The balanced 2-median and 2-maxian problems on a tree
aa r X i v : . [ m a t h . O C ] M a r The balanced 2-median and 2-maxian problems ona tree
Jafar Fathali ∗ Faculty of Mathematical Science, Shahrood University of Technology,University Blvd., Shahrood, Iran, email: [email protected]
Mehdi Zaferanieh
Department of Mathematics, Hakim Sabzevari University, Tovhid town,Sabzevar, Iran, email: [email protected] ∗ Corresponding author bstract This paper deals with the facility location problems with balancingon allocation clients to servers. Two bi-objective models are consid-ered, in which one objective is the traditional p-median or p-maxianobjective and the second is to minimize the maximum demand volumeallocated to any facility. An edge deletion method with time complex-ity O ( n ) is presented for the balanced 2-median problem on a tree.For the balanced 2-maxian problem, it is shown the optimal solutionis two end vertices of the diameter of the tree, which can be obtainedin a linear time. Keywords: facility location; 2-maxian; 2-median; balanced al-location.
The most location problems are concerned about minimization of trans-portation time between servers and clients. However in real applicationsusually the service time is as considerable as transportation time due toattracting clients. So in this paper we consider a bi-objective problem cor-responding to the transportation time as well as service time. The p -medianand p -maxian objective functions have been considered for transportationtime while an avoidance congestion function has been considered for servicetime. We call these problems balanced p -median and p -maxian problems,respectively. To balance the facility servicing times, we propose minimizingthe maximum number of clients that are served by facilities.Let G = ( V, E ) be a given graph, where V is the set of vertices and E is the set of edges. Let | V | = n and | E | = m . The p -median problemasks to find a set of p vertices of G , called facilities, such that the sum ofweighted distances from vertices to the closest facility is minimized. Karivand Hakimi [14] showed that the p -median problem is N P -hard on generalnetworks while it can be solved in polynomial time on tree networks. Theinitial works of the p -median problem is referred to Hakimi [9, 10]. Hakimi[10] showed that at least one optimal solution of the p -median problemis located on vertices. Kariv and Hakimi [14] presented an O ( p n ) timealgorithm for this problem on a tree. The time complexity is improved to O ( pn ) by Tamir [21]. In the case p = 2 on a tree, Gavish and Sridhar [8]presented an O ( nlogn ) algorithm.The obnoxious case of the p -median problem is called p -maxian prob-lem. In the p -maxian problem a set which contains p vertices is sought so2hat the sum of weighted distances of clients to the farthest facility is max-imized. The NP-hardness of this problem on general networks is shown in[12]. Zelinka [23] showed that in the tree graphs an optimal solution of the1-maxian problem is contained on the leaves. Ting [22] proposed a lineartime algorithm to the 1-maxian problem. The 1-maxian problem on generalnetworks is investigated by Church and Garfinkel [5]. They presented an O ( mn log n ) time algorithm, and Tamir [20] improved the time complex-ity to O ( mn ). Burkard et al. [3] showed that the optimal solution of the2-maxian problem on a tree lies on the two end vertices of the diameter,where diameter is the longest path of the tree. They also showed that p -maxian problem on the tree is reduced to the 2-maxian. Based on theseproperties they presented a linear time algorithm for the p -maxian problemon a tree. Kang and Cheng [13] extended the algorithm to the case that theunderlying network is a block graph.The equity location models are introduced in the last two decades. Inthese models, facilities are to be located to maximize the equality betweenthe demand points. Some researchers have been attracted to this subject.Among them Gavalec and Hudec [7] considered an equity model which itsobjective function is the maximum difference in the distance from a de-mand point to its farthest and nearest facility. They called this problemas balancing function. Berman et al. [2] considered the problem of findingthe location of p facilities such that the weights attracted to the differentfacilities are as close as possible. They formulated this problem as minimiz-ing the maximum weight assigned to each facility. Marin [18] consideredthe balanced discrete location problem, in which the objective function isminimizing the difference between the maximum and the minimum weightsallocated to different facilities. Barbati and Piccolo [1] proposed some prop-erties to describe the behavior of the equality measures in an optimizationcontext.Another paper related to balanced facility location models is that ofLejeune and Prasad [16], which propose models to investigate effectiveness-equity tradeoffs in tree network facility location problems. The 1-medianobjective is considerd as a measure of effectiveness, and the Gini index isused as a measure of equity. A bi-criteria problems involving these objec-tives is presented. Landete and Marin [15], considered the spanning treeswith balanced weights, i.e., where the differences among the weights areminimized.In the paper of Lpez-de-los-Mozos et al. [17] the ordered weighted aver-aging operator is applied to define a model which generalizes some inequalitymeasures. In their work, for a location x , the value of the objective function3s the ordered weighted average of the absolute deviations from the averagedistance from the facilities to the location x . We refer the interested readerto [19, 6], two reviews of the literature on equity measurement in locationtheory.In the next section, we formulate the balanced p -median and p -maxianproblems. In Section 3, an O ( n ) time algorithm for balanced 2-medianproblem on a tree is presented. The balanced 2-maxian problem on a treeis investigated in Section 4, and a linear time algorithm is proposed for thisproblem. Let G = ( V, E ) be a graph where V is the set of vertices and E is the setof edges and | V | = n . Each vertex v i has a non-negative weight w i , whichis the number of clients on vertex v i . The weight w i also called the demandat v i . Let d ij be the distance between vertices v i and v j . The p -medianproblem asks to find a set of p vertices, X p = { x , ..., x p } called facilities,so that the sum of the weighted distances from the vertices to the closestfacility in X p is minimized, i.e.min f ( X p ) = n X i =1 w i d ( X p , v i ) , where for any vertex v ∈ V , d ( X p , v ) = min x j ∈ X p d ( x j , v ) . In the p -maxian problem the goal is maximizing the sum of the weighteddistances from the vertices to the farthest facility in X p , i.e.max f ( X p ) = n X i =1 w i max x j ∈ X p d ( x j , v i ) . We define the bi-objective models of balanced p -median and p -maxianproblems as follows. Let t i be the service time by different facilities forserving clients on vertex v i . Also let V j be the set of vertices of V that areallocated to the facility x j . Then, in the balanced p -median problem weshould find a partition { G = ( V , E ) , ..., G p = ( V p , E p ) } of G such that,min f ( X p ) (1)min f ( X p ) = max { X v i ∈ V j w i t i , j = 1 , ..., p } . (2)4imilarly, in the bi-objective function of balanced p -maxian problem wewould find a partition of G such that,max f ( X p ) (3)min f ( X p ) = max { X v i ∈ V j w i t i , j = 1 , ..., p } . Note that the goal in these problems is partitioning graph G = ( V, E )into p subgraphs. Then in the balanced p -median problem, in each partitionthe median should be determined. While in the balanced p -maxian problem,for i = 1 , ..., p , each vertex u ∈ V i is allocated to the facility x i ∈ X p that d ( u, x i ) = max { d ( u, x j ) | x j V i , j = 1 , ..., p } . Therefor, by this partitioning,we can write, f ( X P ) = f ( X p ) = p X j =1 X v i ∈ V j w i d ( x j , v i ) . In the classical p -median problem each client is allocated to the closestfacility and x i ∈ V i for i = 1 , ..., p , while in the p -maxian problem each vertexis allocated to the farthest facility and x i V i for i = 1 , ..., p . However, inthe balanced case either client may be allocated to the closest or farthestfacility.Let z i = w i t i for i = 1 , ..., n then f ( X p ) = max { X v i ∈ V j z i , j = 1 , ..., p } . (4)In the case that the clients’ service times are all equal, i.e. t i = t ≥ i = 1 , ..., n , then the objective function in (2) reduces to the following:min f ( X p ) = max { X v i ∈ V j w i , j = 1 , ..., p } . (5)The problem of minimizing f ( X ) is considered by Berman et al. [2].They showed this problem is N P -hard. Since the p -median and p -maxianproblems are also N P -hard, then the balanced cases are
N P -hard, too.In this paper, we use the weighted sum method to the proposed bi-objective problems. The weighted sum problem of bi-objective p -medianand p -maxian problems, can be written asmin f pmed ( X ) = λf ( X ) + (1 − λ ) f ( X ) , f pmax ( X ) = λf ( X ) − (1 − λ ) f ( X ) , where 0 ≤ λ ≤
1. Note that in these models 1 − λ can be interpreted as theservers’ balanced coefficient.Next, the balanced 2-median and 2-maxian problems are studied. Lemma 1.
In the case p = 2 , the model f ( X p ) can be represented as thefollowing problem, f ( X p ) = | X v i ∈ V z i − X v i ∈ V z i | . (6) Proof
Let Z = P ni =1 z i , then in the case p = 2, the objective function f ( X p ) = max { P v i ∈ V j z i , j = 1 , ..., p } can be represented as f ( X p ) = max { X v i ∈ V z i , X v i ∈ V z i } = | P v i ∈ V z i + P v i ∈ V z i | + | P v i ∈ V z i − P v i ∈ V z i | Z + | P v i ∈ V z i − P v i ∈ V z i | . Since Z is a fixed value, it can be left out from the objective function. ✷ Obviously, by Lemma 1 in the case p = 2, any partition of V to V and V such that P v i ∈ V z i = P v i ∈ V z i provides the optimal solution ofmin f ( X p ). In this section, the balanced 2-median problem on a tree is considered. Wepropose the following weighted objective function to find the Pareto optimalsolutions to the balanced 2-median problem.min f pmed ( X ) = λ ( X v i ∈ T w i d ( v i , x ) + X v i ∈ T w i d ( v i , x )) + (1 − λ ) | X v i ∈ T z i − X v i ∈ T z i | . (7)Where T is the underlaying network and T and T are the two subtrees of T which contain vertices in V and V , respectively.A well-known method for solving the classical 2-median problem on atree is the edge deletion method (see e.g.[4]). In this method by deleting any6 Figure 1:
The tree for Example 1. edge the median of obtained subtrees are found and the best one is chosenas the solution of the 2-median problem. Since in the second part of ourmodel partitioning the tree in two subtrees is necessary, we apply the edgedeletion method to solve the balanced 2-median problem as given below.For every edge e , let T e and T e be two subtrees of T which are ob-tained by deleting edge e . Let m e and m e be the 1-medians of T e and T e ,respectively. We calculate the value of objective function f pmed ( . ) as follows: f pmed ( m e , m e ) = λ ( X v i ∈ T e w i d ( v i , m e ) + X v i ∈ T e w i d ( v i , m e )) + (1 − λ ) | X v i ∈ T e z i − X v i ∈ T e z i | . (8)Then the pairs of medians corresponding to the minimum amounts of Prob-lem (8) are chosen as the solution of the balanced 2-median problem. Thetime complexity of this method is O ( n ).In the following example we show that in the optimal solution of Problem(7) the customers may not be assigned to the closest median. Example 1.
Consider the tree depicted in Fig. 1, where the weights andservice times of all vertices are equal to one. The solution of the balanced2-median problem for the case λ = is { v , v } . If we allocate v to v thenthe value of objective function is (4 + 2) while allocating v to v resultsin the value of objective function equal to (5 + 0) . Therefore, the optimalsolution is obtained by deleting edge ( v , v ) . In this section, we consider the balanced 2-maxian problem on the tree T .Let T and T be the two subtrees of T which contain vertices in V and V ,7espectively. The model of this problem is given as follows:max f pmax ( X ) = λ ( X v i ∈ T w i d ( v i , x ) + X v i ∈ T w i d ( v i , x )) − (1 − λ ) | X v i ∈ T z i − X v i ∈ T z i | . (9)Burkard et al. [3] showed that the solution of the 2-maxian problem istwo end vertices of a diameter which can be obtained in a linear time. For λ ∈ [0 ,
1] the balanced 2-maxian problem also could be solved by such edgedeletion method as in Burkard et al. [4] which is presented for the 2-medianproblem on a tree with positive and negative weights.
Let T a and T b be two subtrees of T which are obtained by deleting edge e = ( a, b ). We construct two trees T e and T e which their vertices are thesame as T . The weight of vertices in T e are defined as w i = (cid:26) w i if v i ∈ T b otherwise, and analogously the weights of vertices in T e are defined as w i = (cid:26) w i if v i ∈ T a otherwise. Then the set X = { x , x } is considered which maximizes the followingobjective function. f pmax ( X ) = λ ( X v i ∈ T e w i d ( v i , x )+ X v i ∈ T e w i d ( v i , x )) − (1 − λ ) | X v i ∈ T e z i − X v i ∈ T e z i | . Note that for each edge we should compute the objective function for anypair { x , x } . Therefore, the time complexity of this method is O ( n ).The following example shows in the optimal solution of the balanced2-maxian problem, clients may not be allocated to the farthest facility. Example 2.
Consider again the tree in Fig. 1, where the weights of edges ( v , v ) and ( v , v ) are replaced by 2 and 1, respectively. The weights andservice times of all vertices are equal to one. The solution of balanced 2-maxian problem for the case λ = is { v , v } (and { v , v } ) which is ob-tained by deleting edge ( v , v ) . The vertices v , v and v are allocated tothe facility in v , and the vertices v , v and v are allocated to the facility n v . Note that the vertex v is allocated to facility in v , but the farthestfacility from v is in v . The optimal value of objective function is (24 − .If the vertex v is allocated to the facility in v then the value of objectivefunction is (25 − . Now we would improve the time complexity of the balanced 2-maxian prob-lem to linear time.
Lemma 2.
Let T be a tree. Then an optimal solution of the balanced 2-maxian problem on T , exist on the leaf nodes of T . Proof
Let X = { x , x } be the solution of balanced 2-maxian problem and V and V be the sets of vertices that are assigned to x and x , respectively.Let T = ( V , E ) and T = ( V , E ) be two subtrees of T which are obtainedby deleting edge e = ( v r , v s ) where v r ∈ T and v s ∈ T . Then x ∈ T and x ∈ T . If either x or x is not leaf node, then we consider its adjacent.Let x be an inner vertex and u ∈ T be its adjacent vertex that is not in thepath connecting x to x . In fact, u is a vertex along the path connecting x to a leaf nodes. Then for all vertices v i ∈ T , d ( v i , u ) = d ( v i , x ) + d ( x , u ) ≥ d ( u, x ) . Therefore, X v i ∈ T w i d ( v i , u ) ≥ X v i ∈ T w i d ( v i , x ) . So by relocation of facilities toward leaf nodes the maxian part of the objec-tive function is not decreased while the balancing part remains unchanged. ✷ As previously mentioned, the optimal solution of 2-maxian problem istwo end vertices of the longest path of the tree. In the following, we showthat this property holds for the balanced case.
Lemma 3.
Let P be the path between two vertices x and x in the tree T . Let T and T be two subtrees of T obtained by deleting edge ( v r , v s ) andcontain x and x , respectively. If P is not the longest path in the tree T ,then there exist either a vertex u ∈ T such that d ( u, x ) ≥ d ( x , x ) or avertex u ′ ∈ T such that d ( u ′ , x ) ≥ d ( x , x ) . Proof
Let P ′ be the diameter of the tree T and a and b be two end verticesof P ′ . Let v a ∈ P and v b ∈ P be the closest vertices in P to a and b ,9espectively (see Fig.2). We consider two cases where the two vertices a and b are in the same or different subtrees. First let a, b ∈ T , then either d ( a, v a ) ≥ d ( v a , x ) or d ( b, v b ) ≥ d ( v b , x ) . Otherwise, the path P ′ is not diameter of the tree. Therefore, either d ( a, x ) = d ( a, v a ) + d ( v a , x ) ≥ d ( v a , x ) + d ( v a , x ) = d ( x , x ) , or d ( b, x ) = d ( b, v b ) + d ( v b , x ) ≥ d ( v b , x ) + d ( v b , x ) = d ( x , x ) . Now consider the other case where a ∈ T and b ∈ T . Then either d ( a, v a ) ≥ d ( v a , x ) or d ( b, v b ) ≥ d ( v b , x ) . Otherwise, the path P ′ is not the longest one. So d ( a, x ) ≥ d ( x , x ) , or d ( b, x ) ≥ d ( x , x ) . The other cases where a, b ∈ T and a ∈ T , b ∈ T would be similarlyproved. ✷ b x x r v T T s v a a v b v Figure 2: Longest and non longest paths on a tree
Theorem 1.
There is an optimal solution of the balanced 2-maxian problemon the leaf nodes of diameter of T . Proof
By Lemma 2 there is an optimal solution on the leaf nodes. Let x and x be two leaf nodes of T that the path connecting them is not diameterof T . Let T and T be two subtrees of T obtained by deleting edge ( v r , v s )which contain x and x , respectively. Let also the objective function of x x respect to this partition be less than or equal to other partitions. Thevertices in T are allocated to x ∈ T and the vertices in T are allocated to x ∈ T (see Fig. 2). Since the path connecting x and x is not diameter,by Lemma 3 either there is a vertex u ∈ T that d ( u, x ) ≥ d ( x , x ) or thereis a vertex u ′ ∈ T that d ( u ′ , x ) ≥ d ( x , x ) . Without loss of generality, let there exist u ∈ T where d ( u, x ) ≥ d ( x , x ).Then d ( u, v r ) + d ( v r , x ) = d ( u, x ) ≥ d ( x , x ) = d ( x , v r ) + d ( v r , x ) , and consequently d ( u, v r ) ≥ d ( x , v r ) . Hence, for each v i ∈ T , d ( v i , x ) = d ( v i , v r ) + d ( v r , x ) ≤ d ( v i , v r ) + d ( v r , u ) = d ( v i , u ) . So X v i ∈ T w i d ( v i , u ) ≥ X v i ∈ T w i d ( v i , x ) . Since the partitions remain unchanged, then the objective function will notbe decreased by choosing u instead of x . ✷ Note that if d ( v i , v j ) > i, j = 1 , ..., n then by Theorem 1 the optimalsolution is two ends of diameter.Since diameter of a tree can be found in a linear time (see e.g. [11]), thenthe optimal solution of balanced 2-maxian problem can be found in O ( n )time. However, to calculate the value of objective function we should com-pute the corresponding objective function by deleting any edge on diameterwhich would be performed in O ( n ) time. To reduce the time complexity,we first consider the computing objective function on a path. Lemma 4.
Let P = v , ..., v n be a path and e = ( v, u ) and e = ( u, s ) betwo adjacent edges on P . Let f e pmax ( v , v n ) and f e pmax ( v , v n ) be the objec-tive function values of balanced 2-maxian problem in vertices v and v n bydeleting edges e and e , respectively. Also let Z = P ni =1 z i and v r be thevertex of P so that P r − i =1 z i < Z and P ri =1 z i ≥ Z . Then f e pmax ( v , v n ) − f e pmax ( v , v n ) = (10) (cid:26) λw u ( d ( u, v ) − d ( u, v n )) + (1 − λ )2 z u if u ∈ { v , ..., v r − } λw u ( d ( u, v ) − d ( u, v n )) − (1 − λ )2 z u if u ∈ { v r +1 , ..., v n } . k v (cid:0) k v ✁ l v l v l v T ✂ l v T ✄ K v T k v T Figure 3: Components of a tree
Proof
For i = 1 ,
2, let T e i and T e i be the subpaths of P obtained bydeleting edge e i . Let T e i and T e i , for i = 1 ,
2, contain the vertices v n and v , respectively. Then T e = T e \ { u } and T e = T e ∪ { u } . Therefore, f e pmax ( v , v n ) = λ ( X v i ∈ T e w i d ( v i , v )+ w u d ( u, v )+ X v i ∈ T e w i d ( v i , v n ) − w u d ( u, v n )) − (1 − λ ) | ( X v i ∈ T e z i + z u ) − ( X v i ∈ T e z i − z u ) | = (cid:26) f e pmax ( v , v n ) + λw u ( d ( u, v ) − d ( u, v n )) + 2(1 − λ ) z u if u ∈ { v , ..., v r − } f e pmax ( v , v n ) + λw u ( d ( u, v ) − d ( u, v n )) − − λ ) z u if u ∈ { v r +1 , ..., v n } . ✷ Let e i = ( v i , v i +1 ) for i = 1 , ..., n −
1, then the optimal objective functionon the path P can be iteratively computed. First f e i pmax ( v , v n ) should becomputed for i = 1 , r . Then by using Lemma 4 the objective functioncorresponding to deletion other edges on path P will be obtained. Therefore,the total time complexity is O ( n ).Now consider the tree T . Let P : v k , ..., v l be the diameter of T . Wecreate a new path b P that the vertices of which are the same as P but theweights and service times of the vertices are varying and defined as follow:ˆ w k = X i ∈ T vk w i , ˆ w k +1 = X i ∈ T vk +1 w i , ..., ˆ w l = X i ∈ T vl w i , z k = X i ∈ T vk z i , ˆ z k +1 = X i ∈ T vk +1 z i , ..., ˆ z l = X i ∈ T vl z i . Where T v i , i = k, k + 1 , ..., l , are the subtrees of T that obtained by deletingonly the edges (not vertices) of P from the tree T so that v i ∈ T v i (see Fig.3). By finding the best deleted edge on b P , the best deleted edge on the tree T is determined. Therefore, the following theorem is concluded. Theorem 2.
The balanced 2-maxian problem can be solved in a linear time.
In the following example the numerical results of the presented methodsare given. The results confirm the validity of the findings in the previoussections.
Example 3.
Consider the tree depicted in Fig. 4, where the weights of itsvertices are given in the Table 1. w w w w w w w w w w w w w w w w w v v v v v v v v v v v v v v v v v
21 1
33 33
Figure 4: A tree with 17 vertices
The solutions of the balanced 2-median problem for varying values of λ are presented in Table 2. In this table the column with heading f indicates he difference between number of clients assigned to each facility. Note that,in the case λ = 0 . , by adding the balanced objective function to medianmodel, we could find a nearly equity solution which its value of the medianobjective function, i.e. f , is not considerably increased. f f deleted edge f pmed medians λ = 1 231 61-16=45 e = ( v , v ) 231 { v , v } λ = 0 . e = ( v , v ) 156.6 { v , v } λ = 0 . e = ( v , v ) 133 { v , v } λ = 0 - 39-38=1 e = ( v , v ) 1 -Table 2: The solutions of the balanced 2-median problem for varying valuesof λ on tree in Fig. 4. Table 3 contains the solutions of balanced 2-maxan problem for varyingvalues of λ . Note that, although for all amounts of λ the optimal solutionis { v , v which are the two end vertices of the diameter. However, thevertices assigned to the facilities are different. Furthermore, in the cases λ = 0 . , . , by adding the balanced objective function, a nearly equity solu-tion is found, which its value of maxian objective function, i.e. f , is notconsiderably decreased. f f deleted edge f pmax optimal solution λ = 1 1266 61-16=45 e = ( v , v ) 1266 { v , v } λ = 0 . e = ( v , v ) 744.6 { v , v } λ = 0 . e = ( v , v ) 618 { v , v } λ = 0 - 39-38=1 e = ( v , v ) -1 -Table 3: The solutions of the balanced 2-maxian problem for varying valuesof λ on tree in Fig. 4. In this section some numerical examples are given for the balanced 2-medianand 2-maxian problems. The algorithms were written in MATLAB andtested for 10 randomly generated problems. The arc lengths are generatedrandomly and taken from the set [0 , , = 0 λ = 0 . λ = 0 . λ = 1Test f pmed f f f pmed f f f pmed f f pmed Table 4: Results for the balanced 2-median problem. λ = 0 λ = 0 . λ = 0 . λ = 1Test f pmax f f f pmax f f f pmax f f pmax Table 5: Results for the balanced 2-maxian problem.the most of test problems with varying values of λ the functions f and f dominated on f . Therefore, the total objective function hasn’t considerablychanged for 0 < λ ≤ λ are presented in Tables 4 and 5. Theresults show that in some test problems the optimal solutions of classical2-median and 2-maxian problems are balanced (see Test No. 4). However,in some other test problems, the balancing is depended on λ . In the most ofcases, the difference number of allocated clients to each facility are jumpedwhen λ is changed from 0.2 to 0.5. We also examined other values of λ , butthe solutions are not considerably changed.Fig. 5 shows the histogram of the changing values of f pmed for testproblem No. 3 respect to λ . The histogram of the changing values of f pmax for test problem No. 5 respect to λ , is given in Fig. 6.15 Figure 5: The histogram of f pmed respect to λ for test NO. 3 In this paper, two bi-objective balanced models of the p -median and p -maxian problems on a tree have been investigated. In the balanced p -medianproblem, the objective function is combination of balance on clients’ allo-cation to the facilities and median problem while in the balanced p -maxianproblem the objective function is balancing on clients’ allocation and maxianproblem. Based on edge deletion method an O ( n ) algorithm is presentedfor the balanced 2-median problem on a tree. Furthermore, it is shown thatthe optimal solution of the balanced 2-maxian problem, is the leaf nodes ofthe diameter of the tree. Then a linear time algorithm is presented to ob-tain the balanced 2-maxian objective function. To illustrate the algorithms,some numerical examples are given. The results of these examples showthat enforcing the balanced objective function to the median and maxianmodels, causes an almost equitable assignment clients to servers.16 × -10123456789 Figure 6: The histogram of f pmax respect to λ for test NO. 5 References [1] Barbati ., Piccolo C., Equality measures properties for location prob-lems, Optimization Letters, 10 (2015), 903-920.[2] Berman O., Drezner Z., Tamir A., Wesolowsky G.O., Optimal locationwith equitable loads, Annals of Operations Research, 167 (2009), 307-325.[3] Burkard R.E., Fathali J., Kakhki H.T., The p-maxian problem on atree, Operations Research Letters, 35 (2007), 331-335.[4] Burkard R.E., C¸ ela E., Dollani H., 2-Median in trees with pos/negweights, Discrete Appl. 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