CDAE 266 - Class 17Oct. 23

Last class: Result of Quiz 4 3. Linear programming and applications

Today: 3. Linear programming and applications Review for the midterm exam Group project 2

Next class: Midterm exam

CDAE 266 - Class 17Oct. 23

Important dates: Problem set 3: due today The answer key will be posted in the website on Oct. 24

Group project 2 (case 1 on page 3-31): due Tuesday, Oct. 30

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?

3.4. How to solve a LP model in Excel?(follow the second LP reading package)

3.4.1. Check the Excel program 3.4.2. Enter the data & formulas

(Example 2 -- Galaxy Industries)

X1 = Number of space ray X2 = Number of zappers

Maximize P = 8X1 + 5X2 subject to 2X1 + 1X2 < 1200 (plastic)

3X1 + 4X2 < 2400 (labor) X1 + X2 < 800 (total) X1 - X2 < 450 (mix) X1 > 0 X2 > 0

3.4. How to solve a LP model in Excel?

3.4.3. Solve the model and obtain computer reports

-- Answer report

-- Sensitivity report

-- Limits report (not needed)

Take home exercise(Tuesday, Oct. 16)

Solve the following LP model in Excel and obtain the “answer report” and “sensitivity report”

X1 = Number of space ray X2 = Number of zappers

Maximize P = 8X1 + 5X2

subject to 2X1 + 1X2 < 1200 (plastic) 3X1 + 4X2 < 2400 (labor) X1 + X2 < 800 (total) X1 - X2 < 450 (mix) X1 > 0 X2 > 0

3.5. How to interpret computer reports and conduct sensitivity analysis?

3.5.1. Answer report

(1) Optimal solution for X1 and X2

(2) Optimal value of the objective function

(3) “Original value”

(4) “Cell value” or LHS value

(5) “Status” of each constraint

(6) “Slack” of each constraint (Relation between “status” & “slack”)

3.5. How to interpret computer reports and conduct sensitivity analysis?

3.5.2. Sensitivity report

(1) Optimal solution for X1 and X2

(2) “Reduced cost”

(3) “Objective coefficient”

(4) “Allowable increase” & “allowable decrease” for each objective coefficient

(5) Calculate the “range of optimality” for each objective coefficient

(6) Interpretation of “range of optimality”

3.5. How to interpret computer reports and conduct sensitivity analysis?

3.5.2. Sensitivity report

(7) Use the “range of optimality” to answer questions regarding a change in an objective coefficient?

(8) What is the “100% rule” and how to use that?

(9) “Final value” = “LHS” value

(10) Interpretation of the “shadow price”

3.5. How to interpret computer reports and conduct sensitivity analysis?

3.5.2. Sensitivity report

(11) “Constraint R.H side”

(12) “Allowable increase” & “allowable decrease” for each constraint

(13) Calculate the “range of feasibility” for each constraint

(14) Use the “range of feasibility” and “shadow” price to answer questions regarding a change in the RHS value

(15) The “100% rule”

3.5. How to interpret computer reports and conduct sensitivity analysis?

3.5.3. Summary of sensitivity analysis

(1) One objective function coefficient changes.

(2) Two objective function coefficients change at the same time -- 100% rule

(3) One RHS value changes

(4) Two RHS values change at the same time -- 100% rule

3.5. How to interpret computer reports and conduct sensitivity analysis?

3.5.4. Practice questions

(1) Practice questions on page 9 of the second LP reading package

(2) Practice quiz on our class website (the answer key to the practice quiz will be posted in the website at 1:00pm, Wed., Oct. 24)

3.6. What are some special cased of LP?

3.6.1. Multiple optimal solutions

3.6.2. Infeasible problems

3.6.3. Unbounded problems

Infeasible LP

No point, simultaneously,

lie above line and below

lines and .

1

2

3 1

2 3

Unbounded LP The Feasible

Region

Maximize

the Objective Function

Midterm exam

1. What will be covered in the exam?

2. What will be the format?

3. What are the study materials and practice problems?

4. What do you need to bring to the exam?

5. Suggestions

Midterm exam

1. What will be covered in the exam?

-- Chapter 1 – Chapter 3-- Approximate distribution:

Chapter 1 10 -- 20%Chapter 2 45 -- 50%Chapter 3 40 -- 45%

Midterm exam

2. What will be the format?

-- Short-answer questions (e.g., quiz questions)

-- Problems (e.g., develop a LP model)

-- Graphical analysis (e.g., solve a LP model graphically)

-- Questions based on computer outputs (e.g., practice questions on page 9 of the second LP reading

package and practice quiz in the website)

Midterm exam

3. What are the study materials and practice problems?

Quiz 1 to Quiz 4Practice quiz in the website

Class examplesClass exercises and take-home exercises

Practice questions on page 9 of the second LP handoutLecture notes (class 1 - class 17)Problem sets 1, 2 and 3 and the answer key to problem set 3Project 1 and project 2*

Reading materials (packages) 1. The Time Value of Money 2. Basic Economic Relations 3. Linear Programming (31 pages) 4. Linear Programming (computer part, 9 pages)

Midterm exam

4. What do you need to bring to the exam?

-- Your calculator

-- A ruler will be helpful

-- TVM tables will be provided

-- Graph paper will be provided

-- No formula sheet is needed

Midterm exam

5. Suggestions a. Go over each quiz and make sure you understand

and know how to answer every question in every quiz

b. Go over each example and exercise and make sure you understand and know how to answer every question

c. Go over group project 1 and make sure you understand the logic and calculation

d. Go over your problem sets and the lecture notes, especially for the classes you missed

e. Go over the reading materials

f. Study hard and feel confident!

Group project 2(2-4 student in each group)

-- Format requirements (see the “Guideline for project reports”)-- Project report (Less than 4 pages of text plus computer outputs)-- What is the business problem of the project?-- Procedures: (1) Develop a LP model (2) Solve the LP model in Excel “answer report” and sensitivity report” (you need to do this ONLY once) (3) Evaluate each of the three options and their combinations (a) More treated straw (b) One more box of handles

(c) More labor (d) (a) + (b)(e) (a) + (c) (f ) (b) + (c)(g) (a) + (b) + (c)

Your project report should address all the six problems listed in the end of the project (it does not need to be the order of the 6 problems).