Linear programming: modelling a transportation problem

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I have a transportation problem with a twist-- I have a set number of managers and a set number of stores (where there are way more stores than managers), and I'm trying to match the managers to stores. The twist is that each manager can have up to a certain number of stores (constraint), and no two managers share the same store. The goal is to minimise the distance between the manager and each store s/he manages. I haven't been able to figure out how to model this problem, or rather if it is even possible to formulate this problem into a linear programming model. Please let me know!




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  • Is this not better thought of as an assignment problem rather than a transportation problem? You will need multiple rows for each manager to represent the multiplicity of stores that can be managed by each. To keep the problem balanced you will also need dummy stores (representing managers not getting a store assignment). The solution is likely to be highly none unique.
    – Paul
    Aug 27 at 18:16














up vote
0
down vote

favorite












I have a transportation problem with a twist-- I have a set number of managers and a set number of stores (where there are way more stores than managers), and I'm trying to match the managers to stores. The twist is that each manager can have up to a certain number of stores (constraint), and no two managers share the same store. The goal is to minimise the distance between the manager and each store s/he manages. I haven't been able to figure out how to model this problem, or rather if it is even possible to formulate this problem into a linear programming model. Please let me know!




optimization linear-programming mathematical-modeling




share|cite|improve this question




















  • Is this not better thought of as an assignment problem rather than a transportation problem? You will need multiple rows for each manager to represent the multiplicity of stores that can be managed by each. To keep the problem balanced you will also need dummy stores (representing managers not getting a store assignment). The solution is likely to be highly none unique.
    – Paul
    Aug 27 at 18:16












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I have a transportation problem with a twist-- I have a set number of managers and a set number of stores (where there are way more stores than managers), and I'm trying to match the managers to stores. The twist is that each manager can have up to a certain number of stores (constraint), and no two managers share the same store. The goal is to minimise the distance between the manager and each store s/he manages. I haven't been able to figure out how to model this problem, or rather if it is even possible to formulate this problem into a linear programming model. Please let me know!




optimization linear-programming mathematical-modeling




share|cite|improve this question












I have a transportation problem with a twist-- I have a set number of managers and a set number of stores (where there are way more stores than managers), and I'm trying to match the managers to stores. The twist is that each manager can have up to a certain number of stores (constraint), and no two managers share the same store. The goal is to minimise the distance between the manager and each store s/he manages. I haven't been able to figure out how to model this problem, or rather if it is even possible to formulate this problem into a linear programming model. Please let me know!





optimization linear-programming mathematical-modeling





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share|cite|improve this question




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asked Aug 27 at 17:37









user3755880

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  • Is this not better thought of as an assignment problem rather than a transportation problem? You will need multiple rows for each manager to represent the multiplicity of stores that can be managed by each. To keep the problem balanced you will also need dummy stores (representing managers not getting a store assignment). The solution is likely to be highly none unique.
    – Paul
    Aug 27 at 18:16
















  • Is this not better thought of as an assignment problem rather than a transportation problem? You will need multiple rows for each manager to represent the multiplicity of stores that can be managed by each. To keep the problem balanced you will also need dummy stores (representing managers not getting a store assignment). The solution is likely to be highly none unique.
    – Paul
    Aug 27 at 18:16















Is this not better thought of as an assignment problem rather than a transportation problem? You will need multiple rows for each manager to represent the multiplicity of stores that can be managed by each. To keep the problem balanced you will also need dummy stores (representing managers not getting a store assignment). The solution is likely to be highly none unique.
– Paul
Aug 27 at 18:16




Is this not better thought of as an assignment problem rather than a transportation problem? You will need multiple rows for each manager to represent the multiplicity of stores that can be managed by each. To keep the problem balanced you will also need dummy stores (representing managers not getting a store assignment). The solution is likely to be highly none unique.
– Paul
Aug 27 at 18:16










1 Answer
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Let $x_ij$ be a binary variable that takes value $1$ if and only if manager $i$ is assigned to store $j$.



You want to minimize the total distance after assignment:
$$
sum_i,jd_ijx_ij
$$
subject to :



  • A manager $i$ cannot have more than $S_i$ stores :
    $$
    sum_jx_ij le S_iquad forall i
    $$

  • No two managers share the same store :
    $$
    sum_ix_ij =1 quad forall j
    $$

  • Variables are binary :
    $$
    x_ijin 0,1 quad forall i,j
    $$





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    1 Answer
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    1 Answer
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    up vote
    0
    down vote













    Let $x_ij$ be a binary variable that takes value $1$ if and only if manager $i$ is assigned to store $j$.



    You want to minimize the total distance after assignment:
    $$
    sum_i,jd_ijx_ij
    $$
    subject to :



    • A manager $i$ cannot have more than $S_i$ stores :
      $$
      sum_jx_ij le S_iquad forall i
      $$

    • No two managers share the same store :
      $$
      sum_ix_ij =1 quad forall j
      $$

    • Variables are binary :
      $$
      x_ijin 0,1 quad forall i,j
      $$





    share|cite|improve this answer
























      up vote
      0
      down vote













      Let $x_ij$ be a binary variable that takes value $1$ if and only if manager $i$ is assigned to store $j$.



      You want to minimize the total distance after assignment:
      $$
      sum_i,jd_ijx_ij
      $$
      subject to :



      • A manager $i$ cannot have more than $S_i$ stores :
        $$
        sum_jx_ij le S_iquad forall i
        $$

      • No two managers share the same store :
        $$
        sum_ix_ij =1 quad forall j
        $$

      • Variables are binary :
        $$
        x_ijin 0,1 quad forall i,j
        $$





      share|cite|improve this answer






















        up vote
        0
        down vote










        up vote
        0
        down vote









        Let $x_ij$ be a binary variable that takes value $1$ if and only if manager $i$ is assigned to store $j$.



        You want to minimize the total distance after assignment:
        $$
        sum_i,jd_ijx_ij
        $$
        subject to :



        • A manager $i$ cannot have more than $S_i$ stores :
          $$
          sum_jx_ij le S_iquad forall i
          $$

        • No two managers share the same store :
          $$
          sum_ix_ij =1 quad forall j
          $$

        • Variables are binary :
          $$
          x_ijin 0,1 quad forall i,j
          $$





        share|cite|improve this answer












        Let $x_ij$ be a binary variable that takes value $1$ if and only if manager $i$ is assigned to store $j$.



        You want to minimize the total distance after assignment:
        $$
        sum_i,jd_ijx_ij
        $$
        subject to :



        • A manager $i$ cannot have more than $S_i$ stores :
          $$
          sum_jx_ij le S_iquad forall i
          $$

        • No two managers share the same store :
          $$
          sum_ix_ij =1 quad forall j
          $$

        • Variables are binary :
          $$
          x_ijin 0,1 quad forall i,j
          $$






        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Aug 28 at 7:02









        Kuifje

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