Chapter 3
Linear Algebra- Preliminaries
Definition 1 Row or Column Vector:
An n vector is a row or column array of n numbers. Each n vector can be represented by a point or
by a line from the origin to the point, with an arrowhead at the end point of the

DEPARTMENT OF MANGEMENT STUDIES
INDIAN INSTITUTE OF TECHNOLOGY, MADRAS
MS 3510 - Fundamentals of Operations Research
Problem Sheet 2 (Simplex Algorithm)
1.
Consider the following linear programming problem
Minimize
Z = X1 + 3X2
Subject to
5X1
+ 4X1 20
3X1

1
Solution for Quiz 1
TTN , DoMS, IIT Madras
2
Problem 1 (1/3)
A company manufactures an assembly consisting of a frame,
a shaft and a ball-bearing. While frames and shafts are
produced in the companys factory, ball-bearings are
procured from an outside s

1
Solution for Quiz 1
TTN , DoMS, IIT Madras
2
Problem 1 (1/3)
A company manufactures an assembly consisting of a frame,
a shaft and a ball-bearing. While frames and shafts are
produced in the companys factory, ball-bearings are
procured from an outside s

1
TTN , DoMS, IIT Madras
2
Problem No.1
Decision variables:
X1 Number of Type A vehicles
X2 Number of Type B vehicles
X3 Number of Type C vehicles
Number of shifts for each vehicle = 3
Capacity of vehicle A = 10*55*18 ton-km/day = 9900 ton-km/day
Capacity

Department of Management Studies
Indian Institute of Technology, Madras
PROBLEM SHEET ON INVENTORY CONTROL
1.
*
Let Q be the optimal order quantity in the simple EOQ model (constant rate of
demand, instantaneous supply, no shortages) Let the actual order

Department of Management Studies
Indian Institute of Technology, Madras
PROBLEM SHEET ON DYNAMIC PROGRAMMING
1. Consider an electronic system consisting of four components each of which must
function in order that the system may function. The reliability

DEPARTMENT OF MANGEMENT STUDIES
INDIAN INSTITUTE OF TECHNOLOGY, MADRAS
MS 3510 - Fundamentals of Operations Research
Problem Sheet 1 (Formulation of linear models)
1.
A trucking company with Rs.20 crores to spend on new equipment is contemplating three ty

OPERATIONS RESEARCH
PROBLEM SHEET
Assignment
1.
M/s Paul Dairy Development Corporation plans to process three products. It owns
three plants, one of which is to be selected to process each product. The following
tables give the estimated processing costs

Chapter 12
Transportation and Assignment
Problem
The Transportation Problem
General description of a transportation problem is specified by the following
1. A set of m supply(origin) points. Origin i, can supply at most si units
2. A set of n demand(desti

DEPARTMENT OF MANGEMENT STUDIES
INDIAN INSTITUTE OF TECHNOLOGY, MADRAS
MS 3510 - Fundamentals of Operations Research
Problem Sheet 2 (Simplex Algorithm)
1.
Consider the following linear programming problem
Minimize
Z = X1 + 3X2
Subject to
5X1
+ 4X2 20
3X1

Chapter 2
Linear Programming
Modelling
In this chapter we provide with a few linear programming formulations of some
well known problems. We have disussed a few more problems in class. I have
not included them here.
Formulation of LP Model:
1. Identify th

Chapter 5
Extreme Points and Basic Feasible
Solutions
Let S = cfw_x : Ax b, x 0 be a polyhedral set. We have already seen an extreme point of S is a point
that cannot be expressed as a strict convex combination of any two distinct points of S.
Definition

Chapter 6
The Simplex Method
In this chapter we develop the simplex method to solve the Linear programming problem. the simplex
method is a procedure that moves from an extreme point(basic feasible solution) to another extreme point
with a better( improve

Chapter 8
Initial Basic Feasible Solution-Big M
Method
The presence of artificial variables at a positive level means that the current point is an infeasible solution
to the original system. One of the ways to get rid of the artificial variables is the Tw

Chapter 11
Sensitivity Analysis
CONTENTS
1. Changing Cost Coefficient of Objective Function
2. Changing the right-hand side vector
3. Changing the Coefficient Matrix
Reference Chapter 6 in BJS
Introduction
Sensitivity analysis is concerned with how change

Chapter 4
LP Geometry in Two Dimensions
Geometric procedure for solving a linear programming problem is only suitable for very small problems. It
provides a great deal of insight into the linear programming problem.
Consider the following problem
0
M inim

Chapter 7
Initial Basic Feasible Solutions -Two
Phase Method
Initial Basic feasible Solution
Simplex Method starts with a basic feasible solution and moves to an improved basic feasible solution, until
an optimal solution is reached or we conclude that th

Department of Management Studies
Indian Institute of Technology, Madras
PROBLEM SHEET ON DYNAMIC PROGRAMMING
1. Consider an electronic system consisting of four components each of which must
function in order that the system may function. The reliability

Department of Management Studies
Indian Institute of Technology, Madras
PROBLEM SHEET ON TRANSPORTATION
1.
A department store wishes to purchase the following quantities of ladies dresses:
Dress Type
A
Quantity
150
B
100
C
74
D
250
E
200
Tenders are submi

Indian Institute of Technology, Madras
Problem Sheet on Duality and Sensitivity Analysis
1.
Consider the Linear Programming problem:
-X1 + 7X2 - 5X3 + 14X4
Maximize
Z =
Subject to
3X1 + 4X2 + 5X3 + 6X4 < 24
-X1 + X2 - 2X3 + 2X4 < 4
Xj > 0
a)
Write the dua

1
Inventory Control
T.T.Narendran
Department of Management Studies,
IIT Madras.
TTN , DoMS, IIT Madras
2
Introduction
Basically two questions are to be answered in inventory control.
1)
How much to order?
2)
When to order?
Notations used:
D - Annual deman

1
TTN , DoMS, IIT Madras
Duality in LP
Consider the L.P:
Maximize Z = 2X1+ X2+ 3X3
Subject to
5X1 2X2 + X3 60
-X1 + 3X2 + 2X3 50
X1, X2, X3 > 0
Call this the PRIMAL
TTN , DoMS, IIT Madras
2
3
This problem has a DUAL written as follows:
Minimize W = 60 y1

1
Assignment Problem
TTN , DoMS, IIT Madras
Assignment
m
Minimize
n
Z = Cij X ij
i =1 j =1
Subject to
m
X
i =1
ij
=1
j =1 2 n
1, 2,.
ij
=1
i = 1, 2,.m
n
X
j =1
X ij (0 1)
(0,1)
i = 1, 2 m and j = 1 2 n
1 2,.
d
1, 2,.
X ijj = Binary integer variable
TTN

Transportation Problem
Part 2
TTN, DoMS, IIT- Madras
Example
1
2
3
4
A
30
7
3
5
6
B
40
8
7
7
3
C
30
2
22
10
27
9
23
4
28
100
Exploring other options
To
1
A
2
22
3
30
8
7
From
4
B
3
19
8
5
40
21
7
C
7
2
2
22
6
10
27
3
9
23
30
28
4
28
100
To
1
A
From
B
2
22

Variations of the
Transportation Problem
TTN, DoMS, IIT- Madras
1. Maximization problem:
VAM: Rewrite matrix with (-ve) sign and proceed.
(OR)
Take difference between largest and second largest and
choose the biggest cell.
2. Degeneracy:
MODI: Cj - ij < 0

1
T.T. Narendran
Department of Management Studies
Indian Institute of Technology Madras
TTN DoMS, IIT Madras, 22-Aug-11
2
Example Problem No. 1
(Linear Programming - Formulation)
A small factory makes three products, soap, shampoo
and liquid soap. The pr

1
T.T. Narendran
Department of Management Studies
Indian Institute of Technology Madras
TTN DoMS, IIT Madras, 29-Aug-11
Complications in LP and their
resolution:
2
There are many deviations from the standard
form of the linear programme that was used to

Transportation Problem
TTN, DoMS, IIT- Madras
Suppose a large industry owns 3 plants at
Jamshedpur, Hosur and Ludhiana and
transports the goods to its 4 warehouses located
in the 4 metros
Weekly production in each plant is known
Weekly demand for each w