03 linear programming

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Operations Research

Introduction toLinear Programming

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Linear Programming

A linear programming (LP) is a tool for solvingoptimization problems.The founders of the subject are Leonid Kantorovich, aRussian mathematician who developed linearprogramming problems in 1939.

George Dantzig (1947) developed an efficientmethod, simplex algorithm for solving LP.

Since the development of the simplex algorithm, LPproven to be one of the most effective operationsresearch tools.

LP has been used to solve optimization problems inthe following areas: military, industry, agriculture,transportation, economics, health systems, and evenbehavioral and social sciences.

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Linear Programming

A linear programming problem (LP) is anoptimization problem for which wedo thefollowing:

1. We attempt to maximize (or minimize) a linearfunction(called the objective function) of the

decision variables.2. The values of the decision variables must satisfy a set

of constraints. Each constraint must be a linearequation or linear inequality.

3. A sign restriction is associated with each variable. Forany variablexi, the sign restriction specifies thatximust be either nonnegative (xi 0) or unrestricted insign (URS).

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Concept of Linear Function and Linear

Inequality

For example,f(x1,x2) = 2x1+ x2is a linear function ofx1andx2,

butf(x1,x2) =x1x2is not a linear function ofx1andx22

Thus, 2x1+ 3x2 3 and 2x1+ x2 3 are linear inequalities,

Butx1x2 3is not a linear inequality2

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Example 1: GiapettosWoodcarving

GiapettosWoodcarving, Inc., manufactures twotypes of wooden toys: soldiers and trains.

Each soldier built: Sells for $27 and uses $10 worth of raw materials.

Labor and overhead costs by $14.

A soldier requires 2 hours of finishing labor and 1 hour ofcarpentry labor.

Each train built: Sells for $21 and uses $9 worth of raw materials.

Labor and overhead costs by $10.

A train requires 1 hour of finishing labor and 1 hour ofcarpentry labor.

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Example 1 continued

Each week, Giapetto can obtain all the neededraw material but only 100 finishing hours and 80carpentry hours.

Demand for trains is unlimited, but at most 40

soldiers are bought each week.Giapetto wants to maximize weekly profit(revenues - costs).

Formulate a mathematical model of Giapettossituation that can be used to maximize Giapettos

weekly profit.

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Example 1 continued

In developing the Giapetto model, we explore characteristics

shared by all linear programming problems.

Decision Variablesx1 = number of soldiers produced each week

x2 = number of trains produced each week

Objective Function

Giapettosweekly revenues and costs can be expressed in terms ofthe decision variablesx1andx2

Giapettosobjective function is:

Maximizez = 3x1+ 2x2

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Example 1 continued

Constraints:Each week, no more than 100 hours of finishing

time may be used.

2x1

+x2

100

Each week, no more than 80 hours of carpentry

time may be used.

x1+x2 80

Because of limited demand, at most 40 soldiersshould be produced.

x1 40

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Nonnegativity/URS

To complete the formulation of a linear programmingproblem, the following question must be answered for eachdecision variable:

Can the decision variable only assume nonnegative

values, or is the decision variable allowed to assume bothpositive and negative values?

oIf a decision variablexican only assume nonnegative values,then we add the sign restrictionxi 0.

oIf a variablexi can assume both positive and negative (orzero) values,then we say thatxiisunrestricted in sign (oftenabbreviated urs).

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Example 1 continued

Complete optimization model for GiapettosWoodcarving:

Maximizez= 3x1+ 2x2

Subject to (s.t.)2x1+x2 100

x1+x2< 80

x1 40

x1, x2 0

(objective function)

(finishing constraint)

(carpentry constraint)

(constraint on demand for soldiers)

(sign restriction)

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Assumptions

The fact that the objective function for an LP must bea linear function of the decision variables has twoimplications:

The contribution of the objective function from each

decision variable is proportional to the value of thedecision variable. For example, the contribution to theobjective function for 4 soldiers is exactly four times thecontribution of 1 soldier.

The contribution to the objective functionfor anyvariable is independent of the other decision variables.For example, no matter what the value ofx2,themanufacture ofx1soldiers will always contribute 3x1dollars to the objective function.

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Assumptions

Analogously, the fact that each LP constraint must bea linear inequality or linear equation has twoimplications:

The contribution of each variable to the left-hand side of

each constraint is proportional to the value of thevariable. For example, it takes exactly 3 times as manyfinishing hours to manufacture 3 soldiers as it does 1soldier.

The contribution of a variable to the left-hand side ofeach constraint is independent of the values of thevariable. For example, no matter what the value ofx1, themanufacture ofx2trains usesx2finishing hours andx2carpentry hours.

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Assumptions

The first item in each list is called theProportionality Assumption of Linear

Programming.

The second item in each list is called theAdditivity Assumption of Linear

Programming.

The divisibility assumption requires that

each decision variable be permitted to

assume fractional values.

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Assumptions

The certainty assumption is that eachparameter (objective function coefficients,right hand side, and technologicalcoefficients) are known with certainty.

The feasible region of an LP is the set of allpoints satisfying all the LPs constraints andsign restrictions.

For a maximization (minimization) problem,an optimal solution to an LP is a point in thefeasible region with the largest (smallest)objective function value.

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Example 2: Diet Problem

My diet requires that all the food I get comefrom one of the four basic food groups.

At present, the following four foods are available

for consumption: brownies, chocolate ice cream,cola and pineapple cheesecake.

Each brownie costs 50, each scoop of ice cream

costs 20 , each bottle of cola costs 30 , and

each piece of pineapple cheesecake costs 80 .

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Example 2 continued

Each day, I need at least 500 calories, 6 oz of chocolate, 10

oz of sugar, and 8 oz of fat.The nutritional content per unit of each food as follows:

Formulate a linear programming model that can be used to

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Example 2 continued

Determining the decisions that must be made

by the decision maker: how much of each type

of food should be eaten daily.

Decision variables:

x1 = number of brownies eaten daily

x2 = number of scoops of chocolate ice cream

eaten daily

x3 = bottles of cola drunk daily

x4 = pieces of pineapple cheesecake eaten daily

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Example 2 continued

Objective is to minimize the cost of diet. The total

cost of any diet may be determined from the

following relation:

(total cost of diet) = (cost of brownies) + (cost of ice

cream) + (cost of cola) + (cost of

cheesecake)

Min 50x1+ 20x2+ 30x3+ 80x4

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Example 2 continued

The decision variables must satisfy the following constraints +nonnegativity (xi 0, i=1,2,3,4)

Daily calorie intake must at least 500 calories

400x1+ 200x2+ 150x3+ 500x4 500Daily chocolate intake must be at least 6 oz.

3x1+ 2x2 6

Daily sugar intake must be at least 10 oz.

2x1+ 2x2+ 4x3+ 4x4 10

Daily fat intake must be at least 8 oz.2x1+ 4x2+x3+ 5x4 819


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Example 3: Work- Scheduling Problem

Many applications of LP involve determining theminimum-cost method for satisfying workforcerequirements.

One type of work scheduling problem is a staticscheduling problem.

In reality, demands change over time, workers

take vacations in the summer, and so on, so thepost office does not face the same situationeach week. This is a dynamic schedulingproblem.

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Example 3 continued

A post office requires different

numbers of full-time employeeson different days of the week.

Union rules state that each

fulltime employee must five

consecutive days and then twodays off.

The post office wants to meet its

daily requirements using only full-

time employees.

Formulate an LP that the post

office can use to minimize the

number of full time employees

who must be hired.

Day # Required

1: Monday 17

2: Tuesday 13

3: Wednesday 15

4: Thursday 19

5: Friday 14

6: Saturday 16

7: Sunday 11

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Example 3 continued

Letxibe the number of employees working on day i.

Minimizez=x1+x2 + x3 + x4+x5 + x6 + x7Subject to (s.t.)

x1 17

x2 13x3 15

x4 19

x5 14

x6 16

x7 11

xi 0, i= 1,2,..,7

Wrong constraints, currentobjective function counts each

Employee five times, not once.

Interrelated variables, working

consecutively

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Example 3 continuedLetxibe the number of employees working on day i.

Minimizez=x1+x2 + x3 + x4+x5 + x6 + x7Subject to (s.t.)

x1 +x4+x5 + x6 + x7 17x1+x2 +x5 + x6 + x7 13

x1+x2 + x3 + x6 + x7 15

x1+x2 + x3 + x4 + x7 19

x1+x2 + x3 + x4+x5 14

x2 + x3 + x4+x5 + x6 16

x3 + x4+x5 + x6 + x7 11

xi 0, i= 1,2,..,7

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Example 3 continued

The optimal solution is x1=4/3, x

2=10/3, x

3=2, x

4=22/3, x

5=0,

x6=10/3, x

7=5

z= 67/3

Minimizez=x1+x2 + x3 + x4+x5 + x6 + x7

Subject to (s.t.)

x1 +x4+x5 + x6 + x7 17

x1+x2 +x5 + x6 + x7 13

x1+x2 + x3 + x6 + x7 15x1+x2 + x3 + x4 + x7 19

x1+x2 + x3 + x4+x5 14

x2 + x3 + x4+x5 + x6 16

x3 + x4+x5 + x6 + x7 11

xi 0, i= 1,2,..,7

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Example 3 continued

If we round up the solution of the LP: x1=2, x

2=4, x

3=2, x

4=4,

x5=0, x

6=4, x

7=5

z= 25


Subject to (s.t.)

x1 +x4+x5 + x6 + x7 17

x1+x2 +x5 + x6 + x7 13

x1+x2 + x3 + x6 + x7 15x1+x2 + x3 + x4 + x7 19

x1+x2 + x3 + x4+x5 14

x2 + x3 + x4+x5 + x6 16

x3 + x4+x5 + x6 + x7 11

xi 0, i= 1,2,..,7

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Example 3 continued

The optimal the solution of the LP: x1=1, x

2=4, x

3=2, x

4=8, x

5=0,

x6=3, x

7=5

z= 23


Subject to (s.t.)

x1 +x4+x5 + x6 + x7 17

x1+x2 +x5 + x6 + x7 13

x1+x2 + x3 + x6 + x7 15x1+x2 + x3 + x4 + x7 19

x1+x2 + x3 + x4+x5 14

x2 + x3 + x4+x5 + x6 16

x3 + x4+x5 + x6 + x7 11

xi 0, and integer i= 1,2,..,7

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Works Scheduling Problem LINGO Model

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Works Scheduling Problem LINGO Model

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Works Scheduling Problem Excel Solver Model

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Works Scheduling Problem Excel Solver Model

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References

Operations Research: Applications and Algorithms,

Wayne L. Winston

Introduction to Operations Research, Hillier & Lieberman,

McGraw-Hill Int.

Operations ResearchAn Introduction, Hamdy A. Taha,Maxwell Macmillan Int. Edition.

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03 linear programming

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