Daria Battini

Andrea Grassi

Alessandro Persona
email: alessandro.persona@unipd.it

Fabio Sgarbossa

A Grassi (#

Consignment Stock Inventory Policy: Methodological Framework and Model

Keywords: SUPPLY CHAIN MANAGEMENT, STOCHASTIC MODELS, PROCESS MODELLING, OUTSOURCING Consignment Stock, Framework

.

This article considers new critical factors (present in several industrial environments) providing a logical extension of the above mentioned study. The proposed methodology addresses, in particular, the design of a new stock in an industrial environment with demand variability, stock-out risk and limited warehouses space.

The analytical model presented demonstrates that a traditional stock policy, such as the Economic Order Quantity approach is always more expensive than the Consignment Stock approach. In addition, the benefits of the CS policy to the supply chain are consistently high including when applied to realities with high demand variation and space limitations.

In a traditional supply process, both partners, buyer and vendor, incur financial disbursement, whether due to stocking or holding costs.

A Consignment Stock Policy (from here on referred as CS policy) leads to a change in the costs structure for both the partners [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF]. For a better understanding of the industrial scenario analyzed, few clarifications are necessary. The holding unit cost, h, is calculated as the sum of two principal components: the financial component, h fin , and the stock component, h stoc . The first refers to the opportunity costs of the company when it invests financial resources in the production of a product. The second, refers to the warehousing costs of the item (stocking, handling, insurance, etc.).

In a CS policy, the buyer incurs in stocking costs, since the materials are located in its plant, but it doesn't sustain financing cost, since the item is purchased only when needed. In fact, since the item is formally purchased upon demand, the vendor sustains the financial costs of the capital immobilized until that moment. The buyer can perceive a reduction in holding cost, which is only equal to h stoc , rather than to h stoc + h fin [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF]. Moreover, [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF] underline that order emission costs are also eliminated.

The vendor also receives a number of advantages. First of all, the average stock is reduced leaving more available space for stocking new items. Moreover, the vendor increases his production management flexibility since he is no longer constricted by consecutive close orders. For these reasons, a CS policy aims to reduce management and logistic complexity of the stocks and to ensure at the same time a constant availability of materials in order to eliminate stock-out risks (Table 1).

[PLACE HERE TABLE 1]

From the examples collected by the authors, consumer items, such as metallic and plastic fasteners, small parts, tools, packaging parts and personal protection equipments (PPE) are some of the items for which a CS policy guarantees the biggest benefits. In light of these considerations, this work analyses and quantifies all the benefits arising from the establishment of a CS policy in a traditional environment, managed by the well known Economic Order Quantity method (EOQ), which is characterised by a reorder point and a fixed order size. This article focuses especially on items characterized by high annual consumption (in number of pieces), low dimension, 3 variable demand and risk of obsolescence, and integrates and completes existing studies by [START_REF] Braglia | Modelling an industrial strategy for inventory management in supply chains: the 'Consignment Stock' case[END_REF] and [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF], touching upon some additional aspects that were not previously considered, such as: 1) the development of an integrated framework to assist the analyst when addressing the problem of switching stock management policy from traditional to CS.

2) an extension of the model developed by [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF], which includes safety stocks, stock-out risk and restricted space availability, issues theat were never addressed in previous studies.

3) a demonstration, for the first time, that Consignment Stock policy is always convenient when compared with the well known EOQ policy for standard components, like consumables materials.

The analytical model developed to support the framework allows to calculate the optimal delivery quantity which is the quantity that minimizes total annual supply chain management costs.

Section 2 presents a state of the art classification and overview on this topic. The new methodological framework is presented and discussed in section 3, while in section 4 the analytical model is developed to support the framework application. Finally, a numerical application of the model is proposed in section 5 where CS policy is compared to a traditional policy, as a prelude to the article's conclusions.

2.State of the art

International literature on CS is still very limited. Table 2 summarizes the main references and classifies them according to the authors' assumptions, such as: market demand modelling, lead time quantification (including production lead time, procurement lead time, transport lead time, etc.), risk of obsolescence of materials in stock (finite or infinite life cycle), type of logistics system considered.

The analytical model for the evaluation of CS policy annual costs developed by [START_REF] Braglia | Modelling an industrial strategy for inventory management in supply chains: the 'Consignment Stock' case[END_REF], refers to the problem of a single vendor and single buyer production situation: the objective of the study is to minimize the total system cost function, that is the sum of buyer and vendor costs. The model considers a constant and fixed product demand during the year, infinite life cycle of products and a fixed delivery quantity q. The same authors highlight some possible inefficiencies of the model such as an increase in inventory levels on buyer's side, even if they are shortterm. A possible solution to prevent this inconvenience is to postpone the last delivery. [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF] develop a model for the Consignment Stock inventory management in case of a single-vendor and single-buyer system. The experiment demonstrates that CS application reduces system costs (for equal service level offered) compared to the Hill model, where performance depends on the maximum stock level S chosen and agreed by both partners. [START_REF] Zanoni | A note on industrial strategy for stock management in supply chains: modelling and performance evaluation[END_REF] extend the solution provided by Braglia and Zavanella optimizing three parameters, through an analytical process: the delivery quantity, the number of deliveries per production batch, the number of deliveries to postpone. [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF] propose an interesting contribution that focuses on the materials obsolescence topic, developing an analytical model, which is an extension of the previous one. Such model could help evaluate obsolescence impact on the supply chain.

[PLACE HERE TABLE 2]

In traditional procurement policies the risk of obsolescence is completely sustained by the buyer, since the stock in his warehouses is his property. In a CS approach, the risk of obsolescence is a critical factor, because the cost associated with it shared costs between buyer and vendor, who is the owner of the materials in consignment stock. The delivery quantity is therefore a critical variable and must be contained. An example for reducing costs by the CS policy application is provided by Piplani and Viswanathan (2003): the authors analyse a single-vendor and multi-buyer system in which the vendor maintains a SOI (Supplier Owned Inventory) accord with only one of his buyers ("the SOI buyer") and the annual costs sustained for managing other buyers are not considered. The results underline that in all combinations the total supply chain cost after the adoption of the SOI policy is less or at least equal to the corresponding traditional approach. Thus, the SOI strategy is defined by the authors as profitable for the supply chain as a whole, however, the vendor does not always obtain direct benefits. [START_REF] Hung | Inventory Management in the Consignment System[END_REF] describe the implementation of the Consignment Stock policy at the "Consumer Electronics Division" of Philips (Taiwan) with the objective to reduce the material handling costs and the space occupied in the warehouse, integrating vendor and client to improve the information and communication level between partners. The authors develop an empirical method to define the minimum stock level 's', equal to an appropriate safety stock quantity, introducing a "fluctuation index", calculated as the ratio between the standard deviation in demand and the forecasted 2: 1) The analysis of a single-vendor and single-buyer system, performed by the majority of the authors results in great simplification of the modern industrial reality: the necessity of future studies to address a systematic approach which considers a single-vendor and multi-buyer system type is hence evident. In this system a number of buyers can start a CS policy agreement with the same vendor, as happens in today's industrial reality.

2) From the examples presented above, CS shows to be a valid stock management policy, especially in the most instable environments with demand variability.

However, a methodological framework to guide the implementation of this approach in traditional contexts has not yet been developed and a complete analytical model able to consider all critical factors (obsolescence, stock out risk and finite warehouse capacity) has not been presented in literature.

3) The most influential factors on the Consignment Stock implementation process are demand variability and finite life cycle of items (risk of obsolescence). However, procurement lead time variability should have been considered in some particular cases.

This paper addresses considerations 2) and 3) developing a new integrated framework for the CS implementation in presence of demand variability, risk of obsolescence and space limitations in the buyer's warehouse.

Methodological framework

As previously mentioned, the work presented develops a methodological framework able to address all the problems emerging when a project for the implementation of CS policy is going to be addressed in actual industrial situations, and offers an analytical model to calculate optimal inventory levels.

Several factors have been included to make the framework as near as possible to actual industrial situations.

Figure 1 shows the proposed framework. Logistics and manufacturing constraints typical of actual industrial systems constitute the base for the model, followed by the 10 phases of the framework. During the implementation of the project, it is necessary to accomplish each task independently from the partners' constraints, which should be individually addressed by both partners. Effective communication from the beginning of the partnership is of the outmost importance to reduce start-up times and inception delays. A CS implementation project should be based on a concurrent engineering paradigm, to ensure that the different activities, developed during the execution of the project, will be carried out by a cross-functional buyer-vendor group, with a continuous reciprocal agreement to prevent any delay in implementation time. Only with an integrated approach the partners will realize annual savings and effective implementation of this policy, without occurring in high start-up costs.

Regardless of who initiates the policy change, whether the vendor proposes the CS policy to the buyer or the buyer suggests a new policy to the vendor, choosing the right partner is crucial for the success of the experiment. From the buyer's point of view, the potential vendors should be those supplying critical components in volumes, value and significance for the final product [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF], those who posses a strong technological know-how and those whose business approach is oriented towards change, collaboration, information sharing, and to Total Quality Management (TQM).

On the other hand, from the supplier's point of view, the potential buyers are certainly those able to guarantee a high annual turnover, and those who are willing to reserve sufficient space for stocking items and to tackle the standardisation of their components.

[PLACE HERE FIGURE 1]

The second step in the implementation of the CS policy consists of selecting the components suitable for such a management policy, which must be jointly carried out by both partners. Often, only a part of the components supplied by a specific vendor are suitable for CS management policy. Table 3, addresses some critical aspects that need to be taken into account and carefully measured for the success of the model.

The most suitable components for a CS agreement are generally the ones characterised by a high consumption [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF], since they can be supplied by open orders, as emphasized by the SOI theory. Since stocking costs must be contained, components and materials requiring particular stocking equipment or stocking environmental condition have to be excluded. In particular, since stocking space is limited and holding cost increases with size, the item size is a determining factor for the success of the trial. Hence, the buyer obtains maximum advantage if the CS policy manages components with a reduced size but high value, because the financial burden of the holding cost is carried by the vendor [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF]. In general, adopting a CS policy with easy to handle components, facilitates supply and replenishment, offering a good starting point.

Another important factor to take into consideration is the expected lifetime of the component, or, in other words, its replaceability ratio or obsolescence. This element is dependent on four different elements: perishability (shelf life), replacement due to internal causes (modifications or replacement of the final product), replacement due to external causes (updates or new versions proposed by the supplier), perishing due to environmental causes [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF].

[PLACE HERE TABLE 3]

In addition, components standardisation and data alignment between vendor and buyer (codes, dimensions, reference regulations, etc.), are of great importance in CS implementation projects, in spite of the timeliness of the task and the stress the operation puts on the resources. All future engineering choices and marketing decisions depend on an effective reduction in the number of items to manage. According to Koudate and Suzue (1992), applying the Variety Reduction Program guidelines to and inventory, might reduce the number of managed items by 35%, with considerable savings. In fact, annual service costs borne by the buyer and agreed with the supplier in the contract drafting stage, are directly proportional to the number of items managed and to the number of locations assigned in the warehouse.

Effective data exchange is paramount for the success of the project, to track daily consumption. For a really efficient data exchange it is necessary to analyze who sends data and how frequently it is sent. A simple kanban system with a double box should be sufficient in many industrial environments, but in some instance a more sophisticated electronic system may sometimes be necessary.

Warehousing methods are to be analysed as well since the physical location of the stock, the centralization/decentralization rate of the stocks into the buyer's plant, the physical location in the plant, might affect supply warehousing and distribution methods. One of the main aspects in drafting a CS agreement is the definition of the lowest and highest stock levels, s and S respectively, for each item. In the definitions of such levels, from a negotiation between the vendor and the buyer, the partners tend to keep opposite positions. On one hand, the buyer generally tends to keep s as high as possible, in order to reduce stock-out risk, and S as low as possible to reduce assigned inventory space. On the other hand, the vendor will prefer a high S level for greater operative flexibility and a low s level to reduce holding costs [START_REF] Grassi (# | Managing stocks in Supply Chains: modelling and issues[END_REF]. Currently, scientific literature does not provide standard formulas for the calculation of s and S levels with the adoption of CS management policy.

It is practical to use the lowest inventory level s as a minimal safety stock determined as a function of the maximum vendor reaction time (physiological and not null) in case of an overrunning under s, and the highest inventory level S as a function of the maximum available inventory space at the buyer's plant. Often, both levels are expressed in weeks (or months) of consumption coverage. Once s and S levels are defined, the optimal shipment quantity q is determined. Optimal shipment quantity is the quantity that allows to minimize the vendor-buyer total annual costs.

Different analytical linear models have been developed over the last year to asses the calculation of the best delivery quantity q, considering different input variables and working conditions: demand variability, obsolescence, lead time variability. The model developed to support this step of the framework is discussed in the next paragraph and allows three general conditions to be combined in a single model: demand variability presence, risk of obsolescence not null and space limitations at buyer's warehouse. The model developed allows calculation of the optimal delivery quantity q and the forecasting of all costs involved per one year period in the whole system and for each partner. The methodological framework is finally concluded by the management costs forecasting for both partners involved, followed by the CS agreement preparation and the setting at zero of inventory levels at buyer's side.

The analytical model

The analytical model developed in this section considers a single-vendor and singlebuyer system, and compares a CS policy with a traditional supply management system.

To determine the economic impact of a CS policy, it is assumed that the buyer switches from an Economic Order Quantity (EOQ) management (traditionally adopted for articles such as fasteners, metallic minutia, packaging, personal protective equipments, ), to a CS policy. The inventory level is constantly monitored and an order is placed each time the inventory reaches a predefined reorder point. Both models take into account logistic and manufacturing constraints on both vendor and buyer side, as reported in the framework of Figure 1, while for the sake of this experiment, total annual costs will be made up of the following cost components:

• Vendor's setup costs;

• Handling and warehousing costs of both partners, including safety stock contribution;

• Financial immobilization of inventories (borne by both of the actors in a traditional policy; only for the vendor in the case of CS policy);

• Stock-out costs (borne by both of the actors in a CS policy);

• Vendor's shipment costs;

• Obsolescence cost (borne by both of the actors in a CS policy);

• Storage res (borne by the vendor in a CS policy);

• Hiring costs of equipment needed for storing and distributing items (i.e. dispensers managed with personalised electronic badges).

a) Analytical model for the traditional inventory management policy

In an traditional inventory management policy, characterised by a reorder point and fixed order size, cost components are expressed as a function of the fixed reorder quantity, and the model will derive the order quantity which minimises total annual costs.

In such a traditional system, some of the strategic characteristics become fundamental and the following hypothesis need to be considered: i) The system is managed by a fixed reorder point;

ii) The fixed reorder quantity t Q is such that total annual costs are minimised;

iii) Buyer's demand is stochastic and normally distributed according to a monthly average demand D m (units/month) and a standard deviation σ (units/month); iv) Item life, T, is limited [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF]; v) Shipment costs are charged to the vendor, whereas warehouse refilling costs are borne by the buyer;

vi) The vendor does not incur obsolescence costs, since the buyer is committed to buy each order issued. Hence, obsolescence costs are borne solely by the buyer. viii) Space limitations at the buyer's plant limit the maximum material quantity to be stored in the buyer's warehouse. The maximum quantity storable is equal to S and it is calculated considering the available space at buyer's plant and material characteristics.

The following notations are adopted: The objective of the model is to minimize the total system cost:

( ) tot C min , which
represents the sum of all the costs borne by vendor and buyer, subject to the following constraints:

1) 0 ≥ t Q 2) S level inventory Maximum ≤ _ _
Inventory holding costs (subdivided in financing and stock component according to [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF] are borne by both the vendor and the buyer, proportionally with time each item is kept in each respective warehouse.

Such costs are defined by the following expressions:

p f f fin c i h ⋅ = , Vendor financial cost (€/ unit in stock) (1) p f f stoc c s h ⋅ = , Vendor stocking cost (€/ unit in stock) (2) p i h c c fin ⋅ = , Buyer financial cost (€/ unit in stock) (3) p s h c c stoc ⋅ = , Buyer stocking cost (€/ unit in stock) (4)
Safety stocks maintained only at the buyer's plant are calculated according to the traditional formula:

c c LT k ss ⋅ ⋅ = σ (units) (5)
where:

LT c is the procurement lead time agreed with the supplier and is generally constant.

The costs borne by the sole vendor (f) are expressed by means of the following formulae:

1. Setup costs:

1 A Q D C t f s ⋅ = (6) 2.
Holding costs:

( )       +       = P D c s i Q C p f f t f m 2 (7) 3. Shipment costs: O Q D C t f tr ⋅ = (8)
The costs borne by the buyer (c) are expressed with the following formulae:

1.

Order emission costs: Holding costs:

( )p s i ss Q C c c c t c m +       + = 2 (10) 3.
Stock-out costs (Vollman et al., 1997[START_REF] Persona | Optimal safety stock levels of subassemblies and manufacturing components[END_REF]:

c m c RS c m p c RS c LS w D c D C , , , ) 1 ( ⋅ - ⋅ ⋅ = ⋅ = (11)
Where:

D p is the annual sale loss, w is the total number of shipments to be carried out in a year, LS is the Service Level = 1-Probability of running out of stock. The Service Level is associated with area under the Gaussian curve therefore this will leads us the corresponding Z value for the normal table [START_REF] Persona | Optimal safety stock levels of subassemblies and manufacturing components[END_REF].

RS are the number of stock-outs in a year, D RS is the average value of unfulfilled demand when a stock-out occurs, i.e. when demand D is greater than the maximum admissible level of stocks (the complete formula derived by normal distribution function is reported in [START_REF] Persona | Optimal safety stock levels of subassemblies and manufacturing components[END_REF].

The following expression are used to compute the above quantities:

) 1 ( , , LS w D RS D D c RS c RS p - ⋅ ⋅ = ⋅ = (12) t Q D w = ( 13
) ) 1 ( LS w RS - ⋅ = (14)
4.

Obsolescence costs:

T D Q D LT ss Q LT D LT Q Q ss LT D LT D Q p C m t m c c t c m c t t c c m c m t c o 1 2 2 ⋅       + + +             - - + + ⋅         - ⋅ = (15)
In formula (15) we express the problem considering a deterministic time-toobsolescence instance. The formula expresses the obsolescence cost as a weighted average of two different cases (as depicted in Figure 2): the amount of leftover stock in buyer's warehouse in case the obsolescence occurs before reaching the point of reorder (Case A in Figure 2) and the amount of leftover stock in buyer's warehouses in case the obsolescence occurs after reaching the reorder level (Case B in Figure 2). In this last case we must consider also the leftover stock of the quantity Q t already ordered to the supplier. The two quantities are weighted in base of the time-to-obsolescence length of both cases. Finally the obsolescence cost expressed in formula (15) has been divided for the item lifetime T (expressed in years).

[PLACE HERE FIGURE 2]

Finally, the whole system costs will be:

c RS c o c m c ord f m f tr f s tot C C C C C C C C + + + + + + = (16)
b) Analytical model for the consignment stock management policy When a consignment stock management policy is adopted, the cost structure undergoes some changes, and the supplier will bear some of the obsolescence, holding and refilling costs.

The model is based on the following hypothesis:

i)

The fixed shipment quantity q is such that total costs are minimized;

ii) The buyer's demand is distributed with an average monthly demand D m (units/month) and standard deviation σ (units/month);

iii) Item life, T, is finite [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF]; iv) Vendor and buyer lead time are known constants, and are expressed in months, coherently with the demand;

v)

The buyer is no longer subjected to order issue costs [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF];

vi) The vendor delivers the order to the buyer as soon as each amount q has been produced, carrying out a production batch

n q Q ⋅ =
. [START_REF] Braglia | Modelling an industrial strategy for inventory management in supply chains: the 'Consignment Stock' case[END_REF]; vii) Shipment and refilling costs are charged to the supplier;

viii) The procurement lead time experienced by the buyer when a stock-out occurs is reduced to the so-called "vendor's maximum reaction time". Hence, it is consistently lower than the lead time experienced in the traditional supply policy (this hypothesis is derived by the fact that in real situations the consideration of a null lead time is too hazardous);

ix) The vendor and the buyer always maintain a safety stock at their plants;

x)

The minimum inventory level s is the safety stock computed as a function of the procurement lead time defined in (viii);

xi) The maximum inventory level in the buyer's plant cannot exceed the maximum agreed level S. This level is calculated as a function of the maximum available space in the buyer's plant as reported in the step 6 of the framework (Figure 1).

The following clarifications are necessary in order to fully understand the model:  

x the lower integer of a variable

x if 0 ≥ x , otherwise 0 if 0 < x .
The variable of the model is q, representing the quantity sent by the vendor to the buyer in each shipment (units/shipment).

The objective of the model is to minimize the system total cost ( ( )

tot C min
), being the sum of the costs borne both by the vendor and the buyer, subject to the same previous constraints.

The financial component of the inventory holding costs at the buyer's plant are charged to the vendor, since he keeps the ownership of the items [START_REF] Valentini | The Consignment Stock of inventories: industrial case and performance analysis[END_REF], hence formula (3) becomes:

0 , = ⋅ = p i h c c fin (17)
In case of a CS policy, the model assumes that demand variability perceived by both partners is equal to market demand variability, since the well known Bullwhip Effect, studied by Forrester under the hypothesis that an OP-EOQ reorder policy (Forrester, 1958) can be drastically reduced under a CS policy with only two echelons. In fact, the different perception of demand variability is reduced by shorter delivery lead time, by the increased reliability of information flow under the CS policy and by a better inventory coordination. Under this assumptions we can express vendor's and buyer's safety stocks using the analytical model formula, based on the supply lead time of component LT (Vollman et al. 1997 and[START_REF] Persona | Optimal safety stock levels of subassemblies and manufacturing components[END_REF], as follows:

f f LT k ss ⋅ ⋅ = σ (units) (18) c c LT k ss ⋅ ⋅ = σ (units) (19)
where:

P Q LT g f =
LTc is constant and expressed according to viii) . The minimum inventory level s in the buyer's warehouse is equal to the safety stock:

c c LT k ss s ⋅ ⋅ = = σ (20)
Moreover, since the item life is limited, we consider the following expression [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF]:

D nq nq TD T t       - = * (21)
where nq/D is the length of a CS cycle, and as a consequence D/nq is the number of CS cycles in a year.

            + - - + - = * * 1 q Dt S P qD nq n n (22)
where qD/P is the inventory level at which the vendor starts the production of the batch Q=np.

            = * n q P t n ; min (23)
We now define the costs for the vendor and the buyer.

Vendor's costs:

1. Setup costs:

      + = 1 1 nq DT T A C f s (24) 2.
Holding cost: a. inventory located in vendor's plant [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF]:

( ) ( ) ( ) ( ) ( )                 +       + +                 +       + =                 +       + +                 +       - + + = * * * * t D nq nq TD ss T h h P q n nq TD P nq q T h h t D nq nq TD ss T h h P q n nq TD P nq ss ss q T h h C f f stoc f fin f stoc f fin f f stoc f fin f f f stoc f fin f m , , , , , , , , 1 , 2 2 (25) b.
inventory located in buyer's plant [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF]:

F o r P e e r R e v i e w O n l y ( ) ( ) ( ) ( )               - ⋅ + ⋅ +       + +                 -             - -       - - +               - - + +       = =         ⋅ +       ⋅ + +                 -             - -         +       - - +               - - + +       - = * * * * * P q n t ss P n q ss D nq nq TD ss T h P q n t P q n t D P D q n q n P n q P D q n P qD D nq nq TD S T h P n q ss D nq nq TD ss T h P q n t P q n t D ss P D q n q n P n q P D q n P qD D nq nq TD ss S T h C c c c f fin f fin c c f fin c c f fin f m 2 1 2 2 1 1 1 2 1 2 1 2 2 1 1 1 2 1 2 , , , , 2 ,
(26)

3.

Obsolescence costs [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF]:

( ) f p p f o ss T c n n T qc C ⋅ + - = * (27) 4.
Stock-out costs (Vollman et al., 1997[START_REF] Persona | Optimal safety stock levels of subassemblies and manufacturing components[END_REF]:

f m f RS f m p f RS c LS w D c D C , , , ) 1 ( ⋅ - ⋅ ⋅ = ⋅ = (28)
Where:

D p is the annual sale loss, w is the total number of shipments to be carried out in a year, LS is the Service Level defined before, RS are the number of stock-outs in a year, D RS is the average value of demand unfulfilled when a stock-out occurs (the complete formula derived by normal distribution function is reported in [START_REF] Persona | Optimal safety stock levels of subassemblies and manufacturing components[END_REF].

The following expression are introduced to express the quantities above:

T n nq TD n w +       ⋅ = ( 29
) ) 1 ( LS w RS - ⋅ = ( 30
) ) 1 ( , , LS w D RS D D f RS f RS p - ⋅ ⋅ = ⋅ = (31)

5.

Shipment and refilling costs:

F o r P e e r R e v i e w O n l y         +       = * n nq TD n T O C f tr (32)
Buyer's costs:

1.

Holding costs, only in the buyer's plant [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF]:

( ) ( ) ( ) ( )               - ⋅ + ⋅ +       + +                 -             - -       - - +               - - + +       = =         ⋅ +       ⋅ + +                 -             - -         +       - - +               - - + +       - = * * * * * P q n t ss P n q ss D nq nq TD ss T h P q n t P q n t D P D q n q n P n q P D q n P qD D nq nq TD S T h P n q ss D nq nq TD ss T h P q n t P q n t D ss P D q n q n P n q P D q n P qD D nq nq TD ss S T h C c c c c stoc c stoc c c c stoc c c c stoc c m 2 1 2 2 1 1 1 2 1 2 1 2 2 1 1 1 2 1 2 , , , , (33) 2.
Obsolescence costs [START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF]: Stock-out costs (Vollman et al., 1997[START_REF] Persona | Optimal safety stock levels of subassemblies and manufacturing components[END_REF]:

c m c RS c m p c RS c LS w D c D C , , , ) 1 ( ⋅ - ⋅ ⋅ = ⋅ = (35)
Where:

) 1 ( , , LS w D RS D D c RS c RS p - ⋅ ⋅ = ⋅ = (36) 4. Equipment hiring costs: c nol C
This cost exists only if the buyer has to hire equipment for computerized/automatic stocking and dispensing of the items from the vendor, as is the case for PPE and small tool automatic dispensers.

The total costs are determined by the following equations:

( ) In this section the model has been applied to a large number of consumable items like metallic and plastic small parts, personal protective equipment, packaging components, etc. All these items are characterized, by low price, high annual consumption, small dimensions and ease of storage, but, they reach a considerable annual monetary volume when considered cumulatively.

c nol c RS c o c m f tr f RS f o f m f m f s c tot f tot tot C C C C C C C C C C C C q C + + + + + + + + + = + = 2 , 1 , , , (37)

Numerical application

The data in this paragraph are relative to the production of a particular personal safety device, a protective glove for manual assembly operations, and are obtained applying the numerical values reported in table 4 to the analytical model presented in previous paragraph.

The sensitivity analysis has been conducted based on the variation of the following parameters:

• Standard deviation in market demand (in % of the average monthly demand)

• Monthly demand (units/month)

• Maximum stock level S (units)

• Item lifetime (years)

For each parameter, the sensitivity analysis carried out by means of MatLab™ and graphically shown as follows, shows the total annual costs of the logistic system (single-vendor and single-buyer) before and after the adoption of the CS policy.

Results provided in this paragraph prove that for each parameters' variable condition the cost of the CS policy is always lower than that related to the traditional policy.

Figure 3 shows inventory level at the buyer's and vendor's warehouses, and Figure 4 shows the total costs trend as a function of the shipment quantity q. For this particular case study, the lowest cost obtained with the traditional policy is 2,469 €/year with an optimal order quantity of 2,800 units/order. The total minimum cost with the CS policy is 1,460 €/year with an optimal shipment quantity of 1,334 units, in accordance with the maximum inventory level S (S= 3,500 units) imposed on both management policies.

The following graphs show how the cost reduction resulting from the adoption of a CS policy can depend on the existing constraints, as reported in Figure 1. The graphs in Figure 5 represent the effect of market demand variability over the total supply chain annual costs, for the two different management strategies, reporting the ratio between the costs under CS policy and the costs under the traditional management policy, if the constraints change:

Condition 1) relaxation of the maximum inventory level S constraint (i.e. warehousing space infinite at buyer's plant, S=infinite); The two graphs reported in Figure 5 show how, for the items studied, the CS policy always results in considerable savings compared to the traditional policy. The expected annual saving for the whole system (vendor and buyer) varies between 20% and 42% of the total annual management costs, depending on the constraints variation.

The graph on the left shows that total costs savings with CS policy decrease as the standard deviation in demand increases, if the constraint on the maximum inventory level at the buyer warehouse is not respected; on the contrary, if condition 2 is applied, the maximum level S constraint is considered. The graph on the right shows how the saving obtained with the CS policy is consistent also when the standard deviation in demand assumes high values.

Figure 6 reports total annual cost and saving analysis as a function of the item lifetime span T and of the maximum inventory level S. The lower annual cost is reached when S equals 3,605 units, that is, about 3.6 times the average monthly consumption of the item (this result is strictly related to items that are not bulky and are easily stored, conditions which are typical of consumables), reaching an annual management cost of 1,420 €/year. The optimal shipment quantity q, when S equals 3,605 units, is 1,501 units (1.5 times the average monthly demand). Total annual costs in figure 7 show an increasingly linear trend over the annual demand D, when the vendor productivity becomes a variable function of the buyer's average monthly demand (P=40*Dm).

[PLACE HERE TABLE 4] [PLACE HERE FIGURE 3]

Moreover figure 7 shows that as the vendor productivity increases proportionally with buyer's demand, the two curves diverge thus the adoption of the CS policy becomes more and more meaningful, resulting in increase savings whenever the demand D rises.

In conclusion, the numerical study demonstrates that total annual supply chain costs under CS decrease as the item lifetime span increases, whereas they increase as demand variability increases. The graphs shows that by setting a constraint S in the maximum inventory level at buyer's plant (due to space limitations in buyer's warehouses), the savings reached by a CS policy increases as the variability in item demand rises. The results obtained in the numerical application have been applied to other simulations and the results have been confirmed for over 100 different items belonging to the following categories: steel and brass small parts, electrical and plastic fasteners, small PPE (protection gloves, earplugs, respirators, protective lenses, etc.), and large consumption tools (wrenches, screwdrivers, metal toes, etc.), all of which present an average consumptions ranging from 500 to 500,000 units per year (a real application is shown in figure 8).

In every case the CS management policy has shown to be preferable than the traditional one, generating an average saving ranging from 20% to 45% of the total annual costs (results of the complete analysis are summarized in table 5) .

[PLACE HERE FIGURES 4, 5, 6, 7, 8]

[PLACE HERE TABLE 5]

Conclusions

In this paper a methodological framework to support the economical and technical feasibility study related to the migration from a traditional inventory management policy to a CS policy is developed and presented. A fully developed model has been presented to support the framework and the numerical results concerning its actual application have been reported.

The model is an extension of the previous studies proposed by [START_REF] Braglia | Modelling an industrial strategy for inventory management in supply chains: the 'Consignment Stock' case[END_REF], Valentini andZavanella (2003), and[START_REF] Persona | Consignment stock of inventories in the presence of obsolescence[END_REF], and for the first time it considers jointly operative environments characterized by obsolescence of materials, variability in demand and finite warehouse capacity at buyer's plant.

The CS management policy results to be always preferable in comparison with the popular EOQ traditional policy (with a reorder point and fixed order quantity), producing savings ranging from 20% to 45% of the total annual management cost.

Moreover, the benefit induced by the CS policy increases as the variability in demand rises, while complying with the maximum inventory level constraint (S) in the buyer's warehouse.

The model can be applied to manage inventories of consumables items, characterized by low unit price, high annual consumption, small dimensions and ease of storage (i.e. metallic, electric, and plastic fasteners, small parts, PPE, small tools). For this kind of items, a CS management policy has been proven to be effective even in presence of variable demand, obsolescence risk and constraints in the space available in buyer's plant.

Limitations and Future Research

As previously discussed at length, the results of this study are largely in accord with our theoretical exceptions. However, like the earlier studies, the present one has its limitations that must be addressed in future researches. First of all, the inventory levels can be controlled by different control policies. Which inventory control policy should be chosen depends on the item characteristics, its use, the consumption, price, availability at the supplier, transport costs and various other aspects. The model developed in this paper aims to present a comparative evaluation of the CS policy performance compared with the EOQ policy with ROP and Safety Stocks calculation. This is justified by experiences: EOQ and ROP are the most used methods to manage inventories of consumables materials which are characterized by low unit price, high annual consumption, small dimensions and ease of storage. Future research in this field should compare the proposed CS policy with results obtained by others policies, in particular with the Fixed Order Cycle (FOC), which aims at economical reordering, by ordering all items supplied by the same supplier together at regular review times or cycle periods, and the MIN-MAX method, which refill the stock to a pre defined maximum level when quantity on stock drops below the minimum level. Moreover, the model presented in this work consider only a single vendor and single buyer system.

The state of the art provided in this work underlines the necessity of future studies to address a systematic approach which considers a single-vendor and multi-buyer system in which several buyers can be managed with a CS policy by the vendor. For this

  This article highlights a topic not yet touched by other authors: the quantification of the strategic parameter 's', based upon the following critical considerations of the researches reported in table

  the buyer reserves a safety stock in his warehouse to protect himself from market demand fluctuations and from stock-out costs;

  presence of the maximum inventory level S constraint (i.e. warehousing space finite at buyer's plant, S=3,500 units);

Figure 3 .

 3 Figure 2. Deterministic obsolescence cost computation for traditional method

Figure 4 .Figure 5 .

 45 Figure 4. Total supply chain annual cost according to variation in shipment quantity q in presence of a maximum inventory level S= 3,500 units.

Figure 6 .

 6 Figure 6. Total annual costs (top-left) and annual savings (top-right) according to variations in the lifetime T of the component (years). Total annual costs under CS (bottom-left) and optimal shipment quantity (bottom-right) according to variations in the maximum inventory level permitted (S) .

Figure 7 .

 7 Figure 7. Safety stocks and total annual costs according to variations in demand standard deviation and in annual demand D under a traditional policy (dashed lines) and under a CS policy (continue lines).

Figure 8 .

 8 Figure 8. Consignment Stock of small PPE materials stored in automated distributors (on the left) and Consignment Stock of steel and brass fasteners stored in shelf (on the right).

F

Table 1 .

 1 reason, the authors are working on developing a new single vendor multiple buyer Consignment Stock inventory model. Consignment Stock policy: benefits

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

2 Figure 1. Methodological framework for a CS policy implementation

Table 3 .

 3 Critical factors analysed in step 2 of the framework

Table 5 .

 5 Results obtained during the testing phase (grouped per item category).

	Page 32 of 32

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uk International Journal of Production Research

http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.ukInternational Journal of Production Research


A A Ball

R J Cripps

J Lin

M Loftus
email: m.loftus@bham.ac.uk

Point-based approach to enhance the quality of geometric data for CAE applications

Keywords:

Commercial CAD surface modelling software is based almost exclusively on Bézier, Bspline and NURBS representations. These methods offer simple interactive shape modification and computationally efficient interrogations, but have some serious practical limitations. The root cause of all these problems is that the parametric polynomial methods are not geometry-based, and the trouble is compounded by the master geometry philosophy which treats the CAD model as if it were absolutely correct. An alternative purely geometric approach to surface modelling has been established, where the surface shape is characterised by a grid of points lying on the surface, but it is recognised that this approach would require major changes in existing CAE systems. This application paper explains the rationale to develop algorithms, which would plug into existing CAE software systems, so that the user could benefit from the point-based construction without disturbing the established mode of operation. The new point-based approach has achieved significant success in ensuring that form tools produce panels within a specified tolerance.

Introduction

Computer Aided Engineering (CAE) is a mature technology which can create photorealistic images of the most complex shaped components, analyse performance using Finite Element Analysis (FEA) or Computational Fluid Dynamics (CFD), manufacture tooling and control production. The technology has transformed engineering over the past thirty years but widening access and increasing expectations of accuracy and reliability are exposing fundamental weaknesses. As ever in engineering, the devil is in the detail and there is a growing awareness that high quality, virtually perfect Computer Aided Design (CAD) surfaces are a pre-requisite for successful CAE integration. Commercial CAD surface modelling software is based almost exclusively on Bézier, B-Spline and Non Uniform Rational B-Splines (NURBS) representations. These methods offer simple interactive shape modification and computationally efficient interrogations but have some serious practical limitations. In particular, it is a highly specialised and time-consuming task to define a totally satisfactory free-form CAD surface [START_REF] Ball | CAD: Master or Servant of Engineering?[END_REF], and any flaw in the definition can subsequently compromise CAD data exchange [START_REF] Wilson | Data exchange and software integration: Interdisciplinary design challenges[END_REF], FEA and CFD mesh generation [START_REF] Farouki | Siam Workshop on Integration of CAD and CFD[END_REF] and manifest itself in the CNC machining paths [START_REF] Klossowski | Requirements of high-end CAD/CAM systems in die and mold manufacturing[END_REF]. The root cause of all these problems is that the parametric polynomial methods are not geometry-based, and the trouble is compounded by the master geometry philosophy which treats the CAD model as if it were absolutely correct [START_REF] Ball | CAD: Master or Servant of Engineering?[END_REF].

A purely geometric approach to surface modelling has been established [START_REF] Cripps | Algorithms to support point-based CADCAM[END_REF], where the surface shape is characterised by a grid of points lying on the surface. The 2 difference between a characterisation of this type and an explicit surface equation is that it does not define a unique surface but rather an equivalence class of surfaces that are, within tolerance, the same shape. It is relatively straightforward to specify a grid of surface points and the format is equally applicable to physical objects, computer models or conceptual designs. In all cases, the interpolating surface is defined by a recursive geometric construction which generates a denser grid of points at each iterative stage. The distinctive features are the accurate estimation of curvature from point data [START_REF] Tookey | Estimation of Curvature from Planar Point Data[END_REF]) and the G 2 Hermite interpolation with a generalised Cornu spiral (GCS) which guarantees monotonic curvature variation between data points [START_REF] Ali | The Generalised Cornu spiral and its application to span generation[END_REF].

Geometric modelling

The point-based surface construction method is well established and guarantees surface definitions of a very high geometric quality, and using these high quality definitions in practical applications gives great benefits. However, it is recognised that the point-base form is not universally accessible and hence the need to convert point-based surfaces into a more accessible form. The intention was to develop a robust interface with the NURBS surfaces in CATIA ® with the expectation that the full generality of the rational form would be used to define an approximation to the point-based construction. Since CATIA ® recognises the rational form only in geometric display and approximates with the non-rational form (NUBS) for subsequent CAE computations, it was decided to concentrate on non-rational piecewise biquintic surfaces assembled with G 2 continuity and to approach the definition as a generalisation of non-rational bicubic B-spline surfaces. This approach is applicable to any NUBS-based CAD system. Given a rectangular grid of points with boundary tangent directions and parametric knot vectors, boundary tangent magnitudes and corner twist vectors have to be determined such that the quality of the resulting interpolated spline surface is comparable to the original point-based data.

In terms of parameterisation, the popular approach is to assign values of the knot vector and specify end tangent magnitudes so that the parameter approximates proportional distance along the NUBS curve. The most commonly used distance-based scheme is chord length [START_REF] Farin | Curves and Surfaces for Computer Aided Geometric Design[END_REF]. The resulting spline will not have constant proportional arc length at the knots of the NUBS, especially for unevenly spaced data sets. This could be addressed by optimising for the given data to improve the length of the curve span between consecutive points and the specified parameterisation. Alternatively, motivated by fitting circular and spiral arcs, a mathematical construction that optimises the parameterisation for the circle is sought. The idea [START_REF] Cripps | Orthogonal C 2 cubic spline curves[END_REF]) is to determine a parameterisation that tries to make the first and second derivative at each data point orthogonal, i.e.

2 2 d (u) du d (u) du 0 ⋅ = C C
. Thus given a string of data points; p i (i=0,…,1) , let a i =p i -p i-1 , b i =a i+1 and c i =a i +b i , then:

( ) ( ) ( ) ( ) 2 2 2 i i i i i i i 1 i i i 1 2 2 2 i i i i i i ½ u u (u u ) ½ . . . . + -   +   = + -   +   b c b c a c a c a c b c ; i=0,…,n-1, (1)
with u 0 =0 and u n =1. The end tangent magnitudes are given by: ( ) ( )

2 0 0 0 0 0 2 1 0 0 0 0 3 d (u ) du u u 2 . .     = -   +   a t a P t a t a , ( )
( ) 2 n n n n n 2 n n 1 n n n 3 d (u ) du u u 2 . . -     = -   +   a t a P t a t a (2)
This parameterisation scheme is optimal with respect to approximating circular arcs and numerical evidence has shown that the approximations match the radius of curvature profiles 3 to within ±5%. Encouragingly, the approach is suitable for both even and unevenly spaced data. For data sampled from a quality curve, the approach performs better than non-optimised chord length parameterisation and is therefore worthy of further attention since it is stable and direct, requiring no iterative methods or optimisation. As expected, it performs less well on spiral segments, but is good enough to suggest that an acceptable approximation to spiral segments can be achieved by using the orthogonal parameterised spline, raising it to quintic and adjusting the internal control points to suit [START_REF] Cripps | Circle approximation for CADCAM using orthogonal C 2 cubic B-splines[END_REF]. When interpolating a surface in CATIA ® , there is a need to 'average' the knot vectors for the respective curves in each parametric direction, but this is not an issue with an appropriate sampling of points from the point-based surfaces.

Existing methods for constructing the corner twist vectors have been evaluated and the Adini method, which simply adopts the twist vectors of a linearly blended Coons patch approximation, gave the best results [START_REF] Barnhill | A new twist in computer aided design[END_REF]. The quality of the Adini twist estimate was surprising because the method seemed naive. The errors in a bicubic spline approximation are typically worst in the four corner bicubic patches. In these regions, the spline surface with Adini twists gives consistently smaller errors than the linearly blended Coons patch and, assuming quadratic variations, it is possible to construct a linear combination with reduced errors. The Adini twist, when applied to all four corners of a single bicubic patch, produces a Coons patch by definition. When an Adini twist is used on the corner of a multi-patch interpolation, the resulting corner patch is not identical to a Coons patch; the non-corner twists of this sub-patch are, in general, different. For quality surfaces, there is an approximate linear relationship between the mid-point of the mutli-patch surface, the mid-point of the corresponding Coons patch and the mid-point that lies on the 'true' surface. As a consequence, there is an approximate linear relationship between the corresponding twist vectors, which can be expressed in the following form for one corner [START_REF] Lockyer | Controlling the interpolation of NURBS curves and surfaces[END_REF] [START_REF] Lockyer | An improved twist in CAGD[END_REF]. Figure 1 shows a typical result for a surface comprising 6X6 bicubic patches. Using zero twists gives a maximum discrepancy of 0.5611 mm. This was reduced to 0.1247 mm with Adini twists and 0.0309 mm with improved twists.

The resulting NUBS interpolated surface based on average orthogonal parameterisation and improved twists minimises any disturbance to the quality of the original point-based definition. Hence the effects of the geometry on downstream processes can be effectively filtered out, enabling such processes to be studied and hence controlled more accurately. This is illustrated in the following section where the CAE application relies on the quality of the geometry to ensure acceptable results.

Finite element modelling and form tool design

Finite element software can be used directly to model a forming process [START_REF] Lin | Selection of material model for predicting necking in superplastic forming[END_REF]) and predict the spring-back in a formed panel, but the real engineering problem is to determine the tool shape so that the panel springs back to the required shape [START_REF] Ho | Modelling of springback in creep forming thick aluminium sheets[END_REF][START_REF] Jeunechamps | A closed form technique to predict springback in creep age-forming[END_REF]). In practice it can be a time-consuming activity not only to iterate to an acceptable triangulation of the tool surface but to tidy the geometry for CNC machining. Consequently, the challenge is to automate the definition of the tool shape and, incidentally, to shift the focus of finite element modelling from analysis to design.

Tool design process

The first stage was to implement a system shell for simulating spring-back using the commercial finite element solver ABAQUS ® for creep age forming (CAF) processes [START_REF] Lin | Modelling of microstructure evolution in hot forming using unified constitutive equations[END_REF], which is used, for example, to form aircraft wing panels. The point based surface definition method was incorporated, through a user defined subroutine in ABAQUS ® , with algorithms to calculate the distance of a node from a tool surface and the tangent plane at a point on the tool surface. Denoting the required panel surface by P, (Figure 2), the tool surface at the n th iteration by T n and the corresponding predicted panel surface by P n , the iterated tool surface is defined by: T n+1 = T n +(P-P n ) n=0,1,…, with T 0 =0. The iteration terminates when = -D P P n is sufficiently small. The methodology is generally applicable to spring-back problems but the principal interest was in creep age forming where the spring-back is high (about 70%) compared to other sheet metal forming processes. The next stage was to construct the tool surface model. The shape quality of the tool surfaces was guaranteed by using a sparse grid of points, refined using GCS recursion. Having determined an appropriate grid for the panel surface P, it was assumed that the same number of points would be appropriate for all tool, T n , and predicted panel, P n , surfaces and the relative geometric spacing of the points across the respective surfaces should be the same. The effectiveness of this iteration method for tool design is directly related the accuracy of spring-back predicted via FE simulation.

Materials modelling and FE simulation

To obtain an accurate prediction of spring-back in creep-age forming, one of the most important aspects is to have an appropriate material model, which enables the creep deformation and thermally-activated stress relaxation to be accurately predicted. A group of creep ageing and stress-relaxation tests were designed and carried out for a 7XXX series aluminium alloy, which was solution heat-treated, quenched and cold-stretched. Precipitates nucleation and growth mechanisms are now understood and the effects of precipitate size on mechanical properties and creep deformation have been investigated (Figure 3). Based on a unified theories approach and ageing kinetics, and assuming isothermal ageing and particle growth at constant volume fraction, a new set of physically-based, unified ageing-creep constitutive equations has been proposed. This material model characterises precipitate formation and growth under the creep ageing condition and their interaction on creep [START_REF] Ho | Constitutive modelling of primary creep for age forming an aluminium alloy[END_REF]). The unified ageing-creep constitutive equations are solved using an implicit numerical integration method and the time step for the integration is automatically controlled. This ensures that the creep curves can be accurately computed. The equations were determined from creep-ageing experimental data using Evolutionary Programming (EP)-based optimisation methods [START_REF] Li | A novel evolutionary algorithm for determining unified creep damage constitutive equations[END_REF]) which aims to minimise the fitness function: the sum of the squares of the shortest distances between the computed and experimental data. The procedures to calculate the shortest distance of an experimental data (2004).

The creep deformation of the aluminium sheets in creep age-forming is governed by the determined unified creep-ageing constitutive equations, which were implemented into ABAQUS ® . In addition to the original constitutive model, the gradients, p p ∆ ∂ ∂ and e p ∂∆ ∂σ were determined accurately, so that the step time could be controlled efficiently through the implicit integration algorithm within ABAQUS ® . Multi-step numerical procedures have been developed to simulate the creep forming process and to evaluate the spring-back using ABAQUS ® . The procedures for the simulation of spring-back in creep age forming were detailed by [START_REF] Lin | An integrated process for modelling of precipitation hardening and springback in creep age-forming[END_REF].

An integrated approach

The point-based approach to tool design using FE analysis within the design process can be summerised by the following stages:

• Tool surface modelled using a commercial CAD package, for example CATIA©.

• Point-based model constructed using the geometric modelling plug-in.

• Point-based model taken as the input to the FE solver, for example ABACUS ® • Iterations within the FE solver alters the tool shape to compensate for springback • Resulting point-based tool surface placed back in host CAD system via the geometric modelling plug-in. The integrated numerical procedure for (a) predicting age hardening and spring-back via FE simulation and (b) offsetting the tool shape to compensate for the spring-back has been applied in a case study and shown to converge with the errors approximately halving at each iterative stage. Tool offsetting is carried out via an application software package developed through the research. The new trial tool is defined using the point based method and implemented into ABAQUS ® . Typically, 5 iterations of FE analysis are required to enable the tool shape to be determined to enable the difference between the designed and predicted panels to be controlled within a specified tolerance.

Conclusion

The major premise in this paper is that commercial CAD surface modelling software, based on Bézier, B-spline and NURBS representations, offer many advantages to the user, but have intrinsic serious practical limitations. The root cause of all these problems is that the parametric polynomial methods are not geometry-based, and the trouble is compounded by the master geometry philosophy which treats the CAD model as if it were absolutely correct. The alternative point-based surface modelling approach creates high quality surfaces that can be used in CAE applications without these practical limitations, but the authors recognise that the major CAD vendors are unlikely to change their valuable software code. The ability to represent the point-based reference in NUBS format delivers the point-based method to existing CAD packages. The CAE application shows that the point-based methodology has considerable potential, and that commercial reluctance to change can be overcome with this approach. Future work includes an investigation of direct modelling using the point-based approach, consideration of the applicability of a point-based approach to solid modelling and, finally, a more robust plug-in to interface with other FE solvers.

  Figure 1. NUBS interpolation using Adini(a) and Improved twists(b).

Figure 2 :

 2 Figure 2: Illustration of offsetting the current tool surface

Figure 3 :

 3 Figure 3: Schematic representation of material ageing stages: (I) Supersaturated solid solution, (II) Nucleation of GP zones (coherent) (III) Precipitate coarsening (semi-coherent) (IV) Over-aged condition (incoherent).

  corresponding numerically integrated creep curve were detailed byHo et al.

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Acknowledgements

The authors wish to acknowledge the financial support of the EPSRC under research grant GR/R35735/01, the contribution of researchers P.L.Chong, K.C.Ho, N.Liu and P.S.Lockyer, and the invaluable guidance provided by the industrial collaborators Abaqus, BAE Systems, Delcam, Landmark Components, Miracle Engineering, R.T.Palmer and Rolls-Royce.