Advanced
CHOOSING THE BEST ROUTE
Because time and money are limited resources, it’s important that we make
well

informed decisions. Engineers go through a decision making process that
involves collecting data, analyzing data, and then making a decision
from
there. This is demonstrated in the following example:
Fastest Route
Predict what you think is the fastest way to get from point A to point B.
Your prediction, from fastest to shortest: ___ ___ ___
Now take the string in front of you and co
mpare how long each one is .
Were you correct?
Notice that on all 3 routes, you start and end at the same two points. The only
thing that changes is the way you get there.
Knowing this, what kind of rule can you write about the shortest distance
betwe
en two points?
Rule:
So in real life, we are not going to use string to measure everything. We can
use math to figure out what the shortest distance is. First we’ll need to learn a
little bit more about numbers and math.
Fun
ctions
Functions are
equations that define rules for numbers. Every function has an
input, a transformation, and an output. If this is not clear

think about your
toaster. You put a piece of bread in, define a setting, and out comes a piece of
toast. You don’t always have to
put bread in a toaster either

you can put in a
pop tart, a bagel, and even a toaster strudel. This is similar to functions. So
here’s what we’re trying to understand then:
Baked Good
Toaster
Toasted Baked Good
Input
Transformation
Output
Here is an example of a function:
f(x) = 5x + 3
f(x) tells us it’s a function
“x” will be our input.
5x + 3 is our “transformation.” In other words, it’s our rule for the
number we put in.
Let’s say that we would like 2 to be our input, and see it
transform. We would
write that like this:
f(2) = 5 * 2 + 3
To solve this function, you simply compute the equation.
f(2) = 5 * 2 + 3 = 13
W
e now can say f(2)=13
Try to do the following problems. The first one has been done for you.
Function
f(x)
Input
x
Rewrite equations
with input
Answer
f(x) = ?
f(x) = 2(x+3)
4
f(2) = 2(4+3)
f(2) = 14
f(x)=x
2

7
3
f(3) = 3
2

7
f(3) = 2
f(x)=
7
昨砩f
㈵
昨砩f
㤠

x
3
2
Functions are extremely important in Industrial and Systems Engineering.
When engineers make decisions for a project, these decisions need to follow
certain rules. The way they define these
rules is often with functions.
F
unctions and Graphing
Functions define a rule for numbers. When it comes to graphing, functions
define the relationship between x and y where one x corresponds to
only one
y.
This is what a graph looks like:
Using
the terms we learned earlier,
you graph your input as “x” and your
output as “y”.
Our function is f(x) = 2x
If x= 2, f(x)= 4, which is also our y

value. When you write a point on a graph,
you write it in the form (x,y). So thi
s point on our graph is written as (2,4). We
first move along the x

axis 2, and then up the y

axis 4.
The more inputs you put into your function, the more points you will get.
Eventually you will be able to just “connect the dots” to the get the line of your
graph. Will in the following table for the function f(x)=2x and graph the points
on the graph below
.
Then connect the dots to form a straight line.
x
f(x)
1
2
4
3
4
5
y
x
A graph consists of a vertical
y

axis, and a horizontal x

axis.
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5
1 2 3 4 5 6 7 8 9 10
Here are some common functions, along with their graphs:
y = x
y = x
2
y = x
3
y=
y =  x 
y =
Math Baseball
Now we are going to play some math baseball!
The class will now be split into two groups. This will be your baseball team. A
baseball diamond is drawn on the board.
Each team will be given a list of math problems.
Whichever team is playin
g defense, or in other words out in the field, will
decide which problems they want to “pitch” to the team at bat. The team who
is up at bat sends a batter to attempt the problem. If this team member gets
the problem right, they go to first base. If they g
et it wrong, it is an out. The
game has the same rules as baseball from here!
CHOOSING
THE BEST SOLUTION
Operations Research: Linear Programming
What is Operations Research?
Operations research (OR) is a scientific approach to analyzing problems and making decisions. OR
professionals aim to provide rational bases for decision making by seeking to understand and structure complex
situations and to use this understanding to pred
ict system behavior and improve system performance. Much of
this work is done using analytical and numerical techniques to develop and manipulate mathematical and
computer models of organizational systems composed of people, machines, and procedures.
In t
he 1950s operations research evolved into a profession with the formation of national societies,
establishment of journals and academic departments in universities. The use of operations research expanded
beyond the military to include both private compani
es and other governmental organizations. The petrochemical
industry was one of the first to broadly embrace operations research to improve the performance of plants,
develop natural resources and plan strategy. Today, operations research plays important ro
les in a variety of
industries such as:
airline

scheduling planes and crews, pricing tickets, taking reservations, and planning the size of the
fleet,
pharmaceutical

R& D management,
logistics companies

routing and planning,
financial services

credit scoring, marketing, and internal operations,
lumber and wood products

managing forests and cutting timber,
local government

deployment of emergency services, and
policy studies and regulation

environmental pollution, air traffic safety, AI
DS, and criminal justice
policy.
The field of operations research is closely rel
ated to Industrial engineering. Industrial engineers
consider
Operations Research (OR) techniques to be a major part of their toolset.
Some of the primary tools used by
operations researchers are statistics, optimization, probability theory, queuing
theory, game theory, graph theory, decision analysis, mathematical modeling and simulation. Because of the
computational nature of these fields, OR also has ties to computer s
cience, and operations researchers use
custom

written and off

the

shelf software.
Linear Programming is a method often used to solve large, complicated problems. These problems often require us to determine
how to use resources most
efficiently. In order
to see how linear programming is used to solve such problems, let’s investigate a problem experienced at the ACME Shoe Factor
y. The
ACME Shoe Factory problem is just a fraction of the size of those actually encountered in the real world.
Industrial Engine
ers use linear algebra to find the best solution possible to the problems given certain limits or constraints. For instance y
ou might want to buy as
many candy bars as possible but you only have $3 dollars to spend. That is an example of a constraint; you
are limited in the number of candy bars you can buy
by how much money you can spend.
Advanced Module: Linear Programming
Step 1: Understanding the problem
The ACME Shoe Factory makes two types of sport shoes, the
Airheads
and the
Groundeds
. Now let’s say
you are the Industrial Engineer at the ACME Shoe
Factory. Your boss tells you to figure out how many pairs of each type of shoe to make each week so that you can make the mos
t money.
The ACME Shoe Factory earns $10 of profit for each pair of
Airheads
and
$8.50 for each pair of
Groundeds
they make. There are two steps to make a pair
of shoes at the ACME Shoe Factory. The first step is to cut all the material to the right size on the Super

Cutter 9000 machine. The second step is to
assemble all the pieces t
ogether using the EZ

Integrator machine.
The number of machines
and factory
working
hours put constraints on the number of pairs of shoes that the ACME Shoe Factory can make. The factory only
has 6 Super

Cutter 9000 machines that cut the materials, 850 EZ

Integrator machines that assemble the shoes, and the factory is only works 40 hours
a week.
Each hour, each Super

Cutter machine can only do 50 minutes of work because it needs 10 minutes each hour to cool off. Each pair of
Airheads
requires 3
minutes of cutting time while
Groundeds
only require 2 minutes.
Each EZ

Integrator machine ta
kes 7 hours to assemble a pair of
Airheads
and 8 hours to
assemble a pair of
Groundeds
.
Your goal is to change the amount of each type of shoe being produced in order to make the most money, which is the same thin
g as saying your goal is to
maximize
your
profits by changing the amount of
Airheads
and
Groundeds
produced.
Decision variables
represent a quantity that
you can change.
Example: the number of
each type of shoe to be
made.
Decision Variables for ACME
Shoe Factory
A
= number of pairs of
Airheads
produced each week
G
= number of pairs of
Groundeds
produced each week
Objective function
the equation that
represents the goal
of either maximizing
profit or minimizing
cost
Constraints
represent limitations
created
by scarce
resources
(time,
equipment, e
tc.).
They are expressed
algebraically by
inequalities.
Step 2: Modeling the problem
Why do we use
≤
12,000 minutes instead of
=
12,000 minutes?
In order to solve this problem you will need to first describe mathematically what is going on. You know what your goal
is. Your goal is to
maximize
your profits by changing the
number of
Airheads
produced each week and by changing
the
number of
Groundeds
produced each week. The
Airheads
produced and the
Groundeds
produced are the only two
quantities you can actually choose. They are the only part of the problem that require you to make a decision about. We
call these
D
ecision variables.
To
represent
your goal mathematically, you need to describe what your profit is.
PROFIT = Profit from
Airheads
+ Profit from
Groundeds
Which is the same thing as saying:
PROFIT = (Price per pair
Airhead
) x (#
Airheads
Produced) + (Price per pair
Grounded
s
) x (#
Groundeds
Produced)
PROFT =
Max.
($10 x A) + ($8.50 x G)
This function is called the
Objective function.
N
ext you will need to describe mathematically the constraints on the machines and factory working hours. The first
thing to do is figure out how much
time during the week the Supper

Cutter 9000 is actually working in minutes. This is
very simple to do mathematically:
(6 Super

Cutter 9000 machines) x (40 work hours per week) x (50 mins per hour) = 12,000 min per week
This means that during the week the
most the Super

Cutter can work is 12,000 mins. The Super

Cutter 9000’s job is just
to cut the necessary material for both the
Airheads
and the
Groundeds
. The total amount of time cutting for the two
shoes cannot exceed total number of minutes the Super

Cutter is actually working each week. Remembering that it takes
the Super

Cutter 9000 3 minutes to cut the material for a single pair of
Airheads
and that
it takes 2 minutes to cut the
material for a single pair of
Groundeds
, we can mathematically represent this as:
(Cutting Time of
Airheads
) x (#
Airheads
) + (Cutting Time of
Groundeds
) x (#
Groundeds
) ≤ (Total Cutting Time)
Which is the same as:
(3 mins x
A) + (2 mins x G) ≤ 12,000 We call this function a
constraint.
0
1000
2000
3000
4000
5000
6000
7000
0
1000
2000
3000
4000
5000
Constraint 1
Constraint 2
At this point, we have mathematically represented the limitations of the Super

Cutter 9000. Now we need to the same thing for the EZ

Integrator. How many hours of work
can 850 EZ

Integrator machines do in a week? This can also be solved fairly easily.
(850 EZ

Integrator machines) x (40 work hours per week) = 34,000 hours per week
This means that during the week the most the EZ

Integrator can work is 34,000 hours. Just
like with the Super

Cutter 9000, the EZ

Integrators job is to assemble the material for both the
Airheads
and the
Groundeds
. The total amount of time assembling the two types of
shoes cannot exceed total number of hours the EZ

Integrator is actually workin
g each week. Remembering that it takes the EZ

Integrator 7
hours to assemble a single pair of
Airheads
and that it takes 8 hours to assemble a single pair of
Groundeds
, we can mathematically
represent this as:
(Assembly Time of
Airheads
) x (#
Airheads
) +
(Assembly Time of
Groundeds
) x (#
Groundeds
) ≤ (Total Assembly Time)
Which is the same as:
(7 hours x A) + (8 hours x G) ≤ 34,000 This is also a
constraint.
It would never be possible to produce negative quantities of either
Airheads
or
Groundeds
, but you could produce zero
Airheads
and
produce all
Groundeds
. Mathematically this is the same as saying:
A≥ 0 and G≥ 0.
This is just saying that A and G cannot be negative. These are also
constraints.
You have now cons
tructed a complete model for what is going on at ACME Shoe Factory. The completed model mathematically looks like:
Objective Function:
PROFT = Max. ($10 x A) + ($8.50 x G)
Constraints:
3A + 2G ≤ 12000
7A + 8G ≤ 34000
A≥ 0
G≥
0
We call this a
linear program
.
Step 3: Graphing a Linear Program

Feasible Region
The graph of this system of constraints appears below. The shaded region represents the set of
points which satisfy all of the constraints. Values of
A
and
G
which satisfy all of the constraints are called
feasible
and
the set of all such feasible points is called the
feasible region.
Feasible Region
All possible solutions to
the problem lie in the
feasible region or on the
boundary
Graph
G
A
0
1000
2000
3000
4000
5000
6000
7000
0
1000
2000
3000
4000
5000
Constraint 1
Constraint 2
Step 4: Best Production Plan: Searching for the Optimal Solution
We now have a list (or rather graph) of feasible solutions to the original problem. Now we need to evaluate the possible solu
tions and
pick the best one. We call this optimizing and we call the best solution the
optimal solution.
To determine the optimal
solution, there are many strategies you could use. One way is to try all the
possible answers. Lets pick the values A = 1000 and G = 3000. Plugging it into the objective (profit)
equation we get:
A=Airheads G=Groundeds Pro
fit = $10.00*A+$8.50*G
A=1,000 G=3,000 Profit = $10.00*(1,000)+$8.50*(3,000)
= $35,500
However, trying to pick points this way is a rather time con
suming and inefficient way to go about
finding the best solution. Fortunately, there is another way of finding optimal solutions. In fact, there are
many ways. We will briefly examine a few of them here.
Optimal Solution
The optimal
solution yields the
best solution (e.g.
the most profit or
the least cost)
The point we picked earlier is represented on the graph above by the circle. As we can see, this point is within the feasible
region. However,
without too much
difficulty we can find a better solution. Lets pick the point (1000,3375), represented on the graph by a square. Plugging thi
s
into the objective function (profit function) yields:
A=Airheads G=Groundeds Profit = $10.00*A+$
8.50*G
A=1,000 G=3,375 Profit = $10.00*(1,000)+$8.50*(3,375)
= $38,687.50
This new solution is greater than the old solution, therefore, it is more
optimal. If we continue with this trend of picking points, we find that
the most optimal solutions lay on the boarders of the feasible region. Thinking back over the last two points we picked, this
makes sense
because in the case of the first point we wer
e only producing 3,000 Groundeds when we had enough resources to build as many as 3,375.
Graph
G
A
0
1000
2000
3000
4000
5000
6000
7000
0
1000
2000
3000
4000
5000
Constraint 1
Constraint 2
If we continue to pick points, but this time only from the border of the feasible region, we
will find that the most optimal solutions occur at the four corners of the feasible region.
This is what we call the
Corner Principle
.
The reason for this is
that once we find a point at one of the borders of the feasible region
any better solutions (more optimal solutions) can only be found by traveling up or down the
border until we run into the next side of the border. The graph below illustrates this as wel
l
as representing the corners of the feasible region with circles.
The Corner Principle
states that the
optimal solution
will always lie
on a corner of
the feasible
We can now check the four corner points to see which one is most optimal (greatest
profit) of the four.
A=Airheads G=Groundeds Profit = $10.00*A+$8.50*G
A=0 G=0 Profit = $10.00*(0)+$8.50*(0)
= $0.00
A=Airheads G=Groundeds Profit = $10.00*A+$8.50*G
A=0 G= 4,250 Profit = $10.00*(0)+$8.50*(4,250)
= $36
,125.00
A=Airheads G=Groundeds Profit = $10.00*A+$8.50*G
A=2,800 G=1,800 Profit = $10.00*(2,800)+$8.50*(1,800)
= $4
3,300.00
A=Airheads G=Groundeds Profit = $10.00*A+$8.50*G
A=4,000 G=0 Profit = $10.00*(4,000)+$8.50*(0)
= $40,0
00.00
From plugging in the corner points we see that the point (A=2,800,G=1,800) is the most optimal that is, it’s the most
profitable. Since the point is in the feasible region, we know it meets the constraints.
What does this mean? What we have shown
is that mathematically, the ACME Shoe Factory can make the most money
by producing 2,800 Airheads and 1,800 Groundeds. With this production plan, the company should be able to $43,300
in profit.
Graph
G
A
Workforce Planning at Pizza
π's:
How to Minimize Costs and Still Get the Job Done
Linear programming is one of the most powerful mathematical modeling tools in the field of Operations Research. It helps mana
gers find the best
ways to allocate limite
d resources in order to maximize profits or minimize costs (optimization). For example, McDonald's franchises have used
linear programming to develop worker schedules which minimize their labor costs. Every
linear programming
model includes:
1.) Decision v
ariables,
2.) An objective function to be maximized or minimized, and
3.) A system of inequalities and equations that represents constraints that restrict the decision maker's options.
Anytime a business or corporation has more than one employee and more than one shift there is a scheduling problem. A
scheduling problem
involves determining the number of employees required to get the work done while minimizing the total daily wages. Many
businesses require
employees with different skills, but some employees may be able to do more than one job. The process of matching up individua
ls with tasks is
called
manpower planning.
This process becomes more complex when you have to determine a speci
fic work schedule. The most complex
manpower planning problem involves the airlines because they have to coordinate flight crews, ground crews, ticket agents and
airport staff for
flights around the world. Problems of this sort can be solved by a linear pr
ogramming process.
Sample problem:
A new Pizza
outlet sells pizza for carry out, dining in and frozen pizzas to local groceries and gourmet shops. The Pizza
is open from noon
to midnight each day. Toni Pepperoni, the manager of the outlet, needs to
figure out how many people should be scheduled to work each of the
two 8 hour shifts. The daytime shift is from noon to 8 p.m., and the evening shift is from 4 p.m. to midnight. The two shifts
overlap during the
busy dinner period, which starts at about 4
p.m. and continues until about 8 p.m. Based on information about the average number of pizzas
ordered at other outlets, Toni has estimated the number of employees she needs in each four hour period to supply those pizza
s:
noon to 4 p.m.
4 p.m. to 8 p.m.
8
p.m. to midnight
6 employees
16 employees
8 employees
Employees are paid $5 per hour for hours worked between noon and 8 p.m. The pay per hour between 8 p.m. and midnight is $7 pe
r hour. Toni
must decide how many people to schedule on the noon to 8 p.m.
shift and how many on the 4 p.m. to midnight shift. There must be enough
workers on duty to complete the workload in each four hour period and the goal is to minimize the amount spent on daily wages
. Answer the
following:
1
2
3
What decisions must Toni mak
e?_______________________________________________________________
Recommend to Toni the number of workers she should schedule each shift. _____________________________
Why did you recommend these
numbers?_________________________________________________________
__________________________________________________________________________________________
Completing the questions which follow will determine if your recommendation is a workable solution
for Toni. You will also find a solution which
minimizes her cost, and will provide enough workers to complete all tasks.
4
Identify all of the variables in this problem. (Hint: read ahead if you get stuck here.)
____________________________________________
______________________________________________
Decision Variables and Constraints:
1
Workforce Planning at Pizza
π's
Two of the variables you should have identified are the number of people working the day
shift and the number of people working the evening shift. These are called
decision variables.
All linear programming problems have decision variables. Decision variables should
completely describe the decisions to be made. The two decision variables in th
e Pizza
problem can be represented by:
Decision variables
:
a set of
variable quantities completely
describing the decisions to be made
D
= number of people scheduled from noon to 8 p.m.;
E
= number of people scheduled from 4 p.m. to midnight.
5
If three
people work the day shift and five people work the evening shift, altogether how much will they make?
_________________________________________________________________________________________
6
Write an expression to represent the total daily wages paid to
employees if Toni schedules
D
workers for the day shift,
and
E
workers for the evening shift? Total daily wages = _______________________________________
In any linear programming problem the decision maker wants to maximize or
minimize some function of
the decision variables. This function is called the
objective function.
In the Pizza
problem Toni wants to minimize the total
daily wages. Therefore, the
objective function.
is: total daily wages = 40D +
48E
Objective Function:
a quantity
to be optimized
which is defined
in terms of the decision variables
Answering the following questions will help you to understand what is meant by a
constraint.
Constraints
restrictions on
the values of
one or more of
the decision
variables
7
8
9
Toni must have
at least
Toni must have at least
workers scheduled from noon to 4 p.m.; therefore,
D
_ _______.
workers scheduled from 8 p.m. to midnight, so
E
________.
Write an inequality to show the minimum number of workers needed from 4 p.m. to 8 p.m.:
Hint: Both d
ay and evening shifts are working. __________________________________.
A System of Inequalities

The Feasible Region:
These inequalities represent restrictions on the decision variables
D
and
E.
These restrictions are called
constraints.
The
optimal solution
is just the best solution to a particular problem. In this case, the values of
D
and
E
which satisfy all of the constraints and
minimize the total daily wages paid to employees is the
optimal solution.
You may recall that the objective fun
ction, total daily wages =
40D + 48E, defines this cost, where
D
represents the number of employees working the day shift, and
E
represents the number of employees
working the evening shift. So how
do
we find the optimal solution for this problem? In other
words, what is the lowest possible total daily wages
that still meets all of the constraints?
First, we need to find all of the values of the decision variables
D
and
E
which satisfy all three of the constraints:
D
>
6,
E
>
8, and
D
+
E
>
16.
2
Workforce Planning at Pizza
π's
One way to do this is to graph each inequality on the same coordinate system. We need to
decide which variable to plot on each axis. Suppose we agree to graph
D
on the horizontal axis
and
E
on the vertical
axis. On a sheet of your own graph paper, graph each of the three
inequalities above and lightly shade the region(s) of the coordinate system containing all of the
points satisfying all three inequalities.
The portion of the graph which you shaded is calle
d the
feasible region.
The feasible region
contains all of the points which
could
be the optimal solution, because they satisfy all of the
constraints.
10
How many points lie in the feasible region? _________________________________
11
Feasible Region:
the set of all
points which satisfy all of
the constraints
Write the coordinates of three points that lie in the feasible region. Include one point that lies on the boundary of the
feasible region. (
,
); (
,
); (
,
) Also mark
and label the points you choose on your graph.
12
Why are boundary points part of the feasible region? ________________________________________________________
13
Does the point (6,8) lie in the feasible region? ______Why or why not?
_________________________________
The Search for Optimality

Finding the Best Schedule:
Now that we know the location of all points satisfying the constraints (the feasible region), we can begin to look for the op
timal solution. The
optimal solution must
be one of the points in the feasible region. But which point? Let's begin by considering a point that we know is feasible.
The point (9,10) lies in the feasible region.
14
What does this point represent? ____________________________________________________
_______________
If there are 9 day employees and 10 evening employees, then we can use the objective function to find the total daily wages o
f these employees:
Total daily wages = 40
D
+ 48
E
= 40(9) + 48(10) = 840.
15
D
E
Find the total daily wages for
each of the three points in the feasible region which you identified in # 11.
Total daily wages = 40
D
+ 48
E
16
Did any of your points produce total daily wages
lower than $840? ______________
17
Are there any points in the feasible region that produce
total daily wages lower than $840? _______________
,
)
18
Which of your points produced the lowest total daily wages? (
19
What were the total daily wages for that point? ____________________
To help find the optimal solution, we will use the graph of the
line 40D + 48
E
= 840. Add the graph of this equation to your graph of the feasible
region, and label it line
l.
Next, using your answer from #18, add the graph of
40
D
+ 48
E
=
your
lowest total daily wages to the previous graph and label it line
m.
20
What do you notice about lines
l
and
m?
________________________________________________________
Check with two of your neighbors to see if they have a similar result. (Recall that parallel lines have the same slope.)
Now compare the structure of these two
equations:
line
l:
40D + 48
E
= 840 line
m:
40
D
+ 48
E
=
your
lowest total daily wages
3
Workforce Planning at Pizza
π's
Angelina used the point (12,9) to generate total daily wages of $912.
21
22
23
What equation did she use to
determine these total daily wages? ______________________________________
Where do you think Angelina's line is with respect to line
l?
________________________________________
Where is her line with respect to
your
line
m?
____________________________________________________
Michael used the point (5,5) to compute total daily wages of $440. Add the graph of Michael's line to your graph
24
25
Is Michael's line parallel to line
l?
________
Could Michael's line contain the opti
mal solution? ________ Why or why not? ______________________________
On your graph, try to draw the line that represents the optimal solution.
Remember that the "optimal line" will intersect the feasible region, yet
have the lowest possible total daily
wages. Notice that any other line
parallel to the optimal line which has
lower total daily wages fails to intersect the feasible region, while
any other line parallel to the optimal line which intersects the
feasible region has higher total daily wages.
Op
timal Solution
:
the set of values of the decision
variables which satisfies all of the constraints and
achieves the goal of minimizing (or maximizing) the
objective function
The
optimal solution
occurs at the point where the optimal line intersects the
fea
sible region.
In this case, that point has the coordinates (8,8).
26
27
What does this point mean in terms of the problem? _____________________________________________________
What is the lowest total daily wages?
_________________________________________________________________
28
How would you describe the location of the point (8,8) in the
feasible region? __________________________________________
If there is a
unique solution
to a linear programming
problem, it
must occur at
one of the corner points
of the feasible
region.
29
Thinking about the original problem situation, why does this
solution make sense? _____________________________________
The point (8,8) is called a
corner point
of the feasible region. The
feasible region
for this problem has one other corner point.
30
What are the coordinates of the other corner point? (
,
)
4
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