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Zorluk: OrtaOptimization and Bounded Constraints

A renewable energy company designs micro-grid installations containing two types of modular components: Battery Storage Units (BB) and Solar Inverter Modules (II). The installation configuration is subject to the following system constraints:

1. Budget Bound: Each Battery Storage Unit costs \4,000andeachInverterModulecosts$2,000.Thetotalinstallationbudgetcannotexceed$32,000,givingtheconstraint4,000 and each Inverter Module costs \$2,000. The total installation budget cannot exceed \$32,000, giving the constraint 2B + I \le 16 .2.∗∗PeakPowerRequirement∗∗:Tomeetgridstabilityrequirements,thecombinedsystemratingmustsatisfy. 2. **Peak Power Requirement**: To meet grid stability requirements, the combined system rating must satisfy 3B + 4I \ge 30 .3.∗∗PhysicalSpaceLimit∗∗:Duetofloorloadcapacity,amaximumof6BatteryStorageUnitscanbeinstalled(. 3. **Physical Space Limit**: Due to floor load capacity, a maximum of 6 Battery Storage Units can be installed ( B \le 6$).

The total daily credit rating generated by the installation is given by the objective function E=12B+5IE = 12B + 5I.

Which combination of Battery Storage Units (BB) and Solar Inverter Modules (II) satisfies all system constraints while maximizing the total daily credit rating EE?

  1. 6 Battery Storage Units and 4 Inverter ModulesCevap
  2. B
    6 Battery Storage Units and 5 Inverter Modules
  3. C
    5 Battery Storage Units and 6 Inverter Modules
  4. D
    4 Battery Storage Units and 8 Inverter Modules
  5. E
    7 Battery Storage Units and 2 Inverter Modules

Cevap

6 Battery Storage Units and 4 Inverter Modules
The combination of 6 Battery Storage Units and 4 Inverter Modules strictly satisfies all three constraints: the space constraint (6≤66 \le 6), the budget constraint (2(6)+4=16≤162(6) + 4 = 16 \le 16), and the peak power requirement (3(6)+4(4)=34≥303(6) + 4(4) = 34 \ge 30). Substituting these values into the credit rating function yields E=12(6)+5(4)=92E = 12(6) + 5(4) = 92, which is higher than any other feasible combination.

Adım Adım Çözüm

1
Identify the feasible values for BB based on bounded constraints.
The physical space constraint limits BB to non-negative integers where B≤6B \le 6.
Establishing individual bounds reduces the search space for integer solutions.
2
Evaluate upper bounds for II using the budget constraint I≤16−2BI \le 16 - 2B for candidate values of BB.
For B=6B = 6, I≤16−12=4I \le 16 - 12 = 4. For B=5B = 5, I≤16−10=6I \le 16 - 10 = 6. For B=4B = 4, I≤16−8=8I \le 16 - 8 = 8.
Determines maximum allowable Inverter Modules for each valid count of Battery Storage Units.
3
Check peak power constraint 3B+4I≥303B + 4I \ge 30 and compute objective function E=12B+5IE = 12B + 5I for feasible candidate pairs.
Pair (6,4)(6, 4): 3(6)+4(4)=34≥303(6) + 4(4) = 34 \ge 30, E=12(6)+5(4)=92E = 12(6) + 5(4) = 92.
Pair (5,6)(5, 6): 3(5)+4(6)=39≥303(5) + 4(6) = 39 \ge 30, E=12(5)+5(6)=90E = 12(5) + 5(6) = 90.
Pair (4,8)(4, 8): 3(4)+4(8)=44≥303(4) + 4(8) = 44 \ge 30, E=12(4)+5(8)=88E = 12(4) + 5(8) = 88.
Testing valid candidates against all joint constraints identifies the configuration yielding maximum credits.
4
Compare total daily credit ratings across all valid choices.
The maximum credit rating is 92, achieved at B=6B = 6 and I=4I = 4.
Confirms the optimal bounded solution.

Anahtar Kavram

Optimization under Bounded Linear Constraints
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Konular

Optimization and Bounded Constraints

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