Mathematical Formulation of Linear Programming Problem




Let x be the number of tables and y be the number of chairs that the dealer buys. Obviously, x and y must be non-negative, i.e.,
x ≥  0                                                                 ...(1)
y ≥ 0         (Non-negative constraints)             ...(2)
The dealer is constrained by the maximum amount he can invest (Here it is Rs 50,000) and by the maximum number of items he can store (Here it is 60). Stated mathematically, 

2500x + 500y ≤ 50000 (investment constraint) 
or     5x + y ≤ 100                                          ... (3) 
and    x + y ≤ 60          (storage constraint)   ... (4) 

The dealer wants to invest in such a way so as to maximise his profit, say, Z which stated as a function of x and y is given by 
Z = 250x + 75y   (called objective function)    ... (5) 

Maximise the linear function Z subject to certain conditions determined by a set of linear inequalities with variables as non - negative. Such problem are called Linear Programing problems.
Linear programing problem is one that is concerned with finding the optimal value (maximum or minimum value) of a linear function (objective function) of several  variables (x and y) , subject to the condition that the variables are non negative and satisfy a set of linear inequalities (linear constraints) .

For example : Z = 250x + 75y        (objective function)
x ≥ 0 and y ≥ 0
2500x + 500y ≤ 50000                  (investment constraint)
x + y ≤  60                                     (storage constraint)
The term linear is all the mathematical relation used in the problem are linear relations where as programming is the method of determining particular programme or plan of action.

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