## 1 Introduction  
In current logistics, supply chain management (SCM) is very important because it ensures that products are moved efficiently and smoothly from their manufacturers to consumers across international boundaries [1,2]. It is therefore essential that these worldwide supply chains are optimized to guarantee both recovery and efficiency in terms of cost and time. In the current condition of Industry 4.0, the dynamic and uncertain supply chain makes the old school of deterministic models limited in handling volatile conditions [3]. These models seldom account for uncertainties like demand changes, disruptions, or variations in lead times, yet all these are critical to the overall performance of the supply chain [4,5]. To overcome the constraints above, recent developments have centered on amalgamating probabilistic and fuzzy decision-making methodologies. Probabilistic models enable deeper insight into uncertainties and variabilities by embracing the chances of different events taking place, while fuzzy logic plays a role in managing imperfect and uncertain input, reflecting real-life situations more appropriately [6]. These approaches represent a significant departure from deterministic models and offer greater robustness and flexibility for supply chain operations. To further enhance decision-making in supply chain management (SCM), it is advisable that uncertainty theory be employed. This principle states that the risks within supply chains should be incorporated into stochastic, fuzzy logic processes to build stronger and more flexible systems [7]. Superior overall performance of the supply chain necessitates this improvement in risk management as well as informed decision-making [8]. As supply chains continue to grow and change, it becomes increasingly important to use optimization models that are capable of managing these complexities [9]. In this changing environment, goal programming (GP) stands out as a powerful multi-objective optimization technique. [PERSON] and [PERSON] [10] first introduced it more than fifty years ago. Its flexibility and effectiveness make it especially valuable within the complex and fluctuating landscape of supply chains. GP enables decision-makers to address multiple, often conflicting objectives simultaneously [10]. To improve optimization models, dependent chance constraints (DCC) are used to capture how different uncertain parameters depend on each other. By considering the joint distribution of uncertain factors, DCC-based models [11] offer holistic risk management, significantly enhancing optimization in supply chains [6]. This integration allows for more accurate and reliable optimization results, reflecting real-world conditions. Metaheuristic algorithms such as the hunger games search algorithm (HGSA) [12] and genetic algorithms (GA) [13] further strengthen optimization. Inspired by natural processes, these algorithms excel in exploring and exploiting multi-dimensional solution spaces, making them ideal for modern supply chains. HGSA mimics foraging behavior among animals, while GA simulates natural selection processes. Both algorithms improve the performance of supply chain optimization by making them more adaptive and efficient ([14; 15]). In this context, the current study has developed a novel optimization model that synergizes goal programming (GP), dependent chance constraints (DCC), and the hunger games search algorithm (HGSA). This model optimizes critical decision variables such as production setups, quantities produced, inventory levels, and backorders to minimize costs and maximize on-time deliveries. The 99-method is used in several numerical analyses and simulations, showing substantial improvements in cost efficiency and delivery performance. This integrated approach, therefore, provides a solid foundation for addressing contemporary supply chain challenges.