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Difficulty: HardOptimization and Bounded Constraints

A commercial greenhouse allocates integer numbers of acres to grow two high-yield crops: Organic Tomatoes (TT) and Hydroponic Cucumbers (CC). The operational parameters and constraints are as follows:

- Each acre of Tomatoes requires 33 units of water per day and 44 units of fertilizer per week, producing a net revenue of $1,200\$1,200 per week.
- Each acre of Cucumbers requires 55 units of water per day and 22 units of fertilizer per week, producing a net revenue of $1,000\$1,000 per week.
- Total daily water usage across both crops cannot exceed 4545 units.
- Total weekly fertilizer usage across both crops cannot exceed 3636 units.
- To fulfill distributor agreements, the total cultivated area (T+CT + C) must be at least 1010 acres.

Which of the following crop allocation pairs (T,C)(T, C) maximizes total weekly net revenue while satisfying all operational constraints?

  1. 77 acres of Tomatoes and 44 acres of CucumbersAnswer
  2. B
    88 acres of Tomatoes and 33 acres of Cucumbers
  3. C
    66 acres of Tomatoes and 55 acres of Cucumbers
  4. D
    55 acres of Tomatoes and 66 acres of Cucumbers
  5. E
    99 acres of Tomatoes and 11 acre of Cucumbers

Answer

The optimal crop allocation is 7 acres of Tomatoes and 4 acres of Cucumbers, yielding a maximum weekly revenue of $12,400.
The allocation of 7 acres of Tomatoes and 4 acres of Cucumbers fully respects all three operational boundaries: daily water (41 out of 45 units), weekly fertilizer (36 out of 36 units), and total cultivated area (11 acres, meeting the minimum threshold of 10 acres). It produces the highest valid weekly revenue of $12,400.

Step-by-Step Solution

1
Formulate the mathematical model from the context
Maximize Revenue R=1200T+1000CR = 1200T + 1000C subject to:
1) Water constraint: 3T+5C≤453T + 5C \le 45
2) Fertilizer constraint: 4T+2C≤36  ⟹  2T+C≤184T + 2C \le 36 \implies 2T + C \le 18
3) Land constraint: T+C≥10T + C \ge 10
4) Integer constraints: T,C∈Z≥0T, C \in \mathbb{Z}_{\ge 0}
Establishing explicit linear inequality bounds and the objective function is required for constrained optimization.
2
Evaluate feasible integer boundary points near the intersection of joint constraints
Testing boundary candidate pairs:
- At T=7,C=4T = 7, C = 4: Water =3(7)+5(4)=41≤45= 3(7)+5(4) = 41 \le 45, Fertilizer =4(7)+2(4)=36≤36= 4(7)+2(4) = 36 \le 36, Area =11≥10= 11 \ge 10. Feasible! Revenue =1200(7)+1000(4)=$12,400= 1200(7) + 1000(4) = \$12,400.
- At T=8,C=2T = 8, C = 2: Water =34≤45= 34 \le 45, Fertilizer =36≤36= 36 \le 36, Area =10≥10= 10 \ge 10. Feasible! Revenue =1200(8)+1000(2)=$11,600= 1200(8) + 1000(2) = \$11,600.
- At T=6,C=5T = 6, C = 5: Water =43≤45= 43 \le 45, Fertilizer =34≤36= 34 \le 36, Area =11≥10= 11 \ge 10. Feasible! Revenue =1200(6)+1000(5)=$12,200= 1200(6) + 1000(5) = \$12,200.
Systematic evaluation of feasible integer boundary points identifies the combination that maximizes net revenue.
3
Verify that higher revenue candidates violate joint system bounds
Checking candidate (8,3)(8, 3): Revenue would be $12,600\$12,600, but Fertilizer =4(8)+2(3)=38>36= 4(8) + 2(3) = 38 > 36 (infeasible).
Ensures that candidates yielding higher revenue are invalid due to joint constraint violations.

Key Concept

Optimization under Bounded Linear Constraints
Estimated Time:2m 30s
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Topics

Optimization and Bounded Constraints

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