# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

All Issues

## Journal of Industrial and Management Optimization

January 2021 , Volume 17 , Issue 1

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2021, 17(1): 1-28 doi: 10.3934/jimo.2019096 +[Abstract](3145) +[HTML](1006) +[PDF](528.59KB)
Abstract:

The internet of things (IoT) is an emerging archetype of technology for the guaranteed quality of services (QoS). The availability of the uninterrupted power supply (UPS) is one of the most challenging criteria in the successful implementation of the service system of IoT. In this paper, we consider a fault-tolerant power generation system of finite operating machines along with warm standby machine provisioning. The time-to-failure for each of the operating and standby machines are assumed to be exponentially distributed. The time-to-repair by the single service facility for the failed machine follows the arbitrary distribution. For modeling purpose, we have also incorporated realistic machining behaviors like imperfect coverage of the failure of machines, switching failure of standby machine, reboot delay, switch over delay, etc. For the evaluation of the explicit expression for steady-state probabilities of the system, the only required input is the Laplace-Stieltjes transform (LST) of the repair time distribution. The step-wise recursive procedure, illustrative examples, and numerical results have been presented for the following different type of repair time distribution: exponential (\begin{document}$M$\end{document}), \begin{document}$n$\end{document}-stage Erlang (\begin{document}$Er_{n}$\end{document}), deterministic (\begin{document}$D$\end{document}), uniform (\begin{document}$U(a, b)$\end{document}), \begin{document}$n$\end{document}-stage generalized Erlang (\begin{document}$GE_n$\end{document}) and hyperexponential (\begin{document}$HE_n$\end{document}). Concluding remarks and future scopes have also been included.

2021, 17(1): 29-50 doi: 10.3934/jimo.2019097 +[Abstract](2271) +[HTML](997) +[PDF](563.88KB)
Abstract:

In this paper, we analyse the Perron vector of an irreducible nonnegative tensor, and present some lower and upper bounds for the ratio of the smallest and largest entries of a Perron vector based on some new techniques, which always improve the existing ones. Applying these new ratio results, we first refine two-sided bounds for the spectral radius of an irreducible nonnegative tensor. In particular, for the matrix case, the new bounds also improve the corresponding ones. Second, we provide a new Ky Fan type theorem, which improves the existing one. Third, we refine the perturbation bound for the spectral radii of nonnegative tensors, from which one may derive a comparison theorem for spectral radii of nonnegative tensors. Numerical examples are given to show the efficiency of the theoretical results.

2021, 17(1): 51-66 doi: 10.3934/jimo.2019098 +[Abstract](2075) +[HTML](887) +[PDF](642.21KB)
Abstract:

This paper proposes an optimization approach to find a set of orthogonal intrinsic mode functions (IMFs). In particular, an optimization problem is formulated in such a way that the total energy of the difference between the original IMFs and the corresponding obtained IMFs is minimized subject to both the orthogonal condition and the IMF conditions. This formulated optimization problem consists of an exclusive or constraint. This exclusive or constraint is further reformulated to an inequality constraint. Using the Lagrange multiplier approach, it is required to solve a linear matrix equation, a quadratic matrix equation and a highly nonlinear matrix equation only dependent on the orthogonal IMFs as well as a nonlinear matrix equation dependent on both the orthogonal IMFs and the Lagrange multipliers. To solve these matrix equations, the first three equations are considered. First, a new optimization problem is formulated in such a way that the error energy of the highly nonlinear matrix equation is minimized subject to the linear matrix equation and the quadratic matrix equation. By finding the nearly global optimal solution of this newly formulated optimization problem and checking whether the objective functional value evaluated at the obtained solution is close to zero or not, the orthogonal IMFs are found. Finally, by substituting the obtained orthogonal IMFs to the last matrix equation, this last matrix equation reduced to a linear matrix equation which is only dependent on the Lagrange multipliers. Therefore, the Lagrange multipliers can be found. Consequently, the solution of the original optimization problem is found. By repeating these procedures with different initial conditions, a nearly global optimal solution is obtained.

2021, 17(1): 67-79 doi: 10.3934/jimo.2019099 +[Abstract](3017) +[HTML](1002) +[PDF](713.12KB)
Abstract:

In this paper, we develop two algorithms for stochastic model predictive control (SMPC) problems with discrete linear systems. Participially, chance constraints on the state and control are considered. Different from the state-of-the-art robust model predictive control (RMPC) algorithm, the proposed is less conservative. Meanwhile, the proposed algorithms do not assume the full knowledge of the disturbance distribution. It only requires the mean and variance of the disturbance. Rigorous computational analysis is carried out for the proposed algorithms. Numerical results are provided to demonstrate the effectiveness and the superior of the proposed SMPC algorithms.

2021, 17(1): 81-99 doi: 10.3934/jimo.2019100 +[Abstract](2268) +[HTML](868) +[PDF](509.6KB)
Abstract:

We study the distributionally robust stochastic optimization problem within a general framework of risk measures, in which the ambiguity set is described by a spectrum of practically used probability distribution constraints such as bounds on mean-deviation and entropic value-at-risk. We show that a subgradient of the objective function can be obtained by solving a finite-dimensional optimization problem, which facilitates subgradient-type algorithms for solving the robust stochastic optimization problem. We develop an algorithm for two-stage robust stochastic programming with conditional value at risk measure. A numerical example is presented to show the effectiveness of the proposed method.

2021, 17(1): 101-116 doi: 10.3934/jimo.2019101 +[Abstract](2319) +[HTML](723) +[PDF](581.31KB)
Abstract:

Iterative methods for nonlinear monotone equations do not require the differentiability assumption on the residual function. This special property of the methods makes them suitable for solving large-scale nonsmooth monotone equations. In this work, we present a diagonal Polak-Ribi\begin{document}$\grave{e}$\end{document}re-Polyak (PRP) conjugate gradient-type method for solving large-scale nonlinear monotone equations with convex constraints. The search direction is a combine form of a multivariate (diagonal) spectral method and a modified PRP conjugate gradient method. Proper safeguards are devised to ensure positive definiteness of the diagonal matrix associated with the search direction. Based on Lipschitz continuity and monotonicity assumptions the method is shown to be globally convergent. Numerical results are presented by means of comparative experiments with recently proposed multivariate spectral Dai-Yuan-type (J. Ind. Manag. Optim. 13 (2017) 283-295) and Wei-Yao-Liu-type (Int. J. Comput. Math. 92 (2015) 2261-2272) conjugate gradient methods.

2021, 17(1): 117-131 doi: 10.3934/jimo.2019102 +[Abstract](2651) +[HTML](785) +[PDF](370.59KB)
Abstract:

In this article, an exact method is proposed to optimize two preference functions over the efficient set of a multiobjective integer linear program (MOILP). This kind of problems arises whenever two associated decision-makers have to optimize their respective preference functions over many efficient solutions. For this purpose, we develop a branch-and-cut algorithm based on linear programming, for finding efficient solutions in terms of both preference functions and MOILP problem, without explicitly enumerating all efficient solutions of MOILP problem. The branch and bound process, strengthened by efficient cuts and tests, allows us to prune a large number of nodes in the tree to avoid many solutions. An illustrative example and an experimental study are reported.

2021, 17(1): 133-149 doi: 10.3934/jimo.2019103 +[Abstract](2071) +[HTML](768) +[PDF](483.44KB)
Abstract:

In this article, we propose a jump diffusion framework to price the power exchange options. We model the price dynamics of assets using a Hawkes jump diffusion model with common factors to describe the correlated jump risk and clustering of asset price jumps. In the proposed model, the jumps, reflecting common systematic risk and idiosyncratic risk, are modeled by self-exciting Hawkes process with exponential decay. A pricing formula for valuation of power exchange option is obtained following the measure-change technique. Existing models in the literature are shown to be special cases of the proposed model. Finally, sensitivity analysis is given to illustrate the effect of jump risk and jump clustering on option prices. We observe that jump clustering significantly effects the option prices.

2021, 17(1): 151-168 doi: 10.3934/jimo.2019104 +[Abstract](2272) +[HTML](859) +[PDF](496.58KB)
Abstract:

This paper proposes a second-order cone programming (SOCP) relaxation for the generalized trust-region problem by exploiting the property that any symmetric matrix and identity matrix can be simultaneously diagonalizable. We show that our proposed SOCP relaxation can provide a lower bound as tight as that of the standard semidefinite programming (SDP) relaxation. Moreover, we provide a sufficient condition under which the proposed SOCP relaxation is exact. Since the standard SDP relaxation suffers from a much heavier computing burden, the proposed SOCP relaxation has a much higher efficiency in solving process. Then we design a branch-and-bound algorithm based on this SOCP relaxation to obtain the global optimal solution for a general problem. Three types of numerical experiments are carried out to demonstrate the effectiveness and efficiency of our proposed SOCP relaxation.

Wei Ouyang and
2021, 17(1): 169-184 doi: 10.3934/jimo.2019105 +[Abstract](1913) +[HTML](711) +[PDF](425.0KB)
Abstract:

In this paper we conduct local convergence analysis of the inexact Newton methods for solving the generalized equation \begin{document}$0\in f(x)+F(x)$\end{document} under the assumption of Hölder strong metric subregularity, where \begin{document}$f : X \rightarrow Y$\end{document} is a single-valued mapping while \begin{document}$F : X \rightrightarrows Y$\end{document} is a set-valued mapping between arbitrary Banach spaces. Our work are proceeded as twofold: we first explore fully the property of Hölder strong metric subregularity by establishing a verifiable necessary and sufficient condition as well as discussing its stability under small perturbations, and secondly, with the help of aforementioned theoretical analysis, we conclude that every sequence generated by the inexact (quasi) Newton method and staying in a neighborhood of the solution \begin{document}$\bar x$\end{document} is convergent (superlinearly) of order \begin{document}$p(1+q)$\end{document} where \begin{document}$p$\end{document} is the order of Hölder strong metric subregularity imposed on the mapping \begin{document}$f+F$\end{document} and \begin{document}$q$\end{document} is the order of Hölder calmness property for the derivative \begin{document}$Df$\end{document} while \begin{document}$p$\end{document} and \begin{document}$q$\end{document} complement each other as long as \begin{document}$p(1+q)\geq 1$\end{document}.

2021, 17(1): 185-204 doi: 10.3934/jimo.2019106 +[Abstract](2067) +[HTML](745) +[PDF](1107.26KB)
Abstract:

In this paper, we analyze the delay performance of queueing systems in which the service rate varies with time and the number of service states may be infinite. Except in some simple special cases, in general, the queueing model with varying service rate is mathematically intractable. Motivated by the P-K formula for M/G/1 queue, we developed a limiting analysis approach based on the connection between the fluctuation of service rate and the mean queue length. Considering the two extreme service rates, we provide a lower bound and upper bound of mean queue length. Furthermore, an approximate mean queue length formula is derived from the convex combination of these two bounds. The accuracy of our approximation has been confirmed by extensive simulation studies with different system parameters. We also verified that all limiting cases of the system behavior are consistent with the predictions made by our formula.

2021, 17(1): 205-220 doi: 10.3934/jimo.2019107 +[Abstract](2417) +[HTML](821) +[PDF](1501.82KB)
Abstract:

The existing method of determining the size of the time series sliding window by empirical value exists some problems which should be solved urgently, such as when considering a large amount of information and high density of the original measurement data collected from industry equipment, the important information of the data cannot be maximally retained, and the calculation complexity is high. Therefore, by studying the effect of sliding window on time series similarity technology in practical application, an algorithm to determine the initial size of the sliding window is proposed. The upper and lower boundary curves with a higher fitting degree are constructed, and the trend weighting is introduced into the \begin{document}$LB\_Hust$\end{document} distance calculation method to reduce the difficulty of mathematical modeling and improve the efficiency of data similarity computation.

2021, 17(1): 221-232 doi: 10.3934/jimo.2019108 +[Abstract](1863) +[HTML](856) +[PDF](377.28KB)
Abstract:

In location theory, group median generalizes the concepts of both median and center. We address in this paper the problem of modifying vertex weights of a tree at minimum total cost so that a prespecified vertex becomes a group 1-median with respect to the new weights. We call this problem the inverse group 1-median on trees. To solve the problem, we first reformulate the optimality criterion for a vertex being a group 1-median of the tree. Based on this result, we prove that the problem is \begin{document}$NP$\end{document}-hard. Particularly, the corresponding problem with exactly two groups is however solvable in \begin{document}$O(n^2\log n)$\end{document} time, where \begin{document}$n$\end{document} is the number of vertices in the tree.

2021, 17(1): 233-259 doi: 10.3934/jimo.2019109 +[Abstract](4102) +[HTML](2456) +[PDF](705.01KB)
Abstract:

Two-stage risk-neutral stochastic optimization problem has been widely studied recently. The goals of our research are to construct a two-stage distributionally robust optimization model with risk aversion and to extend it to multi-stage case. We use a coherent risk measure, Conditional Value-at-Risk, to describe risk. Due to the computational complexity of the nonlinear objective function of the proposed model, two decomposition methods based on cutting planes algorithm are proposed to solve the two-stage and multi-stage distributional robust optimization problems, respectively. To verify the validity of the two models, we give two applications on multi-product assembly problem and portfolio selection problem, respectively. Compared with the risk-neutral stochastic optimization models, the proposed models are more robust.

2021, 17(1): 261-277 doi: 10.3934/jimo.2019110 +[Abstract](2580) +[HTML](800) +[PDF](640.21KB)
Abstract:

To stimulate the capital-constrained manufacturer to produce green products, the government often adopts two incentive mechanisms: green credit (i.e., subsidy offered directly to bank) and subsidy (i.e., subsidy offered directly to manufacturer). This paper examines the optimal interest rate of the bank, and the optimal product green degree and sales price of the manufacturer under the two mechanisms, respectively. Furthermore, we investigate the effects of these mechanisms on the optimal decisions, the profits of players, the social welfare and the environmental benefits. Several important results are obtained. First, when the total government subsidy is low, the green credit mechanism can bring the higher green degree, product sales price and demand, as well as higher profits for the bank and manufacturer, rather than the subsidy mechanism. Otherwise, the result is opposite. Second, the government should adopt the green credit mechanism to support the manufacturer to develop green products when the budget is limited and relatively low. If the government budget is sufficient, the subsidy mechanism is the best choice, which can bring higher economic and environmental benefits.

2021, 17(1): 279-297 doi: 10.3934/jimo.2019111 +[Abstract](1775) +[HTML](719) +[PDF](712.35KB)
Abstract:

We examine the transient behavior of a positioning system with a large number of tags trying to connect to the infrastructure with an exponential backoff policy in case of unsuccessful connection. Using a classic mean-field approach, we derive a system of differential equations whose solution approximates the original process. Analysis of the solution shows that both the solution and the original system exhibits an unusual log-periodic behavior in the mean-field limit, along with other interesting patterns of behavior. We also perform numerical optimization for the backoff policy.

2021, 17(1): 299-316 doi: 10.3934/jimo.2019112 +[Abstract](2059) +[HTML](837) +[PDF](515.85KB)
Abstract:

This paper studies the effect of information suppression on Naor's model as well as on Edelson and Hildebrand's model under geometric distribution. We set the suitable non-cooperative games and search for their Nash equilibria under the observable and unobservable system. In each case, we analyze the effects of information level on the customers' equilibrium and socially optimal balking strategies as well as on the profit maximization of the system manager. The socially optimal behavior and the inefficiency of the equilibrium strategies are quantified via the price of anarchy measure. We discuss a comparison study of the profit maximization and social welfare under an imposed admission fee. Also, the impact of information on the selfish and social optimal joining rates is examined. Numerical results are presented to exemplify the impact of system parameters on the optimal behavior of customers under different information levels.

2021, 17(1): 317-337 doi: 10.3934/jimo.2019113 +[Abstract](2071) +[HTML](725) +[PDF](940.04KB)
Abstract:

In this paper, we consider estimation problems involving constrained nonlinear systems with the unknown time-delays and unknown system parameters. These unknown quantities are to be estimated such that a least-squares error function between the system output and a set of noisy measurements is minimized subject to the characteristic time constraints specifying the restrictions. We first present the classical estimation formulation, where the expectation of the error function is regarded as the cost function. Then, in order to obtain robust estimates against the noises in measurements, we propose a robust estimation formulation, in which the cost function is the variance of the error function and an additional constraint indicates an allowable sacrifice from the optimal expectation value of the classical estimation problem. For these two estimation problems, we derive the gradients of the corresponding cost and constraint functions with respect to time-delays and system parameters by solving some auxiliary time-delay systems backward in time. On this basis, we develop gradient-based optimization algorithms to determine the optimal time-delays and system parameters. Finally, we consider two example problems, including a parameter estimation problem in microbial batch fermentation process, to illustrate the effectiveness and applicability of our proposed algorithms.

2021, 17(1): 339-355 doi: 10.3934/jimo.2019114 +[Abstract](1623) +[HTML](763) +[PDF](390.77KB)
Abstract:

In this paper, we study the optimal investment problem for an insurer, who is allowed to invest in a financial market which consists of \begin{document}$N$\end{document} risky securities modeled by an \begin{document}$N$\end{document}-dimensional Itô process. The surplus of the insurer is modeled by a general risk model. For the insurer's wealth, some money (called liquid reserves) can only be used to cope with risk, and can not be invested in the financial market. We suggest that the liquid reserve is a proportion of the total claim amount. By the martingale approach, we derive the optimal strategies for the CARA and the quadratic utilities, respectively.

2021, 17(1): 357-368 doi: 10.3934/jimo.2019115 +[Abstract](2004) +[HTML](738) +[PDF](1249.59KB)
Abstract:

Shifted symmetric higher-order power method (SS-HOPM) is an effective method of computing tensor eigenpairs. However the point-wise convergence of SS-HOPM has not been proven yet. In this paper, we provide a solid proof of the point-wise convergence of SS-HOPM via Łojasiewicz inequality. In particular, we establish a mapping from the sequence generated by the algorithm to a specially defined sequence. Using Łojasiewicz inequality, we prove the convergence of the new sequence, then the original sequence is convergent based on the relation of two sequences.

2021, 17(1): 369-392 doi: 10.3934/jimo.2019116 +[Abstract](1870) +[HTML](791) +[PDF](1310.39KB)
Abstract:

Angel capital is an important source of fund for start-ups. Based on the characteristics of angel investment market and the emotional factors between angel investor and entrepreneur, we establish two different principal-agent models to study their impact on different exit mechanisms i.e., IPO and acquisition. We find that: 1) In the case of IPO: as the entrepreneur's emotional factor increases, the optimal incentives decrease; but as the investor's emotional factor increases, the optimal incentives and the efforts increase. 2) When it comes to acquisitions: with the rising of entrepreneur's emotional factor, the optimal incentives decline; but the investor's emotional factor does not affect the optimal incentives and efforts. 3) Under certain conditions, the exit decision is influenced only by the entrepreneur's emotional factor. Moreover, IPO will be the best exit mechanism, only if the entrepreneur's emotional factor is greater than a unique threshold.

2021, 17(1): 393-408 doi: 10.3934/jimo.2019117 +[Abstract](2370) +[HTML](748) +[PDF](519.83KB)
Abstract:

Hazardous wastes are likely to cause danger to humans and the environment. In this paper, a new mathematical optimization model is developed for the multi-period hazardous waste collection planning problem. The hazardous wastes generated by each source are time-varying in weight and allow incomplete and delayed collection. The aim of the model is to help decision makers determine the weight of hazardous wastes to collect from each source and the transportation routes of vehicles in each period. In the developed model, three objectives are considered simultaneously: (1) minimisation of total cost over all periods, which includes start-up fee of vehicles, transportation cost of hazardous wastes, and penalty fee for the delayed collection; (2) minimisation of total transportation risk posing to the surrounding of routes over all periods; and (3) even distribution of transportation risk among all periods, also called risk stability. The developed multi-objective model is transformed into a single-objective one based on the weighted sums method, which is finally equated to a mixed 0-1 linear programming by introducing a set of auxiliary variables and constraints. Numerical experiments are computed with CPLEX software to find the optimal solutions. The computational results and parameters analysis demonstrate the applicability and validity of the developed model. It is found that the consideration of the risk stability can reduce the total transportation risk, the uneven distribution of the transportation risk among all periods, and the maximum number of vehicles used, though increasing the total cost to some extent.

2021, 17(1): 409-426 doi: 10.3934/jimo.2019118 +[Abstract](2097) +[HTML](812) +[PDF](4019.57KB)
Abstract:

The purpose of this paper is to investigate the supplier development (SD) in construction industry. As the supplier's production capacity cannot meet the construction requirements, the owner wants to take incentives to encourage the supplier to improve its production capacity. A principal-agent model and a Stackelberg game model are proposed to study the impact of owner's incentives including cost sharing and purchase price incentive on the production capacity improvement in SD. Furthermore, we give a sensitivity analysis of the influence of supplier's internal and external parameters, i.e., purchase quantity, cost structure, market price and market demand, etc., on the production capacity improvement. The findings of this study can help the owner to make a better decision on the incentive mechanisms for SD, resulting in both better SD practices and a win-win situation.

2021, 17(1): 427-445 doi: 10.3934/jimo.2019119 +[Abstract](2471) +[HTML](778) +[PDF](2296.95KB)
Abstract:

The calibration of a 3D laser rangefinder (LRF) and a camera is a key technique in the field of computer vision and intelligent robots. This paper proposes a new method for the calibration of a 3D LRF and a camera based on optimization solution. The calibration is achieved by freely moving a checkerboard pattern in front of the camera and the 3D LRF. The images and the 3D point clouds of the checkerboard pattern in various poses are collected by the camera and the 3D LRF respectively. By using the images, the intrinsic parameters and the poses of the checkerboard pattern are obtained. Then, two kinds of geometric constraints, line-to-plane constraints and plane-to-plane constraints, are constructed to solve the extrinsic parameters by linear optimization. Finally, the intrinsic and extrinsic parameters are further refined by global optimization, and are used to compute the geometric mapping relationship between the 3D LRF and the camera. The proposed calibration method is evaluated with both synthetic data and real data. The experimental results show that the proposed calibration method is accurate and robust to noise.

2021, 17(1): 447-465 doi: 10.3934/jimo.2019120 +[Abstract](1820) +[HTML](792) +[PDF](1015.28KB)
Abstract:

Considering the parameter uncertainty and actuator failure of hypersonic vehicle during maneuvering, this paper proposes a state observer-based hypersonic vehicle fault-tolerant control (FTC) system design method. Because hypersonic vehicles are prone to failure during maneuvering, the state quantity cannot be measured. First, a state observer-based FTC control method is designed for the linear parameter-varying (LPV) model with parameter uncertainty and partial failure of the actuator. Then, the Lyapunov function is used to demonstrate the asymptotic stability of the closed-loop system. The performance index function proved that the system has robust stability under the disturbance condition. Subsequently, the linear matrix inequality (LMI) was used to solve the observer parameters and the corresponding gain matrix in the control system. The simulation results indicated that the designed controller can track the flight command signal stably and has strong robustness, which verified the effectiveness of the design controller.

2021, 17(1): 467-484 doi: 10.3934/jimo.2020043 +[Abstract](1904) +[HTML](685) +[PDF](566.27KB)
Abstract:

The purpose of this paper concentrates on an economic production quantity model with the factors of imperfect quality and quantity discounts, in which the inspection action occurs during the production stage. There is specific consideration of there being a finite production rate, and the quantity discounts offered by the supplier serves the purpose of stimulating buying greater quantities. This is in contrast to EPQ models that do not take these added factors into consideration. The objective of this paper is to determine the setup cost reduction, which is a function of capital investment, and inventory lot size. An alternative solution procedure was developed that does not employ the Hessian Matrix concavity in the expected total profit. We develop an algorithm to determine the optimal solution for this model. Theoretical results are discussed and a numerical example is proposed. Managerial insights are also examined.

2021, 17(1): 485-500 doi: 10.3934/jimo.2020117 +[Abstract](1709) +[HTML](481) +[PDF](452.62KB)
Abstract:

In this paper, we study some interesting properties of nonconvex oriented distance function. In particular, we present complete characterizations of monotonicity properties of oriented distance function. Moreover, the Clark subdifferentials of nonconvex oriented distance function are explored in the solid case. As applications, fuzzy necessary optimality conditions for approximate solutions to vector optimization problems are provided.

2021, 17(1): 501-516 doi: 10.3934/jimo.2020149 +[Abstract](1016) +[HTML](358) +[PDF](376.05KB)
Abstract:

This paper presents modified preventive maintenance policies for an operating system that works at random processing times and is imperfectly maintained. The system may suffer from one of the two types of failures based on a time-dependent imperfect maintenance mechanism: type-Ⅰ (minor) failure, which can be rectified by minimal repair, and type-Ⅱ (catastrophic) failure, which can be removed by corrective maintenance. When the system needs to be maintained, two policies "preventive maintenance-first (PMF) and preventive maintenance-last (PML)" may be applied. In each maintenance interval, before any type-Ⅱ failure occurs, the system is maintained at a planned time \begin{document}$T$\end{document} or at the completion of a working time, whichever occurs first and last, which are called PMF and PML, respectively. After any maintenance activity, the system improves but its failure characteristic is also altered. At the \begin{document}$N$\end{document}-th maintenance, the system is replaced rather than maintained. For each policy, the optimal preventive maintenance schedule (\begin{document}$T$\end{document}, \begin{document}$N$\end{document})\begin{document}$^{*}$\end{document} that minimizes the mean cost rate function is derived analytically and determined numerically in terms of its existence and uniqueness. The proposed models provide a general framework for analyzing the maintenance policies in reliability theory.

2021 Impact Factor: 1.411
5 Year Impact Factor: 1.441
2021 CiteScore: 2.1