A helpful extension for this work are fork-join networks. When multiple concurrent activities are performed in a station, in such a way that a customer leaves when all the activities are finished, the network is no longer of product form. These nodes are known as fork-join nodes. Dietz and Jenkins  extend the MVA to approximate the mean performance measures of these networks, to which we refer as IMVA.
The goal of this work is to improve a cyclic transportation system performance by judicious joint resource assignment for fleet and maintenance capacity. We adopt a business centered multi-criteria analysis, considering production and maintenance areas in an integrated optimization methodology
Three KPIs are considered under a balanced scorecard framework: global cost rate, machine availability and system throughput.The system’s KPI are obtained using an improved version of the Mean Value Analysis (IMVA) presented in D.C. Dietz, R.C. Jenkins , Analysis of Aircraft Sortie Generation with the Use of a Fork-Join Queueing Network Model, Naval Research Logistics, 44(2), pp. 153-164, 1997. Du and Hall  study the fleet size and distribution of transport equipment in complex networks.
Fork-Join Queueing Network ModelA fork-join station offers several kinds of jobs (sub- stations) to an arriving machine. Each job has its own resources and waiting queue. When a machine enters a fork-join station i, it may enter each of the Ki sub-stations with fixed probabilities qik. This defines a set S of required jobs each time a machine enters. A machine leaves the fork-join station j when it has completed all the required jobs.
We consider four cost sources: cost for idle servers at the operating system, cost for idle equipment that is available, idle servers at the repair facility, and unavailable equipmentWhere cb is the idle production servers cost rate (per server), cl the idle equipment cost rate, Nb is the number of production servers, N the number of machines, Db the idle time fraction of production blocks, and Dlb the idle time fraction of available machines due to queues at the production servers. Dlr the machine unavailability. For a job k, crk is the idle resource cost rate, rk are the resource units, and Dkr the idle resource fractionThe mean value analysis is an analytical method that allows an efficient estimation of steady-state expected performance measures of a closed or capacitated queuing network [13, 30]. It is suitable for product-form networks, with symmetric service disciplines (first come, first served exponential, processor sharing, infinite server, or last come, first served preemptive) . The main principle of this method is to calculate the mean queue lengths, and other indicators such as service times, throughput and utilization rates. It does so recursively on the number of customer in the network.
To illustrate the application of the model, we present a case study adapted from Huang et al. . Let us consider an underground mining operation, with sub- level caving mining method (figure 4). There are 10 mining blocks producing fragmented ore. A fleet of 12 LHD has been assigned to the operation. At any given time, a LHD can be in any of the following four states: operating, idle at the block queue, idle at the maintenance queue, and being repaired. We as- sume the time between maintenance and repair actions λ follows an exponential distribution [3, 5, 29].
Pascual Martinez Giesen
Mining Center - Department of Transport Engineering and Logistics Engineering School Pontificia Universidad CatólicaA Joint optimization Approach for Determining Fleet Size and Maintenance Capacity Rodrigo Pascual, Alejandro Martínez, Ricardo Giesen Workshop LAND TRANSLOG II, Puerto Varas, Chile, December, 2011
Outline• Motivation• Model Formulation• Case Study• Conclusions
Model Formulation• Huang and Kumar (1994) consider a similar problem and proposed the sum of the cost of idle operating servers, and the cost due to idle machines that are ready to use• We consider four cost sources: cost for idle servers at the operating system, cost for idle equipment that is available, idle servers at the repair facility, and unavailable equipment• The idle time fractions and machine unavailability can be expressed in terms of mean network metrics using the Improved Mean Value Analysis (IMVA)
Conclusions• An integrated perspective is needed – Physical assets management• Opportunity costs are relevant – It is not only about direct costs• Repair resources flexibility allows to obtain global cost savings and throughput improvements• Bonus: fast analytical model
Mining Center - Department of Transport Engineering and Logistics Engineering School Pontificia Universidad Católica Thanks!A Joint optimization Approach for Determining Fleet Size and Maintenance Capacity Alejandro Martínez – email@example.com Rodrigo Pascual – firstname.lastname@example.org Ricardo Giesen– email@example.com Workshop LAND TRANSLOG II, Puerto Varas, Chile, December, 2011