Linear Programming
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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 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. ___________________________________________________________________________ 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 ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry Linear Programming Blending Problem ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry Linear Programming Applications Blending Problem Inputs (Ingredients) Output (Final blend) ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry Linear Programming Applications Blending Problem Inputs chemicals metal alloys crude oils livestock feeds foodstuffs Output Quality Quantity Cost Restrictions Requirements Objective 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 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 ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry Linear Programming Applications Blending Problem Example – Feed Decision variables Amount of feed F1 in the final blend x1 - || - F2 - || - x2 - || - F3 - || - x3 - || - F4 - || - x4 ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry Linear Programming Applications Blending Problem Example – Feed Optimal solution F1 120 kg F2 - F3 60 kg F4 20 kg Total cost 6 600 CZK ___________________________________________________________________________ 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: Time of interview: with children without children 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 Daytime interview Evening interview Households with children 50 CZK 60 CZK Households without children 40 CZK 50 CZK Objective: minimize total cost ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry Linear Programming Applications Marketing Research Example – MarketQuest, Inc. Decision variables Daytime interview Evening interview Households with children x1 x2 Households without children x3 x4 ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry 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 ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry 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 ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry Linear Programming Portfolio Selection Problem ___________________________________________________________________________ Quantitative Methods of Management 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 ___________________________________________________________________________ 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 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 ___________________________________________________________________________ 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 Bohemian Beer share x1 Moravian Wine share x2 Moravian Brandy share x3 Bohemian Milk share x4 Government bond x5 ___________________________________________________________________________ Quantitative Methods of Management 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 ___________________________________________________________________________ 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 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 ___________________________________________________________________________ Quantitative Methods of Management Jan Fábry
