Optimal maintenance policy for stochastically failing equipment: A diffusion approximation

A system receives shocks at random points of time. Each shock causes a random amount of damage which accumulates over time. The system fails when the accumulated damage exceeds a fixed threshold. Upon failure the system is replaced by a new one. The damage process is controlled by means of a maintenance policy. There are M possible maintenance actions. Given that a maintenance action m is employed, then the cumulative damage decreases at rate r^m. Replacement costs and maintenance costs are considered. The objective is to determine an optimal maintenance policy under the following optimality criteria: (1) long-run average cost; (2) total expected discounted cost over an infinite horizon. For a diffusion approximation, we show that the optimal maintenance expenditure rate is monotonically increasing in the cumulative damage level.     

MAD: Mathematical analysis of downtime

The MAD model presents a mathematic treatment of the relationship between aircraft reliability and maintainability, system manning and inspection policies, scheduling and sortie length, and aircraft downtime. Log normal distributions are postulated for subsystem repair times and simultaneous repair of malfunctions is assumed. The aircraft downtime for maintenance is computed with the distribution of the largest of k log normal distributions. Waiting time for maintenance men is calculated either by using a multiple-channel queuing model or by generating the distribution of the number of maintenance men required and comparing this to the number of men available to determine the probability of waiting at each inspection.     

Scheduling jobs and maintenance activities on parallel machines

Most machine scheduling models assume that the machines are available all of the time. However, in most realistic situations, machines need to be maintained and hence may become unavailable during certain periods. In this paper, we study the problem of processing a set of n jobs on m parallel machines where each machine must be maintained once during the planning horizon. Our objective is to schedule jobs and maintenance activities so that the total weighted completion time of jobs is minimized. Two cases are studied in this paper. In the first case, there are sufficient resources so that different machines can be maintained simultaneously if necessary. In the second case, only one machine can be maintained at any given time. In this paper, we first show that, even when all jobs have the same weight, both cases of the problem are NP-hard. We then propose branch and bound algorithms based on the column generation approach for solving both cases of the problem. Our algorithms are capable of optimally solving medium sized problems within a reasonable computational time. We note that the general problem where at most j machines, 1 ≤ j ≤ m, can be maintained simultaneously, can be solved similarly by the column generation approach proposed in this paper. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 145–165, 2000     

Markov chain analysis of a situation where cannibalization is the only repair activity

This paper considers a group of S identical aircraft, each of which is partitioned into K parts which fail exponentially. The only way in which a failed aircraft can be repaired is by cannibalizing its out-of-commission parts from other failed aircraft. The evolution of the number of good aircraft over time is governed by the transient behavior of an absorbing Markov chain. We can therefore study this behavior by matrix multiplication although the computational problem grows large for K ≥ 3. Some numerical results and some approximations are also provided.     

Sojourn times in G/M/1 fork‐join networks

We consider a processing network in which jobs arrive at a fork‐node according to a renewal process. Each job requires the completion of m tasks, which are instantaneously assigned by the fork‐node to m task‐processing nodes that operate like G/M/1 queueing stations. The job is completed when all of its m tasks are finished. The sojourn time (or response time) of a job in this G/M/1 fork‐join network is the total time it takes to complete the m tasks. Our main result is a closed‐form approximation of the sojourn‐time distribution of a job that arrives in equilibrium. This is obtained by the use of bounds, properties of D/M/1 and M/M/1 fork‐join networks, and exploratory simulations. Statistical tests show that our approximation distributions are good fits for the sojourn‐time distributions obtained from simulations. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimal refueling sequence for a mixed fleet with limited refuelings

Consider a fleet of vehicles comprised of K₁ identical tankers and K₂ identical nontankers (small aircraft). Tankers are capable of refueling other tankers as well as nontankers. The problem is to find that refueling sequence of the tankers that maximizes the range simultaneously attainable by all K₂ nontankers. A recent paper established that the “unit refueling sequence,” comprised of one tanker refueling at each of K₁ refueling operations, is optimal. The same paper also proffered the following conjecture for the case that the number of refueling operations is constrained to be less than the number of tankers: A nonincreasing refueling sequence is optimal. This article proves the conjecture.     

One-facility location with rectilinear tour distances

This article concerns the location of a facility among n points where the points are serviced by “tours” taken from the facility. Tours include m points at a time and each group of m points may become active (may need a tour) with some known probability. Distances are assumed to be rectilinear. For m ≤ 3, it is proved that the objective function is separable in each dimension and an exact solution method is given that involves finding the median of numbers appropriately generated from the problem data. It is shown that the objective function becomes multimodal when some tours pass through four or more points. A bounded heuristic procedure is suggested for this latter case. This heuristic involves solving an auxiliary three-point tour location problem.     

Optimizing usage and maintenance decisions for k‐out‐of‐n systems of moving assets

We consider an integrated usage and maintenance optimization problem for a k‐out‐of‐n system pertaining to a moving asset. The k‐out‐of‐n systems are commonly utilized in practice to increase availability, where n denotes the total number of parallel and identical units and k the number of units required to be active for a functional system. Moving assets such as aircraft, ships, and submarines are subject to different operating modes. Operating modes can dictate not only the number of system units that are needed to be active, but also where the moving asset physically is, and under which environmental conditions it operates. We use the intrinsic age concept to model the degradation process. The intrinsic age is analogous to an intrinsic clock which ticks on a different pace in different operating modes. In our problem setting, the number of active units, degradation rates of active and standby units, maintenance costs, and type of economic dependencies are functions of operating modes. In each operating mode, the decision maker should decide on the set of units to activate (usage decision) and the set of units to maintain (maintenance decision). Since the degradation rate differs for active and standby units, the units to be maintained depend on the units that have been activated, and vice versa. In order to minimize maintenance costs, usage and maintenance decisions should be jointly optimized. We formulate this problem as a Markov decision process and provide some structural properties of the optimal policy. Moreover, we assess the performance of usage policies that are commonly implemented for maritime systems. We show that the cost increase resulting from these policies is up to 27% for realistic settings. Our numerical experiments demonstrate the cases in which joint usage and maintenance optimization is more valuable. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 418–434, 2017     

Optimal (τ, T) opportunistic maintenance of a k‐out‐of‐n:G system with imperfect PM and partial failure

The opportunistic maintenance of a k‐out‐of‐n:G system with imperfect preventive maintenance (PM) is studied in this paper, where partial failure is allowed. In many applications, the optimal maintenance actions for one component often depend on the states of the other components and system reliability requirements. Two new (τ, T) opportunistic maintenance models with the consideration of reliability requirements are proposed. In these two models, only minimal repairs are performed on failed components before time τ and the corrective maintenance (CM) of all failed components are combined with PM of all functioning but deteriorated components after τ; if the system survives to time T without perfect maintenance, it will be subject to PM at time T. Considering maintenance time, asymptotic system cost rate and availability are derived. The results obtained generalize and unify some previous research in this area. Application to aircraft engine maintenance is presented. © 2000 John Wiley & Sons;, Inc. Naval Research Logistics 47: 223–239, 2000     

Scheduling parallel dedicated machines with the speeding‐up resource

We consider a problem of scheduling jobs on m parallel machines. The machines are dedicated, i.e., for each job the processing machine is known in advance. We mainly concentrate on the model in which at any time there is one unit of an additional resource. Any job may be assigned the resource and this reduces its processing time. A job that is given the resource uses it at each time of its processing. No two jobs are allowed to use the resource simultaneously. The objective is to minimize the makespan. We prove that the two‐machine problem is NP‐hard in the ordinary sense, describe a pseudopolynomial dynamic programming algorithm and convert it into an FPTAS. For the problem with an arbitrary number of machines we present an algorithm with a worst‐case ratio close to 3/2, and close to 3, if a job can be given several units of the resource. For the problem with a fixed number of machines we give a PTAS. Virtually all algorithms rely on a certain variant of the linear knapsack problem (maximization, minimization, multiple‐choice, bicriteria). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Heuristics for multicomponent joint replacement: Applications to aircraft engine maintenance

This paper considers the maintenance of aircraft engine components where economies exist for joint replacement because (a) the aircraft must be pulled from service for maintenance and (b) repair of some components requires removal and disassembly of the engine. It is well known that the joint replacement problem is difficult to solve exactly, because the optimal solution does not have a simple structured form. Therefore, we formulate three easy-to-implement heuristics and test their performance against a lower bound for various numerical examples. One of our heuristics, the base interval approach, in which replacement cycles for all components are restricted to be multiples of a specified interval, is shown to be robustly accurate. Moreover, this heuristic is consistent with maintenance policies used by commercial airlines in which periodic maintenance checks are made at regular intervals. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 435–458, 1998     

Heuristics for minimizing mean tardiness for m parallel machines

The concept of parallel operations has been widely used in manufacturing and data processing. However, not many efficient methods have been proposed to reduce job tardiness. This article proposes an efficient heuristic to minimize the mean tardiness of a set of tasks with known processing times and due dates for single and m parallel machines. For the single-machine case, the proposed heuristic is compared with the well-known Wilkerson and Irwin algorithm; for the m parallel machine case, it is compared with an extension of the Wilkerson-Irwin algorithm. We also introduce a simple dispatching rule, and it is compared with some existing dispatching rules. The comprehensive simulation results show that the proposed heuristic performs better than the Wilkerson-Irwin algorithm at a significantly reduced computational time.     

Solving a selected class of location problems by exploiting problem structure: A decomposition approach

For many combinatorial optimization problems that are NP-hard, a number of special cases exist that can be solved in polynomial time. This paper addresses the issue of solving one such problem, the well-known m-median problem with mutual communication (MMMC), by exploiting polynomially solvable special cases of the problem. For MMMC, a dependency graph is defined that characterizes the structure of the interactions between decision variables. A Lagrangian decomposition scheme is proposed that partitions the problem into two or more subproblems, each having the same structure as the original problem, but with simpler dependency graphs. The dual problems are solved using subgradient or multiplier adjustment methods. An efficient method of adjusting the multiplier values is given. Computational results are reported that show the method to be quite effective. In addition, applications of the approach to other difficult location problems is discussed. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 791–815, 1998     

Optimal refueling strategies for a mixed-vehicle fleet

The problem treated here involves a mixed fleet of vehicles comprising two types of vehicles: K₁ tanker-type vehicles capable of refueling themselves and other vehicles, and K₂ nontanker vehicles incapable of refueling. The two groups of vehicles have different fuel capacities as well as different fuel consumption rates. The problem is to find the tanker refueling sequence that maximizes the range attainable for the K₂ nontankers. A tanker refueling sequence is a partition of the tankers into m subsets (2 ≤ m ≤ K₁). A given sequence of the partition provides a realization of the number of tankers participating in each successive refueling operation. The problem is first formulated as a nonlinear mixed-integer program and reduced to a linear program for a fixed sequence which may be solved by a simple recursive procedure. It is proven that a “unit refueling sequence” composed of one tanker refueling at each of K₁ refueling operations is optimal. In addition, the problem of designing the “minimum fleet” (minimum number of tankers) required for a given set of K₂ nontankers to attain maximal range is resolved. Also studied are extensions to the problem with a constraint on the number of refueling operations, different nontanker recovery base geometry, and refueling on the return trip.     

Machine scheduling with an availability constraint and job delivery coordination

In this paper we study the scheduling problem that considers both production and job delivery at the same time with machine availability considerations. Only one vehicle is available to deliver jobs in a fixed transportation time to a distribution center. The vehicle can load at most K jobs as a delivery batch in one shipment due to the vehicle capacity constraint. The objective is to minimize the arrival time of the last delivery batch to the distribution center. Since machines may not always be available over the production period in real life due to preventive maintenance, we incorporate machine availability into the models. Three scenarios of the problem are studied. For the problem in which the jobs are processed on a single machine and the jobs interrupted by the unavailable machine interval are resumable, we provide a polynomial algorithm to solve the problem optimally. For the problem in which the jobs are processed on a single machine and the interrupted jobs are nonresumable, we first show that the problem is NP‐hard. We then propose a heuristic with a worst‐case error bound of 1/2 and show that the bound is tight. For the problem in which the jobs are processed on either one of two parallel machines, where only one machine has an unavailable interval and the interrupted jobs are resumable, we propose a heuristic with a worst‐case error bound of 2/3. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

An extension of the (szwarc) truck assignment problem

In this journal in 1967. Szware presented an algorithm for the optimal routing of a common vehicle fleet between m sources and n sinks with p different types of commodities. The main premise of the formulation is that a truck may carry only one commodity at a time and must deliver the entire load to one demand area. This eliminates the problem of routing vehicles between sources or between sinks and limits the problem to the routing of loaded trucks between sources and sinks and empty trucks making the return trip. Szwarc considered only the transportation aspect of the problem (i. e., no intermediate points) and presented a very efficient algorithm for solution of the case he described. If the total supply is greater than the total demand, Szwarc shows that the problem is equivalent to a (mp + n) by (np + m) Hitchcock transportation problem. Digital computer codes for this algorithm require rapid access storage for a matrix of size (mp + n) by (np + m); therefore, computer storage required grows proportionally to p². This paper offers an extension of his work to a more general form: a transshipment network with capacity constraints on all arcs and facilities. The problem is shown to be solvable directly by Fulkerson's out-of-kilter algorithm. Digital computer codes for this formulation require rapid access storage proportional to p instead of p². Computational results indicate that, in addition to handling the extensions, the out-of-kilter algorithm is more efficient in the solution of the original problem when there is a mad, rate number of commodities and a computer of limited storage capacity.     

The conjugate gradient technique for certain quadratic network problems

We consider a class of network flow problems with pure quadratic costs and demonstrate that the conjugate gradient technique is highly effective for large-scale versions. It is shown that finding a saddle point for the Lagrangian of an m constraint, n variable network problem requires only the solution of an unconstrained quadratic programming problem with only m variables. It is demonstrated that the number of iterations for the conjugate gradient algorithm is substantially smaller than the number of variables or constraints in the (primal) network problem. Forty quadratic minimum-cost flow problems of various sizes up to 100 nodes are solved. Solution time for the largest problems (4,950 variables and 99 linear constraints) averaged 4 seconds on the CBC Cyber 70 Model 72 computer.     

Scheduling for parallel dedicated machines with a single server

This paper examines scheduling problems in which the setup phase of each operation needs to be attended by a single server, common for all jobs and different from the processing machines. The objective in each situation is to minimize the makespan. For the processing system consisting of two parallel dedicated machines we prove that the problem of finding an optimal schedule is N P‐hard in the strong sense even if all setup times are equal or if all processing times are equal. For the case of m parallel dedicated machines, a simple greedy algorithm is shown to create a schedule with the makespan that is at most twice the optimum value. For the two machine case, an improved heuristic guarantees a tight worst‐case ratio of 3/2. We also describe several polynomially solvable cases of the later problem. The two‐machine flow shop and the open shop problems with a single server are also shown to be N P‐hard in the strong sense. However, we reduce the two‐machine flow shop no‐wait problem with a single server to the Gilmore—Gomory traveling salesman problem and solve it in polynomial time. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 304–328, 2000     

Open shop scheduling to minimize the number of late jobs

We develop polynomial algorithms for several cases of the NP-hard open shop scheduling problem of minimizing the number of late jobs by utilizing some recent results for the open shop makespan problem. For the two machine common due date problem, we assume that either the machines or the jobs are ordered. For the m machine common due date problem, we assume that one machine is maximal and impose a restriction on its load. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 525–532, 1998     

A note on combined job selection and sequencing problems

We investigate the solvability of two single‐machine scheduling problems when the objective is to identify among all job subsets with cardinality k,1≤k≤n, the one that has the minimum objective function value. For the single‐machine minimum maximum lateness problem, we conclude that the problem is solvable in O(n²) time using the proposed REMOVE algorithm. This algorithm can also be used as an alternative to Moore's algorithm to solve the minimum number of tardy jobs problem by actually solving the hierarchical problem in which the objective is to minimize the maximum lateness subject to the minimum number of tardy jobs. We then show that the REMOVE algorithm cannot be used to solve the general case of the single‐machine total‐weighted completion time problem; we derive sufficient conditions among the job parameters so that the total weighted completion time problem becomes solvable in O(n²) time. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 449–453, 2013