Consider the Linear Programming formulation and its associated MS Excel solution below. Carefully examine the MS Excel output above.

The dual price associated with constraint two is 0.67. If the right hand side of constraint two is increased by 5 units (i.e., 5×1 + 4×2 >=40 becomes 5×1 + 4×2 >=45) what impact does this have on the optimal objective function value of z = 82.67? Specifically, show any calculations to get a revised z-value.

Similarly, suppose that the right hand side of constraint two is decreased by three (3) units. What impact does the aforementioned change have on the objective function value of z = 82.67? Specifically show any calculations to get a revised z-value.

## Expert Answer

The shadow price of constraint is an amount by which the objective function value will increase or decrease if a unit of the binding constraint is made available or reduced respectively. But the shadow price is valid till the increase in constraint value is within allowable increase and decrease.

The shadow price or dual price of painting hours is 0.67, it means by increasing the RHS by 1 unit the objective function value will increase by 0.67 per unit increase or the 0.67 per unit decrease.

The allowable increase in the RHS of constraint is 12.8 and allowable decrease is 16 units, the shadow price of the constraint is valid between (40+12.8 =) 54.8 to (40-16 =) 24.

If the RHS of the constraint is increased by 5 units, which is within the allowable increase, the objective function value will increase by 0.67 x 5 = 3.35. thus, revised objective function value = 82.67 + 3.35 = 86.02.

If the RHS of the constraint is decreased by 3 units, which is within the allowable decrease, the objective function value will decrease by 0.67 x 3 = 2.01. thus, revised objective function value = 82.67 – 2.01 = 80.66.