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1 Chapter 7: Linear Programming.

7.2: LINEAR PROGRAMMING

Abstarct: We will learn to state the nature ofa linear programming problem along with the intro-duction of terminology associated with it, and then

developing a method for its solution geometrically.

Many business and economic problems are concerned

with optimizing (maximizing or minimizing) a func-

tion subject to a system of equalities or inequalities.

The function to be optimized is called the objective funct

Prot functions and cost functions are examples of

objective functions.

The system of equalities and inequalities to which

the objective function is subjected reects the con-

straints (for example, limitations on resources such


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as materials and labor) imposed on the solution(s)

to the problem. Problems of this nature are calledmathematical programming problems. In particular,

problems in which both the objective function and

the constraints are expressed as linear equations or

inequalities are called linear programming problems.


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A linear programming problem consists of a linear

objective function to maximized or minimized sub-ject to certain constraints in the form of linear equal-

ities or inequalities.

Existence of a Solution. Consider a linear program-

ming problem with the set R of feasible points and

objective function z = Ax + BY:

1: If R is bounded, then z has a maximum and a

minimum value on R:

2: If R is unbounded and A 0; B 0; and theconstraints include x 0 and y 0; then z has a

minimum value on R but not a maximum.

3. If R is the empty set, then the linear program-

ming problem has no solution and z has neither amaximum nor a minimum value.

Fundamental Theorem of Linear Programming


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If a linear programming problem has a solution, it

is located at a corner point of the set of feasiblepoints. If a linear programming problem has multiple

solutions, at least one of them is located at a corner

point of the set of feasible points. In either case

the corresponding value of the objective function is

unique.


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Steps for Solving a Linear Programming Problem (SM1

If a linear programming problem has a solution, fol-low these steps to nd it:

STEP 1 Write an expression for the quantity that isto be maximized or minimized (the objective func-

tion).

STEP 2 Determine all the constraints and graph theset of feasible points.

STEP 3 List the corner points of the set of feasiblepoints.

STEP 4 Determine the value of the objective func-tion at each corner point.

STEP 5 Select the minimum or minimum value ofthe objective function.

Empty Feasible Region. Whenever the feasible re-gion of a linear programming problem is empty, nooptimum solution exists.


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Example. 176SM. Maximize and minimize the ob-

jective function

z = x + 5y

subject to the constraints

8>>>>>>>>>>>:

x + 4y 12 (1)x 8 (2)

x + y 2 (3)x 0 (4)y 0 (5)

Solution. The objective function is z = x + 5y and

the constraints consist of a system of ve linear

inequalities.

We proceed to graph the system of ve linear in-

equalities.


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The shaded portion of Figure 1 illustrates the graph,

the set of feasible points.

Since this set is bounded, we know a solution to

the linear programming problem exists.

Notice in Figure 1 that we have labeled each linefrom the system linear inequalities. We have also

labeled the corner points.

The set of feasible points is bounded. So we know

a solution exists.

GRAPH: y = 0:25x + 3; y = x + 2 ; x = 8:

Now we locate the corner points of the set of feasible

points at the points of intersection of lines.


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Equation of pair of LinesPoints of

Inter sec tionx + 4y = 12; x = 0 (0; 3)x + 4y = 12; x = 8 (8; 1)

x = 8; y = 0 (8; 0)x + y = 2; y = 0 (2; 0)x + y = 2; x = 0 (0; 2)

To nd the maximum and minimum value of z =

x + 5y;

we set up a table:

Corner Point(x; y)

Value of Objective Functionz = x + 5y

(0; 3) z = 0 + 5 (3) = 15(8; 1) z = 8 + 5 (1) = 13(8; 0) z = 8 + 5 (0) = 8(2; 0) z = 2 + 5 (0) = 2

(0;

2)z

= 0 + 5 (2) = 10

The maximum value of z is 15;


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and it occurs at the point (0; 3) :

The minimum value of z is 2;

and it occurs at the point (2; 0) :


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How much nitrogen will be added?

(B) If the grower wants to minimize the amount of

nitrogen added to the grove, how many bags of each

mix should be used?

How much nitrogen will be added?

Solution. Let x = the number of bags of brand A

and y = the number of bags of brand B:

(A) The mathematical model for the problem is :

Maximize N = 8x + 3y

subject to:

8>>>>>:

4x + 4y 10002x + y 400

x 0y 0


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Pair of LinesP nts: I nter:

Corner Pnt:N = 8x + 3y

x = 0;4x + 4y = 1000

(0; 250)N = 8 (0) +

3 (250) = 7504x + 4y = 1000;

2x + y = 400(150; 100)

N = 8(150)+3 (100) = 1500

2x + y = 400;x = 0

(0; 400)N = 8 (0) +

3 (400) = 1200

y = x + 250; y = 2x + 400:

25 020 015 010 0500

40 0

30 0

20 0

10 0

0

x

y

x

y

The feasible region S is the solution set of the sys-

tem of inequalities, and indicate it by the shading in

the graph.

Thus, the maximum occurs when x = 150 and y =

100:


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That is, the grower should use 150 bags of brand A

and 100 bags of brand B:

The maximum number of pounds of nitrogen is 1500:

(B) The mathematical model for this problem is:

Minimize N = 8x + 3y

Subject to:subject to:

8>>>>>:

4x + 4y 10002x + y 400

x 0y 0

The feasible region S and the corner points are thesame as in part (A) :

Thus, the minimum occurs when x = 0 and y =

250:

That is, the grower should use 0 bags of brand Aand 250 bags of brand B:

The minimum number of pounds of nitrogen is 750:


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Example. TB16Lk. A producer grower is purchas-

ing fertilizer containing three nutrients: A;B;and C:

The minimum weekly requirements are 80 units of

A; 120 of B; and 240 of C:

There are two popular blends of fertilizer on the

market. Blend I; costing $4 a bag, contains 2 units

of A; 6 of B; and 4 of C:

Blend II; costing $5 a bag, contains 2 units of A; 2of B; and 12 of C:

How many bags of each blend should the grower

buy each week to minimize the cost of meeting the

nutrient requirements?

Answer: Hint: Minimize Z = 4x + 5y


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Constraints:

8>>>>>>>>>>>:

x 0y 0

2x + 2y 806x + 2y 1204x + 12y 240

y = x + 40; y = 3x + 60; y = 1

3x + 20:

1007550250

100

75

50

25

0

x

y

x

y

C is minimized at the corner point (30; 10) where

C = 170:

Thus each week the grower should buy 30 bags of

blend I and 10 bags of blend II.


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Example. TB18Lk. Restate the question by omit-

ting the last one 0 digit from each number usedin the question.

Answer: Hint:Hint: Minimize C = 2500x + 2000y

Constraints:

8>>>>>>>>>>>:

x 0y 0

200x + 100y 800300x + 200y 1400100x + 100y 500

107.552.50

10

5

0

-5

-10

x

y

x

y


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Example. TB14. A manufacturer produces two types

of barbecue grills.:

Old Smokey and Blaze Away: During production,

the grills require the use of two machines, A and

B:The number of hours needed on both machines

are indicated in the following table:

Machine A Machine B Price/grillOld Smokey 2 hours 4 hours $4Blaze Away 4 hours 2 hours $6

If each machine can be used 24 hours a day, and the

prots on Old Smokey and Blaze Away models are

$4 and $6; respectively,

how many of each type of grill should be made per

day to obtain maximum prot?

What is the maximum prot?


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Solution. Solution. Let x = the number of units of

Old Smokey

and y = the number of units of Blaze Away that are

made per day, respectively.

Then we are to maximize P = 4x + 6y; where

Constraints:

8>>>>>:

x 0y 0

2x + 4y 244x + 2y 24

1512.5107.552.50-2.5

10

7.5

5

2.5

0

-2.5

x

y

x

y


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The point of intersection of the lines 2x + 4y = 24

and 4x + 2y = 24 is (4; 4) :

The corner points are (0; 0) ; (0; 6) ; (4; 4) ; and (6; 0) :

The values of P = 4x + 6y at these points:

Point (0; 0) (0; 6) (4; 4) (6; 0)Value 0 36 40 24

Evaluating P at each corner point, we nd that P

is maximized at corner point (4; 4) where its valueis 40: Thus 4 units of each type of barbecue grills

from Old Smokey and Blaze Away should be made

each day in order to give a maximum prot of $40:

=End of 7.2 Linear Programming Class Lecture Notes=


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Example. TB16. Fertilizer Nutrients. A produce

grower is purchasing fertilizer containing threenutrients: A, B, and C.

The minimum weekly requirements are 80 units of

A; 120 units of B; and 240 units of C:

There are two popular blends of fertilizer on the

market.

Blend I; costing $ 4 a bag, contains 2 units of A; 6

of B; and 4 of C:

Blend II; costing $ 5 a bag, contains 2 units of A;

2 of B; and 12 of C:

How many bags of each blend should the grower

buy each week to minimize the cost of meeting the

nutrient requirements?


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