Introduction to Linear Programming
A linear form is made a mathematical expression of the type a1x1 + a2x2 + .... + anxn, where a1, a2, ..., an are constant and x1, x2 ... xn are variable. The word Programming means the procedure of determining a particular program or plan of action. So Linear Programming (LP) is one of the highly significant optimization (maximization / minimization) techniques which are developed in the field of Operations Research (OR).
The methods applied for answering a linear programming problem are fundamentally simple problems; an answer can be achieved by a set of simultaneous equations. Though an exclusive solution for a set of simultaneous equations in n-variables (x1, x2 ... xn), at least one of them is non-zero, can be obtained if there are precisely n relations. When the number of relations is more than or less than n then a unique solution does not exist but several trial solutions can be found.
In different practical conditions, the queries are seen in which the number of relations is not equal to the number of the number of variables and a lot of relations are in the form of inequalities (≤ or ≥) to maximize or minimize a linear function of the variables subject to like conditions. These problems are called as Linear Programming Problem (LPP).
General form of LPP
We develop a mathematical model for common problem of allocating resources to different activities. In particular, this model is to choose the values for x1, x2 ... xn in order to maximize or minimize
Z = c1x1 + c2x2 +.............+cnxn
subject to limitations or restrictions
a11x1 + a12x2 + .............+a1nxn (≤ or ≥) b1
a21x1 + a22x2 + ...........+a2nxn (≤ or ≥) b2
am1x1 + am2x2 + ..........+amnxn (≤ or ≥) bm
x1 ≥ 0, x2 ≥ 0,..., xn ≥ 0
Z = value of on the whole measure of performance
xj = level of activity (for j = 1, 2, ..., n)
cj = increase in Z that would result from each unit enhance in level of activity j
bi = amount of resource i that is available for allocation to activities (for i = 1,2, ..., m)
aij = amount of resource i used by each unit of activity j
Resource usage per unit of activity
Amount of resource available
1 2 .......................... n
a11 a12 .........................a1n
a21 a22 .........................a2n
am1 am2 .........................amn
Contribution to Z per unit of activity
c1 c2 .............................cn
Data needed for LP model
Assumptions in LPP
Sum of the resources utilized by diverse activities must be equivalent to the total amount of resources used by each activity for all resources collectively or individually.
The contribution or participation of each variable in the main function or its usage of the resources is directly proportional to the value of the variable that is if resource accessibility rises by some percentage, then the output will also rise by same percentage
The variables are not limited to integer values
Variables and constraints are limited in number.
Coefficients in the objective function and constraints are totally known and do not vary during the period under study in all the problems taken.
In LPP, we establish the decision variables in order to optimize the objective function of the LPP.
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