Linear Programming - mcclenahan.info

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1 Winston, Wayne L., Operations Research Applications and Algorithms, 3rd ed. Duxbury Press ... The following example is taken from Operations Research.
Name __________________________________ Period __________ Date:

Essential Question: What is an “objective function”?

Linear Programming Topic:

Standard: A_REI.12 Objective:

Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half planes.

Learn to solve real-world problems using systems of linear inequalities. The following example is taken from Operations Research Applications and Algorithms by Wayne L. Winston.1 Giapetto’s Woodcarving, inc., manufactures two types of wooden toys: soldiers and trains. A soldier sells for $27 and uses $10 worth of raw materials. Each soldier that is manufactured increases Giapetto’s variable overhead and labor costs by $14. A train sells for $21 and uses $9 worth of raw materials. Each train built increases Giapetto’s variable overhead and labor costs by $10. The manufacture of wooden soldiers and trains requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing labor and 1 hour of carpentry labor. A train requires 1 hour of finishing labor and 1 hour of carpentry labor. Each week, Giapetto can obtain all the needed raw material but only 100 finishing hours and 80 carpentry hours. Demand for trains is unlimited, but at most 40 soldiers are bought each week. Giapetto wants to maximize weekly profits (revenues  costs). Formulate a mathematical model of Giapetto’s situation that can be used to maximize Giapetto’s weekly profit.

Summary

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Winston, Wayne L., Operations Research Applications and Algorithms, 3rd ed. Duxbury Press, Belmont, CA (1994)

Step 1:

Giapetto’s wants to determine the number of soldiers and trains that will maximize their weekly profit.

Step 2:

In step 2 we define the decision variables. The decision variables should completely describe the decisions to be made. With this in mind, Let

s = the number of soldiers produced each week.

Let

t = the number of trains produced each week.

The revenue from soldiers is (dollars/soldier)·(soldiers/week)

The revenue from trains is (dollars/train)·(trains/week)

The total revenue is the sum of these two.

The weekly raw material cost from soldiers is (cost/soldier)·(soldiers/week)

The weekly raw material cost from trains is (cost/train)·(trains/week)

The total raw material cost is the sum of these two.

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The weekly variable labor and overhead cost from soldiers is (cost/soldier)·(soldiers/week)

The weekly variable labor and overhead cost from trains is (cost/train)·(trains/week)

The total variable labor and overhead cost is the sum of these two.

Step 3:

The weekly profit is the weekly revenue less the weekly, variable costs. (

)

(

)

By combining like terms we find

Objective This is called the objective function, and Giapetto’s wants to Function maximize this function in order to maximize profits. The coefficients of the variables are the contributions of the variables to the company’s profits.

We must maximize the objective function subject to the constraints. Constraints are mathematical representations of the conditions that limit the company’s options. Constraints arise from the fact that resources are often limited. In this example, the company cannot use more than 80 carpentry hours per week or 100 finishing hours per week.

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The number of carpentry hours used per week for soldiers is, (carpentry hours/soldier)·(soldiers/week)

The number of carpentry hours used per week for trains is, (carpentry hours/train)·(trains/week)

The total carpentry hours used per week is the sum of these two. The constraint due to the carpentry hours is, Contsraint 1: The number of finishing hours used per week for soldiers is, (finishing hours/soldier)·(soldiers/week)

The number of finishing hours used per week for trains is, (finishing hours/train)·(trains/week)

The total finishing hours used per week is the sum of these two. The constraint due to the finishing hours is, Contsraint 2: Since Giapetto’s cannot sell more than 40 soldiers per week, we have a third constraint: Contsraint 3:

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Finally, we have two more constraints to consider. Giapetto cannot produce a negative number of toys. Therefore, we have the following two constraints:

These are called sign restrictions, because in this case, the variables cannot be negative. Finally, we have completed the optimization model. We must maximize the objective function subject to the constraints: (Objective function) Subject to: (Finishing constraint) (Carpentry constraint) (Demand constraint for soldiers) (Sign restriction) (Sign restriction)

Step 4:

We are finally ready to solve this problem. The first step is to graph the above system of linear inequalities on the next page.

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Giapetto's Woodcarving, Inc. 200 190 180 170 160

150 140 130 120

Trains

110 100 90

80 70 60 50 40 30 20 10 0 0

10

20

30

40

50

60

70

80

90

100

Soldiers

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Feasibility Region:

The region where the five linear inequalities overlap is called the feasibility region. All the points that satisfy all the constraint inequalities lie within the feasibility region, and the solution to Giapetti’s problem must be one of these points. Our problem now, is to determine which point. Notice that the feasibility region is a five-sided polygon (a pentagon). This pentagon has five vertices. It turns out that the optimal value for the objective function, that is the largest value of the objective function that satisfies all the constraints, must be one of these vertices. Start at the origin of the graph and label that vertex “O”. Next, move to the right until you come to the vertex at (40, 0). Label that vertex “A”. Next, move up along the line until you come to the next vertex at (40, 20). Label that vertex “B”. Move up and to the left until you come to the vertex at (20, 60). Label that vertex “C”. Finally, move to the last vertex at (0, 80). Label that vertex “D”. The table below summarizes these vertices. Vertex

soldiers

trains

O

0

0

A

40

0

B

40

20

C

20

60

D

0

80

Objective Function

Calculate the value of the objective function for each vertex. Enter your results in the appropriate spaces in the table. Which vertex maximizes the value of the objective function? _________ What is that value? _____________________

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Step 5:

Write the solution of the problem. Giapetto’s should make _____ soldiers each week Giapetto’s should make _____ trains each week By selling this many soldiers and trains each week, Giapetto’s will make a profit of $__________ each week.

Recall the equations for the number of carpentry hours and finishing hours.

Calculate the number of carpentry hours and finishing hours used by the optimal solution. Notice that the optimal solution utilizes all the available carpentry and finishing hours. In this sort of problem, this is often the case, but not always. Also notice that the optimal solution for this problem involves whole numbers of soldiers and trains. This is convenient, since manufacturing half a soldier is problematic. Sometimes the optimal solution might produce a fraction when the real-life situation requires a whole number solution. When this happens, you need to examine the solution to determine the appropriate modification that satisfies the additional criteria.

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