Wyndor Model: Optimality Test and Solving an Example, Lecture notes of Linear Programming

Download Wyndor Model: Optimality Test and Solving an Example and more Lecture notes Linear Programming in PDF only on Docsity! Chapter 4: The Simplex Method Cathal Heavey September 26, 2011 1/35 Introduction Background Simplex Method Geometric Interpretation The Algebra of The Simplex Method Tabular Form Introduction 2/35 . ° ° . . . . ° ° . ° . . Background Simplex Method Introduction Background Simplex Method : Geometric Interpretation $ The Algebra of The Simplex Method : Tabular Form Developed by George Dantzig in 1947 Remarkably efficient solution method Section 4.1 introduces geometric interpretation Sections 4.2-4.4 presents a method to solve standard LP problem (< form) Sections 4.5-4.6 adapt method for other forms Section 4.7 discusses postoptimality analysis Section 4.8 discusses computer implementation issues Finally section 4.9 briefly describes a new solution method interior-point approach 3/35 Background Simplex Method Introduction Background Simplex Method : Geometric Interpretation $ The Algebra of The Simplex Method : Tabular Form Developed by George Dantzig in 1947 Remarkably efficient solution method Section 4.1 introduces geometric interpretation Sections 4.2-4.4 presents a method to solve standard LP problem (< form) Sections 4.5-4.6 adapt method for other forms Section 4.7 discusses postoptimality analysis Section 4.8 discusses computer implementation issues Finally section 4.9 briefly describes a new solution method interior-point approach 3/35 Background Simplex Method Introduction Background Simplex Method : Geometric Interpretation $ The Algebra of The Simplex Method : Tabular Form Developed by George Dantzig in 1947 Remarkably efficient solution method Section 4.1 introduces geometric interpretation Sections 4.2-4.4 presents a method to solve standard LP problem (< form) Sections 4.5-4.6 adapt method for other forms Section 4.7 discusses postoptimality analysis Section 4.8 discusses computer implementation issues Finally section 4.9 briefly describes a new solution method interior-point approach 3/35 Background Simplex Method Introduction Background Simplex Method Geometric Interpretation $ The Algebra of The Simplex Method : Tabular Form Developed by George Dantzig in 1947 Remarkably efficient solution method Section 4.1 introduces geometric interpretation Sections 4.2-4.4 presents a method to solve standard LP problem (< form) Sections 4.5-4.6 adapt method for other forms Section 4.7 discusses postoptimality analysis Section 4.8 discusses computer implementation issues Finally section 4.9 briefly describes a new solution method interior-point approach 3/35 Background Simplex Method Introduction Background Simplex Method Geometric Interpretation $ The Algebra of The Simplex Method : Tabular Form Developed by George Dantzig in 1947 Remarkably efficient solution method Section 4.1 introduces geometric interpretation Sections 4.2-4.4 presents a method to solve standard LP problem (< form) Sections 4.5-4.6 adapt method for other forms Section 4.7 discusses postoptimality analysis Section 4.8 discusses computer implementation issues Finally section 4.9 briefly describes a new solution method interior-point approach 3/35 Introduction Geometric Interpretation Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form . ° ° . . . . ° ° . ° . . Geometric Interpretation 4/35 Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method . Tabular Form Optimality test Optimality Test: Consider any linear programming problem that processes least one optimal solution. If a CPF solution has no adjacent CPF solutions that are better (as measured by Z), then it must be an optimal solution. at 7/35 Optimality test Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form Optimality Test: Consider any linear programming problem that processes at least one optimal solution. If a CPF solution has no adjacent CPF solutions that are better (as measured by Z), then it must be an optimal solution. (2, 6) must be optimal with Z = 36 as (0, 6) gives Z = 30 and (4, 3) gives Z = 27 7/35 Solving the Example Introduction Geometric Interpretation Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form POCO C rere creer reer eer ee ree ee eee eee Deer eee eee Dee Dee Dee reer eee res seereeseeersessenD x2 (0, 6) 8/35 Key Concepts — 1-2 Introduction Solution Concept 1: For any problem with at least one optimal solution, Geometric Interpretation $ . Corner Point Feasible : optimal equal best CPF. Solutions (CPF)) Wyndor Model Solution Concept 2: Simplex method is iterative Optimality test Solving the Example Key Concepts — 1-2 co . Key Concepts — 3-4 Initialization: Setting Up The Simplex Method Augmented (Standard) Model : Optimality Test: Some Definitions : Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form ° 9/35 Key Concepts — 1-2 Introduction Solution Concept 1: For any problem with at least one optimal solution, Geometric Interpretation $ . Corner Point Feasible : optimal equal best CPF. Solutions (CPF)) Wyndor Model Solution Concept 2: Simplex method is iterative Optimality test Solving the Example Key Concepts — 1-2 co . Key Concepts — 3-4 -——— Thnitialization: Setting Up The Simplex Method Augmented (Standard) Model : Optimality Test: Some Definitions : Basic Solution Basic Solution Example If no If yes ———————_—_> Stop Objective Function Equation The Algebra of The Simplex Method . . : -—_————Iteration: Tabular Form ° 9/35 Key Concepts — 3-4 Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form Solution Concept 3: Use the origin (all decision variables are zero) for initial CPF solution Solution Concept 4: Move from CPF solution to CPF solution is computationally quicker. Moves along the edges of the feasible region. Solution Concept 5: Move to CPF solution that gives the /argest rate of improvement of Z Solution 6: Optimality test consists of edge gives positive rate of improvement in Z — if not current solution optimal 10/35 Key Concepts — 3-4 Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form Solution Concept 3: Use the origin (all decision variables are zero) for initial CPF solution Solution Concept 4: Move from CPF solution to CPF solution is computationally quicker. Moves along the eages of the feasible region. Solution Concept 5: Move to CPF solution that gives the /argest rate of improvement of Z Solution 6: Optimality test consists of edge gives positive rate of improvement in Z — if not current solution optimal 10/35 Introduction Geometric Interpretation Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form ° Setting Up The Simplex Method @ Add Slack Variables x1 <4 add slack variable gives gy+a3=40r73 =4—-21 @ x, < 4is equivalent to two constraints: v1+ 23 =4and x73 > 0 11/35 Introduction Geometric Interpretation Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form ° Setting Up The Simplex Method @ Add Slack Variables 1 < 4 add slack variable gives t+%3=40r73 =4-27)1 @ x, < 4is equivalent to two constraints: v1+ 23 =4and x73 > 0 11/35 Introduction Geometric Interpretation Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form Augmented (Standard) Model Original Model Max Z = 37, + 5x9 subject to: v1 < A 2%2 < 12 341 + 2% < 18 and x, 20, %2 20 12/35 Augmented (Standard) Model Introduction Geometric Interpretation Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form Original Model Max Z = 37, + 5x9 subject to: v1 < A 2%2 < 12 341 + 2% < 18 and ey > 0, v2 > 0 Augmented Model Max Z = 341 + 5x2 subject to: v1 + &3 2x2 + 4 341 + 2x2 + 2&5 and x; > 0, forg = 1,2,3,4,5 12 18 12/35 Some Definitions Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form An augmented solution is a solution for the original variables plus augmented variables. Example (1 = 3,2 = 2,73 = 1,274 = 8,25 = 5) A basic solution is an augmented corner point solution. Example (1 = 4,29 = 6,23 = 0,24 = 0,45 = —6) A basic feasible (BF) solution is an augmented CPF solution. Example (x1 = 0,22 = 6,4 —3 = 4,24 = 0,25 = 6) Degrees of freedom: Number of variables — number of equations = 5 — 3 = 2. 2 degrees of freedom for problem, i.e. two variables can be chosen arbitrary always 0 Basic variables not given nonzero values, Nonbasic variables are given zero values 13/35 Some Definitions Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form An augmented solution is a solution for the original variables plus augmented variables. Example (v1 = 3,%2 = 2,43 = 1,24 = 8,25 = 5) A basic solution is an augmented corner point solution. Example (a1 = 4,22 = 6,273 = 0,24 = 0,25 = —6) A basic feasible (BF) solution is an augmented CPF solution. Example (a1 = 0,22 = 6,4 —3 = 4,24 = 0,25 = 6) Degrees of freedom: Number of variables — number of equations = 5-3 = 2. 2 degrees of freedom for problem, i.e. two variables can be chosen arbitrary always 0 Basic variables not given nonzero values, Nonbasic variables are given zero values 13/35 Some Definitions Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form An augmented solution is a solution for the original variables plus augmented variables. Example (v1 = 3,%2 = 2,43 = 1,24 = 8,25 = 5) A basic solution is an augmented corner point solution. Example (a1 = 4,22 = 6,273 = 0,24 = 0,25 = —6) A basic feasible (BF) solution is an augmented CPF solution. Example (a1 = 0,22 = 6,4 —3 = 4,24 = 0,25 = 6) Degrees of freedom: Number of variables — number of equations = 5-3 = 2. 2 degrees of freedom for problem, i.e. two variables can be chosen arbitrary always 0 Basic variables not given nonzero values, Nonbasic variables are given zero values 13/35 Basic Solution Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form A basic solution has the following properties 1. 2. Variables basic or non-basic No of basic variables = functional constraints and nonbasic variables = no of variables — no of functional constraints Nonbasic variables are set to zero Basic variables values obtained by solving set of simultaneous linear equations If basic varables are non-negative then basic feasible solution 14/35 Basic Solution Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form A basic solution has the following properties 1. 2. Variables basic or non-basic No of basic variables = functional constraints and nonbasic variables = no of variables — no of functional constraints Nonbasic variables are set to zero Basic variables values obtained by solving set of simultaneous linear equations If basic varables are non-negative then basic feasible solution 14/35 Basic Solution Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form A basic solution has the following properties 1. 2. Variables basic or non-basic No of basic variables = functional constraints and nonbasic variables = no of variables — no of functional constraints Nonbasic variables are set to zero Basic variables values obtained by solving set of simultaneous linear equations If basic varables are non-negative then basic feasible solution 14/35 Basic Solution Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example ¢ Objective Function Equation The Algebra of The Simplex Method . Tabular Form A basic solution has the following properties 1. 2. Variables basic or non-basic No of basic variables = functional constraints and nonbasic variables = no of variables — no of functional constraints Nonbasic variables are set to zero Basic variables values obtained by solving set of simultaneous linear equations If basic varables are non-negative then basic feasible solution 14/35 Basic Solution Example Introduction Geometric Interpretation $ Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example = Objective Function Equation The Algebra of The Simplex Method . Tabular Form Let (11 = 0,272 = 6,23 = 4,24 = 0,25 = 6), x1 and x4 are nonbasic variables. This gives: v1 + £3 < 4 r3=4 2x9 + 2&4 < 12 r2=6 3%, + 2x9 + a5 < 18 5 =6 H Two BF solutions are adjacent if all but one of their nonbasic variables are the same. m Example, solutions (x; = 0,22 = 0) and (21 = 0,72 = 6) give augmented solutions (7; = 0,22 = 0,23 = 4,44 = 12,25 = 18) and (xy = 0, x2 = 6, 73 = 4, x4 = 0, 25 = 6) 15/35 Introduction Geometric Interpretation Corner Point Feasible Solutions (CPF)) Wyndor Model Optimality test Solving the Example Key Concepts — 1-2 Key Concepts — 3-4 Setting Up The Simplex Method Augmented (Standard) Model Some Definitions Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form Objective Function Equation Model can be expressed as a system of linear equations: 16/35 Objective Function Equation Lioauction —____: Model can be expressed as a system of linear equations: Geometric Interpretation : Corner Point Feasible : Z —321 —_ 5x2 = 0 Solutions (CPF)) Wyndor Model : LY + x3 = 4. Optimality test : — ptimality tes : 2x2 + vA — 12 Solving the Example : Key Concepts — 1-2 : 321 + 222 + U5 1 8 Key Concepts — 3-4 : Setting Up The Simplex : Method : Augmented (Standard) : Model : Some Definitions : Basic Solution Basic Solution Example Objective Function Equation The Algebra of The Simplex Method Tabular Form ° 16/35 Geometric versus Algebraic Introduction Geometric Interpretation The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Method l Geometric Interpretation Algebraic Interpretation Initialization Choose (0,0) to be the initial CPF Choose x1 and x9 to be the nonbasic variables (=0) for the initial BF solution (0,0,4,12,8) Optimality Not optimal, because moving along either edge Not optimal, because increasing either nonbasic test from (0,0) increases Z variable (x1 or x2) increases Z 18/35 Geometric versus Algebraic Introduction Geometric Interpretation The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Method l Geometric Interpretation Algebraic Interpretation Initialization Choose (0,0) to be the initial CPF Choose x1 and x9 to be the nonbasic variables (=0) for the initial BF solution (0,0,4,12,8) Optimality Not optimal, because moving along either edge Not optimal, because increasing either nonbasic test from (0,0) increases Z variable (x1 or x2) increases Z Iteration 1 Move up the edge lying on the x9 axis Increase x 2 while adjusting other variables values Step 1 to satisfy the system of equations 18/35 Introduction Geometric Interpretation The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Geometric versus Algebraic Method l Geometric Interpretation Algebraic Interpretation Initialization Choose (0,0) to be the initial CPF Choose x1 and x9 to be the nonbasic variables (=0) for the initial BF solution (0,0,4,12,8) Optimality Not optimal, because moving along either edge Not optimal, because increasing either nonbasic test from (0,0) increases Z variable (x1 or x2) increases Z Iteration 1 Move up the edge lying on the x9 axis Increase x 2 while adjusting other variables values Step 1 to satisfy the system of equations Step 2 Stop when the first new constraint boundary Stop when the first basic variable (x3, 24, @5) (222 = 12 is reached) drops to zero (x4) 18/35 Introduction Geometric Interpretation The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Geometric versus Algebraic Method l Geometric Interpretation Algebraic Interpretation Initialization Choose (0,0) to be the initial CPF Choose x1 and x9 to be the nonbasic variables (=0) for the initial BF solution (0,0,4,12,8) Optimality Not optimal, because moving along either edge Not optimal, because increasing either nonbasic test from (0,0) increases Z variable (x1 or x2) increases Z Iteration 1 Move up the edge lying on the x9 axis Increase x 2 while adjusting other variables values Step 1 to satisfy the system of equations Step 2 Stop when the first new constraint boundary Stop when the first basic variable (x3, 24, @5) (2xg = 12 is reached) drops to zero (x4) Step 3 Find the intersection of the new pair of constraint With xg now a basic variable and x4 now a boundaries: (0, 6) is the new CPF solution. nonbasic variable, solve the system of equations: (0, 6, 4, 0) is the new BF solution. Optimality Not optimal, because moving along the edge from Not optimal, because increasing one nonbasic vari- test (0, 6) to the right increases Z. able (x 1) increases Z. Iteration 2 Move along this edge to the right Increase x 1 while adjusting other variable values Step 1 to satisfy the systems of equations. 18/35 Introduction Geometric Interpretation The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Geometric versus Algebraic Method l Geometric Interpretation Algebraic Interpretation Initialization Choose (0,0) to be the initial CPF Choose x1 and x9 to be the nonbasic variables (=0) for the initial BF solution (0,0,4,12,8) Optimality Not optimal, because moving along either edge Not optimal, because increasing either nonbasic test from (0,0) increases Z variable (x1 or x2) increases Z Iteration 1 Move up the edge lying on the x9 axis Increase x 2 while adjusting other variables values Step 1 to satisfy the system of equations Step 2 Stop when the first new constraint boundary Stop when the first basic variable (x3, 24, @5) (2xg = 12 is reached) drops to zero (x4) Step 3 Find the intersection of the new pair of constraint With xg now a basic variable and x4 now a boundaries: (0, 6) is the new CPF solution. nonbasic variable, solve the system of equations: (0, 6, 4, 0) is the new BF solution. Optimality Not optimal, because moving along the edge from Not optimal, because increasing one nonbasic vari- test (0, 6) to the right increases Z. able (x 1) increases Z. Iteration 2 Move along this edge to the right Increase x 1 while adjusting other variable values Step 1 to satisfy the systems of equations. Step 2 Stop when the first new constraint boundary Stop when the first basic variable (x1, 73, @5) (321 + 2x2 = 18) is reached. drops to zero 18/35 Introduction Geometric Interpretation The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Geometric versus Algebraic Method l Geometric Interpretation Algebraic Interpretation Initialization Choose (0,0) to be the initial CPF Choose x1 and x9 to be the nonbasic variables (=0) for the initial BF solution (0,0,4,12,8) Optimality Not optimal, because moving along either edge Not optimal, because increasing either nonbasic test from (0,0) increases Z variable (x1 or x2) increases Z Iteration 1 Move up the edge lying on the x9 axis Increase x 2 while adjusting other variables values Step 1 to satisfy the system of equations Step 2 Stop when the first new constraint boundary Stop when the first basic variable (x3, 24, @5) (2xg = 12 is reached) drops to zero (x4) Step 3 Find the intersection of the new pair of constraint With xg now a basic variable and x4 now a boundaries: (0, 6) is the new CPF solution. nonbasic variable, solve the system of equations: (0, 6, 4, 0) is the new BF solution. Optimality Not optimal, because moving along the edge from Not optimal, because increasing one nonbasic vari- test (0, 6) to the right increases Z. able (x 1) increases Z. Iteration 2 Move along this edge to the right Increase x 1 while adjusting other variable values Step 1 to satisfy the systems of equations. Step 2 Stop when the first new constraint boundary Stop when the first basic variable (x1, 73, @5) (321 + 2x2 = 18) is reached. drops to zero Step 3 Find the intersection of the new pair of constraint With x1 now a basic variable and x5 now a boundaries: (2, 6) is the new CPF solution nonbasic variable, solve the system of equations: (2, 6, 2, 0, 0) is the new BF solution. 18/35 Initialization Introduction : m= 0 and t= 0 so: Geometric Interpretation : The Algebra of The : Ly + 23 = 4 rw =4 Simplex Method : Geometric versus : 222 + v4 = 12 v4 = 12 Algebraic : 321 + 2x9 + U5 = 18 t5 = 18 Initialization : Optimality Test Step 1 of Iteration 1 m Above solution is called proper form Gaussian elimination Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian ™@ Gaussian elimination is used in each iteration to convert the system of Elimination : . . Optimality Test : equations to this form. Iteration 2 (Again) : Optimality Test Again Tabular Form 19/35 Initialization Introduction : m= 0 and t= 0 so: Geometric Interpretation : The Algebra of The : LY + x3 = 4 v3 = 4 Simplex Method : Geometric versus : 222 + v4 = 12 v4 = 12 Algebraic : 321 + 2x9 + U5 = 18 rs = 18 Initialization Optimality Test Step 1 of Iteration 1 . . . . . . m Above solution is called proper form Gaussian elimination Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian in =m Gaussian elimination is used in each iteration to convert the system of Imination . Optimality Test : equations to this form. Iteration 2 (Again) ° Optimality Test Again Tabular Form 19/35 Initialization Introduction : m= 0 and t= 0 so: Geometric Interpretation : The Algebra of The : LY + x3 = 4 v3 = 4 Simplex Method : Geometric versus : 222 + v4 = 12 v4 = 12 Algebraic : 321 + 2x9 + U5 = 18 rs = 18 Initialization Optimality Test Step 1 of Iteration 1 . . . . . . m Above solution is called proper form Gaussian elimination Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian in =m Gaussian elimination is used in each iteration to convert the system of Imination . Optimality Test : equations to this form. Iteration 2 (Again) ° Optimality Test Again Tabular Form 19/35 Optimality Test Introduction : M™ Objective function is Z = 3x1 + 2x2 Geometric Interpretation $ The Algebra of The ° Simplex Method @ Note basic variables have coefficient of zero on objective function Geometric versus Algebraic —H Coefficients of nonbasic variables (3 and 5) give the rate of improvement of Z if increased from zero. Initialization Optimality Test Step 1 of Iteration 1 ™ Conclude solution (0,0, 4, 12, 18) is not optimal. Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) ° Optimality Test Again Tabular Form 20/35 Optimality Test Introduction ™@ Objective function is Z = 3x71 + 2x2 Geometric Interpretation The Algebra of The Simplex Method @ Note basic variables have coefficient of zero on objective function Geometric versus Algebraic —H Coefficients of nonbasic variables (3 and 5) give the rate of improvement of Z if increased from zero. Initialization Optimality Test Step 1 of Iteration 1 ™ Conclude solution (0,0, 4, 12, 18) is not optimal. Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) ° Optimality Test Again Tabular Form 20/35 Step 1 of Iteration 1 Introduction : m Which nonbasic variable to increase? Geometric Interpretation Spins ieee “ Z = 3x1 + 5x9 hones Increase 71? Rate of improvement in Z = 3 Initialization : Increase 72? Rate of improvement in Z = 5 Optimality Test : 5 > 3, so choose £2 to increase Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration @ Call x2 the entering basic variable Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) ° Optimality Test Again Tabular Form 21/35 Step 2 of an Iteration Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form ™ How far to increase the entering variable (x2)? @ When variables become nonnegative solution becomes infeasible @ Noting x; = 0 then: Ly + 23 = 4 rz = 4 229 + LA = 12 v4 = 12— 2290 3x1 + 229 + 5 = 18 @5 = 18— 2x9 @ Analysing above gives: x3 =4>0 = noupper bound on x2 12 tq = 12-2%2 >0>%0< > = 6 < minimum 18 5 = 18-22% >05>%2< > =9 @ Thus 72 can be increased to 6 22/35 Step 2 of an Iteration Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form ™ How far to increase the entering variable (x2)? m When variables become nonnegative solution becomes infeasible @ Noting x; = 0 then: Ly + 23 = 4 rz = 4 229 + LA = 12 v4 = 12— 2290 3x1 + 229 + 5 = 18 @5 = 18— 2x9 @ Analysing above gives: x3 =4>0 = noupper bound on x2 12 tq = 12-2%2 >0>%0< > = 6 < minimum 18 5 = 18-22% >05>%2< > =9 @ Thus 72 can be increased to 6 22/35 Step 2 of an Iteration Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form ™ How far to increase the entering variable (x2)? m When variables become nonnegative solution becomes infeasible ® Noting x, = 0 then: “Ly + x3 = 4 23-4 2x9 + LA = 12 v4 = 12-249 3x1 + 229 + v5 = 18 v5 = 18— 2x9 @ Analysing above gives: x3 =4>0 = noupper bound on x2 12 t4 = 12-2472 > 0 => rg < — =6 <— minimum t = 18-20, 270>m<5 7 =9 @ Thus 72 can be increased to 6 22/35 Step 3 of an Iteration Introduction : B%wv=O0>%=6 Geometric Interpretation $ Simpler Method Initial BF Solution New BF Solution Goomelrio versus : Nonbasic variables: 2, =0, x2 =0 z1=0, v4 = 0 Algebraic : basic variables: 7a =4,¢4=12,%75=18 wg =?, v2 =625 =? Initialization : Optimality Test : . . . Step 1 of Kteration 1 : @ Apply Gaussian elimination to: Step 2 of an Iteration : Step 3 of an Iteration P Z —_ 324 —_ 0x2 = Step 3 — Gaussian : Elimination : LY +23 — Optimality Test . 2x9 +24 = 12 Iteration 2 (Again) ° 2... on on _ Optimality Test Again : 3x 1 +24 2 +x 5 a 1 8 Tabular Form F ; m Want coefficients of x2 (—5,0, 2,2) = (0,0, 1,0) 23/35 Step 3 of an Iteration Introduction : B%wv=O0>%=6 Geometric Interpretation $ The Algebra of The ° wy: . . Simplex Method : Initial BF Solution New BF Solution Geometric versus : Nonbasic variables: 2, = 0, x2 =0 v1 =0, v4 = 0 Algebraic : basic variables: wa =4,%4=12,%75=18 «w3=?, 2 =645 =? Initialization : Optimality Test : Step 1 of Kteration 1 : @ Apply Gaussian elimination to: Step 2 of an Iteration : Step 3 of an Iteration : Z — 324 — 5x9 = Step 3 — Gaussian : Elimination : LY +23 — Optimality Test : 229 +24 — 12 Iteration 2 (Again) : « 7 l| a CO 3%, +229 +25 Optimality Test Again Tabular Form m Want coefficients of x2 (—5,0, 2,2) = (0,0, 1,0) 23/35 Step 3 of an Iteration Introduction : B%wv=O0>%=6 Geometric Interpretation $ The Algebra of The ° wy: . . Simplex Method : Initial BF Solution New BF Solution Geometric versus : Nonbasic variables: 2, = 0, x2 =0 v1 =0, v4 = 0 Algebraic : basic variables: wa =4,%4=12,%75=18 «w3=?, 2 =645 =? Initialization : Optimality Test : Step toflterationt m@ Apply Gaussian elimination to: Step 2 of an Iteration : Step 3 of an Iteration : Z — 324 — 5x9 = Step 3 — Gaussian : Elimination : LY +23 — Optimality Test : 229 +24 — 12 Iteration 2 (Again) : « 7 l| a CO 3%, +229 +25 Optimality Test Again Tabular Form m Want coefficients of x2 (—5,0, 2,2) = (0,0, 1,0) 23/35 Step 3 — Gaussian Elimination Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Two elementary operations: O Multiply (or divide) an equation by a nonzero constant O Add (or subtract) a multiple of one equation to (or from) another equation Apply operations to get: Z —32, XY m@ New BF solution (x; = 0,72 = 6,23 = 4,274 = 0,25 = 6) and Z = 30 24/35 Step 3 — Gaussian Elimination Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form H Two elementary operations: LO Multiply (or divide) an equation by a nonzero constant O Add (or subtract) a multiple of one equation to (or from) another equation @ Apply operations to get: Z —32, XY m@ New BF solution (x; = 0,72 = 6,23 = 4,274 = 0,25 = 6) and Z = 30 24/35 Step 3 — Gaussian Elimination Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form H Two elementary operations: LO Multiply (or divide) an equation by a nonzero constant LO Add (or subtract) a multiple of one equation to (or from) another equation @ Apply operations to get: Z —32, XY m@ New BF solution (x; = 0,72 = 6,23 = 4,274 = 0,25 = 6) and Z = 30 24/35 Step 3 — Gaussian Elimination Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form H Two elementary operations: LO Multiply (or divide) an equation by a nonzero constant LO Add (or subtract) a multiple of one equation to (or from) another equation @ Apply operations to get: Z -3x1 +324 = 30 ry +23 = x2 +3204 = 321 —X4 +5 = m@ New BF solution (x1 = 0,22 = 6,273 = 4,24 = 0,25 = 6) and Z = 30 24/35 Optimality Test Introduction : H® Current solution gives the following Obj. Fn Geometric Interpretation $ The Algebra of The ° SimplexMethod Z = 30+ 8 — Sry Geometric versus Algebraic Initialization Optimality Test Step Tot leraton Ty @ Will increasing a nonbasic variable (x; or x4) above 0 improve Obj. Step 2 of an Iteration : En 9 Step 3 of an Iteration Step 3 — Gaussian CD : M™ x, selected as entering variable Optimality Test Iteration 2 (Again) ° Optimality Test Again Tabular Form 25/35 Optimality Test Introduction : H® Current solution gives the following Obj. Fn Geometric Interpretation $ The Algebra of The ° SimplexMethod Z = 30+ 8 — Sry Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 : Will increasing a nonbasic variable (x; or x4) above 0 improve Obj. : Fn.? Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian CD : M™ x, selected as entering variable Optimality Test Iteration 2 (Again) ° Optimality Test Again Tabular Form 25/35 Iteration 2 (Again) Introduction : ™ Current system of equations yields (x4 = 0): Geometric Interpretation : 4 The Algebra of The : 13 4-7, >0 > 4,<-=4 Simplex Method : ~~ ~~ 1 Geomet : Aigebraic : x2 =6 > 0 = noupper bound on x1 Initialization : Optimality Test : x5 => 6 —_— 321 > 0 => LY < == 2 — minimum Step 1 of Iteration 1 : 3 Step 2 of an Iteration : Step 3 of an teration @ Minimum ratio indicates x5 leaving basic variable Step 3 — Gaussian : Elimination : . . . . se . 5 : ™@ x, replacing x5 as basic variable, perform Gaussian Elimination: ptimality Test . Iteration 2 (Again) : 3 Optimality Test Again : Z +524 +25 — 36 Tabular Form F X3 5X4 _— $25 = : x2 + 4 v4 = 1 1 : Ly —3%4 13l5 = @ BF solution is (71 = 2,272 = 6,23 = 2,24 = 0,25 = 0) 26/35 Iteration 2 (Again) Introduction : ™ Current system of equations yields (x4 = 0): Geometric Interpretation : 4 The Algebra of The : 13 4-7, >0 > 4,<-=4 Simplex Method : ~~ ~~ 1 Geomet : Aigebraic : x2 =6 > 0 = noupper bound on x1 Initialization : Optimality Test : x5 => 6 —_— 321 > 0 => LY < == 2 — minimum Step 1 of Iteration 1 : 3 Step 2 of an Iteration : Step 3 of an teration H Minimum ratio indicates x5 leaving basic variable Step 3 — Gaussian : Elimination : . . . . se . 5 : ™@ x, replacing x5 as basic variable, perform Gaussian Elimination: ptimality Test . Iteration 2 (Again) : 3 Optimality Test Again : Z +504 +25 => 36 Tabular Form F X3 5X4 _— $25 = : x2 + 4 v4 = 1 1 : Ly —3%4 13l5 = @ BF solution is (71 = 2,272 = 6,23 = 2,24 = 0,25 = 0) 26/35 Iteration 2 (Again) Introduction ™ Current system of equations yields (x4 = 0): Geometric Interpretation : 4 The Algebra of The : r3= 4 — 2 > 0 => XY <-= 4 Simplex Method : ~~ ~~ 1 G i : Aigebraic : x2 =6 > 0 = noupper bound on x1 Initialization : Optimality Test : %=6-3%7,>0 > %< 3 = 2 <— minimum Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of anteration H Minimum ratio indicates x5 leaving basic variable Step 3 — Gaussian : Elimination : . . . . . . . . Optimality Test F M™ x, replacing v5 as basic variable, perform Gaussian Elimination: Iteration 2 (Again) : 3 Optimality Test Again : Z + 5 LA +25 => 36 Tabular Form F X3 ELA — 55 — 2 F x9 tio, =6 F Ly — $4 + x 5 =2 @ BF solution is (71 = 2,272 = 6,23 = 2,24 = 0,25 = 0) 26/35 Optimality Test Again Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Obj. Fn. Z = 36 — 3x4 — x5 Increasing either x4 or x5 would decrease Z Optimal solution is 7; = 2, X2 = 6 and slack variables v3 = 0,274 = 0,275 = O Demonstration entitled Simple Method —Algebraic Form in OR Tutor Solve Interactively by the Simplex Method in Courseware 27/35 Optimality Test Again Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Obj. Fn. Z = 36 — 3x4 — x5 Increasing either 24 or x5 would decrease Z Optimal solution is 7; = 2, X2 = 6 and slack variables v3 = 0,274 = 0,275 = O Demonstration entitled Simple Method —Algebraic Form in OR Tutor Solve Interactively by the Simplex Method in Courseware 27/35 Optimality Test Again Introduction Geometric Interpretation $ The Algebra of The Simplex Method Geometric versus Algebraic Initialization Optimality Test Step 1 of Iteration 1 Step 2 of an Iteration Step 3 of an Iteration Step 3 — Gaussian Elimination Optimality Test Iteration 2 (Again) Optimality Test Again Tabular Form Obj. Fn. Z = 36 — 3x4 — x5 Increasing either 24 or x5 would decrease Z Optimal solution is x1 = 2, X2 = 6 and slack variables v3 = 0,74 = 0,275 = 0 Demonstration entitled Simple Method —Algebraic Form in OR Tutor Solve Interactively by the Simplex Method in Courseware 27/35