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Zorluk: ZorOptimization and Bounded Constraints

A logistics company operates two types of delivery vehicles: Type X and Type Y.

- Each Type X vehicle carries 4040 packages and costs $300\$300 per trip.
- Each Type Y vehicle carries 2525 packages and costs $150\$150 per trip.

The company operates under a maximum daily budget of $3,300\$3,300. Additionally, due to fleet maintenance regulations, the number of Type Y vehicle trips cannot exceed twice the number of Type X vehicle trips.

Which combination of Type X and Type Y vehicle trips maximizes the total package delivery capacity while satisfying all daily budget and maintenance constraints?

  1. 6 Type X vehicles and 10 Type Y vehiclesCevap
  2. B
    5 Type X vehicles and 12 Type Y vehicles
  3. C
    4 Type X vehicles and 14 Type Y vehicles
  4. D
    7 Type X vehicles and 8 Type Y vehicles
  5. E
    8 Type X vehicles and 6 Type Y vehicles

Cevap

6 Type X vehicles and 10 Type Y vehicles
The combination of 6 Type X vehicles and 10 Type Y vehicles strictly satisfies the budget constraint (300×6+150×10=3300≤3300300 \times 6 + 150 \times 10 = 3300 \le 3300) and the maintenance constraint (10≤2×6=1210 \le 2 \times 6 = 12). It produces a maximum valid package capacity of 490.

Adım Adım Çözüm

1
Formulate the algebraic constraint inequalities
Budget constraint: 300X+150Y≤3300  ⟹  2X+Y≤22300X + 150Y \le 3300 \implies 2X + Y \le 22. Maintenance constraint: Y≤2XY \le 2X.
Linear constraints define the feasible region for vehicle counts.
2
Express total capacity CC in terms of XX along the budget boundary
C=40X+25YC = 40X + 25Y. On the boundary Y=22−2XY = 22 - 2X, C=40X+25(22−2X)=550−10XC = 40X + 25(22 - 2X) = 550 - 10X.
To maximize capacity, smaller values of XX along the boundary yield higher total capacity because Type Y provides more capacity per dollar.
3
Find the intersection of constraints to locate the optimal integer point
Setting Y=2XY = 2X into 2X+Y=222X + Y = 22 gives 4X=22  ⟹  X=5.54X = 22 \implies X = 5.5. The smallest valid integer X≥5.5X \ge 5.5 satisfying Y=22−2X≤2XY = 22 - 2X \le 2X is X=6X = 6.
When X=6X = 6, Y=22−2(6)=10Y = 22 - 2(6) = 10. Checking maintenance: 10≤2(6)=1210 \le 2(6) = 12 holds true.
4
Calculate the maximum capacity
C=40(6)+25(10)=240+250=490C = 40(6) + 25(10) = 240 + 250 = 490 packages.
This point (6,10)(6, 10) yields the maximum capacity satisfying all linear inequality constraints.

Anahtar Kavram

Linear Optimization under Bounded Constraints
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Konular

Optimization and Bounded Constraints

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