Abstract
This paper deals with the orienteering problem (OP) which is a combination of two well-known problems (i.e., travelling salesman problem and the knapsack problem). OP is an NP-hard problem and is useful in appropriately modeling several challenging applications. As the parameters involved in these applications cannot be measured precisely, depicting them using crisp numbers is unrealistic. Further, the decision maker may be satisfied with graded satisfaction levels of solutions, which cannot be formulated using a crisp program. To deal with the above-stated two issues, we formulate thefuzzyorienteering problem (FOP) and provide a method to solve it. Here we state the two necessary conditions of OP of maximizing the total collected score and minimizing the time taken to traverse a path (within the specified time bound) as fuzzy goals and the remaining necessary conditions as crisp constraints. Using the max-min formulation of the fuzzy sets obtained from the fuzzy goals, we calculate the fuzzy decision sets (ZandZ∗) that contain the feasible paths and the desirable paths, respectively, along with the degrees to which they are acceptable. To efficiently solve large instances of FOP, we also present a parallel algorithm on CREW PRAM model.
Highlights
The orienteering problem (OP) is an NP-hard problem, derived from the game of orienteering where the player is required to start from the initial control point and arrive at the final control point within the specified time limit, at the same time collecting the rewards (Score) assigned to each of the checkpoints that link the initial control point to the final control point
The OP can be observed as a combination of two well-known problems, that is, travelling salesman problem (TSP) and the knapsack problem (KP), where the objective of maximizing the score is derived from KP and the objective of minimizing the time taken to travel from the initial control point and final control point is similar to the objective of TSP with the difference that in TSP all the vertices connecting the source to the target should be visited once, but in OP it is not necessary to visit all the intermediate checkpoints [1]
To provide a realistic model, fuzzy numbers can be used as they are capable of modelling the uncertainty present in the parameters and here we use trapezoidal fuzzy numbers to represent the values of the parameters involved
Summary
The orienteering problem (OP) is an NP-hard problem, derived from the game of orienteering where the player is required to start from the initial control point and arrive at the final control point within the specified time limit, at the same time collecting the rewards (Score) assigned to each of the checkpoints that link the initial control point to the final control point. As seen in the stated example, the two parameters involved, that is, time and score, cannot be determined precisely as the priorities differ from one tourist to another and the time taken to travel from one location to another cannot be predicted exactly.
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