inventories in a network

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Optimization Letters (2007) 1:401406DOI 10.1007/s11590-006-0040-3

O R I G I N A L PA P E R

Modelling inventories in a network

B. D. Craven

Received: 10 August 2006 / Accepted: 17 October 2006 / Published online: 18 January 2007 Springer-Verlag 2007

Abstract If each inventory in a supply chain, comprising a network ofproduction and transportation facilities, is optimized separately by some EOQformula, the overall result may be far from optimal. If the objective of inven-tories is to reduce the variability of production and delivery rates, then thiscan be modelled as a network system with feedback links. The propagation ofdemands through the network is described by certain propagation vectors,

and the optimization is with respect to certain feedback factors.

Keywords Inventory Network Feedback

1 Introduction

Consider a supply network, consisting of a number of production facilities.Inventories may be required at various places in the network. If each facilityoptimizes its own inventory independently, the result need not be optimal, inany sense, for the network as a whole. An alternative model is proposed, inwhich the cost of variability in the production rates is balanced against the costof holding the inventories. The decision of what proportion of a variation inforecast demand should be met by immediate production, and what proportiontaken from inventory, depends on certain parameters, describing feedback in acontrol system.

In a supply chain, a demand at a given time t generates demands at earliertimes for intermediate products and raw materials. The demand at time t may beconsidered as propagating backwards in time through a network of production

B. D. Craven (B)University of Melbourne, Melbourne, Australiae-mail: [email protected]


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402 B. D. Craven

facilities. The earlier production must often be made in anticipation of demand,so that what is propagated is information, based on a forecast of demand,which is necessarily uncertain. So some random uncertainty is also propagatedthrough the network. Inventories of raw materials and intermediate products

are needed, to deal with these uncertainties.The amount of inventory is often based on an economic order quantity for-

mula, which describes a balance between the cost of ordering and delivery andthe cost of holding inventory, assuming a given pattern of demand. But thiscriterion may be questioned, when each inventory is part of a larger system, sowhy should each subsystem be optimized independently? The optimal level ofan inventory may depend, not on an assumed cost of placing orders, but ratheron decisions of how much production is required, and the production levels arein turn influenced by the levels of inventories. Also, the cost of delivery may be

a less important factor than some production costs, especially the added costresulting from fluctuations in production level.

In order to approach this question, an aggregated model is studied. Althougha given product may be made by various machines, or factories, etc., the modellumps together its production facilities during a given time interval. The alloca-tion of production between different machines, etc., is left as a separate question,for which techniques are well known, usually using liner programming models.The uncertainties propagated through the system are measured by standarddeviations at different stages. There is some choice of how much uncertainty

is carried by the inventories, and how much by the production facilities; bothinvolve costs, and affect the required level of inventories.Scheduling production from uncertain forecasts is inherently unstable, unless

some negative feedback is used to stabilize the system. The amounts made dur-ing a planning interval must increase as demand increases, and decrease as theprevious inventory increases. A linear model is proposed, with two feedbackparameters and, whose values may be optimized. Only a fraction of theforecast variation of production from a steady state is scheduled to be made,with inventory supplying any shortfall. This approach follows vassian [3] and

Magee [2] for a single product. It becomes more important when there areseveral products in a network. The result is a more stable system, with produc-tion levels less disturbed by the bullwhip effect. The parameterrelates tostabilization of inventory levels.

For comparison, Cohen and Huchzermeier [1] have given mixed-integerlinear programming models for the optimal production levels for differentstages in a supply chain. These models do not discuss in-process inventories,perhaps assuming that just in time deliveries can be assured. But that couldoften require inventories to be held elsewhere, e.g. by subcontractors whosupply components. The question of optimizing the whole system remains fordiscussion.

With the classical EOQ, the optimum is not critical; thus, the cost is notgreatly increased by some substantial departures from the optimum. This couldalso be so for inventories in an integrated system. To check this would requiredetailed numerical data for some real system, to which the author has no access.


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Modelling inventories in a network 403

However, while the main purpose of any materials procurement program is toschedule the production, and not necessarily to optimize it, there may be someadvantage in reconsidering the inventory levels, from the viewpoint of reducingvariability. The supply chain is then regarded as a control system, with feedback

loops.

2 An aggregated model for production and inventory

Consider a firm (or an industry sector), with various production facilities, oper-ating in discrete time t = 0,1,2, . . . Materials include raw materials, itemsbought in from outside the firm, intermediate products, and finished products;index them bys = 1,2, . . . ,N.

Suppose that a quantity Qtis made during the interval(t1, t), so is available

at timet. (HereQtis a column vector ofNcomponents, representing the rawmaterials, partial products, and finished products in the system.) In order tomakeQt, a quantityGQtof raw materials and partial products is required attimet 1, whereG is an input-output matrix (usually a rather sparse matrix).Suppose that an inventory Itis held at time t. Suppose that the demand at time tis Dt. Denote by Q, D and I datum values (to be defined later, for a steady state)for Qt, Dtand It. Denote qt := Qt Q, dt := DtD, and it := It I. Productionat timetrequires contributions from earlier timest 1, t 2, . . . , t m + 1 forsomem, ending when only raw materials are required; henceGm = 0.

Suppose that, at time t1, a forecastD+uj,t1is available for Dt+j= D+dt+jforj= 0,1,2, . . . ,m. Note that the random variableuj,t1 dt+jrepresents theforecasting error, assumed here to have expectation 0. If production duringthe time interval(t 1, t)is based on these forecasts, then a quantity D + yt isrequired, where

yt :=

m

j=1

Gjuj,t1. (1)

Material balance requires the equation:

It It1 = Qt yt Dt. (2)

A steady state satisfies:

0 = Q 0 D, (3)

assuming that theuj,t1are random variables with zero expectation, and

it it1 = qt yt dt. (4)

Assume that the amount made during(t 1, t)is described by:

qt= (yt+ dt) (1 )it1 (5)


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404 B. D. Craven

Here, the required amountyt+ dtis reduced by a stabilization factor < 1,and a feedback term(1 )it1, with 0 <


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Modelling inventories in a network 405

The propagation vectorsK and H are functions of the network only, andnot of the random variables for demand, inventory and production, nor of theparameters and.

4 Objectives

This model provides measures of variability for productionQtand inventoryIt. Changes in the rate of production cost money, and some inventory is held toreduce that variability. Suppose then that costs are assigned to:

(i) the standard deviations of production rates at facilitiesn = 1,2, . . . ,N;(ii) the average inventory levels i;

(iii) the maximum storage capacity at each facility; with attached (row vector)

unit costsc1, c2, c3. The disruption caused by inventory reaching zero, oroverflowing the storage, will be considered in terms of probabilities. Thecost of placing an order for internal inventories may be considered aspart of the production cost; it will be incurred in any case.

The mean inventory level Iis chosen asb1ie, (whereeis a vector ofrones),with a multiplierb1chosen to give a small probability of running out of inven-tory (sayb1 2 for a 5% probability of running out.) The maximum storagecapacity is i + b2ie, withb2 chosen to give a small probability of overflowingthe storage (sayb2 2.). The costs attached to inventory are then

c2(b1ie) + c3(b1ie + b2ie) = [(b1 + b2)(c3e) + b1(c2e)]i gi. (13)

From (12), the cost of changes in production rates is

r

j=1

c1j

2K2r + (1 )2H2r

1/2, (14)

wherec1jis componentjofc1.Consider now the case where theKr= KandHr= Hare the same for each

r. Then (dividing byK) the total cost function, to be minimized with respect tothe parameters and, in [0, 1] is

f(, ):= (1 2)1/2[ (1 ) + (2 + (1 )2]1/2, (15)

in which

= g/K; = H2

/K2

. (16)

Since, at = 0,

f

= (2 + (1 2))1/2(1 + )


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406 B. D. Craven

and f(, ) as 1, f

= 0 at some value of between 0 and 1,depending on . Also

f

=

0 when = [

2 +(

1

+)

2]1/2[

+(

1

+)

], (18)

which requires > (1 )/(1 + ), thus not too small, for a solution.

5 Varying the steady state

This model has considered a fixed steady state Dfor demand, and fluctuationsabout it. However, Ditself may change, on a slower time scale than the short-term fluctuations considered so far. For example, there could be a linear trendof increasing demand. The previous analysis would still apply, with the samefeedback parameters and, but with Dreplaced by a smoothed estimate ofchanged demand level, e.g. by an exponentially weighted moving average.

Acknowledgements The author is much indebted to a referee for careful checking and corrections.

References

1. Cohen, M.A., Huchzermeier, A.: Global supply chain management: a survey of research and

applications. In: Tayur S., Ganeshan R., Magazine M. (eds.) Quantitative Models For SupplyChain Management, Chap. 21, pp 669702. Kluwers International Series in Operations Research& Management Science. Kluwer, London (1999)

2. Magee, J.F.: Production Planning and Inventory Control. McGraw-Hill, New York (1958)3. Vassian, H.J.: Application of discrete variable servo theory to inventory control. J. Oper. Res.

Soc. Am.3(3), 272282 (1955)

inventories in a network

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