1. © 2008 Prentice-Hall, Inc. Chapter 1 Introduction to Quantitative Analysis

2. © 2009 Prentice-Hall, Inc. 1 – 2 Learning Objectives 1.Describe the quantitative analysis approach 2.Understand the application of quantitative analysis in a real situation 3.Describe the use of modeling in quantitative analysis 4.Discuss possible problems in using quantitative analysis 5.Perform a break-even analysis After completing this chapter, students will be able to:

3. © 2009 Prentice-Hall, Inc. 1 – 3 Chapter Outline 1.1Introduction 1.2What Is Quantitative Analysis? 1.3The Quantitative Analysis Approach 1.4How to Develop a Quantitative Analysis Model 1.5The Role of Computers and Spreadsheet Models in the Quantitative Analysis Approach 1.6Possible Problems in the Quantitative Analysis Approach 1.7Implementation — Not Just the Final Step

4. © 2009 Prentice-Hall, Inc. 1 – 4 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

5. © 2009 Prentice-Hall, Inc. 1 – 5 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 disruption

6. © 2009 Prentice-Hall, Inc. 1 – 6 Meaningful Information Quantitative Analysis 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?

7. © 2009 Prentice-Hall, Inc. 1 – 7 Scope of OR in Management Marketing and sales – product selection and competitive strategies, utilization of salesman, their time and territory control, frequency of visits in sales force analysis, marketing advertising decision, forecasting, pricing and market research decision etc. Production Management – product mix, maintenance policies, crew sizing and replacement system, project planning Quality Control decision and material handling facilities etc.

8. © 2009 Prentice-Hall, Inc. 1 – 8 Purchasing, Procurement and Inventory Controls- buying policies levels, negotiations, bidding policies, time and quality of procurement. Finance, Investment & Budgeting – Profit planning, cash flow analysis, investment policy for maximum return, dividend policies, investment decision, risk analysis and portfolio analysis etc.

9. © 2009 Prentice-Hall, Inc. 1 – 9 Quantitative factors Quantitative factors might be different investment alternatives, interest rates, inventory levels, demand, or labor cost Qualitative factors 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?

10. © 2009 Prentice-Hall, Inc. 1 – 10 AlternativeStarting Salary Potential for Advancement Job Location 1. Rochester$ 38,500Average 2. Dallas$ 36,000ExcellentGood 3. Greensboro$ 36,000GoodExcellent 4. Pittsburgh$ 37,000AverageGood Job evaluation decision- making problem

11. © 2009 Prentice-Hall, Inc. 1 – 11 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

12. © 2009 Prentice-Hall, Inc. 1 – 12 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

13. © 2009 Prentice-Hall, Inc. 1 – 13 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 = b 0 + b 1 X Schematic models Scale models

14. © 2009 Prentice-Hall, Inc. 1 – 14 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

15. © 2009 Prentice-Hall, Inc. 1 – 15 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

16. © 2009 Prentice-Hall, Inc. 1 – 16 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 Solving equations Trial and error Trial and error – trying various approaches and picking the best result Complete enumeration Complete enumeration – trying all possible values algorithm Using an algorithm – a series of repeating steps to reach a solution

17. © 2009 Prentice-Hall, Inc. 1 – 17 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

18. © 2009 Prentice-Hall, Inc. 1 – 18 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 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

19. © 2009 Prentice-Hall, Inc. 1 – 19 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

20. © 2009 Prentice-Hall, Inc. 1 – 20 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

21. © 2009 Prentice-Hall, Inc. 1 – 21 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

22. © 2009 Prentice-Hall, Inc. 1 – 22 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

23. © 2009 Prentice-Hall, Inc. 1 – 23 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

24. © 2009 Prentice-Hall, Inc. 1 – 24 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 –$1,000 If sales = 0, profits = –$1,000 If sales = 1,000, profits= [(10)(1,000) – 1,000 – (5)(1,000)] = $4,000

25. © 2009 Prentice-Hall, Inc. 1 – 25 Pritchett’s Precious Time Pieces 0 = sX – f – vX, or 0 = ( s – v ) X – f break-even point 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)

26. © 2009 Prentice-Hall, Inc. 1 – 26 Pritchett’s Precious Time Pieces 0 = sX – f – vX, or 0 = ( s – v ) X – f break-even point 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

27. © 2009 Prentice-Hall, Inc. 1 – 27 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

28. © 2009 Prentice-Hall, Inc. 1 – 28 Mathematical Models Objective Function – a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing cost Constraints – a set of restrictions or limitations, such as production capacities Uncontrollable Inputs – environmental factors that are not under the control of the decision maker Decision Variables – controllable inputs; decision alternatives specified by the decision maker, such as the number of units of Product X to produce

29. © 2009 Prentice-Hall, Inc. 1 – 29 Example : O’Neill Shoe Manufacturing company The O’Neill Shoe Manufacturing company will produce a special-style shoe if the order size is large enough to provide a reasonable profit. For each special-style order, 5 hours are required to manufacture and only 40 hours of manufacturing time are available. The profit is $ 10 per pair. Let x denotes the number of pairs of shoes produced. Develop a mathematics model.

30. © 2009 Prentice-Hall, Inc. 1 – 30 Model Solution The analyst attempts to identify the alternative (the set of decision variable values) that provides the “best” output for the model. The “best” output is the optimal solution. If the alternative does not satisfy all of the model constraints, it is rejected as being infeasible, regardless of the objective function value. If the alternative satisfies all of the model constraints, it is feasible and a candidate for the “best” solution

31. © 2009 Prentice-Hall, Inc. 1 – 31 Model Solution One solution approach is trial-and- error. Might not provide the best solution Inefficient (numerous calculations required) Special solution procedures have been developed for specific mathematical models. Some small models/problems can be solved by hand calculations Most practical applications require using a computer 31

32. © 2009 Prentice-Hall, Inc. 1 – 32 Trial and Error Solution For The O’Neill Shoe Manufacturing Company 32

33. © 2009 Prentice-Hall, Inc. 1 – 33 Example: Iron Works, Inc. Iron Works, Inc. manufactures two products made from steel and just received this month's allocation of b pounds of steel. It takes a 1 pounds of steel to make a unit of product 1 and a 2 pounds of steel to make a unit of product 2. Let x 1 and x 2 denote this month's production level of product 1 and product 2, respectively. Denote by p 1 and p 2 the unit profits for products 1 and 2, respectively. Iron Works has a contract calling for at least m units of product 1 this month. The firm's facilities are such that at most u units of product 2 may be produced monthly. 33

34. © 2009 Prentice-Hall, Inc. 1 – 34 Example: Iron Works, Inc Mathematical Model The total monthly profit = (profit per unit of product 1) x (monthly production of product 1) + (profit per unit of product 2) x (monthly production of product 2) = p 1 x 1 + p 2 x 2 We want to maximize total monthly profit: Max p 1 x 1 + p 2 x 2 34

35. © 2009 Prentice-Hall, Inc. 1 – 35 Example: Iron Works, Inc Mathematical Model (continued) The total amount of steel used during monthly production equals: (steel required per unit of product 1) x (monthly production of product 1) + (steel required per unit of product 2) x (monthly production of product 2) = a 1 x 1 + a 2 x 2 This quantity must be less than or equal to the allocated b pounds of steel: a 1 x 1 + a 2 x 2 < b 35

36. © 2009 Prentice-Hall, Inc. 1 – 36 Example: Iron Works, Inc. Mathematical Model (continued) The monthly production level of product 1 must be greater than or equal to m : x 1 > m The monthly production level of product 2 must be less than or equal to u : x 2 < u However, the production level for product 2 cannot be negative: x 2 > 0 36

37. © 2009 Prentice-Hall, Inc. 1 – 37 Example: Iron Works, Inc. Mathematical Model Summary Max p 1 x 1 + p 2 x 2 s.t a 1 x 1 + a 2 x 2 < b x 1 > m x 2 < u x 2 > 0 37 ObjectiveFunction Constraints “Subject to”

38. © 2009 Prentice-Hall, Inc. 1 – 38 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

39. © 2009 Prentice-Hall, Inc. 1 – 39 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

40. © 2009 Prentice-Hall, Inc. 1 – 40 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

41. © 2009 Prentice-Hall, Inc. 1 – 41 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

42. © 2009 Prentice-Hall, Inc. 1 – 42 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

43. © 2009 Prentice-Hall, Inc. 1 – 43 Summary Quantitative analysis is a scientific approach to decision making The approach includes Defining the problem Acquiring input data Developing a solution Testing the solution Analyzing the results Implementing the results

44. © 2009 Prentice-Hall, Inc. 1 – 44 Summary Potential problems include Conflicting viewpoints The impact on other departments Beginning assumptions Outdated solutions Fitting textbook models Understanding the model Acquiring good input data Hard-to-understand mathematics Obtaining only one answer Testing the solution Analyzing the results

45. © 2009 Prentice-Hall, Inc. 1 – 45 Summary Implementation is not the final step Problems can occur because of Lack of commitment to the approach Resistance to change