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The minisum location problem on an undirected network with continuous link demands
Comput. Opns Res. Vol. 14, No. 5, pp. 369-383, Printed in Great Britain. All rights reserved
THE MINISUM NETWORK
1987 Copyright0
LOCATION PROBLEM WITH CONTINUOUS SAMUEL
Department of Engineering-Economic
S.
0305-0548/87 1987 Pergamon
$3.00 + 0.00 Journals Ltd
ON AN UNDIRECTED LINK DEMANDS*
CHIU?
Systems, Stanford University, Stanford, CA 94305-4025, U.S.A.
(Received October 1985; revised
December 1986)
Scope and Purpose-Planners
for spatially distributed service systems must confront the problem of locating their service facilities. In this paper we address the situation in which one such facility is to be located to serve an entire region represented as a network. Traditional location models assume that users of this facility can only originate from discrete points in the region which one aggregates as nodes of a network. In reality, customers are distributed over the network in a continuous manner (e.g. demands for tow truck service, pavement repairs, residential customers along a street). In this paper, we allow arbitrarily spatial distribution ofcustomers over the entire network. The objective is the minimization of average travel time to the chosen facility location. We present a solution procedure for cyclic networks and also a specialized algorithm for acylic networks. Abstract--In the one-median problem, a facility is to be located on a network minimizing total travel distances from the facility to customer demands restricted to nodal locations. In reality, however, demands do occur on links of a network. Thus, aggregation of or restriction to nodal demands may not be a satisfactory approximation. In this paper, we generalize the one-median problem to a network with discrete nodal as well as general continuous link demands. Properties of the total travel distance, as a function of the facility’s location, are examined. We introduce an exact and a heuristic procedure to find an optimal location for the facility. An efficient algorithm is developed when the network is a tree.
INTRODUCTION
Locating a facility to provide service to spatially distributed customers has been an active area of research for the past 20 years, since the work of Hakimi [ 1,2]. Location theory essentially deals with the spatial allocation of resources in order to optimize certain objective(s). The classical objective is the minimization of total travel distance from the facility to a set of spatially dispersed customers. This is known as the median problem when the spatial topology to travel is restricted on a network, and demands for service originate solely from the nodes of the network. Hakimi [l] shows that, without loss of optimality, one can restrict the search to the set of nodes in the network. This nodal optimality result is also applicable to the p-median problem [ 11, where each nodal customer is served by the nearest of the p facilities. Researchers have extended this minisum (minimizing the sum of travel distances) problem to a plane, where demands for service are distributed probabilistically over a geometric region with travel distance determined by various metric measures [3-51 (see also Larson and Odoni [6] for an excellent tutorial on the subject). A most recent result concerns facility location in the presence of unpenetrable barriers [7,8]. In this paper, we generalize the network minisum problem to a case where one allows demands to arise discretely on the nodes as well as continuously along the links. In reality, demands do occur on links. Aggregation (or restriction) of demands to nodal locations is, quite often, not a satisfactory approximation. There are many real world applications of this generalized formulation, to name a few: tow truck service; emergency service: police, fire, ambulance; public works repair: highway, pavement; utility repair: water pipeline, gas pipeline, electric cable, telephone cable, sewage network; public and private facilities: libraries, post offices, supermarkets; and delivery services. Handler and Mirchandani [9] do formulate the p-median problem on a general network with discrete nodal and continuous link demands. On a tree network, they propose a solution procedure, which is a slight variant of the Goldman algorithm [lo] for the one-median problem. However, no *An earlier version of this paper was presented at the TlMSjORSA Conference, San Diego, Calif., 25 October, 1982. t Professor Samuel S. Chiu received his Ph.D. from MIT in Operations Research and has been with the EngineeringEconomic Systems Department at Stanford University since 1982. He is a recipient of the 1984 Presidential Young Investigator Award administered by the National Science Foundation. His research interests include quantitative methods in urban service systems, locational theory, incentive theory and the transportation of hazardous materials 369
370
SAMUEL S. CHIU
analysis is performed to characterize the behavior of the average travel time as a function of the location of the facility. Minieka [I l] implicitly considers link demands in a surrogate way. Specifically, he defines a general absolute median of a network to be a point on the network that minimizes the sum of (unweighted) distances from that point to the most distant point on each link. We believe that this is the first complete analysis to include continuous link demands in network location problems. Recent facility location results that incorporate continuous link demands on special networks appear in [12-141. We will first formulate this minisum location problem on a general network with discrete nodal as well as continuous link demands. The total (or average) travel distance D(x) is expressed in terms of the location, x, of the service facility. We investigate the properties of D(x) on a general network and then specialize the analysis to a tree. We develop exact and heuristic solution procedures to locate a minsum site on a general as well as a tree network. Numerical examples are given to illustrate procedures in the proposed algorithms.
PROBLEM
FORMULATION
Consider an undirected graph G(N, L) with node set N (INI = n) and link set L. Associated with each node HEN and link 1s (i,j)~L, we define the following quantities: We sometimes use the notation 1~ (ij) or m = (a&) to denote the dual representation of a link in L. We will denote the length of a link 1 by tand the length of (i,j) by d,. hi = fraction of demands originating from node i. fi = fraction of demands originating from link 1~ (i, j). CisNhi+CIPLfizl.
Given that a demand arises on link 1, fi(y) is a general density function for demand y E (0, 0. The integral of fJy) over the link 1 equals to one. Without loss of generality, we do not allow any non-zero probability masses in the density function. This is because we can always place a fictitious node at the point where there is a non-zero probability mass. By introducing f,(y), we have imposed an orientation on link 1.We will discuss such concerns later. When the orientation of a link 1 is well defined, we define the following quantities:
F,(x) =
sx s s
f,(~)
dy,
yfdy)
dy,
0
I
YI=
0 x
Y,(x)=
0
Y~CV)dy.
We identify d(x, y) as the shortest path distance between two points x and y on G(N, L). All travel between x and y assumes the shortest path distance. Without loss of generality, we also assume that d(i, j) = d, when (i, j) E L. If this were not true, one can always place a fictitious node on a link to satisfy this condition. This assumption simplifies our evaluation of the objective function. Our objective is to find a facility on G(N, L) such that the sum of the weighted (by probability) distances from all demands to the facility is minimized. For a facility located at x on G(N, L), the sum of weighted distances is: D(X)
3
C hid(i, x) + C J isN
ISL
s YE1
db, x)hol) dy.
Definition x* on G(N, L) is called the minisum location (or site) if D(x*) d D(x) for all x E G(N, L). Our immediate goal is to characterize the function D(x) on G(N, L).
The minisum location problem on an undirected network with continuous link demands
371
ANALYSIS
Classification of demands
For a facility located on link (a, b) = m, the shortest route between each demand point and the facility will pass through either node a or node b. When we say a facility is located at x on (a, b) we mean a point on link (a, b) at a distance x from node a. We always use (a, b) = m to denote the link in G on which the facility is being considered for possible location. We partition the entire network into a set A(a, b; x) and the complementary set B(a, b; x) as follows: A(a,b;x)=(yEG(N,L)Id(y,a)+x
and B(a, b; x) = G(N, L) - A(a, b; x).
An element in A(a, b; x) will be called an “A” demand. A “B” demand is defined correspondingly. We also define P(x, y) as a shortest path between x and y on G. In other words, the set A(a, b; x) consists of points y on G(N, L) for which a E P(x, y) for at least one P(x, y). And y E B(a, b; x) if b belongs to at least one P(x, y). When the context is clear, we will simply use the notations A and B. For nodal partitions, we have the following classifications: Definition AN(a, b; x) 5 {j E N 1d(j, a) + x < d(j, b) + r%- x}, BN(a, b; x) = N - AN(a, b; x).
We will simply write AN and BN when there is no ambiguity. Definition
A breakpoint is a point on G(N, L) at which there is a change in nodal partitions AN, and, consequently, BN. A primary region is each portion of a link between two consecutive break points. We will now identify each link 1EL by classifying the demands on 1 as “A” demands or “B” demands or a mixture of both. By virtue of the demands on link (i, j) = 1, each link can be categorized as one of the following three types: (i) AL link if both nodes i and j belong to AN; (ii) BL link if both nodes i and j belong to BN; or (iii) ABL link if one node belongs to AN and the other node belongs to BN.
Link (ij) has a special orientation, namely from node i to node j. When link (ij) is an ABL link, without loss of generality, we always assume i to be an AN node and j a BN node. In characterizing each link (i,j) = 1,we consider the following cases (note that we are evaluating the function D(x) where x is the facility being considered on link (a,b) = m): (i) i E AN and j E AN, i.e. (i, j) E AL; the length of (i, j) is E We observe that there is a point 2, on (i, j) such that for 0 6 y < Z,, the shortest route from y to x on (a, b) is via node i; and for Z, < y < L the shortest route from y to x on (a, b) is via node j. Specifically, Z, + d(i, a) + x = T- Z, + d(j, a) + x,
or Z, = (1/2)[d(j, a) - d(i, a) + fl. Note that 0 < Z, 6 1 because d(* , - ) is the shortest distance measure and thus it obeys the triangular inequality. We want to point out that the value Z, remains unchanged throughout the corresponding primary region on link (a, b). When the facility location x moves across breakpoints on (a, b), both nodes may remain AN nodes, node i may turn into a BN node, j may turn into a BN node, or both may turn into BN nodes.
372
SAMUEL
S.CHIU
(ii) i E BN and j E BN, i.e. (i, j) E BL. Exactly the same analysis applies here, except that Z, now takes on the following value: 0 6 Z, = (1/2)[d(j, b) - d(i, b) + lj < 1. A moment’s thought will convince the reader that nodes i and j remain “BN” nodes as the facility moves away from node a, across breakpoints, on link (a, b). (iii) i E AN and j E BN, i.e. (i, j) E ABL. Again, we identify a point Z, on link (i, j). However, this separation point Z, changes as x changes even within the same primary region on link (a, b). Specifically, Z, satisfies the following relationship: Z, + d(i, a) + x = r- Z, + d(j, 6) + ti - x, or Z, = (1/2)[ti + t+ d(j,b) - d(i,a) - 2x]. Also, 0 < Z, 6 rbecause j E BN =E=d(j, b) + ti - x < d(i, a) + I+ x, and ieAN*
d(i, a)+x
b)+C-x.
We see that the point Z, moves continuously as x moves within a primary region on link (a, b). In fact, Z, decreases as x increases. Therefore, more demands on link (i, j) will turn into “B” type demands continuously as would be expected. Before expressing D(x), we pause for some examples illustrating the three types of link classifications we have just introduced. Figure 1 shows link (a,b) serving as a bridge between two sub-networks, A and B. All links/nodes in A are AL/AN links/nodes and all links/nodes in B are BLJBN links/nodes. There are no ABL links for facilities located on link (a, b). Figure 2 shows link (i, j) as an ABL link when the facility is located at x on link (a, b). The length of each link is indicated next to the edge. The point Z, separates link (i, j) into two segments: for y on (i, j), 0 6 y 6 Z, demands in this segment will be served by a facility at x on (a, b) via node i and node a; for y on (i, j), Z, < y 6 2 demands are served via node j, then node b to point x. We calculate Zr = 2 - x. The implication is that more demands on link (i, j) become “B” demands as x moves from node a to node b. Figure 3 shows link (i, j) as an AL link. However, part of link (i, j) has its shortest route to node a via node i and the rest to node a via node j. Here, Z, = 2.5 on link (i, j) from node i.
Fig. 1. No ABL links associated
with locations
on (a,b).
373
The minisum location problem on an undirected network with continuous link demands
Fig. 2. Segmentation of link (i, j), an ABL link.
Fig. 3. Segmentation of link (i, j), an AL link.
Characterization of D(x) on G(N, L)
On a network with only discrete nodal demands (i.e. fi = 0 for all 1E L), the function D(x) is piecewise linear and concave when evaluated on any given link of the network (for example, see [6]). The presence of continuous link demands introduces non-linearity into the objective function, which complicates the analysis. We will now develop an algebraic expression for D(x) by considering the contribution of each type of demand (as classified in the previous section) to the objective function. Within a primary region on link (a, b) E m, we can compute the total expected travel distance, by demand type, as follows: (I) Nodal demands
C hj -
1 jsBN
jsAN
hj
x +
>
1
hjd(u, j) + 2
jeAN
hj[d(b,j) + fi] = CIX + C,.
jcBN
(2) Link demands
(a) for all 1G (i, j) E AL, 1# (a, b)
s I
4=fi
[Y + 44 a) + x]fib) dy +
[r- Y + d(j, a) + xlfiol) dy
Z,
where Z, = (1/2)[d(j, a) - d(i, a) + il. We
simplify D, as follows: D, =fr{x + d(j, a) + f+ [d(i, a) - r- d(j, a)]F,(Z,) - jr + 2jJ(Z,)},
with yl(*) and j, defined in the Problem Formulation
section.
Note that Z1 does not depend on x; the only term in D, dependent on x is fix.
,
SAMUEL S. CHIU
314
(b) I E (i, j) E BL, I # [a, b).
4=X
” + is [Y
&, b) + 61 - x]f,(y) dy +
’
0
[i-y+ d(j,
b) + m - x]f,(y) dy
,
s Zl
where ZI = (1/2)[d(j, b) - d(i, b) + r]. Simplifying D,, we obtain: D, =fi{ -X + ti + i+ d(j, b) + [d(i, b) - i- d(j, b)]F,(Z,) - y, + 2y,(z,)}. Again, D, depends on x only through -xx. (c) I - (i, j) E ABL, I # (a, b). Q=fi
[y + d(i, a) + x]fk.y) dy +
’ [i- y + d(j, b) + 6 - x]fJy) dy , s Zl
where Z, = (1/2)[+1+ i+ d(j, b) - d(i, a)] - x. Carrying out the integral in D,, we obtain: D, = fi{ [ 1 - F,(Z,)][f+
6 + d(j, b) - x] + [d(i, a) + x]F,(Z,) + 2y,(Z,) -
Y,}.
There are many x dependent terms in D,. (d) 1= (a, b) E m. Since we assume that d(a, b) = dab, we can write:
~,=f,
x [s 0
(x - ~lf,W
dy +
=fm{xFm(x) - x[l -F.(x);+
m s-
1
0, - ~.M.Y)dy y, - 2Y,(x)}.
The objective function takes on the following form: D(x)=C,X+C,+
1 IEAL l+m
D,+
c
D,+
IEBL I#m
1
D,+D,.
IEABL lfm
The following theorem establishes the fact that the curvature of D(x) is undetermined even within a primary region: Theorem
1
D(x) is neither concave nor convex over a primary region in a general undirected network with continuous link demands. Proof We will consider each component of D(x):
(1) Nodal contribution: C,x + C, is linear. (2) Link contribution from I E AL or BL: D, is linear in x. (3) Link contribution from ZEABL; 1z (i, j) # (a, b): Recalling Z, = (1/2)[m + i+ d(j, b) - d(i, a)] - x, we compute:
z =[2F,(Z,) - l]fi, and d2D, dX2 = fi( - 2f,(Z,)) d 0 * D, concave.
The minisum location problem on an undirected network with continuous link demands
(4) Link contribution
315
from I = m - (a, b):
yf
=
[27,(x) - l]fm,
and d2D ---!!! = 2f,f,(x) dx2
2 0 * D, convex.
The second derivative of D(x) is:
fi”f@,) IcABL
1 ,
where Z, = (1/2)[m + T+ d(j, b) - d(i, a)] - x. Therefore, the sign of d2D(x)/dx2 is undetermined.
0
Before going into the details of searching for the minimum of D(x) over G(N,L), we give an intuitive explanation for the phenomenon expressed in Theorem 1: even within a primary region where there is no change in the nodal partition, the “A” demands on an ABL link are “defecting” to become “B” demands continuously when the separation point Z, moves with x. This movement of Z, induces concavity in D,. Our definition of a primary region allows infinitesimal shifting of the probability mass from “A” demands to “B” demands on ABL links. This can be seen in the case of nodal-demand-only situation, where the expected travel distance is linear in a primary region and concave across a breakpoint. The contribution to D(x) from link (a, b), on which the facility is being located, however, is convex in x. This is because “B” demands are turning into “A” demands as x moves from node a to node b. The implication of this is that zeros of the first derivative of D(x) do not guarantee the achievement of local minima. Moreover, there may be many zeros of D’(x) = dD(x)/dx over a single primary region. This can be seen by examining D’(x):
D’(x)= Cl + 1 fi-
c fi+2f,LW+2
1sAL
1 f,&(Z,h IEABL
‘CL!
Note that F,(x) is monotonically non-decreasing in x, while F,(Z,), IEABL, is monotonically nonincreasing in x. Therefore, there may be many zeros of D’(x) over a single primary region, without the guarantee of local minima. The search for minimum D(x) involves, first of all, the identification of all the primary regions (or breakpoints) and then the local minima in each primary region. To identify all the breakpoints, one can use a procedure as in Berman et al. [ 151. To find the local minisum in each primary region, it involves the specification of a subroutine to find the zeros of D’(x). This procedure is exhaustive in nature and depends very much on the link density functions we are dealing with. For example, when we have uniform density of demand on each link, D(x) is quadratic over a primary region. Consequently, the search to minimize D(x) is straightforward. One can also develop specialized results for cycle graph with uniform link demands (for example, see Cavalier and Sherali [13]). Specialized results such as the “gated majority theorem” [16,17] can be easily generalized to incorporate continuous link demands. We will only concentrate our remaining effort on the case of tree networks. Before characterizing D(x) on a tree network, we introduce a heuristic to locate the minisum site on a general network. A heuristic to locate the minisum site
Recall that 7d-G) IEABL
1 .
376
SAMUEL
S.CHIU
Fig. 4. A S-node symmetric complete network with 30 breakpoints. Length of links on the perimeter is 1. Length of links inside is 1.618. Breakpoints on short links are (including nodes) (0, 0.191, 0.5, 0.809, 1). Breakpoints on long links are (0, 0.5, 0.809, 1.118, 1.618).
It is likely that D(x) is concave because the contribution to D”(x) comes from, hopefully, many A&!. links. Except for abnormally large values of f, and &,(x), we can heuristically check only the breakpoints for the minisum location. Before focusing our attention on tree networks, we wish to establish a tight upper bound on the number of breakpoints on a general network. Consider a five-node complete network in Fig. 4. There are n - 2 = 3 breakpoints on each link or a total of (n - 2)n(n - 1)/2 = 30 breakpoints. Together with live nodes, we have to compute E)(x) at 35 points in the heuristic. In the case of a symmetry odd-node (number of nodes) complete network, if the length of each link is approp~ately defined, one could have a maximum possibility of (n - 2)&z - 1),/Zbreakpoints. This upper bound is tight. Therefore, the heuristic could make as many as O(n3) evaluations of the function D(x). MINISUM
LOCATION
ON A TREE NETWORK
For a tree network, the partitioning of A and B depends only on the link on which the facility is located, and not on the exact location of x on the link. Also, there are no ABL links on a tree network. As a convention, we include link (a, b) (where the facility is located) in the set BL. We now define an orientation for each link, f, when we integrate its continuous demands along its length.
I(a) = near (link (a, b)) node of link 1. I(y) = far (from link (a, b)) node of link 1. That is, l(x), I(y) satisfy d(l(y), k) = d(l(a), k) + i; for k = a or b. Note also that on a tree network rl(j, b) = d(j, a) + ti for j E:AN, and d(j, a) = d(j, b) + 61 for j E BN. We also define m(a) = a, m(y) = b, and (a, b) = m E BL as a convention. When we integrate the link demand density, we always move from node I(a) to node Z(y).We can write D(x) as follows: D(x) = (C, + CL,)x + 2f,xF,(x) where jeAN
jcBN
- 2j7,jQx) + D,
The minisum location problem on an undirected network with continuous link demands
311
and
~0 =
C hjd(j, a) + C fi+?l+ 1 f,d(‘(a), a). jfN
IPL
lOL
We want to emphasize that AN, BN, AL and BL remain unchanged for all x on (a, b) = m in a tree network. The first result concerns convexity of D(x) on a link. Lemma I D(x) is convex for all x on a link of a tree. Proof = 2f,f,(x)
y
>, 0.
cl
The next result concerns convexity of D(x) along paths of a tree network. To facilitate discussion, we introduce the following notations and definitions pertaining to a tree network. Definition
A(j; i, k) G the set of nodes connected to j (including node j) after the removal of links (i, j) and (j, k), AL(j; i, k) E the set of links connected to node j after the removal of links (i, j) and (j, k), AT(j; i, k) = A(j; i, k)u AL(j; i, k). When there is no ambiguity, we will simply use A, AL, and AT. Along a link (i, j) of length c where a function H(x) is defined from i(x = 0) to j(x = 0, we introduce the concept of in- and out-derivatives as follows : Definition H(i);,, = lim
fw)-H(O)
6
.
d-O+
H(j){, =
lim 6-O+
H(I) - H([- 6) 6.
They are nothing more than the directional derivatives when the derivative is not well defined at the boundary. We now prove the main result for a tree. Theorem
2
D(x) is convex along any path of a tree. Proof. Since D(x) is convex on a link, we need only concern ourselves with the behavior of D(x)
across a node. Consider a path i-j-k. We associate AN’, BN’, AL!, and BL’ with link (i,j) G 1’and AN”, BN”, AL”, and BL!’ with link (j, k) E I”. The theorem is proved if along i-j-k
It is easy to show that AL!‘= AL’u ALU {1’},
and BL’= BL”u ALU (1’).
378
SAMUEL S. CHIC
Then along i-j-k, we can write: D(j);” = c; + CL; f 2J;,!, because on link (i, j), Fls(x)(,,p = 1
WX,, = c; + CL;’ + 2&F,.(x)~,=, = c; + CL;. We can show that c;=c;+2
c h,, P*A
and
Therefore, o(j);“, = c; + CL; + 25 -i-2
1
s, 3 Gus
c3
The above inequality in general holds strictly; otherwise, the subtree AT is uninteresting (i.e. with no demands) and can be disregarded. The implication of this theorem is that there is at most one direction (along a link) of movement with decreasing D(x) at each node. Figure 5 shows a legitimate path i-j-k. Figure 6 shows a contradiction [to convexity of D(x) along a path] if there is more than one direction of movement having decreasing values of D(x) out of node j. Taking advantage of the convexity of D(x) over any path of a tree, we develop an efficient algorithm to find the minisum location, This algorithm is, obviously, applicable to the traditional median problem in which only discrete nodal demands are allowed. In addition, we will introduce a procedure which parallels the Goldman algorithm [lo] applied to a tree network. We will first make several observations before introducing a search procedure for the minisum location on a tree network: 0 We can lump all mass fi (of link I) at a distance jr on link 1 from node i(g); D(x) becomes D(X)
=
1
jeAN'
hj
-
C
X +
2f,XF,(X) - I2f,Y,(X)+ C hjd(j, a) jsN+
jeBN+
where AN+, BN +, and Nf include all the fictitious nodes with mass f on link 1at a distance jjr from node i(a). We have, then, essentially a discrete-nodal-demand version of the classic Hakimi median
Fig. 5. Path i-j--k.
The minisum location problem on an undirected network with continuous link demands
379
Fig. 6. Value of D(x) on i-j-k.
problem except for two non-linear terms. Mirchandani and Handler make a similar observation [9], except that they ignore the non-linear terms. l Another more useful observation is to lump all mass fi on node I(a) for each I # m; D(x) becomes D(x) =
c
hf - 1 hf x - 7,x + 2f,xF,(x) - 2f,j,(x) + constant terms jsBN
jeAN
where hf includes all the link masses 7, with j = I(U) (except for link (a, b) E m). If the minimum of D(x) occurs on the interior of link (a, b), then
l
D(x)’ = 0, OK
c hf -
jeAN
j;N hf -_L + GA(x) = 0,
or c
hf +_LFJx)
= c
hf +f,(1
- F,(x)) = 0.5,
jcBN
jeAN
since c
hf +,zN hf +f,=
1,
jsAN
and O
1, for x~(O,rii).
The point x is truly a median in the probabilistic sense. Intuitively, if we move the facility away from x, we are sacrificing more than half of the demands into making a longer trip to the facility. Thus, we will increase the expected travel time. 0 The minisum location is at node i if D(i)&, 3 0 along all links leading out of node i. We wish to make this inequality operational. We define Tk,,as the subtree connected to node k after the removal of link (i, k), and w(k 1i) as the total fraction of nodal and link weights of Tkiiplus the weight of link (i, k); i.e.
w(k14E 1 fi + C hj +.Tti,k). lCTk1,
jeTk+
Then, D(i)bt 3 0 along all links out of node i is equivalent to w(k 1i) < 0.5 for all k such that (i, k) E L. Convexity of D(x) over path of T implies that there is at most one subtree Tk,,with w(k ( i) > 0.5, at each node i for all (i, k) E L. To specify our algorithm, we define degree of a node as the number of links incident on a node. The following procedure finds the minisum location on a tree network with continuous link demands:
380
SAMUEL
S. C~J
MINISUM i, of maximum degree, say k. If k = 1, go to (3); otherwise, evaluate w(ij 1i), for j = 1,2, . . ., k. If w(ijIi)<0.5, for j= I,2 ,..., k, node i is the minisum site. If w(i, 1i) > 0.5, for some p = 1,2, . . . , k. Go to (2). (2) Update nodal weight for node i
(1) Find a node, say
w(ij 1i)-b
hi + i j=
hi,
1
j#P
delete all subtrees Tjli and links (i, ij), for 1 6 j 6 k, and j # p. Go to (1). (3) If one of the two remaining nodes has weight equalling or exceeding 0.5, it is theminisum location. Otherwise, locate the minisum site on the sole link. We still have to specify the operation in step (3), namely, to locate the minisum point on a link, say of link I from node i to node j, we have to find a point x*, such that
(i, j) G 1. With the orientation
hi + fiF,(X*) = 0.5 where nodes i and j consist of all the updated weights, or X* = F;‘[(1/~~)(0.5 - hi)]. Note that, according to MINISUM, hi < 0.5,
hi+f,~0.5
implying (l/f)(O.S - hi) > 0 and (llyJ(O.5 - hi) 6 1. x* exists because F,(x), being a cumulative probability distribution function, is monotonically non-decreasing. Also there is no discontinuity in F,(x) since no finite probability masses are allowed. Figure 7 shows a graph of F,(x) and x*. A variant of the above procedure is to pick an interior point of a link and look for a direction of decreasing D(x). Convexity of D(x) will provide a unique direction of descent. Note that one only needs to evaluate one derivative (or equivalently the aggregated weights on one side of the point being considered) each time. Another procedure is to start from an end node (a node of degree one) and follow a unique direction of descent (as guaranteed by convexity) until the minimum is reached. At each node, one may have to evaluate many directional derivatives (or equivalently aggregated
I I I
1
a x--+
X*
Fig. 7. Graph
of F,(x) and x*.
The minisum location problem on an undirected network with continuous link demands
381
and its associated
YES
) Node i is the minisum location
The minisum location is at x on (i,j) with: x= F~'[(l/?-&(l,'2-hi)]
r
v
L---j(
I
STOP
hj+fLfhi-
T Fig. 8. Flowchart of a Goldman-like procedure to locate a minisum site on a tree
weights) as in the MINISUM Procedure. We conclude this section with a flow chart (Fig. 8) showing a Goldman-like algorithm to locate a minisum site on a tree. Contrasting to a similar procedure suggested by Handler and Mirchandani, no evaluation of _jlis required. We feel that the MINISUM procedure (or its variant discussed earlier) provides new insights to this problem by exploiting convexity of D(x) over paths of a tree. The beauty of the Goldman procedure lies in its simplicity in programming effort. We make no attempt to compare these two approaches. The next section contains a ‘numerical example illustrating both the MINISUM and the Goldman-like procedures. A NUMERICAL
EXAMPLE
Figure 9(a) is a tree network with nodal weights (hj) shown next to each node, and length/weight indicator next to each link. For illustrative purposes, we assume a uniform distribution of link demands. The algorithm iterates as follows: (1) Pick node 4.
w(5/4)=0.318<
l/2
w(lj4) = 0.554 > l/2. (2) Update h, --+ 0.446. Resulting tree is in Fig. 9(b).
382
SAMUEL S.CHIU .006
.298
.12
(a)
0
(b)
0 4
0.446
(cl
(d) Fig. 9
(3) Pick node 1. w(2 11) = 0.508 > l/2 Update h, --) 0.492 [Fig. 9(c)]. (4) Pick node 2. w(l) 2) = 0.552 > l/2 Update h2 + 0.448 [Fig. 9(d)]. (5) The minisum location is on link (2,l) at a distance x from node 2, where x satisfies 0.06
X(l/2) dy -t 0.448 = 0.5 s0
or x = 1.73. Figure 10 shows the same example with a Goldman-like procedure (flowchart in Fig. 8). The selection of end nodes is in the order of: 7,3, and then 2. After node 2 is selected, a search over link (1,2) is performed to locate the minisum site [same procedure as step (5) above].
The minisum location problem on an undirected network with continuous link demands
383
(a)
.448
p---o
(b)
Fig. 10. Example: minisum location on a tree network with continuous link demands. Acknowledgements-The author wishes to thank Professor Richard C. Larson of MIT for his constant encouragement and support. Comments of an anonymous referee clarifying one assumption on triangular inequality is gratefully acknowledged. This research was supported in part by an NSF grant No. ECS-8307798.
REFERENCES 1. S. L. Hakimi, Optimal location of switching centers and the absolute centers and medians of a graph. Opns Res. 12,45& 459 (1964). 2. S. L. Hakimi, Optimal distribution of switching centers in a communication network and some related graph theoretic problems. Opns Res. 13, 462-475 (1965). 3. P. Kolesar and E. H. Blum, Square root laws for fire engine response distances. Mgmt Sci. 19, 1368-1378 (1973). 4. R. F. Love and J. G. Morris, Solving constrained multi-facility location problems involving I, distances using convex programming. Opns Res. 23, 581-587 (1975). 5. G. 0. Wesolowsky and R. F. Love, The optimal location of new facilities using rectangular distances. Opns Res. 19, 124 130 (1971). 6. R. C. Larson and A. R. Odoni, Urban Operations Research. Prentice Hall, Englewood Cliffs, N.J. (1981). 7. R. C. Larson and G. Sadiq, Facility locations with the L, metric in the presence of barriers to travel. Opns Res. 31,652-669 (1983). 8. A. R. Odoni and G. Sadiq, Two planar facility location problems with high-speed corridors and continuous demand. Reg. Sci. Urban Econ. 12, 467-484 (1982). 9. G. Y. Handler and P. B. Mirchandani, Location on Networks. MIT Press, Cambridge, Mass. (1979). 10. A. J. Goldman, Optimal center location in simple networks. Transportat. Sci. 5, 212-221 (1971). 11. E. Minieka, The centers and medians of a graph. Opns Res. 25, 641-650 (1977). 12. M. B. Brandeau, S. S. Chiu and R. Batta, Locating the two-median ofa tree network with continuous link demands. Ann. Opns Res. 6, 223-253 (1986). 13. T. M. Cavalier and H. D. Sherali, Network location problem with continuous link demands: p-medians on a chain, 2medians of a tree and l-median on isolated cycle graphs. Presented at the ORSAjTlMS Meeting, Orlando (1983). 14. H. D. Sherali and F. L. Nordai, A capacitated, balanced 2-median location-allocation problem on a tree network with continuous link demands. Department of Industrial Engineering and Operations Research Working Paper, Virginia Polytechnic Institute and State University, Blacksburg, Va (1985). 15. 0. Berman, R. C. Larson and S. S. Chiu, Optimal server location on a network operating as an M/G/l queue. Opns Res. 33, 746-771 (1985). 16. A. J. Goldman and P. R. Meyers, A domination theorem for optimal locations. Opns Res. 13, B-147 (abstract) (1965). 17. A. J. Goldman and C. J. Witzgall, A localization theorem for optimal facility placement. Transportat. Sci. 4, 40&409 (1970).
THE MINISUM NETWORK
1987 Copyright0
LOCATION PROBLEM WITH CONTINUOUS SAMUEL
Department of Engineering-Economic
S.
0305-0548/87 1987 Pergamon
$3.00 + 0.00 Journals Ltd
ON AN UNDIRECTED LINK DEMANDS*
CHIU?
Systems, Stanford University, Stanford, CA 94305-4025, U.S.A.
(Received October 1985; revised
December 1986)
Scope and Purpose-Planners
for spatially distributed service systems must confront the problem of locating their service facilities. In this paper we address the situation in which one such facility is to be located to serve an entire region represented as a network. Traditional location models assume that users of this facility can only originate from discrete points in the region which one aggregates as nodes of a network. In reality, customers are distributed over the network in a continuous manner (e.g. demands for tow truck service, pavement repairs, residential customers along a street). In this paper, we allow arbitrarily spatial distribution ofcustomers over the entire network. The objective is the minimization of average travel time to the chosen facility location. We present a solution procedure for cyclic networks and also a specialized algorithm for acylic networks. Abstract--In the one-median problem, a facility is to be located on a network minimizing total travel distances from the facility to customer demands restricted to nodal locations. In reality, however, demands do occur on links of a network. Thus, aggregation of or restriction to nodal demands may not be a satisfactory approximation. In this paper, we generalize the one-median problem to a network with discrete nodal as well as general continuous link demands. Properties of the total travel distance, as a function of the facility’s location, are examined. We introduce an exact and a heuristic procedure to find an optimal location for the facility. An efficient algorithm is developed when the network is a tree.
INTRODUCTION
Locating a facility to provide service to spatially distributed customers has been an active area of research for the past 20 years, since the work of Hakimi [ 1,2]. Location theory essentially deals with the spatial allocation of resources in order to optimize certain objective(s). The classical objective is the minimization of total travel distance from the facility to a set of spatially dispersed customers. This is known as the median problem when the spatial topology to travel is restricted on a network, and demands for service originate solely from the nodes of the network. Hakimi [l] shows that, without loss of optimality, one can restrict the search to the set of nodes in the network. This nodal optimality result is also applicable to the p-median problem [ 11, where each nodal customer is served by the nearest of the p facilities. Researchers have extended this minisum (minimizing the sum of travel distances) problem to a plane, where demands for service are distributed probabilistically over a geometric region with travel distance determined by various metric measures [3-51 (see also Larson and Odoni [6] for an excellent tutorial on the subject). A most recent result concerns facility location in the presence of unpenetrable barriers [7,8]. In this paper, we generalize the network minisum problem to a case where one allows demands to arise discretely on the nodes as well as continuously along the links. In reality, demands do occur on links. Aggregation (or restriction) of demands to nodal locations is, quite often, not a satisfactory approximation. There are many real world applications of this generalized formulation, to name a few: tow truck service; emergency service: police, fire, ambulance; public works repair: highway, pavement; utility repair: water pipeline, gas pipeline, electric cable, telephone cable, sewage network; public and private facilities: libraries, post offices, supermarkets; and delivery services. Handler and Mirchandani [9] do formulate the p-median problem on a general network with discrete nodal and continuous link demands. On a tree network, they propose a solution procedure, which is a slight variant of the Goldman algorithm [lo] for the one-median problem. However, no *An earlier version of this paper was presented at the TlMSjORSA Conference, San Diego, Calif., 25 October, 1982. t Professor Samuel S. Chiu received his Ph.D. from MIT in Operations Research and has been with the EngineeringEconomic Systems Department at Stanford University since 1982. He is a recipient of the 1984 Presidential Young Investigator Award administered by the National Science Foundation. His research interests include quantitative methods in urban service systems, locational theory, incentive theory and the transportation of hazardous materials 369
370
SAMUEL S. CHIU
analysis is performed to characterize the behavior of the average travel time as a function of the location of the facility. Minieka [I l] implicitly considers link demands in a surrogate way. Specifically, he defines a general absolute median of a network to be a point on the network that minimizes the sum of (unweighted) distances from that point to the most distant point on each link. We believe that this is the first complete analysis to include continuous link demands in network location problems. Recent facility location results that incorporate continuous link demands on special networks appear in [12-141. We will first formulate this minisum location problem on a general network with discrete nodal as well as continuous link demands. The total (or average) travel distance D(x) is expressed in terms of the location, x, of the service facility. We investigate the properties of D(x) on a general network and then specialize the analysis to a tree. We develop exact and heuristic solution procedures to locate a minsum site on a general as well as a tree network. Numerical examples are given to illustrate procedures in the proposed algorithms.
PROBLEM
FORMULATION
Consider an undirected graph G(N, L) with node set N (INI = n) and link set L. Associated with each node HEN and link 1s (i,j)~L, we define the following quantities: We sometimes use the notation 1~ (ij) or m = (a&) to denote the dual representation of a link in L. We will denote the length of a link 1 by tand the length of (i,j) by d,. hi = fraction of demands originating from node i. fi = fraction of demands originating from link 1~ (i, j). CisNhi+CIPLfizl.
Given that a demand arises on link 1, fi(y) is a general density function for demand y E (0, 0. The integral of fJy) over the link 1 equals to one. Without loss of generality, we do not allow any non-zero probability masses in the density function. This is because we can always place a fictitious node at the point where there is a non-zero probability mass. By introducing f,(y), we have imposed an orientation on link 1.We will discuss such concerns later. When the orientation of a link 1 is well defined, we define the following quantities:
F,(x) =
sx s s
f,(~)
dy,
yfdy)
dy,
0
I
YI=
0 x
Y,(x)=
0
Y~CV)dy.
We identify d(x, y) as the shortest path distance between two points x and y on G(N, L). All travel between x and y assumes the shortest path distance. Without loss of generality, we also assume that d(i, j) = d, when (i, j) E L. If this were not true, one can always place a fictitious node on a link to satisfy this condition. This assumption simplifies our evaluation of the objective function. Our objective is to find a facility on G(N, L) such that the sum of the weighted (by probability) distances from all demands to the facility is minimized. For a facility located at x on G(N, L), the sum of weighted distances is: D(X)
3
C hid(i, x) + C J isN
ISL
s YE1
db, x)hol) dy.
Definition x* on G(N, L) is called the minisum location (or site) if D(x*) d D(x) for all x E G(N, L). Our immediate goal is to characterize the function D(x) on G(N, L).
The minisum location problem on an undirected network with continuous link demands
371
ANALYSIS
Classification of demands
For a facility located on link (a, b) = m, the shortest route between each demand point and the facility will pass through either node a or node b. When we say a facility is located at x on (a, b) we mean a point on link (a, b) at a distance x from node a. We always use (a, b) = m to denote the link in G on which the facility is being considered for possible location. We partition the entire network into a set A(a, b; x) and the complementary set B(a, b; x) as follows: A(a,b;x)=(yEG(N,L)Id(y,a)+x
and B(a, b; x) = G(N, L) - A(a, b; x).
An element in A(a, b; x) will be called an “A” demand. A “B” demand is defined correspondingly. We also define P(x, y) as a shortest path between x and y on G. In other words, the set A(a, b; x) consists of points y on G(N, L) for which a E P(x, y) for at least one P(x, y). And y E B(a, b; x) if b belongs to at least one P(x, y). When the context is clear, we will simply use the notations A and B. For nodal partitions, we have the following classifications: Definition AN(a, b; x) 5 {j E N 1d(j, a) + x < d(j, b) + r%- x}, BN(a, b; x) = N - AN(a, b; x).
We will simply write AN and BN when there is no ambiguity. Definition
A breakpoint is a point on G(N, L) at which there is a change in nodal partitions AN, and, consequently, BN. A primary region is each portion of a link between two consecutive break points. We will now identify each link 1EL by classifying the demands on 1 as “A” demands or “B” demands or a mixture of both. By virtue of the demands on link (i, j) = 1, each link can be categorized as one of the following three types: (i) AL link if both nodes i and j belong to AN; (ii) BL link if both nodes i and j belong to BN; or (iii) ABL link if one node belongs to AN and the other node belongs to BN.
Link (ij) has a special orientation, namely from node i to node j. When link (ij) is an ABL link, without loss of generality, we always assume i to be an AN node and j a BN node. In characterizing each link (i,j) = 1,we consider the following cases (note that we are evaluating the function D(x) where x is the facility being considered on link (a,b) = m): (i) i E AN and j E AN, i.e. (i, j) E AL; the length of (i, j) is E We observe that there is a point 2, on (i, j) such that for 0 6 y < Z,, the shortest route from y to x on (a, b) is via node i; and for Z, < y < L the shortest route from y to x on (a, b) is via node j. Specifically, Z, + d(i, a) + x = T- Z, + d(j, a) + x,
or Z, = (1/2)[d(j, a) - d(i, a) + fl. Note that 0 < Z, 6 1 because d(* , - ) is the shortest distance measure and thus it obeys the triangular inequality. We want to point out that the value Z, remains unchanged throughout the corresponding primary region on link (a, b). When the facility location x moves across breakpoints on (a, b), both nodes may remain AN nodes, node i may turn into a BN node, j may turn into a BN node, or both may turn into BN nodes.
372
SAMUEL
S.CHIU
(ii) i E BN and j E BN, i.e. (i, j) E BL. Exactly the same analysis applies here, except that Z, now takes on the following value: 0 6 Z, = (1/2)[d(j, b) - d(i, b) + lj < 1. A moment’s thought will convince the reader that nodes i and j remain “BN” nodes as the facility moves away from node a, across breakpoints, on link (a, b). (iii) i E AN and j E BN, i.e. (i, j) E ABL. Again, we identify a point Z, on link (i, j). However, this separation point Z, changes as x changes even within the same primary region on link (a, b). Specifically, Z, satisfies the following relationship: Z, + d(i, a) + x = r- Z, + d(j, 6) + ti - x, or Z, = (1/2)[ti + t+ d(j,b) - d(i,a) - 2x]. Also, 0 < Z, 6 rbecause j E BN =E=d(j, b) + ti - x < d(i, a) + I+ x, and ieAN*
d(i, a)+x
b)+C-x.
We see that the point Z, moves continuously as x moves within a primary region on link (a, b). In fact, Z, decreases as x increases. Therefore, more demands on link (i, j) will turn into “B” type demands continuously as would be expected. Before expressing D(x), we pause for some examples illustrating the three types of link classifications we have just introduced. Figure 1 shows link (a,b) serving as a bridge between two sub-networks, A and B. All links/nodes in A are AL/AN links/nodes and all links/nodes in B are BLJBN links/nodes. There are no ABL links for facilities located on link (a, b). Figure 2 shows link (i, j) as an ABL link when the facility is located at x on link (a, b). The length of each link is indicated next to the edge. The point Z, separates link (i, j) into two segments: for y on (i, j), 0 6 y 6 Z, demands in this segment will be served by a facility at x on (a, b) via node i and node a; for y on (i, j), Z, < y 6 2 demands are served via node j, then node b to point x. We calculate Zr = 2 - x. The implication is that more demands on link (i, j) become “B” demands as x moves from node a to node b. Figure 3 shows link (i, j) as an AL link. However, part of link (i, j) has its shortest route to node a via node i and the rest to node a via node j. Here, Z, = 2.5 on link (i, j) from node i.
Fig. 1. No ABL links associated
with locations
on (a,b).
373
The minisum location problem on an undirected network with continuous link demands
Fig. 2. Segmentation of link (i, j), an ABL link.
Fig. 3. Segmentation of link (i, j), an AL link.
Characterization of D(x) on G(N, L)
On a network with only discrete nodal demands (i.e. fi = 0 for all 1E L), the function D(x) is piecewise linear and concave when evaluated on any given link of the network (for example, see [6]). The presence of continuous link demands introduces non-linearity into the objective function, which complicates the analysis. We will now develop an algebraic expression for D(x) by considering the contribution of each type of demand (as classified in the previous section) to the objective function. Within a primary region on link (a, b) E m, we can compute the total expected travel distance, by demand type, as follows: (I) Nodal demands
C hj -
1 jsBN
jsAN
hj
x +
>
1
hjd(u, j) + 2
jeAN
hj[d(b,j) + fi] = CIX + C,.
jcBN
(2) Link demands
(a) for all 1G (i, j) E AL, 1# (a, b)
s I
4=fi
[Y + 44 a) + x]fib) dy +
[r- Y + d(j, a) + xlfiol) dy
Z,
where Z, = (1/2)[d(j, a) - d(i, a) + il. We
simplify D, as follows: D, =fr{x + d(j, a) + f+ [d(i, a) - r- d(j, a)]F,(Z,) - jr + 2jJ(Z,)},
with yl(*) and j, defined in the Problem Formulation
section.
Note that Z1 does not depend on x; the only term in D, dependent on x is fix.
,
SAMUEL S. CHIU
314
(b) I E (i, j) E BL, I # [a, b).
4=X
” + is [Y
&, b) + 61 - x]f,(y) dy +
’
0
[i-y+ d(j,
b) + m - x]f,(y) dy
,
s Zl
where ZI = (1/2)[d(j, b) - d(i, b) + r]. Simplifying D,, we obtain: D, =fi{ -X + ti + i+ d(j, b) + [d(i, b) - i- d(j, b)]F,(Z,) - y, + 2y,(z,)}. Again, D, depends on x only through -xx. (c) I - (i, j) E ABL, I # (a, b). Q=fi
[y + d(i, a) + x]fk.y) dy +
’ [i- y + d(j, b) + 6 - x]fJy) dy , s Zl
where Z, = (1/2)[+1+ i+ d(j, b) - d(i, a)] - x. Carrying out the integral in D,, we obtain: D, = fi{ [ 1 - F,(Z,)][f+
6 + d(j, b) - x] + [d(i, a) + x]F,(Z,) + 2y,(Z,) -
Y,}.
There are many x dependent terms in D,. (d) 1= (a, b) E m. Since we assume that d(a, b) = dab, we can write:
~,=f,
x [s 0
(x - ~lf,W
dy +
=fm{xFm(x) - x[l -F.(x);+
m s-
1
0, - ~.M.Y)dy y, - 2Y,(x)}.
The objective function takes on the following form: D(x)=C,X+C,+
1 IEAL l+m
D,+
c
D,+
IEBL I#m
1
D,+D,.
IEABL lfm
The following theorem establishes the fact that the curvature of D(x) is undetermined even within a primary region: Theorem
1
D(x) is neither concave nor convex over a primary region in a general undirected network with continuous link demands. Proof We will consider each component of D(x):
(1) Nodal contribution: C,x + C, is linear. (2) Link contribution from I E AL or BL: D, is linear in x. (3) Link contribution from ZEABL; 1z (i, j) # (a, b): Recalling Z, = (1/2)[m + i+ d(j, b) - d(i, a)] - x, we compute:
z =[2F,(Z,) - l]fi, and d2D, dX2 = fi( - 2f,(Z,)) d 0 * D, concave.
The minisum location problem on an undirected network with continuous link demands
(4) Link contribution
315
from I = m - (a, b):
yf
=
[27,(x) - l]fm,
and d2D ---!!! = 2f,f,(x) dx2
2 0 * D, convex.
The second derivative of D(x) is:
fi”f@,) IcABL
1 ,
where Z, = (1/2)[m + T+ d(j, b) - d(i, a)] - x. Therefore, the sign of d2D(x)/dx2 is undetermined.
0
Before going into the details of searching for the minimum of D(x) over G(N,L), we give an intuitive explanation for the phenomenon expressed in Theorem 1: even within a primary region where there is no change in the nodal partition, the “A” demands on an ABL link are “defecting” to become “B” demands continuously when the separation point Z, moves with x. This movement of Z, induces concavity in D,. Our definition of a primary region allows infinitesimal shifting of the probability mass from “A” demands to “B” demands on ABL links. This can be seen in the case of nodal-demand-only situation, where the expected travel distance is linear in a primary region and concave across a breakpoint. The contribution to D(x) from link (a, b), on which the facility is being located, however, is convex in x. This is because “B” demands are turning into “A” demands as x moves from node a to node b. The implication of this is that zeros of the first derivative of D(x) do not guarantee the achievement of local minima. Moreover, there may be many zeros of D’(x) = dD(x)/dx over a single primary region. This can be seen by examining D’(x):
D’(x)= Cl + 1 fi-
c fi+2f,LW+2
1sAL
1 f,&(Z,h IEABL
‘CL!
Note that F,(x) is monotonically non-decreasing in x, while F,(Z,), IEABL, is monotonically nonincreasing in x. Therefore, there may be many zeros of D’(x) over a single primary region, without the guarantee of local minima. The search for minimum D(x) involves, first of all, the identification of all the primary regions (or breakpoints) and then the local minima in each primary region. To identify all the breakpoints, one can use a procedure as in Berman et al. [ 151. To find the local minisum in each primary region, it involves the specification of a subroutine to find the zeros of D’(x). This procedure is exhaustive in nature and depends very much on the link density functions we are dealing with. For example, when we have uniform density of demand on each link, D(x) is quadratic over a primary region. Consequently, the search to minimize D(x) is straightforward. One can also develop specialized results for cycle graph with uniform link demands (for example, see Cavalier and Sherali [13]). Specialized results such as the “gated majority theorem” [16,17] can be easily generalized to incorporate continuous link demands. We will only concentrate our remaining effort on the case of tree networks. Before characterizing D(x) on a tree network, we introduce a heuristic to locate the minisum site on a general network. A heuristic to locate the minisum site
Recall that 7d-G) IEABL
1 .
376
SAMUEL
S.CHIU
Fig. 4. A S-node symmetric complete network with 30 breakpoints. Length of links on the perimeter is 1. Length of links inside is 1.618. Breakpoints on short links are (including nodes) (0, 0.191, 0.5, 0.809, 1). Breakpoints on long links are (0, 0.5, 0.809, 1.118, 1.618).
It is likely that D(x) is concave because the contribution to D”(x) comes from, hopefully, many A&!. links. Except for abnormally large values of f, and &,(x), we can heuristically check only the breakpoints for the minisum location. Before focusing our attention on tree networks, we wish to establish a tight upper bound on the number of breakpoints on a general network. Consider a five-node complete network in Fig. 4. There are n - 2 = 3 breakpoints on each link or a total of (n - 2)n(n - 1)/2 = 30 breakpoints. Together with live nodes, we have to compute E)(x) at 35 points in the heuristic. In the case of a symmetry odd-node (number of nodes) complete network, if the length of each link is approp~ately defined, one could have a maximum possibility of (n - 2)&z - 1),/Zbreakpoints. This upper bound is tight. Therefore, the heuristic could make as many as O(n3) evaluations of the function D(x). MINISUM
LOCATION
ON A TREE NETWORK
For a tree network, the partitioning of A and B depends only on the link on which the facility is located, and not on the exact location of x on the link. Also, there are no ABL links on a tree network. As a convention, we include link (a, b) (where the facility is located) in the set BL. We now define an orientation for each link, f, when we integrate its continuous demands along its length.
I(a) = near (link (a, b)) node of link 1. I(y) = far (from link (a, b)) node of link 1. That is, l(x), I(y) satisfy d(l(y), k) = d(l(a), k) + i; for k = a or b. Note also that on a tree network rl(j, b) = d(j, a) + ti for j E:AN, and d(j, a) = d(j, b) + 61 for j E BN. We also define m(a) = a, m(y) = b, and (a, b) = m E BL as a convention. When we integrate the link demand density, we always move from node I(a) to node Z(y).We can write D(x) as follows: D(x) = (C, + CL,)x + 2f,xF,(x) where jeAN
jcBN
- 2j7,jQx) + D,
The minisum location problem on an undirected network with continuous link demands
311
and
~0 =
C hjd(j, a) + C fi+?l+ 1 f,d(‘(a), a). jfN
IPL
lOL
We want to emphasize that AN, BN, AL and BL remain unchanged for all x on (a, b) = m in a tree network. The first result concerns convexity of D(x) on a link. Lemma I D(x) is convex for all x on a link of a tree. Proof = 2f,f,(x)
y
>, 0.
cl
The next result concerns convexity of D(x) along paths of a tree network. To facilitate discussion, we introduce the following notations and definitions pertaining to a tree network. Definition
A(j; i, k) G the set of nodes connected to j (including node j) after the removal of links (i, j) and (j, k), AL(j; i, k) E the set of links connected to node j after the removal of links (i, j) and (j, k), AT(j; i, k) = A(j; i, k)u AL(j; i, k). When there is no ambiguity, we will simply use A, AL, and AT. Along a link (i, j) of length c where a function H(x) is defined from i(x = 0) to j(x = 0, we introduce the concept of in- and out-derivatives as follows : Definition H(i);,, = lim
fw)-H(O)
6
.
d-O+
H(j){, =
lim 6-O+
H(I) - H([- 6) 6.
They are nothing more than the directional derivatives when the derivative is not well defined at the boundary. We now prove the main result for a tree. Theorem
2
D(x) is convex along any path of a tree. Proof. Since D(x) is convex on a link, we need only concern ourselves with the behavior of D(x)
across a node. Consider a path i-j-k. We associate AN’, BN’, AL!, and BL’ with link (i,j) G 1’and AN”, BN”, AL”, and BL!’ with link (j, k) E I”. The theorem is proved if along i-j-k
It is easy to show that AL!‘= AL’u ALU {1’},
and BL’= BL”u ALU (1’).
378
SAMUEL S. CHIC
Then along i-j-k, we can write: D(j);” = c; + CL; f 2J;,!, because on link (i, j), Fls(x)(,,p = 1
WX,, = c; + CL;’ + 2&F,.(x)~,=, = c; + CL;. We can show that c;=c;+2
c h,, P*A
and
Therefore, o(j);“, = c; + CL; + 25 -i-2
1
s, 3 Gus
c3
The above inequality in general holds strictly; otherwise, the subtree AT is uninteresting (i.e. with no demands) and can be disregarded. The implication of this theorem is that there is at most one direction (along a link) of movement with decreasing D(x) at each node. Figure 5 shows a legitimate path i-j-k. Figure 6 shows a contradiction [to convexity of D(x) along a path] if there is more than one direction of movement having decreasing values of D(x) out of node j. Taking advantage of the convexity of D(x) over any path of a tree, we develop an efficient algorithm to find the minisum location, This algorithm is, obviously, applicable to the traditional median problem in which only discrete nodal demands are allowed. In addition, we will introduce a procedure which parallels the Goldman algorithm [lo] applied to a tree network. We will first make several observations before introducing a search procedure for the minisum location on a tree network: 0 We can lump all mass fi (of link I) at a distance jr on link 1 from node i(g); D(x) becomes D(X)
=
1
jeAN'
hj
-
C
X +
2f,XF,(X) - I2f,Y,(X)+ C hjd(j, a) jsN+
jeBN+
where AN+, BN +, and Nf include all the fictitious nodes with mass f on link 1at a distance jjr from node i(a). We have, then, essentially a discrete-nodal-demand version of the classic Hakimi median
Fig. 5. Path i-j--k.
The minisum location problem on an undirected network with continuous link demands
379
Fig. 6. Value of D(x) on i-j-k.
problem except for two non-linear terms. Mirchandani and Handler make a similar observation [9], except that they ignore the non-linear terms. l Another more useful observation is to lump all mass fi on node I(a) for each I # m; D(x) becomes D(x) =
c
hf - 1 hf x - 7,x + 2f,xF,(x) - 2f,j,(x) + constant terms jsBN
jeAN
where hf includes all the link masses 7, with j = I(U) (except for link (a, b) E m). If the minimum of D(x) occurs on the interior of link (a, b), then
l
D(x)’ = 0, OK
c hf -
jeAN
j;N hf -_L + GA(x) = 0,
or c
hf +_LFJx)
= c
hf +f,(1
- F,(x)) = 0.5,
jcBN
jeAN
since c
hf +,zN hf +f,=
1,
jsAN
and O
1, for x~(O,rii).
The point x is truly a median in the probabilistic sense. Intuitively, if we move the facility away from x, we are sacrificing more than half of the demands into making a longer trip to the facility. Thus, we will increase the expected travel time. 0 The minisum location is at node i if D(i)&, 3 0 along all links leading out of node i. We wish to make this inequality operational. We define Tk,,as the subtree connected to node k after the removal of link (i, k), and w(k 1i) as the total fraction of nodal and link weights of Tkiiplus the weight of link (i, k); i.e.
w(k14E 1 fi + C hj +.Tti,k). lCTk1,
jeTk+
Then, D(i)bt 3 0 along all links out of node i is equivalent to w(k 1i) < 0.5 for all k such that (i, k) E L. Convexity of D(x) over path of T implies that there is at most one subtree Tk,,with w(k ( i) > 0.5, at each node i for all (i, k) E L. To specify our algorithm, we define degree of a node as the number of links incident on a node. The following procedure finds the minisum location on a tree network with continuous link demands:
380
SAMUEL
S. C~J
MINISUM i, of maximum degree, say k. If k = 1, go to (3); otherwise, evaluate w(ij 1i), for j = 1,2, . . ., k. If w(ijIi)<0.5, for j= I,2 ,..., k, node i is the minisum site. If w(i, 1i) > 0.5, for some p = 1,2, . . . , k. Go to (2). (2) Update nodal weight for node i
(1) Find a node, say
w(ij 1i)-b
hi + i j=
hi,
1
j#P
delete all subtrees Tjli and links (i, ij), for 1 6 j 6 k, and j # p. Go to (1). (3) If one of the two remaining nodes has weight equalling or exceeding 0.5, it is theminisum location. Otherwise, locate the minisum site on the sole link. We still have to specify the operation in step (3), namely, to locate the minisum point on a link, say of link I from node i to node j, we have to find a point x*, such that
(i, j) G 1. With the orientation
hi + fiF,(X*) = 0.5 where nodes i and j consist of all the updated weights, or X* = F;‘[(1/~~)(0.5 - hi)]. Note that, according to MINISUM, hi < 0.5,
hi+f,~0.5
implying (l/f)(O.S - hi) > 0 and (llyJ(O.5 - hi) 6 1. x* exists because F,(x), being a cumulative probability distribution function, is monotonically non-decreasing. Also there is no discontinuity in F,(x) since no finite probability masses are allowed. Figure 7 shows a graph of F,(x) and x*. A variant of the above procedure is to pick an interior point of a link and look for a direction of decreasing D(x). Convexity of D(x) will provide a unique direction of descent. Note that one only needs to evaluate one derivative (or equivalently the aggregated weights on one side of the point being considered) each time. Another procedure is to start from an end node (a node of degree one) and follow a unique direction of descent (as guaranteed by convexity) until the minimum is reached. At each node, one may have to evaluate many directional derivatives (or equivalently aggregated
I I I
1
a x--+
X*
Fig. 7. Graph
of F,(x) and x*.
The minisum location problem on an undirected network with continuous link demands
381
and its associated
YES
) Node i is the minisum location
The minisum location is at x on (i,j) with: x= F~'[(l/?-&(l,'2-hi)]
r
v
L---j(
I
STOP
hj+fLfhi-
T Fig. 8. Flowchart of a Goldman-like procedure to locate a minisum site on a tree
weights) as in the MINISUM Procedure. We conclude this section with a flow chart (Fig. 8) showing a Goldman-like algorithm to locate a minisum site on a tree. Contrasting to a similar procedure suggested by Handler and Mirchandani, no evaluation of _jlis required. We feel that the MINISUM procedure (or its variant discussed earlier) provides new insights to this problem by exploiting convexity of D(x) over paths of a tree. The beauty of the Goldman procedure lies in its simplicity in programming effort. We make no attempt to compare these two approaches. The next section contains a ‘numerical example illustrating both the MINISUM and the Goldman-like procedures. A NUMERICAL
EXAMPLE
Figure 9(a) is a tree network with nodal weights (hj) shown next to each node, and length/weight indicator next to each link. For illustrative purposes, we assume a uniform distribution of link demands. The algorithm iterates as follows: (1) Pick node 4.
w(5/4)=0.318<
l/2
w(lj4) = 0.554 > l/2. (2) Update h, --+ 0.446. Resulting tree is in Fig. 9(b).
382
SAMUEL S.CHIU .006
.298
.12
(a)
0
(b)
0 4
0.446
(cl
(d) Fig. 9
(3) Pick node 1. w(2 11) = 0.508 > l/2 Update h, --) 0.492 [Fig. 9(c)]. (4) Pick node 2. w(l) 2) = 0.552 > l/2 Update h2 + 0.448 [Fig. 9(d)]. (5) The minisum location is on link (2,l) at a distance x from node 2, where x satisfies 0.06
X(l/2) dy -t 0.448 = 0.5 s0
or x = 1.73. Figure 10 shows the same example with a Goldman-like procedure (flowchart in Fig. 8). The selection of end nodes is in the order of: 7,3, and then 2. After node 2 is selected, a search over link (1,2) is performed to locate the minisum site [same procedure as step (5) above].
The minisum location problem on an undirected network with continuous link demands
383
(a)
.448
p---o
(b)
Fig. 10. Example: minisum location on a tree network with continuous link demands. Acknowledgements-The author wishes to thank Professor Richard C. Larson of MIT for his constant encouragement and support. Comments of an anonymous referee clarifying one assumption on triangular inequality is gratefully acknowledged. This research was supported in part by an NSF grant No. ECS-8307798.
REFERENCES 1. S. L. Hakimi, Optimal location of switching centers and the absolute centers and medians of a graph. Opns Res. 12,45& 459 (1964). 2. S. L. Hakimi, Optimal distribution of switching centers in a communication network and some related graph theoretic problems. Opns Res. 13, 462-475 (1965). 3. P. Kolesar and E. H. Blum, Square root laws for fire engine response distances. Mgmt Sci. 19, 1368-1378 (1973). 4. R. F. Love and J. G. Morris, Solving constrained multi-facility location problems involving I, distances using convex programming. Opns Res. 23, 581-587 (1975). 5. G. 0. Wesolowsky and R. F. Love, The optimal location of new facilities using rectangular distances. Opns Res. 19, 124 130 (1971). 6. R. C. Larson and A. R. Odoni, Urban Operations Research. Prentice Hall, Englewood Cliffs, N.J. (1981). 7. R. C. Larson and G. Sadiq, Facility locations with the L, metric in the presence of barriers to travel. Opns Res. 31,652-669 (1983). 8. A. R. Odoni and G. Sadiq, Two planar facility location problems with high-speed corridors and continuous demand. Reg. Sci. Urban Econ. 12, 467-484 (1982). 9. G. Y. Handler and P. B. Mirchandani, Location on Networks. MIT Press, Cambridge, Mass. (1979). 10. A. J. Goldman, Optimal center location in simple networks. Transportat. Sci. 5, 212-221 (1971). 11. E. Minieka, The centers and medians of a graph. Opns Res. 25, 641-650 (1977). 12. M. B. Brandeau, S. S. Chiu and R. Batta, Locating the two-median ofa tree network with continuous link demands. Ann. Opns Res. 6, 223-253 (1986). 13. T. M. Cavalier and H. D. Sherali, Network location problem with continuous link demands: p-medians on a chain, 2medians of a tree and l-median on isolated cycle graphs. Presented at the ORSAjTlMS Meeting, Orlando (1983). 14. H. D. Sherali and F. L. Nordai, A capacitated, balanced 2-median location-allocation problem on a tree network with continuous link demands. Department of Industrial Engineering and Operations Research Working Paper, Virginia Polytechnic Institute and State University, Blacksburg, Va (1985). 15. 0. Berman, R. C. Larson and S. S. Chiu, Optimal server location on a network operating as an M/G/l queue. Opns Res. 33, 746-771 (1985). 16. A. J. Goldman and P. R. Meyers, A domination theorem for optimal locations. Opns Res. 13, B-147 (abstract) (1965). 17. A. J. Goldman and C. J. Witzgall, A localization theorem for optimal facility placement. Transportat. Sci. 4, 40&409 (1970).