1. OPSM 301 Operations Management
Koç University
OPSM 301 Operations Management
Class 10:
Introduction to Linear Programming
Zeynep Aksin

2. Assignment 2 due on Monday Midterm 1 next Wednesday
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

3. A Kristen’s like example
2 min/unit
10 min/unit
6 min/unit
Flow time T = = 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%

4. Tools: Gantt Chart
Gantt charts show the time at which different activities are performed, as well as the sequence of activities 1 2 3 4
activities
Resources
time

5. Three Workers R3 R2 R1 R1 2 min/unit R2 10 min/unit R3 6 min/unit W2 R2 R1

6. Three Workers R3 R2 R1 Throughput time for a rush order of 1 unit
System cycle time R3 R2
Throughput time for
an order of 5 units R1

7. Two Workers W2 W1 R3 R2 R1

8. 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

9. Product Mix Decisions: Kristen Cookies offers 2 products
Sale Price of Chocolate Chip Cookies: $5.00/dozen
Cost of Materials: $2.50/dozen
Sale Price of Oatmeal Raisin Cookies: $5.50/dozen
Cost of Materials: $2.40/dozen
Maximum weekly demand of
Chocolate Chip Cookies: 100 dozen
Oatmeal Raisin Cookies: 50 dozen
Total weekly operating expense $270

10. Product Mix Decisions
Total time available in week: 20 hrs

11. Margin per dozen Chocolate Chip cookies = $2.50
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

12. 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?

13. Linear programming

14. 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)

15. We all face decision about how to use limited resources such as:
Introduction
We all face decision about how to use limited resources such as:
time
money
workers/manpower

16. Mathematical Programming...
find the optimal, or most efficient, way of using limited resources to achieve objectives.
Optimization

17. 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

18. 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

19. 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 non-negative

20. 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

21. LP for Optimal Product Mix Selection
xcc: Dozens of chocolate chip cookies sold.
xor: Dozens of oatmeal raisin cookies sold.
Max 2.5 xcc xor
subject to
8 xcc xor < 1200
10 xcc xor < 1200
4 xcc xor < 1200
xcc < 100
xor <
Technology Constraints
Market Constraints

22. Solving the LP using Excel Solver
Number to make
100
Total profit
Unit Profits
2.5
3.1
Constraints
Value
RHS (constraint)
You 8 5
1200
Oven 10 15
Room Mate 4
Market cc 1
Market or 50
Optimal product-mix
Optimal Profit
Constraint not
binding in optimal
solution

23. Reading the variable information
The optimal solution for Kristen’s is to produce, 100 dozen chocolate chip and dozen oatmeal raisin resulting in an optimal profit of $ (This is the maximum possible profit attainable with the current resources)

24. 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

25. 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

26. 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.

27. 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.

28. 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.

29. 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 = 350X X2

30. 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

31. Summary of the LP Model for Blue Ridge Hot Tubs
MAX: 350X X2
S.T.: 1X1 + 1X2 <= 200
9X1 + 6X2 <= 1566
12X1 + 16X2 <= 2880
X1 >= 0
X2 >= 0

32. 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 <= or X1 <=174
3rd constraint: 12X1 <= 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!

33. 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.

34. Let’s Implement a Model for the Blue Ridge Hot Tubs Example...
MAX: 350X X2 } profit
S.T.: 1X1 + 1X2 <= 200 } pumps
9X1 + 6X2 <= 1566 } labor
12X1 + 16X2 <= 2880 } tubing
X1, X2 >= 0 } nonnegativity

35. 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

36. Solve the LP using solver
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.

37. Implementing the Model