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Difficulty: EasyOptimization and Bounded Constraints

A small workshop produces two items, Item X and Item Y. Each unit of Item X requires 22 hours of labor and yields a profit of $30\$30. Each unit of Item Y requires 44 hours of labor and yields a profit of $50\$50. The workshop has a maximum of 1616 labor hours available daily and must produce a combined total of at least 55 units per day. Assuming only whole units can be produced, which combination of (Item X, Item Y) maximizes total daily profit while satisfying all given constraints?

  1. 6 units of Item X and 1 unit of Item YAnswer
  2. B
    4 units of Item X and 2 units of Item Y
  3. C
    2 units of Item X and 3 units of Item Y
  4. D
    0 units of Item X and 4 units of Item Y
  5. E
    1 unit of Item X and 4 units of Item Y

Answer

6 units of Item X and 1 unit of Item Y
The pair consisting of 6 units of Item X and 1 unit of Item Y uses 2(6)+4(1)=162(6) + 4(1) = 16 labor hours (satisfying the maximum 1616-hour limit) and produces 6+1=76 + 1 = 7 units (satisfying the minimum 55-unit limit). It yields a profit of 30(6)+50(1)=$23030(6) + 50(1) = \$230, which is higher than all other feasible options.

Step-by-Step Solution

1
Formulate the constraint inequalities and profit objective function.
Labor constraint: 2X+4Y≤162X + 4Y \le 16. Minimum unit constraint: X+Y≥5X + Y \ge 5, where X,Y≥0X, Y \ge 0 are integers. Profit function: P=30X+50YP = 30X + 50Y.
Setting up the mathematical relations allows systematic evaluation of candidate pairs.
2
Evaluate feasible integer combinations (X,Y)(X, Y) against both constraints.
Combination (6,1)(6, 1): Labor =16≤16= 16 \le 16, Units =7≥5= 7 \ge 5. Profit =30(6)+50(1)=$230= 30(6) + 50(1) = \$230.
Combination (4,2)(4, 2): Labor =16≤16= 16 \le 16, Units =6≥5= 6 \ge 5. Profit =30(4)+50(2)=$220= 30(4) + 50(2) = \$220.
Combination (2,3)(2, 3): Labor =16≤16= 16 \le 16, Units =5≥5= 5 \ge 5. Profit =30(2)+50(3)=$210= 30(2) + 50(3) = \$210.
Evaluating all valid pairs determines which produces the peak profit value.
3
Compare profit outcomes across feasible pairs.
The combination of 66 units of Item X and 11 unit of Item Y achieves the highest valid profit of $230\$230.
The option (6,1)(6, 1) optimizes the objective function under all bounded constraints.

Key Concept

Optimization under linear integer and inequality bounds
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Topics

Optimization and Bounded Constraints

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