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Suppose there is a network of railroads and our aim is to schedule trains from a fixed origin station to the destination one. Of course, there are some cases that there are some intermediate stations between the origin and the destination station depended on the railroad network (the connection between stations). To avoid collisions, between any two stations, trains can be traversed in only one direction and we assume that this direction can carry a limited number of trains at once time. Apart from the origin and destination, the number of trains entering must be equal to the number of trains leaving in each intermediate. Our interest is to maximize the number of trains that can all together leave the origin and reach the destination. Furthermore, one might want to know more about the minimum removal of combination of tracks would prevent any trains moving from the original station to the destination and those combinations of tracks served the smallest sum of trains.

Input themselves are generally not of interest and the outputs that the inputs imply are usually of greater interest. It is apparent that inaccurate judgement about input modelling can lead to more error. Since fitting distributions to available data using statistical framework as maximum likelihood estimation is required to understand their inference, our post will focus more on explaining how to avoid errors in input modelling in some certain cases.

One interesting task that I have met in Optimisation coursework is the “cutting-stock” problem. The problem is that given unlimited standard rods of equal length L meters, we need to cut them into different smaller, yet specified sized pieces in order to produce enough required numbers of the small pieces.

Copula is a fascinating tool for understanding dependence among multivariate outcomes. The origin of the term copula is from Latin word “copulare”, meaning connection or joint. It first used in the work of Sklar in 1959. The main use of copulas is to understand the interrelation of two or more random variables.

Travelling by flights is very popular nowadays and is predicted that in Europe the number of passengers in 2035 travelling by airplanes will be as twice as that in 2012, the number of flights in 2035 will increase 1.5 times the level of traffic in 2012. However, it can create many challenges in terms of sustainability and competitiveness with the current operations of aviation. For example, due to the limited capacity, in 2035 many flights cannot be accommodated or equivalently, a lot of passengers cannot travel by flight. Moreover, the problems about the environmental impact, security and safety are going to grow as well. Therefore, the European Commission has adopted some significant changes in Air Traffic Management (ATM) system.

The single node infinite-server queueing models form the basis of the extension to networks of infinite server queues, which are considered as a network of interconnected queues where customers in the system are able to move from one node to other nodes. Massey and Whitt (1993) addressed many important results on infinite server network models in continuous time with inhomogeneous Poisson arrivals.

Based on records from the NatCatSERVICE of Munich Re, natural disaster for a period 1980-2016 in the EU Member States due to weather and climate-related extremes accounted for approximately EUR 410 billion, which made up 83 % of the total economic losses/damages. However, note that this estimation mainly contains the monetised direct damages such as flooded properties, severed transport routes damages, but doesn’t cover the loss of human life, cultural heritage or ecosystem services. Therefore, having reliable estimates of the frequencies of extreme events is very significant for the public, insurance companies, and government agencies. Moreover, due to the global warming, flood occurred unpredictably and more frequently. For example, Harbertonford was flooded totally 21 times in the last 60 years, but there is six individual occasions of floods from 1998 to 2000. Therefore, this problem raise an issue about how to account the unpredicted occurrence of the extreme events in the model?

Mathematical algorithm can be expressed as a set of steps within a finite amount of time designed to accomplish some calculation, data processing and automated reasoning tasks. Algorithms can give us a deeper insight into mathematics. For instance, the well-known Euclidean algorithm essentially provides the foundation of number theory. Moreover, algorithms are effective tools in several services that impact our day-to-day lives. For example, the growing complexity of algorithms nowadays contribute to more advanced processors on smartphones or computers or the searching algorithms have improved the Google search applications which is very useful in optimizing traffic flow, etc.
Two of many extremely significant concepts in algorithms are invariants and mono-variants. A mono-variant is a quantity that changes monotonically, either non-increasingly or non-decreasingly, and an invariant is a quantity that remains unchanged. Let see the next example and how mono-variants play crucial role in constructing the algorithm.

This is a second post about queue theory, but about the application of infinite server queues in healthcare system. Healthcare is full with delays. Most of us have to wait for days or weeks or even months to get an appointment with a physician, nurse, or doctor … and upon arrival we must wait some more until being served. This is due to the problems that many healthcare facilities are facing are the growing and varying demands, different types of patient (in case of emergency) and tightly constrained resources. However, in fact, there can be waste of resources in elsewhere or at particular time, like overstaff, unused beds available, etc. The crucial question to tackle now is how can we utilize resources (staffing) in the most cost-effective way to reduce delays, number of cancellations or staff hour at time when the demand is low. There are many factors to assess the performance measure such as reducing waiting times of patients, reducing the number of cancellations or rescheduling staff working hours at times when the demand is low. One measure way is to achieve the government target for A&E departments: “98% (now 95%) of patients finish their journey (discharged or be admitted to hospital) in 4 hours”.

19 th February, 2019.

What is the best man and woman athletics performance in running when we have data about the annual best times taken in seven Olympic distances events (100 m, 200 m, 400 m, 800 m, 1500 m, 5000 m and 10000 m) for both male and female athletics? Extreme value theory would be very useful to give the answer to this question. From athletics records, the interest lies in the fastest times taken by athletes, or the minima, which gives more information that only world records because it even includes fast performances that not actually record breaking, then more accurate conclusion can be obtainable.

11th, February, 2019

Flooding can occur in two different ways. Whenever there is too much extra water accumulated over long period in a river's channel because of rain, ice melting which can result in breaking through the river bank and spreading over the land. Other natural disasters such as hurricanes, cyclones, earthquakes etc. can cause flooding immediately and very heavily. In the latter case it is a big observation that causes flooding, which means that partial maxima exceed some threshold. In fact, this kind of intellectual curiosity has motivated the development of the theory of extremes.

Queueing theory is the study of problems involving waiting or queueing and helps to analyse and understand the queueing behaviour and then, make decisions about the resources allocation or utilization.

The demand of air transport grows rapidly, which leads to serious problem due to the inadequate capacity of airport. One of the main problem is the imbalance between demand for airport services and supply of the required airport resources. In more detail, the airports capacity cannot supply enough for demand, however its infrastructure cannot be expanded in the short term to meet its requirement. This airport is called by “coordinated”. In reality, 170 of the major airports around the world are schedule coordinated airports (IATA, 2014a).

Markov decision model is a useful and powerful tool for understanding probabilistic sequential decision processes with an infinite planning horizon. Focusing on a discrete-time MDP model of Admission and Inventory Control in Service Facility Systems, the time of operation of the system is divided into periods of time t >0. Decisions are taken at the beginning of each periods (epochs) to control both admission to service and inventory replenishment. Assume that we have 2 kinds of queue, including eligible queue and potential queue. Customers is transferred by the Admission control system at decision epochs from potential queue to eligible queue, either reject or admit. The demand for the services and the service times are assumed to have time invariant probability distributions g() (the arrival of customer in each periods) and f() respectively. Let the revenue be constant throughout the time period. There are 3 types of cost. The system operator earns R for every completely served customer and there is a charge when holding x items in inventory and when there are y customers in the system, i.e. h(x) and k(y) respectively. The maximum inventory is assumed to be M. The MDP model takes account on average cost to find the optimal policy to be implemented for the system.

The last mile has emerged as a critical source of opportunity for cost efficiency. And the integration of drones into last mile delivery is quickly becoming attractive due to the potential for reducing labor costs and improving fuel efficiency. Immediately, there is two question appearing. The first one is why should we use drone, and the second one is why using truck. Apparently, there is four main plus points of operating a drone for delivery. Firstly, it can run without human. Secondly, the traffic jam in road networks is no longer needed to be considered because it is flying over. Thirdly, it is obvious that it runs faster, and then it will take less time to reach customers than trucks. Finally yet importantly, transportation costs per kilometre when operating drone is considerably lower since truck is very heavy and consume more energy.