Question

1.3 Minimize z = a + b + c
Subject to:
1. a - b - c ≤ 0
2. a + b + c ≥ 4
3. a + b - c = 2
4. a, b ≥ 0

Answer 1

Clearly; min(z)=a+b+c=4

as it is given that $a+b+c \geq4$

One of the feasible solution is (a,b,c)=(1,2,1)

Checking this with the given constraints, we get;

1. $a - b - c ≤ 0 \implies 1-2-1=-2≤0$

2. $a + b + c ≥ 4 \implies 1+2+1=4≥ 4$

3. $a + b - c =2 \implies 1+2-1= 2=2$

4. $a, b ≥ 0 \implies 1 ≥ 0, 2 ≥ 0$

As all the constraints are satisfied, the proposed solution is valid.