Introduction to Operations Research

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Introduction to Operations Research

• Linear Programming

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Linear Programming Review

• Linear Programming provides methods for
  allocating limited resources among competing
  activities in an optimal way.
• Any problem whose model fits the format for the
  linear programming model is a linear programming
  problem.
• Wyndor Glass Co. example
• Two variables Graphical method
• Maximize profit

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Radiation Therapy Example

• Mary diagnosed with cancer of the bladder ? needs
  radiation therapy
• Radiation therapy
• Involves using an external beam to pass radiation
  through the patients body
• Damages both cancerous and healthy tissue
• Goal of therapy design is to select the number,
  direction and intensity of beams to generate best
  possible dose distribution
• Doctors have already selected the number (2) and
  direction of the beams to be used
• Goal Optimize intensity (measured in kilorads)
  referred to as the dose

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Radiation Therapy
Area Fraction of Entry Dose Absorbed by Area (Average) Fraction of Entry Dose Absorbed by Area (Average) Restriction on Total Average Dosage, Kilorads
Area Beam1 Beam2 Restriction on Total Average Dosage, Kilorads
Healthy Anatomy 0.4 0.5 minimize
Critical Tissues 0.3 0.1 2.7
Tumor Region 0.5 0.5 6.0
Tumor Center 0.6 0.4 6.0

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Radiation Therapy

• Graph the equations to determine relationships
• Minimize
• Z 0.4x1 0.5x2
• Subject to
• 0.3x1 0.1x2 2.7
• 0.5x1 0.5x2 6
• 0.6x1 0.4x2 18
• x1 0, x2 0

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Mixture Problem

• In order to ensure optimal health (and thus
  accurate test results), a lab technician needs to
  feed the rabbits a daily diet containing a
  minimum of 24 grams (g) of fat, 36 g of
  carbohydrates, and 4 g of protein. But the
  rabbits should be fed no more than five ounces of
  food a day.
• Rather than order rabbit food that is
  custom-blended, it is cheaper to order Food X and
  Food Y, and blend them for an optimal mix.
• Food X contains 8 g of fat, 12 g of
  carbohydrates, and 2 g of protein per ounce, and
  costs 0.20 per ounce.
• Food Y contains 12 g of fat, 12 g of
  carbohydrates, and 1 g of protein per ounce, at a
  cost of 0.30 per ounce.
• What is the optimal blend?

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Mixture Problem
Daily Amount Food Type Food Type Daily Requirements (grams)
Daily Amount X Y Daily Requirements (grams)
Fat 8 12 24
Carbohydrates 12 12 36
Protein 2 1 4

maximum weight of the food is five ounces X Y
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Minimize the cost Z 0.2X 0.3Y
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Mixture Problem

• Graph the equations to determine relationships
• Minimize
• Z 0.2x 0.3y
• Subject to
• fat 8x 12y 24
• carbs 12x 12y 36
• protein 2x 1y 4
• weight x y 5
• x 0, y 0

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Mixture Problem

• When you test the corners at
• (0, 4), (0, 5), (3, 0), (5, 0), and (1, 2)
• you get a minimum cost of sixty cents per daily
  serving, using three ounces of Food X only.
• Only need to buy Food X

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Investment example

• You have 12,000 to invest, and three different
  funds from which to choose.
• Municipal bond 7 return
• CDs 8 return
• High-risk acct 12 return (expected)
• To minimize risk, you decide not to invest any
  more than 2,000 in the high-risk account.
• For tax reasons, you need to invest at least
  three times as much in the municipal bonds as in
  the bank CDs.
• Assuming the year-end yields are as expected,
  what are the optimal investment amounts?

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Investment example

• Bonds (in thousands) x
• CDs (in thousands) y
• High Risk z
• Um... now what? I have three variables for a
  two-dimensional linear plot
• Use the "how much is left" concept
• Since 12,000 is invested, then the high risk
  account can be represented as
• z 12 x y

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Investment example

• Constraints
• Amounts are non-negative
• x 0
• y 0
• z 0 ? 12 x y 0
• ? y x 12
• High risk has upper limit
• z 2 ? 12 x y 2
• ? y x 10
• Taxes
• 3y x ? y 1/3 x

• Objective to maximize the return
• Z 0.07x 0.08y 0.12z
• ?
• Z 1.44 - 0.05x 0.04y

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Investment example

• When you test the corner points at (9, 3), (12,
  0), (10, 0), and (7.5, 2.5), you should get an
  optimal return of 965 when you invest 7,500 in
  municipal bonds, 2,500 in CDs, and the remaining
  2,000 in the high-risk account.

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Manufacturing Example
Machine data
Product data
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Product Structure
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LP Formulation
Objective Function
xQ

max
45 xP

60





10 xQ
20 xP

1800

Structural
s.t.



constraints
12 xP

28 xQ

1440




15 xP

6 xQ
2040





10 xP

15 xQ
2400



demand
xP ? 100, xQ ? 40
xP 0, xQ 0
nonnegativity
Are we done?
Are the LP assumptions valid for this problem?


81.82
16.36
x
Optimal solution

x


Q
P
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Feasible Region
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Optimal Solution
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Discussion of ResulTS

• Optimal objective value is 4664 but when we
  subtract the weekly operating expenses of 3000
  we obtain a weekly profit of 1664.
• Machines A B are being used at maximum level
  and are bottlenecks.
• There is slack production capacity in Machines C
  D.

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