Nonlinear Programming

Transcription

1 Nonlinear Programming Chapter

2 Chapter Topics Nonlinear Profit Analysis Constrained Optimization Solution of Nonlinear Programming Problems with Excel Nonlinear Programming Model with Multiple Constraints Nonlinear Model Examples 10-2

3 Overview Problems that fit the general linear programming format but contain nonlinear functions are termed nonlinear programming (NLP) problems. Solution methods are more complex than linear programming methods. Determining an optimal solution is often difficult, if not impossible. Solution techniques generally involve searching a solution surface for high or low points requiring the use of advanced mathematics. 10-3

4 Optimal Value of a Single Nonlinear Function Basic Model 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, p Figure 10.1 Linear Relationship of Volume to Price 10-4

5 Optimal Value of a Single Nonlinear Function With fixed cost (c f = $10,000) and variable cost (c v = $8): Profit, Z = 1,696.8p p 2-22,000 Figure 10.2 The Nonlinear Profit Function 10-5

6 Optimal Value of a Single Nonlinear Function Maximum Point on a Curve 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 10.3 Maximum profit for the profit function 10-6

7 Optimal Value of a Single Nonlinear Function Solution Using Calculus Z = 1,696.8p p 2-2,000 dz/dp = 1, p = 0 p = /49.2 = $34.49 v = 1, p v = pairs of jeans Z = $7, Figure

8 Constrained Optimization in Nonlinear Problems Definition A nonlinear problem containing one or more constraints becomes a constrained optimization model or a nonlinear programming (NLP) 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. 10-8

9 Constrained Optimization in Nonlinear Problems Graphical Interpretation (1 of 3) Effect of adding constraints to nonlinear problem: Figure 10.5 Nonlinear Profit Curve for the Profit Analysis Model 10-9

10 Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3) Figure 10.6 A Constrained Optimization Model 10-10

11 Constrained Optimization in Nonlinear Problems Graphical Interpretation (3 of 3) Figure

12 Constrained Optimization in Nonlinear Problems Characteristics 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

13 Western Clothing Problem Solution Using Excel (1 of 3) Exhibit

14 Western Clothing Problem Solution Using Excel (2 of 3) Exhibit

15 Western Clothing Problem Solution Using Excel (3 of 3) Exhibit

16 Beaver Creek Pottery Company Problem Solution Using Excel (1 of 6) Maximize Z = $(4-0.1x 1 )x 1 + (5-0.2x 2 )x 2 subject to: x 1 + 2x 2 = 40 Where: x 1 = number of bowls produced x 2 = number of mugs produced 4 0.1X 1 = profit ($) per bowl 5 0.2X 2 = profit ($) per mug 10-16

17 Beaver Creek Pottery Company Problem Solution Using Excel (2 of 6) Exhibit

18 Beaver Creek Pottery Company Problem Solution Using Excel (3 of 6) Exhibit

19 Beaver Creek Pottery Company Problem Solution Using Excel (4 of 6) Exhibit

20 Beaver Creek Pottery Company Problem Solution Using Excel (5 of 6) Exhibit

21 Beaver Creek Pottery Company Problem Solution Using Excel (6 of 6) Exhibit

22 Western Clothing Company Problem Solution Using Excel (1 of 4) Maximize Z = (p 1-12)x 1 + (p 2-9)x 2 subject to: 2x x 2 6, x x 2 8, x x 2 15,000 where: x 1 = 1, p 1 x 2 = 2, p 2 p 1 = price of designer jeans p 2 = price of straight jeans 10-22

23 Western Clothing Company Problem Solution Using Excel (2 of 4) Exhibit

24 Western Clothing Company Problem Solution Using Excel (3 of 4) Exhibit

25 Western Clothing Company Problem Solution Using Excel (4 of 4) Exhibit

26 Facility Location Example Problem Problem Definition and Data (1 of 2) Centrally locate a facility that serves several customers or other facilities in order to minimize distance or miles traveled (d) between facility and customers. Where: d i = sqrt[(x i - x) 2 + (y i - y) 2 ] (x,y) = coordinates of proposed facility (x i,y i ) = coordinates of customer or location facility i Minimize total miles d = Σ d i t i Where: d i = distance to town i t i =annual trips to town i 10-26

27 Facility Location Example Problem Problem Definition and Data (2 of 2) Coordinates Town x y Annual Trips Abbeville Benton Clayton Dunnig Eden

28 Facility Location Example Problem Solution Using Excel Exhibit

29 Facility Location Example Problem Solution Map Figure 10.8 Rescue Squad Facility Location 10-29

30 Investment Portfolio Selection Example Problem Definition and Model Formulation (1 of 2) Objective of the portfolio selection model is to: minimize some measure of portfolio risk (variance in the return on investment) while achieving some specified minimum return on the total portfolio investment

31 Investment Portfolio Selection Example Problem Definition and Model Formulation (2 of 2) Minimize S = x 12 s 12 + x 22 s x n2 s n2 + Σx i x j r ij s i s j i j where: S = variance of annual return of the portfolio x i,x j = the proportion of money invested in investments i or j s i2 = the variance for investment i r ij = the correlation between returns on investments i and j s i,s j = the std. dev. of returns for investments i and j subject to: r 1 x 1 + r 2 x r n x n r m x 1 + x 2 + x n = 1.0 where: r i = expected annual return on investment i r m = the minimum desired annual return from the portfolio 10-31

32 Investment Portfolio Selection Example Problem Solution Using Excel (1 of 5) Stock (x i ) Annual Return (r i ) Variance (s i ) Altacam Bestco Com.com Delphi Stock combination (i,j) Correlation (r ij ) A,B A,C A,D B,C B,D C,D

33 Investment Portfolio Selection Example Problem Solution Using Excel (2 of 5) Four stocks, desired annual return of at least Minimize Z = S = x 12 (.009) + x 22 (.015) + x 32 (.040) + X 42 (.023) + x 1 x 2 (.4)(.009) 1/2 (0.015) 1/2 + x 1 x 3 (.3)(.009) 1/2 (.040) 1/2 + x 1 x 4 (.6)(.009) 1/2 (.023) 1/2 + x 2 x 3 (.2)(.015) 1/2 (.040) 1/2 + x 2 x 4 (.7)(.015) 1/2 (.023) 1/2 + x 3 x 4 (.4)(.040) 1/2 (.023) 1/2 + x 2 x 1 (.4)(.015) 1/2 (.009) 1/2 + x 3 x 1 (.3)(.040) 1/2 + (.009) 1/2 + x 4 x 1 (.6)(.023) 1/2 (.009) 1/2 + x 3 x 2 (.2)(.040) 1/2 (.015) 1/2 + x 4 x 2 (.7)(.023) 1/2 (.015) 1/2 + x 4 x 3 (.4)(.023) 1/2 (.040) 1/2 subject to:.08x x x x x 1 + x 2 + x 3 + x 4 = 1.00 x i

34 Investment Portfolio Selection Example Problem Solution Using Excel (3 of 5) Exhibit

35 Investment Portfolio Selection Example Problem Solution Using Excel (4 of 5) Exhibit

36 Investment Portfolio Selection Example Problem Solution Using Excel (5 of 5) Exhibit

37 Hickory Cabinet and Furniture Company Example Problem and Solution (1 of 2) Model: Maximize Z = $280x 1-6x x 2-3x 2 2 subject to: 20x x 2 = 800 board ft. Where: x 1 = number of chairs x 2 = number of tables 10-37

38 Hickory Cabinet and Furniture Company Example Problem and Solution (2 of 2) 10-38

39 10-39