Chapter 1
Introduction to Optimization
and Linear Programming
1-1
Introduction
• We all face decision about how to use
limited resources such as:
– Oil in the earth
– Land for waste dumps
– Time
– Money
– Workers
1-2
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
1-3
Applications of Optimization
•
•
•
•
Determining Product Mix
Manufacturing
Routing and Logistics
Financial Planning
1-4
Characteristics of Optimization Problems
• Decisions
• Constraints
• Objectives
1-5
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
1-6
Linear Programming (LP) Problems
MAX (or MIN): c1X1 + c2X2 + … + cnXn
Subject to:
a11X1 + a12X2 + … + a1nXn <= b1
:
ak1X1 + ak2X2 + … + aknXn >=bk
:
am1X1 + am2X2 + … + amnXn = bm
1-7
An Example LP Problem
Blue Ridge Hot Tubs produces two types of hot
tubs: Aqua-Spas & Hydro-Luxes.
Pumps
Labor
Tubing
Unit Profit
Aqua-Spa
1
9 hours
12 feet
$350
Hydro-Lux
1
6 hours
16 feet
$300
There are 200 pumps, 1566 hours of labor,
and 2880 feet of tubing available.
1-8
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
1-9
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
1-10
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
1-11
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!
1-12
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.
1-13
Plotting the First Constraint
X2
250
(0, 200)
200
boundary line of pump constraint
X1 + X2 = 200
150
100
50
(200, 0)
0
0
50
100
150
200
250
X1
1-14
Plotting the Second Constraint
X2
(0, 261)
250
boundary line of labor constraint
9X1 + 6X2 = 1566
200
150
100
50
(174, 0)
0
0
50
100
150
200
250
X1
1-15
Plotting the Third Constraint
X2
250
(0, 180)
200
150
boundary line of tubing constraint
12X1 + 16X2 = 2880
100
Feasible Region
50
(240, 0)
0
0
50
100
150
200
250
X1
1-16
Plotting A Level Curve of the
Objective Function
X2
250
200
(0, 116.67)
objective function
150
350X1 + 300X2 = 35000
100
(100, 0)
50
0
0
50
100
150
200
250
X1
1-17
A Second Level Curve of the
Objective Function
X2
250
(0, 175)
200
objective function
350X1 + 300X2 = 35000
objective function
350X1 + 300X2 = 52500
150
100
(150, 0)
50
0
0
50
100
150
200
250
X1
1-18
Using A Level Curve to Locate
the Optimal Solution
X2
250
objective function
350X1 + 300X2 = 35000
200
150
optimal solution
100
objective function
350X1 + 300X2 = 52500
50
0
0
50
100
150
200
250
X1
1-19
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
1-20
Enumerating The Corner Points
X2
250
obj. value = $54,000
(0, 180)
200
Note: This technique will not work if
the solution is unbounded.
obj. value = $64,000
150
(80, 120)
obj. value = $66,100
(122, 78)
100
50
obj. value = $60,900
(174, 0)
obj. value = $0
(0, 0)
0
0
50
100
150
200
250
X1
1-21
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
1-22
Special Conditions in LP Models
• A number of anomalies can occur in LP
problems:
– Alternate Optimal Solutions
– Redundant Constraints
– Unbounded Solutions
– Infeasibility
1-23
Example of Alternate Optimal
Solutions
X2
250
objective function level curve
450X1 + 300X2 = 78300
200
150
100
alternate optimal solutions
50
0
0
50
100
150
200
250
X1
1-24
Example of a Redundant Constraint
X2
250
boundary line of tubing constraint
200
boundary line of pump constraint
150
boundary line of labor constraint
100
Feasible Region
50
0
0
50
100
150
200
250
X1
1-25
Example of an Unbounded Solution
X2
1000
objective function
X1 + X2 = 600
800
-X1 + 2X2 = 400
objective function
X1 + X2 = 800
600
400
200
X1 + X2 = 400
0
0
200
400
600
800
1000
X1
1-26
Example of Infeasibility
X2
250
200
X1 + X2 = 200
feasible region for
second constraint
150
100
feasible region
for first
constraint
50
X1 + X2 = 150
0
0
50
100
150
200
250
X1
1-27
Important ”Behind the Scenes”
Assumptions in LP Models
1-28
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)
1-29
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.
1-30
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]
1-31
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 aquaspas produced), the production of x2
hydro-luxes uses x2 pumps,
6x2 hours of labor,
16x2 feet of tubing.
1-32
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.
1-33
End of Chapter 1
1-34