process has been developed in order to identify a multi-objective optimal solution.
3.1. Comparison of the Statistical Distributions In order to decide between the potential alternatives, we have to compare their statistical desirability distributions with regard to each PI. Many probabilistic descriptions allow the comparison of these distributions by means of robustness measures (Beyer and Sendhoff 2007). In this work, the robust design objective functions are based on the weighted combination of desirability mean (i.e. performance measure) and variance (i.e. dispersion measure). Each objective function fi of the alternative Xn can be written as below:
X = , − . , (1) where the coefficient is used to derive the representative value from mean and variance (Apley et al. 2006). = 0 corresponds to a risk neutral variant (where uncertainty is not accounted for) while increasing values of are related to a higher risk-aversion. In this paper, we assume = 1.
It is thus possible to identify the alternative corresponding to the best compromise between the performance and the dispersion. It must be noted that two parameters are used to account for risk aversion: r coefficient (§2.4.) assigns the possible range of desirability values while acts on the weight to be given to dispersion of desirability values.
3.2. Multi Objective Optimization
Multi-objective optimization implies simultaneous optimization of several conflicting objectives. In our case, it can be defined as below:
= ଵ, , … (2) ! "1 , > 0 #$ ∈ % (3) where = (B, L, D) is the vector of "design variables" which have to satisfy the constraints 0.3 m ≤ L ≤ 3 m, 0.4 m ≤ D ≤ 2 m and 1 ≤ L/B ≤ 5 which define the search space S; fi are the six objective functions in terms of desirability; "1 , is the first quartile of the cumulative distribution function of desirability values for PI i. The choice of the first quartile is assumed in this work but can be discussed. The lower the percentile, the larger the number of design alternatives, but the lower the robustness of the potential optimal solution.
All "optimal compromises" are located on a Pareto front. They belong to the non-dominated set of solutions, which means that an alternative is an "optimal compromise" if it is not dominated by any feasible solution. A decision vector ଵ is said to Pareto-dominate the decision vector ଶ, in a maximization context, if and only if:
∀ ∈ &1, . . ,6', ଵ ≥ ଶ
#$ ∃ ∈ &1, . . ,6', ௨ଵ > ௨ଶ (4) To simulate the Pareto front, the multiple objective particle swarm optimization (MOPSO) algorithm is computed (Alvarez-Benitez et al. 2005).
Figure 2 illustrates the 6-dimension Pareto front by considering, as an example, four objective functions. Each point represents a particle defined by a set of design variables. The curve corresponds to the Pareto front without discretization of design variables and the crosses represent potential solutions with a discretization step for all dimensions of 0.1 m. The Pareto front is the limit beyond which no feasible solution exists. For 30 iterations and 50 particles in the swarm, we obtained 21 alternatives (crosses) among which the optimal solution can be chosen. All alternatives are maximized (desirability equal to 1) for the objective function against sliding failure. Values taken by the other objective functions can be more scattered, as it can be seen for bearing capacity.
Figure 2. Distribution of particles on the Pareto front
3.3. Multi-Objective Decision Aid
In order to select an alternative from the "optimal compromises" on the Pareto front, a multi-objective decision aid method is needed. The method adopted in this study is based on PROMETHEE II (Preference Ranking Organization Method for Enrichment Evaluation using complete ranking) developed by Brans (Brans and Vincke 1985). The level of complexity of this method is low, thus it is particularly clear and understandable by the decision-maker. This method uses the outranking methodology to rank a finite number of alternatives considering several weighted objectives. The weighting has been determined in accordance with the decision-maker's opinions as shown in Table 3, collected through interviews.
PROMETHEE II provides clear information about the preferences between the alternatives and enables the optimal solution to be selected: [B = L = 0.9 m; D = 0.4 m]. Table 4 gives: (a) the mean and the standard deviation of each PI,
(b) the mean and the standard deviation of each desirability function, (c) the values of objective functions.
Table 3. Weighting of "performance indicators"
PI Weights wi FS_BC 0.15 R 0.15 FS_SR 0.15 s 0.15 CC 0.35 CD 0.05
It can be seen that the desirability mean of the settlement is relatively small (0.406), due to its low sensitivity to the variability of the compressed area of the footing. Thus, to improve the settlement desirability, the width and the length of the footing must be much larger, which would result in an excessive increasing of cost and duration and thus would deteriorate the quality of multi-objective consensus. Conversely, desirability means of the first three PI, which are the ULS objectives, are very close to the value of 1, which corresponds to perfect satisfaction.
Table 4. Description of the optimal solution regarding each “performance indicator”
PI PI mean PI std Desirability mean Desirability std Objective functions
FS_BC 2.25 0.483 0.995 0.044 0.951 R 0.03 0.005 1 0 1.000 FS_SR 4.39 0.811 1 0 1.000 S (mm) 6.42 1.944 0.406 0.195 0.211 CC (k€) 304.73 6.705 0.492 0.011 0.481 CD (days) 39.06 0.413 0.535 0.005 0.530
3.4. Effect of Model Uncertainties
The sensitivity of the optimal solution against three model uncertainties (wi, , r) was analyzed in several simulations. For some preference options and risk-aversion degrees, the change in the optimal solution (defined by a set of design variables) is analyzed.
It can be shown that the values for the model parameters must be carefully selected. Extreme values of r, large values of >0, or a inappropriate weighting can excessively penalize some objectives provided that the constraints are satisfied. The parameter r affects the objective functions that are not very sensitive to changes in design variables, and the parameter affects the objective functions that are very dispersed.
Moreover, it appears that the variability of the optimal design is more sensitive to model uncertainties than to uncertainty on soil variability. However, the optimal solution is certainly very sensitive to uncertainties about the mean values of soil properties.
4. Conclusion and Perspectives
This paper presents a global and innovative approach for the evaluation and the optimization of the optimal design of a spread footing.
Uncertainties are introduced in the input data considering the inherent variability of the soil properties, the uncertain wind load, and the fuzzy economic and productivity data. The limit states, the cost and the construction duration were estimated with Monte-Carlo simulations and they were evaluated regarding the decision-maker's preferences.
In order to optimize the foundation design, a multi-objective particle swarm algorithm is used considering six robust design objective functions which combine a performance measure and a dispersion measure. Finally, considering the weighting of each PI with respect to the decision-maker's preferences, the PROMETHEE II method enabled selection of the optimal solution between the alternatives located on the Pareto front. The model parameters must be carefully selected to obtain a solution respecting the multi-objective problem.
The design process presented in this paper can be extended to other foundation systems and developed for a global optimization foundation/superstructure considering the soil-structure interaction. Then, the optimal solution obtained for each foundation system could be compared.
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