key: cord-0058685-jg5huh9i
authors: Uteshev, Alexei Yu.; Semenova, Elizaveta A.
title: Analytics of the Multifacility Weber Problem
date: 2020-08-20
journal: Computational Science and Its Applications - ICCSA 2020
DOI: 10.1007/978-3-030-58808-3_29
sha: 5c24e9d9c5c55fff7e66f4eabb2301c923855a02
doc_id: 58685
cord_uid: jg5huh9i

For the Weber problem of construction of the minimal cost planar weighted network connecting four terminals with two extra facilities, the solution by radicals is proposed. The conditions for existence of the network in the assumed topology and the explicit formulae for coordinates of the facilities are presented. It is shown that the bifacility network is less costly than the unifacility one. Extension of the results to the general Weber problem is also discussed.

The classical Weber or generalized Fermat-Torricelli problem is stated as that of finding the point (facility, junction) W = (x * , y * ) that minimizes the sum of weighted distances from itself to n ≥ 3 fixed points (terminals) {P j = (x j , y j )} n j=1 in the Euclidean plane:

Hereinafter | · | stands for the Euclidean distance and the weights {m j } n j=1 are assumed to be positive real numbers.

The treatment of the problem in the case n = 3 terminals was first undertaken in 1872 by Launhardt [4] whose interest stemmed from the evident relation to the Economic Geography problem nowadays known as Optimal Facility Location. For instance, one can be interested in minimizing transportation costs for a plant manufacturing one ton of the final product from {m j } n j=1 tons of distinct raw materials located at corresponding {P j } n j=1 . Further development of the problem was carried out in 1909 by Alfred Weber [10] . First, he suggested a different economic interpretation for the three-terminal problem. Let P 3 be a place of consumption of m 3 tons of a product produced from two different types of raw materials: m 1 tons of the first type located at P 1 and m 2 tons of the second type located at P 2 , let m 3 < m 1 + m 2 . Where is the optimal location of the production? In the course of the economic background, Weber formulated the following extension of the problem to the case of four terminals 1 .

"Let us take a simple case, an enterprise with three material deposits and one which is capable of being split, technologically speaking, into two stages. In the first stage two materials are combined into a half-finished product; in the second stage this half-finished product is combined with the third material into the final product. . . Let us suppose that possible location of the split production would be in W 1 and W 2 ; W 1 for the first stage and W 2 for the second stage. What will be the result if the splitting occurs?" [10] .

Mathematically the stated problem can be formulated as that of finding the points W 1 = (x * , y * ) and W 2 = (x * * , y * * ) which yield (2) and the weights {m j } 4 j=1 , m are treated as given positive real numbers. The general Multifacility Weber problem is stated as that of location of the given number ≥ 2 of the facility points (or, simply, facilities)

here some of the weights m ij and m ik might be zero. We will refer to this value as to the minimal cost of the network. This problem can be considered as a natural generalization of the celebrated Steiner minimal tree problem aimed at construction of the network of minimal length linking the given terminals. Dozens of papers are devoted to the Weber problem, its ramifications and applications; we refer to [3, 5, 11] for the reviews. The majority of them are concerned with the problem statement where the objective function (3) is free of the inter-facilities connections. This problem is known as the Multisource Weber problem or the -median problem. The present paper is focused on a solution to the Multifacility Weber problem. The mainstream approach in the treatment of this nonlinear optimization problem is the one based on reducing it to an appropriate iterative numerical procedure. For instance, the unifacility version of the problem (1) can be resolved via the modified Weiszfeld algorithm. The main obstacle of this approach consists in the fact that the objective (or cost) function of the Weber problem is non-differentiable at terminal points, and the iterative procedure might diverge if any of the facilities happens to lie close to a terminal (or, in case of the multifacility problem, if two facilities are about to collide).

The present paper is devoted to an alternative approach for the problem, namely an analytical one. We are looking for the conditions for existence of the network and the explicit expressions for the facility coordinates in terms of the problem parameters, i.e. terminal coordinates and weights. This approach has been originated in the recent paper [6] where the unifacility Weber problem for three terminals had been solved by radicals. Within the framework of this approach, we will focus here on solution to the planar multifacility Weber problem for the case of n = 4 terminals and = 2 facilities (i.e. the problem (2)), and also for the case of n = 5 terminals and = 3 facilities.

Our analytical treatment stems from the geometric solution to the problem originated by Georg Pick and published in the Mathematical Appendix of Weber's book [10] . Nevertheless, Pick did not provide any proof of validity for his algorithm. In the conference paper [8] the present authors have announced without a proof the claim that the Weber problem (2) is solvable by radicals. In a simplified version (and with an extra assumption missed in [8] ), this statement is now proved in Sect. 3. In addition, the conditions for the existence of the desired configuration of the network are provided.

In the case of the problem involving variable parameters, analytics provides one with a unique opportunity to evaluate their influence on the solution. In particular, it gives the means to determine the bifurcation values for these parameters, i.e. those responsible for the degeneracy of the network topology. We discuss these issues in Sect. 4 via investigation of the facilities dynamics under variation of the terminals location or the value of the involved weights. One may imagine a relevant economic optimization problem with a trawler fishing in the ocean and a floating fish processing facility drifting in anticipation of the catch transferred to it. We also prove here that, in the case of existence, the optimal bifacility network has its cost lower than the unifacility one.

In Sect. 5, we briefly discuss an opportunity for extension of the results to the case of n ≥ 5 terminals and ≥ 3 facilities. This extension is based on the reduction of the problem to a similar one with n − 1 terminals and − 1 facilities via a replacement of a pair of terminals by a suitable auxiliary phantom terminal. This trick is just a counterpart of the one utilized in Melzak's algorithm for Steiner tree construction.

Analytical solution for the three-terminal problem, i.e. for finding

is given in [6] . In the present section we assume the vertices P j = (x j , y j ) of the triangle P 1 P 2 P 3 be counted counterclockwise. Thus, the value

is two times the area of this triangle. Denote

by α 1 , α 2 , α 3 the angles of the triangle P 1 P 2 P 3 , while by β 1 , β 2 , β 3 the angles of the so-called weight triangle formed by the triple of weights

is the Heron formula for the area of the weight triangle.

Under this condition, the coordinates of the optimal facility W = (x * , y * ) are given by the formulae

with the cost of the optimal network C = √ d.

Here

providing the stationary points of the objective function 3 j=1 m j |P j W |. The theorem states that the three-terminal Weber problem is solvable by radicals. Geometric meaning of the constants appeared in this theorem is as follows: 1 2 |S| equals the area of the triangle P 1 P 2 P 3 while 1 4 √ k equals (due to Heron's formula) the area of the weight triangle.

We now formulate some technical results to be exploited later.

If the facility W is the solution to the problem (4) for some configuration P 1 P 2 P 3 m 1 m 2 m 3 then this facility remains unchanged for the configuration

with any position of the terminal P 3 in the half-line W P 3 .

The facility W lies in the segment P 3 Q 1 . For any position of the terminal P 3 , the facility W lies in the arc of the circle C 1 passing through the points P 1 , P 2 and Q 1 . Here

For any value of the weight m 3 , the optimal facility W lies in the arc of the algebraic curve of the 4th degree given by the equation

We next treat the four-terminal case.

Hereinafter we will treat the case where the terminals {P j } 4 j=1 , while counted counterclockwise, compose a convex quadrilateral P 1 P 2 P 3 P 4 .

Stationary points of the function 4 j=1 m j |W P j | are given by the system of equations

Though this system is not an algebraic one with respect to x, y, it can be reduced to this form via successive squaring of every equation. This permits one to apply the procedure of elimination of a variable via computation of the resultant. Thereby, the problem of finding the coordinates of the facility W can be reduced to that of resolving a univariate algebraic equation with coefficients polynomially depending on {m j , (x j , y j )} 4 j=1 [9] . The degree of this equation generically equals 12, it is irreducible over Z, and cannot be expected to be solvable by radicals [1] .

Assumption 2. We will assume the weights of the problem (2) to satisfy the restrictions

and

From this follows that the values

are positive. Additionally we assume the fulfillment of the following inequalities:

Theorem 5. Let Assumptions 1 and 2 be fulfilled. Set

and set the values for τ 3 , τ 4 , η 3 , η 4 via the formulae obtained by the cyclic substitution for subscripts 1 2 3 4 3 4 1 2

in the above expressions for τ 1 , τ 2 , η 1 , η 2 correspondingly. If all the values

and

are positive then there exists a pair of points W 1 and W 2 lying inside P 1 P 2 P 3 P 4 that provides the global minimum value for the function (2) . The coordinates of the optimal facility W 1 are as follows:

while those of W 2 :

The corresponding minimum value of the function (2) equals

Here

Proof. (I) We first present some directly verified relations between the values τ -s , η-s and δ-s.

(II) Consider the system of equations for determining stationary points of the objective function (2):

Let us verify the validity of (37). First establish the alternative representations for the coordinates (21) and (22):

Indeed, the difference of the right-hand sides of (21) and (41) equals

and the numerator of the involved fraction can be transformed into

The equivalence of (42) and (22) can be demonstrated in a similar manner. Now express the segment lengths:

and, similarly,

With the aid of relations (21), (41), (43) and (44) one can represent the first two terms in the left-hand side of the equality (37) as

The third summand in the equality (37) needs more laborious manipulations. We first transform its numerator:

Now write down the following modification:

Finally,

Similarly the following equality can be deduced: y * − y * * = δ (η1+η2) Δ , and both formulae yield

Therefore, the last summand of equality (37) takes the form

Summation this with (45) yields 0 and this completes the proof of (37). The validity of the remaining equalities (38)-(40) can be established in a similar way.

(III) We now deduce the formula (25) for the network cost. With the aid of the formulae (43), (44), (46) and their counterparts for the segment lengths |W 2 P 3 | and |W 2 P 4 |, one gets

For the proof of the two last statements we refer to [7] .

(IV) The facilities W 1 and W 2 providing the solution to the problem (2) lie inside the quadrilateral P 1 P 2 P 3 P 4 .

(V) The function (2) is strictly convex inside the convex (due to Assumption 1) domain given as the Cartesian product P 1 P 2 P 3 P 4 × P 1 P 2 P 3 P 4 . Therefore the solution of the system (37)-(40) provides the global minimum value for this function.

The result of Theorem 5 claims that the bifacility Weber problem for four terminals is solvable by radicals, and thus we get a natural extension of the three-terminal problem solution given in Theorem 1. An additional correlation between these two results can be watched, namely that the denominators of all the formulae for the facilities coordinates contain the explicit expression for the cost of the corresponding network. It looks like every facility "knows" the cost of the network that includes this point.

Analytical solution obtained in the previous section provides one with an opportunity to analyze the dynamics of the network under variation of the parameters of the configuration and to find the bifurcation values for these parameters, i.e. those responsible for the topology degeneracy. We first treat the cases where either the coordinates of a terminal or the corresponding weight are variated.

For any position of the terminal P 3 , the facility W 1 lies in the arc of the circle C 1 passing through the points P 1 , P 2 and Q 1 = (q 1x , q 1y ) given by the formula (8) where substitution m 3 → m is made. At the same time, the facility W 2 lies in the arc of the circle C 3 passing through the points Q 1 , P 4 and Q 3 . Here Q 3 is given by (8) where substitution

is applied to. Solution. The trajectory of P 3 does not influence those of W 1 and W 2 , i.e. both facilities do not leave the corresponding arcs for any drift of P 3 until the latter swashes the line L = Q 3 W (Fig. 1) 4 .048287) which stands for the second point of intersection of the circles C 1 and C 3 , and yields a solution to the unifacility Weber problem (1) for the configuration P 1 P 2 P 3 P 4 m 1 m 2 m 3 m 4 . When P 3 crosses the line L, the solution to the bifacility Weber problem (2) does not exist (while the unifacility counterpart (1) still possesses a solution).

The following result gives rise to an alternative geometric construction of the facility points W 1 and W 2 for the optimal network. Theorem 7. In the notation of Theorem 6, the facility W 2 lies at the point of intersection of the circle C 3 with the line Q 3 P 3 . The facility W 1 lies at the point of intersection of the circle C 1 with the line Q 1 W 2 . The minimal cost of the network equals C = m 3 | Q 3 P 3 |. (q 1x , q 1y ) be defined as in Theorem 6. For any value of the weight m 3 , the optimal facility W 1 lies in the arc of the circle C 1 . At the same time, the facility W 2 lies in the arc of the 4th degree algebraic curve passing through the points P 3 , P 4 and Q 1 . It is given by the equation

We finally treat the case of the variation of the parameter directly responsible for the inter-facilities connection; the bifurcation equation is now determined by (20). This scenario demonstrates a paradoxical phenomenon: the weight m increase forces the facilities to a collision, i.e. to a network configuration where its influence disappears completely.

When m decreases from m = 4, the facility W 1 moves towards P 1 while W 2 moves towards P 4 . The first drift is faster than the second one: W 1 approaches P 1 when m coincides with a zero of the equation δ 1 (m) = 0. The latter can be reduced to an algebraic one with a zero m 0,2 ≈ 3.145546. Theorem 9. If the optimal bifacility network exists, it is less costly than the unifacility one.

Proof. If the cost (25) is considered as the function of the configuration parameters then the following identities are valid:

The last one results in ∂C/∂m = δ/(4m 3 C). Therefore for any specialization of the weights {m j } 4 j=1 , the function C(m) increases to its maximal value at the positive zero of δ(m).

In order to extend an analytical approach developed in Sect. 3 to the multifacility Weber problem, we first demonstrate here an alternative approach for solution of the four-terminal problem (2) . It is based on the reduction of this problem to the pair of the three-terminal Weber problems. We will utilize abbreviations {4t2f } and {3t1f } for the corresponding problems.

Assume that solution for the {4t2f }-Weber problem (2) exists. Then the system of equations (37)-(40) providing the coordinates of the facilities could be split into two subsystems. Comparing equations (37) and (38) with (6) and (7) permits one to claim that the optimal facility W 1 coincides with its counterpart for the {3t1f }-Weber problem for the configuration P 1 P 2 W 2 m 1 m 2 m . A similar statement is also valid for the facility W 2 , i.e. it is the solution to the Weber problem for the configuration P 3 P 4 W 1 m 3 m 4 m . From this point of view, it looks like the four-terminal Weber problem can be reduced to the pair of the threeterminal ones. However, this reduction should be modified since the loci of the facilities W 2 or W 1 remain still undetermined. The result of Theorems 2 and 3 permits one to replace these facilities by those with known positions. This theorem claims that the four-terminal Weber problem can be solved by its reduction to the three-terminal counterpart via a formal replacement of a pair of the real terminals, say P 3 and P 4 , by a single phantom terminal Q 2 . This reduction algorithm is similar to that used for construction of the Steiner minimal tree (firstly introduced by Gergonne as early as in 1810, and 150 years later rediscovered by Melzak [2] ). The approach can be evidently extended to the general case of n ≥ 5 terminals as will be clarified by the following example. The reduction procedure illuminated in the previous example, in the general case should be accompanied by the conditions similar to those from Theorem 5.

We conclude this section with formulation of two problems for further research. The first one, for simplicity, is given in terms of the last example:

Find the pair of the weights ( m 1,3 , m 2,3 ) with the minimal possible sum m 1,3 + m 2,3 such that the corresponding optimal network contains a single facility.

The second problem consists in proving (or disproving) of the following Conjecture. The {n terminals facilities}-Weber problem (3) is solvable by radicals if = n − 2 and the valency of every facility in the network equals 3.

We provide an analytical solution to the bifacility Weber problem (2) approving thereby the geometric solution by G. Pick. We also formulate the conditions for the existence of the network in a prescribed topology and construct the solution for five terminals. Several problems for further investigations are mentioned in Section 5. One extra problem concerns the treatment of distance depending functions like F L (P ) = n j=1 m j |P P j | L with different exponents L ∈ Q. The choice L = −1 corresponds to Newton or Coulomb potential. It turns out that the stationary point sets of all the functions {F L } can be treated in the universal manner [9] . We hope to discuss these issues in forthcoming papers.

The algebraic degree of geometric optimization problems. Discrete Comput

On the history of the Euclidean Steiner tree problem

The Weber problem

Kommercielle Tracirung der Verkehrswege

Location analysis: a synthesis and survey

Analytical solution for the generalized Fermat-Torricelli problem

On the multifacility Weber problem for four terminals

Geometry and analytics of the multifacility Weber problem

Stationary points for the family of Fermat-Torricelli-Coulomb-like potential functions

Über den Standort der industrien, Tübingen. English Translation: The Theory of the Location of Industries

The Euclidean facilities location problem

To reduce the problem to the {4t2f }-case, replace a pair of the terminals P 1 and P 2 by the point Q 1 defined by the formula (8) where the substitution m 3 → m 1,3 is made.