J P Bourrières

T Lecompte-Alix

A linear model for production management -Optimal solving policies

Keywords: MANUFACTURING SYSTEMS, PRODUCTION MANAGEMENT, PROCESS MODELLING, MATH Manufacturing systems, Quadratic programming Manufacturing systems, production management, quadratic programming

A linear model for production management considering generic data on stocks, manufacturing processes, resource capacities and costs is presented. Any type of continuous or discrete production process also including assembly and disassembly tasks is taken into account. The model inversion issue is analysed and relevant joint production/inventory management policies are solved by optimizing linear and quadratic programs under resource and stock capacity constraints. Beyond the single period inversion issue which is the focus of the paper, the model is tailored to support multi-period and multi-level planning approaches whose development is the subject of separate publications.

Background

In manufacturing, the lean production and Just-In-Time concepts that appeared in the 70s are still in force to minimize as far as reasonable the stocks and to shorten the production cycles. The last decade has seen the emergence of networked organizations so-called supply chains [START_REF] Lee | The evaluation of supply chain management models and practice at Hewlet -Packard[END_REF], extended or virtual enterprises [START_REF] Jagdev | The extended enterprise-a context for manufacturing[END_REF] providing the enterprise network with the agility required by the market without taking the partners away from their own core business. Combining the traditional need for production planning with cooperative environments leads to the integration of Production Management and Logistics, and requires appropriate advanced ERP and APS solutions. The role of the underlying information system is determining to offer the relevant contextual information to the decision makers and to synchronize partners activities despite the heterogeneity of the physical resources, processes and applications [START_REF] Raghunathan | Impact of demand correlation on the value of and incentives for information sharing in a supply chain[END_REF]. Moreover, a focal enterprise commonly cumulates the activities resulting from various roles in the open market: those directly induced by customer orders, but also by external order makers in the surrounding supply chain(s). A consequence is that the networked production system is largely heterogeneous in terms of production management philosophy, as well as of IT solutions and the coherence of the global operating is most of the time far to be provable. The interoperability of the decision systems in networked manufacturing organizations is becoming a key topic in the research community [START_REF] Chen | Enterprise Interoperability Framework and Knowledge Corpus, Public Deliverable DI[END_REF]. To face the growing mass of surrounding technical data, various multi-echelon [START_REF] Mesarovic | Theory of hierarchical multilevel systems[END_REF], hierarchical or fractal [START_REF] Warnecke | The Fractal Company: a Revolution in Corporate Culture[END_REF]) decision making approaches based on aggregate views of the technical data were proposed [START_REF] Tung | Multiple objective scheduling for the hierarchical control of flexible manufacturing systems[END_REF][START_REF] Axsäter | A framework for decentralized multi-echelon inventory control[END_REF][START_REF] Van Der Vorst | Modelling and simulating multi-echelon food systems[END_REF][START_REF] Fontan | Hierarchical scheduling for decision support[END_REF]. Note that the aggregate planning approaches (Chen andLiao 2003, Kogan and[START_REF] Kogan | Multi-period aggregate production planning in a newsvendor framework[END_REF] are mainly driven by local planning solving issues and are not designed to be applied to networked manufacturing organizations. Production Management and Logistics are largely investigated as an application field of Operations Research [START_REF] Coughan | Action research for operations management[END_REF][START_REF] Eom | A survey of decision support system applications[END_REF][START_REF] Scholl | State-of-the-art exact and heuristic solution procedures for simple assembly line balancing[END_REF], more specifically addressing optimal inventory management, Supply Chain management [START_REF] Lee | The evaluation of supply chain management models and practice at Hewlet -Packard[END_REF], planning and scheduling [START_REF] Uzsoy | Book review " the Planning and Scheduling of Production Systems, Chapman and Hall[END_REF]. Nevertheless most of the models supporting these optimization issues are case-specific and the global modeling of a networked production management is far to be achieved in the literature.

Among the generic issues addressed in Production Management are planning and scheduling. Resource planning, as largely promoted by the MRP approach [START_REF] Orlicky | Material Requirements Planning[END_REF], is the definition of the activity of a production unit seen as one aggregate resource (typically a workshop or a factory) in a sequence of future time periods, and correlatively of a material procurement as well as product delivery plan. Planning solutions appeal to optimization techniques to implement management policies under capacity and cost constraints [START_REF] Olhager | Enterprise Resource Planning survey of Swedish manufacturing firms[END_REF]. The scheduling issue considers the internal resources (typically workstations) of a production system and aims to allocate appropriate sub-activities to those resources, according to a complex decision subject to logical (task precedence), technical (transformation and transport resource capacities) and economical (work and stock costs) constraints. See e.g. (Tharumarajah 2001, Herroelen and[START_REF] Herroelen | Project scheduling under uncertainty: Survey and research potentials[END_REF] for a review of scheduling techniques. Actually, planning and scheduling are two types of decision to organize the activity of networked production organizations at various levels of detail. The need for a joint approach to planning and scheduling is emphasized in the literature [START_REF] Barták | Conceptual Models for Combined Planning and Scheduling[END_REF][START_REF] Smith | Bridging the Gap Between Planning and Scheduling[END_REF].

The goal of our investigations was to develop a formal and generic model allowing to support decision making in activity planning and scheduling within networked organizations [START_REF] Bourrières | Multi-level algebraic modelling of complex manufacturing tasks[END_REF][START_REF] Lecompte | A data model for generalised scheduling for virtual enterprise[END_REF]. As a first step towards this objective, this paper is focused on the elaboration of our generic production model and on its inversion over one single time-period under combined production and stock management strategies. The planning issue as such is not the subject of this paper. Nonetheless the model presented here is tailored to support multi-period planning and multi-level scheduling investigations that are the subject of separate publications. The last section indicates how to extend the results presented in this paper to address the multi-period planning issue. The remainder of the paper is organized as follows: section 2 presents the generic model or direct production model; section 3 focuses on model inversion and on optimal production solving policies; section 4 is dedicated to an illustrative example; and section 5 is a conclusion.

Direct production model

In this section are presented the parameters and variables to capture the technical data, then a production model is established that links the variations of the product amounts to the works generating those variations. We also define the resource capacities and cost data. All vectors in this paper are non-negative column-vectors.

Product modeling

Parameters: The parameters below define the nomenclature of all product items The vectors above depict the virtual stocks of products in the production system.

Work modeling

Parameters: We similarly introduce the parameters and variables referring to the transformation tasks:

T the set of tasks n the length of task list of references; n= T Variables: Vector w addresses quantities of transformation tasks and as such qualifies and quantifies the works carried out in the manufacturing system. w the vector of task amounts with reference to set T

Timing

In production management, time is divided into periods of constant duration and the concatenation of periods covers a finite time horizon. Below are introduced time-related notations.

s(k )

the stock vector at the beginning of period k s(k +1) the stock vector at the end of period k, i.e. beginning of period k +1 w (k ) the vector of work performed during period k

In this paper, the production problem is considered over one period only, turning the stock vector from s(k ) to s(k +1). In the conclusion we mention how to bridge the one-period optimization related in this paper with the multi-period planning issue.

Costing

Cost modelling for production management basically consists in the valuation of product transformation costs and of inventory costs. The data model is here

β(k)
the cost vector of the work performed during period k γ(k) the cost vector of stocks over period k Over period k, the cost of work w(k) is β T (k)w(k) and the cost of stock s(k) is γ T (k)s(k).

Direct production model

Any product transformation requires specific input items and produces output items as well. Considering all item references and all product transformations feasible by the production system, the quantitative impact of a definite work w(k) on stock vector s(k) is defined by the linear model:

s(k + 1) -s(k) = Cw(k) w(k) ≥ 0 s(k) ≥ 0 s(k + 1) ≥ 0 (1)

Resource capacity model

The expression of resource capacities must be homogeneous with the expression of works, in order to be able to assess the load-capacity alignment. We define the capacity of a transformation resource for period k as the maximal work that the resource is able to perform during the period. The resource is potentially polyvalent and the way the resource is able to share its capacity upon the various types of work must also be modeled. One classically distinguishes disjunctive from cumulative resources [START_REF] Mercier | Strong polynomiality of resource constraint propagation[END_REF]. A disjunctive resource is able to perform one task at once among a set of feasible tasks, in other words is constrained by time-sharing. A cumulative resource is able to perform various tasks simultaneously and independently. In the latter case, the parallel tasks are commonly sharing a physical scalar (energy, space, weight, etc.). Nevertheless, if the resource is an aggregate and abstract resource, the various activities may be performed independently (e.g. a workshop can simultaneously and independently cumulate various activities allocated to distinct machines). To summarize, three basic categories of resource cover most of the situations: disjunctive resource, cumulative resource with dependent capacities, cumulative resource with independent capacities, but the two first categories can be merged as they both express the sharing of a scalar (time, energy, etc.). Hence, we finally distinguish two categories of transformation resources only: with dependent capacities or with independent capacities.

Below is defined the working capacity of a resource by a vector a(k), a(k) ≥ 0, the component a j (k), j ∈ {1, • • • , n} of which is the inverse of the maximal number of j -type task occurrences that can be achieved during period k, assumed that the resource does not execute any other task. The work amount w(k) can be achieved by the resource during period k under the following conditions:

a T (k)w(k) ≤ 1 dependent capacities (2a) A(k)w (k) ≤ 1 independent capacities (2b) with A(k) = diag[a 1 (k), • • • , a n (k)].
The graphical representation of capacity models (2a) and ( 2b) for n=2 is shown on Figure 2.

The load rate of the resource can be characterized by a scalar ρ(k) ∈ [0, 1] when the resource is with dependent capacities, and by a vector ρ(k) ∈ [0, 1] n when the resource is with independent capacities, respectively defined by:

ρ(k) = a T (k)w(k) ρ(k) ∈ [0, 1] dependent capacities (3a) ρ(k) = A(k)w(k) ρ(k) ∈ [0, 1] n independent capacities (3b)
A correlated issue is to identify the minimal duration d[w(k)] required to achieve a given load w(k). As the capacities are proportional to the duration of the period, an extrapolation (see Figure 3) can be made that yields: 3. Inverse production model

d[w(k)] = ha T (k)w(k) dependent capacities (4a) d[w(k)] = hM ax[a j (k)w j (k)] j=1,n independent capacities (4b)

Production capacity

A preliminary issue is to identify the maximal amount of products the system is able to achieve in the period, in other words to turn the resource capacities into product capacities : given the resource capacity vector a(k) and the current stock level vector s(k), find the maximal amount of j-type products that the system is able to manufacture in period k. The question may be relevant not only regarding the finished products (set O y ) but also any intermediate items (set O x ). Moreover, considering the particular case s x (k) = 0 leads to determine the number of j-type products the system is per se able to produce in the period, i.e. limited only by its capacity, assumed that the stocks of intermediate products are initially empty.

Let C i be the i-eth line of matrix C. The work w(k) generates the quantity C i w(k) of items o i . The maximal production of items o i under the resource capacity constraint (2a) or ( 2b) is provided by solving the following linear program:

Maximize C i w(k) (5)
subject to :

w(k) ≥ 0 non negative solutions s(k) + Cw(k) ≥ s * (k + 1) stock floor constraint a T (k)w(k) ≤ 1 or A(k)w(k) ≤ 1 resource capacity constraint w(k) ∈ N n natural integer solutions (optional)
The last constraint is optional. The stock floor constraint s * (k + 1) is given non-negative. The initial stock s(k) is assumed to satisfy the floor and ceiling constraints. One may also add a stock ceilling constraint as shown in section 3.3.

The program above solves the maximal production problem. Nevertheless the solution w(k) may be not unique. It is necessary to search for the optimal achievement of the maximal production target stemming from program (5) by additionally considering one of the optimization programs presented in the next section.

Model inversion

The inversion of model ( 1) is interpreted as follows: given the current stock s(k), find any work vector w(k) producing the stock variation δ(k) = s(k + 1)s(k) expected. Practically, the production target δ(k) is only partially specified, addressing at least the items of O y (finished products) but also potentially some items of O x (intermediate products) when a stock regulation policy is applied. The unspecified part of δ(k), i.e. the consumption of components and the unspecified stock variations of intermediate products will result from the solution w(k) chosen. We consequently share set O in two subsets: Ō the list of item references for which a production target is given, and Ô the list of item references for which no production target is specified. Accordingly, the stock cost vector γ is split into vectors γ and γ, the variation vector δ(k) into δ(k) and δ(k), and matrix C into blocks C

and Ĉ whose dimensions are m-by-n and m-by-n respectively, with m the cardinal of Ō and m = m -m the cardinal of Ô. Consequently, the production model ( 1) is refined as follows:

F o r P e e r R e v i e w O n l y δ(k) = Cw(k) (6) δ(k) = Ĉw(k) (7)
Then, to invert the linear form (6) i.e. to identify the work w(k) required by the achievement of the specified stock variation δ(k). Each solution w(k) will thanks to relation ( 7), allow to determine the free stock variation δ(k), in particular the component requirement.

Matrix C being non square, the inversion of the linear model ( 6) falls under classical discussion. The inversion problem has solutions only if all of the mrank C characteristic determinants of equation ( 6) all equate to zero. Under these conditions, the dimension of the solution space, i.e. the order of multiplicity of the solution, is n-rank C. We here assume that C is regular, i.e. rank C = M in( m, n).

If m > n, i.e. rank C = n, the system (6) is overdetermined. The system is invertible when all of the mn characteristic determinants are equal to zero. Under this condition, there is an unique solution w(k) = C+ δ(k) where C+ = CT ( C CT ) -1 is the Moore-Penrose inverse [START_REF] Penrose | A generalized inverse for matrices[END_REF], also called pseudo-inverse, of C. The corresponding free stock variation is δ(k) = Ĉ C+ δ(k). Note that as the target δ(k) is implicated in the characteristic determinants, the existence of solutions results from the compatibility of the production target with the logical structure of the manufacturing process expressed by matrix C. Should the system be non invertible, in other words when the production target cannot be exactly achieved, then w(k) = C+ δ(k) is the 'best fit' (least square) solution. This solution generates a stock variation that is as close as possible to the target δ(k) expected. If m = n, i.e. rank C = m, equation ( 6) is a Cramer system whose unique solution is w

(k) = C-1 δ(k), therefore δ(k) = Ĉ C-1 δ(k).
If m < n, i.e. rank C = m, the solution is not unique and an optimization criterion is required to identify good solutions within the n -m dimensional solution space. In production management, the objective is obviously to minimize both stock and working costs. The stock management policy is here expressed i) by the specification of 'hard' targets s(k + 1) for the items whose stock level at the end of the period is strictly defined and ii) by the definition, in the optimization function, of 'soft' attracting targets ŝ * (k + 1) for all other items. Note that the number m of controlled stocks must not overpass the number n of tasks, otherwise the system (6) is over-determined and conditionally invertible only.

Below is discussed how to solve the optimization problem, using first a linear programming approach then quadratic programming.

Optimal inverse model

An economical way to reach the target δ(k) is defined by the following linear program:

F o r P e e r R e v i e w O n l y M inimize J 1 = β T w(k) + γT [ŝ(k + 1) -ŝ * (k + 1)] (8)
subject to :

w(k) ≥ 0 non negative solutions Cw(k) = δ(k) achieve specified targets ŝ(k + 1) = ŝ(k) + Ĉw(k) production model ŝ(k + 1) ≥ ŝ * (k + 1) stock floor constraint a T (k)w(k) ≤ 1 or A(k)w(k) ≤ 1 respect resource capacity ŝ(k + 1) ≤ ŝMax stock ceiling constraint (optional) w(k) ∈ N n natural integer solutions (optional)
The two last constraints are optional. The stock floor ŝ * (k + 1) and ceilling ŝMax are given non-negative. Should some stock levels remain free of control, the corresponding weights in vector γ will be set to weak values. The linear program formulated above is easily solved using a Simplex algorithm. The solving is made in real or natural integer numbers, depending on the semantics of the production model. Given the initial stock s(k) and the target δ(k), the optimization process finally provides the optimal work load w • (k), the corresponding load rate according to (3a) or (3b), the minimal cost J • 1 and the stock variation δ

(k) = Ĉw • (k) among which is in particular the component requirement δu (k) = C u w • (k).
Instead of solving numerically the optimal production problem, a parametric approach may be also expected to provide a deeper understanding of the problem solving. We will focus on the following quadratic optimization problem:

Minimize J = w T Ψw Ψ symmetric positive definite (9)
subject to:

Cw(k) = δ(k) C regular m-by-n m < n w(k) ≥ 0 a T (k)w(k) ≤ 1 dependent capacities or A(k)w(k) ≤ 1 independent capacities
Matrix Ψ will allow us to express definite production strategies.

Minimizing a quadratic criterion subject to underdetermined linear equality constraints is a classical issue, see e.g. (Ben-Israel 2002):

J = x T Ψx s.t. Mx = N is minimal for x = Ψ -1 M T [MΨ -1 M T ] -1 N (10)
assumed that Ψ is a symmetric positive definite matrix. Adding the inequality constraints (the optimal solution w • (k) must be non-negative and satisfy the capacity constraints) the program has a unique solution when the Karush-Kuhn-Tucker (KKT) conditions are satisfied, see e.g. [START_REF] Jensen | Operations Research Models and Methods: Models and Methods[END_REF]Bard 2002, Shinners 1998). Logarithmic barriers may also be introduced in the objective function to consider positive solutions [START_REF] Frisch | The logarithmic potential method for convex programming[END_REF]. Nevertheless, solving the KKT conditions leads to set up a new linear program itself requiring Simplex-like numerical algorithms. The solving process is thus not fully parametric and there is no more argument to prefer the quadratic optimization approach to the linear program (8). Here we bypass the difficulty of quadratic programming under inequality constraints by previously identifying the 'feasible' production targets for which we are sure that the optimal solution is non-negative and capacity complying. We hereafter make a two-fold study consisting in i) identifying the optimal solution w • (k) for some relevant quadratic objective functions subject to equality constraints only and ii) identifying the conditions of target feasibility which ensure that the inequality constraints are satisfied.

Let first equation (6) be considered. An analytical description of the solution space is provided by the generalised inverse formalism:

w(k, z) = C+ δ(k) + (I - C+ C)z ( 11
)
where z is an arbitrary n-dimensional vector, I the identity matrix and C+ the pseudo-inverse of C [START_REF] Mayer | Generalized inversion of modified matrices[END_REF][START_REF] Varga | Computing generalized inverse systems using matrix pencil methods[END_REF][START_REF] Ben-Israel | The Moore of the Moore-Penrose inverse[END_REF]. According to (7), the corresponding free stock variation is:

δ(k, z) = Ĉ C+ δ(k) + Ĉ(I - C+ C)z (12)
Within the solution domain ( 11) we identify the solutions that optimize some specific quadratic criteria expressing relevant production policies, as shown in Table 1. Hereafter is discussed each of the optimization policies. (J 3 ) Minimize working cost. We have to minimize criterion J 3 = Ew(k) 2 with E=diag(β 1 , • • • β n ) where β j is the cost of task j. Let v be a vector such that w(k)=E -1 v . According to (10), minimizing

J 3 = v 2 s.t. CE -1 v = δ(k) leads to solution v = ( CE -1 ) + δ(k) . Therefore J 3 is minimal for w • (k) = E -1 ( CE -1 ) + δ(k).
If more particularly the task costs are all identical, then E=I and w • (k) = C+ δ(k).

(J 4a ) Load rate control (dependent capacities). Assumed that the resource has dependent capacities defined by (2a), the objective function J 4a aims to engage the resource capacities at a specified level ρ • (k) while achieving the production target δ(k). Here we have to minimize

ρ(k) -ρ • (k) 2 = [w(k) -w • (k)] T a(k)a(k) T [w(k) -w • (k)] where w • (k) is so that a(k) T w • (k) = ρ • (k).
Unfortunately, matrix a(k)a(k) T is singular thus solution (10) is not applicable. To minimize J 4a we follow [START_REF] Fletcher | Semi-definite matrix constraints in optimization[END_REF] by identifying a m-by-n matrix S and a m-by-(n-m) matrix Z complying with CS=I and CZ=0 and [S|Z] invertible. Note that Z T a(k)a(k) T Z is positive definite on the solution space defined by the equality

L(w, λ) = (w -w • ) T aa T (w -w • ) + λ T ( Cw -δ)
which requires to solve :

∇ w = 2aa T (w -w • ) - CT λ = 0 ∇ λ = Cw -δ = 0
By introducing a vector v such as w=S δ+Zv we still have Cwδ = 0 since CS=I and CZ=0. By left multiplying ∇ w by Z T we eliminate λ:

Z T aa T (S δ + Zv -w • ) = 0 As Z T aa T Z is regular, the unique solution is : v = (Z T aa T Z) -1 Z T aa T (w • -S δ)
Finally, w=S δ+Zv provides the optimal solution:

w = Z(a T Z) -1 ρ • + [I -Z(a T Z) -1 a T ]S δ
Note that solution w leads ρ(k)ρ • (k) 2 to be exactly equal to zero hence w = w • . The reason is that the quadratic program is semi-definite. This can be verified by left multiplying w by a T , which leads to a T w = 1ρ • + 0 δ = ρ • .

(J 4b ) Load rate control (independent capacities). If the resource has independent capacities, see (2b), one may wish to spare the resource capacities by minimizing

J 4b = ρ(k) 2 = Aw(k) 2 with A(k) = diag[a 1 (k), • • • , a n (k)]. The optimal solution is here w • (k) = A -1 ( CA -1 )
+ δ(k). As matrix A is regular, the quadratic programme is definite and does not bring J 4b to zero but to its minimal positive value.

* (k + 1) = ŝ(k) + δ * (k). Noting that δ(k) - δ * (k) = ŝ(k + 1) -ŝ * (k + 1), the expression F[ δ(k) - δ * (k)
] 2 is the weighted quadratic error on the final stock and the entries of the diagonal matrix F are the stock costs. The minimization of J 5 with simplified notations is as follows: Here w * is the work generated by the variation δ * . Using the Lagrange multiplier technic, we have L(w, λ) = J 5 + λ T ( Cwδ) and consequently:

J 5 = Ew 2 + F( δ - δ * ) 2 = w T Φw -2w * T ĈT F2 Ĉw + w * T ĈT F2 Ĉw * where Φ = E 2 + ĈT F2 Ĉ.
∇ w = 2Φw -2 ĈT F2 Ĉw * - ĈT λ = 0 ∇ λ = Cw -δ = 0
whose explicit solution after some calculation is :

w • = Φ -1 ĈT ( CΦ -1 ĈT ) -1 δ + [I -Φ -1 ĈT ( CΦ -1 ĈT ) -1 C]Φ -1 ĈT F2 δ *
Hence the error on the final stock is ŝ * (k + 1) -Ĉw • (k). The particular case E = 0 minimizes stock costs only, whereas program J 5 tends to program J 3 (work minimization) when F tends to 0.

Taking into account inequality constraints : target feasibility domain

To ensure that the optimal solutions given by Table 1 are non-negative (real or natural integer) vectors satisfying the capacity constraints, we here identify the restrictions on the production targets.

Looking for real solutions, the optimal solutions from Table 1 must additionally satisfy the following constraints depending on the capacity model ( 2a) or (2b):

w • (k) ≥ 0 a T (k)w • (k) ∈ [0, 1] (dependent capacities) (13a) w • (k) = A -1 ρ • (k) ρ • (k) ∈ [0, 1] n (independent capacities) (13b)
The condition (13a) or (13b) defines the target feasibility domain making δ(k), δ * (k) and ρ • (k) linearly dependent. Note that the target feasibility domain is a linear inequality defining a convex set and may be turned into a standard LMI form to benefit from LMI-based optimization techniques [START_REF] Boyd | Linear Matrix Inequalities in Systems and Control Theory[END_REF]. For example looking for the real solutions of quadratic program J 4a (load rate control), the conditions (13a) applied to J 4a define the target feasibility domain as follows:

Z(a T Z) -1 ρ • (k) + [I -Z(a T Z) -1 a T ]S δ(k) ≥ 0 ρ • (k) ∈ [0, 1] (14a)
whereas (13b) combined with (6) defines the following feasibility domain:

δ(k) = CA -1 ρ • (k) ρ • (k) ∈ [0, 1] n (14b)
Looking for natural integer solutions, the optimal solutions from Table 1 must satisfy the following constraints depending on the capacity model (2a) or (2b):

w • (k) ∈ N n a T (k)w • (k) ∈ [0, 1] (dependent capacities) (15a) w • (k) = A -1 ρ • (k) [ρ • (k)] j = η j a j (independent capacities) (15b)
with η j = 0, 1, . Finally, an appropriate approach to solve the quadratic programs J 2 to J 5 submitted to both equality and inequality constraints is the following: 1) prior define the feasible targets under inequality constraints using (13a), (13b), (15a) or (15b), then 2) apply the optimal solution provided by Table 1. Programs J 2 ,J 3 ,J 5 are usable with both capacity models (2a) and (2b). Program J 4a suits for the dependent capacity model (2a) whereas J 4b suits for the independent capacity model (2b).

Application

This section illustrates the use of our linear model in the determination of singleperiod production scenarios according to the control policies presented in Table 1. We consider the production model shown on Figure 1. We associate the 'hard' production target with items o 4 , o 6 and o 7 whereas the 'soft' target is on items o 3 and o 5 . We consequently note:

δ(k) = (δ 4 δ 6 δ 7 ) T δ(k) = (δ 1 δ 2 δ 3 δ 5 ) T
Let moreover the following data be considered: We have :

s(k) = (
Ō = {o 4 , o 6 , o 7 } C =   1 1 -2 -1 0 0 3 0 0 0 1 2   Ô = {o 1 , o 2 , o 3 , o 5 } Ĉ =     -2 -1 0 0 0 -1 0 0 2 0 -1 0 0 1 0 -3     C+ = 1 6     3 1.5 1.5 3 1.5 1.5 0 2 0 0 -1 3     I - C+ C = 1 2     1 -1 0 0 -1 1 0 0 0 0 0 0 0 0 0 0    
Here m = 3, n = 4, rank C = 3 and the dimension of the solution space is nrank C = 1. To solve the semi-definite program J 4a the following matrices may be used: which verify CS = I and CZ = 0.

S = 1 6     6 0 3 0 3 0 0 2 0 0 -1 3     Z =     1 -1 0 0     F o r P e e r
Maximal production The linear program (5) applied to items o 6 and o 7 provides the production capacities shown in Table 2. Here the constraint w(k) ∈ N n is additionally considered. The production capacities are both computed for the initial stock s(k) previously given and for s x (k) = 0 (the intermediate stocks are empty and the resource has consequently to achieve the whole manufacturing processes). The results are obtained for dependent and for independent resource capacities.

Feasible targets Assumed first that the resource is with dependent capacities, we solve the load rate control problem by previously identifying the feasible targets according to (13a):

    125 -125 0 0     ρ • (k) + 1 24     -6 -3 -33 30 15 45 0 8 0 0 -4 12     δ(k) ≥ 0 ρ • (k) ∈ [0, 1]
The LMI above defines the domain of feasibility of the production target δ(k) and ensures that the corresponding work vector w(k) is non negative and satisfies the capacity constraint. The domain is ρ • -homothetic. It is shown on Figure 4a. Setting the load rate to 1, the production of achieved items is maximal at point P 1 ( δ6 =300; δ7 =100), assumed that the stock level of o 4 is sufficient ( δ4 =-200). If the production of o 7 only is expected, one can obtain up to δ7 =100 with δ4 =-50, see point P 2 . One may also expect to concentrate the activity on the production item o 4 up to δ4 =500, see P 3 .

If the resource is with independent capacities, we consider (13b): δ(k) =   100 500 -200 -50 0 0 300 0 0 0 100 100

  ρ • (k) ρ • (k) ∈ [0, 1] 4
The target feasibility domain is shown on Figure 4b.

Optimal production policies Let e.g. the 'hard' target δ(k)=(0 70 40) T be specified i.e. to produce 70 o 6 and 40 o 7 while strictly maintaining the stock o 4 unchanged. To be feasible, the target must additionally comply with (13a) or (13b), which yields:

0.51 ≤ ρ • (k) ≤ 0.95 (dependent capacity) 20ρ • 1 + 100ρ • 2 = 11 ρ • 3 = 0.23 ρ • 4 = 0.17 (independent capacity)
The optimized solutions to reach the target δ(k) = (0 70 40) T with reference to criteria J 1 to J 5 are presented in Table 3. A Simplex solver was used to minimize J 1 . The other criteria were optimized analytically, applying the results from Table 2. For criterion J 4a we chose ρ • = 0.51, the minimal possible value of the load rate identified above. The 'soft' target δ * (k) = (-100 -100 0 0) T was used for J 5 . The optimization of J 5 was made for the matrix E previously given (work and stock cost minimization) and for E=0 (stock cost minimization only) as well. For each optimization problem, we have computed the optimal solution w • (k) and the corresponding production δ(k). The 'hard' production target δ(k) is strictly reached, whereas the 'free' production δ(k) -of which the component requirementstems from the particular economical and/or stock level control policy expressed by the objective function. The optimization of J 2 is very easily computerized and minimizes the quantity of operations to be performed. Nevertheless the resource capacity is also partly engaged to generate a parasitic production. Minimizing J 3 provides the most economical solution with regard to the working cost. But the parasitic production of items o 3 is very high and over consumes the resource capacity (ρ • =0.86). The programs J 4a and J 4b allow to spare the resource capacity by engaging the minimal load rate. The counterpart is the consumption of the intermediate stock o 3 . The program J 1 and J 5 give the priority to working and stocking costs minimization rather than sparing the load rate.

Conclusion

The linear model presented is this paper is applicable to a large range of discrete or continuous production systems performing various transformation tasks, including assembly/disassembly. The boundaries of the article are the consideration of single-period production and stock management policies formalized as optimization problems subject to linear equalities (the production model) and inequalities (resource capacity constraints). Appealing to linear criteria leads to easy problem solving approaches using popular Simplex algorithms. Being numerical, those algorithms nevertheless do not provide any deep understanding of the solutions obtained. For this reason, we have performed the analytical solving of some quadratic programs expressing the most relevant policies. The inequality constraints were integrated through prior analysis of the target feasibility domain.

On the basis of the model presented in this paper, our investigations are in two directions : i) multi-period resource planning and ii) multi-level scheduling.

The multi-period planning model is derived from the one-period production problem by extending vector w(k) to a vector w(k) such that:

wT (k) = w T (k) π T (k) ξ T (k)
with π T (k) the procurement vector in period k and ξ T (k) the expedition vector in period k. We consequently state an under-determined linear model over a planning horizon model k = 1,H with (w T (k), k =1,H) as decision variables whose inversion under capacity and cost constraints leads to formulate linear or quadratic programs similar to those studied in this paper.

The formulation of the multi-level scheduling problem addresses simultaneously two levels of data : i) aggregate data related to the capacity of a mother resource as well as to the corresponding product/task/processes and ii) detailed data related to the network of daughter-resources encapsulated in the mother resource. The interrelations between the aggregate and detailed views can be formulated in a linear framework and lead to identify aggregation/disaggregation consistency rules. Based on the linear model presented in this paper, our investigations aim to promote unified planning and scheduling generic primitives in a multi-level plan breaking-up approach to iteratively turn a product order book into detailed activities.

Optimization criterion

Optimal solution

J 2 = w(k) 2 w • (k) = C+ δ(k) J 3 = Ew(k) 2 w • (k) = E -1 ( CE -1 ) + δ(k) J 4a = ρ(k) -ρ • (k) 2 w • (k) = Z(a T Z) -1 ρ • (k) + [I -Z(a T Z) -1 a T ]S δ(k) J 4b = ρ(k) 2 w • (k) = A -1 ( CA -1 ) + δ(k) J 5 = Ew(k) 2 + F( δ(k) - δ * (k)) 2 w • (k) = Φ -1 ĈT ( CΦ -1 ĈT ) -1 δ(k) + [I -Φ -1 ĈT ( CΦ -1 ĈT ) -1 C]Φ -1 ĈT F2 δ * (k) Notes: E = diag(β 1 , • • • βn) A = diag(a 1 , • • • an) F diagonal Φ = E 2 + ĈT F2 Ĉ
J 4a and J 4b for respectively dependent and independent resource capacities

(J 2 )

 2 Minimize work amount. Minimizing criterion J 2 = w(k) 2 simply ensures that the target is achieved with the minimal quadratic work amount. The optimal solution is w • (k) = C+ δ(k), provided by (10) with Ψ = I, M = C and N= δ(k).

  Taylor & Francis and I.T. Consultant constraint. With simplified notations, we minimize the following Lagrangian :

(J 5 )

 5 Stock and/or work control. The component Ew(k) 2 of criterion J 5 aims to minimize work cost whereas F[ δ(k) -δ * (k)] 2 is to attract the parasitic production δ(k) to a predefined value δ * (k) expressing a 'soft' stock target ŝ

  Figure 1. Production model

  the number of component references; m u = O u m x the number of intermediate product references; m x = O x m y the number of finished product references; m y = O y m the total number of product references ; m = m u +m x +m y Variables: The following vectors address quantities of products of any possible references: s u the vector of component amounts with reference to set O u s x the vector of intermediate product amounts with reference to set O x s y the vector of finished product amounts with reference to set O y s the global stock vector, i.e. the concatenation of vectors s u , s x , s y

	traced in the manufacturing system: components, intermediate and finished prod-
	ucts.
	O u	the set of component references
	O x	the set of intermediate product references
	O y	the set of finished product references
	O	the global set of product references
	m u
		F o r
		P
		e
		e r
		R
		e
		v i e
		w
		O n l
		y

  , Cx and C y defined with reference to sets O u , O x and O y . The entries of C u are non-positive since the input components can only be consumed by the production processes. The entries of C y are non-negative as the finished products can only be generated by the production processes. Figure1shows an example of production model with O u ={o 1 , o 2 } two references of components, O x ={o 3 , o 4 , o 5 } three references of intermediate products, O y ={o 6 , o 7 } two references of finished products and T ={t 1 , t 2 , t 3 , t 4 } four references of tasks.

	Notation s(k ) and s(k +1) respectively refers to the state of the global stock before
	and after completion of work w (k ). The m-by-n matrix C defines the variation of
	product amounts generated by the achievement of work w (k ). The components of
	vectors w (k ) and s(k ) are non negative. They may be natural integer or real num-
	bers, depending on the application case. Equation (1) links the working activities
	to the consumption and production of stocks, in other words identifies the direct
	production model, i.e. all feasible manufacturing processes. A graphical representa-
	tion of the direct production model is obtained by considering equation (1) as the
	fundamental equation of a Petri net < O, T, C > where the set of places O is the
	concatenation of sets O u , O x and O y of product references, the set of transitions
	T is the set of task references and C is the incidence matrix of the Petri net. The
	weighting of the arcs defines the number of product items consumed and produced
	by the occurrence of each task. Note that all types of product transformations (ma-
	chining, assembly, cutting, etc) without any restriction can be modeled. Matrix C
	is composed of three blocks C u

  ., a j

	April 24, 2009	16:26	International Journal of Production Research	revision1
			F o r
			P
			e
			e r
			R
			e
				v i e
				w
				O n l
				y
			-1

Table 1 .

 1 Optimal solutions s.t. equality constraint Cw(k) = δ(k)

		April 24, 2009
		16:26
	International Journal of Production Research	International Journal of Production Research revision1
	17

Table 2 .

 2 Production capacity

	Resource capacity	dependent dependent independent independent
	Initial stock	s(k)	sx(k)=0	s(k)	sx(k)=0
	Max δ 6	216	165	300	300
	Max δ 7	90	76	200	200

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Table 3 .

 3 Optimal production scenarios for target δ(k) =(0 70 40) T

	Policy		Optimal work		Load rate			Stock variation
		w 1 (k) w 2 (k) w 3 (k) w 4 (k)	ρ(k)	δ1 (k)	δ2 (k)	δ3 (k)	δ4 (k)	δ5 (k)	δ6 (k)	δ7 (k)
						ρ T (k)
						resource with dependent capacities
	J 1	30	25	23.33	8.33	0.75	-85	-25	36.66	0	0	70	40
	J 4a	0	55	23.33	8.33	0.51	-55	-55	-23.33	0	30	70	40
						resource with independent capacities
	J 1	55	0	23.33	8.33	[0.55 0.00 0.23 0.17]	-85	-25	36.66	0	0	70	40
	J 4b	2.12	52.88	23.33	8.33	[0.02 0.11 0.23 0.17]	-57.12	-52.88	-19.10	0	27.88	70	40
					resource with dependent or independent capacities
	J 2	27.5	27.5	23.33	8.33	0.73	-82.5	-27.5	31.66	0	2.5	70	40
						[0.28 0.06 0.23 0.17]
	J 3	44	11	23.33	8.33	0.86	-99	-11	-64.66	0	-14	70	40
						[0.44 0.02 0.23 0.17]
	J 5	35.12	19.88	23.33	8.33	0.79	-90.12	-19.88	46.92	0	5.12	70	40
						[0.35 0.04 0.23 0.17]
	J 5 (E = 0)	6.31	48.69	23.33	8.33	0.56	-61.31	-48.69	-10.71	0	23.69	70	40
						[0.06 0.10 0.23 0.17]

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Taylor & Francis and I.T. Consultant

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Nabil Mikati

Dependence of lead time on batch size studied by a system dynamics model

Keywords: BATCH SIZING, LEAD-TIME REDUCTION, SIMULATION lead time, optimal batch size, system dynamics, simulation, production policies, capacity utilization

Most planning models treat lead time as a constant independent of workload, but the resulting order rate implies capacity utilization which in its turn affects the lead time. An important factor that determines production order rate is the batch size, one expects therefore a relationship between batch-sizing and lead time. This dependency is examined for different operational conditions using system dynamics simulation of a manufacturing model comprising a quality control unit which is also the bottleneck of the system. It is shown that there is an optimal batch size that results in a minimum lead time and that inventory level at optimum matches desired inventory. Below optimal batch, lead time increases sharply due to congestion at the bottleneck. The reported results have implications for production planning and implementation of process improvement.

Introduction

Lead time 1 is a key performance indicator which besides being a crucial measure of service levels is the only parameter in the objectives scheme described by Hopp and Spearman (2000, P. 196) that supports both lower manufacturing costs and high sales. Hence insight on how lead times might vary with factors such as arrival rate, variability and batch size is essential for effective planning and scheduling. The dependency on batch size is especially important given that in many optimising models for example the Economical Order Quantity, Penido 2007, or using Material Requirement Planning (MRP) procedures, lead times are treated as a constant independent of planning policy. In this work system dynamics (SD) simulation is used to investigate lead times in a manufacturing environment where one of the processing units is a bottleneck. Similar workflows are found in the pharmaceutical and chemical industries where semi-finished products have to undergo thorough testing before being packaged and released to the market. It is often the case, especially in pharmaceutical manufacturing, that quality testing is time consuming, labour intensive and administered by a unit independent from production although from the perspective of supply chain it is an integral part of the workflow. When demand increases planers feel the pressure to send more orders into production thereby risking congestion at the bottleneck and ending up in long queues and delays. [START_REF] Karmarker | Manufacturing lead times, order release and capacity loading[END_REF][START_REF] Karmarker | Manufacturing lead times, order release and capacity loading[END_REF] examined the relationship between lot-sizing and lead times from the perspective of queue theory. He showed that as batch sizes are reduced, utilization i.e. the ratio between arrival rate and throughput approaches unity, the average time an item spends in the system increases very rapidly. At the other end of the scale as batch size increases, waiting times dominate the process and the average lead time starts to increase. In between there is a minimum lead time. Hopp and Spearman (2000, P. 305) computed optimal batch sizes for different batching processes. Their results are similar to those of Karmarker cited above. Gung 1999 considered the effect of set up time and batch size reduction on lead times. The author describes a workload balancing model and suggests a minimum set up time reduction ratio. Chandra and Gupta 1997 studied lead time reduction in a semiconductor facility where batch processing is a part of the manufacturing line. They outline a procedure whereby the bottleneck station pulls its requirements from the assembly section. Lee and Chung 1998 investigated batching decisions in a multi product environment. They used a non linear mixed integer program to minimize the flow in a closed job shop. Enns 2001 analysed the relation between planned lead times and batch size in the context of MRP logic. Olinder and Olhager 1998 studied the effect of different lot sizing models controlled by MRP logic on lead times. [START_REF] Ocrun | Using system dynamics simulations to compare capacity models for production planning[END_REF] compared a number of capacity models using system dynamics simulations. The authors conclude that at high utilization many models used in production planning and system dynamics fail to capture the non-linear behaviour involved, while a saturated concave clearing function 2 does. Recently [START_REF] Pahl | Production planning with load dependent lead times: an update of research[END_REF]

Related work

System Dynamics

System Dynamics is a method for analysing policies and solving complex problems by using computer simulation models. The models capture the causal interlinks within the system and project them as a structure of feedback loops. The use of SD in Supply Chain Management is dated back to the seminal work of [START_REF] Forrester | Industrial dynamics: A major breakthrough for decision makers[END_REF][START_REF] Forrester | Industrial dynamics: A major breakthrough for decision makers[END_REF] and it has been applied to a wide range of problems related to manufacturing and supply chain management [START_REF] Sterman | Business dynamics, system thinking and modelling for a complex world[END_REF][START_REF] Angerhofer | System dynamic modelling in supply chain management: research review[END_REF], Akkermans and Dellaert 2005, Min Huang et al. 2007. SD takes an aggregate view on policies, the level of aggregation and the boundary of the model should reflect the time scale for the dynamics of interest and the problem studied.

Production Model

An overview of the manufacturing model is shown in Figure 1, some links are omitted in order to simplify the sketch. The model is adapted from the generic structure by (Sterman 2000 chap. 18), neglecting those elements of the supply chain that deals with procurement or administrative delays since the concern of the study is the effects of batch-sizing on lead time. The simulation experiments were performed using VENSIM software. Further details regarding the underlying equations are given in the appendix.

Figure 1

Orders arrive continuously and batching is accomplished using DELAY BATCH function which takes a continuous input stream and returns it as pulses when the amount accumulated is equal to the specified batch size. Following the first manufacturing stage (production) a given number of samples are sent for quality testing. Meanwhile semi-finished products are kept in quarantine pending release. When quality control is completed, finishing (packaging, labelling etc.) is carried out and products are sent to finished goods inventory for delivery. Unfilled orders are backlogged and lead time is calculated from lead time = backlog/ order fulfilment rate

(1)

For an order rate D and batch size Q, the waiting time to reach Q is Q/D, also the frequency of sample arrivals to quality control is ND/Q, where N is the number of samples sent to quality control for every batch produced. Hence decreasing batch size shortens the waiting time but increases the load on the bottleneck causing congestion and delays. Increasing Q will have the opposite effect. An intermediate batch size results in a minimum lead time.

Delays at the Work Centres

The outflow from stocks is described by first order material delay [START_REF] Sterman | Business dynamics, system thinking and modelling for a complex world[END_REF] The delay time at Production and Finishing is set to 0.5 weeks irrespective of batch size. In the range of batch sizes considered this a reasonable assumption since many facilities may have additional capacity. More importantly, making the delays at these work centres time dependent will have little effect on the main results as long as these delays are kept shorter than the delay at the bottleneck.

The output rate at the bottle neck is taken to be labour constrained which is a common situation in control laboratories. Maximum throughput becomes p is the productivity (samples cleared by unit labour), w the workforce and T 0 is the raw process time, i.e. the time it takes to perform quality control when there is ample capacity and no congestion and W 0 = pw is the critical WIP level.

The delay time at quality control is modelled using the concepts of best-case performance and practical worst-case performance (PWC), two special cases described by Hopp and Spearman 2000 for a balanced line consisting of a number of single machines. Best case performance is the scenario where there is no process variation. The delay time is constant up to a point where load is equal W 0 .

Beyond that point delay time increases proportionally with the load. It should be noted that the absence of variability referred to does not imply a FIFO discipline in the present study since a first order material delay assumes mixing of the units in the stock; it is rather referring to the regularity of processing times of the samples. The equations for the outflow rate and delay time for best-case performance are Practical worst-case performance describes the situation where there is a maximum randomness in the system and every possible state has the same probability, for example presence or absence of labour occurs with the same frequency.

The outflow rate and delay time are

(4) (5) (6)
(3) The delay time in this case reduces to its minimum T 0 when there is only one sample in transient. Equations ( 4) and ( 6) are examples of the clearing functions mentioned earlier used to model capacity versus work load. In the terminology of Orcun et al. 2006 these equations correspond to capacitated constant proportion and concave saturating clearing functions. Karmarker 1993, used an empirical parameter k in Eq. ( 6) instead for W 0 -1 to determine the output rate.

Simulation and Results

The aim of the simulation experiments is to investigate the variation of lead time with batch size under different operational conditions. The design of the simulation runs is given in Table 1.

Table 1

In these runs the productivity is held constant equal to one, but since W 0 the critical WIP is calculated from the product of labour with productivity the results can equally well be interpreted as if workforce is constant and productivity is changing. The demand rate is assigned a value of 5 pallets/week. Random demand is generated from random numbers in the range 3 to 7 resulting in a series that has an average value of 5.1 and standard deviation 1.5 pallets/week. The same sequence of numbers is used for all random demand runs. The purpose of the run with T 0 = 0.5 and w = 5 is to investigate the systems performance when work force and raw process time are reduced simultaneously keeping the bottleneck rate constant at 10 pallets/week. For a given batch size the simulation covers a period of 60 weeks, lead time is taken as the average of the last 40 weeks where transient effects have faded out. For small and large batches, lead times increase systematically with time resulting in average values that depend on the averaging period. However for batch sizes just smaller than optimal and up to roughly 20 pallets, lead times are either constant (constant demand) or vary randomly (random demand). The results of the simulations are summarized in Tables 2a and2b, typical plots of lead time against batch size are shown in Figure 2.

Figure 2

As batch size approahes ND/TH max , that is U comes close to 1 there is a sharp increase in lead time while to the right of the minimum there is a slow and initially almost linear increase. Comparing constant and random demand, the minimum occurs practically at the same batch size but the lead time is longer in the later case. Shortest lead times are obtained when utilization is aproximaely 0.85 or 0.5 depending on whether it is best case or practical worst case. T cal is the waiting time to reach a given batch size plus the shortest delays at the work centers. The values obtained agree with lead times for constant demand. Table 2 lists also the inventoy levels at steady state. These levels are qual to, or for the case of random demand some what above, desired inventory specified by the model. Lower than optimum batch sizes results in lesser inventory but deliveries will not be able to cope with demand causing a steady increase in backlog. Increasing the batch size results in periodic oscillations and raises inventory level. In this sence short lead times are consistent with optimal inventory management.

Table 2a

Relation to Queue theory

Treating quality control as an M/M/1 queuing system (Hopp and Spearman 2000, p. 269) the average lead time can be written as the last two terms are due to production and finishing. Equation ( 8) is graphed in Figure 3, the curve is similar to those obtained by SD simulation but is less flat.

Figure 3 An expression for the batch size that minimizes T(Q) can be found by taking the derivative and setting it to zero giving

The second term is due to randomness because Q* = DT 0 N/W 0 is equivalent to 100% capacity utilization. Eq.( 9) gives results that agree well with practical worst-case scenario, for example substituting the data used to plot Figure 3, Eq. ( 9) gives 12 pallets as in Table 2.

Many Product System

So far the analysis has been limited to a one product system and samples having identical raw process time. To model the general case where process times at the testing unit are different (this may be the case even for a homogenous batch) requires indexing the individual batches as well as the samples, a procedure that considerably complicates the simulation. Moreover identifying the individual items raises issues such as scheduling and prioritisation which are beyond the scope of this work. Nevertheless by expressing the workload in time units it is possible to estimate the degree of capacity utilization and total batch size resulting in smallest average lead time. Let be the total demand and the total number of batches. The proportion of the in individual batches in the production mix should reflect demand i.e. = otherwise there will be an access inventory of one product and shortage of another. This production mix generates samples each having a raw process time . In time units the workload generated is and maximum throughput for sample type j becomes where p and w are productivity and work force as in Eq.(3). Since the output is considered to be constrained only by labour sample analysis is carried out in parallel, hence the maximum total throughput is . If the number of samples exceeds available workforce than some of the in the summation will be zero. This is equivalent to assigning some analysts more than one sample and considering these as a single unit of work with a process time equal to the sum of the individual process times. The summation is then taken over the number of analysts available. With these definitions Eq.( 7) and Eq.( 8) can be rewritten as WL is the work load in time units and is the average raw process time.

From Eq.( 11) one gets

(12)
Which reduces to Eq.( 9) when one item with constant raw process time is considered. Simulation runs after adapting the model to the definitions above and expressing the delay time by Eq.( 10) generates curves as in Figure 2. Numerical results are reported in Table 3.

Table 3 4. Summery

System dynamics is a useful tool for studying the effectiveness of different policies. The model presented above reproduces a simple manufacturing system where product must be tested before being released. The quality control is assumed to be the bottleneck section of the workflow. The maximum output rate at the testing station is constrained by labour, productivity and raw process time. In both cases it is shown that as capacity is heavily exploited WIP levels and lead times increase indefinitely due to congestion at the bottleneck. Also there is an optimum batch size which results in minimum lead times. These insights are important in two ways. Firstly, constantly pushing production rate beyond a certain point results in a viscous circle of missed due dates, increased workload and longer queues. Preliminary Eq.( 9), or a utilization level in the range 0.5 to 0.8 can be used to estimate a passable work load. Secondly, one should take a systemic approach to process improvement. Streamlining and enhancing the workflow in one sector may turn out to be counter productive unless measures are also taken to manage the side effects generated at the bottleneck. Equations ( 5) and ( 7) Constant and random 1 8

Equations ( 5) and ( 7) Constant and random 1 6

Equations ( 5) and ( 7) Constant and random 1 5

Equations ( 5) and ( 7

  reviewed different models of load dependent lead times in

Figure 2 .

 2 Figure 2. Lead time vs. batch size, (a) best-case, (b) practical worst case. For both runs w = 8, P = 1, T 0 = 1

Table 1 .

 1 Design variables and their attributes

	Work force	Delay time	Demand	T 0
	10

  batch size corresponding to shortest lead time T min b average finished goods inventory at Q* c standard deviation finished goods inventory

	Table 2a. Results summery, constant demand
	T 0 = 1	Work force	Q* a	T min	FGI b	Std c	U d		T cal e
	Best case	10	6	3.6	12.5	0.0	0.83		3.2
	Best case	8		8	3.7	12.5	0.1	0.78		3.6
	Best case	6		10	3.9	12.5	0.2	0.83		4.0
	PWC = 5 pallets/week 10 PWC 8 PWC 6 T 0 = 0.5 Best case 5 Productivity p=1 for all samples. 11 12 15 6 a corresponding to shortest lead time T min obtained form model simulation 4.4 12.5 0.2 0.45 4.5 12.5 0.3 0.52 4.8 12.5 0.4 0.56 2.9 10 0.1 0.83 F r o b Optimal obtained from Eq.(12)	4.2 4.4 5.0 2.7
	PWC	5	P	9	3.4	10	0.1	0.56		3.3
			e
			e r
				R
					e
	Table 2b. Results summery, random demand T 0 = 1 Work force Q* a T min v i FGI b e		Std c	U d
	Best case Best case Best case	10 8 6	6 7 9	5.2 5.2 5.4	13.5 13.5 13.5 w	2.2 2.3 2.2	0.83 0.89 0.93
	PWC PWC PWC		10 8 6	10 12 14	5.9 6.0 6.3	13.5 13.6 13.5 O 2.3 2.3 2.1 n l	0.5 0.52 0.64
	T 0 = 0.5						y
	Best case	5	6	4.4	10.8		2.1	0.83
	PWC		5	7	4.9	10.8		2.2	0.71

a d utilization e theoretical lead time calculated by Tcal = Q*/D + T0 + 1

Table 3

 3

							Page 16 of 16
		. Results for many product system
	<T 0 >	Work force	N tot	Q* a	Q* b	T min	U
	1	8	10	12	12	4.5	0.53
	1.5	8	10	17	18	5.5	0.53

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Appendix

Further details regarding the model shipment rate = MIN(desired shipment rate, finished goods inventory/minimum delay time), that is the firm ships either what it wants or what it is able to. finished goods inventory (initial value)= customer order rate * minimum delivery delay order fulfilment rate = shipment rate; formulated as MIN(back log/time step, shipment rate) in order to prevent the backlog becoming negative backlog(initial value) = customer order rate * minimum delivery delay desired shipment rate = customer order rate minimum delivery delay = delay time production + T 0 + delay time finishing delay time production = delay time finishing = 0.5 weeks adjustment for inventory = (desired inventory -finished goods inventory)/inventory adjustment time inventory adjustment time = 4 weeks expected order rate = SMOOTH(order rate, averaging time), this term calculates a time average based on an exponential smoothing of the input averaging time = 4 weeks desired inventory = expected order rate * minimum delivery delay + safety stock safety stock = 2.5 pallets number of samples = 10 samples Runge-Kutta integration method was used, time step 0.016 week