LSA_9-10-11_location_discrete_models.pdf

Maximum covering cover the most customer demands with

Info icon This preview shows pages 2–4. Sign up to view the full content.

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
Image of page 2

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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.
Image of page 3
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern