An earth-observation satellite payload operator allocates integer numbers of channels to t… — GMAT

An earth-observation satellite payload operator allocates integer numbers of channels to two operational payload modes: High-Resolution Imaging ( $I$ ) and Atmospheric Sounding ( $S$ ). Operational parameters and resource bounds are specified as follows:

- Bandwidth: Each imaging channel requires $15\text{ MHz}$ and each sounding channel requires $10\text{ MHz}$ . Total available payload bandwidth is at most $210\text{ MHz}$ .
- Power: Each imaging channel consumes $40\text{ W}$ and each sounding channel consumes $15\text{ W}$ . Total available power budget is at most $440\text{ W}$ .
- Mission Minimums: The satellite must operate at least $2$ imaging channels ( $I \ge 2$ ) and at least $4$ sounding channels ( $S \ge 4$ ).
- Buffer Constraint: The number of sounding channels cannot exceed twice the number of imaging channels ( $S \le 2I$ ).

Match each payload optimization target on the left with its corresponding integer channel count on the right.

Maximum possible number of Imaging channels ( $I$ )9
Maximum possible number of Sounding channels ( $S$ )12
Number of Sounding channels ( $S$ ) that maximizes daily data return ( $R = 70I + 20S$ Gbit/day)5

Cevap

The maximum possible number of Imaging channels (

I

) is 9. The maximum possible number of Sounding channels (

S

) is 12. The number of Sounding channels (

S

) that maximizes daily data return is 5.

Matching the maximum imaging channels (

I

) to 9 is correct because

I = 10

forces

S \le 2

, violating

S \ge 4

, whereas

I = 9

is fully feasible with

S = 5

. Matching the maximum sounding channels (

S

) to 12 is correct because at

(6, 12)

, bandwidth, power, and buffer constraints are simultaneously satisfied. Matching the sounding channels for maximum data return to 5 is correct because the function

R = 70I + 20S

reaches its peak of 730 Gbit/day at

(9, 5)

Adım Adım Çözüm

Formulate the algebraic system of inequality constraints.

Bandwidth:

15I + 10S \le 210 \implies 3I + 2S \le 42

. Power:

40I + 15S \le 440 \implies 8I + 3S \le 88

. Bounds:

I \ge 2

S \ge 4

, and

S \le 2I

Converting narrative bounds into simplified algebraic inequalities facilitates evaluating integer extreme points.

Find the maximum feasible value for Imaging channels (

I

I = 10

, power requires

8(10) + 3S \le 88 \implies 3S \le 8 \implies S \le 2

, which violates the minimum requirement

S \ge 4

. If

I = 9

, power requires

8(9) + 3S \le 88 \implies 3S \le 16 \implies S \le 5

. With

S = 4

S = 5

, bandwidth gives

3(9) + 2(5) = 37 \le 42

and

S \le 2(9)

holds. Thus, max

I = 9

Determining the upper bound of

I

requires testing larger integer values of

I

while ensuring

S \ge 4

is not violated.

Find the maximum feasible value for Sounding channels (

S

Since

S \le 2I

S

is constrained by

I

. Testing

I = 6

: bandwidth gives

3(6) + 2S \le 42 \implies 2S \le 24 \implies S \le 12

. Power gives

8(6) + 3S \le 88 \implies 3S \le 40 \implies S \le 13.33

. Buffer gives

S \le 2(6) = 12

. At

(6, 12)

, all constraints hold (

18+24=42 \le 42

48+36=84 \le 88

12 \le 12

). Thus, max

S = 12

Maximizing

S

requires balancing the ratio constraint

S \le 2I

against bandwidth and power limits.

Maximize the objective function

R(I, S) = 70I + 20S

over all feasible boundary points.

Evaluating candidate integer vertex points:

R(9, 5) = 70(9) + 20(5) = 730

;

R(9, 4) = 710

;

R(8, 8) = 720

;

R(7, 10) = 690

;

R(6, 12) = 660

. The maximum data return occurs at

(9, 5)

, where

S = 5

Comparing total return at all boundary points identifies the global integer maximum.

Anahtar Kavram

Linear Programming and Integer Bounded Optimization

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru