ECON 3300 LEC

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ECON 3300 LEC 3

• 08/28/06

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Outline

• Graphical method solving LP
• Maximization problems
• Minimization problems
• Slack and Surplus variable
• Special cases LP
• Multiple optimal solution
• Infeasible problem
• Unbounded problem
• Limitations Linear programming

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Linear programming

• Beaver Creek Pottery company is a small crafts
  operation run by a Native American tribal
  council. The company employs skilled artisans to
  produce clay bowls and mugs with authentic Native
  American designs and colors. The two primary
  resources used by company are special pottery
  clay and skilled labor. The company wants to know
  how many bowls and mugs to produce each day in
  order to maximize profit

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Linear programming

• Complete model
• Maximize z 40x150x2
• subject to
• 1x12x2lt40
• 4x13x2lt120
• x1, x2gt0

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x12x240 - Put x10 we get x220 - Put x20 we
get x140 This yields (40,0) and (0,20)
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4x13x2120 - Put x10 we get x240 - Put x20
we get x130
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Area common to both constraints
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• Take a arbitrary objective function value to
  start with
• Let Z800
• Solving 80040x150x2 would yield (20,0) and
  (0,16) feasible solutions
• Then move it away from origin to get points max
  profit

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Optimal Point
B
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Optimal Point
B
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24
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Optimal solution point

• Optimal points or solutions always occur at the
  corner points of the feasible region
• Optimal point can be obtained by solving the
  equations forming the corner point
• For example B is a corner point formed by the
  intersection of
• x12x240
• 4x13x2120
• Solving the two equations yields (24,8)

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Solutions - corner points
x10 bowls x220 mugs Z1000
x124 bowls x28 mugs Z1360
A
x130 bowls, x20 mugs Z1200
B
C
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Effect of change in Objective function
coefficients
Z70x120x2
A
Optimal point x130 bowls x20 mugs Z2100
B
C
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Model formulation

• Reddy Mikks produces both interior and exterior
  paints from two raw materials, M1 and M2.
  Exterior paint requires 6 tons of raw material M1
  and 1 ton of raw material M2. Interior paint
  requires 4 tons of raw material M1 and 2 tons of
  raw material M2. The available quantities of raw
  material M1 and M2 are 24 and 6 respectively. The
  profit per ton of producing exterior paint and
  interior paint is 5000 and 4000 respectively

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Model formulation

• Complete model is written as
• Maximize z 5x14x2
• subject to
• 6x14x2lt24
• x12x2lt6
• -x1x2lt1
• x2lt2
• x1gt0
• x2gt0

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Increasing z
z21
z15
Optimum x13 tons x21.5 tons
z2100
z10
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Example

• Ozark Farms uses at least 800 lb of special feed
  daily. The special feed is a mixture of corn and
  soybean meal with the following compositions

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Example

• The dietary requirements of the special feed
  stipulate at least 30 protein and at most 5
  fiber.

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Example
Decision variables x1lb of corn in the daily
mix x2lb of soybean meal in the daily
mix Objective function minimize the total daily
cost (in dollars) of the feed mix Minimize z
.3x1.9x2
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Example

• Constraints
• x1x2gt800 (daily feed requirement)
• .09x1.6x2gt.3(x1x2) (protein requirement)
• .02x1.06x2lt.05(x1x2) (fiber requirement)
• Non negativity restrictions
• x1gt0
• x2gt0

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Example

• Complete model
• Minimize z.3x1.9x2
• subject to
• x1x2gt800
• .21x1-.3x2lt0
• .03x1-.01x2gt0
• x1,x2 gt0

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Slack variable

• Slack variable unused resource
• Slack variables introduced in the constraint
  equations is as follows
• x12x2s140
• 4x13x2s2120
• Here s1 and s2 are slack variables.
• s1gt0 and s2gt0

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Slack variable
x10 bowls x220 mugs s10 s260
x124 bowls x28 mugs s10 s20
A
x130 bowls, x20 mugs s110, s20
B
C
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Linear programming

• Minimization problem
• A farmer is preparing to plant a crop in the
• spring and needs to fertilize a field. There
• are two brands of fertilizer to choose from,
• Super-gro and Crop-quick. Each brand
• yields a specific amount of nitrogen and
• phosphate as follows

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Linear programming

• Complete model
• Minimize z6x13x2
• subject to
• 2x14x2gt16
• 4x13x2gt24
• x1, x2gt0

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x10 x28 Z24
A
Z6x13x2
B
C
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Surplus variable

• Excess above a constraint requirement
• Surplus variables introduced in the constraint as
  follows
• 2x14x2-s116
• 4x13x2-s224
• Here s1, s2 are surplus variables
• s1gt0 and s2gt0

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x10 x28 s116 s20
A
Z6x13x2
x14.8 x21.6 s10 s20
x18 x20 s10 s28
B
C
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Special cases Linear programming problems

• Multiple optimal solutions
• Objective function parallel to one of the
  constraints
• For example, model below
• Maximize Z40x130x2
• subject to
• x12x2lt40
• 4x13x2lt120
• x1gt0, x2gt0
• Provide greater flexibility to decision maker

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Point C x130 x20 Z1200
Point B x124 x28 Z1200
A
B
C
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Infeasible problem

• Solution to this problem not possible
• For example, the model below
• Maximize Z5x13x2
• subject to
• 4x13x2lt8
• x1gt4
• x2gt6
• x1gt0, x2gt0
• No feasible solution area, every point violates
  one or the other constraint

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x1gt4
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Unbounded problem

• Objective function can increase indefinitely
  without reaching maximum value
• Solution space is not completely closed in
• For example, the model below
• Maximize Z4x12x2
• subject to
• x1gt4
• x2lt2
• x1, x2gt0

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Cases of Optima and Feasible
Unique Finite Optimal Bounded Feasible Region
Unique Finite Optimal Unbounded Feasible Region
Alternate Finite Optimal Bounded Feasible Region
Alternate Finite Optimal Unbounded Feasible
Region
Unbounded
Infeasible
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Limitations - Linear programming

• Proportionality
• Objective function and every constraint function
  must be linear
• Measure of effectiveness and resource usage
  proportional to the level of each activity
• Additivity
• Activities should be additive with respect to the
  measure of effectiveness and each resource usage
• For example, in the Beaver pottery model, total
  profit must be equal to the sum of profits earned
  from making bowls and mugs

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Limitations Linear programming

• Divisibility
• Frequently it is required that decision variables
  would have significance only if they have integer
  values.
• Linear programs lead to solutions which could be
  fractional levels of decision variables
• Rounding up can lead to lot of errors
  infeasible solutions
• Feasible solutions also could be far from optimal

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Limitations Linear programming

• Integer programming developed to care of such
  problems
• Deterministic
• All coefficients in the LP model assumed to be
  constants
• In reality they are neither known nor are
  constants
• LPs formulated generally to select some future
  course of action
• Coefficients used very frequently are based on
  prediction of future conditions