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Introduction

Theory of Inventory Control

copyright byMarkus Zizler

May 2007

Faculty of PhysicsUniversity of Regensburg

Prof. Dr. Ingo Morgenstern

Contents

1 Introduction 2

2 One-Item-Models 7

2.1 Deterministic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.1 The Classical Lot Size Model . . . . . . . . . . . . . . . . . . . . 82.1.2 Wagner-Within-Model . . . . . . . . . . . . . . . . . . . . . . . . 102.1.3 Heuristic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Stochastic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2.1 The Newsboy-Problem . . . . . . . . . . . . . . . . . . . . . . . . 122.2.2 The Arrow-Harris-Marschak-Problem . . . . . . . . . . . . . . . 132.2.3 Hadley-Within-Model . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Multi-Item-Models 16

3.1 Flaccidities of Single Item Models . . . . . . . . . . . . . . . . . . . . . . 163.2 Models of Multi-Item-Inventories . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Classical Lot Size Model . . . . . . . . . . . . . . . . . . . . . . . 173.2.2 Dixon - Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Forecasting 20

4.1 Different Types of Forecasting Methods . . . . . . . . . . . . . . . . . . 204.1.1 Stationary Demand . . . . . . . . . . . . . . . . . . . . . . . . . 214.1.2 Demand with Growth and Seasonal Characteristics . . . . . . . . 224.1.3 The Moving Average . . . . . . . . . . . . . . . . . . . . . . . . . 244.1.4 Exponentially Weighted Average . . . . . . . . . . . . . . . . . . 25

4.2 Monitoring Forecast Systems . . . . . . . . . . . . . . . . . . . . . . . . 264.3 (Auto-)Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Bibliography 31

1

Chapter 1

Introduction

Inventory Control is the volitional break of the operative material flow; and thusdeliberately composed stocks develop. Inventory Control needs a storage, that meansa room, building or area to store the item. The in-pouring items are called storage

input, the outpouring items storage output.

Figure 1.1: The elementary storage transaction

Therefore inventory control contains all activities and considers all consequences,which are connected with the storage of items. On the one hand there is the meretechnical and logistical aspect of inventory control, for example the storage layout. Onthe other hand there are general questions, which are related to the total stock of acompany.

One of the most important decisions is about the quantity of inventories. Thereforea lot of mathmatical models have been developed, which are summarized under theconcept of Inventory Control within the scope of Operations Research.

For the stock of a retail market the outflow is induced through customer demandand the replenishment is secured through orders. The stock disposal therefore consists

2

CHAPTER 1. INTRODUCTION 3

of ordering the right quantity (lot size) at the right time. Less orders produce lessorder costs; but for a higher level of order quantity the storage costs rise. Theadvantage of a great inventory is that there is a high level of service and most customerrequirements can be fullfilled. Real (short term) inventory problems are those, who dealwith order costs, storage costs and the service level. Problems of long term inventorycontrol do not belong to this issue, because the order costs are considered global andnot for each order.

The situation is similiar with intermediate storages. This storages are stronglybound to production and we cannot speak of a proper inventory problem. But theresults of inventory control theory can be used for the disposal of intermediate storages.

The areas of application are all inventories of the retail market. But also theinventories of industrial purchasing and selling are pliable to the models of inventorycontrol. Subsequent to the inventory of finished items from industrial selling there isa system of distribution. The disposal of such hierarchical systems is in the domainof multi-echelon inventory control; that is an extension of real inventory controltheory. The problems of inventory control are characterised through the following:

1. Several items are managed in one stock; this means that order handling andstorage occur collaborative. Every item is singular disposed.

2. Demand and delivery time (of the order to the stock) are often stochastic or notknown.

3. Not only the disposal of costs has to be considered, but also non-monetary andnon-quantitative aspects.

For problems of this type inventory control has developed a mass of models. Those canbe defined through the following dynamic decision problem:


(1) yt: Available stock at the beginning of the period t,(t = 0, . . . , N), before the arrival of an ordering

Yt: Range of allwowed stocks: yt ∈ Yt

(2) qt: Ordering at the beginning of the period t

Qt: Range of allowed orders; qt ∈ Qt

xt = atqt−λt(qt): Access at the beginning of period t

λt(qt): Stochastic delivery time with knownProbability distribution

at: Coefficient of order variation; often the delivery doesnot correspond to the order, e.g. by defective goods

(3) dt: Stochastic demand in period (t,t + 1);Demand is modelized as random variable;The distribution shall be known. The period (t,t + 1)is refered to as interval of verification.

(4) dt+1 = Gt(yt, xt, dt): Equation of stock balanceOne differentiates between the Back order caseyt+1 = yt + xt − dt, where demand can be marked andlater satisfied. In case of lost salesyt+1 = max{yt + xt − dt; 0} no demand can be marked.

(5) Quality Criterion The Quality criterion is not necessarilya criterion of cost. The loss of goodwillor the ability to implement a certain order policy issummarized under this idea.

With this common stochastic model many different situations can be included. Thecharacteristic feature is either stochastic-deterministic or stationary-instationary.

The classification in stochastic and deterministic models marks two totally differentdirections of research. Especially in the sixties of the nineteenth century the main focuswas on stochastic models. The underlying theory was named AHM-Theory, labelledafter the authors (Arrow, Harris, Marschak) of a very important work in 1951.

For the determination of cost parameters only those costs have to be considered,which have an affect on date and quantity of the order. As mentioned there are threedifferent cost parameters:

1. Order costs are costs caused through an order transaction. The order trans-


action comprehends all activities from the triggering of an order (storage de-termination, supplier selection, etc.) to the storage and the paying of the bill.Some of the order costs depend on the order quantity, e.g. quantity discountson acquisition prices. Others depend on the order transaction, for example massindependent costs transport or quality control; one speaks of fixed costs. Thiscosts can be indirect or direct; the fixed order costs are mostly indirect costs,because the order handling is carried out collaborative for all items.

2. Storage costs: Analogue to the order costs there is a classification in direct andindirect costs. Direct costs are interest charges of bound assets in storage, im-putable taxes, insurance contributions or costs through damage, loss and ageing.Indirect costs are personell costs, leasing costs, amortisations, etc.

3. Shortage costs arise, when the inventory is not ready for delivery. Direct short-age costs are for example additional costs for an express delivery or penaltieswhen the items are not delivered in due time.

How are the cost parameters determined ? Apparently there is no problem withdirect costs. But the indirect costs can not be allocated according to their origin; thusthere are capacities, which represent restrictions in a model. The standard models donot include this restrictions, but consider them through opportunity costs. The costparameters will be fixed in such a manner, that the optimum order rules for singleitems do not disturb the capacity restrictions.

In spite of great efforts, these models were not used in practice. Only the so calledHadley-Within-Modell with its stationary stochastic demand has reached some influ-ence. For this, one can list two reasons:

1. In order to use stochastic models, the stochastic processes of the demand timeseries have to be identified. And this is a very elaborate task, because often thereare thousands of items in a single stock.

2. In reality the demand processes are frequently instationary and high correlated.Thus the optimum cannot be detected with justifiable costs.

Real storage problems are not one item problems and can not be described througha deterministic demand. Aside this, the cost parameters have to be fixed and severalcriteria determined to find the correct ordering rule.

In practice one assumes that the concerning item is disposed through a special rule:if the disposable inventory yt

d falls below the order point st, the stock filled up to St;otherwise nothing is ordered:

q =

{

St − ytd for yd < st

0 for ytd ≥ st

(1.1)

The order point st is calculated through the forecasted demand during delivery timedt,λ plus the safety stock SBt.


Hereby the temporal variability of delivery time can easily be considered. Also adependency on quantity can be regarded approximately. For a stochastic delivery time,one has to use the current estimate. The order limit St follows from the order pointand the optimum factor Dt of the ordered batch: St = st + Dt. Still undetermined isthe safety stock SBt. Other simple decision rules are:

• (t, S)-policy: fixed order rythm, variable order quantity

• (s, q)-policy: fixed lot size, order when required

• orientation at service levels

In common, multi-item-models are rather complex and seldom to find. In practicetherefore only one-item-models are used. This restriction makes sense insofar as themany practical stock problems are determined by organisational conditions; for examplesometimes each item has to be disposed individually.

Chapter 2

One-Item-Models

Figure 2.1: Scheme of one-item-models

2.1 Deterministic Models

The demand is mostly not known. In case of the standard models it is thereforenecessary to substitute the real existing process of demand by a row of forecasts dt(t +τ), τ = 0, 1, 2 . . . Here dt(t + τ) means the forecast of demand in period t + τ . Thoseforecasts will be repeated as often as necessary, in order to minimise the errors. Thusthe standard models are used in rolling planning : after every forecast the orde rulesare calculated anew. But not every forecast error can be averted; therefore the safetystock SBt is needed.

7

CHAPTER 2. ONE-ITEM-MODELS 8

For a static time series we have the methods of moving average and exponentialsmoothing 1. order. Do we have a trend, then we can use exponential smoothing2. order or the linear regression analysis. For seasonal fluctuation, one can avail theforecast method of Winters.

2.1.1 The Classical Lot Size Model

Instead of the classical lot size model one speaks of the Andler-, Harris- or Wilson-model. Precondition is the assumption that demand is constant and continous. More-over there is no delivery time and no shortage. The typical development of inventorylooks like the following:

Figure 2.2: Inventory process in the classical lot size model

The problem of the lot size is to determine the right size q and the length of theorder interval T so, that the sum of order and storage costs are minimal. The relevantorder costs B(q) are fixed costs, which arise through ordering.

B(q) =

{

cB ∀ q > 00 ∀ q = 0

(2.1)

The storage costs L(q) of the order cycle T are

L(q) = Tq

2cL (2.2)

with q2 as middle inventory and cL as costs per unit and inspection intervall. In order

to determine the optimum order policy one procedes as following: The middle total


costs C per time unit are

C =1

T(cB + T

q

2h)

=cB

T+

q

2cL

=d

qcB + q

cL

2(2.3)

with T = qd

and d as rate of demand. The middle order costs B(q)T

= dqcB and the

middle storage costs L(q)T

= cL

2 can be well illustrated:

Figure 2.3: Costs in the classical lot size model

Whereas the middle storage costs rise linear with the order size, the order costsdiminish. Both cost types are antidromicly, thus there is a minimum of total costs fora certain order quantity. One finds the minimum through differentiating 2.3 after q:

∂C

∂q= −

d

q2cB +

cL

2= 0

=⇒ q∗ =

√

2dcB

cL

(2.4)

q∗ is the classical lot size. For the optimal length of the cycle T ∗ and the optimalaveraged costs following is valid:

T ∗ =q∗

r=

√

2cB

dcL

(2.5)

C∗ =√

2dcBcL (2.6)


The classical lot size model can easily consider many restrictions. Therefore it wasextended in many directions. For example delivery dates, continous access, allowances,shortages and similiar things can be considered; in this way the lot size model is con-tained as a special case in a lot of complex models.

2.1.2 Wagner-Within-Model

The Wagner-Within-Model is marked through a row of specialisations of the abovegeneral deterministic system. It is defined by the following deterministic dynamicdecision problem

(1) yt: Inventory at the beginning of period t; t = 0, 1, . . . , NYt = yt : yt ≥ 0 State domainy0: Initial inventory

The inventories can take every positive value. Thatmeans there is neither a restriction on storagecapacity nor shortfalls.

(2) qt: Ordering in period t; t = 0, 1, . . . , N − 1Qt = qt : qt ≥ 0

The orderings can take every positive value. Sizerestrictions and quantisations do not exist; butthe orderings has to avert shortfalls.

(3) dt: Demand in the intervall of inspection t, t + 1; t = 0, 1, . . . , N − 1

(4) yt+1 = yt + qt − dt: Equation of stock balance

(5) Cost Criterion min C =∑N−1

k=0 (B(qt) + L(yt+1)) with

Cost of Ordering: B(qt) = cB(qt) = cB for qt 6= 0B(qt) = cB(qt) = 0 else

Cost of Storage: L(yt+1) = cL(yt+1 + dt

2 )

In this model the term of delivery was omitted. But that is no restriction; it onlyserves the simplification of notation. For orders there are only fix costs. Quantity basedcosts do not have to be considered, because they don’t influence the moment or the sizeof ordering. Quantity dependent, non-proportional costs are not included in the abovecost criterion. The storage costs evaluate a middle inventory yt+1 + dt

2 with the storagecost rate cL. The special structure of the model implicates two essential simplificationsfor the determination of the optimum order policy:


1. When the stock is empty, there will be an order; or alternatively when the inven-tory is fallen to a minimum level. Otherwise there would be unnecessary storagecosts.

2. The combined demand of future periods will be ordered; otherwise there wouldbe unnecessary storage costs as well.

These two directly evident conditions to an optimum policy lead to a basic restrictionof different policies. The saturation of the conditions 1 and 2 conducts to the so calledWagner-Within-Algorithm. In principle it is about a special forward recursion of thebelonging Dynamical Programming Algorithm.

The forward recursion is used in such a way, that at first the moments 0 and 1, then0, 1, 2, then 0, 1, 2, 3, and so on, are considered; thereby the results of the last shorteroptimization are used. It is decisive, that because of 1. und 2. merely a fraction of thedifferent policies has to be considered.

2.1.3 Heuristic Methods

Although the Wagner-Within-model is a very efficient algorithm, several methods havebeen developed as approximation to this model. Contrary to Wagner-Within thoseheuristic algorithms dont’t consider the whole planning period. Therefore the com-putation time is lower, but also the solution quality. The followings methods check,whether the demand of a period can be satisfied by the last order or whether a neworder has to be dismissed. The first order is determined by the first period, which hasa demand. The next required quantities are satisfied by one order as long as a specialcriterion is fullfilled; otherwise a new order is dismissed.

Least-Unit-Cost:

The order quantity in period t is increased with future material requirements as longas the average costs per quantity unit can be reduced. If there is an order in period τ

and the demand is covered up to period j with j > τ , the average costs are definedthrough:

kunitτj =

cB + cL

∑jt=τ+1(t − τ)dt

∑jt=τ+1 dt

(2.7)

Thus the order quantity of period τ has to be determined, which leads to a minimumof equation 2.8. The decision problem of period τ then can be formulated as:

max{j|kunitτj < kunit

τj−1} (2.8)

This means, that the highest j has to be found, which fullfills the conditionkunit

τj < kunitτj−1. Or in other words: the period has to be found, whose demand can be

satisfied by the order in period τ without an increase in the average costs.


Part-Period-Balancing:

The key note of the Cost-Balancing heuristics is, that an order is stated for so manyperiods till the storage costs are equal to the order fix costs.

cL

j∗∑

ν=k+1

(ν − k)dν ≤ cB (2.9)

whereas j∗ is the period till the order is reaching.

Silver-Meal:

In the style of the classical lot size model the Silver-Meal Method tries to minimise thecosts per time unit or period. If there is an order in period τ , which covers the materialrequirement up to period t, following costs have to be considered:

kperiodτj =

cB + cL

∑jt=τ+1(t − τ)dt

j − τ + 1(2.10)

In a period τ those j is sought-after, which fullfills following condition:

max{j|kperiodτj < k

periodτ,j−1 } (2.11)

This means that the costs per period shall be minimised.

More Algorithms:

Groff and Savings are two more methods with which the right time and quantity of anorder can be determined. They work in a similiar way like the previous algorithms;thus they shall not be presented here. For further reading [Te03] can be recommended.

2.2 Stochastic Models

The optimal policies of the classical lot size model and the Wagner-Within-Model de-pend on exact information about the demand. But those informations are based oninsecure forecasting. The decisive tasks of stochastic lot size models are to clarify theproblem structure and the funding of disposal methods.

2.2.1 The Newsboy-Problem

This problem has a single period model with uncertain demand. The objective functionis defined as follows:

min h

x∑

u=0

(x − u)pu + g

∞∑

u=x+1

(u − x) (2.12)

with parameter


x stock of newspapersU number of sold newspaperspu probability of u sold newspapersφU (u) density function of the number of sold newspapersµ expectation value of U

h loss per non-sold newspaperg loss per missing newspaper (g > h).

We search for the optimum value of x, which minimizes the loss. We find thesolution through a changeover to a steady loss function:

min h

∫ x

0φU (u)du + g

∫

∞

x

(u − x)φU (u)du

= h

∫ x

0φU (u)du + g

∫

∞

0(u − x)φU (u)du − g

∫ x

0(u − x)φU (u)du

= (h + g)

∫ x

0φU (u)du + g(µ − x) (2.13)

With partial integration and ΦU (u) =∫ u

0 φU (z)dz as distribution function of U onereceives:

min (h + g)

∫ x

0ΦU (u)du + g(µ − x) (2.14)

After derivation of x one reveives the restriction for the optimum:

(h + g)ΦU (x)g = 0 ⇔ x = Φ−1(g

h + g) (2.15)

2.2.2 The Arrow-Harris-Marschak-Problem

2.2.3 Hadley-Within-Model

Stochastic models are characterised through the fact, that shortages normally can notbe avoided. Shortages have to be evaluated with costs. So analogue to the storagecost rate h there shall be a shortage cost rate π . Like the lot size model it shall becontinous. The probability density Φ(r) gives the demand at a moment. The time ofdelivery λ shall be deterministic and constant; and there shall be no more than oneordering standing.

One can show generally in the context of the AHM-theory that the optimization ofthe expected costs of such a model leads to a (s,S)-policy:

q =

{

0 if yd > s

S − yd if yd ≤ s(2.16)

whereas yd means the disposable stock. Furthermore is assumed: 1. Due to the com-puter support of the inventory management every demand immediately leads to anextrapolation of the stock. An order can be mailed every time; the inspection interval


has the length zero. Together with the continuity prerequisite of the demand followingcan be written:

q =

{

0 if yd > s

S − s if yd ≤ s(2.17)

If Q := S − s is set, one also speaks of a (s,Q)-model; That means, that as soon asthe disposable stock level falls under the order point s, an order of the size of Q alwayswill be activated. It will be the task of the following considerations to determine thosevalues of s = s∗ and Q = Q∗, which minimise the average of the expected costs over aperiod of several order cycles.

The average annual order costs are B = µQ

cB . Here µ is the average annual need

and therefore µQ

the average number of annual orders. The average annual storage costsL are determined as follows. The minimum inventory level SB is defined by the stock,which has to be available in the stock. That inventory must be available particularlywhen the new order arrives ( 2.4).

Figure 2.4: Calculation of the middle inventory

The average annual inventory is therefore

Q

2+ SB =

Q

2+ s − µλ

with µλ as the middle outflow during time of delivery. So the middle annual storagecosts are:

L = (Q

2+ s − µλ)cL (2.18)

The average annual shortage costs can be received through the calculation of themiddle shortage costs per cycle, which then have to be multiplied with the middleannual number of cycles µ

Q. If x is the (stochastic) accumulated demand during the


time of delivery, there is a shortage of x − s for x > s. The average shortage per cycleis therefore:

η :=

∫

∞

s

(x − s)cLλ(x)dx =

∫

∞

s

xcLλ(x)dx − sH(s) (2.19)

with cLλ(x) as probability density function for the accumulated demand during time of

delivery and H(s) as related distribution function. Therewith one receives the averageannual shortage costs:

F =µ

Qη(s)π

Finally we have for the annual total average costs:

C =µ

QK + (

Q

2+ s − µλ)cL + η(s)π (2.20)

The optimum values of Q and s are determined as follows:

dC

dQ= −

µ

Q2K + (

1

2h − µQ2)η(s)π = 0

⇐⇒ Q∗ =

√

2m(K + ph(s∗))

h(2.21)

dC

ds= h +

µ

qπ(−shλ(s) + shλ(s) − H(s)) = 0

⇐⇒ H(s∗) =Q∗h

πµ(2.22)

And with numerical methods then we can calculate Q∗ and s∗.

Chapter 3

Multi-Item-Models

3.1 Flaccidities of Single Item Models

Till the middle of the sixties, theory of inventory control just dealt with single itemmodels.

Several items in one stock were not object of research. A relationship between itemsis considered only by restrictions effecting all items; e.g. all poducts compete about afixed budget, a limited stockroom or a restricted quantity of order possibilities.

Thereby the reciprocal dependencies because of complementary and substitutionalproperties were neglected. Those relations not only influence the demand of one itembut also the total demand of all items.

A further problem in the multi-item-case are the cost savings, which result from acollective order of several items. The single-item-models try to use this advantages intwo ways: Either one waits until a certain, cost-saving order quantity is reached, orthere is a fixed order time for all items. The results do not satisfy, because with suchmethods the specialties of individual items can not be considered.

All those objections against an isolated approach are aggrevated, when the stock isreplenished through own itemion. Because the relations between items have a greaterinfluence on itemion costs than on costs of external procurement.

The independent treatment of connected items is used because of the complex anddynamic reality of inventory control. Thus the single item solutions are used as firstapproximation; then they have to be improved concerning the total optimum.

In some situations this treatment may lead to a useful solution, for example in ahomogeneous assortment of goods; and in special complex systems such a treatmentmay be necessary. But two grave exceptions cannot be ignored: At first the policyof single item models constitute the frame of the optimum total policy, which shallbe determined empirically. But the one dimensional consideration seldom providesan overview about all possible policies of the total problem. And secondly the singlesolutions can be far away from the global optimum.

16

CHAPTER 3. MULTI-ITEM-MODELS 17

3.2 Models of Multi-Item-Inventories

The models of multi-item inventories deal with several items at one time. They can bedescribed through following objective function:

K = minN

∑

i=1

T∑

t=1

cBxit + cLyit (3.1)

under the restrictions:

yi,t−1 + qit − yit = dit i = 1, . . . , N ; t = 1, . . . , T (a)qit − Mxit ≤ 0 i = 1, . . . , N ; t = 1, . . . , T (b)qit ≥ 0 i = 1, . . . , N ; t = 1, . . . , T (c)yit ≥ 0 i = 1, . . . , N ; t = 1, . . . , T (d)xit = {0, 1} i = 1, . . . , N ; t = 1, . . . , T (e)∑N

i=1 yit ≤ Lj i = 1, . . . , N ; t = 1, . . . , T (f)

with the variables:

T Length of the planning periodN Number of itemsdit : Net material requirement of item i in period t

cL : Storage feeM : Large numberqit : Lot size of item i in period t

T : Length of the planning periodN : Number of itemsxit : Binary variableyit : Inventory at the end of period t

Lj : Maximum storage capacity

The objective function 3.1 consists of order and inventory costs for each item. Thebinary variables xit have the value 1, if the lot size qit is higher than one. This isrealised by restriction (b) in connection with the minimised objective function: thebinary variable xit must be 1, if the lot size qit is larger than zero. Thereby M is alarge number, which has to be higher than the maximum lot size. Equation (a) statesa connection between the demand of a period, the stock at the beginning and the endof a period and the inward stock movement. Equation (f) is responsible for preventingan overrun of inventory capacity.

One way to solve this problem are methods of mathmatical optimisation.

3.2.1 Classical Lot Size Model

A stock of several items j = 1, . . . ,m with deterministic demand dj can be optimizedwith the classical lot size model. Further assumptions are: (1) delivery without delayand (2) restricted stockroom. One item j needs a space of bj; b is the upper limit of


the average inventory. The solution method is:

min C =

m∑

j=1

(1

2qjcLj +

cRjdj

qj

)

(3.2)

u.d.N.

m∑

j=1

1

2qjbj ≥ b

with the Lagrange-function

Λ =

m∑

j=1

(1

2qjcLj +

cRjdj

qj

)

+ λ(

m∑

j=1

1

2qjbj − b

)

(3.3)

and following restriction for the transfer prices λ

λ

{

= 0 for b >

> 0 for b =

} m∑

j=1

1

2qjbj (3.4)

The optimum lot size of every item is received by setting the first derivation to zero

∂Λ

∂qj=

cLj

2−

cRj

qj2

+ λbj = 0 (3.5)

=⇒ qj∗ =

√

2cRjdj

cLj + λbj

(3.6)

The multi item inventory is a difficult problem in combinatorical optimisation. Be-cause there are no relevant solution methods in practice, many heuristics have beendeveloped.

3.2.2 Dixon - Model

In this model the storage capacity as scarce factor is included. It is built on the Silver-Meal Heuristic (section 2.1), which tries to pool the lots as long as the average costsper period are minimal. Because many items compete for the storage capacity, there isno guarantee, that each lot is ordered in those period, which has the minimal averagecosts. Thus it can be necessary, that several items have to be ordered earlier in orderto get a valid plan, which doesn’t overcharge the inventory.

The quality of the solution depends on the sequence of the considered alternatives.Therefore a rule is necessary to determine the sequence, in which the items are dealtwith. Dixon calculates priority numbers from the known average costs of the Silver-Meal-Heuristic:

pτk =k

periodτj − k

periodτ,j+1

dk,j+1(3.7)

The expression in the numerator describes the increase of the average costs per periodwhen the order quantity for item k in period τ is enlarged by the demand dk,j+1. The


denominator expresses the raised usage of the inventory. Through the connection ofboth variables pτk describes the marginal increase in costs per additional used capacityunit. For pτk ≤ 0 the costs rise because of the additional order size.

The basic proceeding of this method is following: At first the order quantities ofall items in period τ = 1 are fixed; then the order quantities in period τ = 2, etc. Thedetermination of the order quantities is carried out similiar to the Silver-Meal method.The sequence of the single items is stated by the priority numbers of Equation 3.7. Aslong as the average costs are decreasing and the storage capacity is not depleted, theorder quantities will rise. If the space of the ordered items exceeds the storage capacity,the order of some demands has to be brought forward.

The initial situation is, that the material requirement of period one is ordered inperiod 1. Then the remaining free capacity of the period is calculated. The next thingto check, whether there are capacity shortages in the whole planning period. Afterthis the ordered quantity in period one is enlargened. Therefore the item with thehighest priority number is chosen; then the Silver-Meal criterion decides whether it isadvantageous to rise the lot size. This is repeated until the storage capacity is depletedor all priority numbers are pτl ≤ 0. After that there is one more test, whether there isa storage overload in the following periods.

If the sum of the later material requirements from period tc on is higher than theavailable storage capacity, the required capacity Q is calculated. Afterwards those itemsare considered, whose order quantities in period τ don’t cover the material requirementstill period tc. On a trial basis then the oder size is enlargened for one period or moreand the associated costs are calculated. The order size is enlargened for item k, whichhas the smallest increase of costs. This procedure is repeated as long as Q ≥ 0.

Chapter 4

Forecasting

Forecasting is a necessary pre-requisite to all inventory control situations. Without anestimate of the future customer demand, it is impossible to plan the levels of inventoriesthat will be required to offer customers a reasonable level of service.

In general terms, forecasting at all levels from long term to short term can be inter-preted as being a deterministic process of estimating a future event by casting forwardpast data. In all these forecasting processes, past data are initially analysed to establishthe basic level of demand (the stationary element) and any underlying trends (such asgrowth and seasonality) which characterise the data. This information is then used in apredetermined way to obtain an estimate of the future. Thus forecasting processes areusually largely computer-automated. In contrast to forecasting, prediction is generallyinterpreted as a process of estimating a future event based primarily on subjective con-siderations; therefore it is not automated but based in manual methods. The forecastis calculated on assumptions that characteristic trends identified in past demand datawill continue into the future. Therefore an automatically produced forecast shouldalways be open to alteration if predictions (changes in market conditions) appear tosuggest that such assumptions could be invalid. Because predictions are predominantlysubjective and involve manual interruption, they are generally far more expensive toimplement on a routine basis than forecasts. For many item lines involved it is thusnormally more effective to operate on the assumption that scientifically produced fore-casts are assumed to be satisfactory unless and until a monitoring procedure indicatesthat the forecast for a particular item line is no longer in control. In order to controlforecasts, several effective monitoring systems are available.

4.1 Different Types of Forecasting Methods

A useful way of classifying demand forecasting methods is to define the type of forecaston the basis of the time period associated with the demand data which are beinganalysed, as illustrated in Figure 4.1.

Although there is no strict demarcation between the various types of forecastingcategorized within 4.2, it is generally assumed that short time forecasting methods are

20

CHAPTER 4. FORECASTING 21

Figure 4.1: Types of demand forecast based on underlying time unit

most suitable in situations where there are many components or item lines as typicallydoes occur in an inventory control environment. Within such an environment is alsooften true, that the demand patterns being analysed are relatively fast moving. Theforecasting models used when operating in such an evironment are therefore necessarilyrequired to be simple and relatively cheap to operate while still being robust.

Inventory control systems are required to cope with a variety of different customerdemand patterns for which forecasts are necessary if an effective overall policy forcontrolling inventory is to be achieved. In practice it is assumed that the followingdemand patterns can exist.

4.1.1 Stationary Demand

This assumes that although customer demand per unit time fluctuates, there is nounderlying growth or seasonal trend. The left part of Figure ?? illustrates the basicstationary character of such data but also identifies the fact that variability in demandexists.

Because no growth or seasonality are assumed in stationary demand patterns, fore-casts ahead are fixed in value and the forecast for one period ahead is the forecast forany number of periods ahead. But it should be accepted, that occasionally fundamentalchanges in the demand pattern may occur, but these are assumed to be short-term innature, such as:

• Impulses - individual demands which are significantly higher or lower than normal.Such impulses are best ignored by a forecasting system linked to an inventory


Figure 4.2: Stationary demand patterns

control policy, since such policies are basically designed to cope with a reasonablylevel of demand with a known, measurable degree of variation.

• Step changes - a series of successive demands which are significantly higher orlower than normal which in effect produces two stationary demand situations:one before the step change followed by another stationary siuation at a differntlevel subsequent to the step change.

The ideal response of a forecast to a step change in demand is that it should reactas quickly as possible in adapting to the post step change level of demand. Should thisnot be feasible, a competent forecasting system should at least identify that such a stepchange has occurred and should also instigate remedial action to ensure that the fore-cast, which will naturally lag behind such a sudden change of level, is corrected. Unlikean impulse, a step change is sustained beyond the period of the initial increase/decreasein demand.

The right part of Figure 4.2 illustrates a demand pattern where a single periodimpulse (a significant, high demand occurring for one period only) is followed by apositive step in demand (a succession of significantly high values). The stationarydemand pattern is the simplest type of demand characteristic to analyse. However,more complex demand patterns do occur as can be evidenced by plotting demandvalues against time to demonstrate trends in either growth/decline or seasonality.

4.1.2 Demand with Growth and Seasonal Characteristics

Where a demand pattern exhibits a growth characteristic over the longer term, the fore-casting models required to accomodate that growth become more complex than thoseused in the stationary demand pattern. In growth situations, stationary forecastingmodels not only produce forecasts which in retrospect lag behind known data, but alsoproduce forecasts ahead which are fixed in value and therefore do not respond to theunderlying growth situation. There are many examples of demand patterns exhibiting


growth, at least in the medium term. A demand pattern exhibiting growth charac-teristics is clearly more complex than a simple stationary process and requires morecomplex forecasting models to . . .

• . . . identify the rate of qrowth of the demand data.

• . . . incorporate the rate of growth in the forecasts.

Many demand series are influenced by the seasons of the year and by other events whichoccur anually (Figure 4.3). In such situations it is possible to establish the degree towhich demand in any particular period of the year is higher or lower than for a typicalaverage period. Hence the aim of forecasting models taking seasonality into accountis to establish this relationship for each and every period within the year and to usethe de-seasonalizing factors that are identified by this process to produce forecasts.For technically reasons it is generally assumed that growth may also exist in demandpatterns characterized by seasonality as is shown on the left side in Figure ??. If thereis no growth, the analysis simply registers actual growth as negligible.

Figure 4.3: Demand patterns with long term growth (right) and seasonal influence (left)

The simplest demand environment within which to produce forecasts occurs whenit can be assumed that the underlying demand process is stationary. The basic assump-tion within a stationary demand process is that there is variation about a relativelystationary average value and that any change in the average value is due to a special,one-off cause rather than to overall growth or seasonality.

Before developing specific forecasting models to be linked with inventory controlpolicies, it is clear that in all forecasting situations it is necessary to define the timingof both forecasts and demand data to the particular time period to which they belongor relate. The convention is normally to regard the current period as present time t andrefer all other timings to present time. Therefore dt defines the demand that occurredin the most recent period under consideration. Past time is considered as negativewithin respect to the current period t, hence dt−1 defines the demand that occurred inthe period immediately previous to the period in which dt occurred. Although demanddata can only occur in the past, forecasts are clearly targeted to the future. Hence,


future time is defined as positive with respect to the current period and ft+1 woulddefine the timing of the forecast for the next period follwing the current period. In astationary demand situation the forecast for one period ahead is the forecast for anynumber of T periods, where T is any specified forecast horizon projecting into thefuture. Hence, in the stationary demand situation only the forecast for T periods ft+T

is given by:ft+T = ft+1 (4.1)

4.1.3 The Moving Average

The general form of the moving average mt as a forecasting model is:

ft+1 = mt =1

ndt +

1

ndt−1 + · · · +

1

ndt−n+1 (4.2)

where n = 2, 3, 4 . . . and so on, and where the sum of the n weights will always sum toone, this being the definition of a true average.

However, in practice, the use of a moving average as a forecasting model has thefollowing significant, practical problems:

• It is difficult to start from a situation where no data exist.

• Relatively large amounts of data have to be stored, what represents a significantdata storage problem if forecasts are to be provided for several thousand stockeditems.

• The sensitivity of the number of periods included can not be varied.

• It imposes a sudden cut off in weighting for data not included.

• All data are weighted equally irrespective of their age; but simple logic wouldsuggest that more recent data should be weighted more heavily than older data.

The final problem of equal weighting could be overcome by developing an one-periodahead forecast on an unequally weighted moving average, such as:

ft+1 = mt = 0.5dt + 0.3dt−1 + 0.2dt−2 (4.3)

which is a valid, average based forecasting model since the sum of the weights doindeed add up to one. It is the extension of this concept of an unequally weightedmoving average which leads to the development of an average with an infinite numberof weights which decrease exponentially with time, that is, an exponentially weightedaverage.


4.1.4 Exponentially Weighted Average

The definition of an average ut with weights declining exponentially with time wouldbe of the general form of an infinite series defined as:

ut = αdt + α(1 − α)dt−1 + α(1 − α)2dt−2 + α(1 − α)3dt−3 . . . (4.4)

where alpha is a constant whose value must be between zero and one, since to producea true average the sum of weights must sum to one. A value of alpha = 0.2 is a goodcompromise figure. On first examination, a forecast based on Equation 4.4 would appearto be relatively complicated to implement and with an infinite number of demandvalues. This approach would not appear to solve the problem of the amount of datathat has to be stored. However, it is possible to show that Equation 4.4 can be modifiedto a much simpler statement such that a one-period ahead forecast ft+1 is of the form:

ft+1 = ut = αdt + α(1 − α)dt−1 + α(1 − α)2dt−2 . . . (4.5)

= αdt + (1 − α)[

αdt−1 + α(1 − α)dt−2 . . . (4.6)

= αdt + α(1 − α)ut−1 (4.7)

which is the equivalent of

ft+1 = ut = ut−1 + α(dt − ut−1) (4.8)

and since the current forecasting error et = dt − ut−1 can be defined as the currentdemand value dt minus the one-period ahead forecast evaluated last period ut−1, then

ft+1 = ut = ut−1 + αet (4.9)

follows.In contrast to the moving average, the simple exponentially weighted average offers

the following advantages:

• It is easy to initialize, since once an estimate for ut−1 is made, forecasting canproceed since all the unknowns on the right hand side of Equation 4.5 are thendefined.

• The data storage is economical since ut−1 embodies all previous data and henceonly the value of ut−1 needs to be retained from one period to the next.

• The sensitivity can be changed at any time by altering the value of α just as longas the value of α is set between zero and one.

• It does not produce a sudden cut off in weighting of demand data irrespective ofage.

For the simple exponentially weighted average, when the value of α is high, a goodresponse to an upward change can be anticipated. However, with such a high value of α


a single high demand value can cause an over-reaction one period late. Conversely, whenthe value of α is low, although the effect of an impulse will be ignored, the response toan upward change will be poor. For the extreme case of α = 0 the forecast is totallyinsensitive to changes in the demand pattern; and for α = 1 the forecast is extremelysensitive to changes and can over-react to relatively small changes. Ideally the bestvalue of α will be that which minimizes the sum of squared forecasting errors, but inthe majority of practical situations values of either 0.1 or 0.2 are useful compromisefigures.

Figure 4.4: Response of a simple exponentially weighted average forecast with α = 0.2;the Tracking signal is explained in section 4.2

The simple exponentially weighted average represents an ideal model for producingrelatively short-term forecasts for inventory control systems when demand is stationary.When more complex demand patterns exist, such as those influenced by growth orseasonality, adaptions of the simple exponentially weighted average are required.

4.2 Monitoring Forecast Systems

Because of the adaptability and flexibility of the family of forecasting models based onthe exponentially weighted average principle, these tend to predominate in inventorycontrol systems. Hence when choosing which particular forecasting model to use thechoice of model in this case simplifies to the choice of the value of the exponentiallysmoothing constant α. Although typical values of α are 0.1 or 0.2 for exponentiallyweighted average forecasting models, it is necessary to have statistical informationavailable, when trying to establish which forecast is best in any particular situation.


The two most used statistics for selecting the suitability of forecasting models are nowdescribed in detail.

The mean squared error (MSE) is the average of the squared forecasting errors.As such it is often the statistic used to ascertain the best forecasting model, it beingassumed that the model with the minimum MSE will be best where:

MSE =1

n

n∑

t=1

e2t (4.10)

The mean absolute percentage error (MAPE) is one of the most commonly usedstatistics in all types of forecasting. It gives an indication of the average size of fore-casting error expressed as a percentage of the relevant demand value, irrespective ofwhether that forecasting error is positive or negative. In computational terms, if theforecasting error et is defined as the demand dt minus the forecast ft, it then followsthat the MAPE is defined as:

MAPE =100

n

n∑

t=1

|et|

dt

(4.11)

where |et| represents the absolute values and n is the number of observations involved.Because the MAPE measures the average relative size of the absolute forecasting erroras a percentage of the corresponding demand value, in practice a value of less than10 per cent would be regarded a very good fit and providing potentially very goodforecasts. If the MAPE is < 20 %, the forecast is potentially good; it is reasonable for< 30 and it is inaccurate for > 50 %.

Because the MSE is not a relative measure and cannot be used to compare the fore-casting effectiveness between different data series, its main application is to determinethe ideal forecasting parameters for a particular data series. The MAPE in contrast isan relative measure and can be used for comparing different data series.

Within any forecasting system it is necessary to monitor the accuracy of the fore-casts being produced and to manually correct those forecasts which go out of controldue to significant changes in the demand pattern. In the following, the monitoring ofshort-term forecasts is discussed with particular emphasis placed on those situationswhere many stocked item forecasts are being produced to establish inventory controlparameters. Most practical forecasting systems which involve many items operate onthe basis that if there is no evidence to the contrary then it is assumed that the forecastis in control; that means there have been no significant changes in the demand patternto make current forecasts invalid. For such a policy of management by exception tooperate successfully, clearly an effective monitoring system is essential. Although sev-eral different approaches have been taken with regard to monitoring forecasts, Trigg’s(1964) proposal for a tracking signal has become an essential part of the majority ofcomprehensive short-term forecasting systems.

The Trigg or smoothed error tracking signal is based on the fact, that if fore-casting errors et are defined as demand minus forecast then the current smoothed error


et is defined as the exponentially weighted average of the forecasting errors et and isproduced by:

et = α′et + (1 − α′)et−1 (4.12)

where et−1 is the value of the smoothed error for the previous time period. The currentvalue of mean absolute deviation (MAD) is then defined as the exponentially weighedaverage of the absolute forecasting errors et using the formula:

MADt = α′et + (1 − α′)MADt−1 (4.13)

where the absolute value signs | | indicate that all errors et are treated as positiveirrespective of their actual polarity, and where MADt−1 is the value of the mean absolutedeviation for the previous time period. In both Equations 4.14 and 4.13 the parameterα′ is an exponential weighting constant (EWC) whose value must be between zero andone. By convention, for monitoring applications α′ is set universally at a fixed valueof 0.2. Having defined the smoothed error et and the mean absolute deviation MADt,the tracking signal Tt is then defined as the ratio of the smoothed error to the meanabsolute deviation, hence:

Tt =et

MADt

(4.14)

Given that the value of α′ used to produce both et and MADt are the same and set at 0.2,then in practice, irrespective of the data involved, the value of the tracking signal canonly vary between +1 and −1. This is because in the extreme case where a significantincrease in demand has occurred, all forecasting errors are positive and effectivelyEquations 4.14 and 4.13 become the same and et → MADt and hence Tt → +1.Contrariwise, in the extreme case where a significant decrease in demand has occorred,all forecasting errors are negative and it follows that et → −MADt and hence Tt → −1.If the value of the tracking signal exceeds 0.7, the user can be 95 % confident in thehypothesis that the accompanying forecast is out of control due to an untypically highset of demand values for which there should be an identifiable, external cause. If thesignal is lower than -0.7 the forecast is also out of control, but this time because of anuntypically low demand.

In an inventory situation a comprehensive method of implementing the smoothederror tracking signal would be: at first calculate the value of the tracking signal for allitems. Then those items should be listed, for which the absolute value of the trackingsignal exceeds a value of 0.7; after that for this listed items the reasons of the forecasterrors have to be investigated. In addition it is necessary to check that the forecastis producing reasonable results and is typically achieving a mean absolute error of lessthan 20 %; besides the tracking signal shall have a value of less than 0.7.

4.3 (Auto-)Correlation

4.3.1 Correlation

In probability theory and statistics, covariance is the measure of how much two randomvariables vary together. The simple variance measures how much a single variable


varies. If two variables tend to vary together in the same direction, then the covariancebetween the two variables will be positive. On the other hand, if the one variable goesdown during the other one rising, then the covariance between the two variables willbe negative.

The covariance between two real-valued random variables X and Y , with expectedvalues E(X) = µ and E(Y ) = ν is defined as

Cov(X,Y ) = E((X − µ) · (Y − ν))

= E(X · Y ) − µν (4.15)

The second equation is valid because of the theorem of Steiner. If X, Y are real-valuedrandom variables and a, b are constant (non-random), then the following facts are aconsequence of the definition of covariance:

Cov(X,Y ) = Cov(Y,X)

Cov(X,X) = V ar(X)

Cov(aX, bY ) = ab · Cov(X,Y ) (4.16)

If X and Y are independent, then their covariance is zero. This follows because underindependence,

E(X · Y ) = E(X) · E(Y )

⇒ Cov(X,Y ) = µν − µν = 0

The converse is not true: if X and Y have covariance zero, they need not be independent.The units of measurement of the covariance Cov(X, Y) are those of X times those of Y.By contrast, correlation, which depends on the covariance, is a dimensionless measureof linear dependence:

σ =Cov(X,Y )

√

V ar(X) ·√

V ar(Y )(4.17)

Random variables whose covariance is zero are called uncorrelated.

4.3.2 Autocorrelation

Empirical time series normally have a partly repeating pattern. Such a pattern can bedescribed by a measurement of the correlation between the values of a time series, whichis called autocorrelation or autocovariance. This is a measure for the relation betweendata, which have a fixed time lag to each other. For example the autocorrelation withtime lag 1 is a measure for the relation between yt and yt+1 for t = 1, . . . , n−1; with timelag 2 the autocorrelation is a measure between yt and yt+2 for t = 1, . . . , n−2; and so on.From the n values of a time series, n−1 successive pairs (x1, x2), (x2, x3) . . . (xn−1, xn)can be formed. Their autocovariance is:

γ =1

n − 1

n−1∑

t=1

(xt − µ)(xt+1 − ν) (4.18)


Thereby µ is the average from the values x1, . . . , xn−1 and ν is the arithmetic averagefrom x2, . . . , xn. This means, that the autocovariance measures the linear relation ofn − 1 values of two time series. According to this, the autocovariance of values, whichare further apart than one time unit, can be determined. Therefore the autocovariancedepends on the time lag τ :

γ(τ) =1

n − τ

n−τ∑

t=1

(xt − µ)(xt+τ − ν) τ = 0, 1, 2, . . . , n − 1 (4.19)

With γ(τ = 0) = V ar(X). The autocorrelation is the standardised form of the auto-covariance:

ρ =γ(τ)

V arµV arν(4.20)

Thereby V arµ, V arν are the variances of the time series belonging to the expectationvalues of ν, ν. The autocorrelation can be seen as a hint, whether there is a regularcomponent in the time series. If there is one, the autocorrelation is near +1 whenthe periodicity is met and near -1 at the half periodicity. If all autocorrelation valuesare near zero, presumably there will be no regular components. The calculation ofthe autocorrelation coefficient makes only sense, when there are enough data available.The maximum time lag is one half or one third of the time series.

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[Br98] J. Britze: Anwendung von Methoden der Statistischen Physik auf Opti-

mierungsprobleme der Materialplanung, Universitat Regensburg, Diplomarbeit,1998

[Ge97] U. Gebauer: Anwendung und Vergleich physikalischer und herkommlicher Op-

timierungsverfahren im Bereich der Materialbeschaffung, Universitat Regensburg,Diplomarbeit, 1997

[He81] A. Herbein: Kostenoptimale Bestellpolitiken im Mehr-Produkt-Lager, Univer-sitat Augsburg, Dissertation, 1981

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