A Hop Constrained Min-Sum Arborescence with Outage Costs

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1 A Hop Consrained Min-Sum Arborescence wih Ouage Coss Rakesh Kawara Minnesoa Sae Universiy, Mankao, MN Absrac The hop consrained min-sum arborescence wih ouage coss problem consiss of selecing links in a nework so as o connec a se of erminal nodes N={,3, n} o a cenral node wih minimal oal link cos such ha (a) each erminal node has exacly one enering link; (b) for each erminal node, a unique pah from he cenral node o exiss; (c) for each erminal node he number of links beween he cenral node and is limied o a predefined number h, and (d) each erminal node has an associaed ouage cos, which is he economic cos incurred by he nework user whenever ha node is disabled due o failure of a link. We sugges a Lagrangian based heurisic o solve he ineger programming formulaion of his nework problem. 1. Inroducion The hop consrained min-sum arborescence (HCMA) problem is frequenly encounered in he nework design, rouing and scheduling problems. I consiss of finding links o connec a se of geographically remoe nodes o a cenral node such ha for each remoe node here is exacly one link enering node, and for each remoe node, a unique pah exiss from he cenral node o node. The soluion is subec o hop consrains which limis he number of links beween he cenral node and any erminal node o a predefined number h. These hop consrains are ofen used o conain he maximum delay beween any erminal node and he cenral node. The hop consrains can also be used o model reliabiliy consrains when designing elecommunicaion neworks as poined ou by LeBlanc and Reddoch [6]. They also sugges ha in many neworks hop consrains can be used o avoid degradaion of he signal qualiy. Gouveia [3] presened an ineger programming model for spanning rees wih hop consrains and suggesed a Lagrangian based heurisic for solving he problem. In his paper, we sudy he HCMA problem where each erminal node has an associaed ouage cos. The ouage cos associaed wih a erminal node is he economic cos incurred by he nework user whenever ha node is disabled due o failure of a link. I is a measure of he opporuniy cos of he equipmen and is user and does no include cos of repairing or replacing he link. The locaion of he cenral node and erminal nodes, and he limi h on he number of hops beween he cenral node and each erminal node is given. Also given are he annual coss of insalling he links and he ouage cos associaed wih each erminal node. We formulae he problem as an ineger programming problem and use a Lagrangian based heurisic mehod o solve i. We used he lower bound given by he Lagrangian mehod o esimae he qualiy of our heurisic soluions. Subgradien opimizaion mehod is used o find good lower bounds. Compuaional resuls are presened o demonsrae he performance of he Lagrangian based heurisic for differen nework srucures.. Model formulaion The hop consrained min-sum arborescence problem wih node ouage cos is formulaed as an ineger-programming problem. Our obecive is o minimize he oal annual cos consising of links coss and he expeced node ouage coss. We use he following noaions in he paper: N: he se of erminal nodes,3...n; Node 1: cenral node; T 1 = family of spanning arborescences (direced rees or branchings) wih roo a node 1, i.e., he family of direced neworks which do no conain a cycle and such ha for every erminal node, here is a pah from node 1 o node. C : annual cos of insalling a link (i, ); D : node ouage cos associaed wih erminal node ; Proceedings of he 36h Hawaii Inernaional Conference on Sysem Sciences (HICSS 03)

2 Q: link failure rae; h : he limi on he maximum number of links beween he cenral node and erminal node. Decision Variables X : a binary variable such ha X = 1 indicaes ha link (i, ) is in he soluion; oherwise X = 0; Y : a variable such ha Y = 1 if he direced link from node i o node in on he pah from node 1 o node ; oherwise Y = 0. The hop consrained min-sum arborescence problem wih node ouage coss is formulaed as: n n n n n Z = IP Minimum CX QD Y (1) i= 1= = i= 1= subec o X T 1 () N N Y Y i = = = 1 Y 1 if i = 1 1 if i = = 0 oherwise for all i N [1], N (3) X for all i N [1], and, N (4) N N Y h for all N (5) i= 1= X {0,1} for all i N [1], and N (6) Y 0,1 for all i N [1], and, N (7) { } In he above model, consrains (3) are flow conservaion consrains. Consrains (4) ensures ha if here is no direc link from node i o node, hen here canno be direc flow from node i o node. Consrains (5) are he hop consrains which for every erminal node, N, limi he number of links beween he cenral node and node o a predefined number h. 3. Soluion mehods We propose a Lagrangian based heurisic mehod o solve his problem. In our approach we firs form a Lagrangian relaxaion of he problem which is solved opimally. Nex, we use a branch exchange heurisic o generae a feasible soluion from he infeasible Lagrangian soluion. We use subgradien opimizaion mehod o find good Lagrangian mulipliers. The bes values of he lower bound and he feasible soluion are reained when he subgradien algorihm sops. The lower bound given by he Lagrangian relaxaion is used o obain a quaniaive esimae of he qualiy of he soluion given by he branch exchange heurisic. 3.1 Lagrangian relaxaion In his sudy we use a Lagrangian relaxaion approach o generae lower bounds for he hop consrained minsum arborescence problem wih node ouage coss. Lagrangian relaxaion can be used o obain igh lower bounds for a variey of ineger programming problems. (See Fisher [] for an applicaion-oriened survey of Lagrangian relaxaion). We form a relaxaion of he hop consrained min-sum arborescence problem by muliplying each consrain (4) by a nonnegaive Lagrange muliplier µ and each consrain (5) by nonnegaive muliplier θ and adding he producs o he obecive funcion. This resuls in he following relaxaion of problem Z IP : n n L ( µ, θ) = Minimize Q( µ ) M ( µ, θ) θh = = s.. (), (3),(6) and (7), where n n n Q( µ ) = Minimize X(C µ ) i=1= = s.. () and (6), and n n M ( µ, θ) = Minimize ( µ θ Q D ) Y i= 1= s.. (3) and (7) Procedure for evaluaing Q(µ) Solving Q(µ) requires finding he min-sum arborescence X ( µ ) rooed a node 1, which for a given vecor of Lagrange mulipliers µ can be accomplished using Fischei and Toh's algorihm [1]. For his algorihm, we use he lengh of n arc (i, ) = ( C µ ). = 3.1. Procedure for solving M ( µ, θ) The funcion M ( µ, θ) is evaluaed by solving a single-commodiy flow problem. In his problem, one uni of a commodiy is o be shipped from he cenral node o node. Since he links are uncapaciaed, he Proceedings of he 36h Hawaii Inernaional Conference on Sysem Sciences (HICSS 03)

3 flow Ŷ ( µ, θ) will be along he shores pah from he cenral node o node, which can be found using Dksra's algorihm [5] wih µ θ Q D as he cos of direcly shipping one uni of commodiy from node i o node. While solving M ( µ, θ) we sop he Dksra's algorihm as soon as a shores pah o node is found. 3. Improving he Lagrange mulipliers I is well known ha for any µ and θ, he value of he Lagrangian relaxaion L(µ,θ) provides a lower bound o Z IP. We wish o find he ighes bound which can be achieved, i.e., we wish o solve he Lagrangian dual problem o obain he bes lower bound, L(µ,θ ) = max {L(µ,θ)}. µ, θ 0 Searching for he opimal Lagrangian muliplier vecors µ and θ is very ime consuming; however, approximae values can be found by using a subgradien opimizaion mehod [4]. This mehod begins wih an iniial vecor of mulipliers µ 0 and θ 0, which a ieraion p is adused using he following rule: p 1 p µ = µ s p V p, N, i N [1] p 1 p θ = θ Sp U p N where V = Y X, N, i N [1] U n n Y h i= 1= = p p λ(z L( µ, θ )) and sp =. p V U N In he compuaion above, V and U are he subgradiens of L(µ,θ), denoes he Euclidean norm, Z is he bes available overesimae of he opimal soluion value, and λ is a scalar muliplier which saisfies he condiion 0<λ. The value of λ is iniially se equal o and is reduced during he course of he search. 3.3 A Lagrangian based branch exchange heurisic In our research, for simpliciy we assumed h = h for each erminal node. Our heurisic can very easily be exended o allow differen values of h for differen erminal nodes. Since Q(µ) was solved independen of hop consrains, he opimal soluion o Q(µ) may have more han h hops in he pah from he cenral node o some of he erminal nodes, which is an infeasible soluion o problem Z IP. Afer every ieraion of he subgradien opimizaion algorihm we use a branch exchange heurisic o generae a feasible soluion o Z IP. The bes feasible soluion is reained when he subgradien opimizaion algorihm is erminaed. This branch exchange heurisic is an ieraive procedure. In each ieraion his heurisic idenifies se of erminal nodes, called B, which are exacly (h1) hops from he cenral node. Nex he heurisic finds a subse C of se B such ha for each node ha belongs o he se C: he node or nodes ha belong o he subree rooed a node and are furhes away from node are exacly (h-1) hops away from his node. For each node belonging o se C, he heurisic removes he link erminaing a his node and replaces i wih link (1,). If se C is empy, hen he heurisic idenifies a erminal node belonging o se B for which a minimal increase in oal annual cos of he nework, he link erminaing a node can be replaced wih link (i, ) where node i is a mos (h-1) hops from he cenral node. If he replacing link violaes any consrain, hen i is ignored. This is coninued unil all he erminal nodes in he nework saisfy he hop consrain. The branch exchange heurisic ends when he curren soluion is feasible. For his heurisic we define he following addiional noaions: g = he origin of he link inciden o node. Level : number of hops beween he cenral node and node. Leaf node: a erminal node wih no links originaing from i. Leaf : se of leaf nodes belonging o he subree rooed a node. B = { Level = h 1for all N}, i.e., he se of nodes ha are (h1) hops from he cenral node. B = { Level < h for all N}, i.e., he se of nodes ha are a mos (h-1) hops from he cenral node. Iniially, X = Xˆ ( µ ). Noe ha, since all of he values above are dependen upon he value of X, hey mus be recompued each ime X is modified by replacemen of one or more links. Sep 1. If B = φ, hen STOP; Else for each B, find max lev = max { Level i }. i Leaf Sep. Find s.. max lev = h 1. Proceedings of he 36h Hawaii Inernaional Conference on Sysem Sciences (HICSS 03)

4 Sep 3. If = { φ}, hen go o Sep 4; Else Se X g = 0 and X = 1, and reurn o Sep 1. 1 Ties if any are broken arbirarily. Sep 4. If B = φ, hen STOP; ELSE for each B and i B, find = { C Cg (Leveli Levelg )QD }; Sep 5. Find ( i, ) arg min{ } = ; i, Sep 6. Se X = 0 and X 1, g i = and reurn o Sep Numerical resuls The effeciveness of he Lagrangian based heurisic was invesigaed by solving a randomly generaed se of es problems. The daa for he compuaional experimens were generaed by drawing he coordinaes of he nodes from a uniform disribuion over a square of size 1000 by The annual cos of link (i, ) was chosen o be he Euclidean disance beween poin i and poin. The node ouage coss were randomly generaed from a uniform disribuion U[0,1000]. We solved he problems for n = 0, 40, and 60; h = 3, 4, and 5; and Q = 0.0, 0.04, and In our experimens we assumed h = h for all є N. For each parameer se we solved 3 insances of he problem and compued he average gap. For purposes of he subgradien opimizaion mehod, we used he bes heurisic soluion obained so far as he overesimae of he opimal obecive funcion value. The iniial value of he scalar λ was se o, and halved whenever L ( µ p, θ p ) did no improve in successive ieraions. The Lagrange mulipliers were iniially se o 0. The sopping crierion in compuaion of he lower bounds was: sop if he oal number of ieraions exceeds 4000 or if he obecive funcion value changes by less han 0. in successive ieraions. The Lagrangian based heurisic mehod was coded in Forran 77 and run on IBM SP compuer wih a maximum processing speed of 888 MHz. Compuaional resuls of he experimen are presened in Table 1. The compuaional resuls presened in Table 1 show ha he average gap beween he heurisic soluion and he Lagrangian lower bound is wihin 14 percen. This gap provides an upper bound for he gap beween he heurisic and opimal soluions. Table 1. Compuaional resuls N h Q Average Gap % % % % % % % % % % % % % % % % % % % % % % % % % % % Gap=(heurisic soluion lower bound)/(heurisic soluion) 5. Conclusions In his paper we presened an ineger programming model of a hop consrained min-sum arborescence problem wih node ouage coss, in which he erminal nodes in he nework mus be conneced o a cenral node wih a consrain ha limis he maximum number of links beween he cenral node and each erminal node o a predeermined number h. We have suggesed a Lagrangian based heurisic o find a low cos feasible soluion. The lower bound found as a byproduc of he soluion procedure is used o esimae he qualiy of he heurisic soluion. Compuaional resuls for a variey of problems are repored. In our compuaional experimen, he average gap beween he heurisic soluion value and he opimal soluion is shown o be wihin 14 percen. Proceedings of he 36h Hawaii Inernaional Conference on Sysem Sciences (HICSS 03)

5 6. References [1] M. Fischei and P. Toh, "An efficien algorihm for he min-sum arborescence problem on complee digraphs," ORSA Journal on Compuing, vol. 5, no. 4, pp , [] M. L. Fisher, "The Lagrangian relaxaion mehod for solving ineger programming problems", Managemen Science, vol. 7, 1-18, [3] L. Gouveia, Mulicommodiy flow models for spanning rees wih hop consrains, European Journal of Operaional Research, vol. 95, , [4] M. Held, P. Wolfe, and H. D. Crowder, "Validaion of Subgradien Opimizaion", Mahemaical Programming, vol. 6, 6-88, [5] R. Larson, and A. Odoni, "Urban Operaions Research", Prenice Hall, Englewood Cliffs, N. J., [6] L. LeBlanc and R. Reddoch, Reliable link opology/capaciy design and rouing in backbone elecommunicaion neworks, paper presened a he Firs ORSA Telecommunicaions SIG conference, Proceedings of he 36h Hawaii Inernaional Conference on Sysem Sciences (HICSS 03)