Operations Research 1 • Lecturer: XIAO Li • Email: xiaoli@sz.tsinghua.edu.cn • Office: 1002, Xinxi Building • Consultation: by appointment • Teaching Assistant: WANG Zhenxiao • Email: wang-zx21@mails.tsinghua.edu.cn • Two breaks: each 5mins • Wednesday afternoonοΌ13:30-14:15; 14:20-15:05; 15:10-15:55 2 • Textbook • Introduction to Operations Research, 10th ed. Frederick S. Hillier and Gerald J. Lieberman • Reference • Introduction to Linear Optimization, Dimitri Bertsimas and John N Tsitsiklis • Fundamentals of queueing theory, 4th ed. G Donald, J F Tompson, J M Harris • Stochastic processes, Sheldon M Ross 3 Grading • Participation (10%) • Answer or ask questions in class • Homework (30%): individual • Around 6 to 10 assignments. • In-class quiz (30%): individual • Around 3 to 6 quiz. Open book, each takes 30-40 minutes. • Final project (30%): group project. • Each group consists of 1 to 4 students. • Present the final project in the last one or two weeks. • A project report should be submitted by each group: no more than 10 pages. 4 Outline • Introduction (chapter 1,2) • Linear programming (chapter 3,4,5,6) • Dynamic program (chapter 11) • Nonlinear programming (chapter 13) • Game theory (chapter 15) • Decision analysis (chapter 16) • Queueing theory (chapter 17) • Inventory theory (chapter 18) 5 Introduction to Operations Research 6 The Origins of Operations Research • Operations research (OR) • Seeks solutions to problems that arise in complex organizations • First used in the military during world war II • Managing allocation of scarce resources in an effective manner • Do research on (military) operations • Early 1950s: OR introduced into business, industry, and government • Factors influencing growth of OR • Improved techniques: simplex method, dynamic programming, queueing theory • Computer revolution: fast solution of complex computational problems 7 The Nature of Operations Research • Research on how to conduct and coordinate an organization’s activities • Operations Research (OR) • Is also called management science • Utilizes the scientific method • Must provide understandable conclusions to the decision maker • Process • • • • • Carefully observe and formulate the problem Gather data Construct a mathematical model Test whether the model represents the actual situation Modify the model as appropriate and validate again 8 The Rise of Analytics • Analytics (business analytics) • Introduced in 2006 by Thomas H. Davenport • Similar in concept to operations research • Scientific process of transforming data into insight to make better decisions • Categories of analytics applications • Descriptive analytics • Identify interesting patterns to describe what is currently happening • Predictive analytics • Using data to predict what will happen in the future • Prescriptive analytics • Using data to prescribe what should be done in the future • OR focuses on predictive and prescriptive analytics 9 The Impact of Operations Research • Institute for Operations Research and the Management Sciences (INFORMS) • Manufacturing and Service Operations Management Society (MSOM) • Production and Operations Management Society (POMS) • A number of journals and conference proceedings in OR: Operations Research, Management Sciences, Manufacturing & Service Operations Management, Production and Operations Management, Interfaces, Annals of Operations Research, IIE Transactions, IIE Annual Conference, Winter Simulation Conference, etc. 10 Introduction to Linear Programming 11 Introduction • Linear programming • Most important scientific advance in mid-20th century • Programming means planning • Model contains linear mathematical functions • An application of linear programming • Allocate limited resources among competing activities in the best possible way • Apply to a wide variety of situations 12 Prototype Example • Wyndor Glass Co. • • • • Produces high-quality glass products, including windows and glass doors Plant 1 makes aluminum frames and hardware Plant 2 makes wood frames Plant 3 produces glass and assembles products • Because of declining earning, management team decides to stop producing unprofitable product and launch two new products • Product 1: 8 ft. glass door with aluminum frame (need plant 1 and 3) • Product 2: 4 x 6 ft. double-hung, wood-framed window (need plant 2 and 3) • Problem: What mix of products would be most profitable? (because product 1 and product 2 compete production capacity of plant 3) • Assuming company could sell as much of either product as could be produced 13 • Products produced in batches of 20 • Data needed • Number of hours of production time available per week in each plant for new products • Production time used in each plant for each batch of each new product • Profit per batch of each new product 14 • Formulating the model x1 = number of batches of product 1 produced per week x2 = number of batches of product 2 produced per week Z = total profit per week (thousands of dollars) from producing these two products • Objective π = 3 π₯1 + 5π₯2 • Constraints π₯1 ≤ 4 2π₯2 ≤ 12 3π₯1 + 2π₯2 ≤ 18 π₯1 ≥ 0 π₯2 ≥ 0 Classic example of resource-allocation problem • Most common type of linear programming problem 15 • Graphical Method. • Two dimensional graph with x1 and x2 as the axes • First step: identify values of x1 and x2 permitted by the restrictions 16 • Next step: pick a point in the feasible region that maximizes value of Z • Slop-intercept form: π₯2 = −(3π₯1 )/5+π/5 17 The linear programming model • General problem terminology and examples • Resources: money, particular types of machines, vehicles • Activities: investing in particular projects, advertising in particular media, or shipping from a particular source • Problem involves choosing levels of activities to maximize overall measure of performance 18 • Standard form 19 • Other legitimate forms • • • • Minimizing (rather than maximizing) the objective function Functional constraints with greater-than-or-equal-to inequality Some functional constraints in equation form Some decision variables may be negative • Feasible solution • Solution for which all constraints are satisfied • Might not exist for a given problem 20 • Infeasible solution • Solution for which at least one constraint is violated • Optimal solution • Has most favorable value of objective function • Might have multiple optimal solutions for a given problem, if so, infinite number of optimal solutions • Might not exist for a given problem. (no feasible solution or unbounded objective) 21 • Corner-point feasible (CPF) solution • Solution that lies at the corner of the feasible region • Linear programming problem with feasible solution and bounded feasible region • Must have CPF solutions and optimal solution(s) • Best CPF solution must be an optimal solution • If one problem has exactly one optimal solution, it must be a CPT solution. • If one problem has multiple optimal solutions, two of them must be CPT solutions. 22 Assumptions of Linear Programming • Proportionality assumption • The contribution of each activity to the value of the objective function (or lefthand side of a functional constraint) is proportional to the level of the activity • If assumption does not hold, one must use nonlinear programming 23 Assumptions of Linear Programming • Additivity • Every function in a linear programming model is the sum of the individual contributions of the activities 24 Assumptions of Linear Programming • Divisibility • Decision variables in a linear programming model may have any values • Including noninteger values • Assumes activities can be run at fractional values • Certainty • Value assigned to each parameter of a linear programming model is assumed to be a known constant • Seldom satisfied precisely in real applications • Sensitivity analysis used 25 Example 1: Personnel scheduling • UNION Airways needs hire service agents to provide services to customers based on the schedule shown in the rightmost column of the table. • Each agent can work for one same 8-hour shift per day for 5 days per week. • Different shifts get different amounts of paid as shown in the last row of the table. • UNION Airways wanted to minimize the total personnel cost. 26 27 Example 2: Distribution network • DISTRIBUTION UNLIMITED Co. produces products at two factories at different locations. • The products must be shipped to two warehouses. • The distribution network is shown in the figure. • F1 and F2 are the two factories. • W1 and W2 are the two warehouses. • DC is a distribution center. • The amounts of units produced at the factories, the capacities of the warehouses, and the capacities and shipping costs of the lanes are all shown in the figure. Unlimited capacity if not specified. • The company wanted to minimize the total shipping cost. 28 29 30 Solving Linear Programming Models on a Spreadsheet • Excel and its Solver add-in • Popular tools for solving small linear programming problems 31 • Read: chapter 1, chapter 2, chapter 3.1, 3.2, 3.3, 3.4, 3.5 • Homework: chapter 3 • 3.1-9 (a)(b), 3.1-10 (a)(b), 3.4-9 (a)(b), 3.4-11(a)(b) • Note, use excel solver to solve 3.4-9(b) and 3.4-11(b) 32
