From 91c94cc4dab7b2a2afeae0fb4c35818be367613a Mon Sep 17 00:00:00 2001
From: Francois <44402056+PH4NT0M78@users.noreply.github.com>
Date: Wed, 30 Jan 2019 23:27:53 +0200
Subject: [PATCH] Added File on the Basics of Simplex Method for Solving Linear
 Programming Problems (#27040)

* Create Basic-Simplex-Solution

Basics on Solving Linear Problems using the Primal Simplex Method and additional information on Alternative Simplex Algorithms.

* Rename guide/english/mathematics/Basic-Simplex-Solution to guide/english/mathematics/linear-equations/basic-simplex-solution/index.md
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+---
+title: Basics of Simplex Solutions
+---
+
+## Reason for Sthe Simplex Method
+
+In linear algebra, some problems have multiple solutions that can be accepted as feasible, but in order to get the Optimal solution it becomes necessary to use a method such as the Simplex method.
+Note: Alternative methods such as a Graphical LP can also be used but this is not always possible.
+
+## Different Simplex Methods:
+
+Here are the 3 most used methods for solving a LP using Simplex:
+1. Primal Simplex - Used to solve MAXIMUM Problems.
+2. Dual Simplex - Used when a derived problem exists within a Primal Simplex Solution (Duality).
+3. 2-Phase Simplex - Used to solve MIN Problems.
+
+
+## Solving Linear Programming Problems using Simplex
+
+In order to use the Simplex method, first the Linear model needs to be converted into canonical form.
+The canonical for is when all inequality expressions are changed into equal expressions.
+
+Example:
+
+Basic LP
+
+MAX Z = 5X + 4Y
+Subject to Constraints:
+                      1. 8X + 6Y <= 120
+                      2. 2X + 1Y <= 50
+                      3. X >= 10
+                      4. X, y >= 0
+                      
+Canonical Form
+
+MAX Z = -5X - 4Y                                      Note: Z Row becomes negative.
+ST:
+    1. 8X + 6Y + S1 = 120                             Note: In order to set a Smaller-than equation to equal a Slack variable is introduced.
+    2. 2X - 1y + S2 = 50
+    3. X - E1 + A1 = 10                               Note: In order to set a Larger-than equation to equal a Excess variable is added and an Artificial variable is subtracted.
+    
+Now the Initial Tablau can be created:
+
+Note: X & Y will be the Non-Basic-Variables as they are Negative and Slack/Axcess/Artificial variables do not count in this case.
+
+      *
+T0  | X  | Y  | S1  | S2  | E1  | A1  ||  RHS | Ratio
+Z   | -5 | -4 | 0   | 0   | 0   | 0   ||  0   | ---
+1   | 8  | 6  | 1   | 0   | 0   | 0   ||  120 | 15
+2   | 2  | 1  | 0   | 1   | 0   | 0   ||  50  | 25
+3   | 1  | 0  | 0   | 0   | -1  | 1   ||  10  | 10    *
+
+Note: * Represents the Column with the Smallest Negative and the Column with the Smallest Positive.
+
+Steps:
+1. Find the Column with the Smallest Negative.
+2. Devide the RHS with the selected column to calculate the Ratio.
+3. Find the Row with the Smallest Positive Ratio
+4. Pivot on the selected Row & Column.
+5. Continue doing this until no more negative NBVs remain.
+
+Optimal Table:
+
+T2  | X  | Y  | S1  | S2  | E1  | A1  ||  RHS 
+Z   | 0  | 0  | 0.67| 0   | 0.3 | -0.3||  76.67   
+1   | 0  | 1  | 1.67| 0   | 1.3 | -1.3||  6.67 
+2   | 0  | 0  | -0.1| 1   | 0.6 | -0.6||  23.33  
+3   | 1  | 0  | 0   | 0   | -1  | 1   ||  10  
+
+Steps:
+1. Identify all Basic Variables:
+  1.1. Only Columns with a single '1' and rest '0' can be a Basic Variable.
+  1.2. Order of identification is determined by Row number.
+2. The RHS value corresponding to the '1' value for each BV is the Value of that variable.
+
+Eg:
+
+cBV:  Y = 6.67 ;  S1 = 23.33  ; X = 10
+
+Thus the Optimal Solution to the LP is Z = 5(10) + 4(6.67)
+
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