The Computer Search for the Optimal Settings of a Multi-Factorial Experiment Using Response Surfaces D-Optimality Design Criterion
OBJECTIVES OF THE STUDY
The main aim and objectives of this dissertation are
(i) to explore and exploit MATLAB 5.3 to derive D-optimal designs for a multi-factorial experiment under each of three alternative second order response surface models,
(ii) to establish D-effeciency and rotatability-effeciency of the optimal designs under each of the three response surface models, and
(iii) to determine the optimum Nitrogen, Phosphorus, Potassium and Sulphur (NPKS) fertilizer applications for grain production of maize, sorghum and millet in Savannah zone of Nigeria.
This chapter considers the history and developments in RSM, where several literatures in the area of RSM were reviewed.
THE EMERGENCE AND DEVELOPMENT OF RESPONSE SURFACE METHODS
The genesis of response surface methodology (RSM) can be traced back to the works of J. Wishart, C.P. Winsor, E.A. Mitscherlich, F. Yates, and others in the early 1930s or even earlier. However it was not until 1951 that RSM was formally developed by G.E.P. Box and K.B. Wilson and other colleagues at Imperial Chemical Industries in England, (Box and Wilson, 1951). Their objective was to explore relationships such as those between the yield of a chemical process and a set of input variables presumed to influence the yield. Since the pioneering work of Box and his co-workers, RSM has been successfully used and applied in many diverse fields such as chemical engineering, industrial development and process improvement, agricultural and biological research, even computer simulation, to mention just a few.
The applications of RSM can be found in Edmondson (1991), Smith et al. (1997), Mountzouris et al. (1999a, 1999b, 2001), Regalado et al. (1994), Rosenthal et al. (2001), Kikafunda et al. 1998, Jauregi et al. (1997), Trinca and Gilmour (1998, 1999, 2000a, 2000b, 2001), Gilmour and Ringrose (1999), Gilmour and Trinca (2003), Gilmour and Mead (2003).
However, there are several procedures that are used in the realization of RSM objectives, especially in the area of criteria for choosing a design. These ranges from the least squares based procedures, the integrated mean square error (IMSE) criterion, the theory of design optimality and the design robustness. In our work we focussed mainly on the theory of design optimality and in particular the D-optimality criterion.
Optimal design theory was developed mainly after World War II. Kiefer (1958, 1959, 1960, 1961, 1962a, 1962b) is attributed to having provided the basic mathematical groundwork for optimal design theory. Presently there are two schools of thought regarding the application of the principles of optimal design theory to the derivation of response surface designs: the “Kiefer school” and the “Box school.” In the latter school, bias suspected of being present in the fitted model plays a significant role. In the Kiefer school, however, bias is regarded as insignificant or it does not exist. The main preoccupation in this school is, therefore, with designs that minimize the variance associated with the fitted model, ŷ(x).
The aim of the optimality theory is selecting an optimum experimental design which, in most cases, needs to be multifaceted. Therefore the problem of selecting a suitable design is thus a formidable one (Box and Draper, 1987). The optimum designs are usually constructed using computer algorithm and are therefore referred to as computer-aided designs. One form of the computer-aided designs is the D-optimal designs. It possesses the most important design criterion in applications (Atkinson and Donev, 1992). These types of computer-aided designs are particularly useful when classical designs do not apply. Unlike standard classical designs such as factorials and fractional factorials, D-optimal design matrices are usually not orthogonal and effect estimates are correlated. The D-optimal designs are always an option regardless of the type of model the experimenter wishes to fit (for example, first order, first order plus some interactions, full quadratic, cubic, etc.) or the objective specified for the experiment (for example, screening designs, response surfaces, etc.). The optimality criterion used in generating D-optimal designs is one of maximizing |X’X|, the determinant of the information matrix X’X.
Several researches related to D-optimum designs have been undertaken. Mitchell (1974a, 1974b) in his two papers gave algorithms for the construction of ‘D-optimal’ experimental designs, Mitchell and Bayne (1978) discussed the D-optimal fractions of three-level factorial designs, Galil and Kiefer (1980) in their paper considered time-and space-saving computer methods, related to Mitchell’s DETMAX, for finding D-optimal designs. Cook and Nachtsheim (1980) compared algorithms for constructing exact D- optimal designs. A recent work by DuMouchel and Jones (1994) gave a simple Bayesian modification of D-optimal to reduce dependence on an assumed model. Johnson and Nachtsheim (1983) gave some guidelines for constructing Exact D-optimal designs on convex design spaces. Some of the other works on D-optimal designs include Cook and Nachtsheim (1989), Dykstra (1971), Harville (1974), Johnson et al. (1990), Mitchell and Miller (1970), Snee (1985), Ogolime and Bamiduro (1998), Chigbu (1998), Vance (1986) and Pronzato (2002) who showed in his paper how it is possible, during the search for an optimum design, to remove from X some design points that cannot be support points of the optimum design in order to accelerate the D-optimum design algorithms. Some of the studies on D-optimum designs are the comparison of Lucas (1976) and of Donev and Atkinson (1988) they showed that central composite designs behave well according to the criteria of D- and G-optimality and also for the average variance criterion, V-optimality. However, the designs are restricted to a few values of N. if a design of a rather different size is required; it will either have to be generated by computer search or looked up in a catalogue. However, for the background theories of optimal experiments see Federov (1972).
The nature of our research problem also calls for generation by a computer search a D-optimal design.
In the next section the various standard experimental designs used in optimality theory are presented.
RSM OPTIMIZATION PROCEDURE (PATH OF STEEPEST ASCENT)
In this chapter we considered one of the RSM optimizations procedures i.e. the Path of Steepest Ascent (PSA). The determination of optimal settings of the experimental factors that produce the maximum (or minimum) value of the response in RSM is achieved using the path of steepest ascent as follows.
COMPUTER-AIDED DESIGNS METHODOLOGY
In this chapter we present the concept of design regions and the conditions necessitating the use of computer-aided designs. The details concerning D-optimal designs, starting from its least squares, properties and sequential algorithms were all discussed. The models to be employed and the measure of rotatability were stated.
In this chapter we considered the MATLAB software which was used in implementing our algorithm for the determination of D-optimal designs.
MATLAB is an acronym for Matrix Laboratory; a powerful fourth generation programming language. It is a high performance language for technical computing it integrates computation, visualization, and programming in an easy-to-use environment where problems and solutions are expressed in familiar mathematical notation. Typical uses include:
CONSTRUCTION OF D-OPTIMUM DESIGNS
In this chapter we implemented the programs given in chapter five with the aim of generating the D-optimum designs.
Observe that our experiment has 5 levels of nitrogen (N), 5 levels of phosphorus (P), 3 levels of sulphur (S), and 2 levels of potassium (K) giving 5×5×3×2 = 150 treatment combinations. A maximum of the best 21 treatment combinations is required.
The first levels of all the factors are zero levels which are regarded as controls. Without the controls the treatment combinations would be 4×4×2×1= 32. We shall be generating our designs based on with and without controls.
The construction of a D-optimum design for the experiment would be for runs or support points or treatment combinations based on property (6) of the D-optimal designs, which states that, n the number of support point should be p ≤ n ≤ p(p+1)/2, where p is the number of parameter estimates. The procedure for searching for the designs (as described in chapter five), using our program in MATLAB is as follows.
- We first transform (code) the levels of the factors to lie between -1 and 1.
- We generate the 150 or 32 treatment combinations as the case may
- Select the best D-optimal designs based on our model.
SUMMARY, CONCLUSION AND RECOMMENDATION
In this chapter we present the summary, conclusion and recommendations based on the results obtained in chapter six.
SUMMARY AND CONCLUSION
We recall that the main objective of our thesis is to generate with the aid of computer algorithms an optimal settings for a four factor experiment involving nitrogen, phosphorus, sulphur and potassium at 5,5,3 and 2 levels respectively if controls are used, and 4,4,2 and 1 respectively, if there is no control. The factors are to be tested on maize, sorghum and millet. It should also be noted that in the initial experiment 21 treatment combinations are required because of limitation of resources. In chapter six, we started first by generating the treatment combinations for both with and without controls using the algorithms given in chapter five. We used the exchange algorithm also given in chapter five to search for D-optimal settings (designs) for maize, sorghum and millet using the interactions, quadratic and pure-quadratic models. Since it isn’t possible plotting the optimal settings for with control, the optimal settings generated for without control were then plotted on a three-dimensional space to depict the distributions of the settings, as shown in fig. 6.3 through 6.8.
The three models interaction, quadratic and pure-quadratic models are compared for the crops. The maize was considered separately while sorghum and millet were considered jointly because they have the same initial and optimal settings.
We favoured the use of quadratic model and evaluated the rotatability of the optimal settings (design) of the model using Khuri’s measure of rotatability. It was found that the design was near rotatable with a value of 83.55%.
Based on our observations in 6.6, we recommend the use of the settings generated by the quadratic model for the initial experiments instead of the ones used. This is because for these type of levels (more than two), the quadratic explains the relationships among factor levels better than the general linear model used for the analysis of the experiment (which only explains relationship of levels for at most two).