3.5 Logistics Facility Location in the Public Sector
3.5.2 The location-covering model
In the location-covering model the aim is to locate a least-cost set of facilities in such a way that each user can be reached within a maximum travel time from the closest facility. The problem can modelled on a complete graphG(V1∪V2, E), where vertices inV1and inV2represent potential facilities and customers, respectively, and each edge(i, j )∈E=V1×V2corresponds to a least-cost path betweeniandj.
112 DESIGNING THE LOGISTICS NETWORK Table 3.17 Travel time (in minutes) on the road network edges in the La Mancha problem.
(i, j ) aij (i, j ) aij (1,2) 12 (1,11) 8
(2,3) 9 (2,9) 8
(3,4) 11 (2,10) 9
(4,5) 9 (3,9) 4
(5,6) 2 (4,8) 10
(6,7) 3 (5,8) 6
(7,8) 4 (6,11) 5
(8,9) 1 (7,10) 5
(8,10) 7 (1,10) 6 (10,11) 4
Table 3.18 Travel times (in minutes)tij, i, j∈V, in the La Mancha problem.
i
j 1 2 3 4 5 6 7 8 9 10 11
1 0 12 18 23 15 13 11 13 14 6 8
2 12 0 9 19 15 16 13 9 8 9 13
3 18 9 0 11 11 12 9 5 4 12 16
4 23 19 11 0 9 11 14 10 11 17 16
5 15 15 11 9 0 2 5 6 7 10 7
6 13 16 12 11 2 0 3 7 8 8 5
7 11 13 9 14 5 3 0 4 5 5 8
8 13 9 5 10 6 7 4 0 1 7 11
9 14 8 4 11 7 8 5 1 0 8 12
10 6 9 12 17 10 8 5 7 8 0 4
11 8 13 16 16 7 5 8 11 12 4 0
Letfi, i ∈ V1, be the fixed cost of potential facilityi;pj, j ∈ V2, the penalty incurred if customerj is unserviced;tij, i ∈ V1,j ∈V2, the least-cost travel time between potential facilityiand customerj;aij, i ∈V1, j ∈ V2, a binary constant equal to 1 if potential facilityiis able to serve customerj, 0 otherwise (given a user- defined maximum timeT,aij =1, iftij ⩽T , i ∈V1, j ∈V2, otherwiseaij =0).
The decision variables are binary:yi, i ∈V1, is equal to 1 if facilityiis opened, 0 otherwise;zj, j∈V2, is equal to 1 if customerj is not served, otherwise it is 0.
The location-covering problem is modelled as follows:
Minimize
i∈V1
fiyi+
j∈V2
pjzj (3.58)
DESIGNING THE LOGISTICS NETWORK 113 Table 3.19 Distances of the local centresp∗hkfrom verticesh(in kilometres) and
maxi∈V{Ti(p∗hk)}(in minutes) in the La Mancha problem.
(h, k) γh(phk∗ ) maxi∈V{Ti(p∗hk)} (h, k) γh(p∗hk) maxi∈V{Ti(phk∗ )}
(1,2) 18.00 19.0 (1,11) 12.00 16.0
(2,3) 6.00 17.0 (2,9) 12.00 14.0
(3,4) 0.00 18.0 (2,10) 13.50 17.0
(4,5) 13.50 15.0 (3,9) 6.00 14.0
(5,6) 0.00 15.0 (4,8) 15.00 13.0
(6,7) 3.75 13.5 (5,8) 9.00 13.0
(7,8) 2.25 12.5 (6,11) 4.50 15.0
(8,9) 0.00 13.0 (7,10) 0.00 14.0
(8,10) 2.25 11.5 (1,10) 9.00 17.0
(10,11) 6.00 16.0
17
3.75 13.50
0.75 3.00 6.00 9.009.75 11.25
max{Ti∈V i(p23)}
2(p23) γ
Figure 3.13 Time maxi∈V{Ti(p23)}versus positionγ2(p23)ofp23in the La Mancha problem.
subject to
i∈V1
aijyi+zj ⩾1, j ∈V2, (3.59) yi ∈ {0,1}, i∈V1,
zj ∈ {0,1}, j ∈V2.
The objective function (3.58) is the sum of the fixed costs of the open facilities and the penalties corresponding to the unserviced customers. Constraints (3.59) impose that, for eachj ∈V2,zjis equal to 1 if the facilities opened do not cover customerj (i.e. if
i∈V1aijyi =0).
114 DESIGNING THE LOGISTICS NETWORK Table 3.20 Distances (in kilometres) between municipalities of
the consortium in Portugal (Part I).
Almada Azenha Carregosa Corroios Lavradio
Almada 0.0 24.4 33.2 4.7 29.9
Azenha 24.4 0.0 13.2 18.2 14.8
Carregosa 33.2 13.2 0.0 27.1 11.1
Corroios 4.7 18.2 27.1 0.0 23.8
Lavradio 29.9 14.8 11.1 23.8 0.0
Macau 39.0 15.4 13.9 32.9 16.1
Moita 29.5 9.5 5.1 23.4 7.4
Montijo 38.8 18.7 7.2 32.7 16.6
Palmela 34.3 24.0 16.4 28.2 26.2
Pinhal Novo 39.1 16.9 10.9 33.0 14.2
If all customers must be served (i.e. if the penaltiespj are sufficiently high for eachj ∈V2), variableszj, j ∈V2, can be assumed equal to 0. Hence, the location- covering problem is a generalization of the well-knownset coveringproblem and is therefore NP-hard.
Several variants of the location-covering model can be used in practice. For exam- ple, if fixed costsfi are equal for all potential sites i ∈ V1, it can be convenient to discriminate among all the solutions with the least number of open facilities the one corresponding to the least total travelling time, or to the most equitable demand distribution among the facilities. In the former case, letxij, i ∈ V1, j ∈ V2, be a binary decision variable equal to 1 if customerj is served by facilityi, 0 otherwise.
The problem can be modelled as follows:
Minimize
i∈V1
Myi+
i∈V1
j∈V2
tijxij (3.60)
subject to
i∈V1
aijxij ⩾1, j ∈V2, (3.61)
j∈V2
xij ⩽|V2|yi, i∈V1, (3.62)
yi ∈ {0,1}, i∈V1, (3.63)
xij ∈ {0,1}, i∈V1, j ∈V2, (3.64) whereM is a large positive constant. Constraints (3.61) guarantee that all the cus- tomersj ∈V2are serviced, while constraints (3.62) ensure that if facilityi∈V1is not set up (yi =0), then no customerj ∈V2can be served by it.
DESIGNING THE LOGISTICS NETWORK 115 Table 3.21 Distances (in kilometres) between municipalities of
the consortium in Portugal (Part II).
Macau Moita Montijo Palmela Pinhal Novo
Almada 39.0 29.5 38.8 34.3 39.1
Azenha 15.4 9.5 18.7 24.0 16.9
Carregosa 13.9 5.1 7.2 16.4 10.9
Corroios 32.9 23.4 32.7 28.2 33.0
Lavradio 16.1 7.4 16.6 26.2 14.2
Macau 0.0 9.1 12.0 6.9 1.7
Moita 9.1 0.0 10.6 11.7 7.2
Montijo 12.0 10.6 0.0 19.0 10.3
Palmela 6.9 11.7 19.0 0.0 8.8
Pinhal Novo 1.7 7.2 10.3 8.8 0.0
In Portugal, a consortium of 10 municipalities (Almada, Azenha, Carregosa, Cor- roios, Lavradio, Macau, Moita, Montijo, Palmela, Pinhal Novo), located in the neigh- bourhood of Lisbon, has decided to improve its fire-fighting service. The person responsible for the project has established that each centre of the community must be reached within 15 min from the closest fire station. Since the main aim is just to provide a first help in case of fire, the decision maker has decided to assign a single vehicle to each station. The annual cost of a station inclusive of the expenses of the personnel is €198 000. It is assumed that the average travelling speed is 60 km/h everywhere. In order to determine the optimal station location the location-covering model (3.60)–(3.64) is used, whereV1 =V2 = {Almada, Azenha, Carregosa, Cor- roios, Lavradio, Macau, Moita, Montijo, Palmela, Pinhal Novo}. Time coefficients tij, i ∈ V1, j ∈ V2, can be easily determined from the distances (in kilometres) reported in Tables 3.20 and 3.21. Coefficientsaij, i ∈ V1, j ∈ V2, were obtained from Tables 3.20 and 3.21. The minimum number of fire stations turns out to be two.
The facilities are located in Almada and Moita. The fire station located in Almada serves Corroios and Almada itself, the remaining ones are served by the fire station located in Moita.
Another interesting variant of the location-covering model arises when one must locate facilities to ensure double coverage of demand points. A classic case is ambu- lance location when users are better protected if two ambulances are located within their vicinity. If one of the two ambulances has to answer a call, there will remain one ambulance to provide coverage.