Machine Constraints and Profit Maximization

CAT 2003 Slot 1 · QA · Medium · Optimization

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A leather factory produces two kinds of bags, standard and deluxe. The profit margin is Rs. 20 on a standard bag and Rs. 30 on a deluxe bag. Every bag must be processed on Machine A and Machine B. The processing times per bag on the two machines are as follows:

Machine AMachine B
Standard Bag46
Deluxe Bag510

The total time available on machine A is 700 hours and on machine B is 1250 hours. Among the following production plans, which one meets the machine availability constraints and maximizes the profit?

  1. A.

    Standard 75 bags, Deluxe 80 bags

  2. B.

    Standard 100 bags, Deluxe 60 bags

  3. C.

    Standard 50 bags, Deluxe 100 bags

  4. D.

    Standard 60 bags, Deluxe 90 bags

Answer

A

Explanation

Let xx be standard bags and yy be deluxe bags. Constraints: 4x+5y≤7004x + 5y \le 700 6x+10y≤1250  ⟹  3x+5y≤6256x + 10y \le 1250 \implies 3x + 5y \le 625 Profit P=20x+30yP = 20x + 30y. Checking options:

  1. (75,80)(75, 80): 4(75)+5(80)=300+400=700≤7004(75)+5(80) = 300+400 = 700 \le 700; 3(75)+5(80)=225+400=625≤6253(75)+5(80) = 225+400 = 625 \le 625. Valid! Profit = 20(75)+30(80)=1500+2400=390020(75)+30(80) = 1500+2400 = 3900.
  2. (100,60)(100, 60): 4(100)+5(60)=7004(100)+5(60) = 700; 3(100)+5(60)=600<6253(100)+5(60) = 600 < 625. Valid. Profit = 2000+1800=38002000+1800 = 3800.
  3. (50,100)(50, 100): 3(50)+5(100)=650>6253(50)+5(100) = 650 > 625. Invalid.
  4. (60,90)(60, 90): 3(60)+5(90)=630>6253(60)+5(90) = 630 > 625. Invalid. Max profit is given by Option 1.

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