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 defining costs and capacities. For the sake of simplicity, the topic will be illustrated in three steps.
Case A. The operating costFi(ui)of a facilityi∈V1is given by Equation (3.7) (see
DESIGNING THE LOGISTICS NETWORK 91 Table 3.8 Fraction of the annual demand of the sales districti∈V2satisfied by the
production planti∈V1(maximum distance of 70 km) in the Goutte problem.
Brossard Granby Sainte-Julie Sherbrooke Valleyfield 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´eal 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
Valleyfield 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(ui)=
fi +giui, ifui >0,
0, ifui =0, i∈V1, where, using Equation (3.8),ui =
j∈V2djxij.
Hence, this problem can be transformed into model (3.9)–(3.14) by rewriting the objective function (3.9) as
i∈V1
j∈V2
tijxij+
i∈V1
fiyi, (3.29)
where
tij =cij +gidj, i∈V1, j∈V2. (3.30) This transformation is based on the following observations. If the sitei ∈ V1
is opened, the first term of the objective function (3.29) includes not only the transportation costcijxij,j ∈V2, but also the contributiongidjxijof the variable cost of the facilityi∈ V1; if, instead, the sitei∈ V1is not opened, the variables xij,j ∈V2, take the value 0 and that does not generate any cost.
Case B. A potential facility cannot be run economically if its activity level is lower than a valueqi−or higher than a thresholdqi+. For intermediate values, the operating cost grows linearly (see Figure 3.6). This case can be modelled as the previous case, provided that in (3.29), (3.10)–(3.14), capacity constraints (3.11) are replaced with the following relations:
qi−yi ⩽
j∈V2
djxij ⩽qi+yi, i∈V1.
92 DESIGNING THE LOGISTICS NETWORK
Fi(ui)
ui fi
Figure 3.5 Operating costFi(ui)of potential facilityi∈V1versus activity levelui(case A).
ui
qi qi
fi
+
− Fi(ui)
Figure 3.6 Operating costFi(ui)of potential facilityi∈V1versus activity levelui(case B).
Case C. The operating costFi(ui)of a potential facilityi∈V1is a general concave piecewise linear function of its activity level because of economies of scale. In the simplest case, there only are two piecewise lines (see Figure 3.7). Then
Fi(ui)=
fi+giui, ifui > ui,
fi+giui, if 0< ui ⩽ui, i∈V1,
0, ifui =0,
(3.31)
wherefi< fiandgi > gi. In order to model this problem as before, each potential facility is replaced by as many artificial facilities as the piecewise lines of its cost function. For instance, if Equation (3.31) holds, facilityi∈V1is replaced by two
DESIGNING THE LOGISTICS NETWORK 93
Fi(ui)
ui fi'
ui' fi''
Figure 3.7 Operating costFi(ui)of potential facilityi∈V1versus activity levelui(case C).
artificial facilitiesiandiwhose operating costs are characterized, respectively, by fixed costs equal tofiandfi, and by marginal costs equal togiandgi. In this way the problem belongs to case A described above since it is easy to demonstrate that in every optimal solution at most one of the artificial facilities is selected.
Logconsult is an American consulting company commissioned to propose changes to the logistics system of Gelido, a Mexican firm distributing deep-frozen food. A key aspect of the analysis is the relocation of the Gelido DCs. A preliminary examina- tion of the problem led to the identification of about|V1| =30 potential sites where warehouses can be open or already exist. In each site several different types of ware- houses can usually be installed. Here we show how the cost function of a potential facilityi∈V1can be estimated. Facility fixed costs include rent, amortization of the machinery, insurance of premises and machinery, and staff wages. They add up to
$80 000 per year. The variable costs are related to the storage and handling of goods.
Logconsult has estimated the Gelido variable costs on the basis of historical data (see Table 3.9).
Facility variable costs (see Figure 3.8) are influenced by inventory costs which generally increase with the square root of the demand (see Chapter 4 for further details). This relation suggests approximating the cost function of potential facility i ∈ V1 through Equation (3.31), where ui = 3500 hundred kilograms per year.
The values offi,fi,gi andgi can be obtained by applying linear regression (see Chapter 2) to each of the two sets of available data. This way the following relations are obtained:
fi=80 000+2252=82 252 dollars per year;
gi=18.5 dollars per hundred kilograms;
94 DESIGNING THE LOGISTICS NETWORK fi=80 000+54 400=134 400 dollars per year;
gi=4.1 dollars per hundred kilograms.
The problem can therefore be modelled as in (3.29), (3.10)–(3.14), provided that each potential facilityi∈V1is replaced by two dummy facilitiesiandiwith fixed costs equal tofiandfi, and marginal costs equal togiandgi, respectively.
By way of example, this approach is applied to a simplified version of the problem where the potential facilities are in Linares, Monclova and Monterrey, each of them having a capacity of 20 000 hundred kilograms per year, and the sales districts are concentrated in four areas, around Bustamante, Saltillo, Santa Catarina and Monte- morelos, respectively. The annual demands add up to 6200 hundred kilograms for Bustamante, 6600 hundred kilograms for Saltillo, 5800 hundred kilograms for Santa Catarina and 4400 hundred kilograms for Montemorelos. Transportation is carried out by trucks whose capacity is 10 hundred kilograms and whose cost is equal to
$0.98 per mile.
The Gelido location problem can be modelled as a CPL formulation:
Minimize
i∈V1
j∈V2
tijxij+
i∈V1
fiyi subject to
i∈V1
xij =1, j ∈V2,
j∈V2
djxij ⩽qiyi, i∈V1,
0⩽xij ⩽1, i∈V1, j ∈V2, yi ∈ {0,1}, i∈V1,
whereV1 = {Linares, Linares, Monclova, Monclova, Monterrey, Monterrey}, V2= {Bustamante, Saltillo, Santa Catarina, Montemorelos}. Linaresand Linares represent two dummy facilities which can be opened up in Linares, with a capacity equal to
qi=3500 hundred kilograms per year, qi=20 000 hundred kilograms per year,
respectively (the same goes for Monclova, Monclova, Monterreyand Monterrey);
xij, i ∈ V1, j ∈ V2, is a decision variable representing the fraction of the annual demand of sales district j satisfied by facility i; yi, i ∈ V1, is a binary decision variable, whose value is equal to 1 if the potential facilityi is open, 0 otherwise.
Costs tij, i ∈ V1, j ∈ V2, reported in Table 3.11, were obtained by means of
DESIGNING THE LOGISTICS NETWORK 95 Table 3.9 Demand entries (in hundreds of kilograms per year) versus facility variable costs
(in dollars) as reported in the past in the Logconsult problem.
Demand Variable cost 1 000 17 579 2 500 56 350 3 500 62 208 6 000 76 403 8 000 85 491 9 000 90 237 9 500 96 251 12 000 109 429 13 500 107 355 15 000 122 432 16 000 116 816 18 000 124 736
Table 3.10 Distances (in miles) between potential facilities and sales districts in the Logconsult problem.
Bustamante Saltillo Santa Catarina Montemorelos
Linares 165.0 132.5 92.7 32.4
Monclova 90.8 118.5 139.0 176.7
Monterrey 84.2 51.6 11.9 49.5
Equations (3.30), where the quantitiescij, i ∈ V1, j ∈ V2, are in turn obtained through Equation (3.15). In other words,cij = ¯cijdj, withc¯ij =(0.98×2×lij)/10, wherelij, i ∈V1, j ∈V2, represents the distance (in miles) between facilityiand marketj (see Table 3.10).
The optimal demand allocation (see Table 3.12) leads to an optimal cost equal to
$569 383.52 per year. Two facilities are located in Linares and Monterrey, with an activity level equal to 3000 hundred kilograms per year and 20 000 hundred kilograms per year, respectively (this means that the Linaresand Monterreydummy facilities are opened).