Maintenance and Two-Way Transshipment Control for Balanced UAV Systems

In many modern engineered systems, the condition of one component affects the role, loading, or operating status of another component, so the system-level response to failure depends on structural relationships as much as on component reliability. Balanced systems represent a clear example of this type of dependency. Their defining requirement is that operating units must remain arranged in symmetric working positions. A failure in one unit therefore does not simply remove that unit from service; it also forces the corresponding unit in the symmetric position to stop operating so that the system remains balanced. This feature gives balanced systems a distinct reliability and maintenance structure. In a conventional multi-component system, corrective maintenance is usually framed around replacing or repairing the failed component. In a balanced system, maintenance planning must account for more than the failed component itself. A failed unit may create a standby unit on the opposite side, and these standby units may later be rearranged into symmetric positions to restore part of the system’s operating capacity. Replacement remains important, but it is no longer the only route to recovery.

The problem becomes broader when spare-parts availability is included. Balanced systems such as unmanned aerial vehicles may operate from distributed bases, while spare parts are supplied through a two-echelon network consisting of a central depot and multiple bases. If a base has insufficient inventory, replacement actions may be delayed or become more costly. Holding too many spare parts, however, increases inventory cost. The practical decision is therefore not only how to maintain the system after shocks occur, but also how to replenish spare parts so that maintenance actions remain feasible without excessive stock accumulation. Traditional maintenance and inventory models are really not designed for this coupled decision environment. Longitudinal transshipment from a depot to a base is commonly considered, but lateral transshipment among bases may also be valuable because nearby bases can provide faster support when local stock becomes low. At the same time, replacement time, rearrangement time, and order completion time may not follow memoryless distributions, which calls for a formulation beyond simpler Markovian assumptions. These features create a methodological gap: balanced-system maintenance, rearrangement, longitudinal replenishment, and lateral replenishment need to be optimized together under random shocks and general action times.

In a recently published research paper in Reliability Engineering & System Safety, Professor Jingjing Wang, Lingyun Luo, and Yuxue Jin from Qingdao University of Technology, working together with Professor Li Yang from Beihang University, addressed this problem through an integrated optimization framework for maintenance and spare-parts transshipment in balanced systems. The new model treats the base state as a joint description of working units, standby units, and spare-parts inventory, which allows maintenance and replenishment decisions to be optimized together. They also developed a semi-Markov decision process formulation and a modified value-iteration algorithm to handle general action times and compute optimal stationary policies. The technically distinct element is that balanced-system rearrangement and two-way spare-parts transshipment are optimized together, allowing maintenance recovery and spare-parts movement to be coordinated within a single long-run cost framework.

The researchers represented the setting through a base-level state model that links working units, standby units, and inventory level. This state definition is important because it ties the physical condition of the balanced systems directly to the spare-parts situation at the base. A maintenance decision can then depend not only on whether a component has failed, but also on whether standby units are available and whether sufficient inventory exists to support replacement. They represented environmental shocks as a homogeneous Poisson process. When a shock causes a unit failure, the corresponding symmetric unit stops working and becomes a cold standby unit. If two standby units become available, a rearrangement action can place them into symmetric positions and return them to operation. If a failed unit is detected at an inspection epoch and spare parts are available, a replacement action can restore the failed unit and its paired standby unit to service.  For inventory control, the authors combined a longitudinal order policy and a lateral order policy. The longitudinal policy follows an (_s1, S) structure, where a base orders from the depot when its inventory drops below the longitudinal order point and replenishes up to a maximum level. The lateral policy follows a (Q, _s2) structure, where a base receives a fixed quantity from other bases when its inventory falls below a lower lateral order point. The condition _s2 < _s1 reflects the operational logic that lateral transshipment is reserved for a more urgent inventory state, while depot replenishment covers the broader replenishment need.

Because the completion times of replacement, rearrangement, and ordering actions do not provide a simple Markov structure, the researchers used a semi-Markov decision process. They derived transition probabilities for cases in which no activity occurs, one action occurs, or a maintenance action and an order action occur together. Costs were also assigned at the state-action level, including inspection cost, holding cost, replacement cost, rearrangement cost, longitudinal order cost, lateral order cost, failure penalty, and revenue associated with dispatching spare parts from the tagged base to another base. The objective was to minimize the long-run average operation and maintenance cost rate while requiring the probability of normal operation to exceed a specified threshold.

To solve the model, the authors developed a modified value-iteration algorithm. The semi-Markov problem was transformed into an equivalent discrete-time decision problem through a data transformation based on expected sojourn times. They also accounted for reducible Markov chains by removing states not connected with the other states, allowing the algorithm to operate on an irreducible chain. This connects the mathematical formulation to a practical computable policy. The numerical example used a UAV setting with two UAVs at each base and six propellers per UAV. The depot supplied eleven bases, and the lateral dispatch probability for a tagged base was set from the number of other bases. Under the specified cost and time parameters, the modified value-iteration algorithm converged within a finite number of iterations.  The team compared between policies which gave them most direct operational finding and found with only longitudinal ordering, increasing the depot order point raised the average cost rate, although it improved normal operation probability. A low order point reduced cost but could fail to meet the operational probability requirement. When lateral transshipment was added, the model identified policies that lowered the average cost rate while maintaining or improving the probability of operation. Afterward, the authors conducted sensitivity which clarified how order costs and order times shift the optimal balance among replenishment frequency, inventory risk, and operating cost.

The findings of Professor Jingjing Wang et al. have direct engineering value for UAV fleets with symmetrically arranged propellers. In such systems, the failure of one unit can force its paired unit to stop operating, so maintenance planning cannot be limited to replacing the visibly failed component. The proposed framework helps engineers decide when failed units should be replaced, when standby units can be rearranged into useful symmetric positions, and when spare parts should be replenished through either depot supply or nearby bases. For UAV fleet operation, the model can support base-level spare-parts planning. A central depot may hold the main inventory, but individual bases still need enough propellers or equivalent components to respond quickly to environmental shocks. This allows a low-stock base to recover spare-part availability without relying only on the longer depot route. The new approach can also be applied to other balanced engineering systems, such as dual-tire assemblies, shock absorber arrangements, balance bikes, or mechanical platforms where paired components must remain operational in symmetric positions. In these cases, the framework offers a way to coordinate maintenance actions with inventory movement rather than treating them as separate management problems. From an operations perspective, the authors’ proposed method is useful for minimizing long-run maintenance and logistics cost while maintaining a required probability of normal operation. It provides engineers and fleet managers with a structured decision tool for selecting inventory thresholds, lateral transfer quantities, and maintenance actions under random shocks and uncertain action times.