Introduction to Optimization: Modeling, Algorithms, and Solutions

1. Chapter 4An Introduction to Optimization

2. A schematic view of modeling/optimization process assumptions, abstraction,data,simplifications Real-world problem Mathematical model makes sense? change the model, assumptions? optimization algorithm Solution to real-world problem Solution to model interpretation

3. Mathematical models in Optimization • The general form of an optimization model: min or maxf(x1,…,xn) (objective function) subject to gi(x1,…,xn) ≥ 0 (functional constraints) x1,…,xn S(set constraints) • x1,…,xn are called decision variables • In words, the goal is to find x1,…,xnthat • satisfy the constraints; • achieve min (max) objective function value.

4. Types of Optimization Models Stochastic (probabilistic information on data) Deterministic (data are certain) Discrete, Integer (S = Zn) Continuous (S = Rn) Linear (f and g are linear) Nonlinear (f and g are nonlinear)

5. Linear Programming • Programming = Planning in this context • Origins go back to military logistics in WWII (1940s). • In a survey of Fortune 500 firms, 85% of those responding said that they had used linear or integer programming. • Why is it so popular? • Many different real-life situations can be modeled as linear programs (LPs). • There are efficient algorithms to solve LPs.

6. Standard form of linear program maximizec1x1+c2x2+…+cnxn (objective function) subject to a11x1+a12x2+…+a1nxn b1 (functional constraints) a21x1+a22x2+…+a2nxn b2 …. am1x1+am2x2+…+amnxn bm x1, x2 , …, xn≥ 0(set constraints) Note: Can also have equality or ≥ constraint in non-standard form.

7. Standard form of linear program • In matrix form: maximizecx (objective function) subject toAxb (functional constraints) x≥ 0(set constraints) Input for IP: 1n vector c, mn matrice A, m1 vector b. Output of IP: n1 vector x .

8. Example of Linear Program(Production Planning-Furniture Manufacturer) • Technological data: Production of 1 table requires 5 ft pine, 2 ft oak, 3 hrs labor 1 chair requires 1 ft pine, 3 ft oak, 2 hrs labor 1 desk requires 9 ft pine, 4 ft oak, 5 hrs labor • Capacities for 1 week: 1500 ft pine, 1000 ft oak, 20 employees (each works 40 hrs). • Market data: • Goal: Find a production schedule for 1 week that will maximize the profit.

9. Production Planning-Furniture Manufacturer: modeling the problem as linear program The goal can be achieved by making appropriate decisions. First define decision variables: Let xt be the number of tables to be produced; xc be the number of chairs to be produced; xd be the number of desks to be produced. (Always define decision variables properly!)

10. Production Planning-Furniture Manufacturer: modeling the problem as integer program • Objective is to maximize profit: max 12xt + 5xc + 15xd • Functional Constraints capacity constraints: pine: 5xt + 1xc + 9xd  1500 oak: 2xt + 3xc + 4xd  1000 labor: 3xt + 2xc + 5xd  800 market demand constraints: tables: xt ≥ 40 chairs: xc ≥ 130 desks: xd ≥ 30 • Set Constraints xt , xc , xd ≥ 0

11. Solutions to linear programs • A solution is an assignment of values to variables. • A solution can hence be thought of as an n-dimensional vector. • A feasible solution is an assignment of values to variables such that all the constraints are satisfied. • The objective function value of a solution is obtained by evaluating the objective function at the given point. • An optimal solution (assuming maximization) is one whose objective function value is greater than or equal to that of all other feasible solutions. • Next handout will discuss solution methods in details