Table 2 Parameter values tested in the calibration
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scores and has a goal of maximizing the total score of the visited destinations. Tsiligirides (1984) approached two heuristics for OP: they are stochastic and deterministic algorithms. OP can be formulated as follows: G=(V, E) is an edge-weighted graph with a score for the destination, s is a starting location, f is a terminal location, T is a positive time limit budget, and the goal is finding a path from s to ¢ (or tour if s=s) with a duration of, at most, T: The objective is collecting a maximum total profit from the visited destinations. Souffriau et al. (2008) said OP is the simplest routing model in which each destination has its own score and the objective is to create a single route that will maximize the collected score within a limited time. Li and Fu (2012) developed an equation with the objective of maximizing total profit (1). Let p; be tourists’ preference value or profit for destination V;, while x; is 1 or 0, depending on it is visited or not. S; is the set of successor destinations of destination V; and P; is set of predecessors. Equation (2.2) and (2.3) explain the flow conversion in model. Equation (2.4) ensures every destination is visited no more than once. Equation (2.5) and (2.6) guarantee that, if one destination is visited, the arrival time in that destina- tion is the sum of preceding arrival time, service time, and travel time between those destinations. v, is visiting time on node V,. Equation (2.7) consists of the start time and end time constraint. If ¢ is starting time, f; is arrival time at node V,. In this equation, fy 1s equal to t, means V, is the starting location and time starts from there. Equation (2.8) and (2.9) are variables constraints. The result for this model is a route that satisfies tourist prefer- ence and limitation, and follows the destinations’ time windows. The second step is analyzing and modifying the previous ILS method. The previous metaheuristics method still uses a random phase at the beginning, which makes the probability of getting feasible results small. For modifying the existing model, we replaced the random phase with a heuristic con- struction that considers all the constraints, to make a more feasible result and become a candidate for optimization. The Adjusted Iterated Local Search (AILS) pseudo code can be read as below. destination 9 in route 1. Time budgets and time windows must be checked for every route, and if they give a better total time the route will change. Figure 3 shows the illus- tration how the route can change. The red pin shows the starting point and ending point. In the left figure, there are two routes and each is distinguish by the line style, solid for route 1 and dashes for route 2. Then in the right figure, the routes are changed. The proposed FTOPTW algorithm and AILS method was implemented in Java by Eclipse. The experiments are carry out on a computer with an Intel Core m51.10 GHz CPU. To demonstrate the applicability of the FTOPTW algorithm, Fig. 3 A visual illustration of the example solution given in The best solutions obtained by AILS, MSA, SA, and ABC for each instance are shown in Table 3. In this table, the instance column displays the name of the problem instance, while the numbers are number of routes, total score, number of visited destinations, and running time in seconds. With this number of tours, it should be feasible to visit every loca- tion, and hence the optimal result equals the sum of all the Table 3 Computational results for AILS, MSA, SA and ABC method Journal of Industrial Engineering International
