1 Chapter 1: Introduction to Mathematical Optimization 1.1 Introduction 1.2 Simple Examples 1.3 The OPTMODEL Procedure

2 Chapter 1: Introduction to Mathematical Optimization 1.1 Introduction 1.2 Simple Examples 1.3 The OPTMODEL Procedure

3 Objectives Explain the role of optimization in the decision-making process. Become familiar with basic terminology and the classification of mathematical optimization problems.

4 What Is Optimization? A major component of operations research / industrial engineering / management science Choosing the available actions that produce the best results Building directive analytics atop descriptive and predictive analytics The payoff from amassing business intelligence Moving from “What happened?” and “What will happen?” to “What should we do?”

5 $ROI Raw Data Standard Reports Ad Hoc Reports and OLAP Descriptive Modeling Predictive Modeling Data Information Intelligence Optimization Modeling Decision Support What happened? What will happen? What should we do? Decision Guidance Toward More Effective Decisions

6 Why Optimization? You can improve the decision-making process by adding the following: Structure: Know how decisions are motivated Consistency: Drive all decisions toward same goals Repeatability: Make decisions in the same manner over time Adaptability: Update, but keep same principles

7 Some Examples of the Use of Optimization Facility Location Production Planning Workforce Planning Product Distribution Delivery Network Configuration Investment Planning Supplier Selection and Evaluation Inventory Replenishment Planning Promotional Marketing Retail Pricing

8 Defining Mathematical Optimization Problems The general form of optimization problems is where x is a set of decision variables. f(x) is an objective function. c i (x) are functions that together with the bounds on x determine the constraints.

9 Defining Mathematical Optimization Problems The general form of optimization problems is where x is a set of decision variables. f(x) is an objective function. c i (x) are functions that together with the bounds on x determine the constraints.

10 Classification of Mathematical Optimization Problems LP: f(x) and c i (x) are linear functions. ILP: f(x) is linear, x must be integer, and c i (x) are linear functions. MILP: f(x,y) is linear, x must be integer, and c i (x,y) are linear functions, but y may be fractional. NLP: f(x) and c i (x) are continuous but not all linear functions.

Example 11 The figure below shows graphs of a linear function (y=0.5x), a continuous function, and a discontinuous function (y=3 for x≤4 and y=7-0.5x for x>4) of a single variable x.

12 A Crash Course in Solving Mathematical Optimization Problems LP: You do not need to understand all of the controls. Not hard to fly after it is in the air.

13 A Crash Course in Solving Mathematical Optimization Problems ILP/MILP: The autopilot can fly most of the way, but the “gas mileage” is much worse than LP.

14 A Crash Course in Solving Mathematical Optimization Problems NLP: There are some new controls to learn, and takeoff and landing are more complicated.

15 This exercise reinforces the concepts discussed previously. Exercise 1

16 Chapter 1: Introduction to Mathematical Optimization 1.1 Introduction 1.2 Simple Examples 1.3 The OPTMODEL Procedure

17 Objectives Develop intuition for the solution techniques for linear and nonlinear programming problems.

18 One-Dimensional Example maximizey = -2x(x-8) subject to 1  x  3 This problem maximizes a nonlinear function with one decision variable upper and lower bounds no other constraints. Calculus can be used to find the solution: y'(x) = -2x(1)+(-2)(x-8) = -4x+16 = 0 ⇒ x = 4 y"(x) = -4 < 0

19 One-Dimensional Example maximizey = -2x(x-8) subject to 1  x  3 This problem maximizes a nonlinear function with one decision variable upper and lower bounds no other constraints. Calculus can be used to find the solution: y'(x) = -2x(1)+(-2)(x-8) = -4x+16 = 0 ⇒ x = 4 y"(x) = -4 < 0

20 Graph of One-Dimensional Example

21 Graph of One-Dimensional Example y(4) = 32

22 Graph of One-Dimensional Example y(4) = 32 y(3) = 30 y(1) = 14

23 Two-Dimensional Example minimize z = x 2 -x-2y-xy+y 2 This problem minimizes a nonlinear function with two decision variables, x and y no bounds or other constraints. Vector calculus can be used to find the solution: z = [∂z/∂x, ∂z/∂y] = [2x-1-y, -2-x+2y] = [0, 0] ⇒ [x, y] = [4/3, 5/3] H(z) = (positive definite)

24 Graph of Two-Dimensional Example

25 Chapter 1: Introduction to Mathematical Optimization 1.1 Introduction 1.2 Simple Examples 1.3 The OPTMODEL Procedure

26 Objectives Solve an unconstrained mathematical optimization problem using the OPTMODEL procedure.

27 This demonstration illustrates the solution of an unconstrained mathematical optimization problem using PROC OPTMODEL. Solving the Simple Polynomial Example Using PROC OPTMODEL polynomial.sas

28 This exercise reinforces the concepts discussed previously. Exercise 2

Object function: Banana function 29 f = (y-x2) 2 + (1-x) 2

Exercise 2 30