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In general, the optimization algorithm is used to optimize (minimize) the objective function by obtaining the optimum value of the variable vector X. To optimize the variable vector X, consist of some continuous and discrete variables, a mixed discrete version of SPBO may be used. For an n-dimensional problem that includes continuous and discrete variables, the variable vector may be represented as in (Equation 1.5).

(1.5)

where [Xcont] and [Xdisc] are the continuous and discrete variable vectors, respectively. For the scenario of m continuous variables and remaining (n-m) discrete variables, the [Xcont] and [Xdisc] maybe expressed as in (Equation 1.6) and (Equation 1.7), respectively.

(1.6)

(1.7)

As mentioned earlier the mixed discrete SPBO is capable to handle both the continuous and discrete variables. In the mixed discrete SPBO, the continuous variables are updated as the conventional SPBO. The updating process of the continuous variables is the same as discussed in the previous section using four categories of students (namely best student, good student, average student, and students who want to improve randomly) with the help of (Equations 1.1–1.4). For the discrete variables, the discretization may be done with the help of the nearest vertex approach (NVA). The NVA method is normally based on finding out the Euclidean norm in the design space. The discrete variables may be expressed in terms of a hypercube, which are represented by the sets of ordered pair and can be represented as in (Equation 1.8)

(1.8)

where, are the lower and upper limits of the discrete variables, in the hypercube. In the hypercube, the lower and upper limit can be defined by floor and ceiling functions as presented in (Equation 1.9) and (Equation 1.10), respectively.

(1.9)

(1.10)

where, ℤ⁺ is the set of integers. In the hypercube H, the closest vertex of the discrete variables can be determined using the NPV method as expressed in (Equation 1.11).

(1.11)

where, is the discrete version of the continuous variable Xij. To restrict the variables within the boundary of minimum and maximum limit the same procedure may be used as used in the conventional SPBO algorithm.