2 introduction to lp

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Ahad Ali, Ph.D.

Lawrence Tech University

Fall 2011

EME 5123: Optimization of Manufacturing Systems

Introduction to Linear

Programming (LP)



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Introduction to LP

Linear Programming (LP) is a technique for optimization of a linear objective function,

subject to linear equality and linear inequality

constraints.

Informally, linear programming determines the

way to achieve the best outcome (such as

maximum profit or lowest cost) in a givenmathematical model and given some list of

requirements represented as linear equations.



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Introduction to LP

A Linear Programming model seeks tomaximize or minimize a linear function,

subject to a set of linear constraints.

The linear model consists of the following

components:

± A set of decision variables.

± An objective function. ± A set of constraints.



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Introduction to LP

The Importance of Linear Programming ±Many real world problems lend themselves to

linear programming modeling.

±Many real world problems can be approximated

by linear models.

±There are well-known successful applications: Manufacturing

Marketing

Finance (investment)

Advertising

Agriculture



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Introduction to LP

The Importance of Linear Programming

±There are efficient solution techniques that

solve linear programming models.

±The output generated from linear programming

packages provides useful ´what if µ analysis.



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Introduction To Linear Programming

Today many of the resources needed asinputs to operations are in limited supply.

Operations managers must understand the

impact of this situation on meeting their objectives.

Linear programming (LP) is one way thatoperations managers can determine how best

to allocate their scarce resources.



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Linear Programming (LP)

There are five common types of decisions inwhich LP may play a role

± Product mix

± Production plan

± Ingredient mix

± Transportation

± Assignment



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LP Problems: Product Mix

ObjectiveTo select the mix of products or services thatresults in maximum profits for the planningperiod

Decision VariablesHow much to produce and market of eachproduct or service for the planning period

ConstraintsMaximum amount of each product or servicedemanded; Minimum amount of product or service policy will allow; Maximum amount of resources available



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LP Problems: Production Plan

ObjectiveTo select the mix of products or services thatresults in maximum profits for the planningperiod

Decision VariablesHow much to produce on straight-time labor and overtime labor during each month of theyear

ConstraintsAmount of products demanded in each month;Maximum labor and machine capacityavailable in each month; Maximum inventory

space available in each month



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Recognizing LP Problems

Characteristics of LP Problems A well-defined single objective must be stated.

There must be alternative courses of action.

The total achievement of the objective must be

constrained by scarce resources or other

restraints.

The objective and each of the constraintsmust be expressed as linear mathematical

functions.



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Steps in Formulating LP Problems

1. Define the objective. (min or max)2. Define the decision variables. (positive, binary)

3. Write the mathematical function for the objective.

4. Write a 1- or 2-word description of each constraint.

5. Write the right-hand side (RHS) of each constraint.

6. Write <, =, or > for each constraint.

7. Write the decision variables on LHS of eachconstraint.

8. Write the coefficient for each decision variable in

each constraint.



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Introduction to LP

Assumptions of the linear programming model ±The parameter values are known with certainty .

±The objective function and constraints exhibit

constant returns to scale.

±There are no interactions between the decision

variables (the additivity assumption).

±The C ontinuity assumption: Variables can take

on any value within a given feasible range.



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Introduction to Linear ProgrammingIntroduction to Linear Programming

A Standard Form of LP:Objective Function:

M a x z = c1x1 + c2x2 + + cnxn

Subject tos.t. a11x1 + a12x2 + + a1nxn e b1

a21x1 + a22x2 + + a2nxn e b2

am1x1 + am2x2 + + amnxn e bm

Sign restriction

xi u 0 i = 1,2 ««.n

X = Decision VariablesC = Constant

n = No. of decision variables

--- columns

m = Constraints -- rows



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Introduction to Linear ProgrammingIntroduction to Linear Programming

Or in matrix form:

max z = cx

Subject to

Ax e b

Sign restriction

x u 0

Matrix tableau form:

x1

X = x2

.

xn

a11 + a12 + + a1n

a21 + a22 + + a2n

A =

am1 + am2 + + amn

b1

b = b2

bm

xi u 0 i = 1, 2, «, n



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Example 1: Giapetto¶s Woodcarving

Giapetto¶s, Inc., manufactures wooden soldiers andtrains.

± Each soldier built:

Sell for $27 and uses $10 worth of raw materials.

Increase Giapetto¶s variable labor/overhead costs by $14. Requires 2 hours of finishing labor. Requires 1 hour of carpentry labor.

± Each train built:

Sell for $21 and used $9 worth of raw materials. Increases Giapetto¶s variable labor/overhead costs by $10. Requires 1 hour of finishing labor. Requires 1 hour of carpentry labor.



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Ex. 1 - continued

Each week Giapetto can obtain: ± All needed raw material. ± Only 100 finishing hours. ± Only 80 carpentry hours.

Demand for the trains is unlimited. At most 40 soldiers are bought each week.

Giapetto wants to maximize weekly profit(revenues ± costs).

Formulate a mathematical model of Giapetto¶ssituation that can be used to maximize weeklyprofit.



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Ex. 1 - Formulation

Identify Decision Variables

Identify Objective Function

Identify Constraints

Formulate Optimization Model/ Mathematical

Model



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Ex. 1 - Formulation continued

Decision Variables:

Giapetto must decide how many soldiers and

trains should be manufactured each week.

x1 = number of soldiers produced each week.

x2 = number of trains produced each week



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Ex. 1 - Formulation continue

The Giapetto solution model incorporates the

characteristics shared by all linear programmingproblems.

± Decision variables should completely describe thedecisions to be made.

x1 = number of soldiers produced each week x2 = number of trains produced each week

± The decision maker wants to maximize (usuallyrevenue or profit) or minimize (usually costs) some

function of the decision variables. This function tomaximized or minimized is called the objectivefunction.

For the Giapetto problem, fixed costs do notdepend upon the the values of x1 or x2.



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Objective Function:weekly revenues and costs can be expressed in

terms of decision variables x1 and x2

Weekly revenues = weekly revenue from soldiers + revenue from trains

= 27x1 + 21x2

Weekly raw material cost = 10x1 + 9x2

Weekly variable costs = 14x1 + 10x2

Gispetto wants to maximize profit (revenue ± costs)

(27x1 + 21x2) ± (10x1 + 9x2) ± (14x1 + 10x2) = = 3x1 + 2x2

Giapetto¶s objective is to choose x1 and x2 to maximum 3x1 + 2x2.

Maximize z = 3x1 + 2x2

The coefficient of an objective function variable

is called an objective function coefficient.



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As x

1 and x

2 increase, Giapetto¶s objectivefunction grows larger.

For Giapetto, the values of x 1 and x 2 are

limited by the following three restrictions

(often called constraints):

± Each week, no more than 100 hours of finishing

time may be used.

± Each week, no more than 80 hours of carpentrytime may be used.

± Because of limited demand, at most 40 soldiers

should be produced.



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Constraints:Constraint 1: Each week no more than 100 hours of finishingtime may be used:

2 x 1 + x 2 e 100

Constraint 2: Each week no more than 100 hours of finishing

time may be used: x 1 + x 2 e 80

Constraint 3: Because of limited demand, at most 40soldiers should be produced:

x 1e

40Sign restriction: xi can assume non-negative value

x 1 u 0

x 2 u 0



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Optimization Model:

Max z = 3 x 1 + 2 x 2 (Objective Function)

Subject to (s.t.)

2x1 + x2 e 100 (Finishing constraint)

x1 + x2 e 80 (Carpentry constraint)

x1 e 40 (Constraint on demand for soldiers)

x1 u 0 (Sign restriction)

x2 u 0 (Sign restriction)

2 introduction to lp

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