*2.5. Response Surface Analysis*

Experimental data was used to develop the second-order response model Equation (2).

$$\mathbf{y} = b\_0 + \sum\_{i=1}^{n} b\_i \mathbf{x}\_i + \sum\_{i=1}^{n} b\_{ii} \mathbf{x}\_i^2 + \sum\_{i=1}^{n} \sum\_{j=1}^{n} b\_{ij} \mathbf{x}\_i \mathbf{x}\_j + \varepsilon \tag{2}$$

where:



Surface plots were further developed using the models. These surface plots were drawn to understand the interactive effect of the process variables on pellet quality and specific energy consumption. Statistica software, version 9.1 (StatSoft. Inc., 2300 East 14th St. Tulsa, OK, USA (www.statsoft.com)), was used to do the response surface analysis [47].

In the case of optimization, the response surface models (Equations (3)–(5)) that were developed using the experimental data were then further used as the objective functions. These objective functions are either minimized or maximized using the hybrid genetic algorithm developed by Tumuluru and McCulloch [48]. As the genetic algorithm is heuristic in nature, and they seldom reach global optimum. So, for better conversion of the optimum values, Tumuluru and McCulloch [48] hybridized a genetic algorithm with a gradient-based method. These authors tested the algorithm on food and bio-engineering problems and concluded that the hybrid genetic algorithm has better convergence compared to a regular genetic algorithm. In the case of bulk density and durability, the objective functions were maximized, whereas, in the case of pellet moisture content, the objective function is minimized to find the optimum process conditions.

$$f(y) = \text{Minimize}\left(PMC \, model\right) \tag{3}$$

$$f(y) = \text{Maximize}\left(\text{BD model}\right) \tag{4}$$

$$f(y) = \text{Maximize (D \ model)}\tag{5}$$

Note: PMC: pellet moisture content (%, w.b.); BD: bulk density (kg/m3); D: durability (%).
