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___________________________________________________________________________ Quantitative Methods of Management Jan Fábry

ApplicationsApplications

Linear ProgrammingLinear Programming




Guideline 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.




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


Blending ProblemBlending Problem





Blending Problem Blending Problem

InputsInputs(Ingredients)(Ingredients)

OutputOutput (Final blend)(Final blend)





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

QuantityQuantity

RestrictionsRestrictionsRequirementsRequirements

ObjectiveObjective





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






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 00

Price (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






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






Optimal solutionOptimal solution

F1F1 120 kg120 kg

F2F2 --

F3F3 60 kg60 kg

F4F4 20 kg20 kg

Total costTotal cost 66 6600 CZK00 CZK


Marketing 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





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





Example – MarketQuest, Inc.Example – MarketQuest, Inc.

Daytime Daytime iinterviewnterview

Evening Evening interviewinterview

Households with childrenHouseholds with children 50 CZK50 CZK 60 CZK60 CZK

Households without childrenHouseholds without children 40 CZK40 CZK 50 CZK50 CZK

CostCost

Objective: minimize total costObjective: minimize total cost






Daytime Daytime iinterviewnterview

Evening Evening interviewinterview

Households with childrenHouseholds with children xx11 xx22

Households without childrenHouseholds without children xx33 xx44






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






Daytime Daytime iinterviewnterviewss

Evening Evening interviewinterviews s

Households with childrenHouseholds with children 195195 105105

Households without childrenHouseholds without children 245245 455455

Total costTotal cost 48 600 CZK48 600 CZK



Portfolio Selection Portfolio Selection ProblemProblem





Portfolio 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





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





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





Example – Drink Invest, Inc.Example – Drink Invest, Inc.


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






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






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







Bohemian Beer shareBohemian Beer share 800 000 CZK800 000 CZK

Moravian Wine shareMoravian Wine share --

Moravian Brandy shareMoravian Brandy share 800 000 CZK800 000 CZK

Bohemian Milk shareBohemian Milk share --

Government bondGovernment bond 400 000 CZK400 000 CZK

Expected annual returnExpected annual return 240 000 CZK240 000 CZK

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Documents

x4linear programming

final blendlinear programming

total cost linear programming

nutritive feed

units of starch

objective function

final blendx1

units of proteinswill