An enhanced genetic algorithm solution for itinerary recommendation considering various constraints

The COX method description

Our proposed COX method diagram is shown in Fig. 10, and its pseudocode is given in Algorithm 2 . Then it is clarified in Fig. 11 using numerical example.

As it is seen in the example in Fig. 11, the middle parts of the parents are not changed in the children, even they were not swapped, in order not to get any duplication problem.

• C12 = P12 = [7 2 9 3 10].

• C22 = P22 = [6 2 5 10 7].

• C11 = Order(P11) in P2 = [4 8 6 1].

• C13 = Order(P13) in P2 = [11 5 12].

• C21 = Order(P21) in P1 = [8 4 3 11].

• C23 = Order(P23) in P1 = [1 9 12].

The utilization of the COX technique in our study ensures the prevention of duplication issues by facilitating the inheritance of specific traits from one parent to each child while incorporating characteristics from the other parent. This methodology effectively promotes the generation of offspring with unique genetic compositions, thereby enhancing the diversity and variability within the population. Notably, unlike other order crossover methods that involve wrapping or filling discrete positions for the offspring, the COX method maintains the first-place position of each individual, preserving the established optimal order within the offspring sequences. As a result, the COX method facilitates both effective exploration and exploitation of the solution space, leading to improved solution quality. Furthermore, it mitigates any concerns related to duplication or deviation from the optimal solution.

Analysis of variances (ANOVA)

Consider the following:

• S₁, S₂, …, S₁₂ represent the different scenarios described in our experiments.

• X_ij represents the fitness values for the i-th scenario (i = 1, 2, …, 12) and the j-th observation within that scenario (j = 1, 2, …, n_i).

We use one-way ANOVA to test the null hypothesis H₀: “The means of the fitness values across all scenarios are equal.”

The overall mean ($\overline{X}$) is calculated as follows: (14)${\displaystyle \overline{X}=\frac{1}{N}\sum _{i=1}^k\sum _{j=1}^{n_i}{X}_{ij}=0.109116,}$

where N = 900 is total number of observations in our experiment, k = 12 is the number of scenarios, and n_i is the number of observations in the i-th scenario.

The Sum of Squares Between Groups (SSB) is calculated as follows: (15)${\displaystyle SSB=\sum _{i=1}^k{n}_i{\left({\overline{X}}_i-\overline{X}\right)}^2=0.234,}$

where ${\overline{X}}_i$ is the mean fitness value for scenario i.

The Sum of Squares Within Groups (SSW) is calculated as follows: (16)${\displaystyle SSW=\sum _{i=1}^k\sum _{j=1}^{n_i}{\left(X_{ij}-{\overline{X}}_i\right)}^2=0.025.}$

Degrees of freedom between groups: (17)${\displaystyle df1=k-1=11.}$

Degrees of freedom within groups: (18)${\displaystyle df2=N-k=900-12=888.}$

The Mean Square Between Groups (MSB): (19)${\displaystyle MSB=\frac{SSB}{df1}.}$

The Mean Square Within Groups (MSW): (20)${\displaystyle MSW=\frac{SSW}{df2}.}$

Then the F-statistic is: (21)${\displaystyle F=\frac{MSB}{SSW}=751.475.}$

Our experiment with degrees of freedom between groups (11) and within groups (888) yielded a critical F-value of 1.799 (Soper, 2024). As it is seen in Table 9, the F-statistic value (751.475) significantly exceeds this critical value, indicating a statistically significant difference between scenarios (p < 0.0001). This rejects the null hypothesis of equal means between scenarios.

Post_hoc analysis

We applied the Tukey Honest Significant Difference (HSD) test to determine which specific scenarios differed significantly from each other. The results are presented in Fig. 20.

Figure 20 indicates that scenarios 6, 5, 4, 3, 2, and 1 are ranked in descending order of performance, with scenario 6 achieving the best results compared with others. Similarly, scenarios 12 through 7 follow the same trend. This supports the validity of the results we obtained regarding the superiority of our proposed method.

MARS for itinerary recommendation

MARS is a non-parametric regression technique developed by Friedman (1991) for modeling complex relationships in data. MARS achieves this by automatically identifying nonlinearities and interactions between variables. MARS-based approach is used in various fields, including addressing data uncertainty and optimizing natural gas consumption models by Özmen, Zinchenko & Weber (2023), approximating data in stochastic systems for studying the mutual relationship between financial processes and investors’ sentiment by Kalaycı, Özmen & Weber (2020), and using magnetic resonance images for detecting diseases (Çevik et al., 2017).

Utilizing MARS for itinerary recommendations has many advantages like:

1. Complex relationship modeling: MARS is adept at identifying non-linearities and interactions between variables, making it ideal for analyzing user data (past trips, preferences, demographics) and travel information (destination details, activity costs, weather patterns). This enables suggestions that go beyond basic user input.

2. Data type versatility: MARS can handle various data types, which is crucial for capturing the subjective and complex nature of travel preferences influenced by factors such as budget constraints, travel style, and weather impact on activities. This allows MARS to generate more nuanced itineraries.

3. Unexpected connections: MARS can uncover unexpected connections within the data. For instance, it might reveal that users who enjoy hiking in a specific region are also interested in local historical sites, even if these interests aren’t explicitly linked. This helps to suggest relevant activities that users might not have considered.

Mathematical insights

MARS constructs models that can capture complex, non-linear relationships within the data by fitting piecewise linear splines. The MARS model, adapted for itinerary recommendation, is formulated as follows: (23)${\displaystyle \hat{S}\left(I\right)={\beta }_0+\sum _{m=1}^M{\beta }_m{h}_m\left(X_I\right),}$

where:

• $\hat{S}\left(I\right)$ represents the predicted user satisfaction for itinerary I.

• β₀ is the intercept term.

• β_m are the coefficients associated with each basis function.

• h_m(X_I) are the basis functions applied to the vector of travel factors X_I relevant to itinerary.

• M is the total number of basis functions used in the model.

The basis functions h_m(X_I) in the MARS model are of two types: Hinge functions and product of hinge functions.

1. Hinge functions: Hinge functions help to identify thresholds, such as user preferences for shorter travel distances for weekends. The mathematical representation is: (24)${\displaystyle h\left(x,t\right)={\left(x-t\right)}_{+}=max\left(0,x-t\right);}$

and (25)${\displaystyle h\left(x,t\right)={\left(t-x\right)}_{+}=max\left(0,t-x\right),}$

where x is the input value (e.g., distance), and t is a knot or threshold value (e.g., a distance value specific to weekend trips).

2. Product of hinge functions: To model interactions between travel vectors, the MARS model can also use the product of hinge functions. This product allows the model to capture the combined effect of multiple factors on user satisfaction. The Product of Hinge Functions is defined as: (26)${\displaystyle h\left(x_1,t_1,x_2,t_2\right)=max\left(0,x_1-t_1\right)\times max\left(0,x_2-t_2\right),}$

where:

• x₁ and x₂ are different travel factors (e.g., cost and time).

• t₁ and t₂ are the corresponding kont points.

For instance, if x₁represents cost and x₂ represents time, $h\left(x_1,t_1,x_2,t_2\right)$ could model how user satisfaction decreases when both the cost and time exceed their respective preferred thresholds t₁ and t₂.

Combining the hinge functions and their products, the basis functions h_m(X_I) in the MARS model can be expressed as: (27)${\displaystyle h_m\left(X_I\right)=\prod _{k=1}^{K_m}max\left(0,x_{i_k}-t_{i_k}\right),}$

where:

• K_m is the number of hinge functions involved in the product for the m-th basis function.

• x_ik is the k-th travel factor in the itinerary I.

• t_ik is the corresponding knot point.

This formulation allows the MARS model to capture both the individual effects of travel factors and their interactions on user satisfaction. It ensures that the model can accurately predict user preferences for itineraries based on a combination of factors like cost, time, and distance.

MARS two-step process: To identify the most relevant basis functions $h_m\left(X_I\right)$, MARS employs a two-step process: forward selection and backward elimination.

1. Forward selection: MARS incrementally adds basis functions that most reduce the residual error and improve the model’s accuracy. The performance of the model at each step is evaluated using the residual sum of squares (RSS): (28)${\displaystyle RSS=\sum _{i=1}^n{\left(y_i-{\hat{y}}_i\right)}^2,}$

where y_i is the actual satisfaction score for itinerary I_i, ${\hat{y}}_i$ is the predicted satisfaction score, and n is the number of itineraries.

The model at this stage is defined as: (29)${\displaystyle {\hat{S}}_{forward}\left(I\right)={\beta }_0+\sum _{m=1}^{M_f}{\beta }_m{h}_m\left(X_I\right),}$

where M_f is the number of basis functions after forward selection.

2. Backward elimination: MARS then removes less significant basis functions to streamline the model while maintaining its predictive power. This step prevents overfitting. The effectiveness of the pruning process is assessed using the generalized cross-validation (GCV) criterion: (30)${\displaystyle GCV=\frac{RSS}{{\left(n-pM\right)}^2},}$

where p is a penalty term (related to the complexity of the model), M is the number of basis functions, and n is the number of data points.

The final model is given by: (31)${\displaystyle \hat{S}\left(I\right)={\beta }_0+\sum _{m=1}^{M_b}{\beta }_m{h}_m\left(X_I\right),}$

where M_b (with M_b ≤ M_f) is the number of basis functions retained after backward elimination.

The RSS measures the difference between the actual and predicted satisfaction scores, helping the model identify the best-fitting basis functions during forward selection. The GCV, on the other hand, helps prevent overfitting during backward elimination by penalizing models that are too complex. These metrics ensure that only the most relevant factors, which significantly affect user satisfaction, are included in the final model.

Integrating MARS with GA

Once MARS has revealed these hidden patterns and potentially generated new features (e.g., ”adjusted_distance” based on weather), GA can optimize these features within the itinerary recommendation process.

1. Itinerary representation: Each potential itinerary I is represented as an individual in the GA population. This includes chosen flights, hotels, activities, and their associated costs C (I), travel times T (I), distances D (I), and satisfaction scores $\hat{S}\left(I\right)$ predicted by MARS.

2. Fitness function: A fitness function F (I) is a weighted combination of factors like: (32)${\displaystyle F\left(I\right)=w_{c}C\left(I\right)+w_{t}T\left(I\right)+w_{d}D\left(I\right)+w_s\hat{S}\left(I\right),}$

where w_c, w_t, w_d, andw_s are weights for cost, travel time, distance, and satisfaction score, respectively.

3. GA iteration: The GA iteratively improves the population of itineraries, selecting those that offer the best combination of high satisfaction and low costs, travel time, and distance.

This integration allows the system to make recommendations that accurately reflect detailed user preferences and uncover hidden patterns in the data, thereby significantly enhancing the travel planning experience.