4 - primal and dual v1
Post on 07-Jul-2018
220 Views
Preview:
TRANSCRIPT
-
8/19/2019 4 - Primal and Dual V1
1/36
REVISED SIMPLEX METHODRevised Simplex Method (5.2,5.3,5.4)
Dual Introduction (6.1)
1/19/2016 1ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
2/36
Revised Simplex Method
• Method was motivated for efficient computational purposes
• Works only with relevant piece of information at each iteration
• Revised Simplex Method uses Matrix Notation and Matrix
Manipulations• The insights it provide us are very useful in dual formulation of
the original LP (which is called primal)
1/19/2016 2ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
3/36
Matrix Notation
1/19/2016 3ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
4/36
Matrix Notation : Augmented Form
1/19/2016 4ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
5/36
Matrix Notation : Augmented Form
Example – WYNDOR Glass Co.
1/19/2016 5ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
6/36
Matrix Notation : Current set of equations
(Simplex Tableau)
1/19/2016 6ME 332 – IEOR 2016 Spring
Current Set of Equations
-
8/19/2019 4 - Primal and Dual V1
7/36
Solving for BFS with Revised Simplex
• Eliminate non basic variables from
to obtain the vector of basic variables
• Eliminate a j columns of Non Basic Variables from
to obtain the basis matrix
We want to work only with Basic Variables
1/19/2016 7ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
8/36
Solving for BFS with Revised Simplex
• Basic Variable Values are obtained by:
• C B is the vector whose elements are objective function
coefficients corresponding to X B then the value of objectivefunction for the current basic solution is
1/19/2016 8ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
9/36
Example - WYNDOR
1/19/2016 9ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
10/36
Getting all the other values of current iteration
through B-1
•
Matrix form of current set of equations:
1/19/2016 10ME 332 – IEOR 2016 Spring
Premultiply both sides with
-
8/19/2019 4 - Primal and Dual V1
11/36
Simplex tableau in Matrix Form
• At any iteration:
1/19/2016 11ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
12/36
Simplex Tableau – Fundamental Insights
1/19/2016 12ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
13/36
Simplex Tableau – Fundamental Insights
• New row(1) = old row (1)
• New row(2) = ½ *old row( 2)
• New row (3) = (-1)* old row (2) + 1*old row (3)
• After any iteration , the coeff. of the slack variables in each equation immediately reveal how that equation hasbeen obtained from the initial equations.
1/19/2016 13ME 332 – IEOR 2016 Spring
Current itr. Slack Coeff Initial rows 1-3
=
New rows 1-3
-
8/19/2019 4 - Primal and Dual V1
14/36
Simplex Tableau – Fundamental Insights
1/19/2016 14ME 332 – IEOR 2016 Spring
=
New row 0
+
Initial row 0 initial rows 1 to 3Currentslack vari.
cost coeff
-
8/19/2019 4 - Primal and Dual V1
15/36
Simplex Tableau – Fundamental Insights
1/19/2016 15ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
16/36
Revised Simplex Algorithm
• Initialize the Problem, Introduce slack variables
• Obtain
• Iteration
• Determine the Entering Basic Variable
• Determine the Leaving Basic Variable
• Determine the new BFS
• Optimality Test
• Current solution is optimal if all the coeff are non negative
1/19/2016 16ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
17/36
Revised Simplex – Summary
1/19/2016 17ME 332 – IEOR 2016 Spring
/ / 18
-
8/19/2019 4 - Primal and Dual V1
18/36
Dual - Introduction
• At any iteration simplex method for primal problem, current
Row 0 is represented as shown in the table
1/19/2016 18ME 332 – IEOR 2016 Spring
1/19/2016 19ME 322 IEOR 2016 S i
-
8/19/2019 4 - Primal and Dual V1
19/36
Dual - Introduction
1/19/2016 19ME 322 – IEOR 2016 Spring
y
1/19/2016 20ME 332 IEOR 2016 S i
-
8/19/2019 4 - Primal and Dual V1
20/36
Dual - Introduction
Iterations Primal Dual
0 Feasible, Not optimal Infeasible
1 Feasible, Not optimal Infeasible
2 Feasible, Optimal Feasible, Optimal
1/19/2016 20ME 332 – IEOR 2016 Spring
1/19/2016 21ME 332 IEOR 2016 S i
-
8/19/2019 4 - Primal and Dual V1
21/36
PRIMAL - DUAL
Dual Formulation (6.1)Primal-Dual Relations (6.1, 6.3)
Dual Solution (Dual Simplex – 7.1)
Sensitivity Analysis (6.5)
Economic Interpretations (6.2)
1/19/2016 21ME 332 – IEOR 2016 Spring
1/19/2016 22ME 332 IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
22/36
Dual - Introduction
• Every Linear Programming problem is associated with it
another LP called the Dual , the original is called the Primal
• When ever we solve LP we are solving two problems
• Primal Resource Allocation Problem
• Dual Resource Valuation Problem
• Primary use of Duality lies in the interpretation and
implementation of sensitivity analysis – How to trade off
resources
• If Primal has n variables and m constraints, its Dual has m
variables and n constraints
1/19/2016 22ME 332 – IEOR 2016 Spring
1/19/2016 23ME 332 IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
23/36
Dual Formulation
1/19/2016 23ME 332 – IEOR 2016 Spring
1/19/2016 24ME 332 IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
24/36
Dual Formulation
• Number of parameters remain the same
• Primal obj. coeff. become constraint right hand sides in Dual
• Primal constraint RHS becomes obj. coeff. in Dual
• Coeff. of variable in functional constraint of primal problem are
the coeff in a functional constraint of the Dual
1/19/2016 24ME 332 – IEOR 2016 Spring
1/19/2016 25ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
25/36
Dual Formulation: Rules *
1/19/2016 25ME 332 – IEOR 2016 Spring
* Taken from Hamdy A. Taha
1/19/2016 26ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
26/36
Dual Formulation: Example 1
• Try to formulate dual of the dual*
1/19/2016 26ME 332 IEOR 2016 Spring
1/19/2016 27ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
27/36
Dual Formulation: Example 2
1/19/2016 27ME 332 IEOR 2016 Spring
1/19/2016 28ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
28/36
Primal – Dual variables correspondence
1/19/2016 28ME 332 IEOR 2016 Spring
1/19/2016 29ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
29/36
Primal – Dual Relationships
Theorem Primal Dual
Weak Duality Property: If x is a
feasible solution for the primal
problem and y is a feasible solution
for the dual problem, then
cx ≤ yb
x1 = 3
x2 = 3
Z = cx = 24
y1 = 1
y2 = 1
y3 = 2
W = yb = 36
Strong Duality Property: if x* is an
optimal solution for the primal
problem and y* is an optimal solution
for the dual problem, then
cx* = y*b
x1 = 2
x2 = 6
Z = cx *= 36
(Primal Maximumfeasible hence optimal)
y1 = 0
y2 = 3/2
y3 = 1
W = y*b = 36
(Dual Minimum
feasible hence
optimal)
1/19/2016 29ME 332 IEOR 2016 Spring
1/19/2016 30ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
30/36
Primal – Dual Relationships
Theorem Primal Dual
Complementary Solutions Property: At each
iteration , the simplex method simultaneously
identifies a CPF solution x for primal and a
complementary solution y for its dual
problemcx = yb
x1 = 0
x2 = 6
Z = cx = 30
y1 = 0
y2 = 5/2
y3 = 0
W = yb = 30
Complementary Optimal Solutions Property:
At the final iteration, the simplex method
simultaneously identifies an optimal solution
x* for the primal problem and a
complementary optimal solution y* for the
dual problem
cx* = y*b
x1* = 2
x2* = 6
Z *= cx* = 36
y1* = 0
y2* = 3/2
y3* = 1
W* = y*b = 36
/ / 30 p g
1/19/2016 31ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
31/36
Primal – Dual Relationships (Basic Solutions)
Theorem Primal Dual
Complementary Basic Solutions Property:
Each basic solution in the primal problem has a
complementary basic solution in the dual
problem, where their respective objectivefunction values (Z and W) are equal.
(Basic)
x3 = 4
x4 = 12
x5 = 18
(Non basic)x1 = 0
x2 = 0
(Non basic)
y1 = 0
y2 = 0
y3 = 0
(Basic)y4 = -3
y5 = -5
Complementary Slackness Property:
The primal basic solution and the complementary
dual basic solution satisfy the complementary
slackness relation
i.e, Basic variables in primal are Non basic
variables in dual and Vice versa
Primal Basic
(0,6,4,0,6)
Basic: x2,x3,x5
Non Basic
x1,x4
Compl. Dual
basic
(0,5/2,0,-3,0)
Non Basic
y1,y3,y5 = 0
Basic
(y1 + 3y3 - 3)
(2y2 + 2y3 - 5)
/ / p g
1/19/2016 32ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
32/36
Primal - Dual Relationships
/ / p g
1/19/2016 33ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
33/36
Simplex on the Dual
Y1 Y2 Y3 Y4 a1 Y5 a2 Z Coeff Mi
Z -1 -9996 12 -29982 10000 0 0 10000 -30000 0
a1 0 1 0 3 -1 1 0 0 3
a2 0 0 2 2 0 0 -1 1 5
Y1 Y2 Y3 Y4 a1 Y5 a2 Z Coeff Mi
Z -1 -9996 -19988 -49982 10000 0 10000 0 -80000 -49982
a1 0 1 0 3 -1 1 0 0 3 1
a2 0 0 2 2 0 0 -1 1 5 2.5
Y1 Y2 Y3 Y4 a1 Y5 a2 Z Coeff Mi
Z -1 6664.667 -19988 0 -6660.666667 16660.67 10000 0 -30018 -19988
Y3 0 0.333333 0 1 -0.333333333 0.333333 0 0 1Y2 0 -0.66667 2 0 0.666666667 -0.66667 -1 1 3
Y1 Y2 Y3 Y4 a1 Y5 a2 Z Coeff Mi
Z -1 2 0 0 2 9998 6 9994 -36 0
Y3 0 0.333333 0 1 -0.333333333 0.333333 0 0 1
Y2 0 -0.33333 1 0 0.333333333 -0.33333 -0.5 0.5 1.5
1/19/2016 34ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
34/36
Dual Simplex
1/19/2016 35ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
35/36
Why Dual?
1/19/2016 36ME 332 – IEOR 2016 Spring
-
8/19/2019 4 - Primal and Dual V1
36/36
TO BE UPDATED…
top related