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Zorluk: OrtaOptimization and Bounded Constraints

A municipal transit agency is installing two types of electric vehicle (EV) charging stations at a new central station: Level 2 Chargers (LL) and Level 3 Fast Chargers (DD). Each Level 2 charger costs $4,000\$4,000 to install and draws 10 kW10\text{ kW} of grid power. Each Level 3 charger costs $15,000\$15,000 to install and draws 50 kW50\text{ kW} of grid power. The agency has a total installation budget of $100,000\$100,000 and a maximum available grid power allocation of 320 kW320\text{ kW}. To meet minimum service grant requirements, the hub must install at least 5 Level 2 chargers (L≥5L \ge 5) and at least 3 Level 3 chargers (D≥3D \ge 3). Each Level 2 charger can serve up to 8 sessions per day, and each Level 3 charger can serve up to 30 sessions per day. Match each operational optimization metric on the left with its corresponding correct value on the right that maximizes the total daily charging sessions served by the hub.

  • Optimal number of Level 2 chargers (LL)10
  • Optimal number of Level 3 chargers (DD)4
  • Maximum total daily charging sessions200

Cevap

The optimal operational configuration consists of 10 Level 2 chargers and 4 Level 3 chargers, achieving a maximum total of 200 daily charging sessions.
To maximize daily sessions S=8L+30DS = 8L + 30D under budget 4L+15D≤1004L + 15D \le 100, power L+5D≤32L + 5D \le 32, and lower bounds L≥5,D≥3L \ge 5, D \ge 3, testing valid integer points shows that setting Level 3 chargers to 4 and Level 2 chargers to 10 satisfies all joint constraints while achieving the peak capacity of 200 total daily sessions.

Adım Adım Çözüm

1
Formulate the linear system of constraints and objective function.
Budget constraint: 4,000L+15,000D≤100,000  ⟹  4L+15D≤1004,000L + 15,000D \le 100,000 \implies 4L + 15D \le 100.
Power constraint: 10L+50D≤320  ⟹  L+5D≤3210L + 50D \le 320 \implies L + 5D \le 32.
Minimum bounds: L≥5L \ge 5 and D≥3D \ge 3.
Objective function to maximize: S=8L+30DS = 8L + 30D.
Define the mathematical boundaries and optimization goal based on problem parameters.
2
Evaluate feasible integer values for Level 3 chargers (DD).
Case 1: If D=3D = 3:
- Budget bound: 4L+15(3)≤100  ⟹  4L≤55  ⟹  L≤134L + 15(3) \le 100 \implies 4L \le 55 \implies L \le 13.
- Power bound: L+5(3)≤32  ⟹  L≤17L + 5(3) \le 32 \implies L \le 17.
- Max feasible L=13L = 13.
- Total sessions S=8(13)+30(3)=104+90=194S = 8(13) + 30(3) = 104 + 90 = 194.

Case 2: If D=4D = 4:
- Budget bound: 4L+15(4)≤100  ⟹  4L≤40  ⟹  L≤104L + 15(4) \le 100 \implies 4L \le 40 \implies L \le 10.
- Power bound: L+5(4)≤32  ⟹  L≤12L + 5(4) \le 32 \implies L \le 12.
- Max feasible L=10L = 10.
- Total sessions S=8(10)+30(4)=80+120=200S = 8(10) + 30(4) = 80 + 120 = 200.

Case 3: If D=5D = 5:
- Budget bound: 4L+15(5)≤100  ⟹  4L≤25  ⟹  L≤64L + 15(5) \le 100 \implies 4L \le 25 \implies L \le 6.
- Power bound: L+5(5)≤32  ⟹  L≤7L + 5(5) \le 32 \implies L \le 7.
- Max feasible L=6L = 6.
- Total sessions S=8(6)+30(5)=48+150=198S = 8(6) + 30(5) = 48 + 150 = 198.

Case 4: If D=6D = 6:
- Budget bound: 4L+15(6)≤100  ⟹  4L≤10  ⟹  L≤24L + 15(6) \le 100 \implies 4L \le 10 \implies L \le 2.
- Violates minimum constraint L≥5L \ge 5.
Systematically check integer combinations of variables within the feasible region to identify maximum daily sessions.
3
Compare total sessions across valid integer pairs.
The maximum session count occurs when D=4D = 4 and L=10L = 10, producing S=200S = 200 daily sessions.
Determine the unique optimal combination satisfying all joint constraints.

Anahtar Kavram

Linear Optimization under Bounded Integer Constraints
Tahmini Süre:2m 0s
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Konular

Optimization and Bounded Constraints

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