3 Graphing Systems of Linear Inequalities in Two Variables Linear Programming Problems Graphical...

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Graphing Systems of Linear Graphing Systems of Linear Inequalities in Two VariablesInequalities in Two Variables

Linear Programming ProblemsLinear Programming Problems Graphical Solutions of Linear Graphical Solutions of Linear

Programming ProblemsProgramming Problems Sensitivity AnalysisSensitivity Analysis

Linear Programming: A Geometric ApproachLinear Programming: A Geometric Approach

3.13.1Graphing Systems of Linear Inequalities Graphing Systems of Linear Inequalities in Two Variablesin Two Variables

–1 1 2 3x

y

4

3

2

1

4x + 3y = 12

12 127 7( , )P 12 127 7( , )P

x – y = 0

4 3 120

x yx y

4 3 120

x yx y

Graphing Linear InequalitiesGraphing Linear Inequalities

We’ve seen that We’ve seen that a linear a linear equationequation in two variables in two variables x x and and y y

has a has a solution setsolution set that may be exhibited that may be exhibited graphicallygraphically as as points points on a straight lineon a straight line in the in the xyxy-plane-plane..

There is also a simple There is also a simple graphical representationgraphical representation for for linear linear inequalitiesinequalities of two variables of two variables::

0ax by c 0ax by c

0ax by c 0ax by c

0ax by c 0ax by c

0ax by c 0ax by c

0ax by c 0ax by c

Procedure for Graphing Linear InequalitiesProcedure for Graphing Linear Inequalities

1.1. Draw the Draw the graphgraph of the of the equationequation obtained for the given obtained for the given inequality by inequality by replacing the inequality sign with an replacing the inequality sign with an equal signequal sign..✦ Use a Use a dashed or dotted linedashed or dotted line if the problem involves a if the problem involves a

strict inequalitystrict inequality, , << or or >>..✦ Otherwise, use a Otherwise, use a solid linesolid line to indicate that to indicate that the line the line

itself constitutes part of the solutionitself constitutes part of the solution..2.2. Pick a test pointPick a test point lying in one of the half-planes lying in one of the half-planes

determined by the line sketched in determined by the line sketched in step 1step 1 and and substitutesubstitute the values of the values of xx and and yy into the given into the given inequalityinequality..✦ Use the Use the originorigin whenever possible. whenever possible.

3.3. If the If the inequality is satisfiedinequality is satisfied, the graph of , the graph of the inequality the inequality includes the half-planeincludes the half-plane containing the containing the test pointtest point..✦ Otherwise, the solution includes the half-plane not Otherwise, the solution includes the half-plane not

containing the test point.containing the test point.

ExamplesExamples

Determine the Determine the solution setsolution set for the for the inequalityinequality 22xx + 3 + 3yy 6 6..SolutionSolution ReplacingReplacing the the inequalityinequality with an with an equalityequality ==, we obtain , we obtain

the equation the equation 22xx + 3 + 3yy = 6 = 6, whose graph is:, whose graph is:

xx

yy

77

55

33

11

–– 1 1 –– 55 –– 33 – – 11 11 33 55

22xx + 3 + 3yy = 6 = 6

Example 1, page 158

ExamplesExamples

Determine the Determine the solution setsolution set for the for the inequalityinequality 22xx + 3 + 3yy 6 6..SolutionSolution Picking the Picking the originorigin as a as a test pointtest point, we find , we find 2(0) + 3(0) 2(0) + 3(0) 6 6, ,

or or 0 0 6 6, which is , which is falsefalse. . Thus, the Thus, the solution setsolution set is: is:

xx

yy

77

55

33

11

–– 1 1 –– 55 –– 33 – – 11 11 33 55

22xx + 3 + 3yy = 6 = 622xx + 3 + 3yy 6 6

(0, 0)(0, 0)

Example 1, page 158

Graphing Systems of Linear InequalitiesGraphing Systems of Linear Inequalities

The The solution setsolution set of a of a system of linear inequalitiessystem of linear inequalities in two in two variables variables x x and and yy is the is the set of all pointsset of all points ((xx, , yy)) that that satisfy satisfy each inequalityeach inequality of the system. of the system.

The The graphical solutiongraphical solution of such a system may be obtained of such a system may be obtained by by graphing the solution set for each inequalitygraphing the solution set for each inequality independently and then independently and then determining the region in determining the region in commoncommon with each solution set. with each solution set.

–– 5 5 – – 33 11 33 55

ExamplesExamples

Graph Graph xx – 3 – 3yy > 0 > 0..SolutionSolution ReplacingReplacing the the inequalityinequality >> with an with an equalityequality ==, we obtain , we obtain

the equation the equation xx – 3 – 3yy = 0 = 0, whose graph is:, whose graph is:

xx

yy

33

11

–– 11

–– 3 3

xx – 3 – 3yy = 0 = 0

ExamplesExamples

Graph Graph xx – 3 – 3yy > 0 > 0..SolutionSolution We use a We use a dashed linedashed line to indicate to indicate the line itself will the line itself will notnot be be

part of the solutionpart of the solution, since we are dealing with a , since we are dealing with a strict strict inequalityinequality >>. .

xx

yy

xx – 3 – 3yy = 0 = 0

–– 5 5 – – 33 11 33 55

33

11

–– 11

–– 3 3

–– 5 5 – – 33 11 33 55

33

11

–– 11

–– 3 3

ExamplesExamples

Graph Graph xx – 3 – 3yy > 0 > 0..SolutionSolution Since the origin lies on the line, we Since the origin lies on the line, we cannot use the origincannot use the origin

as a as a testing pointtesting point: :

xx

yy

xx – 3 – 3yy = 0 = 0

(0, 0)(0, 0)

ExamplesExamples

Graph Graph xx – 3 – 3yy > 0 > 0..SolutionSolution Picking instead Picking instead (3, 0)(3, 0) as a as a test pointtest point, we find , we find (3) – 2(0) > 0(3) – 2(0) > 0, ,

or or 3 > 03 > 0, which is , which is truetrue. . Thus, the Thus, the solution setsolution set is: is:

yy

xx – 3 – 3yy = 0 = 0

xx – 3 – 3yy > 0 > 0

–– 5 5 – – 33 11 33 55

33

11

–– 11

–– 3 3

xx(3, 0)(3, 0)

Graphing Systems of Linear InequalitiesGraphing Systems of Linear Inequalities

The The solution setsolution set of a of a system of linear inequalitiessystem of linear inequalities in two in two variables variables x x and and yy is the is the set of all pointsset of all points ((xx, , yy)) that that satisfy satisfy each inequalityeach inequality of the system. of the system.

The The graphical solutiongraphical solution of such a system may be obtained of such a system may be obtained by by graphing the solution set for each inequalitygraphing the solution set for each inequality independently and then independently and then determining the region in determining the region in commoncommon with each solution set. with each solution set.

ExampleExample Determine the solution set for the systemDetermine the solution set for the system

SolutionSolution The The intersectionintersection of the of the solution regionssolution regions of the two of the two

inequalitiesinequalities represents the represents the solution to the systemsolution to the system::

4 3 120

x yx y

4 3 120

x yx y

xx

yy

44

33

22

11

44xx + 3 + 3yy 12 12

44xx + 3 + 3yy = 12 = 12

–– 11 11 22 33

Example 4, page 159




4 3 120

x yx y

4 3 120

x yx y

xx

yy

xx – – y y 0 0 xx – – y y = 0= 0

44

33

22

11

–– 11 11 22 33

Example 4, page 159




4 3 120

x yx y

4 3 120

x yx y

xx

yy

44xx + 3 + 3yy = 12 = 12

xx – – y y = 0= 0

4 3 120

x yx y

4 3 120

x yx y

44

33

22

11

–– 11 11 22 33

12 127 7( , )P 12 127 7( , )P

Example 4, page 159

Bounded and Unbounded SetsBounded and Unbounded Sets

The The solution setsolution set of a system of linear inequalities of a system of linear inequalities is is boundedbounded if it if it can be enclosed by a circlecan be enclosed by a circle..

Otherwise, it is Otherwise, it is unboundedunbounded..

ExampleExample

The solution to the problem we just discussed is The solution to the problem we just discussed is unboundedunbounded, since , since the solution setthe solution set cannot be cannot be enclosed in a circleenclosed in a circle::

xx

yy

44xx + 3 + 3yy = 12 = 12

12 127 7( , )P 12 127 7( , )P

xx – – y y = 0= 0

4 3 120

x yx y

4 3 120

x yx y

44

33

22

11

–– 11 11 22 33

77

55

33

11

––1 1 11 33 55 99


SolutionSolution The The intersectionintersection of the of the solution regionssolution regions of the four of the four


6 0 2 8 0 0 0 x y x y x y 6 0 2 8 0 0 0 x y x y x y

xx

yy

2 8 0x y 2 8 0x y

6 0x y 6 0x y

(2,4)P(2,4)P

Example 5, page 160


SolutionSolution Note that the solution to this problem is Note that the solution to this problem is boundedbounded, since , since it it

can be enclosed by a circlecan be enclosed by a circle::

6 0 2 8 0 0 0 x y x y x y 6 0 2 8 0 0 0 x y x y x y

––1 1 11 33 55 99xx

yy

77

55

33

116 0x y 6 0x y

(2,4)P(2,4)P

Example 5, page 160

2 8 0x y 2 8 0x y

3.23.2Linear Programming ProblemsLinear Programming Problems

Maximize

Subject to

0

0

x

y

0

0

x

y

1.2P x y 1.2P x y

3 300x y 3 300x y

2 180x y 2 180x y

Linear Programming ProblemLinear Programming Problem

A linear programming problem consists of a A linear programming problem consists of a linear objective functionlinear objective function to be to be maximized or maximized or minimizedminimized subject to certain subject to certain constraintsconstraints in the in the form of form of linear equations or inequalitieslinear equations or inequalities..

Applied Example 1:Applied Example 1: A Production Problem A Production Problem

Ace Novelty wishes to produce Ace Novelty wishes to produce two types of souvenirstwo types of souvenirs: : type-A type-A will result in a profit of will result in a profit of $1.00$1.00, and , and type-Btype-B in a in a profit of profit of $1.20$1.20..

To manufacture a To manufacture a type-Atype-A souvenir requires souvenir requires 2 2 minutes on minutes on machine Imachine I and and 11 minute on minute on machine IImachine II..

A A type-Btype-B souvenir requires souvenir requires 11 minute on minute on machine Imachine I and and 3 3 minutes on minutes on machine IImachine II..

There are There are 33 hours available on hours available on machine Imachine I and and 5 5 hours hours available on available on machine IImachine II..

How many souvenirsHow many souvenirs of each type should Ace make in of each type should Ace make in order to order to maximize its profitmaximize its profit??

Applied Example 1, page 164


SolutionSolution Let’s first Let’s first tabulatetabulate the given information: the given information:

Let Let xx be the number of be the number of type-A type-A souvenirs and souvenirs and yy the number the number of of type-Btype-B souvenirs to be made. souvenirs to be made.

Type-AType-A Type-BType-B Time AvailableTime Available

Profit/UnitProfit/Unit $1.00$1.00 $1.20$1.20

Machine IMachine I 2 min2 min 1 min1 min 180 min180 min

Machine IIMachine II 1 min1 min 3 min3 min 300 min300 min




Then, the Then, the total profittotal profit (in dollars) is given by (in dollars) is given by

which is the which is the objective functionobjective function to be to be maximizedmaximized..

1.2P x y 1.2P x y








The total amount of The total amount of timetime that that machine Imachine I is used is is used is

and must not exceed and must not exceed 180180 minutes. minutes. Thus, we have the Thus, we have the inequalityinequality

2x y2x y

2 180x y 2 180x y








The total amount of The total amount of timetime that that machine IImachine II is used is is used is

and must not exceed and must not exceed 300300 minutes. minutes. Thus, we have the Thus, we have the inequalityinequality

3x y3x y

3 300x y 3 300x y








Finally, neither Finally, neither x x nor nor yy can be can be negativenegative, so, so

0

0

x

y

0

0

x

y







SolutionSolution In short, we want to In short, we want to maximizemaximize the the objective functionobjective function

subject tosubject to the the system of inequalitiessystem of inequalities

We will discuss the We will discuss the solutionsolution to this problem in to this problem in section 3.3section 3.3..

0

0

x

y

0

0

x

y

1.2P x y 1.2P x y

3 300x y 3 300x y

2 180x y 2 180x y


Applied Example 2:Applied Example 2: A Nutrition Problem A Nutrition Problem

A nutritionist advises an individual who is suffering from A nutritionist advises an individual who is suffering from ironiron and and vitamin B vitamin B deficiency to take at least deficiency to take at least 24002400 milligrams (mg) of milligrams (mg) of ironiron, , 21002100 mg of mg of vitamin Bvitamin B11, and , and 15001500

mg of mg of vitamin Bvitamin B22 over a period of time. over a period of time.

Two vitamin pills are suitable, Two vitamin pills are suitable, brand-Abrand-A and and brand-Bbrand-B.. Each Each brand-Abrand-A pill costs pill costs 66 cents and contains cents and contains 4040 mg of mg of ironiron, ,

1010 mg of mg of vitamin Bvitamin B11, and , and 55 mg of mg of vitamin Bvitamin B22..

Each Each brand-B brand-B pill costs pill costs 88 cents and contains cents and contains 1010 mg of mg of ironiron and and 1515 mg each of mg each of vitamins Bvitamins B11 and and BB22..

What combination of pillsWhat combination of pills should the individual purchase should the individual purchase in order to in order to meetmeet the minimum iron and vitamin the minimum iron and vitamin requirementsrequirements at the at the lowest costlowest cost??


Applied Example 2:Applied Example 2: A Nutrition Problem A Nutrition ProblemSolutionSolution Lets first Lets first tabulatetabulate the given information: the given information:

Let Let xx be the number of be the number of brand-A brand-A pills and pills and yy the number of the number of brand-Bbrand-B pills to be pills to be purchasedpurchased..

Brand-ABrand-A Brand-BBrand-B Minimum RequirementMinimum Requirement

Cost/PillCost/Pill 66¢¢ 88¢¢

IronIron 40 mg40 mg 10 mg10 mg 2400 mg2400 mg

Vitamin BVitamin B11 10 mg10 mg 15 mg15 mg 2100 mg2100 mg

Vitamin BVitamin B22 5mg5mg 15 mg15 mg 1500 mg1500 mg



The The costcost CC (in cents) is given by (in cents) is given by

and is the and is the objective functionobjective function to be to be minimizedminimized..






6 8C x y 6 8C x y



The amount of The amount of ironiron contained in contained in x x brand-A brand-A pills and pills and yy brand-Bbrand-B pills is given by pills is given by 4040xx + 10 + 10y y mg, and this must be mg, and this must be greater than or equal to greater than or equal to 24002400 mg. mg.

This translates into the This translates into the inequalityinequality






40 10 2400x y 40 10 2400x y



The amount of The amount of vitamin Bvitamin B11 contained in contained in x x brand-A brand-A pills and pills and yy brand-Bbrand-B pills is given by pills is given by 1010xx + 15 + 15y y mg, and this must be mg, and this must be grater or equal to grater or equal to 21002100 mg. mg.







10 15 2100x y 10 15 2100x y



The amount of The amount of vitamin Bvitamin B22 contained in contained in x x brand-A brand-A pills and pills and yy brand-Bbrand-B pills is given by pills is given by 55xx + 15 + 15y y mg, and this must be mg, and this must be grater or equal to grater or equal to 15001500 mg. mg.







5 15 1500x y 5 15 1500x y


Applied Example 2:Applied Example 2: A Nutrition Problem A Nutrition ProblemSolutionSolution In short, we want to In short, we want to minimizeminimize the the objective functionobjective function


We will discuss the We will discuss the solutionsolution to this problem in to this problem in section 3.3section 3.3..

5 15 1500x y 5 15 1500x y

6 8C x y 6 8C x y

40 10 2400x y 40 10 2400x y

10 15 2100x y 10 15 2100x y

0

0

x

y

0

0

x

y


3.33.3Graphical Solutions Graphical Solutions of Linear Programming Problemsof Linear Programming Problems

200

100

100 200 300x

y

S10 15 2100x y 10 15 2100x y

40 10 2400x y 40 10 2400x y

5 15 1500x y 5 15 1500x y

C(120, 60)

D(300, 0)

A(0, 240)

B(30, 120)

Feasible Solution Set and Optimal SolutionFeasible Solution Set and Optimal Solution

The The constraintsconstraints in a in a linear programming problemlinear programming problem form a form a system of linear inequalitiessystem of linear inequalities, which have a , which have a solution setsolution set SS..

Each point in Each point in SS is a is a candidatecandidate for the for the solutionsolution of the linear of the linear programming problem and is referred to as a programming problem and is referred to as a feasible feasible solutionsolution..

The set The set SS itself is referred to as a itself is referred to as a feasible setfeasible set.. Among all the points in the set Among all the points in the set SS, the point(s) that , the point(s) that

optimizes the objective functionoptimizes the objective function of the linear programming of the linear programming problem is called an problem is called an optimal solutionoptimal solution..

Theorem 1Theorem 1

Linear ProgrammingLinear Programming If a linear programming problem has a If a linear programming problem has a solutionsolution, ,

then it must occur at a then it must occur at a vertexvertex, or , or corner pointcorner point, , of the of the feasible setfeasible set SS associated with the problem. associated with the problem.

If the If the objective functionobjective function PP is is optimizedoptimized at at twotwo adjacent verticesadjacent vertices of of SS, then it is optimized , then it is optimized at at every every pointpoint on the line segment on the line segment joining these vertices, in joining these vertices, in which case there are which case there are infinitely many solutionsinfinitely many solutions to to the problem.the problem.

Theorem 2Theorem 2

Existence of a SolutionExistence of a Solution Suppose we are given a linear programming Suppose we are given a linear programming

problem with a problem with a feasible setfeasible set SS and an and an objective objective functionfunction PP = = axax + + byby..a.a. If If S S is is boundedbounded, then , then PP has both a has both a maximum and maximum and

a minimum value a minimum value on on SS..

b.b. If If S S is is unboundedunbounded and both and both aa and and bb are are nonnegativenonnegative, then , then PP has a has a minimum valueminimum value on on SS provided that the constraints definingprovided that the constraints defining S S include include the inequalities the inequalities xx 0 0 and and yy 0 0..

c.c. If If SS is the is the empty setempty set, then the linear , then the linear programming problem has programming problem has no solutionno solution: that is, : that is, PP has has neither a maximum nor a minimumneither a maximum nor a minimum value. value.

The Method of CornersThe Method of Corners

1.1. GraphGraph the the feasible setfeasible set..2.2. Find the Find the coordinatescoordinates of all of all corner pointscorner points

(vertices) of the feasible set.(vertices) of the feasible set.3.3. Evaluate the Evaluate the objective functionobjective function at at each corner each corner

pointpoint..4.4. Find the Find the vertexvertex that renders the that renders the objective objective

functionfunction a a maximummaximum or a or a minimumminimum..✦ If there is only If there is only one such vertexone such vertex, it constitutes a , it constitutes a

unique solutionunique solution to the problem. to the problem.✦ If there are two If there are two such adjacent verticessuch adjacent vertices, there , there

are are infinitely many optimal solutionsinfinitely many optimal solutions given by given by the points on the line segment determined by the points on the line segment determined by these vertices.these vertices.


Recall Recall Applied Example 1Applied Example 1 from the from the last section (3.2) last section (3.2), which , which required us to find the required us to find the optimal quantitiesoptimal quantities to produce of to produce of type-A type-A and and type-Btype-B souvenirs in order to souvenirs in order to maximize profitsmaximize profits..

We We restatedrestated the problem as a the problem as a linear programming problemlinear programming problem in which we wanted to in which we wanted to maximizemaximize the the objective functionobjective function


We can now We can now solve the problemsolve the problem graphically. graphically.

0

0

x

y

0

0

x

y

1.2P x y 1.2P x y

3 300x y 3 300x y

2 180x y 2 180x y


200200

100100

100100 200200 300300


We first We first graph the feasible setgraph the feasible set SS for the problem. for the problem.✦ Graph the Graph the solutionsolution for the inequality for the inequality

considering only considering only positive valuespositive values for for xx and and yy::

2 180x y 2 180x y

xx

yy

2 180x y 2 180x y

(90, 0)(90, 0)

(0, 180)(0, 180)

2 180x y 2 180x y


200200

100100




3 300x y 3 300x y

100100 200200 300300xx

yy

3 300x y 3 300x y

(0, 100)(0, 100)

(300, 0)(300, 0)

3 300x y 3 300x y


200200

100100


We first We first graph the feasible setgraph the feasible set SS for the problem. for the problem.✦ Graph the Graph the intersectionintersection of the solutions to the inequalities, of the solutions to the inequalities,

yielding the yielding the feasible setfeasible set SS..(Note that the (Note that the feasible setfeasible set SS is is boundedbounded))

100100 200200 300300xx

yy

SS 3 300x y 3 300x y

2 180x y 2 180x y Applied Example 1, page 175

200200

100100


Next, find the Next, find the verticesvertices of the of the feasible setfeasible set SS. . ✦ The The verticesvertices are are AA(0, 0)(0, 0), , BB(90, 0)(90, 0), , CC(48, 84)(48, 84), and , and DD(0, 100)(0, 100)..

100100 200200 300300xx

yy

SS

CC(48, 84)(48, 84)

3 300x y 3 300x y

2 180x y 2 180x y

DD(0, 100)(0, 100)

BB(90, 0)(90, 0)AA(0, 0)(0, 0)


200200

100100


Now, find the Now, find the valuesvalues of of PP at the at the verticesvertices and and tabulate tabulate them:them:

100100 200200 300300xx

yy

SS

CC(48, 84)(48, 84)

3 300x y 3 300x y

2 180x y 2 180x y

DD(0, 100)(0, 100)

BB(90, 0)(90, 0)AA(0, 0)(0, 0)

VertexVertex PP = = x x + 1.2 + 1.2 yy

AA(0, 0)(0, 0) 00

BB(90, 0)(90, 0) 9090

CC(48, 84)(48, 84) 148.8148.8

DD(0, 100)(0, 100) 120120


200200

100100


Finally, Finally, identifyidentify the the vertexvertex with the with the highest valuehighest value for for PP::✦ We can see that We can see that PP is is maximizedmaximized at the vertex at the vertex CC(48, 84)(48, 84)

and has a value of and has a value of 148.8148.8..

100100 200200 300300xx

yy

SS 3 300x y 3 300x y

2 180x y 2 180x y

DD(0, 100)(0, 100)

BB(90, 0)(90, 0)AA(0, 0)(0, 0)

VertexVertex PP = = x x + 1.2 + 1.2 yy

AA(0, 0)(0, 0) 00

BB(90, 0)(90, 0) 9090

CC(48, 84)(48, 84) 148.8148.8

DD(0, 100)(0, 100) 120120

CC(48, 84)(48, 84)



Finally, Finally, identifyidentify the the vertexvertex with the with the highest valuehighest value for for PP::✦ We can see that We can see that PP is is maximizedmaximized at the vertex at the vertex CC(48, 84)(48, 84)

and has a value of and has a value of 148.8148.8..✦ Recalling what the symbols Recalling what the symbols xx, , yy, and , and PP represent, we represent, we

conclude that conclude that ACE NoveltyACE Novelty would would maximize its profitmaximize its profit at at $148.80$148.80 by producing by producing 4848 type-A type-A souvenirs and souvenirs and 8484 type-Btype-B souvenirs.souvenirs.



Recall Recall Applied Example 2Applied Example 2 from the from the last section (3.2) last section (3.2), which , which asked us to determine the asked us to determine the optimal combinationoptimal combination ofof pills pills to to be purchased in order to be purchased in order to meetmeet the minimum the minimum ironiron and and vitaminvitamin requirementsrequirements at the at the lowest costlowest cost..

We We restatedrestated the problem as a the problem as a linear programming problemlinear programming problem in which we wanted to in which we wanted to minimizeminimize the the objective functionobjective function


We can now We can now solve the problemsolve the problem graphically. graphically.

5 15 1500x y 5 15 1500x y

6 8C x y 6 8C x y

40 10 2400x y 40 10 2400x y

10 15 2100x y 10 15 2100x y

, 0x y , 0x y


200200

100100




100100 200200 300300xx

yy40 10 2400x y 40 10 2400x y

40 10 2400x y 40 10 2400x y

(60, 0)(60, 0)

(0, 240)(0, 240)


200200

100100




100100 200200 300300xx

yy

10 15 2100x y 10 15 2100x y

10 15 2100x y 10 15 2100x y

(210, 0)(210, 0)

(0, 140)(0, 140)


200200

100100




100100 200200 300300xx

yy

5 15 1500x y 5 15 1500x y

5 15 1500x y 5 15 1500x y

(300, 0)(300, 0)

(0, 100)(0, 100)


200200

100100


We first We first graph the feasible setgraph the feasible set SS for the problem. for the problem.✦ Graph the Graph the intersectionintersection of the solutions to the inequalities, of the solutions to the inequalities,

yielding the yielding the feasible setfeasible set SS..

(Note that the (Note that the feasible setfeasible set SS is is unboundedunbounded))

100100 200200 300300xx

yy

SS10 15 2100x y 10 15 2100x y

40 10 2400x y 40 10 2400x y

5 15 1500x y 5 15 1500x y


200200

100100


Next, find the Next, find the verticesvertices of the of the feasible setfeasible set SS. . ✦ The The verticesvertices are are AA(0, 240)(0, 240), , BB(30, 120)(30, 120), , CC(120, 60)(120, 60), and , and

DD(300, 0)(300, 0)..

100100 200200 300300xx

yy

SS10 15 2100x y 10 15 2100x y

40 10 2400x y 40 10 2400x y

5 15 1500x y 5 15 1500x y

CC(120, 60)(120, 60)

DD(300, 0)(300, 0)

AA(0, 240)(0, 240)

BB(30, 120)(30, 120)



Now, find the Now, find the valuesvalues of of CC at the at the verticesvertices and and tabulate tabulate them:them:

200200

100100

100100 200200 300300xx

yy

SS10 15 2100x y 10 15 2100x y

40 10 2400x y 40 10 2400x y

5 15 1500x y 5 15 1500x y

CC(120, 60)(120, 60)

DD(300, 0)(300, 0)

AA(0, 240)(0, 240)

BB(30, 120)(30, 120)

VertexVertex CC = 6 = 6x x + 8+ 8yy

AA(0, 240)(0, 240) 19201920

BB(30, 120)(30, 120) 11401140

CC(120, 60)(120, 60) 12001200

DD(300, 0)(300, 0) 18001800



Finally, Finally, identifyidentify the the vertexvertex with the with the lowest valuelowest value for for CC::✦ We can see that We can see that CC is is minimizedminimized at the vertex at the vertex BB(30, 120)(30, 120)

and has a value of and has a value of 11401140..

200200

100100

100100 200200 300300xx

yy

SS10 15 2100x y 10 15 2100x y

40 10 2400x y 40 10 2400x y

5 15 1500x y 5 15 1500x y

CC(120, 60)(120, 60)

DD(300, 0)(300, 0)

AA(0, 240)(0, 240)

VertexVertex CC = 6 = 6x x + 8+ 8yy

AA(0, 240)(0, 240) 19201920

BB(30, 120)(30, 120) 11401140

CC(120, 60)(120, 60) 12001200

DD(300, 0)(300, 0) 18001800

BB(30, 120)(30, 120)



Finally, Finally, identifyidentify the the vertexvertex with the with the lowest valuelowest value for for CC::✦ We can see that We can see that CC is is minimizedminimized at the vertex at the vertex BB(30, 120)(30, 120)

and has a value of and has a value of 11401140..✦ Recalling what the symbols Recalling what the symbols xx, , yy, and , and CC represent, we represent, we

conclude that the individual should conclude that the individual should purchasepurchase 3030 brand-A brand-A pills and pills and 120120 brand-Bbrand-B pills at a pills at a minimum costminimum cost of of $11.40$11.40..


3.43.4Sensitivity AnalysisSensitivity Analysis

Sensitivity AnalysisSensitivity Analysis

Sensitivity analysis consists of studying Sensitivity analysis consists of studying how significantlyhow significantly do do changes changes in thein the parameters parameters of a linear programming of a linear programming problem problem affect its optimal solutionaffect its optimal solution..

We shall apply this analysis to We shall apply this analysis to Applied Example 1Applied Example 1 that we that we discussed in the previous two sections (3.1 and 3.2).discussed in the previous two sections (3.1 and 3.2).


Recall that Recall that Applied Example 1Applied Example 1 required us to find the required us to find the optimal quantitiesoptimal quantities to produce of to produce of type-A type-A and and type-Btype-B souvenirs in order to souvenirs in order to maximize profitsmaximize profits..

In In section 3.2section 3.2 we restated the problem as a we restated the problem as a linear linear programming problemprogramming problem in which we wanted to in which we wanted to maximizemaximize the the objective functionobjective function


0

0

x

y

0

0

x

y

1.2P x y 1.2P x y

3 300x y 3 300x y

2 180x y 2 180x y

Changes in the Coefficients of the Objective Function, page 187


In In section 3.3section 3.3, we found that , we found that PP is is maximizedmaximized at the vertex at the vertex CC(48, 84)(48, 84) and has a value of and has a value of 148.8148.8..

Thus, Thus, ACE NoveltyACE Novelty maximizes its profitmaximizes its profit at at $148.80$148.80 by by producing producing 4848 type-A type-A souvenirs and souvenirs and 8484 type-Btype-B souvenirs. souvenirs.

200200

100100

100100 200200 300300xx

yy

SS 3 300x y 3 300x y

2 180x y 2 180x y

DD(0, 100)(0, 100)

BB(90, 0)(90, 0)AA(0, 0)(0, 0)

CC(48, 84)(48, 84)


How do How do changes in the coefficientschanges in the coefficients of the of the objective functionobjective function affect the affect the optimal solutionoptimal solution??✦ The objective function isThe objective function is

✦ The The coefficientcoefficient of of x x is is 11, which tells us that the , which tells us that the contribution contribution to theto the profit profit for each for each type-Atype-A souvenir is souvenir is $1.00$1.00..

✦ We want to know by how much this We want to know by how much this coefficientcoefficient can can changechange without changingwithout changing the the optimal solutionoptimal solution..

✦ Suppose the contribution to the profit of each Suppose the contribution to the profit of each type-A type-A souvenir is souvenir is $$cc, so that, so that

✦ We need to determine the We need to determine the range of valuesrange of values for which the for which the solutionsolution we determined we determined remains optimalremains optimal..

1.2P x y 1.2P x y

1.2P cx y 1.2P cx y


How do How do changes in the coefficientschanges in the coefficients of the of the objective functionobjective function affect the affect the optimal solutionoptimal solution??✦ We can We can rewriterewrite the profit equation (the the profit equation (the isoprofit lineisoprofit line) )

containing the containing the cc coefficient in coefficient in slope-intercept formslope-intercept form: :

✦ The The slopeslope of the of the isoprofit lineisoprofit line is is ––cc/1.2/1.2..

1.2 1.2

c Py x

1.2 1.2

c Py x


How do How do changes in the coefficientschanges in the coefficients of the of the objective functionobjective function affect the affect the optimal solutionoptimal solution??✦ If the If the slopeslope of the of the isoprofit lineisoprofit line, , ––cc/1.2/1.2, , exceedsexceeds that of the that of the

lineline associated with associated with constraint 2constraint 2, then the , then the optimaloptimal solutionsolution shifts from point shifts from point CC to point to point DD::

150 150

50 50

100100 200200 300300xx

yy

3 300x y 3 300x y

2 180x y 2 180x y

BB

Constraint 1:Constraint 1:


(slope = –2)(slope = –2)

(slope = –1/3)(slope = –1/3)

Isoprofit lineIsoprofit line

(slope = (slope = ––5/6)5/6) CCIsoprofit lineIsoprofit line

(slope > (slope > ––1/3)1/3) SS

AA

DD


How do How do changes in the coefficientschanges in the coefficients of the of the objective functionobjective function affect the affect the optimal solutionoptimal solution??✦ If the If the slopeslope of the of the isoprofit lineisoprofit line, , ––cc/1.2/1.2, is, is less than less than that of that of

the the lineline associated with associated with constraint 1constraint 1, then the , then the optimaloptimal solutionsolution shifts from point shifts from point CC to point to point BB::

150 150

50 50

100100 200200 300300xx

yy

SS

2 180x y 2 180x y

BB

AA

CC


(slope = –2)(slope = –2)

3 300x y 3 300x y Constraint 2:Constraint 2:

(slope = –1/3)(slope = –1/3)


(slope = (slope = ––5/6)5/6)DD


(slope < (slope < ––2)2)


How do How do changes in the coefficientschanges in the coefficients of the of the objective functionobjective function affect the affect the optimal solutionoptimal solution??✦ Thus, as long as Thus, as long as 0.4 0.4 c c 2.4 2.4, the , the optimal solutionoptimal solution of of

CC(48, 84)(48, 84) remains unaffectedremains unaffected..

✦ In other words, as long as the In other words, as long as the contribution to the profitcontribution to the profit of of type-A type-A souvenirs lies between souvenirs lies between $.40$.40 and and $2.40$2.40, Ace , Ace Novelty should still make Novelty should still make 4040 type-A type-A souvenirs and souvenirs and 8484 type-Btype-B souvenirs. souvenirs.

✦ This demonstrates that the This demonstrates that the parameterparameter cc in this problem in this problem is is not sensitivenot sensitive..

✦ A A similarsimilar sensitivity analysissensitivity analysis can be conducted regarding can be conducted regarding the contribution to the profit of the the contribution to the profit of the type-B type-B souvenir.souvenir.

End of End of Chapter Chapter

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