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Linear Programming Optimization

XAT 2010 · Quantitative Ability & Data Interpretation · Medium · Quantitative Ability

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A manufacturer produces two types of products - A and B, which are subjected to two types of operations, viz. grinding and polishing. Each unit of product A takes 2 hours of grinding and 3 hours of polishing whereas product B takes 3 hours of grinding and 2 hours of polishing. The manufacturer has 10 grinders and 15 polishers. Each grinder operates for 12 hours/day and each polisher 10 hours/day. The profit margin per unit of A and B are Rs. 5/ - and Rs. 7/ - respectively. If the manufacturer utilises all his resources for producing these two types of items, what is the maximum profit that the manufacturer can earn in a day?

  1. A.

    Rs. 280/-

  2. B.

    Rs. 294/-

  3. C.

    Rs. 515/-

  4. D.

    Rs. 550/-

  5. E.

    None of the above

Answer

B

Explanation

Total grinder hours = 10×12=12010 \times 12 = 120. Total polisher hours = 15×10=15015 \times 10 = 150. Grinding equation: 2x+3y=1202x + 3y = 120. Polishing equation: 3x+2y=1503x + 2y = 150. Solving simultaneously: 3(2x+3y)−2(3x+2y)=360−300  ⟹  5y=60  ⟹  y=123(2x + 3y) - 2(3x + 2y) = 360 - 300 \implies 5y = 60 \implies y = 12. 2x+36=120  ⟹  2x=84  ⟹  x=422x + 36 = 120 \implies 2x = 84 \implies x = 42. Maximum Profit =5(42)+7(12)=210+84=Rs. 294= 5(42) + 7(12) = 210 + 84 = \text{Rs. } 294.

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