# A Swarm Intelligence Solution for the Multi-Vehicle Profitable Pickup and Delivery Problem
## 1 Introduction  
Over the past five years, delivery apps have experienced substantial growth, particularly food delivery apps, where COVID-19 propelled the industry forward a few years as millions of individuals in lockdown ordered food and other essential services online. The grocery delivery platform called Instacart reported that its objectives for 2022 were met during the third week of lockdown. In 2022, China emerged as the largest food delivery market, with a staggering market value of USD 42.5 billion. By 2029, the market size of the food delivery app industry as a whole is anticipated to reach USD 165 billion [1]. Moreover, 85% of Americans place online food orders via food ordering applications. Moreover, 80% of restaurants in the United States provide customers with the convenience of mobile ordering [2]. Similarly, food delivery applications in Saudi Arabia are providing instantaneous services and a streamlined user experience for individuals to enjoy food. As a result, new applications that vie to provide services, convenience, and comfort to over 35 million individuals emerge periodically. Examples of these applications that have become very popular in Saudi Arabia are Mrsool, Uber Eats, Talabat, Hunger station, Ninja, Jaez, To You, The Chifz, Shgardi, and NANA [3]. The escalating demand for food delivery applications has concurrently heightened the strain on delivery enterprises, which requireexpeditious algorithms to coordinate the transportation of goods in order to generate profits and guarantee client contentment. One of the problems that was presented in the literature that mimics the goal of food delivery apps, increasing profits, achieving user satisfaction, and reducing costs is the multi-vehicle profitable pickup and delivery problem (MVPPDP). [PERSON] et al. [4] introduced the MVPPDP, which is a subtype of the selective pickup and delivery problem (SPDP) and a novel variation of the well-known pickup and delivery problem (PDP). Within the constraints of a one-day planning horizon, the MVPPDP seeks to identify the most profitable routes by minimising travel expenses and maximising revenue collection. The MVPPDP is a static problem characterised by a standardised fleet of vehicles, a central depot, a collection of customers, and a predetermined quantity of requests and products. A customer pair, consisting of the pickup customer and the delivery customer, defines each request. By doing so, the products are conveyed from a pickup customer to a delivery customer in order to generate a revenue stream for the operation. MVPPDP real-world applications include mobile applications for food delivery services and delivery companies. The MVPPDP is one of the combinatorial optimisation (CO) problem applications that receives particular attention from researchers in computer science and operations research. Solving a CO problem mainly requires searching to find the optimal or best possible solution from a limited set of feasible candidate solutions. Numerous algorithms can be used to solve CO problems, and these algorithms can be classified into two categories: exact algorithms, which are guaranteed to obtain optimal solutions in an adequate amount of time, and approximate algorithms, usually called heuristics or metaheuristics, which are not guaranteed to deliver an optimal solution but provide a good solution in a reasonable amount of time. The words heuristic and meta belong to the Greek language, where heuristic is from _heuriskein', meaning _to find' or _to discover' new methods and strategies to solve problems, while the prefix meta means _beyond, in an upper level. Metaheuristic methods can be classified based on the number of solutions used at the same time into two categories: single solution-based metaheuristics (S-metaheuristics) and population-based metaheuristics (P-metaheuristics). In S-metaheuristics, the focus is on modifying and improving a single candidate solution. On the other hand, P-metaheuristics preserve and improve several candidate solutions instead of just a single solution.