<<

. 4
( 15)



>>

values of the smoothing constant ± for the TV sets forecasting problem of Sarath
company. By using MAD12 (see Table 2.27), ± = 0.1 comes out to give the most
precise forecast. With this value of the smoothing constant we obtain MAPD12 =
0.79%, which corresponds to a very good accuracy.



2.9.2 Forecast control

A forecasting method works correctly if the errors are random and not systematic.
Typical systematic errors occur when the demand value is constantly underestimated
or overestimated, and a seasonal variation is not taken into account. Forecasting
control can be done through a tracking signal or a control chart. The tracking signal
St , 1 < t T , is de¬ned as the ratio between the cumulative error and the MADt ,

Et
St = ,
MADt




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66 FORECASTING LOGISTICS REQUIREMENTS
St




SMAX
Interval of
acceptable
values


0
t




’SMAX

Need of a
corrective
action


Figure 2.18 Use of a tracking signal for a forecasting control.


where
t
Et = ek .
k=2

The tracking signal is greater than zero if the forecast systematically underestimates
the demand; vice versa, a negative value of St indicates a systematic overestimate of
the demand. For this reason, a forecast is assumed to be unbiased if the tracking signal
falls in the range ±Smax . The value of Smax is established heuristically, and usually
varies between 3 and 8. If the tracking signal is outside this interval, the parameters of
the forecasting method should be modi¬ed or a different forecasting method should
be selected (see Figure 2.18).
Unlike tracking signals, control charts are based on the plot of single errors et . Under
the hypothesis that the expected value of the errors is zero, a forecast is effective if each
error ek , k = 2, . . . , t, is in the con¬dence interval ±mσt , where σt is the standard
deviation of the errors. An estimate of σt can be obtained as

σt = MSEt .

The parameter m can be related to the probability that the error be in the interval
±mσt . If the errors are normally distributed with zero mean, the error belongs to
the interval ±2σt with a probability around 97.7%, and to the interval ±3σt with
a probability around 99.8%. Finally, it is worth observing that the interval ±3σt
corresponds approximately to the band ±4 of the tracking signal.
In addition to the previous analytical check, it is useful to verify visually whether the
error pattern reveals the possibility of improving the forecast by introducing suitable
modi¬cations. Here are three of the most common pathological situations.
• The errors have an expected value different from zero; this means that the
forecast is biased (see Figure 2.19).




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FORECASTING LOGISTICS REQUIREMENTS 67
et




0
t



Figure 2.19 Nonzero mean error.

• The error pattern shows a positive or negative trend; in this case, the accuracy
of the forecasting method is progressively diminishing.

• The error pattern is periodic; this can happen if an existing seasonal effect has
not been identi¬ed.


By using the tracking signal (±4 band) and the control chart (±3σT ), we can mon-
itor the demand forecast of sports goods for Browns supermarkets. The forecasting
technique is the exponential smoothing method with ± = 0.3. The demand history
for the last few months, and the required forecasts (both in hundreds of dollars), are
reported in Table 2.28. The tracking signal St , t = 2, . . . , T , is always in the inter-
val ±4. On the basis of this preliminary evaluation, we can state that the forecast is
under control. To make a further check, the expected value and the standard deviation
of the error at time period t = T are estimated. The results are ’3.02 and 45.36,
respectively. We observe that, since the average error is much less than the average
demand, we can consider the forecast unbiased. Furthermore, since all errors are in
the interval ±3σT , even this test suggests that the forecast is under control. Finally,
the examination of the control chart (see Figure 2.20) does not show any systematic
error.




2.10 Questions and Problems
2.1 In containerized freight transportation, empty containers have to be periodically
allocated to depots in order to satisfy future customer demands. How would
you forecast the demand for carrier ISO 20 refrigerated containers?




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68 FORECASTING LOGISTICS REQUIREMENTS

Table 2.28 Demand entries and forecasts (in hundreds of dollars) in the Browns problem.

2
|et |
t dt pt et Et St et
MADt MSEt

1 975 ” ” ” ” ” ” ” ”
2 995 975.00 20.00 20.00 20.00 20.00 1.00 400.00 ”
’29.00 ’9.00 ’0.37
3 952 981.00 29.00 24.50 841.00 1241.00
4 982 972.30 9.70 9.70 19.57 0.70 0.04 94.09 667.55
’52.21 ’51.51 ’1.86
5 923 975.21 52.21 27.73 2725.88 1353.66
’26.06 ’0.96
6 985 959.55 25.45 25.45 27.27 647.86 1177.21
’65.18 ’91.24 ’2.72
7 902 967.18 65.18 33.59 4248.81 1791.53
’9.63 ’100.87 ’3.34
8 938 947.63 9.63 30.17 92.7 1508.39
’62.61 ’2.01
9 983 944.74 38.26 38.26 31.18 1463.86 1502.03
’61.22 ’123.83 ’3.59
10 895 956.22 61.22 34.52 3747.61 1782.73
’111.68 ’3.46
11 950 937.85 12.15 12.15 32.28 147.56 1601.04
’33.17 ’0.91
12 1020 941.50 78.50 78.50 36.48 6162.77 2057.21
13 ” 965.05 ” ” ” ” ” ” ”


et

100

80

60

40

20

0

-20

-40

-60

-80
t
1 2 3 4 5 6 7 8 9 10 11 12

Figure 2.20 Control chart for the demand forecasting of
sports goods in the Browns problem.

2.2 Illustrate a logistics system where there is little need for forecasting.
2.3 To what extent are the forecasting practices different in an MTS and in an MTA
system?
2.4 How would you predict the future demand of a new product?
2.5 Your company is planning to add extra capacity to a plant currently manufactur-
ing 110 000 items per year. You are asked to suggest how much capacity should
be added to the factory. After an accurate sales forecast over the next few years




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FORECASTING LOGISTICS REQUIREMENTS 69

Table 2.29 Number of installed heaters by Hot Spot and service requests.

Installed heaters

Less than two More than two Service
Year years ago years ago Total requests

1995 260 000 69 500 329 500 18 672
1996 265 000 74 200 339 200 19 076
1997 287 800 82 850 370 650 20 994
1998 313 750 90 550 404 300 23 249
1999 345 350 97 150 442 500 25 025
2000 379 050 105 950 485 000 28 111
2001 416 950 111 550 528 500 30 985
2002 459 100 117 000 576 100 33 397
2003 502 550 123 200 625 750


you are quite sure that the most likely value of the annual demand is 140 000
items and that the MSE is equal to 108 . You also know that your company
loses $3 for each unit of unused capacity and $7 for each unit of unsatis¬ed
demand. How much capacity should your company buy? (Hint: suppose that
the forecasting error can be assumed to be normally distributed.)

2.6 Hot Spot is a ¬rm based in the USA whose core business is the maintenance
of home heaters. The company usually forecasts service requests on the basis
of the number of installed heaters. Make a forecast of the service requests in
2003 in New Jersey by using data in Table 2.29.
To this purpose use

(a) a single regression analysis (service requests versus total installed heat-
ers);
(b) a multiple regression analysis (service requests versus the number of
heaters installed less than two years ago and at least two years).

Which technique is the most accurate? Why?

2.7 Sunshine Ltd is one of the world™s leading suppliers of fast-moving goods in
household care and personal product categories. According to management, its
facial soap sales depend mainly on the promotion expenditure that the com-
pany and its competitor make. Table 2.30 reports facial soap sales in Canada
versus (a) Sunshine promotion expenditure, and (b) the competitor™s promo-
tion expenditure divided by the Sunshine promotion expenditure. Make a sales
forecast for the next two periods under the hypotheses that Sunshine increases
its promotion expenditure to 14.5 millions of Canadian dollars and the com-




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70 FORECASTING LOGISTICS REQUIREMENTS

Table 2.30 Sunshine Ltd facial soap sales (in millions of Canadian dollars) in Canada.

Sunshine Competitor™s promotion
Period promotion expenditure/Sunshine Sales of facial
(trimester, year) expenditure promotion expenditure soaps in Canada

I 2001 6.0 1.2 46.8
II 2001 6.8 1.2 52.7
III 2001 7.5 1.4 60.5
IV 2001 7.5 1.5 56.6
I 2002 9.0 1.5 64.4
II 2002 10.5 1.7 74.1
III 2002 12.0 1.8 72.2
IV 2002 12.0 1.5 78.0
I 2003 13.5 1.4 87.8
II 2003 13.5 1.5 95.6



Table 2.31 Number of light trucks sold by Mitsumishi.

Year

Month 2000 2001 2002 2003

January 22 882 23 478 24 768 24 765
February 19 981 17 019 19 351 21 739
March 18 811 20 967 23 953 25 153
April 19 352 19 759 18 855 20 515
May 27 226 22 200 28 414 24 038
June 14 932 24 162 18 537 26 151
July 18 531 20 275 22 845
August 8 523 7 949 9 451
September 13 064 14 328 15 842
October 13 733 16 691 16 409
November 12 597 13 784 13 881
December 7 645 10 986 11 230



petitor™s promotion expenditure remains the same as in the second trimester of
2003.

2.8 Mitsumishi is a Korean company whose number of light trucks sold between
January 2000 and June 2003 is reported in Table 2.31.
The company usually carries out a promotion in May and/or in June. The sales
improvement achieved during the last few promotions are shown in Table 2.32.




TLFeBOOK
FORECASTING LOGISTICS REQUIREMENTS 71

Table 2.32 Light trucks sales improvement of Mitsumishi.

Period Sales improvement (%)

May 2000 +30
May 2001 +20
June 2001 +15
May 2002 +25
June 2003 +30


Table 2.33 Sales of the new spiced food (in hundreds of kilograms)
produced by Mare Nostrum.

Year

Month 2001 2002 2003

January 130 000 141 988 156 467
February 129 720 142 376 158 137
March 129 703 143 636 159 140
April 129 633 144 543 161 156
May 129 632 147 534 162 835
June 129 854 148 919 165 479
July 130 436 150 961
August 132 751 152 748
September 133 334 152 977
October 133 761 154 387
November 135 286 156 856
December 136 800 157 349


(a) Using a classical decomposition method, forecast the sales for the next
six months.
(b) Plot a control chart of the error over the last three months. Are you able
to detect any anomaly?

2.9 Mare Nostrum, a canned tuna manufacturer based in Sicily (Italy), has marketed
a new spiced food line in June 2002. Now the management wishes to project
sales for improved planning of production and logistics operations. Sales data
for one and a half years are reported in Table 2.33.

(a) Plot the data on a graph. What important observations can you make
about the demand pattern? Which data are relevant and should be used
for forecasting purposes?




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72 FORECASTING LOGISTICS REQUIREMENTS

Table 2.34 Demand of refrigerated trucks (in europallets) between
Antwerp and Brussels over the last ten weeks.

Week

Day 1 2 3 4 5 6 7 8 9 10

Monday 67 68 76 75 75 82 77 88 84 84
Tuesday 54 57 59 57 58 69 65 57 72 56
Wednesday 47 49 49 52 57 59 52 54 68 59
Thursday 40 45 46 43 48 49 55 50 59 52
Friday 60 63 68 69 72 69 68 66 63 69


(b) Using the classical time series decomposition analysis, predict the expect-
ed sales over the next six months.
(c) Repeat the forecast by applying the Holt method.
(d) Estimate the MAPD of both methods using the last six months. Which
approach seems to work best?

2.10 The Belgian Trucking Company needs to determine the number of refrigerated
trucks to satisfy the transportation demand between Antwerp and Brussels on a
daily basis. The volume of the demand for the last weeks is given in Table 2.34.

(a) Using the Winters method, predict the expected number of pallets to be
transported for the next week.
(b) Estimate the error in the above forecast using the last three weeks.
(c) Construct a 95% con¬dence interval on the forecasting. (Hint: assume a
normal distribution of demand.)


2.11 Annotated Bibliography
An in-depth treatment of the forecasting methods is reported in:
1. Montgomery DC, Johnson LA and Gradiner JS 1990 Forecasting and Time
Series Analysis. McGraw-Hill, New York.
For a detailed description of the statistical methods, the reader can refer to:
2. Sandy R 1990 Statistics for Business and Economics. McGraw-Hill, NewYork.
The results of the survey reported in Table 2.1 are taken from:
3. Sanders NR and Manrodt KB 1994 Forecasting practices in US corporations:
survey results. Interfaces 24(2), 92“100.




TLFeBOOK
3

Designing the Logistics Network

3.1 Introduction
In business logistics the network planning process consists of designing the system
through which commodities ¬‚ow from suppliers to demand points, while in the public
sector it consists of determining the set of facilities from which users are serviced. In
both cases the main issues are to determine the number, location, equipment and size
of new facilities, as well as the divestment, displacement or downsizing of facilities.
Of course, the objectives and constraints vary depending on the sector (private or pub-
lic) and on the type of facilities (plants, CDCs, RDCs, regional and ¬eld warehouses,
retail outlets, dumpsites, incinerators, ambulance parking places, ¬re stations, etc.).
The aim generally pursued in business logistics is the minimization of the annual
total logistics cost subject to side constraints related to facility capacity and required
customer service level (recall the discussion in Chapter 1). As a rule, the cost to
be minimized is associated with facility operations (manufacturing, storage, sorting,
consolidation, selling, incineration, parking, etc.), and to transportation between facil-
ities, or between facilities and users. Also, when designing the logistics network for
a utility company, different objectives, such as achieving equity in servicing users,
may have to be considered.
Research in logistics network design dates back to the early location theories of
the 19th century. Since then a variety of models and solution methodologies has been
proposed and analysed. In this chapter some of the most important facility location
problems are examined. To put this analysis in the right perspective, a number of
relevant issues are ¬rst introduced and discussed.

When location decisions are needed. Facility location decisions must obviously
be made when a logistics system is started from scratch. They are also required as a
consequence of variations in the demand pattern or spatial distribution, or following
modi¬cations of materials, energy or labour cost. In particular, location decisions are
often made when new products or services are launched, or outdated products are
withdrawn from the market.

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)




TLFeBOOK
74 DESIGNING THE LOGISTICS NETWORK

Location decisions may be strategic or tactical. Whereas facilities are purchased
or built, location decisions involve sizeable investments. In this case, changing sites or
equipment is unlikely in the short or medium term. This may be true even if facilities
are leased. On the other hand, if space and equipment are rented (e.g. from a public
warehouses) or operations are subcontracted, location decisions can be reversible in
the medium term.

Location and allocation decisions are intertwined. Location decisions are strictly
related to those of de¬ning facility area boundaries (i.e. allocating demand to facili-
ties). For example, in a two-echelon distribution system (see Figure 3.1), opening a
new RDC must be accompanied by a rede¬nition of the sales districts along with a dif-
ferent allocation of the RDCs to the CDCs and of the CDCs to the production plants.
For this reason location problems are sometimes referred to as location“allocation
problems.

Location decisions may affect demand. Facility location may affect the demand
volume. For example, opening a new RDC may lead to the acquisition of customers
who previously could not be served at a satisfactory level of service because they
lived too far away.


3.2 Classi¬cation of Location Problems
Location problems come in a variety of forms, which can be classi¬ed with respect
to a number of criteria. The classi¬cation proposed below is logistics-oriented.

Time horizon. In single-period problems, facility location decisions must be made
at the beginning of the planning horizon on the basis of the forecasted logistics require-
ments. In multi-period problems one has to decide, at the beginning of the planning
horizon, a sequence of changes to be made at given time instants within the planning
horizon.

Facility typology. In single-type location problems, a single type of facility (e.g.
only RDCs) are located. Instead, in multi-type problems several kinds of facility (e.g.
both CDCs and RDCs) are located.

Material ¬‚ows. In single-commodity problems it can be assumed that a single homo-
geneous ¬‚ow of materials exists in the logistics system, while in multicommodity
problems there are several items, each with different characteristics. In the latter case
each commodity is associated with a speci¬c ¬‚ow pattern.

Interaction among facilities. In complex logistics systems there can be material
¬‚ows among facilities of the same kind (e.g. component ¬‚ows among plants). In this
case, optimal facility locations depend not only on the spatial distribution of ¬nished




TLFeBOOK
DESIGNING THE LOGISTICS NETWORK 75




Production plants

Supply points RDCs

Figure 3.1 A two-echelon single-type location problem.


product demand but also on the mutual position of the facilities (location problems
with interaction).

Dominant material ¬‚ows. Single-echelon location problems are single-type prob-
lems such that either the material ¬‚ow coming out or the material ¬‚ow entering the
facilities to be located is negligible. In multiple-echelon problems, both inbound and
outbound commodities are relevant. This is the case, for example, when DCs have to
be located taking into account both the transportation cost from plants to DCs and the
transportation cost from DCs to customers. In multiple-echelon problems, constraints
aiming at balancing inbound and outbound ¬‚ows have to be considered.

Demand divisibility. In some distribution systems it is required, for administrative
or book-keeping reasons, that each facility or customer be supplied by a single centre,
while in others a facility or a customer may be served by two or more centres. In the
former case demand is said to be divisible while in the latter it is indivisible.

In¬‚uence of transportation on location decisions. Most location models assume
that transportation cost between two facilities, or between a facility and an user, is
computed as a suitable transportation rate multiplied by the freight volume and the
distance between the two points. Such an approach is appropriate if vehicles travel by
means of a direct route. However, if each vehicle makes collections or deliveries to
several points, then a transportation rate cannot easily be established. In such cases the
routes followed by the vehicles should be taken explicitly into account when locating
the facilities (location-routing models). To illustrate this concept, consider Figure 3.2,
where a warehouse serves three sales districts located at the vertices of triangle ABC.
Under the hypothesis that the facility ¬xed cost is independent of the site, there can
be two extreme cases.




TLFeBOOK
76 DESIGNING THE LOGISTICS NETWORK

A




O




B C



Figure 3.2 The optimal location of a warehouse depends on the way customers are serviced.

• Each customer requires a full-load supply and, therefore, the optimal location
of the DC is equal to the Steiner point O.
• A single vehicle can service all points and hence the facility can be located at
any point of the triangle ABC perimeter.
The interdependence between facility locations and vehicle routes is particularly
strong when dealing with mail distribution, solid waste collection or road mainte-
nance, as the users are located almost continuously on the road network.

Retail location. When planning a store network, the main issue is to optimally
locate a set of retail outlets that compete with other stores for customers. In such a
context, predicting the expected revenues of a new site is dif¬cult since it depends
on a number of factors such as location, sales area and level of competition. Retail
location problems can be modelled as competitive location models, the analysis of
which is also beyond the scope of this textbook. The reader should again consult the
references quoted in the last section of this chapter for further details.

Modelling and solving location problems
In the remainder of this chapter, some selected facility location problems are modelled
and solved as MIP problems. An optimal solution can be determined in principle
by means of a general-purpose or tailored branch-and-bound algorithm. Such an
approach only works for relatively simple problems (such as the single-echelon single-
commodity (SESC) problem) whenever instance size is small. For multiple-echelon
multiple-commodity problems, determining an optimal solution can be prohibitive
even if the number of potential facilities is relatively small (less than 100). As a result,
heuristic procedures capable of determining a ˜good™ feasible solution in a reasonable
amount of time can be very useful. To evaluate whether a heuristic solution provides
a tight upper bound (UB) on the optimal solution value, it is useful to determine a




TLFeBOOK
DESIGNING THE LOGISTICS NETWORK 77

lower bound (LB) on the optimal solution value. This yields a ratio (UB ’ LB)/LB
which represents an overestimate of relative deviation of the heuristic solution value
from the optimum.


3.3 Single-Echelon Single-Commodity
Location Models
The SESC location problem is based on the following assumptions:
• the facilities to be located are homogeneous (e.g. they are all regional ware-
houses);
• either the material ¬‚ow coming out or the material ¬‚ow entering such facilities
is negligible;
• all material ¬‚ows are homogeneous and can therefore be considered as a single
commodity;
• transportation cost is linear or piecewise linear and concave;
• facility operating cost is piecewise linear and concave (or, in particular, con-
stant).
The second assumption is the most restrictive. It holds in contexts where locating
production plants whose ¬nished product (e.g. steel) weighs much less than the raw
materials (iron and coal in the example) used in the manufacturing process. Another
application arises in warehouse location for a distribution company (see Figure 3.3),
whereas the goods are purchased at a price inclusive of transportation cost up to the
warehouses. We examine the case where inbound ¬‚ows are negligible, although the
same methodology can be applied without any change to the case where they are
important and outbound ¬‚ows are negligible. Moreover, to simplify the exposition, it
is assumed that the facilities to be located are warehouses and the demand points are
customers.
The problem can be modelled through a bipartite complete directed graph G(V1 ∪
V2 , A), where the vertices in V1 stand for the potential facilities, the vertices in V2
represent the customers, and the arcs in A = V1 — V2 are associated with the material
¬‚ows between the potential facilities and the demand points.
In what follows, we further assume that
• the demand is divisible (see Section 3.2).
Let dj , j ∈ V2 , be the demand of customer j ; qi , i ∈ V1 , the capacity of the potential
facility i; ui , i ∈ V1 , a decision variable that accounts for operations in potential
facility i; sij , i ∈ V1 , j ∈ V2 , a decision variable representing the amount of product
sent from site i to demand point j ; Cij (sij ), i ∈ V1 , j ∈ V2 , the cost of transporting
sij units of product from site i to customer j ; Fi (ui ), i ∈ V1 , the cost for operating




TLFeBOOK
78 DESIGNING THE LOGISTICS NETWORK




RDCs Demand points

Figure 3.3 RDC location.

potential facility i at level ui . Then the problem can be expressed in the following
way.
Minimize
Cij (sij ) + Fi (ui ) (3.1)
i∈V1 j ∈V2 i∈V1

subject to

sij = ui , i ∈ V1 , (3.2)
j ∈V2

sij = dj , j ∈ V2 , (3.3)
i∈V1
i ∈ V1 ,
ui qi , (3.4)
i ∈ V1 , j ∈ V2 ,
sij 0, (3.5)
i ∈ V1 .
ui 0, (3.6)
Variables ui , i ∈ V1 , implicitly de¬ne a location decision since a facility i ∈ V1 is
open if only if ui is strictly positive. Variables sij , i ∈ V1 , j ∈ V2 , determine customer
allocations to facilities. The objective function (3.1) is the sum of the facility operating
costs plus the transportation cost between facilities and users. Constraints (3.2) state
that the sum of the ¬‚ows outgoing a facility equals its activity level. Constraints (3.3)
ensure that each customer demand is satis¬ed, while constraints (3.4) force the activity
level of a facility not to exceed the corresponding capacity.
Model (3.1)“(3.6) is quite general and can be easily adapted to the case where, in
order to have an acceptable service level, some arcs (i, j ) ∈ A having a travel time
larger than a given threshold cannot be used (we remove the corresponding decision
variables sij , for the appropriate i ∈ V1 and j ∈ V2 (see Figure 3.4)). In the remainder




TLFeBOOK
DESIGNING THE LOGISTICS NETWORK 79

1 1




...
...



i j




...
...



|V1| |V2|



Figure 3.4 Graph representation of the single-echelon location problem (note that the arcs
(1, j ) and (|V1 |, 1) are absent, given that the corresponding travel times are longer than the
given threshold).

of this section two particular cases of the SESC location problem are examined in
detail. In the ¬rst case, the transportation cost per unit of commodity is constant, and
facility operating costs consist of ¬xed costs. In the second case, transportation costs
are still linear but facilities operating costs are piecewise linear and concave. Both
problems are NP-hard and can be modelled as MIP problems. Hence they can be
solved through general purpose (or tailored) branch-and-bound algorithms.


3.3.1 Linear transportation costs and facility ¬xed costs
If the transportation costs per unit of ¬‚ow are constant, then
Cij (sij ) = cij sij , i ∈ V1 , j ∈ V2 .
Moreover, if facility costs Fi (ui ) are described by a ¬xed cost fi and a constant
marginal cost gi , then

fi + gi ui , if ui > 0,
Fi (ui ) = i ∈ V1 . (3.7)
if ui = 0,
0,
Equation (3.7) leads to the introduction in problem (3.1)“(3.6) of a binary variable
yi replacing ui , for each i ∈ V1 , whose value is equal to 1 if potential facility i is
open, and 0 otherwise. If gi is negligible, then Equation (3.7) is replaced by
Fi (yi ) = fi yi , i ∈ V1 ,
and constraints (3.2) and (3.4) become
i ∈ V1 .
sij qi yi ,
j ∈V2




TLFeBOOK
80 DESIGNING THE LOGISTICS NETWORK

The new set of variables easily allows the imposition of lower and upper bounds on
the number of open facilities. For instance, if exactly p facilities have to be opened,
then
yi = p.
i∈V1

Finally, introducing xij variables, i ∈ V1 , j ∈ V2 , representing the fraction of
demand dj satis¬ed by facility i, we can write

sij = dj xij , i ∈ V1 , j ∈ V2 ,

(3.8)
ui = dj xij , i ∈ V1 . 

j ∈V2

The SESC problem can hence be formulated as an MIP problem as follows.
Minimize
cij xij + fi y i (3.9)
i∈V1 j ∈V2 i∈V1

subject to

xij = 1, j ∈ V2 , (3.10)
i∈V1

i ∈ V1 ,
dj xij qi yi , (3.11)
j ∈V2

yi = p, (3.12)
i∈V1
i ∈ V1 , j ∈ V2 ,
xij
0 1, (3.13)
yi ∈ {0, 1}, i ∈ V1 , (3.14)

where
cij = cij dj , i ∈ V1 , j ∈ V2 , (3.15)
is the transportation cost incurred for satisfying the entire demand dj of customer
j ∈ V2 from facility i ∈ V1 . It is worth noting that on the basis of constraints (3.10),
variables xij , i ∈ V1 , j ∈ V2 , cannot take a value larger than 1. Therefore, rela-
0, i ∈ V1 , j ∈ V2 .
tions (3.13) can be written more simply in the form xij
1, i ∈ V1 , j ∈ V2 , are fundamental in a context where the con-
Relations xij
straints (3.10) are relaxed, as in the Lagrangian procedure described below.
The SESC formulation (3.9)“(3.14) is quite general and can sometimes be simpli-
¬ed. In particular, if constraint (3.12) is removed, the model is known as capacitated
plant location (CPL). If relations (3.11) are also deleted, the so-called simple plant
location (SPL) model is obtained. If ¬xed costs fi are the same for each i ∈ V1 ,
dj = 1 for each j ∈ V2 and qi = |V2 | for each i ∈ V1 , then formulation (3.9)“(3.14)
is known as a p-median model.




TLFeBOOK
DESIGNING THE LOGISTICS NETWORK 81


Koster Express is an American LTL express carrier operating in Oklahoma (USA).
The carrier service is organized through a distribution subsystem, a group of termi-
nals and a long-haul transportation subsystem. The distribution subsystem uses a set
of trucks, based at the terminals, that pick up the outgoing goods from 1:00 p.m. to
7:00 p.m., and deliver the incoming goods from 9:00 a.m. to 1:00 p.m. The terminals
are equipped areas where the outgoing items are collected and consolidated on pallets,
the incoming pallets are opened up, and their items are classi¬ed and directed to the
distribution. At present the ¬rm has nine terminals (located in Ardmore, Bartlesville,
Duncan, Enid, Lawton, Muskogee, Oklahoma City, Ponca City and Tulsa). The long-
haul transportation subsystem provides the transport, generally during the night, of
the consolidated loads between the origin and destination terminals. For that purpose,
trucks with a capacity between 14 and 18 pallets (and a maximum weight of between
0.8 and 1 ton) are used. During the last six years, the long-haul transportation subsys-
tem has had two hubs in Duncan and Tulsa. The Duncan hub receives the goods from
Ardmore, Lawton, Oklahoma City and Duncan itself, while the Tulsa hub collects
the items coming from the other terminals. In each hub the goods assigned to the
terminals of the other hub are sent on large trucks, generally equipped with trailers.
For example, goods coming from Enid and destined to Oklahoma City are brought to
the Tulsa hub, from where they are sent to the Duncan hub together with the goods
coming from Bartlesville, Muskogee, Ponca City and Tulsa itself and directed to the
terminals served by the Duncan hub; ¬nally, they are sent to Oklahoma City. Goods
destined to other terminals of the same hub are stocked for a few hours until the trucks
coming from the other hub arrive, and only then are they sent to their destinations.
For example, goods to Lawton coming from Ardmore are sent to the Duncan hub,
stocked until the arrival of the trucks from the Tulsa hub and then, sent to Lawton,
together with the goods coming from all the other terminals.
Recently, the ¬rm was offered a major growth opportunity with the opening of new
terminals in Altus, Edmond and Stillwater. At the same time the management decided
to relocate its two hubs. The team hired to carry out a preliminary analysis decided
to consider only the transportation cost from each terminal to the hub and vice versa
(neglecting, therefore, both the transportation cost between the two hubs and the cost,
yet considerable, associated with the possible divestment of the pre-existing hub).
Under the hypothesis that each terminal can accommodate a hub, the problem can be
formulated a p-median problem in the following way.
Minimize
cij xij
i∈V1 j ∈V2

subject to

xij = 1, j ∈ V2 ,
i∈V1




TLFeBOOK
82 DESIGNING THE LOGISTICS NETWORK

Table 3.1 Distances (in miles) between terminals in the Koster Express problem (Part I).

Altus Ardmore Bartlesville Duncan Edmond Enid

Altus 0.0 169.8 291.8 88.2 153.9 208.2
Ardmore 169.8 0.0 248.6 75.9 112.5 199.0
Bartlesville 291.8 248.6 0.0 231.5 146.0 132.4
Duncan 88.2 75.9 231.5 0.0 93.5 137.5
Edmond 153.9 112.5 146.0 93.5 0.0 88.8
Enid 208.2 199.0 132.4 137.5 88.8 0.0
Lawton 54.2 115.8 238.7 34.1 100.7 145.0
Muskogee 274.2 230.4 92.2 213.5 145.7 166.4
Oklahoma City 141.1 100.5 151.4 80.9 14.4 87.6
Ponca City 245.0 202.2 70.2 184.8 91.9 64.5
Stillwater 209.2 162.6 115.0 145.3 53.0 65.8
Tulsa 248.0 204.6 45.6 187.8 102.2 118.4




|V2 |yi , i ∈ V1 ,
xij
j ∈V2

yi = 2,
i∈V1
xij ∈ {0, 1}, i ∈ V1 , j ∈ V2 ,
yi ∈ {0, 1}, i ∈ V1 ,

where V1 = V2 represent the set of old and new terminals; yi , i ∈ V1 , is a binary
decision variable whose value is equal to 1 if terminal i accommodates a hub, 0
otherwise; xij , i ∈ V1 , j ∈ V2 , is a binary decision variable whose value is equal
to 1 if the hub located in terminal i serves terminal j , 0 otherwise; however, due to
the particular structure of the problem constraints, variables xij cannot be fractional
and greater than 1, therefore xij ∈ {0, 1}, i ∈ V1 , j ∈ V2 can be replaced with
xij 0, i ∈ V1 , j ∈ V2 .
Since the goods incoming and outgoing every day from each terminal can be
transferred by a single truck with a capacity of 14 pallets, the daily transport cost (in
dollars) between a pair of terminals i ∈ V1 and j ∈ V2 is given by cij = 2—0.74—lij ,
where 0.74 is the transportation cost (in dollars per mile), and lij is the distance (in
miles) between the terminals (see Tables 3.1 and 3.2).
The optimal solution leads to the opening of two hubs located in Duncan and
Stillwater (the daily total cost being $1081.73). Terminals in Altus, Ardmore, Duncan
and Lawton are assigned to the Duncan hub, while the Stillwater hub serves the
terminals in Bartlesville, Edmond, Enid, Muskogee, Oklahoma City, Ponca City,
Stillwater and Tulsa.




TLFeBOOK
DESIGNING THE LOGISTICS NETWORK 83

Table 3.2 Distances (in miles) between terminals in the Koster Express problem (Part II).

Lawton Muskogee Oklahoma City Ponca City Stillwater Tulsa

Altus 54.2 274.2 141.1 245.0 209.2 248.0
Ardmore 115.8 230.4 100.5 202.2 162.6 204.6
Bartlesville 238.7 92.2 151.4 70.2 115.0 45.6
Duncan 34.1 213.5 80.9 184.8 145.3 187.8
Edmond 100.7 145.7 14.4 91.9 53.0 102.2
Enid 145.0 166.4 87.6 64.5 65.8 118.4
Lawton 0.0 220.6 88.0 191.9 152.5 194.9
Muskogee 220.6 0.0 140.4 142.5 119.2 48.1
Oklahoma City 88.0 140.4 0.0 104.7 66.6 107.6
Ponca City 191.9 142.5 104.7 0.0 41.9 96.5
Stillwater 152.5 119.2 66.6 41.9 0.0 71.2
Tulsa 194.9 48.1 107.6 96.5 71.2 0.0

Demand allocation
¯ ¯
For a given set V1 ⊆ V1 of open facilities, an optimal demand allocation to V1 can be
determined by means of the following LP model.
Minimize
cij xij (3.16)
¯
i∈V1 j ∈V2
subject to
xij = 1, (3.17)
¯
i∈V1
¯
i ∈ V1 ,
dj xij qi , (3.18)
j ∈V2
¯
i ∈ V1 , j ∈ V2 .
xij 0, (3.19)

In an optimal solution of problem (3.16)“(3.19), some xij values may be fractional
(i.e. the demand of a vertex j ∈ V2 can be satis¬ed by more than one vertex i ∈ V1 )
because of capacity constraints (3.18). However, in the absence of a capacity constraint
(as in the SPL and p-median models), there exists at least one optimal solution such
that the demand of each vertex j ∈ V2 is satis¬ed by a single facility i ∈ V1 (single
¯
assignment property). This solution can be de¬ned as follows. Let ij ∈ V1 be a facility
such that
ij = arg min cij .
¯
i∈V1
Then, the allocation variables can be de¬ned as follows:
if i = ij ,
1,

xij =
0, otherwise.




TLFeBOOK
84 DESIGNING THE LOGISTICS NETWORK

A Lagrangian heuristic for the capacitated plant location problem
Several heuristic methods have been developed for the solution of SESC problem
(3.9)“(3.14). Among them, Lagrangian relaxation techniques play a major role. They
usually provide high-quality upper and lower bounds within a few iterations. For
the sake of simplicity, in the remainder of this section, a Lagrangian procedure is
illustrated for the CPL problem, although this approach may be used for the general
SESC model (3.9)“(3.14). The fundamental step of the heuristic is the determination
of a lower bound, obtained by relaxing demand satisfaction constraints (3.10) in a
Lagrangian fashion. Let »j ∈ be the multiplier associated with the j th constraint
(3.10). Then the relaxed problem is:
Minimize
cij xij + fi yi + xij ’ 1
»j (3.20)
i∈V1 j ∈V2 j ∈V2
i∈V1 i∈V1

subject to

i ∈ V1 ,
dj xij q i yi , (3.21)
j ∈V2
i ∈ V1 , j ∈ V2 ,
xij
0 1, (3.22)
y1 ∈ {0, 1}, i ∈ V1 . (3.23)

It is easy to check that problem (3.20)“(3.23) can be decomposed into |V1 | sub-
problems. Indeed, for a given vector » ∈ |V2 | of multipliers, the optimal objective
value of problem (3.20)“(3.23), LBCPL (»), can be determined by solving, for each
potential facility i ∈ V1 , the following subproblem:
Minimize
(cij + »j )xij + fi yi (3.24)
j ∈V2

subject to

dj xij qi yi , (3.25)
j ∈V2
j ∈ V2 ,
xij
0 1, (3.26)
yi ∈ {0, 1}, (3.27)

and then by setting

LBi (») ’
LBCPL (») = »j ,
CPL
j ∈V2
i∈V1

where LBi (») is the optimal objective function value of subproblem (3.24)“(3.27).
CPL
LBi (») can be easily determined by inspection, by observing that
CPL




TLFeBOOK
DESIGNING THE LOGISTICS NETWORK 85

• for yi = 0, Equation (3.25) implies xij = 0, for each j ∈ V2 , and therefore
LBi (») = 0;
CPL

• for yi = 1, subproblem (3.24)“(3.27) is a continuous knapsack problem (with
an objective function to be minimized); it is well known that this problem can
be solved in polynomial time by means of a ˜greedy™ procedure.

By suitably modifying the optimal solution of the Lagrangian relaxation, it is pos-
sible to construct a CPL feasible solution as follows.

Step 1. (Finding the facilities to be activated.) Let L be the list of potential facilities
i ∈ V1 sorted by nondecreasing values of LBi (»), i ∈ V1 (note that LBi (»)
CPL CPL
|V2 | ). Extract from L the minimum number of facilities capable
0, i ∈ V1 , » ∈
¯
to satisfy the total demand j ∈V2 dj . Let V1 be the set of facilities selected. Then
¯
V1 satis¬es the relation:
qi dj .
¯ j ∈V2
i∈V1


Step 2. (Customer allocation to the selected facilities.) Solve the demand allocation
¯
problem (3.16)“(3.19) considering V1 as the set of facilities to be opened. Let
UBCPL (») be the cost (3.24) associated with the optimal allocation.

The heuristic ¬rst selects the facilities characterized by the smallest LBi (»)
CPL
values and then optimally allocates the demand to them.
Thus for each set of multipliers » ∈ |V2 | , the above procedure computes both
a lower and upper bound (LBCPL (») and UBCPL (»), respectively). If these bounds
coincide, an optimal solution has been found. Otherwise, in order to determine the
multipliers corresponding to the maximum possible lower bound LBCPL (») (or at
least a satisfactory bound), the classical subgradient algorithm can be used. This
method also generates, in many cases, better upper bounds, since the feasible solutions
generated from improved lower bounds are generally less costly. Here is a schematic
description of the subgradient algorithm.

0. Set LB = ’∞, UB =
Step 0. (Initialization.) Select a tolerance value µ
∞, k = 1 and »k = 0, j ∈ V2 .
j

Step 1. (Computation of a new lower bound.) Solve the Lagrangian relaxation (3.20)“
(3.23) using »k ∈ |V2 | as a vector of multipliers. If LBCPL (»k ) > LB, set LB =
LBCPL (»k ).

Step 2. (Computation of a new upper bound.) Determine the corresponding feasible
solution. Let UBCPL (»k ) be its cost. If UBCPL (»k ) < UB, set UB = UBCPL (»k ).

Step 3. (Check of the stopping criterion.) If (UB ’ LB)/LB µ, STOP. LB and UB

represent the best upper and lower bound available for zCPL , respectively.




TLFeBOOK
86 DESIGNING THE LOGISTICS NETWORK

Table 3.3 Distances (in kilometres) between potential production plants
and markets in the Goutte problem.

Brossard Granby Sainte-Julie Sherbrooke Valley¬eld Verdun

Brossard 0.0 76.1 30.4 139.4 72.6 11.7
Granby 76.1 0.0 71.0 77.2 144.5 83.7
LaSalle 20.8 92.9 47.2 156.1 47.5 11.7
Mascouche 54.7 113.3 52.9 187.2 93.0 45.2
Montr´ al
e 13.5 85.5 28.0 148.7 67.3 9.3
Sainte-Julie 30.4 71.0 0.0 138.2 94.5 38.1
Sherbrooke 139.4 77.2 138.2 0.0 207.9 146.9
Terrebonne 47.8 106.5 46.2 180.2 86.7 38.9
Valley¬eld 72.6 144.5 94.5 207.9 0.0 63.4
Verdun 11.7 83.7 38.1 146.9 63.4 0.0

Step 4. (Updating of the Lagrangian multipliers.) Determine the subgradient of the
j th relaxed constraint,
k k
sj = xij ’ 1, j ∈ V2 ,
i∈V1
|V2 |
where xij is the solution of the Lagrangian relaxation (3.20)“(3.23) using »k ∈
k

as Lagrangian multipliers. Then set
»k+1 = »k + β k sj ,
k
j ∈ V2 , (3.28)
j
j

where β k is a suitable scalar coef¬cient. Let k = k + 1 and go back to Step 1.
This algorithm attempts to determine an µ-optimal solution, i.e. a feasible solution
with a maximum user-de¬ned deviation µ from the optimal solution.
Computational experiments have shown that the initial values of the Lagrangian
multipliers do not signi¬cantly affect the behaviour of the procedure. Hence multi-
pliers are set equal to 0 in Step 0. Formula (3.28) can be explained in the following
way. If, at the kth iteration, the left-hand side of constraint (3.10) is higher than the
right-hand side ( i∈V1 xij > 1) for a certain j ∈ V2 , the subgradient sj is positive
k k

and the corresponding Lagrangian multiplier has to be increased in order to heavily
penalize the constraint violation. Vice versa, if the left-hand side of constraint (3.10)
is lower than the right-hand side ( i∈V1 xij < 1) for a certain j ∈ V2 , the associated
k
k
subgradient sj is negative and the value of the associated multiplier must be decreased
to make the service of the unsatis¬ed demand fraction 1 ’ i∈V1 xij more attractive.
k

Finally, if the j th constraint (3.10) is satis¬ed ( i∈V1 xij = 1), the corresponding
k

multiplier is unchanged.
The term β k in Equation (3.28) is a proportionality coef¬cient de¬ned as
±(UB ’ LBCPL (»k ))
k
β= ,
|V2 | k 2
j =1 (sj )




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DESIGNING THE LOGISTICS NETWORK 87

Table 3.4 Plant operating costs (in Canadian dollars per year) and capacity
(in hectolitres per year) in the Goutte problem.

Site Fixed cost Capacity

Brossard 81 400 22 000
Granby 83 800 24 000
LaSalle 88 600 28 000
Mascouche 91 000 30 000
Montr´ al
e 79 000 20 000
Sainte-Julie 86 200 26 000
Sherbrooke 88 600 28 000
Terrebonne 91 000 30 000
Valley¬eld 79 000 20 000
Verdun 80 200 21 000


Table 3.5 Demands (in hectolitres per year) of the sales districts in the Goutte problem.

Site Demand

Brossard 14 000
Granby 10 000
Sainte-Julie 8 000
Sherbrooke 12 000
Valley¬eld 10 000
Verdun 9 000


where ± is a scalar arbitrarily chosen in the interval (0, 2]. The use of parameter
β k in Equation (3.28) limits the variations of the multipliers when the lower bound
LBCPL (»k ) approaches the current upper bound UB.
Finally, note that the {LBCPL (»k )} sequence produced by the subgradient method
does not decrease monotonically. Therefore, there could exist iterations k for which
LBCPL (»k ) < LBCPL (»k’1 ) (this explains the lower bound update in Step 1). In
practice, LBCPL (»k ) values exhibit a zigzagging pattern.
The procedure is particularly ef¬cient. Indeed, it generally requires, for µ ≈ 0.01,
only a few thousand iterations for problems with hundreds of vertices in V1 and in V2 .

Goutte is a Canadian company manufacturing and distributing soft drinks. The
¬rm has recently achieved an unexpected increase of its sales mostly because of the
launch of a new beverage which has become very popular with young consumers.
The management is now considering the opportunity of opening a new plant to which
some of the production of the other factories could be moved.




TLFeBOOK
88 DESIGNING THE LOGISTICS NETWORK


The production process makes use of water, available anywhere in Canada in an
unlimited quantity at a negligible cost, and of sugar extracts which compose a modest
percentage of the total weight of the ¬nished product. For this reason, supply costs
are negligible compared to ¬nished product distribution costs, and the transportation
costs are product independent. Table 3.3 provides the distances between potential
plants and markets.
The management has decided to undertake a preliminary analysis in order to eval-
uate the theoretical minimum cost plant con¬guration for different service levels. In
this study, the logistics system is assumed to be designed from scratch.
The annual operating costs and capacities of the potential plants are shown in
Table 3.4, while the annual demands of the sales districts are reported in Table 3.5.
The maximum distance between a potential plant and a market is successively
set equal to ∞ and 70 km. The trucks have a capacity of 150 hectolitres and a cost
(inclusive of the crew™s wages) of 0.92 Canadian dollars per kilometre.
The Goutte problem is then modelled as a CPL formulation:
Minimize
cij xij + fi y i
i∈V1 j ∈V2 i∈V1

subject to

xij = 1, j ∈ V2 ,
i∈V1

i ∈ V1 ,
dj xij qi yi ,
j ∈V2
i ∈ V1 , j ∈ V2 ,
xij 1,
0
yi ∈ {0, 1}, i ∈ V1 ,
where V1 = {Brossard, Granby, LaSalle, Mascouche, Montr´ al, Sainte-Julie, Sher-
e
brooke, Terrebonne, Valley¬eld, Verdun} and V2 = {Brossard, Granby, Sainte-Julie,
Sherbrooke, Valley¬eld, Verdun}; xij , i ∈ V1 , j ∈ V2 , is a decision variable repre-
senting the fraction of the annual demand of market j satis¬ed by plant i; yi , i ∈ V1 ,
is a binary decision variable, whose value is equal to 1 if the potential plant i is
open, 0 otherwise. Transportation costs cij , i ∈ V1 , j ∈ V2 (see Table 3.6), are
calculated through Equations (3.15), observing that trucks always travel with a full
load on the outward journey and empty on the return journey. For example, since
the distance between the potential plant located in Granby and the sales district of
Brossard is 76.1 km, the transportation cost is 2 — 76.1 — 0.92 = 140.06 Cana-
dian dollars for transporting 150 hectolitres. Since the demand of the sales district
of Brossard is 14 000 hectolitres per year, the annual transportation cost for satisfy-
ing the entire demand of the Brossard market from the potential plant of Granby is
(14 000/150) — 140.06 = 13 072.32 Canadian dollars.




TLFeBOOK
DESIGNING THE LOGISTICS NETWORK 89


If the maximum distance between a plant and a market is set equal to in¬nity, the
optimal solution leads to the opening of three production plants (located in Brossard,

Granby and Valley¬eld) and the optimal xij values, i ∈ V1 , j ∈ V2 , are reported in
Table 3.7.
The annual total logistics cost equals to 265 283.12 Canadian dollars. The optimal
solution can be found by using the Lagrangian heuristic (with ± = 1.5). The results
of the ¬rst two iterations can be summarized as follows:
»1 = 0, j = 1, . . . , 6;
j
LBCPL (»1 ) = 0;
LB = 0;
y 1 = [1, 0, 0, 0, 1, 0, 0, 0, 0, 1]T ,

corresponding to the plants located in Brossard, Montr´ al and Verdun. This choice
e
guarantees the minimum number of open plants capable of satisfying the whole
demand (63 000 hectolitres per year) of the market at the lowest cost:
UBCPL (»1 ) = 282 537.24;
UB = 282 537.24;
sj = ’1, j = 1, . . . , 6;
1

β 1 = 70 634.31;
»2 = ’70 634.31, j = 1, . . . , 6;
j

LBCPL (»2 ) = ’529 969.24;
LB = 0;
y 2 = [1, 0, 0, 0, 1, 0, 0, 0, 1, 1]T ;
UBCPL (»2 ) = 353 542.24;
UB = 282 537.24;
s 2 = [’1.00, 2.60, 9.00, ’0.92, 3.50, 8.67]T ;
β 2 = 6887.15.

If the maximum distance between a potential plant and a market is 70 km, the
optimal solution leads to the opening of four plants (located in Brossard, Granby,
Sherbrooke and Valley¬eld). The demand allocation is reported in Table 3.8.
The annual total logistics cost would be 342 784.87 Canadian dollars, with an
increase of 77 501.75 Canadian dollars compared with the ¬rst solution found.




TLFeBOOK
90 DESIGNING THE LOGISTICS NETWORK

Table 3.6 Transportation costs (in Canadian dollars per year) cij , i ∈ V1 , j ∈ V2 ,
in the Goutte problem.

Brossard Granby Sainte-Julie Sherbrooke Valley¬eld Verdun

Brossard 0.00 9 337.37 2 984.80 20 514.58 8 903.08 1 296.97
Granby 13 072.32 0.00 6 964.54 11 370.67 17 727.19 9 238.67
LaSalle 3 565.18 11 390.41 4 627.23 22 978.23 5 823.52 1 296.97
Mascouche 9 396.60 13 897.49 5 195.76 27 550.19 11 410.15 4 992.43
Montr´ al
e 2 321.51 10 482.34 2 747.91 21 888.54 8 251.63 1 030.47
Sainte-Julie 5 223.40 8 705.67 0.00 20 348.76 11 587.82 4 210.70
Sherbrooke 23 933.68 9 475.56 13 565.84 0.00 25 505.04 16 220.97
Terrebonne 8 208.20 13 068.37 4 532.48 26 531.56 10 640.26 4 299.53
Valley¬eld 12 464.31 17 727.19 9 270.25 30 606.05 0.00 7 000.07
Verdun 2 017.50 10 265.19 3 742.85 21 627.96 7 777.85 0.00



Table 3.7 Fraction of the annual demand of the sales district j ∈ V2 satis¬ed by
the production plant i ∈ V1 (no limit on the distance) in the Goutte problem.

Brossard Granby Sainte-Julie Sherbrooke Valley¬eld Verdun

Brossard 1.00 0.00 0.75 0.00 0.00 0.22
Granby 0.00 1.00 0.25 1.00 0.00 0.00
LaSalle 0.00 0.00 0.00 0.00 0.00 0.00
Mascouche 0.00 0.00 0.00 0.00 0.00 0.00
Montr´ al
e 0.00 0.00 0.00 0.00 0.00 0.00
Sainte-Julie 0.00 0.00 0.00 0.00 0.00 0.00
Sherbrooke 0.00 0.00 0.00 0.00 0.00 0.00
Terrebonne 0.00 0.00 0.00 0.00 0.00 0.00
Valley¬eld 0.00 0.00 0.00 0.00 1.00 0.78
Verdun 0.00 0.00 0.00 0.00 0.00 0.00



3.3.2 Linear transportation costs and concave piecewise linear
facility operating costs

In this subsection we show how an SESC location problem with piecewise linear and
concave facility operating costs can still be modelled as an MIP model. Indeed, such a
problem can be transformed into model (3.9)“(3.14) by introducing dummy potential
facilities and suitably de¬ning costs and capacities. For the sake of simplicity, the
topic will be illustrated in three steps.

Case A. The operating cost Fi (ui ) of a facility i ∈ V1 is given by Equation (3.7) (see




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DESIGNING THE LOGISTICS NETWORK 91

Table 3.8 Fraction of the annual demand of the sales district i ∈ V2 satis¬ed by the
production plant i ∈ V1 (maximum distance of 70 km) in the Goutte problem.

Brossard Granby Sainte-Julie Sherbrooke Valley¬eld Verdun

Brossard 1.00 0.00 1.00 0.00 0.00 0.00
Granby 0.00 1.00 0.00 0.00 0.00 0.00
LaSalle 0.00 0.00 0.00 0.00 0.00 0.00
Mascouche 0.00 0.00 0.00 0.00 0.00 0.00
Montr´ al
e 0.00 0.00 0.00 0.00 0.00 0.00
Sainte-Julie 0.00 0.00 0.00 0.00 0.00 0.00
Sherbrooke 0.00 0.00 0.00 1.00 0.00 0.00
Terrebonne 0.00 0.00 0.00 0.00 0.00 0.00
Valley¬eld 0.00 0.00 0.00 0.00 1.00 1.00
Verdun 0.00 0.00 0.00 0.00 0.00 0.00


Figure 3.5):
fi + gi ui , if ui > 0,
Fi (ui ) = i ∈ V1 ,
if ui = 0,
0,
where, using Equation (3.8), ui = dj xij .
j ∈V2
Hence, this problem can be transformed into model (3.9)“(3.14) by rewriting the
objective function (3.9) as

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