CEE
LSA_9-10-11_location_discrete_models.pdf

# Once we find the convergent solution to lr we need to

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Once we find the convergent solution to (LR), we need to find a feasible solution: • For each facility where x j = 1, set z i = 1 for all i ’s that are covered by j . This gives a (feasible) lower bound.

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Logistics Systems Analysis LR Applications – Median I Median Problem Integer Program (IP) relax Lagrangian Relaxation (LR) For any given u , the inner minimization problem can be solved as follows • Let be the value of facility at j . Find the P smallest v j ’s and set the corresponding x j = 1; 0 otherwise y ij = 1 if x j = 1 and ( h i d ij u i ) < 0; 0 otherwise v d h u j ij i i i = min( , ) 0 Then apply the iterative sub-gradient procedure to solve for optimal u. The optimal objective value of (LR) is a lower bound (may not be feasible). Once we find the convergent solution to (LR), we need to find a feasible solution: • use the solution of x j ’s ; allocate customer demands to nearest open facility. This gives an (feasible) upper bound. min . . d y h s t x P ij ij i i j j j = y x i j y i x j y i j ij j ij j j ij = , , , { , }, ; { , }, , 1 0 1 0 1 maxmin ( ) . . , + = u x y ij ij i i j i ij j i j j d y h u y s t x P 1 y x i j x j y i j ij j j ij , , { , }, ; { , }, , 0 1 0 1 The LR objective function equals maxmin ( ) + u y ij i i ij i j i i d h u y u Logistics Systems Analysis LR Applications – Median II Median Problem Integer Program (IP) relax Lagrangian Relaxation (LR) For any given u 0, the inner minimization problem can be separated into two problems (for x and y ) • For each i, calculate ( h i d ij + u ij ) for all j . Find the minimum j and set y ij = 1. • Let . Find the P largest v j ’s and set the corresponding x j = 1; 0 otherwise Then apply the iterative sub-gradient procedure to solve for optimal u. The optimal objective value of (LR) is a lower bound (may not be feasible). Once we find the convergent solution to (LR), we need to find a feasible solution: • use the solution of x j ’s ; allocate customer demands to nearest open facility. This gives an (feasible) upper bound. min . . d y h s t x P ij ij i i j j j = y x i j y i x j y i j ij j ij j j ij = , , , { , }, ; { , }, , 1 0 1 0 1 maxmin ( ) . . , , u x y ij ij i i j ij ij j i j j j d y h u y x s t x P + = 0 y i x j y i j ij j j ij = 1 0 1 0 1 , { , }, ; { , }, , The LR objective function becomes maxmin ( ) , u x y ij i ij ij i j ij i j j d h u y u x + 0 v u j ij i =
Logistics Systems Analysis LR Applications – Uncapacitated Fixed charge Uncapacitated Fixed charge Integer Program (IP) relax Lagrangian Relaxation (LR) For any given u , the inner minimization problem can be separately solved for x and y • Let be the value of facility at j . If v j < 0, then set the corresponding x j = 1; 0 otherwise y ij = 1 if x j = 1 and ( h i d ij u i ) < 0; 0 otherwise v f d h u j j ij i i i = + min( , ) α 0 Then apply the iterative sub-gradient procedure to solve for optimal u. The optimal objective value of (LR) is a lower bound (may not be feasible).
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