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Linear Programming. Applications. ___________________________________________________________________________ Operations Research Jan Fábry. Linear Programming. Applications. Guideline for Model Formulation . 1. Understand the problem thoroughly. . - PowerPoint PPT PresentationTRANSCRIPT
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___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplications
Linear ProgrammingLinear Programming
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsGuideline for Model Formulation Guideline for Model Formulation
5.5. Write the constraints in terms of the decision Write the constraints in terms of the decision variables. variables.
4. 4. Write the objective function in terms of the Write the objective function in terms of the decision variables. decision variables.
3. D3. Define the decision variables. efine the decision variables.
2. 2. Write a verbal statement of the objective function Write a verbal statement of the objective function and eachand each constraint. constraint.
1. 1. Understand the problem thoroughly. Understand the problem thoroughly.
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplications
Cutting Stock Problem Cutting Stock Problem
Production Process Models Production Process Models
Portfolio Selection Problem Portfolio Selection Problem
Marketing Research Marketing Research
Blending Problems Blending Problems
Transportation Problem Transportation Problem
Assignment Problem Assignment Problem
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___________________________________________________________________________ Operations Research Jan Fábry
Blending ProblemBlending Problem
Linear ProgrammingLinear Programming
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsBlending Problem Blending Problem
InputsInputs(Ingredients)(Ingredients)
OutputOutput (Final blend)(Final blend)
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsBlending Problem Blending Problem
InputsInputs
metal alloysmetal alloys
chemicalschemicals
livestock feedslivestock feeds
crude oilscrude oils
foodstuffsfoodstuffs
Decision variables: Decision variables: amount of ingredients amount of ingredients
used in final blendused in final blend
OutputOutput
CostCost
QualityQuality
QuantityQuantityRestrictionsRestrictions
RequirementsRequirements
ObjectiveObjective
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsBlending Problem Blending Problem
Example – FeedExample – 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 costObjective: minimize total cost
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsBlending Problem Blending Problem
Example – FeedExample – Feed
Feed F1Feed F1 Feed F2Feed F2 Feed F3Feed F3 Feed F4Feed F4
Proteins (units)Proteins (units) 00 33 11 22
Starch (units)Starch (units) 11 22 33 00Price (CZK)Price (CZK) 2020 8080 6060 3030
Contents of proteins and starch in 1kg of each nutritive feed and prices for 1 kg of feed
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsBlending Problem Blending Problem
Example – FeedExample – Feed
Decision variablesDecision variables
Amount of feed F1 in the final blendAmount of feed F1 in the final blend xx11
- || - F2 - || - - || - F2 - || - xx22
- || - F3 - || - - || - F3 - || - xx33
- || - F4 - || - - || - F4 - || - xx44
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsBlending Problem Blending Problem
Example – FeedExample – Feed
Optimal solutionOptimal solution
F1F1 120 kg120 kgF2F2 --F3F3 60 kg60 kgF4F4 20 kg20 kg
Total costTotal cost 66 6600 CZK00 CZK
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___________________________________________________________________________ Operations Research Jan Fábry
Marketing ResearchMarketing Research
Linear ProgrammingLinear Programming
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsMarketing ResearchMarketing Research
Example – MarketQuest, Inc.Example – MarketQuest, Inc. EEvaluating consumer’s reaction to new products and servicesvaluating consumer’s reaction to new products and services
PPrepare a campaign with door-to-door personal repare a campaign with door-to-door personal interviews about households’ opinioninterviews about households’ opinion
MQ‘s client introduces a MQ‘s client introduces a new type of washing powdernew type of washing powder
HouseholdsHouseholds: : with childrenwith childrenwithout childrenwithout children
Time of interview: Time of interview: daytimedaytime
eveningevening
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsMarketing ResearchMarketing Research
Example – MarketQuest, Inc.Example – MarketQuest, Inc. Plan: to Plan: to conduct 1000 interviewsconduct 1000 interviews
At least 400 households without children should be interviewed At least 400 households without children should be interviewed
At least 300 households with children should be interviewed At least 300 households with children should be interviewed
NNumber of evening interviews umber of evening interviews number of daytime interviews number of daytime interviews At least 35% of the interviews for households with children At least 35% of the interviews for households with children
should be conducted during evening should be conducted during evening At least 65% of the interviews for households without children At least 65% of the interviews for households without children
should be conducted during evening should be conducted during evening
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsMarketing ResearchMarketing Research
Example – MarketQuest, Inc.Example – MarketQuest, Inc.
Daytime Daytime iinterviewnterview
Evening Evening interviewinterview
Households with childrenHouseholds with children 50 CZK50 CZK 60 CZK60 CZKHouseholds without childrenHouseholds without children 40 CZK40 CZK 50 CZK50 CZK
CostCost
Objective: minimize total costObjective: minimize total cost
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsMarketing ResearchMarketing Research
Example – MarketQuest, Inc.Example – MarketQuest, Inc.
Daytime Daytime iinterviewnterview
Evening Evening interviewinterview
Households with childrenHouseholds with children xx11 xx22
Households without childrenHouseholds without children xx33 xx44
Decision variablesDecision variables
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsMarketing ResearchMarketing Research
Example – MarketQuest, Inc.Example – MarketQuest, Inc.1)1) Plan: to Plan: to conduct 1conduct 1 000 interviews000 interviews
3)3) At least 400 households without children should be interviewed At least 400 households without children should be interviewed
2)2) At least 300 households with children should be interviewed At least 300 households with children should be interviewed
4)4) NNumber of evening interviews umber of evening interviews number of daytime interviews number of daytime interviews
5)5) At least 35% of the interviews for households with children At least 35% of the interviews for households with children should be conducted during evening should be conducted during evening
6)6) At least 65% of the interviews for households without children At least 65% of the interviews for households without children should be conducted during evening should be conducted during evening
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsMarketing ResearchMarketing Research
Example – MarketQuest, Inc.Example – MarketQuest, Inc.
Daytime Daytime iinterviewnterviewss
Evening Evening interviewinterviews s
Households with childrenHouseholds with children 195195 105105Households without childrenHouseholds without children 245245 455455
Total costTotal cost 48 600 CZK48 600 CZK
Optimal solutionOptimal solution
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___________________________________________________________________________ Operations Research Jan Fábry
Portfolio Selection Portfolio Selection ProblemProblem
Linear ProgrammingLinear Programming
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsPortfolio Selection ProblemPortfolio Selection Problem
MMaximization of expected returnaximization of expected return
Alternative investments (shares, bonds, etc.)Alternative investments (shares, bonds, etc.)
MMutual funds, credit unions, banksutual funds, credit unions, banks, i, insurance nsurance companiescompanies
MiniMinimization of mization of risk risk
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsPortfolio Selection ProblemPortfolio Selection Problem
Example – Drink Invest, Inc.Example – Drink Invest, Inc. Investing money Investing money in stocks of companies producing drinksin stocks of companies producing drinks Plan to invest to 4 shares and 1 government bondPlan to invest to 4 shares and 1 government bond
Rate of returnRate of return Risk indexRisk index
Bohemian Beer share Bohemian Beer share 12 %12 % 0.070.07
Moravian Wine shareMoravian Wine share 9 %9 % 0.090.09
Moravian Brandy shareMoravian Brandy share 15 %15 % 0.050.05
Bohemian Milk shareBohemian Milk share 7 %7 % 0.030.03
Government bondGovernment bond 6 %6 % 0.010.01
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsPortfolio Selection ProblemPortfolio Selection Problem
Example – Drink Invest, Inc.Example – Drink Invest, Inc. Plan: to invest 2 000 000 CZKPlan: to invest 2 000 000 CZK
Government bonds should cover at least 20% of all investments Government bonds should cover at least 20% of all investments
No more than 200 000 CZK might be invested in Bohemian Milk shares No more than 200 000 CZK might be invested in Bohemian Milk shares
Because of diversification of portfolio neither alcohol-drink Because of diversification of portfolio neither alcohol-drink company should receive more than 800 000 CZK company should receive more than 800 000 CZK
Risk index of the final portfolio should be maximally 0.05 Risk index of the final portfolio should be maximally 0.05
Objective: maximize annual return of the portfolioObjective: maximize annual return of the portfolio
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsPortfolio Selection ProblemPortfolio Selection Problem
Example – Drink Invest, Inc.Example – Drink Invest, Inc.
Decision variablesDecision variables
Bohemian Beer shareBohemian Beer share xx11
Moravian Wine shareMoravian Wine share xx22
Moravian Brandy shareMoravian Brandy share xx33
Bohemian Milk shareBohemian Milk share xx44
Government bondGovernment bond xx55
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsPortfolio Selection ProblemPortfolio Selection Problem
Example – Drink Invest, Inc.Example – Drink Invest, Inc.
Rate of returnRate of return Risk indexRisk index
Bohemian Beer share Bohemian Beer share 12 %12 % 0.070.07
Moravian Wine shareMoravian Wine share 9 %9 % 0.090.09
Moravian Brandy shareMoravian Brandy share 15 %15 % 0.050.05
Bohemian Milk shareBohemian Milk share 7 %7 % 0.030.03
Government bondGovernment bond 6 %6 % 0.010.01
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsPortfolio Selection ProblemPortfolio Selection Problem
Example – Drink Invest, Inc.Example – Drink Invest, Inc.
1)1) Plan: to invest 2 000 000 CZKPlan: to invest 2 000 000 CZK
3)3) Government bonds should cover at least 20% of all investments Government bonds should cover at least 20% of all investments
2)2) No more than 200 000 CZK might be invested in Bohemian Milk shares No more than 200 000 CZK might be invested in Bohemian Milk shares
4)4) Because of diversification of portfolio neither alcohol-drink Because of diversification of portfolio neither alcohol-drink company should receive more than 800 000 CZK company should receive more than 800 000 CZK
5)5) Risk index of the final portfolio should be maximally 0.05 Risk index of the final portfolio should be maximally 0.05
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Linear ProgrammingLinear Programming
___________________________________________________________________________ Operations Research Jan Fábry
ApplicationsApplicationsPortfolio Selection ProblemPortfolio Selection Problem
Example – Drink Invest, Inc.Example – Drink Invest, Inc.
Optimal solutionOptimal solution
Bohemian Beer shareBohemian Beer share 800 000 CZK800 000 CZKMoravian Wine shareMoravian Wine share --
Moravian Brandy shareMoravian Brandy share 800 000 CZK800 000 CZKBohemian Milk shareBohemian Milk share --
Government bondGovernment bond 400 000 CZK400 000 CZKExpected annual returnExpected annual return 240 000 CZK240 000 CZK