Thierry Baron
email: thierry.baron@anses.fr

Jean-Noël Arsac

Jérémy Verchère

Habiba Tlili

Claire Aufauvre

Dominique Bétemps

Co-expression of APP/PS1 disrupts the distribution of brain lesions in a synucleinopathy transgenic mouse model (M83)

Keywords: alpha-synuclein, beta-amyloid, Parkinson's, strain, aggregation

Human neurodegenerative diseases (HNDs) often involve the aggregation of several proteins, with disease resulting from their interactions rather than their coincidental co-existence [START_REF] Attems | The multi-morbid old brain[END_REF]10].

Dementia with Lewy bodies is characterized by alpha-synuclein (aS) aggregation and most cases show Alzheimer's disease neuropathology involving amyloid plaques and neurofibrillary tangles [START_REF] Irwin | Neuropathological and genetic correlates of survival and dementia onset in synucleinopathies: a retrospective analysis[END_REF], which is associated with earlier onset of dementia. This suggests synergistic interactions of aS with β-amyloid or/and tau proteins.

Here, we performed a pilot study in which we examined aging of a cohort of 13 mice obtained by breeding M83 homozygous (Ho) mice, expressing human A53T mutated aS [START_REF] Giasson | Neuronal alpha-synucleinopathy with severe movement disorder in mice expressing A53T human alpha-synuclein[END_REF], with APPPS1-21 hemizygous (He) mice expressing human mutated amyloid precursor (APP) and presenilin-1 (PS1) proteins [9]. Mice were sacrificed at the age of 22 months, beyond which M83 He mice can develop signs of paralysis, occurring in old M83 mice. None of the mice had, however, developed paralysis, even the 6 APPxM83 mice bearing the APP/PS1 transgene. Surprisingly, Western blot analyses of samples from the cerebral cortices of the 13 mice showed abundant aggregated serine 129 phosphorylated aS (PSer129-aS) in 3 of the 6 APPxM83 mice, instead of only monomeric aS in all the other mice (Fig. 1a). We further examined PSer129-aS neuroanatomical distribution using both Western blot and ELISA, which showed the highest levels in the cerebral cortex, hippocampus and striatum (Fig. 1b and Supplementary Fig. 1), where lesions are scarce or absent in sick M83 mice [START_REF] Giasson | Neuronal alpha-synucleinopathy with severe movement disorder in mice expressing A53T human alpha-synuclein[END_REF]. PSer129-aS was found at lower levels in the midbrain and brainstem, where it is consistently more abundant in sick M83 mice. This striking difference was confirmed by immunohistochemistry, as illustrated by major and unexpected somatodendritic and neuritic PSer129-aS labelling in the hippocampus and cerebral cortex of APPxM83 mice (Fig. 1c-e and Supplementary Fig. 2c), where it is typically absent or scarce in sick M83 mice. Remarkably, the amyloid plaque load was strongly reduced in APPxM83 mice compared to age-matched APPPS1-21 mice (Supplementary Fig. 2a-b). The direct molecular interactions of aS with APP could favor the accumulation of phosphorylated aggregation-prone aS conformers in the brain regions with the highest levels of both aS and APP expression (Supplementary Fig. 3) [10]. An effect of mutated PS1 on aS phosphorylation has also been reported in humans [START_REF] Kaneko | Enhanced accumulation of phosphorylated alpha-synuclein and elevated beta-amyloid 42/40 ratio caused by expression of the presenilin-1 deltaT440 mutant associated with References 1[END_REF].

Additionally, intra-striatal inoculation in M83 (Ho) of a cerebral cortex homogenate from one of the APPxM83 mice containing PSer129-aS triggered accelerated paralysis of M83 (Ho) mice, which was not the case after inoculations of two PSer129-aS negative cerebral cortex homogenates from APPxM83 or M83 (He) mice (Supplementary Fig. 4). Typical distribution of PSer129-aS in the hindbrain was observed in these paralyzed M83 (Ho) mice.

Recent studies suggested that aS "strains" may explain the phenotypic diversity of synucleinopathies [START_REF] Ferreira | Multiple system atrophy-associated oligodendroglial protein p25alpha stimulates formation of novel alpha-synuclein strain with enhanced neurodegenerative potential[END_REF]7,8,11]. One of these studies suggested that "strains" could be differentiated in M83 mice inoculated with distinct sources of aggregated aS, based on the finding of more abundant PSer129-aS deposits in the hippocampus and cerebral cortex [7].

Another study showed increased pathogenicity and lesions in M83 mice challenged by recombinant aS, when its aggregation was initially triggered in the presence of p25α protein involved in the lesions of multiple system atrophy in humans [START_REF] Ferreira | Multiple system atrophy-associated oligodendroglial protein p25alpha stimulates formation of novel alpha-synuclein strain with enhanced neurodegenerative potential[END_REF].

Distinct conformations of these proteins, similar to prion "strains", may help to explain the phenotypic diversity of HNDs. However, another, not mutually exclusive, hypothesis related to the multiple pathologies within the multi-morbid old brain, has been poorly considered [START_REF] Bras | Alpha-Synuclein: Mechanisms of Release and Pathology Progression in Synucleinopathies[END_REF]. Altogether, our data reinforce the idea that interactions of aS with other cerebral components may play a key role in determining its pathogenic behavior.

Corresponding author:

Thierry Baron, ANSES -Laboratoire de Lyon, 31 avenue Tony Garnier, 69364 Lyon cedex 7,

France. Tel: +33(0)478696833, Fax: +33(0)478619145, E-mail: thierry.baron@anses.fr

Materials and Methods

Animals

M83 transgenic mice (B6;C3H-Tg[SNCA]83Vle/J) express A53T mutated human alphasynuclein (aS) protein under the control of the prion protein promoter [START_REF] Bras | Alpha-Synuclein: Mechanisms of Release and Pathology Progression in Synucleinopathies[END_REF]. Homozygous (Ho)

M83 mice develop characteristic motor symptoms between 8 and 16 months of life, beginning with reduced ambulation, balance disorders, partial paralysis of a hind leg and then progressing to prostration, difficulty in feeding, weight loss, hunched back and general paralysis [START_REF] Bras | Alpha-Synuclein: Mechanisms of Release and Pathology Progression in Synucleinopathies[END_REF]. These clinical signs may only occur later, beyond the age of 22 months, in hemizygous (He) mice. At the terminal stage, brain lesions, including intraneuronal deposits of serine 129 phosphorylated aS (PSer129-aS), are consistently prominent in the midbrain, brainstem and deep cerebellar nuclei, whereas they remain scarce or absent in most cortical brain regions or in the hippocampus, including after intracerebral inoculations of aggregated aS [1-3, 5, 6].

APPPS1-21 transgenic mice (B6.Cg-Tg(Thy1-APPSw,Thy1-PSEN1*L166P)21Jckr) contain human transgenes for both amyloid precursor protein (APP), bearing the Swedish mutation, and presenilin 1 (PS1), containing an L166P mutation, both under the control of the Thy1 promoter [START_REF] Giasson | Neuronal alpha-synucleinopathy with severe movement disorder in mice expressing A53T human alpha-synuclein[END_REF]. Amyloid deposits appear at the age of 6 weeks in the neocortex then spread in the hippocampus and widely throughout most of the brain regions.

Homozygous M83 mice and hemizygous APPPS1-21 mice were bred to produce a progeny from which 13 animals were followed up to the age of 22 months. Genotypes were determined with selected forward (GAATTCCGACATGACTCAGG) and reverse Free gels were activated for 1 min on Chemidoc MP apparatus (BioRad, Marnes-La-Coquette, France) and image analysis was done using ImageLab software (BioRad) to control the protein loads per lane. aS was cross-linked to the membranes using 4% paraformaldehyde and 0.01% glutaraldehyde diluted in PBS for 30 minutes and unspecific binding sites were saturated by adding 5% milk for 1 h. aS was detected by the rabbit monoclonal EP1536Y antibody (1:1000;

Abcam, Paris, France) for PSer129-aS, or rabbit monoclonal antibodies MJFR1 (1:4000;

Abcam, Paris, France) or D37A6 (1:1000; Ozyme, Saint-Cyr-l'Ecole, France), for human or murine aS respectively. GAPDH was detected by the mouse monoclonal antibody 6C5 (1:4000;

Sigma, Saint-Quentin Fallavier, France). The antibodies were incubated overnight at +4°C.

After washings, membranes were then incubated with anti-mouse or anti-rabbit HRP (1:10000;

Clinisciences, Nanterre, France), for 1 h at room temperature (RT). HRP was then visualized with chemiluminescent substrate (Supersignal WestDura, 10445345, Fisher, Illkirch, France)

and analyzed using the ChemiDoc system and ImageLab software (Bio-Rad).

For ELISA analyses, PSer129-aS was detected from mouse brain homogenates as previously described [START_REF] Irwin | Neuropathological and genetic correlates of survival and dementia onset in synucleinopathies: a retrospective analysis[END_REF]. plates were incubated for 1 h at 25°C. After washing the plates 5 times with PBST, 100 μL of 3,3,5,5-tetramethylbenzidine solution (T0440, Sigma, Saint-Quentin Fallavier, France) were added to each well and plates were incubated for 15 min with shaking. The reaction was stopped with 100 μL of 1 N H2SO4, and the absorbance was measured at 450 nm with the Clariostar (BMG) microplate reader. The threshold was calculated as the mean + 3xSD of three repeated ELISA tests on two half-brains from 5-month-old M83 mice. For detection of human and murine aS, the following modifications of the above ELISA protocol were performed. Plates were coated with clone 42 antibody (1:2500; BD Biosciences, Pont de Claix, France) in 50mM

Na2CO3/NaHCO3 (pH9.6) at 4°C overnight. Captured aS (0,5 µg of protein for human aS and 2 µg of protein for murine aS) was detected either by rabbit monoclonal antibodies MJFR1

(1:25000; Abcam, Paris, France) or D37A6 (1:1000; Ozyme, Saint-Cyr-l'Ecole, France) for human or murine aS respectively. The threshold was calculated as the mean + 3xSD of three repeated ELISA on half-brains from C57Bl/6 mice devoid of aS.

Immunohistochemical detection of PSer129-aS deposits and amyloid plaques

Brain samples were fixed in 4% paraformaldehyde and paraffin embedded to be cut into serial 5 µm coronal brain sections. Slices were deparaffinized and rehydrated. For the detection of PSer129-aS, brain sections were treated with a citrate solution (C9999, Sigma, Saint-Quentin Fallavier, France) with heat antigen retrieval. Once refrigerated, the brain sections were treated by a 4 M solution of guanidinium thiocyanate (EU0046, Euromedex, Souffelweyersheim, France) during 20 min. For the detection of amyloid plaques we did a pretreatment with 90% formic acid at room temperature (RT) for 20 min, followed by washing in an excess of water.

The endogenous peroxidase activity was blocked by a 5 min immersion in a 3% hydrogen peroxide solution. The non-specific binding was blocked by incubating the sections in a solution of Blocking Reagent (11096176001, Roche, Boulogne-Billancourt, France) for 30 min.

Subsequently, the sections were incubated overnight at 4°C with primary antibodies rabbit anti- old homozygous M83 mice. An additional control experiment was performed by intra-striatal injection of a midbrain homogenate from a 20-month-old sick M83 mouse, as previously described [START_REF] Attems | The multi-morbid old brain[END_REF]. Mice were observed daily and were sacrificed when they developed the first symptoms of the typical M83 disease, including slow ambulation, balance disorders and partial paralysis of a hind limb. The M83 disease was confirmed by PSer129-aS ELISA detection as previous described [START_REF] Irwin | Neuropathological and genetic correlates of survival and dementia onset in synucleinopathies: a retrospective analysis[END_REF], in a sample of the cervical spinal cord.

Fig. 1

 1 Fig. 1 Co-expression of APP/PS1 leads to distinct distribution of brain lesions in M83 mice.

( a ) 1 1

 a1 Western blot detection of PSer129-aS in brain homogenates from cerebral cortices of M83 (He) mice (Nos. 1 to 7) or APPxM83 mice (Nos. 1 to 6). High molecular weight (HMW) aS species are seen (lower panel) in three of the six mice expressing APP (6E10 antibody) (upper panel). (b) Western blot and ELISA detection of PSer129-aS in the brain of an APPxM83 mouse (No. 4) compared to a 19-month-old sick M83 mouse. ELISA threshold (dashed line) represents means + 3xSD of three repeated ELISA tests on two half-brains from 5-month-old M83 mice. OB: olfactory bulbs, CTX: cerebral cortex, STR: striatum, HIP: hippocampus, MB: midbrain, CB: cerebellum, BS: brainstem. (c) Immunohistochemical 6E10 staining in the hippocampus of an M83 (He) mouse (No. 3) or APPxM83 mouse (No. 1). Scale bar: 500 μm. (d, e) pSer129-aS staining in hippocampal regions CA1 (d) and dentate gyrus (e) of an M83 (He) mouse (No. 7) or APPxM83 mouse (No. 3). Scale bar: 50 μm. familial Lewy body disease and variant Alzheimer's disease. J Neurosci 27: 13092-13097 Doi 10.1523/JNEUROSCI.4244-07.2007 7 Lau A, So RWL, Lau HHC, Sang JC, Ruiz-Riquelme A, Fleck SC et al (2020) alpha-Synuclein strains target distinct brain regions and cell types. Nat Neurosci 23: 21-31 Doi 10.1038/s41593-019-0541-x 8 Prusiner SB, Woerman AL, Mordes DA, Watts JC, Rampersaud R, Berry DB et al (2015) Evidence for alpha-synuclein prions causing multiple system atrophy in humans with parkinsonism. Proc Natl Acad Sci U S A 112: E5308-5317 Doi 10.1073/pnas.1514475112 9 Radde R, Bolmont T, Kaeser SA, Coomaraswamy J, Lindau D, Stoltze L et al (2006) Abeta42-driven cerebral amyloidosis in transgenic mice reveals early and robust pathology. EMBO Rep 7: 940-946 Doi 10.1038/sj.embor.7400784 10 Spires-Jones TL, Attems J, Thal DR (2017) Interactions of pathological proteins in neurodegenerative diseases. Acta Neuropathol 134: 187-205 Doi 10.1007/s00401-017-Korczyn AD (2018) A critical review of the prion hypothesis of human synucleinopathies. Cell Tissue Res 373: 213-220 Doi 10.1007/s00441-017-2712-------------------------Thierry Baron 1 , Jean-Noël Arsac 1 , Jérémy Verchère 1 , Habiba Tlili 1 , Claire Aufauvre 1 , Dominique Bétemps ANSES (French Agency for Food, Environmental and Occupational Health & Safety), University of Lyon, Lyon, France

  PSer129-aS EP1536Y (1:300; Abcam, Paris, France) or mouse anti-β-amyloid 6E10(1:200; Biolegend, San Diego, USA ) antibodies. The histological sections were blocked a second time by incubating the sections in the Solution Blocking Reagent (11096176001, Merck, Saint-Quentin Fallavier, France) for 30 min. The HRP-labeled anti-rabbit Ig secondary antibody(1:250; Cliniscience, Nanterre, France) or anti-mouse Ig secondary antibody (1:500; 1010-05, Cliniscience, Nanterre, France) were incubated during 1.5 h at room temperature.HRP enzymatic reaction was revealed using ImmPACT DAB ® peroxidase (HRP) substrate (SK-4105, Eurobio Scientific, Les Ulis, France) for 8 min at room temperature then washed in water. The slides were then counterstained with aqueous hematoxylin. The stained sections were observed under a light microscope BX51 (Olympus) coupled to INFINITY3-6UR camera (Lumenera) and lumenera software (Infinity analyze version 6.5.5) was used.BioassaysTo examine the seeding capacity of the cerebral cortex of APPxM83 mice according to the presence or not of Ser129P-aS, homogenates (2% (wt/vol)) were prepared from the cerebral cortices of two 22-month-old APPxM83 mice containing or not PSer129-aS (APPxM83 mice Nos. 3 and 6 respectively) and from one of the 22-month-old M83 hemizygous mice (M83 No. 7). 2 µl (40 µg brain equivalent) were stereotactically injected into the left striatum (anteroposterior (AP): +0.014, mediolateral (ML): +0.2, dorsoventral (DV): -0.275) of 2 month

Figure S1 .

 S1 Figure S1. Detection of serine 129 phosphorylated alpha-synuclein (PSer129-aS) in APPxM83mice. Brain homogenates of 7 brain regions were analyzed by ELISA using Syn303-EP1536Y antibodies as previously described[START_REF] Irwin | Neuropathological and genetic correlates of survival and dementia onset in synucleinopathies: a retrospective analysis[END_REF], from one of the 7 M83 hemizygous (He) mice (No. 7) (a), a 19-month-old sick M83 homozygous (Ho) mouse (b) and the 6 APPxM83 mice (Nos. 1 to 6 respectively in c-h). The dashed line indicates a cut-off level determined as means + 3xSD measured in samples from half-brains from two 5-month-old M83 mice. The results were obtained from three repeated measures of immunoreactivity (OD 450 nm). OB: olfactory bulbs, CTX: cerebral cortex, STR: striatum, HIP: hippocampus, MB: midbrain, CB: cerebellum, BS: brainstem.

Figure S2 .Figure S3 .

 S2S3 Figure S2. Co-expression of human A53T mutated aS in APPxM83 mice reduces β-amyloid load in the cerebral cortex and hippocampus and promotes widespread deposits of serine 129 phosphorylated aS. (a) Representative images of amyloid plaques (6E10) staining in the APP (He), APPxM83 or M83 (He) mouse brain at the ages of 5 or 18 months, showing reduced amyloid plaques in the cerebral cortex and hippocampus of APPxM83 mice. Abbreviations: CA1 region of the hippocampus (CA1); CA3 region of the hippocampus (CA3); dentate gyrus (DG); cerebral cortex (CTX). Scale bar = 500 μm. (b) Quantification was performed by measuring the percentage-occupied area of the 6E10 labelled surface in the cerebral cortex and hippocampus (n = 3 for APP (He) or n=2 for APPxM83 mice). (c) Representative images of PSer129-aS labelling in the cerebral cortex of 22-month-old APPxM83 (Nos. 1 and 3) and M83 (He) (No. 4) mice or of a 18-month-old APP (He) mouse. Note that in the APP (He) mouse PSer129-aS staining is present, located around amyloid plaques (black arrowheads), which is increased in APPxM83 mice. In an APPxM83 mouse with the highest levels of PSer129-aS (No. 3), abundant PSer129-aS deposits are also present as widespread intracytoplasmic and neuritic deposits (red arrowheads), independently of amyloid plaques. Scale bar = 50 μm.

  The animals were housed per group in enriched cages in a temperature-controlled room on a 12h light/dark cycle, and received water and food ad libitum, in our approved facilities (No. D69387 0801) for breeding and experimental studies, in accordance with EEC Directive 86/609/EEC and French decree No. 2013-118. The experimental studies were authorized by the Ethics Committee CE2A -16, ComEth ANSES/ENVA/UPEC and by the French Ministry of Higher Education, Research and Innovation (refs 19-019 and 19-042).

	Biochemical detection of alpha-synuclein
	Biochemical analyses included analyses of PSer129-aS accumulation and of human/murine aS
	expression in 22-month-old or 5-month-old APPxM83 and M83 (He) mice.
	(GTTCTGCTGCATCTTGGACA) primers in order to identify hemizygous M83 mice
	presenting or not the APP/PS1 genes. The APP expression was further checked by Western blot

analysis with the 6E10 antibody

(1:2000; Biolegend, 803001, San Diego, USA)

.

aS extractions from dissected brain homogenates was obtained in High Salt (HS) Buffer (50 mM Tris-HCl pH 7,5, 750 mM NaCl, 5 mM EDTA, 1 mM DTT, 1% phosphatase and protease inhibitor cocktail).

For Western blot analyses brain homogenates were mixed with TD4215 denaturing buffer (final concentration: 4% SDS, 2% β mercaptoethanol, 192 mM glycine, 25 mM Tris, 5% sucrose) in order to load 20 or 2 µg proteins per lane for PSer129-aS or human/murine aS respectively, after assessment of proteins concentrations with DC protein Assay (Biorad, Marnes-La-Coquette, France). Proteins were separated after heat denaturation (5 min at 100°C) on 12% TGX Stain Free FastCast acrylamide gels or TGX FastCast acrylamide gels (Biorad, Marnes-La-Coquette, France) loaded with denatured proteins, then blotted onto PVDF membranes (Immobilon-P, Sigma, Saint-Quentin Fallavier, France). After migration, Stain

---------------------------

Acknowledgements

We gratefully acknowledge Mathias Jucker for providing the APPPS1-21 mice and for helpful discussions, as well as Damien Gaillard, Muris Humo and Latifa Lakhdar for the follow-up of animal experiments.

Declarations

Conflict of interest

The authors declare no actual or potential conflict of interest.

Ethics approval


Hong-Nguyen Nguyen

Matthieu Godichaud

Lionel Amodeo

Impact of Energy Consumption in a Production Inventory Model with Price-and Carbon Emission-Sensitive Demand

As companies strive to reduce energy consumption and minimize their carbon footprint, there is a growing trend among customers to choose eco-friendly products. In order to address these concerns, the present study puts forth a new economic production quantity (EPQ) model that takes into account various factors such as energy usage, carbon emissions, and market demand in relation to product price and environmental impact. This is achieved through an analysis of the working phases of the manufacturing machine. By maximizing the overall profitability of the system, optimal decisions can be made regarding cycle time, production rate, demand rate, and machine states during non-production phases (standby or powered off). A Mixed integer nonlinear programming (MINLP) problem is proposed and analyzed. A case study demonstrates that meeting customer expectations for sustainable products can lead to lower profits for businesses, but it also results in reduced energy consumption and environmental emissions. The sensitivity analysis demonstrates that market demand, price sensitivity coefficient, and unit production cost have a more substantial impact on profit compared to the other parameters.

I. INTRODUCTION

In recent years, high energy prices and fierce competition in the global supply chain make the cost of operating and purchasing manufacturing systems a major challenge for companies. The efficient use of energy in various industrial processes, including production, warehousing, and transportation, is a significant concern in the context of reducing CO 2 emissions and mitigating climate change. In order to tackle this challenge, scholars have endeavored to integrate these factors into inventory management models with the aim of minimizing both energy consumption and emissionsrelated expenses. In particular, adjusting the production rate has emerged as a popular method for suppliers to reduce inventory costs [START_REF] Glock | The impact of controllable production rates on the performance of inventory systems: A systematic review of the literature[END_REF]. Several inventory models have been proposed, extending the demand assumption of the basic inventory model in different directions, including deterministic and stochastic models [START_REF] Lin | Integrated inventory model with quantity discount and price-sensitive demand[END_REF], [START_REF] Glock | Sustainability strategies in an epq model with price-and quality-sensitive demand[END_REF]. Additionally, some researchers have considered the impact of customer preferences for environmentally friendly products on demand by integrating environmental factors into the demand function and assessing its influence on firms' inventory policies [START_REF] Taleizadeh | A sustainable inventory system with price-sensitive demand and carbon emissions under partial trade credit and partial backordering[END_REF], [START_REF] Hovelaque | The carbon-constrained eoq model with carbon emission dependent demand[END_REF]. Overall, these approaches can help businesses achieve more sustainable and environmentally responsible supply chain management practices. The current literature lacks research that combines *Corresponding author 1 Computer Sciences and Digital Society Laboratory (LIST3N), University of Technology of Troyes, Troyes, France, {hong nguyen.nguyen, matthieu.godichaud, lionel.amodeo} @utt.fr inventory modeling with sustainability considerations and price-dependent demand and carbon emissions. To fill this gap, the paper aims to make specific contributions in this area. Our main contributions can be listed as follows:

(i) Novel inventory model for Economic Production Quantity (EPQ) that takes into account the energy consumed during production. In this model, the costs associated with CO 2 emissions resulting from energy consumption will be factored into a carbon tax, which may impact the demand from environmentally conscious customers.

(ii) Maximize the profits of the enterprise by making optimal decisions related to production rate, demand rate, cycle time, and machine status during non-production phases.

The paper is structured as follows: Section II provides a comprehensive review of the inventory model with a focus on sustainable aspects and demand sensitivity assumptions. Section III outlines the problem studied and details the mathematical modeling employed. In Section IV, the MINLP is analyzed and simplified. Section V presents a case study to illustrate the model and highlights managerial insights. Additionally, sensitivity analysis is performed to further assess the effectiveness of the proposed model.

II. LITERATURE REVIEW

The relevant studies for this paper can be categorized into two streams: (1) Inventory models that consider sustainable development factors, including energy consumption and greenhouse gas emissions, and (2) Sustainable inventory models that incorporate dependent demand.

A. Inventory model with sustainable aspects

Harris's initial inventory model [START_REF] Harris | How many parts to make at once[END_REF], known as the economic order quantity (EOQ) model, was later expanded to the EPQ model for manufacturing enterprises. As supply chains have evolved, inventory models have become more complex and updated to reflect real-world situations. This includes considerations for energy efficiency and GHG reduction, which are now government-mandated for sustainable development. As a result, there has been a growing number of studies on this topic. Jaber et al. [START_REF] Jaber | Supply chain coordination with emissions reduction incentives[END_REF] propose a two-tier supply chain model that incorporates the carbon emission trading mechanism and emissions from the production process. Zanoni et al. [START_REF] Zanoni | Energy implications in a two-stage production system with controllable production rates[END_REF] focus on enhancing energy efficiency in the production process by adjusting the production rate with a two-machine production model. Bazan et al. [START_REF] Bazan | Supply chain models with greenhouse gases emissions, energy usage and different coordination decisions[END_REF] study the single-vendor, single-buyer model with the Vendor-Managed Inventory with Consignment Stock (VMI-CS) agreement policy, taking into account the multi-level emission-taxing scheme. Marchi et al. [START_REF] Marchi | Economic production quantity model with learning in production, quality, reliability and energy efficiency[END_REF] propose an EPQ model that considers the effect of worker training on the production process as well as energy efficiency. Then, the work is extended by integrating inventory modeling with VMI-CS into the supply chain and assessing its impact on GHG emissions [START_REF] Marchi | Supply chain models with greenhouse gases emissions, energy usage, imperfect process under different coordination decisions[END_REF]. Nezami and Heydar [START_REF] Nezami | Energy-aware economic production quantity model with variable energy pricing[END_REF] integrate energy price variations into the EPQ model. Fichtinger et al. [START_REF] Fichtinger | Assessing the environmental impact of integrated inventory and warehouse management[END_REF] analyze the impact of warehouse operations on emissions through inventory modeling. Tiwari et al. [START_REF] Tiwari | Sustainable inventory management with deteriorating and imperfect quality items considering carbon emission[END_REF] propose a two-tier supply chain model that considers integrated deteriorating and imperfect quality items while taking carbon emissions into account. Nguyen et al. [START_REF] Nguyen | Economic production quantity model with energy consideration[END_REF] examine the effects of using different types of Specific Energy Consumption (SEC) functions in EPQ models that consider energy consumption in production. These notable studies highlight the critical need and development of inventory modeling studies that integrate sustainability aspects in the present era.

B. Sustainable inventory model with dependent demand

Several researchers have contributed to bringing basic inventory models closer to reality. One approach is to change the input assumption of demand rate as a known constant. A number of inventory models with price-dependent demand have been proposed [START_REF] Lin | Integrated inventory model with quantity discount and price-sensitive demand[END_REF], [START_REF] Glock | Sustainability strategies in an epq model with price-and quality-sensitive demand[END_REF], [START_REF] Datta | An inventory system with stock-dependent, price-sensitive demand rate[END_REF]- [START_REF] Pour-Massahian-Tafti | Disassembly eoq models with price-sensitive demands[END_REF]. With the shift in consumer spending towards environmental concerns, several authors have studied the integration of demand that is sensitive to the level of emissions of the product. Hovelaque and Bironneau [START_REF] Hovelaque | The carbon-constrained eoq model with carbon emission dependent demand[END_REF] integrate the EOQ model with the demand that depends on both price and emissions. In this model, both carbon tax and carbon trading mechanisms are analyzed to maximize corporate profits. Later, Ruidas et al. [START_REF] Ruidas | A production inventory model with interval-valued carbon emission parameters under pricesensitive demand[END_REF] integrate price-sensitive demand into the EPQ model, which is applied to an imperfect production system. The objective of this paper is to establish an inventory model that incorporates interval numbers for various carbon parameters and considers multiple carbon emission regulatory policies, including simple tax, cap and purchase, cap and reward, and permitted cap policies. De-la-Cruz-Márquez et al. [START_REF] De-La Cruz-Márquez | An inventory model for growing items with imperfect quality when the demand is price sensitive under carbon emissions and shortages[END_REF] introduce an inventory model that accounts for carbon emissions and shortages while considering the price-sensitive demand for growing items (such as farmyard animals). The model determines the optimal policy for the selling price, order quantity, and backordering quantity to maximize the expected total profit per unit of time.

The previous literature reviews indicate a growing trend in research on sustainable inventory models. However, the combination of demand that is dependent on price and emissions during the production process has not been extensively studied. This paper aims to address this gap by expanding the EPQ inventory model. The model considers energy consumption in the production process in detail by examining the different working stages of the machine. Emissions are calculated based on the energy consumed under the carbon tax mechanism. As customers increasingly prefer environmentally friendly products, the model incorporates demand that depends on both market prices and carbon emissions from the production process. The MINLP problem is analyzed and simplified for numerical analysis, and optimal decisions are made to maximize company profits regarding demand rate, production rate, cycle time, and machine working mode. The model is validated through case studies and sensitivity analysis, which provide useful managerial insights. Table I

III. MODEL DEVELOPMENT

A. Problem statement

This study analyzes a single-product, single-machine production system's energy consumption during production and non-production time. The machine is turned off or kept on standby after production time and activated before the next cycle to optimize energy consumption, as illustrated in Fig. 1. Demand is affected by price and customer sensitivity to carbon emissions. The model also integrates carbon tax policies to assess inventory management's effectiveness in managing carbon footprint. The production rate can be controlled and optimized to maximize overall average profit, including production rate, demand rate, and non-production machine state.

The following notations are used to model the problem: Parameters: S: setup cost (C/setup) H: inventory holding cost (C/(unit.h)) W : idle power of the machine (kW) K: energy consumption of the machine to produce one unit (kWh/unit) V : price for on unit of product (C) M i : total carbon emissions for mode i (tonCO 2 /h) E i : power consumption of the machine for mode i (kW) P r i : average profit for mode i ( C/h) Decision variables: P : production rate (unit/h) D: demand rate (unit/h) t c : cycle time (h) i: the binary variable presents for machine mode during the non-production time. i is equal to 0 if the machine is shut down, and 1 if the machine is kept in standby

The model considers the following assumptions:

• The demand rate is smaller than the production rate (P > D > 0); • The production rate is controllable and limited in a specific interval [P min , P max ]; • Shortages are not allowed;

• The lead time is zero.

B. Mathematical modeling

Following the same assumption made in Hovelaque et al. [START_REF] Hovelaque | The carbon-constrained eoq model with carbon emission dependent demand[END_REF], the demand in this study is not constant but depends on both market price and customer concern regarding the product's emission level. The demand is expressed as follows:

D = α -βV -δM (1)
Where the average total carbon emission M is solely associated with the energy consumption during the production process. From (1), the price can be presented by the demand as (2)

V = α -D -δM β (2)
Nguyen et al. [START_REF] Nguyen | Economic production quantity model with energy consideration[END_REF] proposed the concept of consumption power, which is the combined energy consumption during both the production and non-production phases of the machine (expressed in equations ( 3) and ( 4)). The machine's two modes during the non-production time are represented by an index (i) with a value of either 0 or 1.

E 0 (t c , P, D) = W P + K D + F W t on t c (3)
E 1 (D) = W P + K D + 1 - D P W = W + KD (4)
From the energy consumption components in ( 3) and (4) for each case of i, the total carbon emission for each case can be calculated by the function:

M i = E i (t c , P, D)E t .
The average profit functions are defined as

P r i (D, P, t c ) = α -D -δE t E i (t c , P, D) β D -CD - S t c + HDt c 2 1 - D P -E i (t c , P, D)(E e + C tx E t ) (5)
where the components of the equation, in order, represent sales revenue, production cost, traditional inventory cost, energy cost and emissions cost.

IV. RESOLUTION ANALYSIS

The problem now can be summarized as follows: Maximize {P r 0 (D, P, t c ); P r 1 (D, P, t c )} s.t. P min ≤ P ≤ P max t on < 1 -D P t c for the case: i = 0 [START_REF] Nguyen | Economic production quantity model with energy consideration[END_REF]. The objective average profit function for each instance of machine mode in the non-production period is analyzed. The case that generates a higher profit will be selected. The average profit function can be divided into two cases of i.

A. Case i = 0 (the machine is turned off-on)

The average profit function for this case is

P r 0 (D, P, tc) = α -D -δEt W P + K D + F W ton tc β D -CD - S tc + HDtc 2 1 - D P - W P + K D + F W ton tc (Ee + CtxEt) (6)
The second partial derivatives of P r 0 with respect to t c as present in [START_REF] Jaber | Supply chain coordination with emissions reduction incentives[END_REF] is smaller than zero with all t c larger than zero. Therefore, P r 0 is concave with respect to t c for any given value of P and D.

∂ 2 P r 0 ∂t 2 c = - 2S + 2F W t on E e + C tx E t + DEtδ β t 3 c (7)
By set the first partial derivative to zero, we can present the value of t c and P r 0 by D and P as

t * c,0 (P, D) = S + F W t on E e + C tx E t + DEtδ β DH 2 (1 -D P ) (8)
P r0(t * c (P, D), P, D) = α -D -DEtδ( W P + K) β D -CD -2 S + F W ton Ee + CtxEt + DEtδ β DH 2 1 - D P -D W P + K (Ee + CtxEt) (9)
Now the profit function has been simplified by removing the variable t c . Although the convexity of ( 9) cannot be mathematically verified, the optimal solutions that maximize P r 0 (t * c (P, D), P, D) can only be obtained numerically with specific sets of input parameters.

B. Case i = 1 (the machine is kept in standby mode)

In this case, the average profit function can be expressed as

P r 1 (D, P, t c ) = α -D -δE t (W + KD) β D -CD - S t c + HDt c 2 1 - D P -(W + KD)(E e + C tx E t ) (10)
The first partial derivative of P r 1 with respect to P is

∂P r 1 ∂P = - D 2 Ht c 2P 2 < 0, ∀P > 0 (11)
Therefore, P * 1 = P min . The second partial derivative of P r 1 with respect to t c as shown in [START_REF] Nezami | Energy-aware economic production quantity model with variable energy pricing[END_REF] demonstrates that P r 1 is concave with this variable. Hence, t * c,1 and P r 1 now can be presented by the last variable D as ( 13) and ( 14).

∂ 2 P r 1 ∂t 2 c = - 2S t 2 c < 0, ∀t c > 0 (12) t * c,1 (D) = S DH 2 (1 -D Pmin ) (13)
P r1(t * c (D), Pmin, D) = α -DEtδ(W + KD) β D -CD -2 SDH 2 1 - D Pmin -(W + KD)(Ee + CtxEt) (14)
The optimal solutions that maximize P r 1 (t * c (D), P min , D) in ( 14) can also be numerically achieved.

V. NUMERICAL ANALYSIS

This section presents case studies that illustrate the properties of the model. The model is analyzed under various scenarios, considering the impact of emission-dependent demand. The study concludes with a sensitivity analysis that evaluates the role of the parameters in the optimal decisionmaking process.

A. Case studies

The input data used in the study was sourced and adjusted from previous works [START_REF] Zanoni | Energy implications in a two-stage production system with controllable production rates[END_REF] to fit the model. The related sensitive demand parameters and CO 2 emission are taken from [START_REF] Hovelaque | The carbon-constrained eoq model with carbon emission dependent demand[END_REF] and [START_REF] Taleizadeh | Sustainable economic production quantity models for inventory systems with shortage[END_REF]. The data used for the numerical analysis are summarized in Table II. ) values for each scenario of i. The optimal decision of switching off/on the machine during the non-production period is evident (in bold). In this case, the optimal production rate of P max allows for reduced production time and thus, greater energy savings during the non-active period. This is reflected in the average cost of energy (C * e,0 ) and CO 2 emissions (C * CO2,0 ). Notably, emission reduction provides price and demand advantages over the i = 1 scenario.

The difference between the two profit functions is calculated as follows:

∆ = P r1 -P r0 = F W ton tc -W 1 - D P (Ee + CtxEt + δEtD β ) (15)
The parameter ∆'s sign is dependent on the values of two input parameters, F and t on . When the value of these two parameters is high enough, the cost of machine setup after a shutdown becomes significantly high. This phenomenon is observed in various industries, such as industrial printing presses, semiconductor manufacturing equipment, and CNC machines, among others. To obtain the optimal machine mode for i = 1, we aim to increase the values of these two parameters for verification purposes. The results of this effort are also presented in Table III.

The analysis reveals that modifying the values of F and t on has no effect on the optimal results of the case i = 1. Elevating these two parameters leads to higher energy consumption costs, carbon emissions costs (increased 2 times), and lower profit (-4.83%). Additionally, the optimal value of t * c,0 must increase to reduce their impact on the objective function, resulting in higher inventory costs compared to the previous case study. Consequently, the study suggests that keeping the machine in standby mode provides a more significant economic benefit.

B. Effects of emission consideration on the demand

To further analyze the properties of the model, we compare the current model with a simpler version of itself that assumes the rate of demand depends only on market prices and not on production emissions, using the function: D = α -βV . Table IV displays the optimal outcomes of the proposed model (optimal case is in bold). It is evident that the economic benefits of this scenario where customers were assumed to be indifferent toward the emissions produced during the product manufacturing process, are higher than the studying scenario in both cases of i. Where the percentage value results from comparing to the corresponding case of the studying model as presented in Table III. This is validated by the positive difference in the average returns between the two scenarios, as depicted in [START_REF] Datta | An inventory system with stock-dependent, price-sensitive demand rate[END_REF], for all positive values of P , D, and t c .

∆ ′ = P r non,i -P r i = E i δE t β D (16)
Where P r non,i stands for the average profit of the comparing scenario (demand is not dependent on the emission). Although this scenario may achieve a greater advantage in terms of demand ratio and market price in the short term, it comes at the expense of higher emissions and energy consumption costs. These costs are likely to pose a significant challenge in light of changing market trends, such as the growing environmental consciousness among consumers, the current upward trend in energy prices, and the introduction of policies aimed at reducing energy consumption and promoting sustainability. Moreover, governmental incentives and penalties for companies based on their carbon footprint (such as carbon trading and carbon caps) are becoming increasingly prevalent.

C. Sensitivity analysis

The model's sensitivity analysis was conducted by varying each input parameter (as presented in Table II) from -25% to +25% with a step size of 5% to investigate its impact on the maximum average total return, as depicted in Figure 234. Here, the maximal profits are archived for the case i = 0. The results show that most of the parameters have an inverse relationship with the profit, except for variable α. Among them, parameters maximal market demand (α), price sensitivity coefficient (β), and unit production cost (C), have a significant influence on the profit change (varied up to 90%), while the other parameters have a relatively minor effect (varied under 1%). However, given the recent surge in energy prices, the introduction of the carbon tax, and the growing environmental awareness among customers, managers must rely on the model to make informed decisions in the future. Additionally, integrating bonus/penalty mechanisms or trading for carbon emissions in the model can emphasize the importance of sustainability factors in production.

VI. CONCLUSIONS

This study proposes a novel inventory production model that integrates considerations of energy consumption and emissions in the production process while incorporating demand based on market prices and customers' preferences for green products. Energy consumption is analyzed specifically by examining the working stages of the machine. Mathematical modeling is developed by separating the problem into two cases with respect to machine states during the non-production period. The MINLP problem is analyzed to simplify finding the maximum profit and decision variables related to the demand rate, production rate, cycle time, and optimal machine working mode variables during nonproduction phases. Case studies demonstrate that although considering customer awareness of emissions may lead to immediate disadvantages for manufacturers, it is a necessary consideration in the long run for production to deal with increasing energy prices, government mechanisms related to carbon emissions, and shifting consumer consciousness.

Future research may explore emissions from storage and transportation in the supply chain or incorporate additional carbon emission mechanisms into the model, such as carbon trading and carbon caps.

Fig. 1 .

 1 Fig. 1.Inventory level and the feasible machine modes during nonproduction time

Fig. 2 .

 2 Fig. 2.Percentage change in total cost when varying a parameter (S, H, ton, F ).

Fig. 3 .

 3 Fig. 3. Percentage change in total cost when varying a parameter (α, β, δ, K, C).

Fig. 4 .

 4 Fig. 4.Percentage change in total cost when varying a parameter (W, Ctx, Et, Ee).

  provides an overview of the literature review.

TABLE IV OPTIMAL

 IV SOLUTIONS WITH DIFFERENT VALUES OF F AND ton FOR THE

			INDEPENDENT EMISSION DEMAND MODEL
	i	t * c,i	P * i	D * i	W * p,i	P r * i	C * e,i	C * CO 2 ,i
	ton = 0.01; F = 2
	0	18.41 300.00	12.38	157.80	1908.76	25.60	7.68
						120.19% 48.57%	48.57%
	1	18.77	150.00	12.38 157.75 1884.10	44.76	13.43
						125.18% 22.23%	22.23%
	ton = 10; F = 10
	0	97.77 150.00	12.31	158.66	1863.30	46.71	14.01
						125.85% 42.70%	42.70%
	1	18.77	150.00	12.38 157.75 1884.10	44.76	13.43
						125.18% 22.23%	22.23%

Faculty of Project Management, The University of Danang -University of Science and Technology, Danang, Vietnam

ACKNOWLEDGMENT

We thank the department council of Aube (CD10), along with the Grand Est region funding for supporting the research.