Render/Stair/Hanna Chapter 1

© 2008 Prentice-Hall, Inc. 

Chapter 1 

To accompany 
Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna

Power Point slides created by Jeff Heyl 

Introduction to Quantitative Analysis 

© 2009 Prentice-Hall, Inc.

© 2009 Prentice-Hall, Inc.    1 – 2  

Introduction 

• Mathematical tools have been used for thousands of years
• Quantitative analysis can be applied to a wide variety of problems
• It’s not enough to just know the mathematics of a technique
• One must understand the specific applicability of the technique, its limitations, and its assumptions

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Need for Operations Management 

• The increased complexity of running a successful business.
• Many large companies with complex business processes have used OM for years to help executives and managers make good strategic and operational decisions.
• American Airlines and IBM have incredibly complex operations in logistics, customer service and resource allocation that are built on OM technologies.
• As the trend of increased business complexity moves to smaller enterprises, OM will play vital operational and strategic roles.

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Need for OM 

• Lots of information, but no decisions.
• Enterprise resource planning (ERP) systems and the Web have contributed to a pervasive information environment; decision-makers have total access to every piece of data in the organization.
• The problem is that most people need a way to transform this wealth of data into actionable information that helps them make good tactical and strategic decisions.
• The role of OM decision methods is to help leverage a company’s investment in information technology infrastructure by providing a way to convert data into actions.

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Need for OM 

• A large nationwide bank is using OM techniques to configure complicated financial instruments for their customers.
• A process that previously required a human agent and took minutes or hours to perform is now executed automatically in seconds on the bank’s Intranet.
• The resulting financial products are far superior to those produced by the manual process.

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Need for OM 

• A major retail enterprise is using OM methodology for making decisions about customer relationship management (CRM).
• They are using mathematical optimization to achieve the most profitable match between a large number of customer segments, a huge variety of products and services, and an expanding number of marketing and sales channels such

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Need for OM 

• Sears, Roebuck and Company
• Manages a U.S. fleet of more than 1,000 delivery vehicles, some company owned and some not.
• The company makes more than 4 million deliveries a year of 21,000 uniquely different items.
• It has 46 routing offices and provides the largest home delivery service of furniture and appliances in the United States.
• The company also operates a U.S. fleet of 12,500 service vehicles, together with an associated staff of service technicians.
• Service demand is on the order of 15 million calls per year and revenue generated is approximately $3 billion.

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Need for OM 

• OM researchers designed a system to deal with such variables as customer schedules and requested performance times, time estimates for the required service, vehicles and personnel available, skills needed, parts and product availability and so on.
• The system was designed to automatically schedule all facets of performance in such a way as to
• Provide accurate and convenient time windows for the Sears customer
• Minimize costs
• Maximize certain objective measures of task performance, including customer satisfaction.
• This effort generated a one time cost reduction of $9 million as well as ongoing savings of $42 million per year.

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Examples of Quantitative Analyses 

• Taco Bell saved over $150 million using forecasting and scheduling quantitative analysis models
• NBC television increased revenues by over $200 million by using quantitative analysis to develop better sales plans
• Continental Airlines saved over $40 million using quantitative analysis models to quickly recover from weather delays and other disruptions

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Meaningful

Information 

Quantitative

Analysis 

Quantitative analysis is a scientific approach to managerial decision making whereby raw data are processed and manipulated resulting in meaningful information 

Raw Data 

  What is Quantitative Analysis?

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Quantitative factors might be different investment alternatives, interest rates, inventory levels, demand, or labor cost

Qualitative factors such as the weather, state and federal legislation, and technology breakthroughs should also be considered

• Information may be difficult to quantify but can affect the decision-making process

 

  What is Quantitative Analysis?

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Implementing the Results 

Analyzing the Results 

Testing the Solution 

Developing a Solution 

Acquiring Input Data 

Developing a Model 

The Quantitative Analysis Approach 

Defining the Problem 

Figure 1.1

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Defining the Problem 

Need to develop a clear and concise statement that gives direction and meaning to the following steps

• This may be the most important and difficult step
• It is essential to go beyond symptoms and identify true causes
• May be necessary to concentrate on only a few of the problems – selecting the right problems is very important
• Specific and measurable objectives may have to be developed

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Developing a Model 

Quantitative analysis models are realistic, solvable, and understandable mathematical representations of a situation 

There are different types of models 

$ Advertising 

$ Sales 

Y = b₀ + b₁X 

Schematic models 

Scale models

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Developing a Model 

• Models generally contain variables (controllable and uncontrollable) and parameters
• Controllable variables are generally the decision variables and are generally unknown
• Parameters are known quantities that are a part of the problem

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Acquiring Input Data 

Input data must be accurate – GIGO rule 

Data may come from a variety of sources such as company reports, company documents, interviews, on-site direct measurement, or statistical sampling 

Garbage In 

Process 

Garbage Out

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Developing a Solution 

• The best (optimal) solution to a problem is found by manipulating the model variables until a solution is found that is practical and can be implemented
• Common techniques are
• Solving equations
• Trial and error – trying various approaches and picking the best result
• Complete enumeration – trying all possible values
• Using an algorithm – a series of repeating steps to reach a solution

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Testing the Solution 

Both input data and the model should be tested for accuracy before analysis and implementation

• New data can be collected to test the model
• Results should be logical, consistent, and represent the real situation

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Analyzing the Results 

Determine the implications of the solution

• Implementing results often requires change in an organization
• The impact of actions or changes needs to be studied and understood before implementation

 

Sensitivity analysis determines how much the results of the analysis will change if the model or input data changes

• Sensitive models should be very thoroughly tested

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Implementing the Results 

Implementation incorporates the solution into the company

• Implementation can be very difficult
• People can resist changes
• Many quantitative analysis efforts have failed because a good, workable solution was not properly implemented

Changes occur over time, so even successful implementations must be monitored to determine if modifications are necessary

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Modeling in the Real World 

Quantitative analysis models are used extensively by real organizations to solve real problems

• In the real world, quantitative analysis  models can be complex, expensive, and difficult to sell
• Following the steps in the process is an important component of success

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How To Develop a Quantitative Analysis Model 

• An important part of the quantitative analysis approach
• Let’s look at a simple mathematical model of profit

 

Profit = Revenue – Expenses

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How To Develop a Quantitative Analysis Model 

Expenses can be represented as the sum of fixed and variable costs and variable costs are the product of unit costs times the number of units 

Profit = Revenue – (Fixed cost + Variable cost)

Profit = (Selling price per unit)(number of units sold) – [Fixed cost + (Variable costs per unit)(Number of units sold)]

Profit = sX – [f + vX]

Profit = sX – f – vX 

where

   s = selling price per unit v = variable cost per unit

   f = fixed cost X = number of units sold

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How To Develop a Quantitative Analysis Model 

Expenses can be represented as the sum of fixed and variable costs and variable costs are the product of unit costs times the number of units 

Profit = Revenue – (Fixed cost + Variable cost)

Profit = (Selling price per unit)(number of units sold) – [Fixed cost + (Variable costs per unit)(Number of units sold)]

Profit = sX – [f + vX]

Profit = sX – f – vX 

where

   s = selling price per unit v = variable cost per unit

   f = fixed cost X = number of units sold 

The parameters of this model are f, v, and s as these are the inputs inherent in the model

The decision variable of interest is X

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Bagels ‘R Us 

Profits = Revenue - Expenses 

Profits = $1*Number Sold - $100 - $.50*Number Sold 

Assume you are the new owner of Bagels R Us and you want to develop a mathematical model for your daily profits and breakeven point. Your fixed overhead is $100 per day and your variable costs are 0.50 per bagel (these are GREAT bagels). You charge $1 per bagel. 

(Price per Unit)  (Number Sold)  

•   Fixed Cost  
  - (Variable Cost/Unit)    (Number Sold)

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Breakeven Example 

f=$100, s=$1,  v=$.50 

X=f/(s-v) 

X=100/(1-.5) 

X=200 

At this point, Profits are 0

    

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Pritchett’s Precious Time Pieces 

Profits = sX – f – vX 

The company buys, sells, and repairs old clocks. Rebuilt springs sell for $10 per unit. Fixed cost of equipment to build springs is $1,000. Variable cost for spring material is $5 per unit. 

      s = 10 f = 1,000 v = 5

Number of spring sets sold = X 

If sales = 0, profits = –$1,000

If sales = 1,000, profits = [(10)(1,000) – 1,000 – (5)(1,000)]

      = $4,000

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Pritchett’s Precious Time Pieces 

0 = sX – f – vX,     or     0 = (s – v)X – f 

Companies are often interested in their break-even point (BEP). The BEP is the number of units sold that will result in $0 profit. 

Solving for X, we have

f = (s – v)X 

X =  

f

s – v 

BEP =  

Fixed cost

(Selling price per unit) – (Variable cost per unit)

© 2009 Prentice-Hall, Inc.    1 – 29  

Pritchett’s Precious Time Pieces 

0 = sX – f – vX,     or     0 = (s – v)X – f 

Companies are often interested in their break-even point (BEP). The BEP is the number of units sold that will result in $0 profit. 

Solving for X, we have

f = (s – v)X 

X =  

f

s – v 

BEP =  

Fixed cost

(Selling price per unit) – (Variable cost per unit) 

BEP for Pritchett’s Precious Time Pieces 

BEP = $1,000/($10 – $5) = 200 units 

Sales of less than 200 units of rebuilt springs will result in a loss

Sales of over 200 units of rebuilt springs will result in a profit

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Examples 

1. Selling price $1.50, cost/bagel $.80, fixed cost $250 Breakeven point?
2. Seeking a profit of $1,000, selling price $1.25, cost/bagel $.50, 100 sold/day. What is fixed cost?
3. What selling price is needed to achieve a profit of $750 with a fixed cost of $75 and variable cost of $.50

 

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Examples 

Seeing a need for childcare in her community, Sue decided to launch her own daycare service. Her service needed to be affordable, so she decided to watch each child for $12 a day. After doing her homework, Sue came up with the following financial information:

Selling Price (per child per day) $12

Operating Expenses (per month)

Insurance 400 + Rent 200 = Total OE $600

Costs of goods sold $4.00 per unit

Meals 2 @ $1.50 (breakfast & lunch)

Snacks 2 @ $0.50

How many children will she need to watch on a monthly basis to breakeven?

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Examples 

Applying the formula, we have:

$600/($12-$4) = 75 

She has to have a total of 75 children in her program over the month to breakeven.

If she is open only 20 days per month then she needs

75/20=3.75 children per day on the average. 

Expenses per month $600 + 75*$4.00 = $900

Revenue per month 75*$12 = $900

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Advantages of Mathematical Modeling 

1. Models can accurately represent reality
2. Models can help a decision maker formulate problems
3. Models can give us insight and information
4. Models can save time and money in decision making and problem solving
5. A model may be the only way to solve large or complex problems in a timely fashion
6. A model can be used to communicate problems and solutions to others

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Models Categorized by Risk 

• Mathematical models that do not involve risk are called deterministic models
• We know all the values used in the model with complete certainty
• Mathematical models that involve risk, chance, or uncertainty are called probabilistic models
• Values used in the model are estimates based on probabilities

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Computers and Spreadsheet Models 

QM for Windows

• An easy to use decision support system for use in POM and QM courses
• This is the main menu of quantitative models

 

Program 1.1

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Computers and Spreadsheet Models 

Excel QM’s Main Menu (2003)

• Works automatically within Excel spreadsheets

 

Program 1.2A

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Computers and Spreadsheet Models 

Excel QM’s Main Menu (2007)  

Program 1.2B

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Computers and Spreadsheet Models 

Excel QM for the Break-Even Problem 

Program 1.3A

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Computers and Spreadsheet Models 

Excel QM Solution to the Break-Even Problem 

Program 1.3B

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Computers and Spreadsheet Models 

Using Goal Seek in the Break-Even Problem 

Program 1.4

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Computers and Spreadsheet Models 

Using Goal Seek in the Break-Even Problem 

Program 1.4

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Possible Problems in the Quantitative Analysis Approach 

Defining the problem

• Problems are not easily identified
• Conflicting viewpoints
• Impact on other departments
• Beginning assumptions
• Solution outdated

Developing a model

• Fitting the textbook models
• Understanding the model

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Possible Problems in the Quantitative Analysis Approach 

Acquiring input data

• Using accounting data
• Validity of data

Developing a solution

• Hard-to-understand mathematics
• Only one answer is limiting

Testing the solution

Analyzing the results

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Implementation –  
Not Just the Final Step 

Lack of commitment and resistance to change

• Management may fear the use of formal analysis processes will reduce their decision-making power
• Action-oriented managers may want “quick and dirty” techniques
• Management support and user involvement are important

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Implementation –  
Not Just the Final Step 

Lack of commitment by quantitative analysts

• An analysts should be involved with the problem and care about the solution
• Analysts should work with users and take their feelings into account