Sophie Hennequin
email: hennequin@enim.fr

Laura Maria

Ramirez Restrepo
email: laura-maria.ramirez-restrepo@univ-lorraine.fr

A Fuzzy Hedging Point Policy for Sustainable Manufacturing System

Keywords: environmental impacts, economical costs, workforce welfare, hedging point policy, fuzzy logic I

This paper addresses a sustainable manufacturing system, i.e. a manufacturing system on which we try to minimize all economical costs while minimizing environmental impacts and improving workforce welfare. Indeed, on one hand, this system emits pollutants and greenhouse gases during its manufacturing process. On the other hand, the social dimension i.e. the human workforce and its working conditions are also discussed and considered because of the direct and indirect costs associated. A hedging point policy is then applied and adapted to this manufacturing system by using fuzzy logic to ensure sustainability. So, a fuzzy hedging point is defined to take into account pollutant emissions and human factors in order to minimize global costs.

INTRODUCTION

Since the sustainable development has been defined [START_REF]Report of the World Commission on Environment and Development: Our Common Future also called Bruntland Report[END_REF], many countries have adopted strategies to strongly encourage individuals and companies to reduce their environmental impacts [START_REF] Brown | The Many Shades of Green[END_REF] and act ethically. Our work registers in this context and propose to define a production control policy which not only minimizes all production costs but also considers the pollution and greenhouse gases emitted and also the social dimension to obtain sustainability. Indeed, many studies and methodologies have been proposed to include environmental impacts (especially greenhouse gases emissions). We can cite for examples [START_REF] Zhang | Environmentally conscious design and manufacturing: a state-of-art survey[END_REF] and [START_REF] Mckinnon | Green logistics -2 nd Edition -Improving the environmental sustainability of logistics[END_REF] which give good surveys on the subject. However, the social dimension is rarely considered [START_REF] Székely | Responsible Leadership and Corporate Social Responsibility: Metrics for Sustainable Performance[END_REF]. In this paper, we try to consider at the same time the three pillars of sustainable development to define the production control policy. However, we limit our study to the social dimension and do not include the societal responsibility of the enterprise.

The representation of human factors in industry has been the subject of numerous studies conducted by different specialists (designers, engineers, ergonomists, occupational psychologists…) whose goal is generally to find the balance between operator's capacities, whether physical, cognitive or moral, and the requirements of certain tasks [START_REF] Elkosantini | Integration of human behavioral aspects in a dynamic model for a manufacturing system[END_REF]. Thus, the behavior of operator is defined by different factors such as external conditions, that is to say, the requirements and constraints imposed, or the nature of the task at hand, or line managers and also internal conditions (the characteristics of the individual) such as physiological, psychological and psychosocial factors such as motivation, satisfaction, confidence, fatigue, stress, and conflict [START_REF] Elkosantini | Toward a new generic behavior model for human centered system simulation[END_REF]. More and more organizations are addressing the issue of psychological factor at work, because of the direct and indirect costs associated with this issue as higher absenteeism, lost productivity, the behavior of withdrawal, the tensions, ergonomics and musculoskeletal disorders etc. Our objective in this paper is not to clearly identify the human behavior and corresponding costs but only considering this social dimension in the definition of the proposed production policy.

The proposed production control policy is based on the well-known hedging point policy defined for the first time in [START_REF] Kimemia | an algorithm for the computer control of a flexible manufacturing system[END_REF]. In this policy, a nonnegative production surplus of parttypes should be maintained at times of excess capacity in order to hedge against future capacity shortages caused by machine failures. For the case of failure-prone manufacturing system, this policy has been shown to be optimal (see for examples [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF], [START_REF] Song | optimal hedging point control for a failure-prone manufacturing system[END_REF] and [START_REF] Mourani | optimal discrete-flow of a singlestage failure-prone manufacturing system[END_REF]). The hedging point policy has been adapted in [START_REF] Ben-Salem A | An environmental hedging point policy to control production rate and emissions in unreliable manufacturing systems[END_REF] to take into account the pollutant emissions during the production.

In this paper, based on the results obtained in [START_REF] Ben-Salem A | An environmental hedging point policy to control production rate and emissions in unreliable manufacturing systems[END_REF], we redefine the hedging point policy to include at the same time the three pillars of sustainable development. For this, we propose to use fuzzy logic which is a very interesting tool for representing human factors [START_REF] Hennequin | Optimization of imperfect maintenance based on fuzzy logic[END_REF]. Indeed, fuzzy logic defined by L. Zadeh in 1965 is well suited and powerful for vague or qualitative notions [START_REF] Novak | Are fuzzy sets a reasonable tool for modeling vague phenomena?[END_REF]. Then, we propose a fuzzy hedging point policy control based on a Mamdani model [START_REF] Zimmermann | Fuzzy Set. Theory and Its Applications[END_REF] and empirical data in order to easily change and adapt our fuzzy system with actual and real data. This paper is organized as follows. Section 2 addresses the considered problem and presents the sustainable manufacturing system. In section 3, our fuzzy hedging point system is detailed and explained. Numerical results are given in section 4 and a discussion is proposed. We conclude the paper and give some perspectives to our work in section 5.

II. SUSTAINABLE MANUFACTURING SYSTEM

To simplify our study and the presentation, we consider a single-stage single-product manufacturing system composed by a machine M, a buffer B which allows limiting losses and a constant demand by time unit denoted by D (Fig. 1).

The machine can fail at any time, due to external elements such as temperature, humidity, dusts, etc., even if it is not working. When the machine fails it can not work at all, and when it is up, it can work with a production speed u(t) such as U  u(t)  0 where U is the maximal production speed of the machine. The machine state is given by:

    . down is machine the if 0 up, is machine the if 1 ) (t  (1)
All random variables are exponentially distributed which means that when the machine fails, the production should restart to the beginning (we have a Markov process). Mean time between failures (MTBF) is equal to 1/f and mean time to repair (MTTR) is equal to 1/r. The failure/repair process is an independent random process. We denote by x(t) the inventory level given by:

   t Dt ds s u t x 0 ) ( ) ( (2)
x(t) could be negative or positive which respectively represents a backlog or a holding cost. The inventory cost g(x) is then given by:

            0 . 0 . ) ( x if x c x if x c x g ( 3
)
with x+ = max (x ,0), x -= max (-x , 0) and where c + , c -denote the holding and backlog cost respectively (c +  0, c - 0).

We define an environmental cost which corresponds on one hand to the use of fossil energy (then a carbon tax is applied, for example in France, in 2015, the carbon tax is equal to 7 euros by CO2 emitted ton) and on the other hand to the induced pollution (toxic substances, pollutants and waste). This environmental cost is denoted by e(x(t)) and is given by:

e e b t y a t y e   ) ( . )) ( ( (4)
where a e and b e represents respectively fixed and variable parameters of the environmental cost and y(t) represents the total number of produced parts at time t. y(t) is equal to:

  t ds s u t y 0 ) ( ) ( (5)
and could be expressed in function of x(t) since:

Dt t x t y   ) ( ) ( (6)
To take into account the social pillar of sustainable development, we also defined a hardship cost which corresponds to the machine ergonomics (i.e. cost due to musculoskeletal disorders) and to incidents and/or accidents in the manufacturing system (in this case an estimated cost). This hardship cost is denoted by ha(y(t)) and is given by:

ha ha b t y a t y ha   ) ( . )) ( ( (7)
where a ha and b ha represents respectively fixed and variable parameters of the hardship cost. Similarly to the environmental cost, the hardship cost could be expressed in function of x(t).

We can note that generally the inventory cost is convex and nonnegative in function of x(t), and, the other costs increase in function of x(t).

The expected discounted cost J(x), which depends on the inventory level is given by:

                 T s T ds s x ha s x e s x g e E x J 0 )) ( ( )) ( ( )) ( ( lim ) (  (8)
with  > 0,  corresponds to the discount factor.

Hypothesis 1:

x x J   ) ( exists and is continuous in x, indeed J(x) is continuously differentiable in x.
The proof for this hypothesis is similar to the proof given in [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF]. The cost criterion is convex and nonnegative in function of x(t) (see Figure 2). 3), ( 4), ( 6) and ( 7), the discounting cost could be expressed by:

x h h sd 0 J(h) J 0 (x) J(x)
                 T s T ds C s B s x A e E x J 0 . )) ( ( lim ) (  (9)
with A(x(t)) a convex function of x(t) given by the sum of g(x(t)) and a e .x(t) and a ha .x(t), B is equal to (a e +a ha ). The chosen control policy for the machine production speed is of hedging point type. It indicates a nonnegative products surplus should be maintained in order to satisfy the demand when failures occur.

Hypothesis 3: the proposed control policy is supposed to be determinist and stationary. It is defined by: [START_REF] Song | optimal hedging point control for a failure-prone manufacturing system[END_REF] where h sd corresponds to the hedging point for our sustainable manufacturing system. This control policy satisfies the following Hamilton-Jacobi-Bellman equations [START_REF] Oke | Optimal control approach to robust control of linear and non-linear systems[END_REF]:

          else U t and h t x if D h t x or t if t u sd sd 1 ) ( ) ( ) ( 0 ) ( 0 ) (  
0 ) ( ) ( ) ( ) ( ) ( 0 1 0        x rJ x J r x x J D x A  (11)  0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( inf 1 0 1 1 1                        U x J x J f x x J x u x x J D x J x A  (12)
with J 0 (x) corresponds to the fact the machine is down and J 1 (x) corresponds to the fact the machine is up.

From hypothesis 3 and equations ( 11) and ( 12), the hedging point policy given by equation ( 10) is optimal [START_REF] Oke | Optimal control approach to robust control of linear and non-linear systems[END_REF].

We denote by h the exact value of the hedging point which corresponds to the classical problem treated in [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF]. h is the optimal value when the environmental and social costs are not taken into account.

Theorem 1:

The optimal hedging point for our proposed sustainable manufacturing )] ( [ min arg

x J h x sd  is given such that h sd < h.

Proof of theorem 1:

 First, we will prove by contradiction that h is different from

h sd . Suppose )] ( [ n argmi x J h x 
. Then, it implies the cost function studied by Akella and Kumar is the same as ours which is not the case. In [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF], only the inventory cost is considered. Furthermore, the solution of the HJB equations is unique so )]

( [ arg x J n mi h x  =h sd .
 Then, compare the following systems with the same initial state (i.e. the same initial conditions and common parameters): i) the first one denoted by M h corresponds to the manufacturing system defined in [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF] and ii) the second one denoted by M hsd corresponds to our proposed sustainable manufacturing system. From the convexity and precedent remark we have h sd < h [START_REF] Noor | Generalized convexity and integral inequalities[END_REF].

Q.E.D.

From these results, we propose in this paper to estimate the value of the optimal hedging point for our proposed sustainable manufacturing system with the help of fuzzy logic. Indeed, it is difficult to calculate the exact value of this hedging point and generally, authors estimate it by simulations.

III. FUZZY HEDGING POINT

To achieve a sustainable manufacturing system, it is firstly important to identify and to quantify and/or qualify all impact factors for each pillar (economics, environmental and social) and then adopt sustainable development strategies.

In this paper, the objective is not to clearly quantify all these impact factors but only take them into account in the definition of the control policy. To achieve this, we propose to use fuzzy logic which is a very efficient tool when qualitative variables / notions are used without the need of employing precise mathematical analysis and functions. In our study, we define a simple Mamdani fuzzy IF-THEN rules system [START_REF] Zimmermann | Fuzzy Set. Theory and Its Applications[END_REF] which will allow to simply modify it if needed or in function of real data.

Our proposed fuzzy model is composed by two input variables and one output variable. The output variable corresponds to the fuzzy hedging point. It is denoted by "h fuzzy ". This fuzzy output will be given as a percentage of h, the value calculated in [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF]. "h fuzzy " is an estimation of the exact optimal value h sd .

We define the inputs as: "Environmental pillar" which corresponds to a percentage of the total pollution and greenhouse gases emissions, and, "Social pillar" which is based on the work done in [START_REF] Elkosantini | Toward a new generic behavior model for human centered system simulation[END_REF] and corresponds to the operator's stresses and given by a percentage. We do not consider the third pillar which corresponds to economical aspects because it is already taken into account in the calculus of the optimal hedging point value for a classical manufacturing system, denoted in this paper by h.

For each fuzzy input variable, we choose to define empirically five fuzzy sets based on triangular and trapezoidal functions, denoted by "Very Low", "Low", "Medium", "High" and "Very High" see figures 3 and 4. The output variable is defined by also five fuzzy sets: "Very Small", "Small", "Mean", "Important" and "Very Important", see figure 5. All these fuzzy variables (inputs and output) are based on percentages, so the corresponding universes of discourse are given by: U U = {U|0 < K < 100%} where U represents the fuzzy variable. The choice of the membership functions ensures to clearly describe all the universes of discourse [START_REF] Zimmermann | Fuzzy Set. Theory and Its Applications[END_REF].

In this paper, the classical Mamdani's method ("Min-Max" method) is used as the inference scheme. In general, it permits to obtain good results without the need of complicated calculations, and then it is widely adopted.

For the defuzzification, which consists in the calculation of a crisp output from the obtained fuzzy output set, we choose the centroid method [START_REF] Zimmermann | Fuzzy Set. Theory and Its Applications[END_REF] because of its simple definition, relative little PCU time needs and the good results obtaining.

We propose, in this paper, empirical fuzzy rules based on personal assessment without given any weights to input variables or rules. This can of course easily be changed depending on the considered area in question. Our proposed fuzzy rules are based on the fact more important are the environmental and social costs, lesser the hedging point value is. Indeed, in this case, we need to produce lesser parts in order to reduce the total cost. Our fuzzy system is composed by 25 rules.

In what follows, we give two examples of rules for our proposed fuzzy system, denoted by R i and R i+1 :

R i : If "Environmental pillar" is "Very Low"
and "Social pillar" is "Very Low" Then "h fuzzy " is "Very Important", R i+1 : If "Environmental pillar" is "Very High" and "Social pillar" is "Very High" Then "h fuzzy " is "Very Important".

We give in figure 6 the obtained surface for the output variable which is an approximation of the sustainable hedging point value calculated by fuzzy logic. These obtained results could be easily changed according to real data or different problems.

In what follows, we propose numerical results based on our fuzzy hedging point

IV. NUMERICAL RESULTS

We implement our fuzzy system in Matlab to simulate our model. The obtained fuzzy hedging point value is used in a discrete event system based algorithm in order to calculate the cost criterion.

The simulation parameters used are given in Table I. We first compare our results with the ones obtained in [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF]. They are given in Table II. Table II provides simulation results for the hedging point and the optimal cost for both systems. The obtained results are calculated from a discrete event based algorithm. Confidence intervals at 95 percent are also provided and numbers in parenthesis correspond to the 95% confidence half-width. These results highlight our theoretical results (see section II).

We also estimate the optimal hedging point value by using fuzzy logic and results given in [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF]. The obtained results from the Fuzzy Toolbox in Matlab are then implemented in a Simulink program and obtained results are given in Figure 7, with the cost function of the classical manufacturing system [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF], denoted 'JMH' given by a blue line, and the cost function of our proposed manufacturing system, denoted 'J', given by a red line. Figure 8 corresponds to a zoom of figure 7 to show the optimal value estimated by simulation.

We can remark that the obtained value of our proposed fuzzy hedging point is somehow different from the numerical results obtained with a discrete event based algorithm. However, our proposed solution is very interesting in the sense that it allows simply different kinds of adaptation such as the fact to consider all pollutants based for example on life cycle assessment or obtained results even if all the impacting elements are not quantified.

From figure 9, we can notice that the higher the discount factor, the smaller the value of the hedging point for both cost functions, with in blue line the hedging point calculated for the classical manufacturing system [START_REF] Akella | optimal control of production rate in a failure prone manufacturing system[END_REF], denoted 'h', and in red line the hedging point calculated for our proposed manufacturing system, denoted 'hsd'. V. CONCLUSION In this paper, we propose a fuzzy system to estimate the hedging point value in case of sustainable manufacturing system. By sustainable manufacturing system, we try to act to reduce the global economic costs while minimizing environmental impacts during the manufacture of parts and including social dimension, especially operators involved in manufacturing parts. Our fuzzy system is defined with empirical data in order to be simply adapted to real and industrial data or by experts.

In the future, we have to clearly identify and quantify for an industrial case study all the elements for each sustainable development pillar. Then, it is necessary to take them into account in our fuzzy system and realize a sensibility analysis to underscore key parameters that influence the results and allow managers to decide and act in case of problems. We also want to propose other control policies, such as echelon base stock, CONWIP, etc., also based on fuzzy logic. Furthermore, periodic upkeeps could be implemented to the machine to limit pollution and to improve quality and human resources strategies could be carried out to improve the operator's welfare.

Fig. 1 .

 1 Fig. 1. Manufacturing system

Fig. 2 .

 2 Fig. 2. Discounted cost function Hypothesis 2: the constant demand D is independent of x(t) for each t in [0;+[. Consequently, from equations (3), (4), (6) and (7), the discounting cost could be expressed by:

  D and C is equal to b e + b ha . B D M

Fig. 3 .

 3 Fig. 3. "Environmental pillar" input variable.

Fig. 4 .Fig. 5 .

 45 Fig. 4. "Social pillar" input variable.

Fig. 6 .

 6 Fig. 6. Surface results of "hfuzzy".

Fig. 7 .

 7 Fig. 7. Cost function calculated for the case of classical and sustainable manufacturing systems.

Fig. 8 .

 8 Fig. 8. Zoom of the cost function calculated for the case of classical and sustainable manufacturing systems.

Fig. 9 .

 9 Fig. 9. Cost function versus discount factor for both cases.

TABLE I

 I

	.	SIMULATION PARAMETERS
	Parameter	Value
	Repair rate (r)	20
	Failure rate (f)	100
	Demand rate D	3 parts by time unit
	Discount factor 	0.6
	Holding cost c +	5 monetary unit by part
	Backlog cost c -	250 monetary unit by part
	Maximal production speed U	4 parts by time unit
	Variable environmental cost ae	1 monetary unit by part
	Fixed environmental cost be	0.5 monetary unit
	Variable social cost aha	25 monetary unit by part
	Fixed social cost bha	2 monetary unit

TABLE II

 II

		.	COMPARISON
	Parameter	Classical manufacturing	Sustainable manufacturing system
	Optimal hedging point value	h = 530.605 parts (3.055)	h sd = 486.729 parts (6.613)
		J Mh = 2677.807	J=2858.419 monetary
	Optimal cost	monetary units	units
		(68.850)	(27.2245)

ACKNOWLEDGMENT

The developed work is performed with the financial support of the Regional Council of Lorraine.


Edward G Belaga
email: belaga@math.u-strasbg.fr

Effective Polynomial Upper Bounds to Perigees and Numbers of (3x + d)-Cycles of a Given Oddlength

Keywords: 1991 Mathematical Subject Classification : Primary, 11K31, 11K38, 11K55; Secondary, 11B85 3x+1 and 3x+d functions, 3x+1 and 3x+d conjectures, divergent trajectory, cycle, perigee, length, oddlength, odd frame, Collatz signature of a cycle

For any positive odd integer d not divisible by 3, the arith-

otherwise generates on the set N of natural numbers a dynamical system D d . The 3x + d hypothesis, generalizing the well-known 3x + 1 conjecture, asserts that D d has a finite number of cycles and no divergent trajectories. We study here the cyclic structure of the system D d , and prove in particular an effective and sharp polynomial upper bound to the number of cycles in D d with a given number of odd members.

Introduction.

Let d be a positive odd integer not divisible by 3, and let T d be the function defined on the set of positive integers, as follows : T j d (m) = m (note that > 1, since the mapping T d (1:1) has no fixed points). The minimal member of a T d -cycle C is odd, and is called its perigee, n 0 = prg(C). Thus, the number k of odd members of a T d -cycle, called here its oddlength, is a positive integer, k ≥ 1. The length and oddlength of a cycle are related by the inequality ≥ k • log 2 3 [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x + d Context[END_REF]] (see Theorem 3.2(1) below). Note also that no member of a T d -trajectory (1:2), excluding possibly the first one, is divisible by 3, and thus, all odd members of a T d -cycle belong to D.

∀m ∈ N T d (m) = 3m+d 2 , if m is odd ,
It has been conjectured that the dynamical system D d = N, T d has no divergent T d -trajectories (1:2), and that the number ς(d) of cyclic T d -trajectories is finite [Lagarias 1990], [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x + d Context[END_REF]]. In the particular case d = 1, the well-known 3x + 1 conjecture [Lagarias 1985], [START_REF] Günther | The Dynamical System Generated by the 3n+1 Function, LNM 1681[END_REF]] is even more specific : any trajectory τ 1 (m) enters ultimately the (only

) 3x + 1 cycle {1 → 2 → 1}.
The present paper is concerned with the cyclic part of the above 3x + d conjecture, and more generally, with quantitative (and when available, numerical) characteristics of the cyclic structure of systems D d . Let C(d) and C k (d) be the sets of all T d -cycles and, respectively, of all such cycles with k odd members, or, in our terminology, of the oddlength k ≥ 1.

Technically, our main result is the following general upper bound to the perigee of a T d -cycle of the length and oddlength k :

∀d ∈ D ∀C ∈ C(d) , length(C) = , oddlength(C) = k, =⇒ n 0 = prg(C) ≤ d 2 k -3 .
(1 : 3)

The inequality (1:3) has three important implications. The first one is an upper bound to the ratio of the length of a T d -cycle to its oddlength, which, together with the well-known lower bound (2:5), Theorem 2.1(1), confines this ratio to the interval :

∀d ∈ D ∀C ∈ C(d) , log 2 3 ≤ ρ d (C) = length(C) oddlength(C) ≤ log 2 (d + 3) .
(1 : 4)

Second, the inequality (1:3) implies the following general and uniform upper bound to perigees of T d -cycles of the oddlength k ≥ 1 :

∀d ∈ D ∀k ∈ N ∀C ∈ C k (d) n 0 = prg(C) ≤ U d,k = d 2 k•log 2 3 k -3 .
(1 : 5)

The bound (1:5) has an effective polynomial numerical equivalent (see the estimate (1:8) below). It is also sharp in the following natural sense (Theorem 3.2, (3:11(1))) : the average value of an odd member of a T d -cycle of the oddlength k ≥ 1 is bigger than U d,k . Thus, for example, the T 5 -cycle C = {23 → 37 → 58 → 29 → 46} has 3 odd members, n 0 = prg(C) = 23 < U 5,3 ≈ 28.6038 < 29 < 37.

Third, since no two T d -cycles have a common member, any such cycle is fully determined by its perigee. Thus, the upper bound (1:5) not only implies that the set C k (d) is finite, but supplies us with an effective general upper bound to the number ς k

(d) = #C k (d) of T d -cycles of the oddlength k ≥ 1 : ∀d ∈ D ∀k ∈ N , ς k (d) ≤ 1 3 • U d,k = 1 3 • d 2 k•log 2 3 k -3
(1 : 6) (the factor 1 3 is due to the aforementioned inclusion n 0 ∈ D). Any numerical evaluation of the expression U d,k depends on the state of our knowledge of effective lower bounds to diophantine approximations of linear combinations of logarithms log 2 and log 3. The best known at present lower bound belongs to [Rhin 1987], whose techniques lacks the generality of Baker's method (cf. [Baker, Wüstholz 1993] and the references there), but is more effective in the evaluation of linear forms of a few specific logarithms, including our case :

∀r, s ∈ Z , t = max(|r|, |s|) , |r + s log 2 + log 3| > t -13.3 , (1 : 7)
The inequality (1:7) implies the following effective polynomial upper bound to U d,k :

∀d ∈ D ∀k ∈ N , U d,k ≤ 102 • d • k 14.3 . (1 : 8)
Historical Remarks. The present author is not aware of any previous effective (and in any sense sharp) upper bound to the minimal odd member of a T d -cycle. The following general exponential upper bound to the number ς k (d) of T d -cycles of the oddlength k ≥ 1 has been actually proved in [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x + d Context[END_REF]] (even if not explicitly articulated) and refined in [START_REF] Belaga | Cyclic Structure of Dynamical Systems Associated with 3x + d Extensions of Collatz Problem[END_REF] :

∀d ∈ D ∀k ∈ N , ς k (d) < d • 3 2 k • 2 k 2 k -1 , k = k • log 2 3 -k • log 2 3 . (1 : 9)
The bound (1:9) has been derived from an identical upper bound to the maximal odd member of a cycle, the corresponding numerical upper bound being based on the estimate of [Baker, Wüstholz 1993] :

∀d ∈ D ∀k ∈ N , ς k (d) < d • k C • 3 2 k ,
(1 : 10) with an effective but enormous constant C. Refining the proof of the inequality (1:9) and applying the estimate of [Rhin 1987] to (1:9), [START_REF] Belaga | Cyclic Structure of Dynamical Systems Associated with 3x + d Extensions of Collatz Problem[END_REF]] have proved a better, but still exponential effective upper bound

∀d ∈ D ∀k ∈ N , ς k (d) ≤ 204 • d • 3 2 k • k 13.3 .
(1 : 11)

Comments and Future Prospects.

(1) The upper bound (1:4) to the ratio ρ d (C) implies in the 3x + 1 case that the length of a cycle with k odd members does not exceed 2k. Note that the only known at present 3x + 1 cycle {1 → 2 → 1} has one odd member, is of the length two, and has ratio two. More generally, the series of T d -cycles of the length ∈ N, d = 2 -3, n 0 = 1 and of the oddlength k = 1 demonstrates that the upper bound (1:4) is sharp.

(2) The bounds (1:5) and, especially, (1:6) can be apparently improved. In fact, the experimental discovery of 843 T 14303 -cycles of the oddlength 17, with perigees varying from 385057 to 1391321 < U 14303,17 = 2099280, suggests that the bound (1:5) is apparently sharp up to a one-digit constant, whereas the bound (1:6), ς 17 (14303) = 843 < 1 3 U 14303,17 = 699760, far from being sharp, is at least realistic : for some d, k, the dynamical system D d has "many" cycles of the oddlength k.

(3) At present, the bounds (1:3,4,6) look useless, or at least insufficient, for an eventual proof of the cyclic part of the 3x + d conjecture, i. e., of the finiteness of the number ς(d) of T d -cycles. At its best, the straightforward application of these bounds yields the trivial infinite upper bound :

∀d ∈ D ς(d) = k≥1 ς k (d) ≤ 102 • d • k≥1 k 14.3 = ∞ .
(1 : 12)

However, the obstacle of the infinite summation in (1:12) could be possibly circumvented by a refinement of the above scheme, to fit the purpose of yielding directly an absolute (i. e., not depending on k) upper bound to the number ς(d).

Acknowledgments. The author is grateful to Professor Maurice Mignotte for the permission to use here the aforementioned experimental results of our common project [START_REF] Belaga | Cyclic Structure of Dynamical Systems Associated with 3x + d Extensions of Collatz Problem[END_REF]]. We remind the reader that, according to (1:1), the minimal member, or perigee of a T d -cycle, n 0 = prg(C), is odd, and that the total number k ≥ 1 of odd members of a cycle is called its oddlength. Moreover, if n is an odd member of a cycle, then n ∈ D (1:2), since no number divisible by 3 can belong to a cycle.

Exponential Diophantine

Note that if m = m is a member of a T d -cycle C = τ d (m), or in other words, if C meets m , one should view C = C(m , d) as just another name for the same cycle C = C(m, d). Since a T d -cycle is fully characterized by its minimal member, the following notation can be accepted as the canonical one :

C = C(n 0 , d) = C[n 0 , d] = τ d (n 0 ), n 0 = prg(C) .
(2 : 1)

In this case, we also say that C starts at n 0 .

For any positive integer m ∈ N, let odd (m) be the number obtained by factoring out of m the highest possible power of 2, say 2 j , and let ν 2 (m) = j. Thus odd (m) is odd and

m = odd (m) • 2 ν 2 (m) . Define S : D × D → D ; ∀n, d ∈ D . S d (n) = odd (3n + d) .
(2 : 2)

The function S d speeds up the action of T d (1:1), skipping even members of T d -trajectories. In particular, m = 1 becomes the fixed point of the function S 1 = odd (3n + 1), S 1 (1) = 1, corresponding to the (according to the 3x + 1 conjecture, only)

T -cycle C(1, 1) = {1 → 2 → 1}.
We associate with any T d -cycle C = C[n 0 , d] its odd frame, F = Odd (C), the list of odd members of the cycle, in the order of their appearance in τ d (n 0 ) (1:2), as the T d -iterations of n 0 proceed. By definition, the frame is a S d -cycle (2:2) starting at n 0 , and its length is called the oddlength of the cycle C :

           ∀C = C[n 0 , d] , n 0 , d ∈ D , = length(C) = min i ∈ N m i = T i d (n 0 ) = n 0 ; k = oddlength(C) = min j ∈ N n j = S j d (n 0 ) = n 0 ; F = Odd (C) = n 0 , n 1 , . . . , n k-1 ∈ D k .
(2 : 3)

The even members of the T d -cycle C = C[n 0 , d] can be recovered from its frame with the help of the cycle Collatz signature P = θ(C), the vector of exponents of 2 factoring out from the values of the function T d (1:1) at odd members of C, as follows :

                           ∀C = C[n 0 , d] , n 0 , d ∈ D , F = Odd (C) = n 0 , n 1 , . . . , n k-1 ; p 1 = ν 2 T d (n 0 ) , p 2 = ν 2 T d (n 1 ) , . . . , p k = ν 2 T d (n k-1 ) ; P = θ(C) = p 1 , p 2 , . . . , p k ∈ N k ; = length(C) = |P| = p 1 + p 2 + . . . + p k ; ∀j ∈ [1, k -1] ,
m p 1 +...+p j = n j ;

p j > 1 =⇒ ∀i ∈ [1, p j -1] , m p 1 +...+p j -i = 2 i • n j .
(2 : 4) Moreover, the Collatz signature

P = θ(C) of a cycle C = C[n 0 , d], n 0 , d ∈ D, com- pletely characterizes it :
Theorem 2.1. [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x + d Context[END_REF]] (1) The Collatz signature P = θ(C) satisfies the inequality :

= |P| = p 1 + . . . + p k ≥ k • log 2 3 .
(2 : 5)

(2) Define the exponential diophantine function A = a k : N k -→ N, as follows :

         ∀k ≥ 1 ∀P = p 1 , . . . , p k ∈ N k , A = a k (P) = 1, if k = 1 ; 3 k-1 + 2 p 1 • 3 k-2 + . . . + 2 p 1 +...+p k-2 • 3 + 2 p 1 +...+p k-1 , otherwise .
(2 : 6) Let σ = σ k be the circular (counterclockwise) permutation on k-tuples :

∀P = p 1 , . . . , p k ∈ N k . σ(P) = σ k (P) = p 2 , . . . , p k , p 1 .
(2 : 7) d], n 0 , d ∈ D of the length , oddlength k ≥ 1, and with the frame F = n 0 , n 1 , . . . , n k-1 , then

If now P = θ(C) is the Collatz signature of a cycle C = C[n 0 ,
           (1) B = b k (P) = B k, = 2 -3 k > 0 ; (cf. (2 : 5)) (2) n 0 = d • A B , A = a k (P) ; (cf. (2 : 6)) (3) ∀j ∈ [1, k -1] , n j = d • a k σ j (P) B . (cf. (2 : 7))            (2 : 8)
3. Upper Bound to the Number of 3x + d Cycles of a Given Oddlength.

According to the formulae (2:8(2,3)), the odd members of a T d -cycle of the oddlength k satisfy the inequality

∀j ∈ [1, k -1] , n j ≤ W d,k = d• sup P∈N k |P|≥k log 2 3 a k (P) 2 |P| -3 k = d• sup ≥k log 2 3 max P∈D k |P|= a k (P) 2 -3 k . (3 : 1)
Simple calculations show that (cf. (1:9) above)

W d,k = d • 3 2 k • 2 k 2 k -1 , k = k • log 2 3 -k • log 2 3 . (3 : 2)
An analogue of this general upper bound and its more explicit version (1:11), have been, respectively, proved in [START_REF] Belaga | Embedding the 3x+1 Conjecture in a 3x + d Context[END_REF]] and refined in [START_REF] Belaga | Cyclic Structure of Dynamical Systems Associated with 3x + d Extensions of Collatz Problem[END_REF]].

We will be able to improve these bounds thanks, first, to a more careful analysis of the formulae (2:8), and then, to a remarkable inequality (3:5) proved below, Theorem 3.1. Namely, instead of evaluating from above all members of a T d -cycle of the oddlength k, we evaluate here its minimal member

n 0 = prg(C). Since the different T d -cycles have different perigees n 0 ∈ D, an upper bound n 0 ≤ V d,k would imply the bound ς k (d) ≤ 1 3 • V d,k to the number of T d -cycles of the oddlength k. More formally, if P = θ(C) (2:4) is the Collatz signature of the cycle C = C[n 0 , d], n 0 , d ∈ D, of the length = |P| and oddlength k ≥ 1, then, according to (2:8), n 0 ≤ min{n 0 , n 1 , . . . , n k-1 } = d • min j∈[0,k-1] a k σ j (P) 2 -3 k . (3 : 3)
Define for any k-tuple of positive integers P its average P, the arithmetical mean of all its counterclockwise permutations. This k-tuple of positive (generally speaking, rational) numbers depends only on the dimension k and length = |P| of P :

∀k ≥ 1 ∀P = {p 1 , . . . , p k } ∈ N k , = |P| , P = 1 k j∈[0,k-1] σ j (P) = k , . . . , k . (3 : 4)
Extending the definition of the function a k (2:6) to k-tuples of positive reals, we will prove below (Theorem 3.2) the inequality

∀k ≥ 1 ∀P ∈ N k , ãk (P) = min j∈[0,k-1] a k σ j (P) ≤ a k (P) = 2 -3 k 2 k -3 . (3 : 5)
The inequalities (3:3) and (3:5) imply the general upper bound (1:5), depending only on d and k, to the minimal member n 0 = prg(C) of any T d -cycle of the oddlength k :

∀n, d ∈ D , C = C[n, d] =⇒ n ≤ U d,k = d 2 k -3 , (3 : 6)
and, finally, the upper bound (1:6).

Definition 3.1. (1) Let Λ be the set of pairs of positive integers (k, ) satisfying the inequality implied by (2:5),

Λ = (k, ) ∈ N 2 λ(k, ) = -k • log 2 3 ≥ 0 ; (3 : 7)
Extend the definition of the function A = a k (2:6) to k-tuples of positive reals from the (k -1)-dimensional tetrahedron T k, ,

∀(k, ) ∈ Λ , T k, = X ∈ R k |X| = x 1 + . . . + x k = ∧ ∀j ∈ [1, k] , x j ≥ 1 , (3 : 8) with k vertices V 1 , . . . S k , V 1 = { -k + 1, 1, . . . , 1}, V 2 = {1, -k + 1, . . . , 1}, . . . , V k = {1, 1, . . . , -k + 1} (3 : 9)
(2) The permutation σ (2:7) induces on T k, the rotation σ, with the center O of the tetrahedron being the only fixed point :

                 ∀X = x 1 , x 2 , . . . , x k-1 , x k ∈ T k, , σ(X) = x 2 , x 3 , . . . , x k , x 1 ; X = 1 k • j∈[1,k] σ j (X) = k , . . . , k = O , σ(O) = O ; a k (O) = k j=1 3 k-j • 2 k = 2 -3 k 2 k -3 .
(3 : 10) Theorem 3.2. For any k-tuple X from T k, , the inequalities hold : (3 : 11) with the equalities holding only in the case X = O.

∀X ∈ T k,            (1) a k (X) = 1 k • k-1 j=0 a k σ j (X) ≥ a k (O) = 2 -3 k 2 k -3 ; (2) ãk (X) = min j∈[0,k-1] a k σ j (X) ≤ a k (O) = 2 -3 k 2 k -3 ,
4.Proof of Theorem 3.2.

Note that, according to (2:6), if k = 1, then ≥ 2 and

∀X ∈ T 1, , X = X = O , a k (X) = ãk (X) = 2 -3 2 -3 = 1 . (4 : 1)
Thus, we can assume henceforth that k ≥ 2.

(1) The inequality (3:10(1)) is implied by the standard inequality

1 k (a + b + . . .) ≥ k √ a • b • . . ., as follows : ∀k ≥ 2 ∀(k, ) ∈ Λ ∀X ∈ T k, , 1 k 0≤j≤k-1 a k σ j (X) = 3 k-1 + 1≤j≤k-1 3 k-j-1 k 0≤r≤k-1 2 σ r (x 1 +...+x j ) ≥ 3 k-1 + 1≤j≤k-1 3 k-j-1 • 2 1 k 0≤r≤k-1 σ r (x 1 +...+x j ) = a k (O) = 2 -3 k 2 k -3 . cf. (3 : 10) (2) If X = O ∈ T k, , then (3:10(2)) becomes a trivial identity. Otherwise, O = X ∈ T k, (k ≥ 2, (k, ) ∈ Λ)
, and between the k k-tuples σ j (X), 0 ≤ j ≤ k -1, there exists at least two different ones :

∀k ≥ 2 ∀(k, ) ∈ Λ ∀X ∈ T k, \ {O} ∀j ∈ [0, k -1] , σ j (X) = X = O ; ∃j ∈ [1, k -1] , X = σ j (X) . (4 : 2)
Now the proof proceeds ad absurdum : the assumption a k σ j (X) > a k (O) for all j ∈ [0, k -1] would imply that a k (X) > a k (O) as well, -a contradiction, since X = O (3:10), (4:2).

The equation a k (X) = a k (O) induces a break up of the (k -1)-dimensional tetrahedron T k, (3:8) into three disjoint subsets : the closed (k -2)-dimensional submanifold T 0 = T 0 k, defined by this equation, and two (k -1)-dimensional and open in T k, submanifolds T + = T + k, , and T -= T - k, , defined by the inequalities a k (X) > a k (O) and a k (X) < a k (O), respectively :

∀k ≥ 2 ∀(k, ) ∈ Λ ,              T 0 = T 0 k, = X ∈ T k, a k (X) = a k (O) = 2 -3 2 k -3 ; T + = T + k, = X ∈ T k, a k (X) > a k (O) ; T -= T - k, = X ∈ T k, a k (X) < a k (O) .
(4 : 3)

We shall prove below the following properties of these three submanifolds : (A) T 0 is a smooth (in fact, analytical) submanifold.

(B) The submanifolds T 0 , T + , T -are connected and simply connected sets.

(C) The closed set T 0+ = T 0 ∪ T + is strictly convex : the convex closure P(S) of a finite set S of k-tuples from T 0+ belongs to T + , excluding, if necessary, the tuples from S belonging to T 0 .

The last property immediately implies the validity of the above argument ad absurdum.

To prove (A-C), one needs to look at the first and second partial derivaties of the function a k (X) :

                                         ∀k ≥ 2 ∀(k, ) ∈ Λ ∀X = x 1 , . . . , x k ∈ T k, , A k (X) = ∂a k ∂x 1 , . . . , ∂a k ∂x k-1 , ∂a k ∂x k (X) =
ln 2 • 2 x 1 a k-1 (x 2 , . . . , x k ), . . . , 2 x 1 +...+x k-1 a 1 (x k ), 0 ;

∂A k (X)

∂x i = ∂ 2 a k ∂x i ∂x j i,j∈[1,k] (X) ; ∀i, j ∈ [1, k] , r = max(i, j), 1 (ln 2) 2 • ∂ 2 a k ∂x i ∂x j
(X) = 2 x 1 +...+x r a k-r (x r+1 , . . . , x k ) .

(4 : 4)

The properties (A,B) of the submanifolds T 0 , T + , T -are immediately implied by the character of the first derivative A k (X). To prove the property (C), consider the second differential of the function a k , the quadratic form

                       d 2 a k (X) = i,j∈[1,k] ∂ 2 a k ∂x i ∂x j (X)dx i dx j = (ln 2) 2 r∈[1,k]
2 x 1 +...+x r a k-r (x r+1 , . . . , x k )

i,j∈ [1,r] dx i dx j = (ln 2) 2 r∈[1,k] 2 x 1 +...+x r a k-r (x r+1 , . . . , x k ) dx 1 + . . . + dx r 2 > 0 .

(4 : 5)

End of the Proof.

  the function T d generate (3x + d)-(or T d -) trajectories ∀d ∈ D = {1, 5, 7, 11, 13, . . .} ∀m ∈ N , τ d (m) = m, T d (m), T 2 d (m), . . . , (1 : 2) By definition, a trajectory τ d (m) is a cycle of the length , C = C(m, d) = τ d (m), length(C) = , if T d (m) = m and, for any j ∈ [1, -1],

  Formulae for 3x + d Cycles. Let, as above, C = C(m, d) = τ d (m) (1:2) be a T d -cycle of the length , length(C) = .