A pharmaceutical laboratory synthesizes two custom therapeutic compounds, Compound X and C… — GMAT

A pharmaceutical laboratory synthesizes two custom therapeutic compounds, Compound $X$ and Compound $Y$ , in integer batch quantities during a single production cycle. Production is subject to the following resource and operational constraints:

- Bioreactor Time: Each batch of Compound $X$ requires $4$ hours and each batch of Compound $Y$ requires $3$ hours. The laboratory has at most $48$ total hours of bioreactor time available.
- Specialized Solvent: Each batch of Compound $X$ requires $3$ liters and each batch of Compound $Y$ requires $5$ liters. The total solvent supply is capped at $55$ liters.
- Catalyst Stability Limit: To prevent reactive degradation, the number of batches of Compound $Y$ produced cannot exceed twice the number of batches of Compound $X$ plus $1$ ( $Y \le 2X + 1$ ).

Each batch of Compound $X$ generates a net profit of $\$12,000$ , and each batch of Compound $Y$ generates a net profit of $\$14,000$ .

What is the maximum total net profit, in dollars, that the laboratory can achieve within these combined production constraints?

A
$\$184,000$
$\$172,000$ Cevap
C
$\$170,000$
D
$\$168,000$
E
$\$154,000$

Cevap

The maximum total net profit achievable under all constraints is

\$172,000

, obtained by producing

5

batches of Compound

X

and

8

batches of Compound

Y

The production plan of

5

batches of Compound

X

and

8

batches of Compound

Y

satisfies all given constraints: Bioreactor time (

4(5) + 3(8) = 44 \le 48

hours), Solvent volume (

3(5) + 5(8) = 55 \le 55

liters), and Catalyst stability (

8 \le 2(5) + 1 = 11

). This combination yields the maximum valid profit of

5 \times \$12,000 + 8 \times \$14,000 = \$172,000

Adım Adım Çözüm

Formulate the linear inequalities representing the system constraints.

Bioreactor constraint:

4X + 3Y \le 48

; Solvent constraint:

3X + 5Y \le 55

; Catalyst constraint:

Y \le 2X + 1

; Non-negativity and integer constraints:

X, Y \in \mathbb{Z}_{\ge 0}

. Objective function: Maximize

P = 12000X + 14000Y

Establishing the mathematical boundaries of the feasible region for integer optimization.

Find the intersection point of the two primary resource constraint boundaries (

4X + 3Y = 48

and

3X + 5Y = 55

Multiplying

4X + 3Y = 48

5

yields

20X + 15Y = 240

. Multiplying

3X + 5Y = 55

3

yields

9X + 15Y = 165

. Subtracting the two equations gives

11X = 75 \implies X = 75/11 \approx 6.82

, leading to

Y = 76/11 \approx 6.91

Identifies the continuous optimal vertex of the system to narrow down nearby integer search points.

Test nearby integer pairs

(X, Y)

within the feasible region.

For

X = 6

, max

Y

under solvent limit is

\lfloor(55 - 18)/5\rfloor = 7

. Check

(6, 7)

4(6)+3(7)=45 \le 48

3(6)+5(7)=53 \le 55

7 \le 2(6)+1=13

. Profit:

12000(6)+14000(7) = \$170,000

.
For

X = 5

, max

Y

under solvent limit is

\lfloor(55 - 15)/5\rfloor = 8

. Check

(5, 8)

4(5)+3(8)=44 \le 48

3(5)+5(8)=55 \le 55

8 \le 2(5)+1=11

. Profit:

12000(5)+14000(8) = \$172,000

.
For

X = 4

, max

Y

under solvent limit is

\lfloor(55 - 12)/5\rfloor = 8

. Profit:

12000(4)+14000(8) = \$160,000

.
For

X = 7

, max

Y

under bioreactor limit is

\lfloor(48 - 28)/3\rfloor = 6

. Check

(7, 6)

4(7)+3(6)=46 \le 48

3(7)+5(6)=51 \le 55

. Profit:

12000(7)+14000(6) = \$168,000

Evaluates candidate lattice points on the boundary of the feasible integer region.

Compare profit totals among all valid integer combinations.

The combination

(5, 8)

yields the maximum valid profit of

\$172,000

Confirms the global maximum integer solution under all joint bounded constraints.

Anahtar Kavram

Optimization under Integer and Joint Bounded Constraints

Soru

Cevap

Adım Adım Çözüm

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