Introduction to Optimization and Linear Programming

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Introduction to Optimization and Linear
Programming
Chapter 1
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Introduction

• We all face decision about how to use limited
  resources such as
• Oil in the earth
• Land for waste dumps
• Time
• Money
• Workers

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Mathematical Programming...

• MP is a field of operations research that finds
  the optimal, or most efficient, way of using
  limited resources to achieve the objectives of an
  individual or a business.
• a.k.a. Optimization

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Applications of Optimization

• Determining Product Mix
• Manufacturing
• Routing and Logistics
• Financial Planning

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Characteristics of Optimization Problems

• Decisions
• Constraints
• Objectives

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General Form of an Optimization Problem

• MAX (or MIN) f0(X1, X2, , Xn)
• Subject to f1(X1, X2, , Xn)ltb1
• fk(X1, X2, , Xn)gtbk
• fm(X1, X2, , Xn)bm
• Note If all the functions in an optimization are
  linear, the problem is a Linear Programming (LP)
  problem

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Linear Programming (LP) Problems

• MAX (or MIN) c1X1 c2X2 cnXn
• Subject to a11X1 a12X2 a1nXn lt b1
• ak1X1 ak2X2 aknXn gtbk
• am1X1 am2X2 amnXn bm

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An Example LP Problem
Blue Ridge Hot Tubs produces two types of hot
tubs Aqua-Spas Hydro-Luxes.
There are 200 pumps, 1566 hours of labor, and
2880 feet of tubing available.
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5 Steps In Formulating LP Models

• 1. Understand the problem.
• 2. Identify the decision variables.
• X1number of Aqua-Spas to produce
• X2number of Hydro-Luxes to produce
• 3. State the objective function as a linear
  combination of the decision variables.
• MAX 350X1 300X2

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5 Steps In Formulating LP Models(continued)

• 4. State the constraints as linear combinations
  of the decision variables.
• 1X1 1X2 lt 200 pumps
• 9X1 6X2 lt 1566 labor
• 12X1 16X2 lt 2880 tubing
• 5. Identify any upper or lower bounds on the
  decision variables.
• X1 gt 0
• X2 gt 0

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LP Model for Blue Ridge Hot Tubs
MAX 350X1 300X2 S.T. 1X1 1X2 lt 200 9X1
6X2 lt 1566 12X1 16X2 lt 2880 X1 gt 0 X2
gt 0
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Solving LP Problems An Intuitive Approach

• Idea Each Aqua-Spa (X1) generates the highest
  unit profit (350), so lets make as many of them
  as possible!
• How many would that be?
• Let X2 0
• 1st constraint 1X1 lt 200
• 2nd constraint 9X1 lt1566 or X1 lt174
• 3rd constraint 12X1 lt 2880 or X1 lt 240
• If X20, the maximum value of X1 is 174 and the
  total profit is 350174 3000 60,900
• This solution is feasible, but is it optimal?
• No!

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Solving LP ProblemsA Graphical Approach

• The constraints of an LP problem define the
  feasible region.
• The best point in the feasible region is the
  optimal solution to the problem.
• For LP problems with 2 variables, it is easy to
  plot the feasible region and find the optimal
  solution.

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Plotting the First Constraint
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Plotting the Second Constraint
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Plotting the Third Constraint
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X2
Plotting A Level Curve of the Objective Function
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A Second Level Curve of the Objective Function
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Using A Level Curve to Locate the Optimal
Solution
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Calculating the Optimal Solution

• The optimal solution occurs where the pumps and
  labor constraints intersect.
• This occurs where
• X1 X2 200 (1)
• and 9X1 6X2 1566 (2)
• From (1) we have, X2 200 -X1 (3)
• Substituting (3) for X2 in (2) we have,
• 9X1 6 (200 -X1) 1566
• which reduces to X1 122
• So the optimal solution is,
• X1122, X2200-X178
• Total Profit 350122 30078 66,100

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Enumerating The Corner Points
Note This technique will not work if the
solution is unbounded.
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Summary of Graphical Solution to LP Problems

• 1. Plot the boundary line of each constraint
• 2. Identify the feasible region
• 3. Locate the optimal solution by either
• a. Plotting level curves
• b. Enumerating the extreme points

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Special Conditions in LP Models

• A number of anomalies can occur in LP problems
• Alternate Optimal Solutions
• Redundant Constraints
• Unbounded Solutions
• Infeasibility

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Example of Alternate Optimal Solutions
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Example of a Redundant Constraint
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Example of an Unbounded Solution
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Example of Infeasibility
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Important Behind the Scenes Assumptions in LP
Models
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Proportionality and Additivity Assumptions

• An LP objective function is linear this results
  in the following 2 implications
• proportionality contribution to the objective
  function from each decision variable is
  proportional to the value of the decision
  variable. e.g., contribution to profit from
  making 4 aqua-spas (4_at_350) is 4 times the
  contribution from making 1 aqua-spa (350)

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Proportionality and Additivity Assumptions (cont.)

• Additivity contribution to objective function
  from any decision variable is independent of the
  values of the other decision variables. E.g., no
  matter what the value of x2, the manufacture of
  x1 aqua-spas will always contribute 350 x1
  dollars to the objective function.

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Proportionality and Additivity Assumptions (cont.)

• Analogously, since each constraint is a linear
  inequality or linear equation, the following
  implications result
• proportionality contribution of each decision
  variable to the left-hand side of each constraint
  is proportional to the value of the variable.
  E.g., it takes 3 times as many labor hours
  (9_at_327 hours) to make 3 aqua-spas as it takes to
  make 1 aqua-spa (9_at_19 hours) No economy of
  scale

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Proportionality and Additivity Assumptions (cont.)

• Additivity the contribution of a decision
  variable to the left-hand side of a constraint is
  independent of the values of the other decision
  variables. E.g., no matter what the value of x1
  (no. of aqua-spas produced), the production of x2
  hydro-luxes uses x2 pumps,
• 6x2 hours of labor,
• 16x2 feet of tubing.

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More Assumptions

• Divisibility Assumption each decision variable
  is allowed to assume fractional values
• Certainty Assumption each parameter (objective
  function coefficient cj, right-hand side constant
  bi of each constraint, and technology coefficient
  aij) is known with certainty.

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End of Chapter 1