2. Department of Business Administration SPRING 200 7 -0 8 Management Science by Asst. Prof. Sami Fethi © 2007 Pearson Education

3. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 2 Nonlinear Profit Analysis Constrained Optimization Nonlinear Programming Model with Multiple Constraints Nonlinear Model Examples Chapter Topics

4. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 3 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) can be used in this NPL. Overview

5. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 4 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 Optimal Value of a Single Nonlinear Function Basic Model Figure 1 Linear Relationship of Volume to Price

6. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 5 With fixed cost (c f = $10,000) and variable cost (c v = $8): Profit, Z = 1,696.8p - 24.6p 2 - 22,000 Optimal Value of a Single Nonlinear Function Expanding the Basic Model to a Nonlinear Model Figure 2 The Nonlinear Profit Function

7. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 6 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

8. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 7 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

9. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 8 If a nonlinear problem contains one or more constraints it 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. Constrained Optimization in Nonlinear Problems Definition

10. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 9 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)

11. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 10 Figure 6 A Constrained Optimization Model Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3)- First constrained p<= 20

12. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 11 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

13. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 12 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

14. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 13 Furniture Company Problem Solution (1 of 3) A company makes chairs. FC per month of making chair $ 7,500 and VC per chair is $ 40. Price is related to demand according to the following linear equation: V= 400 -1.2 p. (a) Develop the nonlinear profit function and (b) Determine the price that will maximize profit, the optimal volume and the maximum profit per month. (c) Graphically illustrate the profit curve and indicate the optimal price and maximum profit per month.

15. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 14 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 = 400 - 1.2p Figure 8 Furniture Company Problem Solution (2 of 3)

16. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 15 With fixed cost (cf = $7,500) and variable cost (cv = $40): Profit, Z = 448p - 1.2p 2 - 23,500 Z = 448p – 1.2p 2 -23,500 dZ/dp = 448 – 2.4p = 0 p = 448/2.4 = $186.66 v = 400 – 1.2p v = 175.996 chairs Z = $18,313.87 Figure 9 Furniture Company Problem Solution (3 of 3)

17. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 16 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 Determine the optimal solution to this nonlinear programming model Beaver Creek Pottery Company Problem Solution (1 of 1)

18. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 17 Beaver Creek Pottery Company Problem Solution (2 of 2) Maximize Z = $(4 - 0.1x 1 )x 1 + (5 - 0.2x 2 )x 2 Z = 4x 1 - 0.1x 2 1 + 5x 2 - 0.2x 2 2 Subs x 1 = 40-2x 2 into the eq above Z = 13x 2 - 0.6x 2 2 dZ/dx 2 = 13 – 1.2x 2 = 0 →x 2 =10.83 and x 1 =18.34 subs these into following profit fuction: Z = 4x 1 - 0.1x 2 1 + 5x 2 - 0.2x 2 2 Z =$ 70.42

19. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 18 Maximize Z = (p 1 - 12)x 1 + (p 2 - 9)x 2 subject to: 2x 1 + 2.7x 2  6,000 3.6x 1 + 2.9x 2  8,500 7.2x 1 + 8.5x 2  15,000 where: x 1 = 1,500 - 24.6p 1 x 2 = 2,700 - 63.8p 2 p 1 = price of designer jeans p 2 = price of straight jeans Western Clothing Company Problem Solution Using Excel (1 of 4)

20. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 19 Figure 10 Western Clothing Company Problem Solution Using Excel (2 of 4)

21. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 20 Figure 11 Western Clothing Company Problem Solution Using Excel (3 of 4)

22. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 21 Figure 12 Western Clothing Company Problem Solution Using Excel (4 of 4)

23. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 22 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)

24. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 23 Facility Location Example Problem Problem Definition and Data (2 of 2)

25. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 24 Figure 13 Facility Location Example Problem Solution Using Excel

26. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 25 Figure 14 Rescue Squad Facility Location Facility Location Example Problem Solution Map

27. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 26 Facility Location Example Problem Solution Map di = sqrt[(xi - x) 2 + (yi - y) 2 ] d A =sqrt[(20- 21.72) 2 + (20- 14.14) 2 ] d A =6.11........................................ d E =6.22 d =  diti d = 6.11(75)+................................+90(6.22) d = 4432.53 total annual distance

28. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 27 Model: Maximize Z = $280x 1 - 6x 1 2 + 160x 2 - 3x 2 2 subject to: 20x 1 + 10x 2 = 800 board ft. Where: x 1 = number of chairs x 2 = number of tables Hickory Cabinet and Furniture Company Example Problem and Solution (1 of 2)

29. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 28 Hickory Cabinet and Furniture Company Example Problem and Solution (2 of 2)

30. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. Ch 7: Nonlinear programming 29 End of chapter