1. Nonlinear Programming (NLP) Operation Research December 29, 2014 RS and GISc, IST, Karachi

2. Introduction In LP, the goal is to maximize or minimize a linear function subject to linear constraints But in many real-world problems, either –objective function may not be a linear function, or –some of the constraints may be nonlinear Functions having exponents, logarithms, square roots, products of variables, and so on are nonlinear

3. NLP Optimization problems that involve nonlinear functions are called nonlinear programming (NLP) optimization Solution methods are more complex than linear programming methods Solution techniques generally involve searching a solution surface for high or low points requiring the use of advanced mathematics NLPs that do not have any constraints are called unconstrained NLPs

4. Optimality Conditions: Unconstrained optimization Can be solved using calculus For Z=f(X), the optimum occurs at the point where f '(X) =0 and f’''(X) meets second order conditions –A relative minimum occurs where f '(X) =0 and f’''(X) >0 –A relative maximum occurs where f '(X) =0 and f’''(X) <0

5. Concavity and Second Derivative f’’(x) 0 local max and global max local max local min local min and global min

6. Solution process is straightforward using calculus: f'(x) = -2x + 9 Set this equal to zero and obtain x = 4.5 f''(x) = -2 which is negative at x = 4.5 (or at any other x-value) so we have indeed found a maximum rather than a minimum point So the function is maximized when x = 4.5, with a maximum value of -4.5 2 + 9(4.5) + 4 = 24.25. 6 Example: An unconstrained problem

7. Problem One problem is difficulty in distinguishing between a local and global minimum or maximum point Local maximum Global maximum This is trickier: a value x whose first derivative is zero and whose second derivative is negative is not necessarily the solution point! It could be a local maximum point rather than the desired global maximum point.

8. Feasible region Solution point In the case of this constrained optimization problem basic calculus is of no value, as the derivative at the solution point is not equal to zero 8

9. Problems Solutions to NLPs are found using search procedures Search can fail!!!

10. NLP Example: Searches Can Fail! Maximize f(x) = x 3 - 30x 2 + 225x + 50 10  The correct answer is that the problem is unbounded. There is no solution point!  Solvers may converge to a local maximum

11. 11 Profit function, Z, with volume independent of price: Z = vp - c f - vc v where v = sales volume p = price c f = unit fixed cost c v = unit variable cost Add volume/price relationship: v = 1,500 - 24.6p Figure 1 Linear Relationship of Volume to Price Example

12. 12 Profit, Z = 1,696.8p - 24.6p 2 - 22,000 Figure 2 The Nonlinear Profit Function With fixed cost (c f = $10,000) and variable cost (c v = $8):

13. 13 The slope of a curve at any point is equal to the derivative of the curve’s function. The slope of a curve at its highest point equals zero. Figure 3 Maximum profit for the profit function Optimal Value of a Single Nonlinear Function= Maximum Point on a Curve

14. 14 Z = 1,696.8p - 24.6p 2 -22,000 dZ/dp = 1,696.8 - 49.2p = 0 p = 1696.8/49.2 = $34.49 v = 1,500 - 24.6p v = 651.6 pairs of jeans Z = $7,259.45 Figure 4 Maximum Profit, Optimal Price, and Optimal Volume Optimal Value of a Single Nonlinear Function Solution Using Calculus

15. 15 If a nonlinear problem contains one or more constraints it becomes a constrained optimization model A nonlinear programming model has the same general form as the linear programming model except that the objective function and/or the constraint(s) are nonlinear. Solution procedures are much more complex and no guaranteed procedure exists for all NLP models. Constrained Optimization in Nonlinear Problems Definition

16. 16 Effect of adding constraints to nonlinear problem: Figure 5 Nonlinear Profit Curve for the Profit Analysis Model Constrained Optimization in Nonlinear Problems Graphical Interpretation (1 of 3)

17. 17 Figure 6 A Constrained Optimization Model Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3)- First constrained p<= 20

18. 18 Figure 7 A Constrained Optimization Model with a Solution Point Not on the Constraint Boundary Constrained Optimization in Nonlinear Problems Graphical Interpretation (3 of 3) Second constrained p<= 40

19. 19 Unlike linear programming, solution is often not on the boundary of the feasible solution space. Cannot simply look at points on the solution space boundary but must consider other points on the surface of the objective function. This greatly complicates solution approaches. Solution techniques can be very complex. Constrained Optimization in Nonlinear Problems Characteristics

20. 20 Centrally locate a facility that serves several customers or other facilities in order to minimize distance or miles traveled (d) between facility and customers. d i = sqrt[(x i - x) 2 + (y i - y) 2 ] Where: (x,y) = coordinates of proposed facility (x i,y i ) = coordinates of customer or location facility i Minimize total miles d =  d i t i Where: d i = distance to town i t i =annual trips to town i Facility Location Example Problem Problem Definition and Data (1 of 2)

21. 21 Facility Location Example Problem Problem Definition and Data (2 of 2)

22. Facility Location Example Problem: Using Excel Solver

23. Excel Solver

24. 24 Figure 13 Facility Location Example Problem Solution Using Excel

25. 25 Facility Location Example Problem Solution Map di = sqrt[(xi - x) 2 + (yi - y) 2 ] d A =sqrt[(20- 20.668) 2 + (20- 15.473) 2 ] d A =4.57........................................ d E =6.22 d =  diti d = 4.57(75)+................................+90(13.02) d = 5583.8 total annual distance

26. 26 Rescue Squad Facility Location Facility Location Example Problem Solution Map X = 20.668, Y = 15.473

30. More Examples