Koç University
OPSM 301 Operations Management
Class 10:
Introduction to Linear
Programming
Zeynep Aksin
zaksin@ku.edu.tr
Announcements
 Assignment 2 due on Monday
 Midterm 1 next Wednesday
 On Monday October 31, OPSM 301 class
will be held in the computer lab SOS 180
– Graded class participation activity
– Will show how to use Excel Solver to solve
linear programs
– You will need this for assignment 3
A Kristen’s like example
R1
R2
2 min/unit
10 min/unit
•Flow time T = 2+10+6 = 18 min.
•System cycle time 1/R= 10 min.
•Throughput rate R= 6 units / hour
•Utilizations: R1: 2/10=20%
R2=100% (bottleneck)
R3=6/10=60%
R3
6 min/unit
Tools: Gantt Chart
Gantt charts show the time at which different
activities are performed, as well as the sequence of
activities
Resources
1
activities
2
3
4
time
Three Workers
R2
R1
2 min/unit
10 min/unit
W2
W1
R3
R2
R1
1
1
R3
6 min/unit
2
2
W3
3
3
4
4
5
5
1 2 3 4 5
10
20
30
40
50
60
Three Workers
Throughput time for
a rush order of 1 unit
R3
R2
System cycle time
1
1
2
2
3
3
4
4
5
5
Throughput time for
an order of 5 units
R1
1 2 3 4 5
10
20
30
40
50
60
Two Workers
W2
W1
R3
R2
R1
1
1
2
2
3
3
4
4
5
5
1 2 3 4 5
10
20
30
40
50
60
Continue Kristen’s Cookie story..
 The business matures
 Demand information is available
 You and your roommate decide to focus
on chocolate chip or oatmeal raisin
cookies
Product Mix Decisions:
Kristen Cookies offers 2 products
Sale Price of Chocolate Chip Cookies:
Cost of Materials:
$5.00/dozen
$2.50/dozen
Sale Price of Oatmeal Raisin Cookies:
Cost of Materials:
$5.50/dozen
$2.40/dozen
Maximum weekly demand of
Chocolate Chip Cookies:
Maximum weekly demand of
Oatmeal Raisin Cookies:
Total weekly operating expense
100 dozen
50 dozen
$270
Product Mix Decisions
Total time available in week:
20 hrs
Processing
Times
Mix &
Dish
Chocolate
Chip
8 mins.
(6+2/tray)
1
9
Oatmeal
Raisin
5 mins.
(3+2/tray)
1
14
Resource
You
RM+Oven
Oven
Load Oven Bake Cool
& Set
Pack
Receive
Total
5
2/tray
1
26
2
2/tray
1
25
RM
RM
Product Mix Decisions
Margin per dozen Chocolate Chip cookies =
$2.50
Margin per dozen Oatmeal Raisin cookies =
$3.10
Margin per oven minute from Chocolate
Chip cookies = $2.50 / 10 = $ 0.250
Margin per oven minute from Oatmeal
Raisin cookies = $3.10 / 15 = $ 0.207
Baking only one type
 If I bake only chocolate chip:
– In 20 hours I can bake 120 dozen
– At a margin of 2.50 I will make 120*2.5=300
– But my demand is only 100 dozen!
 If I bake only oatmeal raisin:
– In 20 hours I can bake 80 dozen
– At a margin of 3.10 I will make 80*3.10=248
– But my demand is only 50 dozen!
 What about a mix of chocolate chip and
oatmeal raisin? What is the best product
mix?
Linear programming
Announcement
 Linear programming: Appendix A from
another book-copy in course pack
 Skip graphical solution, skip sensitivity
analysis for now
 You can use examples done in class,
example A1, solved problem 1, Problem 3
as a study set (and all other problems if
you like)
Introduction
 We all face decision about how to use
limited resources such as:
– time
– money
– workers/manpower
Mathematical Programming...
 find the optimal, or most efficient, way of
using limited resources to achieve
objectives.
 Optimization
Example Applications
 OPSM: Product mix problem-how much of
each product should be produced given
resource constraints to maximize profits
 Finance: Construct a portfolio of securities
that maximizes return while keeping "risk"
below a predetermined level
 Marketing: Develop an advertising strategy
to maximize exposure of potential
customers while staying within a
predetermined budget
Components of Linear Programming
 A specified objective or a single goal, such as the
maximization of profit, minimization of machine idle
time etc.
 Decision variables represent choices available to
the decision maker in terms of amounts of either
inputs or outputs
 Constraints are limitations which restrict the
alternatives available to decision makers
Conditions for Applicability of
Linear Programming
 Resources must be limited
 There must be an objective function
 There must be linearity in the constraints and in
the objective function
 Resources and products must be homogeneous
 Decision variables must be divisible and nonnegative
Components of Linear Programming
 There are three types of constraints:
– (=<) An upper limit on the amount of some scarce
resource
– (>=) A lower bound that must be achieved in the
final solution
– (=) An exact specification of what a decision
variable should be equal to in the final solution
 Parameters are fixed and given values which
determine the relationships between the decision
variables of the problem
LP for Optimal Product Mix Selection
xcc:
xor:
Dozens of chocolate chip cookies sold.
Dozens of oatmeal raisin cookies sold.
Max
2.5 xcc + 3.1 xor
subject to
8 xcc +
5 xor <
10 xcc +
15 xor <
4 xcc +
4 xor <
xcc
<
xor <
1200
1200
1200
Technology
Constraints
100
50
Market
Constraints

Solving the LP using Excel Solver
Optimal product-mix
Number to make
100
13.33333
Unit Profits
2.5
3.1
Constraints
You
Total profit
291.3333333
Value
RHS (constraint)
8
5
866.6667
1200
10
15
1200
1200
Room Mate
4
4
453.3333
1200
Market cc
1
0
100
100
Market or
0
1
13.33333
50
Oven
Optimal Profit
Constraint not
binding in optimal
solution
Reading the variable information
 The optimal solution for Kristen’s is to produce, 100
dozen chocolate chip and 13.33 dozen oatmeal raisin
resulting in an optimal profit of $291.33. (This is the
maximum possible profit attainable with the current
resources)
Follow me using the file on the network
drive
 Go to STORAGE
 E:\COURSES\UGRADS\OPSM301\SHARE
 Copy KristensLPexample.xls to your
desktop
 Open the spreadsheet and click on first
worksheet
How Solver Views the Model
 Target cell - the cell in the spreadsheet
that represents the objective function
 Changing cells - the cells in the
spreadsheet representing the decision
variables
 Constraint cells - the cells in the
spreadsheet representing the LHS
formulas on the constraints
Goals For Spreadsheet Design
 Communication - A spreadsheet's primary business
purpose is that of communicating information to managers.
 Reliability - The output a spreadsheet generates should be
correct and consistent.
 Auditability - A manager should be able to retrace the steps
followed to generate the different outputs from the model in order to
understand the model and verify results.
 Modifiability - A well-designed spreadsheet should be easy to
change or enhance in order to meet dynamic user requirements.
Lets consider a slightly different version
 Unit profits from Aqua-Spas is $325
 Available hours of labor is 1500
 Make the appropriate changes in your
spreadsheet and resolve.
An Example LP Problem
Blue Ridge Hot Tubs produces two types of hot
tubs: Aqua-Spas & Hydro-Luxes. Find profit
maximizing product-mix.
Aqua-Spa
Hydro-Lux
Pumps
1
1
Labor
9 hours
6 hours
Tubing
12 feet
16 feet
Unit Profit
$350
$300
There are 200 pumps, 1566 hours of labor,
and 2880 feet of tubing available.
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: Profit = 350X1 + 300X2
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
Summary of the 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
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:
• 2nd constraint:
• 3rd constraint:
1X1 <= 200
9X1 <=1566 or X1 <=174
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!
The Steps in Implementing an LP Model in a
Spreadsheet
1. Organize the data for the model on the spreadsheet.
2. Reserve separate cells in the spreadsheet to represent each
decision variable in the model.
3. Create a formula in a cell in the spreadsheet that corresponds to
the objective function.
4. For each constraint, create a formula in a separate cell in the
spreadsheet that corresponds to the left-hand side (LHS) of the
constraint.
Let’s Implement a Model for the
Blue Ridge Hot Tubs Example...
MAX: 350X1 + 300X2
S.T.: 1X1 + 1X2 <= 200
9X1 + 6X2 <= 1566
12X1 + 16X2 <= 2880
X1, X2 >= 0
} profit
} pumps
} labor
} tubing
} nonnegativity
Preparing Excel
 You need the Solver add-in
 First check whether you have this add-in
– Click on the DATA tab
– Check if you have Solver under Analysis (far right)
 If not
– Click on the Office Button (far left top)
– Click on Excel Options (bottom of dialogue box)
– Select Add-Ins from menu on the left
– Add Solver add-in from the right menu
In-class exercise
 Prepare a spreadsheet for the Blue Ridge
Hot Tubs product mix problem we just
formulated
 Solve the LP using solver
 Save the file with your name_lastname in
E:\COURSES\UGRADS\OPSM301\HOMEWORK
 Consider the following changes
– Unit profits from Aqua-Spas is $325
– Available hours of labor is 1500
– Make the appropriate changes in your spreadsheet and resolve.
Implementing the Model