NCTM Standards: 1, 2, 6, 7, 8, 9, 10

Constraints: Limitation placed upon the variables.

Example 1

}){,(4530 profityxPyx

1. Define the variables:

Let x = the units of lumber produced

Let y = the units of plywood produced.2. Write the

constraints:x + y ≤ 600

x ≥ 150

y ≥ 2253. Graph the system:

4. Write the profit equation:

5. Substitute the values of the vertices:

150, 450

30(150) + 45(450)=

24,750

150, 225

30(150) + 45(225)=

14,625

375, 225

30(375) + 45(225)=

21,375

6. Determine whether the situation calls for a maximum value or a minimum value & answer the question. The mill wants

to maximize profit, so they should produce 150 units of lumber & 450 units of plywood for a maximum profit of $24,750

Infeasible ProblemWhen the constraints of a linear programming problem cannot be solved simultaneously.

The constraints do not define a region with any points in common.

Ex:

UnboundedThe region formed is not a polygon. Ex:

The graph has a minimum value at (5, 3), but there is no maximum value.

Alternate Optimal SolutionsWhen a linear programming application has two or more optimal solutions.This usually occurs when the graph of the function to be maximized or minimized is parallel to one side of the polygonal convex set.

Example 2

701015 cb

365.49 cb

0

0

c

b

1. Define the variables:

2. Write the constraints: & the profit equation:Time available for woodworking:

Time available for finishing:

Two obvious ones we almost miss:

cbcb 4060),(Profit

Let b = # bookcases

Let c = # of cabinets

701015 cb

365.49 cb

0

0

c

b

cbcb 4060),(Profit

3. Graph the constraints:

4. Test the values in the profit equation: (0,

0)60(0) + 40(0)$0

(0, 7)

60(0) + 40(7)$280

(2, 4)

60(2) + 40(4)$280 (4,

0)60(0) + 40(0)$24

0

5. Answer the question:The shop will make a

maximum of $280 if they make 0 bookcases & 7 cabinets or 2 bookcases & 4 cabinets.

Homework: Page 115

Homework: Page 115

Homework: Page 115