MBA SEMESTER II MB0048 Operation Research- 4 Credits (Book ID:
B1301) Assignment Set- 1 (60 Marks) Note: Each question carries 10
Marks. Answer all the questionsa .Define O.R and discuss its
characteristics. [ 5 marks] Operations research, or Operational
Research in British usage, is a discipline that deals with the
application of advanced analytical methods to help make better
decisions[1]. It is often considered to be a sub-field of
Mathematics[2]. The terms management science and decision science
are sometimes used as more modern-sounding synonyms[3]. Employing
techniques from other mathematical sciences, such as mathematical
modeling, statistical analysis, and mathematical optimization,
operations research arrives at optimal or near-optimal solutions to
complex decision-making problems. Because of its emphasis on
human-technology interaction and because of its focus on practical
applications, operations research has overlap with other
disciplines, notably industrial engineering and operations
management, and draws on psychology and organization science.
Operations Research is often concerned with determining the maximum
(of profit, performance, or yield) or minimum (of loss, risk, or
cost) of some real-world objective. Originating in military efforts
before World War II, its techniques have grown to concern problems
in a variety of industries. Characteristics of Operations Research
By an eHow Contributor Operations research, an interdisciplinary
division of mathematics and science, uses statistics, algorithms
and mathematical modeling techniques to solve complex problems for
the best possible solutions. This science is basically concerned
with optimizing maxima and minima of the objective functions
involved. Examples of maxima could be profit, performance and
yield. Minima could be loss and risk. The management of various
companies has benefited immensely from operations research.
Operations research is also known as OR. It has basic
characteristics such as systems orientation, using
interdisciplinary groups, applying scientific methodology,
providing quantitative answers, revelation of newer problems and
the consideration of human factors in relation to the state under
which research is being conducted
b. Explain the nature of Operations Research and its
limitations. Operations research has been used to solve only a
fairly limited number of managerial problems. Its limitations
should not be overlooked. In the first place, there is the sheer
magnitude of the mathematical and computing aspects. The number of
variables and interrelationships in many managerial problems, plus
the complexities of human relationships and reactions, calls for a
higher order of mathematics than nuclear physics does. The late
mathematical genius John von Neumann found, in his development of
the theory of games, that his mathematical abilities soon reached
their limit in a relatively simple strategic problem. Managers are,
however, a long way from fully using the mathematics now
available.
In the second place, although probabilities and approximations
are being substituted for unknown quantities and although
scientific method can assign values to factors that could never be
measured before, a major portion of important managerial decisions
still involves qualitative factors. Until these can be measured,
operations research will have limited usefulness in these areas,
and decisions will continue to be based on non-quantitative
judgments. Related to the fact that many management decisions
involve un-measurable factors is the lack of information needed to
make operations research useful in practice. In conceptualizing a
problem area and constructing a mathematical model to represent it,
people discover variables about which they need information that is
not now available. To improve this situation, persons interested in
the practical applications of operations research should place far
more emphasis on developing this required information. Still
another limitation is the gap between practicing managers and
trained operation researchers. Managers in general lack a knowledge
and appreciation of mathematics, just as mathematicians lack an
understanding of managerial problems. This gap is being dealt with,
to an increasing extent, by the business schools and, more often,
by business firms that team up managers with operations research.
But it is still the major reasons why firms are slow to use
operations research. A final drawback of operations research at
least in its application to complex problems is that analyses and
programming are expensive and many problems are not important
enough to justify this cost. However, in practice this has not
really been a major limitation.
a. What are the essential characteristics of a linear
programming model? marks]
[ 5
b. Explain the graphical method of solving a LPP involving two
variables. [ 5 marks] Linear programming (LP, or linear
optimization) is a mathematical method for determining a way to
achieve the best outcome (such as maximum profit or lowest cost) in
a given mathematical model for some list of requirements
represented as linear relationships. Linear programming is a
specific case of mathematical programming (mathematical
optimization). More formally, linear programming is a technique for
the optimization of a linear objective function, subject to linear
equality and linear inequality constraints. Its feasible region is
a convex polyhedron, which is a set defined as the intersection of
finitely many half spaces, each of which is defined by a linear
inequality. Its objective function is a real-valued affine function
defined on this polyhedron. A linear programming algorithm finds a
point in the polyhedron where this function has the smallest (or
largest) value if such point exists. Linear programs are problems
that can be expressed in canonical form: where x represents the
vector of variables (to be determined), c and b are vectors of
(known) coefficients and A is a (known) matrix of coefficients. The
expression to be maximized or minimized is called the objective
function (cTx in this case). The equations Ax b are the constraints
which specify a convex polytope over which the objective function
is to be optimized. (In this context, two vectors are comparable
when every entry in one is less-than or equal-to the corresponding
entry in the other. Otherwise, they are incomparable.) Linear
programming can be applied to various fields of study. It is used
in business and economics, but can also be utilized for some
engineering problems. Industries that use linear programming models
include transportation, energy, telecommunications, and
manufacturing.
It has proved useful in modeling diverse types of problems in
planning, routing, scheduling, assignment, and design. Linear
Programming Graphical Method Learn about linear programming by
graphical method. If the objective function Z is a function of two
variables only, then the Linear Programming Problem can be solved
effectively by the graphical method. If Z is a function of three
variables, then also it can be solved by this method. But in this
case the graphical solution becomes complicated enough. The linear
programming problems are solved in applied mathematics models.
Method of Linear Programming Graph Draw the graph of the
constraints. Determine the region which satisfies all the
constraints and non-negative constraints (x > 0, y > 0). This
region is called the feasible region. Determine the co-ordinates of
the corners of the feasible region. Calculate the values of the
objective function at each corner. Select the corner point which
gives the optimum (maximum or minimum) value of the objective
function. The co-ordinates of that point determine the optimal
solution. Below you can see the problems linear programming by
graphical method -
Problem 1:Use graphical method to solve the following linear
programming problem. Maximize Z = 2x + 10 y Subject to the
constraints 2 x + 5y < 16, x < 5, x > 0, y > 0.
Solution:
Since x > 0 and y > 0 the solution set is restricted to
the first quadrant.| i) 2x + 5y < 16 Draw the graph of 2x + 5y =
16 2x + 5y = 16 y= x 8 0
3
0 3.2 2 y Determine the region represented by 2x + 5y < 16
ii) x < 5 Draw the graph of x = 5 Determine the region
represented by x < 5. Shade the intersection of the two regions.
The shaded region OABC is the feasible region B(5, 1.2) is the
point of intersection of 2x + 5y = 16 and x = 5. The corner points
of OABC are O(0,0), A(5,0), B(5,1.2) and C(0,3.2). Corners O(0,0)
A(5,0) B(5,1.2) C(0,3.2) 0 Z = 2x + 10 y 10 22 32
Z is maximum at x = 0, y = 3.2 Maximum value of Z = 32. Problem
2: Use graphical method to solve the following linear programming
problem. Maximize Z = 20 x + 15y Subject to 180x + 120y < 1500,
x + y < 10, x > 0, y > 0 Solution:
Since x > 0 and y > 0, the solution set is restricted to
the first quadrant. i) 180x + 120 y < 1500 180x + 120y < 1500
=> 3x + 2y < 25. Draw the graph of 3x + 2y = 25 3x + 2y = 25
y= x 0 0 y Determine the region represented by 3x + 2y < 25. ii)
x + y < 10 Draw the graph of x + y = 10 x + y = 10 y =10 - x x 0
10 5 10 0 y 5
5 5
Determine the region represented by x + y < 10 Shade the
intersection of the two regions. The shaded region OABC is the
feasible region. B(5,5) is the point of intersection of 3x + 2y =
25 and x + y = 10. The corner points of OABC are O(0,0), A( Corners
, 0), B (5,5) and C(0,10). O(0,0) A( ,0) 0 166.67
B(5,5) 175
C(0,10) 150
Z = 20x + 15 y
Z is maximum at x = 5 and y = 5. Maximum value of Z = 175. a.
Explain the simplex procedure to solve a linear programming
problem. [ 5 marks] b. Explain the use of artificial variables in
L.P [ 5 marks] About Artificial Variables in Linear Programming
(LP) Another frequently asked question by students is related to
use of artificial variables while preparing the initial basic
feasible solution table. The common flow of discussion forces the
student to think in a logical way as he has been thinking about
slack and surplus variables but the artificial variables can not be
considered in the same logical category as the previous two. Just
to recall, slack and surplus variables are used in LP to convert
inequality constraints to that of equality. If the constraint is
of