Michael I. Vidalis

Exact analysis of a discrete material three-station one-buffer merge system with unreliable machines

In this paper, a model of a discrete material flow line consisting of three unreliable machines and one buffer of limited capacity is analysed. A similar system, but with continuous flow of material was examined by Helber and Mehrtens (2001) and Tan (2001). In our system it is assumed that the buffer has two immediate preceding machines, performing the same operations and one immediate succeeding machine that receives material from the buffer. For the case where the buffer reaches its own capacity, one of the two preceding machines has priority over the other to dispose its processed part into the buffer. Processing times are assumed to be deterministic and identical for all machines and are taken as the time unit. Geometrically distributed operation dependent failures at the machines are assumed. All possible transition equations for the examined model are derived and a recursive algorithm that generates the transition matrix for any value N of the storage level is developed. Once the transition matrix is known the performance measures of the model under consideration can be easily evaluated. This model may be used as a building block in a decomposition method to evaluate large production systems with split/merge operations (for example, flow lines with quality inspections and rework loops).

Optimal buffer allocation in short m -balanced unreliable production lines

Abstract In this work, we investigate the optimal buffer allocation in short μ -balanced production lines consisting of machines that are subject to breakdown. Repair times and times to failure are assumed exponential, whereas service times are allowed to follow the Erlang- k distribution (with k =1, 2, 4 and 8). By an improved enumeration procedure and applying the evaluative algorithm of Heavey et al. (European Journal of Operational Research 1993;68:69–89) for the calculation of throughput, we have examined in a systematic way several systems with 3, 4, 5, 6 and 7 stations and with a different total number of buffer slots. We have been able to give answers to some critical questions. These include the effect of the distribution of the service and repair times, the availability of the stations and the repair rates on the optimal buffer allocation and the throughput of the lines.

Performance evaluation of a merge supply network: A distribution centre with multiple reliable random suppliers

In this study a two echelon discrete flow convergent supply network system is examined. Multiple reliable suppliers with different processing rates feed a distribution centre with a shared buffer. The first echelon consists of a distribution centre and a shared buffer. The second echelon includes multiple non-identical suppliers that feed the distribution centre. Each supplier has their service rate. The active lead times (production & delivery time) are exponentially distributed. The supply network is modeled as a continuous time Markov process with discrete states. The structure of the transition matrices of these specific systems is explored and a computational algorithm is developed to generate stationary distribution for different values of the system characteristics. The effect of certain system characteristics, such as the capacity of the distribution centre, the buffer capacity and the number of suppliers, on the system performance measures, such as the output rate (throughput), the work in process in the system and the distribution centre are traced through numerical experiments. The computational algorithm is used as a design tool to optimise the systems performance, i.e. to achieve the maximisation of throughput, the minimisation of wip, etc.

Markovian Analysis of a Push-Pull Merge System with Two Suppliers, An Intermediate Buffer, and Two Retailers

This paper examines a push-pull merge system with two suppliers, two retailers and an intermediate buffer distribution centre. Two reliable non identical suppliers performing merge operations feed a buffer that is located immediately upstream of two non-identical reliable retailers. External customers arrive to each retailer with non-identical inter-arrival times that are exponentially distributed. The amount ordered from each retailer by a customer is exactly one unit. The material flows between upstream stages suppliers is push type, while between downstream stages retailers it is driven by continuous review, reorder point/order quantity inventory control policy s,S. Both suppliers and retailers have exponential service rates. The considered system is modelled as a continuous time Markov process with discrete states. An algorithm that generates the transition matrix for any value of the parameters of the system is developed. Once the transition matrix is known the stationary probabilities can be computed and therefore the performance measures of the model under consideration can be easily evaluated.

Performance evaluation of a push–pull merge system with multiple suppliers, an intermediate buffer and a distribution centre with parallel machines/channels

This paper examines a push–pull merge system with external demand. Multiple reliable non-identical suppliers feed a buffer that is located immediately upstream a distribution centre (DC) with parallel identical reliable machines. The DC performs another operation on the items stored in the preceding buffer and the finished products are stored in another buffer (the finished products buffer) immediately downstream the DC. Customers arrive to the system according to a Poisson process with given intensity λ and remove a finished product from the buffer of finished products. The size of a customer demand is equal to one. Both suppliers and the identical machines at DC have exponential service rates. The considered system is modelled as a continuous-time Markov process with discrete states. An algorithm that generates the transition matrix for any value of the parameters of the system is developed and all possible transition equations are derived and solved analytically. Once the transition matrix is known the performance measures of the model under consideration can be easily evaluated.

Performance Evaluation of a Push Merge System with Multiple Suppliers, an Intermediate Buffer and a Distribution Center with Parallel Channels: The Erlang Case

In this paper, the most important performance measures of a two stage, push merge system are estimated. More specifically, S non identical and reliable suppliers send material to a distribution center (DC) of one product type. The DC has N possible parallel identical reliable machines. Between the suppliers and the DC a buffer with unlimited capacity, is located in order the material flow to be controlled. All stations processing and replenishment times are assumed stochastic and follow the Erlang distribution. The considered model is developed as Markov Process with discrete states where the Matrix Analytic method is used to calculate the system stationary probabilities and performance measures.

Performance Evaluation of a Lost Sales, Push-Pull, Production-Inventory System Under Supply and Demand Uncertainty

A three stages, linear, push-pull, production-inventory system is investigated. The system consists of a production station, a finished goods buffer, and a retailer following continuous review (s, Q) policy. Exponentially distributed production and transportation times are assumed. External demand is modeled as a compound Poisson process, and a lost sales regime is assumed. The system is modeled as a continuous time—discreet space Markov process using Matrix Analytic methods. An algorithm is developed in MatLab to construct the transition matrix that describes the system for different parameters. The resulting system of linear equations provides the vector of the stationary probabilities, and then key performance measures such as customer service levels, average inventories etc. are computed. The proposed model can be used as a descriptive model to explore the dynamics of the system via different scenarios concerning structural characteristics. Also, it may be used as an optimization tool in the context of a prescriptive model.

Coordinating push and pull flows in a lost sales stochastic supply chain.

In this paper a serial, three echelon, push-pull supply chain is investigated. The supply chain consists of a provider, a distribution centre (buffer) and a retailer. The material flow between upstream stages is push type, while between downstream stages it is driven by continuous review, reorder point/order quantity inventory control policy. Exponentially distributed lead times between stages are assumed. External demand occurs according to pure Poisson, while the demand that cannot be met is lost. The system is modelled using matrix analytic methods as a Markov birth-and-death process. An algorithm is developed to generate the transition matrix for different parameters of the system. Then, the corresponding system of stationary linear equations is generated and the solution of the stationary probabilities is provided. Key performance metrics such as average inventories and customer service levels at each echelon of the system can be computed. The algorithm is programmed in Matlab© and its validity is tested using simulation, with the two approaches giving practically identical results. The contribution of our work is an exact algorithm for a lost sales push-pull supply network. This algorithm can be used to evaluate different scenarios for supply chain design, to explore the dynamics of a push-pull system, or as an optimization tool.