A cloud analytics company configures daily data processing operations using two types of v… — GMAT

A cloud analytics company configures daily data processing operations using two types of virtual server instances: Compute-Optimized ( $C$ ) and Memory-Optimized ( $M$ ). The operational parameters and constraints are as follows:

- Each $C$ instance processes 50 batch jobs per hour and incurs an operating cost of $15 per hour. - Each$ M $instance processes 30 batch jobs per hour and incurs an operating cost of$ 10 per hour.
- The system must process at least 360 batch jobs per hour in total.
- The total hourly operating budget for server instances cannot exceed $115. - Due to active database caching policies, at least 3$ M $instances must be active at all times ($ M \ge 3 $). -$ C $and$ M$ must both be non-negative integers.

Which of the following combinations of Compute ( $C$ ) and Memory ( $M$ ) server instances maximizes the total hourly job throughput while satisfying all operational constraints?

5 Compute instances and 4 Memory instancesCevap
B
3 Compute instances and 7 Memory instances
C
6 Compute instances and 2 Memory instances
D
4 Compute instances and 6 Memory instances
E
2 Compute instances and 9 Memory instances

Cevap

5 Compute instances and 4 Memory instances

The combination of 5 Compute instances and 4 Memory instances satisfies all stated system bounds (

15 \times 5 + 10 \times 4 = 115 \le 115

50 \times 5 + 30 \times 4 = 370 \ge 360

, and

4 \ge 3

) and delivers the maximum feasible throughput of 370 jobs per hour.

Adım Adım Çözüm

Formulate the linear system of inequalities from the given constraints.

Throughput:

50C + 30M \ge 360 \Rightarrow 5C + 3M \ge 36

.
Budget:

15C + 10M \le 115 \Rightarrow 3C + 2M \le 23

.
Minimum

M

M \ge 3

, where

C, M \in \mathbb{Z}_{\ge 0}

Converting narrative conditions into simplified algebraic inequalities makes constraint checking straightforward.

Determine the set of integer pairs

(C, M)

that satisfy all three inequalities.

Combining

3C + 2M \le 23

and

5C + 3M \ge 36

with

M \ge 3

yields only two valid integer pairs:
- Pair 1:

(C=3, M=7)

with cost

15(3)+10(7)=115

and throughput

50(3)+30(7)=360

- Pair 2:

(C=5, M=4)

with cost

15(5)+10(4)=115

and throughput

50(5)+30(4)=370

Systematic testing of integer values bounded by the linear inequalities identifies all feasible solutions.

Compare total throughput across feasible solutions to find the maximum.

Pair 2

(C=5, M=4)

produces 370 jobs per hour, which exceeds Pair 1's 360 jobs per hour.

The question specifically asks to maximize throughput among all valid configurations.

Anahtar Kavram

Optimization under linear integer constraints and bounded systems

Soru

Cevap

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