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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
satisfy my daily nutritional requirements at minimum cost. 16
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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
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
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
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, 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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