Simplex Method Overview

00:06:29
https://www.youtube.com/watch?v=5fcbjxZbAXo

摘要

TLDRThe video provides a detailed overview of the Simplex method, used in linear programming to find optimal solutions efficiently. Initially, the system is set up in Tableau format, arranging columns for the objective function and constraints. Basic variables, usually slack variables in the initial step, provide a basic feasible solution. Testing for optimality involves examining the objective row; positive entries indicate optimality, while non-positive indicate possible improvements. The method involves cycles of choosing entering and exiting variables, pivoting, and checking for improvements until the optimal solution is reached. Under certain conditions, such as negative pivot columns denoting unbounded solutions, the problem may be ill-posed or lack constraints. The overview concludes by emphasizing the cyclic nature of the Simplex algorithm and encourages further study of the method.

心得

  • 📊 Set up the system in Tableau format, including slack variables and objective function.
  • 🔍 Check the objective row for non-negative entries to determine if the solution is optimal.
  • 🔄 Cycles of choosing entering/exiting variables are key in the Simplex process.
  • 🔗 The ratio test helps select the exiting variable by finding the smallest positive ratio.
  • 🚫 Unbounded solutions indicate possible issues with system constraints.
  • ⚙️ Pivoting adjusts the system to improve towards an optimal solution.
  • ✔️ Repeat steps in the Simplex cycle until optimal conditions are met.
  • 🔗 Multiple optimal solutions may exist if non-basic variables have zero objective value.
  • 📈 Unbounded solutions suggest infinite improvement potential without constraints.
  • 📚 Understanding the Simplex overview aids in grasping linear programming solutions.

时间轴

  • 00:00:00 - 00:06:29

    The video provides an overview of the Simplex method, outlining its process from a high-level perspective. Initially, the system is set up in a Tableau format, organizing columns for the objective function Z, original variables, slack variables, and the right-hand side. The initial basic variables are chosen to form a basic feasible solution, often beginning with slack variables. The next step involves testing for optimality by evaluating the objective row; if any entries are negative, the corresponding variable can be brought into the basis. The process continues with a ratio test to determine the exiting variable, focusing on the smallest positive ratio. The pivot operation follows, where the tableau is adjusted by focusing on the entering variable column and the exiting variable row. The method iterates through these steps until an optimal solution is found, while also discussing scenarios like multiple optimal solutions, degeneracy, and unbounded solutions. Lastly, the video encourages viewers to engage with additional resources for a deeper understanding.

思维导图

Mind Map

常见问题

  • What is the first step of the Simplex method?

    The first step is setting up the system in the Tableau format.

  • How do you check for optimality in the Simplex method?

    Check the objective row; if all entries are non-negative, the solution is optimal.

  • What happens if all pivot column values are zero or negative?

    This indicates an unbounded solution, suggesting the problem may be ill-posed.

  • What is the purpose of the ratio test in the Simplex method?

    The ratio test helps choose the exiting variable by finding the smallest positive ratio.

  • What are basic variables in the Simplex method?

    Initially, slack variables are chosen as basic variables providing a feasible solution.

  • Why might a problem be unbounded in the Simplex method?

    An unbounded problem might be lacking constraints or be ill-posed.

  • What do you do after pivoting in the Simplex method?

    Repeat steps of checking for optimality, choosing entering and exiting variables until optimality is reached.

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  • 00:00:02
    okay in this video we're going to give
  • 00:00:03
    an overview of the Simplex method so we
  • 00:00:05
    saw an example of it and sort of worked
  • 00:00:07
    through it now let's talk about it from
  • 00:00:09
    a bird's eyee View and sort of discuss
  • 00:00:11
    the steps along the way so uh first step
  • 00:00:15
    is setup put your system into the
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    Tableau format um which means that we're
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    sort of going to arrange um columns for
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    Z each of our original variables which
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    I'm kind of writing an X Vector here to
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    kind of say all those X variables all
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    your slack variables and your right hand
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    side and so columns for each of them um
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    and sort of if this is sort of our setup
  • 00:00:40
    for our maximization LP and augmented
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    form uh then this is what that Tableau
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    is going to look like okay initially
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    we're going to like our constraint
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    Matrix is going to show up down here
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    we're going to have an identity Matrix
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    we're going to have our right hand side
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    and then this line here is essentially
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    our objective row which is going to have
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    a one for Z all of our coefficients of
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    our x uh of of that objective function
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    are going to get negatives when we move
  • 00:01:08
    them over to the left side of the
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    equation and then slack variables don't
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    show up in our objective row and so
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    they're all going to have zeros um in
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    the objective
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    row so that's uh the setup and we'll
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    we'll see that a couple times um next
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    step is to choose an initial set of
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    basic variables that give a basic
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    feasible solution okay in practice this
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    is difficult but to begin with we'll say
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    that we want to choose just the slack of
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    variables as basic variables okay and um
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    at least for these first few problems
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    that we do that will work every time and
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    it will sort of started us off at 000000
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    in the original um you in the original
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    variables and as long as that 0000 is a
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    feasible solution um that's a good place
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    to start okay next step is we test for
  • 00:02:01
    optimality we look at the objective row
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    and if any of the entries are negative
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    we can bring that variable into the uh
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    into the basis okay um I use the kind of
  • 00:02:14
    rule of thumb of choosing the most
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    negative um because that will sort of
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    lead to the fastest Improvement but
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    there's other heris that are used in
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    practice that's just the simplest one to
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    remember so any negative variable would
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    work you choose which ever one you
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    want um if all the entries in that
  • 00:02:34
    objective row are positive for a
  • 00:02:36
    maximization or I should say if you're
  • 00:02:38
    doing minimization the the signs get
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    flipped here um if any of those if all
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    of them are positive then we've reached
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    the optimal value we're we're at a
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    stopping point we're good to
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    go um if all of them are non- negative
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    and we got some non-basic variables that
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    have zero objective value then there
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    might actually be or there will be
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    multiple equivalent Optimal Solutions
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    okay and so sometimes we get a a tie
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    where multiple Corner points have the
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    same objective value and then actually
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    the entire Edge between them uh actually
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    also has the same objective value um
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    that happens there might be multiple uh
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    optimal values okay and and if we wanted
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    to we could do another iteration with
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    that zero cost column as the entering
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    variable um to find the other Optimal
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    Solutions okay once we check for
  • 00:03:33
    optimality choose an entering variable
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    in the same step then we need to choose
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    an exit uh variable and to do that we do
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    use a ratio test where we take the right
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    hand side divided by um the values in
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    our entering variable column okay if the
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    entering variable column has zeros or
  • 00:03:55
    negatives ignore those rows um but
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    otherwise take right hand side to
  • 00:03:59
    divided by um entering variable column
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    okay we want to choose the smallest uh
  • 00:04:06
    ratio uh that's positive
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    um as our exiting variable as our pivot
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    row
  • 00:04:19
    um yeah if the smallest ratio is zero uh
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    then you're at a degenerate solution
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    where sort of U multiple or some of the
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    basic variables were zero um then choose
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    that and use use that row as the exiting
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    variable um if all the pivot column
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    values are negative or zero um then
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    you've actually got an unbounded
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    solution in that direction and you've
  • 00:04:45
    got a objective that's unbounded which
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    means that you're moving in a direction
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    that has Improvement for your problem
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    and you can keep moving in that
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    direction off to infinity and so usually
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    this means that your problem was ill
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    posed in some way and that you were
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    maybe missing a constraint when you
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    formulated or set up the problem okay or
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    sometimes it yeah just says that you can
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    you can uh make your objective
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    arbitrarily large which um sounds nice
  • 00:05:15
    usually there's some actual limiting in
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    in
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    practice uh next step is we pivot on our
  • 00:05:21
    entering variable column and exiting
  • 00:05:23
    variable row um so we Circle that row
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    that column we want to turn that uh
  • 00:05:29
    entering variable column into a standard
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    basis Vector with one in the exiting
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    variable row and zero everywhere else
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    okay use Elementary row operations to do
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    that using that exiting row that pivot
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    row as your sort of tool and adding or
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    subtracting multiples of it to other
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    rows then we get into a loop uh and so
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    weeat steps three through five
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    in a
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    loop okay uh we Loop through steps three
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    to five until finally when we come back
  • 00:06:05
    to step three we test for optimality and
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    find that we've uh reached optimality
  • 00:06:11
    okay so this was an overview video the
  • 00:06:14
    Simplex method I encourage you take a
  • 00:06:16
    while to read through this handout um
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    and kind of understand what all is going
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    on um but that's the overview I'll stop
  • 00:06:23
    this video and in the next video we'll
  • 00:06:25
    go through another example
标签
  • Simplex method
  • linear programming
  • Tableau format
  • optimality
  • basic variables
  • ratio test
  • pivoting
  • unbounded solution