# CaAIS: Cellular Automata-Based Artificial Immune System for Dynamic Environments
## 1 Introduction  
Real-world optimization problems often exhibit a dynamic nature, which leads to their modeling as Dynamic Optimization Problems (DOPs). In such problems, a model's parameters change over time due to the changing environment. Thus, finding an optimal solution is challenging because the objective function and constraints vary with time. As a result, although numerous successful optimization algorithms have been developed for static optimization problems, traditional optimization algorithms could not be effective at reaching an appropriate solution in such scenarios as they do not account for changes in real-time data. Thus, researchers tried to develop suitable algorithms to adapt to changes in dynamic environments. Moreover, designing suitable optimization algorithms for solving real-world applications is not easy due to these environments' limitations and constraints. To name a few applications in dynamic environments, one can mention some examples of scheduling problems. These problems are in such a way that stochastic jobs may be inserted/deleted over time. Another example is routing problems, in which routers may fail or change status (from on to off or vice versa) in the whole routing network. Time-based pricing [1] is a common problem in financial planning. In this problem, customers are divided into multiple groups based on their demand curves, and differentprices are charged to each group at different times. This approach allows businesses to optimize their revenue by charging higher prices when demand is strong and lower prices when demand is weak. In channel assignment and multicast routing in multi-channel wireless mesh networks [2], these networks require dynamically efficient multicast routing protocols to ensure data delivery to multiple receivers simultaneously while minimizing network congestion and delay. Dynamic optimization techniques can be applied to address dynamic multicast problems in mobile ad hoc networks (MANETs) [3]. These networks are characterized by their dynamic topology and mobility, which makes multicast routing protocol optimization challenging. In dynamic multicast problems, the network topology is changed frequently due to node mobility or link failures. This requires an adaptive routing protocol to ensure reliable data delivery. Dynamic optimization techniques can improve multicast routing protocols' efficiency and adaptability in MANETs. Dynamic optimization techniques can be applied to dynamic vehicle routing problems (VRPs) [4] to improve routing solutions' efficiency and adaptability. In dynamic VRPs, the number and routes of vehicles change dynamically over time due to various factors such as traffic conditions, weather, and customer demands. By using dynamic optimization techniques, it is possible to achieve more efficient and cost-effective routing solutions, reduce delivery times, and improve overall customer satisfaction. Dynamic job shop scheduling [5] is a scheduling problem in which jobs are processed on different machines in a factory. The goal is to determine the optimal sequence of jobs to be processed on each machine. This is carried out while considering some factors such as machine availability, job release times, and processing times. These problems are particularly challenging because the optimal schedule may change with time due to job requirements or machine availability changes. As a result, these problems require complex algorithms that adapt to changing conditions in real time. For more applications, it can be addressed in aerospace design [6], car distribution systems [7], object detection [8] and pollution control [7], electric vehicle dispatch optimization [9], cold chain logistics scheduling [10], and railway junction rescheduling [11]. The purpose of optimization in dynamic problems has shifted from simply identifying the stationary optimal solution(s) to precisely monitoring the trajectories of the optimal solution(s) over time [12; 13; 14].