CHAPTER 4 Linear Programming with Two Variableshpayne/Math 13/Lecture Notes/Chapter 4.pdf · CHAPTER 4 Linear Programming with Two Variables In this chapter, we will study systems

CHAPTER 4

Linear Programming with Two Variables

In this chapter, we will study systems of linear inequal-

ities. They are similar to linear systems of equations, but

have inequalitites instead of equalities. We will optimize

(maximize or minimize) a linear function under certain con-

ditions, given in the form of linear inequalities. Such prob-

lems are called linear programming problems and the

corresponding mathematical formulation is called a linear

program. In chapter 4, we solve linear programming prob-

lems in two variables by graphing. In chapter 5, we will

solve linear programming problems with two or more vari-

ables using a matrix method.

First, we will model linear programming problems, but

not solve them.

175

176 HELENE PAYNE, FINITE MATHEMATICS

Exercise 162. An appliance dealer, Andrea Kurtz, wants

to purchase a combined total of no more than 100 refrig-

erators, and dishwashers for inventory. Refrigerators weigh

200 pound each, and dishwashers weigh 100 pounds each.

Suppose that the dealer is limited to a total of 12, 000

pounds for these two items. If a profit of $35 for each

refrigerator and $20 on each dishwasher is projected, how

many of each should be purchased and sold to make the

largest profit?

(a) Organize the data given into a table.

(b) From the table, formulate the linear programming

problem by writing the objective function and con-

straints.

CHAPTER 4. LINEAR PROGRAMMING WITH TWO VARIABLES 177

The inequalities in the above problem,

x+ y ≤ 100

200x+ 100y ≤ 12000

are called structural constraints. There will always be

some structural constraints in linear programming prob-

lems.

The other two inequalities,

x ≥ 0, y ≥ 0

are called non-negativity constraints and they are al-

ways added to the structural constraints in linear program-

ming problems.

The profit function,

P = 35x+ 20y,

is called the objective function. That is the function we

want to optimize, in this case maximize.


Exercise 163. Write the mathematical formulation of the

linear programming problem, and identify the objective func-

tion and all the constraints.

A particular salad contains 4 units of vitamin A, 5 units of

vitamin B complex, and 2 mg of fat per serving. A nutri-

tious soup contains 6 units of vitamin A, 2 units of vitamin

B complex, and 3 mg of fat per serving. If a lunch con-

sisting of these two foods is to have at least 10 units of

vitamin A and at least 10 units of vitamin complex, how

many servings of each should be used to minimize the total

number of milligrams of fat?


Exercise 164. Write the mathematical formulation of the

linear programming problem, and identify the objective func-

tion and all the constraints.

An investment club has at most $30, 000 to invest in ei-

ther junk bonds or premium-quality bonds. Each type of

bond is bought in $1, 000 denominations. The junk bonds

have an average yield of 12%, and the premium-quality

bonds yield 7%. The policy if the club is to invest at least

twice the amount of money in premium-quality bonds as in

junk bonds. How should the club invest its money in these

bonds to receive the maximum return on its investment?


4.1. Systems of Linear Inequalities

In linear programming problems in two variables, the struc-

tural constraints consist of a system of inequalities. So-

lutions of a system of inequalities, when graphing, corre-

sponds to a shaded region in the plane. In this section,

we will learn how to graph a system of linear inequalities,

finding which region to shade. In the following section, we

will solve linear programming problems graphically.

Any linear inequality on two variables has one of the

forms

ax+ by < c,

ax+ by ≤ c,

ax+ by > c,

ax+ by ≥ c,

where a, b, and c are real numbers, where a and b are not

both zero. Examples of linear inequalities in two variables

are, 3x + 2y ≤ 5, x − 3y > 7, and y ≥ 0. The laws of

inequalities are listed below.

Laws of Inequalities

1. If a < b and c is any number, then a+ c < b+ c.

2. If a < b and c is any positive number, then ac < bc.

3. If a < b and c is any negative number, then ac > bc.


Exercise 165. Solve the inequality 2x− y ≤ 4 for y.

Solutions of a Linear Inequality

The solutions of a linear inequality in two variables consists

of all ordered pairs (x, y) that, when substituted into the

inequality, result in a true statement.

Exercise 166. For the inequality in the above problem,

y ≥ 2x − 4. Check which of the following points are solu-

tions to the inequality.

(a) (3, 3)

(b) (2, 1)

(c) (4, 5)


Exercise 167. Graph the line y = 2x − 4 and the points

from the previous exercise, (3, 3), (2, 1), (4, 5).

Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/

(a) y > 2x− 4 represents points above the line.

(b) y = 2x− 4 represents points on the line.

(c) y < 2x− 4 represents points below the line.

Use this information to shade the area which corresponds

to the solutions of the inequality y ≥ 2x− 4


Graphing the Solution of a Linear Inequality

1. Replace the inequality symbol with = to obtain the

equation of the boundary line.

2. Plot the line represented by the boundary line. If the

inequality is ≤ or ≥, plot a solid line. If the inequality is

> or <, plot the line dashed.

3. Select a test point, not on the boundary line and sub-

stitute it into the original inequality. If the resulting in-

equality is true, shade the side of the line where the test

point is located. If the resulting inequality is false, shade

side of the line opposite to where the test point is located.


Exercise 168. Graph the following linear inequalities care-

fully following the steps listed above.

(a) 3x+ y ≤ 9

Free Multi-Width Graph Paper from http://incompetech.com/graphpaper/multiwidth/(b) 3x− 2y < 12



Exercise 169. Graph the solution of the following sys-

tem of linear inequalities. List all the corner points of the

shaded area.

x+ 2y ≥ 4

x− y ≤ 1

x ≥ 0

y ≥ 0



The Feasible Region

For any system of linear inequalities, the set of points that

makes all inequalities true is called the feasible region.

When graphing the solution of a system of linear inequali-

ties, we shade the feasible region.

Corner Point of a Feasible Region

For a feasible region in the plane, defined by a system of

linear inequalities, a point of intersection of two bound-

ary lines that is also part of the feasible region is called a

corner point or vertex of the feasible region.

For any system of linear inequalities, the set of points that

makes all inequalities true is called the feasible region.

When graphing the solution of a system of linear inequali-

ties, we shade the feasible region.


Exercise 170. Graph the feasible region given by the sys-

tem of linear inequalities

2x+ y ≤ 6

x+ 2y ≤ 6

x ≥ 0

y ≥ 0.

List all the corner points of the feasible region.



4.2. Solving Linear Programming Problems Graph-

ically

In section we will learn how to solve linear programming

problems graphically.

Consider the linear programming problem,

Maximize f = 2x + 3y

subject to x+ y ≤ 9

3x+ 2y ≤ 24

x+ 2y ≤ 16

x ≥ 0; y ≥ 0

Below, we can see what different values of the objective

function correspond to graphically on the feasible region

arising from the structural constraints.


f=6

f=12

f=18

f=24f=25

In the corner point, (0, 8) we have f = 2(0) + 3(8) = 24.

There are many points for which f = 24, for example in the

point, (32 , 7), which is in the feasible region. In fact, there is

a whole line for which f = 24, namely the line 2x+3y = 24.

We can also see in the graph that the smaller the values

of f , the lower down in the graph the corresponding line

is located, and the larger the value of f , the higher up the

corresponding line is located. The largest value of the func-

tion f on the feasible region is assumed in the corner point

(2, 7), where f = 25. This example suggests that if a linear

objective function, f has a maximum on the feasible region

bounded by the constraints, then such a maximum will oc-

cur at a corner point of the feasible region.


Feasible regions can be bounded (i.e. completely en-

closed) or unbounded. They can also be convex or not

convex. Here we will only work with convex feasible re-

gions.

unbounded bounded bounded

and convex and convex and not convex


The Fundamental Theorem of Linear Programming

If the feasible region for a linear programming problem is

nonempty and convex, and if the objective function has a

maximum (or minimum) value within that set, then that

maximum (or minimum) will always correspond to at least

one corner point of the region.

For a linear programming problem with a nonempty feasi-

ble region, R and an objective function f ,

1. If R is bounded, then f has both a minimum and a

maximum value at some corner point of R.

2. If R is unbounded and if f has a maximum or a min-

imum value, on R, then that value will occur at a corner

point. However, the nature of f and the shape of R will

determine whether a maximum or a minimum exists.

The graph below shows a function f = 2x − y, which

has no maximum value on the unbounded feasible region.

(a) unboundedand convex

4.3 SOLVING LINEAR PROGRAMMING PROBLEMS GRAPHICALLY 183

(c) boundedand not convex

FIGURE 12An Unbounded Feosible Region

The Fundamental Theorem of Linear Programming

If the feasible region for a linear programming problem is nonempty and convex, andif the objective function has a maximum (or minimum) value within that set, then thatmaximum (or minimum) will always correspond to at least one corner point of theregion.

The fundamental theorem does not guarantee a solution to any linear programming prob-lem: it just says that f the objective function has a maximum or minimum value in the fea-sible region, then that value must occur at some corner point of the feasible region. Fromthe discussion of Example 1 of this section, however, we can assert that if the feasibleregion has at least one point andis bounded, lhen any objective function will have both a

maximum and a minimum value at a corner point. On the other hand, if the feasible regionts unbounded, then the existence of a solution depends upon both the nature of the objec-tive function and the shape of the feasible region. To elaborate on this last statement, con-sider the unbounded region in Figure 12 described by t - ) < 1, x 2 0, and y > 0 andthe following three objective functions:

a. The function f : 2* - y has no maximum or minimum value over this region. (See

Figure 12 for a visual justification of this statement.) Along the boundary given byx - )- : I (y : r - 1), f : 2x - y is reducedto 7 : ;r * 1 by substitution' Thisfunction has no maximum; therefore f : 2x - t has no maximum over the entireregion. In contrast, along the y-axis (x : 0), "f :2x - y becomes / : -y, fromwhich we see that / has no minimum. As a consequence, / has no minimum over theentire region.

b. The function I : x + y has a minimum, but no maximum value over the region. Theminimum value is 0 and occurs at the point (0, 0). Inspection leads to the conclusionthat no maximum exists.

),1

I

14,IVIl.

(b) boundedand convex

FIGURE 11

Bou n d ed ness o n d Convexity

v,8-'7-

6-5-4-,

7-

vizil 4 5 6'1 8 r


Exercise 171. Solve the linear programming problem be-

low by following the steps below.

Maximize P = 4x + 4y

subject to x+ 3y ≤ 30

2x+ y ≤ 20

x ≥ 0; y ≥ 0

(a) Graph the feasible region.

(b) Find the coordinates of the corner points.

(c) Evaluate P at each corner point, and find the ex-

treme value requested.

Free Plain Graph Paper from http://incompetech.com/graphpaper/plain/




Minimize C = 5x + 3y

subject to x+ y ≥ 6

6x+ y ≥ 16

x+ 6y ≥ 16

x ≥ 0; y ≥ 0



(c) Evaluate C at each corner point, and find the ex-






Minimize C = x − 2y

subject to x ≥ 2

x ≤ 4

y ≥ 1

y ≤ 5



(c) Evaluate C at each corner point, and find the ex-




4.3. Models Utilizing Linear Programming with Two

Variables

Exercise 174. Two grains, barley and corn, are to be

mixed for animal food. Barley contains 1 unit of fat per

pound, and corn contains 2 units of fat per pound. The to-

tal number of units of fat in the mixture are not to exceed

12 units. No more than 6 pounds of barley and no more

than 5 pounds of corn are to be used in the mixture. If

barley and corn each contain 1 unit of protein per pound,

how many pounds of each grain should be used to maxi-

mize the number of units of protein in the mixture?



Exercise 175. A sales representative covers territory in

Iowa and Kansas. Her daily travel expenses average $120 in

Iowa and $100 in Kansas. Her company provides an annual

travel allowance of $18, 000. Her company also stipulates

that she must spend at least 50 days in Iowa, and 60 days

in Kansas per year. If sales average $3, 000 per day in Iowa,

and $2, 500 per day in Kansas, how many days should she

spend in each state to maximize sales?


Documents

CHAPTER 4 Linear Programming with Two Variableshpayne/Math 13/Lecture Notes/Chapter 4.pdf · CHAPTER 4 Linear Programming with Two Variables In this chapter, we will study systems