An Introduction to

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Chapter 2

• An Introduction to
• Linear Programming

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Components of Linear Programming

• A goal (to maximize or minimize something)
• An objective (To determine..)
• Decision variables (what the manager can adjust)
• Constraints (subject to these things that the
  manager cannot adjust)

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General Form of an LP Model

• xs are the decision variables
• cs, as and bs are constants
• as are the amount of constraint used or supplied
  by each x
• bs are the total amount of a constraint
  available or required
• cs are the value (cost or benefit) of each x

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General Form of an LP Model
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Assumptions of the LP Model

• Divisibility - basic units of xs are divisible
• Proportionality - as and cs are strictly
  proportional to the xs
• Additivity - each term in the objective function
  and constraints contains only one variable
• Deterministic - all cs, as and bs are known
  and measured without error
• Non-Negativity (caveat)

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Sherwood Furniture Company
Recently, Sherwood Furniture Company has been
interested in developing a new line of stereo
speaker cabinets. In the coming month, Sherwood
expects to have excess capacity in its Assembly
and Finishing departments and would like to
experiment with two new models. One model is the
Standard, a large, high-quality cabinet in a
traditional design that can be sold in virtually
unlimited quantities to several manufacturers of
audio equipment. The other model is the Custom,
a small, inexpensive cabinet in a novel design
that a single buyer will purchase on an exclusive
basis. Under the tentative terms of this
agreement, the buyer will purchase as many
Customs as Sherwood produces, up to 32 units.
The Standard requires 4 hours in the Assembly
Department and 8 hours in the Finishing
Department, and each unit contributes 20 to
profit. The Custom requires 3 hours in Assembly
and 2 hours in Finishing, and each unit
contributes 10 to profit. Current plans call
for 120 hours to be available next month in
Assembly and 160 hours in Finishing for cabinet
production, and Sherwood desires to allocate this
capacity in the most economical way.
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Sherwood Furniture Company Linear Equations
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Sherwood Furniture Company Graphical Solution
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Sherwood Furniture Company Graph Solution
Constraint 1
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Sherwood Furniture Company Graph Solution
Constraint 1
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Sherwood Furniture Company Graph Solution
Constraint 2
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Sherwood Furniture Company Graph Solution
Constraint 1 2
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Sherwood Furniture Company Graph Solution
Constraint 3
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Sherwood Furniture Company Graph Solution
Constraint 1, 2 3
Feasible region (solution set)
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Sherwood Furniture Company Graph Solution
Trial Objective function
Set the objective function equal to some
arbitrary number (well, not totally arbitrary
try to make it somewhere in the realm of
reasonableness and making it evenly divisible by
both objective coefficients makes it easier to
graph).
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Sherwood Furniture Company Graph Solution
Move the objective function toward or away from
the origin (keeping it at the same slope) until
it just touches the point farthest from the origin
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Sherwood Furniture Company Solve Linear
Equations
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Sherwood Furniture Company Solve Linear
Equations
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Sherwood Furniture Company Solve Linear
Equations
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Sherwood Furniture Company Graph Solution
Optimal Point (15, 20)
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Sherwood Furniture Company Slack Calculation
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Sherwood Furniture Company - Slack Variables

• Max
• 20x1 10x2 0S1 0S2 0S3
• s.t.
• 4x1 3x2 1S1 0S2 0S3 120
• 8x1 2x2 0S1 1S2 0S3 160
• 0x1 1x2 0S1 0S2 1S3 32
• and x1, x2, S1 ,S2 ,S3 gt 0

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Sherwood Furniture Company Graph Solution
3
2
1
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Sherwood Furniture Company Slack Calculation
Point 1
Point 1
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Sherwood Furniture Company Graph Solution
3
2
1
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Sherwood Furniture Company Slack Calculation
Point 2
Point 2
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Sherwood Furniture Company Graph Solution
3
2
1
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Sherwood Furniture Company Slack Calculation
Point 3
Point 3
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Sherwood Furniture Company Slack Calculation
Points 1, 2 3
Point 1
Point 2
Point 3
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Sherwood Furniture Company Slack Variables

• For each constraint the difference between the
  RHS and LHS (RHS-LHS). It is the amount of
  resource left over.
• Constraint 1 S1 0 hrs.
• Constraint 2 S2 0 hrs.
• Constraint 3 S3 12 Custom

31
Pet Food Company
A pet food company wants to find the optimal
mix of ingredients, which will minimize the cost
of a batch of food, subject to constraints on
nutritional content. There are two ingredients,
P1 and P2. P1 costs 5/lb. and P2 costs 8/lb. A
batch of food must contain no more than 400 lbs.
of P1 and must contain at least 200 lbs. of P2. A
batch must contain a total of at least 500 lbs.
What is the optimal (minimal cost) mix for a
single batch of food?
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Pet Food Company Linear Equations
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Pet Food Company Graph Solution
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Pet Food Company Graph Solution Constraint 1
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Pet Food Company Graph Solution Constraint 1
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Pet Food Company Graph Solution Constraint 2
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Pet Food Company Graph Solution Constraint 1 2
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Pet Food Company Graph Solution Constraint 3
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Pet Food Company Graph Solution Constraint 1, 2
3
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Pet Food Company Solve Linear Equations
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Pet Food Company Graph Solution
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Pet Food Company Solve Linear Equations
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Pet Food Company Solve Linear Equations
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Pet Food Company Graph Solution
Optimal Point (300, 200)
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Pet Food Company Slack/ Surplus Calculation
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Pet Food Co. Linear Equations Slack/ Surplus
Variables

• Min 5P1 8P2 0S1 0S2 0S3
• s.t. 1P1 0P2 1S1 0S2 0S3 400
• 0P1 1P2 0S1 - 1S2 0S3 200
• 1P1 1P2 0S1 0S2 - 1S3 500
• and P1, P2, S1 ,S2 ,S3 gt 0

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Pet Food Co. Slack Variables

• For each constraint the difference between the
  RHS and LHS (RHS-LHS). It is the amount of
  resource left over.
• Constraint 1 S1 100 lbs.

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Pet Food Co. Surplus Variables

• For each constraint the difference between the
  LHS and RHS (LHS-RHS). It is the amount bt which
  a minimum requirement is exceeded.
• Constraint 2 S2 0 lbs.
• Constraint 3 S3 0 lbs.

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Special Cases

• Alternate Optimal Solutions
• No Feasible Solution
• Unbounded Solutions

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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Alternate Optimal Solutions
A
B
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Alternate Optimal Solutions
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Alternate Optimal Solutions
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Special Cases

• Alternate Optimal Solutions
• No Feasible Solution
• Unbounded Solutions

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No Feasible Solution
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No Feasible Solution
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Unbounded Solutions
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No Feasible Solution
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Special Cases

• Alternate Optimal Solutions
• No Feasible Solution
• Unbounded Solutions

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Unbounded Solutions
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Unbounded Solutions
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Unbounded Solutions
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Unbounded Solutions