Prezentace aplikace PowerPoint

1
Linear Programming
Applications
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____________________________ Operations
Research ? Jan Fábry
2
Linear Programming
Applications
Guideline for Model Formulation
1. Understand the problem thoroughly.
2. Write a verbal statement of the objective
function and each constraint.
3. Define the decision variables.
4. Write the objective function in terms of the
decision variables.
5. Write the constraints in terms of the decision
variables.
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
3

Linear Programming
Applications

• Production Process Models

• Blending Problems

• Marketing Research

• Portfolio Selection Problem

• Cutting Stock Problem

• Transportation Problem

• Assignment Problem

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____________________________ Operations
Research ? Jan Fábry
4
Linear Programming
Blending Problem
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____________________________ Operations
Research ? Jan Fábry
5
Linear Programming
Applications
Blending Problem
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____________________________ Operations
Research ? Jan Fábry
6
Linear Programming
Applications
Blending Problem
Decision variables amount of ingredients used
in final blend
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____________________________ Operations
Research ? Jan Fábry
7
Linear Programming
Applications
Blending Problem
Example Feed

• Design the optimal composition of nutritive mix
  that

• will contain at least 100 units of proteins
• will contain at least 300 units of starch
• will weigh at least 200 kg

• Objective minimize total cost

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____________________________ Operations
Research ? Jan Fábry
8
Linear Programming
Applications
Blending Problem
Example Feed

• Contents of proteins and starch in 1kg of each
  nutritive feed and prices for 1 kg of feed

  Feed F1 Feed F2 Feed F3 Feed F4
Proteins (units) 0 3 1 2
Starch (units) 1 2 3 0
Price (CZK) 20 80 60 30
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____________________________ Operations
Research ? Jan Fábry
9
Linear Programming
Applications
Blending Problem
Example Feed
Decision variables
Amount of feed F1 in the final blend x1
- - F2 - - x2
- - F3 - - x3
- - F4 - - x4
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
10
Linear Programming
Applications
Blending Problem
Example Feed
Optimal solution
F1 120 kg
F2 -
F3 60 kg
F4 20 kg
Total cost 6 600 CZK
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____________________________ Operations
Research ? Jan Fábry
11
Linear Programming
Marketing Research
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____________________________ Operations
Research ? Jan Fábry
12
Linear Programming
Applications
Marketing Research
Example MarketQuest, Inc.

• Evaluating consumers reaction to new products
  and services

• MQs client introduces a new type of washing
  powder

• Prepare a campaign with door-to-door personal
  interviews about households opinion

• Households with children without children

• Time of interview daytime evening

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
13
Linear Programming
Applications
Marketing Research
Example MarketQuest, Inc.

• Plan to conduct 1000 interviews

• At least 300 households with children should be
  interviewed

• At least 400 households without children should
  be interviewed

• Number of evening interviews ? number of daytime
  interviews

• At least 35 of the interviews for households
  with children should be conducted during evening

• At least 65 of the interviews for households
  without children should be conducted during
  evening

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
14
Linear Programming
Applications
Marketing Research
Example MarketQuest, Inc.

• Cost

Daytime interview Evening interview
Households with children 50 CZK 60 CZK
Households without children 40 CZK 50 CZK

• Objective minimize total cost

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____________________________ Operations
Research ? Jan Fábry
15
Linear Programming
Applications
Marketing Research
Example MarketQuest, Inc.
Decision variables
Daytime interview Evening interview
Households with children x1 x2
Households without children x3 x4
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____________________________ Operations
Research ? Jan Fábry
16
Linear Programming
Applications
Marketing Research
Example MarketQuest, Inc.
1)   Plan to conduct 1 000 interviews
2)   At least 300 households with children should
be interviewed
3)   At least 400 households without children
should be interviewed
4)   Number of evening interviews ? number of
daytime interviews
5)   At least 35 of the interviews for
households with children should be conducted
during evening
6)   At least 65 of the interviews for
households without children should be conducted
during evening
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
17
Linear Programming
Applications
Marketing Research
Example MarketQuest, Inc.
Optimal solution
Daytime interviews Evening interviews
Households with children 195 105
Households without children 245 455
Total cost 48 600 CZK 48 600 CZK
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____________________________ Operations
Research ? Jan Fábry
18
Linear Programming
Portfolio Selection Problem
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____________________________ Operations
Research ? Jan Fábry
19
Linear Programming
Applications
Portfolio Selection Problem
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
20
Linear Programming
Applications
Portfolio Selection Problem
Example Drink Invest, Inc.

• Investing money in stocks of companies producing
  drinks

• Plan to invest to 4 shares and 1 government bond

Rate of return Risk index
Bohemian Beer share 12 0.07
Moravian Wine share 9 0.09
Moravian Brandy share 15 0.05
Bohemian Milk share 7 0.03
Government bond 6 0.01
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____________________________ Operations
Research ? Jan Fábry
21
Linear Programming
Applications
Portfolio Selection Problem
Example Drink Invest, Inc.

• Plan to invest 2 000 000 CZK

• No more than 200 000 CZK might be invested in
  Bohemian Milk shares

• Government bonds should cover at least 20 of all
  investments

• Because of diversification of portfolio neither
  alcohol-drink company should receive more than
  800 000 CZK

• Risk index of the final portfolio should be
  maximally 0.05

• Objective maximize annual return of the
  portfolio

_______________________________________________
____________________________ Operations
Research ? Jan Fábry
22
Linear Programming
Applications
Portfolio Selection Problem
Example Drink Invest, Inc.
Decision variables
Bohemian Beer share x1
Moravian Wine share x2
Moravian Brandy share x3
Bohemian Milk share x4
Government bond x5
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
23
Linear Programming
Applications
Portfolio Selection Problem
Example Drink Invest, Inc.
Rate of return Risk index
Bohemian Beer share 12 0.07
Moravian Wine share 9 0.09
Moravian Brandy share 15 0.05
Bohemian Milk share 7 0.03
Government bond 6 0.01
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
24
Linear Programming
Applications
Portfolio Selection Problem
Example Drink Invest, Inc.
1)   Plan to invest 2 000 000 CZK
2)   No more than 200 000 CZK might be invested
in Bohemian Milk shares
3)   Government bonds should cover at least 20
of all investments
4)   Because of diversification of portfolio
neither alcohol-drink company should receive more
than 800 000 CZK
5)   Risk index of the final portfolio should be
maximally 0.05
_______________________________________________
____________________________ Operations
Research ? Jan Fábry
25
Linear Programming
Applications
Portfolio Selection Problem
Example Drink Invest, Inc.
Optimal solution
Bohemian Beer share 800 000 CZK
Moravian Wine share -
Moravian Brandy share 800 000 CZK
Bohemian Milk share -
Government bond 400 000 CZK
Expected annual return 240 000 CZK
_______________________________________________
____________________________ Operations
Research ? Jan Fábry