1. 1 1 Slide © 2009 South-Western, a part of Cengage Learning Quantitative Business Analysis

2. 2 2 Slide © 2009 South-Western, a part of Cengage Learning Chapter 1 Introduction n Body of Knowledge n Problem Solving and Decision Making n Quantitative Analysis and Decision Making n Quantitative Analysis n Models of Cost, Revenue, and Profit n Quantitative Methods in Practice

3. 3 3 Slide © 2009 South-Western, a part of Cengage Learning Body of Knowledge n The body of knowledge involving quantitative approaches to decision making is referred to as Management Science Management Science Operations Research Operations Research Decision Science Decision Science n It had its early roots in World War II and is flourishing in business and industry due, in part, to: numerous methodological developments (e.g. simplex method for solving linear programming problems) numerous methodological developments (e.g. simplex method for solving linear programming problems) a virtual explosion in computing power a virtual explosion in computing power

4. 4 4 Slide © 2009 South-Western, a part of Cengage Learning n 7 Steps of Problem Solving (First 5 steps are the process of decision making) 1. Identify and define the problem. 2. Determine the set of alternative solutions. 3. Determine the criteria for evaluating alternatives. 4. Evaluate the alternatives. 5. Choose an alternative (make a decision). --------------------------------------------------------------------- --------------------------------------------------------------------- 6. Implement the selected alternative. 7. Evaluate the results. Problem Solving and Decision Making

5. 5 5 Slide © 2009 South-Western, a part of Cengage Learning Quantitative Analysis and Decision Making DefinetheProblemDefinetheProblemIdentifytheAlternativesIdentifytheAlternativesDeterminetheCriteriaDeterminetheCriteriaIdentifytheAlternativesIdentifytheAlternativesChooseanAlternativeChooseanAlternative Structuring the Problem Analyzing the Problem n Decision-Making Process Problems in which the objective is to find the best solutionProblems in which the objective is to find the best solution with respect to one criterion are referred to as single-criterion decision problems. Problems that involve more than one criterion are referred toProblems that involve more than one criterion are referred to as multicriteria decision problems.

6. 6 6 Slide © 2009 South-Western, a part of Cengage Learning Step 1 Define Problem At the end of this stage, managers should have a clear understanding of the problem they are tackling, its context and the requirements of their solution. For this stage they might: Do an initial investigation – looking at operations, identifying difficulties and recognizing that there really is a problem. Define the problem – adding details to the initial investigation, saying exactly what the problem is (and not just its symptoms), its context, scope, boundaries and any other relevant details. Set objectives – identifying the decision-makers, their aims, improvements they want, effects on the organization and measures of success. Identify variables, possible alternatives and courses of action. Plan the work – showing how to tackle the problem, schedule activities, design timetables and check resources.

7. 7 7 Slide © 2009 South-Western, a part of Cengage Learning Step 2 Identify the alternatives and criteria for alternatives At the end of this stage, managers should have a clear understanding of their options and the consequences Consider different approaches to solving the problem. Check work done on similar problems and see if they can use the same approach. Study the problem more closely and refine the details. Identify the key variables and relationships between them. Build a model of the problem and test its accuracy. Collect data needed by the model and analyze it. Run more tests on the model and data to make sure that they are working properly, are accurate and describe the real conditions. Experiment with the model to find results in different circumstances and under different conditions. Analyze the results, making sure that they are accurate and consistent.

8. 8 8 Slide © 2009 South-Western, a part of Cengage Learning Step 4 & 5 Evaluate and choose and alternative Make a Decision This is where managers consider the results from analyses, review all the circumstances and make their decisions. There are three steps: (a) Compare solutions, looking at all aspects of their performance. (b) Find solutions that best meet the decision-makers’ objectives. (c) Identify and agree the best overall solution.

9. 9 9 Slide © 2009 South-Western, a part of Cengage Learning Step 5 and 6 Implement the Decisions and Evaluate the Results At this point managers turn ideas into practice, moving from ‘we should do this’ to actually doing it. For this they: Check that the proposed solution really works and is an improvement on current performance. Plan details of the implementation. Change operations to introduce new ways of doing things. Monitor actual performance – after implementing their decisions, managers still have to monitor operations using feedback to compare actual performance with plans to make sure that predicted results actually occur.

10. 10 Slide © 2009 South-Western, a part of Cengage Learning n Analysis Phase of Decision-Making Process Qualitative Analysis Qualitative Analysis based largely on the manager’s judgment and experiencebased largely on the manager’s judgment and experience includes the manager’s intuitive “feel” for the problemincludes the manager’s intuitive “feel” for the problem is more of an art than a scienceis more of an art than a science Quantitative Analysis and Decision Making

11. 11 Slide © 2009 South-Western, a part of Cengage Learning

12. 12 Slide © 2009 South-Western, a part of Cengage Learning n Analysis Phase of Decision-Making Process Quantitative Analysis Quantitative Analysis analyst will concentrate on the quantitative facts or data associated with the problemanalyst will concentrate on the quantitative facts or data associated with the problem analyst will develop mathematical expressions that describe the objectives, constraints, and other relationships that exist in the problemanalyst will develop mathematical expressions that describe the objectives, constraints, and other relationships that exist in the problem analyst will use one or more quantitative methods to make a recommendationanalyst will use one or more quantitative methods to make a recommendation Quantitative Analysis and Decision Making

13. 13 Slide © 2009 South-Western, a part of Cengage Learning Quantitative Analysis and Decision Making n Potential Reasons for a Quantitative Analysis Approach to Decision Making The problem is complex. The problem is complex. The problem is very important. The problem is very important. The problem is new. The problem is new. The problem is repetitive. The problem is repetitive.

14. 14 Slide © 2009 South-Western, a part of Cengage Learning Quantitative Analysis n Quantitative Analysis Process Model Development Model Development Data Preparation Data Preparation Model Solution Model Solution Report Generation Report Generation

15. 15 Slide © 2009 South-Western, a part of Cengage Learning Model Development n Models are representations of real objects or situations n Three forms of models are: Iconic models - physical replicas (scalar representations) of real objects Iconic models - physical replicas (scalar representations) of real objects Analog models - physical in form, but do not physically resemble the object being modeled Analog models - physical in form, but do not physically resemble the object being modeled Mathematical models - represent real world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses Mathematical models - represent real world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses

16. 16 Slide © 2009 South-Western, a part of Cengage Learning Advantages of Models n Generally, experimenting with models (compared to experimenting with the real situation): requires less time requires less time is less expensive is less expensive involves less risk involves less risk n The more closely the model represents the real situation, the more accurate the conclusions and predictions will be.

17. 17 Slide © 2009 South-Western, a part of Cengage Learning Mathematical Models n Objective Function – a mathematical expression that describes the problem’s objective, such as maximizing profit or minimizing cost Consider a simple production problem. Suppose x denotes the number of units produced and sold each week, and the firm’s objective is to maximize total weekly profit. With a profit of $10 per unit, the objective function is 10 x. Consider a simple production problem. Suppose x denotes the number of units produced and sold each week, and the firm’s objective is to maximize total weekly profit. With a profit of $10 per unit, the objective function is 10 x.

18. 18 Slide © 2009 South-Western, a part of Cengage Learning Mathematical Models n Constraints – a set of restrictions or limitations, such as production capacities n To continue our example, a production capacity constraint would be necessary if, for instance, 5 hours are required to produce each unit and only 40 hours are available per week. The production capacity constraint is given by 5 x < 40. n The value of 5 x is the total time required to produce x units; the symbol indicates that the production time required must be less than or equal to the 40 hours available.

19. 19 Slide © 2009 South-Western, a part of Cengage Learning Mathematical Models n Uncontrollable Inputs – environmental factors that are not under the control of the decision maker n In the preceding mathematical model, the profit per unit ($10), the production time per unit (5 hours), and the production capacity (40 hours) are environmental factors not under the control of the manager or decision maker.

20. 20 Slide © 2009 South-Western, a part of Cengage Learning Mathematical Models n Decision Variables – controllable inputs; decision alternatives specified by the decision maker, such as the number of units of a product to produce. n In the preceding mathematical model, the production quantity x is the controllable input to the model.

21. 21 Slide © 2009 South-Western, a part of Cengage Learning Mathematical Models n A complete mathematical model for our simple production problem is: Maximize10 x (objective function) subject to:5 x < 40(constraint) x > 0(constraint) x > 0(constraint) [The second constraint reflects the fact that it is not possible to manufacture a negative number of units.]

22. 22 Slide © 2009 South-Western, a part of Cengage Learning Mathematical Models n Deterministic Model – if all uncontrollable inputs to the model are known and cannot vary n Stochastic (or Probabilistic) Model – if any uncontrollable are uncertain and subject to variation n Stochastic models are often more difficult to analyze. n In our simple production example, if the number of hours of production time per unit could vary from 3 to 6 hours depending on the quality of the raw material, the model would be stochastic.

23. 23 Slide © 2009 South-Western, a part of Cengage Learning Mathematical Models n Cost/benefit considerations must be made in selecting an appropriate mathematical model. n Frequently a less complicated (and perhaps less precise) model is more appropriate than a more complex and accurate one due to cost and ease of solution considerations.

24. 24 Slide © 2009 South-Western, a part of Cengage Learning Transforming Model Inputs into Output Uncontrollable Inputs (Environmental Factors) Uncontrollable Inputs (Environmental Factors) ControllableInputs(DecisionVariables)ControllableInputs(DecisionVariables) Output(Projected Results) Results)Output(Projected MathematicalModelMathematicalModel

25. 25 Slide © 2009 South-Western, a part of Cengage Learning Data Preparation n Data preparation is not a trivial step, due to the time required and the possibility of data collection errors. n A model with 50 decision variables and 25 constraints could have over 1,300 data elements! n Often, a fairly large database is needed. n Information systems specialists might be needed.

26. 26 Slide © 2009 South-Western, a part of Cengage Learning Model Solution n The analyst attempts to identify the alternative (the set of decision variable values) that provides the “best” output for the model. n The “best” output is the optimal solution. n If the alternative does not satisfy all of the model constraints, it is rejected as being infeasible, regardless of the objective function value. n If the alternative satisfies all of the model constraints, it is feasible and a candidate for the “best” solution.

27. 27 Slide © 2009 South-Western, a part of Cengage Learning ProductionProjected Total Hours FeasibleQuantityProfit of Production Solution 0 00Yes 2 20 2010Yes 4 40 4020Yes 6 60 6030Yes 8 80 8040Yes 1010050No 1212060No Model Solution  Trial-and-Error Solution for Production Problem

28. 28 Slide © 2009 South-Western, a part of Cengage Learning Model Solution n A variety of software packages are available for solving mathematical models. Microsoft Excel Microsoft Excel The Management Scientist The Management Scientist LINGO LINGO

29. 29 Slide © 2009 South-Western, a part of Cengage Learning Model Testing and Validation n Often, goodness/accuracy of a model cannot be assessed until solutions are generated. n Small test problems having known, or at least expected, solutions can be used for model testing and validation. n If the model generates expected solutions, use the model on the full-scale problem. n If inaccuracies or potential shortcomings inherent in the model are identified, take corrective action such as: Collection of more-accurate input data Collection of more-accurate input data Modification of the model Modification of the model

30. 30 Slide © 2009 South-Western, a part of Cengage Learning Report Generation n A managerial report, based on the results of the model, should be prepared. n The report should be easily understood by the decision maker. n The report should include: the recommended decision the recommended decision other pertinent information about the results (for example, how sensitive the model solution is to the assumptions and data used in the model) other pertinent information about the results (for example, how sensitive the model solution is to the assumptions and data used in the model)

31. 31 Slide © 2009 South-Western, a part of Cengage Learning Implementation and Follow-Up n Successful implementation of model results is of critical importance. n Secure as much user involvement as possible throughout the modeling process. n Continue to monitor the contribution of the model. n It might be necessary to refine or expand the model.

32. 32 Slide © 2009 South-Western, a part of Cengage Learning Models of Cost, Revenue, and Profit 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. 33 Slide © 2009 South-Western, a part of Cengage Learning Example: Iron Works, Inc. n Mathematical Model The total monthly profit = The total monthly profit = (profit per unit of product 1) x (monthly production of product 1) x (monthly production of product 1) + (profit per unit of product 2) + (profit per unit of product 2) x (monthly production of product 2) x (monthly production of product 2) = p 1 x 1 + p 2 x 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. 34 Slide © 2009 South-Western, a part of Cengage Learning Example: Iron Works, Inc. n Mathematical Model (continued) The total amount of steel used during monthly production equals: The total amount of steel used during monthly production equals: (steel required per unit of product 1) (steel required per unit of product 1) x (monthly production of product 1) x (monthly production of product 1) + (steel required per unit of product 2) + (steel required per unit of product 2) x (monthly production of product 2) x (monthly production of product 2) = a 1 x 1 + a 2 x 2 = a 1 x 1 + a 2 x 2 This quantity must be less than or equal to the allocated b pounds of steel: This quantity must be less than or equal to the allocated b pounds of steel: a 1 x 1 + a 2 x 2 < b a 1 x 1 + a 2 x 2 < b

35. 35 Slide © 2009 South-Western, a part of Cengage Learning Example: Iron Works, Inc. n Mathematical Model (continued) The monthly production level of product 1 must be greater than or equal to m : The monthly production level of product 1 must be greater than or equal to m : x 1 > m x 1 > m The monthly production level of product 2 must be less than or equal to u : The monthly production level of product 2 must be less than or equal to u : x 2 < u x 2 < u However, the production level for product 2 cannot be negative: However, the production level for product 2 cannot be negative: x 2 > 0 x 2 > 0

36. 36 Slide © 2009 South-Western, a part of Cengage Learning Example: Iron Works, Inc. n Mathematical Model Summary Max p 1 x 1 + p 2 x 2 Max p 1 x 1 + p 2 x 2 s.t. a 1 x 1 + a 2 x 2 < b s.t. a 1 x 1 + a 2 x 2 < b x 1 > m x 1 > m x 2 < u x 2 < u x 2 > 0 x 2 > 0 ObjectiveFunction “Subject to” Constraints

37. 37 Slide © 2009 South-Western, a part of Cengage Learning Example: Iron Works, Inc. n Question: Suppose b = 2000, a 1 = 2, a 2 = 3, m = 60, u = 720, p 1 = 100, p 2 = 200. Rewrite the model with these specific values for the uncontrollable inputs.

38. 38 Slide © 2009 South-Western, a part of Cengage Learning Example: Iron Works, Inc. n Answer: Substituting, the model is: Max 100 x 1 + 200 x 2 Max 100 x 1 + 200 x 2 s.t. 2 x 1 + 3 x 2 < 2000 s.t. 2 x 1 + 3 x 2 < 2000 x 1 > 60 x 1 > 60 x 2 < 720 x 2 < 720 x 2 > 0 x 2 > 0

39. 39 Slide © 2009 South-Western, a part of Cengage Learning Example: Iron Works, Inc. n Question: The optimal solution to the current model is x 1 = 60 and x 2 = 626 2/3. If the product were engines, explain why this is not a true optimal solution for the "real-life" problem. n Answer: One cannot produce and sell 2/3 of an engine. Thus the problem is further restricted by the fact that both x 1 and x 2 must be integers. (They could remain fractions if it is assumed these fractions are work in progress to be completed the next month.)

40. 40 Slide © 2009 South-Western, a part of Cengage Learning Example: Iron Works, Inc. Uncontrollable Inputs $100 profit per unit Prod. 1 $200 profit per unit Prod. 2 2 lbs. steel per unit Prod. 1 3 lbs. Steel per unit Prod. 2 2,000 lbs. steel allocated 60 units minimum Prod. 1 720 units maximum Prod. 2 0 units minimum Prod. 2 $100 profit per unit Prod. 1 $200 profit per unit Prod. 2 2 lbs. steel per unit Prod. 1 3 lbs. Steel per unit Prod. 2 2,000 lbs. steel allocated 60 units minimum Prod. 1 720 units maximum Prod. 2 0 units minimum Prod. 2 60 units Prod. 1 60 units Prod. 1 626.67 units Prod. 2 60 units Prod. 1 60 units Prod. 1 626.67 units Prod. 2 Controllable Inputs Profit = $131,333.33 Steel Used = 2,000 Profit = $131,333.33 Steel Used = 2,000 OutputOutput Mathematical Model Max 100(60) + 200(626.67) s.t. 2(60) + 3(626.67) < 2000 60 > 60 60 > 60 626.67 < 720 626.67 < 720 626.67 > 0 626.67 > 0 Max 100(60) + 200(626.67) s.t. 2(60) + 3(626.67) < 2000 60 > 60 60 > 60 626.67 < 720 626.67 < 720 626.67 > 0 626.67 > 0

41. 41 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. Ponderosa Development Corporation (PDC) is a small real estate developer that builds only one style house. The selling price of the house is $115,000. Land for each house costs $55,000 and lumber, supplies, and other materials run another $28,000 per house. Total labor costs are approximately $20,000 per house.

42. 42 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. Ponderosa leases office space for $2,000 per month. The cost of supplies, utilities, and leased equipment runs another $3,000 per month. The one salesperson of PDC is paid a commission of $2,000 on the sale of each house. PDC has seven permanent office employees whose monthly salaries are given on the next slide.

43. 43 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. Employee Monthly Salary President $10,000 President $10,000 VP, Development 6,000 VP, Development 6,000 VP, Marketing 4,500 VP, Marketing 4,500 Project Manager 5,500 Controller 4,000 Office Manager 3,000 Receptionist 2,000

44. 44 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Question: Identify all costs and denote the marginal cost and marginal revenue for each house. n Answer: The monthly salaries total $35,000 and monthly office lease and supply costs total another $5,000. This $40,000 is a monthly fixed cost. The total cost of land, material, labor, and sales commission per house, $105,000, is the marginal cost for a house. The selling price of $115,000 is the marginal revenue per house.

45. 45 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Question: Write the monthly cost function c ( x ), revenue function r ( x ), and profit function p ( x ). n Answer: c ( x ) = variable cost + fixed cost = 105,000 x + 40,000 r ( x ) = 115,000 x r ( x ) = 115,000 x p ( x ) = r ( x ) - c ( x ) = 10,000 x - 40,000 p ( x ) = r ( x ) - c ( x ) = 10,000 x - 40,000

46. 46 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Question: What is the breakeven point for monthly sales of the houses? n Answer: r ( x ) = c ( x ) 115,000 x = 105,000 x + 40,000 Solving, x = 4.

47. 47 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Question: What is the monthly profit if 12 houses per month are built and sold? n Answer: p (12) = 10,000(12) - 40,000 = $80,000 monthly profit p (12) = 10,000(12) - 40,000 = $80,000 monthly profit

48. 48 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. 0 200 400 600 800 1000 1200 012345678910 Number of Houses Sold (x) Thousands of Dollars Break-Even Point = 4 Houses Total Cost = Total Cost = 40,000 + 105,000x 40,000 + 105,000x Total Revenue = Total Revenue = 115,000x 115,000x

49. 49 Slide © 2009 South-Western, a part of Cengage Learning Using Excel for Breakeven Analysis n A spreadsheet software package such as Microsoft Excel can be used to perform a quantitative analysis of Ponderosa Development Corporation. n We will enter the problem data in the top portion of the spreadsheet. n The bottom of the spreadsheet will be used for model development.

50. 50 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Formula Spreadsheet

51. 51 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Question What is the monthly profit if 12 houses are built and sold per month?

52. 52 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Spreadsheet Solution

53. 53 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Question: What is the breakeven point for monthly sales of the houses? n Spreadsheet Solution: One way to determine the break-even point using a spreadsheet is to use the Goal Seek tool. One way to determine the break-even point using a spreadsheet is to use the Goal Seek tool. Microsoft Excel ‘s Goal Seek tool allows the user to determine the value for an input cell that will cause the output cell to equal some specified value. Microsoft Excel ‘s Goal Seek tool allows the user to determine the value for an input cell that will cause the output cell to equal some specified value. In our case, the goal is to set Total Profit to zero by seeking an appropriate value for Sales Volume. In our case, the goal is to set Total Profit to zero by seeking an appropriate value for Sales Volume.

54. 54 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Spreadsheet Solution: Goal Seek Approach Using Excel ’s Goal Seek Tool Step 1: Select the Tools menu Step 2: Choose the Goal Seek option Step 3: When the Goal Seek dialog box appears: Enter B9 in the Set cell box Enter B9 in the Set cell box Enter 0 in the To value box Enter 0 in the To value box Enter B6 in the By changing cell box Enter B6 in the By changing cell box Click OK Click OK

55. 55 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Spreadsheet Solution: Goal Seek Approach Completed Goal Seek Dialog Box Completed Goal Seek Dialog Box

56. 56 Slide © 2009 South-Western, a part of Cengage Learning Example: Ponderosa Development Corp. n Spreadsheet Solution: Goal Seek Approach

57. 57 Slide © 2009 South-Western, a part of Cengage Learning Quantitative Methods in Practice n Linear Programming n Integer Linear Programming n PERT/CPM n Inventory Models n Waiting Line Models n Simulation n Decision Analysis n Goal Programming n Analytic Hierarchy Process n Forecasting n Markov-Process Models n Dynamic Programming

58. 58 Slide © 2009 South-Western, a part of Cengage Learning n Linear programming is a problem-solving approach developed for situations involving maximizing or minimizing a linear function subject to linear constraints that limit the degree to which the objective can be pursued. n Integer linear programming is an approach used for problems that can be set up as linear programs with the additional requirement that some or all of the decision recommendations be integer values. Quantitative Methods in Practice

59. 59 Slide © 2009 South-Western, a part of Cengage Learning n Project scheduling: PERT (Program Evaluation and Review Technique) and CPM (Critical Path Method) help managers responsible for planning, scheduling, and controlling projects that consist of numerous separate jobs or tasks performed by a variety of departments, individuals, and so forth. n Inventory models are used by managers faced with the dual problems of maintaining sufficient inventories to meet demand for goods and, at the same time, incurring the lowest possible inventory holding costs. Quantitative Methods in Practice

60. 60 Slide © 2009 South-Western, a part of Cengage Learning n Waiting line (or queuing) models help managers understand and make better decisions concerning the operation of systems involving waiting lines. n Simulation is a technique used to model the operation of a system. This technique employs a computer program to model the operation and perform simulation computations. n Decision analysis can be used to determine optimal strategies in situations involving several decision alternatives and an uncertain or risk-filled pattern of future events. Quantitative Methods in Practice

61. 61 Slide © 2009 South-Western, a part of Cengage Learning n Goal programming is a technique for solving multi- criteria decision problems, usually within the framework of linear programming. n Analytic hierarchy process is a multi-criteria decision- making technique that permits the inclusion of subjective factors in arriving at a recommended decision. n Forecasting methods are techniques that can be used to predict future aspects of a business operation. n Markov-process models are useful in studying the evolution of certain systems over repeated trials (such as describing the probability that a machine, functioning in one period, will function or break down in another period). Quantitative Methods in Practice

62. 62 Slide © 2009 South-Western, a part of Cengage Learning Methods Used Most Frequently n Linear programming n Integer programming n Network models (such as transportation and transshipment models) n Simulation

63. 63 Slide © 2009 South-Western, a part of Cengage Learning The Management Scientist Software n 12 Modules

64. 64 Slide © 2009 South-Western, a part of Cengage Learning End of Chapter 1