A Queueing Theoretic Approach to Decoupling Inventory

This paper investigates the performance of different hybrid push-pull systems with a decoupling inventory at the semi-finished products and reordering thresholds. Raw materials are ‘pushed’ into the semi-finished product inventory and customers ‘pull’ products by placing orders. Furthermore, production of semi-finished products starts when the inventory goes below a certain level, referred to as the threshold value and stops when the inventory attains stock capacity. As performance of the decoupling stock is critical to the overall cost and performance of manufacturing systems, this paper introduces a Markovian model for hybrid push-pull systems. In particular, we focus on a queueing model with two buffers, thereby accounting for both the decoupling stock as well as for possible backlog of orders. By means of numerical examples, we assess the impact of different reordering policies, irregular order arrivals, the set-up time distribution and the order processing time distribution on the performance of hybrid push-pull systems

Numerical methods for queues with shared service

A queueing system is a mathematical abstraction of a situation where elements, called customers, arrive in a system and wait until they receive some kind of service. Queueing systems are omnipresent in real life. Prime examples include people waiting at a counter to be served, airplanes waiting to take off, traffic jams during rush hour etc. Queueing theory is the mathematical study of queueing phenomena. As often neither the arrival instants of the customers nor their service times are known in advance, queueing theory most often assumes that these processes are random variables. The queueing process itself is then a stochastic process and most often also a Markov process, provided a proper description of the state of the queueing process is introduced. This dissertation investigates numerical methods for a particular type of Markovian queueing systems, namely queueing systems with shared service. These queueing systems differ from traditional queueing systems in that there is simultaneous service of the head-of-line customers of all queues and in that there is no service if there are no customers in one of the queues. The absence of service whenever one of the queues is empty yields particular dynamics which are not found in traditional queueing systems. These queueing systems with shared service are not only beautiful mathematical objects in their own right, but are also motivated by an extensive range of applications. The original motivation for studying queueing systems with shared service came from a particular process in inventory management called kitting. A kitting process collects the necessary parts for an end product in a box prior to sending it to the assembly area. The parts and their inventories being the customers and queues, we get ``shared service'' as kitting cannot proceed if some parts are absent. Still in the area of inventory management, the decoupling inventory of a hybrid make-to-stock/make-to-order system exhibits shared service. The production process prior to the decoupling inventory is make-to-stock and driven by demand forecasts. In contrast, the production process after the decoupling inventory is make-to-order and driven by actual demand as items from the decoupling inventory are customised according to customer specifications. At the decoupling point, the decoupling inventory is complemented with a queue of outstanding orders. As customisation only starts when the decoupling inventory is nonempty and there is at least one order, there is again shared service. Moving to applications in telecommunications, shared service applies to energy harvesting sensor nodes. Such a sensor node scavenges energy from its environment to meet its energy expenditure or to prolong its lifetime. A rechargeable battery operates very much like a queue, customers being discretised as chunks of energy. As a sensor node requires both sensed data and energy for transmission, shared service can again be identified. In the Markovian framework, "solving" a queueing system corresponds to finding the steady-state solution of the Markov process that describes the queueing system at hand. Indeed, most performance measures of interest of the queueing system can be expressed in terms of the steady-state solution of the underlying Markov process. For a finite ergodic Markov process, the steady-state solution is the unique solution of $N-1$ balance equations complemented with the normalisation condition, $N$ being the size of the state space. For the queueing systems with shared service, the size of the state space of the Markov processes grows exponentially with the number of queues involved. Hence, even if only a moderate number of queues are considered, the size of the state space is huge. This is the state-space explosion problem. As direct solution methods for such Markov processes are computationally infeasible, this dissertation aims at exploiting structural properties of the Markov processes, as to speed up computation of the steady-state solution. The first property that can be exploited is sparsity of the generator matrix of the Markov process. Indeed, the number of events that can occur in any state --- or equivalently, the number of transitions to other states --- is far smaller than the size of the state space. This means that the generator matrix of the Markov process is mainly filled with zeroes. Iterative methods for sparse linear systems --- in particular the Krylov subspace solver GMRES --- were found to be computationally efficient for studying kitting processes only if the number of queues is limited. For more queues (or a larger state space), the methods cannot calculate the steady-state performance measures sufficiently fast. The applications related to the decoupling inventory and the energy harvesting sensor node involve only two queues. In this case, the generator matrix exhibits a homogene block-tridiagonal structure. Such Markov processes can be solved efficiently by means of matrix-geometric methods, both in the case that the process has finite size and --- even more efficiently --- in the case that it has an infinite size and a finite block size. Neither of the former exact solution methods allows for investigating systems with many queues. Therefore we developed an approximate numerical solution method, based on Maclaurin series expansions. Rather than focussing on structural properties of the Markov process for any parameter setting, the series expansion technique exploits structural properties of the Markov process when some parameter is sent to zero. For the queues with shared exponential service and the service rate sent to zero, the resulting process has a single absorbing state and the states can be ordered such that the generator matrix is upper-diagonal. In this case, the solution at zero is trivial and the calculation of the higher order terms in the series expansion around zero has a computational complexity proportional to the size of the state space. This is a case of regular perturbation of the parameter and contrasts to singular perturbation which is applied when the service times of the kitting process are phase-type distributed. For singular perturbation, the Markov process has no unique steady-state solution when the parameter is sent to zero. However, similar techniques still apply, albeit at a higher computational cost. Finally we note that the numerical series expansion technique is not limited to evaluating queues with shared service. Resembling shared queueing systems in that a Markov process with multidimensional state space is considered, it is shown that the regular series expansion technique can be applied on an epidemic model for opinion propagation in a social network. Interestingly, we find that the series expansion technique complements the usual fluid approach of the epidemic literature

Back-order lead time behaviour in (s,Q)-inventory models with compound renewal demand

In the practice of inventory management customer-oriented performance characteristics as opposed to availability-oriented performance characteristics have received more and more attention. Instead of measuring the probability of a stockout one measures the probability that a customer's demand is satisfied within one week. In this paper we derive approximate expressions for the customer waiting time distribution in case a single stockpoint is replenished according to an (s,Q)-policies. The demand process is compound renewal, i.e. customer has a demand distributed according to some arbitrary distribution funtion. Extensive simulations show that the approximations yield excellent results. The results can be applied to analyze distribution networks where each stockpoint is controlled according to an (s,Q)-model

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide 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. It should be used as a point of reference or first-port-of-call for 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

How to Make the Most Productive Intervention in a Complex Economic System

Information about supply and demand propagates through supply chains in a queueing network with people and computers as batch information processors. As each batch processor delays propagation of information whilst pursuing optimal local decisions, the effect is delay and distortion of the information that is used to commit resources to actions in the supply chain. This thesis investigates the effect of delay and imperfect information as a source of error, to establish the case for change in research focus from optimal exploitation of physical constraints to optimal exploitation of information. In the context of real world supply chains, the thesis asks "How does one make the most productive intervention in a complex economic system?" and pursues a meta-intervention which perpetually minimises the discovered error-term. Evidence from literature indicates that agent-based modelling permits real-time peer-to-peer communication and distributed optimisation. Based on the literature the research project designs and develops an agent-based model which operates in real-time without batch-processes and can perform incremental multi-objective optimisation under realistic (chronologically progressive) conditions for decision making. The agent based model is then used to investigate two real-world supply chains, as case studies, which reveals a significant improvement of profitability and order-fulfilment. The thesis concludes that agent-based modelling is a very promising direction for "making the most productive intervention" as it reduces delay to a minimum. Finally it recommends that continuous improvement of decision making methods is a role better suited for humans, rather than operational decision making where computers cope much better with the high amount of detailed information

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide 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. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order

IOBPCS based models and decoupling thinking

The inventory and order based production control system (IOBPCS) is mainly a model of a forecast driven production system where the production decision is based on the forecast in combination with the deviation between target inventory and actual inventory. The model has been extended in various directions by including e.g. WIP feedback but also by interpreting the inventory as an order book and hence representing a customer order driven system. In practice a system usually consists of one forecast driven subsystem in tandem with a customer order driven subsystem and the interface between the two subsystems is represented by information flows and a stock point associated with the customer order decoupling point (CODP). The CODP may be positioned late in the flow, as in make to stock systems, or early, as in make to order systems, but in any case the model should be able to capture the properties of both subsystems in combination. A challenge in separating forecast driven from customer order driven is that neither the inventory nor the order book should be allowed to take on negative values, and hence non-linearities are introduced making the model more difficult to solve analytically unless the model is first linearized. In summary the model presented here is based on two derivatives of IOBPCS that are in tandem, and interfaces between them related to where the demand information flow is decoupled and the position of the CODP