Vasco ; Giovagnetti

Maria Letizia Cataldo

Fabio Conversano

Christophe Brunet
email: christophe.brunet@szn.it

Zoologica A Stazione

Dorhn

Functional relation between growth, photosynthetic rate and regulation in the coastal picoeukaryote Phaeomonas sp. RCC 503 (Pinguiophyceae, Stramenopiles)

Keywords: photoacclimation, picoeukaryote, xanthophyll cycle photoacclimation, picoeukaryote, xanthophyll cycle Ax, antheraxanthin, DPS, de-epoxidation state LHC, light-harvesting complex, Neox, neoxanthin, NPQ, non-photochemical quenching, PAM, pulse amplitude modulation, PFD, photon flux density, Vx, violaxanthin, Zx, zeaxanthin

come

INTRODUCTION

In the field, maximum light intensity available for photosynthesis varies predictably over daily and annual cycles. Yet, superimposed on these cycles are such factors as cloud cover that affect the amount of incoming light in an unpredictable way. For phytoplankton, and especially coastal phytoplankton, mild turbulence of the water column adds even further to this unpredictability [START_REF] Lewis | Turbulent motions may control phytoplankton photosynthesis in the upper ocean[END_REF][START_REF] Macintyre | The effect of water motion on short-term rates of photosynthesis by marine phytoplankton[END_REF]Lichtman and Klausmeier, 2001); one moment a cell finds itself at depth where its photosystems must cope with low light intensity, whereas the next moment it may be transported up to the surface where its photosystems have to adjust itself rapidly to conditions approaching full sunlight.

Phytoplankton cells possess a number of physiological acclimation and regulatory responses to cope with these unpredictable, potentially extreme, and often rapid changes in light intensity [START_REF] Macintyre | Photoacclimation of photosynthesis irradiance response curves and photosynthetic pigments in microalgae and cyanobacteria[END_REF][START_REF] Raven | Adaptation, acclimation, and regulation in algal photosynthesis[END_REF].

Acclimation involves net synthesis or breakdown of macromolecules and so changes in pigment quota, stoichiometry of cell components, and ultrastructural changes in the cell.

Regulation refers to faster processes than acclimatory ones (i.e. time constants of seconds to minutes) acting through the control of enzyme activities and energy dissipation pathways [START_REF] Raven | Adaptation, acclimation, and regulation in algal photosynthesis[END_REF]. Regulatory processes therefore allow photosynthesis to react instantaneously and reversibly the environmental changes occurring. These mechanisms include the Rubisco activity, state transition mechanism, and the xanthophyll cycle [START_REF] Raven | Adaptation, acclimation, and regulation in algal photosynthesis[END_REF]. The xanthophyll cycle (XC) is one of the main processes involved in the dissipation of excessively absorbed light energy through the nonphotochemical quenching (NPQ, [START_REF] Finazzi | Non-photochemical quenching of chlorophyll fluorescence in Chlamydomonas reinhardtii[END_REF]. In green algae and higher plants, and in other algal groups (i.e. Pinguiophyceae), the XC mechanism is based on the reversible de-epoxidation of violaxanthin (Vx) into zeaxanthin (Zx) through antheraxanthin (Ax). In most chlorophyll c (Chl c) containing phytoplankton species (e.g. Diatoms), it involves the inter-conversion between diadinoxanthin (Dd) and diatoxanthin (Dt; [START_REF] Stransky | Das Carotinoidmuster und die verbreitung des lichtinduzierten Xanthophyllcyclus in verschiedenen Algenklassen[END_REF][START_REF] Lavaud | Photoprotection capacity differs among diatoms: possible consequences on the spatial distribution of diatoms related to fluctuations in the underwater light climate[END_REF]Dimier et al., 2007 b). Accumulation of Zx or Dt is triggered by the formation of a pH gradient across the thylakoid membrane (e.g., see the review by Goss and Jakob, in press). Little is known about the picoeukaryotes with respect to their ecophysiological features [START_REF] Timmermans | Physiological responses of three species of marine picophytoplankton to ammonium, phosphate, iron and light limitation[END_REF]Dimier et al., 2007Dimier et al., a, 2009;;[START_REF] Six | Contrasting photoacclimation costs in ecotypes of the marine eukaryote picoplankter Ostreococcus[END_REF]Six et al., , 2009)). These small planktonic photolithotrophs usually dominate oligotrophic water ecosystems, reaching up to almost 50 percent of total biomass and production, while in eutrophic waters their contribution to the phytoplankton community is lower [START_REF] Not | Late summer community composition and abundance of photosynthetic picoeukaryotes in Norwegian and Barents Sea[END_REF][START_REF] Worden | Ecology and diversity of picoeukaryotes[END_REF]. Their small size explains many of their biological and ecological features such as the large surface area per unit volume and the minimal diffusion boundary layer (i.e. higher nutrient consumption rates), the low sinking rate, the low packaging effect and the efficient light utilization [START_REF] Raven | The twelfth Tansley lecture. Small is beautiful: the picophytoplankton[END_REF][START_REF] Raven | Picophytoplankton: Bottom-up and topdown controls on ecology and evolution[END_REF].

High plasticity in photosynthetic regulation has been described in picoeukaryotes. For instance, Six and collaborators [START_REF] Six | Contrasting photoacclimation costs in ecotypes of the marine eukaryote picoplankter Ostreococcus[END_REF](Six et al., , 2009) )

METHOD

Algal model and culture conditions

The strain RCC503 of Phaeomonas sp. (Pinguiophyceae) was isolated along the Mediterranean Spanish coast and was provided by the Roscoff Culture Collection [START_REF] Vaulot | The Roscoff Culture Collection (RCC): a collection dedicated to marine picoplankton[END_REF]. The strain was cultivated nonaxenically at 20 °C at 100 µmol photons m -2 s - 1 with a 11:13 hours light : dark photoperiod in locally obtained, and sterilized seawater amended with f/2 medium [START_REF] Guillard | Studies of marine planktonic diatoms. I. Cyclotella nana Hustedt and Detonula confervacea (Cleve) Gran[END_REF] using 225 cm 2 polystyrene canted-neck flasks (Corning Incorporated, NY, 14831, USA). Cultures were continuously aerated and maintained in exponential phase by daily and semi-continuous dilution (50% of the total volume) over a period of more than two weeks before the experiments. Before performing experiments, we checked pre-acclimation of the strain to the light condition, by assessing stability of pigment content and growth rate.

Experimental design and sampling strategy

After 15 days acclimation period at 100 µmol photons m -2 s -1 , cultures were subjected to 5 different sinusoidal light (SL, Fig. 1a) regimes, each with a different maximal photon flux density (PFD) of 10 (LL1), 50 (LL2), 100 (ML), 250 (HL1), 500 (HL2) µmol photons m -2 s -1 , respectively. The mean PFD over the 11 h of illumination ranged between 344 and 7 µmol photons m -2 s -1 , which is likely similar to the light experienced by cells at a depth of 10 m and at the Deep-Chlorophyll maximum layer in a Mediterranean water column, respectively (see Dimier et al., 2009 a).

Population growth was followed over five days while photophysiological and biochemical properties of Phaeomonas sp. were measured during the exponential growth phase. Under the LL conditions, where no exponential phase was observed, sampling of photophysiological and biochemical parameters lasted five days.

Each experiment was performed in triplicate. Flasks were illuminated by the light-bulb kit ACLS (Advanced Control Lighting System) and Infinity XR4 (Aquarium Technologies -Sfiligoi S.r.l., Bassano del Grappa, Italy), simulating light-dark sinusoidal cycles, reproducing dawn and sunset. XR4 is equipped with metallic iodur HQI bulbs Cultures were sampled three times per day (Fig. 1a). First sampling was carried out during dawn (for HL2 condition, incident irradiance was below 40 µmol photons m -2 s -1 ), while the second corresponded with the irradiance peak. The third one was carried out during sunset (HL2 incident irradiance was below 40 µmol photons m -2 s -1 ). Twenty to fifty milliliters of culture were rapidly taken with a 50 mL syringe during each sampling time, for measuring variable fluorescence, NPQ, electron transport rate, and for further analysis of pigments, absorption spectrum, cell size and volume (by Coulter Counter analysis) and cell concentrations (by microscopic counts).

Temperature and pH were controlled daily with a HI-9214-Stick pH meter (Hanna Instruments, Woonsocket, RI, USA). The growth rate was estimated daily from cell abundance measurements using the following equation:

µ = ln [N t2 : N t1 ] : [t 2 -t 1 ] ( 1
)
where µ is the growth rate (d -1 ) and N t is the mean cell concentration at time t, where t 1 and t 2 correspond to the morning sampling time of days 1 and 2, respectively. From the growth rate, the number of cell divisions (n) per day was estimated with the following equation:

n = µ : [ln(2)] ( 2
)
where n is the number of cell divisions per day and µ is the growth rate.

Pigment analysis

One aliquot of algal culture (10 mL) was filtered onto GF/F glass-fiber filters (Whatman)

and immediately stored in liquid nitrogen until analysis. Triplicate samples were taken during each sampling time. Analysis was carried out following the procedure described in Dimier et al. (2007 a). The pigment extract was injected in a Hewlett Packard series 1100 HPLC (Hewlett-Packard) with a C8 BDS 3 mm Hypersil, IP column (Thermo Electron). where I is the incident irradiance (in µmol photons m -2 s -1 ). The mean absorption value (a* ph ) of phytoplankton was normalized by Chl a (m 2 mg Chl a -1 ) and divided by 2 assuming that half of the absorbed light was distributed to PSII.

The ETR-light curves were fitted with the equation of [START_REF] Eilers | A model for the relationship between light intensity and the rate of photosynthesis in phytoplankton[END_REF] to estimate the photosynthetic parameters α B , Ek, abs ETR max . The relative Electron Transport Rate ( rel ETR max ) was estimated with equation ( 4) without Chl a normalization.

Organic Carbon and Nitrogen

Samples for the determination of particulate organic carbon (POC) and particulate organic nitrogen (PON) were filtered on pre-combusted (450°C, 5h) glass fiber filters (Whatman GF/F) and immediately stored at -20 °C. The analyses were performed with a FlashEA 1112 Series Thermo Electron CHN elemental analyzer [START_REF] Hedges | Carbon and nitrogen determination of carbonatecontaining solids[END_REF]. Cyclohexanone-2,4-dinitrophenyl hydrazone was used as standard.

Absorption spectrum

Ten milliliter sample were filtered onto Whatman GF/F filters and immediately frozen.

Measurements were carried out following the procedure described in Dimier et al. (2007 a). Absorption was measured between 280 nm and 800 nm with 1-nm increments on a spectrophotometer (Hewlett Packard HP-8453E) equipped with an integrating sphere (Labsphere RSA-HP-53). The mean absorption value (a*) was thus normalized by the Chl a concentration of the sample to obtain the Chl a specific absorption coefficient (a* ph , m 2 mg Chl a -1 ).

RESULTS

Growth

Phaeomonas sp. was able to grow under the entire range of light conditions with the exception of LL1 (10 µmol photons m -2 s -1 ) and showed the highest growth rate under the highest light regime (Fig. 1b). At the end of the exponential phase for the HL and ML conditions or at day 5 for LL conditions, where no exponential phase was found, cell concentrations ranged between 1.77 × 10 4 ± 1.98 ×10

Photosynthetic pigments

Together with Chl a, many accessory pigments were present in the LHCs of Phaeomonas sp. and the main ones were the following: chlorophyll (Chl) c 2 , fucoxanthin (Fuco), cisfucoxanthin (cis-Fuco), neoxanthin (Neox), Chl c 3 and ß-carotene (ß-Car).

The mean Chl a content per cell was significantly higher in LL conditions compared to the ML or HL ones ranging between 9.96 × 10 -17 ± 1.93 × 10 -17 and 2.82 × 10 -16 ± 7.13 × 10 -17 mol Chl a cell -1 (Fig. 2a). It can be seen that a higher Chl a cell -1 value was measured under HL2 relative to HL1.

Lowering of the light harvesting capacity under high light was revealed by the decrease of almost all main accessory pigments (Fig. The uncoupling between Chl a and the main photosynthetic accessory pigments in response to light change could be a signal of a special photoacclimative strategy developed by this species under low light stress.

Organic Carbon and Nitrogen

Cellular carbon and nitrogen concentrations were significantly correlated (p<0.001) and both strongly related to PFD regimes with lower concentrations under high light and vice-versa (Fig. 3a). The Chl a : POC ratio varied slightly, increasing a little from LL to HL regimes (Fig. 3b), mainly due to the decrease of POC along the light gradient.

The POC : PON ratio, ranging between 4.80 ± 0.24 and 6.82 ± 0.08, increased from low to high PFD regime (Fig. 3c) and was significantly correlated to growth rate (r 2 = 0.98, Fig. 3d).

Photosynthetic parameters

The maximal photosynthetic rate ( abs ETR max ) increased from the LL to HL regimes, reaching a maximum at HL1 and then decreasing a little under HL2 (Fig. 4a). A significantly correlated trend between abs ETR max and growth rate was observed (Fig. 4b).

Under HL2, Chl a enhancement without any related increase in electron transport rate prevented this light condition fitting a linear regression. Indeed, when ETR was not reported in relation to Chl a concentration, the relationship between rel ETR max and growth rate was highly significant and linear over all the PFD range (r 2 = 0.92; Fig. 4c).

The photosynthetic efficiency (α B , Fig. 4d) showed a maximum under the ML condition in agreement with the 100 µmol photons m -2 s -1 pre-acclimation period of cells, and decreased with higher PFDs. Low values of α B under low LL1 and LL2 (similar to the HL1 and HL2 ones) might reflect a limited capacity for photosynthetic acclimation at low PFDs. Indeed, the light saturation index (Ek) did not decrease under the value of 200 µmol photons m -2 s -1 , whereas it increased to almost 1000 µmol photons m -2 s -1 in agreement with the light-acclimated state of this species (Fig. 4e). The high values of Ek indicated that cells were subjected to limiting or sub-optimal irradiance during light periods in all the five experimental conditions. The amplitude of light limitation, estimated with the Ek : PFD ratio, ranged between 24 and 2 (Fig. 4f), with LL1 and LL2 having values higher than 10.

Photophysiology and the xanthophyll cycle

The PSII maximum photochemical efficiency (Fv : Fm) showed a small range of variation, with the highest value under LL1 (0.61, Fig. 5a) and the lowest under HL1 (0.56, 250 µmol photons m -2 s -1 ). Under HL1 and HL2, Phaeomonas sp. developed non- The highest values of Zx were found in cultures grown under HL1 and HL2 (Fig. 6b).

The two pigments Vx and Ax, expressed per cell (data not shown), followed the same trend of Chl a and other accessory photosynthetic pigments cell -1 (see Fig. 2).

Daily evolution of Zx (Fig. 6c) showed that only under the HL2 condition did this pigment increase significantly at midday. Even though NPQ was similar between the two HLs, Zx concentration differed between the two conditions suggesting that a proportion of zeaxanthin was not involved in NPQ development. The morning value of Zx under the HL2 regime was higher than the values measured during each of the HL1 sampling times, revealing a daily-persisting photoprotective state.

DISCUSSION

Chl a content in Phaeomonas sp. RCC503 (~ 2 × 10 -16 mol Chl a cell -1 , cell size ~ 3.3 µm) is in the range of values measured in other small size algal species (Dimier et al., 2007(Dimier et al., , 2009 b) b). Combining this study with data from Dimier et al. (2009 b), a highly significant correlation between size and Chl a cell -1 content was obtained (6 species with size ranged between 1.8 and 4.0 µm, r 2 = 0.92) proving the direct relevance of size in determining pigment content in small species. Taxonomical difference does not affect the relation between size and Chl a, while processes such as photoacclimation have a small effect on this relation in picoeukaryotes (Dimier et al., 2009 b, this study). Indeed, variations in cell size are so small over the irradiance range that any allometric effect (see [START_REF] Finkel | Phytoplankton in a changing world: cell size and elemental stoichiometry[END_REF] Increase in Chl c 3 under low light has been found in many species both in laboratory [START_REF] Rodriguez | Photoacclimation in phytoplankton: implications for biomass estimates, pigment functionality and chemotaxonomy[END_REF]Dimier et al., 2009 a) and in field experiments, where higher Chl c 3 values were found in the Deep-Chlorophyll Maximum layer [START_REF] Brunet | Vertical variability and diel dynamics of picophytoplankton in the Strait of Sicily (Mediterranean Sea) in summer[END_REF], in agreement with its efficient role in light harvesting [START_REF] Johnsen | Bio-optical characteristics and photoadaptive responses in the toxic and bloomforming dinoflagellates Gyrodinium aureolum, Gymnodinium galatheanum, and two strains of Prorocentrum minimum[END_REF]. Thus, our study reinforces the hypothesis of a photoacclimative role of this pigment and demonstrates that changes in Chl c 3 are independent of other pigments, such as fucoxanthin, Chl c 2 and Chl a. Since Chl c 3 is broadly present in phytoplankton species, and in many picoeukaryotes, this result would need further investigation to understand the ecological advantage in synthesizing such a pigment and how its synthetic pathway would work.

The significant relation between photosynthetic rate, cell biochemistry changes and growth rate suggests a direct and functional link between PFD, photoacclimation, photosynthetic and growth rate. In many studies, the variability of the relationship between photosynthetic organic matter production and cell growth has been highlighted [START_REF] Litchman | Growth rate of phytoplankton under fluctuating light[END_REF], growth and photosynthetic parameters being poorly coupled [START_REF] Van Leeuwe | Taxonspecific differences in photoacclimation to fluctuating irradiance in an Antarctic diatom and a green flagellate[END_REF]. This variability has been attributed to factors such as variations in respiration rates, dissolved organic matter loss from cells, and changed extent of metabolites storage such as polysaccharides [START_REF] Van Leeuwe | Taxonspecific differences in photoacclimation to fluctuating irradiance in an Antarctic diatom and a green flagellate[END_REF].

The observed result is probably linked to the experimental conditions we applied.

Simulating what occurs in nature, the sinusoidal irradiance evolution causes cells to experience a low or moderate daily-integrated PFD with a short irradiance peak. Our experimental conditions prevented either continuous high light or continuous limited light stress as occurs under quadratic light evolution. Small cell size limits the intracellular storage of energy [START_REF] Raven | The twelfth Tansley lecture. Small is beautiful: the picophytoplankton[END_REF] Growth rate is linearly related to both Ek : PFD (r 2 = 0.96, p<0.001) and the POC : PON ratio (r 2 = 0.98, p<0.001), these two ratios being correlated (r 2 = 0.95, p<0.001). These results indicate a functional and linear link between PFD and growth rate through photosynthetic rate adjustments and consequently changes in cellular biochemistry. The great range of variation of the POC : PON ratio (35%, between 4.80 to 6.50) reveals significant biochemical modifications related to irradiance and growth rate variations (e.g., [START_REF] Finkel | Phytoplankton in a changing world: cell size and elemental stoichiometry[END_REF].

Higher values of POC : PON obtained under the higher PFD conditions are close to the Redfield ratio, a result that led us to reinforce the hypothesis of an optimal physiological state of cells under high light.

Growth rate being the highest under HL2 condition, the photoacclimative strategy adopted by Phaeomonas sp. is efficient. This species belongs to the "light adapted" functional category [START_REF] Falkowski | Light-shade adaptation: two strategies in marine phytoplankton[END_REF][START_REF] Falkowski | Light-shade adaptation and vertical mixing of marine phytoplankton: a comparative field study[END_REF], as also suggested by the high light saturation for photosynthesis (Ek) estimated from the ETR-light curves.

Under the HL2 condition, the small decrease of maximal photosynthetic rate per Chl a unit ( abs ETR max ) compared to the HL1 condition while growth rate still increased relatively could be due to the Chl a cell -1 increase. Indeed, the relation between rel ETR max and growth rate is linear over the light gradient (Fig. 4c). Increase in both Chl a and accessory photosynthetic pigments under the HL2 condition suggests the development of a σ-type photoacclimation strategy aiming to increase the size of light harvesting antennae [START_REF] Six | Contrasting photoacclimation costs in ecotypes of the marine eukaryote picoplankter Ostreococcus[END_REF], Dimier et al., 2009 a). Increase in photosynthetic pigments under the highest PFD condition was surprising since cells developed photoprotection when irradiance reached its maximum. Comparison with HL1 condition, where no pigment increase was noticeable, pigment enhancement under HL2 condition might be related to the strongest photoprotection. Both the higher level of Zx and its persistent presence in cells growing under the HL2 condition (even before the irradiance peak) indicate a photoprotective state of PSII.

The photoprotective state of HL2 and HL1 cells would also lead to a decreased photosynthetic efficiency, which is demonstrated by the decrease of α B . Since the gradual PFD increase, cells quickly perform photophysiological regulation processes explaining both the low value of DPS (<15%) and its efficiency in protecting PSII (see Zx molecules are able to indirectly protect PSII through changes in thylakoidal membrane fluidity, or preventing lipid peroxidation (see Goss and Jakob, in press). The neoxanthin trend is similar to the one of Zx either over the irradiance range (Fig. 2d) or over the day under HL2 (data not shown). This feature might be related to a role of Neox in preserving PSII from photo-inactivation and protecting membrane lipids as recently demonstrated in plants [START_REF] Dall'osto | The Arabidposis aba4-1 mutant reveals a specific function for neoxanthin in protection against photooxidative stress[END_REF].

This photoprotective status found in cells grown under HL2 might be explained by the rapid kinetics of PFD increase, also linked to the higher irradiance reached, or also by the n-type photoacclimation strategy [START_REF] Six | Contrasting photoacclimation costs in ecotypes of the marine eukaryote picoplankter Ostreococcus[END_REF]. The heavy nutrient cost for low-light growth required by the n-type photoacclimation strategy and its higher-energy cost [START_REF] Six | Contrasting photoacclimation costs in ecotypes of the marine eukaryote picoplankter Ostreococcus[END_REF] can limit or inhibit growth (Dimier et al., 2009 b).

The same photoacclimation strategy has been observed in other picoeukaryotes, the lagoon ecotype of Ostreococcus tauri [START_REF] Six | Contrasting photoacclimation costs in ecotypes of the marine eukaryote picoplankter Ostreococcus[END_REF] and the low-light-adapted

Pelagomonas calceolata (Dimier et al., 2009 a). The latter (Dimier et al., 2009 a) as well as the present study have shown how some species can develop either the σ-or n-type photoacclimation strategy in relation to the light conditions experienced. The n-type photoacclimation strategy would evolve when daily the PFD dose is very low or when cells spend a long time under low light in order to increase their photochemistry (Dimier et al., 2009 a). This result differs from what was shown by [START_REF] Six | Contrasting photoacclimation costs in ecotypes of the marine eukaryote picoplankter Ostreococcus[END_REF] in two

Ostreococcus ecotypes [START_REF] Six | Contrasting photoacclimation costs in ecotypes of the marine eukaryote picoplankter Ostreococcus[END_REF], where the n-type photoacclimation evolved in the lagoon ecotype with decreasing growth irradiance. The inconsistency between these studies could be related to the differences of experimental design, since the present study mol pigment cell -1 (b), fucoxanthin cell -1 (white squares), cis-fucoxanthin cell -1 (white triangles) and β-carotene cell -1 (grey dots) expressed in 10 -16 mol pigment cell -1 (c), and neoxanthin cell -1 content in 10 -17 mol pigment cell -1 (d), over the five PFD conditions.

Chl a, chlorophyll a; Chl c 2 , chlorophyll c 2 ; Fuco, fucoxanthin; cis-Fuco, cis-fucoxanthin;

β-Car, β-carotene, Neox, neoxanthin. Data are the average of three measurements per three days (corresponding to the exponential growth phase for HL and ML conditions), error bars are SD.

  photoacclimation strategies in three ecotypes of Ostreococcus related to their ecological niches. Intraspecific variability has been shown in Pelagomomans calceolata (Dimier et al., 2009 a), demonstrating different photosynthetic regulation strategies in response to irradiance dynamics. All these properties make picoeukaryotes interesting models for physiological studies aiming to analyze the functional responses of algae to environmental changes, and for instance to photon flux density (PFD) variability. The purpose of our experimentation was to investigate the relation between growth rate, photosynthetic capacity and photoacclimative process efficiency in the coastal picoeukaryote Phaeomonas sp. (strain RCC503, Pinguiophyceae, Stramenopiles). Physiological response curve (PRC, Peek et al., 2002) experiments were performed plotting physiological variables and growth rate against a range of PFD. The questions behind this work were: what kind of photoresponses are developed by this species when low and high light conditions are experienced? How do these responses explain photosynthetic efficiency changes? Population growth was investigated over five days under five different sinusoidal light regimes set to peak at maximal photon flux densities of 10, 50, 100, 250 and 500 µmol photons m -2 s -1 . Pigments, variable fluorescence, NPQ, electron transport rate, absorption spectra, cell carbon and nitrogen content together with size, volume and cell concentrations were measured one to three times per day during the exponential phase.

  ). PFD was measured with a 4π quantameter (sensor QSL 2101, Biospherical Instruments Inc., San Diego, CA, USA).

  was composed of two solvent mixtures: methanol and aqueous ammonium acetate (70:30) and methanol. Pigments were detected spectrophotometrically at 440 nm using a Model DAD, Series 1100 Hewlett Packard photodiode array detector. Fluorescent pigments were detected in a Hewlett Packard standard FLD cell series 1100 with excitation and emission wavelengths set at 407 and 665 nm, respectively. Determination and quantification of pigments used pigment standards from D.H.I. Water & Environment (Hørsholm,Denmark). Variable fluorescence and NPQ Only during the irradiance peak sampling time wer samples for PAM measurements taken. Photochemical efficiency of PSII was estimated by a Phyto-PAM fluorometer (Heinz Walz GmbH, Effeltrich, Germany). Triplicate measurements of variable fluorescence were performed on both light-and 15-min dark-acclimated samples. The maximum photochemical efficiency (Fv : Fm, with Fv = Fm -Fo) was measured on 15min dark-acclimated sample during the irradiance peak sampling time. Fm and Fm′ were measured after a saturating pulse of red light (2400 µmol photons m -2 s -1 for 450 ms) causing a complete reduction of the PSII acceptor pool. The non-photochemical quenching of fluorescence was quantified by the Stern-(ETR) vs. light curves were determined on 15-min dark-exposed samples applying ten increasing irradiances (I from 1 to 1500 µmol photons m -2 s -1 , for 2 min each). The absolute Electron Transport Rate (expressed in e -Chl a -1 h -1

  2b, c), in agreement with the lower Chl a specific absorption coefficient (a* ph , data not shown). Following their distribution over the different PFD gradient, three groups of accessory pigments could be defined. The first group was formed by Chl c 2 and fucoxanthin (Fig.2b and c, respectively) as well as violaxanthin and antheraxanthin (see xanthophyll cycle paragraph) showing similar values at LL1 and LL2. Minor pigments such as Chl c 3, cis-fucoxanthin and ß-carotene (Fig.2b and c, respectively) belonged to the second group characterized by an increase from LL2 to LL1, as the Chl a pool. Pigments of the third group (neoxanthin and zeaxanthin, Figs.2d and 6b, respectively) showed a strong increase under HL conditions.

  photochemical quenching (Fig.5b), while it did not under lower PFD regimes, with the exception of LL1.A significant linear correlation was found between NPQ and the photoprotective pigment zeaxanthin [Zx : (Vx + Ax + Zx)], suggesting a relevant role of Zx in the development of NPQ in Phaeomonas sp. (Fig.6a). Zx contribution to the xanthophyll cycle was low (less than 10%) revealing the limited need for photoprotection, in relation with the gradual PFD increase and the not so extreme irradiance peak applied during the experimental conditions.

  that might enhance the coupling of photosynthesis and growth rate, as already discussed byDimier et al. (2009 a) for the picoeukaryote Pelagomonas calceolata.

  photoprotective pigment content. Further studies are needed to test this hypothesis and to obtain clearer knowledge on the ecophysiological capacity of coastal phytoplankton, such as many diatoms, to efficiently photoacclimate and photoprotect.Phaeomonas sp. is not able to grow under strong light limitation (e.g., for a Ek : PFD ratio > 10). The fact that the increase in Chl a is not coupled with an increase in photosynthetic accessory pigments might suggest an enhancement in number of PSII

(

  as Dimier et al., 2009 a) applied a sinusoidal irradiance regime, while the work done by Six et al. (2008) employed a continuous irradiance over the light period. Supporting such a hypothesis, Dimier et al. (2009 a) showed that the main factor responsible for developing the n-or σ-type photoacclimation was the frequency of irradiance variation during the daylight period.

Fig. 1 .

 1 Fig. 1. (a) The five sinusoidal light regimes applied during the study, with different maximal PFD peaks (10, 50, 100, 250 and 500 µmol photons m -2 s -1 , respectively). Arrows indicate sampling. (b) Evolution of growth rate of Phaeomonas sp. over the light conditions. Data are the average of three measurements per three days (corresponding to the exponential growth phase for HL and ML conditions), error bars are SD.

Fig. 2 .

 2 Fig. 2.Evolution of: chlorophyll a cell -1 expressed in 10 -16 mol Chl a cell -1 (a), chlorophyll c 2 cell -1 (white dots) and chlorophyll c 3 cell -1 (black dots) expressed in 10 -16

Fig. 3 .

 3 Fig. 3. Evolution of: cellular organic carbon (POC, white squares) and nitrogen (PON, black squares) concentrations (pg cell -1 ) (a), Chl a : POC ratio expressed in 10 -18 mol Chl a -1 pg C -1 (b), and POC : PON ratio (c), over the five PFD conditions. Relationship between POC : PON ratio and growth rate (r 2 = 98) (d). Data are the average of three measurements per three days (corresponding to the exponential growth phase for HL and ML conditions), error bars are SD.

Fig. 4 .Fig. 5 .

 45 Fig. 4. (a) Evolution of the maximal photosynthetic rate abs ETR max (in mol e -g Chl a -1 h - 1 ) over the five PFD conditions. Relationship between growth rate and abs ETR max (b) or rel ETR max (c). Evolution of α B (mol e -g Chl a -1 h -1 [µmol photons m -2 s -1 ] -1 , d), Ek (in

Fig. 6 .

 6 Fig. 6. (a) Relationship between Zx : (Vx + Ax + Zx) and NPQ (non-photochemical quenching). (b) Evolution of zeaxanthin Chl a -1 (Zx Chl a -1 in 10 -2 mol : mol Chl a) over the five PFD conditions. (c) Mean of zeaxanthin cell -1 in the three samples of the dawn, irradiance peak and sunset under the HL1 and HL2 regimes. Data are the average of three measurements per three days (corresponding to the exponential growth phase for HL and ML conditions), error bars are SD.

  Dimier et al., The highest NPQ value has been reached already under HL1 and no further increase can be observed under HL2 probably in relation to the development of other photoprotective mechanisms such as PSII conformational changes. Additional molecules of Zx under HL2 condition do not have a direct role in NPQ functioning. It is known that

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  The very low-light condition surprisingly induces both NPQ generation and little increase in Zx and Neox. Zx and NPQ formation reveals the development of a trans-thylakoidal proton gradient probably caused by chlororespiration(Goss and Jakob, in press, and references therein) or by a putative water-water cycle (Mehler's reaction,[START_REF] Badger | Electron flow to oxygen in higher plants and algae: rates and control of direct photoreduction (Mehler reaction) and rubisco oxygenase[END_REF]. It is noteworthy that Neox has a role against photooxidation and would be particularly active when Mehler's reaction takes place[START_REF] Dall'osto | The Arabidposis aba4-1 mutant reveals a specific function for neoxanthin in protection against photooxidative stress[END_REF]. Vasco Giovagnetti acknowledges the SZN for his PhD grant. We would like to thank Dr. D. Campbell and the other anonymous referee for their comments and helping suggestions improving the quality of the manuscript.

	F o o F o F o Six C., Sherrard, R., Lionard M., Roy S., Campbell D.A. (2009) Photosystem II and pigment dynamics among ecotypes of the Green Alga Ostreococcus. Plant F o probe. F FIGURE LEGENDS
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http://mc.manuscriptcentral.com/jplankt Journal of Plankton Research

ACKNOWLEDGEMENTS

The authors gratefully acknowledge Ferdinando Tramontano for his help during experiments and HPLC analysis. Enzo Saggiomo and Francesca Margiotta are acknowledged for using the spectrophotometer and the CHN and for sharing the PAM


Thorsten Timm

Alexander Blecken
email: alexander.blecken@hni.upb.de

A method for the hierarchical planning of structure, dimension and material requirements of manufacturing systems

Keywords: MATERIAL FLOW, MRP, MANUFACTURING SYSTEMS, MASTER PRODUCTION SCHEDULE, PRODUCTION PLANNING user): Hierarchical Planning, Coordination Process Material Requirements Planning, Manufacturing Systems, Hierarchical Planning, Master Planning, Coordination Process

Growing competition creates pressure among companies to run their production in a cost optimal way and yet adapt immediately to changes in their environmental conditions. Decisions must always consider the integration of the companies in globalised markets on both the demand and supply side. This paper deals with the optimal planning of decisions concerning the structure and dimension of production facilities. The methods used for material requirements planning (MRP) are selected and configured according to these decisions. These decision problems are represented in the form of four hierarchically layered partial models, each of them realised and iteratively solved by a mathematical optimisation model. The hierarchical planning method is evaluated using a practical case example based on a valve production system of a manufacturing company.

Introduction

Restrictions of logistical and regulative nature between countries and enterprises have been continuously diminishing over the past years. On the regulative layer, this is reinforced by the introduction of bilateral as well as multi-lateral agreements to reduce trade barriers and harmonize international norms and regulations. The scale effects resulting from the increasing traffic of goods between those countries as well as the technological progress lead to reduced transport costs and thus a further reduction in logistical barriers. Hence, enterprises which traditionally have focused on spatially restricted markets or regions become increasingly international and extend their activities. A number of implications follow for enterprises which serve as customers as well as suppliers on globalised markets: As customer, the enterprise not only purchases the materials and auxiliary means of production but also requests human resource capacity in a labour market. These resources need to be efficiently allocated under consideration of market demand.

One result of this development is the trend to concentrate on core competencies, which can be observed across industries at manufacturing companies. Manufacturing companies which assemble the final product, so-called original equipment manufacturers (OEMs), do so based on pre-fabricated modules. These modules are manufactured by other companies such that an ever more complex enterprise network becomes responsible for the production of the final product. Value creation is distributed over more and more participants in the production network; the fraction of the OEM has reduced approximately to only a mere 10% of overall value creation (see Kühn 2004, p. 33).

¿From the perspective of a company, it first needs to be established what an efficient resource allocation under these restrictions is. The foundation for a decision needs to be laid which clarifies where the respective sections of value creation are executed and whether this will be done in-house or outsourced to a supplier. Among various other factors, currently existing infrastructure such as production facilities, know-how and skills of the employees, as well as existing supplier relationships need to be incorporated in the decision making. Concurrently, it needs to be guaranteed that the temporal delays due to the selected links in the supply chain do not endanger the targeted service level and delivery reliability. Framework requirements concerning production lots und times need to be coordinated internally as well as with suppliers and required or implicitly derived stock levels need to be ascertained. In particular, the variability of products and modules on the separate production levels have to be incorporated in the planning.

This paper is structured as follows. Section 2 separates specific planning tasks and identifies their inter-dependencies in order to be able to construct hierarchical decision layers. In Section 3 relevant related contributions on hierarchical planning of manufacturing systems are analysed. The hierarchical planning method with the respective partial models is then constructed in Section 4. In Section 5 a case example shows how our method can be employed in practice with the example of a valve production system of a manufacturing company. Section 6 summarises the findings and concludes the paper.

Hierarchical Planning of Manufacturing Systems

The hierarchical manufacturing planning method developed here is geared to set the manufacturing structure and layout (structuring task), to determine the throughput of all entities under consideration of time (dimensioning task), and to select and configure planning methods and procedures (mid-term material requirements planning). The developed models describe the delimited choice of manufacturing planning tasks in the form of a number of quantitative partial models. These partial models are hierarchically connected by process definitions and can thus be used as manufacturing planning methods when sequentially computed by a solver.

Alternatively, it would be conceivable to solve the prior manufacturing planning tasks in an aggregate planning model (see Kistner and Steven 2001). Aggregate production planning simultaneously establishes optimal production, inventory and employment levels over a given finite planning horizon to meet the total demand for all products that share the same limited resources [START_REF] Nam | Aggregate production planning -A survey of models and methodologies[END_REF]Logendran 1992, Buffa and[START_REF] Buffa | Production-Inventory Systems: Planning and Control[END_REF]. In an early work [START_REF] Nam | Aggregate production planning -A survey of models and methodologies[END_REF] provide a structured overview of existing aggregate production planning approaches. The authors note that despite the numerous sources, very few aggregate production planning techniques have been implemented in industrial situations. Here, a hierarchical planning method is chosen as most promising approach. The partitioning of the manufacturing planning tasks in several partial models is sensible due to a number of reasons:

• Due to the high overall complexity of comprehensively planning the structure, dimension and material requirements of a manufacturing system, the use of a holistic model would in practice lead to intolerably long computation times when using an optimisation software (see McKay et al. 1995, Silver and[START_REF] Silver | Decision Systems For Inventory Management And Production Planning[END_REF]. • A number of time horizons can be examined on the various layers of the hierarchical planning model. Hence, different levels of aggregation can be used to efficiently solve the problem also for partial models with a longer time horizon. When analysing shorter time horizons, a more detailed data basis is used. This results in a reduction of the number of data to be collected and maintained as well as to improvements in computation time for the individual partial models. For a longer time horizon, a detailed data basis is infeasible, in particular due to forecasting problems. • Moreover, integrated holistic planning of all aspects of a manufacturing system is infeasible since the control of such a system requires the continuous adaptation of some parts of the system, while others remain constant for a longer period of time.

When using partial models, this sort of control is enabled by the separate replanning of individual partial models.

The partitioning of the entire planning task in a set of hierarchically layered partial models is a two-stage process. In a first step, the relevant manufacturing planning tasks are identified (Section 2.1). Thereafter, these planning tasks are aggregated according to a number of criteria such that the simultaneous solution of the separate groups by one single model is enabled and thus, partial models for each aggregated group can be built (Section 2.2).

The separate partial models are then connected by coordination processes. A solution for the entire planning system is retrieved by the coordinated solution of the partial models as well as the dynamic integration of feedback loops, i.e. information is transmitted from the subordinate to the super ordinate layers. While this method does not always yield the optimal solution, it is tried to sensibly partition the planning tasks in partial models and to integrate feedback loops to find an overall good solution.

Manufacturing Planning Tasks

The following manufacturing planning tasks can be identified (see [START_REF] Hayes | Restoring our Competitive Edge : Competing Through Manufacturing[END_REF]Wheelwright 1984, Günther andTempelmeier 2005, p. 8):

• Development of the asset (machine) inventory: Machines can be acquired or sold. Implicitly, location decisions are also derived here. The aspects capacity and production structure are thus determined. • Development of the labour force: Employees can be hired or dismissed. The availability of employees (both initially and during the course of time) and their respective skills and qualifications need to be considered when planning the aspect human resources. • Make-or-Buy decisions: For all parts which can either be produced in-house or purchased externally, a decision needs to be taken by which MRP procedure these parts are scheduled for the manufacturing process. This task defines the manufacturing or production depth. • Selection of alternative manufacturing processes: A process alternative defines the number, nature and sequence of process steps to manufacture a specific part.

According to [START_REF] Günther | Produktion und Logistik[END_REF], this is equivalent to the selection of a production technology. In our case example as presented in Section 5, for instance, the alternative manufacturing processes to produce a valve body are chipping and casting. • Selection and configuration of disposition strategies: A consumption-oriented or demand-oriented disposition strategy needs to be selected for each part.1 Thereupon, planning and control methods are determined. Configuration of consumption-oriented disposition is achieved by the parameters reorder level, the decision whether to use order cycles, and the determination of lot sizes. • Definition of stock echelons: In order to guarantee a pre-determined lead time for customers, stock echelons need to be defined enabling independent planning between any two manufacturing processes. This task is of elevated importance since the definition and allocation of stock echelons has implications on the selection and parameterization of planning and control concepts.

Construction of Partial Models

Since the partial models are to be used independently from each other for planning manufacturing systems, tasks which are mutually dependent need to be solved in one common partial model. Figure 1 displays the decisive dependencies between the tasks on the first planning layer. Decisions concerning the manufacturing processes, make-orbuy as well as machine (asset) stock are tightly coupled. Machines are only required for the manufacturing processes actually executed. Conversely the selection of manufacturing processes needs to be based on costs. Here, the required machines are a decisive factor. Furthermore, the make-or-buy decision is directly connected with the machine stock. Purchased items do not require capacities in the manufacturing system. External purchasing, however, competes with the acquisition of machines to produce that item.

While the decisions concerned with the labour force and the employee qualifications are also coupled with the other decisions taken on this layer, it may be assumed that the dependency of this coupling is uni-directional. A prerequisite for this simplifying assumption is that the fraction of the costs incurred by the wages and salaries associated with the manufacturing process are relatively low compared to further costs such as capital costs, holding costs and depreciation of machines and equipment etc.

Two partial models can be distinguished here. The first model deals with decisions related to alternative manufacturing processes, make-or-buy and machine stock. This model will be referred to as partial model I. The second model is based on the decisions of partial model I and optimises the labour force and the development of the skills and qualifications. It will be referred to as partial model II. On a second layer with shorter time horizons, two further partial models can be identified. Partial model III comprises the decision whether to use a consumption-oriented or demand-oriented disposition strategy, the definition of stock echelons, the decision whether to use order cycles, and sets the reorder level. Partial model IV eventually defines lot sizes for production and call-offs from external suppliers.

This paper focuses on the construction of partial models I and II. While these partial models will be elaborated in detail, partial models III and IV will only be described functionally. Their development has been carried out analogously to partial models I and II.

State of the Art

The handling of complex and extensive planning tasks leads to infeasible lead times for practical applications when employing optimal solution methods. The simultaneous planning of a comprehensive model in combination with a solver constitutes one solution method for the overall planning task. However, when dealing with planning tasks of considerable size, this method leads to long execution times despite ever-increasing computing power. Another problem encountered in the industrial practice is that it may only be required to come up with the solution for a limited number of planning tasks. When using comprehensive models, this always requires a new solution for the entire planning task. In order to circumvent these problems, hierarchical planning methods are employed.

A historical review of hierarchical production planning with its origins in scientific management and the applicability of hierarchical planning in manufacturing is provided by [START_REF] Mckay | A review of hierarchical production planning and its applicability for modern manufacturing[END_REF]. The authors elaborate the origins of hierarchical production planning and its underlying principles in order to support the understanding and where hierarchical production planning is suitable and warranted. They assert that the hierarchical planning paradigm is a descriptive model that serves to guide organizational design, structure information flows, and decompose large problems into manageable components. [START_REF] Schneeweiss | Hierarchical planning in organizations: Elements of a general theory[END_REF]

C = E N t=0 C t (z t+1 , a t ) (1)
z t+1 = G t (z t , a t , r t ) (2)
It holds that N is the number of stages in the dynamic program, C t is the optimisation criterion, z t the state, a t the decision and r t the stochastic interference for each stage t. Hence, C constitutes the expected value over the sum of the values of the optimisation criteria over all stages. The state transition relationship reveals how the state of the sequent stage can be deducted from the state of stage t, the decision on this stage as well as the stochastic interference.

¿From this mapping of hierarchical planning on dynamic programming, Schneeweiss (1998) deducts the general scheme of hierarchical planning as depicted in Figure 2. The information on the base model used in top level for its anticipation is represented as feed forward influence. The outcome a T * 1 of the top level is indicated as instruction. Additionally to the top-down influence, a feedback loop is introduced. The results of the planning on the base level can thus lead to replanning on the top level. The final decisions are taken in the form of a multi-staged negotiation process between the levels. A feedback loop is not mandatory in a hierarchical planning system.

When developing a conceptual framework of hierarchical enterprise systems, Rieper (1979) develops a three-layered hierarchical system. The basic two-layered structure of this approach (one of the layers is subsequently split up into two sub layers) does not allow for a direct effect of decisions taken on the super ordinate system, i.e. the enterprise information and decision system, on the system environment as envisioned by the conceptual framework of [START_REF] Schneeweiss | Hierarchical planning in organizations: Elements of a general theory[END_REF]. The enterprise information and decision system simply determines control values for the system on the subordinate level, i.e. the enterprise realisation system. To the largest degree, the enterprise realisation system in this approach is equivalent to the layer Operational Control in the model of [START_REF] Anthony | Planning and Control Systems : A Framework for Analysis[END_REF]. [START_REF] Boysen | Produktionsplanung bei Variantenfliessfertigung : Planungshierarchie und Elemente einer Hierarchischen Planung[END_REF] introduce a hierarchical planning method for variant flow produc- tion. This approach is geared directly towards the requirements of variant flow production as faced by the automotive or electronics industries. The planning tasks tackled by the model of [START_REF] Boysen | Produktionsplanung bei Variantenfliessfertigung : Planungshierarchie und Elemente einer Hierarchischen Planung[END_REF] differ from the planning tasks as derived in Section 2.1. [START_REF] Gebhard | Robuste hierarchische Produktionsplanung mit Bedarfsszenarien[END_REF] design a two-step hierarchical planning method for manufacturing planning. On the super ordinate planning layer (aggregated planning) decisions with medium-term time horizons are taken concerning seasonal stock levels and extension of available capacities with overtime. On the subordinate layer short-term (operational) production plans are developed. A mathematical model is presented which is used to solve each planning task.

A number of other studies on hierarchical planning of manufacturing and production systems exist. [START_REF] Hatchuel | Towards a new planning and scheduling approach for multistage production systems[END_REF] show that significant lead time performance improvements can result from integrated strategies for planning and scheduling decisions. [START_REF] Omar | Revising the master production schedule in a HPP framework context[END_REF] and [START_REF] Omar | Hierarchical production planning and scheduling in a multi-product, batch process environment[END_REF] derive mathematical models for master production scheduling in process industries. The effects of demand pattern, replanning periodicity, setup costs and unit production cost on the performance are examined. [START_REF] Kanyalkar | An integrated aggregate and detailed planning in a multi-site production environment using linear programming[END_REF] develop a linear programming model which derives a production and disposition plan in a multi-site production facility scenario with substitutable capacities serving multiple selling locations. The authors use varying time grids and planning horizons for the planning levels to address the computational complexity.

The majority of existing approaches to hierarchical planning of manufacturing systems focus on the short-term time horizon and address lot sizing or scheduling problems, see for instance [START_REF] Ebadian | Hierarchical production planning and scheduling in make-toorder environments: reaching short and reliable delivery dates[END_REF], [START_REF] Toledo | Multi-population genetic algorithm to solve the synchronized and integrated two-level lot sizing and scheduling problem[END_REF], [START_REF] Sawik | Hierarchical approach to production scheduling in make-to-order assembly[END_REF], [START_REF] Dumoulin | Tactical models for hierarchical capacitated lotsizing problems with setups and changeovers[END_REF]. However, integrated approaches incorporating both short-term and long-term time horizons which enable partial (re-) planning of tactical and operational decisions and which integrate with existing business information systems, remain scarce.

Hierarchical Planning Method

The objective is to develop a method which can be integrated into Advanced Planning System (APS) frameworks as defined by [START_REF] Reuter | Coordination and Integration[END_REF]. Hence, it becomes possible that our system can be used together with modules of existing software. On the other hand, the question tackled by our research goes beyond the functionality and planning horizon of APS systems. In particular, the planning and definition of asset (machine) inventory levels or the planning of human resources including training measures exceed the features of APS systems. Figure 3 shows how the hierarchical planning method is integrated into Rohde's APS framework. The shaded areas are addressed by the method developed here.

Based on the procedure model by [START_REF] Gebhard | Robuste hierarchische Produktionsplanung mit Bedarfsszenarien[END_REF] which considers the uncertainty of future development of demand, the future is represented in scenarios in the mathematical models developed here. Each scenario is attributed with a certain likelihood proportional to the probability of occurrence. These likelihoods will be used as weighting factors in the objective function. Thus, various sources can be used to support the planning decisions, multiple perspectives of stakeholders can be integrated, and even exceptional circumstances and events can be accounted for by assigning a respective (small) probability of occurrence.

The future evolution of the system environment (especially the development of demand) is not given deterministically, but depends on factors which are not susceptible to the decision makers of an enterprise. Thus, various scenarios are devised which deliver The scenarios are arranged in a tree structure, in which one scenario is always the successor of one other scenario (with the exception of the root scenario which does not have a predecessor scenario). When constructing the scenario tree, n P + 1 periods and n S + 1 scenarios are defined, whereas the root of the tree (the root scenario) is always scenario 0.

All partial models as presented in the following sections have been developed based on the Multi Level Capacitated Lot-Sizing Problem (MLCLSP) model. The MLCLSP assumes a finite planning horizon divided into discrete time periods. There are a number of items with period-specific external demands. The items are produced in a multi-stage production process on non-identical resources with limited period-specific capacities. A review of research streams on dynamic lotsizing with capacity constraints including a thorough introduction of the basic MLCLSP model is given by [START_REF] Buschkuehl | Dynamic capacitated lot-sizing problems: a classification and review of solution approaches[END_REF]. For each of the partial models, the basic MLCLSP model has been extended appropriately. Below, the specific extensions are elaborated in detail for each of the partial models.

The four partial models used in our planning method are described in Sections 4.1 to 4.4. Indices are standardised for all models and shown in Table 1. All further notation is introduced in the respective sections. Hence, Tables 2 and3 present symbols and functions of partial model I.

Partial Model I

This partial model integrates the manufacturing planning tasks selection of manufacturing processes, optimisation of machine (asset) stock and make-or-buy decisions into a single mathematical programme. Implicitly, location decisions are taken (via the choice of machines) and the core competences and thus the optimal manufacturing depth of the enterprise and the individual locations are determined. Parts which promise a cost reduction at the same quality when production is outsourced cannot represent a core competence of the enterprise. Such parts are identified as purchase parts by the model and marked accordingly.

The outcome of partial model I is presented in Figure 4. The variable K t is a performance indicator which depends on the results of partial model I and partial model II. The output variables resulting from the planning of partial model I and are used for the entire planning system are indicated beneath the model. An input variable for the model is the number of machines from which those are selected that are used to manufacture the required parts in each period. Alternatively, purchasing terms of possible suppliers are represented. The various manufacturing processes of parts are modelled as technologies. A technology represents manufacturing processes as input-output-processes. It describes a process in which one or more input factors are consumed and one or more machines are employd in the production of one or more outputs. Through the extension of the MLCLSP model with technologies, it becomes possible to model processes which employ more than one machine only. Other limitations of the MLCLSP model such as that the capacity needed to produce a part using a technology must be allocated within one time period, are not avoided by this approach. Also, the modelling of the required labour force for executing the manufacturing processes in a mathematical form is enabled. The latter extension will only be introduced in partial model II, since the availability of employees to execute the technologies is implicitly presumed here. Further input variables for this model are the different demand forecasts separated by scenarios, the limitation of the number of set-ups, machine capacity as well as manufacturing alternatives with their respective technologies.

Symbol

: 0 • • • n S → 0 • • • n S
Function assigning a period to every scenario ω : 0

• • • n S → ]0 • • • 1]
Function assigning a probability of occurrence to every scenario Table 3.: Notation: Functions partial model I Equations (3) to ( 16) represent the mathematical model of the selection of production processes and the asset (machine) stock level. The model extends the MLCLSP model by the following notions:

(1) Selection of different production processes to produce parts (2) Availability of machines in each period is not given deterministically. Rather, acquisition or disposal is part of the planning task and thereupon availability is derived (3) Use of technologies for modelling the production processes (4) Integration of planning uncertainty by parallel scenarios for each period (5) Determination of machine utilisation for each machine in each period min Expression (3) represents the objective function which minimises the expected value of the aggregated cost over all scenarios. Each scenario j is weighted with a likelihood ω(j) and costs are devalued with the calculatory interest factor c Z . The summands represent the costs for changing process alternatives, variable production and setup costs, inventory holding and penalty costs for delays. Additionally to the cost factors considered in the MLCLSP model, costs for external purchasing of parts, the procurement of machines and fixed costs for existing machines are considered in order to enable the optimisation of the machine stock level over the scenario tree. Costs for external purchasing are assumed to be linear. The objective function is subject to a number of restrictions.

n S j=0 ω(j) • 1 + c Z -π(j) • n W n=1 n T m=1 b RT mnj • t RT mn • c R n + x T mj • t T W mn • c P n (3) + n E k=1 max(x L kj , 0) • c L k -min(x L kj , 0) • c V k + n E k=1 x K kj • c K k + n W n=1 b W nπ(j) • c F n + b W N nπ(j) • c N n + b P N A π(j) • c P N A
n RTmax m ≥ n W n=1 b RT mnj ∀m ∈ 1 . . . n T , ∀j ∈ 0 . . . n S (4) x T mj ≤ M • b RT mnj ∀m ∈ 1 . . . n T , ∀n ∈ 1 . . . n W , ∀j ∈ 0 . . . n S (5)
Constraint ( 4) limits the number of setups. Setup actions are represented by the binary variable b RT mnj which is 1, if machine n is setup to execute technology m in scenario j. Constraint (5) restricts the executable technologies to those for which the machines can be setup.

n T m=1 b RT mnj • t RT mn + x T mj • t T W mn ≤ t KW nπ(j) • b W nπ(j) ∀n ∈ 1 . . . n W , ∀j ∈ 0 . . . n S (6) x L kj = x L kν(j) + n T m=1 x T mj • n T E km -n T B km • x T mj + x K kj -n B kj (7) ∀k ∈ 1 . . . n E , ∀j ∈ 0 . . . n S
Constraint (6) restricts the up-time of machines. On the one hand, the sum over all technologies is calculated such that the setup and production time in the new model can be calculated. On the other hand, the capacity of each machine is multiplied with the binary variable b W ni for each scenario in order to incorporate optimised machine availability. Constraint (7) calculates the inventory available at the end of each period for every scenario. Noteworthy is the second summand on the right side, which adds the manufactured quantity of the respective part. The third summand serves to model the external procurement of parts. Transport times are not included here since the time grid in this partial model is assumed to be large enough to allow the omission of transport delays. In Constraint (8) the binary variable b W N ni is adjusted. This variable signals the introduction of machine n in period i. Likewise, the binary variable b W ni is adjusted which signals the availability of machine n in period i. Constraint (9) limits the volume of purchase parts in such a way that these can only be purchased in a scenario if they are marked as purchase parts by the binary variable b K ki . Constraint (10) ensures in the other direction that purchase parts cannot be manufactured in-house. In order to achieve that, the output sum over all technologies is calculated and multiplied by the inverse value of b K ki and limited with a sufficiently large number M .

b W ni ≤ b W n(i-1) + b W N ni ∀n ∈ 1 . . . n W , ∀i ∈ 1 . . . n P (8) x K kj ≤ b K kπ(j) • M ∀k ∈ 1 . . . n E , ∀j ∈ 0 . . . n S (9) n T m=1 x T mj • n T E km ≤ 1 -b K kπ(j) • M ∀k ∈ 1 . . . n E , ∀j ∈ 0 . . . n S (10)
x

T mj ≤ M • n P A l=1 b P lm • b P N lπ(j) ∀m ∈ 1 . . . n T , ∀j ∈ 0 . . . n S ( 11
) n P A l=1 b P N li = 1 ∀i ∈ 0 . . . n P (12) b P N A i ≥ b P N li -b P N l(i-1)
∀l ∈ 1 . . . n P A , ∀i ∈ 1 . . . n P (13) Constraint ( 11) ensures that exclusively those technologies are used which are available in the selected process alternative. Through the multiplication with a sufficiently large number M the part can serve as the upper limit for x T mj . Constraint ( 12) limits the number of employable process alternatives in a single period to 1. Constraint (13) adjusts the binary variable which signals a change in the selected process alternative.

x L k0 = 0 ∀k ∈ 1 . . . n E (14) b W n0 ≤ b W 0 n ∀n ∈ 1 . . . n W (15) b P N l0 = b P N 0 l ∀l ∈ 1 . . . n P A (16)
Constraint ( 14) initialises the inventory in the original scenario with 0 (alternatively any current inventory level). Constraints ( 15) and ( 16) initialise the availability of machines and the usage of process alternatives in the original scenario. Finally, a number of non-negativity constraints which are not shown in detail here, complete partial model I.

Partial Model II

Partial model II deals with supporting decisions related to the planning of human resources, decisions concerned with determining which employees shall acquire new skills or qualifications and how the human resource structure can be optimally planned taking into account the temporal dynamic of external factors. The mathematical model concerned with asset planning, as presented in the previous section, determined the availability of machines for each period which serves as input parameter for partial model II. The number of technologies (and thus the volume of in-house production) is not considered as fixed input parameter since the optimisation in partial model II also incorporates the planning of human resources and an efficient resource allocation differing from the plain asset (machine) stock optimisation is conceivable.

The upshot of partial model II is presented in Figure 5. The matrix b M oi displays whether 17) to ( 33) represent the mathematical model of the human resource planning. This model extends the MLCLSP model by the following notions:

(1) Usage of varying scenarios in order to enable the presentation of alternative developments, especially with respect to demand (2) Availability of employees (3) Skills and qualifications of employees and acquisition of new qualifications min 17) is the objective function of partial model II. In comparison to the MLCLSP model, the objective function is extended with the last sum over all employees n M by wage costs, costs for hirings and dismissals of employees and the costs for acquiring new skills and qualifications. Furthermore, every scenario j is weighted with a likelihood ω(j). Costs are devalued with the calculatory interest factor c Z . The first summand represents the base wages of the employees as product of the matrix b M oi (indicating employee availability) and the base wage vector c M o . Further costs are incurred by overtime represented by the product of required overtime x U oj and the respective costs c U o . The following two summands add the costs for hiring and dismissals of employees. The last summand represents costs of training measures. The objective function is subject to a number of restrictions, presented in the following.

n S j=0 ω(j) • 1 + c Z -π(j) • n W n=1 n T m=1 b RT mnj • t RT mn • c R n + x T mj • t T W mn • c P n (17) + n E k=1 x K kj • c K k + n E k=1 max(x L kj , 0) • c L k -min(x L kj , 0) • c V k + n M o=1 b M oπ(j) • c M o + x U oj • c U o + b M E oπ(j) • c M E + b M R oπ(j) • c M R + n Q p=1 b M QN opπ(j) • c Q p Expression (
n RTmax m ≥ n W n=1 b RT mnj ∀m ∈ 1 . . . n T , ∀j ∈ 0 . . . n S (18) x T mj ≤ M • b RT mnj ∀m ∈ 1 . . . n T , ∀n ∈ 1 . . . n W , ∀j ∈ 0 . . . n S (19)
Constraint ( 18) is identical to Constraint (4) and limits the number of setups. Likewise, Constraint ( 19) is identical to Constraint (5) and limits the executable technologies to those for which the machines can be setup.

n T m=1 b RT mnj • t RT mn + x T mj • t T W mn ≤ t KW nπ(j) • b W nπ(j) ∀n ∈ 1 . . . n W , ∀j ∈ 0 . . . n S (20) x L kj = x L kν(j) + n T m=1 x T mj • n T E km -n T B km • x T mj + x K kj -n B kj (21) ∀k ∈ 1 . . . n E , ∀j ∈ 0 . . . n S
Constraint (20) limits machine up-time. It needs to be considered that b W ni is not a variable in this model but a matrix of given constants; the availability of machines which has been determined by the preceding partial model is considered an input to this model. The inventory at the end of the period as calculated in Constraint ( 21) is equivalent to the model of the asset (machine) inventory planning, see Constraint (7).

n T m=1 x T mj • t T Q mp ≤ n M o=1 x QM poj ∀j ∈ 0 . . . n S , ∀p ∈ 1 . . . n Q (22) n Q p=1 x QM poj + b M QN opπ(j) • t L p ≤ b M oπ(j) • t M oπ(j) + x U oj ∀o ∈ 1 . . . n M , ∀j ∈ 0 . . . n S (23)
The mapping of employees to the usage of their qualifications is achieved through the variable x QM poj . This variable represents the number of time units spent by employee o applying qualification p in scenario j. Constraint (22) ensures that the number of time units spent by employees performing a skill equals the time needed for the technology being used. Constraint (23) limits the available working time of each employee for each period. The first summand is the accumulated time required by the training of skills for each employee. The second summand adds the times required for training measures, where the vector t L p determines the time required for each specific training. The upper limit of working time is the sum of the contractual base working time t M oi and the amount of overtime x U oj . The binary variable b M oi considers whether an employee is actually available in the respective period.

x QM poj ≤ M • b M Q opπ(j) ∀p ∈ 1 . . . n Q , ∀o ∈ 1 . . . n M , ∀j ∈ 1 . . . n S (24) b M Q opi ≤ b M Q op(i-1) + b M QN opi ∀o ∈ 1 . . . n M , ∀p ∈ 1 . . . n Q , ∀i ∈ 1 . . . n P (25) x U oj ≤ n Umax • b M oπ(j) ∀o ∈ 1 . . . n M , ∀j ∈ 1 . . . n S (26)
Constraint (24) ensures that an employee only uses time applying a qualification if he actually possesses this qualification in the respective period. The skill or qualification matrix is updated according to the newly learned skills in Constraint (25). Constraint (26) limits the amount of overtime to a maximum value of n Umax or 0 if the employee is not hired in the respective period.

n T E km • x T mj ≤ M • 1 -b K kπ(j) ∀k ∈ 1 . . . n E , ∀j ∈ 0 . . . n S , ∀m ∈ 1 . . . n T (27) x K kj ≤ M • b K kπ(j) ∀k ∈ 1 . . . n E , ∀j ∈ 0 . . . n S (28) b M E oi ≥ b M oi -b M o(i-1) ∀o ∈ 1 . . . n M , ∀i ∈ 1 . . . n P (29) b M R oi ≥ b M o(i-1) -b M oi ∀o ∈ 1 . . . n M , ∀i ∈ 1 . . . n P (30)
Constraints ( 27) and ( 28) ascertain that the parts marked as purchase parts in partial model I are not produced in-house and parts marked as in-house production cannot be externally purchased. Constraints ( 29) and ( 30

M o0 = b M 0 o ∀o ∈ 1 . . . n M (31) b M Q op0 ≤ b M Q0 op ∀o ∈ 1 . . . n M , ∀p ∈ 1 . . . n Q (32) x L k0 = 0 ∀k ∈ 1 . . . n E (33)
In Constraints ( 31) and ( 32) the initial values for the number of employees and the skills matrix are mapped to the binary variables of the original scenario. Constraint (33) sets the inventory in the original scenario to 0. Finally, a number of non-negativity constraints which are not shown in detail here, complete partial model II.

Partial Model III

The definition of stock echelons and the involved determination of reorder levels for parts planned with a consumption-oriented disposition strategy is an MRP procedure and will be tackled in an integrated mathematical model. The objective of this model is to position stock echelons in such a way that, on the one hand, the required lead time can be guaranteed. On the other hand, inventory costs including capital holding costs need to be kept minimal. The required maximum lead time is an input parameter to this partial model. It limits the critical path of production and replenishment times of all parts which are required for a final product and not served from a stock echelon.

Model III: Definition of stock echelons and determination of reorder levels

Output and subordinate partial models

Coordination Processes

Coordination processes connecting partial models I through IV are defined in order to use them for the hierarchical planning of structure, dimension and material requirements of manufacturing systems. By complementing planning processes starting at partial models III and IV, the manufacturing system can also be planned solely based on these partial models and hence adapt to changing environmental conditions. If required, the planning can be extended on the layer of the super ordinate partial models I and II. The coordination processes determine at which time or event which optimisation model replans one of the partial models and how the results of the replanning will be used subsequently.

The coordination processes are controlled by indicators derived from the partial models. These indicators allow deciding whether a solution fulfils all requirements of the strategic goals. It is conceivable to guarantee the compliance with these requirements already when formulating the planning models by adding respective restrictions. However, this could cause problems in finding feasible solutions for the partial models and the planning would consequently not come to a final result. Therefore, the indicators are used afterwards by the coordination processes. Since the focus of this paper is put on partial models I and II, a detailed description of the coordination processes is omitted here.

Case Example

Scenario

A case example served to assess the applicability, feasibility and performance of the developed hierarchical planning method. Here, pneumatic valve modules (in the following called valve modules) geared to control braking systems of rail vehicles are manufactured in a two-stage production process. 1 These valve modules exist in a number of variants resulting from the various intended usages. Based on properties such as performance, robustness, durability, and general layout 40 different products (or product variants) can be distinguished. For each product variant primary demand from the final customer can exist.

The two-stage production process is currently structured in a machining (chipping) production section and an assembly section as illustrated in Figure 8. Both sections of the production process are carried out in the same plant. The assembly section consists of four pre-assembly lines and one final assembly line which are directly connected. The pre-assembly lines mount the required components for the final assembly line based on actual demand. Inventory between these lines only exists in the form of small buffers. This has been achieved by the application of the One-Piece-Flow idea (see [START_REF] Lander | The Toyota Production System and art: making highly customized and creative products the Toyota way[END_REF], the consistent balancing of lines and the use of small lot sizes at the assembly lines. Thus, the assembly lines can be planned integratedly for the purposes of this analysis.

The parts production is carried out in the same location. Three processing centres can be considered for manufacturing the items required as input for the assembly lines. These parts include the valve bodies to which further components are subsequently assembled. While some further items are produced in-house, these are mainly parts which require advanced manufacturing competence and can thus be considered a core competence of 1 The case example presented here draws from a case study conducted when carrying out this research. While the production structure and according data are drawn from the case study, some estimations and approximations had to be made. Some of the data has been intentionally altered in order to respect confidentiality of the company and sensitive data. However, these alterations did not have an effect on the results themselves which appropriately reflect the outcome obtained with the original data. the company. The allocation in the machining section is performed on a demand-oriented basis. This leads to partially extremely long delivery times and problems with the delivery reliability for the products of the company since long cycle and setup times lead to long replenishment lead times.

On a strategic level, the company has decided that all customer orders are to be filled within a lead time of five days. This delivery time has to be achieved with a delivery reliability of 95%. This decision triggered a comprehensive analysis of the entire planning of structure, dimension and material requirements of the production processes. Specifically, it needs to be examined whether the chipping production, i.e. the machining section of the overall production process, is the appropriate production technology and geared to suffice future customer requirements with respect to delivery time, delivery reliability and costs.

The valve bodies can alternatively be produced by using a different production technology. This can also be done in spatial separation if a respective disposition strategy ensures permanent availability. Hence, an alternative product structure has been developed in which the bodies for the valve modules are not produced by the current processing centres in a chipping production process, but cast. This process cannot be executed by the company itself. A supplier has been identified which would be able to deliver the parts in the required quality. Two central questions arise:

• Shall the valve bodies be produced in-house in the current assembly stations or be outsourced and procured from a supplier who uses an alternative production technology (casting)? • Which disposition methods are to be used for the respective products and to what value need the parameters be set in order to ensure a delivery reliability of 95%?

Implementation

The models have been built in the CPLEX LP format, which served as input format for the optimisation software ILOG CPLEX, which was used in version 10.1.1. The choice for this optimisation software has been made since CPLEX requires the least computational time for solving mathematical models among the currently available systems. Standard settings have been used for CPLEX except the parameter 'mipgap', which has been set to a value of 0.0005 in order to achieve shortest execution time. The hardware used for optimisation was a standard 2 GHz dual core notebook with 2 GB RAM.

Results

Partial Model I

The optimisation of partial model I required 89.95 seconds computation time. The values of the respective variables (not listed in detail here) can be interpreted in the following way:

(1) The binary variable b P N li which denotes whether a process alternative l is used in period i is set to the value l = 1 for all periods. Thus, production is continued at the current location using the currently employed technology. The production of the valve bodies is continued at the processing centres in the current assembly section of the manufacturing plant. (2) The binary b W ni which denotes whether machine n is available in period i is set to a value of 1 for machine 1 over the entire planning horizon. The final assembly line is required continuously since it is part of process alternative 1. This variable is set to a value of 0 for machine 3 over the entire planning horizon. This means that with the chosen combination of in-house production and outsourcing one of the three processing centres is not required anymore. For machines 2 and 4, the binary variable b W ni is first 1 before it switches to 0 towards the end of the planning horizon. This may not be interpreted as a disestablishment of the respective processing centres in those periods. Rather, the reasons why these are no longer required lies in the fact that the required parts for the assembly line are produced in the beginning periods of the planning interval. The existing inventory is then used in the remaining periods. The processing centres should remain for the entire time horizon.

The outcome of the optimisation of partial model I is the production layout as displayed in Figure 9. Production and assembly are continued at their present locations. One processing centre can be omitted which creates additional free space in the production plant.

Partial Model II

The optimisation of partial model II required 72.08 seconds computation time. The values of the respective variables (not listed in detail here) can be interpreted in the following way:

(1) The binary variables b M Q opi and b M QN opi that determine which employees are trained to gain certain skills or qualifications show that one employee currently working in the assembly section is trained to acquire the necessary qualifications to be able to work in the processing centres. Thus, this employee can be flexibly used either in the assembly section or in the machining production section and variances in the labour supply can be balanced. This decision goes in accordance with the development of the labour force, which foresees a reduction of currently employed unskilled workers in the processing centres. determine the further evolution of the labour force. The outcome of the optimisation is that three employees from assembly, one unskilled worker from the processing centres as well as three employees from the processing centres (two instead of three processing centres) are no longer required in their present field of responsibility. This reduction in labour force is significant. During the comprehensive structure, dimension and material requirements planning project carried out in this case example, all target times had been re-established. This has been achieved through actual time measurement during running production. Furthermore, employees in the production section and assembly section were relieved of auxiliary tasks such that they could concentrate on the tasks for which they are skilled. Since this measure requires workers for the centralised material disposition, the reduction in labour force does not necessarily lead to dismissals.

Partial Model III

The optimisation of partial model III required only 0.19 seconds computation time. The values of the respective variables (not listed in detail here) can be interpreted in the following way:

(1) The binary variable b BZ k , which determines the parts which are planned with an order cycle, defines an order cycle for all in-house production parts. This results from the high load of the processing centres and assembly lines in the considered time horizon. Through the use of order cycles, an optimised sequence planning can be carried out in regular intervals for all work stations (determined at five time units in this example).

(2) The binary variable b LS k , which defines for which parts inventory shall be held, determines stock echelons for all parts.

(3) The reorder level x M B k is set to values between 7 (for valve modules which are only ordered in small volumes) and 211 (for add-on parts which are integrated in various valve modules).

Partial Model IV

The optimisation of partial model IV required 7.39 seconds computation time. The output of this partial model is only one variable:

(1) The lot sizes x LG k are adjusted in the optimisation model for each period. The limitation to parts planned with a consumption-oriented disposition strategy is irrelevant here, since partial model III defined a stock echelon for all parts considered. The lot sizes are chosen in such a way that the desired value for the delivery reliability is kept under the parameters set by partial model III. The defined lot sizes vary between 10 and 342.

Conclusions and Outlook

The hierarchical planning method in this paper has been developed in cooperation with a manufacturing company. The case example described in Section 5 showed that the planning tasks identified in this company can be solved with the method in time intervals suitable for practical implementation. To integrate the method into the planning processes of the company, the next steps are to develop user and data interfaces to ex- isting systems and thus, further improving the usability and applicability of the method. Moreover, a fifth partial model is currently under development. It separates the decisions concerning establishing and closing locations from partial model I into a dedicated partial model in order to determine the manufacturing footprint. Hence, the time model for these decisions can be chosen according to the long-term time horizon of these decisions.

  Figure 1.: Dependencies of planning tasks

Figure 3 .

 3 Figure 3.: Integration of the hierarchical planning method into APS framework

0 nnkknn

 0 if part k is a purchase part in period i b P lm Binary variable showing if process alternative l allows the use of technology m b P N li Binary variable showing if process alternative l is used in period i b P N 0 l Binary variable showing if process alternative l is used in the original scenario b P N A i Binary variable, which signals the change in the process alternative used in period i b RT mnj Binary set-up variable showing if machine n is equipped for development of technology m in scenario j b W ni Binary variable showing if machine n is available in period i b W Binary variable showing if machine n is available in the original scenario b W N ni Binary variable showing if machine n is procured in period i c F Fixed costs which accrue in every scenario in which machine n is available c K Unit purchasing price per part k c L Holding cost rate per unit quantity of part k at the end of a period c N Cost of new acquisition of machine n c P Production cost rate per time unit of machine n c P N A Cost of changing the use of a process alternative c R n Set-up cost rate of time unit for machine n c V k Delay cost rate per unit quantity for backorders from part k at the end of one period c Z Calculatory interest factor of the enterprise per period in the tactical planning models M Sufficiently large number n B kj Primary demand for part k in scenario j n E Number of parts n P Number of observed periods (planning horizon) (without the initial period with index 0) n P A Number of process alternatives n RTmax m Maximum number of set-up operations for execution of technology m per period n S Number of observed scenarios (without the root scenario with index 0) n T Number of technologies n T B km Required input of part k for executing one unit of technology m n T E km Output of part k for executing one unit of technology m n in period i in time units t RT mn Time required to set up machine n in order to execute technology m t T W mn Processing time at machine j for the execution of one unit of technology m x K kj Purchase volume of part k in scenario j x L kj Inventory or backorder of part k at the end of scenario j x T mj Number of units of technology m executed in scenario j Table 2.: Notation: Symbols partial model I Function Definition ν : 0 • • • n S → 0 • • • n S ∪ ∅ Function mapping each period in the time model of the operative planning to a period in the time model of the tactical planning π

Figure 5 . 0 op

 50 Figure 5.: Outcome of partial model II

  ) adjust the binary variables b M E oi and b M R oi for employee hirings and dismissals.

  b

Figure 6 .

 6 Figure 6.: Outcome of partial model III

4. 4 .Figure 7 .

 47 Figure 7.: Outcome of partial model IV

Figure 8 .

 8 Figure 8.: Structure of production (as-is state)

  Figure 9.: Structure of production (after optimisation)

•

  Development of employee qualifications: Each employee possesses a unique set of initial qualifications. This set can be expanded by training measures to acquire new qualifications.

	F o r
	P
	e
	e r
	R
	e
	v i e
	w
	O n l
	y

Table 1

 1 Table 4 introduces additional symbols of partial model II.

			Model I: Planning of processes,
			locations, and machine stock
			ni	ni	li	i	b K ki
	Process control	Output and subordinate partial models
			Figure 4.: Outcome of partial model I
	F o r Index Definition
	i j k l m n o p	Period Scenario P Part Process alternative e Technology Machine e Employee r Qualification
			R
				e
				v i e
					w
						O n l
						y

.: Notation: Indices

In a consumption-oriented strategy orders are triggered only by the current invontory level, in a demand-oriented strategy orders are triggered by the (planned) production programme.

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