BUS 3700 Practice Problems for shadow prices
For each problem below, you are given the two binding constraints
at the optimal solution for a linear programming problem.
Determine the shadow prices for each of the two constraints.
Also answer this question: what is the shadow price for the
non-binding constraints?
Problem 1
Constraint 1: 6x + 4y = 144
Constraint 2: 10x + 5y = 220
(a) Solve for the point where two constraints intersect.
(b) Determine the profit at this point, where the profit
of x is 18 and the profit from y is 10
(c) Determine the point where the two constraints intersect if
the capacity of constraint 1 increases by 1.
(d) Determine the profit at this new point.
(e) Determine the shadow price of constraint 1, which is
the difference in profit from step (d) and step (b).
(f) Determine the point where the two constraints intersect if
the capacity of constraint 2 increases by 1.
(g) Determine the profit at this new point.
(h) Determine the shadow price of constraint 2, which is
the difference in profit from step (g) and step (b).
Problem 2
Constraint 1: 10x + 2y = 200
Constraint 2: 3x + 12y = 240
(a) Solve for the point where two constraints intersect.
(b) Determine the profit at this point, where the profit
of x is 7 and the profit from y is 8
(c) Determine the point where the two constraints intersect if
the capacity of constraint 1 increases by 1.
(d) Determine the profit at this new point.
(e) Determine the shadow price of constraint 1, which is
the difference in profit from step (d) and step (b).
(f) Determine the point where the two constraints intersect if
the capacity of constraint 2 increases by 1.
(g) Determine the profit at this new point.
(h) Determine the shadow price of constraint 2, which is
the difference in profit from step (g) and step (b).
Problem 3
Constraint 1: 8x + 10y = 740
Constraint 2: 9x + 12y = 870
(a) Solve for the point where two constraints intersect.
(b) Determine the profit at this point, where the profit
of x is 40 and the profit from y is 52
(c) Determine the point where the two constraints intersect if
the capacity of constraint 1 increases by 1.
(d) Determine the profit at this new point.
(e) Determine the shadow price of constraint 1, which is
the difference in profit from step (d) and step (b).
(f) Determine the point where the two constraints intersect if
the capacity of constraint 2 increases by 1.
(g) Determine the profit at this new point.
(h) Determine the shadow price of constraint 2, which is
the difference in profit from step (g) and step (b).
Problem 4
Constraint 1: 28x + 4y = 82
Constraint 2: 10x + 32y = 96
(a) Solve for the point where two constraints intersect.
(b) Determine the profit at this point, where the profit
of x is 40 and the profit from y is 50
(c) Determine the point where the two constraints intersect if
the capacity of constraint 1 increases by 1.
(d) Determine the profit at this new point.
(e) Determine the shadow price of constraint 1, which is
the difference in profit from step (d) and step (b).
(f) Determine the point where the two constraints intersect if
the capacity of constraint 2 increases by 1.
(g) Determine the profit at this new point.
(h) Determine the shadow price of constraint 2, which is
the difference in profit from step (g) and step (b).
Answers
The shadow price of each non-binding constraint is zero.
Problem 1
Constraint 1: 6x + 4y = 144
Constraint 2: 10x + 5y = 220
(a) Solve for the point where two constraints intersect.
x= 16.000 y= 12.000
(b) Determine the profit at this point, where the profit
of x is 18 and the profit from y is 10
profit = 408.000
(c) Determine the point where the two constraints intersect if
the capacity of constraint 1 increases by 1.
x= 15.500 y= 13.000
(d) Determine the profit at this new point.
profit = 409.000
(e) Determine the shadow price of constraint 1, which is
the difference in profit from step (d) and step (b).
shadow price= 1.000
(f) Determine the point where the two constraints intersect if
the capacity of constraint 2 increases by 1.
x= 16.400 y= 11.400
(g) Determine the profit at this new point.
profit = 409.200
(e) Determine the shadow price of constraint 2, which is
the difference in profit from step (g) and step (b).
shadow price= 1.200
Problem 2
Constraint 1: 10x + 2y = 200
Constraint 2: 3x + 12y = 240
(a) Solve for the point where two constraints intersect.
x= 16.842 y= 15.789
(b) Determine the profit at this point, where the profit
of x is 7 and the profit from y is 8
profit = 244.211
(c) Determine the point where the two constraints intersect if
the capacity of constraint 1 increases by 1.
x= 16.947 y= 15.763
(d) Determine the profit at this new point.
profit = 244.737
(e) Determine the shadow price of constraint 1, which is
the difference in profit from step (d) and step (b).
shadow price= 0.526
(f) Determine the point where the two constraints intersect if
the capacity of constraint 2 increases by 1.
x= 16.825 y= 15.877
(g) Determine the profit at this new point.
profit = 244.789
(e) Determine the shadow price of constraint 2, which is
the difference in profit from step (g) and step (b).
shadow price= 0.579
Problem 3
Constraint 1: 8x + 10y = 740
Constraint 2: 9x + 12y = 870
(a) Solve for the point where two constraints intersect.
x= 30.000 y= 50.000
(b) Determine the profit at this point, where the profit
of x is 40 and the profit from y is 52
profit = 3800.000
(c) Determine the point where the two constraints intersect if
the capacity of constraint 1 increases by 1.
x= 32.000 y= 48.500
(d) Determine the profit at this new point.
profit = 3802.000
(e) Determine the shadow price of constraint 1, which is
the difference in profit from step (d) and step (b).
shadow price= 2.000
(f) Determine the point where the two constraints intersect if
the capacity of constraint 2 increases by 1.
x= 28.333 y= 51.333
(g) Determine the profit at this new point.
profit = 3802.667
(e) Determine the shadow price of constraint 2, which is
the difference in profit from step (g) and step (b).
shadow price= 2.667
Problem 4
Constraint 1: 28x + 4y = 82
Constraint 2: 10x + 32y = 96
(a) Solve for the point where two constraints intersect.
x= 2.617 y= 2.182
(b) Determine the profit at this point, where the profit
of x is 40 and the profit from y is 50
profit = 213.785
(c) Determine the point where the two constraints intersect if
the capacity of constraint 1 increases by 1.
x= 2.654 y= 2.171
(d) Determine the profit at this new point.
profit = 214.696
(e) Determine the shadow price of constraint 1, which is
the difference in profit from step (d) and step (b).
shadow price= 0.911
(f) Determine the point where the two constraints intersect if
the capacity of constraint 2 increases by 1.
x= 2.612 y= 2.215
(g) Determine the profit at this new point.
profit = 215.234
(e) Determine the shadow price of constraint 2, which is
the difference in profit from step (g) and step (b).
shadow price= 1.449