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Throughout its history, Operational Research has evolved to include methods, models and algorithms that have been applied to a wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first summarises the up-to-date knowledge and provides an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion and used as a point of reference by a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes. [ABSTRACT FROM AUTHOR]

We consider a two-stage service system with two types of servers, namely subordinates who perform the first-stage service and supervisors who have their own responsibilities in addition to collaborating with the subordinates on the second-stage service. Rewards are earned when first- or second-stage service is completed and when supervisors finish one of their own responsibilities. Costs are incurred when impatient customers abandon without completing the second-stage service. Our problem is to determine how the supervisors should distribute their time between their joint work with the subordinates and their own responsibilities. Under the assumptions that service times at both stages are exponentially distributed and that the customers waiting for second-stage service abandon after an exponential amount of time, we prove that one of two policies will maximize the long-run average profit. Namely, it is optimal for supervisors to start collaborating with subordinates either when subordinates can no longer serve new customers or as soon as there is a customer ready for second-stage service. Furthermore, we show that the optimality condition is a simple threshold on the system parameters. We conclude by proving that pooling supervisors (and their associated subordinates) improves system performance, but with limited returns as more supervisors are pooled. [ABSTRACT FROM AUTHOR]

Standard server assignment policies for multi-server queueing stations include the noncollaborative policy, where the servers work in parallel on different jobs; and the fully collaborative policy, where the servers work together on the same job. However, if each job can be decomposed into subtasks with no precedence relationships, then we consider a form of server coordination named task assignment where the servers work in parallel on different subtasks of the same job. We identify the task assignment policy that maximizes the long-run average throughput of a queueing station with finite internal buffers when blocked servers can be idled or reassigned to either replace or collaborate with other servers on unblocked subtasks. We then compare the server coordination policies and show that the task assignment is best when the servers are highly specialized; otherwise, the fully collaborative or noncollaborative policies are preferable depending on whether the synergy level among the servers is high or not. We also provide numerical results that quantify our previous comparison. Finally, we address buffer allocation in longer lines where there are precedence relationships between some of the tasks, and present numerical results that suggest our comparisons for one queueing station generalize to longer lines. [ABSTRACT FROM AUTHOR]

Cross‐training is an important tool to improve the performance of manufacturing and service systems through dynamic worker allocation. This study investigates how cross‐training can be leveraged when worker collaboration is not possible and frequent worker reassignment is undesirable. We consider a tandem queueing system with finite buffers between tasks. We show that if each worker is the fastest at a different task, then the optimal policy is always dedicated, regardless of the magnitude of the reassignment costs. Otherwise, if the system is Markovian with two homogeneous tasks and a faster and a slower worker, we completely characterize how the profit‐optimal policy depends on the (constant) reassignment cost. We also prove that for any given reassignment cost, dedicated worker allocation policies are strictly suboptimal for large enough buffer sizes. Instead, the faster worker should move to the downstream task only if there are enough jobs in the intermediate buffer and return to the upstream task when the buffer is sufficiently empty. Furthermore, the benefit of task switching increases as the buffer size becomes larger. We use our theoretical results to develop worker allocation heuristics both for more general systems with two tasks and for systems with more tasks. Numerical experimentation provides insights on how the profit depends on the buffer sizes and reassignment costs, shows that policies that are optimal when workers are collaborative result in excessive switching for non‐collaborative workers, and indicates that our pick‐the‐best heuristics perform well in all settings. [ABSTRACT FROM AUTHOR]

(Max, +) linear systems can be used to represent a class of queueing networks such as acyclic or cyclic fork-and-join queueing networks, finite or infinite capacity tandem queueing networks with various types of blocking, synchronized queueing networks and so on. In this paper, we derive explicit expressions for the tail probability of transient and stationary waiting times in Poisson driven (max, +) linear systems. As an application of our results, we consider the problem of maximizing the long run average throughput subject to a probabilistic constraint on stationary waiting times in (max, +) linear systems with deterministic service times. Index Terms--Input control, queueing networks, series expansions, stochastic event graphs.

The Riemann-Stieltjes integration concept was used to create a novel model for computing the lower and upper bounds of renewal functions. Compared to conventional computing frameworks, the proposed approach provides relatively simple expressions for steady state solutions since it obviates the need for renewal functions. Results derived from an error analysis involving the proposed approach revealed that the computer numerical bounds generated are highly in agreement with standardized values.

In this paper, the authors develop a state variable feedback control for a class of assembly operations. The objective of this control is to increase the number of jobs completed within a time horizon T while keeping the job cycle times within an acceptable range with a prespecified probability. The assembly operations are modeled as stochastic timed-event graphs which in the (max, +) algebra are characterized by a set of linear stochastic difference equations. Index Terms - Assembly, cycle time, event graphs, job flow control, (max, +) algebra.

A new model of system availability in a maintained system shows that for a given mean inspection rate, the system availability is highest when inspections are carried out at fixed intervals. The model assumes that the system deteriorates because of random shocks with random magnitudes and that inspection is necessary to reveal failure. Protection systems, atom systems, and other sensoring devices with high availability requirements are suitable for applying this model.