#### Transcript nonlinear programming

Department of Business Administration SPRING 2007-08 Management Science by Asst. Prof. Sami Fethi © 2007 Pearson Education Ch 7: Nonlinear programming Chapter Topics Nonlinear Profit Analysis Constrained Optimization Nonlinear Programming Model with Multiple Constraints Nonlinear Model Examples Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. 2 Ch 7: Nonlinear programming 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) can be used in this NPL. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. 3 Optimal Value of a Single Nonlinear Function Basic Model Ch 7: Nonlinear programming Profit function, Z, with volume independent of price: Z = vp - cf - vcv where v = sales volume p = price cf = unit fixed cost cv = unit variable cost Add volume/price relationship: v = 1,500 - 24.6p Operations Research Figure 1 Linear Relationship of Volume to Price © 2007/08, Sami Fethi, EMU, All Right Reserved. 4 Optimal Value of a Single Nonlinear Function Expanding the Basic Model to a Nonlinear Model Ch 7: Nonlinear programming With fixed cost (cf = $10,000) and variable cost (cv = $8): Profit, Z = 1,696.8p - 24.6p2 - 22,000 Operations Research Figure 2 The Nonlinear Profit Function © 2007/08, Sami Fethi, EMU, All Right Reserved. 5 Optimal Value of a Single Nonlinear Function Maximum Point on a Curve Ch 7: Nonlinear programming 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. Operations Research Figure 3 Maximum profit for the profit function © 2007/08, Sami Fethi, EMU, All Right Reserved. 6 Optimal Value of a Single Nonlinear Function Ch 7: Nonlinear programming Solution Using Calculus Z = 1,696.8p - 24.6p2 -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 Operations Research Figure 4 Maximum Profit, Optimal Price, and Optimal Volume © 2007/08, Sami Fethi, EMU, All Right Reserved. 7 Ch 7: Nonlinear programming 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 (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. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. 8 Constrained Optimization in Nonlinear Problems Graphical Interpretation (1 of 3) Ch 7: Nonlinear programming Effect of adding constraints to nonlinear problem: Figure 5 Nonlinear Profit Curve for the Profit Analysis Model Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. 9 Ch 7: Nonlinear programming Constrained Optimization in Nonlinear Problems Graphical Interpretation (2 of 3)- First constrained p<= 20 Operations Research Figure 6 A Constrained Optimization Model © 2007/08, Sami Fethi, EMU, All Right Reserved.10 Ch 7: Nonlinear programming Constrained Optimization in Nonlinear Problems Graphical Interpretation (3 of 3) Second constrained p<= 40 Figure 7 A Constrained Optimization Model with a Solution Point Not on the Constraint Boundary Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved. 11 Ch 7: Nonlinear programming 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. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.12 Furniture Company Problem Solution (1 of 3) Ch 7: Nonlinear programming 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. Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.13 Ch 7: Nonlinear programming Furniture Company Problem Solution (2 of 3) Profit function, Z, with volume independent of price: Z = vp - cf - vcv where v = sales volume p = price cf = unit fixed cost cv = unit variable cost Add volume/price relationship: v = 400 - 1.2p Figure 8 Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.14 Ch 7: Nonlinear programming Furniture Company Problem Solution (3 of 3) With fixed cost (cf = $7,500) and variable cost (cv = $40): Profit, Z = 448p - 1.2p2 - 23,500 Z = 448p – 1.2p2 -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 Operations Research Figure 9 © 2007/08, Sami Fethi, EMU, All Right Reserved.15 Ch 7: Nonlinear programming Beaver Creek Pottery Company Problem Solution (1 of 1) Maximize Z = $(4 - 0.1x1)x1 + (5 - 0.2x2)x2 subject to: x1 + 2x2 = 40 Where: x1 = number of bowls produced x2 = number of mugs produced 4 – 0.1X1 = profit ($) per bowl 5 – 0.2X2 = profit ($) per mug Determine the optimal solution to this nonlinear programming model Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.16 Beaver Creek Pottery Company Problem Solution (2 of 2) Ch 7: Nonlinear programming Maximize Z = $(4 - 0.1x1)x1 + (5 - 0.2x2)x2 Z = 4x1 - 0.1x21 + 5x2 - 0.2x22 Subs x1 = 40-2x2 into the eq above Z = 13x2 - 0.6x22 dZ/dx2 = 13 – 1.2x2 = 0 →x2 =10.83 and x1 =18.34 subs these into following profit fuction: Z = 4x1 - 0.1x21 + 5x2 - 0.2x22 Z =$ 70.42 Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.17 Ch 7: Nonlinear programming Western Clothing Company Problem Solution Using Excel (1 of 4) Maximize Z = (p1 - 12)x1 + (p2 - 9)x2 subject to: 2x1 + 2.7x2 6,000 3.6x1 + 2.9x2 8,500 7.2x1 + 8.5x2 15,000 where: x1 = 1,500 - 24.6p1 x2 = 2,700 - 63.8p2 p1 = price of designer jeans p2 = price of straight jeans Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.18 Western Clothing Company Problem Solution Using Excel (2 of 4) Operations Research Ch 7: Nonlinear programming Figure 10 © 2007/08, Sami Fethi, EMU, All Right Reserved.19 Western Clothing Company Problem Solution Using Excel (3 of 4) Operations Research Ch 7: Nonlinear programming Figure 11 © 2007/08, Sami Fethi, EMU, All Right Reserved.20 Western Clothing Company Problem Solution Using Excel (4 of 4) Operations Research Ch 7: Nonlinear programming Figure 12 © 2007/08, Sami Fethi, EMU, All Right Reserved.21 Ch 7: Nonlinear programming 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. di = sqrt[(xi - x)2 + (yi - y)2] Where: (x,y) = coordinates of proposed facility (xi,yi) = coordinates of customer or location facility i Minimize total miles d = diti Where: di = distance to town i ti =annual trips to town i Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.22 Ch 7: Nonlinear programming Facility Location Example Problem Problem Definition and Data (2 of 2) Town Abbeville Benton Clayton Dunnig Eden Operations Research Coordinates y x 20 20 35 10 9 25 15 32 8 10 Annual Trips 75 105 135 60 90 © 2007/08, Sami Fethi, EMU, All Right Reserved.23 Facility Location Example Problem Solution Using Excel Operations Research Ch 7: Nonlinear programming Figure 13 © 2007/08, Sami Fethi, EMU, All Right Reserved.24 Facility Location Example Problem Solution Map Operations Research Ch 7: Nonlinear programming Figure 14 Rescue Squad Facility Location © 2007/08, Sami Fethi, EMU, All Right Reserved.25 Ch 7: Nonlinear programming Facility Location Example Problem Solution Map di = sqrt[(xi - x)2 + (yi - y)2] dA=sqrt[(20- 21.72)2 + (20- 14.14)2] dA=6.11........................................ dE=6.22 d = diti d = 6.11(75)+................................+90(6.22) d = 4432.53 total annual distance Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.26 Ch 7: Nonlinear programming Hickory Cabinet and Furniture Company Example Problem and Solution (1 of 2) Model: Maximize Z = $280x1 - 6x12 + 160x2 - 3x22 subject to: 20x1 + 10x2 = 800 board ft. Where: x1 = number of chairs x2 = number of tables Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.27 Hickory Cabinet and Furniture Company Example Problem and Solution (2 of 2) Operations Research Ch 7: Nonlinear programming © 2007/08, Sami Fethi, EMU, All Right Reserved.28 Ch 7: Nonlinear programming End of chapter Operations Research © 2007/08, Sami Fethi, EMU, All Right Reserved.29