Experiments with Mixtures

Scheffé, Henry

doi:10.1111/j.2517-6161.1958.tb00299.x

Cited by 723 publications

(410 citation statements)

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“…. , w N } (Scheffe 1958(Scheffe , 1963 and calculate these neighbor weight vectors ). 1.2 Initialize the evolutionary population by generating…”

Section: General Algorithm Descriptionmentioning

confidence: 99%

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MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem

Liu

et al. 2015

Soft Comput

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Related to the safety of public lives and property in the lower area of reservoirs, flood control is a priority for most large reservoirs. Considering both dam safety and downstream flood control, reservoir flood control is a multi-objective problem (MOP). To meet the needs of irriCommunicated by V. Loia. B Fang Liu gation and generating electricity after the flood, the decision maker usually has his/her preferred final scheduling water level. To deal with this kind of MOP with user-preference information, we incorporate user-preference information into the framework of MOEA/D (multi-objective evolutionary algorithm-based decomposition). The widely used preference information is mainly composed of reference points and preference directions. Compared with the Pareto dominancebased multi-objective evolutionary algorithms (MOEAs), MOEA/D can naturally include two kinds of preference information since MOEA/D is directly based on the reference point and the preference direction. The weight vector of a subproblem in MOEA/D is just its preference. Aiming to obtain uniformly distributed solutions on the objective space, one of innovation points in this paper is using modified Tchebycheff decomposition instead of Tchebycheff decomposition as the decomposition method. To focus the search on the interesting regions of decision maker, the other innovation point in this paper is to integrate biased subproblem (weight vector) adjustment into the framework of MOEA/D. The distribution of subproblems (weight vectors) are adjusted periodically so that the subproblems are re-distributed adaptively to search the interesting regions. Some subproblems, which are far away from the preference regions, are deleted. And then some new subproblems, which are expected to search the preference regions, are added into the current evolutionary population. The efficiency and the effectiveness of the proposed algorithm are assessed through multi-objective reservoir flood control problem and two-to ten-objective test problems.Keywords Multi-objective optimization · Evolutionary algorithm · Decomposition · Preference · Biased weight vector adjustment · Reservoir flood control 123 X. Ma et al.

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“…. , w N } (Scheffe 1958(Scheffe , 1963 and calculate these neighbor weight vectors ). 1.2 Initialize the evolutionary population by generating…”

Section: General Algorithm Descriptionmentioning

confidence: 99%

“…9. Firstly, we use simplex lattice design (Scheffe 1958(Scheffe , 1963 to generate uniformly distributed solutions S 1 . Secondly, scale and translate S 1 as S 2 to approach preferred region.…”

Section: Performance Metricmentioning

confidence: 99%

MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem

Liu

et al. 2015

Soft Comput

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“…Therefore, the diversity of PS depends on whether the aggregation coefficient vectors can be distributed uniformly in the whole space. Some literatures [4,22,30,31] show that the aggregation coefficient vectors are usually generated by the simplex method [32] proposed by Scheffe in 1958. However, the vectors number is controlled by the parameter H, which cannot be set at will.…”

Section: Symmetric Latin Hypercube Samplingmentioning

confidence: 99%

Multi-objective artificial bee algorithm based on decomposition by PBI method

Bai

Liu

2016

Appl Intell

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This paper proposes a multi-objective artificial bee colony (MOABC) algorithm based on decomposition by penalty-based boundary intersection (PBI) method. It decomposes a MOP into a number of single-objective problems. The objective of every single-objective problem is based on the distance and angle from the current solution to their own ideal point so as to obtain the good convergence and distribution of the optimal solutions. In this way, the aggregate function is proposed by PBI method. Then the three phases of Artificial Bee Colony (ABC) algorithm are adopted, which are the employed bees sending, the unemployed bees following, and the scout bees converting. Among these phases, the improvement degree of every aggregate function is proposed as the fitness function, which could overcome the two shortcomings in the weighted sum function usually applied in the conventional MOABC. Besides, Boltzmann selection mechanism is used to obtain the probability of unemployed bees following the employed bees so that the selection pressure of unemployed bees in the optimizing process could be adjusted dynamically. The algorithm is validated on CEC2009 problems and the problems with complicated Pareto set shapes in terms of four indicators: IGD, HV, SPR, and EPS. Experimental

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“…3c) [25,83,88,89]. Geometrically, the N mixtures represent N points in a q-component space which degenerates into a (q−1)-dimensional simplex network (Scheffé's network) (Fig.…”

Section: Combination Of Chromatogramsmentioning

confidence: 99%

A New Approach to Plant Diversity Assessment Combining HPLC Data, Simplex Mixture Design and Discriminant Analysis

Semmar

Jay

Farman

et al. 2007

Environ Model Assess

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The quantitative assessment of plant diversity and its monitoring with time represent a key environmental issue for management and conservation of natural resources. Assessment of plant diversity could be based on chemical analyses of secondary metabolites (e.g. flavonoids, terpenoids), because of the substantial quantitative and qualitative between-individual variability in such compounds. At a geographical scale, the plant populations become widely dispersed, and their monitoring from numerous routine individual analyses could become restricting. To overcome such constraint, this study develops a multivariate calibration model giving the relative frequency of a particular taxon from a simple high-performance liquid chromatography (HPLC) analysis of a plant mixture. The model was built from a complete set of mixtures combining different taxons, according to an experimental design (Scheffé's matrix). For each mixture, a reference HPLC pattern was simulated by averaging the individual HPLC profiles of the constitutive taxons. The calibration models, based on Bayesian discriminant analysis (BDA), gave statistical relationships between the contributions of each taxon in mixtures and reference HPLC patterns of these mixtures. Finally, these models were validated on new mixtures by using outside plants. This new biodiversity survey approach is illustrated on four chemical taxons (four chemotypes) of Astragalus caprinus (Fabaceae). The more differentiated the taxon, the better predicted its contributions (in mixtures) were by BDA calibration model. This new approach could be very useful for a global routine survey of plant diversity.

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Experiments with Mixtures

Abstract: SUMMARY A theory is developed for experiments with mixtures of q components whose purpose is the empirical prediction of the response to any mixture of the components, when the response depends only on the proportion of the components and not on the total amount.

Cited by 723 publications

References 2 publications

MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem

MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem

Multi-objective artificial bee algorithm based on decomposition by PBI method

A New Approach to Plant Diversity Assessment Combining HPLC Data, Simplex Mixture Design and Discriminant Analysis

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