1introduction to linear programminglesson 2 introduction to linear programming

1 Introduction to Linear Programming Lesson 2

Introduction to Introduction to Linear Linear

ProgrammingProgramming


A Linear A Linear Programming model Programming model seeks to maximize seeks to maximize or minimize a linear or minimize a linear function, subject to function, subject to a set of linear a set of linear constraints.constraints.


The linear model consists The linear model consists

of the following of the following

components:components:

A set of decision A set of decision variables.variables.

An objective function.An objective function. A set of constraints.A set of constraints.


Many problems lend Many problems lend themselves to linear themselves to linear

programming formulations.programming formulations.

Many problems can be Many problems can be approximated by linear models.approximated by linear models.

The output generated by linear The output generated by linear programs provides useful programs provides useful “what-if” information.“what-if” information.


Galaxy manufactures Galaxy manufactures two toy models:two toy models:

Space Ray. Space Ray. Zapper. Zapper.


Resources are limited toResources are limited to

1000 pounds of special 1000 pounds of special plastic.plastic.

40 hours of production 40 hours of production time per week.time per week.


Marketing requirementMarketing requirement

Total production cannot Total production cannot exceed 700 dozens.exceed 700 dozens.

Number of dozens of Number of dozens of

Space Rays cannot exceed Space Rays cannot exceed

number of dozens of number of dozens of

Zappers by more than 350.Zappers by more than 350.


Technological inputTechnological inputSpace Rays requires 2 Space Rays requires 2 pounds of plastic and pounds of plastic and 3 minutes of labor per 3 minutes of labor per dozen.dozen.

Zappers requires 1 Zappers requires 1 pound of plastic and 4 pound of plastic and 4 minutes of labor per minutes of labor per dozen.dozen.


Current production plan calls Current production plan calls for: for:

Producing as much as Producing as much as possible of the more possible of the more profitable product, Space profitable product, Space Ray ($8 profit per dozen).Ray ($8 profit per dozen).

Use resources left over to Use resources left over to produce Zappers ($5 profit produce Zappers ($5 profit per dozen).per dozen).


The current production plan The current production plan consists of:consists of:

Space Rays = 450 dozensSpace Rays = 450 dozens

Zapper = 100 dozensZapper = 100 dozens

Profit = 4100 dollars per Profit = 4100 dollars per weekweek


A Linear Programming Model A Linear Programming Model

can provide an intelligent can provide an intelligent

solution to this problemsolution to this problem


Decisions variables:Decisions variables:X1 = Production level of Space Rays X1 = Production level of Space Rays (in dozens per week).(in dozens per week).

X2 = Production level of Zappers (in X2 = Production level of Zappers (in dozens per week).dozens per week).

Objective Function:Objective Function:

Weekly profit, to be maximizedWeekly profit, to be maximized


Max 8XMax 8X11 + 5X + 5X22 (Weekly profit)(Weekly profit)subject tosubject to2X2X11 + 1X + 1X22 < = 1000 (Plastic) < = 1000 (Plastic)3X3X11 + 4X + 4X22 < = 2400 (Production < = 2400 (Production Time)Time) XX11 + X + X22 < = 700 (Total < = 700 (Total production)production) XX11 - X - X22 < = 350 (Mix) < = 350 (Mix) XXjj> = 0, j = 1,2> = 0, j = 1,2 (Nonnegativity)(Nonnegativity)


The set of all points that satisfy all the constraints

of the model is called

FEASIBLE FEASIBLE REGIONREGION


Using a graphical Using a graphical presentation presentation

we can represent all the we can represent all the constraints, constraints,

the objective function, and the objective function, and the three the three

types of feasible points.types of feasible points.

1000

500

The Plastic constraint

Feasible

The plastic constraint: 2X1+X2<=1000

X2

Infeasible

Production Time3X1+4X2<=2400

Total production constraint: X1+X2<=700

600

700

Production mix constraint:X1-X2<=350

• There are three types of feasible points

Interior points.Boundary points. Extreme points.

X1

700

800

Solving Graphically for an Optimal SolutionSolving Graphically for an Optimal Solution

Recall the feasible Region

600600

800800

12001200

400400 600600 800800

X2X2

X1X1

We now demonstrate the search for an optimal We now demonstrate the search for an optimal solution:solution: Start at some arbitrary profit, say profit = $2,000...Start at some arbitrary profit, say profit = $2,000...

Profit = $ 000 2,

Then increase the profit, if possible...Then increase the profit, if possible...

3,4,

...and continue until it becomes infeasible...and continue until it becomes infeasible

Profit =$4360

600

800

1200

400 600 800

X2

X1

Let’s take a closer look at Let’s take a closer look at the optimal pointthe optimal point

FeasibleregionFeasibleregion

InfeasibleInfeasible


Space Rays = 320 dozensSpace Rays = 320 dozens

Zappers = 360 dozensZappers = 360 dozens

ProfitProfit = $4360 = $4360

Why is this production schedule Why is this production schedule is able to attain better profit is able to attain better profit than the original, considering than the original, considering that there was no increase in that there was no increase in any input resources?any input resources?


Extreme Points and Optimal SolutionsExtreme Points and Optimal Solutions

If a linear programming problem has an If a linear programming problem has an optimal solution, an extreme point is optimal solution, an extreme point is optimal.optimal.

However, not all extreme points are However, not all extreme points are optimaloptimal

Multiple Optimal SolutionsMultiple Optimal Solutions

For multiple optimal solutions to exist, For multiple optimal solutions to exist, the objective function must be parallel to the objective function must be parallel to a part of the feasible region.a part of the feasible region.


The optimal solution will remain The optimal solution will remain unchanged as long as unchanged as long as

An objective function coefficient An objective function coefficient lies within its range of lies within its range of optimality optimality

There are no changes in any There are no changes in any other input parameters. other input parameters.


The value of the The value of the objective function objective function will change if the will change if the coefficient multiplies coefficient multiplies a variable whose a variable whose value is nonzero.value is nonzero.

600

800

1200

400 600 800

X2

X1

The effects of changes in an objective function The effects of changes in an objective function coefficient on the optimal solutioncoefficient on the optimal solution

Max 8x1 + 5x2

Max 4x1 + 5x2Max 3.75x1 + 5x2 Max 2x1 + 5x2

600

800

1200

400 600 800

X2

X1

The effects of changes in an objective The effects of changes in an objective function coefficients on the optimal function coefficients on the optimal solutionsolution

Max8x1 + 5x2

Max 3.75x1 + 5x2

Max8x1 + 5x2

Max 3.75 x1 + 5x2M

ax 10 x1 + 5x23.75

10

Range of optimality

1introduction to linear programminglesson 2 introduction to linear programming

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set of linear constraints