<<

. 2
( 15)



>>

(a) if the goods required by the outlet are available at the associated RDC, the
merchandise will be delivered shortly;
(b) otherwise, the RDC has to resupply its stocks by placing an order to the CDC,
in which case the shipment to the retailer will be further delayed;
(c) if the goods are not available even at the CDC, the plants will be requested to
produce them.
Let pa , pb and pc be the probabilities of events a, b and c, and let fa (t), fb (t), fc (t) be
the (conditional) probability density functions of the order-cycle time in case events
a, b and c occur, respectively. The probability density function of the order-cycle time
is then
f (t) = pa fa (t) + pb fb (t) + pc fc (t).




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INTRODUCING LOGISTICS SYSTEMS 15

pa fa(t)


pb fb(t)


pc fc(t)


f (t)




t

Figure 1.7 Probability density function of the order-cycle time.

Cost versus level of service relationship. Different logistics systems can be classi-
¬ed on the basis of classical multi-objective analysis concepts. Each logistics system
is characterized by a level of investment, a cost and a level of service. For example,
a system with privately owned warehouses and ¬‚eets can be characterized by a high
level of investment, a relatively low cost and a high level of logistics service. In what
follows the focus will be on the cost“service relationship. System A is said to be
dominated by a system B (see Figure 1.8)) if the cost of A is higher or equal to the
cost of B, the level of service of A is less or equal to the level of service of B and at
least one of these two inequalities holds strictly. For example, in Figure 1.8, alterna-
tive con¬gurations 2, 3, 4 and 5 are dominated by system 1, while 3, 4, 5 and 7 are
dominated by 6. The undominated alternatives are called ef¬cient (or Pareto optimal)
and de¬ne the cost versus level of service curve.

Sales versus service relationship. The level of logistics service greatly in¬‚uences
sales volume (see Figure 1.9). If service is poor, few sales are generated. As service
approaches that of the competition, the sales volume grows. As service is further
improved, sales are captured from competing suppliers (provided that other companies
do not change their logistics system). Finally, if service improvements are carried too
far, sales continue to increase but at a much slower rate. The sales versus service
relationship can be estimated by means of buyer surveys and computer simulations.

Determining the optimal service level. The cost versus level of service and sales
versus level of service relationships can be used to determine the level of service that
maximizes the pro¬t contribution to the ¬rm, as shown in Figure 1.9. The optimal
service level usually lies between the low and high extremes. In practice, a slightly
different approach is often used: ¬rst, a customer service level is set; then the logistics
system is designed in order to meet that service level at minimum cost.




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16 INTRODUCING LOGISTICS SYSTEMS
Cost




2
Dominated
solutions


Efficient
3
4 solutions
5
1
7

6




Service level
0 25 50 75 100


Figure 1.8 Cost versus level of service curve (the level of service is de¬ned as the percentage
of orders having an order-cycle time less than or equal to a given number of working days
(e.g. four days)).


Sales


Total cost


Maximum revenue




Service level
0 25 50 75 100


Figure 1.9 Determination of the optimum service level.

1.4 Emerging Trends in Logistics
In recent years, several strategic and technological changes have had a marked impact
on logistics. Among these, three are worthy of mention: globalization, new informa-
tion technologies and e-commerce.

Globalization. An increasing number of companies operate at the world level in
order to take advantage of lower manufacturing costs or cheap raw materials avail-
able in some countries. This is sometimes achieved through acquisitions or strategic
alliances with other ¬rms. As a result of globalization, transportation needs have
increased. More parts and semi-¬nished products have to be moved between produc-
tion sites, and transportation to markets tends to be more complex and costly. The




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INTRODUCING LOGISTICS SYSTEMS 17

Table 1.4 Main differences between traditional logistics and e-logistics.

Traditional logistics E-logistics

Type of load High volumes Parcels
Customer Known Unknown
>$1000 <$100
Average order value
Destinations Concentrated Highly scattered
Demand trend Regular Lumpy



increase in multimodal container transportation is a direct consequence of globaliza-
tion. Also, as a result of globalization, more emphasis must be put on the ef¬cient
design and management of supply chains, sometimes at the world level.

Information technologies. Suppliers and manufacturers make use of EDI. This
enables them to share data on stock levels, timing of deliveries, positioning of in-
transit goods in the supply chain, etc. At the operational level, geographic information
systems (GISs), global positioning systems (GPSs) and on-board computers allow
dispatchers to keep track of the current position of vehicles and to communicate
with drivers. Such technologies are essential to ¬rms engaged in express pick-up and
delivery operations, and to long-haul trucking companies.

E-commerce. An increasing number of companies make commercial transactions
through the internet. It is common to distinguish between business-to-business (B2B)
and business-to-consumers (B2C) transactions. The growth of e-commerce parallels
that of globalization and information technologies. As a result of e-commerce the
volume of goods between producers and retailers should go down while more direct
deliveries should be expected between manufacturers and end-users.
E-commerce leads to a more complex organization of the entire logistics system
(e-logistics), which should be able to manage small- and medium-size shipments to a
large number of customers, sometimes scattered around the world. Furthermore, the
return ¬‚ow of defective (or rejected) goods becomes a major issue (reverse logistics).
Table 1.4 reports the main differences between traditional logistics and e-logistics.
In an e-logistics system different approaches for operating warehouses and distri-
bution are generally adopted. The virtual warehouse and the Points Of Presence In
The Territory (POPITT) are just a few examples. A virtual warehouse is a facility
where suppliers and distributors keep their goods in stock in such a way that the
e-commerce company can ful¬l its orders. A POPITT is a company-owned facility
where customers may go either for purchasing and fetching the ordered goods, or for
returning defective products. Unlike traditional shops, a POPITT only stores already
sold goods waiting to be picked up by customers and defective products waiting to be
returned to the manufacturers. This solution simpli¬es distribution management but
reduces customer service level since it does not allow for home deliveries.




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18 INTRODUCING LOGISTICS SYSTEMS

1.5 Logistics Decisions
When designing and operating a logistics system, one needs to address several fun-
damental issues. For example, should new facilities (manufacturing and assembly
centres, CDCs, RDCs, etc.) be opened? What are their best con¬guration, size and
location? Should any existing facility be divested, displaced or sized down? Where
should materials and components be acquired and stored? Where should manufactur-
ing and assembly take place? Where should ¬nished goods be stored? Should ware-
houses be company-owned or leased? Where should spare parts be stocked? How
should production be planned? How should warehouses operate? (Should goods be
stored in racks or should they be stacked? Should goods be retrieved by a team of
human order pickers or by automated devices?) When and how should each stock-
ing point be resupplied? What mode of transportation should be used to transport
products? Should vehicles be company-owned or leased? What is the best ¬‚eet size?
How should shipment be scheduled? How should vehicles be routed? Should some
transportation be carried out by common carriers?
Logistics decisions are traditionally classi¬ed as strategic, tactical and operational,
according to the planning horizon.

Strategic decisions. Strategic decisions have long-lasting effects (usually over
many years). They include logistics systems design and the acquisition of costly
resources (facility location, capacity sizing, plant and warehouse layout, ¬‚eet sizing).
Because data are often incomplete and imprecise, strategic decisions generally use
forecasts based on aggregated data (obtained, for example, by grouping individual
products into product families and aggregating individual customers into customer
zones).

Tactical decisions. Tactical decisions are made on a medium-term basis (e.g. month-
ly or quarterly) and include production and distribution planning, as well as resource
allocation (storage allocation, order picking strategies, transportation mode selection,
consolidation strategy). Tactical decisions often use forecasts based on disaggregated
data.

Operational decisions. Operational decisions are made on a daily basis or in real-
time and have a narrow scope. They include warehouse order picking as well as
shipment and vehicle dispatching. Operational decisions are customarily based on
very detailed data.


1.5.1 Decision support methods
Quantitative analysis is essential for intelligent logistics decision-making. Operations
research offers a variety of planning tools.
There are three basic situations in which quantitative analysis may be helpful.




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INTRODUCING LOGISTICS SYSTEMS 19

• If a logistics system already exists, one may wish to compare the current system
design (or a current operating policy) to an industry standard.

• One may wish to evaluate speci¬ed alternatives. In particular, one may wish
to answer a number of what-if questions regarding speci¬ed alternatives to the
existing system.

• One may wish to generate a con¬guration (or a policy) which is optimal (or at
least good) with respect to a given performance measure.

Benchmarking. Benchmarking consists of comparing the performance of a logis-
tics system to a ˜best-practice™ standard, i.e. the performance of an industry leader in
logistics operations. The most popular logistics benchmarking is based on the supply
chain operations references (SCOR) model. The SCOR model makes use of several
performance parameters that range from highly aggregated indicators (named key
performance indicators (KPIs)) to indicators describing a speci¬c operational issue.

Simulation. Simulation enables the evaluation of the behaviour of a particular con-
¬guration or policy by considering the dynamics of the system. For instance, a sim-
ulation model can be used to estimate the average order retrieval time in a given
warehouse when a speci¬c storage policy is used. Whenever a different alternative
has to be evaluated, a new simulation is run. For instance, if the number of order pick-
ers is increased by one, a new simulation is required. Simulation models can easily
incorporate a large amount of details, such as individual customer ordering patterns.
However, detailed simulations are time consuming and can be heavy when a large
number of alternatives are considered.

Optimization. The decision-making process can sometimes be cast as a mathe-
matical optimization problem. ˜Easy™ (polynomial) optimization problems can be
consistently solved within a reasonable amount of time even if instance size is large.
This is the case, for example, in linear programming (LP) problems and, in partic-
ular, of linear network ¬‚ow (NF) problems (linear programs with tens of thousands
of variables and constraints can be optimized quickly on a personal computer). NP-
hard optimization problems can be solved consistently within a reasonable amount
of time only if instance size is suf¬ciently small. Most integer programming (IP),
mixed-integer programming (MIP), and nonlinear programming (NLP) problems are
dif¬cult to optimize. Unfortunately, several classes of logistics decisions (production
planning, location decisions, vehicle routing and scheduling, etc.) can only be mod-
elled as IP or MIP problems. This has motivated the development of fast heuristic
algorithms that search for good but not necessarily the best solutions. Popular exam-
ples of heuristics include rounding the solution of the continuous relaxation of an IP
or MIP model, local search, simulated annealing and tabu search. In order to work
properly, such procedures must be tailored to the problem at hand. As a result, a slight
change to problem features may entail a signi¬cant modi¬cation to the heuristic.




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20 INTRODUCING LOGISTICS SYSTEMS

Table 1.5 Annual sales forecast and total cost (in millions of dollars)
for different service levels.

Percentage of orders ¬lled within three working days

70% 75% 80% 85% 90% 95% 100%

Annual sales 4.48 6.67 8.17 9.34 9.87 10.56 11.52
Annual cost 4.41 5.55 5.99 6.22 6.87 7.44 12.84


When using an optimization model, a key aspect is to keep model size as small as
possible. As a result, unlike simulation models, optimization models do not custom-
arily consider systems dynamics issues.

Continuous approximation methods. Continuous approximation methods can be
used whenever customers are so numerous that demand can be seen as a continuous
spatial function. Approximation often yields closed-form solutions and can be used
as a simple heuristic.
This textbook presents the main mathematical optimization and simulation meth-
ods used for decision-making in logistics management. Other approaches such as the
SCOR model and the continuous approximation method are described in the refer-
ences listed at the end of the chapter.


1.5.2 Outline of the book
The remainder of this textbook describes the main quantitative methods used for the
planning, organizing and controlling of logistics systems. The material is divided
into ¬ve major streams: forecasting logistics requirements (Chapter 2); designing
the logistics network (Chapter 3); managing inventories (Chapter 4); designing and
operating warehouses and crossdocks (Chapter 5); planning and controlling long-haul
and short-haul freight transportation (Chapters 6 and 7). Finally, Chapter 8 provides
supplementary material as well as some case studies that show how ef¬cient logistics
plans can be devised by applying or adapting the quantitative methods presented in
Chapters 2 to 7.


1.6 Questions and Problems
1.1 Why does a push-based supply chain react more slowly to changing demand
than a pull-based system?

1.2 Discuss the impact of product diversi¬cation (the increase in the number of
product variants) on logistics systems planning and control.




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INTRODUCING LOGISTICS SYSTEMS 21


Factory
Factory DC Retailer Customers
warehouse

Customer
demand
+10%




Time

Retailer
demand
+18%




Time
Distribution
centre
+34%
demand




Time
Factory
+51%
warehouse
demand




Time
Factory +45%
production
+12%
output


-3%

Time

Figure 1.10 The bullwhip effect.


1.3 CalFruit is an emerging Californian distributor of high-quality fresh fruits and
vegetables, and packaged food. Because the company operates in a very com-
petitive market, the crucial factor in¬‚uencing sales volume is the time required
to meet orders. On the basis of the historical data, the logistician of the com-
pany has estimated that the service level (expressed as the percentage of orders
¬lled within three working days) in¬‚uences annual sales volume and total cost
as reported in Table 1.5. Determine the service level that maximizes revenue.

1.4 Norsk is a Danish producer of dairy products with ¬ve subsidiaries in the EU
countries and a large network of distributors in North America. The company
has recently decided to redesign its Scandinavian distribution network where
140 warehouses have been transformed into pure stocking points, while admin-
istrative activities have been concentrated in 14 new regional logistics centres




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22 INTRODUCING LOGISTICS SYSTEMS

and quantitative forecasting data have been centralized at the company™s head-
quarters. List and classify the decisions faced by Norsk management in the
logistics system redesign.
1.5 The bullwhip effect is an unwanted increase in variability of material ¬‚ows over
time through the supply chain as a consequence of small variations in customer
demand. This phenomenon, which was ¬rst recognized by Procter and Gamble
managers when examining the demand for Pampers disposal diapers, depends
mainly on the fact that individuals managing the different facilities of the supply
chain make decisions based on a limited amount of information. For instance,
the decision to replenish a factory warehouse is usually based on its current
inventory level and the orders actually issued by its immediate successors in
the supply chain (e.g. the CDCs) without any knowledge of end-user demand.
Traditionally, successor orders are used to develop forecasts of the average
value and the standard deviation of the demand perceived by the facility. Then,
such estimates serve as a basis for reorder decisions. For example, in the (s, S)
method (see Section 4.8), an order is issued any time the inventory level falls
below a given reorder level s; the inventory level is then increased to an order-
up-to-level S. As the perceived demand varies, the parameters S and s are
updated and order quantities also changed. Show that the typical bullwhip
effect for a supply chain made up of a factory, a factory warehouse, a DC and
a retailer is like the one reported in Figure 1.10 (where it is assumed that a
sudden 10% increase in end-user demand occurs).
1.6 How can the bullwhip effect be reduced by sharing information among the
facilities of a supply chain?
1.7 Discuss the role of transportation mode selection, allocation of transportation
cost among subsidiaries, and international taxation when operating a global
supply chain.
1.8 Illustrate how a distribution company can take advantage of on-vehicle GPSs.
1.9 Which are the most relevant issues when selecting a company supplier?
1.10 What are the main issues in reverse logistics?


1.7 Annotated Bibliography
A detailed introduction to business logistics is:
1. Ballou R 1998 Business Logistics Management: Planning, Organizing, and
Controlling the Supply Chain. Prentice Hall, New York.
Statistics reported in Table 1.1 are derived from the following survey:
2. Kearney AT 1993 Logistics Excellence in Europe. European Logistics Associ-
ation.




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INTRODUCING LOGISTICS SYSTEMS 23

Table 1.4 is taken from:
3. Bayles DL 2000 E-commerce Logistics and Ful¬llment: Delivering the Goods.
Prentice Hall, New York.
The reference manual to the SCOR approach can be found on the website:
4. Supply Chain Council homepage, http://www.supply-chain.org.
An introductory text to exact and heuristic algorithms for IP and MIP problems is:
5. Wolsey LA 1998 Integer Programming. Wiley, New York.
Continuous approximation methods are surveyed in the book:
6. Daganzo CF 1996 Logistics System Analysis. Springer, Berlin.




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2

Forecasting Logistics
Requirements

2.1 Introduction
Forecasting is an attempt to determine in advance the most likely outcome of an
uncertain variable. Planning and controlling logistics systems need predictions for
the level of future economic activities because of the time lag in matching supply to
demand. Typical decisions that must be made before some data are known consider
virtually every aspect of the network planning process (including facility location and
capacity purchasing) as well as production scheduling, inventory management and
transportation planning.
Logistics requirements to be predicted include customer demand, raw material
prices, labour costs and lead times. In this chapter forecasting techniques are described
with respect to demand although they are equally applicable to other kinds of data.
Forecasting methods are equally relevant to every kind of logistics system, although
they are crucial for MTS systems (see Section 1.1), where inventory levels have to be
set in every facility.

Forecasting is based on some hypotheses. No forecasting method can be deemed
to be superior to others in every respect. As a matter of fact, in order to generate a
forecast the demand must show some degree of regularity. For instance, the demand
pattern must remain nearly the same in the future or the demand entries must depend to
some extent on the past values of a set of variables. Items for which these hypotheses
hold are said to have a regular demand. This is often the case when there are many
customers that individually purchase a small fraction of the whole sales volume.

Lumpy demand. When demand is lumpy or irregular (see Figure 2.1), there is
so much randomness in the demand pattern that no reliable prediction can be made.
This is typically the case when large and rare customer orders dominate the demand
pattern or when the volume of each item is low (this often happens when the degree of

Introduction to Logistics Systems Planning and Control G. Ghiani, G. Laporte and R. Musmanno
© 2004 John Wiley & Sons, Ltd ISBN: 0-470-84916-9 (HB) 0-470-84917-7 (PB)




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26 FORECASTING LOGISTICS REQUIREMENTS
Demand

1200



1000



800



600



400



200



0
Time
1 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Figure 2.1 Irregular demand pattern of an item. Although the demand pattern is discrete we
use a continuous graph representation. This convention will be used throughout the book.

product diversi¬cation is high). When dealing with such items, two alternatives should
be explored. If demand is low, accuracy is not usually a key issue and an overestimate
can be used (this could lead, for example, to a higher safety stock). As an alternative,
the processes of the supply chain (namely, manufacturing and transportation) could
be made more ¬‚exible in order to obtain a quick response. If this is feasible, an MTO
system is able to satisfy promptly each customer request.

Long-term, medium-term and short-term forecasts. Demand forecasts are orga-
nized by periods of time into three general categories. Long-term forecasts span a
time horizon from one to ¬ve years. Predictions for longer periods are very unreli-
able, since political and technological issues come into play. Long-term forecasts are
used for deciding whether a new item should be put on the market, or whether an old
one should be withdrawn, as well as in designing a logistics network. Such forecasts
are often generated for a whole group of commodities (or services) rather than for a
single item (or service). Moreover, in the long term, sector forecasts are more com-
mon than corporate ones. Medium-term forecasts extend over a period from a few
months to one year. They are used for tactical logistical decisions, such as setting
annual production and distribution plans, inventory management and slot allocation
in warehouses. Short-term forecasts cover a time interval from a few days to several
weeks. They are employed to schedule and re-schedule resources in order to meet
medium-term production and distribution targets. As service requests are received,
there is less need for forecasts. Consequently, forecasts for a shorter time interval (a
few hours or a single day) are quite uncommon (see Figure 2.2).

The role of the logistician in generating forecasts. Medium- and long-term de-
mand forecasts are hardly ever left to the logistician. More frequently, this task is




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FORECASTING LOGISTICS REQUIREMENTS 27
Demand




Order
Forecasts
receipts


Shortest Short Medium Long Time
period period period period

Figure 2.2 Demand pattern of an item. In the shortest period the forecasts are
replaced by the order receipts.


assigned to marketing managers who try to in¬‚uence demand, for example, by launch-
ing an advertising campaign for those items whose sales are in decline. On the other
hand, the logistician will often produce short-term demand forecasts.

Spatial location of demand. Since in most cases customers are geographically
dispersed, it is worth estimating not only when, but also where demand volume will
occur. This is because decisions such as warehouse location and inventory level setting
are affected by the spatial location of demand. To this end, a top“down or a bottom“
up approach can be utilized. In the top“down method, the entire demand is globally
forecasted and then divided heuristically among geographic areas (e.g. on the basis
of the most recent sales quotas). On the contrary, in the bottom“up technique, the
demand pattern of an item is estimated in each geographical area, and then aggregated
if necessary.

Derived versus independent demand. The demand for certain items (e.g. the ¬n-
ished goods of a manufacturing ¬rm) cannot be related to the demand of some other
commodities. However, there are some products (like the raw materials and the com-
ponents required by a production schedule) whose demand can be derived determinis-
tically from the requirements of some other items (e.g. ¬nished goods). For example,
the number of loudspeakers needed when assembling a TV set can be easily calcu-
lated as a multiple of the number of ¬nished items. Since even moderately complex
products can contain several hundreds of different components, such calculations are
often performed through computerized procedures, such as manufacturing resource
planning (MRP).




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28 FORECASTING LOGISTICS REQUIREMENTS

2.2 Demand Forecasting Methods
Before introducing a classi¬cation of demand forecasting methods, it is worth observ-
ing the following.
• Short-term forecasts are as a rule more accurate than those for medium and long
time periods. This is simply because the longer the time interval, the greater
the probability of unexpected events.
• Aggregate demand forecasts are generally more precise than those of single
items. This can be explained as follows. Suppose the demands of n items can
be modelled as independent random variables x1 , x2 , . . . , xn , having the same
expected value µx and the same standard deviation σx . Then, the aggregate
demand y is a random variable,

y = x1 + x2 + · · · + xn ,

whose expected value and standard deviation are, respectively,

µy = nµx σy = nσx .
2 2
and

It follows that the ratio between the standard deviation σy and the average µy
is
σy 1 σx
=√ . (2.1)
µy n µx

Equation (2.1) indicates that the relative dispersion of the aggregate demand
around the correspondent expected value µy is less than the relative dispersion
of a single item demand.
Forecasting approaches can be classi¬ed in two main categories: qualitative and
quantitative methods.


2.2.1 Qualitative methods
Qualitative methods are mainly based on workforce experience or on surveys, al-
though they can also make use of simple mathematical tools to combine different
forecasts. Qualitative methods are usually employed for long- and medium-term fore-
casts, when there is insuf¬cient history to use a quantitative approach. This is the case,
for example, when a new product or service is launched on the market, when a product
packaging is changed, or when the future demand pattern is expected to be affected
by political changeovers or by technological advances.
The most widely used qualitative methods are sales force assessment, market
research and the Delphi method. In the ¬rst approach, a forecast is developed by
company salesmen. As a rule, the workforce can provide accurate estimates since it is
close to customers. Market research is based on interviews with potential consumers
or users. It is time consuming and requires a deep knowledge of sampling theory.




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FORECASTING LOGISTICS REQUIREMENTS 29

For these reasons it is used only occasionally, for example, when deciding whether
a new product should be launched. In the Delphi method, a series of questionnaires
is submitted to a panel of experts. Every time a group of questions is answered, new
sets of information become available. Then a new questionnaire is prepared in such
a way that every expert is faced with the new ¬ndings. This procedure eliminates
the bandwagon effect of majority opinion. The Delphi method terminates as soon as
all experts share the same viewpoint. This technique is mainly used to estimate the
in¬‚uence of political or macro-economical changes on an item demand.


2.2.2 Quantitative methods
Quantitative methods can be used every time there is suf¬cient demand history. Such
techniques belong to two main groups: causal methods and time series extrapolation.
Causal methods are based on the hypothesis that future demand depends on the past or
current values of some variables. They include regression, econometric models, input“
output models, life-cycle analysis, computer simulation models and neural networks.
Most of these approaches are dif¬cult to implement, even for larger companies. In
practice, only single or multiple regression is used for logistics planning and control.
Time series extrapolation presupposes that some features of the past demand time pat-
tern will remain the same. The demand pattern is then projected in the future. This can
be done in a number of ways, including the elementary technique, moving averages,
exponential smoothing techniques, the decomposition approach and the Box“Jenkins
method. The choice of the most suitable quantitative forecasting technique (see also
Section 2.9) depends on the kind of historical data available and the type of product
(or service). However, as a rule, it is best to select the simplest possible approach.
This principle is based on the following observations.
• Forecasts obtained by using simple techniques are easier to understand and
explain. This is a fundamental aspect when large sums of money are involved
in the decision-making process.

• In a business context, complex forecasting procedures seldom yield better
results than simple ones.
This rule is often kept in mind by logisticians, as con¬rmed by several surveys
carried out in North America and in the EU (see, for example, Table 2.1).
The usage frequencies reported in columns 2 and 3 of Table 2.1 should be adjusted
in order to take into account the variable levels of familiarity of the decision makers
with different forecasting methods (column 4). For example, when comparing the
decomposition technique and the more complex Box“Jenkins method in the medium
term, one should consider the different level of familiarity that the decision makers
have (57% and 37%, respectively) with such approaches. This can be done by com-
puting the values that the quotas of use would likely have if all the decision makers
knew both techniques (12/0.57 = 21% and 5/0.37 = 13.5%, respectively).




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30 FORECASTING LOGISTICS REQUIREMENTS

Table 2.1 Quota of use of the main quantitative forecasting methods in USA (1994). Reprinted
from Interfaces 24(2), 92“100, Sanders NR and Manrodt KB 1994 Forecasting practices in US
corporations: survey results, ©1994, with the permission of INFORMS.

Use (%) in Use (%) in Level (%) of
Forecasting method short term medium term familiarity

Decomposition 7 12 57
Elementary technique 19 14 84
Moving average 33 28 96
Exponential smoothing 20 17 83
Regression 25 26 83
Box“Jenkins 2 5 37


2.2.3 Notation
In the remainder of this chapter, we will assume, as usual, that the time horizon has
been divided into a ¬nite number of time periods and that all periods have the same
duration. Moreover, we will use the following notation. Let dt , t = 1, . . . , T , be the
demand of a given product (or a service) at time period t, where T indicates the time
period in correspondence of the latest demand entry available. Moreover, let
„ = 1, 2, . . . ,
pt („ ),
be the „ periods ahead forecast made at time t (i.e. the forecast of demand dt+„
generated at time t). If „ = 1, a one-period-ahead forecast has to be generated and
the notation can be simpli¬ed to
pt (1) = pt+1 .
As explained later, it is worth de¬ning a forecast error in order to evaluate, a poste-
riori (i.e. once the forecasted demands become known), the deviation of the demand
from its forecast. The error made by using forecasting pi („ ) instead of demand dt is
given by
ei („ ) = dt ’ pi („ ), i + „ = t.
As before, the notation can be simpli¬ed if „ = 1:
et’1 (1) = et .


2.3 Causal Methods
Causal methods exploit the strong correlation between the future demand of some
items (or services) and the past (or current) values of some causal variables. For
example, the demand for economy cars depends on the level of economic activity
and, therefore, can be related to the GDP. Similarly, the demand for spare parts can
be associated with the number of installed devices using them.




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FORECASTING LOGISTICS REQUIREMENTS 31

Table 2.2 Monthly exports of Italian avicultural meat (in hundreds of kilograms)
to Germany during 1994 and 1995.

Month Quantity Month Quantity

Jan 94 8257 Jan 95 9 443
Feb 94 8659 Feb 95 9 671
Mar 94 8906 Mar 95 11 624
Apr 94 8601 Apr 95 11 371
May 94 8084 May 95 10 627
Jun 94 8669 Jun 95 11 141
Jul 94 8608 Jul 95 10 993
Aug 94 9186 Aug 95 10 572
Sep 94 9162 Sep 95 10 817
Oct 94 9475 Oct 95 11 133
Nov 94 9196 Nov 95 10 761
Dec 94 9283 Dec 95 10 560


The major advantage of causal methods is their ability to anticipate variations
in demand. As such, they are very effective for medium- and long-term forecasts.
Unfortunately, in several cases, it is dif¬cult to identify any causal variable having
a strong correlation with future demands. Moreover, it is even more dif¬cult to ¬nd
a causal variable that leads the forecasted variable in time. For these reasons, causal
methods are less popular than those based on the time series extrapolation.
As explained in the previous section, a number of different techniques can be
classi¬ed as causal method although only regression is widely used by logisticians. In
this section regression-based forecasting is described while the other causal methods
are outlined in Section 2.8.
Regression is a statistical method that relates a dependent variable y (representing,
for example, future demand dT +1 ) to some causal variables x1 , x2 , . . . , xn whose
value is known or can be predicted:
y = f (x1 , x2 , . . . , xn ).
Such a relation can be linear,
y = a0 + a1 x1 + a2 x2 + · · · + an xn ,
or even nonlinear. It is assumed that a set of observed values of the causal variables
and the corresponding values of the dependent variable are available. A function f (·)
is then selected as the one that interpolates best such observations (if f (·) is linear,
this amounts to applying the least squares method in order to estimate a0 , a1 , . . . , an ).

PAI is an association of Italian farmers which, at the end of 1995, ordered market-
ing research to estimate future export levels of some Italian goods to the other EU




TLFeBOOK
32 FORECASTING LOGISTICS REQUIREMENTS

Table 2.3 Average monthly LIT/DM exchange rate
from November 1993 to December 1995.

Month Exchange rate Month Exchange rate

Nov 93 980.619 52 Dec 94 1039.120 00
Dec 93 987.036 82 Jan 95 1051.678 10
Jan 94 975.963 50 Feb 95 1078.278 00
Feb 94 971.129 00 Mar 95 1201.790 43
Mar 94 986.006 96 Apr 95 1239.064 44
Apr 94 957.251 05 May 95 1172.814 09
May 94 961.928 64 Jun 95 1170.013 64
Jun 94 977.984 09 Jul 95 1158.960 95
Jul 94 996.441 43 Aug 95 1111.316 36
Aug 94 1011.230 45 Sep 95 1104.762 86
Sep 94 1010.387 27 Oct 95 1135.130 00
Oct 94 1018.733 33 Nov 95 1124.658 10
Nov 94 1028.189 05 Dec 95 1106.748 89



countries. The estimate of Italian avicultural meat demand in Germany in January
1996 was based on the data reported in Table 2.2.
It was assumed that the export volume was affected by the exchange rate between
Italian lira and German mark (LIT/DM) (see Table 2.3).
More speci¬cally, it was assumed that the exports in a given month depended on
the exchange rate in the two months before. This hypothesis was con¬rmed by the
correlation indices (ρ1 and ρ2 ) between the corresponding time series (see Table 2.4):

ρ1 = 0.881 ρ2 = 0.822.
and
Therefore, the required demand was estimated as
y = f (x1 , x2 ) = a0 + a1 x1 + a2 x2 + a3 x1 x2 + a4 x1 + a5 x2 ,
2 2


where x1 and x2 are the average monthly exchange LIT/DM rates in December 1995
and November 1995, respectively (x1 = 1106.748 89 and x2 = 1124.658 10). The
coef¬cients a0 , a1 , . . . , a5 were estimated through multiple regression applied to data
in Table 2.4. The results were as follows:
a0 = ’75 811.605, a1 = ’0.014, a2 = 0.038,
a3 = ’0.088, a4 = 135.354, a5 = 13.435.

As a consequence, the estimated exports in January 1996 were
y = 10 682.11 hundred kg.




TLFeBOOK
FORECASTING LOGISTICS REQUIREMENTS 33

Table 2.4 Monthly exports of avicultural meat (in hundreds of kilograms) and
average monthly LIT/DM exchange rate during the two previous months.

Exchange rate Exchange rate
Month Quantity one month before two months before

Jan 94 8 257 987.036 82 980.619 52
Feb 94 8 659 975.963 50 987.036 82
Mar 94 8 906 971.129 00 975.963 50
Apr 94 8 601 986.006 96 971.129 00
May 94 8 084 957.251 05 986.006 96
Jun 94 8 669 961.928 64 957.251 05
Jul 94 8 608 977.984 09 961.928 64
Aug 94 9 186 996.441 43 977.984 09
Sep 94 9 162 1011.230 45 996.441 43
Oct 94 9 475 1010.387 27 1011.230 45
Nov 94 9 196 1018.733 33 1010.387 27
Dec 94 9 283 1028.189 05 1018.733 33
Jan 95 9 443 1039.120 00 1028.189 05
Feb 95 9 671 1051.678 10 1039.120 00
Mar 95 11 624 1078.278 00 1051.678 10
Apr 95 11 371 1201.790 43 1078.278 00
May 95 10 627 1239.064 44 1201.790 43
Jun 95 11 141 1172.814 09 1239.064 44
Jul 95 10 993 1170.013 64 1172.814 09
Aug 95 10 572 1158.960 95 1170.013 64
Sep 95 10 817 1111.316 36 1158.960 95
Oct 95 11 133 1104.762 86 1111.316 36
Nov 95 10 761 1135.130 00 1104.762 86
Dec 95 10 560 1124.658 10 1135.130 00



2.4 Time Series Extrapolation
Time series extrapolation methods assume that the main features of past demand
pattern will be replicated in the future. A forecast is then obtained by extrapolating
(projecting) the demand pattern. Such techniques are suitable for short- and medium-
term predictions, where the probability of a changeovers is low.
As explained in Section 2.2, time series extrapolation can be done in a number
of ways. The classical decomposition method is depicted in this section, while the
elementary technique, moving averages and exponential smoothing techniques are
described in Sections 2.5, 2.6 and 2.7. Moreover, the Box“Jenkins method is outlined
in Section 2.8.




TLFeBOOK
34 FORECASTING LOGISTICS REQUIREMENTS
Demand




Time
Growth Maturity Decline

Figure 2.3 Life cycle of a product or service.


2.4.1 Time series decomposition method

The time series decomposition method is based on the assumption that the demand
pattern of a product (or a service) can be decomposed into the following four effects:
trend, cyclical variation, seasonal variation and residual variation.

Trend. The trend is the long-term modi¬cation of demand over time; it may depend
on changes in population and on the product (or service) life cycle (see Figure 2.3).

Cyclical variation. Cyclical variation is caused by the so-called business cycle,
which depends on macro-economic issues. It is quite irregular, but its pattern is roughly
periodic.

Seasonal variation. Seasonal variation is caused by the periodicity of several hu-
man activities. Typical examples are the ups and downs in the demand of some items
over the year. This type of effect can also be observed on a weekly basis (e.g. some
product sales are higher on weekends than on working days).

Residual variation. Residual variation is the portion of demand that cannot be
interpreted as trend, cyclical or seasonal variation. It is often the result of numerous
causes, each of which has a small impact. If there are no other predictable variations
in the demand, the residual effect is a random variable with unit expected value
(assuming that demand is modelled as the product of the four effects).
In the sequel we assume that the way the four components are combined together
is multiplicative,
dt = qt vt st rt , t = 1, . . . , T ,




TLFeBOOK
FORECASTING LOGISTICS REQUIREMENTS 35

where qt represents the trend at time period t (expressed in the same units as the
demand), vt is the cyclical effect at time period t, st is the seasonal variation at time
period t, and rt is the residual variation at time period t. It is worth noting that all
factors are greater than or equal to 0. Also note that if M is the periodicity of the
seasonal variation, then the average of the seasonal effects over M consecutive time
periods is equal to 1:
j +M
t=j +1 st
= 1, j = 0, 1, . . . , T ’ M. (2.2)
M
In Figure 2.4 a typical demand pattern is reported. The decomposition method is
made up of three steps: in the ¬rst phase, the demand time series dt , t = 1, . . . , T ,
is decomposed into the four components qt , vt , st , rt , t = 1, . . . , T ; in the second
phase, the time series of q, v and s are projected into one or more future time periods
(it is worth noting that the residual variation cannot be predicted); ¬nally, in the third
phase the projected values are combined,

pT („ ) = qT („ )vT („ )sT („ ), „ = 1, 2, . . . , (2.3)

to obtain the required demand forecasts. The decomposition phase is carried out as
follows.


Evaluation of the product (qv)t

The product (qv)t is obtained by removing from the time series dt , t = 1, . . . , T ,
the seasonal effect and the random ¬‚uctuation. This can be done by observing that
the average value of the demand over M consecutive time periods is not affected by
the seasonal ¬‚uctuations. Furthermore, by so doing we also remarkably reduce the
in¬‚uence of the random ¬‚uctuations, especially if M is relatively high (see also the
next section). Therefore, the computation of the following quantities,

d1 + · · · + dM
,
M
d2 + · · · + dM+1
,
M
.
.
.
dT ’M+1 + · · · + dT
,
M
allows us to determine a series of demand entries without the seasonal and residual
effects.




TLFeBOOK
36 FORECASTING LOGISTICS REQUIREMENTS

If M is odd, each average can be associated with the central period of the corre-
sponding time interval. Thus,
d1 + · · · + dM
=
(qv) ,
M/2
M
d2 + · · · + dM+1
=
(qv) ,
M/2 +1
M
dT ’M+1 + · · · + dT
=
(qv)T ’ .
M/2 +1
M
Hence the required time series is (qv)t , t = 2 M , . . . , T ’ 2 M + 1.
1 1

If M is even, one can use a weighted average of M + 1 demand entries in which
1
the ¬rst and the last ones have a weight of 2 and all other values have a unit weight.
Then, the time series (qv)t , t = 2 M + 1, . . . , T ’ 2 M is given by
1 1


+ dt’M/2+1 + · · · + 2 dt+M/2
1 1
2 dt’M/2
(qv)t = t = 2 M + 1, . . . , T ’ 2 M.
1 1
,
M
Evaluation of qt and vt
In most cases, it can be assumed that the trend is described by a simple functional
relation, such as a linear or quadratic function. Then the trend is obtained by applying
a simple regression to the time series (qv)t , t = 1, . . . , T (for the sake of simplicity,
we assume that (qv)t spans t = 1, . . . , T , although, as we have seen previously, it
is de¬ned over a shorter time interval). Once qt , t = 1, . . . , T , is determined, the
cyclical effect vt , t = 1, . . . , T can be computed for each t = 1, . . . , T as follows:
(qv)t
vt = .
qt

Evaluation of st and rt
The time series (sr)t , t = 1, . . . , T , which includes both the seasonal variation and
the random ¬‚uctuation, can be computed for each t = 1, . . . , T as follows:
dt
(sr)t = .
(qv)t
¯ ¯
The seasonal effect can then be expressed by means of M indices s1 , . . . , sM ,
de¬ned as
skM+t = st , t = 1, . . . , M, k = 0, 1, . . . .
¯
Each index st , t = 1, . . . , M, represents the average of the values (sr)t , t =
¯
1, . . . , T , associated with homologous time periods (i.e. st , t = 1, . . . , M, is the aver-
¯
age of (sr)t , (sr)M+t , (sr)2M+t , . . . ). This procedure is correct because, as explained
previously, the average calculation reduces greatly the random ¬‚uctuation. Finally,
we observe that, on the basis of the de¬nition of seasonal index, we have
M
¯
t=1 st
= 1. (2.4)
M




TLFeBOOK
FORECASTING LOGISTICS REQUIREMENTS 37
dt

1200



1000



800



600



400



200



0
t
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160

Figure 2.4 Demand pattern of electrosurgical equipment in France.


If Equation (2.4) is not satis¬ed, the following normalized indices st , t = 1, . . . , M,
˜
are used:
¯
M st
st = M
˜ , t = 1, . . . , M.
¯
st
t =1

It is easy to show that the indices st , t = 1, . . . , M, verify the relation,
˜
M
˜
t=1 st
= 1.
M
The random values rt , t = 1, . . . , T , can be obtained by dividing each term of
the time series (sr)t , t = 1, . . . , T , by the correspondent seasonal index st , t =
1, . . . , T , that is
(sr)t
rt = .
st
If the decomposition has been executed correctly, the time series rt , t = 1, . . . , T ,
has an expected value close to 1.
The second phase of the decomposition method amounts to projecting the effects
q, v and s previously determined over one or more future time periods; this is very
easy to accomplish for the trend and the seasonal effect. However, the cyclical trend is
much harder to extrapolate in a quantitative fashion. As a result, it is often estimated
qualitatively, on the basis of the macro-economic forecasts. If no such information is
available, it can be assumed that

vT („ ) = vT , „ = 1, 2, . . . .

Finally, in the third phase, a forecast is generated by combining the projections
obtained in step two, according to Equation (2.3).




TLFeBOOK
38 FORECASTING LOGISTICS REQUIREMENTS

Table 2.5 Demand history (in thousands of euros) of electrosurgical
equipment in France (Part I).

Year Month Period Demand Year Month Period Demand

1986 Jan 1 511.70 1989 May 41 848.40
1986 Feb 2 468.30 1989 Jun 42 820.40
1986 Mar 3 571.90 1989 Jul 43 795.90
1986 Apr 4 648.20 1989 Aug 44 774.90
1986 May 5 705.60 1989 Sep 45 750.40
1986 Jun 6 709.10 1989 Oct 46 759.50
1986 Jul 7 676.90 1989 Nov 47 740.60
1986 Aug 8 661.50 1989 Dec 48 809.90
1986 Sep 9 611.80 1990 Jan 49 603.40
1986 Oct 10 640.50 1990 Feb 50 558.60
1986 Nov 11 611.10 1990 Mar 51 711.20
1986 Dec 12 697.20 1990 Apr 52 760.90
1987 Jan 13 548.80 1990 May 53 840.00
1987 Feb 14 492.10 1990 Jun 54 835.80
1987 Mar 15 613.20 1990 Jul 55 777.00
1987 Apr 16 692.30 1990 Aug 56 727.30
1987 May 17 721.70 1990 Sep 57 714.00
1987 Jun 18 672.00 1990 Oct 58 744.80
1987 Jul 19 670.60 1990 Nov 59 723.10
1987 Aug 20 635.60 1990 Dec 60 770.70
1987 Sep 21 611.80 1991 Jan 61 581.00
1987 Oct 22 686.00 1991 Feb 62 555.80
1987 Nov 23 630.70 1991 Mar 63 665.70
1987 Dec 24 750.40 1991 Apr 64 770.70
1988 Jan 25 515.20 1991 May 65 836.50
1988 Feb 26 498.40 1991 Jun 66 779.10
1988 Mar 27 627.20 1991 Jul 67 745.50
1988 Apr 28 741.30 1991 Aug 68 739.20
1988 May 29 760.90 1991 Sep 69 676.20
1988 Jun 30 754.60 1991 Oct 70 710.50
1988 Jul 31 733.60 1991 Nov 71 711.90
1988 Aug 32 704.90 1991 Dec 72 731.50
1988 Sep 33 709.80 1992 Jan 73 598.50
1988 Oct 34 733.60 1992 Feb 74 578.90
1988 Nov 35 714.70 1992 Mar 75 675.50
1988 Dec 36 831.60 1992 Apr 76 756.00
1989 Jan 37 586.60 1992 May 77 865.20
1989 Feb 38 536.90 1992 Jun 78 819.00
1989 Mar 39 654.50 1992 Jul 79 800.80
1989 Apr 40 767.90 1992 Aug 80 758.10




TLFeBOOK
FORECASTING LOGISTICS REQUIREMENTS 39

Table 2.6 Demand history (in thousands of euros) of electrosurgical
equipment in France (Part II).

Year Month Period Demand Year Month Period Demand

1992 Sep 81 737.80 1996 Mar 123 721.00
1992 Oct 82 774.90 1996 Apr 124 877.10
1992 Nov 83 728.00 1996 May 125 959.70
1992 Dec 84 817.60 1996 Jun 126 916.30
1993 Jan 85 618.10 1996 Jul 127 870.80
1993 Feb 86 565.60 1996 Aug 128 832.30
1993 Mar 87 691.60 1996 Sep 129 760.20
1993 Apr 88 768.60 1996 Oct 130 833.70
1993 May 89 903.00 1996 Nov 131 827.40
1993 Jun 90 847.70 1996 Dec 132 864.50
1993 Jul 91 830.90 1997 Jan 133 705.60
1993 Aug 92 772.10 1997 Feb 134 619.50
1993 Sep 93 755.30 1997 Mar 135 723.10
1993 Oct 94 779.10 1997 Apr 136 847.70
1993 Nov 95 770.00 1997 May 137 942.90
1993 Dec 96 844.20 1997 Jun 138 917.00
1994 Jan 97 671.30 1997 Jul 139 897.40
1994 Feb 98 607.60 1997 Aug 140 859.60
1994 Mar 99 737.80 1997 Sep 141 821.80
1994 Apr 100 863.10 1997 Oct 142 872.20
1994 May 101 908.60 1997 Nov 143 795.90
1994 Jun 102 891.10 1997 Dec 144 824.60
1994 Jul 103 853.30 1998 Jan 145 669.90
1994 Aug 104 836.50 1998 Feb 146 618.10
1994 Sep 105 797.30 1998 Mar 147 756.00
1994 Oct 106 840.70 1998 Apr 148 901.60
1994 Nov 107 816.90 1998 May 149 968.80
1994 Dec 108 872.20 1998 Jun 150 968.80
1995 Jan 109 613.90 1998 Jul 151 921.20
1995 Feb 110 595.00 1998 Aug 152 891.10
1995 Mar 111 744.10 1998 Sep 153 882.00
1995 Apr 112 812.00 1998 Oct 154 887.60
1995 May 113 941.50 1998 Nov 155 840.00
1995 Jun 114 940.10 1998 Dec 156 935.90
1995 Jul 115 863.10 1999 Jan 157 763.70
1995 Aug 116 829.50 1999 Feb 158 700.00
1995 Sep 117 808.50 1999 Mar 159 844.20
1995 Oct 118 800.10 1999 Apr 160 989.10
1995 Nov 119 836.50 1999 May 161 1045.80
1995 Dec 120 870.80 1999 Jun 162 1012.90
1996 Jan 121 684.60 1999 Jul 163 970.90
1996 Feb 122 644.70




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40 FORECASTING LOGISTICS REQUIREMENTS

Table 2.7 Computation of combined trend and cyclical effects (qv)t ,
t = 1, . . . , 163, in the P&D problem.

t (qv)t t (qv)t

1 152 864.65
2 153 871.73
3 154 879.05
4 155 885.91
5 156 890.95
6 157 894.86
7 627.70 158
8 630.23 159
9 632.95 160
10 636.50 161
11 639.01 162
12 638.14 163
... ...




P&D is a French consulting ¬rm which was entrusted in July 1999 to estimate the
future demand of electrosurgical equipment in France for the subsequent six months.
The sales over the past 13 years and seven months are available (Tables 2.5 and 2.6,
Figure 2.4).
The duration M of the seasonal cycle was assumed to be equal to 12 and the
trend was assumed to be linear. Then the decomposition method was applied. The
intermediate and ¬nal results are summarized in Tables 2.7“2.12 and in Figures 2.5“
2.12. The trend equation is qt = 638.51 + 1.43t. The seasonal indices s1 , . . . , s12
¯ ¯
(see Table 2.10 and Figure 2.9) satisfy Equation (2.4). As the expected value of the
random variation is approximately 1, the demand decomposition can be deemed to
be satisfactory. The demand forecasts from August 1999 to January 2000 (the ¬rst
six months ahead) were obtained by combining the projections of the trend with the
seasonal and cyclical effects. The latter was estimated (see Figure 2.11) by using a
quadratic regression curve de¬ned on the basis of vt , t = 150, . . . , 157. In particular,
it was assumed that
vt = f (t ’ 149)2 + g(t ’ 149) + h, t = 150, 151, . . . .
The values of the coef¬cients f , g and h that best ¬t the cyclical effect for t =
150, . . . , 157 are
f = ’0.0004, g = 0.0094, h = 0.9856.




TLFeBOOK
FORECASTING LOGISTICS REQUIREMENTS 41

Table 2.8 Trend qt , t = 7, . . . , 157, and cyclical effect vt , t = 7, . . . , 157,
in the P&D problem.

t (qv)t qt vt

7 627.70 648.52 0.97
8 630.23 649.95 0.97
9 632.95 651.39 0.97
10 636.50 652.82 0.98
11 639.01 654.25 0.98
12 638.14 655.68 0.97
··· ··· ··· ···
152 864.65 856.02 1.01
153 871.73 857.45 1.02
154 879.05 858.88 1.02
155 885.91 860.31 1.03
156 890.95 861.74 1.03
157 894.86 863.17 1.04


Table 2.9 Evaluation of combined seasonal and residual effects (sr)t , t = 1, . . . , 157,
in the P&D problem.

t dt (qv)t (sr)t

7 676.90 627.70 1.08
8 661.50 630.23 1.05
9 611.80 632.95 0.97
10 640.50 636.50 1.01
11 611.10 639.01 0.96
12 697.20 638.14 1.09
... ... ... ...
152 891.10 864.65 1.03
153 882.00 871.73 1.01
154 887.60 879.05 1.01
155 840.00 885.91 0.95
156 935.90 890.95 1.05
157 763.70 894.86 0.85


2.5 Further Time Series Extrapolation Methods:
the Constant Trend Case
We ¬rst analyse the case in which the past demand pattern does not show any relevant
cyclical and seasonal effects, and the trend is constant. We suppose initially that a
forecast must be generated only for the next period ahead.




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42 FORECASTING LOGISTICS REQUIREMENTS

Table 2.10 Computation of the seasonal indices st , t = 1, . . . , 12, in the P&D problem.
¯

¯ ¯
t st t st

1 0.82 7 1.07
2 0.76 8 1.02
3 0.92 9 0.98
4 1.05 10 1.02
5 1.15 11 0.99
6 1.11 12 1.08


Table 2.11 Computation of the residual variations rt , t = 7, . . . , 157, in the P&D problem.

t (sr)t st rt

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