Use the two phase simplex method to identify an initial BFS$.$ Select the NBV with the most negative

reduced cost to enter the basis if there are multiple potential entering NBVs$.$

$$: $3+8$

s$.$t$.$

$+<=6(1)$

$2=4(2)$

$,>=0$

$($a$)$ Convert to standard form and identify an initial basis in phase $1,$ adding artificial variables where

necessary.

Convert the problem into standard form and add an artificial variable for $($c$2)$:

$$: $$

$++=6$

$2+=4$

$,,,>=0$

The initial basis is $$ and $$ and the initial bfs is $(0,0,6,4)$

$($b$)$ Complete phase one of the simplex method.

Phase $1$ RC$($reduced cost$)$

Objective

function

coefficients

$0001$

Constraint

matrix

coefficients

$-11106$

$2-1014$

$$

$$

$1$

$,$

$$

$=$

$0$

N

N

B

B Write value of

current

solution in this

row

$$

$($

$$

$)$

$0$

$0$

$6$

$4$

$$

$$

$($

$$

$)$

$=$

$4$ Use B or N to

represent

basic or

nonbasic in

this row.

$$

$$

$$

$$

$1$

$0$

$1$

$-$

$2$

$$

$$

$$

$$

$=$

$$

$2$

$$

$$ enter the

basis $($since the RC

is most negative, per

directions$)$

$$

$$

$$

$$

$0$

$1$

$-$

$1$

$1$

$$

$$

$$

$$

$=$

$1$

$$

$$

$61$

$=$

$6$

$$

$42$

$=$

$2$

$$

$=$ min

$\{$

$6$

$,$

$2$

$\}$

$=$

$2$

$$

$$leave the

basis

$$

$$

$1$

$,$

$$

$=$

$1$

B

N

B

N

$$

$($

$$

$)$

$2$

$0$

$8$

$0$

$0$ All artificial

variables have

left the basis $$

we can stop

here.

$$

$$

$$

$1/21$

$-1/2$

$0$

$$

$$

$$

$$

$=$

$0$ All reduced

costs are

greater than $0,$

Optimal

solution

reached for

phase $1$ all

artificial

variables have

left the basis

$$

$$

$$

$$

$-1/2$

$0$

$-1/2$

$1$

$$

$$

$$

$$

$=$

$1$

$($c$)$ Identify what you would conclude and how you would proceed to phase $2($but you don$$t actually

have to complete phase $2)$