## 2 Related Works  
The cost-efficient application of an inventory management policy requires setting appropriate parameter values [24]. In the case of the OUT policy, one needs to adjust the reference stock level (RSL) such that excessive operational costs and a loss of customer satisfaction are avoided, which is challenging in networked systems, as the RSLs should be chosen for all the nodes simultaneously. Until now, the policy tuning has been analyzed mostly in the systems with serial and arborescent topologies [25]. For instance, Basten and Houtum [26] studied near-optimal heuristics in serial production systems with backordering. Chang and Lu [27] investigated a similar model, yet with the loss of unsatisfied demand being accounted for. They applied a phase-type approximation with a given cost structure to determine the optimal node RSLs under various fill-rate requirements. Albrecht [28], in turn, presented a decomposition methodology to reconfigure the networked structure into a collection of two-stage subsystems, and in this way, simplified the selection of RSLs. The proposed solution, however, did not take the fixed costs into account. Next, Wang [29] analyzed a two-echelon inventory system that was subjected to a Poisson demand and proposed a heuristic with a reduced computational footprint. However, he considered identical retailers (nodes with the same demand imposed on them). Guerrero et al. [30] extended that model to cover multi-product distribution in the context of the healthcare industry. Then, Eruguz et al. [31] proposed a sequential optimization procedure for five-echelon chains. However, their methodology needed substantial computational effort to tackle the intricacies of the general acyclic multi-echelon systems. Afterward, Avci and Selim [32] developed a numerical framework that enables one to determine safety stock levels in a two-echelon configuration, while allowing for flexibility in choosing the produced goods quantity. However, that framework ensures meeting the holding cost objectives and premium freight performance only for a one-supplier-multiple-retailers model without trans-shipments. Horng and Lin [33] proposed an ordinal optimization-based metaheuristic algorithm for a similar system with multiple suppliers, which displayed improved computational performance. Yet again, the interconnection structure did not allow for the flow of resources between retailers. Afterward, Feng et al. [34] studied a two-stage system with trans-shipments between retailers under uncertain demand, albeit with the investigated topology enforcing a single supplier that replenishes all the retailers at the second stage. Dominguez et al. [35] considered the impact of various structural configurations on the supply chain performance. They revealed the sensitivity of complex distribution networks to rapid and unpredictable demand changes. However, the numerical studies were limited to serial and arborescent configurations only. Ignaciuk and Bartoszewicz [36] investigated optimal control with a quadratic performance index in multi-supplier systems with an arborescent topology. Their control law is an improvement over the OUT policy with respect to demand variance smoothening in the markets that are subjected to significant fluctuations. Recent publications in this category target mostly real-life cases where arborescent models may be applied [37]. However, in the face of globalization, modern logistic systems are organized into mesh-type topologies with arbitrary interconnection architectures. The structural simplifications were found to be ill-suited for these kinds of systems [8], as they lead to undesirable side-effects, e.g., loss of economic gain and customer trust. The policy tuning in more elaborate cases of practical significance, especially in multi-echelon networks, is at an early stage though [38], and requires a detailed study [39].