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A general model for the undesirable single facility location problem

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A general model for the undesirable single facility location problem
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  Operations Research Letters 34 (2006) 427–436 OperationsResearchLetters www.elsevier.com/locate/orl Ageneralmodelfortheundesirablesinglefacilitylocationproblem  J.J. Saameño Rodríguez a , ∗ , C. Guerrero García a ,J. Muñoz Pérez b , E. Mérida Casermeiro a a  Dept. Matemática Aplicada, ETS Telecomunicaciones, Universidad de Málaga, Campus de Teatinos, s/n., 29071 Málaga, Spain b  Dept. L. y Ciencias de la Computación, Universidad de Málaga, Spain Received 12 August 2004; accepted 12 July 2005Available online 22 September 2005 Abstract In this paper, a finite set in which an optimal solution for a general Euclidean problem of locating an undesirable facilityin a polygonal region, is determined and can be found in polynomial time. The general problem we propose leads us, amongothers, to several well-known problems such as the  maxisum ,  maximin ,  anticentdian  or  r-anticentrum  problem.© 2005 Elsevier B.V. All rights reserved. Keywords:  Location; Undesirable facilities; Obnoxious facilities; Maximin; Maxisum 1. Introduction The facilities may be categorized in a general fash-ion as being either desirable, in which case the fa-cilities should be closer to the users, or undesirablewhen they should be far away. Many facilities pro-vide benefits or services to their users while havingan adverse effect on the people near by. Such facil-ities include chemical plants, garbage disposal sites,  This research has been supported by the Spanish Science andTechnology Ministry and FEDER Grant no. TIC2003-03067. ∗ Corresponding author.  E-mail address:  jjsaame@ctima.uma.es (J.J. SaameñoRodríguez).0167-6377/$-see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.orl.2005.07.007 nuclear power stations, plants for treatment of residualwaters, airports, etc. The growing interest in locationmodelling for undesirable facilities may be attributedto our growing concerns over the environment. In fact,any type of modern facility will have some detrimen-tal effects on the quality life, as a result of humanperceptions evidenced by the physical effects of dif-ferent forms of pollution such as air, water and noisepollution.Ingeneral,theobjectiveistolocatethefacilityasfaraway as possible from an identified set of populationcenters; the  maxisum  criterion attempts to maximizethe sum of the distances from the undesirable (ob-noxious) facility to the population centers and the op-timal facility location will always be on the boundary  428  J.J. Saameño Rodríguez et al. / Operations Research Letters 34 (2006) 427–436  of the feasible region. It is more appropriate whenan aggregate measure of quality is desired and eachpopulation center has a measurable contribution inthe objective function. The  maximin  criterion attemptsto maximize the minimum level of quality amongthe population centers and it guarantees that the opti-mal location will not be too close to any populationcenter.Using the weighted Voronoi diagram in the plane,Melachrinoudis and Smith [6] have developed aO (nm 2 )  algorithm for the Euclidean weighted max-imin problem in a polygonal region with  m  edges,being  n  the number of existing facilities. Romero etal. [12] models semi-obnoxious facility location takeninto account both the environmental impact and thetransportation costs and they have proposed a solutionmethod based on the well-known big square–smallsquare (BSSS). Brimberg and Juel [1] have describeda trajectory method for constructing an efficient fron-tier of points for a bicriteria model for locating asemi-desirable facility in the plane (the first crite-rion is used to measure the transportation costs andthe second one estimates the social or environmen-tal cost) by varying the relative weights, where theweighted sum of the two criteria is minimized. Welchand Salhi [14] have proposed three heuristics to solvethe maximin formulation for siting  p  facilities on anetwork considering a pollution dispersion equationand facility interaction. Plastria and Carrizosa [10]have proposed a bicriteria model seeking the lowestaffection of population at the highest level of protec-tion. They have developed fast polynomial algorithmsto construct the complete trade-off curve betweenboth objectives together with corresponding efficientsolutions. Fernández et al. [5] have presented a modelthat minimizes the global repulsion of the inhabitantsof the region while taking into account environmen-tal concerns which make some areas suitable for thelocation of the facility. Rodríguez-Chía et al. [11]have described the ordered Weber problem which is ageneralization of the Weber problem. They study theminimum of this function and obtain some proper-ties that they use for suggest an algorithm and otherapplications.In this paper, we study the weighted problem of locating an undesirable facility in a polygonal regionthat includes several existing population centers. Thisproblem was proposed by Saameño [13] and a charac-terization of the solution set for the unweighted modelcan be seen in Muñoz and Saameño [9].The present paper is organized as follows: Section2 states the proposed model and describes its proper-ties. Section 3 contains the results that identify the fi-nite dominating set of the polygonal region. Section 4contains some results to reduce the candidate point setin the search process for the optimal solution. Section5 describes the proposed algorithm and some compu-tational results. Finally, Section 6 contains some re-marks and conclusions. 2. Problem statement and properties The problem is defined as follows: let  S   be a closedpolygonal region of   R 2 and  { P  1 ,P  2 ,...,P  m }  a finitesubset of points of the plane  R 2 , corresponding tothe population centers. Let   k  =  (k 1 ,k 2 ,...,k m )  be agiven vector of   m  non-negative components and let w i  be the positive weight associated to  P  i , that is, theimportance given to the existing facility  P  i .Let  ·  be the Euclidean norm and, for each  x  ∈  S  ,let    be the permutation of the set M ={ 1 , 2 ,...,m } such that w  ( 1 )  x  −  P   ( 1 )   w  ( 2 )  x  −  P   ( 2 )   ···  w  (m)  x  −  P   (m)  .If we consider the objective function defined as f(x)  =  mi = 1 k i w  (i)  x  −  P   (i)  ,then our problem ( P ) is formulated asmax x ∈ S  f(x)  =  max x ∈ S m  i = 1 k i w  (i)  x  −  P   (i)  .Note that this problem includes well-known problems: •  If   k i  = 1 ,  ∀ i  ∈ M , we obtain the  maxisum  problem. •  If   k 1  =  1 and  k i  =  0 for  i > 1 then we obtain the maximin  location problem. •  If   k r  =  1 and  k i  =  0 for  i  =  r  then the problemis to maximize the  r  th closest distance between theundesirable facility and the affected centers. It is anextension of the maximin location problem calledthe  r  th quantile location problem.   J.J. Saameño Rodríguez et al. / Operations Research Letters 34 (2006) 427–436   429 •  If   k i  = 1 for  i  r  and  k i  = 0 for  i >r  then the prob-lem is to maximize the sum of distances betweenthe undesirable facility and the  r   closest affectedcenters. It is an extension of the maxisum locationproblem called the  r  -anticentrum problem. •  If   k 1 = 1 and  k i  =  ,  ∀ i > 1, we obtain the called   -anticentdian problem (see [8] or [2]): maximize the following weighted objective function which com-bines both maximin and maxisum criteria in a con-vex way:max x ∈ S   ( 1  −   )  min i { w i  x  −  P  i }+  m  j  = 1 w j   x  −  P  j    .The location of undesirable facilities is clearly a mul-tiobjective problem, where the preference of each af-fected point is clear: the undesirable facilities shouldbe sufficiently far away. The decision-maker could beinterested in use of other different criteria from themaximin or the maxisum since if the maximin crite-rion is used, then many centers could be near to facil-ity and inside its influence area. In Fig. 1, 21 popula-tion centers are represented by stars and the maximinsolution by a small square. Note that there are 11 pop-ulation centers inside its influence area. In this case,the criterion “maximize the sum of distance between Fig. 1. A location problem with 21 population centers. the undesirable facility and the  r   closest populationcenters” could be used. It is an intermediate solutionbetween the maxisum and maximin criteria. Note thatthe maxisum criterion locates the optimal solution insome vertex of the polygonal region. Fig. 1 showsthat if the  r  -maxisum criterion is used with  r  = 4 thenthe optimal solution corresponds to the point on theleft boundary marked by a small diamond since thegray tones are the values of the objective functionaccording to the right bar. This solution should bepreferred to the “maximin” solution, since there areonly six population centers inside its influence area.The weights represent the relative incompatibilityor level of repulsion between a given point and thefacility to be located. However, it not clear how theweights should be selected. Morell [7] points out thattheprincipalforcebehindpublicoppositiontoundesir-able facility sitting proposals comes from perceptionsof injustice and unfairness; a sense of inequity whichoccurs when one group is forced to bear a dispropor-tionate fraction of the burdens, while all of the citizensshare in the benefits. The magnitude of the oppositionmay depend on the particular location selected, thesize of the facility, as well as more “political” factors.Erkut and Oncu [4] have used weighted Euclidean dis-tances to reflect the relative importance of the existingsites.When a polluting plant or a radioactive waste de-pository have to be located the maximin criterion canbe used and one may model the social opposition tothe facility by using weights that are inversely propor-tional to sizes of populations or to some root of them,that is,  w i  =  N  − 1 /qi  , where  N  i  is the size of center  i and  q  1, in order to locate the facility far from bigpopulation centers. Note that as  q  goes to infinity theeffect of the weights is cancelled out and our problem( P ) tends to the unweighted version. So  q  = 2 can beseen as an intermediate solution between the weightedand unweighted problem (see [4]). The magnitude of the opposition is measured according to distances withdifferent scales since  a · x  − P  = a · x  − a · P    andthe weights are the scales used (unit of measure). Thefunction  d(x,P)  =  a  ·  x  −  P    is called an adapteddistance. Moving closer to a small population centerhas less opposition than to a big population center.When the annual demands for the service providedby the undesirable facility such as a waterworks, apower-station or a rubbish dump are known and the r  -maxisum criterion is adopted then one may model  430  J.J. Saameño Rodríguez et al. / Operations Research Letters 34 (2006) 427–436  the social opposition to the facility by using weightsthat are inversely proportional to the annual demandsat population centers or to some root of them, that is, w i  = D − 1 /qi  , where  D i  is the annual demand of center i  and  q  =  1, in order to locate the facility far fromcenters with small demand (disutility). In this case,the objective function is a utility function which isassumed to depend on the distance between a facilityand the surrounding population centers and the annualdemands.Our aim is to construct a finite set of points  C which contains a maximum of   f(x)  for any vec-tor  (k 1 , k 2 ,...,k m )  ∈  R m . Note that we have in-finite criteria since we have a criterion for each (k 1 , k 2 ,...,k m ) . If   k i  ∈ { 0 , 1 }  then we have 2 m different objective functions. So, we can find a so-lution in  C  for anyone of them. Moreover, if sev-eral criteria are adopted, which can be expressed by (k i 1 , k i 2 ,...,k im )  for  i  =  1 ,  2 ,...,L , then a multi-criteria decision-making (MCDM) procedure can beused. A standard procedure in MCDM problems is toconstruct an objective function which is a weightedsum of the individual criteria. The preference weightswould reflect the relative importance of each crite-rion as perceived by management. In practice, theapproach suggested for finding such a location is tomaximize the convex combination of this objectivefunctions with suitable preference weights assignedto each of them. This problem can also be expressedby our model, where the vector  k  =   L  i = 1 p i  ·  k i 1 , L  i = 1 p i  ·  k i 2 ,..., L  i = 1 p i  ·  k im  ,and so an optimal solution is found in the set  C .Although  f(x) , in general, is not a convex function,it is a continuous one (because it is a Lipschitzianfunction as can be seen in Theorem 9), and, since  S  is a closed and bounded set, the problem ( P ) has asolution in  S  .The following results describe the properties of ourobjective function  f(x) . Proposition 1.  f(x)  is an isotone function ,  i.e. ,  if   x  −  P  i    y  −  P  i  ,  i  ∈ M then  f(x)  f(y) . Proof.  It is sufficient to prove that if   x  and  y  are twopoints of the plane  R 2 , such that  x  −  P  i    y  −  P  i  ∀ i  ∈ M and    and    are the permutations of the set  M  suchthat w  ( 1 )  x  −  P   ( 1 )   w  ( 2 )  x  −  P   ( 2 )   ···  w  (m)  x  −  P   (m)  and w  ( 1 )  y  −  P   ( 1 )   w  ( 2 )  y  −  P   ( 2 )   ···  w  (m)  y  −  P   (m)  then w  (i)  x  −  P   (i)   w  (i)  y  −  P   (i)  ∀ i  ∈ M .We may assume, without loss of generality, that    isthe identity, so w 1  x  −  P  1   w 2  x  −  P  2   ···  w m  x  −  P  m  and w  ( 1 )  y  −  P   ( 1 )   w  ( 2 )  y  −  P   ( 2 )   ···  w  (m)  y  −  P   (m)  .If    (i)  i  then w i  x  −  P  i   w  (i)  x  −  P   (i)   w  (i)  y  −  P   (i)  .If    (i)<i  then there exists  k <i  such that   (k)  i  and w i  x  −  P  i   w  (k)  x  −  P   (k)   w  (k)  y  −  P   (k)   w  (i)  y  −  P   (i)  .   The following proposition permits to eliminate theset of interior points of   S   that do not belong to theconvex hull of   P  1 ,P  2 ,...,P  m  as candidate points. Proposition 2.  LetXbetheconvexhullof  P  1 ,P  2 ,...,P  m ,  and T the border of   S  .  The set   (S   ∩ X) ∪ (T   − X) contains at least one optimal solution to problem  ( P ) . Proof.  Let  x 0  be a solution to the problem ( P ). If  x 0  ∈  S  ∩ X  the proposition is proved.Thus, we assumethat  x 0  ∈  S   −  X . Hence, there exists a point  x ∗ ∈  X such that   x ∗ −  P  i    x 0  −  P  i  ,  ∀ i  ∈ M (see [15]).   J.J. Saameño Rodríguez et al. / Operations Research Letters 34 (2006) 427–436   431 Then we take   1  =  sup {   :   x 0  +  ( 1  −   )x ∗ ∈  S  } and consider  x 1 =  1 x 0 + ( 1 −  1 )x ∗ . Clearly, the point x 1  ∈  T   −  X  because  S   is a bounded set and   1  1.If    1  =  1 the proposition is proved. So, suppose that  1  > 1. Then  ∀ i  ∈ M we have  x 0  −  P  i  =  1  1 x 1  +   1  −  1  1 x ∗ −  P  i   1  1  x 1  −  P  i  +   1  −  1  1  x ∗ −  P  i   1  1  x 1  −  P  i  +   1  −  1  1  x 0  −  P  i ⇒  x 0  −  P  i    x 1  −  P  i  ,and by Proposition 1 the result is proved.   3. Finite dominating set of candidate points In the previous section, we have shown that there isan optimal solution of the problem which is a point of  S   ∩  X  or a point of the set  T   −  X . In this section, weprove that the set of candidate points can be reducedto a finite one. Theorem 3.  If the maximum of   f(x)  is attained inthe border of   S  ,  then there exists a point   x ∗ ∈  V   ∪ W   which is also an optimal solution ,  where  V   is theset of vertices of   S   and   W   is the set of points  x  of the border of   S   such that there exists  i,j   ∈  M  with w i  x  −  P  i  =  w j   x  −  P  j   . Proof.  Let  S    be the set defined as follows: S    = { x  ∈  S  ; w  (i)  x  −  P   (i)   w  (i + 1 )  x  −  P   (i + 1 )  , i  =  1 , 2 ,...,m  −  1 } and let  L 1 , L 2 ,...,L k  be the sides of the polygonalregion  S  .For all    and  k, S    ∩  L k  =  I   k  is a segment or anempty set, and  f(x)  is a convex function over  I   k ,so the maximum of   f(x)  occurs at an extreme pointof   I   k . Moreover, an extreme point of   I   k  is either avertex of   S   or a point of the border of   S   that verifies w i  x  −  P  i  =  w j   x  −  P  j    for some  i,j   ∈ M .   Given two positive numbers  w i  and  w j   and a pairof different points  P  i  and  P  j  , we call the weightedbisector associated to  P  i  and  P  j   with weights  w i  and w j  , respectively, to the set L ij   = { x  ∈ R 2 :  w i  x  −  P  i  =  w j   x  −  P  j  } .It is a straight line (the bisector of the segment  P  i P  j  )if   w i  =  w j   and, otherwise, it is the circumference of center C  =  w 2 i w 2 i  −  w 2 j  ·  P  i  − w 2 j  w 2 i  −  w 2 j  ·  P  j  and radius R  =  w i  ·  w j   ·  P  i  −  P  j  | w 2 i  −  w 2 j  |  .Now, we will characterize the solutions that belongsto the interior of   S  . Lemma 4.  The objective function  f(x)  is not con-stant over any no collinear segment   [ a,b ]  with any  P  i . Proof.  The weighted bisectors define a partition of the set  S   in a finite number of subsets  S   , and  f(x) =  mi = 1  k i w  (i)  x  −  P   (i)  , for all  x  ∈  S   .The segment  [ a,b ]  includes a subsegment of it insome  S   . Let  [ c,d  ]  be this subsegment.If   f(c)  =  f(d)  the proposition is proven. So, weassume that  f(c)  =  f(d) . Taking 0 <  < 1, the point z  =   c  +  ( 1  −   )d   ∈  (c,d)  ⊂  S    and f(z)  =  f(  c  +  ( 1  −   )d) = m  i = 1 k i w  (i)   c  +  ( 1  −   )d   −  P   (i) = m  i = 1 k i w  (i)   (c  −  P   (i) ) +  ( 1  −   )(d   −  P   (i) )  < m  i = 1 k i w  (i) (   c  −  P   (i) +  ( 1  −   )  d   −  P   (i)  ) =  f(c)  +  ( 1  −   )f(d) ,where   (c  −  P   (i) )  +  ( 1  −   )(d   −  P   (i) )  <   c  −  P   (i)  +  ( 1  −   )  d   −  P   (i) 
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