Business 260: Managerial Decision Analysis

1
Business 260 Managerial Decision
Analysis Professor David Mease Lecture
5 Agenda 1) Reminder about Homework 2
(due Thursday 4/2) 2) Reminder about Exam 2 3)
Linear Programming (QBA Book Chapters 7-10)
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Homework 2 Homework 2 will be due Thursday
4/2 We will have an exam that day after we
review the solutions The complete homework
assignment is now posted on the class web
page http//www.cob.sjsu.edu/mease_d/bus260/260h
omework.html The solutions are also now posted
so you can check your answers http//www.cob.sjs
u.edu/mease_d/bus260/260homework_solutions.html
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Midterm Exam 2 We will have the 2nd midterm
exam on Thursday 4/2 after we go over the
homework It will cover Lecture 4 and Lecture 5
material It is worth 30 total points (30 of
your grade) It is closed notes and closed book
but you will have the Exam 2 formula sheet posted
here http//www.cob.sjsu.edu/mease_d/bus260/260f
ormula_sheets.html You will have 2 hours to
complete the exam Remember to bring a pocket
calculator Questions will be similar to those on
Homework 2
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Quantitative Business Analysis
Linear Programming (Chapters 7-10)
5
Chapter 7 Topics

• The Linear Programming Problem
• Problem Formulation
• The Graphical Solution Procedure
• Extreme Points and the Optimal Solution
• Special Cases
• Also, we will cover linear programming with more
  than 2 variables as in Chapters 8, 9 and 10.

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The Linear Programming (LP) Problem The
maximization or minimization of some quantity is
the objective in all linear programming
problems. All LP problems have constraints that
limit the degree to which the objective can be
pursued. A feasible solution satisfies all the
problem's constraints. An optimal solution is a
feasible solution that results in the largest
possible objective function value when maximizing
(or smallest when minimizing).
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The Linear Programming (LP) Problem A graphical
solution method can be used to solve a linear
program with two variables.
8
The Linear Programming (LP) Problem If both the
objective function and the constraints are
linear, the problem is referred to as a linear
programming problem. (Linear functions are
functions in which each variable appears in a
separate term raised to the first power and is
multiplied by a constant (which could be
0). (Linear constraints are linear functions
that are restricted to be "less than or equal
to", "equal to", or "greater than or equal to" a
constant.)
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Problem Formulation Problem formulation is the
process of translating a verbal statement of a
problem into a mathematical statement. Guidelines
for Model Formulation - Understand the
problem thoroughly. - Describe the objective. -
Describe each constraint. - Define the decision
variables. - Write the objective in terms of the
decision variables. - Write the constraints in
terms of the decision variables.
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In class exercise 75 For the following, write
down the correct and complete formulation of the
linear programming problem. Write all
inequalities with the variables in alphabetical
order on the left side and the constant on the
right side. Get-Toe Apartments is renovating a
new building to put apartments in. They will
have both 2 bedroom and 3 bedroom apartments.
They do not want the number of 2 bedroom
apartments to exceed 6. Due to zoning
regulations, they can not have more than 8 total
apartments or more than 19 total bedrooms. They
charge 500/month for a 2 bedroom apartment and
700/month for a 3 bedroom apartment. How many
of each type of apartment should they have to
maximum their monthly rent?
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(ANSWER)

• In class exercise 75
• For the following, write down the correct and
  complete formulation of the linear programming
  problem. Write all inequalities with the
  variables in alphabetical order on the left side
  and the constant on the right side.

Max 500x1 700x2 s.t. x1
lt 6 2x1 3x2 lt
19 x1 x2 lt 8
x1 gt 0 x2 gt
0
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The Graphical Solution Procedure The problems in
Chapter 7 all deal with only two variables
(dimensions) so you can find the solution
graphically The Graphical Solution Procedure is
illustrated for ICE 75 on the next few
slides The first step is to graph the feasible
region After that, you must determine the point
or points in that region that maximize (or
minimize) your objective function For more than
two variables (dimensions), software packages are
usually used
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The Graphical Solution Procedure The
x2
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
x1 lt 6
(6, 0)
x1
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The Graphical Solution Procedure The
x2
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
(0, 6 1/3)
2x1 3x2 lt 19
(9 1/2, 0)
x1
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The Graphical Solution Procedure The
x2
(0, 8)
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
x1 x2 lt 8
(8, 0)
x1
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The Graphical Solution Procedure The
x2
x1 x2 lt 8
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
x1 lt 6
2x1 3x2 lt 19
x1
17
The Graphical Solution Procedure The
x2
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
Feasible Region
Feasible Region
x1
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The Graphical Solution Procedure The
x2
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
(0, 5)
Objective Function 500x1 700x2 3500
(7, 0)
x1
19
The Graphical Solution Procedure The
x2
Objective Function 500x1 700x2 4600
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
Optimal Solution (x1 5, x2 3)
x1
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Summary of the Graphical Solution
Procedure Prepare a graph of the feasible
solutions for each of the constraints. Determine
the feasible region that satisfies all the
constraints simultaneously. Draw an objective
function line. Move parallel objective function
lines toward larger objective function values
without entirely leaving the feasible
region. Any feasible solution on the objective
function line with the largest value is an
optimal solution.
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Extreme Points and the Optimal Solution The
corners or vertices of the feasible region are
referred to as the extreme points. An optimal
solution to an LP problem can be found at an
extreme point of the feasible region. When
looking for the optimal solution, you do not have
to evaluate all feasible solution points. You
have to consider only the extreme points of the
feasible region. To find extreme points, you
often need to determine the point at which two
lines intersect as in the following in class
exercise.
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In class exercise 76 Determine the extreme
points for the Get-Toe Apartments exercise and
evaluate the objective function at each extreme
point.
x2
x1 x2 lt 8
8 7 6 5 4 3 2 1 1 2
3 4 5 6 7 8 9
10
x1 lt 6
5
4
2x1 3x2 lt 19
Feasible Region
3
1
2
x1
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• In class exercise 77
• Solve the two-dimensional (minimization) linear
  programming problem below. Be sure to include a
  graph of the feasible region in your solution.

Min 5x1 2x2 s.t. 2x1 5x2 gt
10 4x1 - x2 gt 12
x1 x2 gt 4 x1
gt 0 x2 gt 0
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(ANSWER)

• In class exercise 77
• Solve the two-dimensional (minimization) linear
  programming problem below. Be sure to include a
  graph of the feasible region in your solution.

Min z 5x1 2x2 4x1 - x2 gt 12 x1 x2 gt
4
x2
5 4 3 2 1

2x1 5x2 gt 10 Optimal x1 16/5
x2 4/5 which gives z
88/5 as the minimum
1 2 3 4 5
6
x1
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The Feasible Region The feasible region for a
two-variable LP problem can be nonexistent, a
single point, a line, a polygon, or an unbounded
area. Any linear program falls in one of three
categories -is infeasible -has a unique
optimal solution or alternate optimal
solutions -has an objective function that can be
increased without bound A feasible region may
be unbounded and yet there may be an optimal
solutions as in the previous in class exercise.
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Special Cases Alternative Optimal Solutions If
the objective function line is parallel to a
boundary constraint in the direction of
optimization, there are alternate optimal
solutions, with all points on this line segment
being optimal. Infeasibility A linear program
which is overconstrained so that no point
satisfies all the constraints is said to be
infeasible. There is an example on the homework
and in ICE 78 on the following
slide. Unboundedness The previous in class
exercise would be an example of this if it were
maximization instead of minimization. There is
an example on the homework and in ICE 79.
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• In class exercise 78
• Solve the two-dimensional linear programming
  problem below. Be sure to include a graph of the
  feasible region in your solution.

Max 2x1 6x2 s.t.
4x1 3x2 lt 12 2x1 x2 gt 8
x1, x2 gt 0
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(ANSWER)

• In class exercise 78
• Solve the two-dimensional linear programming
  problem below. Be sure to include a graph of the
  feasible region in your solution.

(No points that satisfy both constraints, so
there is no feasible region, and thus no optimal
solution.)
Max 2x1 6x2 s.t.
4x1 3x2 lt 12 2x1 x2 gt 8
x1, x2 gt 0
x2
2x1 x2 gt 8
8
4x1 3x2 lt 12
4
x1
3
4
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• In class exercise 79
• Solve the two-dimensional linear programming
  problem below. Be sure to include a graph of the
  feasible region in your solution.

Max 3x1 4x2 s.t. x1
x2 gt 5 3x1 x2 gt 8 x1, x2 gt 0
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(ANSWER)

• In class exercise 79
• Solve the two-dimensional linear programming
  problem below. Be sure to include a graph of the
  feasible region in your solution.

(The feasible region is unbounded and the
objective function can be increased infinitely.)
Max 3x1 4x2 s.t. x1
x2 gt 5 3x1 x2 gt 8 x1, x2 gt 0
x2
3x1 x2 gt 8
8
Max 3x1 4x2
5
x1 x2 gt 5
x1
5
2.67
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Linear Programming with More Than 2
Variables Chapters 8, 9 and 10 have a number of
examples of linear programming with more than 2
variables. For these, instead of a graphical
solution it makes sense to use software packages.
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Linear Programming with More Than 3
Variables Using software, most large LP problems
can be solved with just a few minutes of computer
time. Small LP problems usually require only a
few seconds. Linear programming solvers are now
part of many spreadsheet packages, such as
Microsoft Excel. In the case of more than 2
variables, we will concentrate on being able to
formulate the problems instead of learning
specific software packages as in the following in
class exercises.
33

• In class exercise 80 (p. 356)
• For the following, write down the correct and
  complete formulation of the linear programming
  problem. Write all inequalities with the
  variables in a consistent order on the left side
  and the constant on the right side.

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• In class exercise 81
• For the following, write down the correct and
  complete formulation of the linear programming
  problem. Write all inequalities with the
  variables in a consistent order on the left side
  and the constant on the right side. (The
  objective for this one is to minimize the cost.)