Section 3.3 Graphical Solutions of Linear Programming Problems

Theorem 1: Solutions of Linear Programming Problems

1. If a linear programming problem has a solution, then it must occur at a corner point of the feasible

set, S, associated with the problem.

2. If the objective function, P , is optimized at two adjacent corner points of S, then it is optimized

at every point on the line segment joining the two points (infinitely many solutions).

Theorem 2: Existence of a Solution

Suppose we are given a linear programming problem with a feasible set S and an objective funtion

P = ax+ by.

1. If S is bounded then P has both a maximum and a minimum value on S.

2. If S is unbounded and both a and b are nonnegative, then P has a minimum value on S provided

that the constraints defining S include the inequalities x � 0 and y � 0.

3. If S is empty, then the linear programming problem has no solution; that is, P has neither a

maximum nor a minimum value. We say that the problem is infeasible.

The Method of Corners

1. Graph the feasible set.

2. If the feasible set is nonempty, find the coordinates of all corner points of the feasible set. In

this class we will use the “rref” calculator function to find corner points whenever the points are

where two lines are crossing.

3. Evaluate the objective function at each corner point.

4. Find the corner point(s) that renders the objective function a maximum (or minimum).

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1. Find the corner points for the feasible region graphed below.

5x+ 2y 30

x+ 2y 12

x � 0, y � 0

2. Solve the linear programming problem by the method of corners.

Maximize P = 3x+ 5y

subject to 2x+ y 16

2x+ 3y 24

y 7

x � 0, y � 0

2 Fall 2019, Maya Johnson

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3. Solve the linear programming problem by the method of corners.

Minimize C = 2x+ y

subject to 4x+ y � 42

2x+ y � 30

x+ 3y � 30

x � 0

y � 0

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4. Perth Mining Company operates two mines for the purpose of extracting gold and silver. The

Saddle Mine costs $15, 000/day to operate, and it yields 50 oz of gold and 3000 oz of silver each

day. The Horseshoe Mine costs $19, 000/day to operate, and it yields 75 oz of gold and 1000 oz

of silver each day. Company management has set a target of at least 650 oz of gold and 18, 000

oz of silver. How many days should each mine be operated so that the target can be met at a

minimum cost? (Let x be the # of days Saddle Mine operates and y be the # of days Horseshoe

Mine operates.)

4 Fall 2019, Maya Johnson

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5. National Business Machines manufactures x model A fax machines and y model B fax machines.

Each model A costs $100 to make, and each model B costs $150. The profits are $40 for each

model A and $35 for each model B fax machine. If the total number of fax machines demanded

per month does not exceed 2500 and the company has earmarked no more than $600, 000/month

for manufacturing costs, how many units of each model should National make each month to

maximize its monthly profit?

5 Fall 2019, Maya Johnson

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