# Optimizing Perishable Product Supply Chain Network Using Hybrid Metaheuristic Algorithms
## 1 Introduction
Before delving into the review of previous studies on the LIRP, it is important to first examine the studies that address two key categories of decisions in supply chain management. The location-inventory problem (LIP) involves determining optimal facility locations and corresponding inventory levels within the SCN to meet customer demand while minimizing costs. Researchers have examined various types of this problem to address real-world scenarios in the PPSCN [12; 13; 14; 15; 16]. For instance, [PERSON], [PERSON], [PERSON] and [PERSON] [3] incorporated a LIP into the PPSCN considering fuzzy capacity and carbon emission constraints. They proposed two heuristic approaches, namely the hybrid genetic algorithm (HGA) and the hybrid harmony search (HHS), to solve the model. Numerical experiments demonstrated that HHS obtained higher-quality solutions compared to HGA, although HGA exhibited faster computational performance. The inventory-routing problem (IRP) refers to optimization of inventory and routing plans within a supply chain to improve logistics efficiency and cost effectiveness. When it comes to perishable products, it is crucial to develop optimal joint policies [17; 18; 19; 20]. [PERSON] et al. [21] proposed an IRP model that incorporates transshipment and a single perishable product. They also devised a method based on GA to address this problem. [PERSON] et al. [22] presented a two-echelon IRP for perishable products and utilized an adaptive large neighborhood search metaheuristic combined with a simplified formula to solve the problem. They compared three variations of the heuristic on randomly generated instances. In the location-routing problem (LRP), the objective is to determine the optimal location and capacity of a logistics distribution center (DC). Subsequently, an optimal distribution route and vehicle arrangement are designed to enhance operational efficiency and minimize logistics costs. [PERSON] et al. [23] introduced a three-echelon LRP that considers time windows and time budget restrictions for perishable food products to high-speed railways. A hybrid cross-entropy algorithm was applied to solve a case study conducted in China. In the pharmaceutical industry, [PERSON] and [PERSON] [24] proposed an LRP that incorporates time windows and direct delivery constraints. Having explored the fundamental concepts, we are now ready to delve into the main subject. Several scholars have already conducted research on the logistics LIRP specifically focused on perishable products. [PERSON] et al. [25] conducted an analysis on a two-stage LIRP with time window constraints specifically for food products. They utilized a hybrid heuristic algorithm to solve a mixed-integer programming model. In a similar vein, [PERSON] et al. [26] proposed a bi-objective mixed-integer linear programming (MILP) model to address a perishable pharmaceutical supply chain network problem, aiming to minimize total cost and lost demand amount. [PERSON] et al. [27] presented a mathematical model for the LIRP involving multi-period and multi-perishable products considering stochastic demands from retailers. Their study incorporated suppliers, DCs, and retailers while accounting for factors such as vehicle schedules, fuel consumption, driver cost, and product waste cost. The Lagrange relaxation method and a heuristic algorithm were employed to solve the model. [PERSON] et al. [28] developed a GA and a novel chromosome representation approach to tackle the LIRP for a single perishable product in a single period. [PERSON] et al. [29] proposed a revised solution method for a bi-level LRP in a two-echelon supply chain with perishable products considering uncertain demand and capacity constraints. Their model incorporated scenario-based programming and employed multi-choice goal programming to minimize deviations from ideal solutions. [PERSON] et al. [30] devised a two-phase hybrid mathematical model for the LIRP concerning perishable products. In the first phase, they formulated the LRP considering stochastic demands and travel time.