Modeling following constraints in MILPI want to know how I should formulate the following constraints in my MIP problem? x = x 1 z 1 + ⋯ + x n z n and y 1 ≤ y ≤ y n and z 1 + ⋯ + z n = 1OR y = y 1 w 1 + ⋯ + y n w n and x 1 ≤ x ≤ x n and w 1 + ⋯ + w n = 1x and y are continuous variables. z 1 , … , z n and w 1 , … , w n are binary decision variables. x 1 , … , x n and y 1 , … , y n are parameters.

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Modeling following constraints in MILPI want to know how I should formulate the following constraints in my MIP problem?  x  =      x    1        z    1    +  ⋯  +      x    n        z    n     and       y    1    ≤  y  ≤      y    n     and       z    1    +  ⋯  +      z    n    =  1OR  y  =      y    1        w    1    +  ⋯  +      y    n        w    n     and       x    1    ≤  x  ≤      x    n     and       w    1    +  ⋯  +      w    n    =  1x and y are continuous variables.       z    1    ,  …  ,      z    n   and       w    1    ,  …  ,      w    n   are binary decision variables.       x    1    ,  …  ,      x    n   and       y    1    ,  …  ,      y    n   are parameters.

Szulikto · Accepted Answer

Step 1Brace yourself, the notation is about to get a little ugly.Let me list the variables first. In addition to x and y (continuous), we will have       z    1    ,  …  ,      z    n    ∈  {  0  ,  1  } and       w    1    ,  …  ,      w    n    ∈  {  0  ,  1  }, plus             z      ^        ,            z      ~        ,            w      ^        ,            w      ~        ∈  [  0  ,  1  ] and one more binary variable   t  ∈  {  0  ,  1  }. The constraints will be as follows:                    x                            =                  ∑                      i                                    x                      i                                    z                      i                          +                  x                      1                                                z            ^                          +                  x                      n                                                z            ~                                              y                            =                  ∑                      i                                    y                      i                                    w                      i                          +                  y                      1                                                w            ^                          +                  y                      n                                                w            ~                                                        ∑                      i                                    z                      i                                              =        1        −        t                                                  z            ^                          +                              z            ~                                              =        t                                      ∑                      i                                    w                      i                                              =        t                                                  w            ^                          +                              w            ~                                              =        1        −        t        .            Step 2If   t  =  0, the fourth equation zeroes out the last two terms of the first equation and the third equation plus the first equation result in x being one of the       x    i  . Meanwhile, the fifth equation zeroes out the summation in the second equation and the sixth equation and what&#039;s left of the second equation make y a convex combination of the endpoints       y    1  ,       y    n  , so basically any value in the interval   [      y    1    ,      y    n    ]. If   t  =  1, the reverse occurs.

varsa1m · Accepted Answer

Step 1                    (1)                              x                      min                          ≤        x                            ≤                  x                      max                                              (2)                              y                      min                          ≤        y                            ≤                  y                      max                                              (3)                              (                      x                          min                                −                      ∑            i                                x            i                    )                t        ≤        x        −                  ∑          i                          x          i                          z          i                                    ≤                  x                      max                          ⋅        t                            (4)                    −        t        ≤                  ∑          i                          z          i                −        1                            ≤        (        n        −        1        )        t                            (5)                              (                      y                          min                                −                      ∑            i                                y            i                    )                (        1        −        t        )        ≤        y        −                  ∑          i                          y          i                          w          i                                    ≤                  y                      max                          (        1        −        t        )                            (6)                    −        (        1        −        t        )        ≤                  ∑          i                          w          i                −        1                            ≤        (        n        −        1        )        (        1        −        t        )            Step 2Constraints (1) and (2) are valid for both sides of the desired disjunction. Constraints (3) and (4) enforce  t  =  0    ⟹        (    x    =          ∑      i              x      i              z      i        ∧          ∑      i              z      i        =    1    )    .Constraints (5) and (6) enforce  t  =  1    ⟹        (    y    =          ∑      i              y      i              w      i        ∧          ∑      i              w      i        =    1    )    .

Modeling following constraints in MILP. I want to know how I should formulate the following constraints in my MIP problem? x=x_1z_1+⋯+x_n z_n and y_1 <= y <= y_n and z_1+ cdots +z_n=1

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