Existing reliability models for complex systems mostly focus on system configuration, phased operation, and failure modes, but not on the complicated mission environments. This paper contributes by modeling the system engaged in a primary mission encompassing multiple operation phases with different mission requirements, resource constraints, and random shock environments. The system may fail due to external shocks and the depletion of limited resources, incurring phase-dependent damage costs. To mitigate the risk of system failure, the mission may be aborted before its completion when certain deterioration conditions defined by the number of shocks appears by a certain time. The abort policy may greatly impact the mission success probability (MSP) and expected mission losses (EML). The performance choices of the system in different phases affect resource consumption and task completion time, and consequently also impact MSP and EML. We propose a probabilistic model to derive MSP and EML and further optimize the performance and abort policy to minimize the EML. A case study of a system crossing five zones with four available performance levels is provided to demonstrate the model as well as effects of key parameters (cost, shock rate, shock resistance, performance level, and available resource level) on the mission performance and the optimal aborting and performance policy.

Motived by abundant real-world examples in diverse domains such as drone-based target detection, sensor network-based environment monitoring, and cybersecurity defense, this paper pioneers the modeling of a weighted voting system (WVS) in which voting units (VUs) operate under distinct random shock environments and contribute to the system decision with varying weights. To mitigate the risk of VU loss, each VU follows an individual mission abort policy (MAP) determined by its shock exposure and operating time. We jointly model and optimize the MAP and the voting rule (defined by the VU weights and system voting threshold) to balance the risk of VU loss against the need to maintain sufficient active VUs for effective decision-making. A universal generating function-based method is suggested to evaluate the reliability as well as the expected cost of damage (ECD) of the considered WVS. An optimization problem is further formulated and solved to determine the optimal aborting and voting rules, minimizing the ECD. An N-version programming cybersecurity system is analyzed to demonstrate the proposed model. Detailed case studies are conducted to examine the effects of several cost and shock parameters on the ECD and optimization solutions, leading to important managerial insights.

Existing aborting models typically assume "all or nothing", meaning a system component either completes its required operation, contributing fully to the mission requirement, or contributes nothing due to failure or mission abort. Motivated by practical applications like scientific computing and payload delivery, which allow task reduction in the event of deterioration, this paper pioneers the joint modeling and optimization of task reduction and abort policies (TRAP) to minimize expected mission losses (EML). Both policies depend on the number of shocks occurring during the task execution and on the operation time elapsed from the beginning of the mission. Different components may have varying TRAPs. A universal generating function-based approach is proposed for assessing the EML of the considered multi-component system. The genetic algorithm is further implemented to determine the optimal TRAP that minimizes the EML. A delivery mission performed by multiple unmanned aerial vehicles is analyzed under homogeneous and heterogeneous scenarios to demonstrate the proposed model. The case study also investigates influences of key parameters, including mission demand, per unit score deficiency penalty, shock rates, shock resistance parameters, and component loss cost, on the EML and the optimal solutions. The interplay between task reduction and task aborting policies is also illustrated.

Motivated by practical examples in diverse application domains (e.g., power systems, electric and autonomous vehicles, propulsion engines, data centers), this paper presents a novel model of an asymmetric dual-component system subject to random external shocks and secondary shocks due to the cascading effect, causing deterioration and even failures of system components. To combat those shocks and alleviate the risk and cost of failures, the system is subject to periodic inspections and preventive component replacements based on shock occurrence statuses, incurring non-negligible additional cost. To minimize the expected mission cost (EMC), we formulate and solve a new optimization problem that determines the optimal inspection and replacement policy (IRP) defined by the number of inspections and values of three shock-based replacement rule parameters. A probabilistic method based on system state transitions is put forward for deriving the EMC of the considered system, based on which an EMC evaluation algorithm using a forward procedure is developed. The proposed algorithm is verified using Monte Carlo simulations. A case study of a dual battery powered sensor system is conducted to demonstrate the proposed model and influences of several model parameters on the EMC and the optimal IRP solutions, leading to some useful managerial recommendations. Parameters investigated include cost parameters linked to the system downtime, inspections as well as component replacements and failures, shock occurrence and resistance parameters, and mission time.

Shock count is a key parameter used in designing mission abort policies (MAPs) for systems executing their operations under random shock conditions. Existing models mostly assume a perfect mechanism of detecting shocks. In practice, the shock detection system may fail to detect shocks that have occurred (false negative) or flag nonexistent shocks (false positive), both leading to wrong shock count and misleading MAP designs. This article's contribution lies in modeling a single-attempt mission system with a fault-tolerant shock detection system that applies threshold voting among multiple imperfect detectors to contribute to the mission abort decision based on shock count and system operation time. A probabilistic approach is put forward for assessing mission performance of the considered system in the form of task success probability (TSP), survival probability of system (SPS), and expected losses of mission (ELM). An ELM minimization problem is further formulated and solved, which aims to determine the optimal triparametric MAP, achieving a balance between TSP and SPS. We analyze a drone-based surveillance system to showcase the suggested model. We also examine the impact of key parameters (cost, shock occurrence rate and detection probability) on mission performance metrics and on the best-obtained MAPs, leading to important managerial recommendations.

At many instances, mission abort in systems performing important tasks is followed by a rescue procedure that is mostly aimed at survival of costly systems. Existing mission aborting models assume that when more than one component is involved in accomplishing a mission, the rescue procedures for individual components are performed independently. However, systems often have no sufficient resources to perform the rescue procedures for several/all components simultaneously. Then the mission abort for some components can be delayed or cancelled. This paper considers the two-component system in which simultaneous rescue of both components is prohibited. It suggests algorithm for evaluating the expected mission losses for this setting. In addition, it formulates and solves the mission abort policy optimization problem and presents practical examples that show that the operational dependence between components effects the mission success metrics and the corresponding mission abort policy.

Extensive studies have been dedicated to the modeling and optimization of abort policies (AP) with the aim of balancing mission’s success probability (MSP) with potential losses of valuable systems or components. Existing models primarily focus on individual component aborts. This paper introduces a general model that combines individual and common mission aborts, enhancing the balance between MSP and component losses. Under the proposed model, a five-parameter AP policy is formulated and optimized for a homogeneous H-out-of-N mission system working in random shock settings. A new probabilistic modeling method is put forth to assess the MSP, the expected number of failures, the expected number of successful rescues following the abort, as well as the normalized expected mission losses (NEML) and expected profit. Based on the evaluation of these mission metrics, we formulate and solve the optimal AP problem to achieve a minimization of NEML or a maximization of the expected profit. A case study on a cargo delivery mission carried out by multiple drones is presented to showcase the suggested model and analyze the impacts of critical model parameters, yielding valuable managerial insights. The advantage of using the combined individual and common AP over a pure individual AP is also showcased.

Mission abort decisions require balancing mission success against system survival, often under economic constraints that necessitate profit-driven policies. Distinct from previous profit-driven models that either neglect external shocks or are limited to single-component systems, this paper develops a profit driven abort and inspection policy for multi-component systems subject to random shocks. Each component undergoes deterioration and potential failure under an independent shock process. At inspection, both the component state and the cumulative number of shocks are observed, enabling the evaluation of task success and rescue completion probabilities for surviving components using a proposed probabilistic modeling method. These assessments inform an optimal selection of components to continue or abort as well as the inspection time, such that the expected profit of mission (EPM) is maximized. The proposed model is demonstrated and validated through case studies involving a swarm of five unmanned aerial vehicles (UAVs) that carry out a payload delivery mission from different source locations along distinct routes to a common destination. Both homogeneous and heterogeneous UAV configurations are considered. Sensitivity analyses are performed to examine the impacts of mission success reward, mission demand requirement, component capacity, and shock rate on the optimal inspection timing and the corresponding EPM, offering managerial insights into profit-driven abort decision-making.

Existing mission abort policies for systems operating in random environments typically use the number of experienced shocks as a key decision parameter and assume that the shock detection mechanism is perfect. In practice, the shock monitoring system is failure prone, leading to wrong detections including both false negative (real shocks are not detected) and false positive (non-existent shocks are flagged) detections. This work advances the state of the art by modeling mission success probability (MSP) and system survival probability (SSP) of a system subject to imperfect shock detections and mission aborting. A dual-parameter abort policy (AP) is considered, which triggers the mission abort and starts a rescue procedure to survive the system when a predefined number of shock detections take place before a predetermined time. The AP optimization problem is solved to minimize the expected mission losses, balancing MSP and SSP. A case study of an unmanned aerial vehicle performing a surveillance mission in shock environments is provided to demonstrate the proposed model. The influences of shock detection errors and several other model parameters related to shock resistance, shock rate and costs on mission metrics and optimal APs are also investigated, leading to concrete managerial recommendations for controlling failure risks.

Mission abort policy (MAP) has been widely studied in systems subject to random shocks. Most existing models assume either individual shocks degrading a single component or common shocks simultaneously impacting multiple components. A few recent studies address both types of shock processes, but are limited by restrictions on the timing of abort decisions. In this paper, we relax these timing restrictions by allowing aborting decisions to be made at any time during a mission performed by an asynchronous system with two heterogeneous components, offering more responsive decision-making. A new probabilistic modeling procedure is proposed for deriving the mission success probability and the expected cost of component losses, which are further used for calculating the normalized expected damage (NED). The optimal MAP that minimizes NED is then determined. A bi-sensor monitoring system is analyzed to illustrate the proposed model accommodating both individual and common shocks as well as flexible abort decision timing.

Systems operating in random shock environments face a high risk of mission failure that can result in substantial damages and economic losses. To ensure operational safety and cost-effectiveness, it is crucial to design profit-aware mission abort policies that balance operation costs and profits, penalties from mission aborts and system failures, and rewards for successful mission completion. However, few models incorporate such profit-oriented considerations and those that do fail to consider the effects of external shocks on system reliability. This paper contributes by modeling and optimizing a joint expected mission profit (EMP)-based abort and inspection policy for systems that deteriorate and fail due to external shocks. The inspection reveals the number of experienced shocks, based on which the EMP-based decision on whether to continue or abort the mission is made. The EMP evaluation and optimization are investigated for three scenarios (missions without or with partial work profit and missions with imperfect shock detection) and validated using case studies on an unmanned aerial vehicle delivery mission and a drilling mission. Sensitivity analyses are conducted to quantify the effects of cost parameters, shock resistance and detection uncertainty on the optimal policy, providing engineering insights into the design of EMP-maximized inspection and abort policies.

A counterterrorism model is developed where a government and a terrorist allocate resources over two periods. Escalation to period 2 occurs if a threshold for the government’s period-1 damage is exceeded. Without escalation four scenarios exist, including deterrence and nonprovocation. With escalation and unitary contest intensity, both players’ fractions of their resources allocated to period 1 equal the sum of their potential period-1 damages divided by the sum of their potential damages in both periods. As the government’s resource superiority increases, the terrorist allocates all its resources to the period-1 attack, and the government deters escalation. Uniform distributions of the contest intensity and the government’s resource superiority over various intervals are considered. Observing that the terrorist’s utility may be U-shaped in the escalation threshold, the government is enabled to determine both its resource allocation and escalation threshold. The government prefers no threshold when it lacks resources and should always escalate, and when it has abundant resources and can deter. For intermediate resource superiority, the government prefers an intermediate threshold. Six game outcomes are shown where escalation is deterred for two disjoint intervals of the government’s resource superiority.

Mission systems (e.g., drones, aircraft) often operate in random shock environments. Existing reliability studies have mostly assumed either individual or common shock processes, which may deteriorate a single component or multiple components simultaneously. This paper advances the state of the art by modeling a mission system with multiple components that undergo both individual and common shocks, operating in parallel and contributing cumulatively to the mission success. To avoid excessive components losses, a shock-based, interval-dependent abort policy is proposed and designed to minimize the expected mission losses (EML). A new numerical procedure is put forward to assess the mission success probability and the expected number of lost components, which are then used to determine the EML. Based on the EML evaluation and a suggested string solution representation, the genetic algorithm is implemented to solve the EML minimization problems under fixed and dynamic possible aborting times (PAT). A detailed case study of a multi-drone target destruction mission system is conducted to demonstrate the proposed model. The impacts of several model parameters (mission failure penalty, common shock rate, number of system components, and number of PAT) on mission performance metrics and optimization solutions are also investigated.

This paper contributes by modeling a new class of repairable, dynamic m-out-of-n standby systems operating in random shock environments. Operating components are exposed to a common shock process and can fail due to external shocks and/or internal deterioration, causing the failure of the entire mission. Therefore, it is pivotal to implement an operation and maintenance schedule (OMS), according to which any operating component may be preventively replaced by a standby component to undergo perfect maintenance during the mission. Due to heterogeneity of system components, different OMSs incur different expected mission cost (EMC) and mission success probability (MSP). We formulate a new optimization problem to determine the optimal OMS that minimizes the EMC while satisfying a certain level of MSP. The solution methodology encompasses a new recursive procedure to evaluate MSP and the realization of genetic algorithm. A case study of a chemical reactor cooling system is conducted to showcase the proposed model and study the effects of component heterogeneity as well as several key model parameters on the system performance. The mission cost sensitivity analysis is also demonstrated, providing insights on the most cost-effective component performance or shock resistance improvement. The proposed model extends the OMS study of standby systems in literature from non-shock to shock operating environments.

Existing mission aborting models assume that a single-phase rescue procedure (RP) is activated and executed upon the termination of the primary mission to save the asset. In some real-world applications, multiple phases of RP may be engaged. During different RP phases, the system may operate with different performance under different random environments modeled by different shocks arrival processes. This paper pioneers the modeling of such multi-phase RP in the mission aborting systems, where the primary mission and different RP phases may be subject to different aborting policies. A probabilistic approach is put forward for evaluating the mission success probability (MSP) and system survival probability (SSP) of the considered system under any given phase aborting policies, based on which the expected mission losses (EML) is further derived. An optimization problem is then formulated and solved, which finds the optimal aborting policies of all phases, minimizing the EML. An unmanned aerial vehicle (UAV) payload delivery mission system is analyzed to demonstrate the multi-phase RP and the proposed methodology. Impacts of different RP phase sequences, mission failure penalty, payload cost and shock rates on the MSP, SSP, and EML and on the optimal solutions are also investigated through the UAV case study.

A minmax zero-sum game between a government and a terrorist is developed. They allocate their resources over two periods. Escalation to period 2 occurs if a threshold t for the government's period 1 damage is exceeded. If damage caused to the terrorist exceeds a threshold Tb, the terrorist is disabled. If collateral damage caused by the government's actions exceeds a threshold T, the government experiences pressure from the world community, which reduces its utility. The outcomes are classified so that the terrorist is deterred, the terrorist is disabled, the government escalates to period 2, and pressure is imposed on the government due to too much damage caused to the terrorist and its sympathizers. The game is solved with double loop optimization to illustrate seven possible outcomes with real-world interpretations based on combinations of the parameter values. In two of the outcomes, the government does not escalate to period 2, i.e., if the incurred damage is low or the government resourcefully disables the terrorist. A resourceful government can also disable the terrorist over two periods. In the remaining four outcomes, which depend on how resourceful the government and the terrorist are, the period 2 damage to the terrorist may be below or above the threshold where the government experiences pressure, and the damage to the terrorist over both periods may be below or above the threshold where the terrorist is disabled. Finally, comparison is made with the government choosing endogenously the threshold for when to escalate to period 2.

As an effective risk control method, mission aborting has recently received significant research attention. However, majority of the models failed to address the effects of loading on mission work progress and system loss risk. The very few models considering loading assumed single-attempt missions or single operating component. This work contributes by modeling the scheduling, loading and aborting policy (SLAP) for a multi-attempt mission system where multiple components are activated one by one according to a certain schedule (allowing overlapped operations) to accomplish a specified amount of work. The mission success depends on cumulative work accomplished by different components. We formulate and solve a new optimization problem that determines the SLAP to minimize the expected mission losses (EML) incurred from uncompleted mission work and losses of components. A new numerical algorithm is proposed to assess the EML of the considered multi-attempt loading-dependent mission system under any SLAP. Based on the EML evaluation, the genetic algorithm is implemented to solve the proposed optimization problem. A case study of a fleet of aerial vehicles performing a delivery mission is provided to showcase the proposed model and explore the impacts of several key model parameters on the EML and optimal SLAP solutions.

Motivated by practical applications like data storage, product defect detection, medical imagining, and sensing, this paper puts forward a new inspected standby system model where only one element can be online operating and only one element can stay in the standby mode at any time. Both operating and standby elements are exposed to random common shocks, causing their deterioration and even failures. The operating element may also be deteriorated by random operational shocks. The system undergoes periodic inspections to determine the refill of the operating or standby element. A new optimization problem is formulated and solved to determine the inspection and standby element addition policy with the objective to minimize the expected mission cost (EMC) attributed to factors including system downtime, number of inspections, element modes and failures, element activation and mode transitions. A new and efficient system state transition-based numerical algorithm is proposed to evaluate the EMC. A case study of a standby sensor system is provided to demonstrate the proposed model and impacts of several cost parameters as well as shock rates on the EMC and the optimal inspection and standby element addition policy, leading to insightful managerial guidelines for the system design and operation.