1. CDAE 266 - Class 12 Oct. 4 Last class: 2. Review of economic and business concepts Today: 3. Linear programming and applications Quiz 3 (sections 2.5 and 2.6) Next class: 3. Linear programming Important date: Problem set 2: due Tuesday, Oct. 9

2. One more application of TVM (Take-home exercise, Sept. 27) Mr. Zhang in Beijing plans to immigrate to Canada and start a business in Montreal and the Canadian government has the following two options of “investment” requirement: A. A one-time and non-refundable payment of $120,000 to the Canadian government. B.A payment of $450,000 to the Canadian government and the payment (i.e., $450,000) will be returned to him in 4 years from the date of payment. (1)How do we help Mr. Zhang compare the two options? (2)If the annual interest rate is 12%, what is the difference in PV? (3)If the annual interest rate is 6%, what is the difference in PV? (4)At what interest rate, the two options are the same in PV?

3. 2. Review of Economics Concepts 2.1. Overview of an economy 2.2. Ten principles of economics 2.3. Theory of the firm 2.4. Time value of money 2.5. Marginal analysis 2.6. Break-even analysis

4. 2.5. Marginal analysis 2.5.1. Basic concepts 2.5.2. Major steps of using quantitative methods 2.5.3. Methods of expressing economic relations 2.5.4. Total, average and marginal relations 2.5.5. How to derive derivatives? 2.5.6. Profit maximization 2.5.7. Average cost minimization

5. 2.5.6. Profit maximization (4) Summary of procedures (a) If we have the total profit function: Step 1: Take the derivative of the total profit function  marginal profit function Step 2: Set the marginal profit function to equal to zero and solve for Q* Step 3: Substitute Q* back into the total profit function and calculate the maximum profit (b) If we have the TR and TC functions: Step 1: Take the derivative of the TR function  MR Step 2: Take the derivative of the TC function  MC Step 3: Set MR=MC and solve for Q* Step 4: Substitute Q* back into the TR and TC functions to calculate the TR and TC and their difference is the maximum total profit

6. 2.5.6. Profit maximization (4) Summary of procedures (c) If we have the demand and TC functions Step 1: Demand function  P = … Step 2: TR = P * Q = ( ) * Q Then follow the steps under (b) on the previous page

7. Class Exercise 4 (Thursday, Sept. 27) 1.Suppose a firm has the following total revenue and total cost functions: TR = 20 Q TC = 1000 + 2Q + 0.2Q 2 How many units should the firm produce in order to maximize its profit? 2. If the demand function is Q = 20 – 0.5P, what are the TR and MR functions?

8. 2.5.7. Average cost minimization (1) Relation between AC and MC: when MC < AC, AC is falling when MC > AC, AC is increasing when MC = AC, AC reaches the minimum level (2) How to derive Q that minimizes AC? Set MC = AC and solve for Q

9. 2.5.7. Average cost minimization (3) An example: TC = 612500 + 1500Q + 1.25Q 2 MC = 1500 + 2.5Q AC = TC/Q = 612500/Q + 1500 + 1.25Q Set MC = AC Q 2 = 490,000 Q = 700 or -700 When Q = 700, AC is at the minimum level

10. 2.6. Break-even analysis 2.6.1. What is a break-even? TC = TR or  = 0 2.6.2. A graphical analysis -- Linear functions -- Nonlinear functions 2.6.3. How to derive the beak-even point or points? Set TC = TR or  = 0 and solve for Q.

11. Break-even analysis: Linear functions Costs ($) Quantity FC TC TR B A Break-even quantity

12. Break-even analysis: nonlinear functions Costs ($) Quantity TC TR Break-even quantity 1Break-even quantity 2 

13. 2.6. Break-even analysis 2.6.4. An example TC = 612500 + 1500Q + 1.25Q 2 TR = 7500Q - 3.75Q 2 612500 + 1500Q + 1.25Q 2 = 7500Q - 3.75Q 2 5Q 2 - 6000Q + 612500 = 0 Review the formula for ax 2 + bx + c = 0 x = ? e.g., x 2 + 2x - 3 = 0, x = ? Q = 1087.3 or Q = 112.6

14. Class Exercise 5 (Tuesday, Oct. 2) 1.Suppose a company has the following total cost (TC) function: TC = 200 + 2Q + 0.5 Q 2 (a) What are the average cost (AC) and marginal cost (MC) functions? (b) If the company wants to know the Q that will yield the lowest average cost, how will you solve the problem mathematically (list the steps and you do not need to solve the equation) 2.Suppose a company has the following total revenue (TR) and total cost (TC) functions: TR = 20 Q TC = 300 + 5Q How many units should the firm produce to have a break-even?

15. 3. Linear programming & applications 3.1. What is linear programming (LP)? 3.2. How to develop a LP model? 3.3. How to solve a LP model graphically? 3.4. How to solve a LP model in Excel? 3.5. How to do sensitivity analysis? 3.6. What are some special cases of LP?

16. 3.1. What is linear programming (LP)? 3.1.1. Two examples: Example 1. The Redwood Furniture Co. manufactures tables & chairs. Table A on the next page shows the resources used, the unit profit for each product, and the availability of resources. The owner wants to determine how many tables and chairs should be made to maximize the total profits.

17. Table A (example 1): --------------------------------------------------------------- Unit requirements Resources ----------------------Amount TableChairavailable --------------------------------------------------------------- Wood ( board feet ) 30 20 300 Labor ( hours ) 5 10 110 ===================================== Unit profit ($) 6 8 ---------------------------------------------------------------

18. 3.1. What is linear programming (LP)? 3.1.1. Two examples: Example 2. Galaxy Industries (a toy manufacture co.) 2 products: Space ray and zapper 2 resources: Plastic & time Resource requirements & unit profits: Table B on the next page.

19. Table B (example 2): --------------------------------------------------------------- Unit requirements Resources ----------------------Amount Space ray Zapper available --------------------------------------------------------------- Plastic (lb.) 2 1 1,200 Labor (min.) 3 4 2,400 ===================================== Unit profit ($) 8 5 ---------------------------------------------------------------

20. 3.1. What is linear programming (LP)? 3.1.1. Two examples: Example 2. Galaxy Industries: Additional requirements (constraints): (1) Total production of the two toys should be no more than 800. (2) The number of space ray cannot exceed the number of zappers plus 450. Question: What is the optimal quantity for each of the two toys?

21. Management is seeking a production schedule that will maximize the company’s profit.

22. Linear programming (LP) can provide intelligent solution to provide intelligent solution to such problems such problems

23. 3.1. What is linear programming (LP)? 3.1.2. Mathematical programming: (1) Linear programming (LP) (2) Integer programming (3) Goal programming (4) Dynamic programming (5) Non-linear programming ……

24. 3.1. What is linear programming (LP)? 3.1.3. Linear programming (LP): (1) A linear programming model: A model that seeks to maximize or minimize a linear objective function subject to a set of linear constraints. (2) Linear programming: A mathematical technique used to solve constrained maximization or minimization problems with linear relations.

25. 3.1. What is linear programming (LP)? 3.1.3. Linear programming (LP): (3) Applications of LP: -- Product mix problems -- Policy analysis -- Transportation problems ……

26. 3.2. How to develop a LP model? 3.2.1. Major components of a LP model: (1) A set of decision variables. (2) An objective function. (3) A set of constraints. 3.2.2. Major assumptions of LP: (1) Variable continuity (2) Parameter certainty (3) Constant return to scale (4) No interactions between decision variables

27. 3.2. How to develop a LP model? 3.2.3. Major steps in developing a LP model: (1) Define decision variables (2) Express the objective function (3) Express the constraints (4) Complete the LP model 3.2.4. Three examples: (1) Furniture manufacturer (2) Galaxy industrials (3) A farmer in Iowa

28. Table A (example 1): --------------------------------------------------------------- Unit requirements Resources ----------------------Amount TableChairavailable --------------------------------------------------------------- Wood ( board feet ) 30 20 300 Labor ( hours ) 5 10 110 ===================================== Unit profit ($) 6 8 ---------------------------------------------------------------

29. Develop the LP model Step 1. Define the decision variables Two variables: T = number of tables made C = number of chairs made Step 2. Express the objective function Step 3. Express the constraints Step 4. Complete the LP model

30. Example 2. Galaxy Industries (a toy manufacturer) 2 products: Space ray and zapper 2 resources: Plastic & time Resource requirements & unit profits (Table B) Additional requirements (constraints): (1) Total production of the two toys should be no more than 800. (2) The number of space ray cannot exceed the number of zappers plus 450.

31. Table B (example 2): --------------------------------------------------------------- Unit requirements Resources ----------------------Amount Space ray Zapper available --------------------------------------------------------------- Plastic (lb.) 2 1 1,200 Labor (min.) 3 4 2,400 ===================================== Unit profit ($) 8 5 ---------------------------------------------------------------