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Exercise 7.3 · Q32
Q.

A company manufactures two types of chemicals A and B. Each chemical requires two types of raw material P and Q. The table below shows number of units of P and Q required to manufacture one unit of A and one unit of B and the total availability of P and Q.

Raw MaterialABAvailability
P32120
Q25160

The company gets profits of Rs.350 and Rs.400 by selling one unit of A and one unit of B respectively. (Assume that the entire production of A and B can be sold). How many units of the chemicals A and B should be manufactured so that the company get maximum profit? Formulate the problem as L.P.P. to maximize the profit.

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✓ Free question

Let xx = units of chemical A and yy = units of chemical B manufactured. Let xx = units of A, yy = units of B. The non-negativity constraints are x≥0,y≥0x\ge0, y\ge0 since a negative quantity produced has no meaning. Raw material P (3 units/A, 2 units/B) is capped at 120 units and Q (2 units/A, 5 units/B) at 160 units; profit is Rs.350 per unit of A and Rs.400 per unit of B, sold entirely as assumed. (The exact production quantities that maximize profit would be found by the corner-point method of section 7.2.3, which this section has not yet introduced — only the formulation is asked here.)

✓Final answer

The L.P.P. is — Maximize: Maximize z=350x+400y\text{Maximize } z = 350x + 400y subject to 3x+2y≤120 (raw material P), 2x+5y≤160 (raw material Q), x≥0, y≥03x+2y\le120 \text{ (raw material P)},\ 2x+5y\le160 \text{ (raw material Q)},\ x\ge0,\ y\ge0.

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