Simplex algorithm - Revision history

62.167.89.174: then -> than

2025-05-18T02:24:53Z

then -> than

← Previous revision		Revision as of 02:24, 18 May 2025
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	</math>		</math>

	This is, fortuitously, already optimal and the optimum value for the original linear program is −130/7. This value is "worse" ~~then~~ -20 which is to be expected for a problem which is more constrained.		This is, fortuitously, already optimal and the optimum value for the original linear program is −130/7. This value is "worse" than -20 which is to be expected for a problem which is more constrained.

	==Advanced topics==		==Advanced topics==

2A02:3037:30D:381C:F9A0:4155:BF70:5308: Contrast examples and make clear that they are different problems.

2025-04-20T09:51:03Z

Contrast examples and make clear that they are different problems.

← Previous revision		Revision as of 09:51, 20 April 2025
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	\end{align}</math>		\end{align}</math>

	It differs from the previous example by having equality instead of inequality constraints. The previous solution <math>x=0, y=0 ~~and~~ z=5\</math> violates the first constraint.		It differs from the previous example by having equality instead of inequality constraints. The previous solution <math>x=y=0\, , z=5</math> violates the first constraint.
	This new problem is represented by the (non-canonical) tableau		This new problem is represented by the (non-canonical) tableau
	:<math>		:<math>
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	</math>		</math>

	This is, fortuitously, already optimal and the optimum value for the original linear program is −130/7.		This is, fortuitously, already optimal and the optimum value for the original linear program is −130/7. This value is "worse" then -20 which is to be expected for a problem which is more constrained.

	==Advanced topics==		==Advanced topics==

2A02:3037:30D:381C:F9A0:4155:BF70:5308: /* Example */

2025-04-20T09:45:03Z

Example

← Previous revision		Revision as of 09:45, 20 April 2025
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	\end{align}</math>		\end{align}</math>

			It differs from the previous example by having equality instead of inequality constraints. The previous solution <math>x=0, y=0 and z=5\</math> violates the first constraint.
	This is represented by the (non-canonical) tableau		This new problem is represented by the (non-canonical) tableau
	:<math>		:<math>
	\begin{bmatrix}		\begin{bmatrix}

INFINITASIUM X: Change example matrix to normalize the row instead. This follows the procedure in the pivot operations section of the article and makes the operation clearer at the cost of visual clutter

2024-12-04T18:46:01Z

Change example matrix to normalize the row instead. This follows the procedure in the pivot operations section of the article and makes the operation clearer at the cost of visual clutter

← Previous revision		Revision as of 18:46, 4 December 2024
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	:<math>		:<math>
	\begin{bmatrix}		\begin{bmatrix}
	3 & -2 & -11 & 0 & 0 & -4 & -60 \\		1 & -\frac{2}{3} & -\frac{11}{3} & 0 & 0 & -\frac{4}{3} & -20 \\
	0 & 7 & 1 & 0 & 3 & -1 & 15 \\		0 & \frac{7}{3} & \frac{1}{3} & 0 & 1 & -\frac{1}{3} & 5 \\
	0 & 2 & 5 & 3 & 0 & 1 & 15		0 & \frac{2}{3} & \frac{5}{3} & 1 & 0 & \frac{1}{3} & 5
	\end{bmatrix}		\end{bmatrix}
	</math>		</math>
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	For the next step, there are no positive entries in the objective row and in fact		For the next step, there are no positive entries in the objective row and in fact
	:<math display="block">Z = \frac{~~-60+2x+11y+4t~~}{3} ~~= -20~~ + \frac{2x+~~11y+4t~~}{3}</math>		:<math display="block">Z = -20 + \frac{2}{3}x+\frac{11}{3}y+\frac{4}{3}t</math>
	so the minimum value of ''Z'' is −20.		so the minimum value of ''Z'' is −20.

Suzannejb: /* See also */

2024-07-05T14:18:06Z

Suzannejb: /* See also */

2024-07-05T14:17:22Z

Suzannejb: added objective function

2024-07-05T14:16:48Z

added objective function

← Previous revision		Revision as of 14:16, 5 July 2024
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	==See also==		==See also==
	{{div col}}		{{div col}}
		⚫	* [[Bland's rule\|Pivoting rule of Bland]], which avoids cycling
	* [[Criss-cross algorithm]]		* [[Criss-cross algorithm]]
	* [[Cutting-plane method]]		* [[Cutting-plane method]]
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	* [[Karmarkar's algorithm]]		* [[Karmarkar's algorithm]]
	* [[Nelder–Mead method\|Nelder–Mead simplicial heuristic]]		* [[Nelder–Mead method\|Nelder–Mead simplicial heuristic]]
			* [[Objective Function]]
⚫	* [[Bland's rule\|Pivoting rule of Bland]], which avoids cycling
	{{colend}}		{{colend}}

185.132.137.43: /* Example */ correcting variables in objective function from a,b,c to x,y,z

2024-07-05T12:24:00Z

Example: correcting variables in objective function from a,b,c to x,y,z

← Previous revision		Revision as of 12:24, 5 July 2024
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	Consider the linear program		Consider the linear program
	:Minimize		:Minimize
	::<math>Z = -2 a - 3 b - 4 c\,</math>		::<math>Z = -2 x - 3 y - 4 z\,</math>

	:Subject to		:Subject to

Michaelnikolaou: Formatted some math

2024-06-20T15:39:41Z

Formatted some math

← Previous revision		Revision as of 15:39, 20 June 2024
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	\begin{bmatrix}		\begin{bmatrix}
	1 & -\mathbf{c}^T & 0 \\		1 & -\mathbf{c}^T & 0 \\
	0 & \mathbf{A} & \mathbf{b}		\mathbf{0} & \mathbf{A} & \mathbf{b}
	\end{bmatrix}		\end{bmatrix}
	</math>		</math>

	The first row defines the objective function and the remaining rows specify the constraints. The zero in the first column represents the zero vector of the same dimension as vector ''b'' (different authors use different conventions as to the exact layout). If the columns of A can be rearranged so that it contains the [[identity matrix]] of order ''p'' (the number of rows in A) then the tableau is said to be in ''canonical form''.<ref>{{harvtxt\|Murty\|1983\|loc=section 2.3.2}}</ref> The variables corresponding to the columns of the identity matrix are called ''basic variables'' while the remaining variables are called ''nonbasic'' or ''free variables''. If the values of the nonbasic variables are set to 0, then the values of the basic variables are easily obtained as entries in ''b'' and this solution is a basic feasible solution. The algebraic interpretation here is that the coefficients of the linear equation represented by each row are either <math>0</math>, <math>1</math>, or some other number. Each row will have <math>1</math> column with value <math>1</math>, <math>p-1</math> columns with coefficients <math>0</math>, and the remaining columns with some other coefficients (these other variables represent our non-basic variables). By setting the values of the non-basic variables to zero we ensure in each row that the value of the variable represented by a <math>1</math> in its column is equal to the <math>b</math> value at that row.		The first row defines the objective function and the remaining rows specify the constraints. The zero in the first column represents the zero vector of the same dimension as the vector <math>\mathbf{b}</math> (different authors use different conventions as to the exact layout). If the columns of <math>\mathbf{A}</math> can be rearranged so that it contains the [[identity matrix]] of order <math>p</math> (the number of rows in <math>\mathbf{A}</math>) then the tableau is said to be in ''canonical form''.<ref>{{harvtxt\|Murty\|1983\|loc=section 2.3.2}}</ref> The variables corresponding to the columns of the identity matrix are called ''basic variables'' while the remaining variables are called ''nonbasic'' or ''free variables''. If the values of the nonbasic variables are set to 0, then the values of the basic variables are easily obtained as entries in <math>\mathbf{b}</math> and this solution is a basic feasible solution. The algebraic interpretation here is that the coefficients of the linear equation represented by each row are either <math>0</math>, <math>1</math>, or some other number. Each row will have <math>1</math> column with value <math>1</math>, <math>p-1</math> columns with coefficients <math>0</math>, and the remaining columns with some other coefficients (these other variables represent our non-basic variables). By setting the values of the non-basic variables to zero we ensure in each row that the value of the variable represented by a <math>1</math> in its column is equal to the <math>b</math> value at that row.

	Conversely, given a basic feasible solution, the columns corresponding to the nonzero variables can be expanded to a nonsingular matrix. If the corresponding tableau is multiplied by the inverse of this matrix then the result is a tableau in canonical form.<ref>{{harvtxt\|Murty\|1983\|loc=section 3.12}}</ref>		Conversely, given a basic feasible solution, the columns corresponding to the nonzero variables can be expanded to a nonsingular matrix. If the corresponding tableau is multiplied by the inverse of this matrix then the result is a tableau in canonical form.<ref>{{harvtxt\|Murty\|1983\|loc=section 3.12}}</ref>

189.122.84.88: /* History */ made link

2024-06-19T00:18:29Z

History: made link

← Previous revision		Revision as of 00:18, 19 June 2024
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	==History==		==History==
	[[George Dantzig]] worked on planning methods for the US Army Air Force during World War II using a desk calculator. During 1946, his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of [[Wassily Leontief]], however, at that time he didn't include an objective as part of his formulation. Without an objective, a vast number of solutions can be feasible, and therefore to find the "best" feasible solution, military-specified "ground rules" must be used that describe how goals can be achieved as opposed to specifying a goal itself. Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized.<ref>{{Cite journal\|url = https://apps.dtic.mil/sti/pdfs/ADA112060.pdf\|archive-url = https://web.archive.org/web/20150520183722/http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA112060\|url-status = live\|archive-date = May 20, 2015\|title = Reminiscences about the origins of linear programming\|date = April 1982\|journal = Operations Research Letters\|doi = 10.1016/0167-6377(82)90043-8\|volume = 1\|issue = 2 \|pages=43–48\|last1 = Dantzig\|first1 = George B.}}</ref> Development of the simplex method was evolutionary and happened over a period of about a year.<ref>{{Cite journal\|url = http://www.phpsimplex.com/en/Dantzig_interview.htm\|title = An Interview with George B. Dantzig: The Father of Linear Programming\|last = Albers and Reid\|date = 1986\|journal = College Mathematics Journal\|volume = 17\|issue = 4\|doi = 10.1080/07468342.1986.11972971\|pages = 292–314}}</ref>		[[George Dantzig]] worked on planning methods for the US Army Air Force during World War II using a [[Mechanical_calculator#1900s_to_1970s\|desk calculator]]. During 1946, his colleague challenged him to mechanize the planning process to distract him from taking another job. Dantzig formulated the problem as linear inequalities inspired by the work of [[Wassily Leontief]], however, at that time he didn't include an objective as part of his formulation. Without an objective, a vast number of solutions can be feasible, and therefore to find the "best" feasible solution, military-specified "ground rules" must be used that describe how goals can be achieved as opposed to specifying a goal itself. Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized.<ref>{{Cite journal\|url = https://apps.dtic.mil/sti/pdfs/ADA112060.pdf\|archive-url = https://web.archive.org/web/20150520183722/http://www.dtic.mil/cgi-bin/GetTRDoc?Location=U2&doc=GetTRDoc.pdf&AD=ADA112060\|url-status = live\|archive-date = May 20, 2015\|title = Reminiscences about the origins of linear programming\|date = April 1982\|journal = Operations Research Letters\|doi = 10.1016/0167-6377(82)90043-8\|volume = 1\|issue = 2 \|pages=43–48\|last1 = Dantzig\|first1 = George B.}}</ref> Development of the simplex method was evolutionary and happened over a period of about a year.<ref>{{Cite journal\|url = http://www.phpsimplex.com/en/Dantzig_interview.htm\|title = An Interview with George B. Dantzig: The Father of Linear Programming\|last = Albers and Reid\|date = 1986\|journal = College Mathematics Journal\|volume = 17\|issue = 4\|doi = 10.1080/07468342.1986.11972971\|pages = 292–314}}</ref>

	After Dantzig included an objective function as part of his formulation during mid-1947, the problem was mathematically more tractable. Dantzig realized that one of the unsolved problems that [[George Dantzig#Education\|he had mistaken]] as homework in his professor [[Jerzy Neyman]]'s class (and actually later solved), was applicable to finding an algorithm for linear programs. This problem involved finding the existence of [[Lagrange multipliers]] for general linear programs over a continuum of variables, each bounded between zero and one, and satisfying linear constraints expressed in the form of [[Lebesgue integral]]s. Dantzig later published his "homework" as a thesis to earn his doctorate. The column geometry used in this thesis gave Dantzig insight that made him believe that the Simplex method would be very efficient.<ref>{{Cite encyclopedia\|url = http://apps.dtic.mil/dtic/tr/fulltext/u2/a182708.pdf\|archive-url = https://web.archive.org/web/20150529003047/http://www.dtic.mil/dtic/tr/fulltext/u2/a182708.pdf\|url-status = live\|archive-date = May 29, 2015\|title = Origins of the simplex method\|last = Dantzig\|first = George\|date = May 1987\|encyclopedia = A History of Scientific Computing\|editor-last=Nash\|editor-first=Stephen G.\|publisher=Association for Computing Machinery\|pages = 141–151\|doi = 10.1145/87252.88081\|isbn = 978-0-201-50814-7}}</ref>		After Dantzig included an objective function as part of his formulation during mid-1947, the problem was mathematically more tractable. Dantzig realized that one of the unsolved problems that [[George Dantzig#Education\|he had mistaken]] as homework in his professor [[Jerzy Neyman]]'s class (and actually later solved), was applicable to finding an algorithm for linear programs. This problem involved finding the existence of [[Lagrange multipliers]] for general linear programs over a continuum of variables, each bounded between zero and one, and satisfying linear constraints expressed in the form of [[Lebesgue integral]]s. Dantzig later published his "homework" as a thesis to earn his doctorate. The column geometry used in this thesis gave Dantzig insight that made him believe that the Simplex method would be very efficient.<ref>{{Cite encyclopedia\|url = http://apps.dtic.mil/dtic/tr/fulltext/u2/a182708.pdf\|archive-url = https://web.archive.org/web/20150529003047/http://www.dtic.mil/dtic/tr/fulltext/u2/a182708.pdf\|url-status = live\|archive-date = May 29, 2015\|title = Origins of the simplex method\|last = Dantzig\|first = George\|date = May 1987\|encyclopedia = A History of Scientific Computing\|editor-last=Nash\|editor-first=Stephen G.\|publisher=Association for Computing Machinery\|pages = 141–151\|doi = 10.1145/87252.88081\|isbn = 978-0-201-50814-7}}</ref>

← Previous revision		Revision as of 14:18, 5 July 2024
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	* [[Karmarkar's algorithm]]		* [[Karmarkar's algorithm]]
	* [[Nelder–Mead method\|Nelder–Mead simplicial heuristic]]		* [[Nelder–Mead method\|Nelder–Mead simplicial heuristic]]
	* [[Loss ~~Function~~]] - a type of Objective Function		* [[Loss Functions]] - a type of Objective Function
	{{colend}}		{{colend}}

← Previous revision		Revision as of 14:17, 5 July 2024
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	* [[Karmarkar's algorithm]]		* [[Karmarkar's algorithm]]
	* [[Nelder–Mead method\|Nelder–Mead simplicial heuristic]]		* [[Nelder–Mead method\|Nelder–Mead simplicial heuristic]]
	* [[~~Objective~~ Function]]		* [[Loss Function]] - a type of Objective Function
	{{colend}}		{{colend}}