During the Second World War (1947), an American mathematician George Dantzig developed an optimization technique which was used in dealing with the massive logistical issues caused by large armies having millions of men and machines (Chenneck; 2000). This techniques developed during world war 11 is the origin of optimization process. Immediately after world war 11, when the firs electric computers were developed, optimization process were made more perfect than what it was used to be before. Today various optimization technique are available, often stimulated by fascinating insight from other fields, but for this thesis only statistical experimental design approach will be used. In recent times various optimization methods have been used in concrete mix design to predict and optimize certain desired qualities (compressive strength, flexural strength, slump, etc) without the conventional methods which involve trial and error (Onuamah; 2015, Obam and Osadebe;
2007, Orie and Osadebe; 2009, Ukamaka; 2007). There are various type of optimization processes which include;
37 a) Statistical experimental design process
In the industries were products such as gasoline, food, detergent etc is to be optimized, the statistical experimental design approach is used. In the case of concrete which is the combination of several components, the performance criteria include setting time, temperature, viscosity, mechanical properties such as strength, elastic modulus, creep and shrinkage etc (Simon; 2003). The application involves the use of theory of statistics (ANOVA) and some specified laboratory results from practical experiment to formulate the mathematical model (equation) which will be used later to predict the strength and other parameters with assumed mix ratio (Anukworji et al; 2011).
b) The mixture approach
George Dantzig is the first man to introduce the mixture approach during the Second World War, but Scheffes; (1958) improved the method by introducing the simplex lattice design and later in 1963 introduced the simplex centroid design. Most real-world linear programming problems have more than two variables and thus too large for a graphical solution procedure, so simplex method is used instead to find the optimal solution. The simplex method is actually an algorithm (or et of instruction) which examines corner points in a methodical fashion until the best solution-higher profit or lower cost is found. The use of mixture experiment in the design of concrete mix is relatively a new area in concrete production (Γzlem et al; 2008), various work have been done on the use of factorial and statistical experiment to develop rapid-set high strength cement and medium strength self-compacting concrete (Srinivasan, et al; 2002, Bajorski, et al; 2007, Snobi;2003, Anyaogu and Ezeh;2013, Umeonyiagu and Adinna; 2014). Simplex centroid design by Scheffes have been the most used mathematical method by researchers in the determination of the compressive strength of concrete and other desired parameters (Mbadike and Osadebe; 2013, Eze and Ibearugbulem;
2009, Anya and Osadebe; 2015, Gamil and Bakar; 2016).
c) Mathematical independent variable
The mathematical independent variable approach is also known as factorial design method.
In this type of design approach, if there exist π components materials (where π is the number of component material) the π components of a mixture are reached to π β 1 independent variable using the two components as an independent variables (Simon and others; 1997). In the case of concrete, water/cement ratio is a natural choice of this ratio variable. For the situation with π β 1 independent variable, a 2(πβ1) factorial design forms the backbone of
38
the experiment (Anukworji and others 2011). Further more in mixture approach, empirical models are fit to the data and polynomial model (linear or quadratic) are used.
d) Regression method
For a given mixture a set of parameters π1, π2, π3, β¦ β¦ β¦ , ππ known as predictors can be used to predict the probable value of a dependent variable π with a particular degree of certainty (Mandenball; 2003). Osadebe in 2003 assumed that the response function πΉ(π) is continuous and differentiable with respect to its predictors ππ. The two researchers presumed that so long as the values of the predictors are known, the corresponding value of the dependent variable can be predicted with some degree of certainty (compressive strength, cost etc). In this design approach few points of observation will be used to formulate a model.
Once the model have been formulated and validated it can be used to predict the future values of independent variables. Regression method have been used extensively in mix design for concrete production (Okere et al; 2013, Okere; 2006, Chijioke et al; 2015, Onwuka et al;
2013, Egbe and Orie; 2016).
e) Neural network approach
The use of computers in recent times has been very useful in the scientific world, especially in the area of accuracy and precision. The use of computer in the implementation of different complex statistical method cannot be over emphasis. With the increasing accuracy and precision of analytical measuring method, it become clear that all effects that are of interest cannot be described by simple uni-variant and even not by the linear multi-variant correlation precise, a set of methods that have recently found very intensive use among engineers are the artificial Neural Networks (Zupan; 1994). These methods have been used in the development of simulator and intelligent system to predict the compressive strength and the workability of high performance concrete. In this type of method, the problem to be solved is first identified;
this will determine the type of network topology to be selected (Vijay and Yogesh; 2013, AcuΓ±a-Pinaud et al; 2017, Vahid and Mohammad 2013, Rasa et al; 2009, Alilou; 2009, Teshnehlab and Alilou;2008). The neural network is defined by its topology, leaning paradigm and learning topology, then effort is made to identify the types of input data whether itβs is all binary (0/1), bipolar (-1/+1) or the data contains real-value inputs. These types of data might disqualify some of the network architecture which use certain function in their learning algorithm, and finally the number of input and output units and the hidden nodes that gives the best performance is determined. The network has to map the features of the inputs and produce the desired output and also solve the problem of classification
39
assuming the study is to classify the mixture proportioning of high performance concrete that can give the best strength based on various factors (Struchenkov; 1999, Struchenkov; 2009, Adulhaq; 2015). Neural network has is used in the prediction of compressive strength of concrete and other desired parameter in concrete (Bilgehan and Turgut; 2010, Kewalramani and Gupta; 2006, Kisi; 2005).
f) Genetic algorithm method
Genetic algorithm methods are family of computational modes inspired by evolution. These algorithms encode a potential solution to a specific problem on simple chromosome-like data structure and apply recombination operators to these structures so as to preserve critical information. Genetic algorithms are often viewed as function optimizers, although the range of problems to which genetic algorithms have been applied is quite broad (Whitley; 2012).
These methods was developed by John Holland in the 1970s with the aim of i) Understanding the adaptive processes of natural systems
ii) Designing artificial system software that retain the robustness of natural systems
ii) Providing efficient, effective techniques for optimization and machine learning application.
Genetic algorithm therefore identifies the individuals with optimum fitness value and those with lower fitness will naturally get discarded from the population. Ultimately the search procedure finds a set of variable that optimizes the fitness of individual or of the whole population. This method has advantages over traditional non-linear solution techniques that cannot always achieve an optimal solution. The process involved in solving problem using genetic algorithm is achievable through the process of evaluation, the process of selection, the process of cross-over and mutation process. Genetic algorithm has been used in the prediction of various desired parameter like compressive strength, flexural strength etc in concrete productions and concrete structures (Ahsanul et al; 2012, Hasan and Kabir; 2011, Garg; 2003, Hasan; 2012,Hamid-Zadeh et al; 2006). It has also been used in the design of low-cost reinforced frames structures (Camp et al; 2003,Ghodrati et al; 2008, Aggarwal et al;
2015). The use of conventional linear regression cannot give satisfactory solution to predicting some desired parameters in concrete, this is where genetic algorithm method comes in, to takes care of these anomalies (Juncai et al; 2017).
40 2.11.3Statistical method
There are so many statistical methods used in the analysis of concrete parameters and this is called Response Surface Methodology (RSM). Response surface methodology is a collection of mathematical and statistical methods used to develop, improve or optimize products to achieve the desired parameter needed (Simon et al; 1999). The RSM is used effective when it comes to products in which every component has an effect on the product. This objective of this method is to be used for the optimization of one or more response like compressive strength, flexural strength etc (Simon; 2003). Concrete is a mixture of various components like water, cement, fine aggregate, coarse aggregate and sometimes admixtures are added to achieve the desired parameter needed. The various components that made up concrete have a direct influence on it. There are three types of procedures used in RSM method they are;
experimental design, modelling and optimization.
2.11.2.1 Design experiments
Design experiments is an important aspect of RSM, and were initially developed to used in model fitting of physical experiments, although it can sometimes be used in numerical experiments. Let us look at q component materials say concrete, here q is the total number of components in the mixture. There two experimental design approach that can be used in this case namely the classic mixture approach, in which the q mixture components are the variables and the mathematically independent variable (MIV) approach, in which q mixture components are transformed into q-1 independent mixture-related variables ( ). The mixture approach the total amount of product is fixed and the settings of each of the q components are proportions. The q-l of the factors can be chosen independently due to the total amount is constrained to sum up to one. For the mathematical independent variable approach the q components of a mixture are reduced to q-1 independent variables by means of the ratio of two components as an independent variable. Relating this to concrete, w/c is a natural choice for this ratio variable. For the situation with q-1 independent variables, a 2q-l factorial design forms the backbone of the experiment. This design consists of several factors (variables) set at two different levels (Simon; 2003).
2.11.1.2 Modelling
Modelling is a theoretic framework that allows us to define a process or relationships existing between various representative elements of a system. Here we are dealing with concrete, the
41
responses and all the components that made up the product.. The scales of modelling are multiple (Torrenti et al; 2010). A model ca be inform of linear, polynomial or quadratic in nature. For polynomial models can be fit to data by the means of variance (ANOVA) and least square methods. Presently there are many statistical software packages that can be used to carry out these entire tasks as describe above. Once a model has been fit, it is important to verify the adequacy of the chosen model quantitatively and graphically (Simon; 2003).
2.11.2.3 Optimization
Concrete mixture optimization involves the adaptation of available resources to meet varying engineering criteria, construction operations, and economic needs. Economic considerations include materials, delivery, placement, and progress time related costs. A lot of responses can be optimized simultaneously any time the appropriate models have been developed. This can be done by the use of mathematical (numerical) or graphical (contour plots) methods. The problem with graphical approach is that it deals with few responses. In numerical optimization the function objective have to be define, that reflects the levels of each response in terms of minimum (zero) to maximum (one) desirability (Simon; 2003)