Question

Find the minimum value of w=2r+10s+8t, subject to the following constraints.
r+s+t≥6
s+2t≥8
-r+2s+2t≥4
r,s,t≥0

Answer 1

Question #78955, Engineering / Other

Find the minimum value of $w = 2r + 10s + 8t$, subject to the following constraints.

$$\begin{array}{l} r + s + t \geq 6 \quad (1) \\ s + 2t \geq 8 \quad (2) \\ -r + 2s + 2t \geq 4 \quad (3) \\ r, s, t \geq 0 \quad (4) \end{array}$$

Solution

(1)+(3):

$$\begin{array}{l} 3s + 3t \geq 10 \\ s + t \geq \frac{10}{3} \\ s \geq \frac{10}{3} - t \\ \end{array}$$

(2):

$$\begin{array}{l} 2t \geq 8 - \left(\frac{10}{3} - t\right) \\ t \geq \frac{14}{3}. \end{array}$$

So,

$$s \geq \frac{10}{3} - \frac{14}{3} = -\frac{4}{3}.$$

But, we have (4):

$$s \geq 0.$$

(1):

$$r \geq 6 - 0 - \frac{14}{3} = \frac{4}{3}.$$

The minimum value of $w$ is

$$w \left(\frac{4}{3}, 0, \frac{14}{3}\right) = 2 \left(\frac{4}{3}\right) + 10(0) + 8 \left(\frac{14}{3}\right) = 40.$$

Answer: 40.

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