CDAE Class 11 Oct. 3 Last class: Result of Quiz 2 2. Review of economic and business concepts Today: Result of Quiz 2 3. Linear programming and applications Quiz 3 (sections 2.5 and 2.6) Next class: 3. Linear programming and applications Readings: “Basic Economic Relation” and “Linear Programming”

CDAE Class 11 Oct. 3 Important date: Problem set 2 due Tuesday, Oct. 10

Result of Quiz 2 N = 44 (take home) Range = 4 – 10Average = 8.62 N = 50 Range = 4 – 10Average = PV, r and n  FVn 2. FVn, r and n  PV 3. Annual interest rate  effective annual interest rate 4. (a) Annual interest rate  effective annual interest rate (b) PV, r and n  FVn when interest is paid semiannually 5. Present value of a bond

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

2.5. Marginal analysis Basic concepts Major steps of using quantitative methods Methods of expressing economic relations Total, average and marginal relations How to derive derivatives? Profit maximization Average cost minimization

Class Exercise 3 (Tuesday, Sept. 26) 1.Suppose a firm has the following total revenue and total cost functions: TR = 20 Q TC = Q + 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?

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

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

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

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

Class Exercise 4 (Thursday, Sept. 28) 1.Suppose a company has the following total cost (TC) function: TC = Q 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, describe how you could solve the problem mathematically (just list the step or steps and you do not need to solve it) 2.Suppose a company has the following total revenue (TR) and total cost (TC) functions: TR = 20 Q TC = Q How many units should the firm produce to have a break-even?

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?

3.1. What is linear programming (LP)? 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.

Table A (example 1): Unit requirements Resources Amount TableChairavailable Wood ( board feet ) Labor ( hours ) ===================================== Unit profit ($)

3.1. What is linear programming (LP)? 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.

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 ($)

3.1. What is linear programming (LP)? 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?

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

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

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

3.1. What is linear programming (LP)? 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.

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

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

3.2. How to develop a LP model? 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 Three examples: (1) Furniture manufacturer (2) Galaxy industrials (3) A farmer in Iowa

Table A (example 1): Unit requirements Resources Amount TableChairavailable Wood ( board feet ) Labor ( hours ) ===================================== Unit profit ($)

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

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.

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 ($)

Example 3. A farmer in Iowa has 500 acres of land which can be used to grow corn and/or soybeans. The per acre net profit is $20 for soybeans and $18 for corn. In addition to the land constraint, the farmer has limited labor resources: 200 hours for planting and 160 hours for cultivation and harvesting. Labor required for planting is 0.6 hour per acre for corn and 0.5 hour per acre for soybean. Labor required for cultivation and harvesting is 0.8 hour per acre for corn and 0.3 hour per acre for soybeans. If the farmer’s objective is to maximize the total profit, develop a LP model that can be used to determine how many acres of soy and how many acres of corn to be planted.

Class Exercise 5 (Tuesday, Oct. 3) Best Brooms is a small company that produces two difference brooms: one with a short handle and one with a long handle. Suppose each short broom requires 1 hour of labor and 2 lbs. of straw and each long broom requires 0.8 hour of labor and 3 lbs. of straws. We also know that each short broom brings a profit of $10 and each long broom brings a profit of $8 and the company has a total of 500 hours of labor and 1500 lbs of straw. Develop a LP model for the company to maximize its total profit.