Exact Solution Method Research Articles

Exact solution methods for the Resource Constrained Project Scheduling Problem with a flexible Project Structure

Exact solution methods for the Resource Constrained Project Scheduling Problem with a flexible Project Structure

Dynamic analysis of high-rise residential structures through the cone method

In this article, to analyze the behavior of surface foundations based on reinforced soil, a simple physical method based on material resistance called (cone method) has been used. Which is used as an alternative to the exact solution methods that are based on the three-dimensional elastodynamic theory. The purpose of this research is to introduce a simple physical method and analytical tool to analyze the surface foundation based on geocell-reinforced soil. With the cone method, it is possible to examine the effects of layered soil. Obtaining the dynamic stiffness of the foundation based on reinforced soil using the cone method for the vertical degree of freedom and also investigating the effect of various parameters related to geocell reinforcement on the dynamic stiffness of the surface foundation are the goals of this research. In general, the longer the geocell cushion has, the smaller the holes, and the higher the modulus of elasticity, the better the results. By increasing the number of geocell layers, the amount of spring stiffness coefficient and damping coefficient and as a result the dynamic stiffness coefficient increases. On the other hand, the closer the geocell is placed to the soil surface, the higher the stiffness of the spring and the lower the damping coefficient. In general, among these two parameters related to dynamic stiffness, the role of the damping coefficient has been more prominent and decisive in the investigation of foundations based on reinforced soil. One of the most obvious characteristics of geosynthetic materials in the soil, regardless of the greater hardness it gives to the soil, is the damping percentage of its constituent materials.

OPTIMIZATION OF LOADING AND UNLOADING WORKS AND PLACEMENT OF TARED AND PIECE CARGOES IN VEHICLES AND WAREHOUSES USING THE METHODS OF ERGONOMICS AND THEORY OF SYSTEMS

Purpose. The tasks of optimizing the placement of tared and piece cargoes (or the task of loading) in a limited volume (capacity) are typical for cases of loading with tared and piece cargoes (mainly large-sized) of transport containers, car bodies, ship holds in the merchant fleet, transport aircraft, railway wagons, areas of the warehouse complex. At the same time, the safe and efficient performance of loading and unloading operations with tared and piece cargoes, including large-sized ones, in a limited space is directly related to taking into account the requirements and using ergonomic methods. The purpose of this paper is to develop a mathematical model for the optimization of the placement of tared and piece cargoes (or about loading) in a limited volume (capacity), which is used to select a subset of cargo units with the maximum relative utility, while observing the limit on the total volume (or mass) of cargo. Methods. The analysis of literary sources made it possible to establish that for modeling and solving the problem of optimizing the placement of cargo (or loading) in a limited volume (capacity), it is rational to use one of the exact solution methods (dynamic programming method), the number of stages of which is the number of types of cargo. The work uses the method of dynamic programming, methods of theory of systems, and linear optimization. Results. The article presents a mathematical model of the problem of optimal placement of tared and piece cargoes (or about loading) in a limited volume, and for the case of placing weapons, ammunition, weapons, it is proposed to maximize not the function of the total cost (value) of the cargo, but the function that would take into account the relative usefulness of each cargo. The relative usefulness of each cargo is proposed to be determined by expert methods of systems theory, for example, the Delphi method, which is well formalized and suitable for computer processing of results. Scientific novelty. The scientific novelty consists in the development of a mathematical model that allows solving the problem of optimal placement of container loads (or loading) in a limited volume, and the target function of the optimization model is proposed to be the total relative usefulness of the cargo. Practical significance. Defined ergonomic requirements for cargo operations with tared and piece cargoes in a limited space (vehicles, warehouses). A method of numerical calculation based on the model is proposed (the case of loading a car body with tared and piece cargoes). In further research, the obtained results will be used for computer simulation according to the developed model using the Solver tool of Microsoft Excel.

Modelling and Analysis of Evacuated Tube Solar Collector Working with Hybrid Nanofluid

This research study is concerned with developing a mathematical model for an Evacuated Tube Solar Collector (ETSC) that utilizes a Hybrid Nanofluid flow in a manifold. The model consists of a set of the partial differential equations that, using the dimensional analysis approach, are reduced to a dimensionless system. The exact solution method, assisted by the Mathematica program, is used to solve the system. The study examines the impact of certain factors on flow and heat transmission, including horizontal velocity, temperature variation, heat source, magnetic field, Prandtl Number, and Eckert Number, and provides a detailed analysis of the results. The key findings of the analysis revealed that an increase in hybrid nanoparticle volume effects in a reduce in velocity and heat transfer coefficient. Additionally, elevated heat source and Prandtl number correspond to an increased heat transfer coefficient.

Machine Learning based Algorithm Selection and Genetic Algorithms for serial-batch scheduling

Whenever combinatorial optimization problems cannot be solved by exact solution methods in reasonable time, tailor-made algorithms (heuristics, meta-heuristics) are developed. Often, these heuristics exploit structural properties and perform well on selected subsets of the problem space. For example, this is how the two best-known construction heuristics solve the scheduling problem investigated in this study (i.e., the scheduling of parallel serial-batch processing machines with incompatible job families, restricted batch capacities, arbitrary batch capacity demands, and sequence-dependent setup times). However, when the properties change, the performance of one algorithm might decrease, and another algorithm might have been the better choice. To resolve this issue, we propose using Machine Learning to exploit the strengths of different algorithms and to select the probably best-performing algorithm for each problem instance individually. To that, we investigate a variety of methods from the “learning-to-rank” literature and propose several adaptations. Furthermore, because there is no algorithm for the considered scheduling problem that is capable to explore the entire solution space, we developed two Genetic Algorithms for the improvement of initial solutions computed by the selected algorithms. Here, we put special emphasis on ensuring that the solution representation (encoding) reflects the entire solution space and that the operators (e.g., for recombination and mutation) are appropriate to explore and exploit this space completely. Our computational experiments show an average increase of 39.19% in solution quality.

Capacitated Mobile Facility Location Problem with Mobile Demand: Efficient Relief Aid Provision to En Route Refugees

As a humanity crisis, the tragedy of forced displacement entails relief aid distribution efforts among en route refugees to alleviate their migration hardships. This study aims to assist humanitarian organizations in cost-efficiently optimizing the logistics of capacitated mobile facilities utilized to deliver relief aid to transiting refugees in a multi-period setting. The problem is referred to as the Capacitated Mobile Facility Location Problem with Mobile Demands (CMFLP-MD). In CMFLP-MD, refugee groups follow specific paths, and meanwhile, they receive relief aid at least once every fixed number of consecutive periods, maintaining continuity of service. To this end, the overall costs associated with capacitated mobile facilities, including fixed, service provision, and relocation costs, are minimized. We formulate a mixed integer linear programming (MILP) model and propose two solution methods to solve this complex problem: an accelerated Benders decomposition approach as an exact solution method and a matheuristic algorithm that relies on an enhanced fix-and-optimize agenda. We evaluate our methodologies by designing realistic instances based on the Honduras migration crisis that commenced in 2018. Our numerical results reveal that the accelerated Benders decomposition excels MILP with a 46% run time improvement on average while acquiring solutions at least as good as the MILP across all instances. Moreover, our matheuristic acquires high-quality solutions with a 2.4% average gap compared to best-incumbents rapidly. An in-depth exploration of the solution properties underscores the robustness of our relief distribution plans under varying migration circumstances. Across several metrics, our sensitivity analyses also highlight the managerial advantages of implementing CMFLP-MD solutions.

Neural network assisted branch and bound algorithm for dynamic berth allocation problems

One of the key challenges in maritime operations at container terminals is the need to improve or optimize berth operation schedules, thus allowing terminal operators to maximize the efficiency of quay usage. Given a set of vessels and a set of berths, the goal of the dynamic berth allocation problem is to determine the allocation of each vessel to a berth and the berthing time that minimizes the total service time. This problem can be solved using exact solution methods such as branch and bound (BB) algorithms or heuristic methods, however, exact methods do not scale to large-scale terminal operations. To this end, this paper proposes a BB algorithm in which branching decisions are made with a deep neural network. The proposed exact algorithm utilizes the search order of nodes based on the output of the neural network, with the goal of speeding up the search. Three types of solution representations are compared, along with machine learning models are created for each of them. Computational results confirm the effectiveness of the proposed method, which leads to computation times that are on average around half of those without the neural network.

Optimizing multiple equipment scheduling for U‐shaped automated container terminals considering loading and unloading operations

AbstractU‐shaped automated container terminals (ACTs) represent a strategic design in port infrastructure that facilitates simultaneous loading and unloading operations. This paper addresses the challenges of scheduling multiple types of equipment, such as dual trolley quay cranes (DTQCs), automated guided vehicles (AGVs), double cantilever rail cranes (DCRCs), and external trucks (ETs) in U‐shaped ACTs. This paper proposes a mixed integer linear programming model for optimizing the multiple equipment scheduling, aiming to minimize container completion time and AGV waiting time simultaneously. This paper customizes a hybrid genetic‐cuckoo optimization algorithm (HGCOA) with double‐point crossover and Lévy flight Cuckoo search strategies. Extensive numerical results show that the proposed HGCOA outperforms the benchmark genetic algorithms in terms of solution quality and computational time while significantly improving efficiency without substantial sacrifices in solution quality compared with the exact solution method. Overall, this study presents a promising solution for enhancing coordination and operation efficiency in U‐shaped ACTs.

Efficient decomposition shooting method for tackling two-point boundary value models

AbstractThis research aims to present an efficient numerical method to find approximate numerical solutions for ordinary differential equations (ODEs), specifically, the class of two-point boundary value models. This approach is predicated on coupling the shooting method and the decomposition method, which the method is named as Efficient Decomposition Shooting Method (EDSM). Thus, we provide the complete outline of EDSM, and thereafter, apply it to some test models, comprising linear and nonlinear models. Further, the results are then validated by comparing them to the exact solution and other computational methods. Thus, the proposed method performs excellently over its peers, and is found to converge rapidly to the available exact solution with a good precision.

New exact optical soliton solutions of the derivative nonlinear Schrödinger equation family

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton.

Accelerated Benders decomposition and local branching for dynamic maximum covering location problems

The maximum covering location problem (MCLP) is a key problem in facility location, with many applications and variants. One such variant is the dynamic (or multi-period) MCLP, which considers the installation of facilities across multiple time periods. To the best of our knowledge, no exact solution method has been proposed to tackle large-scale instances of this problem. To that end, in this work, we expand upon the current state-of-the-art branch-and-Benders-cut solution method in the static case, by exploring several acceleration techniques. Additionally, we propose a specialised local branching scheme which exploits the separability of the problem by time period. This scheme uses a novel distance metric in its definition of subproblems and features a new method for efficient and exact solving of the subproblems. These methods are then compared through extensive computational experiments, highlighting the strengths of the proposed methodologies.

Benders decomposition algorithms for minimizing the spread of harmful contagions in networks

The COVID-19 pandemic has been a recent example for the spread of a harmful contagion in large populations. Moreover, the spread of harmful contagions is not only restricted to an infectious disease, but is also relevant to computer viruses and malware in computer networks. Furthermore, the spread of fake news and propaganda in online social networks is also of major concern. In this study, we introduce the measure-based spread minimization problem (MBSMP), which can help policy makers in minimizing the spread of harmful contagions in large networks. We develop exact solution methods based on branch-and-Benders-cut algorithms that make use of the application of Benders decomposition method to two different mixed-integer programming formulations of the MBSMP: an arc-based formulation and a path-based formulation. We show that for both formulations the Benders optimality cuts can be generated using a combinatorial procedure rather than solving the dual subproblems using linear programming. Additional improvements such as using scenario-dependent extended seed sets, initial cuts, and a starting heuristic are also incorporated into our branch-and-Benders-cut algorithms. We investigate the contribution of various components of the solution algorithms to the performance on the basis of computational results obtained on a set of instances derived from existing ones in the literature.

A Review of Robust Machine Scheduling

Robust optimization (RO) has been recognized as an effective means to deal with unanticipated events in highly uncertain and risky environments. This paper systematically reviews two types of emerging RO machine scheduling approaches—robust machine scheduling (R-MS) and distributionally R-MS (DR-MS) methods—which usually offer tractable formulations and analytical results for machine scheduling problems under uncertainty. First, after highlighting the advantages of RO methods over the stochastic approach in terms of tractability and robustness, we use the bibliometric method to analyze the literature related to R-MS/DR-MS problems and classify them from the following aspects: (1) uncertain factors, (2) uncertainty descriptions, (3) robustness criteria, (4) machine environments and (5) solution methods. Second, we discuss the uncertainty descriptions, and the robust feasibility and robust optimality criteria. We further provide a state-of-the-art review of R-MS/DR-MS models in different machine environments and discuss the performance of the R-MS/DR-MS models. Third, we review and discuss the existing exact, approximation, online, and heuristic solution methods for solving R-MS/DR-MS models. Finally, we present future research opportunities in two promising areas: green machine scheduling problems and machine learning-enabled algorithms. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Machine scheduling plays an essential role in industrial and service systems, such as manufacturing, power generation, transportation and medical systems. However, in practice, scheduling systems usually operate in highly uncertain environments due to noisy measurements, prediction errors, and implementation deviations. To ensure robust feasibility and robust optimality, robust machine scheduling (R-MS) and distributionally R-MS (DR-MS) approaches have been recently proposed to hedge against the uncertainties related to processing time, release time, due date, machine breakdown, etc. This paper provides a comprehensive review of the R-MS/DR-MS models and algorithms in different machine environments from the aspects of uncertainty descriptions, robustness criteria and solution methods. This paper further highlights the challenges of R-MS problems and provides promising and valuable research opportunities in terms of problem formulations and algorithm designs.

A Two-Objective Model for the Multilevel Supply Chain of Blood Products with the Approach of Reducing the Rate of Contagion under the (COVID-19) Epidemic Outbreak Conditions

The conditions of the coronavirus epidemic have put much pressure on the healthcare system. This disease has hurt the blood supply through the reduction of blood donation and the reduction of access to suitable collection facilities due to dysfunction. Considering the importance of the subject, the purpose of this paper is to design a two-level supply chain network for blood products with the approach of reducing costs and the rate of contagion under the conditions of epidemic outbreaks (COVID-19). After examining the solution methods for multilevel supply chain networks of blood products under the conditions of the spread of the COVID-19 virus, three exact solution methods, including LP-metric, an improved version of the augmented ε-constraint (AUGMECON2), and an improved weighted Chebyshev, are proposed. They are used to solve the model in small dimensions. In order to compare the methods in the obtained solutions, several numerical examples of different sizes are generated and solved. Then, using the statistical assumption test, the obtained results are compared in all numerical examples by Tukey’s technique. Also, the TOPSIS is applied to select the best method. Finally, in order to investigate the reaction of the objectives to the changes in the contagion probability parameter, a sensitivity analysis has been performed. The results emphasize that improving the performance of the blood supply chain (BSC) can lead to a reduction in BSC costs and improved service to patients. Also, the adaptation of different components of the BSC and regular coordination between them play an efficient role in controlling and improving this disease and reducing the costs of the BSC. Also, receiving the plasma product of recovered people from type (II) donors can play a vital role in reducing the percentage of disease transmission.

A branch, bound and remember algorithm for maximizing the production rate in the simple assembly line balancing problem

The simple assembly line balancing problem is a classical optimization problem. It aims at efficiently assigning elementary operations to the workstations of a paced assembly line to optimize some efficiency measure. In this paper, we deal with the type-2 version of the problem in which the efficiency measure is the maximization of throughput with a given number of workstations. This problem has been scarcely studied compared to its type-1 counterpart, in which the throughput is fixed and the objective is to minimize the number of workstations. This has led to the development of indirect methods using type-1 exact solution methods to tackle the type-2 case.In this paper, we provide lower bounds, logical tests, and dominance rules for the type-2 problem and adapt the state-of-the-art method for the type-1 problem to develop a direct approach for solving the type-2 problem with a direct approach. Computational results show the effectiveness of the direct method, which can outperform recent type-1-based indirect approaches. The proposed method is able to close all instances from the classical instance set and solves to optimality over 90% of the large (100 tasks) and 76.5% of the very large (1000 tasks) instances derived from the current reference set for its type-1 counterpart. The average gaps between the upper and lower bounds on the optimal cycle time are also tight and are less than 0.13% (0.4%) for large (very large) instances. Finally, our results show that direct and indirect methods are comparable in terms of solution quality, but the direct method outperforms its indirect counterpart in terms of running time and optimality gaps. Moreover, we identify that instances averaging two to three tasks per workstation and featuring large variability between task times pose significant challenges to solve to optimality.

Method of exact solutions code verification of a superelastic constitutive model in a commercial finite element solver

The superelastic constitutive model implemented in the commercial finite element code ABAQUS is verified using the method of exact solutions (MES). An analytical solution for uniaxial strain is first developed under monotonic loading. Numerical simulations are then performed, and simulation predictions are compared to the exact analytical solutions. Results reveal the superelasticity model agrees with the analytical solution to within one ten-thousandth of a percent (0.0001%) or less for stress and strain quantities of interest when using displacement-driven boundary conditions. Full derivation of the analytical solution is provided in an Appendix, and simulation input files and post-processing scripts are provided as supplemental material.

Incorporating Holding Costs in Continuous-Time Service Network Design: New Model, Relaxation, and Exact Algorithm

The continuous-time service network design problem (CTSNDP) occurs widely in practice. It aims to minimize the total operational cost by optimizing the schedules of transportation services and the routes of shipments for dispatching, which can occur at any time point along a continuous planning horizon. In order to be cost-effective, shipments often wait to be consolidated, which incurs a holding cost. Despite its importance, the holding cost has not been taken into account in existing exact solution methods for the CTSNDP because introducing it significantly complicates the problem and makes solution development very challenging. To tackle this challenge, we develop a new dynamic discretization discovery algorithm, which can solve the CTSNDP with holding cost to exactly optimum. The algorithm is based on a novel relaxation model and several new optimization techniques. Results from extensive computational experiments validate the efficiency and effectiveness of the new algorithm and also demonstrate the benefits that can be gained by taking into account holding costs in solving the CTSNDP. In particular, we show that the significance of the benefits depends on the connectivity of the underlying physical network and the flexibility of the shipments’ time requirements. Funding: This work was partially supported by National Natural Science Foundation of China [Grant 71831008] and the Hong Kong Polytechnic University [Project P0043872]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/trsc.2022.0104 .

New features for customer classification in the Flying Sidekick Traveling Salesman Problem

The increasing interest in the deployment of truck and drone delivery systems leads to the definition of new and complex vehicle routing problems. In this context, the flying sidekick traveling salesman problem (FS-TSP) is the first truck-and-drone routing problem defined in the literature. Several variants appeared in the last years differing in the operating conditions and the structure of the hybrid truck-and-drone delivery system. Exact and heuristic solution methods have been proposed in the literature to solve these problems effectively. However, the exact solution methods can generally solve only small-size instances due to the complexity of these problems. On the other hand, heuristic solution methods are able to find feasible solutions with an acceptable computational burden but without any guarantee of the quality of the solution. This work aims to investigate the possibility of using data science and machine learning techniques to reduce the complexity of solving an FS-TSP instance. The idea is to determine a good/optimal customer-to-vehicle assignment apriori to reduce the number of decisions involved in the FS-TSP solution. The assignment is determined through the classification of customers based on a subset of features specifically defined for the FS-TSP. This information can be exploited by existing solution approaches for the FS-TSP to improve their performance. An extensive computational campaign on benchmark instances is carried out with a twofold objective. On the one hand, we aim to evaluate the impact of the features on customer classification. On the other hand, we intend to show the relationship between classification and combinatorial optimization results by observing the effect of the customer classification on the FS-TSP solution.

Properties of ground state and quench dynamics of one-dimensional contact repulsive single-spin flipped Fermi gases

Based on the exact solution method, the ground state and quench dynamics properties of one-dimensional single-spin flipped Fermi gas with repulsion interaction are studied. With the Bethe wave function, the single-body correlation function and two-body correlation function of the ground state and those between different eigen-states can be reduced into a summation of simple functions, thereby greatly reducing the computational difficulty. For the system in the ground state, the single-body correlation functions and two-body correlation functions as well as momentum distributions for spin-up particles are investigated in real space with different interaction strengths. As the interaction strength increases, the number of nodes in the single-body correlation function remains unchanged, while the amplitude of oscillation decreases. Meanwhile, the number of peaks in the two-body correlation function increases by one due to interaction, indicating that the spin-down particle behaves as a spin-up particle. The momentum distribution becomes more smooth around Fermi surface with the interaction strength increasing. The interaction quench dynamics is investigated. The system is prepared in the ground state of ideal Fermi gas, and then the interaction strength is quenched to a finite positive value. The system evolves under time-dependent Schrödinger equation. The overlap between the initial state and eigen-state of post-quench interaction strength is expressed in the form of continued multiplication. The square of the modulus of this overlap, which represents the occupation probability, is calculated. We find that the occupation probabilities of the ground state and doubly degenerated excited state always have the first and the second largest value for an arbitrary interaction strength, respectively, which means that the difference in eigenenergy between these two states gives the primary period of oscillation. For relatively large particle number (&lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ N\geqslant10$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;), the primary period always does not change under different interaction strengths.It is found that in the case of interaction quenching, the momentum distribution and the correlation function show periodic oscillations. When the interaction strength is adjusted to a relatively small value, the oscillation periodicity is well-defined and the oscillation amplitude is small. The system can be approximated by a two-level model. When the interaction strength increases to a very large value, the oscillation periodicity worsens and the amplitude increases, but a primary period remains unchanged. Although the overall deviation is far from the initial state, it is very close to the initial state at time &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ t=mL^2/(2\pi\hbar)$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;. This is because the difference between most energy eigenvalues is almost an integral multiple of energy unit &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ 2\times\left(2\pi/L\right)^{2}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="2-20231425_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;.

Optimal Selective Maintenance for Complex Systems Under Stochastic Maintenance and Break Durations

The selective maintenance problem (SMP) arises in many settings where systems perform sequences of missions separated by maintenance breaks. The SMP aims to identify the optimal subset of maintenance actions that will maximize the system reliability for the next mission. This paper addresses the SMP when the durations of the break and maintenance actions are stochastic. An exact solution method is proposed which uses the saddlepoint approximation in the computation of the system reliability. Several experiments are used to demonstrate the effectiveness of the proposed solution method for large and complex systems.