Introduction to Operations Research Z Max Shen IEOR

Introduction to Operations Research Z. Max Shen IEOR, Berkeley

What is Operations Research? What is Management Science? • World War II: British military leaders asked scientists and engineers to analyze several military problems – Deployment of radar – Management of convoy, bombing, antisubmarine, and mining operations. • The result was called Military Operations Research, later Operations Research 2

What is Management Science (Operations Research)? • • Today: Operations Research and Management Science mean “the use of mathematical models in providing guidelines to managers for making effective decisions within the state of the current information, or in seeking further information if current knowledge is insufficient to reach a proper decision. ” c. f. Decision science, systems analysis, operational research, systems dynamics, operational analysis, engineering systems, systems engineering, and more. 3

Voices from the past • • • Waste neither time nor money, but make the best use of both – Benjamin Franklin Obviously, the highest type of efficiency is that which can utilize existing material to the best advantage. – Jawaharial Nehru It is more probable that the average man could, with no injury to his health, increase his efficiency fifty percent. – Walter Scott 4

Operations Research Over the Years • • • 1947 – Project Scoop (Scientific Computation of Optimum Programs) with George Dantzig and others. Developed the simplex method for linear programs. 1980’s – Widespread availability of personal computers. Increasingly easy access to data. Widespread willingness of managers to use models. 1990’s – Improved use of O. R. systems. Further inroads of O. R. technology, e. g. , optimization and simulation add-ons to spreadsheets, modeling languages, large scale optimization. More intermixing of A. I. and O. R. 5

Operations Research in the 00’s • LOTS of opportunities for OR as a field • Data, data – E-business data (click stream, purchases, other transactional data, E-mail and more) – The human genome project and its outgrowth • Need for more automated decision making • Need for increased coordination for efficient use of resources (Supply chain management) 6

Optimization is Everywhere • The more you know about something, the more you see where optimization can be applied. • Some personal decision making – – Finding the fastest route home (or to class) Optimal allocation of time for homework Optimal budgeting Selecting a major 7

Optimization is Everywhere • Some Berkeley decision making – Setting exam times to minimize overlap – Assigning classes to classrooms and time slots while satisfying constraints – Figuring out prices for parking and subsidies for public transportation so as to maximize fairness, and permit sufficient access 8

Optimization Tools • Optimization is everywhere, but optimization tools are not applied everywhere. • Goals in E 10: present some fundamental of tools for optimization, and illustrate some applications. • When you see optimization problems arise in business (and you will), you will know that there are tools to help you out. 9

Addressing managerial problems: A management science framework • • • Determine the problem to be solved • Verify the model and use the model for prediction or analysis • • • Select a suitable alternative Observe the system and gather data Formulate a mathematical model of the problem and any important subproblems Present the results to the organization Implement and evaluate 10

Some Success Stories • Optimal crew scheduling saves American Airlines $20 million/year. • Improved shipment routing saves Yellow Freight over $17. 3 million/year. • Improved truck dispatching at Reynolds Metals improves on-time delivery and reduces freight cost by $7 million/year. • GTE local capacity expansion saves $30 million/year. 11

Other Success Stories (cont. ) • • Optimizing global supply chains saves digital Equipment over $300 million. Restructuring North America Operations, Proctor and Gamble reduces plants by 20%, saving $200 million/year. Optimal traffic control of Hanshin Expressway in Osaka saves 17 million driver hours/year. Better scheduling of hydro and thermal generating units saves southern company $140 million. 12

Success Stories (cont. ) • • • Improved production planning at Sadia (Brazil) saves $50 million over three years. Production Optimization at Harris Corporation improves on-time deliveries from 75% to 90%. Tata Steel (India) optimizes response to power shortage contributing $73 million. Optimizing police patrol officer scheduling saves police department $11 million/year. Gasoline blending at Texaco results in saving of over $30 million/year. 13

Define the Problem • • • Many practical problems do NOT have clearly defined objectives The first step is to develop a well-defined statement of the problem – Objective – Constraints – Time frame for the study It involves making appropriate assumptions, requires good understanding of the problem being studied 14

Formulate a Mathematical Model • Mathematical models are idealized representations of real world problems • The mathematical models we are going to see and discuss in this class have the following components: – Decision variables • Whose values will need to be determined – Parameters • Constants we already know in the problem – Objective function • An appropriate measure of performance (for instance, cost or profit) – Constraints • Restriction on the values that the decision variables may take 15

What is Mathematical Programming • “Mathematical Programming” describes the minimization or maximization of an objective function, subject to constraints on the decision variables • The term “programming” describes the planning or scheduling of related activities within a large operation – it does not have direct connections with “computer programming” • The solution obtained are often referred to as OPTIMAL SOLUTION 16

Formulation of LP -- Example A Diet Problem Let’s assume that you can buy the following precooked meals: 1 Beef 2 Chicken 3 Macaroni w/ Cheese And you are interested in getting adequate amounts of the following three nutrients: Protein, Vitamin C, and Iron You want to find the lowest-cost combination of food that will meet the requirements for the nutrients during one week. 17

Formulation of LP To solve the problem, you first need to know • • • The cost of one package of each meal The required amount for each nutrient How much of each nutrients is in one package of each food Beef Chicken Macaroni w/ Cheese Required Protein 60 55 20 700 Vitamin C 0 10 35 250 Iron 20 15 10 300 Cost 2. 98 2. 09 1. 49 18

Formulation of LP • • Let j be the index for food – 1 – beef, 2 – chicken, 3 – macaroni w/ cheese Let i be the index for the nutrients – 1 – Protein, 2 – Vitamin C, 3 – Iron • Define Decision Variables – The number of packages of each food you are going to consume • Let x(j), j=1, 2, 3, be the number of packages of beef, chicken, macaroni w/ cheese to buy, respectively 19

Formulation of LP • Define Parameters – The cost of different food • Let c(j), j=1, 2, 3, represent the cost of one package of beef, chicken, macaroni w/ cheese, respectively – The amount of each nutrient required • Let b(i), i=1, 2, 3, be the required amount of protein, VC, Iron, respectively – The amount of each nutrient in one package of each food • Let a(i, j), i=1, 2, 3, j=1, 2, 3, be the amount of nutrient i in food j 20

Formulation of LP • Formulate the Objective Function – We want to minimize the cost of food Minimize 2. 98 x 1+2. 09 x 2+1. 49 x 3 – In more general form, it can be written as: Minimize c 1 x 1+c 2 x 2+c 3 x 3 Also: 21

Formulation of LP • Formulate the Constraints – Get at least required amount of each type of nutrients • • • For protein: 60 x 1+55 x 2+20 x 3>=700 For Vitamin C: 0 x 1+10 x 2+35 x 3>=250 For iron: 20 x 1+15 x 2+10 x 3>=300 – In more general form, it can be written as: • • • For protein: a 11 x 1+a 12 x 2+a 13 x 3>=b 1 For Vitamin C: a 21 x 1+a 22 x 2+a 23 x 3>=b 2 For iron: a 31 x 1+a 32 x 2+a 33 x 3>=b 3 That is: 22

Formulation of LP The LP formulation for the diet problem: Subject to: 23

An Assignment Problem Consider a company in which there are m groups of workers who can be assigned to n different jobs. Each group has a particular cost per worker to do each job. The data has the following form: ai number of workers in group i, i=1, 2, …, m bj number of workers required in job j, j=1, 2, …, n cij cost per worker in group i performing job j The objective is to assign appropriate numbers of workers from each group to each job so that the cost is minimized 24

An Assignment Problem The decision variable for this problem: xij number of workers from group i assigned to job j Objective function: minimize s. t. 25