Linear Programming

Linear programming is a nonstatistical mathematical technique whereby
the maximization or minimization of a linear expresion of variables, call
the objective function, is determined in the presence of known or assumed
restrictions, call constraint.


In essence, it’s a procedure for solving the problems in which there
are more variables than simultaneous equations in which the variables are
expressed.

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No probability or statistics are needed to study linear programming.

The mathematics involved in linear programming is relatively easy to
understand and to manipulate in contrast to calculus. Linear equations and
inequations form the mathematical skeleton around which linear programming
is built.


A linear function called the object function is to be maximized or
minimized in some sense, like optimzed. Most real world problems have many
possible solutions. The purpose of optimization is to choose from among
many possible solutions the “best” possible solutions. Some example of
“best” are highest profit, lowest cost, largest sales, lowest production
time, etc.


The optimization of the objective functions take place in teh presence
of known or assumed restriction. The technical term constraints is used to
describe the restrictions present in linear programming problem. The
constraints are expressed mathemically as inequalities. In a practical
real-world situation, the constraints are generated by the presence of
limited resources or commodities such as capital manpower and raw material.

Mathematically, inequations can be converted to equations by the
introduction of slack variables.


Linear programming can be dated from the year 1947 when G.B. Dantzing
evolved an efficent technique call the Simplex Method, for solving linear
programming problems. The following decades, the rapid development of both
the theory and applications of linear programming which were aided by the
simultaneous introduction of the electronic computers. One of the first
probelm to be solved by the simplex method was Stigler’s diet problem
(1945). Here is the diet problem
??????????????????????????????????????????????????????????????????
ProteinFatCarbohydrateCost
??????????????????????????????????????????????????????????????????
100g bread 4052052.2p
100g cheese60 3806012p
Minimal daily requirement300 790 1350
The problem is determine how much bread and cheese Mrs. Jones
should buy each day in order to minimize the cost of the diet, whilst
fulfilling the calorie requirements. Suppose shy buys x’ * 100g of
cheese and x” * 100g of cheese, then the mathematical problem, known
as a linear programme is as follows.

Minimizez=2.2x’ + 12x”(Cost Of Diet)
Subject To 40x’ + 60x” ; or = 300 (At least 300 cal of protein)
5x’ + 380x” > or = 790 (At least 790 cal of fat)
205x’ + 60x” ; or = 1350 (At least 1350 cal of carbo.)
x’ > or = 0, x2 > = 0(quantites must be non-negative)
The easiest and most illustrative method of solving problems in two
unknowns is the graphical method. The value of x’ and x” satisfying 40x’
+ 60x” > or = 300 lies in the upper half-plane bounded by the straight
line 40x’ + 60x” = 300, so the x’ and x” satistying all the above
inequalities lie in the intersection of their respective half – plane.


Interger Solutions
Provided the supplies and demands are positive intergers, the matrix
minimum method always leads to and optimal solution with integer values as
the method only involves operations on integers which results in integers.

Obviously a non-integer optimal solution would be useless.


Uniqueness
It can happen that two or more differnet allocations of ships between
ports give rise to thesame minimum cost. However, if v’=u’

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