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Related work

 with some applications to an integrated production-distribution planning model under supply and demand uncertainties. The elastic p-robustness increases flexibility of the relative deviation constraint by adding some variables reflecting the constraint violation whose values are penalised in the objective function. For yet another measure of robustness under the discrete uncertainty representation, the so-called (b, w)-robustness, that does not focus only on the worst case and is able to better capture the robustness, we refer the reader to Roy (2010).

. The fuzzy intervals are regarded as possibility distributions, describing the sets of more or less plausible values of demands. Consequently, we obtain a joint possibility distribution, induced by these fuzzy intervals, on the set of scenarios U. Clearly, this approach is a generalisation of the interval uncertainty representation. In this context, the planning processes of MRPII were adapted in Fargier and Thierry (

 [START_REF] Tavakkoli-Moghaddam | A Fuzzy Aggregate Production Planning Model for Make-to-Stock Environments[END_REF][START_REF] Mula | The Effectiveness of a Fuzzy Mathematical Programming Approach for Supply Chain Production Planning with Fuzzy Demand[END_REF][START_REF] Guillaume | A Robust Lot Sizing Problem with Ill-known Demands[END_REF]. Interestingly, it is shown in [START_REF] Guillaume | A Robust Lot Sizing Problem with Ill-known Demands[END_REF] that there exists a link between the interval uncertainty with the minmax criterion and the fuzzy uncertainty representation with the possibility-based criteria. It turns out that choosing a plan in fuzzy-valued lot sizing problems with backorders is not harder than their interval-valued counterparts, since they are reduced to examining a small number of interval problems with the minmax criterion.

If some additional information about uncertain parameters is provided, then less conservative criteria can be used to choose a solution that enable to model various preferences with respect to a risk. For instance, information about importance of each scenario. In this case, such importance may be expressed by assigning some weights to scenarios in U and one can apply the Ordered Weighted Averaging aggregation operator (OWA for short) proposed by [START_REF] Yager | On Ordered Weighted Averaging Aggregation Operators in Multi-criteria Decision Making[END_REF] that utilises this information while computing a solution. It is worth pointing out that the OWA criterion generalises the classical criteria such as the maximum, minimum, average and Hurwicz. If probability distributions of parameters or a probability distribution in U are available, then the value at risk and the conditional value at risk criteria are worthy of consideration (see [START_REF] Pflug | Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk[END_REF][START_REF] Rockafellar | Optimization of Conditional Value-at-Risk[END_REF]. In contrast to the minmax and minmax regret criteria, which consider all scenarios, the above risk criteria take into account a subset of scenarios aggregated with respect to a certain confidence level. In consequence they are less conservative (seeAlem and Morabito 2013 for an application of the risk criteria in production planning under stochastic demands and setup times). A criterion that exploits information about both importance and probability of occurrence of each scenario is the Weighted Ordered Weighted Averaging aggregation operator proposed in [START_REF] Torra | The Weighted OWA Operator[END_REF] (see also [START_REF] Ogryczak | On Efficient WOWA Optimization for Decision Support under Risk[END_REF]. It is a generalization of the expected value and OWA.

Another way of hedging against uncertainty of parameters that does not fall into the aforementioned frameworks of uncertainty handling was given in [START_REF] Bertsimas | The Price of Robustness[END_REF]. It models uncertain parameters as symmetric random variables, with unknown distributions, centred at their nominal values. The main advantage of this framework is its computational tractability and probabilistic guaranties for constraint violation. Moreover, it flexibly controls the level of conservativeness of a resulting solution by assigning to each constraint a parameter, the so-called budget of uncertainty that is an upper bound on the total (scaled) variability of parameters in the constraint from their nominal values. The budgets of uncertainty allow the decision-maker to take his/her attitude towards a risk into account. It is worth pointing out that the approach is a generalisation of a conservative framework earlier proposed in [START_REF] Soyster | Convex Programming with Set-inclusive Constraints and Applications to Inexact Linear Programming[END_REF] which assumes that all parameters may realise at their worst-case values -(the case holds for large values of the budgets of uncertainty). The first direct application of the above-mentioned method to solve lot sizing problems with backorders under uncertain demand with nondecreasing uncertainty budgets in periods was given in [START_REF] Bertsimas | A Robust Optimization Approach to Inventory Theory[END_REF]. Along the same line as the previous application, the method was adapted to periodic inventory control and production planning with uncertain product returns and demand in [START_REF] Wei | Robust Optimal Policies of Production and Inventory with Uncertain Returns and Demand[END_REF], to lot sizing combined with cutting stock problems under uncertain cost and demand in [START_REF] Alem | Production Planning in Furniture Settings via Robust Optimization[END_REF] and to planning in agriculture and processing industry in [START_REF] Bohle | A Robust Optimization Approach to Wine Grape Harvesting Scheduling[END_REF], [START_REF] Alvarez | Application of Robust Optimization to the Sawmill Planning Problem[END_REF], [START_REF] Roccoa | Robust Optimisation Approach Applied to the Analysis of Production/Logistics and Crop Planning in the Tomato Processing Industry[END_REF]. It was also used in [START_REF] Bienstock | Computing Robust Basestock Levels[END_REF] to obtain a good approximation of optimal basestock policies under uncertain demand in the minmax setting. Furthermore, the above ideas are introduced in a continuous time fluid model for dynamic pricing and inventory control with demand uncertainty by Adida and Perakis (2006).

Our results

In this paper, we focus on the tactical level by examining the MPS and MRP planning processes. Namely, we deal with the CLSP (for the MPS process) and MLCLSP (for the MRP process), both with backordering, with uncertain cumulative demand under the interval uncertainty representation of scenario set U with the minmax criterion for choosing a robust production plan (see, e.g. [START_REF] Zangwill | A Backlogging Model and a Multi-Echelon Model of a Dynamic Economic Lot Size Production System -A Network Approach[END_REF][START_REF] Pochet | Production Planning by Mixed Integer Programming[END_REF][START_REF] Mula | Material Requirement Planning with Fuzzy Constraints and Fuzzy Coefficients[END_REF][START_REF] Albrecht | Supply Chain Coordination Mechanisms[END_REF] for models for their deterministic counterparts). Moreover, we assume that deterministic counterparts of the problems under consideration are polynomially solvable and we would like to preserve their computational tractability. Therefore we do not study production processes, for instance, with set-up costs/times, which become NP-hard problems even for many special cases (see, e.g. [START_REF] Florian | Deterministic Production Planning: Algorithms and Complexity[END_REF][START_REF] Chen | Analysis of Relaxations for the Multi-item Capacitated Lot-sizing Problem[END_REF] and with other production parameters under uncertainty, since it may lead to computationally harder problems.

To the best of our knowledge, such a model of uncertainty in cumulative demand has not been investigated in the literature on production planning so far. We distinguish two interval representations of set U of cumulative demand scenarios, namely for the MPS problems and for MRP ones. They are more realistic than the model of uncertainty in demand in periods (see Section 3 for a detailed explanation). Moreover, we show that they lead to the polynomial solvability of the planning problems under consideration in the minmax setting in contrast to the same problems with uncertainty in demand in periods for which there are few polynomial methods. Accordingly, in Section 4.1, we show that computing a worst-case scenario (the adversarial problem) can be done in O(T ) time for the MPS under L4L and POQ rules -note that its computational complexity in the case of uncertainty in demand in periods is still an open problem (see [START_REF] Guillaume | A Robust Lot Sizing Problem with Ill-known Demands[END_REF] and also other hard adversarial problems [START_REF] Bienstock | Computing Robust Basestock Levels[END_REF]). We provide a linear programme for the MPS under L4L and a strongly polynomial combinatorial algorithm for the MPS under POQ for determining an optimal robust production plan. Interestingly, under uncertainty on demand in periods there is only one iterative algorithm given in [START_REF] Guillaume | A Robust Lot Sizing Problem with Ill-known Demands[END_REF] based on Benders' decomposition. A situation is much more involved under the cumulative demand uncertainty for the MRP problems. In Section 4.2 we prove, by characterising optimal scenarios, that computing a worst-case scenario (the adversarial problem) can be done in O(P • T 3 ) time, where P is the number of products. This allows us to propose a linear programming model for determining an optimal robust production plan. In Section 5, we provide and experimental evaluation of the proposed methods, which show, among others, that the decrease in profit of optimal robust production plans is consistently linear when uncertainty increases.

Deterministic MPS/MRP problems

In this section, we recall the deterministic counterparts of the MPS and MRP problems (see, e.g. Albrecht 2010; [START_REF] Pochet | Production Planning by Mixed Integer Programming[END_REF]. We provide slightly modified linear programmes and some compact formulations for them that allow us to propose and analyse, in Section 4, methods and linear programming models for these problems under uncertainty.

Single-item model (MPS) -Lot for Lot rule (L4L)

We now discuss a single-item problem under capacity and cumulative capacity constraints. Given T periods, for period t ∈ [T ] ([T ] denotes the set {1, . . . , T }), let d t ≥ 0 be the demand in period t, let x t ≥ 0 be the production amount in period t. Set D t = t i=1 d i and X t = t i=1 x i , D t and X t stand for the cumulative demand up to period t and the cumulative production up to period t, respectively. Obviously, X t-1 ≤ X t and D t-1 ≤ D t , t = 2, . . . , T . The production x t in each period t ∈ [T ] can be under two kinds of limits: capacity limits l t and u t such that l t ≤ x t ≤ u t and cumulative capacity limits L t and U t on the cumulative production up to period t such that L t ≤ t i=1 x i ≤ U t . Thus the set of feasible production amounts X ⊆ R T + can be defined as follows:

X = {x = (x 1 , . . . , x T ) : l t ≤ x t ≤ u t , L t ≤ t i=1 x i ≤ U t , t ∈ [T ]}. (1)
Moreover, at this (tactical) level of planning for MPS, we assume that production, inventory and backordering cost coefficients and the selling price coefficient do not depend on the period t. These cost and selling price coefficients are denoted, respectively, by c I , c B and b P . We do not consider in the production process setup times and costs. Thus the problem, here studied, is a version of the capacitated single-item lot sizing model with backordering (see, e.g. [START_REF] Pochet | Production Planning by Mixed Integer Programming[END_REF] and it can be formulated as a linear programming model as follows -the decision variables and parameters in model ( 2)-( 7) are defined in Table 1:

min T t=1 (c I I t + c B B t -b P s t ) (2) s.t. B t -I t = D t - t i=1 x i t ∈ [T ] (3) t i=1 s i = D t -B t t ∈ [T ] (4) l t ≤ x t ≤ u t t ∈ [T ] (5) L t ≤ t i=1 x i ≤ U t t ∈ [T ] (6) B t , I t , s t , x t ≥ 0 t ∈ [T ] (7)
Since we have only one product and production cost coefficients are constant over the planning horizon, they can be ignored in the formulation (2)-(7). Constraints (3) are the flow constraints of production with backordering and (4) specify the potential sales of the product in the planning horizon. It is easily seen that ( 4) can be also rewritten as: t i=1 s i = t i=1 x i -I t . Constraints (5) are the capacity limits in each period and (6) are the limits on the cumulative production up to period t. We have assumed that an initial inventory and an initial backorder are equal to zero. This assumption is not particularly restrictive.

An easy computation shows that ( 2)-( 7) may be written in an equivalent compact form. Indeed, define C and P to be the nonnegative real functions that represent either the cost of storing inventory from period t to period t + 1 or the cost of backordering the product amount from period t + 1 to period t, and the benefit from selling the product in the planning horizon, respectively. The functions have the forms: 2) and ( 3)) and 3) and ( 4)). We can now write (2)-( 7) in the following equivalent form:

C(X t , D t ) = max{c I (X t -D t ), c B (D t -X t )} (see (
P(X T , D T ) = b P min{X T , D T } (see (2), (
MPS-L4L: min x∈X F(x) = min x∈X T t=1 C(X t , D t ) -P(X T , D T ) (8)
Therefore, our production planning consists in finding a feasible production plan x ∈ X, subject to the conditions of satisfying each demand, which minimises the total cost of storage and backordering minus the benefit from selling the product. The compact form (8) allows us to formulate (2)-( 7) as the following linear programme which will be exploited in Section 4:

min T t=1 π t (9) s.t. π t ≥ c I t i=1 x i -D t t = 1, . . . , T -1 (10) π t ≥ c B D t - t i=1 x i t = 1, . . . , T -1 ( 1 1 ) π T ≥ c I T i=1 x i -D T -b P D T (12) π T ≥ c B D T - T i=1 x i -b P T i=1 x i (13) π t unrestricted t ∈ [T ] (14) x x x ∈ X (15)
where the optimal values of π t , t = 1, . . . , T -1, are such that π t = C(X t , D t ), π T = C(X T , D T ) -P(X T , D T ) and the value of T t=1 π t equals F(x) for an optimal robust production plan x x.

Single-item model (MPS) -periodic order quantity rule (POQ)

We now turn to a single-item problem under periodic capacity constraints in which u t = ∞ if t is the production period; and u t = 0 otherwise. Obviously, it is a special case of the single product problem with the Lot-For-Lot rule (see problem (8)). Furthermore, we are given a periodicity being a positive integer such that: 1 < < T and x t ≥ 0 if t mod = 1;

x t = 0 otherwise (if t mod = 1) for t ∈ [T ].
We assume without loss of generality that (T -1) mod = 0. Now, the set of feasible production amounts X ⊆ R T + can be defined as follows:

X = {x = (x 1 , . . . , x T ) : x t ≥ 0 for t mod = 1, x t = 0 for t mod = 1, t ∈ [T ]}. ( 16
)
The optimisation problem consists in finding a feasible production plan x ∈ X, subject to the conditions of satisfying each demand, which minimises the total cost of storage and backordering, that is:

MPS-POQ: min x x∈X F(x x) = min x∈X T t=1 C(X t , D t ). ( 17
)
Clearly, problem ( 17) is a version of the uncapacitated single-item lot sizing model with backordering (see, e.g. [START_REF] Zangwill | A Backlogging Model and a Multi-Echelon Model of a Dynamic Economic Lot Size Production System -A Network Approach[END_REF]) with the periodic order quantity policy. Thus, it can be formulated as follows:

min T t=1 (c I I t + c B B t ) (18) s.t. B t -I t = D t - t i=1 x i t ∈ [T ] ( 19
)
x t = 0 t mod = 1, t ∈ [T ] (20) B t , I t , x t ≥ 0 t ∈ [T ] (21)
Taking into account the periodicity in the problem MPS-POQ and its network nature, one can adapt the results of Ahuja, Magnanti, and Orlin (1993, Chapter 19.9) for problem ( 18)-( 21) and obtain the following proposition:

Proposition 1 The problem (18)-( 21) can be solved in O(T ).

Proof. See Appendix 1.

Multi-Item, multi-level model (MRP)

We focus on the most involved problem. We are given T periods, P products and R resources. For period t ∈ [T ] and product p ∈ [P] let d t, p be the external demand for product p in period t, d t, p ≥ 0, let x t, p be the production amount of product p in period t, x t, p ≥ 0. Set D t, p = t i=1 d i, p and X t, p = t i=1 x i, p , D t, p and X t, p stand for the cumulative demand for product p up to period t and the cumulative production of product p up to period t, respectively. Obviously, X t-1, p ≤ X t, p and D t-1, p ≤ D t, p , t = 2, . . . , T . In this problem, making products requires resources. Let a p,r be the required amount of resource r, r ∈ [R], to produce one unit of product p, p ∈ [P]. Hence the production of product p in period t, x t, p , will satisfy two kinds of resource constraints: the capacity limits of each resource r in each period t, denoted by l t,r and u t,r , respectively, such that l t,r ≤ P p=1 a p,r x t, p ≤ u t,r and the cumulative capacity limits of each resource r in each period t, denoted by L t, p and U t, p , respectively, such that L t,r ≤ P p=1 a p,r X t, p ≤ U t,r . Besides the external demand d t, p , a given amount of product p ∈ [P] is required to make one unit of product p ∈ [P], denoted by b p, p (b p, p = 0 if p = p ), and so P j=1 b p, j x t+Ld j , j is the internal demand for product p in period t, where Ld j is the lead-time of product j. Furthermore, if one accepts backordering, this means that an external demand for product p in period t can be fulfilled in period t such that t > t. Of course the backordering concerns only external demands. Therefore, the set of feasible production amounts X ⊆ R T ×P + has the following form:

X = {x x = (x t, p ) : l t,r ≤ P j=1 a j,r x t, j ≤ u t,r , L t,r ≤ P j=1 a j,r t i=1 x i, j ≤ U t,r , t i=1 ⎛ ⎝ x i, p - P j=1 b p, j x i+Ld j , j ⎞ ⎠ ≥ 0, t ∈ [T ], r ∈ [R], p ∈ [P]}, (22)
Table 2. The parameters and decision variables in model ( 23)-( 28). External demand of product p ∈ where t i=1 (x i, p -P j=1 b p, j x i+Ld j , j ) is the cumulative production of product p up to period t minus the internal demand for product p. At the (tactical) level of planning, one assumes that the inventory, production and backordering cost coefficients and selling prices only depend on product p and not on period t. We are now ready to give the linear programming model of the problem under consideration, the decision variables and parameters are defined in Table 2 (we do not take into account setup times and costs in the production process):

Definition

Set of indices

[P] in period t ∈ [T ], cumulative demand D t, p = t i=1 d i, p l t,r Lower capacity limit of resource r ∈ [R] in period t ∈ [T ] u t,r Upper capacity limit of resource r ∈ [R] in period t ∈ [T ] L t,r Lower cumulative capacity limit of resource r ∈ [R] in period t ∈ [T ] U t,
min P p=1 T t=1 (c I p I t, p + c B p B t, p + c P p x t, p -b P p s t, p ) (23) s.t. B t, p -I t, p = D t, p - t i=1 ⎛ ⎝ x i, p - P j=1 b p, j x i+Ld j , j ⎞ ⎠ t ∈ [T ], p ∈ [P] (24) t i=1 s i, p = D t, p -B t, p t ∈ [T ], p ∈ [P] (25) l t,r ≤ P j=1 a j,r x t, j ≤ u t,r t ∈ [T ], r ∈ [R] (26) L t,r ≤ P j=1 a j,r t i=1 x i, j ≤ U t,r t ∈ [T ], r ∈ [R] (27) B t, p , I t, p , s t, p , x t, p ≥ 0 t ∈ [T ], p ∈ [P] (28)
Note that the model ( 23)-( 28) admits that each product can have external and internal demands simultaneously, thus it is a generalisation of ones proposed in [START_REF] Albrecht | Supply Chain Coordination Mechanisms[END_REF], [START_REF] Mula | Material Requirement Planning with Fuzzy Constraints and Fuzzy Coefficients[END_REF], Pochet and Wolsey model, the practitioners are able to express different types of uncertainty on the demand effortlessly. Indeed, the uncertainty can be constant all over a planning horizon, or can increase proportionally to the demand or can be due to shifts of a part of the demand between two consecutive periods. The above three cases can be expressed as uncertainty on the cumulative demand. In the first one, this uncertainty is expressed as an uncertainty of ±K on the cumulative demand. More formally, t = t = K , where K is a constant. In the second case, t = t = αD n t , where α ∈ [0, 1] is a prescribed uncertainty parameter being the maximum percentage deviation of cumulative demands in periods from their nominal values. In the third case t = αd t , and t = αd t+1 , where α ∈ [0, 1], which is the maximum percentage deviation of the demand that can be moved to its adjacent periods. For instance, α = 1 means that all the demand in a period can be moved to its adjacent periods and α = 0.5 means that half of the demand in a period can be moved to its adjacent periods. Moreover, in contrast with the period-specific uncertainty that cannot model uncertainty reductions in parts of a planning horizon, defining uncertainty on the cumulative demand allows the decision-maker to capture a variation of uncertainty throughout a planning horizon.

Model of uncertainty for the MPS problems

For MPS-L4L, MPS-POQ problems studied in Sections 2.1 and 2.2, we adopt a model in which uncertainty in the cumulative demand is small with respect to the minimum demand. In other words, a demand in a period can be equal to 0 only if the previous cumulative demand is at its maximum value. In fact, at this level of planning, the lengths of time periods are big enough (for example, one month) to have the minimum periodic demand greater than zero. Thus this assumption is realistic. More formally, cumulative demand intervals are such that D t-1 ≤ D t for every t = 2, . . . , T . Hence the set of cumulative demand scenarios U is the Cartesian product of the cumulative demands intervals, i.e.

U = [D 1 , D 1 ] × • • • × [D T , D T ], where D t-1 ≤ D t .
(30)

Model of uncertainty for the MRP problem

For the MRP problem, formulated in Section 2.3, one assumes that the length of periods is smaller than the length of periods of the MPS problems, for instance, the length of a period for MPS is one month while the length of a period for MRP is one week. Moreover, the demand has more versatility in the short range. For such a length of periods of MRP, it is possible that the demand is delayed by a period, what is not possible for a longer length of period. For both reasons, it is hard to assume that for MRP problem, uncertainty in the cumulative demand is small with respect to the minimum demand. Furthermore, we extend now the previously described model of uncertainty for the single-item model to the one for the multi-item model.

U = {S = (D(S) t, p ) : D(S) t, p ∈ [D t, p , D t, p ], , t ∈ [T ], D(S) t-1, p ≤ D(S) t, p , t = 2, . . . , T, p ∈ [P]}. ( 31
)

Robust MPS/MRP problems

Assuming that demand uncertainty is represented as in Section 3, we wish to find a robust production plan x ∈ X in MPS-L4L, MPS-POQ, MRP problems with the best worst-case cost guarantee. In order to this, one of common robust criteria called the minmax can be adopted (see, e.g. [START_REF] Kouvelis | Robust Discrete Optimization and its Applications[END_REF]. In other words, we seek a production plan that minimises the maximum production plan cost F(x, S) over all the cumulative demand scenarios from a fixed set U (see ( 30), ( 31)). This leads to the following general robust problem P:

ROB P: min x∈X A(x x) = min x∈X max S∈U F(x, S),
where P ∈ {MPS-L4L, MPS-POQ, MRP}, F(x, S) is the cost of production plan x x (see ( 8), ( 17), ( 29)) under scenario S ∈ U, A(x) is the maximum cost of production plan x ∈ X, A(x x) = max S∈U F(x x, S), and the set of feasible production plans X is the one of the forms: (1), ( 16) and ( 22). An optimal solution x r to the problem ROB P is called optimal robust production plan for P.

We are aware of the conservativeness of the minmax. Fortunately, using this criterion, with the assumed cumulative demand uncertainty representation (see ( 30), ( 31)), enables us to construct polynomial methods for the problems under consideration. Moreover, applying other less conservative criteria may not guarantee the preservation of polynomial solvability and requires additional information about uncertain demands (see Section 1 for comments).

A natural optimisation problem associated with ROB P, and simpler than ROB P, is the one of evaluating a given production plan x x * ∈ X. A scenario S o ∈ U that minimises the total cost F(x x * , S) of the production plan x x * is called optimistic scenario. A scenario S w ∈ U that maximises the total cost F(x * , S) of the production plan x * is called the worstcase scenario. The problem of computing a worst-case scenario is often called adversarial problem (see, e.g. [START_REF] Bienstock | Computing Robust Basestock Levels[END_REF][START_REF] Nasrabadi | Robust Optimization with Incremental Recourse[END_REF], in which an adversary maliciously wants to increase the total cost F(x * , S) of the production plan x * . Therefore in order to evaluate a given production plan x * , we look for the interval,

F x * = [ f x x * , f x x * ],
containing possible values of costs of x * , such that

f x * =F(x x * , S o ) = min S∈U F(x x x * , S), ( 32
)
f x * =F(x * , S w ) = max S∈U F(x * , S). ( 33
)

Robust MPS problems

In this section, we focus on the problem of evaluating a fixed production plan and the problem of computing an optimal robust production plan to ROB P, where P ∈ {MPS-L4L, MPS-POQ}, under the uncertainty representation U defined in (30). We start by proving the following proposition that we substantially use in this section.

Proposition 2 The objective function F(x x * , S) for any fixed production plan x x * ∈ X in ROB P, P ∈ {MPS-L4L, MPS-POQ}, is convex in U.

Proof. See Appendix 1.

The convexity of F(x * , S) in U is a crucial fact. For problem (32), it implies that a locally optimal scenario is also globally optimal. For problem (33), it implies the maximum of F is attained at a vertex of convex polytope U. We recall that a scenario S ∈ U is a vertex of U if it is not the strict convex combination of scenarios of U.

Evaluating a given production plan

We now examine the problem of evaluating a given production plan x x * ∈ X. Namely, we wish to find the interval F x x * = [ f x x * , f x * ] (see problems (32) and ( 33)). Interval F x * determined provides valuable information for the decision-maker, since it contains possible values of costs of x * . It may be useful, for instance, in a risk analysis in the setting of possibility theory [START_REF] Dubois | Possibility Theory: An Approach to Computerized Processing of Uncertainty[END_REF], where it helps in evaluating degrees of possibility and necessity that a cost of a given plan x * does not exceed a given threshold. Thus we need to compute its optimistic and worst-case scenarios. We consider the case ROB MPS-L4L. All the results presented in this section apply also to ROB MPS-POQ -it is sufficient to set b p to zero.

We first deal with the problem of determining an optimistic scenario S o ∈ U for a production plan x x x * , i.e. the problem (32). Fortunately, for every x x * ∈ X one can give the explicit form of an optimistic scenario S o :

D(S o ) t = ⎧ ⎪ ⎨ ⎪ ⎩ D t if X * t < D t , X * t if D t ≤ X * t ≤ D t , D t if X * t > D t , t ∈ [T ]. (34)
It is easy to check that S o in the form of ( 34) is feasible and locally optimal for the problem (32). Hence and from Proposition 2, scenario S o is also globally optimal, f x x * = F(x * S o ). Thus computing an optimistic scenario can be done in O(T ) time.

We now study the problem of computing a worst-case scenario S w ∈ U for a given production plan x x * ∈ X (the adversarial problem), i.e. the problem (33). It turns out that, for problem (33), the set U can be reduced to the set of extreme scenarios denoted by

U ext , U ext ⊆ U, U ext = {D 1 , D 1 } × • • • × {D T , D T }, where D t-1 ≤ D t . ( 35
)
The following proposition characterises a worst-case scenario S w .

Proposition 3 A worst-case scenario S w is an extreme one, i.e. S w ∈ U ext .

Proof. See Appendix 1.

Using Proposition 3 and the fact that the bounds are such that D t-1 ≤ D t , we can rewrite problem (33), equivalently, as

f x x * = F(x x x * S w ) = max S∈U F(x x * , S) = max S∈U ext F(x x x * , S) = T -1 t=1 max D(S) t ∈{D t ,D t } C(X * t , D(S) t ) + max D(S) T ∈{D T ,D T } C(X * T , D(S) T ) -P(X * T , D(S) T ). ( 36
)
Thus a special form of optimisation problem (36) and the forms of functions C and P allow us to give an explicit formula for an optimal solution -a worst-cost scenario S w for every x * ∈ X. That is for t ∈ [T ]:

D(S w ) t = ⎧ ⎪ ⎨ ⎪ ⎩ D t if c I (X * t -D t ) > c B (D t -X * t ) and t = 1, . . . , T -1, D t if c I (X * t -D t ) -b P D t > c B (D t -X * t ) -b P X * t and t = T , D t otherwise. (37)
Therefore, a worst-cost scenario S w for a given production plan x x x * can be computed in O(T ) time.

Computing an optimal robust production plan

In this section, we address the problem of computing an optimal robust production plan x x r ∈ X to ROB P, P ∈ {MPS-L4L, MPS-POQ}. We first consider the problem ROB MPS-L4L. It is easy to see that the problem contains as subproblem, the one of computing a worst-case scenario for a fixed x x x ∈ X -problem (36). Hence, we get the decomposition of ROB MPS-L4L:

min x∈X max S∈U F(x x, S) = min x x∈X max S∈U ext F(x, S) = min x∈X T -1 t=1 max D(S) t ∈{D t ,D t } C(X t , D(S) t ) + max D(S) T ∈{D T ,D T } C(X T , D(S) T ) -P(X T , D(S) T ). ( 38
)
Using model ( 9)-( 14) and the fact that D(S) t ∈ {D t , D t }, t ∈ [T ], we may formulate problem (36) as the following linear programme:

min T t=1 π t (39) s.t. π t ≥ c I (X * t -D t ) t = 1, . . . , T -1 π t ≥ c B (D t -X * t ) t = 1, . . . , T -1 π T ≥ c I (X * T -D T ) -b P D T π T ≥ c B (D T -X * T ) -b P X * T π t unrestricted t ∈ [T ]
For an optimal solution to (39), the value of T t=1 π t equals max S∈U F(x * , S) (problem (36)), the values of

π t , t ∈ [T ], are such that π t = max D(S) t ∈{D t ,D t } C(X * t , D(S) t ) and π T = max D(S) T ∈{D T ,D T } C(X * T , D(S) T ) -P(X * T , D(S) T ).
Adding now the constraints describing the feasible production amounts (1) to (39) and developing X t , t ∈ [T ] yield a linear programming model for computing an optimal robust production plan x r to ROB MPS-L4L:

min T t=1 π t (40) s.t. π t ≥ c I t i=1 x i -D t t = 1, . . . , T -1 (41) π t ≥ c B D t - t i=1 x i t = 1, . . . , T -1 (42) π T ≥ c I T i=1 x i -D T -b P D T (43) π T ≥ c B D T - T i=1 x i -b P T i=1 x i (44) π t unrestricted t ∈ [T ] (45) x x ∈ X (46)
The vector of decision variables x x = (x t ) t∈[T ] describes an optimal robust production plan x r to ROB MPS-L4L and T t=1 π t provides max S∈U F(x x, S). The interpretation of variables π t , t ∈ [T ], is the same as for model (39). A worst-case scenario S w for x x r can be determined by formula (37) and corresponding inventory and backordering levels and sales at the end of each period can be easily computed by ( 3) and ( 4). The problem ROB MPS-L4L with no capacity and cumulative capacity limits is a trivial one. Namely, from results presented in Guillaume, Kobylański, and Zieliński (2012, Section 4.2) and the decomposed form (38) of the problem, we immediately deduce the following O(T ) method for computing an optimal robust production plan x r , for t ∈ [T ]:

X r t = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ c B D t +c I D t c B +c I if t = 1, . . . , T -1, c B D t +(c I +b P )D t c B +c I +b P if t = T. ( 47
)
Of course, X r t-1 ≤ X r t for t = 2, . . . , T . Thus one can set x r 1 = X r 1 and x r t = X r t -X r t-1 for t = 2, . . . , T . It is worth noting that linear programme ( 40)-( 46) for ROB MPS-L4L can be also obtained if we adapt a more general approach proposed in [START_REF] Soyster | Convex Programming with Set-inclusive Constraints and Applications to Inexact Linear Programming[END_REF] to model ( 9)-( 15) for MPS-L4L under cumulative demand uncertainty. Indeed, this approach copes with uncertainty in cumulative demands by considering their worst-case values. Furthermore, it is equivalent to the robust approach to linear optimisation problems in the sense of [START_REF] Bertsimas | The Price of Robustness[END_REF] if cumulative demand D t in each period t is admitted to be uncertain. In order to maximise the values of the right-hand side of constraints ( 10)-( 13) over cumulative demand scenario set U, we set cumulative demand D t in (10), ( 12) and ( 11), ( 13) at its lower possible value D t and its upper possible value D t , respectively, and get constraints ( 41)-( 44). The above equivalence between Soyster's method applied to MPS-L4L under cumulative demand uncertainty and ROB MPS-L4L holds only under scenario set U defined by (30), when D t-1 ≤ D t , t = 2, . . . , T . If D t-1 ≤ D t is dropped then Soyster's approach considers infeasible cumulative demand scenarios, i.e. scenarios S which do not satisfy D(S) t-1 ≤ D(S) t , t = 2, . . . , T . This follows from the fact that the approach assumes that the value of each cumulative demand may vary independently of the values of the remaining cumulative demands. Therefore, Soyster's method and in consequence the method proposed in [START_REF] Bertsimas | The Price of Robustness[END_REF] can no longer be applied to problem MRP under cumulative demand scenario set U defined by (31) considered in Section 4.2.

We proceed with the study of the problem ROB MPS-POQ. An approach similar to ( 40)-( 46) can be applied to obtain a linear programming model for computing an optimal robust production plan x r to ROB MPS-POQ. However, a more efficient combinatorial algorithm for determining x x r can be proposed. From Proposition 3, the fact that the bounds fulfil inequalities D t-1 ≤ D t and the periodicity in the problem ROB MPS-POQ, it follows that one can decompose the problem into N + 1 separate subproblems, where N = (T -1)/ ,that is:

min x x∈X max S∈U ext T t=1 C(X t , D t (S)) = N k=1 min X (k-1)• +1 ∈X k max S∈U k ext k• t=(k-1)• +1 C(X (k-1)• +1 , D t (S)) + min X T ∈X T max S∈U T ext C(X T , D T (S)), ( 48
)
where

X k = [D (k-1)• +1 , D (k-1)• +1 ] ∪ • • • ∪ [D k•P , D k• ], U k ext = {D (k-1)• +1 , D (k-1)• +1 } × • • • × {D k•P , D k• }, k ∈ [N ], X T = [D T , D T ] and U T ext = {D T , D T }; U ext = U 1 × • • • × U N × U T .
Obviously, the above possible cumulative production levels are nondecreasing sequence of their values, X 1 ≤ X +1 ≤ • • • ≤ X T . Therefore, we need only to solve the N + 1 separate subproblems. The last subproblem is trivial, i.e. an optimal cumulative production level X r T can be computed by formula:

X r T = c B D T + c I D T c B + c I . ( 49
)
Hence problem (33) has the equivalent separable form:

f x x * = max S∈U F(x x * , S) = max S∈U P p=1 T t=1 C p (X * int t, p , D(S) t, p ) -P p (X * int T, p , D(S) T, p ) + c P p X * T, p = P p=1 c P p X * T, p + max S∈U p T t=1 C p (X * int t, p , D(S) t, p -P p (X * int T, p , D(S) T, p )). ( 53
)
Accordingly, the problem of computing a worst-case scenario S w ∈ U for a given production plan x x * ∈ X boils down to solving the P separated problems. Namely, max

S∈U p T t=1 C p (X * int t, p , D(S) t, p ) -P p (X * int T, p , D(S) T, p ) + c P p X * T, p , p ∈ [P]. (54)
Let us regard p as fixed. It is evident that the objective function in ( 54) is convex in U p for a given production plan x * ∈ X (see the proof of Proposition 4). Consequently, it attains its maximum value at a vertex of convex polytope U p (see, e.g. [START_REF] Martos | Nonlinear Programming Theory and Methods[END_REF]). Hence it suffices to examine only vertices of U p . We now characterise vertices (vertex scenarios) of scenario set U p . We recall first that v ∈ U p is a vertex (vertex scenario) of U p if v v cannot be the strict convex combination of elements (scenarios) of U p . Thus the following lemma is obvious.

Lemma 1 Let v ∈ U p ∩ t∈[T ] {D t, p , D t, p }. Then v is a vertex of U p . Lemma 2 Suppose that v (v t ) t∈[T ] ∈ U p and there exits at least one component v k of v such that v k ∈ (D k, p , D k, p ). Then v v is a vertex of U p if and only if for each component v k of v, v k ∈ (D k, p , D k, p ), in the subsequence (v i , . . . , v k-1 , v k , v k+1 , . . . , v l ) of v, where i = min{t ∈ [T ] : v t = v k , t ≤ k} and l = max{t ∈ [T ] : v t = v k , k ≤ t}, the first and the last elements are such that v i = D i, p or v l = D l, p . Proof. See Appendix 1.
The above lemma shows, among others, that there may exist a vertex scenario S = (D(S) t, p ) ∈ U p , a candidate for an optimal solution to (54), with at least one cumulative demand D(S) k, p in some period, say k, such that D(S) k, p ∈ (D k, p , D k, p ). Therefore, for instance, Soyster's framework [START_REF] Soyster | Convex Programming with Set-inclusive Constraints and Applications to Inexact Linear Programming[END_REF], which is a particular case of the method for robust linear problems proposed in [START_REF] Bertsimas | The Price of Robustness[END_REF] and focuses also on a worst-case analysis, cannot be applied here. This framework assumes that every cumulative demand D t, p , t ∈ [T ] has to be taken its worst-case value in the interval [D t, p , D t, p ]. Thus it may omit the vertex scenario described above or lead to a scenario S that violates conditions D(S) t-1, p ≤ D(S) t, p , t = 2, . . . , T .

Using Lemmas 1 and 2, we can construct a layered graph G p = (V p , A p ) that represents a subset of scenarios of set U p that contains all its vertex scenarios and during examining no vertex scenario can be omitted. The set

V p is partitioned into T + 2 disjoint layers V p 0 , V p 1 , . . . , V p T , V p T +1 in which V p 0 = {s}
p u , v p u ∈ [D t, p , D t, p ] ∩ t∈[T ] {D t, p , D t, p }, of all the vertices of U p . Obviously, |V p t | = |[D t, p , D t, p ] ∩ t∈[T ] {D t, p , D t, p }|, t ∈ [T ]. Let A p = A p 1 ∪ • • • ∪ A p T ∪ A p T +1 . Arc (u, w) ∈ A p 1 if u ∈ V p 0 and w ∈ V p 1 ; (u, w) ∈ A p T +1 if u ∈ V p T and w ∈ V p T +1 ; and arc (u, w) ∈ A p t , t = 2, . . . , T , if u ∈ V p t-1 , w ∈ V p t and v p u ≤ v p w .
An easy computation shows that A p has O(T 3 ) arcs and V P has O(T 2 ) nodes. We associate with each arc (u, w) ∈ A p length l p u,w in the following way:

l p u,w = ⎧ ⎪ ⎨ ⎪ ⎩ C p (X * int t, p , v p w ) if (u, w) ∈ A p t , t = 1, . . . , T -1, C p (X * int T, p , v p w ) -P p (X * int T, p , v p w ) + c P p X * T, p if (u, w) ∈ A p T , 0 i f (u, w) ∈ A T +1 (55)
for a fixed x * ∈ X. The graph G p models a subset of

U p . Indeed, if s = u 0 ; u 1 ; • • • ; u T ; u T +1 = t is a path from s to t, then it corresponds to v = (v p u t ) t∈[T ] ∈ U p , u t ∈ V p t , t ∈ [T ]
, and the length of this path, t∈[T ] l p u t-1 ,u t , is equal to the value of the objective function in (54) for v. Since G p has been built according to Lemmas 1 and 2, the subset modelled contains all the vertices of U p . Thus by the convexity of the objective function in (54) over U p , this subset contains all the optimal scenarios to problem (54). Hence solving (54) boils down to finding a longest path σ p from s to t in G p , which can be done in O(T 3 ) (see, e.g. [START_REF] Ahuja | Network Flows: Theory, Algorithms, and Applications[END_REF]. From the decomposition of (53), we have an O(P • T 3 ) algorithm for the determining a worst-case scenario S w ∈ U for a given production plan x x * ∈ X (the problem (33)). Worst-case scenario S w corresponds to the concatenation σ 1 ; Making use of (56), combining ( 58)-( 68) with the definition of the set of feasible production amounts X (see ( 22)) and developing X int t, p , X T, p , t ∈ [T ], p ∈ [P], we obtain a linear programing model for computing an optimal robust production plan x r to ROB MRP:

min P p=1 π p t (69) s.t. B p w -I p w = v p w - t i=1 (x i, p - P j=1 b p, j x i+Ld j , j ) w ∈ V p t , t ∈ [T ], p ∈ [P] π p w -π p u ≥ c I p I p w + c B p B p w (u, w) ∈ A p t , t = 1, . . . , T -1, p ∈ [P] π p w -π p u ≥ c I p I p w + c B p B p w -b P p z p w + c P p T i=1 x i, p (u, w) ∈ A p T , p ∈ [P] π p t -π p u ≥ 0 u ∈ V p T , p ∈ [P] π p s = 0 p ∈ [P] z p w ≤ v p w w ∈ V p T , p ∈ [P] z p w ≤ t i=1 (x i, p - P j=1 b p, j x i+Ld j , j ) w ∈ V p T , p ∈ [P] B p w , I p w ≥ 0 w ∈ V p t , t ∈ [T ], p ∈ [P] z p w ≥ 0 w ∈ V p T , p ∈ [P] π p u unrestricted u ∈ V p , p ∈ [P]
x x ∈ X

The decision variables x j, p , p ∈ [P], t ∈ [T ], describe an optimal robust production plan x r to ROB MRP and P p=1 π p t provides max S∈U F(x x r , S). Meanings of the constraints and the rest of decision variables are the same as for model ( 58)-(68).

Experimental results

In this section, we present experiments illustrating our robust approach to the MRP problem with uncertain cumulative demand (ROB MRP, see Section 4.2) under the uncertainty representation U defined in (31). We are given 12 products and 3 resources, P = 12 and R = 3. The bill of materials (the product structures) with the product costs and the lead times of the products are presented in Figure 2. The selling prices of the products are as follows: b P 1 = 10, 000, b P 2 = 5500 and b P p = 0 for p = 3, . . . , 12. The backorder cost c B p of each product p is equal to 15% of its selling price. The inventory cost c I p of each product p is equal to 5% of the sum of the production cost of product p and all the components consumed when product p is produced. For instance, using the product structures shown in Figure 2, the inventory cost of product 5 is computed as follows: c I 5 = 0.05(c P 5 + b 12,5 c P 12 + b 11,5 c P 11 ) = 22.5. The planning horizon consists of 23 periods, T = 23, with the capacity limits: l t,1 = l t,2 = l t,3 = 0 for t = 1, 2 and l t,1 = 1000, l t,2 = l t,3 = 2000 for t = 3, . . . , 23; u t,1 = 10, 000 for t = 1, . . . , 23, u t,2 = 9100 for t = 1, . . . , 14, u t,2 = 11, 200 for t = 15, . . . , 23 and u t,3 = 10, 300 for t = 1, . . . , 8, u t,3 = 9300 for t = 9, . . . , 14, u t,3 = 6800 for t = 15, . . . , 17 u t,3 = 10, 800 for t = 18, . . . , 23. There are no cumulative capacity limits in this example.

Regarding the uncertainty representation (31) of scenario set U, cumulative demands are uncertain and modelled by intervals (see Section 3). In our experiments, the cumulative demand intervals have the following form [D n t, pt, p , D n t, p + t, p ], where D n t, p = t i=1 d i, p is the nominal cumulative demand and d t, p is given demand in period t. We assume that product 1 and 2 have external demand so we only provide in Table 3 demands d t, p for p = 1, 2. The rest of d t, p , and in consequence D n t, p , for t = 1, . . . , 23 and p = 3, . . . , 12 are equal to zero. The values of t, p and t, p are determined according to two kinds of uncertainty on the cumulative demand. In the first type t, p = t, p = αD n t, p , t = 1, . . . , 23, p = 1, 2, where α ∈ [0, 1] is a prescribed uncertainty parameter being the maximum percentage deviation of cumulative demands in periods from their nominal values. In the second type t, p = αd t, p , t = 1, . . . , 23, and t, p = αd t+1, p , t = 1, . . . , 22 and 23, p = 0, p = 1, 2, where α ∈ [0, 1], which is the maximum percentage deviation of demands in periods from their nominal values. This type reflects uncertainty about occurring demands in periods, i.e. some of them may advanced or delayed. For instance, α = 0.5 means half demand in a period can be moved to its adjacent periods. Therefore, optimistic and chooses the nominal production plan x n , instead of an optimal robust plan x r , then its profit may decreases when a worst case scenario reveals. The relative reduction of the minimum profit of optimal robust production plans red r approximately equals twice and 1/5 of the percentage uncertainty introduced for the first and the second types, respectively (see Figure 5). For the latter type, the reduction red n is twice higher than red r (see Figure 5(b)). For instance, for the first type fora = 0.05 the minimum profit of an optimal robust production plan 6.6609 x 10 7 is reduced by 11 % to 5.8980 x 10 7

.

For the second type for a 0.22, the minimum profit of an optimal robust production plan 7.2367 x 10 7 is reduced by 4% to 6.9449 X 10 7 .

Ali the experiments, including the methods runs (see Section 4.2), were implemented using the modelling package JuMP [START_REF] Lubin | Computing in Operations Research Using Julia[END_REF] embedded in the prograrnming language Julia. Within our implementations, we called IBM ILOG CPLEX 12. 5 optimiser. The experiments were executed on a computer equipped with Intel Core 2 Duo 2. 4

GHz. Computation time required to find nominal production plan x n was 9.06 s. Average computation times for computing optimal robust production plans x r under the first and second types of cumulative demand uncertainty were 10.6 and 9.88 s, respectively ( over 30 problem instances for each type of uncertainty ). Thus the ROB MRP problems are still computationally tractable.

Conclusions

In the context of supply chains, the problem of tactical production and capacity planning under uncertainty in the demand is a central concern for the production planning systems. In this paper, we have dealt with the MPS (CLSP) under small uncertainty in the cumulative demand and with MRP (MLCLSP) problems under uncertainty in the cumulative demand. The mode! of uncertainty in the cumulative demand enables to take into account jointly the imprecision on order quantities and dates. For both problems, linear prograrnming models, which include backordering and the cumulative demand, have been presented. Further on, we have proposed efficient methods for evaluating the impact of uncertainty on production plans and linear programmes for computing optimal robust production plans for MPS and MRP problems under the uncertainty in the cumulative demand. Thus we have shown that introducing uncertainty in the cumulative demand to the above optimisation processes with the minmax criterion does not significantly increase their computational complexity, comparing with the deterministic counterparts, -they still remain polynomially solvable. Therefore, they can be applied in industrial context, i.e. in the manufacturing planning tools using linear prograrnming solvers.

There are some opportunities for further research in production planning in the robust optimisation setting. Namely, analysing the above problems under Jess conservative robust criteria or considering the second source of uncertainty in production planning, pointed out in the literature, that is the uncertainty in the process (uncertain lead time and scrap of components) are good topics for further research.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

Pawel Zielinsk:i was supported by Wroclaw University of Science and Technology [grant number 0401/0086/16).

ppp

  Set of products {1, . . . , P} [R] Set of resources {1, . . . , R} [T ] Set of periods in the planning horizon {1, . . . , Required quantity of resource r ∈ [R] to produce one unit of product p ∈ [P] b i, j Amount of product i ∈ [P] required to produce one unit of product j ∈ [P] b P Selling price of one unit of product p ∈ [P] c P Production cost of one unit of product p ∈ [P] c I Inventory cost of one unit of product p ∈ [P] c B p Backorder cost for one unit of product p ∈ [P] d t, p

  and V p T +1 = {t} contain two distinguished nodes, s and t, and each node u ∈ V p t , t ∈ [T ], corresponds to exactly one possible value of the tth component, denoted by v

  p w ). Constraint (59) together with the right-hand side of (61) and constraints (64), (65) represent the value of C p (X * int T, p , v p w ) -P p (X * int Ta possible value of the cumulative demand in period T of product p, v p w ∈ [D T, p , D T, p ]∩ t∈[T ] {D t, p , D t, p }, (64) and (65) help to model the value of P p (X * int T, p , v p w ). Solving linear programming model (58)-(68) for each p ∈ [P] yields P longest paths σ 1 , . . . , σ P whose concatenation σ 1 ; • • • ; σ P corresponds to worst case scenario S w for a fixed production plan x * . Namely, if node w ∈ σ p and w ∈ V p t , t ∈ [T ], then D(S w ) t, p = v p w and the values of decisions variables I p w and B p w are corresponding inventory and backordering levels, respectively, at the end of period t for product p, . Such a longest path in G p can be identified (after solving (58)-(68)), if needed, by performing a breadth-first search of G p using the arcs satisfying the equalities: * int T, p } + c P p X * T, p or π p t -π p u = 0.

Fi

  gur e 5. Relative reductions as functions of uncertainty parameter a. (a) The first type of uncertainty. (b) The second type of uncertainty.

  • • • ; σ P of each longest path σ p in G p , p ∈ [P].4.2.2 Computing an optimal robust production plannow deal with the problem of computing an optimal robust production plan x x r to ROB MRP. Using (53), it can be decomposed in the following way:The inner problem in (56) corresponds to the problem of computing a worst-case scenario for a fixed x * ∈ X. The latter problem can be reduced to find the value of a longest path σ p in each of the P layered weighted graphs G p = (V p , A p ), p ∈ [P] (see Section 4.2.1). Finding the length of a longest path σ p from s to t in G p = (V p , A p ) for a fixed p may be modelled by the following linear programming problem: , (u, w) ∈ A p , are determined according to (55) for a given x x * ∈ X. The optimal value of decision variable π p t is the length of longest path σ p from s to t in G p , i.e. π . Substituting (55) into model (57), we obtain an equivalent linear programming model of the longest path problem in G p = (V p , A p ) for fixed p and x x x * ∈ X: ] {D t, p , D t, p }, of all the vertices of U p . In other words, v p w is a possible value of the cumulative demand in period t of product p. Constraint (59) and the right-hand side of (60) express the value of C p (X * int t, p , v

									P	T
		min x∈X	max S∈U	F(x, S) = min x∈X	p=1	c P p X T, p + max S∈U p	t=1	C t (X int t, p , D(S) t, p ) -P p (X int T, p , D(S) T, p ).	(56)
							min π	p t	(57)
							s.t. π p w -π	p u ≥ l	p u,w	(u, w) ∈ A	p t , t ∈ [T ]
							π	p t -π	p u ≥ 0	p T u ∈ V
							π	p s = 0
							π	p u unrestricted	u ∈ V p
	p u,w p t = max S∈U p T t=1 C p (X T, p min π where lengths l c P p X * p t	* int t, p , D(S) t, p ) -P p (X	* int T, p , D(S) T, p ) + (58)
		s.t. B p w -I p w = v p w -X t, p * int	w ∈ V t , t ∈ [T ] p	(59)
		π p w -π	p u ≥ c I p I p w + c B p B p w	(u, w) ∈ A	p t , t = 1, . . . , T -1	(60)
		π p w -π	p u ≥ c I p I p w + c B p B p w -b P p z p w + c P p X	* T, p	(u, w) ∈ A	p T	(61)
		π	p t -π	p u ≥ 0		u ∈ V T p	(62)
		π	p s = 0			(63)
		z p w ≤ v p w		w ∈ V T p	(64)
		z p w ≤ X	* int T, p		w ∈ V T p	(65)
		B p w , I p w ≥ 0		w ∈ V t , t ∈ [T ] p	(66)
		z p w ≥ 0					w ∈ V T p	(67)
		π	p u unrestricted	u ∈ V p	(68)
	where v	p w , ∈ V t , t ∈ [T ] is a parameter, i.e. it is a possible value of the tth component, v p	p w ∈ [D t, p , D t, p ] ∩
	t∈[T

Appendix 1. Some proofs

Proof of Proposition 1 It is clear that problem (18)-( 21) is a network flow problem. Thus using the results presented in Ahuja, Magnanti, and Orlin (1993, Chapter 19.9) one can model problem ( 18)-( 21) as the shortest path problem in a network that has T + 1 nodes, where node 0 represents the source and the tth node for each t ∈ [T ] corresponds to the tth period. There are zero cost production arcs (0, t) if t mod = 1, t ∈ [T ]; the inventory carrying arcs (t -1, t), t = 2, . . . , T , with cost c I ; and the backorder carrying arcs (t, t -1), t = 2, . . . , T , with cost c B in the network. It turns out (see Ahuja, Magnanti, and Orlin 1993, Chapter 19.9) that an optimal production plan to a tree of shortest paths from the source to every node t ∈ [T ]. Indeed, for each k ∈ [N ], where N = (T -1)/ , we find period nodes:

The node h k determines two paths. The first one 0 ; (k -1) + 1 ; (k -1) + 2 ; • • • ; h k that goes through production and inventory carrying arcs and the second path 0 ; k + 1 ; k P ; • • • ; h k + 1 that traverses production and backorder carrying arcs. Using a tree of shortest paths in the network, induced by h k , k ∈ [N ], an optimal production plan to ( 18)-( 21) can be determined by the following formula:

Proof of Proposition 2 We will show the convexity of F(x * , S) in U for ROB MPS-L4L. Similar arguments apply to the objective function F(x x * , S) in ROB MPS-POQ. Observe that the objective function for production plan x * under scenario S = (D(S) t ) t∈[T ] ∈ U can be rewritten as follows (see ( 8)

and -D(S) T are convex in U. Hence the maximum and the sum of them in U is also a convex function.

Proof of Proposition 3 Proposition follows from the same as in [START_REF] Guillaume | A Robust Lot Sizing Problem with Ill-known Demands[END_REF] reasoning. Scenarios S ∈ U ext are easily seen to be the vertices of convex polytope U . Hence and Proposition 2, we get that F(x x x * , S) attains the maximum value at a vertex of U (see, e.g. [START_REF] Martos | Nonlinear Programming Theory and Methods[END_REF].

Proof of Proposition 4 Note that term c P p X * T, p , p ∈ [P], in F(x x * , S) (see ( 29)) is a constant. Applying Proposition 2 for each p ∈ [P] (taking into account slightly modified functions C p and P p ), we deduce that F(x x * , S) is the sum of P convex functions in U and so F(x x * , S) is also convex in U.

Proof of Lemma 2 Suppose contrary that there exists a component

) and v l ∈ (D l, p , D l, p ). We need consider four cases: (2) such that v lies between them. Set v (1) (2) . Moreover, it is easily seen that there exist λ ∈ (0, 1), (2) . It easy to check that there exist λ ∈ (0, 1) such that v = λ v (1) + (1 -λ )v (2) , a contradiction with the fact that v is a vertex of U p . (iii) 1 = i ≤ l < T and so andv (2) t = D l, p for t = 1, . . . , l. Again a trivial verification shows that v lies between v v (1) and v (2) , a contradiction.

. Similarly to the above cases, we obtain a contradiction. Suppose that v is not a vertex of U p . Thus there exist v (1) , v (2) ∈ U p and v = v v (1) , v v = v (2) and λ ∈ (0, 1) such that v v = λ v v v (1) + (1 -λ )v v (2) . By assumption, for each sequence (v i , . . . , v k-1 , v k , v k+1 , . . . , v l ) associated with a component v k ∈ (D k, p , D k, p )

(2) t = v t for t = i, . . . , l. The other components of v v are at their boundary values. Therefore v v = v (1) and v v = v (2) , a contradiction.


Keywords: Open Archive Toulouse Archive Ouverte piezoelectric transformer, atmospheric plasma jet, decontamination I

This paper deals with the proof of concept and the preliminary evaluation of decontamination performances obtained with a plasma jet generated by a piezoelectric transformer. This low voltage supply solution (<10V) is investigated as a plasma jet device, compact and safe solution for the decontamination of medical thermo-sensitive devices. The principle of the piezoelectric generator is presented, followed by the optical spectroscopy of the plasma jet, the protocol conditions for the bactericidal effect observations and finally the reduction rates obtained on Pseudomonas aeruginosa and Staphylococcus aureus bacteria strains with an argon plasma jet at atmospheric pressure about 2.5W electrical input power.

INTRODUCTION

istorically the studies on plasma for biomedical issues started focusing on the disinfection capabilities of low temperature plasma in the early 90's [START_REF] Laroussi | Nonthermal decontamination of biological media by atmospheric-pressure plasmas: Review, analysis and prospects[END_REF][START_REF] Laroussi | Low temperature plasma-based sterilization: Overview and state-of-the-art[END_REF][START_REF] Laroussi | Perspective: The physics, diagnostics and applications of atmospheric pressure low temperature plasma sources used in plasma medicine[END_REF][START_REF] Gay-Mimbrera | Clinical and biological principles of cold atmospheric plasma application in skin cancer[END_REF]. Thanks to the wide range of plasma interactions with living cells, the recent research directions have been oriented towards therapy considerations [START_REF] Keidar | Plasmas for treating cancer: Opportunities for Adaptive and Self-Adaptive Approaches[END_REF][START_REF] Lu | Transcutaneous plasma stress: From soft-matter models to living tissues[END_REF]. Numerous plasma generators have been developed to satisfy the disinfection requirements of thermosensitive or chemo-sensitive tools. Some examples of plasma jet (or needles) solutions for different applications are briefly described in Table 1. Many difficulties such as a high disinfection level (particularly on high aspect ratio parts), a minimal duration of treatment, still have to be overcome [START_REF] Leggett | Bacterial spore structures and their protective role in biocide resistance[END_REF]. [START_REF] Weltmann | Antimicrobial treatment of heat sensitive products by miniaturized atmospheric pressure plasma jets (APPJs)[END_REF] 4-6kV pulsed source 100ns pulse width 2kHz repetition Gas flow: 5 slm, He / 1% O2 B. Atrophaeus [START_REF] Jiang | Pulsed Atmospheric-Pressure Cold Plasma for Endodontic Disinfection[END_REF] Wound healing Array of µplasma jets 20 kHz 1 kV RMS Gas flow: 1.5 slm, He 2nd degree burns [START_REF] Lee | An experimental burn wound-healing study of nonthermal atmospheric pressure microplasma jet arrays[END_REF] kINPen (neoplastools) 1MHz 1 kV / 3.5 W Gas flow: 3 slm, Ar Chronic wounds on domestic animals, clinical observations on leg ulcers [START_REF] Bekeschus | The plasma jet kINPen -A powerful tool for wound healing[END_REF] Risk assessment kINPen Genotoxicity and mutagenic potential of argon plasma [START_REF] Wende | Risk assessment of a cold argon plasma jet in respect to its mutagenicity[END_REF] 25 kHz 3 kV / 2mA Gas flow: A few slm, He Skin damage on mouse [START_REF] Kos | Safety aspects of atmospheric pressure helium plasma jet operation on skin: In vivo study on mouse skin[END_REF] Although these different sources operate at various frequencies ranging from kHz up to MHz with different DBD designs (most of the time a metal needle inserted in a quartz tube), they all require a high voltage source as input in order to initiate and sustain the discharge (depending on the gas, flow, dielectric configurations…). In some applications and depending on the required output power, such high voltage generators can be inappropriate. The proposed technology in this paper may be an alternative.

II. DESCRIPTION OF THE PLASMA SOURCE

A. Principle of a piezoelectric plasma generator This section briefly introduces the piezoelectric properties and then focuses on the use of a piezoelectric transformer device for plasma discharge. The piezoelectric property is the ability of some (mono or poly crystalline) materials to develop an electric potential when they undergo a mechanical stress [START_REF] Yang | An introduction to the theory of piezoelectricity[END_REF][START_REF] Ikeda | Fundamentals of piezoelectricity[END_REF]. Moreover this property is reversible.

H

The research on this topic really started in 2000's leading to a significant understanding of this technology [START_REF] Itoh | Plasma Sources Science and Technology, Discharge plasmas generated by piezoelectric transformers and their applications[END_REF][START_REF] Teschke | Piezoelectric low voltage atmospheric pressure plasma sources[END_REF]. Since then, the piezoelectric transformer as plasma generator has been explored in various applications such as an ozonizer [START_REF] Teranishi | A novel generation method of dielectric barrier discharge and ozone production using a piezoelectric transformer[END_REF], an excimer lamp VUV light source [START_REF] Teranishi | A compact excimer lamp constructed by piezoelectric transformer[END_REF], an ignitor for pulsed plasma thrusters [START_REF] Carazo | Piezoelectric transformers for space applications[END_REF], a micro-thruster for satellite positioning control [START_REF] Hutsel | Chargedparticle emission and self-biasing of a piezoelectric transformer Plasma source[END_REF], an ionic wind source [START_REF] Johnson | Piezoelectric transformers for low-voltage generation of gas discharges and ionic[END_REF], a source for deuterium or hydrogen neutron production [START_REF] Baxter | Hydrogen and deuterium ion extraction from a piezoelectric transformer plasma source[END_REF], or a patented source for ambient mass spectrometry at atmospheric pressure [START_REF] Neidholdt | Switched ferroelectric plasma ionizer[END_REF].

The first patent of piezoelectric transformer appears in 1931, explaining the concept [START_REF] Mclean Nicolson | Piezo electric crystal transformer[END_REF]. More than 20 years later, in 1954, C.A. Rosen et al. have described some designs which are still used at the present time [START_REF] Rosen | Electromechanical transducer[END_REF]. The piezoelectric technology is particularly suitable for compact devices as demonstrated by their use in micro-sensors and micro-actuators. Most of piezoelectric transformers are made with hard PZT ceramic (Lead Zirconate Titanate). The maximal power density of such hard piezoceramic can be estimated to 300W/cm 3 [START_REF] Flynn | Fundamental Limits on Energy Transfer and Circuit Considerations for Piezoelectric Transformers[END_REF] which suggests a wide range of possibilities. Nevertheless, it is also bounded by a set of technical limitations; -The Curie temperature limitation. Over this limit (which could be around 200°C depending on the ceramic composition), the remanent polarization vanishes and consequently the piezoelectric capability as well, -The mechanical break of the ceramic, induced by excessive mechanical stress.

The principle of the piezoelectric transformer relies on the association of two (or more) piezoelectric elements that are mechanically coupled. The first is excited by an external alternative electric supply applied by electrodes, resulting in mechanical vibrations (the reverse piezoelectric effect). This latter is qualified as the primary side, later referred to as "primary". The second element also vibrates, inducing an electric field (direct piezoelectric effect) and it is qualified as the secondary side. Thus, the operation relies on two successive electromechanical conversions, enabling a voltage/current adaptation, an impedance adaptation, a high insulation level. Since this conversion results from the conversion of mechanical energy (propagating vibrations), the performances of such a component are strongly dependent on the geometry, the operating frequency, the electrode distribution and the piezoelectric material properties. Significant description and modelling of piezoelectric transformer designs and particularities are further discussed in literature [START_REF] Yang | Piezoelectric transformer structural modeling -a review[END_REF][START_REF] Erhart | Bulk piezoelectric ceramic transformers[END_REF]. For optimal efficiency and a maximum output voltage, piezotransformers should be supplied at an operating frequency close to a resonant frequency of the structure. When the primary side is supplied close to one of the longitudinal resonant modes, a standing wave vibration occurs in the material and an electric potential distribution is obtained on secondary.

In the frame of plasma generation, the most interesting architectures are the high step-up voltage transformers: they can reach a voltage gain higher than 100 giving the opportunity to design a compact generator supplied with a very low input voltage.

The present piezo plasma generator uses a classical so-called Rosen-type transformer as illustrated in Fig. 1.

Fig. 1 Design of the step-up Rosen type transformer [START_REF] Laroussi | Inactivation of Bacteria by the Plasma Pencil[END_REF].

The various dimensions and material properties of the Rosen type transformer used in this study are summarized in Table 2. For plasma generation, the piezoelectric transformers can be used in two different ways: ▪ As a simple two-port step-up voltage component. The electric charges are collected on the secondary side by a metal needle [START_REF] Johnson | Characterization of a compact, low-cost atmospheric-pressure plasma jet driven by a piezoelectric transformer[END_REF], or between two electrodes with a narrow gap and separated by a dielectric layer [START_REF] Kartashev | Regimes of piezoelectric transformer operation[END_REF].

▪

Another configuration directly involves the secondary surface as a dielectric layer which supports the discharge [START_REF] Teranishi | Luminous phenomenon of silent discharge using a piezoelectric transformer[END_REF][START_REF] Itoh | Observation of light emissions around a piezoelectric transformer in various gases[END_REF]. In such a configuration, there is no need for the secondary electrode. This particular configuration prevents sparks (DBD-like configuration) and it benefits from the high dielectric constant of the material. Let us remember that the relative permittivity of PZT ceramic is about several hundred units (𝜀𝑟≈1000 for hard PZT ceramic). This latter implementation has been selected for this study.

Contrary to a passive dielectric layer in conventional DBD, the electric potential on the piezo surface is not uniform and depends on the excited resonant vibratory mode. Consequently, for the first longitudinal vibratory modes, the theoretical electric potential distribution is illustrated in Fig. 2(a). This distribution drives the plasma discharge location surrounding the transformer surface (Fig. 2(b)). Moreover, the electric potential amplitude is strongly a function of the input voltage supply, the mechanical damping coefficient and the operating angular pulsation. With the appropriate operating conditions this maximum potential value can reach several kV. The Piezoelectric Transformer (PT) is a resonant device which can be relevantly simulated by an equivalent electrical circuit. This latter is valid in the vicinity of the resonant mode and element values are dependent on the geometry and the material properties.

Fig. 3 The electric equivalent circuit of piezo transformer

As emphasized in Fig. 3, the PT is essentially capacitive due to the primary side. Its electrical behavior changes when supplied at its mechanical resonant frequency simulated by the two parameters Lm and Cm. The voltage gain is depicted by factor ψ. The expressions of the equivalent circuit parameters are not given in detail in this paper. However there is a considerable body of research devoted to the modelling [START_REF] Erhart | Piezoelectric Ceramic Resonators[END_REF].

As a consequence of the non-linear behavior of the plasma discharge and the load dependency of the voltage gain, the output voltage on the transformer termination is rather complicated to evaluate. This constitutes the main difference with conventional plasma generators and prevents the direct measurement of the plasma power. In the present case, the experimental output voltage can be evaluated from indirect methods such as laser interferometry. These considerations will be the purpose of future work and the present paper only mentions the electrical measurements from the primary side.

The output voltage of the transformer highly depends on the concordance of the device's supply and the resonant frequencies. Moreover, the resonant frequency tends to slightly shift with the steady state electric load (plasma load) and the transformer temperature rising. As a consequence, a control feedback is essential to track the operating point and to maintain a stable plasma jet. This optimal operating point is maintained by the admittance phase tracking [START_REF] Kahalerras | Analyses of temperature influence in piezoelectric transformers dedicated to plasma generation[END_REF]. Then, the used generator requires an input voltage of 2.7Vrms. Even if the streamer plasma regime implies a nonperiodic behavior, the electrical consumption can be evaluated by the measurement of the input current, resulting in an average input power of 2.5W. It must be noticed that this input power does not correspond to the power transmitted to the discharge (due to power losses), nevertheless it informs on the maximum order of magnitude.

For the preliminary bactericidal tests carried out with this piezoelectric technology, an elementary plasma stylus design has been made as illustrated in Fig. 4.

B. Plasma Diagnostics

The characterization of piezo-plasma jet was made by optical emission spectroscopy and by bidimensional emission imaging. An optical spectrometer (HR2000+, Ocean Optics, Grating 1800mm-1, slit 10μm, 200 to 1100nm, maximal optical resolution 0.5nm) coupled to an optical fiber (core diameter 600μm) was used to identify the plasma emission lines [START_REF] Kone | Investigation of the interaction between a helium plasma jet and conductive (metal)/ non-conductive (dielectric) targets[END_REF]. The measurement was realized in order to identify the typical optical emissions of the carrier gas and the reactive species due to the ambient air. The optical fiber was placed perpendicular to the piezo-plasma jet on a micrometric translation stage. Moving the fiber along the piezo-plasma jet, the linear translation system allowed to characterize the axial distribution of the species excited or ionized by the plasma. Moreover, a fast ICCD camera (Princeton PIMAX-2K-RB, 1024x1024) was placed beside the jet, perpendicularly to the source, in order to assess the 2D distribution of plasma emissions within the plasma jet. The jet length was determined from the ICCD camera data. The experimental setup has been described previously in [START_REF] Kone | Investigation of the interaction between a helium plasma jet and conductive (metal)/ non-conductive (dielectric) targets[END_REF][START_REF] Chauvet | Characterization of an asymmetric DBD plasma jet source at atmospheric pressure[END_REF].

In the standard operating conditions of the piezo-plasma stylus (1slm Ar gas flow in open air, 1.5Vrms, 53kHz, 2.5W input power), the emission spectra of the discharge were measured. Fig. 5(b) highlights the main emission lines of the plasma jet flowing in air, commonly observed by many authors [START_REF] Erhart | Piezoelectric Ceramic Resonators[END_REF][START_REF] Kahalerras | Analyses of temperature influence in piezoelectric transformers dedicated to plasma generation[END_REF][START_REF] Kone | Investigation of the interaction between a helium plasma jet and conductive (metal)/ non-conductive (dielectric) targets[END_REF][START_REF] Chauvet | Characterization of an asymmetric DBD plasma jet source at atmospheric pressure[END_REF][START_REF] Pereira | Study of a cold atmospheric pressure plasma jet device for indirect treatment of squamous cell carcinoma[END_REF] on a reduced wavelength range (300-800nm). Two spectra were recorded in this work. The first one corresponds to the plasma emissions produced inside the source, close to the piezo transformer edge (Fig. 5(a)). The plasma emission is mainly composed by Ar emission lines. The second one was obtained 15mm away from the stylus outlet (Fig. 5(b)), near the tip of the plasma jet. The UV and visible emissions correspond to the radiative de-excitations of argon, hydroxyl (OH radical) [START_REF] Wanga | Comparison between helium and argon plasma jets on improving the hydrophilic property of PMMA surface[END_REF], molecular nitrogen and oxygen [START_REF] Yi | Characteristic study of cold atmospheric argon plasma jets with rod-tube/tube high voltage electrode[END_REF]. The presence of emission lines relative to ambient air species demonstrates the interaction of the plasma jet with its environment. The presence of hydroxyl in the UV-B region (308 nm) is related to processes involving water molecules. The production of the excited nitrogen molecule (second positive system situated in the 300 to 450 nm range) can come from electron collisions from the fundamental as well as from Ar metastable excitation transfer processes to the nitrogen molecules. Oxygen (777 nm) is the last sign of plasma and ambient air interaction [START_REF] Yi | Characteristic study of cold atmospheric argon plasma jets with rod-tube/tube high voltage electrode[END_REF]. Additionally, the plasma spatial distribution is measured by fast ICCD camera (Fig. 6). Based on the lines of interest, determined from the emission spectra, two filters that allowed the characterization of species production at the outlet of the nozzle in contact with ambient air have been used. The ICCD camera was also coupled to bandpass interference filters (Thorlabs, 10nm as bandwidth) centered on specific central wavelength values. The first one, centered on 750nm, allowed the assessment of argon distribution. The second one, centered on 380nm, showed the nitrogen distribution. The highest argon emission intensity corresponds to the interior of the source. From the exit of the nozzle, argon emission intensity decreases until plasma extinction at several tens of millimeters distant from the exit. Nitrogen emissions start at the exit of the source. The emission intensity increases, reaches a maximum and then decreases. This maximum emission is located in the intermediate region of the plasma jet. Before this maximum, argon-air ratio is too high and plasma is dominated by argon emissions [START_REF] Volkov | Cold plasma interactions with plants: Morphing and movements of venus flytrap and mimosa pudica induced by argon plasma jet[END_REF]. After this maximum value is reached, argon-air proportion is too small and the plasma jet is not able anymore to excite ambient air species [START_REF] Wattieaux | Optical emission spectroscopy for quantification of ultraviolet radiations and biocide active species in microwave argon plasma jet at atmospheric pressure[END_REF]. The methodology relies on the adaptation of the EN 13697 protocol (bactericidal effect on non-porous surface). This standard protocol implies operating at ambient temperature (~20°C) and a significant effect is considered when the reduction rate is over 4log compared to the control sample. Each test has been duplicated at least twice to four times to ensure robustness of the experimental results.

The samples are prepared as follow: a 50μL droplet of the initial suspension (that contains between 1.510 8 CFU / mL and 5.010 8 CFU/ mL of bacteria in tryptone salt diluent) is deposited on a 2cm-diameter stainless steel substrate. Then samples are dried at 37°C in an incubator during 20 minutes before being treated with the plasma jet. It consists in directly and vertically applying the plasma jet on the surface carrying the bacteria inoculum, as illustrated in Fig. 7. The operating plasma discharge regime is comparable to a glow discharge, obtained by simply placing the plasma jet in contact with the inoculum without any particular layout. The experimental details of the procedure are presented in Table 3. The log reduction is finally evaluated for different exposure times for both bacteria strains. The authors are aware that several bactericidal actions operate during the exposure period, leading to the slope variation of the log reduction [START_REF] Barbeau | Low-temperature sterilization using gas plasmas : a review of the experiments and an analysis of the inactivation mechanisms[END_REF].

In this preliminary study, the purpose is to observe if the piezoelectric generator can provide significant bactericidal effect (4 log reduction and beyond) during a sufficient exposure period. Due to the fact that the presence of water in any plasma activated water (PAW) significantly favors the biocidal effect [START_REF] Hahnel | Influence of the air humidity on the reduction of bacillus spores in a defined environment at atmospheric pressure using a dielectric barrier surface discharge[END_REF][START_REF] Kamgang-Youbi | Microbial inactivation using plasma-activated water obtained by gliding electric discharges[END_REF], tests are carried out in two configurations: dry conditions and with the addition of 100μl-distilled-water droplet on the inoculum surface. Log reduction on both bacteria strains is presented in Fig. 8. C. Bactericidal performances Several observations can be deduced from results presented in Fig. 8. Firstly, the exposure of the dried inoculum to an argon gas flow without the plasma discharge (control sample) does not result in significant bactericidal effect: whatever the duration of the exposure period, the log reduction does not change in these conditions. On the contrary, after a few minutes exposure period to plasma discharge, log reduction is observable for both bacteria strains. Therefore, the following conclusions can be drawn: ▪

The longer the exposure time, the higher the reduction of bacteria, which is consistent with any other plasma treatments presented in research.

▪

As already demonstrated in research the Staphylococcus aureus (gram-positive) is more resistant than the Pseudomonas aeruginosa (gram-negative) [START_REF] Volkov | Cold plasma interactions with plants: Morphing and movements of venus flytrap and mimosa pudica induced by argon plasma jet[END_REF].

▪

The presence of water in the decontamination process radically improves the efficiency, including those observed with the S. aureus. After a 15-minutes-exposure, both strains have reached the maximum measurable reduction factor (over 6.73 log). Please note that the droplet water has completely evaporated after 10 minutes of exposure.

In spite of the low input power applied to the piezoelectric device (about 2.5W at 53kHz), the disinfection capacity is clearly visible. Obviously, the exposure time treatment has to be reduced to generate more interest in this technology. The piezoelectric component can be highly improved to be adapted to this new use. However these results are quite promising with regard to the maximum measurable value obtained.

D. The technical limits and improvement perspectives Even if the piezoelectric technology can bring some advantages in terms of product safety, compactness, low voltage supply, it is also subjected to some technical limits which require attention. Firstly, the electric potential significantly depends on the amplitude vibration produced in the piezo material. The vibration implies mechanical stress in a precise location. All necessary precautions should be taken to prevent brittleness, for example by a specific transformer design or by limiting the mechanical stress during the operation [START_REF] Benwell | A high voltage piezoelectric transformer for active interrogation[END_REF]. Moreover, depending on the plasma location and its regime, excessive discharge power density is likely to occur at an unsuitable location of the transformer surface. This can damage the material by erosion and local heating. This phenomenon is observed at the edge of the primary electrode [START_REF] Mai-Prochnow | Gram positive and gram negative bacteria differ in their sensitivity to cold plasma[END_REF]. Such a sparking event can be prevented by depositing a thin coating of a dielectric material on the piezo surface. However this dielectric layer should also be able to withstand the vibration of its substrate. Another solution can be to use other piezoelectric materials more resistant to erosion s(ex: single crystal lithium niobate). Finally, the piezoelectric property can vanish making the transformer inoperative if its temperature exceeds the Curie temperature: indeed at this temperature a reversible change in the crystalline occurs and results in the depolarization of ferroelectric material. The temperature rise is essentially due to the self-heating induced by the internal friction force. The typical Curie temperature limit is below 250°C for standard PZT hard ceramic. All these technical precautions actually limit the power of such a piezo plasma generator to a few watts. However, specific design and precautions may enhance the power availability and thus the resulting bactericidal properties. Finally, the modification of the feedgas mixture will also improve the bactericidal effect as widely demonstrated in literature.

V. CONCLUSION

Preliminary decontamination tests have been carried out with an atmospheric pressure plasma jet generated by a piezoelectric transformer. An elementary plasma stylus has been designed with a standard off the shelf transformer, supplied by few volt voltage source. To prevent the frequency discrepancy due to the thermal effect, a simple phase locked loop is added to maintain plasma discharge over time. This plasma jet relies on 1slm argon gas flow and is finally applied to two different bacteria strains following the recommendations of EN 13697 protocol. The resulting reduction rate obtained with this argon plasma jet has reached significant bacteria reduction. This low voltage solution gives rise to the opportunity to develop safe low voltage plasma generators. Nevertheless at this development stage, a specific design of the piezoelectric element is required. This should be associated to an appropriate feedback control and feedgas mixture to significantly improve these promising preliminary results.

Fig. 2 .

 2 Fig. 2. (a) Theoretical electric potential on the transformer surface for the first four longitudinal vibratory modes. (b) Plasma discharges for first three modes obtained at a pressure of 10mbar in air [37].

Fig. 4

 4 Fig. 4 Design of the piezo-plasma jet stylus (a) : (1) PT (2) electric connectors (3) stylus nozzle (4) stylus body (5) gas connector (6) gas tube. The piezoplasma jet in operation (b) with a 1slm Ar gas flow

Fig. 5 .

 5 Fig. 5. Main emission lines of the plasma, (a) inside the source, 8 mm before the stylus outlet, and (b) in the plasma jet, 15 mm away from the source outlet.

Fig. 6 .

 6 Fig. 6. 2D normalized imaging of the plasma inside and outside the source, 750nm argon filter (left) and 380 nm nitrogen filter (right).

  the piezo-plasma jet efficiency used for the first time as a disinfection device is carried out with two different bacteria strains (Gram-positive and Gram-negative): ▪ Pseudomonas.aeruginosa -Institut Pasteur -CIP 103 467 ▪ Staphylococcus.aureus -Institut Pasteur -CIP 4 83

Fig. 7 .

 7 Fig. 7. The plasma jet treatment protocolAfter the plasma treatment, the stainless steel samples are transferred into a 10mL container of the tryptone-salt diluent where the microorganisms are removed from the disk by shaking with glass beads. This mixture is 10 to 100 diluted and 1mL is grown in TSA (15ml to 20ml) in an incubator at 36 ° C during 224h. Only Petri dishes ranging from 15 up to 300 CFU + -10% are used for counting. The operating plasma discharge regime is comparable to a glow discharge, obtained by simply placing the plasma jet in contact with the inoculum without any particular layout. The experimental details of the procedure are presented in Table3. The log reduction is finally evaluated for different exposure times for both bacteria strains. The authors are aware that several bactericidal actions operate during the exposure period, leading to the slope variation of the log reduction[START_REF] Barbeau | Low-temperature sterilization using gas plasmas : a review of the experiments and an analysis of the inactivation mechanisms[END_REF]. In this preliminary study, the purpose is to observe if the piezoelectric generator can provide significant bactericidal effect (4 log reduction and beyond) during a sufficient exposure period.

Fig. 8 .

 8 Fig. 8. The log reduction according to the exposure time and operational conditions IV.RESULTS AND DISCUSSION

TABLE 1 :

 1 EXAMPLES OF PLASMA JET SOURCES

	Application field	Operating conditions	Target / issue
		Pulsed RF source	Melanoma cells (in
	Oncology	1.2 kV RMS / 3 W Gas flow: 3 slm ,	vitro & in vivo tests) [8]
		98% Ne / 2% Ar
		2.6 kHz	E. Faecalis [9]
	Dentistry	1.1 W
		Air
		27.12MHz RF source	B. Atrophaeus
	Desinfection	20 W	E. coli
		Gas flow: 20 slm Ar

TABLE 2 :

 2 PIEZO TRANSFORMERS CHARACTERISTICS

	ceramic	Hard PZT
	total length	30.4 mm
	width	7.6 mm
	thickness	2.3 mm
	operational vibratory mode	/2
	operating frequency	53kHz
	relative permittivity in poling axis at constant stress 𝜀 33 𝑇	1600
	density 𝜌	7900 kg/m3
	𝐸 transverse compliance at constant 〈𝐸〉 field 𝑠 11 longitudinal compliance at constant 〈𝐸〉 field 𝑠 33 𝐸	1.19×10 -11 m 2 /N 1.39×10 -11 m 2 /N
	transverse piezoelectric coefficient 𝑑 31	-155×10 -12 m/N
	longitudinal piezoelectric coefficient 𝑑 33	3640×10 -12 m/N

TABLE 3

 3

	: EXPERIMENTAL CONDITIONS
	Bacteria strain	Operating conditions
	S. Aureus	▪ ▪	ON / OFF device without / with 100 µL droplet water
		▪	1.5 cm target distance
	P. Aeruginosa	▪ ▪	Gaz flow rate :  1slm argon Exposure duration: 2, 5, 10, 15, 20 min

Acknowledgements: This work has been supported by the French National research agency < ANR-15-CE19-0025 >.