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This applies equally well to strategic, tactical and operational decisions. A strategic
location-routing decision is the location of airline hubs whose choice bears on routing
costs. The location of depots and warehouses in a supply chain is a tactical decision
in¬‚uencing delivery costs to customers. A simple example arising at the operational
level is mail box location. Locating a large number of mail boxes in a city will
improve customer convenience since average walking distance to a mail box will be
reduced. At the same time, the cost of emptying a larger number of mail boxes on a
regular basis will be higher. Unfortunately, integrated location-routing mathematical
models combining these two aspects will often contain too many integer variables and
constraints to be solvable optimally. Heuristics based on a decomposition principle
are often used instead. Facilities are ¬rst located, customers are assigned to facilities,
and routing is then performed. These three decisions are then iteratively updated until
no signi¬cant improvement can be reached. A detailed account of location-routing
applications and methods can be found in the survey by Laporte.


7.9 Vendor-Managed Inventory Routing
VMR refers to situations where decisions about the timing and level of customer
replenishment is determined not by the customer, but by the supplier. Traditionally,
VMR has been the practice in the gas, petroleum and heating oil distribution but it
is now becoming more frequent in automobile parts distribution and in the food and
beverage industries. It requires the supplier to be aware of the customers™ stock levels,
which is not a major dif¬culty in cases where consumption is easily predictable, as




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SHORT-HAUL FREIGHT TRANSPORTATION 295

in heating oil distribution. It is now possible to link vending machines to automatic
dialing systems in order to inform suppliers of their stock level. There are two distinct
advantages to VMR: by deciding when to deliver, the supplier can better plan its
routes and delivery times by suitably combining deliveries to customers located in
the same geographical zone; VMR also relieves customers of the costs associated
with inventory monitoring and ordering.
The particular problem of combining routing and resupply decisions is known as
the inventory-routing problem (IRP). The IRP consists of deciding which customers
to visit during each period (e.g. one day) of a given time horizon (e.g. one week), and
how much to deliver to each of them at each visit. In the simplest of cases, consider
a planning horizon of t days, a single customer whose initial inventory is zero and
whose usage rate (per day) is r. Denote by Q the customer capacity, by q the vehicle
capacity, and by c the delivery cost. Then it is optimal to make deliveries only when
customer stock is zero, which generates a cost of
tr tr
z = c max , .
Q q
The term within the braces is the number of deliveries per period of t days. It is
driven by the smallest of the two capacities Q or q. To provide an idea of just how
dif¬cult the problem becomes when there is more than one customer, even with initial
inventories of zero, consider two customers i = 1, 2, each of them with a capacity
Qi , usage rate ri , and delivery cost ci . Two extreme policies are possible here. Under
the ¬rst policy, each customer is visited separately, generating a cost,
tr1 tr1 tr2 tr2
z1 = c1 max + c2 max
, , . (7.31)
Q1 q Q2 q
Under the second policy, the two customers are always visited jointly, yielding a cost,
t (r1 + r2 )
tr1 tr2
z2 = c1,2 max , , , (7.32)
Q1 Q2 q
where c1,2 is the combined routing cost through the two customers. The ¬rst two terms
in braces apply if the number of deliveries is driven by Q1 or Q2 . In the third case,
vehicle capacity is the most binding constraint. In this case, one must also decide how
much to deliver to each customer. Matters complicate when the number of customers
is larger than two since all possible ways of making joint deliveries must be considered
and the joint routing cost through a given set of customers is the solution value of a
TSP.
Another level of dif¬culty arises from the fact that usage rates are not constant in
practice and must be treated as random variables. As a result, safety margins must be
incorporated into the models and the cost of stockout must be explicitly accounted
for.
With respect to the standard VRP, research on the IRP is still in its early stages
and it is doubtful whether this problem can be solved exactly for any meaningful




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296 SHORT-HAUL FREIGHT TRANSPORTATION

size. A practical and relatively simple heuristic is to select at each period a subset of
customers having a high degree of urgency, i.e. a low inventory level, and solve a VRP
associated with that set of customers, using one of the known VRP heuristics. The
same process can be applied to each period. Solution improvements can be obtained
by rescheduling some visits between sets of consecutive days and reoptimizing the
individual VRP solutions.
An interesting account of VRM and IRP can be found in the article by Campbell
et al. listed in the references.

Monsanto is a company based at Guimares (Portugal) distributing soft drinks in
the Costa Verde and Montanhas regions. The main two customers are located in Porto
and Braga and have demand rates equal to 42 and 26 cartons per day, respectively.
The planning horizon consists of 30 days. The vehicle capacity is 100 cartons. Both
a round trip from Guimares to Porto and a round trip from Guimares to Braga cost
‚¬180, while a tour Guimares“Porto“Braga“Guimares costs ‚¬210. The customer in
Porto can hold at most 70 cartons while the customer in Braga can stock at most 85
cartons. If each customer is visited separately, the cost computed by using Equation
7.31 is z1 = 5040 euros/month, while, if the two customers are visited jointly, the
costs given by Equation 7.32 is z2 = 4410 euros/month.



7.10 Questions and Problems
7.1 Show that if the costs associated with the arcs of a complete directed graph
G satisfy the triangle inequality property, then there exists an ATSP optimal
solution which is a Hamiltonian circuit in G .
7.2 You have an algorithm capable of solving the capacitated NRP with no ¬xed
vehicle costs and you would like to solve a problem where a ¬xed cost f is
attached to each vehicle. Show how such a problem can be solved using the
algorithm at hand.
7.3 Show that, if there are no operational constraints, there always exists an optimal
NRP solution in which a single vehicle is used. (Hint: least-cost path costs
satisfy the triangle inequality.)
7.4 Show that the two alternative subcircuit elimination constraints (7.4) and (7.5)
are equivalent.
7.5 Demonstrate that the optimal solution value of MSrTP is a lower bound on the
optimal solution value of STSP.
7.6 Show that the NRPSC formulation is correct.
7.7 Explain why the distances in Tables 7.8 and 7.9 do not necessarily satisfy the
triangle inequality.




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SHORT-HAUL FREIGHT TRANSPORTATION 297

7.8 Modify the savings algorithm for the case in which the same vehicle may be
assigned to several routes during a given planning period (node routing problem
with multiple use of vehicles).
7.9 Devise a local search for the capacitated ARP.
7.10 Illustrate how the Christo¬des and Frederickson heuristics can be adapted to the
undirected general routing problem, which consists of determining a least-cost
cycle including a set of required vertices and edges.


7.11 Annotated Bibliography
Statistics reported in Section 7.1 are taken from:
1. Golden BL, Wasil EA, Kelly JA and Chao IM 1998 The impact of metaheuristics
on solving the vehicle routing problem: algorithms, problem sets and compu-
tational results. In Fleet Management and Logistics (ed. Laporte G and Crainic
TG). Kluwer, Boston.
A survey on node routing problems is:
2. Fisher ML 1995 Vehicle routing. In Network Routing (ed. Ball MO, Magnanti
TL, Monma CL and Nemhauser GL). North-Holland, Amsterdam.
A description of a good tabu search heuristic for node routing can be found in:
3. Cordeau JF, Laporte G and Mercier A 2001 A uni¬ed tabu search heuristic
for vehicle routing problems with time windows. Journal of the Operational
Research Society 52, 928“936.
Two reviews on arc routing problems are:
4. Assad A and Golden BL 1995 Arc routing methods and applications. In Hand-
books in Operations Research and Management Science, 8: Network Routing
(ed. Ball MO, Magnanti TL, Monma CL and Nemhauser GL), pp. 375“483.
Elsevier Science, Amsterdam.
5. Eiselt HA, Gendreau M and Laporte G 1995 Arc routing problems. Part I. The
Chinese postman problem. Operations Research 43, 231“242.
6. Eiselt HA, Gendreau M and Laporte G 1995 Arc routing problems. Part II. The
rural postman problem. Operations Research 43, 399“414.
Two recent surveys on the real-time VRDP are:
7. Psaraftis HN 1995 Dynamic vehicle routing: status and prospects. Annals of
Operations Research 61, 143“164.
8. Gendreau M and Potvin JY 1998 Dynamic vehicle routing and dispatching.
In Fleet Management and Logistics (ed. Laporte G and Crainic TG). Kluwer,
Boston.




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298 SHORT-HAUL FREIGHT TRANSPORTATION

A survey of location-routing problems is provided in:
9. Laporte G 1988 Location-routing problems. In Vehicle Routing: Methods and
Studies (ed. Assad and Golden BL). North-Holland, Amsterdam.
A survey on quantitative methods for operating vendor-managed distribution systems
is:
10. Campbell A, Clarke L, Kleywegt A and Savelsberg M 1998 The inventory
routing problem. In Fleet Management and Logistics (ed. Laporte G and Crainic
TG). Kluwer, Boston.




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8

Linking Theory to Practice


8.1 Introduction
Many real-world logistics problems possess slightly different features from those
found in the problems we have discussed so far. Some of these features can easily be
taken into account, while some others require more complex modi¬cations, both in
models and algorithms. Two examples will help illustrate this statement.

• When designing vehicle routes in practice, a meal break may have to be inserted
in each route. In the single vehicle case, this can be simply accomplished by
introducing a dummy customer with a service time equal to the required rest
time, and a service time window equal to the interval within which meals are
allowed.

• When designing vehicle routes in garbage collection applications, the street
network is often modelled as a mixed graph. As a result, the single vehicle
case amounts to solving a mixed RPP (see Section 7.6). This can be done,
in principle, by suitably assigning a traversal direction to each edge and then
applying the ˜balance-and-connect™ heuristic for the directed RPP. However,
while this approach is easy to implement, it is not clear whether it yields a good
quality solution for the mixed RPP. Devising a good heuristic for this problem
may indeed require a substantial effort and is a research topic on its own.

The aim of this chapter is to link theory to practice in logistics management planning
and control by providing supplementary material and cases. In Sections 8.2 to 8.6 a few
real-world logistics systems are depicted while in Sections 8.7 to 8.15 some studies,
taken from the scienti¬c literature, illustrate the adaptation of basic techniques to more
complex settings. Finally, further insightful case studies are listed in Section 8.17.

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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300 LINKING THEORY TO PRACTICE

ExxonMobil
Corporation


ExxonMobil ExxonMobil
Refining and Supply Exploration
ExxonMobil ExxonMobil
Chemical Coal and Minerals
ExxonMobil ExxonMobil
Fuels Marketing Production

ExxonMobil
ExxonMobil
Lubrificants & Petroleum ExxonMobil Gas Marketing
Imperial Oil Ltd
Specialties Global Services

ExxonMobil ExxonMobil
Research and Engineering Development


ExxonMobil
Upstream Research


Figure 8.1 ExxonMobil corporation functional companies.

8.2 Shipment Consolidation and Dispatching at
ExxonMobil Chemical
ExxonMobil Chemical is a functional company of ExxonMobil corporation. Exxon-
Mobil is a US corporation formed in 1999 by the combination of Exxon and Mobil,
two companies whose roots can be traced back to the late 19th century. ExxonMobil
is an industry leader in almost every aspect of the energy and petrochemical busi-
ness. Its activities range from the exploration and production of oil and gas to coal
and copper mining, from the re¬ning of petroleum products to the marketing of fuels
(under the Exxon, Mobil and Esso brands), waxes, asphalt and chemicals. In addition,
ExxonMobil is active in electric power generation. Figure 8.1 depicts the organization
of ExxonMobil into several functional companies.
ExxonMobil Chemical is one of the largest petrochemical companies in the world.
Its products include ole¬ns, aromatics, synthetic rubber, polyethylene, polypropylene
and oriented polypropylene packaging ¬lms. The company operates its 54 manufac-
turing plants in more than 20 countries and markets its products in more than 150
countries (see Table 8.1).
At many sites, the ExxonMobil Chemical operations are integrated with re¬ning
operations within a single complex. The ExxonMobil Chemical plant in Brindisi
(Italy) is devoted to the manufacturing of oriented polypropylene packaging ¬lms for
the European market. Oriented polypropylene (OPP) is a ¬‚exible material derived
from melting and orienting (i.e. stretching) a polymer called polypropylene. This raw
material is unaffected by most chemical agents encountered in everyday life. It meets
the requirements of the US Food and Drug Administration and other relevant author-
ities throughout the world. By orienting polypropylene, one can improve its physical
properties, such as water vapour impermeability, stiffness, dimensional stability and
optics. OPP ¬lms are used as ¬‚exible packages for food (e.g. biscuits, bakery prod-
ucts and frozen food) and as high strength ¬lms for garbage bags and liners. Every




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LINKING THEORY TO PRACTICE 301

Table 8.1 ExxonMobil Chemical manufacturing plants.

Site Country Site Country

Adelaide Australia Karlsruhe Germany
Al-Jubayl Saudi Arabia Kashima Japan
Altona Australia Kawasaki Japan
Kerkrade—
Amsterdam The Netherlands The Netherlands
LaGrange—
Antwerp Belgium Georgia (USA)
Augusta Italy Managua Nicaragua
Baton Rouge Louisiana (USA) Meerhout Belgium
Baytown Texas (USA) Mont Belvieu Texas (USA)
Bayway New Jersey (USA) Newport United Kingdom
Beaumont Texas (USA) Notre-Dame-
de-Gravenchon France
Belleville— Canada Panyu China
Botany Bay Australia Paulina Brazil
Brindisi— Italy Pensacola Florida (USA)
Campana Argentina Plaquemine Louisiana (USA)
Chalmette Louisiana (USA) Rotterdam The Netherlands
Cologne Germany Sakai Japan
Dartmouth Canada San Antonio Chile
Edison New Jersey (USA) Sarnia Canada
Shawnee—
Fawley United Kingdom Arkansas (USA)
Fife United Kingdom Singapore Singapore
Fos-sur-Mer France Sriracha Thailland
Stratford—
Geleen The Netherlands New Jersey (USA)
Harnes France Trecate Italy
Virton—
Houston Texas (USA) Belgium
Ingolstadt Germany Wakayama Japan
Jeffersonville Indiana (USA) Yanbu™ al Bahr Saudi Arabia
Jinshan China Yosu South Korea

— = oriented polypropylene ¬lm plant

packaging application is different. For example, special OPP ¬lms are needed for
complex products containing chocolate, sugar and cream that are more sensitive and
need special protection, particularly against oxidation, odour loss and uptake of off-
odours. OPP ¬lms may be transparent, opaque or metallized. ExxonMobil Chemical
rigorously tests every packaging product before it is used commercially.
Although overall market growth is slow, indicating maturity across most sectors,
there remain signi¬cant growth prospects within niche markets, such as those for
individually wrapped biscuits. ExxonMobil Chemical produces more than 230 000
tonnes of OPP annually in seven plants in the USA, Canada, Belgium, Italy and The
Netherlands (shown by asterisks in Table 8.1). The OPP process begins with pellets
of polypropylene resin derived from crude oil or natural gas. Resins are transported
to the Brindisi harbour by boat and then moved to the plant by a local dedicated train.




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302 LINKING THEORY TO PRACTICE


Raw material inventory


Extrusion


WIP inventory


Coating


WIP Inventory


Cutting


Finished product inventory




Figure 8.2 OPP ¬lm manufacturing process.


The manufacturing process is made up of three main stages (see Figure 8.2). First,
the pellets are fed into an extruder, where they are melted by heat and friction from a
continuously revolving screw. At the end of this stage the molten plastic is cast into a
sheet form. Then, sheets are stretched lengthwise or crosswise and an acrylic coating
is applied on one or both sides. Finally, large rolls are cut into smaller rolls to meet
customers™ speci¬cations. At the end of this process, the custom slit ¬lm is shipped to
an end-user or to third-party plants to be metallized or printed. Films manufactured in
Brindisi needing to be metallized are sent either to the Metalvuoto plants in Termoli
and Roncello (Italy), to the Neograf plant in Cuneo (Italy), or to the Metlux plant
in Luxembourg, where a very thin coating of aluminium is applied to one side (see
Figure 8.3).
As a rule, Italian end-users are supplied directly by the Brindisi plant, while cus-
tomers and third-party plants outside Italy are replenished through the DC located
in Milan (Italy). In particular, this warehouse supplies three DCs located in Herstal,
Athus and Zeebrugge (Belgium), which in turn replenish customers in Eastern Europe,
Central Europe and United Kingdom, respectively.


8.3 Distribution Management at P¬zer
The P¬zer Pharmaceuticals Group is the largest pharmaceutical corporation in the
world. Its mission is ˜to discover, develop, manufacture and market innovative, value-
added products that improve the quality of life of people around the world and help
them enjoy longer, healthier, and more productive lives™. The P¬zer range of products
includes a broad portfolio of human pharmaceuticals meeting essential medical needs,
a wide range of consumer products in the area of self-care and well-being, and health
products for livestock and pets.




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LINKING THEORY TO PRACTICE 303
Raw material suppliers



ExxonMobil Chemical plant
(Brindisi, Italy)


Distribution centre
(Milan, Italy)


Neograf plant Metalvuoto plant Metalvuoto plant Metlux
(Cuneo, Italy) (Roncello, Italy) (Termoli, Italy) (Luxembourg)


DC DC DC
(Herstal, Belgium) (Athus, Belgium) (Zeebrugg, Belgium)


Eastern Europe Central Europe European customers
Great Britain
Italian customers
customers customers (urgent request)
customers

Figure 8.3 OPP ¬lm distribution patterns at ExxonMobil Chemical
(dotted lines represent metallized OPP ¬lm ¬‚ows).

Founded in 1849 by Charles P¬zer, the company was ¬rst located in a modest red-
brick building in the Williamsburg section of Brooklyn, New York (USA), that served
as of¬ce, laboratory, factory and warehouse. The ¬rm™s ¬rst product was santonin, a
palatable antiparasitic, which was an immediate success. In 1942 P¬zer responded
to an appeal from the US Government to expedite the manufacture of penicillin,
the ¬rst real defence against bacterial infection, to treat Allied soldiers ¬ghting in
World War II. Of the companies pursuing mass production of penicillin, P¬zer alone
used the innovative fermentation technology. Building value for its shareholders,
P¬zer manufactures and markets some of the most effective and innovative medicines
including atorvastatin calcium, the most prescribed cholesterol-lowering medicine
in the USA, amlodipine besylate, the world-leading medicine for hypertension and
angina, azithromycin, the most-prescribed brand-name oral antibiotic in the USA, and
sildena¬l citrate, a breakthrough treatment for erectile dysfunction.
With a portfolio that includes ¬ve of the world™s 20 top-selling medicines, P¬zer
sets the standard for the pharmaceutical industry. Ten of its medicines are ranked ¬rst
in their therapeutic class in the US market, and eight earn a revenue of more than one
billion dollars annually. Research and development is the lifeblood of P¬zer business.
To pursue its heritage of innovation, P¬zer supports the world™s largest privately
funded biomedical research organization, employing 12 000 scientists worldwide.
The research investment was 5.3 billion dollars in 2002.

8.3.1 The Logistics System
The P¬zer logistics system comprises 58 manufacturing sites around the world (see
Table 8.2), producing medicines for more than 150 countries.




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304 LINKING THEORY TO PRACTICE

Table 8.2 Manufacturing sites of P¬zer.

Location Number of sites

Africa 7
Asia 13
Australia 2
Europe 16
North America 16
South America 4


Table 8.3 Features of some P¬zer plants in Europe.

Number of Number of Items
Country plants Articles (millions per year)

Belgium 1 29 6.5
France 1 14 2.4
Germany 1 3 11.4
Italy 3 182 87.1
United Kingdom 1 8 5.0


Because manufacturing pharmaceutical products requires highly specialized and
costly machines, each P¬zer plant produces a large amount of a limited number of
pharmaceutical ingredients or medicines for an international market (see Table 8.3).
In order to illustrate the main characteristics of a typical P¬zer supply chain, we
will examine the supply chain of a cardiovascular product, named ALFA10. The
product form is a 5 or 10 mg tablet, packaged in blisters. ALFA10 is based on a patent
owned by P¬zer, and every plant involved in its manufacturing is P¬zer™s property.
ALFA10 is produced in a unique European plant (EUPF plant) for an international
market including 90 countries (see Figure 8.4). Every year the plant produces over
117 million blisters. The product expires 60 months after its production and must be
stored at a temperature varying between 8 and 25 —¦ C.
The main component of ALFA10 is an active pharmaceutical ingredient (API),
based on a P¬zer property patent, manufactured in a North American plant. APIs are
transferred by air to the European Logistics Center (ELC) located in Belgium which
in turn replenishes the European plants on a monthly basis (see Figure 8.5). Freight
transportation between the ELC and the manufacturing sites is performed by overland
transport providers such as Danzas. The EUPF plant manufactures ALFA10 tablets
that are subsequently packaged into 120 blister boxes and sent weekly to a third-party
CDC.
The CDC has ¬ve docks, a 1250 m2 receiving zone, a 7700 m2 storage zone (where
10 000 pallets can be kept in stock), and a 2000 m2 shipping zone. Products are pal-




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LINKING THEORY TO PRACTICE 305




Figure 8.4 P¬zer ALFA10 supplied markets (grey area).




ELC (Belgium) EUPF
plant



Outside
markets


CDC
Wholesalers

North American
API plant
Regional
warehouses


Hospitals

Pharmacies

Figure 8.5 ALFA10 supply chain.


letized, and pallets are moved by forklifts and stored onto shelves. The transportation
of ¬nished goods is performed in refrigerated trucks by accredited haulers.


8.3.2 The Italian ALFA10 distribution system
ALFA10 sales are fairly stable in Italy, as shown in Figure 8.6. The distribution
system is made up of two channels. Hospitals are supplied directly by P¬zer while
pharmacies are replenished through wholesalers (see Figure 8.6). P¬zer plants, third-
party DCs and wholesalers communicate through a dedicated information system
named Manugistics.




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306 LINKING THEORY TO PRACTICE
dt

1,200 000



1,000 000



800 000



600 000



400 000



200 000



0
t
10 20 30 40

Figure 8.6 P¬zer ALFA10 monthly demand pattern in Italy.

The hospital distribution channel. In order to supply 2000 Italian hospitals in a
timely manner, P¬zer makes use of a CDC and seven regional warehouses. Hospitals
may be supplied by more than one warehouse, depending on stock levels. Transporta-
tion is performed by specialized haulers in refrigerated vans.

The pharmacy distribution channel. Pharmacies are supplied through whole-
salers. There are almost 16 000 pharmacies in Italy. Pharmacy locations are revised
every two years by the Minister of Health in such a way that citizens located in
rural areas can reach the nearest pharmacy within a given amount of time (indeed,
5000 rural pharmacies are helped by state subsidies). Pharmacies sell both prescribed
medicines (86% of their entire business) and over-the-counter products. Pharmacies
have a high contractual power on wholesalers. Their average revenue margin is 27%
for prescribed medicines and 33% for over-the-counter products. Wholesaler orders
are collected directly by P¬zer and shipped weekly by the CDC. The maximum lead
time between order receipt and shipment is 24 hours. Again ALFA10 transportation
is performed by specialized haulers. The CDC is able to deliver the product in any
Italian location within at most 60 hours. Wholesalers receive orders from pharmacies
very frequently (up to four times a day). Pharmacies expect the wholesalers deliver
medicines within 4“12 hours.
Compared with other EU countries, the number of Italian pharmaceutical whole-
salers is very high (see Table 8.4). In addition, the pharmaceutical distribution business
is very fragmented, as shown in Table 8.5. In particular, four pan-European companies
have a 42% share.
Each wholesaler has an extensive network of RDCs in order to provide a high level
of service to pharmacies.As a result, their average revenue margin is low.As illustrated
in the above description, P¬zer makes use of 3PL. The relationship between P¬zer
and its partners is regulated by contracts. Audits on a regular basis are performed by
P¬zer on all its logistics partners.




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LINKING THEORY TO PRACTICE 307

Table 8.4 Features of pharmaceutical wholesalers in four major EU countries.

Country Wholesalers Wholesaler warehouses

France 14 203
Germany 17 102
Italy 198 302
Spain 101 189



Table 8.5 Classi¬cation of Italian pharmaceutical wholesalers.

Companies Warehouses Share

Pan-European groups 4 86 42%
Local wholesalers 164 184 36%
Others (cooperatives, etc.) 30 32 22%


Further distribution channels. Unlike prescribed medicines (such as ALFA10),
over-the-counter product distribution is not very critical and is performed directly by
P¬zer. Due to the increasing popularity of the Internet among patients and physicians,
prescribed medicines are expected to be delivered directly by P¬zer to the pharmacies
in years to come, resulting in large savings.


8.4 Freight Rail Transportation at Railion
Railion is an international carrier, based in Mainz (Germany), whose core business
is rail transport. Railion is the result of a merger involving DB Cargo AG and NS
Cargo NV, and is Europe™s ¬rst truly international rail company. In The Netherlands,
Belgium and Luxembourg, Railion operates under the name Railion Benelux, while
in Germany it will continue to operate as DB Cargo for the time being.
Railion transports a vast range of products, such as steel, coal, iron ore, paper,
timber, cars, washing machines, computers as well as chemical products. In 2001 the
company moved about 500 000 containers. Besides offering high-quality rail trans-
port, Railion is also engaged in the development of integrated logistics chains. This
involves close cooperation with third parties, such as trucking ¬rms, maritime trans-
porters, as well as forwarding and transshipment companies. This approach allows
Railion to meet the increasingly complex demands of a market which is no longer
prepared to settle for the mere carriage of goods, but requires a complete logistics
package including all the service aspects this entails.
Railion Benelux offers a variety of transport services, tailored to the type of product
to be carried, the destination and the customer™s logistics requirements.




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308 LINKING THEORY TO PRACTICE

• Scheduled services. For the transport of smaller volumes of cargo (a few car-
loads at a time), the scheduled service concept is generally the best and least
expensive option. Individual wagons are delivered to and collected from cus-
tomers, whereupon they are coupled together at the marshalling yards to form
complete trains. From there, they are transported directly to destinations (gener-
ally outside the Benelux), where individual wagonloads are sorted for delivery
to the recipients. If required, Railion can also arrange terminal road services by
truck. European scheduled services generally take between 24 and 48 hours,
depending on the distance involved and the facilities available at the destination
station. The range of different wagon types available includes bulk goods wag-
ons, ¬‚at wagons for machinery and plant, special car transporters, tank wagons
for chemical products, and covered wagons for palletized goods, industrial and
consumer products.
• Charter trains. A charter train is often the perfect solution for the transport of
large quantities of goods, especially if such shipments take place on a regular
basis. A complete train offers tailor-made solutions enabling the delivery of up
to several thousand tonnes of cargo. A charter train requires a so-called branch
line to reach its ¬nal destination. Many industrial plants throughout Europe
already have their own branch lines, often as a result of government subsidies.
• Intermodal trains (container shuttles). Railion Benelux operates as a sort of
wholesaler in the intermodal market, supplying complete trains consisting of
various types of container-carrying wagon to the intermodal operators. Inter-
modal trains almost invariably operate on the basis of a shuttle service with
the same composition, and running back and forth between the same desti-
nations. Intermodal operators can either utilize train capacity for their own
containers (e.g. shipping lines) or sell ˜slots™ to other shippers, thereby operat-
ing as a ˜forwarding agent™. Railion Benelux currently offers shuttle services
to 24 European destinations. Table 8.6 lists the main features of the intermodal
shuttles.



8.5 Yard Management at the Gioia Tauro
Marine Terminal
The Gioia Tauro marine terminal is the largest container transshipment hub on the
Mediterranean Sea and one of the largest in the world (see Table 8.7). Medcenter, the
company set up to manage container handling at Gioia Tauro, is owned by Contship
SpA, which controls 90% of the equity, and by Maersk, a leading international sea
carrier.
The terminal is situated on the western coast of the Calabria region (Southern Italy),
along major deep-sea vessel routes. Deep sea vessels are large highly automated
container ships capable of transporting up to 6000 containers. They include both




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LINKING THEORY TO PRACTICE 309

Table 8.6 Features of the intermodal shuttles provided by Railion
(˜r™ and ˜s™ stand for roundtrip and single trip, respectively).

Frequency TEUs
Origin Destination per week

Rotterdam Antwerp (Belgium) 11r 81/86
Rotterdam Athus (Belgium) 5r 30
Rotterdam Mouscron (Belgium) 5r 43
Rotterdam Muizen (Belgium) 5r 10
Born Antwerp (Belgium) 5r 60
Rotterdam Germersheim (Germany) 6r 81
Rotterdam Mainz (Germany) 3r 68
Rotterdam Mannheim/Munich (Germany) 5r 81
Rotterdam Neuss (Germany) 5r 81
Rotterdam Milan“Melzo (Italy) 9r 74.5
Rotterdam Novara (Italy) 12r 78
Rotterdam Brescia (Italy) 5r 78
Rotterdam Padova (Italy) 6r 74.5
Rotterdam Bettembourg (Luxembourg) 4r 75
Rotterdam Wels (Austria) 2r 77
Rotterdam Malaszewicze (Poland) 3s 80
Rotterdam Poznan/Warsaw (Poland) 3r 78
Rotterdam Prague (Czech Republic) 6r 70
Rotterdam Basel SBB (Switzerland) 5r 75
Rotterdam Zurich (Switzerland) 5r 75
Rotterdam Basel Bad (Switzerland) 2r 75



Table 8.7 The ¬rst 20 largest containerized ports in the world.

Traf¬c in 1999 Traf¬c in 1999
Port (TEUs) Port (TEUs)

Hong Kong 16 100 000 New York 2 863 000
Singapore 15 900 000 Dubai 2 844 000
Kaohsiung 6 985 000 Felixstowe 2 700 000
Pusan 6 439 000 Tokyo 2 700 000
Rotterdam 6 400 000 Port Klang 2 550 000
Long Beach 4 400 000 Tanjung Priok 2 273 000
Shanghai 4 200 000 Gioia Tauro 2 253 000
Los Angeles 3 828 000 Kobe 2 200 000
Hamburg 3 750 000 Yokohama 2 200 000
Antwerp 3 614 000 Brema 2 180 000




TLFeBOOK
310 LINKING THEORY TO PRACTICE




Figure 8.7 Hub and spoke sea transportation system (bold and dotted lines represent deep-sea
vessel and feeder routes, respectively; grey and white vertices are hubs and spokes, respec-
tively).

container ships performing around-the-world trips (through the Panama Canal) and
Post-Panamax vessels performing North America“Europe, Europe“Asia Paci¬c and
Asia Paci¬c“North America trips (the Post-Panamax vessel name is due to the fact that
they are so large that they cannot traverse the Panama Canal). Because the operating
costs of deep-sea vessels are very high, these ships stop at very few transshipment
terminals (hubs), where they pick up and deliver traf¬c originating from or arriving at
end-of-line ports (spokes). Then, smaller vessels (feeder or short sea vessels) transport
goods between hubs and end-of-line ports (hub and spoke system, Figure 8.7).
The Gioia Tauro hub is linked to nearly 50 end-of-line ports on the Mediterranean
Sea. When Gioia Tauro began trading in 1996, its traf¬c amounted to a modest 570 736
TEUs, followed by a dazzling 1.44 million TEUs in 1997, 2.12 million TEUs in 1998,
and 2.25 million TEUs in 1999.
Like other hubs, the Gioia Tauro sea terminal (see Figure 8.8) is made up of
• a harbour, where vessels can wait for an available berth;
• a set of quays, where ships can be tied up and loaded or unloaded;
• a yard, where containers and bulk goods can be stored after being unloaded
from incoming vehicles and before being loaded onto outgoing vehicles;
• a railway station, where wagons can be loaded or unloaded and convoys can
be formed;
• some docks where trucks can be loaded or unloaded;
• a material handling system.
At the Gioia Tauro port, the yard can store nearly 50 000 TEUs (1100 of them can
be refrigerated). The storage area is divided into bays. Each bay is made up of 32
rows, each having 16 slots. In each slot, up to three containers are stacked. Empty
containers (which occupy approximately 40% of the storage area) have an 8“10 day




TLFeBOOK
LINKING THEORY TO PRACTICE 311

Trains


Bay
Railway station




Quay

Harbour
Trucks




Yard Straddle Straddle
carriers carriers


Portainer
buffer

Figure 8.8 A sea container terminal layout.


average dwell time (much more than a full container) and are located in the more
remote positions.
The railway station has six tracks where 20 convoys are formed every day (400 000
TEUs are handled annually). The Gioia Tauro port is close to the Salerno“Reggio
Calabria highway traversing southern Italy from north to south. The material handling
system is made up of 14 portainers, three Gottwald cranes, 51 straddle carriers, ¬ve
multitrailers, six reach stackers, as well as 11 tractors and 60 trailers. Portainers
and Gottwald cranes are used for unloading containers from the vessels. Portainers
are cranes moving along tracks parallel to the quayside. Each portainer has a buffer
where up to six containers can be stored. When the buffer is full, the portainer has
to stop. Gottwal cranes are wheeled vehicles also used for moving containers in the
yard. Straddle carriers are usually utilized for moving full containers over relatively
short distances (less than 500 m) between railroad, yard and berth. These are wheeled
vehicles capable of transporting one or two containers at a time. As a rule, for longer
container transfers, different vehicles are used, namely multitrailers. Empty containers
are stacked and moved ¬ve at a time by reach stackers. In addition, reach stackers
are used for moving containers from the yard to the railway station and vice versa.
Portainers and straddle carriers are the most important pieces of equipment and can
handle seven and 24 containers per hour on average, respectively. Terminal operating
cycles consist of a series of container movements, each carried out by several different
movers. Once a ship is tied up, sea-side cranes load straddle carriers (or similar
movers) that transfer outgoing containers to the terminal yard beside the assigned slot.
Then dedicated yard movers insert containers in the right slot. Alternatively, prime
movers can perform board“board, board“train or board“truck movements depending
on the stowage plan. Similarly, incoming containers can be picked up from the yard
or transferred directly from a train or a truck.




TLFeBOOK
312 LINKING THEORY TO PRACTICE

8.6 Municipal Solid Waste Collection and Disposal
Management at the Regional Municipality of
Hamilton-Wentworth
The regional municipality of Hamilton-Wentworth is situated in south-central Ontario
(Canada), approximately 50 miles west of Niagara Falls. The region has an area of
1100 km2 , includes six cities and towns (Ancaster, Dundas, Flamborough, Glanbrook
Hamilton and Stoney Creek), and has a population of 450 000 inhabitants. Every year,
more than 300 000 tons of residential, industrial and commercial waste are produced
in the region. The waste management system is made up of two major subsystems:
the solid waste collection system and the regional disposal system. Each city or town
is in charge of its own kerbside garbage collection, using either its own workforce
or a contracted service. The regional municipality is responsible for the treatment
and disposal of the collected waste. The primary reason for this is the existence
of economies of scale (i.e. the decline of average cost as scale increases) in refuse
transportation and disposal.
For the purpose of solid waste management, the region is divided into 17 districts.
In 1992, the total cost was approximately 21.7 million dollars. The regional manage-
ment is made up of a waste-to-energy facility, a recycling facility, a 550 acre land¬ll,
a hazardous waste depot, and three transfer stations located in Dundas, Kenora and
Hamilton Mountain. Transfer stations receive waste from municipal collection (or
individual deliveries) and move it either to the waste-to-energy facility, to the recy-
cling facility, or to the land¬ll. The waste picked up through kerbside collection from
Flamborough, Dundas and northwest Ancaster goes to the transfer station in Dundas,
garbage from Glanbrook, Hamilton Mountain and southeast Ancaster is delivered
to the transfer station in Hamilton Mountain, while waste from lower Hamilton and
Stoney Creek is delivered directly to the waste-to-energy facility. The transfer stations
in Dundas and Hamilton Mountain also receive individual deliveries from local indus-
tries and institutions, while the transfer station in Kenora accepts only truckloads of
industrial, commercial and institutional waste. The 1992 waste ¬‚ow allocation pattern
is shown in Figure 8.9.


8.7 Demand Forecasting at Adriatica Accumulatori
AdriaticaAccumulatori is an electromechanical ¬rm, headquartered in Termoli (Italy),
manufacturing car spare parts for the Italian market. In 1993 the results of a
survey showed that, although Adriatica Accumulatori car battery sales constantly
increased during the previous decade, the company progressively lost market share
(see Table 8.8). Until 1993, the company had traditionally based its production and
marketing plans on sales forecasts provided by a time series extrapolation technique
(see Section 2.4). If applied to the data in Table 8.8, this technique would result in the




TLFeBOOK
LINKING THEORY TO PRACTICE 313
Waste to
energy
1740
facility

Municipal 554
85
collector Box
1052

997
Landfill
Transfer
stations in
2.5
Dundas, Kenora
and Hamilton
33
Mountain

Recycling
436
facility
Blue box

Figure 8.9 The waste ¬‚ow allocation pattern in the regional municipality of
Hamilton-Wentworth (all numbers in the ¬gure are average waste ¬‚ows in tons per week).


following regression equation (the trend is linear),

y = 126 364.184 + 15 951.091t, t = 1, 2, . . . ,

which would provide the following demand forecasts: 301 826 units in 1994 (t = 11)
(with a 9.41% increase with respect to 1993), and 317 777 units in 1995 (t = 12)
(with a 15.19% increase with respect to 1993). However, the results of the survey
convinced the company™s management that during the previous decade Adriatica
Accumulatori had lost an opportunity to sell more, mainly because its forecasts were
not related to market demand. Based on this reasoning, it was decided to predict sales
by ¬rst estimating the Italian market demand and then evaluating different scenarios
corresponding to the current market share and increased shares achievable through
appropriate marketing initiatives. In order to forecast the Italian market sales, a causal
method was used (see Section 2.3). The historical series of national sales of batteries
was correlated to the number of cars sold two years before (see Table 8.9). Then the
following linear regression model was used,

y = a0 + a1 x,

where y is the Italian demand of spare batteries in a given year, x represents
car sales two years before, a0 and a1 are two parameters. These parameters were
estimated through the least-squares error method, yielding the regression equation
y = 52 429.797 + 1.924x, with a correlation index ρ equal to 0.95. Using this equa-
tion, the demand of spare batteries in the Italian market in 1994 and 1995 was estimated
to be 2 396 003 and 2 676 295 units, respectively. Then, the company™s management
generated several scenarios based on different market shares. In the case where the
¬rm maintained a market share equal to 11%, the demand would be equal to 263 560
units in 1994 (with a 4.46% decrease with respect to 1993), and 294 392 units in 1995
(with a 6.71% increase with respect to 1993).




TLFeBOOK
314 LINKING THEORY TO PRACTICE

Table 8.8 Number of spare batteries sold.

Italian Adriatica Market
Year market sales Accumulatori sales share

1984 693 326 138 665 20%
1985 803 666 152 696 19%
1986 947 243 170 503 18%
1987 1 136 433 193 192 17%
1988 1 406 432 210 964 15%
1989 1 666 011 233 241 14%
1990 1 869 683 243 058 13%
1991 2 136 463 256 375 12%
1992 2 316 402 266 386 11%
1993 2 507 929 275 872 11%


Table 8.9 Car sales in Italy.

Year Number Year Number

1982 253 321 1988 886 297
1983 381 385 1989 1 014 975
1984 491 755 1990 1 162 246
1985 634 706 1991 1 167 614
1986 951 704 1992 1 217 929
1987 830 175 1993 1 363 594


The time series technique and the casual method resulted in quite different forecasts.
The company therefore decided to analyse in greater detail the logic underlying the
two approaches. Because the Italian economy was undergoing a period of quick and
dramatic change, the latter method was deemed to provide more accurate predictions
than the former technique, which is more suitable when the past demand pattern is
likely to be replicated in the future.


8.8 Distribution Logistics Network Design at
DowBrands
In 1985 Dow Consumer Products, Inc. acquired a division of Morton Thikol, Inc.
giving rise to DowBrands, which produces and markets more than 80 convenience
goods all over North America. On that occasion, the management of the new-born
company decided to redesign the distribution network. After a preliminary analysis, it
was decided that the new distribution system should be made up of CDCs and RDCs.




TLFeBOOK
LINKING THEORY TO PRACTICE 315
TL transportation
LTL transportation




Production plants
...

CDCs

Demand
RDCs points

Figure 8.10 Distribution system of DowBrands.

In the proposed system, CDCs receive TL shipments from the production plants
and supply the RDCs as well as a restricted number of major supermarkets. RDCs
are suburban warehouses from which customers are replenished (see Figure 8.10).
Shipments originating from a CDC are TL, while shipments from an RDC may be TL
or LTL. In all cases freight transportation is performed by common carriers. Each RDC
can be served by a single CDC and each customer can be assigned to a single CDC
or RDC. Thirteen potential CDCs and 23 potential RDCs were selected. The demand
points were aggregated into 93 sales districts while the products were combined in
two macro-products (home products, HP, and food products, FP).
Because customers issue their orders within short notice (a single day or even a
few hours) the management of DowBrands decided to impose an upper bound L
on the maximum distance of an LTL shipment, but no limit was imposed for TL
transportation which is much faster and reliable (see Section 1.2.3).
The distribution system redesign was designed in two stages: ¬rst, the curve of the
total logistics cost as a function of the service level (represented by L) was de¬ned;
then, an ef¬cient con¬guration was selected on the basis of a qualitative analysis
(see Section 1.3). The cost versus level of service curve was drawn as follows. For
a pre-established set of values of L, the least-cost con¬guration was determined by
solving an IP model. The outcome was the number and the locations of the CDCs and
of the RDCs, the allocation of the RDCs to the CDCs, the assignment of the demand
points to the RDCs and to the CDCs, as well as freight routes through the distribution
network.
In order to simplify the formulation, for each sales district and for each macro-
product, a dummy macro-customer TL and a dummy macro-customer LTL were
de¬ned. Therefore, each demand point was represented by four macro-customers:
TL-HP, LTL-HP, TL-FP, LTL-FP. Finally, a virtual RDC for each CDC was introduced
so that, in the next modelling representation, all macro-customers would be served
by an RDC.




TLFeBOOK
316 LINKING THEORY TO PRACTICE

Let V1 be the set of the CDCs; V2 the set of the RDCs; V3 the set of the macro-
customers; fi , i ∈ V1 , the ¬xed cost of the ith potential CDC (inclusive of all the
¬xed expenses connected to the site and to the expected value of the stock); gj ,
j ∈ V2 , the ¬xed cost of the j th potential RDC (inclusive of all the ¬xed expenses
connected to the site and to the expected value of the stock); tij k , i ∈ V1 , j ∈ V2 ,
k ∈ V3 , the unit transportation cost from the production plant to the demand point k
through the ith CDC and the j th RDC; dk , k ∈ V3 , the demand of macro-customer
k; cij k = dk tij k , i ∈ V1 , j ∈ V2 , k ∈ V3 , the transportation cost whether macro-
customer k is serviced through the ith CDC and the j th RDC. Moreover, let zi , i ∈ V1 ,
be a binary decision variable equal to 1 if the ith CDC is selected, and 0 otherwise;
yij , i ∈ V1 , j ∈ V2 , a binary decision variable equal to 1 if the j th RDC is opened
and supplied by the ith potential CDC, and 0 otherwise; xij k , i ∈ V1 , j ∈ V2 , k ∈ V3 ,
a variable representing the fraction of the total demand of the customer k served
through the ith CDC and the j th RDC.
The problem was formulated as follows.
Minimize
fi z i + yij +
gj cij k xij k (8.1)
j ∈V2 i∈V1 j ∈V2 k∈V3
i∈V1 i∈V1

subject to

xij k = 1, k ∈ V3 , (8.2)
i∈V1 j ∈V2
zi , i ∈ V1 , j ∈ V2 ,
yij (8.3)
j ∈ V2 ,
yij 1, (8.4)
i∈V1
yij , i ∈ V1 , j ∈ V2 , k ∈ V3 ,
xij k (8.5)

zi ∈ {0, 1}, i ∈ V1 , (8.6)
yij ∈ {0, 1}, i ∈ V1 , j ∈ V2 , (8.7)
xij k ∈ {0, 1}, i ∈ V1 , j ∈ V2 , k ∈ V3 , (8.8)

where constraints (8.2) establish that each customer k ∈ V3 must be served by one and
only one CDC“RDC pair, constraints (8.3) impose that a CDC must be opened if an
RDC is assigned to it; constraints (8.4) require that each RDC is assigned to a single
CDC; constraints (8.5) impose that the transportation service between a CDC“RDC
pair is activated if it is used by at least one macro-customer.
Because no capacity constraint is imposed, problem (8.1)“(8.8) satis¬es the single
assignment property (see Section 3.3.1). In order to satisfy the service level constraint,
the LTL services j “k between RDC“customer pairs distant by more than a pre-
established threshold L are discarded by setting the associated xij k variables equal
to 0.




TLFeBOOK
LINKING THEORY TO PRACTICE 317
Cost



23 500

23 300

23 100

22 900

22 700

22 500

22 300




Distance
300 370 430 500 560 620 680 750 1250

Figure 8.11 Total cost (in thousands of euros)-service level curve at DowBrands.


The solution of problem (8.1)“(8.8) was evaluated through a general-purpose MIP
solver for various values of L between 300 and 1200 km (see Figure 8.11). It is worth
noting that, as L decreases, at ¬rst the cost increases slowly, then it increases sharply.
Also, when L becomes very large, there is no need for RDCs. On the basis of these
evaluations, the company™s management set L equal to 430 km. By implementing
this solution the company achieved a saving of about 1.5 million dollars per year
compared to the previous con¬guration.


8.9 Container Warehouse Location at Hardcastle
Hardcastle is a North European leader in intermodal transportation. In 2001 the com-
pany operated nearly 240 000 containers, with an annual transportation cost of about
50 million euros.
Like other intermodal transportation companies, Hardcastle manages both full and
empty containers. When a customer places an order for freight transportation, Hard-
castle sends one or several empty containers of the appropriate type in terms of size,
refrigeration, etc., to the pick-up point (see Figure 8.12). The containers are then
loaded and sent to destination using a combination of modes (e.g. railway and sea
transportation). At the destination, the containers are emptied and sent back to the
company unless there is an outgoing load requiring the same kind of container (corre-
sponding to compensation between the demand and the supply of empty containers).
Unless compensation is possible, empty containers are then moved to a new pick-
up point. Relocating empty containers is a resource-consuming activity whose cost
should be kept at minimum. Unfortunately, compensation between the demand and
the supply of empty containers is seldom possible for three main reasons:
• the origin“destination demand matrix is strongly asymmetrical (some loca-




TLFeBOOK
318 LINKING THEORY TO PRACTICE
Independent transfer
Novara
Vercelli
Consolidated load transfer
Collection
Distribution
Milan

Genoa
(Depot)
San Paolo
(Depot)

Rio de
Janeiro




Buenos Aires
Brasilia



Figure 8.12 Freight transportation at Hardcastle.


tions are mainly sources of materials while some others are mainly points of
consumption);

• at a given location, the demand and supply for empty containers do not usually
occur at the same time;

• containers may have a large number of sizes and features; as a result, it is
unlikely that the containers incoming at a customer facility are suitable for
outgoing goods.
For these reasons, the compensation between demand and supply is neglected in
the following.
Because of the economies of scale in transportation, it is not convenient to move
containers directly from supply to demand points. Instead, containers are sent to a
nearby warehouse. Then, on a weekly basis, convoys of empty and full containers are
moved between warehouses (see Figure 8.13). Warehouses are often public so that
their location can easily be changed if necessary. Prior to its redesign, the logistics
system contained 87 depots (64 close to a railway station and 23 close to a sea
terminal). Moreover, empty container movements accounted for nearly 40% of the
total freight traf¬c.
The management of the empty containers is a complex decision process made up
of two stages (see Figure 8.14):
• at a tactical level, one has to determine, on the basis of forecasted origin“
destination transportation demands, the number and locations of warehouses,
as well as the expected container ¬‚ows among warehouses;

• at an operational level, shipments are scheduled and vehicles are dispatched on
the basis of the orders collected and of short-term forecasts.




TLFeBOOK
LINKING THEORY TO PRACTICE 319

Pavia
Full containers (Producer)
Empty containers
Milan
(Depot)

Genoa
(Origin port)




Boston
(Customer)
New York
(Destination port)




New Haven
(Depot)


Figure 8.13 Empty container transportation at Hardcastle.


Medium-term
forecasts




Depot location
Customer assignment
to depots




Short-term
Orders
forecasts


Empty container allocation

Full and empty container routing


Figure 8.14 Main decisions when managing containers at Hardcastle.


In order to redesign its logistics system, Hardcastle aggregated its customers into
300 demand points. Further, containers types were grouped into 12 types. Let C be
the set of customers, D the set of potential depots, P the set of different types of
containers, fj , j ∈ D, the ¬xed cost of depot j , aijp , i ∈ C, j ∈ D, p ∈ P , the
transportation cost of a container of type p from customer i to depot j ; bijp , i ∈ C,
j ∈ D, p ∈ P , the transportation cost of a container of type p from depot j to
customer i; cj kp , j ∈ D, k ∈ D, p ∈ P , the transportation cost of an empty container
of type p from depot j to depot k; dip , i ∈ C, p ∈ P , the number of containers of
type p requested by the customer i; oip , i ∈ C, p ∈ P , the supply of containers of
type p from customer i. Furthermore, let yj , j ∈ D, be a binary decision variable




TLFeBOOK
320 LINKING THEORY TO PRACTICE

equal to 1 if the depot j is selected, and 0 otherwise; xijp , i ∈ C, j ∈ D, p ∈ P , the
¬‚ow of empty containers of type p from customer i to depot j ; sijp , i ∈ C, j ∈ D,
p ∈ P , the ¬‚ow of empty containers of type p from depot j to customer i; wj kp ,
j ∈ D, k ∈ D, p ∈ P , the ¬‚ow of empty containers of type p from depot j to depot
k. The problem was formulated as follows.
Minimize

fj yj + (aijp xijp + bijp sijp ) + cj kp wj kp (8.9)
j ∈D p∈P i∈C j ∈D j ∈D k∈D

subject to

xijp = oip , i ∈ C, p ∈ P , (8.10)
j ∈D

sijp = dip , i ∈ C, p ∈ P , (8.11)
j ∈D

xijp + wkjp ’ sijp ’ wj kp = 0, j ∈ D, p ∈ P , (8.12)
i∈C k∈D i∈C k∈D


(xijp + sijp ) + (wj kp + wkjp )
p∈P i∈C p∈P k∈D

(oip + dip + 2M), j ∈ D,
yj (8.13)
p∈P i∈C


i ∈ C, j ∈ D, p ∈ P ,
xijp 0, (8.14)
i ∈ C, j ∈ D, p ∈ P ,
sijp 0, (8.15)
j ∈ D, k ∈ D, p ∈ P ,
wj kp 0, (8.16)
yj ∈ {0, 1}, j ∈ D, (8.17)
where M is an upper bound on the wj kp ¬‚ows, j ∈ D, k ∈ D, p ∈ P . The objec-
tive function (8.9) is the sum of warehouse ¬xed costs and empty container vari-
able transportation costs (between customers and warehouses, and between pairs of
warehouses). Constraints (8.10)“(8.12) impose empty container ¬‚ow conservation.
Constraints (8.13) state that if yj = 0, j ∈ D, then the incoming and outgoing ¬‚ows
from site j are equal to 0. Otherwise, constraints (8.13) are not binding since
oip , i ∈ C, j ∈ D, p ∈ P ,
xijp
dip , i ∈ C, j ∈ D, p ∈ P ,
sijp
M, j ∈ D, k ∈ D, p ∈ P .
wj kp
The implementation of the optimal solution of model (8.9)“(8.17) yielded a reduction
in the number of warehouses to 48 and a 47% reduction in transportation cost.




TLFeBOOK
LINKING THEORY TO PRACTICE 321

8.10 Inventory Management at Wolferine
Wolferine is a division of the industrial group UOP Limited, which manufactures

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