**6. Experimentation and analysis**

We utilize FA, PSO, and SFS algorithms to establish optimal regimes based on the proposed mathematical model in [5] where *θ* ¼ 0*:*67 and *σ* ¼ 1*:*34. The WHO recommendations concerning the nutrients daily needs were token into considerations [6, 25, 26]. We work on 176 aliments considered as the most consumed in Morocco. The linear part of our model is estimated using the means glycemic load of the considered foods. From now on, we adopt the symbols: TGL for Total Glycemic Load, FTG for Favorable Totale Gap, and UFTG for UFavorable Totale Gap.

a. We used the SFS algorithm to solve problem (D). We tested this algorithm for different values of the parameters: walk probability, maximum diffusion, and the number of iterations. Th **Table 1** gives TG, FTG, and UFTG of diets produced by SFS for max diffusion equals to 5, start points equals to 50, number of iterations of 200, and different values of walk probability from the interval [0.3 0.9] adopting 0.1 as step.

The best diet is the one produced by SFS for walk probability value equals to 0.7 with glycemic load in the interval [82.2152 92.5292] and nutrients requirements gaps


#### **Table 1.**

*TG, FTG, and UFTG of diets produced by SFS for max diffusion = 5, start points = 50, number of iterations of 200, and different values of walk probability.*


#### **Table 2.**

*TG, FTG, and UFTG of diets produced by SFS for start points = 50 (45), number of iterations of 200, walk probability of 0.7, and different values of diffusion.*

143.3103 mg (for positive nutrients) and 30.9554 mg (for negative nutrients). These diets still bad considering the considered three criterions. To investigate possible improvements, we set the walk probability to 0.7 and, start points to 45, and number of iterations to 200, and we variate the value of diffusion.

The **Table 2** give TGL, FTG, and UFTG of diets produced by SFS for start points equals to 50(45), number of iterations of 200, walk probability of 0.7, and different values of diffusion from [5 10] by adopting 1 as step. The obtained diets become to be acceptable and the best diet is the one who's TGL is in [**68.8041 76.3373**], FTG = **52.0240**, and UFTG = **47.8260.**

To investigate more improvements, we set max diffusion to 10, walk probability to 0.7, start points to 45, and we vary different number of iterations; see **Table 3**.

Indeed, we detect a very good diet (produced by SFS) for 600 number of iterations with TG is in [53.8780 66.0715], FTG = 50.1917, and UFTG = 28.5891. The **Figure 6** illustrates the behavior of (D) objective function when solving the diet problem using SFS for max diffusion equals to10, walk probability equals to 0.7, start points equals to 45, and the number of iterations equals to 600; it is clear that the algorithm has not yet


#### **Table 3.**

*TG, FTG, and UFTG of diets produced by SFS for max diffusion 10, walk probability 0.7, start points 45, and different number of iterations.*

#### **Figure 6.**

*Evolution of the model (D) fitness with iterations by SFS for walk probability of 0.7, maximum diffusion of 10, and number of iteration equals to 600.*

converged and an additional number of iterations will allow more improvement, but we compare the algorithms for a very small number of iterations to get a good diet in real time.

b. We used the FA algorithm to solve problem (D). We tested this algorithm for different values of the parameter's population, attraction coefficient base value, iterations, and of Mutation coefficient damping ratio.

The **Table 4** give TG, FTG, and UFTG of diets produced by FA for: population 40, attraction coefficient base value of 2.25, iterations of 300, variation of mutation coefficient damping ratio from in [0.1 0.9] with 0.1 as step.

All the produced diets are acceptable and the best diet is the one produced for Mutation Coefficient Damping Ratio equals to 0.4. To investigate more improvements


#### **Table 4.**

*Diet produced by FA for population equals to 40, attraction coefficient base value of 2.25, iterations equals to 300, and variation of mutation coefficient damping ratio.*


#### **Table 5.**

*Diets produced by FA for attraction coefficient base value equals to 2.25, iterations equals 300, mutation coefficient damping ratio = 0.4, and variation of population.*

of this diets, we variate the number of iterations will setting the mutation coefficient damping ratio to 0.4; see **Table 5**.

In fact, the quality of diets were improveded and the best one is obtained for attraction coefficient base value equals to 2.25, iterations equals to 300, mutation coefficient damping ratio equals to 0.4, size population = 50 with TGL is in [**53.5439 56.3875**], FTG = **10.6000 mg**, and UFTG = **7.8365 mg.**

The **Figure 7** illustrates the behavior of (D) objective function when solving the diet problem using FA for coefficient base value equals to 2.25, iterations equals to 200, mutation coefficient damping ratio equals to 0.4, and size of population equals to 50. We remark that FA algorithm reaches early a very good local solution.

c. We used the PSO algorithm to solve problem (D). We tested this algorithm for different values of iterations, self-adjustment weight, social-adjustment weight, and population size.

#### **Figure 7.**

*Behavior of (D) objective function when solving by FA for: Firefly attraction coefficient base value = 2.25, iterations = 200, mutation coefficient damping ratio = 0.4, variation of population = 50.*


#### **Table 6.**

*Diets produced by PSO for number of iterations = 200, self-adjustment weight = social-adjustment weight = 2, and variation of the population size.*


**Table 7.**

*Diets produced by PSO for number of iterations = 200, variation of self-adjustment weight = SocialAdjustmentWeight, and population size =50.*

The **Table 6** give TG, FTG, and UFTG of diets produced by FA for number of iterations equals to 200, self-adjustment weight = social-adjustment weight = 2, and population size variation between 20 and 80 particles.

The best diet is the one produced by PSO for population size of 50 with TG in [70.8154 80.1564], FTG = 61.6584 mg, and UFTG = 19.1466 mg. To investigate more improvements of this diets, we vary the Adjustment Weight coefficients in [1 2] will setting the population size to 50 (**Table 7**).

#### **Figure 8.**

*The behavior of (D) objective function when solving the diet problem using PSO for number of iterations = 200, self-adjustment weight = social-adjustment weight = 2, and population size = 50.*

*Intelligent Local Search Optimization Methods to Optimal Morocco Regime DOI: http://dx.doi.org/10.5772/intechopen.105600*


#### **Table 8.**

*Comparison between the diets produced by PSO, FA, and SFS.*

Indeed, the quality of diets were improveded and the best one is obtained for PSO with iterations = 200, variation of self adjustment weight = social adjustment weight = 2, and population size =50; the Diet total glycemic load is in [70.8154 80.1564] and FTG = 61.6584 mg, and UFTG = 19.1466 mg, which meets the recommandations given in [24] . It should be noted that the first height diets are unacceptable.

The **Figure 8** illustrates the behavior of (D) objective function when solving the diet problem using PSO for self-adjustment weight = social-adjustment weight = 2, and population size =50. We remark that PSO was attracted very early to a very bad diet.

d. We compared the best diets produced by SFS, FA, and PSO based on the considered three criteria: TGL, FTG, and UFTG; see **Table 8**.

We remark that the best diet is the one produced by firefly algorithm for the configuration shown by the column 2 of the **Table 8** for a small number of iterations.

We can repeat all this study will consider additional quality measures such as the satiety rate and the applicability of the considered diets.
