Lecture Introduction to Management Science with Spreadsheets: Chapter 4 - Stevenson, Ozgur

Chapter 4 Applications and solutions of linear programming, after completing this chapter, you should be able to: Formulate linear programming problems with different objectives, set up constraints that have unique structures, translate statements into constraint formulas, utilize excel to solve a variety of linear programming problems,...

Stevenson and Ozgur

First Edition

Introduction to Management Science with Spreadsheets

2 Set up constraints that have unique structures.

3 Translate statements into constraint formulas.

4 Utilize Excel to solve a variety of linear programming

problems.

5 Interpret the Excel output of linear programming problems.

6 Make managerial conclusions based on computer (Excel)

output.

7 Explain at least two applications of linear programming.

After completing this chapter, you should be able to:

Elements of Linear Programming

Elements of Linear Programming

• Issues in formulating problems

–Degree of simplification of the real-world problem

–Violation of the assumptions of linear programming (additivity, divisibility, certainty, and proportionality)

• Linear programming problems may include:

–Different types of decision variables definitions

–Different types of constraints

–Different types of objective functions

Example 4-1 Product-Mix Problem

Example 4-1 Product-Mix Problem

The Style and Comfort Furniture Manufacturing Company wishes to

determine its production schedule for the next quarter The company

produces four types of furniture, including sofas, love seats, recliners, and coffee tables The profit contribution from selling one sofa is $120, one love seat is $105, one recliner is $150, and one coffee table is $73

The quarterly production budget is set at $180,000 Each unit of a sofa, love seat, recliner, and coffee table costs $400, $300, $500, and $150,

respectively The sales forecasts indicate that the potential sales volume is limited to 200 units of sofas, 150 units of love seats, 100 units of recliners, and 400 units of coffee tables

There are an aggregate of 800 machine hours available and 1,200 labor hours available Table 4-1 summarizes the number of machine hours and the number of labor hours required per unit of each product.

Table 4–1 Per Unit Machine and Labor Hour Required for Each Product Table 4–1 Per Unit Machine and Labor Hour Required for Each Product

Style of Comfort Furniture Example

Style of Comfort Furniture Example

Copyright © 2007 The McGraw­Hill 

Exhibit 4–2 Solver Dialog Box

Exhibit 4–2 Solver Dialog Box

Exhibit 4–3 Add Constraint Dialog Box

Exhibit 4–3 Add Constraint Dialog Box

Exhibit 4–4 Completed Solver Dialog Box

Exhibit 4–4 Completed Solver Dialog Box

There are three types of Solver cells:

1 Target cell shows the overall performance measure (i.e., the maximum profit or

the minimum cost)

2 Output cell shows the output or quantity of the decision variables based on the

changing cells.

3 Data cells simply show the data of the problem (constraint coefficients, objective

function coefficients, and right-hand-side constraints).

Exhibit 4-5 Solver Options Dialog Box

Exhibit 4-5 Solver Options Dialog Box

Copyright © 2007 The McGraw­Hill 

Table 4–2 Cost per Ounce and Dietary Requirements for Diet Problem Table 4–2 Cost per Ounce and Dietary Requirements for Diet Problem

Example 4-2 Diet Problem

Example 4-2 Diet Problem

Example 4-3 Blending Problem

Example 4-3 Blending Problem

Formulate the appropriate model for

the following blending problem:

The sugar content of three juices—

orange, banana, and pineapple—is

10, 15, and 20 percent, respectively

How many quarts of each must be

mixed together to achieve one

gallon (four quarts) that has a sugar

content of at least 17 percent to

minimize cost? The cost per quart is

20 cents for orange juice, 30 cents

for banana juice, and 40 cents for

pineapple juice.

Solution

Variable definitions

O = quantity of orange juice in quarts

B = quantity of banana juice in quarts

P = quantity of pineapple juice in quarts

Marketing Applications

Marketing Applications

• Applications of linear programming in marketing

–Media selection—how to allocate an advertising

budget among different media of advertising

–Determination of the optimal assignment of

salespeople that work for the company among the

sales territories

–Marketing research which the objective is to determine the best number of interviews, mailings, or phone

calls, given a number of client-specified constraints

and the cost considerations

Example 4-5 Media Selection

Example 4-5 Media Selection

The Long Last Appliance Sales Company is in the business of selling

appliances such as microwave ovens, traditional ovens, refrigerators,

dishwashers, washers, dryers, and the like The company has stores in the

greater Chicagoland area and has a monthly advertising budget of $90,000.

Among its options are radio advertising, advertising in the cable TV

channels, newspaper advertising, and direct-mail advertising A 30-second

advertising spot on the local cable channel costs $1,800, a 30-second radio ad costs $350, a half-page ad in the local newspaper costs $700, and a single

mailing of direct-mail insertion for the entire region costs $1,200 per mailing The number of potential buying customers reached per advertising medium

Due to company restrictions and availability of media, the maximum number

of usages of each medium is limited to the following:

advertising allocated to TV and direct mail cannot exceed 60 percent of the

total advertising budget.

What is the optimal allocation of the budget among the four media? What is the total maximum audience contact?

Example 4-5 (cont’d)

Example 4-5 (cont’d)

Exhibit 4–8 Output Screen for the Long Last Appliance Sales Company Exhibit 4–8 Output Screen for the Long Last Appliance Sales Company

Exhibit 4–9 Excel Answer Report for the Long Last Appliance Sales

Company Exhibit 4–9 Excel Answer Report for the Long Last Appliance Sales

Company

–Conduct marketing survey.

–Analyze data and obtain results

–Make recommendations based on the results

Example 4-6 Market Research

Example 4-6 Market Research

Market Facts Inc is a marketing research firm that works with client

companies to determine consumer reaction toward various products and

services A client company requested that Market Facts investigate the

consumer reaction to a recently developed electronic device

Market Facts and the client company agreed that a combination of telephone interviews and direct-mail questionnaires would be used to obtain the

information from different type of households.

The households are divided into six categories:

1 Households containing a single person under 40 years old and without children

under 18 years of age.

2 Households containing married people under 40 years old and without children

under 18 years of age.

3 Households containing single parents with children under 18 years of age.

4 Households containing married families with children under 18 years of age.

5 Households containing single people over 40 years old without children under

18 years of age.

6 Households containing married people over 40 years old without children under

18 years of age.

1 At least 60 percent of the phone interviews must be conducted at

households with children.

2 At least 50 percent of the direct-mail questionnaires must be mailed to

households with children.

3 No more than 30 percent of the phone interviews and mail-in questionnaires must be conducted at households with single people.

4 At least 25 percent of the phone interviews and mail-in questionnaires must

be conducted at households that contain married couples.

Financial Applications

Financial Applications

• Planning Problems for Banks

–Linear programming can be very beneficial in banking decisions

–Financial planning: bankers must decide how a bank wants to allocate its funds among the various types of loans and investment securities

–Portfolio management: decisions are based on

maximizing annual rate of return subject to state and federal regulations, and bank policies and restrictions

Example 4-7 Financial Planning

Example 4-7 Financial Planning

First American Bank issues five types of loans In addition, to

diversify its portfolio, and to minimize risk, the bank invests in

risk-free securities The loans and the risk-risk-free securities with their annual rate of return are given in Table 4-3.

Table 4-3 Rates of Return for Financial Planning Problem

Type of Loan or Security Annual Rate of Return (%)

Example 4-7 Financial Planning (cont’d)

Example 4-7 Financial Planning (cont’d)

The bank’s objective is to maximize the annual rate of return on

investments subject to the following policies, restrictions, and

regulations:

1 The bank has $90 million in available funds

2 Risk-free securities must contain at least 10 percent of the total funds available for investments.

3 Home improvement loans cannot exceed $8,000,000.

4 The investment in mortgage loans must be at least 60 percent

of all the funds invested in loans.

5 The investment in first mortgage loans must be at least twice as much as the investment in second mortgage loans.

6 Home improvement loans cannot exceed 40 percent of the

funds invested in first mortgage loans.

7 Automobile loans and home improvement loans together may not exceed the commercial loans.

8 Commercial loans cannot exceed 50 percent of the total funds invested in mortgage loans.

Example 4-7 Financial Planning (cont’d)

Example 4-7 Financial Planning (cont’d)

Example 4-7 Financial Planning (cont’d)

Example 4-7 Financial Planning (cont’d)

Example 4-8 Portfolio Selection

Example 4-8 Portfolio Selection

A conservative investor has $100,000 to invest The investor has

decided to use three vehicles for generating income: municipal

bonds, a certificate of deposit (CD), and a money market account

After reading a financial newsletter, the investor has also identified several additional restrictions on the investments:

1 No more than 40 percent of the investment should be in bonds.

2 The proportion allocated to the money market account should be

at least double the amount in the CD.

The annual return will be 8 percent for bonds, 9 percent for the CD, and 7 percent for the money market account Assume the entire amount will be invested

Formulate the LP model for this problem, ignoring any transaction costs and the potential for different investment lives Assume that the investor wants to maximize the total annual

return.

Example 4-8 Portfolio Selection (cont’d)

Example 4-8 Portfolio Selection (cont’d)

Example 4-9 Multiperiod Production Scheduling

Example 4-9 Multiperiod Production Scheduling

Morton and Monson Inc is a small manufacturer of parts for the aerospace industry The production capacity for the next four months is given as follows:

Production Capacity in Units

The regular cost of production is $500 per unit and the cost of overtime production is

$150 per unit in addition to the regular cost of production The company can utilize inventories to reduce fluctuations in production, but carrying one unit of inventory costs the company $40 per unit per month Currently there are no units in inventory

However, the company wants to maintain a minimum safety stock of 100 units of

inventory during the months of January, February, and March The estimated demand for the next four months is as follows:

Example 4-9 Multiperiod Production Scheduling (cont’d)

Example 4-9 Multiperiod Production Scheduling (cont’d)

Continued on next slide.

Example 4-9 Multiperiod Production Scheduling (cont’d)

Example 4-9 Multiperiod Production Scheduling (cont’d)

Example 4-9 Multiperiod Production Scheduling (cont’d)

Example 4-9 Multiperiod Production Scheduling (cont’d)

Example 4-10 Workforce Scheduling

Example 4-10 Workforce Scheduling

Example 4-11 Make-or-Buy Decisions

Example 4-11 Make-or-Buy Decisions

Example 4-12 Agriculture Applications

Example 4-12 Agriculture Applications

A farm owner in Des Moines, Iowa, is interested in determining how to divide the farmland among four different types of crops The farmer owns two farms in

separate locations and has decided to plant the following four types of crops in these farms: corn, wheat, bean, and cotton The first farm consists of 1,450 acres

of land, while the second farm consists of 850 acres of land Any of the four

crops may be planted on either farm However, after a survey of the land, based

on the characteristics of the farmlands, Table 4-7 shows the maximum acreage restrictions the farmer has placed for each crop.

Data Envelopment Analysis

Data Envelopment Analysis

• Data envelopment analysis (DEA)

–The use linear programming to measure the relative efficiency of the departments within an organization or relative efficiencies of two different organizations

• Fast food restaurants, hospitals, schools, and banks.

–Measuring the efficiency of departments within the

same organization is difficult because departments

have multiple input measures (number of workers,

cost of labor, cost of machine operations, pay scale for employees, and cost of advertising) and multiple

output measures (profit, sales, and market share)

Example 4-12 Airline Efficiency

Example 4-12 Airline Efficiency

A consumer advocacy group would like to evaluate the efficiency of three

airline companies with respect to several criteria that are important to

consumers The three companies are Beta, Unified, and Southeast The output measures are identified as average minutes late arriving, number of luggage loser per 10,000 customers, waiting time for check-in, friendliness, and quality

of air service (result of a questionnaire that measured this output on a

100-point scale) The input measures were labor cost per day, number of flights per day, and over head cost per day Table 4-9 summarizes the results.

• Assignment problems

–Assigning jobs to machines in such a way that the

total cost of performing the jobs is minimized

• Project management problems

–Finding the minimum project duration for project

competition to shorten various project tasks and

minimize total project costs