Probability masses fitting in the analysis of manufacturing flow lines

A new alternative in the analysis of manufacturing systems with finite buffers is presented. We propose and study a new approach in order to build tractable phase-type distributions, which are required by state-of-the-art analytical models. Called "probability masses fitting" (PMF), the approach is quite simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval, leading to a discrete distribution. PMF shows some interesting properties: it is bounding, monotonic and it conserves the shape of the distribution. After PMF, from the discrete phase-type distributions, state-of-the-art analytical models can be applied. Here, we choose the exactly model the evolution of the system by a Markov chain, and we focus on flow lines. The properties of the global modelling method can be discovered by extending the PMF properties, mainly leading to bounds on the throughput. Finally, the method is shown, by numerical experiments, to compute accurate estimations of the throughput and of various performance measures, reaching accuracy levels of a few tenths of percent.stochastic modelling, flow lines, probability masses fitting, discretization, bounds, performance measures, distributions.

A tight bound on the throughput of queueing networks with blocking

In this paper, we present a bounding methodology that allows to compute a tight lower bound on the cycle time of fork--join queueing networks with blocking and with general service time distributions. The methodology relies on two ideas. First, probability masses fitting (PMF) discretizes the service time distributions so that the evolution of the modified network can be modelled by a Markov chain. The PMF discretization is simple: the probability masses on regular intervals are computed and aggregated on a single value in the orresponding interval. Second, we take advantage of the concept of critical path, i.e. the sequence of jobs that covers a sample run. We show that the critical path can be computed with the discretized distributions and that the same sequence of jobs offers a lower bound on the original cycle time. The tightness of the bound is shown on computational experiments. Finally, we discuss the extension to split--and--merge networks and approximate estimations of the cycle time.queueing networks, blocking, throughput, bound, probability masses fitting, critical path.

How stochasticity and emergencies disrupt the surgical schedule

In health care system, the operating theatre is recognized as having an important role, notably in terms of generated income and cost. Its management, and in particular its scheduling, is thus a critical activity, and has been the sub ject of many studies. However, the stochasticity of the operating theatre environment is rarely considered while it has considerable effect on the actual working of a surgical unit. In practice, the planners keep a safety margin, let’s say 15% of the capacity, in order to absorb the effect of unpredictable events. However, this safety margin is most often chosen sub jectively, from experience. In this paper, our goal is to rationalize this process. We want to give insights to managers in order to deal with the stochasticity of their environment, at a tactical–strategic decision level. For this, we propose an analytical approach that takes account of the stochastic operating times as well as the disruptions caused by emergency arrivals. From our model, various performance measures can be computed: the emergency disruption rate, the waiting time for an emergency, the distribution of the working time, the probability of overtime, the average overtime, etc. In particular, our tool is able to tell how many operations can be scheduled per day in order to keep the overtime limited.health care, surgical schedule, emergencies, Markov chain.

Balancing partner preferences for logistics costs and carbon footprint in a horizontal cooperation

Horizontal cooperation in logistics has gathered momentum in the last decade as a way to reach economic as well as environmental benefits. In the literature, these benefits are most often assessed through aggregation of demand and supply chain optimization of the partnership as a whole. However, such an approach ignores the individual preferences of the participating companies and forces them to agree on a unique coalition objective. Companies with different (potentially conflicting) preferences could improve their individual outcome by diverging from this joint solution. To account for companies preferences, we propose an optimization framework that integrates the individual partners’ interests directly in a cooperative model. The partners specify their preferences regarding the decrease of logistical costs versus reduced CO2 emissions. Doing so, all stakeholders are more likely to accept the solution, and the long-term viability of the collaboration is improved. First, we formulate a multi-objective, multi-partner location-inventory model. Second, we distinguish two approaches for solving it, each focusing primarily on one of these two dimensions. The result is a set of Pareto-optimal solutions that support the decision and negotiation process. Third, we propose and compare three different approaches to construct a unique solution which is fair and efficient for the coalition. Extensive numerical results not only confirm the potential of collaboration but, more importantly, also reveal valuable managerial insights on the effect of dissimilarities between partners with respect to size, geographical overlap and operational preferences

The bounding discrete phase-type method

Models of production systems have always been essential. They are needed at a strategic level in order to guide the design of production systems but also at an operational level when, for example, the daily load and staffing have to be chosen. Models can be classified into three categories: analytical, simulative and approximate. In this paper, we propose an approximation approach that works as follows. Each arrival or service distribution is discretized using the same time step. The evolution of the production system can then be described by a Markov chain. The performances of the production system can then be estimated from the analysis of the Markov chain. The way the discretization is carried on determines the properties of the results. In this paper, we investigate the 'grouping at the end' discretization method and, in order to fix ideas, in the context of production lines. In this case, upper and lower bounds on the throughput can be derived. Furthermore, the distance between these bounds is proved to be related to the time step used in the discretization. They are thus refinable and their precision can be evaluated a priori. All these results are proved using the concept of critical path of a production run. Beside the conceptual contribution of this paper, the method has been successfully applied to a line with three stations in which three buffer spaces have to be allocated. Nevertheless, the complexity and solution aspects will require further attention before making the method eligible for real largescale problems

Modelling queueing networks with blocking using probability mass fitting

In this thesis, we are interested in the modelling of queueing networks with finite buffers and with general service time distributions. Queueing networks models have shown to be very useful tools to evaluate the performance of complex systems in many application fields (manufacturing, communication networks, traffic flow, etc.). In order to analyze such networks, the original distributions are most often transformed into tractable distributions, so that the Markov theory can then be applied. Our main originality lies in this step of the modelling process. We propose to discretize the original distributions by probability mass fitting (PMF). The PMF discretization is simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval. PMF has the advantage to be simple, refinable, and to conserve the shape of the distribution. Moreover, we show that it does not require more phases, and thus more computational effort, than concurrent methods. From the distributions transformed by PMF, the evolution of the system can then be modelled by a discrete Markov chain, and the performance of the system can be evaluated from the chain. This global modelling method leads to various interesting results. First, we propose two methodologies leading to bounds on the cycle time of the system. In particular, a tight lower bound on the cycle time can be computed. Second, probability mass fitting leads to accurate approximation of the performance measures (cycle time, work-in-progress, flow time, etc.). Together with the bounds, the approximations allow to get a good grasp on the exact measure with certainty. Third, the cycle time distribution can be computed in the discretized time and shows to be a good approximation of the original cycle time distribution. The distribution provides more information on the behavior of the system, compared to the isolated expectation (to which other methods are limited). Finally, in order to be able to analyze larger networks, the decomposition technique can be applied after PMF. We show that the accuracy of the performance evaluation is still good, and that the ability of PMF to accurately estimate the distributions brings an improvement in the application of the decomposition. In conclusion, we believe that probability mass fitting can be considered as a valuable alternative in order to build tractable distributions for the analytical modelling of queueing networks.(FSA 3) -- UCL, 200

Probability masses fitting in the analysis of manufacturing flow lines

A new alternative in the analysis of manufacturing systems with finite buffers is presented. We propose and study a new approach in order to build tractable phase-type distributions, which are required by state-of the- art analytical models. Called "probability masses fitting" (PMF), the approach is quite simple: the probability masses on regular intervals are computed and aggregated on a single value in the corresponding interval, leading to a discrete distribution. PMF shows some interesting properties: it is bounding, monotonic and it conserves the shape of the distribution. After PMF, from the discrete phase-type distributions, state-of-the-art analytical models can be applied. Here, we choose the exactly model the evolution of the system by a Markov chain, and we focus on flow lines. The properties of the global modelling method can be discovered by extending the PMF properties, mainly leading to bounds on the throughput. Finally, the method is shown, by numerical experiments, to compute accurate estimations of the throughput and of various performance measures, reaching accuracy levels of a few tenths of percent

How stochasticity and emergencies disrupt the surgical schedule

In health care system, the operating theatre is recognized as having an important role, notably in terms of generated income and cost. Its management, and in particular its scheduling, is thus a critical activity, and has been the sub ject of many studies. However, the stochasticity of the operating theatre environment is rarely considered while it has considerable effect on the actual working of a surgical unit. In practice, the planners keep a safety margin, let’s say 15% of the capacity, in order to absorb the effect of unpredictable events. However, this safety margin is most often chosen sub jectively, from experience. In this paper, our goal is to rationalize this process. We want to give insights to managers in order to deal with the stochasticity of their environment, at a tactical–strategic decision level. For this, we propose an analytical approach that takes account of the stochastic operating times as well as the disruptions caused by emergency arrivals. From our model, various performance measures can be computed: the emergency disruption rate, the waiting time for an emergency, the distribution of the working time, the probability of overtime, the average overtime, etc. In particular, our tool is able to tell how many operations can be scheduled per day in order to keep the overtime limited