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Linear Programming. Applications. ___________________________________________________________________________ Quantitative Methods of Management  Jan Fábry. Linear Programming. Applications. Guideline for Model Formulation. 1. Understand the problem thoroughly.

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Linear Programming

Applications

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Guideline for Model Formulation

1. Understand the problem thoroughly.

2. Write a verbal statement of the objective function and eachconstraint.

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.

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

• Production Process Models
• Blending Problems
• Marketing Research
• Portfolio Selection Problem
• Cutting Stock Problem
• Transportation Problem
• Assignment Problem

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Quantitative Methods of Management  Jan Fábry

Linear Programming

Blending Problem

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Quantitative Methods of Management  Jan Fábry

Inputs(Ingredients)

Output(Final blend)

Linear Programming

Applications

Blending Problem

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Quantitative Methods of Management  Jan Fábry

Inputs

Output

• chemicals

Quality

Quantity

RestrictionsRequirements

• metal alloys
• crude oils
• livestock feeds
• foodstuffs

Cost

Objective

Linear Programming

Applications

Blending Problem

Decision variables: amount of ingredients used in final blend

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

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

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

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

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Blending Problem

Example – Feed

Decision variables

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Blending Problem

Example – Feed

Optimal solution

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Marketing Research

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Marketing Research

Example – MarketQuest, Inc.

• Evaluating consumer’s reaction to new products and services
• MQ‘s client introduces a new type of washing powder
• Prepare a campaign with door-to-door personal interviews about households’ opinion
• Households: with childrenwithout children
• Time of interview: daytime evening

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

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

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Marketing Research

Example – MarketQuest, Inc.

• Cost
• Objective: minimize total cost

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Marketing Research

Example – MarketQuest, Inc.

Decision variables

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Marketing Research

Example – MarketQuest, Inc.

1)Plan: to conduct 1000 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

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Marketing Research

Example – MarketQuest, Inc.

Optimal solution

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Portfolio Selection Problem

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Quantitative Methods of Management  Jan Fábry

Alternative investments (shares, bonds, etc.)

Mutual funds, credit unions, banks, insurance companies

Maximization of expected return

Minimization of risk

Linear Programming

Applications

Portfolio Selection Problem

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

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

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

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

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Portfolio Selection Problem

Example – Drink Invest, Inc.

Decision variables

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Portfolio Selection Problem

Example – Drink Invest, Inc.

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

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

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry

Linear Programming

Applications

Portfolio Selection Problem

Example – Drink Invest, Inc.

Optimal solution

___________________________________________________________________________

Quantitative Methods of Management  Jan Fábry