A Crash Course in “Operations Research”

Archis GhateFeb 7, 2008

HSERV/EPI 539

What is Operations Research (OR)? � OR is a discipline that applies advanced mathematical

techniques to help institutions (private, public, non-profit) and individuals make better decisions.

� OR has recently been called “the science of better”.� http://www.scienceofbetter.org

� OR focuses on finding ways to allocate scarce resources to activities often in an uncertain environment in order to “optimize one or more objectives”.

� Institute for Operations Research and the Management Sciences (INFORMS) - http://www.informs.org

Some Examples of Typical Decision Problems

� How many cars to produce? What intensity of radiation to use in cancer treatment? How many outpatients to schedule every day? Whether to undertake a new project? Whether to invest in a particular stock? Which retirement plan to choose? How many medical service centers to open in a county? How many warehouses to open? What and how much to stock in these warehouses? How many nurses to employ? Which doctors to have on call? How to schedule flights? How to route road traffic / air traffic? Whether to use two-finger screening or ten-finger screening? How to schedule multiple projects? Which classes to take? Which PhD advisor to choose? When to replenish inventory?

Origins and Applications of OR� Initial research in OR is typically attributed to World War

II.� Since the early 1940, OR has grown to be a vast field of

study with applications to healthcare, emergency/disaster management, telecommunications, finance, business modeling, medicine, logistics/transportation, inventory management, manufacturing, sports, engineering design, economics, natural sciences, e-commerce, forest management, supply chain management, …….

� This lecture is a 60 minute crash-course on a 50-year old field !!

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The OR Approach to Modeling� Not different from other quantitative fields of study

1. Define the problem and gather data.2. Formulate a mathematical model to represent the problem3. Develop a computer-based procedure for deriving

solutions to the model4. Test/refine the model5. Implement.

(Hillier and Lieberman, 2005)

1. Defining the Problem and Gathering Data

� Unlike textbook problems, practical decision problems are often “vague”.

� Important to identify the key decision makers and understand their viewpoint/gain insights into the problem.

� What are the different decision alternatives?� What is the objective of interest? Are there multiple

objectives? Do these objectives conflict with one another?� What are the constraints?� What are the sources of uncertainty?

� Data gathering is key.

2. Mathematical Model� After a problem description is available from the decision

makers, we need to build a mathematical model of the problem

� A mathematical model is an “idealized representation” that uses � variables to represent attributes in the problem.� parameters/constants that represent problem data.� simplifying assumptions to make the solution/analysis tractable.

� It is important to build a mathematical model that can be “solved” and yet represents the actual problem to a “reasonable accuracy”.

� Since it is typically hard to assign appropriate values to the parameters, we often perform “sensitivity analysis”

� How does our decision change if we change the parameter values.

3. Computer-based procedure for deriving solutions.

� In some cases, standardized software may be available to solve your model.� Applied research (most relevant for you)

� In some cases, you may have to design new algorithms and write your own programs � Methodological research

A Short List of OR Journals� Operations Research, Management Science,

Manufacturing and Service Operations Management, Mathematical Programming, Interfaces, INFORMS Journal on Computing, Mathematics of Operations Research, Healthcare Management Science

A Partial List of OR Topics� Linear Programming, Non-linear

Programming, Integer Programming, Dynamic Programming, Decision Analysis, Markov chains, Markov Decision Problems, Queueing Theory, Game Theory, Simulation, Network Optimization, Stochastic Programming

A Diet Problem1. Problem description and data

� To decide the quantity of different food itmes to consume every day so as to meet the Minimum Daily Requirement (MDR) of several nutrients at minimum cost.

� What type of data do we need?� What are the different food items under

consideration? Nutrient information and cost of food?

Diet Problem contd..� Simple toy example - two food items and three nutrients

Wheat Rye MDR

Carbs/unit 5 7 8Proteins/unit 4 2 15Vitamins/unit 2 1 3

Cost/unit 0.6 0.35

Diet Problem contd..2. Mathematical model. What variables do we need?

Let XW and XR be the quantities of wheat and rye we consume respectively.

€

minimize 0.6XW +0.35XR

subject to 5XW + 7XR ≥8

4XW +2XR ≥15

2XW + XR ≥ 3

XW ≥0 XR ≥0

cost

vitamin MDRprotein MDR

carb MDR

Diet Problem contd..� Computer solution using EXCEL

XW XR TOTAL COST0 0 0

per unit WHEAT RYE MDR DAILY INTAKECARB 5 7 8 0

PROTEIN 4 2 15 0VITAMIN 2 1 3 0

COST 0.6 0.35

Transportation Problem (adapted from Hillier and Lierberman, 2005)

� A non-profit organization manages three warehouses and four healthcare centers. The organization has estimated the requirements for a specific vaccine at each healthcare center in units of boxes of vials. The organization also knows the number of boxes of vials available at each warehouse. They want to decide how many boxes of vials to ship from the warehouses to the healthcare centers so as to meet the demand for the vaccine at minimum total shipping costs.

� Decision variables - Xij be the number of boxes shipped from warehouse i to healthcare center j.

HC1 HC2 HC3 HC4 AVAILABILITY

W1 464 513 654 867 75

W2 352 416 690 791 125

W3 995 682 388 685 100

REQUIREMENT 80 65 70 85

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A Shortest Path Problem (taken from Denardo, 1982)

� You bike from home (A) to work (G) everyday along a network of bike routes shown in the picture with nodes and links. The travel time along these links is shown next to the links. Which route should you take to minimize total travel time?

Distributing Medical Teams to Counties (adapted from Hillier and Lieberman, 2005)

� A non-profit institution provides health related services to three counties. These services include medical care, health education and training. The institution has five medical teams and wants to decide how many (if any) medical teams to assign to each of these counties so as to maximize additional person-years of life.

� Decision variables - xj be the number of medical teams assigned to county j.

thousands of additionalperson-years of life

countyMedical Teams 1 2 3

0 0 0 01 45 20 502 70 45 703 90 75 804 105 110 1005 120 150 130

Distributing Medical Teams contd..

Distributing Medical Teams to Counties contd..

� Can be solved the same way as the shortest path problem working backward choosing the option with the highest reward.

A Non-linear Problem � We can also tackle non-linear objective

functions/constraints.

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Maximize 126X-9X2 +182Y −13Y2

X ≤4

2Y ≤12

3X+2Y ≤18

X ≥0, Y≥0

X Y OBJ2.667 5 857

1 0 2.667 40 2 10 123 2 18 18

A Crop Farming Problem (taken from Hillier and Lieberman, 2005)

� A farmer wants to decide which crop to grow in order to maximize his expected income. He is considering four crops and feels that the weather this year is going to be either dry, moderate or damp. The farmer has estimated the probabilities for these events and the corresponding net income.

Crop Farming problem contd..

DRY MODERATE DAMP EXP. INCOMECROP1 20 35 40 31.5CROP2 22.5 30 45 30.75CROP3 30 25 25 26.5CROP4 20 20 20 20PROB. 0.3 0.5 0.2

WEATHER

What Did We Learn?� Saw a few toy examples from

“Operations Research”, which takes a systematic mathematical approach to decision making.

� There is a lot more out there and is covered in classes such as INDE 410, 411, 412, 513, 599 etc.

References� Hillier, F. and Lieberman, G.,

Introduction to Operations Research, McGraw Hill, 8th edition, 2005.

� Denardo, E., Dynamic Programming: Models and Applications, Prentice-Hall, 1982.