Sufficient conditions for coincidence in ℓ1 multifacility location problems

S EeEEWe[IXi(e) --Xt(e) -~- ue][

ifxEC,

I +~

otherwise,

where u (Ue)eEE is a tuple of perturbation vectors Ue E ~d. For the fl-minisum location problem, the dual capacity constraint involves the E~-norm. In such a case, problem (D) decomposes into d one-dimensional minimum cost network flow problems, which may be efficiently solved by several known algorithms (see [15, 1]). We also refer the reader to [12, 2] for an interesting cut approach solving the primal problem (P) in polynomial time. From [14, Corollary 30.5.2], it follows that the optimum in both (P) and (D) are attained. Moreover, using [14, Corollary 30.5.1] (see also [10] or [8]) we get =

Theorem 1. L e t x : ( x i ) i ~ v E C a n d u * = ( U *e)eSe E Z be feasible solutions f o r (P) and (D) respectively. Then the followin9 are equivalent: (a) x and u* are optimal solutions o f ( P ) and (D). (b) (Ue* ,Xi(e)--Xt(e) ) ~ W e ]]Xi(e)--Xt(e)]] f o r each e E E. (c) xi(e) - X~(e) E NB*(O,we)(u* ) f o r each e E E. Here NB.(O,w,)(u e* ) denotes the normal cone to the dual ball B*(O, we) at u e .

We remark that once a solution to (D) is known, the set of conditions (c) completely characterizes the optimal solutions to (P). We observe in particular that when line* tl* < We for some arc e E E, every optimal solution for (P) is such that Xt(e) = Xi(e), that is, new facilities associated with the initial and terminal nodes of arc e must be located at exactly the same place. At optimality, this "collapsing" property may force the

R. Cominetti, C Michelot/Operations Research Letters 20 (1997) 179-185

181

coincidence of a complete group of facilities creating what we shall call a cluster. As the size of the graph increases, it is useful to detect such clusters since it permits a decrease in the dimension of the primal and dual problems, as well as a reduction of the nondifferentiability in (P) induced by the coincidence of nodes.

,,eKe,K2) 2. Sufficient conditions for coincidence Fig. 2. Forces involved in the attraction cluster property.

General sufficient conditions for clustering are known [3, 6-8, 13, 16] which are applicable for location problems involving different norms and/or general distance functions. Our goal in this paper is to present a new sufficient condition for clustering which is specific for location problems involving the (1-norm. The following definition describes the structure on which it is based. Definition 1. Given two subsets of nodes V1 C V and V2 C V, we denote by E(V1, V2) the set of edges in E which have one adjacent node on each subset. The weight of E( 1:1, V2) is the quantity w( Vl, V2 ) =

We.

Theorem 2. Let us consider the multifacility location problem (P) for the fl-norm, and assume that K C V is an attraction cluster. Then, there exists an optimal location for (P) in which the positions of all nodes in K coincide. Proof. Let (£i)icvx and ( e )eEEK be primal and dual optimal solutions obtained by solving the location problem on the clustered graph GK. We will show that the location (xi)iev defined by

xi=

{£i ifi~K, ~?k i f i E K ,

eEE( Vt, V2)

A set of nodes K C V \ F is called an attraction cluster if for every partition of K into disjoint sets Kt,Kz we have w(Kb 1(2) >t min{w(Kl, K c), w(K2, K c )} where K c : = V\K. If these inequalities are strict we call K a strict attraction cluster. If we think of the weights w e as forces acting between the adjacent nodes of e, we may interpret this condition by saying that the internal forces of the cluster K are strong enough to resist the external attempts to split it into two pieces (Fig. 2). In the next proof we use the notion of clustered graph GK = ( VK, Er ) which is constructed by identifying all the nodes in the cluster K to a single super-node k, and replacing every edge e adjacent to one node in K by a new edge g adjacent to the super-node k. We remark that edges connecting two nodes in K are disregarded. Notice also that, except for very special situations, the graph Gx will contain multiple edges connecting a given node i ¢~K with the super-node k.

is an optimal solution for the location problem in the original graph G. To this end it suffices to exhibit a dual feasible solution ( u*e )e~e satisfying the optimality condition (b) of Theorem 1. For edges having at most one adjacent node in K we define the dual variables in the obvious way, namely *

Ue =

U-* e -* U~

ifeEEK, ifgcEK,

so that (b) is automatically satisfied. The remaining edges have both adjacent nodes in K and condition (b) will hold no matter how we choose ue* . Therefore, it suffices to find vector flows {u* : e E E(K,K)} which are compatible with the capacity constraints and the flow conservation equations. We construct u e one component at a time by solving a max-flow problem over an auxiliary graph. The procedure is the same for each component so we develop the argument just for one-dimensional ue*'s (d = 1).

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R. Cominetti, C. Michelot / Operations Research Letters 20 (1997) 179-185 K

of this cut is the sum of the upper capacities of the "forward" arcs (those from U to W) minus the lower capacities of the "backward" arcs (from W to U). Denoting U0 = U\{s}, Wo = W\{p} it follows that

e(U,W)=w(Uo, Wo)+

E

fk+

kEWNK +

Fig. 3. The auxiliary capacited graph Gsp.

Let us define the net incoming flow of k E K as

sk:Eu:-Euo i(e)q[K t(e)=k

i(e)=k t(e)f[K

and distinguish the nodes with surplus K + = {k E K : fk > 0} from those with deficit K - = {k E K : fk < 0}. We also denote by K ° = {k E K : fk = 0} the set of balanced nodes. We remark that the flow conservation in GK implies that )--~k~xfk = 0. If all nodes in K are balanced, that is K ° = K, then we may define ue* = 0 for all e E E(K,K) and we are done. Otherwise, we must find appropriate flows on the internal arcs e E E(K, K) in order to re-establish the equilibrium at the nodes in K. In some sense, we must "push" the surplus flow from K + towards K - by using the edges in E(K,K), while respecting the capacity constraints on the edges and the flow conservation at the nodes. To prove the existence of such an internal flow on the edges E(K,K) we build an auxiliary capacitated graph Gsp by considering the nodes in K together with two extra nodes: a source s and a sink p (Fig. 3). We connect the source s to each node k E K + by an arc with capacity interval [0,fk], and each node k E K - to the sink p by an arc with capacity interval [0, - f k ] . The edges e E E(K,K) are assigned capacity interval [-we, We]. The whole question is thus reduced to proving that the maximal flow from s to p is

m:-- ~-~ ik= E (-fk). kEK +

kEK-

We do this with the help of the max flow-min cut theorem [15]. Indeed, let (U, W) be an arbitrary spcut in Gsp separating s from p. The capacity c(U, W)

E

(-fk).

kEUNK-

We claim that e(U, W) >~m. Indeed, from the definition of m and taking into account that fk = 0 for all k E K °, this inequality is equivalent to

Z

w(Uo,Wo)>. Z kEUo

kEWo

Now observe that for each k E K we have

l/kl~< ~

w~=w({kI,X~)

eEE({k},K ¢)

from which we get

E

fk <<.w(Uo, K c)

kEUo

and

~--~ ( - A ) ~ w(Wo, XC). kEWo From these inequalities and the attraction property satisfied by K, we deduce

m:Efk=E(-fk) kEUo

kE~

<~min{w(Uo, KC),w(Wo,KC)} <~w(Uo, Wo) which proves our claim. To conclude we remark that the capacities of the cuts ({s},K U {p}) and also ({s} UK, {p}) attain the lower bound m. Hence the minimum cut, and afortiori the maximal flow from s to p, is exactly m as was to be proved. [] We observe that the attraction cluster technique may be applied recursively, that is to say, once an attraction cluster K is found and after reducing the problem to the clustered graph GK, a new attraction cluster may appear allowing a new reduction of the problem and so on. The proof of the above theorem gives us in fact a constructive way to recover a dual optimal solution for the original problem from the optimal solution on

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R. Cominetti, C Michelot / Operations Research Letters 20 (1997) 179-185

aI

a2

al

a2

LT~ --N~X[[]a4

a3

a4

l+~+g

a3 [ ~

Fig. 4. An examplewith a strict attractioncluster. the clustered graph. From this dual optimal solution, the complete set of optimal locations may be obtained using Theorem 1. Notice also that any attraction tree T as defined in [7,8] is an attraction cluster since for each partition of K = T into disjoint sets K1,/£2 we have W(Kl, 1£2) >~w(K2, K c) (resp. w(K1,1£2) >>w(K1, K c )) as soon as the root of T is contained in K1 (resp. K2). The previous theorem shows the existence of at least one optimal solution where the nodes in K coincide. There may be however other solutions in which the nodes in K stay apart. Our next result establishes a sufficient condition for coincidence in every optimal solution. Theorem 3. Let K be an attraction cluster for the (1 version of problem (P). (a) If e E E(K,K) is such that K remains an attraction cluster when the weight we is reduced by a small amount, then in every optimal location for (P) the positions of the nodes ie and te must coincide. (b) I l K is a strict attraction cluster then in every optimal location for (P) all the positions of the nodes in K must coincide. Proof. We apply the previous theorem to a perturbed location problem (P~) where we is replaced by We -for a sufficiently small e > 0 so that K remains an attraction cluster. Let (x~)i~v be an optimal solution to (P~) in which the positions of the nodes in K coincide and let u* be a corresponding dual optimal solution. We observe that both solutions are also feasible for the unperturbed problem, and since the optimality condition (b) of Theorem 1 still holds, we deduce that both solutions are optimal for (P) and (D). Now, invoking condition (c) of that theorem and observing that Ue* satisfies the capacity constraint Ilue*I]* < we with

strict inequality, we deduce that Xi(e) and Xt(e) must coincide in every optimal location for (P). To prove (b) it suffices to remark that under the assumption of strict attraction we may reduce every weight We by a sufficiently small amount preserving the attraction property. Since every strict attraction cluster must be connected, the conclusion follows inductively using part (a). [] Remark. A particular case of (a) which is sometimes useful is "arc removal": if e E E(K,K) is such that K remains an attraction cluster when e is removed, then in every optimal location the positions of the nodes i(e) and t(e) must coincide. Example 2. In Fig. 4 we have represented by a square the fixed nodes (these are identified with the positions of the pre-existing facilities) and by a circle the free nodes. The set of free nodes is a strict attraction cluster so that the problem may be reduced to locating a single facility. Intuitively, the optimal location should be at the center of the square formed by the fixed facilities. To check this it suffÉces to find dual variables u l*,...,u 4. associated with the edges (k, al ),..., (k, a4), satisfying the optimality condition (b) of Theorem 1. The unique solution is given by u~ = ( 1 , - 1 ) ;

uff = ( - 1 , - 1 ) ;

u~

(1, 1);

u2 = ( - 1 , 1). Once this dual optimal solution has been found we may use condition (c) in Theorem 1 to describe the complete set of optimal locations. In this case it turns out that the optimal solutions are obtained by locating the four new facilities at the same position, which may be chosen as any point in the square defined by the pre-existing facilities.

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R. Cominetti, C. Michelot / Operations Research Letters 20 (1997) 179-185

a2 al~

a2

l+e~2l+¢z a3~ original graph

II III optimal location

II II2 optimal location

Fig. 5. A counterexample with the Euclidean norm.

Example 3. The attraction cluster technique is specific to :l-location problems and may fail for problems involving a different norm. To illustrate this fact, consider the example of Fig. 5, with three fixed nodes (facilities) placed at the vertices of an equilateral triangle. The three free nodes form a strict attraction cluster, so that their positions must coincide in every f~ optimal location. However, if we consider the location problem with distance measured in the usual euclidean norm and with e > 0 small enough, then the unique optimal solution is to locate each free facility at the position of the fixed facility to which it is attached. It should also be observed that none of the older results by Juel and Love [6], Lefebvre et al. [7, 8] and Fliege [3], yields any coincidence conditions for this example. In the definition of an attraction cluster we excluded from K the fixed nodes. As it would also be interesting to detect a set of nodes which is attracted towards a fixed node (whose position is known!), we slightly modify the notion of attraction cluster to cover this case. Obviously, such a cluster may only contain one fixed node.

Definition 2. A set K c V containing a unique fixed node is called a fixed attraction cluster for problem (P) if for every K1 C K not containing this fixed node,

w(Kl, K\KI ) >~w(K1, K c). If these inequalities are strict we shall call K a strict

fixed attraction cluster.

Theorem 4. The conclusions of Theorems 2 and 3 are still valid if we replace "attraction cluster" by "fixed attraction cluster"

~"]ikwl12k2 ~ Fig. 6. The trick for constructing the augmented graph. a2

1+¢

a4 Fig. 7. An example with a fixed attraction cluster.

Proof. The result follows from the previous theorems by using the trick of Fig. 6. We duplicate the fixed node k E K fq F leaving one copy kl as a fixed node and the other copy k2 as a free node. The edges which were adjacent to k are attached to k2 and we add a new edge linking kl and k2 with a sufficiently large weight wk,k2 ensuring that kl and k2 coincide at optimality (wk,k2 > Et(e)=k We will suffice) and ensuring also the equivalence of the fixed attraction property with the attraction property on this augmented graph (wk,k2 > w(K,K c) is enough). []

Example 4. If we add a single edge to the graph of Example 2 as shown in the figure below, the set formed by al and the free nodes becomes a fixed attraction cluster. Hence the location which puts all the free nodes at the position al is optimal (Fig. 7).

R. Cominetti, C. Michelot/ Operations Research Letters 20 (1997)179 185

3. Conclusion and open questions W e presented a sufficient condition for clustering in fl-location problems, based on the concept o f attraction cluster. To make the results interesting from a practical point o f view, it would be important to give an algorithm for detecting these clusters or at least to classify the p r o b l e m in terms o f its complexity. In this respect, it should be noted that the closely related p r o b l e m o f detecting a t t r a c t i o n trees [ 11 ] is an NPcomplete problem. A second question is how should we m o d i f y the n o t i o n o f attraction cluster to make it applicable to location problems with other norms. F r o m the point o f view o f its potential applications the cases o f the (2 and fo~ n o r m s deserve special attention. A possible starting point for further research is the following intuitive modification o f the attraction cluster: for every partition o f K into p pieces, there exists always a pair o f such pieces which resist the attempts to separate them.

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[4] R.L. Francis and A.V. Cabot, "Properties of a multifacility location problem involving Euclidean distances", Nay. Res. Logist. Quaterly 19, 335-353 (1972). [5] R.L. Francis and J.A. White, Facility Layout and Location: An Analytical Approach, Prentice-Hall, Englewood Cliffs, NJ, 1974. [6] H. Juel and R.F. Love, "Sufficient conditions for optimal facility locations to coincide", Transportation Sci. 14, 125129 (1980). [7] O. Lefebvre, C. Michelot and F. Plastria, "Geometric interpretation of the optimality conditions in multifacility location and applications", .L Optim. Theory Appl. 65, 85 101 (1990). [8] O. Lefebvre, C. Michelot and F. Plastria, "Sufficient conditions for coincidence in minisum multifacility location problems with a general metric", Oper. Res. 39, 437-442 (1991). [9] R.F. Love, J.G. Morris and G.O. Wesolowsky, Facilities Location: Models and Methods, North-Holland, New York, 1988. [10] C. Michelot,"The mathematics of continuous location", Stud. Location. Anal. 5, 59-83 (1993). [11] C. Michelot and F. Plastria, "Attraction tree detection is strongly NP-complete", Working paper, Universit6 de Paris l, 1991. [12] J.C. Picard and H.D. Ratliff, "'A cut approach to the rectilinear distance facility location problem", Oper. Res. 26, 422-433, 1978. [13] F. Plastria, "When facilities coincide: exact optimality conditions in multifacility location", £ Math. Anal. Appl. 169, 476-498 (1992). [14] R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970. [15] R.T. Rockafellar, Network Flows and Monotropic Optimization, Wiley, New York, 1984. [16] C. Witzgall, "Optimal location of a central facility: mathematical models and concepts", National Bureau of Standards Report 8388, Washington DC, 1964.