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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)<=b1
:fk(X1, X2, …, Xn)>=bk
: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 <=
b1
:ak1X1 + ak2X2 + … + aknXn >=bk
: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.
Aqua-Spa Hydro-LuxPumps 1 1Labor 9 hours 6 hoursTubing 12 feet 16 feetUnit Profit $350 $300
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5 Steps In Formulating LP Models:
1. Understand the problem.2. Identify the decision variables.
X1=number of Aqua-Spas to produce
X2=number 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 <= 200 } pumps
9X1 + 6X2 <= 1566 } labor
12X1 + 16X2 <= 2880 } tubing
5. Identify any upper or lower bounds on the decision variables.
X1 >= 0
X2 >= 0
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LP Model for Blue Ridge Hot Tubs
MAX: 350X1 + 300X2
S.T.: 1X1 + 1X2 <= 200
9X1 + 6X2 <= 1566
12X1 + 16X2 <= 2880
X1 >= 0
X2 >= 0
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Solving LP Problems: An Intuitive Approach
• Idea: Each Aqua-Spa (X1) generates the highest unit profit ($350), so let’s make as many of them as possible!
• How many would that be?
– Let X2 = 0
• 1st constraint: 1X1 <= 200
• 2nd constraint: 9X1 <=1566 or X1 <=174
• 3rd constraint: 12X1 <= 2880 or X1 <= 240
• If X2=0, the maximum value of X1 is 174 and the total profit is $350*174 + $300*0 = $60,900
• This solution is feasible, but is it optimal?• No!
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Solving LP Problems:A 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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X2
X1
250
200
150
100
50
0 0 50 100 150 200 250
(0, 200)
(200, 0)
boundary line of pump constraint
X1 + X2 = 200
Plotting the First Constraint
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X2
X1
250
200
150
100
50
0 0 50 100 150 200 250
(0, 261)
(174, 0)
boundary line of labor constraint
9X1 + 6X2 = 1566
Plotting the Second Constraint
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X2
X1
250
200
150
100
50
0 0 50 100 150 200 250
(0, 180)
(240, 0)
boundary line of tubing constraint
12X1 + 16X2 = 2880
Feasible Region
Plotting the Third Constraint
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X2Plotting A Level Curve of the
Objective Function
X1
250
200
150
100
50
0 0 50 100 150 200 250
(0, 116.67)
(100, 0)
objective function
350X1 + 300X2 = 35000
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A Second Level Curve of the Objective FunctionX2
X1
250
200
150
100
50
0 0 50 100 150 200 250
(0, 175)
(150, 0)
objective function 350X1 + 300X2 = 35000
objective function 350X1 + 300X2 = 52500
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Using A Level Curve to Locate the Optimal SolutionX2
X1
250
200
150
100
50
0 0 50 100 150 200 250
objective function 350X1 + 300X2 = 35000
objective function
350X1 + 300X2 = 52500
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, X1=122, X2=200-X1=78
Total Profit = $350*122 + $300*78 = $66,100
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Enumerating The Corner PointsX2
X1
250
200
150
100
50
0 0 50 100 150 200 250
(0, 180)
(174, 0)
(122, 78)
(80, 120)
(0, 0)
obj. value = $54,000
obj. value = $64,000
obj. value = $66,100
obj. value = $60,900obj. value = $0
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 region3. Locate the optimal solution by
either:a. Plotting level curvesb. 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 SolutionsX2
X1
250
200
150
100
50
0 0 50 100 150 200 250
450X1 + 300X2 = 78300objective function level curve
alternate optimal solutions
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Example of a Redundant ConstraintX2
X1
250
200
150
100
50
0 0 50 100 150 200 250
boundary line of tubing constraint
Feasible Region
boundary line of pump constraint
boundary line of labor constraint
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Example of an Unbounded SolutionX2
X1
1000
800
600
400
200
0 0 200 400 600 800 1000
X1 + X2 = 400
X1 + X2 = 600
objective function
X1 + X2 = 800objective function
-X1 + 2X2 = 400
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Example of InfeasibilityX2
X1
250
200
150
100
50
0 0 50 100 150 200 250
X1 + X2 = 200
X1 + X2 = 150
feasible region for second constraint
feasible region for first constraint
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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@$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@3=27 hours) to make 3 aqua-spas as it takes to make 1 aqua-spa (9@1=9 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