RESPONSE SURFACE METHODOLOGY VIA DESIRABILITY FUNCTION TECHNIQUES FOR OPTIMIZING CORRELATED RESPONSES OF ELECTRICAL CONDUCTIVITY AND TOTAL DISSOLVED SOLIDS OF SELECTED BOREHOLE WATER

The health benefits in the description and observation of quantitative contents of quality parameters present or contained in any water source cannot be underestimated as they determine selection of best choice from available water sources for different intended uses as well as resource consumption. It also helps to compare the observed quantity of the quality with the acceptable standards or limits to get desired results. Physical parameters like pH, temperature, electrical conductivity (EC) and total dissolved solids (TDS) among others are determined by present of other chemical properties like Cations (Mg, Ca, Na, etc), Anions (Cl, NO3, SO4, etc), heavy metals and other dissolved materials during the course of its formation in different proportions and amounts. This study observed EC and TDS of 20 selected boreholes as two close and correlated water quality parameters as well as two of the major water quality parameters that account for overall quality of any water source, despite their different quantitative contents and physical features, they are likely determined by the same set of cations and anions with similar constraint equations. In contrast to linear programming, multiple criteria optimization models were fitted for EC and TDS using Response Surface Methodology via desirability techniques, optimal values obtained in this case measured against several criteria are found to lie between acceptable standards limits for drinking water, other numerical values and descriptive features in the final results reflect that the response equations obtained were well fitted.


INTRODUCTION
Product optimization is an essential process in the fields of Science, Engineering and Technology as optimization of yields and productivities has been a major goal in biological and physical components since the very beginning of decision processes on industrial and production systems. Optimization is a technique that explores possible and likely behaviors of systems with numerous input responses with goal of identifying the best possible outcome, Cornell (1996). In mathematical terms, the ''outcome'' is the value of some functions, and ''best possible'' often is the maximum or minimum of the function which is the point with highest or lowest possible value. The function itself is often called objective function and its arguments are called control variables or at times input variables observed from various input responses that determine or account for quality of interest, Cornell (1996). In practices, if one multiplies the objective function by -1, the former maximum becomes minimum and vice versa. Thus, finding a maximum or a minimum is basically the same, and we talk generally about finding extremum or optimum. Raymond, et al. (2016) affirmed that the objective function is controlled by some conditions of input factors or variable as defined by set of constraint functions that define boundaries or limits that guarantee each response viability or acceptable standards in the system. Thus, the constraints are dictated by system component, technical and economical factor among others to have final equilibrium output from objective function. A complete optimization to optimize a response of interest involves solving response of interest involved and validating of objective and constraint equations. Summarily, the optimization task with several system components typically reads; 12 ( , ,..., ) h f X X X

Multi-response Design and Model
Multi-response experiment requires careful consideration of the multivariate nature of data observed together in a process. In fitting model for multi-response observation, Khuri and Cornell (1996) stated that response variables should not be investigated individually and independently of one another as interrelationships that may exist among them can render such investigation meaningless. Hill and Hunter (1966) cited several papers in which multiple response are investigated with desire to optimize several response functions simultaneously with a completely defined constraint equations with full information about the condition of constraints optimal solution has to satisfy.

Representation of General Multi-response Model
In (Cornell (1996), Way Kuo, et al. (2001)), it was supported that the design problem in the multi-response case is more complex than in the case of a single response as each of the response values has different set of input variables, at times when set of input variables are the same for some responses, the linear relationship or contrast may differ.
Where i Y is an N  1 vector of observations on the th i response, i X is an N  P matrix of rank P of known function of the setting of coded variables,  is a P  1 vector of unknown constant parameters and i  is a random error vector associated with th i response ( 1, 2,..., ) ir  . The assumptions on i  are that ( ) 0 ( ) 1, 2,..., The r  r matrix whose ( , ) th ij element is , ( , Also, the r equations in 3 can be represented by YX   (5)

Water Quality Parameters As A Multi-response Experiment
As opposite to experiments with single response variables which are reffered to as single-response experiments. However, numerous experiments involve measurements associated with several response variables, in such cases number of responses are measured simultaneuosly for each setting of group of input variables which are reffered to as multi-response experiment by Khuri and Cornell (1996). There are numerous number of multiresponse experiments where researchers' interest is to determine optimum combinations of various system components on the basis of acceptability, nutritional and economic value among other considerations. Like many other systems or processes that quality performance are determined by numerical contents or compositions of different or specific components of that system or process in which no single quality parameter can perform better in isolation. Water is natural substance that during the process of its formation, its contents, components and quality is determined by different organic and inorganic matters that have contact with during and after its formation e.g. different types of soluble and insoluble rocks, soils, atmosherical and biological matters. Different sources of water account for unequal compositions and contents of quality parameters and as a result, there are needs for its different users to make decisions on selected responses and factors of choice to meet their desired limits of acceptance for optimal condition subject to different response requirement with respect to their specification limits as constraints and to analzye the feasibility of optimal values to give desired product quality.

An Overview of Multi-Response Optimization Approaches
In the work of Raissi and Eslami (2009), this process is regarded as a multi-response optimization problem. Most of the common methods are incomplete in such a way that a response variable is selected as the primary one and is optimized by adhering to the other constraints set by the criteria. Among methodologies developed to resolve the multi-response problems. Khuri and Cornell (1987) surveyed the multiresponse problem using a response surface method. Tai, et al. (1992) assigned a weight for each response to resolve the problem. Pignatiello (1993) made use of a squared deviationfrom-target and a variance to and from an expected loss function. Layne (1995) used a procedure capable of considering three functions: weighted loss function, desirability function and distance function. Myers and Montgomery (1997) referred to this as a popular approach to formulate and solve the problem as constrained optimization problem. Kim, et al. (2001) classified it as a priority based as is similar to bounded objective in the multi objective decision making problems where response with highest performance is chosen as the objective and the rest of the functions are considered as constraints. Myers and Carter (1973) first suggested the idea where he assumed two responses as a 'Primary response' and a 'Constraint response' with goal to find condition on a set of design variables which maximize the primary response function subject to the constraint response function. Biles (1975) considered multiple process responses and extended the Myers and Carter's idea. Del Castillo and Montgomery (1993) studied the approach later year. On designs with multiple response, Logothestis and Haigh (1988) discussed a manufacturing process with five responses, one of the five response variables was selected as primary and optimized the objective function while ignoring the possible correlations among the responses and considered the constraints of other determining input variables. Derringer and Suich (1980) proposed that a particular response i Y among other responses in the same experimental set up may be maximized or

Optimizing Through the Desirability Function Approach
An analytic technique for optimization of multi-response design based on the concept of utility or desirability of a property associated with a given response objective function introduced by Harrington in 1965. This approach uses an estimated response such as ( ) transformed to a scale free value i d that is called desirability which ranges from 0 to 1and also allows users to specify minimum and maximum acceptable values for each response. In Harrington (1965) desirability () D is also in the [0,1] interval is obtained by combining all desirabilities () i d . Derringer and Suich (1980) extended the idea and presented a method to construct an overall desirability. There are three scenerios as in the case of response surface work and any one serves as a specific goals for each of the input variables. As sugested by Taguchi (1987) an approach that provides information about the mean and variance of observations, the summary statistic is computed across for observations which is called Signal-to-Noise Ratio (SNR) and emphasis on variance reduction. Each of the three scenarios depends on the choice of experimenter and or the nature or goal of the variable to optimised which can be as; The Smaller The Better (STB), the experimenter wishes to minimize the response, in this case the SNR is given by Nominal The Better (NTB) or The Target is Best, the experimenter wishes to achieve a particular value for the response, in this case we are attempting to determine value of x that achieves a target value for the response, the SNR used by Taguchi is given by (9) Thus s 2 is the sample variance.
The desirability function i d for the three scenarios of optimization problems are illustrated as this approach over other approaches are that easy to use, understand and model.
The overall desirability D is maximized with respect to the controllable factors using a geometric mean function as The approach is based on the idea that the quality of a product or process must meet all k quality characteristics. In adopting the approach for solving the problems of optimization of several responses is the use of a multicriteria methodology which is applied when various responses have to be considered at the same time and it is necessary to find optimal compromises between the total number of input variables taken into account at a time. The Derringer function or desirability function of Harrington, (1965) is the most important and most currently used multicriteria methodology in the optimization of analytical procedures by constructing a desirability for each individual response. In summary, the measured properties related to each response are tranformed into dimensionless individual desirability (

Second Order Response Surface Model
If all factors represent quantitative variables, the most informative model to assist analysis of the yields or response is a function of input variables, i.e. 12 ( , ,..., ) h Y f X X X  (12) the ordinary polynomials, the second order in particular have been extensively employed in exploring response surfaces. This is because it is generally accepted for its simple computation, easy to work with, easy to locate the optimum response. However, they exhibit the undesirable problems of unboundness, symmetry about the optimum. These polynomial models have been used in many biometry researches. In applying the response surface methodology, the dependent is viewed as a surface to which a mathematical model is fitted. For the development of regression equation related to various quality characteristics, the second order response surface may be assumed as Where r e is a random error, the parameter ' Bsare called regression coefficients which are to be estimated and obtained by the design technique. The assumed surface Y contains linear, squared and cross-product terms of variables ' i X s. In order to estimate the regression coefficients, a number of experimental design techniques are available. Box and Hunter (1957) proposed that the scheme based on central composite rotatable, design fit second-order response surfaces very accurately. Many literatures on multi-response experiments utilized a second-order models. Also when restricting the response surface problem to response optimizaton, to select a design that will provide a good fitted model to the data, and in particular provide reliable parameter estimates, which can be used for prescise prediction, second-order models are primarily used for these purposes.  . 3, September, 2020, pp 333 -342 monitor quality of water parameters as it recognizes existing commercial limits as adherence to quality ptoduct is becoming National prioroty in health sector.

Electrical Conductivity (EC) and Total Dissolve solutes (TDS)
Electrical conductivity of water is the measurement of its ability to carry an electric current and can be regarded as a crude indicator of water quality for primary purposes. It reflects the extent of solubility of mobile cations and anions and is related to the sum of ionised solutes or total dissolved solids which is the sum of cations and anions as well as organic and inorganic substances in water that can pass through a 2 micron filter. The relationship between EC and TDS is directly proportional and anyone can be estimated fairly accurately from other via a linear correlation and regression equation. However, as a rough approximation, the relationship between EC and TDS commonly used is High content of any of the parameters in drinking water posses serious health dangers, this is the main reason why they are used to monitor quality in drinking water through their acceptable limits guideline for use. For borehole water, EC value greater than 500 µScm -1 indicate that the water may be polluted, although, values as high as 2000 µScm -1 may be acceptable for farming, but for drinking water EC should not be more than 500 µScm -1 as water with higher value may have quality problem and be unpleasant to drink. For TDS, water with TDS greater than 1200mg/l is very unusable.
Summary of ANOVA Table 3 shows that the model is significant at 0.05 with all single factors found significant (p<0.05), two factor interaction were also found significant except for one while quadratic term was significant were retained. The three R 2 statistics values in Table 5 also explain the significant of the fitted model as predicted R 2 of 82.0% is in reasonable agreement with adjusted R 2 of 96.5%.

RESPONSE SURFACE…
From equation 19, it may be observed that from ANOVA table 4 that all the single terms are significant while all interaction terms not significant, no quadratic term found significant as the model has only 0.01% chance that its value could due to noise, the three R 2 statistics in Table 5 indicate suitability of the model.

Optimization Results and Validation
During the optimization stage, the desirability function approach was used to obtain the best compromise with respect to each response acceptable limits as constraints. The second order polynomial model was fitted to each response observed data to obtain optimal values shown in Table 6. The goal was to minimize i.e. Smaller The Better (STB) scenario which reflects the optimum condition values for EC and TDS containing minimum amount of Calcium, Magnesium and Chloride.

Numerical Optimization Objective and Constraint Functions
As the process includes both dependent and independent variables, the optimal solutions are characterized by objective function represented or obtained by fitted response model and set of constraint functions represented or obtained by acceptable conditions of the process or process equilibrum state. The optimization task in this process components typically reads.

CONCLUSION AND RECOMMENDATION
RSM utilizes regression techniques to study experimental products and process, however, optimization of multiple response design depends too heavily on the assumptions of well estimated models fitted for the responses of interest. It can be seen that optimal values obtained for responses in Table 6 are within the acceptable standards with other model parameters which make the method to be reliable. The optimal values of 80.3 µScm -1 and 210 mg/l of EC and TDS respectively are obtained by the same optimal values of their same set of of input factors of 51.2 mg/l, 9.77 mg/l and 10 mg/l for Calcium, Magnesium and Chloride respectively. The result will be beneficial to water users most especially for drinking to improve the quality of the product for health reasons and benefits.