Linear Programming Applications Operations Research Jan Fbry Linear
Linear Programming Applications ______________________________________ Operations Research Jan Fábry
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
Linear Programming Applications Ø Production Process Models Ø Blending Problems Ø Marketing Research Ø Portfolio Selection Problem Ø Cutting Stock Problem Ø Transportation Problem Ø Assignment Problem ______________________________________ Operations Research Jan Fábry
Linear Programming Blending Problem ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Blending Problem Inputs (Ingredients) Output (Final blend) ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Blending Problem Inputs Ø Ø Ø chemicals metal alloys crude oils livestock feeds foodstuffs Output Quality Quantity Cost Restrictions Requirements Objective Decision variables: amount of ingredients used in final blend ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Blending Problem Example – Feed Ø 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 Ø Objective: minimize total cost ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Blending Problem Example – Feed Ø Contents of proteins and starch in 1 kg of each nutritive feed and prices for 1 kg of feed Feed F 1 Feed F 2 Feed F 3 Feed F 4 Proteins (units) 0 3 1 2 Starch (units) 1 2 3 0 Price (CZK) 20 80 60 30 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Blending Problem Example – Feed Decision variables Amount of feed F 1 in the final blend x 1 - || - F 2 - || - x 2 - || - F 3 - || - x 3 - || - F 4 - || - x 4 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Blending Problem Example – Feed Optimal solution F 1 120 kg F 2 - F 3 60 kg F 4 20 kg Total cost 6 600 CZK ______________________________________ Operations Research Jan Fábry
Linear Programming Marketing Research ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Marketing Research Example – Market. Quest, Inc. Ø Evaluating consumer’s reaction to new products and services Ø MQ‘s client introduces a new type of washing powder Ø Prepare a campaign with door-to-door personal interviews about households’ opinion Ø Households: with children without children Ø Time of interview: daytime evening ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Marketing Research Example – Market. Quest, Inc. Ø Plan: to conduct 1000 interviews Ø At least 300 households with children should be interviewed Ø At least 400 households without children should be interviewed Ø Number of evening interviews number of daytime interviews Ø At least 35% of the interviews for households with children should be conducted during evening Ø At least 65% of the interviews for households without children should be conducted during evening ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Marketing Research Example – Market. Quest, Inc. Ø Cost Daytime interview Evening interview Households with children 50 CZK 60 CZK Households without children 40 CZK 50 CZK Ø Objective: minimize total cost ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Marketing Research Example – Market. Quest, Inc. Decision variables Daytime interview Evening interview Households with children x 1 x 2 Households without children x 3 x 4 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Marketing Research Example – Market. Quest, 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
Linear Programming Applications Marketing Research Example – Market. Quest, Inc. Optimal solution Daytime interviews Evening interviews Households with children 195 105 Households without children 245 455 Total cost 48 600 CZK ______________________________________ Operations Research Jan Fábry
Linear Programming Portfolio Selection Problem ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Portfolio Selection Problem Alternative investments (shares, bonds, etc. ) Mutual funds, credit unions, banks, insurance companies Maximization of expected return Minimization of risk ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Portfolio Selection Problem Example – Drink Invest, Inc. Ø Investing money in stocks of companies producing drinks Ø 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 ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Portfolio Selection Problem Example – Drink Invest, Inc. Ø Plan: to invest 2 000 CZK Ø No more than 200 000 CZK might be invested in Bohemian Milk shares Ø Government bonds should cover at least 20% of all investments Ø Because of diversification of portfolio neither alcohol-drink company should receive more than 800 000 CZK Ø Risk index of the final portfolio should be maximally 0. 05 Ø Objective: maximize annual return of the portfolio ______________________________________ Operations Research Jan Fábry
Linear Programming Applications Portfolio Selection Problem Example – Drink Invest, Inc. Decision variables Bohemian Beer share x 1 Moravian Wine share x 2 Moravian Brandy share x 3 Bohemian Milk share x 4 Government bond x 5 ______________________________________ Operations Research Jan Fábry
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
Linear Programming Applications Portfolio Selection Problem Example – Drink Invest, Inc. 1) Plan: to invest 2 000 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
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
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