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Chapter 4
Location models
Within the Operations Research and Management Science community there is a strong interest in location analysis and modeling due to:
- location decisions are frequently made at all levels of human organizations;
- location decisions are strategic: in fact, they involve large sums of capital re- sources, with long term economic effects, in the private and in the private sectors;
- location decisions often determine economic externalities such as pollution, con- gestion... ;
- location models are often very difficult to solve (at least to optimality), so great interest is aimed toward clever formulations and efficient implementations;
- usually location models are application specific, i.e. their structural form depends on the specific application context (no general location model exists); see Table 3.1 in Z. Drezner and H. Hamacher (2004) for a list of possible applications; such a variety of applications has stimulated the interest in location modeling.
4.1 Basic facility location models
There exist eight basic location models, sharing some common characteristics:
- the underlying logistic network : it is a directed graph whose nodes represent the locations of the clients (also denoted demands) to be served by the facilities and also the locations of the existing facilities (if any): the general location problem is to locate new facilities in the logistic network by optimizing a certain objective;
- the concept of distance (or some measures related to distance such as travel time) is at the basis of the eight models. Depending on how “distance” is taken into account, in the literature the eight models are classified into:
32 4.2. Maximum distance models
- models based on maximum distance (four models);
- models based on total or average distance (four models).
4.2 Maximum distance models
In some location problems, a maximum distance exists “a priori”. For example, ele- mentary school students within a mile of their school must walk to school. As another example, some businesses guarantee a service within a pre-determined time, e.g. 20 minutes.
In the facility location terminology, a priori maximum distances are called covering or coverage distances: the demand of a client is considered fully satisfied if its nearest facility is within the coverage distance, and it is not satisfied (uncovered ) if the closest facility is beyond the coverage distance. Please observe that being closer to a facility than the coverage distance does not improve the satisfaction level of the clients.
facilities
covered client
uncovered client
coverage distance
coverage distance
Figure 4.1: Covered vs. uncovered clients
4.2.1 Set covering location model
The Set Covering location problem is to locate the minimum number of facilities required to “cover” all clients, by considering the following input data:
- I = set of clients (or demand nodes), indexed by i;
- J = set of candidate facility locations, indexed by j;
- d (^) ij = distance between i 2 I and j 2 J, 8 i 2 I, j 2 J;
- D (^) C = coverage distance;
- N (^) i = {j 2 J : d (^) ij D (^) C }, i.e. the set of all candidate facility locations that can cover i, 8 i 2 I.
Example Consider Figure 4.2, where I = { 1 , 2 , 3 , 4 }, J = { 5 , 6 }, D (^) C = 3
Observe that to locate only in 5 (node 4 is not covered) or to locate only in 6 (1 and 2 are not covered) are not feasible solutions. Therefore we have to locate both in 5 and in 6.
34 4.2. Maximum distance models
The (SCLP) model assumes that facilities can be located only at nodes of the logis- tics network (Discrete facility location). However, a lower cost could be achieved in some cases by locating also along the arcs. For example, refer to Figure 4.3. By allowing facility location along the arcs (Continuous facility location), we could locate only one facility (at ), whereas with (SCLP) we have to locate one facility at node A and one either at B or C. A possible (discrete) approach is to augment the network with a finite number of additional nodes along the arcs (i.e. enlarging set J).
A B C
DC = 10, I = A, B, C
Figure 4.3: Example of continuous facility location
4.2.2 Maximal covering location model
In many facility planning situations (e.g. school districts) there is an upper limit on the number of facilities to be opened (budget constraint); therefore, not all demand nodes can be covered.
The problem is to locate a given number of facilities, say p, in such a way as to maximize the covered demand. We augment the definitions used in (SCLP) as follows:
- input data: h (^) i is the demand at node i, 8 i 2 I, and p is the number of facilities to locate;
- decision variables:
z (^) i =
1 if node i is covered 0 otherwise
, 8 i 2 I.
ILP model
max
X
i 2 I
h (^) i z (^) i
X
j 2 J
x (^) j = p
X
j 2 N (^) i
x (^) j z (^) i , 8 i 2 I
x (^) j 2 { 0 , 1 }, 8 j 2 J
z (^) i 2 { 0 , 1 }, 8 i 2 I
(MCLP)
Chapter 4. Location models 35
The Maximal covering location problem is NP -hard. Please observe that, without loss of generality, we can replace the constraints z (^) i 2 { 0 , 1 } with z (^) i 1 , 8 i 2 I (so sparing us some of the boolean variables).
A possible extension to the Maximal covering location problem is to add constraints about closeness between demand nodes and opened facilities. So denoting by D (^) m the maximum acceptable distance of any demand node from an opened facility, and defining M (^) i = {j 2 J : d (^) ij D (^) m }, we can state the Maximal covering location problem with mandatory closeness constraints as:
(MCLP)
X
j 2 M (^) i
x (^) j 1 , 8 i 2 I closeness contraints
4.2.3 p-center model
(SCLP) and (MCLP) assume that the covering distance, D (^) C , is a fixed standard. How- ever, in many situations D (^) C is a target rather than a fixed standard. For example, in fire station planning we may want to minimize the maximum distance that a citizen has from such kind of facilities, for equity reasons.
The p-center problem (Hakimi, 1964–1965) is related to this issue. Given p facilities to be located, we want to minimize the maximum distance that a demand node is from its closest facility. Some variations of the basic model exist:
vertex p-center: facilities can be located only at the nodes;
absolute p-center: facilities can be located anywhere along the arcs;
unweighted: all demand nodes are treated equally;
weighted: distances d (^) ij are multiplied by a weight associated with demand node i.
Hereafter, we shall consider the weighted vertex p-center model. Define the following decision variables (in addition to x (^) ij ):
y (^) ij =
1 if demand node i is assigned to a facility at j 0 otherwise
, 8 i 2 I, j 2 J.
Define also w as the maximum distance between a demand node and the facility to which it is assigned. Therefore we can state the following model:
Chapter 4. Location models 37
i j
I J
d (^) ij
Figure 4.4: Bipartite graph in (SCLP), (MCLP) and (p-center) problems
I J
Figure 4.5: Bipartite graph for the (SCLP) example
where d (^) ij are expressed in minutes. Remember that N (^) i = {j 2 J : d (^) ij D (^) C }. So in this case it is N 1 = { 5 }, N 2 = N 3 = { 6 }, N 4 = { 6 , 7 }.
The ILP model is therefore the one in Model 4.1.
The optimal solution can be easily found by inspection and is depicted in Figure 4.6: x ⇤ 5 = 1, x ⇤ 6 = 1, x ⇤ 7 = 0; the objective function value is 2. Please observe that facility 7 would be closer than 6 for node 4 (10 minutes instead of 15 minutes); however, in (SCLP) the target is just to guarantee a distance D (^) C = 20 minutes.
Suppose now that, for budget reason, we can open only one facility (p = 1), and so we want to maximize the covered demand. Assume that h (^) i = 1, i = 1, 2 , 3 , 4. In such a case we get a Maximal covering location problem, and the model to be solved is the one in Model 4.2.
The new optimal solution, found by inspection, is depicted in Figure 4.7 and it is:
x ⇤ 6 = 1, x ⇤ 5 = 0, x ⇤ 7 = 0 z ⇤ 1 = 0, z ⇤ 2 = 1, z 3 ⇤ = 1, z 4 ⇤ = 1
Now, assume to have an additional information, i.e. the service is not acceptable beyond D (^) m = 25 minutes. Recall that M (^) i = {j 2 J : d (^) ij D (^) m } and so in this example
38 4.2. Maximum distance models
min x 5 + x 6 + x (^7) x 5 1 covering node 1 x 6 1 covering node 2 x 6 1 covering node 3 (redundant!) x 6 + x 7 1 covering node 4 x 5 , x 6 , x 7 2 { 0 , 1 }
Model 4.
Figure 4.6: Optimal solution for the (SCLP) and p-center examples
max z 1 + z 2 + z 3 + z (^4) x 5 z (^1) x 6 z (^2) x 6 z (^3) x 6 + x 7 z (^4) x 5 + x 6 + x 7 = 1 x 5 , x 6 , x 7 2 { 0 , 1 } z 1 , z 2 , z 3 , z 4 2 { 0 , 1 }
Model 4.
uncovered 1
Figure 4.7: Optimal solution for the (MCLP) example
40 4.2. Maximum distance models
D (^) C = 20 minutes, and is depicted in Figure 4.6:
x ⇤ 5 = 1, x ⇤ 6 = 1, x ⇤ 7 = 0 y 15 ⇤ = 1, y ⇤ 26 = 1, y ⇤ 36 = 1, y ⇤ 46 = 1, y (^) ij⇤ = 0 otherwise w ⇤^ = 20 minutes
The optimal solution for p = 1 would be to open a facility at site 6 (w ⇤^ = 25).
4.2.5 p-dispersion model
The previous models address the distance between demand nodes and new facilities; the assumption is that being close to a facility is desirable. The new problem differs from them in that it is concerned only with distances between facilities, since the objective is to maximize the minimum distance between any pair of facilities.
Possible applications of this kind of problem are siting of military installations (the separation makes them more difficult to attack) and outlet locations (separation may help storage management).
To express the objective function, we introduce a new decision variable D, that models the minimum separation distance between pairs of facilities.
Using D, the problem can be modelled as follows:
max D X
j 2 J
x (^) j = p
D + (M d (^) ij )x (^) i + (M d (^) ij )x (^) j 2 M d (^) ij , 8 i, j 2 J : i < j
x (^) j 2 { 0 , 1 }, 8 j 2 J
(p-dispersion)
where d (^) ij is the distance between i and j, and M denotes a very large value (often referred to as big M ). For example we can set M = max i,j 2 J
{d (^) ij }. Let us explain the
meaning of the complicating constraints:
- if x (^) i = x (^) j = 0 (i.e. no facility in i and j), then the constraint related to i and j becomes D 2 M d (^) ij , and therefore D + d (^) ij 2 M , which is always satisfied since both D, d (^) ij M ;
- if x (^) i = 1 and x (^) j = 0 (i.e. only one facility at i), then D + M (^) d (^) ij 2 M (^) d (^) ij , i.e. D M , which is always satisfied;
- if x (^) i = 0 and x (^) j = 1, then the same explanation of case 2 applies;
Chapter 4. Location models 41
- if x (^) i = x (^) j = 1 (i.e. this is a pair of opened facilities), then the complicating constraint related to i and j becomes
D + (^) ⇢M⇢ (^) d (^) ij + (^) ⇢M⇢ d (^) ij (^) 2 M (^) d (^) ij () D d (^) ij.
In this last case, the constraint imposes that D be a lower bound on the distance between i and j; since there is a constraint of this type for any pair of opened facilities, D must be a lower bound on the smallest inter-facility distance: maximizing D has the effect of forcing this smallest distance to be as large as possible.
Please observe that the condition i < j is added to take into account symmetries so as to avoid redundant constraints (e.g. the constraint for the pair (3, 5) is equal to the one for (5, 3); so, only one of the two constraints can be introduced within the model).
Example of p-dispersion p = 4
Consider the logistics network depicted in Figure 4.8.
p 2
p 2
p 2
p 2
Figure 4.8: Example of p-dispersion
Note that the graph is not complete: for every i, j 2 J, d (^) ij is the cost of the shortest path linking i and j; e.g. d 14 = 2
p
- The optimal solution is marked with thick nodes in the picture, and the optimum objective function value is D ⇤^ = min{ 2 , 2
p 2 } = 2. The ILP model is presented in Model 4.
Let’s explicit some constraints for the optimal solution, x ⇤ 1 = x ⇤ 2 = x ⇤ 3 = x ⇤ 4 = 1, x ⇤ 5 = 0:
- pair (1, 2) : D + (M 2) + (M 2) 2 M 2 () D 2 = d 12 ;
- pair (1, 3) : analogous to the previous one: D 2 = d 13 ;
- pair (1, 4) : D + (M 2
p
p
- 2 M 2
p 2 () D 2
p 2 = d 14 ;
- pair (1, 5) : D + (M (^) ⇢⇢
p
⇠⇠ (M
p 2)x ⇤ 5 2 M
p 2 () D M , which is always true if M is big.
Chapter 4. Location models 43
min
X
i 2 I
X
j 2 J
h (^) ij d (^) ij y (^) ij
X
j 2 J
x (^) j = p
X
j 2 J
y (^) ij = 1, 8 i 2 I
y (^) ij x (^) j , 8 i 2 I, j 2 J
x (^) j 2 { 0 , 1 } 8 j 2 J
y (^) ij 2 { 0 , 1 } 8 i 2 I, j 2 J
(p-median)
The problem is NP -hard for variable values of p.
An instance of p-median
Consider the graph depicted in Figure 4.9, and assume p = 2. Let h (^) i = 1, i = 1, 2 , 3 , 4 , and
d (^) ij =
The corresponding ILP model is presented in Model 4.6.
An optimal solution for this toy example can be found by inspection and is depicted in Figure 4.10:
x ⇤ 5 = 1, x ⇤ 7 = 1, x ⇤ 6 = 0 y ⇤ 15 = 1, y 27 ⇤ = 1, y 37 ⇤ = 1, y 47 ⇤ = 1, y ⇤ ij = 0 otherwise optimal value = minimum total distance = 50
Observe that this solution is also optimal for the p-center problem defined on the same instance: the minimum maximum distance is 20! However, (p-center) and (p-median) are not equivalent. Let us present a counterexample where an optimal solution to p-median is not necessarily optimal to p-center.
44 4.3. Total or average distance models
5 x 5 2 6 x 6 3 7 x^7 4
I J
y (^) ij
Figure 4.9: Bipartite graph for the p-median example
min 20y 15 + 25y 16 + 60y 17 + 30y 25 +
- 15y 26 + 10y 27 + 30y 35 + 15y 36 +
- 10y 37 + 30y 45 + 15y 46 + 10y (^47) x 5 + x 6 + x 7 = 2 y 15 + y 16 + y 17 = 1 y 25 + y 26 + y 27 = 1 allocation y 35 + y 36 + y 47 = 1 constraints y 45 + y 46 + y 47 = 1 y 15 x (^5) y 16 x 6 linking between location y 17 x 7 and allocation (same for the other links) .. . x 5 , x 6 , x 7 2 { 0 , 1 } y (^) ij 2 { 0 , 1 } 8 i 2 I, j 2 J
Model 4.
Figure 4.10: Optimal solution for the p-median example
46 4.3. Total or average distance models
The ILP model is the following:
min
X
j 2 J
f (^) j x (^) j + ↵
X
i 2 I
X
j 2 J
h (^) ij d (^) ij y (^) ij
X
j 2 J
y (^) ij = 1 8 i 2 I assignment constraints
y (^) ij x (^) j 8 i 2 I, j 2 J linking between location–allocation constraints X
i 2 I
h (^) i y (^) ij C (^) j x (^) j 8 j 2 J “new” constraints
x (^) j 2 { 0 , 1 } 8 j 2 J
y (^) ij 2 { 0 , 1 } 8 i 2 I, j 2 J (FCLP)
The “new” constraints are highlighted within the model. The constraint related to j can be interpreted in the following way:
- if x (^) j = 1 (i.e. locate a facility at j) then
P
i 2 J h^ i^ y^ ij^ ^ C^ j^ , that is we put an upper bound on the total demand served by j;
- if x (^) j = 0 (i.e. no facility at j) then
P
i 2 I h^ i^ y^ ij^ ^0 , that is^ y^ ij^ = 0,^8 i^2 I, so no client is assigned to j. This was already fixed by the linking constraints between location and allocation, therefore y (^) ij x (^) j , 8 i 2 I, j 2 J are redundant, and can be eliminated (although they enhance the LP relaxation).
Variants of the model
We present here two variants of the (FCLP) model:
- by replacing (relaxing) y (^) ij 2 { 0 , 1 } with 0 y (^) ij 1 , 8 i 2 I, j 2 J, we allow node i to be served (partially) by multiple facilities (no more “single-source allocation”, as imposed by (FCLP));
- if we remove the capacity constraints then we get the Uncapacitated fixed charge location problem (UFCLP): in this case each client will be served by its closest (open) facility. Allocation then becomes an easy task.
An instance of (FCLP)
Consider now the same instance of (p-median), but with different values of h (^) i (see Figure 4.12). Recall that distances are:
Chapter 4. Location models 47
d (^) ij =
Differently than (p-median), we have:
- facility capacities C (^) j (specified in Figure 4.12);
- fixed costs: f 5 = 50, f 6 = f 7 = 100;
- unit cost per demand per distance: ↵ = 1.
Observe that since the total demand is 15, we need to open at least two facilities (due to their capacities). Figure 4.13 presents an optimal solution for the example, found by inspection:
x ⇤ 5 = 1, x ⇤ 6 = 0, x ⇤ 7 = 1 y 15 ⇤ = 1, y ⇤ 25 = 1, y ⇤ 35 = 1, y ⇤ 47 = 1, yij = 0 otherwise.
The minimum cost is
z^ fixed cost }| { 100 + 50 +1 ·
z transportation cost}| { (40 + 150 + 90 + 50) = 480. Observe that nodes 2 and 3 are not served by their closest facility (as it would happen in p-median), due to capacity restrictions. The ILP model is presented in Model 4.7.
min 50x 5 + 100x 6 + 100x 7 + 2 · 20 y 15 + 2 · 25 y 16 +... y 15 + y 16 + y 17 = 1 y 25 + y 26 + y 27 = 1 y 35 + y 36 + y 37 = 1 y 45 + y 46 + y 47 = 1 2 y 15 + 5y 25 + 3y 35 + 5y 45 10 x (^5) 2 y 16 + 5y 26 + 3y 36 + 5y 46 5 x (^6) 2 y 17 + 5y 27 + 3y 37 + 5y 47 5 x (^7) x 5 , x 6 , x 7 2 { 0 , 1 } y (^) ij 2 { 0 , 1 } 8 i 2 I, j 2 J
Model 4.
4.3.3 Hub location problems
Logistics systems such as the ones related to airline networks use hub and spoke systems, in order to exploit larger capacity or faster vehicles during the delivery from an origin
Chapter 4. Location models 49
to a destination, so reducing the overall transportation cost. Figure 4.14 depicts an example of a delivery from an origin i to a destination j using two intermediate hubs. We call ↵ the discount factor for transportation between hubs.
i origin
k hub
m hub
j destination
c (^) ik ↵ · c (^) km c^ mj
Figure 4.14: Example of a delivery using intermediate hubs
The basic p-hub location problem is to locate p hubs and assign each demand node to exactly one hub, so as to minimize the total transportation cost between the origin– destination pairs. As in the previous models, J denotes the set of the candidate facility locations, where hubs can be activated. On the other hand, the set of demand nodes I induces a set of origin–destination pairs (i, j), and we know the amount of flow h (^) ij that the origin i must send to j along the network (whereas the previous models assume to know the request of each demand nodes versus a facility, and not versus other demand nodes).
The input data consist of:
- h (^) ij are the units of product (flow) to be sent from i to j, 8 i, j 2 I (note that h (^) ij = 0 if no sending is required);
- c (^) ij is the unitary transportation cost from i to j, 8 i, j 2 J;
- ↵ is the discount factor for inter-hub sending.
Consider the following decision variables:
1 if we locate a hub at j 0 otherwise
, 8 j 2 J;
1 if node i is assigned to hub j 0 otherwise
, 8 i 2 I, j 2 J.
The mathematical model is the following:
50 4.3. Total or average distance models
min
X
i 2 I
X
j 2 I
h (^) ij
X
k 2 J
c (^) ik y (^) ik +
X
m 2 J
c (^) mj y (^) jm + ↵
X
k 2 J
X
m 2 J
c (^) km
nonlinear z }| { y (^) ik y (^) jm
X
j 2 J
x (^) j = p exactly p hubs
X
j 2 J
y (^) ij = 1 8 i 2 I allocation
y (^) ij x (^) j 8 i 2 I, j 2 J linking between location–allocation
x (^) j 2 { 0 , 1 } 8 j 2 J
y (^) ij 2 { 0 , 1 } 8 i 2 I, j 2 J
(p-hub)
Two assumptions were made in formulating the problem: first, the hub portion of the logistics network is a complete graph, so the flow between any pair (i, j) will pass through at most two hubs. Second, each demand node is assigned to exactly one hub (we could relax
P
j 2 J y^ ij^ = 1, y^ ij^2 {^0 ,^1 },^8 i^2 I, j^2 J, to allow that nodes be served by more than one hub).
Some characteristics of the (p-hub) model are worth being pointed out:
- the model take into consideration node-to-node flows rather than simply demands at nodes (versus facilities);
- the model is not ILP since the objective function is nonlinear (it is quadratic);
- in the optimal solution, nodes are not necessarily assigned to the nearest hub, since the cost is measured in terms of origin–destination flows.
Example of p-hub location
Consider the network partially depicted in Figure 4.15 (the graph is in fact complete), where I = { 1 , 2 , 3 , 4 }, J = { 5 , 6 , 7 }, c (^) ij = 100, 8 i, j 2 J, i 6 = j, ↵ = 101.
Let h 13 = 5, h 24 = 15, hij = 0 otherwise. Furthermore, let c (^) ij =
For p = 2, a feasible solution is the one depicted in Figure 4.16:
x ⇤ 5 = 1, x ⇤ 6 = 1, x ⇤ 7 = 0 y 15 ⇤ = 1, y ⇤ 25 = 1, y ⇤ 36 = 1, y ⇤ 46 = 1, y (^) ij⇤ = 0 otherwise
