Artificial Variable Technique
(The Big-M Method)
ATISH KHADSE
Big-M Method of solving LPP
The Big-M method of handling instances with artificial
variables is the commonsense approach. Essentially, the
notion is to make the artificial variables, through th
IE 400:
Homework #2
Spring 2015
Due: February 23, 17:00
Note: Solve question 3 and only one of the (question 1 or 2).
Question 1. Solve the following linear programming model graphically:
a)
M in
s.t.
4x1 + 5x2
2x1 + 3x2 30
x1 + x2 12
2x1 + x2 20
x1 , x2
IE 400 Principles of Engineering Management
The Simplex Algorithm-I:
Set 3
Bilkent University
Spring 2015
1
So far, we have studied how to solve two-variable
LP problems graphically.
However, most real life problems have more than
two variables!
Theref
IE 400 Principles of Engineering Management
The Simplex Algorithm-II:
Set 4
Bilkent University
Spring 2015
1
The Simplex Algorithm:
To begin the simplex algorithm, convert the LP into
a standard form,
Convert the objective function z=c1x1+c2x2+ cnxn
to
IE 400: Principles of Engineering Management
QUIZ 1
Spring 2015
February 13, 2015
NAME:
ID:
Question: The Southern Confederation of Kibbutzim is a group of three kibbutzim (farming
communities). Overall planning for this group is done by an office and thi
IE 400: Principles of Engineering Management
QUIZ 2
Spring 2015
February 27, 17:4018:40
NAME:
ID:
SECTION:
(a) Solve the following problem graphically. Illustrate the feasible region clearly.
X2
min z = x1 + 2x2
s.t.
x1 + x2 3
x1 + 2x2 6
x1 0
x2 : urs
.
X
IE 400: Principles of
Engineering Management
Lecture 1
1
Any questions?
What is Operations Research (OR)?
OR is a decision making tool based on
mathematical modeling and analysis to aid in
determining how best to design and operate a
system usually under
IE 400 Principles of Engineering Management
Integer Programming: Modeling Examples
Lecture Notes 6
1
IE 400 Principles of Engineering Management
Integer Programming:
Set 5
2
3
4
5
6
Integer Programming:
So far, we have considered problems under the follow
IE 400 STUDY SET
Question 1) The Primo Insurance Company is introducing two new product lines: special risk
insurance and mortgages. The expected profit is $5 per unit on special risk insurance and $2
per unit on mortgages. Management wishes to establish
IE400 Spring 2015 Homework I - Due: 09.02.2015 13:00
Question 1) Young MBA Erica Cudahy may invest up to $1000. She can invest her money in stocks
and loans. Each dollar invested in stock yields 10 cent profit, and each dollar invested in a loan yields
15
07.03.2012
IE 400 Principles of Engineering Management
The Simplex Algorithm-II:
Lecture Notes 5
The Simplex Algorithm:
To begin the simplex algorithm, convert the LP into
a standard form,
Convert the objective function z=c1x1+c2x2+ cnxn
to the row-0 fo
Dorusal Programlama
BYK M YNTEM
VE DUYARLLK ANALIZI
Byk M Yntemi: Eer bir DP'de > veya = kstlar varsa, Simpleks yntemi
kullanlarak bir balang temel olurlu zm (bfs) oluturulamaz.
Bu durumda Byk M (Big M) yntemi veya ki Evreli (Two Phase) Simpleks
yntemi ku
IE 400-01: Principles of Engineering Management - Spring 2012
Instructor:
Section 1: Ouzhan Vcl, e-mail: oguzhan.vicil@bilkent.edu.tr
g
Course Web Page: http:/www.ie.bilkent.edu.tr/ ie400
Teaching Assistant:
Cem Ozdemir, EA 323, cemo@bilkent.edu.tr,
Oce H
IE 400: Homework 2 Solution
Due April 2, 17.30 PM
1) Consider the following LP:
Min z= 4 X1 + 2 X2
s.t.
3X1 + X2 27
-X1 - X2 -21
X1 + 2X2 30
X10
X2 unrestricted
a) Solve the above LP by using the graphical method.
Since this is straightforward, the soluti
IE 400: HW 4
Due By May 18 2012, Friday
1) A project consists of ten activities with the following precedence constraints and durations:
Activity
A
B
C
D
E
F
G
H
I
J
Time (days)
4
5
2
5
4
4
3
7
6
9
Immediate Predecessor
A
B,C
B
D
E,F
D
G,H
C
a) Construct
Agenda
Excel
Solver
Xpress
Excel Solver
Simple is Better
Set-up Excel Solver Add-In
Transform optimization model into Excel Sheet
Assume we have the following model
max
s.t.
x +y+z
x+y 20
2x+3z =12
x+y+2z 10
3x+4y+2z 20
Excel Sheet
Construct the model
IE 400: Homework 1
Due March 15, 17.30 PM
1) A farmer has 60 acres of land and he is planning to plant the fields with grain. The available
options are wheat, corn, barley and oat. Each acre of land planted with wheat yields 1 ,500
TL revenue, each with c
IE 400: Homework 2
Due April 2, 17.30 PM
1) Consider the following LP:
Min z= 4 X1 + 2 X2
s.t.
3 X1 + X2 2 7
-X1 - X2 -21
X1 + 2X 2 3 0
X10
X2 unrestricted
a) Solve the above LP by using the graphical method.
b) Solve the above LP by using Big M Method, a
IE 400: HW 3
Due By April 26 2012, Thursday
1) Use the Branch-and-Bound method to find the optimal solution to the following IP:
a) Min z = 4X1 + 5X2 - 2X3
s.t.
X1 + 4 X2 + X3 5
3X1 + 2X2 - 2X3 7
X1, X2 0; X1, X2 integer
X3cfw_0,1
(Hint: Without solving P
IE 400: Homework 1 Solution
Due March 15, 17.30 PM
1) Let us define the decision variables:
X1= number of acres of land planted with wheat
X2= number of acres of land planted with corn
X3= number of acres of land planted with barley
X4= number of acres of
IE 400: HW 3
Due By April 26 2012, Thursday
1) Use the Branch-and-Bound method to find the optimal solution to the following IP:
a) Min z = 4X1 + 5X2 - 2X3
s.t.
X1 + 4X2 + X3 5
3 X1 + 2 X2 - 2 X3 7
X1, X2 0; X1, X2 integer
X3cfw_0,1
(Hint: Without solving
2/16/2012
IE 400: Principles of
Engineering Management
Lecture Notes: 1
1
What is Operations Research (OR)?
OR is a decision making tool based on
mathematical modeling and analysis to aid in
determining how best to design and operate a
system usually unde
2/21/2012
IE 400: Principles of
Engineering Management
Lecture Notes: 1&2
1
What is Operations Research (OR)?
OR is a decision making tool based on
mathematical modeling and analysis to aid in
determining how best to design and operate a
system usually un
27.02.2012
IE 400 Principles of Engineering Management
The Simplex Algorithm-I:
Lecture Notes 4
So far, we have studied how to solve two-variable
LP problems graphically.
However, most real life problems have more than
two variables!
Therefore, we need