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## General form of LLP

Introduction to Linear ProgrammingA linear form is made a mathematical expression of the type a

_{1}x_{1}+ a_{2}x_{2}+ .... + a_{n}x_{n}, where a_{1}, a_{2}, ..., a_{n}are constant and x_{1}, x_{2 }... x_{n}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 (x

_{1}, x_{2 }... x_{n}), at least one of them is non-zero, can be obtained if there are preciselynrelations. When the number of relations is more than or less thannthen 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 x

_{1}, x_{2 }... x_{n}in order to maximize or minimizeZ = c

_{1}x_{1}+ c_{2}x_{2}+.............+c_{n}x_{n }subject to limitations or restrictions

a

_{11}x_{1}+ a_{12}x_{2}+ .............+a_{1}nx_{n}(≤ or ≥) b_{1}a

_{21}x_{1}+ a_{22}x_{2}+ ...........+a_{2}nx_{n}(≤ or ≥) b_{2}.

.

.

a

_{m1}x_{1}+ a_{m2}x_{2}+ ..........+a_{mn}x_{n}(≤ or ≥) b_{m}and

x

_{1}≥ 0, x_{2 }≥ 0,..., x_{n }≥ 0Where

Z = value of on the whole measure of performance

x

_{j}= level of activity (for j = 1, 2, ..., n)c

_{j}= increase in Z that would result from each unit enhance in level of activity jb

_{i}= amount of resource i that is available for allocation to activities (for i = 1,2, ..., m)a

_{ij}= amount of resource i used by each unit of activity jResource

Resource usage per unit of activity

Amount of resource available

Activity

1 2 .......................... n

1

2

.

.

.

m

a

_{11}a_{12}.........................a_{1n}a

_{21}a_{22}.........................a_{2n}.

.

.

a

_{m1}a_{m2}.........................a_{mn}b

_{1}b

_{2}.

.

.

b

_{m}Contribution to Z per unit of activity

c

_{1}c_{2}.............................c_{n}Data needed for LP model_{1}, x_{2}.........x_{n}are known asdecision variables._{j},_{ }b_{i, }a_{ij}(for i=1, 2 ... m and j=1, 2 ... n) are theinput constantsfor the model. They are known asparametersof the model._{1}x_{1}+ c_{2}x_{2}+.... +c_{n}x_{n }is known asobjective function.constraints. The constraint a_{i1}x_{1}+ a_{i2}x_{2}... a_{in}x_{n}are at times knownasfunctional constraint(L.H.S constraint). x_{j}≥ 0 restrictions are referred to asnon-negativity constraint.Assumptions in LPP

AdditivitySum 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.

ProportionalityThe 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

DivisibilityThe variables are not limited to integer values

FinitenessVariables and constraints are limited in number.

DeterministicCoefficients in the objective function and constraints are totally known and do not vary during the period under study in all the problems taken.

OptimalityIn LPP, we establish the decision variables in order to optimize the objective function of the LPP.

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