William Gousseau
email: william.gousseau@insa-lyon.fr

Jérôme Antoni
email: jerome.antoni@insa-lyon.fr

François Girardin
email: francois.girardin@insa-lyon.fr

Julien Griffaton
email: julien.griffaton@snecma.fr

Analysis of the Rolling Element Bearing data set of the Center for Intelligent Maintenance Systems

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Motivations

Todays, vibration-based diagnosis and prognostics of rolling element bearings attracts more and more attentation in the scientific community. Therefore, providing open-access databases is of prime importance. There are so far few available datasets. The Case Western Reserve University data [START_REF] Smith | Rolling element bearing diagnostics using the Case Western Reserve University data: A benchmark study[END_REF] are one example dedicated to diagnostics which has been widely used in numerous works. The IMS dataset is another example which constitutes a precious database for prognostics. Indeed, endurance tests with natural degradation are rare. In most cases, damage is artificially initiationed in order to accelerate the damage growth. Whereas for diagnosis studies seeded faults may be acceptable, prognostis requires an evolution of the damage starting ideally from a natural initiation. The IMS dataset has therefore interesting potential both for diagnosis and (mostly) for prognostics.

As opposed to several studies based on the direct application of some ad hoctrending features (such as statistical moments), this paper follows an explicative approach where the vibration signals of the IMS dataset are thoroughly analysed by dedicated signal processing tools.

IMS dataset in the literature

This IMS database has been so far used in different research works published in the specialized literature. Many papers used it for illustrating remaining useful life (RUL) estimation based on statistical indicator [START_REF] Mahamad | Predicting remaining useful life of rotating machinery based artificial neural network[END_REF][START_REF] Soualhi | Prognosis of bearing failures using hidden Markov models and the adaptive neuro-fuzzy inference system[END_REF][START_REF] Ali | Accurate bearing remaining useful life prediction based on Weibull distribution and artificial neural network[END_REF][START_REF] Ali | A new enhanced feature extraction strategy for bearing Remaining Useful Life estimation[END_REF] (Mahamad & all 2010) (Soualhi & all 2013) (Ben Ali & all 2015) (Ben Ali & all 2014). Other papers used it to test new signal denoising techniques [START_REF] Guo | Machinery vibration signal denoising based on learned dictionary and sparse representation[END_REF] (Guo & all 2015) often with the goal of acheiving an early detection of the damage [START_REF] Qiu | Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics[END_REF][START_REF] Mortada | Diagnosis of rotor bearings using logical analysis of data[END_REF] (Qiu & all 2006) (Mortada & all 2011). Paper of Qiu and all [START_REF] Qiu | Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics[END_REF] is the reference paper concerning this dataset.

Dataset description

The test rig

The IMS bearing dataset has been collected on an endurance test rig of the University of Cincinnati and relased in 2014 [START_REF]IMS bearings dataset[END_REF] . The test rig (shown in Fig. 1) has the following characteristics:

• 4 double row bearings of type Rexnord ZA-2115,

• 2000 rpm stationary speed,

• 6000 lbs load applied onto the shaft and bearing by a spring mechanism,

• PCB 253B33 High sensitivity Quart ICP ® accelerometers.

An AC motor, coupled by a rub belt, keeps the rotation speed constant. The four bearings are in the same shaft and are forced lubricated by a circulation system that regulates the flow and the temperature. It is announced on the provided "Readme Document for IMS Bearing Data" in the downloaded file, that the test was stopped

Figure 1. IMS test rig layout

when the accumulation of debris on a magnetic plug exceeded a certain level indicating the possibility of an impending failure.

The four bearings are all of the same type. There are double range pillow blocks rolling elements bearing. Their characteristics are given in table 1.

Data description

Three datasets are provided on the downloaded file, composed of numerous files of one second each. Each file is made of 20,480 samples. Althougth it is mentioned that the sampling frequency is 20 kHz, it is thus believed that it was actually 20.48 kHz(and the spectral analyses presented in this paper all seem to support this assumption, since more consistent results are then obtained with respect to expected fault frequencies). A one second acquisition has been made every ten minutes. There is an exception for the first dataset for which the first forty-three files have been acquired every five minutes.

The timeline of the signals files gives important information about the recording. In particular time intervals are identified in the first dataset where recording was switched off.

The description of the datasets is given the table 2 Concerning dataset 1, there are two accelerometers on each bearing (x and y positions). The two other datasets only have one accelerometer on each bearing. The fault frequencies required for the diagnosis of the rolling element bearing have been calculated from the bearing characteristics. They are reported in table 3.

Time analysis

Before proceeding to sophisticated signal processing tools, a simple analysis in the time domain can already provide interesting information. Another point concerns the announced faults. There is no fault on bearing 1 and 2. Bearing 1 and 2 are considered as healthy and their analyses are not shown hereafter.

Dataset 2 and 3 have been recorded without interruption. Records of 1s -hereafter denoted as frames -have been captured every 10 min all along the endurance test.

Methodology

This section introduces the signal processing methods which are used to extract the diagnostic information contained in the three endurance datasets. Two preprocessing methods are first presented which have been used to enhance the signal.

Preprocessing

Figure 2. Raw signal and its real time history scale for dataset 1

In order to enhance the results, pre-whitening and blind deconvolution are used in a pre-processing stage.

The pre-whitening operation used here consists in normalizing the spectrogram by its averge value taken over fifty frames. This is supposed to remove the constant baseline endured all along the test, which includes structural resonances of the system as well as s of the machinery (belt, motor). This operation has been found important to better enhance the contrast between the beginning and the end of the test. For example, a spectral peak at 236 Hz appears in all frames all along the endurance, which in all likelihood corresponds to the ball passage frequency seen by the sensors which are placed very close to the bearing outer race. This frequency is therefore not to be confused with that of an outer race fault, the reason why it is important to remove from the beginning. Applying pre-whitening will thus avoid confusion in the diagnosis.

Blind deconvolution is applied to the time signal in order to enhance the emergence of impulses in the signal due to the initiation of the fault. The blind deconvolution algorithm consists in maximizing the Kurtosis [START_REF] Endo | Enhancement of autoregressive model based gear tooth fault detection technique by the use of minimum entropy deconvolution filter[END_REF] .

Signal processing

This section introduces the signal processing tools used to detect the possible presence of damage as early as possible in the endurance test. They pertain to the current state-of-the-art in the domain [START_REF] Randall | Rolling element bearing diagnostics -a tutorial[END_REF] . Time-frequency analysis is based on the Short Time Fourier Transform on each bloc of one second, with a 1 Hz frequency resolution without overlap (because of time discontinuity between each frame). Results are represented by means of the spectrogram on a logarithmic scale. This allows easy tracking of the frequencies along the tests.

The Squared Envelop Spectrum (SES) is used as indicated in Ref. (11) after filtering the signal in an optimal band that maximized the kurtosis as found by application of the Kurtogram [START_REF]Fast computation of the kurtogram for the detection of transient faults[END_REF] . As demonstrated in the literature, bearing defaults produce cyclostationary signals. The spectral coherence [START_REF] Randall | The relationship between spectral correlation and envelope analysis in the diagnostics of bearing faults and other cyclostationary machine signals[END_REF] has been found as anefficient tool to detect the presence of cyclostationarity and to identify the bearing fault frequency.

Results

This section presents diagnostic results based on the signal processing tools introduced above. Results are presented by means of tables and figures for each dataset. The goal is to confirm that diagnosis of the bearing faults is feasible and, in certain cases, to explain some unexpected results.

The following categorization of results is used:

Dataset 1, inner-race damage on bearing 3, ball damage on bearing 4

This dataset contains 2 accelerometers per bearing. Since analyses carried out on sensors placed on the same bearing are found quite similar, only results obtained on one of the two accelerometers are shown. This observation could indeed explain why there is only one accelerometer available per bearing for the other datasets.

Weak harmonics are visible at twice and three times the BPFI (table 3) with modulation sidebands spaced by the shaft speed(33.3 Hz). This signature is typical of an inner race fault. In the reference paper [START_REF] Qiu | Wavelet filter-based weak signature detection method and its application on rolling element bearing prognostics[END_REF] is also shown a BPFO fault at the end of endurance on bearing 4. It is also visible on the analyses but not shown in this paper.

In the spectrogram (Fig. 4) the inner race fault signature is visible from 33,8 days, as evidenced by a spectral comb. Twice and three times the BPFI are the dominant frequencies. The shaft modulations, typical of an inner race fault, are also easily identified. This is so considered as a TP.

For test 1, the SES has been computed in the entire frequency bandwidth -see Fig. 5. Results are close to those of the spectrogram, with dominant frequencies at twice and three times the BPFI. However, the fault signature is visible slightly sooner, from 32 days. The sideband due to shaft modulations are also visible at 34 days.

The spectral coherence probably returns the most interesting results. Although the detected frequencies are the same (twice and three times the BPFI), the signature of the inner race fault is visible from 29.2 days (Fig. 6). The shaft modulation sideband is also visible later, around 32 days. The BSF (table 3) and harmonics are difficult to find and appear late in the endurance test.

The signature of the fault in the spectrogram is only visible at the very end of the test, around 34 days, with clear modulation sidebands ate the shaft speed (Fig. 7). Only the sixth BSF harmonics can be identified from 18 days, with FTF modulations. This is enough the diagnose the ball fault, which is thus categorized as a TP.

The SES has been computed in the entire frequency bandwidth. Although the signature is weaker than in the spectrogram, its is more consistant with a ball fault: a peak is visible at twice the BSF beginning around 25 days. Some modulations at shaft speed are also visible at the very end, from 34 days. The spectral coherence returns similar results as the SES, with more pronounced FTF modulations. In addition, there is a higher increase of the the peak magnitudes at the BSF and its harmonics from 23 days. This case is also considered as a TP.

Table 5 summarizes the results concerning dataset 1. It reports the categorization of each analysis and the time from which the fault diagnosis seems possible.

Table 5. Summary of dataset 1 results

Inner race fault

Ball fault accelerometer acc3x acc3y acc4x acc4y acc3x acc3y acc4x acc4y

T-F analysis TP (33.

Dataset 2, outer race damage on bearing 1

This case is easy to diagnose, and all methods give relevant results. The spectrogram (see Fig. 10) provides enough information to make a first diagnosis. The fundamental frequency is not the most relevant ones to monitor here, in particular because if corresponds to the ball pass frequency seen by the sensor placed close to the outer-race, even in the absence of a fault. The three-time BPFO harmonic appears from around 3,5 The Squared Envelop Spectrum (see Fig. 11) has been computed after bandpass filtering in the optimal band between 6830 Hz and 10240 Hz as returned by the Kurtogram. Even if the BPFO harmonics are visible they appear later --from 4,8 days -than in the spectrogram. Once again,because of its dominance, the twice BPFO harmonic seems the most efficient to follow.

The spectral coherence returns the best diagnostics results. Again, the BPFO frequency appears to not be the best one to track; although it is clearly visible, it appears from the beginning of the test and increases only after 3,5 days. Twice and three times BPFO seem again more relevant because of their dominance This case is clearly considered as a TP. The three other accelerometers of dataset 2 show similar results which confirm the diagnosis made from accelerometer 1. Although they are located on other bearings (see Fig. 1), they are well able to sense the BPFO fault in bearing 1.. This is categorized here as a TP2. Having the possibility to monitor bearing from distant location can be considered as very useful.

Analysis of the signals reveals that the fault signature appears at the same time whatever the distance to the faulty bearing. The spectrogram, the SES and the spectral coherence of accelerometer 4 are nearly identical to those of accelerometer 1 shown in Fig. 10, 11, and 12.

Table (6) summarizes the results concerning dataset 2. It reportes the categorization of each analysis and the time from which the fault diagnosis seems possible.

Table 6. Summary of dataset 2 results

OUTER

Dataset 3, Outer-race damage on bearing 3

The description of dataset 3 is similar to dataset 2 except for two differences. First the outer race on announced inbearing 3 instead of bearing 1 at the end of the endurance test and second the signals are longer (about 30 days of 1s records every 10 minutes resulting in 72 min of signals). Although similar results as with dataset 2 are expected, they eventually turn out very different. None of thethe accelerometers are able to detect the fault.

All other accelerometers return identical results. These results probably explain why dataset 3 has never been used in the literature. It so considers as a FN.

Conclusion and discussion

This IMS database is frequently used in the specialized literature to test and validate the proposal of new diagnostic and prognositc algorithms. The goal of this paper was to establish in a comprehensive way which diagnostic information can be extracted from it. This should turn out most useful for its future use by other researchers.

The IMS dataset constitutes an interesting database for several reasons. First, signals are provided from a long endurance test with a natural degradation of the damage, without artificial initiation. Second, it includes three different types of fault with different expected signatures. Third, it contains signals simultaneous recordered by at least four accelerometers placed at different locations. Some of the main findings of the paper are the following.

1) It is not advisable to detect and track the fundamental frequency of the BPFO for diagnostic and prognostic purposes in Dataset 1 2 (outer-race fault); this frequency is present from the beginning of the test and corresponds to the natural passage of the balls as seen by accelerometer 1 mounted close to the outer race.

2) The spectral signature of the inner-race fault in dataset 1 is perhaps unusual as it mainly contains the second and thirs harmonics of the BPFI.

3) It is possible to detect the inner-race in bearing 3 fault and the roller in bearing 4 fault from accelerometers 1 and 2 mounted on the other bearings even if ot shown in this paper (even the outer-race in bearing 4). 4) It is possible to detect the outer-race in bearing 1 fault from accelerometers 2, 3 and 4 mounted on the other bearings.

  Dataset 1 has been subjected to a series of interruption during the recording, which makes the time history not continuous. This is illustrated in Fig 2 for the time signal of accelerometer 1 displayed as a function of the real time (and not recording time which makes the signal continuous).

Figure 3 .

 3 Figure 3. a) raw spectrogram, b) pre-whitened spectrogram

Figure 4 . 3 Figure 5 .

 435 Figure 4. Spectrogram, dataset 1 accelerometer 3

Figure 6 . 3 Figure 7 .

 637 Figure 6. Spectral Coherence, dataset 1 accelerometer 3

Figure 8 . 4 Figure 9 . Spectral Coherence, dataset 1 accelerometer 4 (

 8494 Figure 8. Squared Envelop Spectrum, dataset 1 accelerometer 4

Figure 10 . 1 Figure 11 .

 10111 Figure 10. Spectrogram, dataset 2 accelerometer 1

  Figure 12. Spectral Coherence, dataset 2 accelerometer 1

5 )

 5 Contrary to what is announced in Ref.(1), it seems that Dataset 3 does not contain any signature of an outer-race fault in bearing 3 (or any other bearing). 6) It seems that the sampling frequency announced in Ref.(1) should be fixed to 20.48 kHz (instead of 20kHz).

Table 1 . Bearing characteristics

 1

	Rexnord ZA-2115 Characteristics
	Pitch diameter	2.815 inch 71.5mm
	Rolling element diameter	0.331 inch 8.4mm
	Number of rolling element per row	16	16
	Contact angle	15.17°	15.17°
	Static load	6000 lbs	26690 N

Table 2 . Datasets description

 2

		Number	Number	Endurance	Duration of	Announced damages at
		of files	of	duration	recorded signal	the end of the endurance
			channels
	Dataset	2156	8	49680 min	36 min	Bearing 3: inner race
	1			34 days 12h		Bearing 4: rolling element
	Dataset	984	4	9840 min	16 min	Bearing 1: outer race
	2			6 days 20h
	Dataset	4448	4	44480 min	74 min	Bearing 3: outer race
	3			31 days 10h

Table 3 . Characteristic frequencies of the test rig

 3

	Characteristic frequencies
	Shaft frequency	33.3 Hz
	Ball Pass Frequency Outer race (BPFO)	236 Hz
	Ball Pass Frequency Inner race (BPFI)	297 Hz
	Ball Spin Frequency (BSF)	278Hz (2x139 Hz)
	Fundamental Train Frequency (FTF)	15 Hz

Table 4 . Categorisation of diagnosis

 4

	Category Explanation
	TP	« True Positive », when a fault has been announced and is confirmed by the
		diagnosis
	TP2	« True positive 2 », confirmation of what has been announced needing some
		more explanation.
	FN	« False Negative » when a fault has been announced but is not confirmed by
		the diagnosis
	TN	« True Negative » Nothing has been announced and the diagnosis confirms it
	FP	« False Positive » Nothing has been announced but the diagnosis detects
		some signatures

Table 7 .Summary of dataset 3 results

 7

			OUTER RACE FAULT
		acc1	acc2	acc3	acc4
	T-F analysis	FN	FN	FN	FN
	Envelop	FN	FN	FN	FN
	Spectral Coherence	FN	FN	FN	FN


Andrei P Kirilyuk
email: andrei.kirilyuk@gmail.com

OBJECTIVE FUNDAMENTAL REALITY STRUCTURE BY THE UNREDUCED COMPLEXITY DEVELOPMENT

We explain why exactly the simplified abstract scheme of reality within the standard science paradigm cannot provide the consistent picture of "truly fundamental" reality and how the unreduced, causally complete description of the latter is regained within the extended, provably complete solution to arbitrary interaction problem and the ensuing concept of universal dynamic complexity. We emphasize the practical importance of this extension for both particular problem solution and further, now basically unlimited fundamental science development (otherwise dangerously stagnating within its traditional paradigm).

Objective fundamental reality cannot rely on postulated "principles", "mathematical reality", or "observation results" Despite the growing verbal popularity of "emergent" reality and concepts, the standard, positivistic science and knowledge structure is deeply and rigorously fixed, devoid of any intrinsic emergence dynamics and thus, basically, evolution, which is only formally simulated by artificial insertion of the notorious time-parameter (cf. e. g. [START_REF] Prigogine | From Being to Becoming: Time and Complexity in the Physical Sciences[END_REF]). That is exactly why "more is different" [START_REF] Anderson | More Is Different: Broken symmetry and the nature of the hierarchical structure of science[END_REF] in that, actually quite special (see below) and now desperately "ending" kind of science, where no structure or behaviour is directly, explicitly obtained from simpler components and should therefore be considered as given, by inevitable invention of ad hoc postulates, technical "principles", and equally abstract new entities. This is the true reason behind all the difficulties and ambiguities around the "what is fundamental?" question, again only within that kind of positivistic, or "unitary", science and its simplification of reality, which can be rigorously specified within the extended and intrinsically complete knowledge basis (see below). Characteristic examples include the relation between quantum and classical behaviour (both of persistently ambiguous origin), relativity, gravity, and cosmology issues. And although the observed structure and dynamics for each case and level of reality can be technically postulated as an empirically based "model", with various (but always limited) degree of success, the dynamic emergence links between the truly fundamental entities of unreduced reality (and therefore their genuine origin) remain poorly specified, thus compromising the objectivity of the resulting order of the world.

P a g e | 2
The fact that these are not only "philosophical" problems of little practical or scientific importance is strongly supported by the accumulating new and persistent old "unsolvable problems" and "mysteries" of modern scholar science, which even falls in a strangely critical state, in the middle of its technical omnipotence, where the yet quite recent situation of science ending because of its quasi-total completeness is "suddenly" replaced by those growing numbers of "difficult problems" without solution in various fields, in striking similarity to the situation of imminent scientific revolution a hundred years ago. This quite practical importance of accumulated "general" problems of the official knowledge paradigm becomes even more evident and globally vital at higher levels of "truly complex" systems, in biology, medicine, ecology, and the entire civilisation development, critically depending on the growing unitary science impasse (in ever more striking contrast to the spectacular technological progress, strongly influencing also the fundamental science development).

And while the empirically based intuition underlies the attempts of positivistic science to put the broken unity of nature together again by arbitrary guesses and respective artificial adjustment of separated world components in terms of abstract postulated "principles", redundant entities and their ambiguous "observation results", the obtained "Frankensteinian" construction of mechanically fixed "mathematical reality" [START_REF] Tegmark | Our Mathematical Universe: My Quest for the Ultimate Nature of Reality[END_REF] does not and cannot show the intrinsic, interaction-driven coherence and life-supporting efficiency of unreduced world dynamics and evolution, thus confirming ever more the importance of genuine, dynamically emerging fundamental levels of reality. The latter can be unambiguously derived only within the intrinsically unified and causally complete image of reality, obtained with the help of essential, qualitative extension of simplified unitary science projection to the unreduced hierarchy of dynamically multivalued interaction results, or universal dynamic complexity [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | Extended Mathematics of Unreduced Dynamic Complexity: The Exact Image of Unified Reality, from the Electron to Consciousness[END_REF][START_REF] Kirilyuk | The Unified Complex-Dynamic Origin of Time, Intention, Life, and Everything[END_REF], providing consistent solutions to old and new problems, within the rigorously specified objective structure of fundamental reality emerging as the multilevel hierarchy of unreduced dynamic complexity.

Dynamically emerging reality of unreduced interaction complexity

In accord with the above programme of emergent and therefore objective fundamental reality, one should start its causally complete description with the unreduced solution of the underlying (arbitrary) interaction problem, beginning at its simplest possible configuration of the most fundamental level of reality that forms its material origin as such and then directly producing all its higher-level fundamental entities, in full agreement with their natural emergence, with all their intrinsic properties, dynamical laws and "fundamental principles", now unambiguously derived in their natural unity [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF] (instead of being only mathematically and separately "postulated" in usual theory description).

The mentioned simplest (and thus most fundamental) configuration of the global interaction process is obviously provided by two initially homogeneous continuous media, or "protofields", uniformly attracted to each other (and eventually giving rise to the electromagnetic and gravitational interaction forces respectively). If we analyse this interaction process rigorously and without any "model" or "perturbative" simplification (thus without any assumption of interaction weakness or special configuration), as permitted by the generalised effective potential method [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF], then we obtain the explicit dynamic emergence of P a g e | 3 elementary world structures, recognised as (massive) elementary particles, in the form of quantum beat processes. The latter are obtained as localised, highly nonlinear and internally chaotic self-oscillation processes of the homogeneously attracting protofields, which perform nonstop cycles of their local interaction-driven squeeze (or "reduction") and extension due to the intrinsic dynamic instability of the unreduced interaction process. It is important that each dynamic reduction of the particle's quantum beat process is centred on a new physical "point" of thus emerging space, chosen in a dynamically random fashion among many other, equally probable and incompatible, or "redundant", reduction centres.

We thus discover here, already at this lowest, most fundamental level of reality, the origin of the universal dynamic complexity within every real world structure, in the form of those dynamically multivalued, or redundant, results of any real, unreduced interaction process (which should, of course, be strong enough for noticeable structure formation, beyond insignificant fluctuations) [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF]. Those explicitly emerging and mutually incompatible configurations of a system or interaction process, called realisations, are forced, by the driving interaction itself, to permanently replace one another in truly and causally random order thus dynamically obtained and rigorously defined. Dynamic complexity C (intrinsically coupled to chaoticity) is universally specified then as a growing function of the number of system realisations N  (for integral complexity measures) or rate of their change (for dif- ferential complexity measures) equal to zero for the unrealistic case of only one system realisation, ()

C C N   , 0 dC dN   , (1) 0 C  (e. g. 0 ln C C N   or 0 ( 1) C C N   ).
In fact, it is only the latter unrealistic case of effectively zero-dimensional (point-like) projection of dynamically multivalued reality that is exclusively considered in usual, dynamically single-valued, or unitary, theory and science paradigm (including its complexity imitations), thus underlying all its specific features, with relative successes, unsolvable mysteries, and ambiguities of abstract postulated "fundamental" entities and properties (which cannot be consistently defined within this severely limited approximation).

While the detailed mathematical formalism of the universal complexity science can be found elsewhere [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF], leading to the new mathematics of complexity, we can summarise it here by a symbolic relation, expressing the emerging complex-dynamical (multivalued) structure of any real interaction process 01 ( , ,..., )

n V q q q :           0 1 0 1 , ,..., , , , ,..., n r r r n V q q q q q q      D , (1)
where 01 , ,..., n q q q are the degrees of freedom of interacting system components forming the causally complete, multivalued, or complex, system dynamics D , which includes a well-defined set of system realisations {} r  , with their respective, dynamically determined emergence probabilities {} r  , measured system density distributions 01 { ( , ,..., )} rn q q q  , and other relevant features. The dynamically entangled degrees of freedom of the resulting system structure on the right-hand side of eq. (1) (within every realisation) form the physically real space of the emerging complexity/structure level, 01 { ( , ,..., )} rn x x q q q  .

As realisations are permanently changing (appearing and disappearing) in dynamically random order, the observed system density () x  is obtained as the dynamically probabilistic sum of individual realisation densities ()

r x  : P a g e | 4     1 r N r xx       , (2)
where the sign  marks the special, dynamically probabilistic meaning of the sum, implying the unstoppable probabilistic emergence and disappearance of individual realisation contributions () r x



, with dynamic probabilities r  causally determined as

1,..., ; , 1 r r r r r rr N N N N N N                 , (3)
where N  is the total number of elementary realisations (determined by the number of in- teracting system mode combinations) and r N the number of elementary realisations grouped within the r-th actually observed, compound realisation (which is the unified selforganisation mechanism, see below).

The eventual multilevel, probabilistically fractal structure of the general interaction problem solution is implied in ( 1)-( 3), so that any real system structure (including the world as a whole) emerges as permanently probabilistically evolving, multilevel branches of the dynamically probabilistic fractal [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF], whose integral dynamic complexity C is always defined by a growing function of realisation number N  , ()

C C N   , 0 dC dN   , (1) 0
C  .
Returning to the first level of world structure emergence in the form of quantum beat processes within every elementary particle (to be compared with abstract "state vectors" in usual theory), we note that already this lowest level of fundamental world structure is made by complex-dynamical, multivalued processes with complexity values ( ) 0 CN   (and e. g. 100 N  for the electron). This leads, in particular, to the intrinsic and universally defined property of inertial mass, emerging without any additional entity (like the Higgs field and boson in the Standard Model), but simply due to the dynamic (spatial) randomness, or chaoticity, of the quantum beat process [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF]. Other intrinsic particle properties (electric charge, spin, interactions) are also explicitly obtained as dynamically emerging features of the protofield interaction result. The related dynamical, e. g. quantum and relativistic, laws are equally consistently derived as results of interaction complexity development, without any abstract postulates or formally imposed "principles". All those properties and laws emerge in their natural unity, simply by their clearly specified dynamic origin, thus solving the "unsolvable" problems of usual theory.

The fundamental entities of physically real space and time are obtained as emergent forms of unreduced dynamic complexity, reproducing the multilevel hierarchy of the latter [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF]. The naturally discrete space emerges as the material tissue of dynamically entangled interaction components (the two protofields at the most fundamental level), in the form of physical space "points" given by the size of localised realisation structure (its eigenvalue separation in the unreduced interaction problem solution) and elementary length given by the separation of neighbouring realisations (i. e. their eigenvalues). The unstoppably and irreversibly flowing time emerges as the non-material realisation change process, specified by its intensity, i. e. frequency of realisation change events, where both the flow and irreversibility of this physically real time are due to the dynamic multivaluedness of unreduced interaction results. It becomes clear why these physically real entities of time and space cannot be consistently specified within the unitary science description and paradigm. P a g e | 5

Fundamental particle interactions that naturally emerge, with their observed properties, in this interaction-driven structure formation process [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF] give rise to emergence of higher complexity levels, with their own, specific in detail, but universally described multivalued structures and processes, in the long general sequence of dynamic complexity development, or unfolding, determined eventually by the initial protofield interaction.

The entire world structure hierarchy, including its highest known levels of life and consciousness, is thus explicitly obtained as emergent, interaction-driven, physically real and dynamically unified fractal of realisation sets (1)-( 3), containing the clearly specified variety of dynamic regimes and obeying the single law of the universal symmetry, or conservation and transformation, of complexity, which includes the extended, causally complete versions of all (correct) particular laws and principles [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | Extended Mathematics of Unreduced Dynamic Complexity: The Exact Image of Unified Reality, from the Electron to Consciousness[END_REF][START_REF] Kirilyuk | The Unified Complex-Dynamic Origin of Time, Intention, Life, and Everything[END_REF]. The mentioned diversity of complex dynamic regimes includes the entire range of observed internally chaotic structures, from the externally quasi-regular multivalued self-organisation case (with multiple and constantly changing, but very similar realisations, thus solving the entropy-growth problem) to the ultimately irregular regime of global, or uniform, chaos (with strongly differing and "entangled" realisations), with the rigorously specified criterion of transition between them.

While the unified mathematical expression of this emergent world structure formation, dynamics and evolution can be found elsewhere [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | Extended Mathematics of Unreduced Dynamic Complexity: The Exact Image of Unified Reality, from the Electron to Consciousness[END_REF][START_REF] Kirilyuk | The Unified Complex-Dynamic Origin of Time, Intention, Life, and Everything[END_REF], what we want to emphasize here is the related natural, physically and mathematically complete solution to all "fundamentality problems", simply because all world's explicitly emerging structures, features and laws, obtained in the unreduced completeness of their observed properties, are both fundamental by their clearly specified dynamic origin and equally causally ordered in this fundamentality, according to respective values of rigorously and universally defined unreduced dynamic complexity (as outlined above).

As a particular example, we obtain the classical, permanently localised kind of system and behaviour as a higher level of unreduced dynamic complexity emerging from interaction of essentially quantum components (usually elementary particles) in the form of elementary bound systems, such as atoms [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF], without any ambiguous "decoherence" of usual theory, just arbitrarily breaking the fundamental natural order of emergent world structures. It is confirmed by many other examples from "physical" and higher complexity levels, including genuine quantum chaos in nanosystems and the complex-dynamic origin of life, intelligence and consciousness, remaining "mysteries" in unitary science [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF].

Beyond the horizon: The breakthrough research agenda

There is much more than mere "philosophical" consistency to the described naturally emergent hierarchy of unfolding (unreduced) complexity of the world: in addition to causally complete solution of all "difficult", old and new, problems of unitary science (e. g. in cosmology [START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF]), it provides the fundamentally unlimited prospects for such intrinsically complete knowledge development, which are just badly missing in modern official science state that shows the growing and already dominating signs of its inevitable "end", either in the form of ontological saturation towards the independently progressing technologies or as a strangely increasing number of internal contradictions and "unsolvable" problems. P a g e | 6

Indeed, the described way of progressive development of the unreduced interaction complexity doesn't need to stop at any its known level and can have various probable links and issues towards larger and deeper reality of other complexity levels and structures. While they can be separated from the explicitly known world structures by natural complexity gaps (actually omnipresent in the observed complexity hierarchy and contributing to its reasonable efficiency), nothing prevents us from their progressive discovery and understanding, similar to previous, although often spontaneous and fundamentally blind search for deeper levels of now "well-known" material reality. Even the latter is very far from being perfectly understood within the dominating unitary science projection, as we show in detail in various applications of the qualitatively extended and provably complete framework of the universal science of complexity, from elementary particles to consciousness and modern development problems [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | Extended Mathematics of Unreduced Dynamic Complexity: The Exact Image of Unified Reality, from the Electron to Consciousness[END_REF][START_REF] Kirilyuk | The Unified Complex-Dynamic Origin of Time, Intention, Life, and Everything[END_REF].

It becomes evident therefore that still much greater and fundamentally unlimited novelties can be waiting for us in the infinitely larger, dynamically multivalued space of unreduced complexity dimensions, and there are clear and rigorously specified indications in favour of their reality [START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF], far beyond the most advanced dreams of unitary science paradigm.

It is important that now this fascinating search for genuine, qualitative novelties, both within and beyond the "ordinary" material reality, doesn't need to remain empirically intuitive and conceptually blind (which is the inevitable way of traditional unitary, or positivistic, science), since the unified hierarchy of naturally developing world complexity outlined above and constituting the objectively specified and causally complete fundamental reality provides both the consistent, well-defined "order of the world" and the guiding line for its further exploration.

In summary, we show how and why the unreduced, now truly rigorous mathematical solution to the arbitrary interaction problem (illustrated by eqs. (1)-( 3)), with its qualitatively extended (dynamically multivalued) structure as compared to the artificially restricted, dynamically single-valued projection of standard positivistic "models", not only provides intrinsically consistent solutions to various stagnating problems of the latter, but leads to the objectively determined and causally complete picture of multilevel fundamental reality, where all "impossible miracles" of unitary science become possible and natural, including the total, physically real unification of all, now causally complete knowledge and its basically unlimited development towards new levels and dimensions of unreduced, universally defined dynamic complexity [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Consistent Cosmology, Dynamic Relativity and Causal Quantum Mechanics as Unified Manifestations of the Symmetry of Complexity[END_REF][START_REF] Kirilyuk | Complex-Dynamical Solution to Many-Body Interaction Problem and Its Applications in Fundamental Physics[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Complex-Dynamical Nanobiotechnology Paradigm and Intrinsically Creative Evolution[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | Extended Mathematics of Unreduced Dynamic Complexity: The Exact Image of Unified Reality, from the Electron to Consciousness[END_REF][START_REF] Kirilyuk | The Unified Complex-Dynamic Origin of Time, Intention, Life, and Everything[END_REF].


Sarah Ben

Othman * Slim

Hammadi * Alain Quilliot
email: alain.quilliot@isima.fr

Multi-Objective Evolutionary for Multi-Skill Health Care Tasks Scheduling

Keywords: Planning Method, Pediatric Emergency Department, Assignment, Fuzzy Logic, Evolutionary Method, Medical Staff Scheduling, Performance

Emergency health care services have a highly complex patient treatment system. This system is characterized by stochastic arrivals of patients which can lead in the case of activity to its services overload. In fact, the complexity of these medical exercises relays on assigning health care operations to medical staff members respecting constraints related to the uncertain environment. The objective is to minimize costs and delays and to increase the quality of care as well as patient satisfaction. In this article, a planning method is applied in the Pediatric Emergency Department of the Regional University Hospital of Lille (Northern France).The proposed approach is composed of two phases: the first one is an assignment procedure based on fuzzy logic and the second phase is based on an evolutionary method to solve the problem of medical staff scheduling. This approach improves the performance of the scheduling system in order to help physicians to better manage their organization and anticipate the overcrowding feature.

INTRODUCTION

Health care facilities, which represent a growing sector, are confronted in recent decades to a mutation imposed by their providers and their consumers. It is essential that the offered services are of a good quality to best meet patients demand. Indeed, there is some patient dissatisfaction generally due to long waiting times or the mismatch between their needs and skills of human resources providing these services. These institutions have therefore a real need to develop a more efficient and rigorous human resources management system to improve the productivity and efficiency of their organization while ensuring the quality of health care provided to patients.

The main issue related to scheduling in health care institutions is to allocate human resources to health care tasks considering their availabilities and their skills. Human resources dimensioning is of highest importance for hospitals. These resources, usually related to high costs should be able to meet the objectives. A resource undersizing is a handicap to achieve these goals while oversizing leads to underutilization of available resources and therefore to additional costs. Besides, additional information about patients' pathologies received by the medical staff members during the health care process makes it difficult to determine the necessary medical treatments at the beginning of the handling process. Further, the duration of diagnosis and treatments are stochastic due to the specific characteristics of each patients. The immediate need for treatment in case of severe emergencies may also cause disturbances in the schedule and complications which can occur during a treatment are able to engender long waiting times and modify pathways for other patients.

Health care systems are thus complex systems facing a huge number of challenges related to production functions below the optimal and problems of information flow.

In this work, we consider the scheduling problem in emergency departments characterized by chaotic arrivals of patients. Several studies have shown that one of two emergency services works in overdrive, which means that all patients do not have the privilege of being supported in optimal conditions with extended waiting times [START_REF] Bertrand | Press Kit[END_REF]. Planning and resources scheduling has an impact on performance management and system control, typically based on a global evaluation of the existing system [START_REF] Pham | Care of the patient in the hospital to better manage the complexity of the coordination of actors[END_REF]. The objective is to provide diagnosis and anticipation of possible changes likely to affect its current functioning by the adjustment of some parameters [START_REF] Canelon | Intelligent supervision of petroleum processes based on multi-agent systems[END_REF]]. We are particularly interested in human resources planning in healthcare organizations. This planning aims to determine the best balance between patient demand and involved human resources in order to meet needs for treatment while minimizing waiting times, optimizing the quality of health care services and reducing costs. This work belongs to HOST project (Hospital: Optimization, Simulation and Crowding Avoidance) supported and financed by the French National Research Agency (ANR). It targets to optimize the Pediatric Emergency Department functioning characterized by stochastic arrivals of patients which can lead to its overcrowding. It aims to better manage health care organizations, anticipate the overcrowding feature and establish avoidance proposals for it.

Our concerns focus on modeling the human resources allocation in the scheduling process and also searching for the existence of an assignment allowing the realization of a health care plan. This paper is organized as follows: an analogy between jobshop scheduling and multi-skill health care tasks scheduling is presented in the second section followed by a mathematical modelling of the studied problem. Then, the proposed method will be described in the fourth section. A scenario of simulation is given is the fifth section. Finally, the last section is for conclusion and prospects.

ANALOGY BETWEEN JOB-SHOP SCHEDULING AND MULTI-SKILL HEALTH CARE TASKS SCHEDULING

Health care systems are similar to production systems which always try to meet demands (patients for the hospital and customers for industry). A health care system can be considered as a full production system, constrained by limited material and human capacity in order to deliver the best care at the lowest cost. Human resources planning is a central element of health institutions' management because of the cost and the constraints related to it (skills and availability). The numerous hazards such as the stochastic arrivals of patients and the complications that can occur during treatment process, the difficulty of standardization and the coordination of medical acts with the high number of actors make the planning a complex process. The implementation of generated plans requires sustained efforts.

The search for industrial excellence in corporate comes near to the concept of optimization of the quality of health care facilities. The specific feature of health care systems is that they cannot speculate on human suffering and for which the objective of profit must be dismissed.

The most commonly discussed programming policies in previous works are those included in workshops. A classification scheduling problems in a workshop can be made according to the number of machines and the order of their use to make a product, which depends on the nature of the considered workshop. A workshop is characterized by the number of machines it contains and its type. We distinguish three types of workshops: flow-shop, job shop and open-shop. We compare in this paper, the scheduling problem in the PED to a job-shop scheduling problem in industry.

Job-shop problems called also Multipath workshops are workshops where operations are carried out in a fixed order, vary according to the task. The flexible job-shop is an extension of the classical job-shop model. Its uniqueness lies in the fact that several machines are potentially capable of achieving a subset of operations. Basically, it is a problem of planning and organization of a set of tasks to be performed on a set of resources with variable performance [Gotha, 1993].

Similarly, multi-skill health care tasks scheduling corresponds to assign health care tasks to medical staff members who are characterized by their skills and availabilities in order to satisfy patients' needs while respecting their emergency degrees and taking into account their length of stay in the PED.

Given a set of tasks and a set of resources, resolving a scheduling problem corresponds to program tasks and allocate resources to optimize one or more goals (corresponding to objective performance criteria), respecting a set of constraints.

The problem of job-shop scheduling consists in organizing the realization of N jobs on M machines and a job j represents a number of nj non preemptable ordered operations. In multiskill health care tasks scheduling, jobs corresponds to treatment tasks (a treatment task for each patient), machines are the medical staff members and health care operations being executed are also non preemptable. In both cases, the execution of each operation involves one resource selected from the set of available resources and at a given time, a resource can only execute one operation: it becomes available to other operations once the operation currently assigned is accomplished (resource constraints). The assignment of an operation to a resource entails the occupancy of this resource during a processing time. So, for each flexible job-shop problem, we can associate a processing time to each operation. However, because the most important target of health organizations is to ensure a high quality of care, durations are not given importance while scheduling in health care process.

In fact, durations of health care operations are uncertain and not known in advance. They depend on care providers' skills and patients' states. On the other hand, health care demands can occur unexpectedly. We cannot, in any circumstances know when the request arises. Requests are prioritized and scheduled according to emergency degrees. Some requests are less urgent than others; they can be delayed without endangering patients' lives. However, most of the requests in the PED and the emergency departments in general require immediate intervention. According to the legal structure of the hospital, this latter has to accept all patients. In such a situation, it is necessary to insert urgent patients in the planning already established, which sometimes causes malfunctions and usually additional operating costs. This makes the duration of a treatment operation, which is determinist in job-shop scheduling problems, undergoes significant variations depending on the type of operation, the level of expertise of care provider, the patient and his pathology, etc. The variability of processing times often leads to changes in the schedule of the PED activity which may cause a decline in quality concerning services provided to patients. The consequences include long waiting times and additional costs due to overtime.

Hence, treatment tasks durations should be disintegrated while resolving human resources affectation for health care operations execution. In fact, durations of treatment tasks in emergency department are difficult to calculate and cannot be known in advance. So we chose to define the Demand Load (DL) of healthcare treatment in the PED to quantify patient treatment load. The more progression is, the less healthcare treatment demand will be. This reflects the progression of health care process for each patient and gives us an idea about health care operations already executed and the ones which remain to be done.

In job-shop scheduling a resource (machine) becomes available to other tasks once the task which is currently assigned to is completed. Task's completion time can be calculated in advance. But, in health care treatment process, the availabilities of medical staff members depend on their skills and experience, patient's health state, pathology severity degree and the evolution of the current medical treatment task.

To set the availability date, it is hard for experts to afford exact values due to the uncertainty involved. Besides, the evaluation is not the same in the eyes of the decision-makers [Issai and Singh, 2000], it depends on human feeling and recognition. So, health care providers cannot make the single judgment [START_REF] Abbod | Survey of utilization of fuzzy technology in Medicine and Healthcare[END_REF].

For the objectives to achieve, it is to minimize the overall completion time (makespan), the total workload of the machines in job-shop scheduling, etc. However, in our study, we aim also to minimize patients waiting time in addition to balancing workload between all medical staff members and minimizing the response time as well as the workload of all the medical staff in the PED. The problem is to organize the execution of N health care operations by M medical staff members. The set of medical staff members is U. Each task Tj corresponds to a patient waiting for treatment in the PED and is a sequence of n j health care operations. Each operation i of a task Tj (noted O i,j ) may be performed by a set of medical staff members U i,j  U.

MATHEMATICAL MODELING

Notation

The affectation of an operation O i,j to a medical staff member MS k  U i,j leads to the occupancy of this medical staff member for a period d i,j,k (we assume that d i,j,k IN*).

In this issue, we make the following assumptions:  Each treatment task Tj can be started at the time t = rj;  Each medical staff member can execute at a time a single operation (resource constraint);  The total number of operations to be performed is higher than that of medical staff members.

Performance Indicators

To evaluate functioning quality of the PED, we define three performance indicators:  Minimize the total load of all medical staff members:

The total load is equal to the sum of the lengths of all health care operations executed according to any assignment. However, for each operation Oi,j, the length of execution time is greater than the minimum length j , i  where:

) ( min , , , k j i k j i d  
With i,j the minimum length of the health care operation execution for a treatment task Tj and Oi,j is the i th operation.

METHOD

The studied multi-skill health care tasks scheduling presents two difficulties. The first one is to assign each operation O i,j to a medical staff member M k (selected from the set U). The second one is the calculation of the starting times t i,j . The proposed method consists in two stages of resolution.

First stage

 Assignment Algorithm

It allows us to assign each health care operation to the suitable medical staff member taking into account his availability date and workloads of health care providers to whom operations have been already assigned. To compute the availability date of each MSk a simple application of fuzzy logic is proposed. Calculation is based on analysing the affordable skills, the evolution of the current treatment task, patient's health state and pathology severity degree. These are the inputs. Then, we define for each input three sub-sets {"Low", "Medium" and "High"}.

Each subset is characterized by its trapezoidal Membership Functions (MF) meanwhile the state varies gradually. After the definition of MF of the variables or Inference which is based on decision rules depending on experts' views and historical data. Example of rules: if (the medical staff is "high qualified") and (the evolution of the current act is "high") and (the pathology is "serious") and (patient's health state is "improves") then (the medical staff is "highly available").

The result which is a fuzzy value undergoes a defuzzification to obtain an exact number as final output using the Center Of Area method (COA). For the assignment, we choose to assign the health care operation to the medical staff member who corresponds to the highest fuzzy value which reflects his availability rate. If two medical staff members have the same fuzzy value, we make the choice while balancing the workload between all health care providers. This assignment procedure allows us to construct a set E of assignments (E = {S z / 1≤z≤cardinal(E)}) and balance the medical staff members workload. Each assignment is represented in a table S z , S z = {S z i,j,k / 1≤j≤N; 1≤i≤n j ; 1≤k≤M}. For each i, j, k, the value of S z i,j,k can take 0 or 1. The value "S z i,j,k =1" means that Oi,j is assigned to MSk. The value "S z i,j,k = 0" means that Oi,j isn't assigned to MSk.

 Scheduling Algorithm

For each assignment, it calculates starting times ti,j by considering medical staff availabilities and precedence constraints. Conflicts are resolved by applying conventional priority rules (SPT, LPT, FIFO, LIFO, FIRO [START_REF] Boucon | Scheduling Workshop: assistance to the choice of priority rules[END_REF], [START_REF] Bel | Production Scheduling, Chp 6: Simulating Approach, under the supervision of Lopez, P. and Roubellat Hermes[END_REF]), so we get a set of plans according to the applied priority rules. In emergency department priority is given at first to the most urgent cases, then to the patient who has arrived first. The scheduling procedure is as follows: according to the availability date of a medical staff members and the availability of the corresponding health care operation, the starting time of the operation is the minimum date among the two availabilities' dates.

Second stage

The scheduling approach described in the previous paragraph can contribute to a multi-objective optimization by combining it with genetic algorithms and make develop an initial set of solutions to a final one while improving the performance of the whole system according to criteria we have fixed at the beginning [START_REF] Michalewicz | Genetic Algorithms + Data Structures = Evolution Programs[END_REF], [START_REF] Yamada | A genetic algorithm application to large scale job shop problems[END_REF]]- [START_REF] Ono | A genetic algorithms for job-shop scheduling problems using job-based order crossover[END_REF].

Genetic algorithms are the most popular variant of evolutionary algorithms. Many designate and continue to designate the evolutionary approaches as "genetic algorithms". As their name suggests, genetic algorithms are based on the genetic inheritance of an individual (genotype) represented by its chromosomes. The interaction of the genotype of an individual with its environment determines its phenotype which can be modified by mutation. Phenotype is evaluated by coding the genotype, which is often a binary symbol in order to provide a usable performance value by the selection of operators. The variation operators (crossover and mutation) presented above are related to the binary representation since they act on binary genotypes. In a simple genetic algorithm, the search is set by the successive application of the variation operators. The cross is the phase of cooperation between individuals while the mutation corresponds to the individual adaptation phase.

We consider the important characteristics of evolutionary algorithms and their relevance to solve NP-hard problems. We present some key points for solving approach:

-A genetic representation (coding) appropriate to the problem to determine possible solutions of the optimization problem; -Genetic operators that transform the composition of children during reproduction. Because a task must be treated by a single medical staff member selected from the set of members who are able to provide the corresponding health care service, we choose to correct the solutions generated by another operator to meet this requirement; -Parents are randomly selected from the current population for the crossover and mutation with a probability of crossover pc (0 <pc <1) and a mutation probability pm (0 <pm <1); -We take a non-elitist replacement technique to build the new population;

-Fitness functions evaluate solutions based on two criteria: medical staff workload and response time.

In this stage, we generate from the set E constructed in the first stage, an assignment scheme to control the genetic algorithm. This scheme is going therefore to represent a constraint that must be respected by the new created individuals. This method consists in considering the assignments S Z given by the earlier scheduling method and to determine (for each operation) the set of possible medical staff members using a genetic algorithm.

Table 1. Medical Staff Skills

C = {C i,j,k / 1≤j≤N ; 1≤i≤n j ; 1≤k≤M} MS1 MS2 MS3 MS4 T 1 O 1 ,1 0.2 0.6 0.5 0.2 O 2 ,1 0.4 0.9 0.3 0.2 O 3 ,1 0.4 0.6 0.5 0.8 T 2 O 1 ,2 0.5 0.2 0.2 0.5 O 2 ,2 0.3 0.4 1 0.4 O 3 ,2 1 0.2 0.3 0.3 T 3
O 1 ,3 0.9 0.7 0.4 0.6 O 2 ,3 0.5 0.6 0.9 0.2

A possible scheduling plan related to medical staff skills is given in Table 1. We consider that the assignment of a health care operation Oi,j to a medical staff member is possible when the competence C i,j,k >=0.5.

Table 2. Assignment S ch

S ch = { S ch i,j,k / 1≤j≤N ; 1≤i≤n j ; 1≤k≤M}

MS1 MS2 MS3 MS4 T 1 O 1 ,1 0 * *
The symbol: " * " indicates that the assignment is possible i.e. (C i,j,k >=0.5). We cannot have the value "1" and the symbol " * " in the same line.

The application of the assignment procedure described in the first stage may give as result the following scheduling S (Table 3).

MS1 MS2 MS3 MS4 T 1 O 1 ,1 0 0 1 0 O 2 ,1 0 1 0 0 O 3 ,1 0 1 0 0 T 2 O 1 ,2 0 0 0 1 O 2 ,2 0 0 1 0 O 3 ,2 1 0 0 0 T 3 O 1 ,3 0 0 0 1 O 2 ,3 1 0 0 0 a. Coding
We represent the scheduling in the same assignment table S.

Each Si,j,k=1 and Si,j,k = * are replaced by the couple ( ti,j, tfi,j ) where ti,j is the starting time and tfi,j is the completion time. Si,j,k = 0 remain the same.

b. Crossover

It consists in combining elements from two parent chromosomes into one or more child chromosomes. [START_REF] Michalewicz | Genetic Algorithms + Data Structures = Evolution Programs[END_REF]. It allows to create new combinations and enlarge our chance to find a better solution. Our operator uses a Crossover Mask. We apply an efficient coding inspired from [START_REF] Kacem | Direct Chromosome Representation and Advanced Genetic Operators for Job-shop Problems[END_REF] which respects our problem constraints.

Cossover Algorithm

-Select 2 parents S 1 et S 2 randomly; -Select randomly 2 integers j and j' such that j≤j'≤N ; -Select randomly 2 integers i and i' such that i≤nj and i'≤nj' (in the case where j=j', i≤i') ; -The assignment in f 1 must match the same assignments in S 1 for the set of operations between the line (i,j) and the (i',j'); -The rest of assignments in f 1 must match the same assignments in S 2 ; -The assignment in f 2 must match the same assignments in S 2 for the set of operations between the line (i,j) and the (i',j'); -The rest of assignments in f 2 must match the same assignments in S 1 ; -Call to "Scheduling_Algorithm" to calculate the starting and completion times;

c. Mutation

The genetic operators should be therefore able to ensure this optimization according to the criteria presented in the previous section. Thus, we propose two artificial mutation operators: the first one is responsible for reducing the response time for patients' treatment.

Operator of mutation reducing the response time for treatment for patient j (RTLj)

Mutation 1 -Select randomly an individual S ; -Choose the treatment task j whose response time the most long : (Max j { RTLj such that RTLj =∑i∑k Si,j,k.d i,j,k }) ; -i=1; r = 0 ; -WHILE (i≤nj And r = 0)  Find K0 such that Si,j,K0 =1;  FOR (k=1, k≤M) IF (d i,j,k < d i,j,k0 ) Then {Si,j,K0 =0; Si,j,K =1; r=1 ;} End IF End FOR  i=i+1 ; End WHILE -Call to "Scheduling_Algorithm" to calculate the starting and completion times;

The second mutation is responsible for the workload balancing:

Operator of mutation balancing workloads of medical staff members

Mutation 2 -Select randomly an individual S ; -Find the medical staff member who has the highest workload Mk1 (Maxk { Wk / Wk=∑j∑i Si,j,k.di,j,k }) ; -Find the medical staff member who has the lowest workload Mk2 (Min k { Wk }) ; -Choose randomly an operation Oi, j such that S i ,j,k1 =1 ; -Assign this operation to the medical staff member who has the lowest workload: S i ,j,k1 =0 ; S i ,j,k2=1 ; -Calculate the starting and completion times according to the algorithm "Scheduling_Algorithm";

SIMULATION AND RESULTS

To better understand the proposed approach we present in this section a scenario of a clinical case in the PED. We suppose the arrival of 3 patients at time t = 0 to the health care institution with 4 medical staff members mastering 3 types of skills, the degree is between 0 and 1. After applying the evolutionary approach, the final assignment table is as follows: We notice that in table 7, the treatment task T1 has the longest treatment time. We have therefore reduce it to 50. Besides, the highest workload was 50, by applying the evolutionary approach it became 40. The medical staff members have as a result balanced workload.

Figure 1: Comparison between results of the first step and results given by the evolutionary approach

Figure 1 shows that the evolutionary approach contributes to the improvement of our system performance. Balancing the workload between the health care providers leads to a less stressed medical staff and then to a higher performance and minimizing response time reassures patients.

CONCLUSION

In this paper, we have presented an efficient approach for multi-skill health care tasks scheduling problems. This method evolves two steps: the first one is to apply the assignment procedure to solve the resource allocation problem and generate the assignment plans and the second one is to apply a controlled evolutionary algorithm. The initial population is constructed starting from the results of the first phase. This approach helps us to improve the performance of the proposed medical staff scheduling system.

In our future works we will consider the rest of the performance indicators and we will integrate our approach in an intelligent agent-based system.

So

  Reduce waiting time of each patient It corresponds to minimize Cr2 = ∑ max l j=1 (0, cj -dj) With: cj = the completion time of the treatment task Tj, dj = the theoretical treatment time for the task Tj, l = the total number of treatment tasks.



  Minimize response time to health care tasksIt corresponds to minimizeThe proposed optimization method presented in this work will focus on balancing the workload of medical staff members (k € [1…M]) as well as minimizing the response time for patients' treatment.

,

  𝑑 𝑖,𝑗,𝑘 : execution time of the operation Oi,j, 𝐶 𝑖,𝑗,𝑘 : the skill of the medical staff member 𝑀k needed for the execution of the operation 𝑂𝑖, 𝑗 ,

	M :	number of available medical staff members,
	k :	index of medical staff member Mk,
	j :	index of treatment task Tj,
	n j :	number of operations of treatment task Tj,
	r j :	earliest starting date of task Tj,
	i :	index of a health care operation,
	Oi,j :	i th operation of task Tj,
	r i,j :	the earliest availability date of operation O i,j,
	Nt :	total number of operations to execute,	N	t		j 	n	j	,
	Pt :	total number of patients waiting for health care
		treatment	t P		 j	P	j

Table 3

 3

	. Assignment S
	S: possible scheduling

Table 4 : Medical staff members

 4

	Medical Staff	Description
	MS1	Paediatrician
	MS2	Nurse1
	MS3	Nurse 2
	MS4	Care assistant
	Patient 1 suffers from a mild concussion without loss of
	consciousness, patient 2 suffers from cardiopulmonary arrest
	and Patient 3 from an uncomplicated pneumonia.
	The assignment table of medical staff members is given in the
	table below:

Table 5 : Assignment table with medical staff skills

 5

			MS1 MS2 MS3 MS4
		O 1 ,1	0.9	0	0	0.9
	T1	O 2 ,1	0	0	0.8	1
		O 3 ,1	0.7	0	0.6	0
		O 1 ,2	0	1	1	0
	T2	O 2 ,2	0.7	0.6	0	0
		O 3 ,2	0	1	0.8	0
		O 1 ,3	0	0	0.6	0
	T3	O 2 ,3	0	0	0	1

Table 6 : Assignment using only the first step

 6

			MS1	MS2	MS3	MS4
		O 1 ,1	0	0	0	0, 10
	T 1	O 2 ,1	0	0	30, 50 0
		O 3 ,1	50, 80 0	0	0
		O 1 ,2	0	0, 10	0	0
	T 2	O 2 ,2	0	10, 40 0	0
		O 3 ,2	0	0	0	40, 60
		O 1 ,3	0	0	0, 30	0
	T 3	O 2 ,3	0	0	0	30, 40

Table 7 : Final assignment

 7

O 2 ,1 0

0 0 O 3 ,1 0 * * * T

O 1 ,2 * 0 0 * O 2 ,2 0 0 1 0 O

,2 1 0 0 0 T 3 O 1 ,3 * * 0 * O 2 ,3 * * * 0The value " S ch i,j,k = 0" indicates that the medical staff member MSk is not enough qualified for this health care operation so we cannot assign it to him. The value " S ch i,j,k = 1" indicates that the assignment of the operation Oi,j to the medical staff member MSk is obligatory because he is the only one whose C i,j,k > =0.5, in this case, all values of the rest of the line (i, j) are equal to "0".

This work is integrated into HOST project (supported and financed by the French National Agency (ANR