CHAPTER ONE INTRODUCTION TO OPERATIONS RESEARCH AND MODELLING Mulugeta K. (PhD) Department of Management 1.1. Introduction to Operations Research (OR) • Operations Research (OR) came in to existence as a discipline during World War II: – when there was a critical need to manage scarce resources. • The term “Operations research” was coined as a result of research on military operations during this war. • The objective was the most effective utilization of most limited military resources by the use of quantitative techniques. Cont… OR/Management Science Defined • OR is a scientific approach to decision making, which seeks to determine how best to design and operate a system, usually under conditions requiring the allocation of scarce resources.” -Winston • Management science is the application of a scientific approach to solving management problems in order to help managers make better decisions. -Taylor and Bernard Cont… • It is a scientific method of providing executive departments with a quantitative basis for decisions regarding the operations under their control.” - Kimball & Morse • In general, • OR is a systematic application of quantitative methods, techniques and tools to the analysis of problems involving the operation of systems. – A quantitative approach to decision making Cont… • Other names used for this discipline include: – Management science (MS) – OR/MS or just ORMS – Industrial engineering (IE), – Decision Science (DS) and Problem solving • In OR, problems are: – Decomposed into basic components and – Solved via mathematical analysis. Significance Operations Research • Helps to balance conflicting objectives (goals or interests) of sub units in the organization. – where there are many alternative courses of action available to the decision-makers. • OR Balances the following: # Global Optimum: the decision that is best for the organization as a whole. # Suboptimum decision: A decision that is best for one or more sections of the organization Benefits of OR • OR helps companies to: – Minimize Cost of Investment – Increase Revenue or ROI – Increase Market Share – Manage and Reduce Risk – Improve Quality – Increase Throughput while Decreasing Delays – Improves Utilization of Limited Resources – Demonstrate Feasibility and Workability Features of Operations Research Approach • Features of OR approach to any decision and control problems include: Inter-disciplinary approach Methodological Approach Holistic Approach/ Systems Orientate Objectivistic Approach Decision Making Approach Use of Computers Human factors Cont… • A system is an organization of interdependent components that work together to accomplish the goal of the system. • The components may be either physical or conceptual or both but they share a unique relationship with each other and with the overall objective of the system. Cont… • We have said that OR is a quantitative and rational approach to decision making. • The scientific approach to decision making requires the use of one or more mathematical models. • A mathematical model is a mathematical representation of the actual situation that may be used to make better decisions or clarify the situation. 1.2. An Introduction to Modeling in OR • What is a model? – A representation of an object, a system, or an idea in some form other than that of the entity itself. - Shannon – A model is a limited approximation of reality. – Models do not and cannot represent every aspect of reality. • The reason being the innumerable and changing characteristics of the real life problems Cont… • For a model to be effective, it must: – be representative of those aspects of reality that are being investigated and – have a major impact on the decision situation. • A model is constructed to analyze and understand the given system for the purpose of improving its performance. • It allows an examination of the behavioral changes of a system w/o disturbing the on-going operations. Types of OR Models • There are many ways to classify models: Based on structure a. Physical Models: provide a physical appearance of the real object under study either reduced in size or scaled up. – Example: Scale models, prototype plants,… b. Symbolic models: use symbols and functions to represent variables and their relationships to describe the properties of the system. Cont… Based on function or purpose a. Descriptive models: describe some aspects of a situation, based on observation, survey, interview and questionnaire results or available data. – They do not recommend anything. b. Predictive Models: indicate “If this occurs, then that follow”. – They relate dependent and independent variables and permit trying out, “what if” questions. Cont… • For example: S = a + bA +cI • is a model that describes how the sales (S) of a product changes in advertising expenditures (A) and disposal personal income (I). • Here, a, b, and c are parameters whose values must be estimated. • These models do not have an objective function as a part of the model. Cont… c. Normative (Optimization) models: provide the “best” or “Optimal” solution to problems subject to certain limitations on the use of resources. – They provide recommended courses of action. – These models are also called prescriptive models, because they prescribe what the decision maker ought to do. – Example: Mathematical programming models (LP, IP, TP, AP etc) Cont… Based on Time Reference a. A static model is one in which the decision variables do not involve sequences of decisions over multiple periods. [do not account for changes over time.] – Example: EOQ (Inventory model) b. A dynamic model is a model in which the decision variables do involve sequences of decisions over multiple periods. – Example: Dynamic programming Cont… Based on degree of certainty a. Deterministic Models: If all the parameters, constants and functional relationships are assumed to be known with certainty – Examples: Linear programming models b. Probabilistic (Stochastic) models: Models in which at least one parameter or decision variable is random. – But it is possible to predict a pattern of values of the variables by their probability distribution. Cont… Based on Method of solution or Quantification a. Heuristic Models: employ some sets of rules which, though not optimal, do facilitate solutions of problems b. Analytical Models: have a specific mathematical structure and can be solved by known analytical techniques. [Eg: Optimization models] c. Simulation Models: have a mathematical structure but are not solved by applying known mathematical techniques. Cont… • What benefits do you think do models have? Seven Steps to Good OR Analysis 1. Observe and Identify the Problem – Define the problem (Is it too narrow or too broad?) – Specify objectives – Determine parts of the organization to be studied. 2. Understand the System – Determine parameters affecting the problem. – How different components of the system ınteract wıth each other? – Collect data to estimate values of the parameters. Cont… 3. Formulate a Mathematical Model • Basic components required in decision problem model are: – – – – – – Controllable (decision) Variables Uncontrollable variable Objective function Constraints or Limitations Functional relationships Parameters Cont… • A general decision problem model might take the form: Cont… ꙮ A model is referred to as a linear model if all functional relationships among decision variables X1, X2, Xn in f(x) and g(x) are of a linear form. ꙮ But if one or more of the relationships are non – linear, the model is said to be a non-linear model. ꙮ If one or more of the decision variables must be integer, then the model is an integer model. ꙮ If all the decision variables are free to assume fractional values, then the model is a non-integer model. Cont… 4. Verify the Model and Use the Model for Prediction – Do outputs match current observatıons for current ınputs? – Are outputs reasonable? – Could the model be erroneous? 5. Select a Suitable Alternative – Given a set of alternative solutions, determine which solution best meets the objectives. – Inherently the most difficult step. Cont… 6. Present the Results of the Analysis – Present the results to the decision maker(s) • If necessary, prepare several alternative solutions and permit them to choose. – Any non-approval of the study’s solution may have stemmed from an incorrect problem definition or failure to involve the decision maker(s) from the start of the project. • In such a case, return to step 1, 2, or 3. Cont… 7. Implement and Evaluate Recommendations • Assist in implementing the recommendations. • Monitor and dynamically update the system as the environment and parameters change to ensure that recommendations enable the organization to meet its goals. Real Life Example: GE Capital • GE Capital provides credit card service to 50 million accounts with an average outstanding balance of $12 billion. • GE Capital led by Makuch et al. (1989) developed the PAYMENT system to reduce delinquent accounts and the cost of collecting from delinquent accounts. # Step 1: Its goal was to reduce the delinquent accounts and the cost of processing them. – To do this, GE capital needed to come up with a method of assigning scarce labor resources to delinquent accounts. Cont… # Step 2: The key to modeling delinquent accounts is the concept of a delinquency movement matrix (DMM). – The DMM determines how the probability of the payment on a delinquent account during the current month depends on the following factors: • size of unpaid balance • action taken • performance score Cont… • Step 3 GE developed a linear optimization model. – The objective function was to maximize the expected delinquent accounts collected during the next six months. – The decision variables represented the fraction of each type of delinquent account. – The constraints ensure that available resources are not overused. – Constraints also relate the number of each type of delinquent account present in, say, January to the number of delinquent accounts for each type present in the next month. – This dynamic aspect is crucial to the model’s success. Cont… # Step 4: PAYMENT was piloted on a $62 million portfolio for a single department store. – GE Capital managers came up with their own strategies for allocating resources (called CHAMPION). – Store’s accounts were randomly assigned to the CHAMPION and PAYMENT strategies. – PAYMENT used more live calls and “no action”. – PAYMENT showed a 5-7% improvement over CHAMPION. # Step 5: For each type of account PAYMENT tells the credit managers the fraction that should receive each type of contact. Cont… # Steps 6 and 7: PAYMENT was next applied to the 18 million accounts of the $4.6 billion Montgomery Ward store portfolio. – Comparing accounts to the previous year, PAYMENT increased collections by $1.6 million per month. – Overall, GE Capital estimated PAYMENT increased collections by $37 million per year and used fewer resources than previous strategies. Some other Successful Applications of OR Company Year Problem Designing buffers into production line Techniques Used Annual Savings Hewlett Packard 1998 Queuing models $280 million Taco Bell 1998 Employee scheduling IP, Forecasting, Simulation $13 million Proctor & Gamble 1997 Redesign production & distributon system Transportation models $200 million Delta Airlines 1994 Assigning planes to routes Integer Programming $100 million AT&T 1993 Call center design Queuing models, Simulation $750 million Yellow Freight Systems, Inc. 1992 Design trucking network Network models, Forecasting, Simulation $17.3 million San Francisco Police Dept. 1989 Patrol Scheduling Linear Programming $11 million Bethlehem Steel 1989 Design an Ingot Mold Stripper Integer Programming $8 million North American Van Lines 1988 Assigning loads to drivers Network modeling $2.5 million Citgo Petroleum 1987 Refinery operations & distribution Linear Programming, Forecasting $70 million United Airlines 1986 Scheduling reservation personnel LP, Queuing, Forecasting $6 million Dairyman's Creamery 1985 Optimal production levels Linear Programming $48,000 Phillips Petroleum 1983 Equipment replacement Network modeling $90,000 [Some] Application Areas of OR Transportation and Travel • OR techniques are used by airlines and rail companies to offer varying fares and make higher revenues by filling more seats at different prices - an OR technique known as Yield Management. All airlines depend on the effective use of OR techniques to make them operate at a profit. Financial Services • Used to address issues such as portfolio and risk management, planning and analysis of customer service. They are widely employed in Credit Risk Management—to find optimum balance of risk and revenue. Cont… Retailing • In supermarkets, data from store loyalty card schemes is analyzed by OR groups to advise on merchandising policies & profitability improvement. OR methods are also used to decide when and where new store developments should be made. Health • Hospital managers use OR to make decisions such as determining the optimal utilization of operating rooms and personnel, assessing the risks posed by patients with various medical conditions, and deciding necessary levels of perishable medicine in stock. Cont… Government • OR is a key contributor in modernizing government services and making them more efficient. Some OR applications include modeling the impact of performance related paying for employees, evaluating government call-centers, projecting the size of the prison population, and national defense….. Production Systems • Such problems may arise in settings that include, but are not limited to, manufacturing, telecommunications, health-care delivery, facility location, layout, and staffing Common Operations Research Models 1. Allocation Models: Optimization models • • Helps to solve both: Linear programming problems, and Non-linear programming problems Allocation models include – Linear programming – Integer programming – Goal programming – Stochastic programming – Transportation problems – Assignment problems Cont… 2. Probabilistic Models – – Decision Analysis models Queuing Theory 3. Inventory Model – – EOQ EPQ 4. Network Models – – CPM PERT 5. Competitive (Game Theory) Model 6. Simulation Models Algorisms and OR Software Packages • Popular tools/ modeling systems, approaches and software packages include: – – – – – – Excel Solver LINDO/LINGO MATLAB OR Tutor CPLEX MPL Recommended OR Journals • The following academic journals contain many useful articles that reflect state of the art applications of OR. 1. Operations Research 2. Management Science 3. European Journal of Operational Research 4. Journal of the Operational Research Society 5. Mathematical Programming 6. Networks 7. Naval Research Logistics 8. Interfaces • The first seven of the above are mainly theoretical whilst the eighth (Interfaces) concentrates upon case studies. 41