: the optimal value of  : the expected profit function

We consider a single-product and single-period (newsvendor) problem with the following sequence of events:

 Let Q be the quantity that the inventory manager orders from his supply system.  He will receive goods, update the information system by scanning products and store them in the warehouse.  Because of errors, , i.e. what is physically available in inventory, may not be equal to , i.e. what the information system shows as being available.



Just before the beginning of the selling period, the inventory manager will receive a cumulative online order from the final customers. He will compare the total quantity requested by the customers with the IS inventory record to accept or decline orders. If the cumulative order is less than , the e-retailer will accept all the orders. If not, he will only accept orders summing up to the IS inventory.  Later on, products will be shipped to the customers. All the orders that the e-retailer has committed himself to should in principle be satisfied. However, this may not always be the case due to inventory inaccuracies. Such a situation occurs for instance when is larger than , and the demand is larger than .

The expected profit formulation

Based on the sequence of events previously described the commitment C and the period sales could be written as follows:

For a given vector , the profit achieved by the inventory manager could be deduced as follows:

The Inventory inaccuracy issue under a multiplicative error structure Selma Khader*, Yacine Rekik**, Jean-Pierre Campagne* and Aris Syntetos***

					
		*DISP Lab, INSA Lyon, 20, avenue Albert Einstein -69621 Villeurbanne, France
		(e-mail: selma-afakh.khader@insa-lyon.fr , jean-pierre.campagne@insa-lyon.fr )
		**DISP Lab, EMLYON Business School, 23 av. Guy de Collonges, 69134 Ecully, France
					(e-mail: rekik@em-lyon.com)
	*** Centre for Operational Research and Applied Statistics, Salford Business School, University of Salford, Manchester M5
					4WT, UK
				(e-mail: A.Syntetos@salford.ac.uk)
	The notations used in this paper are as follows:
		: the random variable representing demand
		: the PDF (CDF respectively) characterizing D
		: the average of D
		: the standard deviation of D
		: the random variable representing IS (PH) errors
		: the average of	(	)
		: the standard deviation of	(	)
		: the PDF (CDF respectively) characterizing
		: the PDF (CDF respectively) characterizing
		: the unit selling price
		: the unit purchase cost
		: the unit salvage cost
		: the unit cost paid for a non-satisfied commitment
		: the ordering quantity
	

The first and the second terms of the profit correspond to the margin achieved from sales and salvages respectively. The third term is associated with purchase costs and the last one corresponds to the penalty paid when a commitment is not satisfied, i.e., when . The profit function could also be written as follows:

After some basic algebra, the profit could be rewritten as:

The previous expression of the profit could be interpreted as follows:

: represents the expected sales revenue : represents the cost paid in an overstocking situation : represents the underage penalty incurred if a demand is not satisfied when answering customers' requests : represents the underage penalty incurred when a commitment is made and then not respected Based on the previous expression explanation, we could define the following unit costs:

, the unit type 1 shortage cost paid when the IS level is not enough to satisfy a demand , the unit type 2 shortage cost paid when the PH level is not enough to satisfy a commitment , the unit overage cost

In the classical inventory literature, only and are considered since the IS level is implicitly assumed to be equal to the PH inventory level. Considering the multiplicative error structure ( and ) and applying the expectation to profit function permits us to deduce as follows:

After some simplification of profit equation we obtain the final expression

The optimal inventory policy The concavity of the expected profit function is not always verified. Next we provide the conditions of the existence of an optimal ordering policy permitting to minimize the expected profit.

Let's first defined three constants , and that could be easily calculated if the costs as well as the errors' parameters (average and standard deviation) are used: solves the following equation Calculating the first derivative of the expected profit function and setting it equal to zero permits us to deduce that the optimal ordering quantity , as well as the conditions of its existence, is as it is stated in the following theorem:

Theorem:

The optimal ordering quantity maximizing the expected profit function when errors are taken into account should satisfy (1):

We decomposed this equation on three parts (X 1 , X 2 , X 3 ):

The behaviors of and are similar to the cumulative function F (fig. 1)

Fig; 1 The behaviors of and So the equation (1) will have this form: with Equation (1) might have zero, one or two solutions depending on the values taken by the constants , and . The following figures represent the behaviors of After studying these three functions, can have almost six behaviors, and the number of solutions will depend on the number of intersection between and Here we present these six behaviors:

Q So as we can see the equation can have 1, 2 or 0 solutions and this number depends on the sign and order of superiority of C1, C2 and C3. And in order to choose the right solution, we have to find the sign of .

C1


Wilma ; Polini
email: polini@unicas.it

Massimiliano Marziale

A review of two models for tolerance analysis of an assembly: jacobian and torsor

Keywords: TOLERANCE ANALYSIS, TOLERANCING user): jacobian model, torsor model tolerance analysis, jacobian model, torsor model

come

A review of two models for tolerance analysis of an assembly: jacobian and torsor

Introduction

There is a strong need for industries to produce high precision assemblies at lower costs.

Therefore, there is a strong need to use tolerance analysis to predict the effects of the tolerances that have been assigned to the components of an assembly on the functional requirements of the assembly itself. The aim of the tolerance analysis is to study the accumulation of dimensional and/or geometric variations resulting from a stack of dimensions and tolerances. The results of the analysis are meaningfully conditioned by the adopted mathematical model. Some are the models proposed by the literature to carry out a tolerance analysis of an assembly constituted by rigid parts. The foremost works are found in Requicha that introduced the mathematical definition of the tolerance's semantic [START_REF] Requicha | Toward a theory of geometric tolerancing[END_REF][START_REF] Requicha | Mathematical definitions of tolerance specifications[END_REF]) and proposed a solid offset approach for this purpose. Since then, a lot of models are proposed by the literature: the vector loop uses vectors to represent relevant dimensions in an assembly [START_REF] Chase | General 2-D tolerance analysis of mechanical assemblies with small kinematic adjustments[END_REF][START_REF] Chase | Including geometric feature variations in tolerance analysis of mechanical assemblies[END_REF][START_REF] Gao | Generalized 3-D Tolerance Analysis of mechanical assemblies with small kinematic adjustments[END_REF]; the variational uses homogeneous transformation matrix to represent the variability of an assembly due to tolerances and assembly constraints [START_REF] Martino | Application of variational geometry to the analysis of mechanical toleranced[END_REF][START_REF] Boyer | Modeling spaces for toleranced objects[END_REF][START_REF] Gupta | Variational solid modelling for tolerance analysis[END_REF]; the matrix uses displacement matrix to describe any roto-translational variation a feature may be subjected to [START_REF] Salomons | A computer aided tolerancing tool II: Tolerance analysis[END_REF][START_REF] Desrochers | A Matrix Approach to the Representation of Tolerance Zones and Clearances[END_REF]; the tolerance-map uses hypothetical solid including points in n-dimensions which represent all possible variations of a feature or an assembly [START_REF] Davidson | A new mathematical model for geometric tolerances as applied to round faces[END_REF][START_REF] Mujezinovic | A new mathematical model for geometric tolerances as applied to polygonal faces[END_REF][START_REF] Ameta | Influence of form on Frequency Distribution for 1-D clearance which is generated from Tolerance-Maps[END_REF][START_REF] Shen | Automatic generation of min/max tolerance charts for tolerance analysis from CAD models[END_REF][START_REF] Singh | Worst-Case Tolerance Analysis of a Self-Aligning Coupling Assembly using Tolerance-Maps[END_REF]; the jacobian uses an approach derived from the description of kinematic chains in robotics to formulated the displacement matrices; and the torsor uses screw parameters to model three dimensional tolerance zones. In the literature, some studies compare these models for tolerance analysis by dealing with their general features [START_REF] Hong | A comprehensive review of tolerancing research[END_REF][START_REF] Shen | A comparative study of tolerance analysis methods[END_REF]. Other studies compare the main Computer Aided Tolerancing softwares that implement some of the models of the tolerance analysis [START_REF] Salomons | Current Status of CAT Systems[END_REF][START_REF] Prisco | Overview of Current CAT Systems[END_REF]; but these studies focus the attention only on the general features of the considered models. Moreover, there does not exist a paper that compares the different analytical methods on the basis of a case study that underlines in a clear way all the advantages and the weakness; therefore, no guidelines exist to select the method more appropriate to the specific aims.

The purpose of this work is to analyse two of the most significant models for tolerance analysis of rigid-parts assembly: the model called jacobian and the model called torsor. The comparison of the models starts from their application to a case study. Dimensional and geometrical tolerances have been considered as part of stack-up functions. The worst and the statistical approaches have been taken into account. The application of the Envelope Principle (ASME 1994) and of the Independence Principle (ISO 1985) has been deeply investigated.

Finally, the evolution of these two models, called unified jacobian-torsor model, is presented too; it should overcome the limits of the two compared models. Two further works of the authors compare the other main developed models of the literature: the matrix and the vector loop models [START_REF] Marziale | A Review of two models for tolerance analysis of an assembly -Vector Loop and MATRIX[END_REF] and the variational models (Marziale and Polini 2010).

Section 2 gives an overall explanation of the jacobian and the torsor models. Section 3 gives a comprehensive comparison of the two models by means of a case study that is characterized by 2D tolerance stack-up functions. It offers some guidelines for those who will have to make the choice too. Finally, Section 4 presents the evolution of the two models, the unified jacobian-torsor, that seems to overcome some of the limits of the jacobian and torsor models.

Tolerance analysis models

Jacobian model

Jacobian model uses pairs of Functional Elements (FEs) to represent both the dimensions and the variations of an assembly. The Functional Elements pairs are arranged in chains representing those dimensions that stack together to determine the resultant assembly Functional Requirement (FR). This model is based on the idea that the tolerances allow small linear and angular dispersions to a functional element with respect to its nominal position, orientation and form (this last is usually neglected). These dispersions are expressed by pairs of Functional Elements used to model a stack around the Functional Requirement of an assembly. Two types of FE pairs in a tolerance stack can be distinguished: internal pairs, which are pairs of the same part, and kinematic pairs, which are pairs of different parts in contact; however these two types of elements are schematized in the same way.

Based on the results sought in the kinematic of robot, a generic dispersion in a pair of Functional Elements can be expressed by a set of six virtual joints and coordinate frames (see Figure 1). In this way the spatial relationship relating the position and the orientation of the terminal FE, expressed in the Datum Reference Frame (DRF) of the base FE, is given by [START_REF] Lafond | Jacobian-based modeling of dispersions affecting predefined functional requirements of mechanical assemblies[END_REF], Laperrière et al. 1999): where j i D is the variable quantifying the translation from frame i to frame j (linear dimension of the part), and j i Ω is the variable quantifying the rotation from frame i to frame j (angular dimension of the part). It follows that the relative position and orientation of any FE in the stack of "n" elements, with respect to the DRF of the base FE, as a result of both nominal dimensions and degrees of freedom of the FEs involved in the tolerance stack, is given by:

F o r P e e r R e v i e w O n l y ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )               Ω ⋅ Ω - Ω Ω ⋅ Ω - - Ω ⋅ Ω - Ω ⋅ Ω ⋅ Ω - Ω ⋅ Ω - Ω ⋅ Ω ⋅ Ω - Ω ⋅ Ω Ω ⋅ Ω - Ω ⋅ Ω ⋅ Ω Ω ⋅ Ω Ω ⋅ Ω ⋅ Ω = =
n n n T T T T 6 6 6 12 6 6 0 6 0 ... - ⋅ ⋅ ⋅ = (2)
where "n" represents the total number of FE pairs (both internal and kinematic) involved in the tolerance stack. The global deviation of the FR, expressed in DRF of the first feature R 0 , can be expressed as: For small rotational virtual joints, the i th column of the Jacobian matrix J i is computed as:

[ ] [ ] [ ]               ⋅ =       - - n FE i FE n FE n n n i FE J J J J J J
( )         - = - - - 1 0 1 0 0 1 0 i i n i i z d d x z J r r (4)
while for small translational virtual joints, there is no contribution to small rotational displacements of the point of interest and the i th column of the Jacobian matrix J i is computed simply as:

        = - 0 1 0 r r i i z J (5)
where

1 0 i z - r is the 3 rd column of 1 0 i T -; and 1 0 i d - r is the last column of 1 0 i T -. Equation (3) relates
the small dispersions of the assembly FR to the virtual joints of the FE pairs, that form the stack, by a Jacobian matrix that is easily to evaluate from the nominal position of the features. Despite this advantage, the virtual joints displacements, that are the independent variable in equation (3), are difficult to relate to the tolerances applied to the assembly components.

To better understand this method, the basic steps to build a stack of Functional Element's pairs and to carry out a tolerance analysis are given below:

(1) Identify the assembly Functional Requirement and the Functional Element's pairsthe first step is to identify the Functional Requirement of the assembly and the Functional Elements that are involved in the stack. Therefore, the Functional Elements are arranged in consecutive pairs to form the Functional Requirement stack.

(2) Locate the virtual joints and the DRFs of each FE -the next step is to locate the virtual joints and the DRFs of each FE. Therefore, the transformation matrix (1) is calculated for each FEs.

(3) Create the chain and the Jacobian matrix -The Functional Requirement chain (2) is evaluated by means of the transformation matrix evaluated in the step 2. Therefore, the Jacobian matrix of the assembly Functional Requirement is evaluated too by equations ( 4) and ( 5).

(4) Derive the Functional Requirement equation -once calculated the Jacobian matrix of the Functional Requirement, the dispersion of the virtual joints, due to the applied tolerances, have to be derived. Therefore, the displacement of the Functional Requirement may be evaluated and its equation (i.e. the stack-up function) may be formalized.

(5) The stack-up function may be solved by means of the usual methods of the literature (Whitney 2004) for both the worst case or statistical case approaches.

A fundamental consideration about this model is that it is based on both a Technologically and Topologically Related Surfaces (TTRS) [START_REF] Clément | The TTRSs: 13 Constraints for Dimensioning and Tolerancing[END_REF]) and a positional tolerancing [START_REF] Clément | Cotation tridimensionelle des systèmes mécaniques, théorie & pratique[END_REF]) criteria; therefore, the tolerances of a generic drawing need to be translated according with the previously described criteria before performing the analysis. Moreover, this model may be applied to assemblies involving joints which makes a linear structure among the parts (stack-up function, see Figure 2a), while joints which makes a complex structure among the parts (network function, see Figure 2b) are not supported.

Torsor model

The torsor model uses screw parameters to model three dimensional tolerance zones [START_REF] Desrochers | Modeling three dimensional tolerance zones using screw parameters[END_REF][START_REF] Legoff | Geometrical tolerancing in process planning: a tridimensional approach[END_REF]. Each actual surface of a part is modelled by a substitute surface. A substitute surface is a surface that has the shape of the nominal surface and it is used to model the actual surface. A substitute surface is characterized by a set of screw parameters which are the deviation of the substitute surface from the nominal one. For each of the seven types of tolerance zone, there are the correspondent screw parameters obtained by annulling the ones that leave the surface invariant in its local frame. The obtained screw parameters are arranged in a particular mathematical operator called "torsor".

Considering a generic feature, if u A , v A , w A are the translation parameters of the point A, and α, β, γ are the rotation angles (considerated small) as regards to the nominal position, the torsor of point A is given by: where R is the DRF where the screw parameters are evaluated. Once known the torsor of point A, the torsor of point B may be evaluated as:

R A A A A w v u T           = γ β α (6)
R B B B B w v u T           = γ β α (7)
where:

          ⋅ + ⋅ - ⋅ - ⋅ + ⋅ + ⋅ - =           ⋅           - - - +           =           y x A z x A z y A z y x A A A B B B AB AB w AB AB v AB AB u AB AB AB w v u w v u α β α γ β γ α β α γ β γ 0 0 0 (8)
where AB x , AB y and AB z qre the vectors of the distance between points A and B along the x, y and z axes respectively.

To model the interaction between the parts of an assembly, three kinds of torsors (or Small Displacement Torsor SDT) may be defined [START_REF] Ballot | Geometrical behaviour laws for computer aided tolerancing[END_REF][START_REF] Ballot | A computational method for the consequences of geometric errors in mechanisms[END_REF]): a part SDT for each part of the mechanism (A, B, …) to model the displacement allowed to the part; a deviation SDT for each surface (A1, A2, …) of each part to model the geometrical variations of the surface; a gap SDT between two surfaces linking two parts to model the joint properties.

Therefore, a union of the set of SDTs that are involved at the joints is used in order to obtain the global behaviour of the mechanism. This may be bone by considering that, with the worst case approach, the cumulative effect of a simple chain of n-elements is simply expressed by adding the single components of the torsors:

n n n T T T T / 1 2 / 1 1 / 0 / 0 ... - + + + = (9)
(it is to observe that to compute this sum is necessary that the components of all the single torsors are referred to the same point B and in the same datum reference frame R).

The basic steps of the torsor model are (Villeneuve et al. 2001):

(1) Identify the elements of the parts and the relations among them -the first step is to identify the elements of the parts and the relations among them; these information are (2) Define the parameters of the mechanism -a deviation torsor has to be associated to each surface of the parts, therefore, a characterization of the global SDT of each part has to be done. The shape of the gap torsor, that is associated to each joint according to the functional conditions required by the assembly, has to be defined too.

(3) Computate the cumulative effect of the torsors involved in each stack in order to evaluate the Functional Requirements by equation ( 9).

Finally, some fundamental considerations are needed. The first is that this model is developed under the hypothesis to use the TTRS and the positional tolerancing criteria. The second is that the solution of stacks arranged in a network is not completely developed in spite of the different works produced in the last years (Teissandier et al. 2007[START_REF] Franciosa | Modelling and Simulation of Assembly Constraints in Tolerance Analysis of Rigid Part Assemlies[END_REF]). The third is that the torsor components assume a double meaning. In a first approach the small displacement torsor components are simple parameters and they are computed by means of common algebraic rules. An example of this approach is in [START_REF] Legoff | Geometrical tolerancing in process planning: a tridimensional approach[END_REF] where a tolerance problem involving network functions is solved. In the second approach the small displacement torsor components are admissible intervals according with the applied tolerance ranges. An example of this approach is in (Teissandier et al. 1999). The first approach gives a solution to the tolerance analysis problems very similar to the other approaches of the literature; while the second approach may theoretically relate the variability range of the Functional Requirements of the assembly to the assigned tolerance ranges. However, this second approach needs to compute the small displacement torsor components, that are intervals, by means of the arithmetic by interval, that is not fully developed yet. In the following we make reference only to this second approach, since its potentialities seem more interesting in tolerance analysis.

Models comparison

Case Study

To compare the two models previously described the case study shown in Figure 3 has been used. The two-dimensional geometry of the example assembly is made of a rectangular box containing two disk-shaped parts. The width g of the gap between the top disk and the upper surface of the box is assumed as the functional requirement to be investigated by the analysis.

Goal of the tolerance analysis problem is to identify the tolerance stack-up function that defines the variability of g, and describes it as a function of the geometries and tolerances of the components involved in the assembly.

Tolerance analysis is based on the dimensional and geometrical tolerances illustrated in Figure 4. The example is adapted from a real-life industrial application and properly simplified to make it easier to be presented and discussed in this context. The applied tolerancing scheme, which may not appear as entirely rigorous under the viewpoint of a strict application of standardized tolerancing rules, is directly derived from the current practice adopted for the actual industrial product.

The case study is representative of all the main aspects and critical issues involved in a typical tolerance analysis problem, and it is simple enough to allow for applying a simplified manual computation procedure for obtaining the extreme values of the gap g, for the special case where only dimensional tolerances are considered. The manual computation is based on searching for the worst-case conditions, i.e. the combinations of part dimensions that give origin to the maximum and minimum gap values; since no geometric tolerances are considered, part geometries are assumed at nominal states.

The maximum value of the gap has been calculated by considering the maximum height and width of the box, together with the minimum value of the radius of the disks:

(

) ( )

mm
= - - - - ⋅ - - = (11)
The variability of the gap is the difference between the maximum or the minimum values and the nominal one:

( ) ( ) ( ) ( ) mm g g g mm g g g N N 78 . 0 4909 . 0 2702 . 1 84 . 0 2702 . 1 1064 . 2 dim min 2 dim dim max 1 dim - ≅ - - = - - = ∆ + ≅ - = - + = ∆ (12)
Although operating on a simplified problem (geometrical tolerances are neglected) the manual computation of the gap boundary values provides a useful support for the quantitative comparison of the five methods, at least when they are applied by considering dimensional tolerances only. The manually-obtained, extreme gap values will be used as reference values later on, then the results of the five methods will be discussed.

Jacobian model solution

Dimensional tolerances only

The jacobian model of the case study is made under the simplified hypothesis to consider as fixed at 90° the orientation of the four sides of the box. This simplification is needed to avoid the network in the assembly. Indeed it is to observe (see Figure 3) that the functional requirement g has to be measured between the top side of the box and the second disk. Being the second disk assembled with the sub-assembly box-disk 1, firstly the assembly between part 1 and part 2 has to be solved. Therefore, the assembly between the part 3 and the subassembly box-disk 1 has to be solved. Once indicated with x 1 and x 2 the dimensions of the box, with x 3 , and x 4 the diameter of the two disks and with U 1 , U 2 , U 3 , U 4 the assembly variables (see Figure 5), the simplification adopted makes it possible to directly solve the assembly problem as:

3 1 x U = 3 2 x U = (13) F o r P e e r R e v i e w O n l y 4 1 3 x x U - = ( ) ( ) 2 4 3 1 2 4 3 3 4 x x x x x x U - - - + + =
Therefore, the functional requirement g and the functional elements pairs of the case study are identified (Figure 5). The first functional elements pair is associated with the points G and Ω and it is an internal one. The second functional elements pair is associated with the points Ω and O 1 , while the third functional elements pair is associated with the points O 1 and O 2 ; they are both externals. The last internal functional elements pair is associated with the points O 2 and H. The required functional requirement g is correspondent to the functional elements pair is associated with the points G and H, and it is evaluated as the chain of the four functional elements pairs just identified. At this point, is possible for each FE to locate the virtual joints and the reference frames, and to evaluate the transformation matrices 24 0

3 0 2 0 1 0 ,..., , , T T T T
with the equations ( 1) and ( 2). The matrix of the total transformation is equal to:

( ) ( )             - - - + + + + - = 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 2 4 3 1 2 4 3 4 3 2 24 0 x x x x x x x x T (14)
Therefore, the Jacobian matrix of the functional requirement is evaluated using equations ( 4) and ( 5

                                                                            - - - - - - - - - - - - - - - = 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 4 4 4 3 1 3 3 4 3 1 1 4 4 1 x x x x x x h x h x x x x x h h x x J T (15) where ( ) ( ) 2 4 3 1 2 4 3 4 3 x x x x x x x h - - - + + + = .
Once calculated the Jacobian matrix of the functional requirement pair, the stack-up function may be formalized considering that the requirement ∆g must be evaluated as the translation of point H long the -Z 0 axis (Figure 5), and then as the third component of equation ( 3). It is: , 1, 2, 6, 7, 8, 12, 13, 14, 18, 19, 20) is the translation along the i th axis and i z δφ (i=3, 4, 5, 9, 10, 11, 15, 16, 17, 21, 22, 23) is the rotation around the i th axis. This equation relates the functional requirement ∆g to the virtual joints displacements assigned to ) ( )

( ) ( ) 18 12 3 1 11 6 4 1 5 0 24 z z x x x z z x x z z z g δ δ δφ δ δφ δ δ - - - - ⋅ + - - ⋅ + - = - = ∆ (16) where i z δ (i=0
4 3 1 2 4 3 1 2 4 3 12 2910 . 1 2910 . 1 2582 . 0 x x x x x x x x z δ δ δ δ ⋅ + ⋅ + ⋅ - =       - - - + ∇ = 4 18
x z δ δ = and then is:

4 3 2 1 2910 . 2 2910 . 2 2582 . 0 x x x x g δ δ δ δ ⋅ - ⋅ - + ⋅ = ∆ (19)
Once obtained the required stack-up function, it can be solved with the usual methods of the literature. For example, for the worst case approach (Whitney 2004):

( )

± ≅ ± = = ⋅ + ⋅ + + ⋅ ± = ⋅ ∑ ± = ∆ = ( 20
)
while for statistical case approach:

( )

mm t S g i i i SC 53 . 0 5281 . 0 4 1 2 ± ≅ ± = ∑ ⋅ ± = ∆ = (21)

Dimensional and geometrical tolerances

If the dimensional and the geometrical tolerances are considered, none changes as regards to the previous case, since it has been adopted the simplification to consider fixed the angles in order to avoid the network. In fact, to solve the stack-up function, it is needed to relate the virtual joints displacements to the tolerances assigned on the components. However, the form Therefore, the simplification to consider fixed the angles of the box, due to the need to avoid network, causes the assigned geometrical tolerances do not produce any effects and the results are the same as the previous case where are considered only dimensional tolerances.

Moreover, the application of the Envelope Principle or of the Independence Principle does not produce any effect for the jacobian model.

Torsor model solution

Dimensional tolerances only

The required functional characteristic g has to be evaluated by considering that there is a network among the components of the assembly. At the state of the art, the solution of parallel chains through the torsor method is not completely developed. Therefore, the simplification to consider fixed the angles of the box has been used in order to avoid the network. This simplification may solve the assembly problem as showed in equations ( 13).

The first step of the method is to identify the elements of the parts (see Figure 6), and the relations among them; these information are reported in the surfaces graph of the case study (see Figure 7). Considering the angles of the box as fixed, the network can be solved and, therefore, the surfaces graph is simplified as showed in Figure 8. The cumulative torsor G is expressed as:

R H H H H g v u T T T G           ∆ = + + - = γ β α 1 , 3 3 / 1 1 , 1 (22)
where T 1,1 is the torsor of feature 1 of part 1 (the box); T 3,1 is the torsor of feature 1 of part 3 (the circle 2); and T 1/3 is the torsor of the link between part 1 and part 3. Therefore, the functional requirement ∆g is expressed by the translation component along the global Z axis. The next step is to evaluate the components of the torsors (indeed it is enough to evaluate the third components due to translation). For the T 1,1 torsor, with reference to Figure 9 and Table 1 and considering that the case study is a 2D problem on x-z plane (i.e. α=0, γ=0, v=0), it is:

1 , 1 1 , 1 1 , 1 1 , 1 R M M w T           - - - - = β (23)
where M is the median point of the feature. Considering that the point of interest is H, from equation ( 8):

          - - - =           - - ⋅           - - - - - +           - - =           1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 5 2702 . 1 2703 . 1 5 0 0 0 β β β β M M H H H w w w v u (24)
and then from equation ( 7):

1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 5 2702 . 1 R M M w T           - - - - - = β β β (25)
This torsor (note that only three of its components are not null) is expressed in the local frame of the feature and it has to be expressed in the global reference frame R (Figure 9). In this simply case it is possible to note that the local y-axis coincides with the global Y-axis, therefore, the angle β 1,1 is the same too. The local x and z axes are the inverse of the global X and Z axes respectively, therefore, the correspondent translation needs to be inverted. It is:

( ) R M H w T           - - - - - - = 1 , 1 1 , 1 1 , 1 1 , 1 1 , 1 5 2702 . 1 β β β (26)
Considering the simplification of considering fixed the angles of the box, β 1,1 = 0 and therefore:

R M H w T           - - - - - = 1 , 1 1 , 1 0 (27)
( ) ( ) R H t t t t T                     + + + - - - - - - = 2 , 2 0 2 1 , 1 2 1 , 1 1 , 1 (28)
where t 1,1 is the thickness of the tolerance zone of S 1,1 , and t 2 is the dimensional tolerance of the dimension x 2 . Note that this torsor shows the admissible range of variations of the small displacements associated to the feature. In the same way the torsor T 3,1 of feature 1 of part 3 may be computed; it results:

R H t t t t T                     + - - -       + - - = 2 , 2 0 2 , 2 4 4 4 4 1 , 3 (29)
where t 4 is the dimensional tolerance of the dimension x 4 .

The evaluation of the torsor linking part 1 and part 3 (T 1/3 ) is very difficult because it needs the solution of the networks among the components of the assembly. By adopting the simplification that the angles of the box are fixed, it is possible to solve this problem as the sum of two terms:

3 / 2 2 / 1 3 / 1 T T T + = (30)
The first term (T 1/2 ) is the torsor of the link between part 1 and part 2. With reference to Figure 9 it is:

( ) ( ) ( ) ( ) R H t t t t t t t t T                     + + + - - -       + + + - - = 2 , 2 0 2 , 2 3 3 , 1 3 3 , 1 3 2 , 1 3 2 , 1 2 / 1 (31)
The second term (T 2/3 ) is the torsor of the link between part 1+2 and part 3. With reference to Figure 9 it is:

F o r P e e r R e v i e w O n l y ( ) ( ) ( ) ( ) R H t t t t t t t t t t t t t t T                     ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ - - -       + + + + + + + - - = 2 2910 . 1 2910 . 1 2582 . 0 , 2 2910 . 1 2910 . 1 2582 . 0 0 2 , 2 4 3 1 4 3 1 4 3 1 4 , 1 4 3 1 4 , 1 3 / 2 (32)
where t 1,2 , t 1,3 , and t 1,4 are the thicknesses of the tolerance zones of S 1,2 , S 1,3 and S 1,4 ; and t 1 , t 3 , and t 4 are the tolerances on the dimensions x 1 , x 3 and x 4 . Therefore, the functional requirement is:

( ) 2 2910 . 2 2910 . 2 2582 . 0 13 11 4 3 2 1 t t t t t t g + + ⋅ + ⋅ + + ⋅ ± = ∆ (33)
Now, it is necessary to relate the thicknesses of each tolerance zone assigned to each feature to the tolerances required on the components. This is another critical step of the torsor model. However, under the simplified hypothesis adopted (i.e. fixed angles of the box) and by considering only dimensional tolerances, it may have:

mm t t
± ≅ ± = ⋅ + ⋅ + + ⋅ ± = ∆ (35)
as obtained with the jacobian approach. It may be added that the torsor method does not allow to evaluate the results due to a statistical approach, since the torsor's components are considered the extreme possible intervals of the small displacements, and this is not compatible with the statistical approach where a probably density function is to each parameter.

Dimensional and geometrical tolerances

If the dimensional and the geometrical tolerances are considered, equation ( 30) is still valid (under the hypothesis of fixed angles of the box), and it is always needed to relate the thickness of the tolerance zones to the tolerances required on the components. Moreover, by using the simplification to consider fixed the angles of the box, none changes as regards the case of only dimensional tolerances. Therefore, the simplification to consider fixed the angles of the box causes the geometrical tolerances do not have effects on the results of the case study.

Moreover, the application of the Envelope Principle or the Independence Principle does not produce any effect for the torsor model too.

Comparison

Table 1 shows the results due to the application of the two considered models to the same case study. If only the dimensional tolerances are applied, the worst case approach gives small under estimated results of about 4%, when compared with the geometrical exact solution. This is probably due to the same way the dimensional tolerances are schematize (i.e. the first datum is nominal, the variability due to the dimensional tolerance is considered applied only on one of the two features delimiting the dimension). If the geometrical tolerances are applied too, the worst case approach gives the same result, since it has been adopted the simplification to consider fixed the angles of the box in order to avoid the network. In fact, to solve the stack-up function, it is needed to relate the virtual joints displacements to the tolerances assigned on the components. However, the form tolerances (the planar one applied on the bottom side of the box and the two circularities applied on the circles) do not produce any effect because in the jacobian model the features are considered with nominal shape; the other ones (the perpendicularity applied on the left side of the box, and the two parallelisms applied on the other sides of the box) cannot produce any orientation gives similar results, when only dimensional tolerances or both dimensional and geometrical tolerances are applied. In this case the variability range is smaller than that of the worst case approach, as it is foreseen.

The jacobian model has the advantage, as regards the torsor model, to be able to perform the analysis by both the worst case and the statistical approaches. Moreover, the jacobian model allows to evaluate the jacobian matrix from the nominal conditions and, therefore, it is possible to directly relate the displacements of the functional requirements to the virtual joints displacements. Another advantage of the jacobian model is that it uses the usual algebraic rules to evaluate the displacements of the functional requirements, while the torsor model needs intervals algebraic rules which are much complex; this aspect is fundamental to approach the solution of the network functions which need to be developed for both the models.

Despite these advantages of the jacobian model, its virtual joints displacements are difficult to relate to the tolerances applied on the components. This step is critical for the torsor model too, but it seems easier to approach, despite with this second model the computation of the displacement components of the points of interest in the same global datum reference frame is very difficult and it has to be developed completely yet.

Unified Jacobian-Torsor model

As just underlined, the jacobian model takes its best advantage in the simplicity to evaluate the jacobian matrix from the nominal conditions; equations (4) and ( 5). This makes it possible to directly relate the displacements of the functional requirements to the virtual joints displacements; equation (3). The solution of the network functions seems easier to approach than the torsor one too. Despite this advantage, the virtual joint displacements are difficult to relate to the tolerances applied on the components. The torsor model allows to easily evaluate the variability ranges of the small displacements from the tolerances applied on the components, but it is very difficult to relate these ranges to the variability ranges of the functional requirements of the assembly.

In the last years the idea of the Unified Jacobian-Torsor model has been presented in order to evaluate the virtual joint displacements from the tolerances applied on the components by the torsors and, therefore, to relate the displacements of the functional requirements to the virtual joint displacements by the jacobian matrix (Laperrière et al. 2002, Desrochers et al. 2003); it is theoretically possible since the deviations are usually small and the equations may be linearized. The proposed unified model expands the functionalities of the jacobian model under two important aspects [START_REF] Ghie | A Unified Jacobian-Torsor model for analysis in Computer Aided Tolerancing[END_REF]. First, the punctual small displacement variables of the former jacobian formulation are now considered as intervals formulated and solved using interval-base arithmetic. The equations describing the bounds within which the feature is permitted to move, which are the constraint equations of the torsor formulation, are applied on the unified model. Second, some of the small displacement variables used in the model are eliminated due to the invariant nature of the movements they generate with respect to the toleranced feature. This standard result of the torsor formulation is applied to the unified model. The effect of this is to significantly reduce the unified model size. This new model enables to perform tolerance analysis and tolerance synthesis (Laperrière et al. 2002) or to redesign the assembly tolerances [START_REF] Ghie | Re-design of Mechanical Assemblies using the Unified Jacobian-Torsor Model for Tolerance Analysis[END_REF]). The Unified Jacobian-Torsor model has been developed for deterministic (worst-case) computer aided-tolerancing. Recently, the same set of interval-based deterministic equations has been applied to a statistical context [START_REF] Ghie | Statistical tolerance analysis using the unified Jacobian-Torsor model[END_REF] and the model has been used to develop a method for obtaining the functional requirement cost for product [START_REF] Ghie | Functional requirement cost for product using Jacobian-Torsor model[END_REF].

Conclusions

This paper firstly makes a brief review of two state of the art tolerance analysis models for rigid-parts assembly, the jacobian and the torsor. The two models are compared, in order to The two considered models have some common limits. The first two deal with the assembly cycle: the two models have not developed yet an approach to consider functional requirement functions arranged in a network and to correctly represent the coupling with clearance between two parts. The third deals with the representation of the tolerances applied on the assembly's components: the two models does not give a complete correspondence among the model variables (displacements) and the part's tolerances. In other words the translation of the part's tolerances into model variables does not satisfy the standards (ASME or ISO). The last deals with the form deviations and the Independence Principle: the two models do not allow to consider form deviations (the features are considered with nominal shape) and to apply the Independence and/or the Envelope Rule to different tolerances of the same parts.

However, the jacobian model has the advantage to directly relate the displacements of the functional requirements to the virtual joints displacements by the evaluation of the jacobian matrix from the nominal conditions. Moreover, it can be used to perform both a worst case and a statistical case approaches.

The torsor model has an easily evaluation of the variability ranges of the small displacements from the tolerances applied to the components.

The experimental results of the case study show how both the jacobian and the torsor models give the same underestimated result; this is due to the adopted simplification to consider fixed the angles of the box.

To overcome these limits in the last years a new model called Unified Jacobian-Torsor Model has been proposed, but it is still object of research.

r

  is the vector of the 3 small translations of the considered point, expressed in the DRF of the first feature R 0 ; δα r is the vector of the 3 small rotations of the considered point, expressed in the DRF of the first feature R 0 ; FE of the i th FE pair (internal or kinematic) to which the tolerances are applied, with i = 1 to n; vector of small dispersions associated with the FE of the i th FE pair (internal or kinematic) to which the tolerances are applied, expressed in the local DRF, with i = 1 to n.

  a Surfaces Graph. The Functional Requirements and the stacks to relate these FRs are identified too.

  pairs; now it is necessary to relate these virtual joints displacements to the tolerances assigned on the components. This step is the critical task of the jacobian model. However considering the simplification adopted reference to Figure5of the four functional requirement pairs and to equations (13), and considering the nominal dimensions:

(

  planar one applied on the bottom side of the box and the two circularities applied on the circles) do not produce any effect because in the jacobian model the features are considered with nominal shape; the other ones (the perpendicularity applied on the left side of the box, and the two parallelisms applied on the other sides of the box) cannot produce any orientation deviation, since the angles of the box are considered fixed.

  This torsor has constrained by considering that the extreme points F and Q remain into the tolerance zone. It results:

  the angles of the box are considered fixed. The statistical approach

  and the weakness of each model, on the basis of the experimental results and the information available from the literature.

Further

  researches include the definition of a new and original model able to overcome the limits that have been highlighted in this work.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .Figure 9 .

 123456789 Figure captions

  Figure 2

  Figure 1 199x256mm (96 x 96 DPI)

Table captions Table 1 .

 captions1 Case study results

	F o r
	P
	e
	e r
	R
	e
	e i v Figure 1
	w
	O n l
	y

URL: http://mc.manuscriptcentral.com/tandf/tcim Email:ijcim@bath.ac.ukInternational Journal of Computer Integrated Manufacturing

Acknowledgements

This research was partially funded by the Italian Ministry of University and Research.


Dr N Botsoglou

Keywords: Turkeys, Plants, Nutrition, Microbiology

Oxidative stability and microbial growth of turkey breast fillets during refrigerated storage as influenced by feed supplementation with olive leaves, oregano and/or α-tocopheryl acetate

Evropi Botsoglou, Alexandros Govaris, Anargyros Moulas, N Botsoglu

tissues, to accumulate dietary tocopherol in their tissues (Botsoglou et al., 2003a). To 83 delay lipid oxidation, additives, especially of synthetic origin, are added to muscle 84 foods. However, consumers and health authorities increasingly dictate that use of 85 synthetic food additives should be phased out and, where possible, only natural products 86 should be used. Hence, alternative substances from natural sources such as 87 phytochemicals are investigated for achieving meat of higher quality.

88

When the additive is the lipid soluble antioxidant vitamin E, its dietary 89 supplementation allows uniform incorporation of α-tocopherol into the subcellular 90 membranes [START_REF] Packer | Vitamin E: The antioxidant harvesting center of 512 membranes and lipoproteins[END_REF], where it can effectively inhibit the oxidative 91 reactions at their localised sites and improve meat quality through a delay in lipid 92 oxidation. The efficacy of relatively high doses of dietary supplemented α-tocopheryl 93 acetate against lipid oxidation of turkey muscle tissues has been well demonstrated 94 (Botsoglou et al., 2003a,b;[START_REF] Botsoglou | a) The effects of dietary oregano essential oil and α-456 tocopheryl acetate on lipid oxidation in raw and cooked turkey during 457 refrigerated storage[END_REF]Govaris et al. 2004[START_REF] Govaris | Dietary supplementation of oregano essential oil and α-tocopheryl 494 acetate on microbial growth and lipid oxidation of turkey breast fillets during 495 storage[END_REF] [START_REF] Kubo | Antimicrobial activity of the olive oil flavor 497 compound[END_REF][START_REF] Markin | In vitro antimicrobial activity of 507 olive leaves[END_REF], whereas 131 other non-phenolic components of the olive leaves, such as aldehydes, have also been 132 studied for their antimicrobial properties [START_REF] Kubo | Antimicrobial activity of the olive oil flavor 497 compound[END_REF][START_REF] Bisignano | On the 'in vitro' antimicrobial activity of oleuropein and hydroxytyrosol[END_REF]. The amount of total phenolics in olive leaves and oregano was determined using the 193 Folin-Ciocalteau reagent according to a slightly modified procedure (Singleton et al., 194 1999). In this analysis, the sample (0.5 g) was transferred to a 50-ml centrifuge tube 195 where 25 ml of 80% aqueous methanol was added. The mixture was homogenised 196 (Ultra-Turrax T25-S5, Janke &Kunkel GmbH, Staufen, Germany) at high speed for 30 s 197 and the homogenate was filtered. Filtrate aliquots of 0.05 ml were transferred to 15 ml 198 tubes and mixed (Vortex mixer K-550-GE, Scientific Industries, Inc., New York, USA) 199 with 6.45 ml of distilled water and 0.5 ml of the Folin-Ciocalteau reagent (Sigma- Aldrich GmbH, Sternheim, Germany). The mixture was allowed to stand for 3 min and, 201 after the addition of 3 ml of a 7.5% aqueous Na 2 CO 3 solution, allowed to incubate for 202 additional 60 min at room temperature. Following incubation, the mixture was 203 centrifuged at 2000 g for 5 min and its absorbance measured at 760 nm 204 (Spectrophotometer Shimadzu, Model UV-160A, Tokyo, Japan).

205

As the concentration of total phenolics was calculated in gallic acid equivalents per 206 g of sample, the calibration curve was constructed using gallic acid as a reference 207 standard. This standard solution was prepared by weighing 20 mg of gallic acid into a 208 50-ml volumetric flask and diluting to volume with 80% aqueous methanol. Volumes of 209 0.05-2.5 ml from this solution that corresponded to 20-100 µg of gallic acid were placed 210 in 15-ml tubes, diluted to 6.5 ml with water, mixed with 0.5 ml of Folin-Ciocalteau could not be directly demonstrated, due to lack of adequate analytical methodology and 327 equipment, their antioxidant activity towards the ABTS radical cation and their total 328 phenolic content was determined. Analyses showed that the total phenolic content of 329 olive leaves was 26.2 mg of gallic acid equivalents/g, whereas the antioxidant activity 330 expressed in TEAC values was 1.61 mM/g. This TEAC value is in agreement with that 331 reported in a previous study in which it was also reported that the olive leaves had an 332 antioxidant activity higher than vitamin C and E, due to a synergy between flavonoids, 333 oleuropeosides and substituted phenols [START_REF] Benavente-Garcia | Antioxidant activity of phenolics extracted from Olea europea L. leaves[END_REF]. Analyses also 334 showed that the total phenols content of oregano was 40.7 mg of gallic acid 335 equivalents/g, whereas its antioxidant activity expressed in TEAC values was 1.26 mM 336 Trolox equivalents/g.

337

The results of the present study are difficult to compare with previous reports since 338 there have not been pertinent in vivo studies with both olive leaves and oregano. Some 339 in vivo studies with oregano dietary supplemented to turkeys (Florou-Paneri et al., 340 2005;[START_REF] Govaris | Dietary supplementation of oregano essential oil and α-tocopheryl 494 acetate on microbial growth and lipid oxidation of turkey breast fillets during 495 storage[END_REF] or rats (Botsoglou et al., 2008) have shown the protective 341 effect of long-term dietary administration of oregano against lipid oxidation of tissues.

342

Other in vitro studies with olive leaves have shown that their main components, 343 oleuropein and hydroxytyrosol, were much more effective than BHT or vitamin E at 344 limiting the oxidation of methyl linoleate in model systems (Le Tutour and Guedon, 345 1992). Use of olive leaf juice has been also found suitable for extending the stability of 346 sunflower oil during heating (Farag et al., 2007).

347

Considering the fact that the greater the amount of α-tocopherol in tissues, the better 348 the protection against oxidative attack, fillet samples from all groups were analysed for 349 their α-tocopherol content at d 0 of refrigerated storage. Mean α-tocopherol level in the 350 CONT group was found to be 0.74±0.12 µg/g tissue. Supplementing the diets with olive (Botsoglou et al., 2003a,b;[START_REF] Govaris | Dietary supplementation of oregano essential oil and α-tocopheryl 494 acetate on microbial growth and lipid oxidation of turkey breast fillets during 495 storage[END_REF]Govaris et al., 357 2005). The concentrations of α-tocopherol in the turkey fillets appear low compared 358 with those found in chicken breast (8.4 µg/g) after dietary supplementation with 200 mg 359 α-tocopheryl acetate/kg diet [START_REF] Botsoglou | Effects of dietary oregano essential oil on performance of chickens 452 and on iron-induced lipid oxidation of breast, thigh and abdominal fat tissues[END_REF]; however, α-tocopherol is not 360 efficiently absorbed and deposited in cell membranes of turkeys (Botsoglou et al., 361 2003a).

362

As the shelf life of stored muscle food is generally associated with the extent and 363 type of microbial growth that may occur during storage, TVC, LAB, ENB and PSY, 364 bacteria that constitute the major causes of poultry meat spoilage during refrigerated 365 storage, were determined in fillet samples.

366

Figure 2 shows that TVC, LAB, ENB and PSY bacterial counts were all 367 significantly increased in samples from all groups throughout the refrigerated storage.

368

The CONT, T150 and T300 groups presented TVC, LAB, ENB and PSY counts that did 369 not differ significantly from each other, whereas the groups supplemented with olive 370 leaves or oregano presented bacterial counts that were significantly lower than the 371 CONT, T150 and T300 groups at d 2 of storage period and thereafter. During this 372 period, ORE group presented TVC, LAB, ENB and PSY counts that were significantly 373 lower than OLE group.

374

It may be of interest to note that at d 10 and thereafter, the CONT group with TVC 375 counts around 7 log CFU/g, showed initial signs of spoilage with the development of (2000), who reported that meat spoilage is visible when bacteria counts reach about 7 378 log CFU/g. Other workers also found similar results for the onset of spoilage in turkey 379 fillets under refrigerated storage [START_REF] Mano | Growth/survival 504 of natural flora and Aeromonas Hydrophila on refrigerated uncooked pork and 505 turkey packaged in modified atmospheres[END_REF]. No signs of spoilage were noticed 380 in the other groups up to the end of storage, although the T150 and T300 groups (Capasso et al., 1995;[START_REF] Markin | In vitro antimicrobial activity of 507 olive leaves[END_REF]. Olive leaf extracts and their constituents 389 have shown an inhibitory effect against psychrotrophic bacteria such as B. subtilis, 390 Pseudomonas spp. and Proteus vulgaris [START_REF] Kubo | Antimicrobial activity of the olive oil flavor 497 compound[END_REF][START_REF] Markin | In vitro antimicrobial activity of 507 olive leaves[END_REF], 391 whereas oleuropein has been reported to act against lactic acid bacteria such as 392 Lactobacillus plantarum [START_REF] Ruiz-Barba | Bactericidal 522 action of oleuropein extracted from green olives against Lactobacillus 523 plantarum[END_REF] and food-borne pathogens such as 393 Salmonella enteriditis (Tassou and Nychas, 1995) and Bacillus cereus (Tassou et al., 394 1991). Oleuropein has also shown to have an inhibitory effect against the growth and 395 enterotoxin production by Staphylococcus aureus (Tassou and Nychas, 1994), whereas 396 both hydroxytyrosol and oleuropein have shown an inhibitory effect against several 397 bacterial strains (ATCC) including Staphylococcus aureus, Vibrio parahaemolyticus 398 and Salmonella spp that were causal agents of intestinal or respiratory tract infections in 399 humans [START_REF] Bisignano | On the 'in vitro' antimicrobial activity of oleuropein and hydroxytyrosol[END_REF]. In addition, the results of this study lend support to previous reports on the in vitro 401 antimicrobial activity of oregano and its major biologically active components carvacrol 402 and thymol against pathogenic and food spoilage bacteria [START_REF] Aligiannis | Composition 432 and antimicrobial activity of the essential oils of two Origanum species[END_REF] 403 Skandamis and Nychas, 2001b). Oregano essential oil has also shown considerable

  Laboratory of Hygiene of Foods of AnimalOrigin, School of Veterinary Medicine, 10 University of Thessaly, 431 00 Karditsa, Greece, 1 Department of Animal Production, 11 Technological Education Institute, Larissa, Greece and 2 Aristotle University, 12 Thessaloniki, Greece

  , oxidative deterioration of muscle tissues has been of increasing concern 76 because novel feeding strategies have further raised the concentrations of 77 polyunsaturated fatty acids in meat. However, increases in the concentrations of 78 polyunsaturated fatty acids in tissues accelerates lipid oxidation, leading to quality 79 losses and a reduction in consumer acceptability. Turkey muscle tissues are particularly 80 susceptible to lipid oxidation during refrigerated storage due to their relatively high 81 content of polyunsaturated fatty acids and their weaker ability, compared with chicken 82

95Florou-

  Paneri et al., 2005) 96 In the search for phytochemicals with potential antioxidant activity, much attention 97 has been focused in recent years on aromatic plants, especially those of the Labiatae 98 family, which are traditionally used in folk medicine. Among them, oregano, a 99 oleuropeosides, such as oleuropein and verbascoside, flavonoid 125 compounds, such as luteolin, luteolin-7-glucoside, apigenin-7-glucoside, diosmetin, 126 diosmetin-7-glucoside, rutin and catechin, and simple phenolic compounds such as 127 tyrosol, hydroxytyrosol, vanillin, vanillic acid and caffeic acid. Oleuropein, the most 128 prominent compound, may reach concentrations of 60-90 mg/g dried leaves (Ryan et 129 al., 2002). Most of these phenolic compounds have been shown in vitro to possess 130 antimicrobial and antioxidant activity

  133Nevertheless, in vivo investigations on the antioxidant and antimicrobial activity of 134 olive leaves supplemented to turkey diets have not yet been carried out. 135 The use of nutritional strategies to improve the oxidative stability of animal 136 products can extend the shelf life and increase the acceptability of muscle food during 137 retail display. However, a major concern with extending the shelf life of muscle food is 138 associated to the extent and type of microbial growth that may occur during the 139 additional time of product acceptability. Delayed lipid oxidation caused by use of 140 supplemental dietary antioxidants to the growing animal may mask microbial growth 141 without warning consumers; some of the initial signs of meat spoilage like off-flavours 142 or discolouration may be hidden due to decreased lipid oxidation, although populations 143 of bacteria may be high. This fact may be an adverse factor in the health of consumers 144 of such meat. The present study was designed to evaluate and compare the inhibitory 145 potentials of feed supplementation with olive leaves, oregano and/or α-tocopheryl 146 acetate on microbial growth and lipid oxidation of turkey breast fillets 40 female 8-week-old turkeys of the local black strain, were purchased from a 153 farm in the region of central Greece and divided into 5 equal groups. Each of the groups 154 was housed in a separate floor pen 2 m x 2 m and allowed to acclimate for a period of 2 155 weeks. During the acclimatisation period, they were fed ad libitum on a common 156 commercial maize-soybean based turkey diet supplemented with 10 mg α-tocopheryl 157 acetate/kg. The ingredients and the composition of the commercial diet are presented in 158

162

  At the end of the acclimatisation period, feeding of the commercial diet was 163 discontinued to all but one of the groups. The birds within this control group (CONT) 164 were given the commercial diet for a further 4 weeks. The experimental diets given to 165 the remaining 4 groups were based on the same commercial diet but contained an 166 additional 10 g olive leaves/kg (OLE), 10 g oregano/kg (ORE), 150 mg α-tocopheryl 167 acetate/kg (T150) or 300 mg α-tocopheryl acetate/kg feed (T300). Supplementation of 168 the commercial diet with olive leaves or oregano was achieved at the expense of wheat 169 bran. Feeding of the experimental diets to turkeys lasted 4 weeks. 170 The α-tocopheryl acetate was from Roche Products Ltd. (Hertfordshire, UK) and the 171 olive leaves from Olea europaea L trees cultivated under organic farming practices. 172 Collected leaves were first air-oven dried at 45 o C for 3 consecutive d, and then ground 173 to pass 2 mm screen. Proximate analysis (AOAC, 2000) showed that they contained 7.3 tops, leaves, stems and stalks of cultivated Origanum vulgare 176 subsp. hirtum plants. It was collected during summer, air-oven dried at 45 o C for 3 177 consecutive d and ground to pass 2 mm screen. Steam distillation of the ground oregano 178 using a Clevenger type apparatus resulted in an essential oil yield of 2.3 ml/100 g. 179 Analysis of this essential oil was performed by gas chromatography (ThermoQuest 180 Trace GC, Italy) on a fused silica column DB-5, 30 m x 0.25 mm i.d. (J & W Scientific 181 Inc., Folsom, CA) using an injection temperature of 260°C, helium as the carrier gas, a 182 split ratio 1:30, a flame ionisation detector and a temperature programme of 55-120°C 183 at a rate of 3°C/min, 120-200°C at 4°C/min, 200-220°C at 6°C/min and 220°C for 5 184 min. Analyses showed that the major components were carvacrol at 812 g/kg, thymol at 185 49 g/kg, p-cymene at 29 g/kg and γ-terpinene at 22 g/kg. of age, turkeys were slaughtered and carcases trimmed for breast fillets by 188 removing skin, bones and connective tissue. Following trimming, breast fillets from 189 each bird were individually divided into two sections. One section was used for 190 microbiological analysis and the other for oxidative stability studies 191 Determination of total phenolics in olive leaves and oregano 192

Figure

  Figure 1 near here

  at 10 g/kg, changed the concentrations of α-tocopherol to 0.84±0.28 352 or 0.91±0.24 µg/g respectively; these values did not differ significantly from each other 353 or from the CONT group. Supplementing the diet with 150 and 300 mg α-tocopheryl 354 acetate/kg increased the concentrations of α-tocopherol in fillets to 2.24±0.22 and 355 2.88±0.31 µg/g respectively. The observed values of α-tocopherol agreed well with 356 literature data

  376

Figure

  Figure 2 near here

  377

  381 presented bacterial counts similar to the CONT group. This retardation in visible signs 382 of spoilage of these samples might endanger consumer health due to increased numbers 383 of bacterial counts. 384 These results indicated that the dietary supplementation with either olive leaves or 385 oregano at 10 g/kg feed exerted an inhibitory activity on the growth of TVC, LAB, ENB 386 and PSY bacteria in the stored turkey breast samples. This lends support to previous 387 reports on the in vitro antimicrobial activity of olive leaves and their constituents 388

Figure 2

 2 Figure2shows that dietary supplementation of 150 and 300 mg α-tocopheryl

Figure 1 .

 1 Figure 1. Effect of refrigerated storage on lipid oxidation of raw breast fillets as a

Figure 2 .

 2 Figure 2. Changes in total viable counts (TVC), lactic acid bacteria (LAB),

Table 1 .

 1 Room temperature was controlled, continuous lighting was provided, and

	159
	160	conventional breeding and management procedures were employed throughout the
	161	experiment. Birds were cared for in accordance with the guide for the care and use of
		laboratory animals (Committee on Care and Use of Laboratory Animals, 1996).

Table 1 .

 1 Proceedings of the 44th International Congress of Meat Science and 559 Technology, Barcelona, pp. 490-491. Nutrients composition of commercial diet for turkeys

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

% moisture, 8.4 % protein, 4.4 % fat, 17.8 % crude fibre and 5.0 % ash. Oregano

E-mail: br.poultsci@bbsrc.ac.uk URL: http://mc.manuscriptcentral.com/cbps British Poultry Science

utilised in novel applications as nutritional supplements or a functional food 426 components.