ISE 195 Introduction to Industrial Engineering

Lecture 3 Mathematical Optimization (Topics in ISE 470 Deterministic Operations Research Models)

3 What is “OR”? “Operations Research” = “Study of Mathematical Optimization”  “OR” is short for Operations Research  Home professional society: the Institute for Operations Research and the Management Sciences (INFORMS)  Researchers in “OR” focus on how to improve the theory (mathematical) and algorithm (computational) aspects of formulating and solving mathematical optimization problems  Basic “OR” tools are helpful when a decision problem has many variables and constraints that can be described with a linear function  Web Site,

4 INFORMS

5 ISE and “OR” “Industrial & Systems Engineering” = “Branch of Engineering Concerned with Integrating and Improving Systems”  ISEs can use “OR” tools to do this, usually with the help of a computer  ISEs focus on problems in Logistics, Scheduling, Healthcare, etc. that have an optimization focus and that have a “scale” large enough to utilize OR tools  ISEs use “OR” to formulate design problems and generate solutions

6 Why the Comparison? Pure Operations Research has a heavy mathematical and computational orientation  There are many mathematical details to formulating problems successfully  There are many computational (computer programming, algorithmic) details to successfully finding “optimal” solutions to a stated problem ISE applications of OR do not have as high a theoretical mathematical or algorithmic content ISEs try to use the correct technique to improve the integrated system under investigation, including OR when appropriate

7 Model Formulation and Solution Mathematical optimization model formulation and solution  Represent the system or phenomena in some set of algebraic structures  Uses the “decision-makers” view, usually different from the “real-world” view  Simulation models have a closer mapping to real world details  Encode the resulting model in a computer via some modeling language  GAMS, X-Press, Excel  Find a “solution” to the model (hopefully “optimal”) Solution algorithms vary for linear, nonlinear and integer decision variables Solutions generated suggest new designs for a system  A “prescriptive” decision technique  Trying to find a “best” solution with which to prescribe how to make the best use of limited resources

8 Mathematical Modeling Describe system with set of algebraic equations  Capture key relationships within the system  Capture key behaviors in system Decisions for which insight needed are decision variables Goal embedded within the objective function Limitations/restrictions in constraints  Physical constraints  Logical constraints

9 General Form of Math Model

10 General Parametric Form MAX (or MIN): c 1 X 1 + c 2 X 2 + … + c n X n Subject to:a 11 X 1 + a 12 X 2 + … + a 1n X n <= b 1 : a k1 X 1 + a k2 X 2 + … + a kn X n <= b k : a m1 X 1 + a m2 X 2 + … + a mn X n = b m

11 A Simple Example Blue Ridge Hot Tubs produces two types of hot tubs: Aqua-Spas & Hydro-Luxes. There are 200 pumps, 1566 hours of labor, and 2880 feet of tubing available. How many of each type should be produced to maximize profits? Aqua-SpaHydro-Lux Pumps11 Labor 9 hours6 hours Tubing12 feet16 feet Unit Profit$350$300

12 Model Formulation Process 1. Understand the problem 2. Identify the decision variables X 1 =number of Aqua-Spas to produce X 2 =number of Hydro-Luxes to produce 3.State the objective function as a linear combination of the decision variables MAX: 350X X 2

13 Model Formulation Process 4. State the constraints as linear combinations of the decision variables 1X 1 + 1X 2 <= 200} pumps 9X 1 + 6X 2 <= 1566} labor 12X X 2 <= 2880} tubing 5. Identify any upper or lower bounds on the decision variables X 1 >= 0 X 2 >= 0

14 Model Formulation Process MAX: (Objective function) 350X X 2 S.T.: (Constraint set) 1X 1 + 1X 2 <= 200 9X 1 + 6X 2 <= X X 2 <= 2880 Non-negativity X 1 >= 0 X 2 >= 0

15 Formulation in Excel Solver

16 2 nd Simple Example Model Objective: Determine production mix that maximizes the profit under the raw material constraint and other production requirements (detailed next). Maximize 50D + 30C + 6 M Subject to 7D + 3C + 1.5M 100 (contract ) C 0(Non-negativity) D and C are integers

17 What We are Looking For Want to find the best solution Solution must satisfy each of the constraints Constraints must be satisfied simultaneously Common area satisfying all the constraints is called the “feasible region” ANY point in the feasible region is a possible solution to the problem What we want is that feasible solution that provides the largest value of the objective function (the “optimal solution”)

18 Feasible Region of an LP Profit =$ X2X2 X1X1

19 Linear Programming Assumptions of the linear programming model  The parameter values are known with certainty.  The objective function and constraints exhibit constant returns to scale.  There are no interactions between the decision variables (the additivity assumption).  The Continuity assumption: Variables can take on any value within a given feasible range.

20 How Do We Solve an LP? “Solving an LP” = “Finding the best solution possible” = “Finding the optimal solution” In ISE 470, you will learn the “Simplex Method” of “solving an LP”  Uses ideas from “matrix algebra” (i.e. pay attention in MTH 235)  Can be performed by hand using matrix operations (“pivots”) The simplex algorithm is available in many forms in software  Excel “Solver” Tool  Many commercial solver packages (CPLEX, XPRESS, etc)

21 Integer Programming Many real life problems call for at least one integer decision variable. There are three types of Integer models:  Pure integer (AILP)  Mixed integer (MILP)  Binary (BILP) (zero-one variables, on-off) Unfortunately, these get quite hard to solve  Real-world problems can have hundreds or thousands of variables and constraints  Some problems are “theoretically hard”, even when they seem to have small numbers of elements Real benefit of these types of models is that we can use binary variables to represent a host of logical conditions within the mathematical formulation

22 Y 1 + Y 2 + Y 3 + Y 4 + Y 5 + Y 6 > 3 1. Three of six projects must be selected. 2. No more than three of six projects can be selected. Y 1 + Y 2 + Y 3 + Y 4 + Y 5 + Y 6 < 3 Logical Contraints (1)

23 Y 2 + Y 4 + Y 6 = 1 3. Either Project 2 or Project 4 or Project 6 must be selected 4. Production level for Product 1 cannot exceed the production level for Product 5. X 1 < X 5 Logical Contraints (2)

24 X 4 < MY 4 X 4 > 120Y 4 5. If Product 4 is produced, then at least 120 units of Product 4 must be produced. 6. Four projects are numbered in ascending order. If any project selected, all lower numbered projects must also be selected. Y 2 < Y 1 Y 3 < Y 2 Y 4 < Y 3 Logical Constraint (3)

25 Y 3 < Y 4 + Y 5 7. If Project 3 selected, either Project 4 or Project 5 must be selected. 8. If Product 6 produced, production must be 50 or 100 units. X 6 = 50Y Y 62 Y 61 + Y 62 < 1 Logical Constraint (4)

26 Basic Solution Methods Linear models – Simplex algorithm  Fast in practice  Lots of good sensitivity analysis information Integer models – Branch and bound  Can take very long time  No sensitivity information  Can be sped up with specific knowledge Nonlinear models  Variety of solution methods, usually based on the “derivative” of the objective function  Use “Hill Climbing” and other methods

27 Applications of Mathematical Optimization Budgeting, capital budgeting Resource allocation, how much to make How much to make, how much to contract Assignment of personnel to jobs  Any other kind of assignment application Funding allocation, investment planning Inventory planning Facility layout and planning

28 Applications of Mathematical Optimization Routing  Ever use Google Maps?  Pick-up and delivery of items Selection of items for loading onto delivery trucks Scheduling  Jobs onto machines  Patients to doctors  Doctors to shifts, workers to shifts Packing problems, cutting problems  How to cut patterns from material And the list goes on and on…

29 Other Related Areas Non-linear optimization  These involve non-linear functions within the formulations  Most applicable solution methods are gradient-based local search procedures Modern heuristic optimization  These involve integer programming models that are particularly difficult to solve  Also involve non-linear models with forms for which the existing non-linear solution codes have a particularly hard time solving  These are essentially search methods with some pretty strange analogies to nature  Genetic Algorithms  Ant-Colony Algorithms  Simulated Annealing Algorithms

30 Large-Scale Applications Airline Crew Scheduling  Scenario: A late-day storm has canceled many flights for American Airlines in and out of Chicago O’Hare Airport. Many planes scheduled to come to Chicago and leave Chicago have not made it to their planned destination for the end of the day  Problem: How should we reassign crews and planes to the routes for the next day, to optimize our use of people, aircraft and other resources? Companies such as American Airlines, IBM, Hewlett-Packard, UPS, FedEx, Toyota have large-scale problems that are well suited to mathematical optimization

Questions?