Optimization Techniques

Transcription

1 Optimization Techniques Methods for maximizing or minimizing an objective function Examples Consumers maximize utility by purchasing an optimal combination of goods Firms maximize profit by producing and selling an optimal quantity of goods Firms minimize their cost of production by using an optimal combination of inputs PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 1

2 Concept of the Derivative The derivative of Y with respect to X is equal to the limit of the ratio ΔY/ΔX as ΔX approaches zero dy dx = lim Δ X 0 ΔY ΔX PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 2

3 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 3

4 Total Revenue Equation Equation: TR = 100Q - 10Q 2 Table: Q TR TR Graph: Q PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 4

5 Total, Average, and Marginal Revenue TR = PQ AR = TR/Q MR = ΔTR/ΔQ PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 5

6 Total Revenue Schedule of a Firm PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 6

7 Total Revenue Curve of a Firm PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 7

8 300 TR 250 Total Revenue Q AR, MR 120 Average and Marginal Revenue Q PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 8

9 Total, Average, and Marginal Cost AC = TC/Q MC = ΔTC/ΔQ Q TC AC MC PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 9

10 Total, Average, and Marginal Cost PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 10

11 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 11

12 Geometric Relationships The slope of a tangent to a total curve at a point is equal to the marginal value at that point The slope of a ray from the origin to a point on a total curve is equal to the average value at that point PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 12

13 Geometric Relationships A marginal value is positive, zero, and negative, respectively, when a total curve slopes upward, is horizontal, and slopes downward A marginal value may be negative, but an average value can never be negative PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 13

14 Profit Maximization Q TR TC Profit PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 14

15 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 15

16 Steps in Optimization Define an objective function of one or more choice variables Define the constraint on the values of the objective function Determine the values of the choice variables that maximize or minimize the objective function while satisfying the constraint PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 16

17 New Management Tools for Optimization Benchmarking (tool for improving productivity and quality) Total Quality Management (constantly improving the quality of products and the firm s processes to deliver more value to customers; e.g. Six Sigma) Reengineering (radical redesign of all the firm s processes to achieve major gains) Learning Organization (values continuing learning, both individual and collective) PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 17

18 Other Management Tools for Optimization Broad-banding (elimination of multiple salary grades to foster movement among jobs within the firm and lower cost) Direct Business Model (eliminating the time and cost of third-party distribution) Networking (forming of temporary strategic alliances among firms as per their core competence) Performance Management (holding executives and their subordinates accountable for delivering the desired results) PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 18

19 Other Management Tools for Optimization Pricing Power (ability of a firm to raise prices faster than the rise in its costs and vice-versa) Small-World Model (linking well-connected individuals from each level of the organization to one another to improve flow of information and the operational efficiency) Strategic Development (continuous review of strategic decisions) Virtual Integration (treating suppliers and customers as if they were part of the company which reduces the need for inventories) Virtual Management (ability of a manager to simulate consumer behavior using computer models) PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 19

20 Univariate Optimization Given objective function Y = f(x) Find X such that dy/dx = 0 Second derivative rules: If d 2 Y/dX 2 > 0, then X is a minimum If d 2 Y/dX 2 < 0, then X is a maximum PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 20

21 Example 1 Given the following total revenue (TR) function, determine the quantity of output (Q) that will maximize total revenue: TR = 100Q 10Q 2 dtr/dq = Q = 0 Q* = 5 and d 2 TR/dQ 2 = -20 < 0 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 21

22 Example 2 Given the following total revenue (TR) function, determine the quantity of output (Q) that will maximize total revenue: TR = 45Q 0.5Q 2 dtr/dq = 45 Q = 0 Q* = 45 and d 2 TR/dQ 2 = -1 < 0 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 22

23 Example 3 Given the following marginal cost function (MC), determine the quantity of output that will minimize MC: MC = 3Q 2 16Q + 57 dmc/dq = 6Q - 16 = 0 Q* = 2.67 and d 2 MC/dQ 2 = 6 > 0 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 23

24 Example 4 Given TR = 45Q 0.5Q 2 TC = Q 3 8Q Q + 2 Determine Q that maximizes profit (π): π = 45Q 0.5Q 2 (Q 3 8Q Q + 2) PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 24

25 Method 1 Example 4: Solution dπ/dq = 45 Q 3Q Q 57 = Q 3Q 2 = 0 Method 2 MR = dtr/dq = 45 Q MC = dtc/dq = 3Q 2 16Q + 57 Set MR = MC: 45 Q = 3Q 2 16Q + 57 Use quadratic formula: Q* = 4 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 25

26 Quadratic Formula Write the equation in the following form: ax 2 + bx + c = 0 The solutions have the following form: ± b b 2 4ac 2a PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 26

27 Multivariate Optimization Objective function Y = f(x 1, X 2,...,X k ) Find all X i such that Y/ X i = 0 Partial derivative: Y/ X i = dy/dx i while all X j (where j i) are held constant PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 27

28 Example 5 Determine the values of X and Y that maximize the following profit function: π = 80X 2X 2 XY 3Y Y Solution π/ X = 80 4X Y = 0 π/ Y = X 6Y = 0 Solve simultaneously X = and Y = PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 28

29 Constrained Optimization Substitution Method Substitute constraints into the objective function and then maximize the objective function Lagrangian Method Form the Lagrangian function by adding the Lagrangian variable and constraint to the objective function and then maximize the Lagrangian function PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 29

30 Example 6 Use the substitution method to maximize the following profit function: π = 80X 2X 2 XY 3Y Y Subject to the following constraint: X + Y = 12 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 30

31 Example 6: Solution Substitute X = 12 Y into profit: π = 80(12 Y) 2(12 Y) 2 (12 Y)Y 3Y Y π = 4Y Y Solve as univariate function: dπ/dy = 8Y + 56 = 0 Y = 7 and X = 5 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 31

32 Example 7 Use the Lagrangian method to maximize the following profit function: π = 80X 2X 2 XY 3Y Y Subject to the following constraint: X + Y = 12 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 32

33 Example 7: Solution Form the Lagrangian function L = 80X 2X 2 XY 3Y Y + λ(x + Y 12) Find the partial derivatives and solve simultaneously dl/dx = 80 4X Y + λ = 0 dl/dy = X 6Y λ = 0 dl/dλ = X + Y 12 = 0 Solution: X = 5, Y = 7, and λ = -53 PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 33

34 Interpretation of the Lagrangian Multiplier, λ Lambda, λ, is the derivative of the optimal value of the objective function with respect to the constraint In Example 7, λ = -53, so a one-unit increase in the value of the constraint (from -12 to -11) will cause profit to decrease by approximately 53 units PowerPoint Slides Prepared by Robert F. Brooker, Ph.D. Slide 34