Linear Programming
Agenda
1. Forecasting projects due
2. Interfaces presentations
3. Plan for manufacturing chairs and tables using Legos (legoexercise.ppt)
4. Formulate the problem algebraically (Page 4)
Variable definition
Objective
Linearity
Proportionality (see Pages 11 – 17 for bad proportionality)
Additivity
Constraints
Linearity
Proportionality
Additivity
Nonnegativity
5. Solve using QM for Windows (displayed in QM and on Page 5)
Solution
Variable values
Total Profit
Dual Values
Ranging Table (page 5 – 7)
Constraints
Dual values & range (upper and lower bounds)
Slack
Objective
Reduced cost
Objective function ranging (upper and lower bounds)
Graph
6. Possible outcomes
Unique solution (original example)
Multiple solutions (change profit from 17 to 20)
Unbounded solution (change <= constraints to >= constraints)
No feasible solution (add constraint # tables >=10)
7. More examples (pages 8-10)
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Overview of next five chapters and next four lectures
For the next three weekends we will be talking about optimization models. Chapters 7, 8 and 9 present
linear programming. Chapter 7 introduces linear programming and explains how to solve a problem
graphically. Since we use software to solve problems our main concern is with the formulation and
structure of linear programming. Thus you should pay particular attention to sections 7.1, 7.2 & 7.3.
Chapter 8 contains a very wide set of examples of how linear programming models can be used.
Through the examples and the homework problems in this chapter you will hopefully appreciate the
usefulness of linear programming. Chapter 9, which is not on the syllabus, contains the algebraic
solution method for solving linear programming problems. Since we use software we are not interested
in this chapter. Chapter 10 contains two special cases of linear programming models – the transportation
model and the assignment model. Finally, Chapter 11 contains integer models that look identical to
linear programs but have computational difficulties that need to be explained.
Assignments
The only way to both learn how to formulate linear programs and to appreciate the wide variety of
problems that can be approached using linear programming is to formulate as many problems as
possible. Chapter 7 has relatively simple formulations. Our main interest is in Chapter 8, which presents
many of the applications of linear programming.
In addition to being able to formulate linear programs so that they can be solved it is important to
understand the meaning and understanding of the complete solution. The book discusses this in the
chapter but I think you will find this week’s lecture and the materials I have posted to be clearer than the
textbook. You have an individual assignment and a team assignment due for me that will help you to
learn about the interpretation of the results.
In order to keep the workload reasonable, I am splitting up the formulations over several weeks as shown
below. Formulate each of the problems listed below. As a partial check of your formulation, you should
solve the problem and see if the answers seem reasonable. I do not collect homework but looking at
these problems will give you an appreciation of the wide variety of problems for which linear
programming is useful.
Also, please note that the bulk of the homework over the coming weeks is represented by the problems
below but that additional homework will be assigned in the coming weeks. These problems represent
only the formulations.
It is quite possible (maybe even likely) that some of these formulations will give you trouble. However,
for each of these problems you should be able to identify the (decision) variables. Limit your formulation
work (no more than 15 minutes per problem) but be sure that even if you can not derive the algebraic
formulation that you are very familiar with the problems in order that the posted solution will be
meaningful.
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Due
Sun (Feb 6)
Sun (Feb 6)
Assignment
Chapter 7, problems 14, 15, 16, 17, 18, 19, 20, 21, 25
First (individual) Linear Program interpretation is due (by email by 8pm)
Fri Feb 11
Be sure Excel’s Solver is an option underTools in Excel 2003 and on the Data
tab in Excel 2007/2010. We will be using Solver in our next class.
Sun (Feb 20) Chapter 8, problems 1, 2, 3, 4, 5 (LP or Assignment), 6, 7
Sun (Feb 20) Second (team) linear programming interpretation is due (by email by 8pm)
Sat Feb 26
Chapter 8, problems 8, 9 (LP or transportation), 10, 11, 12, 13, 14
Sat Feb 26
Chapter 8, problems 15, 16, (17), 18, 19, 21, 22, 23
Reminder: No more than 15 minutes per problem.
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Linear Programming Formulation Examples
Example 1: Resource Allocation/Product Mix
A furniture manufacturer produces chairs and tables from the same raw materials - small blocks and
large blocks. The resource requirements for each product and the total resources available are as follows:
Resources/unit
Small blocks
Large blocks
Chair
2
1
Table
2
2
AVAILABLE (per day)
8
6
Each chair produced results in a profit of $10, and each table earns $17 in profit. Determine the number
of chairs and tables to produce each week in order to maximize profit.
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Output screens from QM for Windows
Linear Programming Results Screen
Tables
Maximize
Small
Large
Solution->
Chairs
17
2
2
2
RHS
10
2 <=
1 <=
2
Dual
8
6
54
1.5
7
Ranging Screen
Variable Value
Reduced Cost Original Val
Lower Bound Upper Bound
Tables
2 tables
$0
$17
$10
$20
Chairs
2 chairs
$0
$10
$8.5
$17
Constraint Dual Value Slack/Surplus Original Val
Lower Bound Upper Bound
Small
$1.5 0 small blocks 8 small blocks 6 small blocks
12 small blocks
Large
$7 0 large blocks 6 large blocks 4 large blocks
8 large blocks
Revised Profit on Tables
Linear Programming Results Screen
Maximize
Small
Large
Solution->
Tables
21
2
2
3
Chairs
RHS
10
2
1
0
<=
<=
Dual
8
6
63
0
10.5
Ranging Screen
Variable
Tables
Chairs
Constraint
Small
Large
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Reduced
Cost
Value
3
0
Dual
Value
0
10.5
0
0.5
Slack/Surplus
2
0
Original
Val
21
10
Original
Val
8
6
Lower
Bound
20
=-Infinity
Lower
Bound
Upper
Bound
Infinity
10.5
6
0
Upper
Bound
Infinity
8
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Output screens from QM for Windows
Linear Programming Results Screen
Tables
Chairs
Maximize
17
Small
2
Large
2
Solution->
2
make 2 tables and 2 chairs
RHS
10
2
1
2
Dual
<=
<=
8
6
54
Total profit is 54
Lower
Bound
Slack/Surplus
Original
Val
17
10
Original
Val
0
0
8
6
1.5
7
Ranging Screen
Variable
Tables
Chairs
Constraint
Small
Large
Reduced
Cost
Value
2
2
Dual
Value
1.5
7
0
0
Upper
Bound
10
8.5
Lower
Bound
20
17
Note: Total profit does not appear in the
ranging table but can be computed by
12
8
multiplying the values*original profits and
summing
.
54
=SUMPRODUCT(B15:B16,D15:D16)
Upper
Bound
6
4
The reduced costs apply only to items we should NOT make. Since we should make tables and chairs they
do not apply here.
If the profit per unit on the table is between $10 and $20 and the profit per unit remains the same for the chairs
then our product mix (2 tables, 2 chairs) should not change. (The total profit would change).
Similarly, if the profit per unit on the chair is between $8.50 and $17 and the profit per unit remains the same
for the tables then our product mix (2 tables, 2 chairs) should not change. (The total profit would change).
If we had one MORE small block our profit would INCREASE by $1.5 (to $55.5)
If we had one LESS small block our profit would DECREASE by $1.5 (to $53.5)
The above statements holds true for a total number of small blocks between 6 and 12. Since we started with
8 blocks the statements hold for an increase of 4 (12-8) small blocks or a decrease of 2 (8-6)small blocks
If we had one MORE large block our profit would INCREASE by $7 (to $61)
If we had one LESS large block our profit would DECREASE by $7 (to $$47)
The above statements holds true for a total number of large blocks between 4 and 8 Since we started with
6 blocks the statements hold for an increase of 2 (8-6) large blocks or a decrease of 2 (6-4) large blocks
We have no remaining small or large blocks. That is, we use all of the materials to make the products.
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Lego revised
Output screens from QM for Windows
Linear Programming Results Screen
Tables
Chairs
21
Maximize
Small
2
Large
2
Solution->
3
make 3 tables and 0 chairs
RHS
Dual
10
2
1
0
<=
<=
8
6
63
Total profit is 63
Lower
Bound
0
0.5
Slack/Surplus
Original
Val
21
10
Original
Val
2
0
8
6
0
10.5
Ranging Screen
Variable
Tables
Chairs
Constraint
Small
Large
Reduced
Cost
Value
3
0
Dual
Value
0
10.5
20
=-Infinity
Lower
Bound
Upper
Bound
Infinity
10.5
Upper
Bound
6
0
Infinity
8
Note: Total profit does not appear in the
ranging table but can be computed by
multiplying the values*original profits and
summing
.
63
That is, 21*3 +10*0
The reduced cost does not apply to tables but for chairs the meaning is that if we make a chair (even though
we should not) our profit will be reduced by $.50
If the profit per unit on the table is $20 or higher and the profit per unit remains the same for the chairs
then our product mix (3 tables, 0 chairs) should not change. (The total profit would change).
Similarly, if the profit per unit on the chair is less than $10.50 and the profit per unit remains the same
for the tables then our product mix (3 tables, 0 chairs) should not change. (The total profit would change).
We have 2 small blocks left over. If we had one more small block our profit would not change.
Since we have 2 small blocks left over we could get rid of the 2 (8-6) blocks and our profit would not change
The above statements holds true for a total number of small blocks greater than or equal to 6.
We have no remaining large blocks.
If we had one more large block our profit would INCREASE by $10.50 (to $74.50)
If we had one less large block our profit would DECREASE by $10.50 (to $52.50)
The above statements holds true for a total number of large blocks between 0 and 8. Since we started with
6 blocks the statements hold for an increase of 2 (8-6) large blocks or a decrease of 6 (6-0) largel blocks
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Example 2: Diet problem
Andy Reid has decided to trim down and is about to begin the Fourth and 15 diet. In this diet one
should eat only Oatmeal, Pasta, Peanut Butter and Green Beans. His doctors have told him that to
maintain his stamina he should be sure to have 100% of the minimum daily requirements of four
nutrients – Protein, Iron, Fiber and Niacin. The calories and nutrients according to
http://www.nutritiondata.com are listed on the next page. An adult male should have at least 40
grams of protein per day. (e.g., see http://www.indoorclimbing.com/Protein_Requirement.html)
In addition, to the requirements Andy insists on having at least one serving of peanut butter per
day and on not having more servings of green beans than servings of pasta.
How many servings of each food should Coach Reid eat in order to meet the minimum daily
requirements with a minimum total number of calories?
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Oatmeal
Pasta
Peanut Butter
Green Beans
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Example 3 – Advertising (including ratios)
A manager of a store is attempting to decide on the types and amounts of advertising to purchase
each week. Given below is the information about the three types of ads.
Television
Newspaper
Radio
Male Reach
10,000
6,000
6,000
Female Reach
15,000
4,000
9,000
Cost per ad
$15,000
$4,000
$6,000
The store has the following policy:
a.
b.
c.
Reach at least 200,000 customers
Use at least twice as many newspaper ads as radio commercials.
Make sure that at least 30% of the audience is women.
Available space limits the number of newspaper ads to 10.
1. Formulate the problem as a linear program.
a. Define the (decision) variables
b. Define the objective function
c. Define the constraints
d. Remember the non-negativity restrictions
2. Use software to find the optimal number of each type of ads that minimizes total cost.
3. Interpret the ranging table.
4. (Formulate the problem in Excel)
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Examples of Lack of Proportionality
Bad Math
Jack’s Restaurant and Bar
Lancer’s Diner
U.S. Food and Drug Administration
Sales
Best Buy
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Check prices of
tapas platters
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Check prices of
juices
/
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