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Commission, Brokerage and Discount
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Insurance and Annuity
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Linear Regression
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Linear Programming
Assignment Problem and Sequencing
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Probability Distributions
Notes
There are many examples to a mathematical formulation of the problem in two variables.In example ,
(i) The dealer can invest his money in buying tables or chairs or combination thereof. Further he would earn different profits by following different investment strategies.
(ii) There are certain overriding conditions or constraints viz., his investment is limited to a maximum of Rs 50,000 and so is his storage space which is for a maximum of 60 pieces.
Suppose he decides to buy tables only and no chairs, so he can buy 50000 ÷ 2500, i.e., 20 tables. His profit in this case will be Rs (250 × 20), i.e., Rs 5000.
Suppose he chooses to buy chairs only and no tables. With his capital of Rs 50,000, he can buy 50000 ÷ 500, i.e. 100 chairs. But he can store only 60 pieces. Therefore, he is forced to buy only 60 chairs which will give him a total profit of Rs (60 × 75), i.e., Rs 4500.
There are many other possibilities, for instance, he may choose to buy 10 tables and 50 chairs, as he can store only 60 pieces. Total profit in this case would be Rs (10 × 250 + 50 × 75), i.e., Rs 6250 and so on.
We, thus, find that the dealer can invest his money in different ways and he would earn different profits by following different investment strategies.
Notes
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.
