Lecture 11_Chap 6 Duality Theory_after class, updated (1).pdf

IE310. Operations Research
Chapter 6. Duality Theory and Sensitivity Analysis
Lecture 11 Duality Theory (2)
Ruoyu Sun
ISE Department
1

2
Review Questions of Lecture 11
• What is a dual problem of a standard LP?
How to interpret the dual problem, using Burger/Pizza restaurant as an
example?
• How to interpret the dual problem of the primal problem, using Ford/Tesla as
an example?
• How to interpret the dual problem, using Burger/Pizza restaurant as an
example?
y

3
This Lecture
• Duality theory: weak duality and strong duality
After today’s lecture, you should be able to
• Describe the relation between primal and dual LP
• Find a complementary basic solution for a given primal BF solution

4
Homework 3 correction
• homework#3, question 4.6-5, part (b) asks to use a computer package based
on the simplex method to determine that the problem has no feasible solution.
• Skip this part.
• You still need to answer other parts, (a), (c), (d).

5
Question on Software
• Will we learn software? Very very briefly later; not in detail.
• Benefit of understanding the procedure of simplex method (by hand):
• 1) More interpretable.
--Counter-example: blackbox: deep learning
e.g. after reading images and descriptions of symptoms, the computer
generates output: this patient has got appendicitis. But the patient may or may
not believe the computer’s output.
--OR methods, like LP are often much more interpretable
• E.g. If you just run software to solve how to assign the numbers of
burgers/pizzas to produce, and make recommendation to your boss. Your
boss may not believe you
• You can tell your boss: let me show you a toy example, run 2 iterations of
simplex method to get the solution. Explain each step.
• 2) another benefit : debug. On GANs, quite hard to debug; for LP, you know
much better how to debug
• 3) post-optimality analysis

6
Basic
Var
Z Original Variables Slack Variables RHS
Z
1 c
B
T
B
-1
A – c
T
= y
T
A-c
T
c
B
T
B
-1
= y
T
c
B
T
x
B
=c
B
T
B
-1
b
=y T b
x
B
0
B
-1
A
B -1 x
B
=B -1 b
Its dual LP :
min b
T
y
s.t. A
T
y ≥ c
y ≥ 0
Relation of Dual LP and Simplex Tableau?
Simplex Tableau of the standard form