1. NEW MEXICO INSTITUTE OF MINING AND TECHNOLOGY Department of Management Management Science for Engineering Management (EMGT 501) Instructor : Toshi Sueyoshi (Ph.D.) HP address : www.nmt.edu/~toshi E-mail Address : toshi@nmt.edu Office : Speare 143-A

2. 1. Course Description: The purpose of this course is to introduce Management Science (MS) techniques for manufacturing, services, and public sector. MS includes a variety of techniques used in modeling business applications for both better understanding the system in question and making best decisions.

3. MS techniques have been applied in many situations, ranging from inventory management in manufacturing firms to capital budgeting in large and small organizations. Public and Private Sector Applications

4. The main objective of this graduate course is to provide engineers with a variety of decisional tools available for modeling and solving problems in a real business and/or nonprofit context. In this class, each individual will explore how to make various business models and how to solve them effectively.

5. 2. Texts -- The texts for this course: (1) Anderson, Sweeney, Williams & Martin An Introduction to Management Science: Quantitative Approaches to Decision Making, Thomson South-Western (Required)

6. 3. Grading: In a course, like this class, homework problems are essential. We will have homework assignments. Homework has significant weight. The grade allocation is separated as follows: Homework 20% Mid-Term Exam 40% Final Exam 40% The usual scale (90-100=A, 80-89.99=B, 70- 79.99=C, 60-69.99=D) will be used. Please remember no makeup exam.

7. 4. Course Outline: Week Topic(s) Text(s) 1 Introduction and Overview Ch. 1&2 2 Linear Programming Ch. 3&17 3 Solving LP and Dual Ch. 4&18 4 DEA Ch. 5 5 Game Theory Ch.5 6 Project Scheduling: PERT-CPM Ch. 9 7 Inventory Models Ch. 10 8 Review for Mid-Term EXAM

8. Week Topic(s) Text(s) 9Waiting Line Models Ch. 11 10Waiting Line Models Ch. 11 11Decision Analysis Ch. 13 12Multi-criteria Decision Ch. 14 13Forecasting Ch. 15 14Markov Process Ch. 16 15Slack (for Class Delay) 16Review for FINAL EXAM

9. Assessment Please indicate the current level of your knowledge. (1: no idea, 2: little, 3: considerable, 4: very well). Topic Your Assessment (1) Linear Programming (2) Dual and Primal Relationship (3) Simplex Method (4) Data Envelopment Analysis (5) PERT/CPM (6) Inventory Return the assessment by Sep 1 (noon) to toshi@nmt.edu

10. Model Development Models are representations of real objects or situations –Mathematical models - represent real world problems through a system of mathematical formulas and expressions based on key assumptions, estimates, or statistical analyses

11. Advantages of Models Generally, experimenting with models (compared to experimenting with the real situation): –requires less time –is less expensive –involves less risk

12. Mathematical Models Cost/benefit considerations must be made in selecting an appropriate mathematical model. 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.

13. Mathematical Models Relate decision variables (controllable inputs) with fixed or variable parameters (uncontrollable inputs) Frequently seek to maximize or minimize some objective function subject to constraints Are said to be stochastic if any of the uncontrollable inputs is subject to variation, otherwise are deterministic Generally, stochastic models are more difficult to analyze. The values of the decision variables that provide the mathematically-best output are referred to as the optimal solution for the model.

14. Body of Knowledge The body of knowledge involving quantitative approaches to decision making is referred to as –Management Science –Operations research –Decision science It had its early roots in World War II and is flourishing in business and industry with the aid of computers

15. Transforming Model Inputs into Output Uncontrollable Inputs (Environmental Factors) Uncontrollable Inputs (Environmental Factors) ControllableInputs(DecisionVariables)ControllableInputs(DecisionVariables) Output(Projected Results) Results)Output(Projected MathematicalModelMathematicalModel

16. Example: Project Scheduling Consider the construction of a 250-unit apartment complex. The project consists of hundreds of activities involving excavating, framing, wiring, plastering, painting, land-scaping, and more. Some of the activities must be done sequentially and others can be done at the same time. Also, some of the activities can be completed faster than normal by purchasing additional resources (workers, equipment, etc.).

17. Example: Project Scheduling Question: What is the best schedule for the activities and for which activities should additional resources be purchased? How could management science be used to solve this problem? Answer: Management science can provide a structured, quantitative approach for determining the minimum project completion time based on the activities' normal times and then based on the activities' expedited (reduced) times.

18. Example: Project Scheduling Question: What would be the decision variables of the mathematical model? The objective function? The constraints? Answer: –Decision variables: which activities to expedite and by how much, and when to start each activity –Objective function: minimize project completion time –Constraints: do not violate any activity precedence relationships and do not expedite in excess of the funds available.

19. Example: Project Scheduling Question: Is the model deterministic or stochastic? Answer: Stochastic. Activity completion times, both normal and expedited, are uncertain and subject to variation. Activity expediting costs are uncertain. The number of activities and their precedence relationships might change before the project is completed due to a project design change.

20. Example: Project Scheduling Question: Suggest assumptions that could be made to simplify the model. Answer: Make the model deterministic by assuming normal and expedited activity times are known with certainty and are constant. The same assumption might be made about the other stochastic, uncontrollable inputs.

21. Data Preparation Data preparation is not a trivial step, due to the time required and the possibility of data collection errors. A model with 50 decision variables and 25 constraints could have over 1300 data elements! Often, a fairly large data base is needed. Information systems specialists might be needed.

22. Model Solution 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.

23. Computer Software A variety of software packages are available for solving mathematical models. a) Management Scientist Software (attached to the text book) b) QSB and Spreadsheet packages such as Microsoft Excel

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

25. Report Generation A managerial report, based on the results of the model, should be prepared. The report should be easily understood by the decision maker. The report should include: –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)

26. Implementation and Follow-Up Successful implementation of model results is of critical importance. Secure as much user involvement as possible throughout the modeling process. Continue to monitor the contribution of the model. It might be necessary to refine or expand the model.

27. Linear Programming (LP): A mathematical method that consists of an objective function and many constraints. LP involves the planning of activities to obtain an optimal result, using a mathematical model, in which all the functions are expressed by a linear relation.

28. Maximize subject to A standard Linear Programming Problem Applications: Man Power Design, Portfolio Analysis

29. Simplex method: A remarkably efficient solution procedure for solving various LP problems. Extensions and variations of the simplex method are used to perform postoptimality analysis (including sensitivity analysis).

30. (0) (1) (2) (3) (0) (1) (2) (3) (a) Algebraic Form (b) Tabular Form Coefficient of: Right Side Basic Variable Z Eq. 1 -3 -5 0 0 0 0 0 1 0 1 0 0 0 0 2 0 0 1 0 12 0 3 2 0 0 1 18

31. Duality Theory: An important discovery in the early development of LP is Duality Theory. Each LP problem, referred to as ” a primal problem” is associated with another LP problem called “a dual problem”. One of the key uses of duality theory lies in the interpretation and implementation of sensitivity analysis.

32. PERT (Program Evaluation and Review Technique)-CPM (Critical Path Method): PERT and CPM have been used extensively to assist project managers in planning, scheduling, and controlling their projects. Applications: Project Management, Project Scheduling

33. A 2 B C E M N START FINISH H G D J I F L K 4 10 4 7 6 7 9 8 54 6 2 5 0 0 Critical Path 2 + 4 + 10 + 4 + 5 + 8 + 5 + 6 = 44 weeks

34. Decision Analysis: An important technique for decision making in uncertainty. It divides decision making between the cases of without experimentation and with experimentation. Applications: Decision Making, Planning

35. Oil 0.5 0.3 Favorable 0.75 Dry 0.85 Dry a e d c b f g h Drill Sell Drill Sell Drill Oil 0.14 Oil 0.25 0.5 Dry Do seismic survey Unfavorable 0.7 No seismic survey decision fork chance fork

36. Markov Chain Model: A special kind of a stochastic process. It has a special property that probabilities, involving how a process will evolve in future, depend only on the present state of the process, and so are independent of events in the past. Applications: Inventory Control, Forecasting

37. Queueing Theory: This theory studies queueing systems by formulating mathematical models of their operation and then using these models to derive measures of performance.

38. This analysis provides vital information for effectively designing queueing systems that achieve an appropriate balance between the cost of providing a service and the cost associated with waiting for the service.

39. S S Service S facility S CCCCCCCC Served customers C C C Queueing system Customers Queue Applications: Waiting Line Design, Banking, Network Design

40. Inventory Theory: This theory is used by both wholesalers and retailers to maintain inventories of goods to be available for purchase by customers. The just-in-time inventory system is such an example that emphasizes planning and scheduling so that the needed materials arrive “just-in-time” for their use. Applications: Inventory Analysis, Warehouse Design

41. Economic Order Quantity (EOQ) model Time t Inventory level Batch size

42. Forecasting: When historical sales data are available, statistical forecasting methods have been developed for using these data to forecast future demand. Several judgmental forecasting methods use expert judgment. Applications: Future Prediction, Inventory Analysis

43. 1/99 4/99 7/99 10/99 1/00 4/00 7/00 The evolution of the monthly sales of a product illustrates a time series Monthly sales (units sold) 10,000 8,000 6,000 4,000 2,000 0

44. Introduction to MS/OR MS: Management Science OR: Operations Research Key components: (a) Modeling/Formulation (b) Algorithm (c) Application

45. Management Science (MS) (1) A discipline that attempts to aid managerial decision making by applying a scientific approach to managerial problems that involve quantitative factors. (2) MS is based upon mathematics, computer science and other social sciences like economics and business.

46. General Steps of MS Step 1: Define problem and gather data Step 2: Formulate a mathematical model to represent the problem Step 3: Develop a computer based procedure for deriving a solution(s) to the problem

47. Step 4: Test the model and refine it as needed Step 5: Apply the model to analyze the problem and make recommendation for management Step 6: Help implementation

48. Linear Programming (LP)

49. Linear Programming (LP) Problem The maximization or minimization of some quantity is the objective in all linear programming problems. All LP problems have constraints that limit the degree to which the objective can be pursued. A feasible solution satisfies all the problem's constraints. An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing). A graphical solution method can be used to solve a linear program with two variables.

50. Linear Programming (LP) Problem If both the objective function and the constraints are linear, the problem is referred to as a linear programming problem. Linear functions are functions in which each variable appears in a separate term raised to the first power and is multiplied by a constant (which could be 0). Linear constraints are linear functions that are restricted to be "less than or equal to", "equal to", or "greater than or equal to" a constant.

51. Problem Formulation Problem formulation or modeling is the process of translating a verbal statement of a problem into a mathematical statement.

52. [1] LP Formulation (a) Decision Variables : All the decision variables are non-negative. (b) Objective Function : Min or Max (c) Constraints s.t. : subject to

53. Guidelines for Model Formulation Understand the problem thoroughly. Describe the objective. Describe each constraint. Define the decision variables. Write the objective in terms of the decision variables. Write the constraints in terms of the decision variables.

54. [2] Example A company has three plants, Plant 1, Plant 2, Plant 3. Because of declining earnings, top management has decided to revamp the company’s product line. Product 1: It requires some of production capacity in Plants 1 and 3. Product 2: It needs Plants 2 and 3.

55. The marketing division has concluded that the company could sell as much as could be produced by these plants. However, because both products would be competing for the same production capacity in Plant 3, it is not clear which mix of the two products would be most profitable.

56. The data needed to be gathered: 1. Number of hours of production time available per week in each plant for these new products. (The available capacity for the new products is quite limited.) 2. Production time used in each plant for each batch to yield each new product. 3. There is a profit per batch from a new product.

57. Production Time per Batch, Hours Production Time Available per Week, Hours Plant Product Profit per batch 123123 4 12 18 1 2 1 0 0 2 3 2 $3,000 $5,000

58. : # of batches of product 1 produced per week : # of batches of product 2 produced per week : the total profit per week Maximize subject to

59. 0 2 4 6 8 2 4 6 8 10 Graphic Solution Feasible region

60. 0 2 4 6 8 2 4 6 8 10 Feasible region

61. 0 2 4 6 8 2 4 6 8 10 Feasible region

62. 0 2 4 6 8 2 4 6 8 10 Feasible region

63. 0 2 4 6 8 10 2 4 6 8 Maximize: The optimal solution The largest value Slope-intercept form:

64. Summary of the Graphical Solution Procedure for Maximization Problems Prepare a graph of the feasible solutions for each of the constraints. Determine the feasible region that satisfies all the constraints simultaneously.. Draw an objective function line. Move parallel objective function lines toward larger objective function values without entirely leaving the feasible region. Any feasible solution on the objective function line with the largest value is an optimal solution.

65. Maximize s.t. [4] Standard Form of LP Model

66. [5] Other Forms The other LP forms are the following: 1. Minimizing the objective function: 2. Greater-than-or-equal-to constraints: Minimize

67. 3. Some functional constraints in equation form: 4. Deleting the nonnegativity constraints for some decision variables: : unrestricted in sign

68. [6] Key Terminology (a) A feasible solution is a solution for which all constraints are satisfied (b) An infeasible solution is a solution for which at least one constraint is violated (c) A feasible region is a collection of all feasible solutions

69. (d) An optimal solution is a feasible solution that has the most favorable value of the objective function (e) Multiple optimal solutions have an infinite number of solutions with the same optimal objective value

70. and Maximize Subject to Example Multiple optimal solutions:

71. 0 2 4 6 8 10 2 4 6 8 Feasible region Every point on this red line segment is optimal, each with Z=18. Multiple optimal solutions

72. (f) An unbounded solution occurs when the constraints do not prevent improving the value of the objective function.

73. Case Study - Personal Scheduling UNION AIRWAYS needs to hire additional customer service agents. Management recognizes the need for cost control while also consistently providing a satisfactory level of service to customers. Based on the new schedule of flights, an analysis has been made of the minimum number of customer service agents that need to be on duty at different times of the day to provide a satisfactory level of service.

74. * * * * * * * * * * Shift Time Period Covered Minimum # of Agents needed Time Period 6:00 am to 8:00 am 8:00 am to 10:00 am 10:00 am to noon Noon to 2:00 pm 2:00 pm to 4:00 pm 4:00 pm to 6:00 pm 6:00 pm to 8:00 pm 8:00 pm to 10:00 pm 10:00 pm to midnight Midnight to 6:00 am 1 2 3 4 5 48 79 65 87 64 73 82 43 52 15 170 160 175 180 195 Daily cost per agent

75. The problem is to determine how many agents should be assigned to the respective shifts each day to minimize the total personnel cost for agents, while meeting (or surpassing) the service requirements. Activities correspond to shifts, where the level of each activity is the number of agents assigned to that shift. This problem involves finding the best mix of shift sizes.

76. : # of agents for shift 1 (6AM - 2PM) : # of agents for shift 2 (8AM - 4PM) : # of agents for shift 3 (Noon - 8PM) : # of agents for shift 4 (4PM - Midnight) : # of agents for shift 5 (10PM - 6AM) The objective is to minimize the total cost of the agents assigned to the five shifts.

77. Min s.t. all

78. Total Personal Cost = $30,610

79. Slack and Surplus Variables A linear program in which all the variables are non- negative and all the constraints are equalities is said to be in standard form. Standard form is attained by adding slack variables to "less than or equal to" constraints, and by subtracting surplus variables from "greater than or equal to" constraints. Slack and surplus variables represent the difference between the left and right sides of the constraints. Slack and surplus variables have objective function coefficients equal to 0.

80. Example 1: Standard Form Max 5x 1 + 7x 2 + 0s 1 + 0s 2 + 0s 3 s.t. x 1 + s 1 = 6 2x 1 + 3x 2 + s 2 = 19 x 1 + x 2 + s 3 = 8 x 1, x 2, s 1, s 2, s 3 > 0

81. Interpretation of Computer Output In this chapter we will discuss the following output: –objective function value –values of the decision variables –reduced costs –slack/surplus In the next chapter we will discuss how an optimal solution is affected by a change in: –a coefficient of the objective function –the right-hand side value of a constraint

82. Example 1: Spreadsheet Solution Partial Spreadsheet Showing Solution

83. Example 1: Spreadsheet Solution Reduced Costs