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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.
Video Tutorials
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Related QuestionsVIEW ALL [52]
A firm manufactures 3 products A, B and C. The profits are Rs 3, Rs 2 and Rs 4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine on each product :
| Machine | Products | ||
| A | B | C | |
| M1 M2 |
4 | 3 | 5 |
| 2 | 2 | 4 | |
Machines M1 and M2 have 2000 and 2500 machine minutes respectively. The firm must manufacture 100 A's, 200 B's and 50 C's but not more than 150 A's. Set up a LPP to maximize the profit.
A small manufacturing firm produces two types of gadgets A and B, which are first processed in the foundry, then sent to the machine shop for finishing. The number of man-hours of labour required in each shop for the production of each unit of A and B, and the number of man-hours the firm has available per week are as follows:
| Gadget | Foundry | Machine-shop |
| A | 10 | 5 |
| B | 6 | 4 |
| Firm's capacity per week | 1000 | 600 |
The profit on the sale of A is Rs 30 per unit as compared with Rs 20 per unit of B. The problem is to determine the weekly production of gadgets A and B, so that the total profit is maximized. Formulate this problem as a LPP.
