MATHEMATICAL FORMULATION

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OEE is formulated as a function of a number of mutually exclusive components, such as availability efficiency, performance efficiency, and quality efficiency in order to quantify various types of productivity losses.

OEE is a value variable from 0 to 100%. An high value of OEE indicates that machine is operating close to its maximum efficiency. Although the OEE does not diagnose a specific reason why a machine is not running as efficiently as possible, it does give some insight into the reason1. It is therefore possible to analyze these areas to determine where the lack of efficiency is occurring: breakdown, set-up and adjustment, idling and minor storage, reduced speed, and quality defect and rework2 3.

In literature exist a meaningful set of time losses classification related to the three reported efficiencies (availability, performance and quality). Grando et al.4 for example provided a meaningful and comprehensive classification of the time-losses that affect a single equipment, considering its interaction in the interaction system. Waters et al.5 and Chase et al.6 showed a variety of acknowledged possible efficiency losses schemes, while Nakajima7 defined the most acknowledged classification of the “6 big losses”.

In accordance with Nakajima notations, the conventional formula for OEE can be written as follow8:

$OEE= A_{eff} Pe_{eff} Q_{eff}$

$A_{eff}= \frac{T_{u}}{T_{t}}$

$Pe_{eff}= \frac{T_{p}}{T_{u}}\ast \frac{R_{avg}^{\left ( a \right )}}{R_{avg}^{\left ( th \right )}}$

$Q_{eff}= \frac{P_{g}}{P_{a}}$

Table 3.4 summarizes briefly each factor.

Table 3.4 OEE factors description
Factor	Description
A_eff	Availability efficiency. It considers failure and maintenance downtime and time devoted to indirect production task (e.g. set up, changeovers).
Pe_eff	Performance efficiency. It consider minor stoppages and time losses caused by speed reduction
Q_eff	Quality efficiency. It consider loss of production caused by scraps and rework.
T_u	Equipment uptime during the T_t. It is lower that T_t because of failure, maintenance and set up.
T_t	Total time of observation.
T_p	Equipment production time. It is lower than T_t because of minor stoppages, resets, adjustments following changeovers.
$R_{avg}^{\left ( a \right )}$	Average actual processing rate for equipment in production for actual product output. It is lower than theoretical (R(th)avg) because of speed/production rate slowdowns.
$R_{avg}^{\left ( th \right )}$	Average theoretical processing rate for actual product output.
P_g	Good product output from equipment during T_t.
P_a	Actual product units processed by equipment during T_t. We assume that for each product rework the same cycle time is requested.

The OEE analysis, if based on single equipment data, is not sufficient, since no machine is isolated in a factory, but operates in a linked and complex environment9. A set of inter-dependent relations between two or more equipments of a production system generally exists, which leads to the propagation of availability, performance and quality losses throughout the system.

Mutual influence between two consecutive stations occurs even if both stations are working ideally. In fact if two consecutive stations (e.g. station A and station B) present different cycle times, the faster station (eg. Station A = 100 pcs/hour) need to reduce/stop its production rate in accordance with the other station production rate (e.g. Station B = 80 pcs/hour).

Station A	Station B
100 pcs/hour	80 pcs/hour

In this case, the detected OEE of station A would be 80%, even if any efficiency loss occurs. This losses propagation is due to the unbalanced cycle time.

Therefore, when considering the OEE of equipment in a given manufacturing system, the measured OEE is always the performance of the equipment within the specific system. This leads to practical consequence for the design of the system itself.

A comprehensive analysis of the production system performance can be reached by extending the concept of OEE, as the performance of individual equipment, up to factory level10. In this sense OEE metric is well accepted as an effective measure of manufacturing performance not only for single machine but also for the whole production system11 and it is known as Overall Throughput Effectiveness OTE12 13.

We refer to OTE as the OEE of the whole production system.

Therefore we can talk of:

Equipment OEE, as the OEE of the single equipment, which measures the performance of the equipment in the given production system.
System OEE (or OTE), which is the performance of the whole system and can be defined as the performance of the bottleneck equipment in the given production system.

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