1. ___________________________________________________________________________ Operations Research  Jan Fábry Applications Linear Programming

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

3. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications  Cutting Stock Problem  Production Process Models  Portfolio Selection Problem  Marketing Research  Blending Problems  Transportation Problem  Assignment Problem

4. ___________________________________________________________________________ Operations Research  Jan Fábry Blending Problem Linear Programming

5. ___________________________________________________________________________ Operations Research  Jan Fábry Applications Blending Problem Inputs (Ingredients) Output (Final blend)

6. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Blending Problem Inputs  metal alloys  chemicals  livestock feeds  crude oils  foodstuffs Decision variables: amount of ingredients used in final blend OutputCost QualityQuantity Restrictions Requirements Objective

7. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry 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

8. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Blending Problem Example – Feed Feed F1 Feed F2 Feed F3 Feed F4 Proteins (units) 0312 Starch (units) 1230 Price (CZK) 20806030  Contents of proteins and starch in 1kg of each nutritive feed and prices for 1 kg of feed

9. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Blending Problem Example – Feed Decision variables Amount of feed F1 in the final blend x1x1x1x1 - || - F2 - || - x2x2x2x2 - || - F3 - || - x3x3x3x3 - || - F4 - || - x4x4x4x4

10. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Blending Problem Example – Feed Optimal solution F1 120 kg F2- F3 60 kg F4 20 kg Total cost 6 600 CZK

11. ___________________________________________________________________________ Operations Research  Jan Fábry Marketing Research Linear Programming

12. ___________________________________________________________________________ Operations Research  Jan Fábry Applications Marketing Research Example – MarketQuest, Inc.  Evaluating consumer’s reaction to new products and services  Prepare a campaign with door-to-door personal interviews about households’ opinion  MQ‘s client introduces a new type of washing powder  Households: with children without children  Time of interview: daytime evening

13. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Marketing Research Example – MarketQuest, Inc.  Plan: to conduct 1000 interviews  At least 400 households without children should be interviewed  At least 300 households with 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

14. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Marketing Research Example – MarketQuest, Inc. Daytime interview Evening interview Households with children 50 CZK 60 CZK Households without children 40 CZK 50 CZK  Cost  Objective: minimize total cost

15. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Marketing Research Example – MarketQuest, Inc. Daytime interview Evening interview Households with children x1x1x1x1 x2x2x2x2 Households without children x3x3x3x3 x4x4x4x4 Decision variables

16. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Marketing Research Example – MarketQuest, Inc. 1) Plan: to conduct 1 000 interviews 3) At least 400 households without children should be interviewed 2) At least 300 households with 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

17. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Marketing Research Example – MarketQuest, Inc. Daytime interviews Evening interviews Households with children 195105 Households without children 245455 Total cost 48 600 CZK Optimal solution

18. ___________________________________________________________________________ Operations Research  Jan Fábry Portfolio Selection Problem Linear Programming

19. ___________________________________________________________________________ Operations Research  Jan Fábry Applications Portfolio Selection Problem Maximization of expected return Alternative investments (shares, bonds, etc.) Mutual funds, credit unions, banks, insurance companies Minimization of risk

20. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry 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

21. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Portfolio Selection Problem Example – Drink Invest, Inc.  Plan: to invest 2 000 000 CZK  Government bonds should cover at least 20% of all investments  No more than 200 000 CZK might be invested in Bohemian Milk shares  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

22. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Portfolio Selection Problem Example – Drink Invest, Inc. Decision variables Bohemian Beer share x1x1x1x1 Moravian Wine share x2x2x2x2 Moravian Brandy share x3x3x3x3 Bohemian Milk share x4x4x4x4 Government bond x5x5x5x5

23. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry 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

24. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry Applications Portfolio Selection Problem Example – Drink Invest, Inc. 1) Plan: to invest 2 000 000 CZK 3) Government bonds should cover at least 20% of all investments 2) No more than 200 000 CZK might be invested in Bohemian Milk shares 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

25. Linear Programming ___________________________________________________________________________ Operations Research  Jan Fábry 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