OPSM 301 Operations Management

1
OPSM 301 Operations Management
Koç University

• Class 10
• Introduction to Linear Programming

Zeynep Aksin zaksin_at_ku.edu.tr
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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 Kristens like example
R1
R2
R3
10 min/unit
2 min/unit
6 min/unit

• Flow time T 2106 18 min.
• System cycle time 1/R 10 min.
• Throughput rate R 6 units / hour
• Utilizations R1 2/1020
• R2100 (bottleneck)
• R36/1060

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 6 min/unit
R2 10 min/unit
R1 2 min/unit
W2
W3
W1
R3
1 2 3
4 5
R2
1 2 3
4 5
R1
1 2 3 4 5
10 20 30
40 50 60

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Three Workers
Throughput time for a rush order of 1 unit
System cycle time
R3
1 2 3
4 5
R2
1 2 3
4 5
Throughput time for an order of 5 units
R1
1 2 3 4 5
10 20 30
40 50 60

7
Two Workers
W2
W1
R3
1 2 3
4 5
R2
1 2 3
4 5
R1
1 2 3 4 5
1 2 3 4 5
10 20 30
40 50 60

8
Continue Kristens 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
• Maximum weekly demand of
• Oatmeal Raisin Cookies 50 dozen
• Total weekly operating expense 270

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Product Mix Decisions
Total time available in week 20 hrs
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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

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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 1202.5300
• 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 803.10248
• But my demand is only 50 dozen!
• What about a mix of chocolate chip and oatmeal
  raisin? What is the best product mix?

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Linear programming
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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)

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Introduction

• We all face decision about how to use limited
  resources such as
• time
• money
• workers/manpower

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Mathematical Programming...

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

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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

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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

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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

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Components of Linear Programming

• There are three types of constraints
• (lt) An upper limit on the amount of some scarce
  resource
• (gt) 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

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LP for Optimal Product Mix Selection

• xcc Dozens of chocolate chip cookies sold.
• xor Dozens of oatmeal raisin cookies sold.
• Max 2.5 xcc 3.1 xor
• subject to
• 8 xcc 5 xor lt 1200
• 10 xcc 15 xor lt 1200
• 4 xcc 4 xor lt 1200
• xcc lt 100
• xor lt 50

Technology Constraints
Market Constraints
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Solving the LP using Excel Solver



Number to make 100 13.33333 Total profit
Unit Profits 2.5 3.1 291.3333333

Constraints Value RHS (constraint)
You 8 5 866.6667 1200
Oven 10 15 1200 1200
Room Mate 4 4 453.3333 1200
Market cc 1 0 100 100
Market or 0 1 13.33333 50





Optimal product-mix
Optimal Profit
Constraint not binding in optimal solution
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Reading the variable information

• The optimal solution for Kristens 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)

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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

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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

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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.

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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.

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An Example LP Problem
Blue Ridge Hot Tubs produces two types of hot
tubs Aqua-Spas Hydro-Luxes. Find profit
maximizing product-mix.
There are 200 pumps, 1566 hours of labor, and
2880 feet of tubing available.
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5 Steps In Formulating LP Models

• 1. Understand the problem
• 2. Identify the decision variables
• X1number of Aqua-Spas to produce
• X2number of Hydro-Luxes to produce
• 3. State the objective function as a linear
  combination of the decision variables
• MAX Profit 350X1 300X2

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5 Steps In Formulating LP Models(continued)

• 4. State the constraints as linear combinations
  of the decision variables.
• 1X1 1X2 lt 200 pumps
• 9X1 6X2 lt 1566 labor
• 12X1 16X2 lt 2880 tubing
• 5. Identify any upper or lower bounds on the
  decision variables.
• X1 gt 0
• X2 gt 0

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Summary of the LP Model for Blue Ridge Hot Tubs
MAX 350X1 300X2 S.T. 1X1 1X2 lt 200 9X1
6X2 lt 1566 12X1 16X2 lt 2880 X1 gt 0 X2
gt 0
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Solving LP ProblemsAn Intuitive Approach

• Idea Each Aqua-Spa (X1) generates the highest
  unit profit (350), so lets make as many of them
  as possible!
• How many would that be?
• Let X2 0
• 1st constraint 1X1 lt 200
• 2nd constraint 9X1 lt1566 or X1 lt174
• 3rd constraint 12X1 lt 2880 or X1 lt 240
• If X20, the maximum value of X1 is 174 and the
  total profit is 350174 3000 60,900
• This solution is feasible, but is it optimal?
• No!

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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.

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Lets Implement a Model for the Blue Ridge Hot
Tubs Example...
MAX 350X1 300X2 profit S.T. 1X1 1X2 lt
200 pumps 9X1 6X2 lt 1566 labor 12X1
16X2 lt 2880 tubing X1, X2 gt 0
nonnegativity
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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

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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.

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Implementing the Model