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21 Pages

### Facility

Course: MSANDE 310, Fall 2009
School: Stanford
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Word Count: 631

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Location Facility using Linear Programming Duality Yinyu Ye Department if Management Science and Engineering Stanford University Facility Location Problem Input A set of clients or cities D A set of facilities F with facility cost fi Connection cost Cij, (obey triangle inequality) Output A subset of facilities F' An assignment of clients to facilities in F' Objective Minimize the total cost...

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Location Facility using Linear Programming Duality Yinyu Ye Department if Management Science and Engineering Stanford University Facility Location Problem Input A set of clients or cities D A set of facilities F with facility cost fi Connection cost Cij, (obey triangle inequality) Output A subset of facilities F' An assignment of clients to facilities in F' Objective Minimize the total cost (facility + connection) Facility Location Problem location of a potential facility (opening cost) (connection cost) client Facility Location Problem location of a potential facility (opening cost) client (connection cost) min opening cost + connection cost R-Approximate Solution and Algorithm An algorithm found a feasible (integral) solution of UFLP, with the total cost, Cost , that satisfies the following : Cost R Cost * for some constant R 1. Hardness Results yNP-hard. Cornuejols, Nemhauser & Wolsey [1990]. y1.463 polynomial approximation algorithm implies NP =P. Guha & Khuller [1998], Sviridenko [1998]. ILP Formulation Min s.t. C iF jD ij ij xij + f i yi iF x iF =1 j D j D, i F xij yi xij , yi {0,1} j D, i F Each client should be assigned to one facility. Clients can only be assigned to open facilities. LP Relaxation and its Dual Min s.t. C x + f y i F j D ij ij ij i F i i Max s.t. j D j x i F =1 j D j D, i F j D, i F j - ij cij j D, i F xij yi xij 0 j D ij fi i F j D, i F ij 0 Interpretation: clients share the cost to open a facility, and pay the connection cost. ij = max{0, j - cij } is the contribution of client j to facility i. Bi-Factor Dual Fitting Suppose an algorithm found a feasible (integral) solution of FLP, with the total cost j , where j satisfies the following : jD (1) (2) - ij Rc cij j , D i F j i F ij R f f i j D for some constant Rc , R f 1 and ij 0, then we have : F + C = j R f F * + Rc C * . jD A bi-factor (R ,R )-approximate algorithm is a Simple Greedy Algorithm Jain et al [2003] Introduce notion a of time, such that each event can be associated with the time at which it happened. The algorithm start at time 0. Initially, all facilities are closed; all clients are unconnected; all set to 0. Let C=D j j C j C While , increase simultaneously for all , until one of the following events occurs: i such that j = j C (1). For some client , and a open facility , cij then connect client j to facility i and remove j from C; max(0, j - cij ) = fi (2). For some closed facility i, , then open jC j cij j C facility i, and connect client with to facility i, and remove j from C. Time = 0 F1=3 F2=4 3 5 4 3 6 4 Time = 1 F1=3 F2=4 3 5 4 3 6 4 Time = 2 F1=3 F2=4 3 5 4 3 6 4 Time = 3 F1=3 F2=4 3 5 4 3 6 4 Time = 4 F1=3 F2=4 3 5 4 3 6 4 Time = 5 F1=3 F2=4 3 5 4 3 6 4 Time = 5 F1=3 F2=4 Open the facility on left, and connect clients "green" and "red" to it. 3 5 4 3 6 4 Time = 6 F1=3 F2=4 Continue increase the budget of client "blue" 3 5 4 3 6 4 Time = 6 F1=3 ...

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