OPSM 301 Operations Management

Class 10:

Introduction to Linear Programming

Koç University

Zeynep Aksinzaksin@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

10 min/unit2 min/unit 6 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%

R1 R2 R3

Tools: Gantt Chart

Gantt charts show the time at which different activities are performed, as well as the sequence of activities

Res

ourc

es

1

2

3

4

time

activities

R2

10 min/unitR1

2 min/unitR3

6 min/unit

R1

R2

R3

10 20 30 40 50 60

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

Three Workers

W1 W2 W3

R1

R2

R3

10 20 30 40 50 60

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

Three Workers

Throughput time for an order of 5 units

System cycle time

Throughput time for a rush order of 1 unit

R1

R2

R3

10 20 30 40 50 60

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

W1

W2

Two Workers

1 2 3 4 5

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: $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

Product Mix Decisions

ProcessingTimes

Mix &Dish

Load Oven& Set

Bake Cool Pack Receive Total

ChocolateChip

8 mins.(6+2/tray)

1 9 5 2/tray 1 26

OatmealRaisin

5 mins.(3+2/tray)

1 14 2 2/tray 1 25

Resource You RM+Oven Oven RM RM

Total time available in week: 20 hrs

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 non-

negative

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: 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 < 1200

10 xcc + 15 xor < 1200

4 xcc + 4 xor < 1200

xcc < 100

xor < 50

Technology Constraints

Market Constraints

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

Constraint not binding in optimalsolution

Optimal Profit

Optimal product-mix

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 ProblemBlue 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.

Aqua-Spa Hydro-LuxPumps 1 1Labor 9 hours 6 hoursTubing 12 feet 16 feetUnit Profit $350 $300

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: 1X1 <= 200

• 2nd constraint: 9X1 <=1566 or X1 <=174

• 3rd constraint: 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 } profit

S.T.: 1X1 + 1X2 <= 200} pumps

9X1 + 6X2 <= 1566 } labor

12X1 + 16X2 <= 2880 } tubing

X1, X2 >= 0 } 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