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Linear Programming

Developed by Dantzig in the late 1940s

A mathematical method of allocating scarce resources

to achieve a single objective

The objective may be profit, cost, return on investment,

sales, market share, space, time

LP is used for production planning, capital budgeting,

manpower scheduling, gasoline blending

By companies such as AMD, New York Life, Chevron,

Ford


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Linear Programming

Decision Variables

symbols used to represent an item that can take onany value (e.g., x1=labor hours, x2=# of workers)

Parametersknown constant values that are defined for each

problem (e.g., price of a unit, production capacity)

Decision Variables and Parameters will bedefined for each unique problem


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Linear Programming

A method for solving linear mathematical models

Linear Functions

f(x) = 5x + 1g(x1, x2) = x1 + x2

Non-linear functions

f(x) = 5x2 + 1

g(x1, x2) = x1x2 + x2


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Formulating an LP

Define the decision variables

identifying the key variables whose values we wish

to determine

Determine the objective function

determine what we are trying to do

Maximize profit, Minimize total cost

Formulate the constraintsdetermine the limitations of the decision variables


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3 Parts of a Linear Program

Objective Function

Constraints

Non-negativity assumptions


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Objective Function

linear relationships of decision variables describing

the problems objectivealways consists of maximizing or minimizing some

value

e.g., maximize Z = profit, minimize Z = cost


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Constraints

linear relationships of decision variables

representing restrictions or rules

e.g., limited resources like labor or capital

Non-negativity assumptions

restricts decision variables to values greater than or

equal to zero


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Objective Function

Always a Max or Min statement

Maximize Profit

Minimize Cost A linear function of decision variables

Maximize Profit = Z = 3x1 + 5x2

Minimize Cost = Z = 6x1 - 15x2


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Constraints

Constraints are restrictions on the problem

total labor hours must be less than 50

x1 must use less than 20 gallons of additive Defined as linear relationships

total labor hours must be less than 50

x1 + x2 < 50

x1 must use less than 20 gallons of additive

x1 < 20


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Non-negativity Assumptions

Negative decision variables are inconceivable in

most LP problems

Minus 10 units of production or a negativeconsumption make no sense

The assumptions are written as follows

x1, x2 > 0


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LP Standard Form

Each LP must contain the three parts

A standard form for a LP is as follows:

Maximize Z = c1x1 + c2x2 + ... + cnxnsubject to a11x1 + a12x2 + ... + a1nxn < b1

.

am1x1 + am2x2 + ... + amnxn < bm

x1, x2, xn > 0


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Methods of Solving LP Problems

Two basic solution approaches of linear

programming exist

The graphical Methodsimple, but limited to two decision variables

The simplex method

more complex, but solves multiple decision variable

problems


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Graphical Method

1. Construct an x-y coordinate plane/graph

2. Plot all constraints on the plane/graph

3. Identify the feasible region dictated by the constraints

4. Identify the optimum solution by plotting a series of

objective functions over the feasible region

5. Determine the exact solution values of the decision

variables and the objective function at the optimumsolution


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Production Planning: Given several products with varying

production requirements and cost structures, determine howmuch of each product to produce in order to maximize profits.

Scheduling: Given a staff of people, determine an optimalwork schedule that maximizes worker preferences while

adhering to scheduling rules.Portfolio Management: Determine bond portfolios thatmaximize expected return subject to constraints on risk levelsand diversification.

And an incredible number more.

Some Applications of LPs + IPs


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Graphing 2-Dimensional LPs

Example 2:

FeasibleRegion

x0 y

0

-2 x + 2 y 4x 3

Subject to:

Minimize ** x - y

MultipleOptimal

Solutions!4

1

x31 2

y

0

2

0

3

1/3 x + y 4


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Example 3:

FeasibleRegion

x

0 y

0

x + y 20

x 5-2 x + 5 y 150

Subject to:

Minimize x + 1/3 y

OptimalSolution

x

3010 20

y

0

10

20

40

0

30

40


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y

x0

1

2

3

4

0 1 2

3

x3010 20

y

0

10

20

40

0

30

40

Do We Notice Anything From These3 Examples?

x

y

0

1

2

3

4

0 1 2

3


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A Fundamental Point

If an optimal solution exists, there is always acorner point optimal solution!

y

x0

1

2

3

4

0 1 2

3

x3010 20

y

0

10

20

40

0

30

40x

y

0

1

2

3

4

0 1 2

3


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Example 1:

x30 1 2

y

0

1

2

4

3

FeasibleRegion

x

0 y

0

x + 2 y 2

y 4x 3

Subject to:

Maximize x + y

OptimalSolution

Initial Cornerpt.

SecondCorner pt.

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