## 1 Introduction  
As a transport mode enabling reduction of the number of vehicles in use, energy consumption and emissions of greenhouse gas in cities, shared mobility has drawn researchers' and practitioners' attention around the world. Many transport service providers such as Uber [1] and Lyft [2] have implemented shared mobility systems. Shared mobility has led to the creation of a lot of studies in the ridesharing literature [3, 4, 5]. In the past years, various ridesharing problems with different models were studied. Recently, ridesharing with multiple platforms was considered in [6] to match supply and demand across multiple platforms to improve the overall revenue of platforms through coordination. The study [7] aims to determine efficient routes and schedules for drivers and passengers with flexible pick-up and delivery locations. Motivated by the need to reduce CO2 emissions, a graph-theoretical algorithm-based approach was proposed to achieve higher levels of reduction in CO2 emissions in the study [8]. A review of the challenges and opportunities due to shared mobility is available in [9; 10; 11].  
Financial benefits provide one strong incentive for the prevailing ridesharing model. In the literature, cost savings due to ridesharing make it possible for drivers and riders to reduce their travel cost and enjoy the monetary incentives in shared mobility systems. In the literature, different mechanisms have been developed to incentivize ridesharing drivers and riders. These include cost sharing [12], discounting [13] and cost allocation [14; 15]. In [12], a general cost sharing framework to create cost sharing mechanisms with desirable properties and compensation for inconvenience costs due to detours was proposed and validated based on a real case study. In [13], a non-linear mixed integer optimization problem to maximize the profit and offer shared trips to riders with varying price discounts was proposed. The problem was decomposed into two sequential sub-problems to achieve computation efficiency. Results of experiments confirm that the dynamic discount strategy produces substantial economic benefits.  
As cost savings are a benefit of shared mobility that provides one financial incentive to riders, many studies focus on maximization of cost savings in shared mobility systems. However, if the amount of cost savings is insufficient to attract riders, riders will not accept the ridesharing mode [14]; proper allocation of cost savings can help improve the acceptability of ridesharing [15]. For this reason, the study [16] formulated a discount-guaranteed ridesharing problem (DGRP) to ensure a minimal discount for drivers and riders in ridesharing systems. Due to the complexity of a DGRP, metaheuristic algorithms have been developed in [16] to find solutions.  
Metaheuristic algorithms refer to algorithms that are characterized by imitating natural phenomena or human intelligence and are able to tackle the complexity issue in solving optimization problems. Natural phenomena or human intelligence can be applied to develop effective strategies to guide the process to search solutions efficiently. In the metaheuristics literature, many metaheuristic algorithms were proposed by drawing inspiration from natural phenomena, biological sciences, laws of physics, rules of games and human activities. In the study [17], metaheuristic algorithms are classified into five categories: physics-based, swarm-based, evolutionary-based, game-based and human-based methods. In the remainder of this section, we will first give a brief review of these categories. Then we will focus on the issue to create metaheuristic algorithms based on the saying \"Two heads are better than one\".