Applying Numerical Optimization Technique in the Development of Valueadded Powdered Custard

Materials

Custard can be developed from different grains. The raw materials used for this study were millet, Tiger Nuts, soybean (Glycine max), sweet potato (Ipomoea batatas) extract, date (Phoenix dactylifera), clove (Syzygium aromaticum), cinnamon (Cinnamomum verum), turmeric, and ginger (Zingiber officinale). These materials were purchased in Kure market in Minna, Niger State, Nigeria.

Processing of Millet Flour, Soybean, and Tigernut Meals

Properly cleaned, dry millet grains, soybean and tigernut were milled into fine powder and stored under room temperature prior to use.

Methods

Experimental Design for the Formulation of Value- Added Breakfast Cereal Meals: The formulations of value-added custard were carried out using constrained optimal (custom) mixture experimental design and response surface methodology, employing Design-Expert software (version 11, Stat-Ease Inc., Minneapolis, U.S.A), which gave 69 randomized experimental runs. D-optimal designs are one form of design provided by a computer algorithm. They are often used in researches where 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. These types of 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, response surface, etc.). D-optimal designs are straight optimizations based on a chosen optimality criterion and the model that will be fit. The

optimality criterion used in generating D-optimal designs is one of maximizing |X’X|, the determinant of the information matrix X’X. This ‘optimality’ of a given D-optimal design is model dependent.

The formulation design constraints were: millet flour ( ) 1 10% 70% , x ≤ ≤ soybean ( ) 2 10% 70% , x ≤ ≤ and tigernut ( ) 3 10% 70% . x ≤ ≤ Other constant components of the formulation were sweet potato extract (3%), date (2 %), clove (1%), cinnamon (1%), turmeric (1%), and ginger (2%). The design matrix used for the formulation experiment was presented in Table 1. The samples were prepared based on the design matrix. The other minor components were added to each of the sixty-nine samples and mixed together thoroughly, to obtained homogeneous mixture.

Proximate Analysis and Sensory Evaluations: The quality characteristic of custard which was measured using the method described by the Association of Analytical Chemist [9]; include ash, crude protein, moisture content, fat, crude fibre, and carbohydrate. The sensory evaluation of the samples was conducted using thirty trained panelists. The samples were evaluated for taste, flavour, sweetness, colour, texture and overall acceptability. A 9-point hedonic scale ranging from 9 = like extremely and 1= dislike extremely was used to evaluate the samples.

Statistical Analysis

The experimental data were analyzed and appropriate Scheffe canonical models were fitted to the mean proximate property data. The statistical significance of the terms in the Scheffe canonical regression models were tested using analysis of variance (ANOVA) for each response, and the adequacy of the models were evaluated by coefficient of determination, F-value, and model p-values at the 5% level of significance. The models were also subjected to lack- of-fit and adequacy tests. The fitted models for each of the response were used to generate 3-D response surface as well as the contour plots for the proximate properties using the DESIGN EXPERT 11.0 statistical software.

Numerical Optimization: A Numerical optimization approach, exploiting the desirability function technique, was utilized to generate the optimal formulation with the anticipated responses. Numerical optimization becomes essential when investigating many components with many responses. Numerical optimization maximizes, minimizes, or targets desired response based on set criteria for all variables, including components proportions. Components can be allowed to range within their pre-established constraints in the design or they can be set to desired goals. Also, components can be set equal to specified levels. Optimization goals are assigned to parameters and these goals were used to construct desirability indices (di). Desirability index range from zero to one for any given response and individual desirability for all the responses, in the case of multi-response optimization, are combined into a single number known as overall desirability index. A value of one represents the case where all goals are met perfectly. A zero indicates that one or more responses fall outside the set desirable limits.

Under this approach, each ith response is assigned a desirability function, id , where the value of id varies between 0 and 1. The function is defined differently based on the objective of the response. If the response is to be maximized, then id is defined as follows:

0 y L <     −     = ≤ ≤     −       >  

i w i i i y L d L y T T L y T

( )

1

1 i Where T represents the target value of the ith response, iy , L represents the acceptable lower limit value for this response and w represents the weight. When 1 w = the function is linear. If 1 w > then more importance is placed on achieving the target for the response, iy . When 1 w < , less weight is assigned to achieving the target for the response, iy . If the response is to be minimized, as in the case when the response is cost, is then id is defined as follows:

1 y T <     −     = ≤ ≤     −       >  

i w i i i U y d T y U U T y U

( )

2

0 i Where U represents the acceptable upper limit for the response. There may be times when the experimenter wants the response to be neither maximized nor minimized, but instead stay as close to a specified target as possible. In such cases, the desirability function is defined as follows:

0 L

y <     −     ≤ ≤     −    =   −     ≤ ≤     −     >     i w i i y L y T T L d U y T y U U T y U

1 L 3

( ) i w i i

2

0 i

Once a desirability function is defined for each of the responses, assuming that there are m responses, an overall desirability function is obtained as follows:

( ) ( ) ( ) 3 1 2 1 2 3 ........ 1 1 2 3 . . ......... 4 m m r r r r r r r r m D d d d d + + + + = Where the ir s represent the importance of each response. The greater the value of ir , the more important the response with respect to the other responses. The objective is to now find the settings that return the maximum value of . D the rationale for using a geometric rather than an arithmetic mean is that if any individual desirability di is equal to zero, then the overall desirability will also be equal to zero.

Numerical optimization solutions are given as a list, in their order of desirability, detailing the components Run 1_x_ %

2_x_ %

3_x_ %

3_x_ %

3_x_ %

3_x_ %

3_x_ %

3_x_ %

3_x_ % ycp yac % %

proportions and process variables values that satisfies the set criteria and the overall desirability. The numerical solution can be presented in the form of desirability contour and 3-D Surface plots, optimal bar graph and graphical optimization overlay contour plot; showing the optimized formulation compositions and/or regions that meet specifications [10, 11, 12, 13].

Empirical Modeling of Proximate Compositions of Formulated Value-Added Custard

Empirical models in terms of L-pseudo components were fitted to the proximate and physicochemical properties of the formulated composite gari. The ash content fitted model in terms of L_Pseudo Components is presented in equation (5).

} ( ) 5 4 1 2 3 2.98 .09 3.21 y x x x ac + = +

This model can be used to peruse the design space and to make predictions about ash content for given levels of each factor. The contours and 3-D plots for the ash content are summarized in Figure 1.

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The crude protein fitted model in terms of L_Pseudo Components is presented in equation (6). The model can be used to navigate the design space and to make predictions about crude protein for given levels of each factor.

} ( ) 6 3 1 2 3 22.8 0.4 24.8 y x x x cp + + =

The contours and 3-D plots for the crude protein are summarized in Figure 2.

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The moisture content fitted model in terms of L_Pseudo Components is presented in equation (7):

} ( ) 7 1 1 2 3 9.56 1.0 8.41 y x x x mc + + =

This model can still be used to navigate the design space and to make predictions about the moisture content for given levels of each factor. The contours and 3-D plots for the moisture content are summarized in Figure 3.

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The fat content fitted model in terms of L_Pseudo Components is presented in equation (8) } ( ) 8 1 1 2 3 18.1 7.9 24.4 y x x x fat + + = This model can be used to navigate the design space and to make predictions about fat content for given levels of each factor. The contours and 3-D plots for the fat content are summarized in Figure 4.

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The crude fibre fitted model in terms of L_Pseudo Components is presented in equation (9):

4.13 3.74 4.01 35.93 2.78 2.61

1 2 3 y x x x x x x x c

+ + − + + = 

f

4 5 1 4 ( )

The crude fibre model can still be used to navigate the design space and to make predictions about crude fibre for given levels of each factor. The contours and 3-D plots for the crude fibre are summarized in Figure 5.

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The carbohydrate fitted model in terms of L_Pseudo Components is presented in equation (10):

} ( ) 10 3 1 2 3 42.8 3.2 35.4 y x x x cho + + =

This model can be used to navigate the design space and to make predictions about carbohydrate for given levels of each factor. The contours and 3-D plots for the carbohydrate are summarized in Figure 6.

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The overall acceptability fitted model in terms of L_ Pseudo Components is presented in equation (10):

} ( ) 1 2 1 3 2 3 6.64 6.31 5.73 4.01 5.99 1 1. 1 42 1 2 3 y x x x o x x x x x x a = + + + − − The overall acceptability model can still be used to navigate the design space and to make predictions about crude fiber for given levels of each factor. The contours and 3-D plots for the overall acceptability are summarized in Figure 7.

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Generation of Optimal Formulation

A Numerical optimization approach, exploiting the desirability function technique, was utilized to generate the optimal formulation with the anticipated responses. Table 2 presents the summary of the optimization constraints employed in the optimization module. The four desirability solutions that were found were as presented in Table 3. The numerical solution desirability contour plot and 3-D Surface were presented in Figure 2. The numerical solution, presented in the form of optimal instant cereal meals bar graph and the graphical optimization overlay contour plot, showing the optimized formulation compositions with the respective quality parameters, are summarized in Figure 3.

( ) % mc Moisture y Content = , ( ) % pc Protein y Content = , ( ) % fat Fat Content y = ( ) % ac Ash Content y = , ( ) % cho Carbo y hydrate = , ( ) % fc Fibre Co y ntent = , ta Taste y = flav y Flavour = , sw Sweetness y = , cl Colour y = , tx Texture y = , oa Overall Acceptability y =

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The result of the optimization gave optimized formulated value-added custard with overall desirability index of 0.535, based on the set optimization goals and individual quality desirability indices. The optimal formulated value-added custard was gotten from 10.0 % millet flour, 49.3 % soybean meal, 30.7 % tigernut. The quality properties of this optimal custard are: 3.78 % ash content, 28.5 % crude protein, 10.1 % moisture content, 20.1 % fat content, 4.43 % crude fibre, 33.9 % carbohydrate, and 5.79 overall acceptability. The box in the overlay contour plot (i.e., Figure 4) indicates the properties of the optimal custard and the component proportions to obtain it.