A high-tech manufacturing firm produces custom drone components: Micro-Sensors (S) and Con… — GMAT

A high-tech manufacturing firm produces custom drone components: Micro-Sensors ( $S$ ) and Control Units ( $U$ ). Each Micro-Sensor requires 2 hours of precision calibration and 1 GB of firmware memory allocation. Each Control Unit requires 5 hours of precision calibration and 4 GB of firmware memory allocation. For an upcoming production batch, the facility has a maximum of 40 hours of calibration time available and a maximum total memory allocation limit of 30 GB. If the firm earns a profit of $100 per Micro-Sensor and$ 350 per Control Unit, which of the following combinations of Micro-Sensors ( $S$ ) and Control Units ( $U$ ) maximizes total batch profit while satisfying all operational bounds?

2 Micro-Sensors and 7 Control UnitsCevap
B
6 Micro-Sensors and 6 Control Units
C
0 Micro-Sensors and 8 Control Units
D
5 Micro-Sensors and 6 Control Units
E
20 Micro-Sensors and 0 Control Units

Cevap

The combination of 2 Micro-Sensors and 7 Control Units maximizes total profit ($2,650) under the given calibration and memory constraints.

The pair with 2 Micro-Sensors and 7 Control Units fully satisfies both constraints (

2(2) + 5(7) = 39 \\leq 40

hours and

2 + 4(7) = 30 \\leq 30

GB) and yields the maximum profit of $2,650.

Adım Adım Çözüm

Set up the linear programming model with objective function and system of inequalities.

Maximize

P = 100S + 350U

subject to:

2S + 5U \\leq 40

(Calibration limit),

S + 4U \\leq 30

(Memory limit), and non-negative integers

S, U \\geq 0

Establishing explicit bounds is necessary to evaluate boundary integer points.

Solve for the intersection of the boundary equations

2S + 5U = 40

and

S + 4U = 30

Substituting

S = 30 - 4U

into the first equation yields

2(30 - 4U) + 5U = 40 \\implies 60 - 3U = 40 \\implies U = 20/3 \\approx 6.67

and

S = 3.33

The continuous optimal point occurs at the intersection of the binding constraints.

Evaluate feasible integer coordinates near the intersection boundary.

For

U = 7

S \\leq \\min(\\frac{40 - 35}{2}, 30 - 28) = \\min(2.5, 2) = 2 \\implies (2, 7)

. Profit =

100(2) + 350(7) = \\

2,650

. For

U = 6

:

S \\leq \\min(\\frac{40 - 30}{2}, 30 - 24) = \\min(5, 6) = 5 \\implies (5, 6)

. Profit =

100(5) + 350(6) = \\

2,600

Since production quantities must be integers, testing feasible boundary pairs determines the global optimum.

Anahtar Kavram

Optimization under Bounded Linear Constraints

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Cevap

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