Use Branch and Bound technique to solve the following problem Maximise z = 3x1 + 3x2 + 13 x3 Subject to – 3x1 + 6x2 + 7x3 = 8 6x1 – 3x2 + 7x3 = 8 0 = xj = 5 And xj are integer j = 1, 2, 3 Document...

Set 1
1 a) What is a Linear Programming Problem?
A linear program (LP) is a minimization problem where we are asked to minimize
a given linear
function subject to one or more linear inequality constraints. The linear function is
also called
the objective function.
Formulation:
Minimize CiXi (where Ci is real and constant and Xi are real and variables)
i=1 to n
Subject to constraints:
a11x1 + a12x2 + ........ + a1nxn _ b1
a21x1 + a22x2 + ........ + a2nxn _ b2
a31x1 + a32x2 + ........ + a3nxn _ b3
...
an1x1 + an2x2 + ........ + annxn _ bn
b) Let X1 and X2 be the number of dolls produced per day of type A and
B,respectively
Maximize (Z) = 3x1+5x2
Subject to constraints x1+2x2≤2000,
x1+2x2≤1500,
x2≤600 Non Negative Restrictions x1, x2≥0

Solve these equations we get,
Z is maximum at(1000,500) and the maximum value is 5500
Hence for maximun 1000 dolls of type A and 500 dolls of type B should be produced

2)Linear Programming:
Answer:
The term was introduced in 1950 to refer to plans or schedules for training,
logistical supply and for deployment of men in the service. A linear programming is
a subset of mathematical programming, and the later field is part of operations
research.
A linear programming problem differs from the general variety in that a
mathematical mode or description of the problem can be stated using relationships
which are “straight-line” or linear. The mathematical statement of the linear-
programming problem includes a set of linear equation which represent the
conditions of the problem.
Linear Programming is the part of mathematics deals with the study of
optimization problems with required number of constraints and objective.

Advantages of Linear Programming:
Some of the real time applications are in production scheduling, production
planning and repair, plant layout, equipment acquisition and replacement,logistic
management and fixation. Linear programming has maintained special structure
that can be exploited to gain computational advantages. Some of the advantages of
Linear Programming are:

 Utilized to analyze numerous economic, social, military and industrial
problem.
 Linear programming is most suitable for solving complex problems.
 Helps in simplicity and productive management of an organization which
gives better outcomes.
 Improves quality of decision: A better quality can be obtained with the
system by making use of linear programming.
 Provides a way to unify results from disparate areas of mechanism design.
 More flexible than any other system, a wide range of problems can be solved
easily.



3) What is Monte Carlo simulation?
Answer:
Monte Carlo simulation is a computerized mathematical technique that allows people to
account for risk in quantitative analysis and decision making. The technique is used by
professionals in such widely disparate fields as finance, project management, energy,
manufacturing, engineering, research and development, insurance, oil & gas,
transportation, and the environment.

Monte Carlo simulation furnishes the decision-maker with a range of possible outcomes
and the probabilities they will occur for any choice of action.. It shows the extreme
possibilities—the outcomes of going for broke and for the most conservative decision—
along with all possible consequences for middle-of-the-road decisions.

The technique was first used by scientists working on the atom bomb; it was named for
Monte Carlo, the Monaco resort town renowned for its casinos. Since its introduction in
World War II, Monte Carlo simulation has been used to model a variety of physical and
conceptual systems.

Monte Carlo simulation working.

Monte Carlo simulation performs risk analysis by building models of possible
results by substituting a range of values—a probability distribution—for any factor that
has inherent uncertainty. It then calculates results over and over, each time using a
different set of random values from the probability functions. Depending upon the
number of uncertainties and the ranges specified for them, a Monte Carlo simulation
could involve thousands or tens of thousands of recalculations before it is complete.
Monte Carlo simulation produces distributions of possible outcome values.

By using probability distributions, variables can have different probabilities of different
outcomes occurring. Probability distributions are a much more realistic way of
describing uncertainty in variables of a risk analysis. Common probability distributions
include:

Normal – Or ―bell curve.‖ The user simply defines the mean or expected value and a
standard deviation to describe the variation about the mean. Values in the middle near
the mean are most likely to occur. It is symmetric and describes many natural
phenomena such as people’s heights. Examples of variables described by normal
distributions include inflation rates...