Rekha Guchhait
email: rg.rekhaguchhait@gmail.com

Mitali Sarkar
email: mitalisarkar.ms@gmail.com

Biswajit Sarkar
email: bsbiswajitsarkar@gmail.com

How Much Green Investments Are Efficient for a Smart Production System?

Keywords: Smart Production, Green Investment, Customer Satisfaction, System Reliability, Imperfect Production

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Introduction

The changing relation of the production industry with the environment is the concern of this study. The carbon emission and energy consumption are obvious for a production-inventory system. The issue is the increasing rate of the average temperature of the earth, one of the main reasons is carbon emission. The carbon emission from the energy consumption of the system is high due to the use of fossil fuels as a resource. The failure rate within a production-inventory model is related to the working hours and the capacity of the machine. The fact is the prediction of the breakdown period is impossible and the consideration of the random time is more realistic than a fixed time interval. Then, it is common to produce imperfect during the breakdown period.

There are several studies discussing the machine breakdown and its consequences. A random defective rate of production was investigated by Sana [?]. The warranty policy is one of the business strategies for imperfect products. The rework process for the imperfect products was discussed by Kang et al. [?]. Now, the quality improvement for the products was considered by Guchhait et al. [?]. They discussed the setup cost reduction and warranty policy. Iqbal et al. [?] discussed the preservation technology for deteriorating products. Sarkar et al.

[?,?] discussed a variable production model. Gong et al. [?] studied a green production system with the workers' flexibility. The system was energy-efficient and an evolutionary algorithm was used to solve the mathematical problem. These studies are about the traditional imperfect production system with no effect on energy.

There are very few studies that admit the energy consumption issue of the production quantity model or the supply chain model. Sun et al. [?] discussed the electricity demand reduction between the energy-saving things and the production loss. They tried to find out the inventory control policy with reduced electricity demand. But, Bhuniya et The above discussions give ideas on how demand varies with the greening factor and emission from the production system. This study finds out the economic benefits of greening technology. Meanwhile, the greening process has some effect on customers. The following questions are addressed here: (i) The industry tries to reduce the emission from the system due to the energy consumption. How customer will react to this initiative of the industry? (ii) What will be the customer satisfaction level? (iii) How does the industry make a profit by carbon reduction strategy? The rest of the study is decorated as Section 2 provides the problem definition and mathematical model. Section 3 gives the theoretical and numerical results. Discussion about the sensitivity analysis and the managerial insights are described in section 4. Section 5 gives the conclusions about the study.

Methods

This section gives the problem definition and detail of the study.

Problem Definition

Green investment and customer satisfaction are discussed within an imperfect production system. The imperfect production system encounters the failure rate of the system which is proportional to the system reliability. Multiple products are produced from the system and the system goes to the out-of-control state at a random time ν i , i stands for the multiple products. The production system is complex in nature as the reworking process occurs within the same cycle. The imperfect products are found by an error-free automated inspection process and sent for reworking. The perfect products are sent to the market directly and imperfect products are sent to the market after reworking. Each stage of the production consumes energy and emits carbon dioxide. Thus, all the costs related to the production system are under the effect of energy. The defective rate is random as it depends upon the random ν i .

Mathematical Modelling

A mathematical model for a production-inventory system is derived here for multiple types of products that are the single production system produces n types of different products. The production system goes to the out-of-control state at a random time ν i and starts to produce imperfect products. The production rate is P i (P i > ∆ i ) and the lot size is q i at time t i . Therefore, t i = qi Pi . For no system failure, the production continues up to the time qi

Pi . Here, one failure occurs during the time duration of qi Pi and the inventory accumulates at a rate P i -∆ i . The breakdown starts a point M and the reworking process starts at a time t 1i . After t 1i , the inventory deploy rate is ∆ i . The governing differential equations of the inventory are

dI 1 i (t i ) dt i = P i -∆ i , 0 ≤ t i ≤ t 1i
with the initial condition I 1 i (0) = 0, and

dI 2 i (t i ) dt i = -∆ i , t 1i < t i ≤ with the boundary conditionI 2 i (T ) = 0.
The on-hand inventory at the time t i is

I 1 i (t i ) = (P i -∆ i ) t i , 0 ≤ t i ≤ t 1i , andI 2 i (t i ) = ∆ i (T -t i ) , t 1i < t i ≤ T.
The relation between T and t 1i is t 1i = ∆iT Pi . The demand is variable and dependent upon the price and customer satisfaction. Then, the demand of the product i is ∆ i = pmax-pi pi-pmin + ηy µ i , where y i is the customer satisfaction, p i is the unit selling price of the product i, p max is the maximum selling price of the product i, and p min is the minimum selling price of the product i, i.e., p i ∈ [p min , p max ]. Then, the revenue

is p i ∆ i = p i pmax-pi pi-pmin + ηy µ i . The carbon reduction function for green invest- ment (GI c ) is GI c = αφ i -βφ 2
i . φ i is the investment for the greening of the environment for product i. The investment αφ i is used for carbon emission reduction. βφ 2 i is used for the carbon emission due to consumed energy within the system. GI c is an increasing quadratic function. If e is the unit carbon emission cost, then the savings from the carbon emission reduction is e αφ i -βφ 2 i . If the investment for the customer satisfaction for the product i is x for satisfaction y i , then the investment function for the customer satisfaction for all products is n i=1

xy 2 i 2 .
The carbon emission from the production system is one of the causes of the global warming issue. All the cost components are therefore bared a carbon emission cost under the effect of energy. If the unit setup cost for the product i is S c i , unit energy consumption cost is S e i , and the unit carbon emission cost for the system is e, then the setup cost of all product for each cycle is n i=1 (S c i + S e i + e) ∆i qi . All products are going through the automated inspection process which ensures the error-free result that is inspection error is negligible. The imperfect products are sent for the reworking. The production cost S(τ, P i ) is dependent upon the system reliability, production rate, and the development cost A(τ ), where the development cost is system reliability dependent. The system reliability parameter τ is defined as τ = number of failure total number of working hours . τ has its upper and lower limit as τ max is the maximum value and τ min is the minimum value of the τ . The development cost A i (τ ) of the system is

A i (τ ) = (L c + L e ) + B i e u τmax-τ
τ -τ min , where L c and L e are the fixed costs for labor and energy, u is the parameter for the difficulties of the reliability, and B i is the resource cost for the corresponding product i. A i (τ ) is inversely proportional to the system reliability τ i.e., the development cost is maximum when τ = τ min and is minimum when τ = τ max . The minimum development cost is (L c + L e ) + B i . The unit production cost is S(τ, P i ) = (M i + e) + Ai(τ )

Pi + ζP ω i , where M i is the per unit material cost of product i, A i (τ ) is the development cost, ζ is the tool/die variation constant, and P i is the production rate. From the second term, it is found that the production cost is proportional to the development cost and the distribution cost is distributed over the production at time t i . The third term represents the tool/die cost of the production system. The production cost for all products per cycle is

n i=1 (M i + e) + Ai(τ ) Pi + ζP ω i ∆ i .
The average inventory within the interval [0, T ] is

n i=1 ti1 0 I 1 i (t i )dt i + T ti1 I 2 i (t i )dt i T .
If the unit holding cost per unit time is H c i for product i and the energy cost is H e i , then the total holding cost of all products is given by n i=1

(H c i +H e i +e)qi 2
1 -∆i Pi . The machine produces imperfect products at a random time ν i . If the unit inspection cost for the product i is C c i and the energy cost is C e i , then the total inspection cost is n i=1 (C c i + C e i + e) ∆ i . After inspection, imperfect products are sent to the reworking process to make those as perfect. The duration of the imperfect production is [ν i , t 1i ] for the product i. After detecting the imperfect products, the reworking process starts immediately and continued until the time t 1i . t 1i is the maximum time of the production as well as reworking.

After t 1i , the system goes through corrective maintenance until the time T and the maintenance is completed during the time (Pi-∆i)qi

Pi∆i

. As there is no imperfect production during [0, ν i ], therefore the total imperfect inventory IN i of product i is

θ ξ+1 P λ+1 i (t 1i -ν i )
ξ+1 . Now, the random time ν i has a random variable X i which follows the exponential distribution i.e., X i ∼ Exp(τ ). As the reworking process occurs within the same cycle, the time for the reworking is [0, t 1i ]. Thus, the total number of imperfect products for reworking is

n i=1 IN T i (τ, q i , P i ) = n i=1 τ θ ξ + 1 P λ+1 i e -τ q i P i Ψ i (τ, q i , P i ), t 1i = q i P i
, where

Ψ i (τ, q i , P i ) = ∞ j=1 τ j-1 t ξ+j+1 1i (j -1)!(ξ + j + 1
) .

If the unit reworking cost is R c i for the product i and the energy cost is R e i , then the total reworking cost of all products is n i=1 (R c i +R e i +e)∆iIN Ti(τ,qi,Pi) qi . There the total profit of the system is given by subtracting all costs from the revenue, i.e,

ET P (P

i , p i , q i , y i , τ, φ i ) = n i=1 ∆ i [p i -S(τ, P i )] + e αφ i -βφ 2 i -φ i - xy 2 i 2 - (S c i + S e i + e) q i ∆ i - (H c i + H e i + e) q i 2 1 - ∆ i P i -(C c i + C e i + e) ∆ i - (R c i + R e i + e) ∆ i IN T i (τ, q i , P i ) q i .

Results

This section provides the theoretical and numerical results of the objective function which is a profit function.

Solution Methodology

The objective function is a complex non-linear function that is solved by the classical optimization technique. The optimum values are in quasi-closed form. The values are

q * i = A i Y i , P * i = Z i ωζ ω-1 , p * i = p min + E i ∆ i , y * i = x ηµF i 1 µ-2 , τ * = τ min + H i J i , φ * i = eα -1 2βe .
[See Appendix for the value of A i , Y i , E i , F i , H i , and J i ]. * stands for the optimal value.

Lemma 1. The selling price p * i of the product i exits if Ei ∆i ≥ 0. τ * gives the value of the system reliability if Hi Ji > 0 and J i = 0. Proposition 1. The total profit ET P at P * i , p * i , q * i , y * i , τ * , and φ * i will be optimum if all the principal minors of the Hessian matrix of order 7 × 7 are alternative in sign.

Numerical Experiment

This section provides the numerical study and its results. Table 1 provides the corresponding values of the parameters. The optimum values of the decision variable and the total profit are given in Table 2. The total profit of the system is $12, 476.1 per cycle. Two types of products are considered for the testing of the mathematical model.

Discussion

The revenue from the carbon reduction facility for green technology GI c is $86.1.

If only the investment exists in the system as general investment and there could not a specific carbon reduction strategy based on the green investment φ i , then the profit of the system could be less than the present profit by $86.1 i.e., $12, 390. This means not only the investment is important but also the execution in a proper way is important. This profit is made by the general carbon emission reduction and carbon emission reduction from the energy sources, as it is considered that the entire production-inventory system is measured under the energy effect.

No Investment for Customer Satisfaction

If the system only consists of the green investment and the reduction of carbon and carbon from energy consumption, there is no customer satisfaction, then η = 0 and γ = 0. Then the total profit is ET P 1 = $23.84, which is 99.8% less than the original profit ET P of the system. That is the system survives anyhow and capable to pretend the loss. That is, without this investment, the production system able to maintain the revenue of the system instead of the profit.

Managerial Insights

The major benefit of this research to the industry is that the green investment for the carbon emission will save the carbon emission cost and add more revenue to the system. This serves two advantages at a time: reduction of the carbon from the system which is an environmental benefit and revenue generation which is an economical benefit. Thus, it establishes beneficial for the industry. The greening process of the industry faces several burdens and one of the reasons is customer feedback. The product which is launched by the industry should be acceptable by the customers. Thus, customer satisfaction feedback is important for the industry manager. From the special case of the discussion, it is found that if the industry manager wants to discard the customer satisfaction investment, then the profit of the system is significantly low. As the random breakdown time exists within the system, the industry manager needs to take precautions about the delivery time and the quantity.

Sensitivity Analysis

From the sensitivity analysis of key parameters, it has been found that the investment for customer satisfaction is the most sensitive parameter for the total profit of the system (Table 3). Whenever the system cost reduces due to reduction of the investment by 50%, total profit increases 3.90%. The negative percentage changes are less than the positive changes. Setup cost, labor cost, reworking cost, and resource cost have a similar pattern that the profit increases whenever the cost decreases. Changing of holding cost during -50% to +50% changes never gives more profit than the global optimum profit. If the holding cost decreases 51% or more than that, the total profit increases 0.44% or more.

Conclusions

An imperfect production system with random breakdown was discussed under the effect of energy and greening effect. Green investment and carbon emission reduction were explained in detail. The improvement of environmental health took equal priority as the economic benefit. Results found that the investment cost for the emission reduction from both the system and energy is beneficial for the system as it could save the emission cost and ultimately increased the profit of the system. The development cost, as well as the production cost, could be readjusted and the production rate also, based on the value of the reliability parameter. The mathematical model can be extended by assuming all reworked products are brand new and thus, the secondary market concept and warehouse are possible ways. Another possible extension is to use a different cycle for the reworking of defective products. The recycling of used products is very phenomenal to the recent time of era which can be introduced within the production system.

Appendix

A i = q 2 i (H c i + H e i + e) 2
1 -∆ i P i + (R c i + R e i + e) P λ+1 i τ θ ξ + 1 ∞ j=1 τ j-1 (ξ + j + 1)q ξ+j+1 i (j -1)!(ξ + j + 1)P j+ξ+1 i -{(S c i + S e i + e) + (R c i + R e i + e)

IN T i (τ, q i , P i )} ∆ i Y i = (R c i + R e i + e) P λ+1 i τ θτ (ξ + 1)P i e -τ q i P i Ψ i (τ, q i , P i )

Z i =A i (τ ) - H c i + H e i + e 2 +
R c i + R e i + e q i τ θ ξ + 1 e -τ q i P i P λ+2 i Ψ i (τ, q i , P i ) +Ψ i (τ, q i , P i )P λ+1 i τ q i e -τ q i P i + e -τ q i P i P λ+3 i + ∞ j=1 τ j-1 (ξ + j + 1)q ξ+j+1 i (j -1)!(-1)(ξ + j + 1)P

J i = R c i + R e
i + e q i P λ+1 i e -τ q i P i θ ξ + 1 Ψ i (τ, q i , P i ) -θτ q i (ξ + 1)P i e -τ q i P i Ψ i (τ, q i , P i ) +e -τ q i P i Ψ i (τ, q i , P i ) ∞ j=1 qi Pi j+ξ+1 (j -1)τ j-2

(j -1)!(ξ + j + 1)

   .

  al. [?] studied the energy utilization within the traditional production-inventory model for variable market demand. Toptal et al. [?] discussed the emission reduction by another investment under emission regulations. The integrated inventory management is now turned towards the greening (Raza et al. [?]) process and other traditional enforcements. The carbon emission scenario for the defective production system was studied by Sangal et al. [?]. Thus, the economic perspective of an industry is changed now (Adhikari and Bisi [?]) based on green involvement in terms of collaboration and bargaining. Dong et al. [?] illustrated some strategic investments for green development.

Table 1 .

 1 Values of parameters.

	Parameters Values	Parameters Values	Parameters Values
	pmax, pmin $2,000, $300 (/unit)	n, λ	2, 0.8	R e 1 , R e 2	$0.2, $0.2 (/unit)
	x	$1,000,	η, µ	10, 0.185	R c 1 , R c 2	$9.7, $14.7 (/unit)
	S c 1 , S c 2	$599.9, $499.9 (/setup) α, β	18, 0.13	H c 1 , H c 2	$1.9, $2.95 (/unit/unit time)
	S e 1 , S e 2	$0.2, $0.2 (/setup)	u	0.06	M1, M2	$99.8, $101.8 (/unit)
	L c , L e	$199.8, $0.2	ζ, ω	0.2, 0.7	τmax, τmin 0.9, 0.1
	B1, B2	$30, $32 (/unit)	θ, ξ	0.05, 3	C c 1 , C c 2	$0.9, $1.5 (/unit)
	e	$0.1	C e 1 , C e 2	$0.2, $0.2 (/unit) H e 1 , H e 2	$0.2, $0.2 (/unit/unit time)

Table 2 .

 2 Optimum values of decision variables.

	Decision Values	Decision Values	Decision Values
	variables		variables		variables
	p * 1 , p * 2	$493.81, $492.35 (/unit) q * 1 , q * 2	100.38, 76.13 units/year P * 1 , P * 2	229.94, 201.51 units/year
	y * 1 , y * 2	0.83, 0.82	τ *	0.47	φ * 1 , φ * 2	$30.77, $30.77

Table 3 .

 3 Sensitivity of cost parameters.

	Decision Changes (%) Changes in Decision Changes (%) Changes in
	variables		ET P	variables		ET P
	x	+50	-2.19	L c	+50	-0.11
		+25	-1.21		+25	-0.06
		-25	+1.59		-25	+0.07
		-50	+3.90		-50	N.A.
	H c 1	+50	-0.48	R c 1	+50	-0.002
		+25	-0.41		+25	-0.001
		-25	-0.26		-25	+0.001
		-50	-0.18		-50	+0.003
	S c 1	+50	-0.38	B c 1	+50	-0.009
		+25	-0.20		+25	-0.005
		-25	+0.22		-25	+0.005
		-50	+0.49		-50	+0.010

  = (p max -p min ) p i -S i (τ, P i ) -Ψ i (τ, q i , P i ) F i =p i -S i (τ, P i ) -

		-(C c i + C e i + e) -P i S c R c i + R e i + e q i P λ+1 i τ θ ξ + 1 -τ q i e i + S e i + e q i + (H c i + H e i + e) q i 2P i	-(C c i + C e i + e)
		-	R c i + R e i + e q i	P λ+1 i	e	-τ q i P i	τ θ ξ + 1	Ψ i (τ, q i , P i )
	H i =	uB i P i	(τ max -τ min ) e	u τmax -τ τ -τ min
								
								ξ+j+2	
								i
	E i S c i + S e i + e q i	+	(H c i + H e i + e) q i 2P i

This research was supported by the Yonsei University Research Fund of 2020 (Project Number 2020-22-0509).


Zhe Yuan
email: zhe.yuan@devinci.fr

Mingyang Chen
email: mychen@henu.edu.cn

Quantity-flexibility contract models for the supply chain with green-sensitive demand in the automotive manufacturing industry

Keywords: Sustainable supply chain, quantity-flexibility contract, green-sensitive demand, game theory

This paper considers a quantity-flexibility contract with greensensitive demand in the automotive industry. The automobile manufacturer determines the green level, and the retailer determines the retail price. The authors apply game theory to build the models, optimize the green level for maximizing the automobile manufacturer's profit, and optimize the retail price for maximizing the retailer's profit. We consider the equilibrium decision between the green level and the retail price to maximize the supply chain's profit. Furthermore, we conduct experiments to verify our analysis results. Interestingly, this study finds that it is optimal for the retailer in the decentralized case to charge a higher price when the green sensitivity increases. However, the optimal price in the centralized case is not necessarily monotonic when the demand is lower. The manufacturer should set a higher green level as the greening investment parameter increases.

Introduction

In Paris, the government divides vehicles into six environmental classes. The air quality certificate increases customers' demand for green automobiles, including electric vehicles, natural gas vehicles, or hybrid vehicles. This phenomenon can reflect that more customers prefer buying environmentally-friendly automobiles and are sensitive to green products, which results in many automobile manufacturers and retailers begin to produce and sell green automobiles to meet customers' needs. Meanwhile, retailers require their automobile manufacturers to participate in carbon reduction disclosure projects, primarily from the necessary environmental certifications [START_REF] Lee | Corporate carbon strategies in responding to climate change[END_REF]. Therefore, facing the green-conscious consumer market, retailers cooperate with automobile manufacturers to initiate green product and process changes. This paper is motivated by this evidence and considers an automotive supply chain, including an automobile manufacturer and a retailer with green-sensitive customers' demand, and explores the influences of green-sensitive consumer demand on the supply chain members and the collaboration among them.

The greening improvement of green automobiles implies some attribute modifications, such as the use of electric or hydrogen-powered, and the combination of an internal combustion engine and an electric engine that reduces CO 2 emissions compared with the traditional gasoline or diesel automobiles [START_REF] Emadi | Fuel cell vehicles: opportunities and challenges[END_REF]. After the air quality certificate starts working to raise the green level through green innovation, traditional automobile manufacturers need to invest in new technologies to produce plug-in hybrid automobiles as the green level improves. For example, producing 100% pure electric or hydrogen automobiles requires more managerial and technological investment to improve the green level. The automobile manufacturer may charge a green level and a higher unit cost than the condition where the order quantity is constant over time. The retailer must balance between the purchase, inventory holding, and shortage costs.

Automotive industries commonly use quantity-flexibility contracting as a necessary mechanism because the quantity-flexibility contract is a downstream protection contract, which can share the retailer's risk and increase the order quantities. Furthermore, the quantity-flexibility contract can coordinate the green supply chain of automotive manufacturing [START_REF] Tsay | Quantity flexibility contracts and supply chain performance[END_REF][START_REF] Knoblich | Modeling supply contracts in semiconductor supply chains[END_REF][START_REF] Kim | Order quantity flexibility as a form of customer service in a supply chain contract model[END_REF]. For example, Toyota Motor Corporation [START_REF] Höhn | Relational supply contracts: optimal concessions in return policies for continuous quality improvements[END_REF] currently uses the quantity-flexibility contract to deal with the variability in its production [START_REF] Niemann | Development of a method to increase flexibility and changeability of supply contracts in the automotive industry[END_REF]. Although more practice about quantityflexibility contracts appears in the automotive industries, few scholars research quantity-flexibility contracts in the green supply chain [START_REF] Ghosh | Supply chain analysis under green sensitive consumer demand and cost sharing contract[END_REF][START_REF] Giri | Analysing a closed-loop supply chain with selling price, warranty period and green sensitive consumer demand under revenue sharing contract[END_REF][START_REF] Song | Green supply chain game model and analysis under revenuesharing contract[END_REF]. In this work, we propose the research question: How to optimize the green level and the retail price with green-sensitive demand to maximize the profits in the automotive industry?

To answer this question, we model a quantity-flexibility contract with greensensitive demand between an automobile manufacturer and a retailer in the decentralized and centralized decision-making supply chain. Our work makes several contributions: First, this is one of the earliest researches to study quantityflexibility contracts considering green-sensitive customers since a few previous studies have considered. Second, we combine quantity-flexibility contracts with green problems in the automobile industry because few types of research consider quantity-flexibility contracts in the automotive industry. Third, we find it optimal for the retailer in the decentralized case to charge a higher price when the green sensitivity increases. However, the optimal price in the centralized case is not necessarily monotonic when the demand is lower.

The model

We consider a practical problem between an automobile manufacturer and a retailer. The production plans of the assembly lines determine the demand for a component. However, the production plans may vary due to green-sensitive customer demand. The automobile manufacturer and retailer establish a quantityflexibility contract in the green supply chain. Two key parameters of such a contract are nominal quantity Q and variation rate β ∈ (0, 1), where β = 0 represents the inflexibility contract, and we also analyze this contract to validate the model, and Q is the quantity reserved by the retailer [START_REF] Li | Supply chain coordination through capacity reservation contract and quantity flexibility contract[END_REF]. These parameters determine the interval [Q(1 -β), Q(1 + β)] that the actual order quantities from the retailer must in [START_REF] Barnes-Schuster | Coordination and flexibility in supply contracts with options[END_REF]. When the order quantities are within this interval, the automobile manufacturer is committed to fulfilling them. The retailer commits herself to purchase, and the allowable range is determined as 1, we provide the sequence of events and decisions as follows: First, the automobile manufacturer offers the retailer a quantity-flexibility contract, and the retailer decides whether to accept it or not. Second, the retailer makes a nominal order quantity of green automobiles Q before the period starts if she accepts this contract. Then, she determines the retail price p. Third, the nominal order becomes the basis for the automobile manufacturer to determine his green level, and he needs to decide the green level θ. In automotive manufacturing, retailers intend to release orders as close as possible to the demand because of the limited storage space in the assembly plant and the high holding cost. On the other hand, demand variability prevents them from ordering a constant quantity in every period. The unit purchase cost is a linear or piecewise linear convex function for the quantity-flexibility contract. We give the expression of the automobile manufacturer's unit wholesale price w(β) = c 0 + c 1 β, where c 0 represents the minimal possible cost with zero flexibility and c 1 is a fixed rate [START_REF] Chu | Flexibility analysis on a supply chain contract using a parametric linear programming model[END_REF]. We consider that the supply chain players' green-sensitive demand x is a linear function of retail price p and green level θ, where θ is a continuous variable. The demand function is x = a -bp + αθ [START_REF] Ghosh | Supply chain analysis under green sensitive consumer demand and cost sharing contract[END_REF], where a > 0 expresses the potential market size of green automobiles and b > 0 denotes the customer sensitiveness to the retail price of green automobiles, and α > 0 represents customer's sensitivity to greening improvement level of green automobiles.

[Q(1 -β), Q(1 + β)].According to Figure
Referring to [START_REF] Ghosh | Supply chain analysis under green sensitive consumer demand and cost sharing contract[END_REF], we use an increasing and convex function to represent the cost incurred in improving the green level. The cost of greening is Iθ 2 , where I indicates the greening investment parameter of green automobiles. For example, after the air quality certificate starts, traditional automobile manufacturers need to invest in new technologies to produce plug-in hybrid automobiles (Crit'Air 1 sticker) to raise the green level through green innovation. As a result, the initial green investment is relatively costly, and then raising the green level might be relatively inexpensive. When the green level improves, more managerial and technological investment is required for further improvement in the green level, making it more expensive to reach the goal of a higher green level comparable with that at the early stage. To avoid trivial cases, let I > θ 2 -4b. In the view

Remaining items

Fig. 1. Sequence of events under a quantity-flexibility contract of consumers' sensitivity to greening and price, green innovation requires higher investment, consistent with the actual situation of product innovation [START_REF] Ghosh | Supply chain analysis under green sensitive consumer demand and cost sharing contract[END_REF]. The automobile manufacturer bears the cost of greening.

Decentralized decision-making policy

In the decentralized decision-making supply chain, the automobile manufacturer chooses the product green level θ for his profit maximization using the retailer's response function. Then, the retailer decides the retail price p to maximize her profit given the green level.

If the demand of green-sensitive customers is less than Q(1 -β), the retailer buys at the minimum commitment Q(1 -β), and thus, the automobile manufacturer sells to the retailer. If the demand of green-sensitive customers is in the interval [Q(1 -β), Q(1 + β)], the retailer buys green automobiles in quantity to the demand of affected areas. If the demand of green-sensitive customers is greater than Q(1 + β), the retailer buys at maximum the automobile manufacturer's commitment level and will have x -Q(1 + β) product shortage to satisfy. As x = a -bp + αθ, b > 0, and p > 0, we can get x < a + αθ. We give the order quantity in Equation ( 1) by following the principle of quantity-flexibility contract.

q =      Q(1 -β), 0 ≤ x < Q(1 -β), x, Q(1 -β) ≤ x ≤ Q(1 + β), Q(1 + β), Q(1 + β) < x < a + αθ. (1)
According to Equation (1), we can get the profit of the automobile manufacturer as follows:

πM (θ) =        wQ(1 -β) -(hM + c f + ct + cm + c l )Q(1 + β) -Iθ 2 , 0 ≤ x < Q(1 -β), wx -(hM + c f + ct + cm + c l )Q(1 + β) -Iθ 2 , Q(1 -β) ≤ x ≤ Q(1 + β), wQ(1 + β) -(hM + c f + ct + cm + c l )Q(1 + β) -Iθ 2 , Q(1 + β) < x < a + αθ. (2)
The first term of Equation ( 2) is derived from demand satisfaction and the second term represents manufacturing and inventory costs for the automobile manufacturer, and the last term specifies the green cost of the automobile manufacturer. Then, the profit function of the retailer is:

π R (p) =      px -wQ(1 -β) -h R [Q(1 -β) -x], 0 ≤ x < Q(1 -β), px -wx, Q(1 -β) ≤ x ≤ Q(1 + β), pQ(1 + β) -wQ(1 + β) -s[x -Q(1 + β)], Q(1 + β) < x < a + αθ. (3)
Similar to Equation (2), the first term of the function is derived from demand satisfaction and the second term represents the retailer's purchasing cost, and the last term specifies the retailer's inventory or shortage cost.

First, the retailer determines the optimal retail price, and we can derive the following proposition: Proposition 1. If the demand is less than Q(1-β), the retailer's profit function is concave with p, and the optimal p * is p . If the demand is more than Q(1 + β), the retailer's profit is increasing in p and the optimal p * = a+αθ-Q(1+β) b .

= a+αθ-h R b 2b . If Q(1 -β) ≤ x ≤ Q(1 + β),
From Proposition 1, we find that when green-sensitive demand is less than Q(1-β), the retailer must purchase Q(1-β), which results in holding Q(1-β)-x inventory. The profit is affected by the retail price, the green level, and unit inventory holding cost. So, we get the optimal p * (θ

) = a+αθ-h R b 2b . When green- sensitive demand is in the interval [Q(1 -β), Q(1 + β)],
the retailer can purchase automobiles to satisfy all demand without inventory holding cost. The profit is affected by the retail price, the green level, and the automobile manufacturer's wholesale price (the variation rate). So, we get the optimal p * (θ) = a+αθ+(c0+c1β)b 2b

. When green-sensitive demand is more than Q(1 + β), since the maximum order quantity is Q(1 + β), the retailer's profit is increasing in p. However, the retail price is limited by the green level and the variation rate and the optimal p * (θ) = a+αθ-Q(1+β) b .

Then, the retailer decides the retail price in line with Proposition 1. After getting this response, the automobile manufacturer optimizes his own profit and determines the optimal green level, as shown in the following proposition: Proposition 2. If the demand of green-sensitive customer 0 ≤ x < Q(1 -β) or Q(1 + β) < x < a + αθ, the automobile manufacturer's profit function is decreasing in θ. If the demand is less than Q(1 -β), the optimal θ * = 0. If the demand is more than

Q(1 + β), the optimal θ * = Q(1+β)-a+bp α . If Q(1 - β) ≤ x ≤ Q(1 + β)
, the profit function of the automobile manufacturer with θ is concave. The optimal θ is θ * = α(c0+c1β)

2I

and the optimal retail price is

2Ia+(c0+c1β)(α 2 +2Ib) 4Ib
.

From Proposition 2, we know that when green-sensitive demand is less than Q(1 -β) or more than Q(1 + β), no need appears for the automobile manufacturer to increase the green level. Because when green-sensitive demand is less than Q(1 -β), customers are less sensitive to green automobiles, which increases that the green level will not attract more customers to purchase green automobiles and eventually leads to lower profits. When green-sensitive demand is more than Q(1 + β), customers are more sensitive to green automobiles. However, due to the maximum order quantity is Q(1 + β), the increase of the green level will not improve the profit but will increase the green investment cost. When green-sensitive demand is in the interval [Q(1 -β), Q(1 + β)], the automobile manufacturer's profit will first increase and then decrease as the green level increases, and we get the optimal green level θ * = α(c0+c1β) 2I .

Centralized decision-making policy

In this centralized decision-making policy, we consider the automobile manufacturer and the retailer as a whole to maximize the channel profit by choosing the green level and the retail price. Thus, the profit function of the supply chain is as follows,

πSC (θ, p) =        px -(hM + c f + ct + cm + c l )Q(1 + β) -hR[Q(1 -β) -x] -Iθ 2 , 0 ≤ x < Q(1 -β), px -(hM + c f + ct + cm + c l )Q(1 + β) -Iθ 2 , Q(1 -β) ≤ x ≤ Q(1 + β), pQ(1 + β) -(hM + c f + ct + cm + c l )Q(1 + β) -s[x -Q(1 + β)] -Iθ 2 , Q(1 + β) < x < a + αθ. ( 4
)
In the centralized decision-making supply chain, we solve the supply chain's profit function and get the following proposition: Proposition 3. If the demand of greens-sensitive customer is less than Q(1β), the supply chain's profit function is jointly concave in θ and p, and the optimal θ * = aα 4Ib-α 2 and p

* = 2Ia-h R α 2 -2Ih R b 4Ib-α 2 . If the demand of green-sensitive customer is in [Q(1 -β), Q(1 + β)],
the supply chain's profit function is jointly concave in θ and p, and the optimal θ * = aα 4Ib-α 2 and p * = 2Ia 4Ib-α 2 .

From Proposition 3, we find that when green-sensitive demand is less than Q(1 -β), customers are less sensitive to green automobiles, the retailer must purchase Q(1 -β), which results in holding Q(1 -β) -x inventory. The profit of the supply chain is affected by the retail price, the green level, and unit inventory holding cost. So, we get equilibrium values for the supply chain. The optimal θ * = aα 4Ib-α 2 and

p * = 2Ia-h R α 2 -2Ih R b 4Ib-α 2 . When green-sensitive demand is in the interval [Q(1 -β), Q(1 + β)],
the retailer can purchase automobiles to satisfy all demand without inventory holding cost. The profit is affected by the retail price, the green level, and the automobile manufacturer's wholesale price (the variation rate). The optimal θ * = aα 4Ib-α 2 and p * = 2Ia 4Ib-α 2 . When green-sensitive demand is more than Q(1 + β), the supply chain's profit function is decreasing in θ and increasing in p, and equilibrium values do not exist.

Equilibrium analysis

In this subsection, we analyze the equilibrium strategies derived by Propositions 2 and 3, and find the condition that the optimal solutions can sustain. The primary purpose is to solve the coordination problem of the members in the supply chain. First, we can obtain the condition that the equilibrium strategies exist according to the demand function because the quantity in the centralized case must be not less than that in the decentralized case based on the principle of the members' coordination. Since the optimal quantity in the range of Q(1-β) ≤ x ≤ Q(1 + β) under the decentralized decision-making policy is q * = 2abI 4bI-α 2 . By comparing the optimal quantities in the decentralized case and that in the centralized case, we can obtain the following proposition:

Proposition 4. When c 0 + c 1 β ≥ 2aα 2 I
-α 4 -8b 2 I 2 +6α 2 bI , the supply chain coordination can be sustained in the range of x ∈ [Q(1 -β), Q(1 + β)], otherwise, the optimal solutions may not exist.

Then, we analyze the equilibrium strategies in the decentralized case. By computing these equilibrium solutions in Proposition 2, we can examine the impact of α, c, and I on the performance measures associated with different members and obtain the following proposition: Proposition 5. As α or β increases, the optimal price p * or the optimal green level θ * also increases, i.e., dp * dα ≥ 0 and dp * dβ ≥ 0. Moreover, when c 0 or c 1 increases, the optimal green level θ * increases, i.e., dθ * dc0 ≥ 0 and dθ * dc1 ≥ 0. When I increases, the optimal green level θ * decreases, i.e., dθ * dI ≤ 0. For dp * dα ≥ 0 in Proposition 5, we find that both p * and θ * are increasing in α. The reason is that consumers are more sensitive to the green level in the automobile manufacturer with the increases of α, which indicates that they are willing to pay more for a higher greening improvement level and the manufacturer can set a higher price. For dθ * dc0 ≥ 0 and dθ * dc1 ≥ 0, we can explain them as follows: The optimal green level θ * will be affected by the minimal possible cost with zero flexibility and the minimal possible cost with the fixed rate because the wholesale price includes them. Thus, the increased wholesale price means more investment and efforts for green automobiles, which improves the green level. The reason for dp * dβ ≥ 0 is similar to some explanations for dθ * dc0 ≥ 0 and dθ * dc1 ≥ 0 because the wholesale price is also increasing in β. Interestingly, from Proposition 5, we can find that dθ * dI ≤ 0 represents that the manufacturer who has a lower marginal cost for any given quality in the quantity-flexibility contract is an inefficient firm [START_REF] Jiang | Quality and pricing decisions in a market with consumer information sharing[END_REF]. Now, we turn to the equilibrium solutions in the centralized case. We can establish the following monotonicity results for the centralized decision-making policy after computing the optimal solutions in Proposition 3: Proposition 6. Under the centralized case, the optimal solutions exhibit the following characteristics:

(i) If the demand is less than Q(1 -β), the optimal price p * is nonmonotonic in α or b, i.e., dp * dα ≤ 0 when a ∈ (0, 3bh R ) and dp * dα ≥ 0 when a ≥ 3bh R ;

dp * db ≤ 0 when a ≥ 3α 2 h R

4I

and dp * db ≥ 0 when a ∈ (0, 3α 2 h R 4I ). As I increases, the optimal green level θ * decreases, i.e., dθ * dI ≤ 0.

(ii) Unlike x < Q(1 -β), the optimal price p * is increasing in α and decreasing in b if the demand is in the range of [Q(1 -β), Q(1 + β)], i.e., dp * dα ≥ 0 and dp * db ≤ 0.
From Proposition 6 (i), we can find that the optimal price p * could first decrease and then increase in α given a > 0. This is because when the market potential is smaller, fewer consumers are concerned about EVs, which causes the retailer to lower the price to attract customers to buy. When the market potential is larger, more consumers are starting to pay attention to greenness. Thus, the retailer can raise the price to obtain more profits. The same explanation also applies to dp * db and we also find that the optimal price p * first increases and then decreases in b. The reason is that when the retailer is indifferent about the pricesensitivity of consumer when the market potential is smaller. When the market potential is larger, the retailer may lower the price because customers' higher price sensitivity means the higher demand reduction. Interestingly, Proposition 6 (ii) tells us that if x ∈ [Q(1 -β), Q(1 + β)], the monotonicity of parameters α and b is different from that in x < Q(1 -β). An important reason is that as the retailer's order quantity increases, the demand in the market also increases, which influences the market potential that is larger than 3bh R or 3α 2 h R 4I . So, the monotonicity of α and b has changed.

Concluding remarks

This paper studies a quantity-flexibility contract between a manufacturer and a retailer with green-sensitive demand in the automotive industry. In a decentralized supply chain, we use game theory to solve the problem. The automobile manufacturer acts as the Stackelberg leader to set the green level, and the retailer acts as the follower to set the retail price. We determine the optimal green level to maximize the automobile manufacturer's profit and the retail price to maximize the retailer's profit. Under centralized decision-making, we consider the equilibrium decision between the automobile manufacturer and the retailer. We use the game theory to analyze the model to maximize the supply chain's profit. We analyze the equilibrium strategies and find the condition that the optimal solutions can sustain. This paper is one of the first researches to study quantity-flexibility contract considering green-sensitive customers. We propose a novel collaboration model in quantity-flexibility contract considering green issues. It is optimal for the retailer in the decentralized case to charge a higher price when green-sensitivity increases; however, the optimal price in the centralized case is not necessarily monotonic when the market potential is low. Interestingly, we also find that the supply chain with the quantity-flexibility contract does not always generate more profits than its inflexibility counterpart; it mainly depends on the retail price and greenness improvement level. Our results can be applied to maximize the profits on both upstream and downstream partners with quantity-flexibility contracts in the green supply chain.

  the retailer's profit function is concave with p and the optimal p * = a+αθ+(c0+c1β)b 2b

Yeming Gong is partially supported by Business Intelligence Center (BIC)


Canan Pehlivan
email: canan.pehlivan@yeditepe.edu.tr

Thomas G Yeung
email: thomas.yeung@imt-atlantique.fr

Aline Suzette Alvarado Munguia

Crop selection and scheduling for green production with intercropping and rotation

Keywords: Sustainable agricultural production, Crop schedule, Crop rotation, Intercropping

In this study, we address a crop selection and scheduling problem employing two cropping systems, crop rotation and intercropping, simultaneously on a given parcel of land. Moreover, we also consider other ecological practices such as the use of fallow and green manure. We propose a 0-1 linear programming model to maximize revenue by determining the optimal combinations of crops in space (crops to be planted as neighbors) and in time (crop rotation schedule) while meeting the yearly demand of each crop. A realistic numerical example is presented to demonstrate the performance of the model.

Introduction

Conventional agricultural production systems are generally based on monoculture (i.e., growing only one crop) as this practice is generally lowest in production cost. However, monoculture often has important adverse side-effects that are not accounted for. These include the environmental costs of water contamination by pesticides and other polluting inputs, or the social costs associated with the exclusion of small farmers due to the large-scale capital requirement [START_REF] Altieri | Agroecology: the science of sustainable agriculture. boulder. Westview Press[END_REF], [START_REF] Gliessman | Agroecology: ecological processes in sustainable agriculture[END_REF]. To avoid these adverse effects, more sustainable agricultural production systems are needed that are economically profitable, environmentally safe, and socially equitable. To this purpose, several strategies based on increasing spatial and temporal crop diversification have been proposed such as intercropping (combination in space), crop rotation (combination in time), agroforestry, composting, and green manuring [START_REF] Lichtfouse | Agronomy for sustainable agriculture: a review[END_REF], [START_REF] Vandermeer | Global change and multi-species agroecosystems: concepts and issues[END_REF].

To compete with monoculture, the proposed farming practices should be combined in such a way that fewer inputs and resources are used while yields are not drastically reduced and may even be improved. This requires thorough planning and solving complex problems where, both technical and ecological production aspects of those farming practices need to be considered.

In this study, we address a crop selection and scheduling problem employing two cropping systems, crop rotation and intercropping, simultaneously on a given land, while considering other ecological practices such as the regular employment of fallows and green manures. Our objective is to determine the optimal combinations of crops in space (crops to be planted as neighbors) and in time (crop rotation schedule) to maximize revenue while meeting the yearly demand of each crop. We propose an integer linear programming (ILP) model to select the best combination of crops in space and time. A numerical experiment is conducted based on real crop data and an optimal crop schedule is provided to demonstrate the performance of the model.

In the next section, we first give a primer on sustainable cropping systems. In Section 3, we present the literature review and highlight the contribution of our work. In Section 4, we present the ILP formulation for our problem. Section 5 presents the numerical example followed by a conclusion.

Sustainable Agriculture Production Practices

Below we explain the sustainable agricultural practices that have been employed in this research. We present as well our assumptions, depending on the practical use of those practices.

Crop Rotation is the practice of planting different types of crops sequentially on the same plot of land to improve soil health, optimize nutrients in the soil, and combat pest and weed pressure [START_REF] Altieri | Agroecology: the science of sustainable agriculture. boulder. Westview Press[END_REF]. It is generally undesirable to grow two crops of the same botanical family in sequence on the same piece of land to prohibit the propagation of pests [START_REF] Dos Santos | Sustainable vegetable crop supply problem[END_REF]. Crop Rotations are most effective when combined with practices such as composting, intercropping, cover cropping, green manuring and short fallow periods [START_REF] Baldwin | Crop rotations on organic farms[END_REF].

Intercropping is the practice of growing two or more crops in the same field simultaneously or sequentially within the same season [START_REF] Altieri | Agroecology: the science of sustainable agriculture. boulder. Westview Press[END_REF]. The key advantage of intercropping is the greater yield generated by the crop interactions. It has many other beneficial effects like increased biodiversity, nitrogen fixations by legumes, nutrient recycling in the soil, better control of pests and diseases, more efficient use of environmental sources, soil coverage and stable yields [START_REF] Lithourgidis | Annual intercrops: an alternative pathway for sustainable agriculture[END_REF]. In this work, we limit intercropping to two crops for tractability and simplicity. There exist several types of intercropping systems. Most common ones such as mixed cropping, row intercropping, strip intercropping and relay intercropping are illustrated in Figure 1. In this work, we consider row intercropping where different crops are grown at the same time in the same piece of land with a distinct row arrangement [START_REF] Andrews | The importance of multiple cropping in increasing world food supplies[END_REF]. Within the row layout, we also employ relay intercropping where intercrops were sown when the first crop reach maturity but before being harvested. We choose to work with row intercropping due to its structured layout and suitability for a large-scale farming, and relay intercropping due to the time flexibility it provides.

Green manuring is the practice of leaving uprooted or sown crop parts to wither on a field so that they serve as a mulch and soil amendment [START_REF] Altieri | Agroecology: the science of sustainable agriculture. boulder. Westview Press[END_REF], [START_REF] Gliessman | Agroecology: ecological processes in sustainable agriculture[END_REF]. The function of a green manure crop is to add organic matter to the soil, increasing the nitrogen and nutrient supply of the soil thereby increasing its productivity.

In crop rotation, crops are categorized into three groups with respect to their nutrient demand. Green manures are heavy givers that provide necessary nutrients to heavy feeders or light feeders. Since heavy feeders consume most of the nutrient in soil, in our schedule, we consider employing green manure (planting a heavy giver) after a fixed number of heavy feeders, not necessarily sequential.

Fallow is the practice of leaving the land without being sown for one or more vegetative cycles to allow the land to recover and store organic matter and reduce pest damage [START_REF] Altieri | Agroecology: the science of sustainable agriculture. boulder. Westview Press[END_REF], [START_REF] Gliessman | Agroecology: ecological processes in sustainable agriculture[END_REF].

Literature Review

The standard crop rotation problem is defined as determining the sequence of crops to plant, one after the other, in a given area to optimize a certain objective such as maximizing yield, profit, or minimizing area, etc. The problem is modeled as a linear program by many researchers such as [START_REF] El-Nazer | The choice of crop rotation: a modeling approach and case study[END_REF], [START_REF] Clarke | Combinatorial aspects of cropping pattern selection in agriculture[END_REF], [START_REF] Haneveld | Crop succession requirements in agricultural production planning[END_REF], [START_REF] Detlefsen | Modelling optimal crop sequences using network flows[END_REF]. However, none of them consider environmental concerns and sustainability practices.

In the recent literature, there exists a number of studies that address the problem in a sustainable development context, taking into account environmental constraints or objectives related with increasing soil quality, reducing the use of resources, increasing diversity. [START_REF] Dogliotti | Rotat, a tool for systematically generating crop rotations[END_REF] developed a software program called RO-TAT that generates all possible rotations for a given set of crops, however, they do not optimize the rotations. Later, a group of researchers, [START_REF] Dos Santos | Sustainable vegetable crop supply problem[END_REF], [START_REF] Alfandari | A branch-and-price-and-cut approach for sustainable crop rotation planning[END_REF] focus on managing the area to be planted while satisfying the demand through reasonable crop schedules rather than optimizing the crop selection itself. [START_REF] Dos Santos | Sustainable vegetable crop supply problem[END_REF] considered a sustainable vegetable crop demand supply problem. They determine the division of the available arable areas in plots and, for each plot, obtain an appropriate crop rotation schedule respecting ecological constraints. Their objective is to maximize the plots occupation considering demand constraints, proposing a 0-1 linear optimization model. [START_REF] Alfandari | A branch-and-price-and-cut approach for sustainable crop rotation planning[END_REF] address Minimum-Space Crop Rotation Problem (MSCRP) which is defined as constructing crop rotations to minimize the total space area required to meet demand. [START_REF] Dos Santos | Crop rotation scheduling with adjacency constraints[END_REF] is the first to introduce adjacency constraints in crop rotation problem. Adjacency constraints prevent crops of the same botanical family to be cultivated at the same time in neighboring plots. They propose a 0-1 optimization model for crop rotation in different areas, with an objective again to maximize the land use subject to neighborhood and succession restrictions for crops of the same botanic family. Later, [START_REF] Aliano Filho | Metaheuristics for a crop rotation problem[END_REF] adapted the model of [START_REF] Dos Santos | Crop rotation scheduling with adjacency constraints[END_REF] and introduced demand constraints as well as an objective function which maximizes the profit of planted crops. Recently, [START_REF] Mauri | Improved mathematical model and bounds for the crop rotation scheduling problem with adjacency constraints[END_REF] proposed some improvements in the model presented by [START_REF] Dos Santos | Crop rotation scheduling with adjacency constraints[END_REF] and [START_REF] Aliano Filho | Metaheuristics for a crop rotation problem[END_REF] by considering both objectives of maximizing plot occupation and the profit.

We are the first to employ two cropping systems (crop rotation and intercropping) simultaneously to determine the crop schedule. Intercropping can be considered similar to adjacency constraints, but they differ in several key ways. In [START_REF] Dos Santos | Crop rotation scheduling with adjacency constraints[END_REF], [START_REF] Aliano Filho | Metaheuristics for a crop rotation problem[END_REF], and [START_REF] Mauri | Improved mathematical model and bounds for the crop rotation scheduling problem with adjacency constraints[END_REF], adjacency relations are modeled as constraints which state adjacent plots cannot grow, simultaneously, crops belonging to the same botanical family. Thus, it defines strictly which group of crops cannot be planted, and ignore the yield information. This might be misleading considering that individual crops within the same family might have different effects on productivity when intercropped with a specific crop [START_REF] Lithourgidis | Annual intercrops: an alternative pathway for sustainable agriculture[END_REF].

We consider both the family of crops (in determining the crop sequence) and the yield information (in determining the crops to intercrop). Additionally, intercropping involves dividing the land into many adjacent rows or strips and defines the two-way interaction between specific crops in terms of yield. Hence, our model chooses to plant the specific crops (not a family of crops) simultaneously or in sequence that have the highest positive outcome on the soil, and thus also on the yield. Moreover, [START_REF] Dos Santos | Crop rotation scheduling with adjacency constraints[END_REF], [START_REF] Aliano Filho | Metaheuristics for a crop rotation problem[END_REF] mainly focused on determining the area used for plantation or the best division of land to meet the demand through a feasible crop schedule. They do not address finding the optimal sequence of crops to be planted in a crop rotation problem.

Methodology

We consider a planning horizon of |L| years, where the planning time unit is one month. We assume that each crop is planted at the beginning of a month. Each crop has a production time h i , which includes the time required to sow, grow, and harvest. Each crop also has a suitable sowing (planting) time window. To disincentivize monocropping the highest revenue crop, we assume that there is a maximum and minimum annual demand (d max il and d min il ) for each crop. C is the set of crops that can be selected for planting, including the ones for green manuring and a hypothetical crop associated with fallow. Crops are also categorized according to their nutrient demands; the sets C f , C v and C g correspond to the set of heavy feeders, heavy givers and cover crops (green manure), respectively. Note that the sets C v and C g might contain the same crops, as heavy givers are generally also used as cover crops. However, cover crops differ in terms of production time in green manure (we assume half time) and they do not have any yield as they are not harvested.

To consider intercropping effects, the arable land is divided into S number of rows, consisting of an equal number (S/2) of rows a and rows b. The parameter e ijk models the intercropping effect on the yield which is dependent not only on the type of crops intercropped but also the timing of plantation. x a it = 1 (x b it = 1) and z a ijtk = 1 (z b ijtk = 1) are the binary decision variables described in the notation list.

The crop rotation problem consists in determining the types of crops to plant and their sequence at rows a and rows b over a finite planning horizon in order to maximize the revenue subject to ecological and demand constraints. The integer linear programming model formulation is given as:

Maximize |T | t=0   N i=1 p i y i   x a it + x b it + N j=1 hi-1 k=-hj +1 e ijk (z a ijtk + z b ijtk )     (1) subject to i∈C hi-1 r=0 x α i(t-r) ≤ 1 α = {a, b} ∀t ∈ T (2) i∈C\C m(t) x α it = 0 α = {a, b} ∀t ∈ T (3) i∈F (δ) hi r=0
x α i(t-r) ≤ 1 α = {a, b} ∀t ∈ T and δ = 1, 2, . . . N F (4)

x a it + x b j(t+k) ≥ 2z a ijtk ∀i, j ∈ C, ∀t ∈ T, k = -h j + 1, . . . , h i -1 (5) x a it + x b j(t+k) -1 ≤ z a ijtk ∀i, j ∈ C, ∀t ∈ T, k = -h j + 1, . . . , h i -1 (6) x b it + x a j(t+k) ≥ 2z b ijtk ∀i, j ∈ C, ∀t ∈ T, k = -h j + 1, . . . , h i -1 (7) x b it + x a j(t+k) -1 ≤ z b ijtk ∀i, j ∈ C, ∀t ∈ T, k = -h j + 1, . . . , h i -1 (8) t-1 r=0 i∈C f x α ir + x α jt ≤ ω t+hj r=0 k∈Cg x α kr + (ω -1) α = a, b ∀j ∈ C f , ∀t ∈ T (9) ω t r=0 k∈Cg x α kr ≤ t r=0 i∈C f x α ir α = {a, b}, ∀t ∈ T (10)
In the objective function, we maximize the total revenue obtained from each crop pair throughout the planning horizon. Constraint [START_REF] Alfandari | A branch-and-price-and-cut approach for sustainable crop rotation planning[END_REF] ensure that at most one crop is planted at a given time t in rows a and b. Additionally, constraints do not allow to plant any other crop during the production time (h i ) of the cultivated crop i. Constraint (3) ensure only the crops that are possible to be planted during that month are planted. Constraint (4) forbid crops of the same botanic family to be planted in sequence (immediately one after the other). Constraints [START_REF] Altieri | Agroecology: the science of sustainable agriculture. boulder. Westview Press[END_REF][START_REF] Andrews | The importance of multiple cropping in increasing world food supplies[END_REF] and [START_REF] Baldwin | Crop rotations on organic farms[END_REF][START_REF] Clarke | Combinatorial aspects of cropping pattern selection in agriculture[END_REF] keep the intercropping information for the crops in rows a and rows b, respectively. Constraint (9) force a green manure after ω heavy feeders are planted in rows a and b (regardless if they are consecutive or not). Constraint [START_REF] Dogliotti | Rotat, a tool for systematically generating crop rotations[END_REF] ensure that green manures are not done preemptively to allow successive heavy feeders greater than ω. Constraints [START_REF] El-Nazer | The choice of crop rotation: a modeling approach and case study[END_REF] ensure that we do either fallow (the crop N is fallow) or green manure in a 12-month rolling time intervals. The last constraint enforces the minimum and maximum demands.

Numerical Example

We demonstrate our model on a hypothetical tropical land based on real data as research in agro-ecology and intercropping is very popular in the regions with this climate. In our numerical example, we consider 11 crops grown in tropical regions, e.g. Brazil and India. Table 1 presents the type of crops and their parameters. The yield of sole-planted crops is obtained from FAO Statistics [1]. The price of the crops are determined based on average prices in France in 2020. Table 2 presents the percentage yield effect of intercropping one crop with another. The intercropping data is obtained through extensive literature review; some of the references are [START_REF] Monim | Effect of intercropping different vegetables with groundnut[END_REF], [START_REF] Ali | Intercropping maize with different vegetables[END_REF] and [START_REF] Raji | Intercropping soybean and maize in a derived savanna ecology[END_REF]. A value of 0 indicates no effect, while a value of -100 denotes the crops are not compatible for intercropping and thus forbidden. We consider a planning horizon of two years (|L| = 2), divided into months (|T | = 24). We set ω = 2 to allow only two heavy feeders before forcing a green manure to heal the soil.

The model is implemented in the PuLP library for Python and solved using ILOG CPLEX 12.10. The example was run on a PC with an Intel(R) Core (TM) i7-10510U CPU @ 1.80 GHz with 16GB of RAM. The problem was solved to optimality in 2134 seconds yielding the policy in Figure 2 with an objective value of 615,797.33. Alternate shading of light and dark grey is used for successive crops with white space corresponding to nothing planted at the time. The solid black shows a forced fallow period. Green manure are diagonally hatched.

Table 1 reveals that the highest revenue per row (yield x price) comes from Coriander, Tomatoes, Eggplant, Ginger and Chilli, respectively, with Coriander almost three times the revenue of Tomatoes. The model would normally seek to plant Coriander whenever possible, however we limit it to once per year by setting its demand equal to its yield. We similarly limit Tomatoes and Eggplant to once per year by limiting their demand. To promote the planting of Chilli and Beans which have comparatively lower revenues, we set a minimum demand to be met. The example policy shows that the highest revenue crops are indeed selected as much as possible, while respecting all of their constraints regarding placement and scheduling.

Constraint (1) is obviously respected as two crops are never planted at the same time and Constraint ( 2) is respected by only planting crops in the period in which they are allowed. Constraint (3) is seen through the spacing (fallow) between Chilli and Eggplant and then between Chilli and Tomatoes in Row B as they are from the same botanical family and can not be planted immediately following one another. Since this is not a forced fallow due to Constraint [START_REF] Dogliotti | Rotat, a tool for systematically generating crop rotations[END_REF], but for family scheduling purposes, it is shown as white space. Constraints (8) and ( 9) are respected by forcing a green manure (Groundnut GM) after the two heavy feeders Eggplant and Chilli in Row B and again after Tomatoes and Eggplant in Row A. Constraint [START_REF] Dogliotti | Rotat, a tool for systematically generating crop rotations[END_REF] is seen in Row A where a Fallow is forced at t = 8 and t = 17 as there is no green manure in Row A until the very end and we must have at least one fallow within any 12-month interval.

Fig. 2: Optimal Crop Schedule

A strong impact of intercropping effects can also be seen in this policy. Eggplant planted with beans will decrease its yield by 16.25%, which is unfortunate, but is chosen by the model as it is still highly desirable due to its still high revenue with respect to the other crops, and beans are required by the demand constraints. Beans will, however, receive a 13% increase in yield which will help to offset the yield loss of Eggplant. Eggplant is also forbidden to be planted with Chilli (-100 intercropping effect) which also has a minimum demand to be met. The policy is able to satisfy these intercropping constraints.

Conclusion and Future Work

We have formulated a 0-1 linear program to optimize crop selection and scheduling while considering ecological aspects such as intercropping and rotation. This model is an initial work in trying to promote (economically, ecologically, and socially) a combination of green agro-ecology principles in a practical way. It will be a valuable tool in moving away from destructive monoculture habits that currently widely exist in large-scale agriculture. While the model is efficient for small sizes, it quickly becomes intractable for CPLEX as the number of crops and the time horizon increases. Specialized algorithms and/or heuristics will need to be developed to solve larger instances to (near) optimality. Moreover, the structure of the policies could be studied and demonstrated to develop more efficient solution procedures.

Fig. 1 :

 1 Fig. 1: Schematic illustrations of alternative intercropping systems

  1 α = {a, b}, t = 6, . . . , |T |and ∀l ∈ L

Table 1 :

 1 Crop Data

	Crops	Botanical	Nutrition	Plantation Prod.time Yield	Price Demand
		Family	Demand	Periods	(months) (kg/row) (e/kg) (min,max)
	1 Onion	Allium	Light Feeder 2-3, 5-6, 8-11	4	40	160	(0, 100)
	2 Coriander	Apiaceae	Light Feeder	6-7, 10-11	2	7	25000	(0, 7)
	3 Beans	Fabaceae	Heavy Giver	2-3, 5-6	3	1,75	2430	(1, 100)
	4 Soybeans	Fabaceae	Heavy Giver	6-7, 2-3	4	2	344	(0, 100)
	5 Groundnut	Fabaceae	Heavy Giver	6, 10-1	4	2,5	1410	(0, 100)
	6 Maize	Poaceae	Heavy Feeder 6-7, 10-11, 2	3	7,5	160	(0, 100)
	7 Tomatoes	Solanaceae Heavy Feeder	all year	4	55	1120	(0, 55)
	8 Chilli	Solanaceae Heavy Feeder 1-2, 5-6, 9-10	4	12,5	1700 (12, 100)
	9 Eggplant	Solanaceae Heavy Feeder 5-6, 8-9, 12-1	4	22,5	2170	(0, 25)
	10 Potato	Solanaceae Heavy Feeder 6-7, 10-11	4	20	130	(0, 100)
	11 Ginger	Zingiberaceae Heavy Feeder	2, 3, 5	6	14,08	2300	(0, 20)

Table 2 :

 2 Percentage Yield Effect (e ij. ) of Intercropping crop i ∈ C with j ∈ C

		1	2	3	4	5	6	7	8	9	10	11
	1	0	0	-100 -100	0	-82,67 -3,22 -24,9 -75,81 -100	0
	2	0	0	-100 -100 -24,74 -90,85 -73,54	0	-65,72 -100	0
	3 -100 -100	0	0	-100	-58	-25	0	13	-100 27,42
	4 -100 -100	0	0	-100 -13,25	0	-26,31	0	-100	0
	5	0	-14,43 -100 -100	0	-55,55	0	0	-63,92 -100 -100
	6 -8,84 -3,3	9,25 -20,39 -2,29	0	-13,96 -21,49 -28,51 -44,68 79,27
	7 -15,16 -16,71 -3	0	0	-52,3	0	-100 -100 -100 -12,83
	8 -73,51	0	0	-40	0	-42,96 -100	0	-100 -100 -100
	9 -0,46 -21,58 -16,25	0	20,97	2	-100 -100	0	-100 -100
	10 -100 -100 -100 -100 -100 -61,72 -100 -100 -100	0	-100
	11	0	0	27,2	0	-100 -23,01 -18,64 -100 -100 -100	0


Loïc Perrin
email: perrin@emse.fr

Valérie Laforest
email: laforest@emse.fr

Marie De Roy
email: marie.de_roy@univ.bpclermont.fr

Claude Forano
email: claude.forano@univ-bpclermont.fr

Jacques Bourgois
email: bourgois@emse.fr

Adsorption of Chromium (VI) by metal hydroxide sludge from the metal finishing

Keywords: valorisation, sorption, polymetallic hydroxide sludge, Cr VI

Metal finishing industries produce an important tonnage of metal hydroxide sludge (MHS) during the treatment of their liquid effluents charged with heavy metals. Generally, a small part of these sludge is valorized because of their important metal fickleness. Consequently, the majority of these metal hydroxide sludge is sent to landfill centers. We propose to valorize this sludge by using them as pollutant sorbent in order to retain the polluting species contained in the industrial aqueous effluents. The chosen pollutant for this preliminary study is Chromium (+VI). Starting from an experimental well defined protocol, various parameters have been studied (chemical nature of the sludge, solid/liquid ratio, concentration of the pollutant, particle size) in order to determine the experimental parameters which influence the sorption yield of these sludge and thus to validate the feasibility of an industrial application of this new way of valorisation. First results point out that the sorption yield is more than 95% independently from the experimental conditions. Moreover, experiments highlight that the sludge mass (and so sorbent sites) is THE only real parameter which influences the yield. In this article, a review of the patents on different aspects of the valorisation of polymetallic hydroxide sludge as pollutant trappers is presented.

Introduction

Industrial aqueous pollution (heavy metals) accounts for 30 to 40% of industrial pollution. Metal finishing is one of the sectors which contributes mostly to this pollution. Due to its unique production, metal finishing consumes and discharges wastewater containing metallic ions, cyanide, COD, etc.. Therefore, this activity is bound to the European Integrated Pollution Prevention and Control Directive since 1996, known as the "IPPC Directive" [1]. In France, this directive is translated "as is" for metal finishing workshops through the ministerial order of June 30 th , 2006. Before 1996, this text was already part of the legislation of the officially identified sites for the protection of the environment codification 2565 [START_REF]Arrêté du 30 juin 2006 relatif aux installations de traitement de surfaces soumises à autorisation au titre de la rubrique 2565 de la nomenclature des installations classées[END_REF]. This ministerial order defines thresholds in terms of pollutants in the effluents and in terms of flow yield and concentration of specific chemical species for this branch of activity. These workshops must implement curative or preventive Best Available Techniques (BAT) which is another major principle of the IPPC Directive [START_REF] Cikankowitz | Méthodologie d'évaluation des performances environnementales de procédés en vue de les comparer et les valider aux meilleures techniques disponibles[END_REF]. For about 65% of workshops, initiatives and actions designed to limit environmental impacts are typically curative. Concerning wastewater composed mainly of heavy metals, the detoxication process of wastewater is the co-precipitation of metal ions in hydroxide. Some species such as chromates may require an oxidation/reduction treatment. After filtration, hydroxides obtained as sludge are disposed of in various sectors. This sequence of processes is described in the BAT reference documents (BREF), on "Best Available Techniques for metal finishing treatment and plastic finishing treatment" [START_REF] Bref | Bureau européen pour l'IPPC. IPPC BREF outline and guide[END_REF]. It is important to take into account that France produces 180 000 tons of metal sludge per year compared to the overall European production which averages to 950 000 tons per year (source CETIM). Currently, despite the existence of processes such as hydrometallurgy for sludge rich in zinc, pyrometallurgy for sludge rich in nickel, thermal reconditioning for sludge with specific characteristics or finally added to cement for sludge riches in aluminium or iron, polymetallic hydroxide sludge (predominance of non-metal) are mostly eliminated in hazardous waste landfills. For these reasons, this research project's aim is to propose a new method of polymetallic hydroxide sludge valorisation that can not be otherwise valorised by the previously mentioned methods of treatment because of their characteristics.

Another key problem of the metal finishing industry deals with chromium. Indeed, it is a very widespread metal finishing application. It gives different properties such as hardness, aesthetics, resistance to temperature, corrosion or wear. This versatility justifies that the chromium is an essential element for various applications. The most widespread properties of the chromium metal deposit are their protective and decorative characteristics. It is estimated, at the European level, that 30 000 to 32 000 tons of chromic acid per year are used in metal finishing. Because of drag-out all along the production line, 20 to 60% are found in industrial aqueous effluents. This ratio depends on the local context of each workshop namely its size and recycling measures implemented. So, it appears that the decontamination of chromium is very important for the metal finishing industry.

Research Aims

For several years, many studies [START_REF] Ting | Hydrogen sulfide removal from coal gas by the metal ferrite sorbents made from the heavy metal wastewater sludge[END_REF][START_REF] Zhang | Mercury removal from water using activated carbons derived from organic sewage sludge[END_REF][START_REF] Santos | Waste metal hydroxide sludge as sorbent for a reactive dye[END_REF][START_REF] Zhu | Removal of an anionic dye by sorption/precipitation processes using alkaline white mud[END_REF][START_REF] Bhattacharya | Sorption,kinetics and equilibrium studies on removal of Cr (VI) form aqueous solutions using different low cost sorbents[END_REF][START_REF] Bhanagar | Vanadium removal form water by waste metal sludge and cement immobilization[END_REF][START_REF] Wei | Phosphorus removal by acid mine drainage sludge from secondary effluents of municipal wastewater treatment plants[END_REF][START_REF] Sirinuntapiboon | Biological removal of cyanide compounds from electroplating wastewater (EPWW) by sequencing batch reactor (SBR) system[END_REF][START_REF] Ju | Biosportion of reactive dye (Rhodamine-B) from an aqueous solution using dried biomass of activated sludge[END_REF][START_REF] Thawornchaisit | Application of dried sewage sludge as phenol. Biosorbent[END_REF][START_REF] Rozada | Sorption of heavy metals onto sewage sludge derived materials[END_REF][START_REF] Bouzid | A study on removal characteristics of copper from aqueous solution by sewage sludge and pomace ashes[END_REF][START_REF] Iddou | Waste activated sludge (WAS) as Cr (III) sorbent biosolid from wastewater effluent, colloids and surfaces B[END_REF] have been carried out concerning the valorisation of different sludge which are either industrial (metal hydroxide sludge) or urban (activated sludge). The main application under development related to this kind of waste is the use of sludge for sorption of pollutants whether the composition is organic or mineral. It is possible to combine these applications in three categories:

The less used or in any case the less referenced in the bibliography consists in the production of a sub-product with good sorption property for heavy metals (Hg) or toxic gas (H 2 S) from a mineral sludge (heavy metals) [START_REF] Ting | Hydrogen sulfide removal from coal gas by the metal ferrite sorbents made from the heavy metal wastewater sludge[END_REF] or an organic sludge [START_REF] Zhang | Mercury removal from water using activated carbons derived from organic sewage sludge[END_REF]. The metal hydroxide sludge generated, for example by metal finishing treatment or metallurgy, could constitute the second category of application in the pollutant sorption whether they are organic [START_REF] Santos | Waste metal hydroxide sludge as sorbent for a reactive dye[END_REF][START_REF] Zhu | Removal of an anionic dye by sorption/precipitation processes using alkaline white mud[END_REF] or mineral (oxometallic anions) [START_REF] Bhattacharya | Sorption,kinetics and equilibrium studies on removal of Cr (VI) form aqueous solutions using different low cost sorbents[END_REF][START_REF] Bhanagar | Vanadium removal form water by waste metal sludge and cement immobilization[END_REF]. The last application is linked to the biological activity contained in the activated sludge (organic compound, etc.). This application is highly prized by the enthusiasm of the scientific community in view of numerous publications on this subject, and is applied to all types of pollutants in water such as non-metallic anions [START_REF] Wei | Phosphorus removal by acid mine drainage sludge from secondary effluents of municipal wastewater treatment plants[END_REF][START_REF] Sirinuntapiboon | Biological removal of cyanide compounds from electroplating wastewater (EPWW) by sequencing batch reactor (SBR) system[END_REF], organic molecules [START_REF] Ju | Biosportion of reactive dye (Rhodamine-B) from an aqueous solution using dried biomass of activated sludge[END_REF][START_REF] Thawornchaisit | Application of dried sewage sludge as phenol. Biosorbent[END_REF], heavy metals in the form of oxometallic anions [START_REF] Rozada | Sorption of heavy metals onto sewage sludge derived materials[END_REF][START_REF] Bouzid | A study on removal characteristics of copper from aqueous solution by sewage sludge and pomace ashes[END_REF][START_REF] Iddou | Waste activated sludge (WAS) as Cr (III) sorbent biosolid from wastewater effluent, colloids and surfaces B[END_REF].

Generally, the versatility of applications of heavy metals in industry is such that it is difficult to generalize properties of this complex material (sludge) because chemical composition and chemical nature are different from one sector to the other. However, according to the industrial sector, it is easier to identify metals mainly used. In our case study, three transition metals are commonly used: zinc (+II), nickel (+II) and chromium (+III). The sludge used for this study is coming from two metal finishing industries whose application resides essentially in zinc, nickel and chromium plating. The metal hydroxide sludge is mainly composed of these three metals with a variation of metal composition depending on the SME activity. The aim of this research is to allow the use of sludge as a trapping matrix of pollutants, without any other specific treatment in advance (for example incineration) and independently of their metal compositions. Considering its importance in the metal finishing treatment and in the views of its toxic properties, the pollutant chosen is chromium (+VI). Then, the aim of this project is to evaluate the influence of various parameters such as contact time, the solid / liquid ratio, temperature, grain size and chemical nature of the sludge (origin) on the sorption yield. Unlike most previous studies, the pH will be a free parameter in the sense that it will not be fixed by outside intervention (adding base or acid). Indeed, our matrix is composed of metals which can be amphoteric (Zn, Al); as a result, it could be dissolved during the experiment time if a pH adjustment was to be realized. Hence, it is better to let the pH of the solution come into balance and so to avoid a more important preferential dissolution.

Experimental protocol

Methods

A sludge mass "m" (mg) ranging from 50 mg to 3125 mg is put in suspension in a beaker containing a volume "V" (ml) (25 or 50 mL) of a potassium dichromate solution (Cr 2 O 7 2-) to a given concentration "C i " (mg.L -1 ). The system is under magnetic stirring at a temperature T (°C) (room temperature, 25°C and 35°C), for a time "t" (min) (ranging from 0 to 180 min). After this time "t", the solid is separated from the liquid phase by filtration. The liquid fraction is then analyzed by ICP (Inductively Coupled Plasma), to get the concentration "C e " in equilibrium in total ion Cr remaining in solution. The range of initial concentrations in Cr 2 O 7

2-(C i ) was chosen so as to comply to chromate concentrations in industrial wastewater metal finishing. This range of concentrations is from 0 to 360 mg.L -1 .

Raw Sludge (sorbents)

The sludge used in experimental sorption were collected in both a local SME (sludge A taken at the exit of the filter press i.e with 35% average of dry content) and a larger company (sludge B heat-treated at 1000°C beforehand). Infrared analysis, ICP analysis and X-rays diffraction have been carried out on this sludge to define precisely the chemical nature. The ICP analysis of the sample B revealed that it was composed of several metals including three predominant ones (Zn, Ni, Cu), representing approximately 75% of metal composition. Analyzes of sludge A unveiled the presence of Zinc, Nickel and Chromium in majority. A sampling study has shown that for the sludge A the zinc composition fluctuates between 40% and 65%, the nickel composition between 5% and 10% and the chromium composition between 5% and 35%.

Experimental results and Discussion

Sorption kinetics of the two sludge

Sludge nature

According to the protocol defined in 3.1, a sorption kinetic study was realized on the two samples A and B. The experiment is conducted with 25 mL of a solution containing 30 mg.L -1 of Cr VI and for a sludge mass of 1250 mg at room temperature (Figure 1). Figure 1 shows that the nature of the tested sludge does not influence the kinetic sorption. The maximum sorption yield is obtained very quickly and for an experimental duration of 20min. Moreover, it is possible to observe that the chemical nature of the sludge influences it. Indeed, sludge B achieves an average sorption yield of 95% whereas for sludge A the yield is 90%. This slight difference could be explained by experimental errors. These errors are due to the mass variation of each sample (50±2 mg that is equivalent to an experimental error of 4%).

Sludge mass

A kinetic study based on the variation of the mass of the sludge B was realized in order to evaluate the influence of sorbent mass on the sorption yield (Figure 2). The experimental conditions are the same as 3.3.1 : 25ml of a solution containing 30 mg.L -1 of Cr VI and at room temperature. Figure 2 reveals that the sludge mass has a great influence on the sorption yield but not the kinetic which is always quickly. If the sorbent mass in suspension is multiplied by 30 (from 50 mg to 1500 mg), the sorption yield average goes from 75% to over 95%. Moreover, for a weight of sludge higher than 500 mg, the sorption yields are similar (average 95%). This can be explained by the fact that adding a too much important quantity of sludge in the beaker leads to a non-suspension conducting to a loss of surface contact between the liquid and solid phase.

[Cr VI ]

Part 3.3.2 highlighted that the optimal sludge mass for an optimal yield equal to 500 mg. In fact, beyond this mass, no improvement of sorption is observed. In this perspective, the protocol described before is used in setting the mass of the sludge B to 500 mg. Thus, the aim is to define the influence of a solution of Cr VI on the sorption kinetics. For this, three concentrations are tested: 16, 28.8 and 73 mg.L -1 (Figure 3). whereas for a concentration of 73 mg.L -1 , it reaches 90%. For a mass of 500 mg in 25 ml of solution, a fourfold increasing of the concentration of Cr VI (from 16 to 73 mg.L -1 ) leads to a loss of 5% of the sorption yield. In fact, if sludge mass is more important than 500 mg, it could be possible to increase the sorption yield for the solution of 73 mg.L -1 . In fact, by adding more sludge, sorbent sites will increase and then sorption yield may reach more than 90%.

Influencing sorption parameters study

Temperature

The sorption experiments are done with a mass of sludge B of 50 mg in a volume of 50 mL of Cr VI at concentrations ranging from 0 to 370 mg L -1 . The beakers are kept at a constant temperature of 25°C and 35°C and are magnetically stirred for 24 hours. The sludge B has a grain size ranging from 1 to 500 μm (Figure 4). The temperature of 35°C was chosen because it is one of the temperatures encountered in metal finishing for hot baths. Figure 4 shows that the temperature has limited effects on the sorption yield. In some studies [START_REF] Hsu | The removal and recovery of Cr (VI) by Li/Al layered double hydroxide (LDH)[END_REF], it was highlighted that temperature only increased sorption kinetics and not sorption yield. The results obtained are normal because in our case, sorption kinetics are quick. Finally, temperatures, which are tested, (close to temperatures used in metal finishing) justify that it would not be necessary to increase the reactor temperature for an industrial application in order to improve the process.

Grain size

The influence of sludge grain size is tested by means of a sorption experiment on 50 mg of sludge B in 50mL of solution of Cr VI at different concentrations (ranging from 0 to 370 mg L - 1 ). The beakers are kept at constant temperature of 25°C and magnetically stirred for 24 hours. The grain sizes are detailed below: a preparation was made with a mortar without sifting, another with a grain size between 1 and 500 µm and finally one with a size between 1 and 75 µm (Figure 5). Figure 5 points out that sifting has a significant effect on the sorption yield whatever the concentration of Cr VI maybe; this is especially true for the most concentrated of solutions. The difference in sorption yield between non-sifted and sifted sludge reaches about 20% for concentrated solutions. However, the sifting effect has a limit. Indeed, the range of a maximal grain size of 500 µm through 75 µm has virtually no impact on the sorption yield. So, it would be interesting to determine the optimal grain size is in order to obtain the best sorption yield because the greater the limit of grain size, the easier the sifting. This leads to a significant gain of time in the sludge preparation.

Moisture

The protocol used is described in 3.1. The masses of sludge A used (no dry i.e. sludge at the exit of filter press with 35% average of dry content and dry i.e. sludge dried during for seven days under extractor hood) without sifting range from 50 mg to 3125 mg and the concentration of Cr VI is 30 mg L -1 . The sorption time is 30 minutes and takes place at room temperature under magnetic stirring (Figure 6). Figure 6 shows that, for a given mass (2500 mg), the moisture does not influence any more the sorption yields. In fact, for a low mass of sludge, the bigger the effect of moisture, the higher the sludge mass. This result is normal because, for a low mass, the more significant the moisture, the more the yield of dry content (sorbent) is low. Therefore, this brings to a decrease of sorption yield. This effect is offset in the case of important masses because, as Figure 4 shows, there is a specific mass (or sorbent sites) beyond which no effect will be visible on the sorption yield.

Effects of various parameters

The whole parameters described throughout this article are presented in Figure 7. They concern only the sludge B. Figure 7 focuses on the fact that whatever the concentration of Cr VI , the temperature, and the size of grains of sludge, for the sludge mass "m" (1500 mg here for sludge B), sorption yields are similar. This result is very important because it shows that the sludge mass (and then sorbent sites) is THE only real parameter which influences the yield. Besides, the results of a suitable sorption yield (95%) are closely linked to the sludge mass in suspension.

Conclusion

Throughout this study, it was point out that the polymetallic hydroxide sludge, from metal finishing treatment, have great sorption properties for the detoxification of an aqueous solution containing Cr VI . The sorption yield may reach 95% depending on the experimental parameters. First of all, it was highlighted that the crystallinity, temperature, concentration of pollutant, humidity and the grain size have a very limited influence on sorption compared to the influence of the sludge mass in suspension. Moreover, greater the added mass, greater the mitigation of the effect of the other parameters. The kinetic sorption is quick: it is possible to consider that after 20 minutes at least, the sorption yield reaches its' maximum. The minimal time is interesting because it avoids an eventual dissolution of the sludge.

An assumption was made to justify such sorption properties of the matrix (mixed metal hydroxide) about LDH presence in sludge. At the laboratory scale, during the co-precipitation of divalent or trivalent metal at constant pH and according to experimental conditions, Layered Double Hydroxides (LDH) could be synthesized. Their general chemical formula is M 2+ 1-x M 3+ x (OH) 2 X q- x/q .nH 2 O (at least one divalent and one trivalent metal). This is a compound family made of hydroxide sheets [M 2+

1-x M 3+ x (OH) 2 ] x+ where interlayered space X q- x/q .nH 2 O can be formed by different chemical species (Oxometallic anions (CrO 4 2-, Cr 2 O 7 2-, ...), non-metallic anions,...). The sorption property of LDH stems from their structure which promotes the trapping of various pollutants (transfer of the pollutant from the aqueous phase to a solid phase). More importantly, this property is well known to the scientist community given the large number of publications on the subject [START_REF] Alvarez-Ayuso | Purification of Chromium (VI) finishing wastewaters using calcined and uncalcined Mg-Al-CO3-hydrotalcite[END_REF][START_REF] Hsu | The removal and recovery of Cr(VI) by Li/Al layered double hydroxide (LDH)[END_REF][START_REF] Wang | Sorption and thermal desorption of Cr (VI) on Li/Al layered double hydroxide[END_REF][START_REF] Lochan | Sorption of Cr (VI) in layered double hydroxides[END_REF][START_REF] Terry | Characterization of Cr ion exchange with hydrotalcite[END_REF]. The industrial conditions of coprecipitation are one of the major arguments in favour of the following hypothesis : in the industrial sludge production, it is possible to obtain LDH phases [START_REF] Rives | Layered double hydroxides-present and future[END_REF] with the metal hydroxide mixture. In spite of the lack of validation, it will be planned to identify the presence of LDH and quantify it in the industrial sludge. Moreover, we will also study the best precipitation settings in industrial conditions to yield the higher LDH in the industrial sludge in order to maximize sorption capacity. The sorption mechanism (adsorption or absorption) is being studied.

Figure 1 :

 1 Figure 1 : Sorption kinetics of sludge A and B

Figure 2 :

 2 Figure 2 : Kinetics in function on sorbent mass

Figure 3 :Figure 3

 33 Figure 3 : kinetics according to [Cr VI ] to a sludge mass 500mg Figure 3 underlines that the concentration has a slight influence on the sorption yield. For concentrations of 16 and 28.8 mg.L -1 , the values of the yield are quite similar and about

Figure 4 :

 4 Figure 4 : Evolution of sorption yield in function of temperature

Figure 5 :

 5 Figure 5 : Evolution of the sorption yield depending on the grain sizes

evolution of adsorption yield to the sludge mass and the moisture yield of sludgeFigure 6 :

 6 Figure 6 : Evolution of sorption yield depending on the sludge nature

Figure 7 :

 7 Figure 7 : Evolution of sorption yield according to some parameters