1. Another example Max z=5x1+12x2+4x3-MR S.t. x1+2x2+x3+x4=10
2x1-x2+3x3+R=8
X1,x2,x3,x4,R >=0.
The optimum table is in next slide, find the dual problem and its optimal solution

2. basic x1 x2 x3 x4 R
sol z
3/5
29/5
-2/5+M
54+4/5 1
-1/5
2/5
12/5
7/5
1/5
26/5
Answer y1=29/5 y2= -2/5

3. Dual Price
Z=W Z is the dollars and W should also be the dollars. W= ∑bi yi bi represents the number of units available of resource i. Therefore Dollars= unit of resource i X yi Hence yi= dollars/unit of resource I So yi or dual price or shadow price of a resource I is the worth per unit of resource i.

4. Dual Price
Each dual price is associated with a constraint. It is the amount of improvement in the objective function value that is caused by a one-unit increase in the RHS of the constraint.
It is also called Shadow Price.

5. More on Dual Price:
A dual price can be negative, which shows a negative ( or worse off) contribution to the objective function value by an additional unit of RHS increase of the constraint.

6. Primal and Dual in LP
Each linear program has another associated with it. They are called a pair of primal and dual.
Primal and dual have equal optimal objective function values.
The solution of the dual is the dual prices of the primal, and vice versa.