The Fundamental Insight of the Simplex Method

00:18:51
https://www.youtube.com/watch?v=ANPWXQGFfbY

Ringkasan

TLDRThis video serves as an introduction to duality in linear programming, with a focus on the Simplex method. It explains linear programs in matrix form, highlighting objective values, constraints, and slack variables. Initial tableau transformation through row operations is discussed, involving the selection of entering and exiting variables to achieve the standard basis vectors in columns. Dual variables, or shadow prices, are introduced alongside their significance in terms of reduced costs. The concept of a basis being optimal is linked to the reduced costs being non-negative, providing a foundation for duality, which will be further explored in subsequent videos.

Takeaways

  • 🤔 Duality is crucial for understanding linear programming.
  • 🧮 The Simplex method relies on transforming the tableau via row operations.
  • 🔄 Slack variables convert inequalities into equalities in linear programs.
  • 🔗 Identifying the basis matrix helps in solving linear programs effectively.
  • 💡 Dual variables provide insights into changes in objective value.
  • 📉 Reduced costs help in determining optimization efficiency.
  • ✅ A basis is optimal when reduced costs are non-negative.
  • 🔍 Understanding matrix inverses is key to the Simplex method.
  • 🔄 Duality connects primal and dual problems in optimization.
  • 📊 Shadow prices offer a perspective on constraint impacts.

Garis waktu

  • 00:00:00 - 00:05:00

    The video introduces the concept of Duality in the context of linear programming and the Simplex method, based on the Hillier textbook. It starts by outlining the formulation of a linear program in matrix form, where the objective function is represented as a product of the objective coefficients vector and the variable vector X. The constraints are introduced both in terms of less than or equal conditions and non-negativity of the variables. The starting point for solving this problem using the Simplex method is the initial tableau, which organizes the objective function and constraints into a structured format. The tableau is explained with columns for the objective function Z, original variables, slack variables, and the right-hand side. The process of selecting entering and exiting variables and performing row operations to pivot from one basic solution to another is introduced. The elementary row operations are conceptualized as matrix multiplications, emphasizing the role of the basis matrix and its inverse in these transformations.

  • 00:05:00 - 00:10:00

    The second section elaborates on the transformations applied to the initial tableau in the Simplex method. The concept of multiplying the tableau by the inverse of a basis matrix is explained, using block matrices for illustration. This transformation aims to convert blocks of the tableau into standard basis vectors and the identity matrix. The discussion involves verifying the matrix multiplication process and the role of block matrices in achieving the desired transformation. The importance of the basis matrix inverse in determining solutions and performing row operations in the Simplex method is highlighted. The section further explains how the right-hand side of the tableau represents the current solution for the basic variables. The introduction of dual variables or shadow prices as part of the tableau transformation process is mentioned, setting the stage for further discussion on their significance.

  • 00:10:00 - 00:18:51

    The final part focuses on interpreting the modified tableau and identifying optimal solutions within the framework of duality. Key concepts such as the current objective value, reduced costs, and dual variables are introduced. The reduced costs are described as measures to determine the optimality of a basis by assessing if they are non-negative, adhering to the conditions of the maximization problem. The video outlines the conditions under which the tableau or basis is considered optimal, relating this to non-negativity constraints on the dual variables and other constraints involving the original variables and the objective function coefficients. The suggestion that these conditions parallel constraints in a broader duality framework is noted, leading to the concluding remarks that set up the next video’s focus on shadow prices, which would further explore the relationship between these dual variables and the original linear programming problem.

Peta Pikiran

Mind Map

Pertanyaan yang Sering Diajukan

  • What is the fundamental insight of the Simplex method?

    The fundamental insight of the Simplex method involves understanding how to apply row operations to transition from one tableau to another while managing constraints, objective functions, and slack variables.

  • What are slack variables?

    Slack variables are additional variables introduced in linear programming to transform inequality constraints into equality constraints in the Simplex method.

  • Why is the inverse of a matrix important in the Simplex method?

    The inverse of the basis matrix is used to perform row operations that transform the tableau. It aids in obtaining the identity matrix, which helps in solving the linear program.

  • What are dual variables and what do they represent?

    Dual variables, also known as shadow prices, appear in the dual problem of a linear program. They can be interpreted as the rate of change of the objective value with respect to the constraints.

  • What are reduced costs in linear programming?

    Reduced costs indicate how much the objective function coefficient must improve before a non-basic variable can enter the basis. They are derived from the original cost minus the product of dual variables and the constraint matrix.

  • When is a basis considered optimal in linear programming?

    A basis is considered optimal if all the reduced costs are greater than or equal to zero for a maximization problem and if the dual variables are greater than or equal to zero.

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Gulir Otomatis:
  • 00:00:02
    welcome to another video today's uh
  • 00:00:05
    video is the first one in talking about
  • 00:00:07
    Duality um so first topic we're going to
  • 00:00:10
    talk about with Duality is uh what
  • 00:00:12
    you're the text the Hillier textbook
  • 00:00:14
    calls the fundamental Insight of the
  • 00:00:16
    Simplex method so uh let's think about a
  • 00:00:21
    linear program in Matrix form so uh
  • 00:00:25
    looking at this Matrix uh sort of see th
  • 00:00:28
    that those are the coeffic ients of the
  • 00:00:31
    objective vector and so we'll call Z our
  • 00:00:35
    objective value these are going to be
  • 00:00:37
    the uh
  • 00:00:39
    objective
  • 00:00:42
    coefficients and we can Co collect them
  • 00:00:45
    together
  • 00:00:46
    into uh a vector that we're multiplying
  • 00:00:49
    by our X Vector
  • 00:00:52
    X is our
  • 00:00:56
    variables and so a * X that's our con
  • 00:01:00
    that a is the constraint
  • 00:01:08
    Matrix and then the
  • 00:01:10
    B is the right hand side essentially the
  • 00:01:13
    limits of each of the
  • 00:01:15
    constraints uh and so typical
  • 00:01:18
    maximization problem in standard form uh
  • 00:01:22
    we with less than or equal to
  • 00:01:24
    constraints and then each of our
  • 00:01:26
    variables is constrained to be non-
  • 00:01:28
    negative
  • 00:01:30
    okay once we introduce slack variables
  • 00:01:34
    uh this is the starting Tableau for um
  • 00:01:38
    this problem so uh kind of labeling the
  • 00:01:40
    columns this First Column will be for Z
  • 00:01:44
    this is actually a chunk of columns
  • 00:01:46
    several columns and those are all the
  • 00:01:48
    columns corresponding to our original
  • 00:01:51
    variables these columns here and there's
  • 00:01:54
    going to be several of them uh those are
  • 00:01:57
    all the columns that correspond to our
  • 00:01:59
    slack VAR variables and then this here
  • 00:02:02
    is our right hand
  • 00:02:04
    side okay and then we could I guess
  • 00:02:06
    think of uh labeling the rows this top
  • 00:02:09
    row that I've got put here that's the
  • 00:02:13
    the zeroth row the row zero uh
  • 00:02:15
    corresponding to the objective and then
  • 00:02:19
    um all the rest of the rows and and this
  • 00:02:21
    is a whole Matrix here so there's
  • 00:02:23
    usually um M constraints and so this is
  • 00:02:26
    this part here actually represents all
  • 00:02:28
    the other rows um and those correspond
  • 00:02:31
    to our our um basis and we'll sort of
  • 00:02:34
    abbreviate all the variables that are in
  • 00:02:37
    our basis as x with a subscript of B for
  • 00:02:40
    for our basic
  • 00:02:42
    variables okay so this is our initial
  • 00:02:45
    tblo and then uh what normally happens
  • 00:02:49
    in the Simplex method is that uh we
  • 00:02:51
    choose an entering uh basic variable and
  • 00:02:53
    an exiting basic variable and then we do
  • 00:02:56
    row operations to change from this
  • 00:02:57
    Tableau to another Tableau and when we
  • 00:03:00
    move to another Tableau uh we always
  • 00:03:02
    make it so that our basic variables uh
  • 00:03:05
    have the standard basis vectors showing
  • 00:03:08
    up in those
  • 00:03:09
    columns okay well the elementary row
  • 00:03:12
    operations that we use to move from this
  • 00:03:15
    uh Tableau to the next Tableau uh we can
  • 00:03:19
    think of that as actually multiplying on
  • 00:03:22
    the left by a matrix okay and so when we
  • 00:03:27
    choose our basis um remember we always
  • 00:03:31
    have Z in our basis and then all the
  • 00:03:33
    columns that correspond to um our basis
  • 00:03:37
    we'll call those XB representing all the
  • 00:03:40
    variables in our
  • 00:03:41
    basis if you were to take those columns
  • 00:03:44
    out of this uh Tableau up here and
  • 00:03:48
    collect them all together into a matrix
  • 00:03:51
    and this will be this will have um m +
  • 00:03:54
    one columns and m+ one rows um because
  • 00:03:57
    there's M basic variables one for each
  • 00:03:59
    constraint um as well as the Z kind of
  • 00:04:02
    is the bonus basic variable um that's
  • 00:04:05
    always
  • 00:04:08
    there okay so uh we take The Columns of
  • 00:04:12
    the basis collect them here uh We've
  • 00:04:14
    called B basis Matrix um these are the
  • 00:04:17
    columns uh of a and potentially of I um
  • 00:04:22
    that show up in the basis uh negative CB
  • 00:04:26
    so um remember when we get our
  • 00:04:30
    objective we've got Z equals c transpose
  • 00:04:34
    X um but we want to get all of our
  • 00:04:37
    variables on to the right uh to the left
  • 00:04:39
    side and so we switch this to be Z minus
  • 00:04:42
    cpose x equals z um and that's why we
  • 00:04:46
    got a zero right here um at the
  • 00:04:48
    beginning so um we get the negative
  • 00:04:52
    coefficients showing up in the top row
  • 00:04:54
    and then any of the basic variables
  • 00:04:56
    they're either original variables and
  • 00:04:58
    then they'd have sort of a negative
  • 00:05:00
    whatever their original coefficient was
  • 00:05:02
    or if they were slack variables they
  • 00:05:04
    started off with zero as their uh
  • 00:05:07
    coefficient in the objective row and so
  • 00:05:10
    this CB is the negative coefficients of
  • 00:05:13
    the cost Vector that's the objective
  • 00:05:15
    Vector um corresponding to the basis
  • 00:05:18
    okay and if it's a slack variable in the
  • 00:05:21
    basis it's got zero for the initial cost
  • 00:05:24
    or for the for the cost Vector
  • 00:05:26
    value okay so what we need to do is uh
  • 00:05:30
    when we do our row operations we change
  • 00:05:33
    all of these columns to the standard
  • 00:05:36
    basis vectors okay so one Z z0 and0 One
  • 00:05:40
    z z z and so forth okay where there's
  • 00:05:44
    only one one in each column and um yeah
  • 00:05:49
    and and whichever row that one uh ends
  • 00:05:51
    up in is the row that corresponds to
  • 00:05:54
    that um basic variable okay so what we
  • 00:05:58
    need to do is essentially turn this into
  • 00:06:01
    the identity okay and uh if we want to
  • 00:06:03
    turn a matrix into an the identity
  • 00:06:06
    Matrix we need to multiply by the
  • 00:06:08
    inverse of that Matrix so uh there are
  • 00:06:11
    some nice formulas for how to do an
  • 00:06:14
    inverse of a block Matrix so this is
  • 00:06:17
    sort of a matrix of matrices so this B
  • 00:06:19
    is a full Matrix here um there's some
  • 00:06:22
    nice formulas out there um I'm not going
  • 00:06:24
    to drive them but I'll I'll essentially
  • 00:06:25
    show you what the the inverse Matrix is
  • 00:06:28
    in this case
  • 00:06:30
    and so if we think about here's the
  • 00:06:33
    Matrix we're going to use and we want to
  • 00:06:35
    multiply by uh one this is actually a
  • 00:06:39
    zero Vector here's a negative CB
  • 00:06:43
    transpose and
  • 00:06:46
    B and we want to choose a matrix here so
  • 00:06:49
    that we end up with a one here a
  • 00:06:53
    zero Vector transposed essentially a
  • 00:06:56
    zero row there a zero Vector here here
  • 00:06:59
    and an identity M uh Matrix
  • 00:07:03
    there that's our that's our goal okay um
  • 00:07:08
    and so let's let's choose something
  • 00:07:09
    that's going to get us there um we're
  • 00:07:12
    going to need a one here a zero here um
  • 00:07:18
    to get rid of B we're going to need a b
  • 00:07:21
    inverse here and then the tricky part
  • 00:07:24
    here is that we multiply by CB transpose
  • 00:07:27
    B inverse
  • 00:07:30
    okay let's verify that this when we
  • 00:07:33
    multiply these together gives us this so
  • 00:07:36
    when multiplying block
  • 00:07:37
    matrices it's almost as long as the
  • 00:07:41
    blocks are sort of consistent with each
  • 00:07:42
    other it's just like multiplying
  • 00:07:44
    matrices so I'll multiply first row by
  • 00:07:47
    First Column and so 1 * 1 and this times
  • 00:07:51
    the zero Vector that will give us 1 + 0
  • 00:07:58
    uh and that's not a vector that's number
  • 00:08:00
    okay and so that gives us the one that
  • 00:08:03
    we want that's good uh let's do this uh
  • 00:08:07
    row times this column to get this entry
  • 00:08:09
    over here and so we get 1 * negative CB
  • 00:08:12
    transpose is going to be CB transpose
  • 00:08:16
    and then we'll get uh
  • 00:08:19
    plus this times B and so
  • 00:08:22
    CB transpose B inverse
  • 00:08:28
    B okay
  • 00:08:30
    uh then let's do this row times this
  • 00:08:32
    column so 0 * 1 plus b inverse * 0 and
  • 00:08:36
    that's going to give us a zero
  • 00:08:39
    Vector zero Vector plus another zero
  • 00:08:41
    vector and then finally we'll do this
  • 00:08:44
    row times this column and so 0 time
  • 00:08:48
    negative CB transposed is going to give
  • 00:08:50
    us a zero technically it's going to give
  • 00:08:52
    us a zero Matrix uh but then we'll get
  • 00:08:55
    plus uh B inverse time B so uh sort of a
  • 00:09:00
    zero Matrix plus b inverse time
  • 00:09:05
    B okay so 1 + 0 is 1 that's what we want
  • 00:09:10
    there we get a zero Vector here that's
  • 00:09:12
    good up here B inverse time B is the
  • 00:09:15
    identity Matrix so that essentially
  • 00:09:17
    disappears and it leaves us with
  • 00:09:18
    negative CB transpose plus CB transpose
  • 00:09:22
    and those will then cancel each other
  • 00:09:24
    out and give us zero finally down here
  • 00:09:27
    we got 0 + B inverse * B is the identity
  • 00:09:30
    and so 0 plus the identity is just the
  • 00:09:32
    identity Matrix so this is going to
  • 00:09:37
    be or uh this is our inverse of this uh
  • 00:09:41
    since they multiply together to give us
  • 00:09:43
    the identity Matrix okay so now what we
  • 00:09:47
    want to do is our row operations are
  • 00:09:50
    equivalent to multiplying on the left by
  • 00:09:52
    this
  • 00:09:54
    Matrix so uh let's go ahead and and do
  • 00:09:58
    that and let's take our initial Tableau
  • 00:10:00
    oopsie our initial Tableau and multiply
  • 00:10:03
    on the left by this
  • 00:10:04
    Matrix so we'll have the Matrix one CB
  • 00:10:09
    transpose B inverse to and zero and B
  • 00:10:13
    inverse and we want to multiply by our
  • 00:10:18
    Tableau so
  • 00:10:21
    onega C transpose zero transpose and
  • 00:10:27
    zero and then zero
  • 00:10:31
    Vector um a
  • 00:10:33
    matrix identity Matrix and a rightand
  • 00:10:37
    side
  • 00:10:40
    Vector okay let's go ahead and do that
  • 00:10:42
    multiplication uh we'll do the top top
  • 00:10:44
    row
  • 00:10:48
    first uh using the top row of here and
  • 00:10:50
    then multiplying down each of the
  • 00:10:52
    separate columns there and
  • 00:10:55
    so this times this will give us 1 * 1
  • 00:10:58
    plus this time Z that * Z will cancel
  • 00:11:01
    that out and so we'll get just
  • 00:11:04
    one now let's do this times this and
  • 00:11:08
    we'll get 1 * negative C
  • 00:11:10
    transpose is negative C transpose
  • 00:11:14
    plus CB transpose B inverse
  • 00:11:20
    a and then we'll do this times this 1 *
  • 00:11:24
    0 plus this * the identity Matrix
  • 00:11:27
    remember that the identity Matrix is
  • 00:11:29
    like multiplying by one it doesn't
  • 00:11:31
    change anything and so that times that
  • 00:11:34
    will give us just CB transpose B inverse
  • 00:11:39
    and then this times this will give us 1
  • 00:11:43
    * 0 + CB transpose B
  • 00:11:47
    inverse
  • 00:11:49
    B okay that's our top row our objective
  • 00:11:52
    row now let's go ahead and do the bottom
  • 00:11:54
    row so we're multiplying by 0 and B
  • 00:11:56
    inverse 0 * 1 is 0 B inverse time 0 is 0
  • 00:12:01
    and so we get zero there and that
  • 00:12:03
    corresponds with what we expect we
  • 00:12:05
    expect the First Column that corresponds
  • 00:12:07
    to our um
  • 00:12:10
    Z to always be one followed by
  • 00:12:14
    zeros now we'll take this times this and
  • 00:12:18
    uh this zero will essentially zero out
  • 00:12:20
    everything in the top row and we'll be
  • 00:12:22
    left with just B inverse a and then this
  • 00:12:26
    times this will give us just B inverse
  • 00:12:29
    and this times this will give us B
  • 00:12:33
    inverse
  • 00:12:35
    B okay let's label those columns uh so
  • 00:12:39
    this is going to be all of our original
  • 00:12:41
    variables X these are all our slack
  • 00:12:44
    variables and this is our right hand
  • 00:12:48
    side okay let's uh call out and label
  • 00:12:52
    what these are um kind of maybe I'll go
  • 00:12:56
    ahead and kind of draw separators in
  • 00:13:02
    here and think about what each of these
  • 00:13:06
    are okay so first
  • 00:13:09
    off
  • 00:13:12
    um kind of thinking through this is what
  • 00:13:15
    we'll notice is we get B inverse right
  • 00:13:18
    here and so if you're doing your row
  • 00:13:23
    operations uh the B inverse Matrix is
  • 00:13:27
    the Matrix directly under your SL slack
  • 00:13:29
    variables and so uh I'll say this is
  • 00:13:32
    going to be
  • 00:13:33
    useful later on and that we can always
  • 00:13:36
    find our B inverse Matrix after doing
  • 00:13:38
    our row operations and know what is that
  • 00:13:41
    b inverse Matrix
  • 00:13:44
    okay here is the right hand side uh and
  • 00:13:48
    it's the right hand side and it's uh the
  • 00:13:51
    right hand side after transformation and
  • 00:13:53
    if and if you think about what we do
  • 00:13:55
    that right hand side tells us what our
  • 00:13:57
    current solution is for our basic
  • 00:13:59
    variables and so we're going to call
  • 00:14:01
    this XB it is the value of our basic
  • 00:14:06
    variables at our current solution right
  • 00:14:09
    and then remember all of our non-basic
  • 00:14:11
    variables are set to zero so this is
  • 00:14:13
    essentially our current solution it's
  • 00:14:15
    going to be a vector that's not quite
  • 00:14:17
    long enough because it it will only have
  • 00:14:19
    the basic variables in there but all the
  • 00:14:21
    other uh variables are than zero
  • 00:14:26
    okay let's uh introduce this quantity
  • 00:14:30
    here so underneath the slack
  • 00:14:34
    variables uh this quantity here we'll
  • 00:14:37
    call this y
  • 00:14:38
    transpose and these are what we call the
  • 00:14:42
    Dual variables we're going to talk about
  • 00:14:45
    duality in just a bit um these are the
  • 00:14:49
    Dual variables um they have a little bit
  • 00:14:52
    of a um interpretation as Shadow prices
  • 00:14:55
    that we'll talk about in just a bit um
  • 00:14:58
    and and yeah so we can kind of
  • 00:15:01
    abbreviate uh essentially the quantity
  • 00:15:04
    CB transpose B inverse as uh we'll call
  • 00:15:08
    that y y transpose it's going to be the
  • 00:15:12
    values in the objective row beneath the
  • 00:15:15
    slack variables and we'll talk a little
  • 00:15:17
    bit more about what those mean in a
  • 00:15:19
    little
  • 00:15:21
    bit okay uh let's uh look at this value
  • 00:15:26
    here
  • 00:15:29
    this is the
  • 00:15:32
    current objective
  • 00:15:40
    value okay and so if we take the costs
  • 00:15:44
    or the yeah the objective values
  • 00:15:46
    corresponding to the basic variables we
  • 00:15:49
    multiply it by B inverse and then by B
  • 00:15:52
    um our right hand side we get our
  • 00:15:55
    objective Uh current objective value um
  • 00:16:02
    but we just introduced the fact that
  • 00:16:04
    we're calling this quantity here our y
  • 00:16:07
    transpose and so we can think of this as
  • 00:16:09
    also y transpose B so if we take these
  • 00:16:13
    dual variables and multiply them by our
  • 00:16:16
    original right hand side um that gets us
  • 00:16:19
    our current objective
  • 00:16:22
    value okay uh and then maybe the last
  • 00:16:25
    thing uh I'll point out here
  • 00:16:32
    is the objective row value as underneath
  • 00:16:36
    our original variables um so this is
  • 00:16:39
    going to be let me actually um relabel
  • 00:16:42
    this as being y transposed because we
  • 00:16:43
    see another copy of that CB transpose B
  • 00:16:47
    inverse and so this is going to be uh y
  • 00:16:52
    transpose a minus C
  • 00:16:56
    transpose okay uh these are called uh
  • 00:17:00
    reduced
  • 00:17:05
    cost uh of the original variable um sort
  • 00:17:08
    of thinking of uh they had their
  • 00:17:11
    original cost is this negative C
  • 00:17:13
    transpose and then uh it's reduced by
  • 00:17:16
    whatever y transpose a is and we'll talk
  • 00:17:19
    a little bit about what that means in a
  • 00:17:21
    little bit later on too okay um this
  • 00:17:26
    it's optimal or current basis is optimal
  • 00:17:31
    if y transpose a minus cpose the
  • 00:17:35
    quantities uh underneath the original
  • 00:17:38
    variables
  • 00:17:40
    there let me slide this up a little bit
  • 00:17:43
    it's optimal if all of those reduced
  • 00:17:46
    costs are greater than or equal to zero
  • 00:17:48
    for an OP for a maximization problem as
  • 00:17:52
    well as we also need the values
  • 00:17:54
    underneath s to to be uh greater than or
  • 00:17:58
    equal to Z Z so we also need that y
  • 00:18:00
    transpose is greater than or equal to
  • 00:18:01
    zero okay so our
  • 00:18:04
    our Tableau or our basis is
  • 00:18:08
    optimal exactly
  • 00:18:10
    if uh this condition holds and this
  • 00:18:14
    condition holds and if you look at that
  • 00:18:16
    that almost looks like
  • 00:18:18
    constraints uh for like a non-
  • 00:18:22
    negativity constraint on my dual
  • 00:18:24
    variables as well as another uh
  • 00:18:27
    constraint involving my a Matrix on um
  • 00:18:30
    on my dual variables as well so um
  • 00:18:33
    interesting and that's going to be
  • 00:18:35
    essentially the heart of duality in just
  • 00:18:37
    a
  • 00:18:43
    second that's it for this video in the
  • 00:18:46
    next video we'll talk about Shadow
  • 00:18:48
    prices
Tags
  • Duality
  • Simplex Method
  • Linear Programming
  • Slack Variables
  • Basis Matrix
  • Inverse Matrix
  • Dual Variables
  • Reduced Costs
  • Optimization
  • Shadow Prices