Introduction to Management Science (8th Edition, Bernard W. Taylor III) Chapter 11 Nonlinear Programming. Chapter 11 - Nonlinear Programming 1

Transcription

1 Introduction to Management Science (8th Edition, Bernard W. Taylor III) Chapter 11 Nonlinear Programming Chapter 11 - Nonlinear Programming 1

2 Chapter Topics Nonlinear Profit Analysis Constrained Optimization Solution of Nonlinear Programming Problems with Excel A Nonlinear Programming Model with Multiple Constraints Nonlinear Model Examples Chapter 11 - Nonlinear Programming 2

3 Overview Many business problems can be modeled only with nonlinear functions. 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. Often difficult, if not impossible, to determine optimal solution. Solution techniques generally involve searching a solution surface for high or low points requiring the use of advanced mathematics. Computer techniques (Excel) are used in this chapter. Chapter 11 - Nonlinear Programming 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 = fixed cost c v = variable cost Add volume/price relationship: v = 1, p Figure 11.1 Linear Relationship of Volume to Price Chapter 11 - Nonlinear Programming 4

5 Optimal Value of a Single Nonlinear Function Expanding the Basic Model to a Nonlinear Model With fixed cost (c f = $10,000) and variable cost (c v = $8): Z = 1,696.8p p 2-22,000 Figure 11.2 The Nonlinear Profit Function Chapter 11 - Nonlinear Programming 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 11.3 Maximum profit for the profit function Chapter 11 - Nonlinear Programming 6

7 Optimal Value of a Single Nonlinear Function Solution Using Calculus Z = 1,696.8 p p 2-22,000 dz/dp = 1, p p = $34.49 v = 1, p v = pairs of jeans Z = $7, nonlinearity! Figure 11.4 Maximum Profit, Optimal Price, and Optimal Volume Chapter 11 - Nonlinear Programming 7

8 Constrained Optimization in Nonlinear Problems Definition If a nonlinear problem contains one or more constraints it becomes a constrained optimization model or a nonlinear programming 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. Chapter 11 - Nonlinear Programming 8

9 Constrained Optimization in Nonlinear Problems Graphical Interpretation (1 of 3) Effect of adding constraints to nonlinear problem: Figure 11.5 Nonlinear Profit Curve for the Profit Analysis Model Chapter 11 - Nonlinear Programming 9

10 Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3) Figure 11.6 A Constrained Optimization Model Chapter 11 - Nonlinear Programming 10

11 Constrained Optimization in Nonlinear Problems Graphical Interpretation (3 of 3) Figure 11.7 A Constrained Optimization Model with a Solution Point Not on the Constraint Boundary Chapter 11 - Nonlinear Programming 11

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. Chapter 11 - Nonlinear Programming 12

13 Western Clothing Problem Solution Using Excel (1 of 3) Exhibit 11.1 Chapter 11 - Nonlinear Programming 13

14 Western Clothing Problem Solution Using Excel (2 of 3) Exhibit 11.2 Chapter 11 - Nonlinear Programming 14

15 Western Clothing Problem Solution Using Excel (3 of 3) Exhibit 11.3 Chapter 11 - Nonlinear Programming 15

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 + x 2 = 40 Where: x 1 = number of bowls produced x 2 = number of mugs produced Chapter 11 - Nonlinear Programming 16

17 Beaver Creek Pottery Company Problem Solution Using Excel (2 of 6) Exhibit 11.4 Chapter 11 - Nonlinear Programming 17

18 Beaver Creek Pottery Company Problem Solution Using Excel (3 of 6) Exhibit 11.5 Chapter 11 - Nonlinear Programming 18

19 Beaver Creek Pottery Company Problem Solution Using Excel (4 of 6) Exhibit 11.6 Chapter 11 - Nonlinear Programming 19

20 Beaver Creek Pottery Company Problem Solution Using Excel (5 of 6) Exhibit 11.7 Chapter 11 - Nonlinear Programming 20

21 Beaver Creek Pottery Company Problem Solution Using Excel (6 of 6) Exhibit 11.8 Chapter 11 - Nonlinear Programming 21

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 6,00 3.6x x 2 8, x x 2 15,000 where: x 1 = 1, p 1 x 2 = 2, p p 1 = price of designer jeans p 2 = price of straight jeans Chapter 11 - Nonlinear Programming 22

23 Western Clothing Company Problem Solution Using Excel (2 of 4) Exhibit 11.9 Chapter 11 - Nonlinear Programming 23

24 Western Clothing Company Problem Solution Using Excel (3 of 4) Exhibit Chapter 11 - Nonlinear Programming 24

25 Western Clothing Company Problem Solution Using Excel (4 of 4) Exhibit Chapter 11 - Nonlinear Programming 25

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. d i = sqrt((x i - x) 2 + (y i - y) 2 ) (= straight-line distance) 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 Chapter 11 - Nonlinear Programming 26

27 Facility Location Example Problem Problem Definition and Data (2 of 2) Coordinates Town x y Annual Trips Abbeville Benton Clayton Dunnig Eden Chapter 11 - Nonlinear Programming 27

28 Facility Location Example Problem Solution Using Excel Exhibit Chapter 11 - Nonlinear Programming 28

29 Facility Location Example Problem Solution Map Figure 11.8 Rescue Squad Facility Location Chapter 11 - Nonlinear Programming 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. Since variance is the sum of squares of differences, it is mathematically identical to the straight-line distance! Thus, it is possible to visualize variances as such distances, and minimizing the overall variance is then mathematically identical to minimizing such distances. Chapter 11 - Nonlinear Programming 30

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 + x i x j r ij s i s j where: straight-line distance 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 Chapter 11 - Nonlinear Programming 31

32 Investment Portfolio Selection Example Problem Solution Using Excel (1 of 5) Four stocks, desired annual return of at least Minimize Z = S = x A2 (.009) + x B2 (.015) + x C2 (.040) + X D2 (.023) + x A x B (.4)(.009) 1/2 (0.015) 1/2 + x A x C (.3)(.009) 1/2 (.040) 1/2 + x A x D (.6)(.009) 1/2 (.023) 1/2 + x B x C (.2)(.015) 1/2 (.040) 1/2 + x B x D (.7)(.015) 1/2 (.023) 1/2 + x C x D (.4)(.040) 1/2 (.023) 1/2 + x B x A (.4)(.015) 1/2 (.009) 1/2 + x C x A (.3)(.040) 1/2 + (.009) 1/2 + x D x A (.6)(.023) 1/2 (.009) 1/2 + x C x B (.2)(.040) 1/2 (.015) 1/2 + x D x B (.7)(.023) 1/2 (.015) 1/2 + x D x C (.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 0 Chapter 11 - Nonlinear Programming 32

33 Investment Portfolio Selection Example Problem Solution Using Excel (2 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.4 A,C.3 A,D.6 B,C.2 B,D.7 C,D.4 Chapter 11 - Nonlinear Programming 33

34 Investment Portfolio Selection Example Problem Solution Using Excel (3 of 5) Exhibit Chapter 11 - Nonlinear Programming 34

35 Investment Portfolio Selection Example Problem Solution Using Excel (4 of 5) Exhibit Chapter 11 - Nonlinear Programming 35

36 Investment Portfolio Selection Example Problem Solution Using Excel (5 of 5) Exhibit Chapter 11 - Nonlinear Programming 36

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 Chapter 11 - Nonlinear Programming 37

38 Hickory Cabinet and Furniture Company Example Problem and Solution (2 of 2) Chapter 11 - Nonlinear Programming 38

39 Chapter 11 - Nonlinear Programming 39