**1. Introduction**

The lives of ordinary consumers have changed almost beyond recognition in the past 20 years. First, with the introduction of high-speed internet access; but, more recently, with the arrival of mobile computing devices such as smartphones and tablets. According to data from the 2017 Gallup World Survey [1], 93 of adults in high-income economies have their cell phones, while 79% in developing economies. In India, 69% of adults have a cell phone, as well as 85% in Brazil and 93% in China [1]. Smartphones and the internet have created a novel digital ecosystem where the adoption of new paradigms is increasingly fast, and each innovation that appears and presents itself to the market can disrupt an entire segment.

In the transportation segment, a central theme is how the digital revolution has created opportunities to consider new models of delivering services under the paradigm of Mobility as a Service (MaaS) [2]. There is a growing interes%t in *MaaS* due to the notion of a sharing economy. Millennials own fewer vehicles than previous generations [3]. As evidenced by the ascension of on-demand mobility

platforms, they are quickly adopting car sharing as a mainstream transportation solution. Investments in new travel patterns have become a priority to enable the transformation of opportunities in the urban mobility segment into new revenue streams.

The main contributions of this chapter are summarized in the following.

*Multi-Strategy* MAX-MIN *Ant System for Solving Quota Traveling Salesman…*

quality solutions achieved;

*DOI: http://dx.doi.org/10.5772/intechopen.93860*

in [11];

QTSP-PIC.

**2. Problem definition**

the minimum quota.

**79**

Section 6.

• The extension of the MS concept proposed in [5] with a roulette mechanism that orients the ants to choose their heuristic information based on the best

• Improvement of the *MM*AS design with a memory based technique proposed

• Presentation of a novel *MM*AS variant that combines the improved MS concept

• Experiments on a set of QTSP-PIC instances ranging: 10 to 500 cities; and 30 to 10.000 travel requests. The results showed that the proposed *MM*AS variant is competitive regarding the other three ACO variants presented in [5] for the

The remainder of this chapter is organized as follows. Section 2 presents the QTSP-PIC and its formulation. Section 3 presents the Ant Colony Optimization metaheuristic and the implementation design of the MS-*MM*AS. Section 4 presents experimental results. The performance of the proposed ant-based algorithm is discussed in Section 5. Conclusions and future research directions are outlined in

The TSP can be formulated as a complete weighted directed graph *G* ¼ ð Þ *N*, *A* where *<sup>N</sup>* is the set of vertices and *<sup>A</sup>* <sup>¼</sup> f g ð Þj *<sup>i</sup>*, *<sup>j</sup> <sup>i</sup>*, *<sup>j</sup>*<sup>∈</sup> *<sup>N</sup>* is the set of arcs. *<sup>C</sup>* <sup>¼</sup> *cij* is

The QTSP-PIC is a QTSP variant in which the salesman is the driver of a vehicle and can reduce travel costs by sharing expenses with passengers. There is a travel request, associated with each person demanding a ride, consisting of a pickup and a drop off point, a budget limit, a limit for the travel duration, and penalties associated with alternative drop-off points. There is a penalty associated with each point different from the destination demanded by each person. The salesman can accept or decline travel requests. This model combines elements of ride-sharing systems [14] with alternative destinations [4], and the selective pickup and delivery problem [15]. Let *G N*ð Þ , *A* be a connected graph, where *N* is the set of vertices and *A* ¼ f g ð Þj *i*, *j i*, *j*∈ *N* is the set of arcs. Parameter *qi* denotes the quota associated with vertex *i*∈ *N* and *gi* the time required to collect the quota. *cij* and *tij* denote,

respectively, the cost and time required to traverse edge ð Þ *i*, *j* ∈ *A*. Let *L* be the set of passengers. List *li* ⊆*L* denotes the subset of passengers who depart from *i* ∈ *N*. Let *org l*ð Þ and *dst l*ð Þ∈ *N* be the pickup and drop-off points requested by passenger *l*. The

the arc-weight matrix such that *cij* is the cost of arc ð Þ *i*, *j* . The objective is to determine the shortest Hamiltonian cycle in *G*. Due to its applicability, many TSP variants deal with specific constraints [12]. Awerbuch et al. [13] presented several quota-driven variants. One of them, called Quota Traveling Salesman Problem (QTSP), is the basis for the problem investigated in this study. In the QTSP, there is a bonus associated with each vertex of *G*. The salesman has to collect a minimum quota of bonuses in the visited vertices. Thus the salesman needs to figure out which cities to visit to achieve the minimum quota. The goal is to find a minimum cost tour such that the sum of the bonuses collected in the visited vertices is at least

and memory-based principles and assessment of its performance;

This study deals with a novel optimization model that can improve the services provided by on-demand mobility platforms, called Quota Traveling Salesman Problem with Passengers, Incomplete Ride, and Collection Time (QTSP-PIC). In this problem, the salesman is the vehicle driver and can reduce travel costs by sharing expenses with passengers. He must respect the budget limitations and the maximum travel time of every passenger. Each passenger can be transported directly to the desired destination or an alternate destination. Lira et al. [4] suggest pro-environmental or money-saving concerns can induce users of a ride-sharing service to agree to fulfill their needs at an alternate destination.

The QTSP-PIC can model a wide variety of real-world applications. Cases related to sales and tourism are the most pertinent ones. The salesman must choose which cities to visit to reach a minimum sales quota, and the order to visit them to fulfill travel requests. In the tourism case, the salesman is a tourist that chooses the best tourist attractions to visit during a vacation trip and can use a ride-sharing system to reduce travel expenses. In both cases, the driver negotiates discounts with passengers transported to a destination similar to the desired one.

The QTSP-PIC was introduced by Silva et al. [5]. They presented a mathematical formulation and heuristics based on Ant Colony Optimization (ACO) [6]. To support the ant algorithms, they proposed a Ride-Matching Heuristic (RMH) and a local search with multiple neighborhood operators, called Multi-neighborhood Local Search (MnLS). They tested the performances of the ant algorithms on 144 instances up to 500 vertices. One of these algorithms, the Multi-Strategy Ant Colony System (MS-ACS), provided the best results. They concluded that their most promising algorithm could improve with learning techniques to choose the source of information regarding the instance type and the search space.

In this study, a *MAX-MIN* Ant System (*MM*AS) adaptation to the QTSP-PIC, called Multi-Strategy *MAX-MIN* Ant System (MS-*MM*AS), is discussed. *MM*AS improves the design of Ant System [6], the first ACO algorithm, with three important aspects: only the best ants are allowed to add pheromone during the pheromone trail update; use of a mechanism for limiting the strengths of the pheromone trails; and, incorporation of local search algorithms to improve the best solutions. Plenty of recent studies proved good effectiveness of the *MM*AS in correlated problems to QTSP-PIC [7–10]. However, none of these explored the Multi-Strategy (MS) concept.

In the traditional ant algorithms applied to Traveling Salesman Problem (TSP), ants use the arcs' cost as heuristic information [6]. The heuristic information adopted is called visibility. When solving the QTSP-PIC, different types of decisions must be considered: the accomplishment of the minimum quota, management of the ride requests, and minimization of travel costs. The MS idea is to use different mechanisms of visibility for the ants to improve diversification. Every ant decides which strategy to use at random with uniform distribution. The MS proposed in this study extends the original implementation proposed in [5]. MS-*MM*AS also incorporates RMH and MnLS and uses a memory-based technique proposed in [11] to avoid redundant work. In MS-*MM*AS, a hash table stores every solution constructed and used as initial solutions to a local algorithm. When the algorithm constructs a new solution, it starts the local search phase if the new solution is not in the hash table.

The benchmark for the tests consisted of 144 QTSP-PIC instances. It was proposed by Silva et al. [5]. Numerical results confirmed the effectiveness of the MS-*MM*AS by comparing it to other ACO variants proposed in [5].

*Multi-Strategy* MAX-MIN *Ant System for Solving Quota Traveling Salesman… DOI: http://dx.doi.org/10.5772/intechopen.93860*

The main contributions of this chapter are summarized in the following.


The remainder of this chapter is organized as follows. Section 2 presents the QTSP-PIC and its formulation. Section 3 presents the Ant Colony Optimization metaheuristic and the implementation design of the MS-*MM*AS. Section 4 presents experimental results. The performance of the proposed ant-based algorithm is discussed in Section 5. Conclusions and future research directions are outlined in Section 6.
