LSA_9-10-11_location_discrete_models.pdf

# Maximum covering cover the most customer demands with

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Maximum covering Cover the most customer demands with no more than P facilities Data: h i = demand at customer i . Variable: x j = 1 if we locate a facility at j , 0 otherwise z i = 1 if customer i is covered, 0 otherwise max . . , { , }, ; { , }, h z s t x P a x z i I x j J z i I i i i j j ij j j i j i ∀ ∈ ∀ ∈ ∀ ∈ 0 1 0 1 Logistics Systems Analysis Center models P-Center problem (minimax): Locate P centers to minimize the maximum distance between each customer and its nearest facility (facilities now called centers). Formulation: Customers set I , candidate facility location set J ; d ij = distance from customer i to candidate location j . Variable: x j = 1 if we locate a facility at j , 0 otherwise y ij = 1 if customer i is covered by a facility at j , 0 otherwise min . . , , , , { , }, { , }, , D s t x P y x i j y i d y D i x j y i j j j ij j ij j ij ij j j ij = = 1 0 1 0 1 …………..…… Maximum distance btwn customer and its nearest facility Relationship between set covering models and center models: If ( X* , D* ) is the optimal locations and maximum distances for a P-center problem, then ( P , X* ) is the optimal solution to the related set covering problem with covering distance D=D* ; and vice versa. Thus, we can use set covering model solution techniques to solve center problems through a binary search for D* --- find the minimum D=D* that will make ( P , X* ) a feasible solution to the set covering problem. …………..…… P facilities …………..…… to cover i by j , facility must exist at j …..…… demand at i be covered by some (nearest) facility ………..…… D is the maximum distance for all customers y i j ij 0, , …. or if we allow split of demand Number of facility P Coverage distance D

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Logistics Systems Analysis Set covering / Maximum covering / Center models Adapted from Daskin (1995). The set covering, maximum covering, and center models are closely related. Set Covering Given: customer set I , candidate location set J , distances, and coverage distance D Find: Minimum number (and location) of facilities to cover ALL customers (May require a lot of facilities) Maximum Covering Given: set covering inputs, PLUS number of facilities P , demand at each customer h i . Find: Locations of P facilities to MAXIMIZE covered demand P-Center Given: customer set I , candidate location set J , distances, and number of facilities P . Find: Locations of P facilities to cover all demand while coverage distance D is MINIMIZED (relax total coverage requirement) (relax coverage distance) Logistics Systems Analysis Median models Locate P facilities to minimize the total (or average) distance between the customers and their nearest facilities. Formulation: Customers set I , candidate facility location set J ; h i is the demand at i , d ij is the distance between customer i and location j.
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