Labwork 4

Project-4

QBA775- Quantitative Methods

Due-05-09-2023 at 11:59 PM

• You can discuss this lab-work with any other student in the class.

• You can get help or clarification from the instructor.

• For each question, you need to fully interpret your results. Your report should be typed.

• You can upload your lab-work file under lab assignment folder in Blackboard.

• Late submission is not accepted.

• You are supposed to upload LINGO output in blackboard.

1. Better Fitness, Inc. (BFI), manufactures exercise equipment at its plant in Freeport, Long

Island. It recently designed two universal weight machines for the home exercise market.

Both machines use BFI-patented technology that provides the user with an extremely wide

range of motion capability for each type of exercise performed. Until now, such capabilities

have been available only on expensive weight machines used primarily by physical therapists.

At a recent trade show, demonstrations of the machines resulted in significant dealer interest.

In fact, the number of orders that BFI received at the trade show far exceeded its manufac-

turing capabilities for the current production period. As a result, management decided to

begin production of the two machines. The two machines, which BFI named the BodyPlus

100 and the BodyPlus 200, require different amounts of resources to produce.

The BodyPlus 100 consists of a frame unit, a press station, and a pec-dec station. Each

frame produced uses 4 hours of machining and welding time and 2 hours of painting and

finishing time. Each press station requires 2 hours of machining and welding time and 1 hour

of painting and finishing time, and each pec-dec station uses 2 hours of machining and welding

time and 2 hours of painting and finishing time. In addition, 2 hours are spent assembling,

testing, and packaging each BodyPlus 100. The raw material costs are $450 for each frame,

$300 for each press station, and $250 for each pec-dec station; packaging costs are estimated

to be $50 per unit.

The BodyPlus 200 consists of a frame unit, a press station, a pec-dec station, and a leg-press

station. Each frame produced uses 5 hours of machining and welding time and 4 hours of

painting and finishing time. Each press station requires 3 hours of machining and welding time

and 2 hours of painting and finishing time, each pec-dec station uses 2 hours of machining and

welding time and 2 hours of painting and finishing time, and each leg-press station requires 2

hours of machining and welding time and 2 hours of painting and finishing time. In addition,

2 hours are spent assembling, testing, and packaging each BodyPlus 200. The raw material

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costs are $650 for each frame, $400 for each press station, $250 for each pec-dec station, and

$200 for each leg-press station; packaging costs are estimated to be $75 per unit.

For the next production period, management estimates that 600 hours of machining and

welding time, 450 hours of painting and finishing time, and 140 hours of assembly, testing,

and packaging time will be available. Current labor costs are $20 per hour for machining and

welding time, $15 per hour for painting and finishing time, and $12 per hour for assembly,

testing, and packaging time. The market in which the two machines must compete suggests

a retail price of $2400 for the BodyPlus 100 and $3500 for the BodyPlus 200, although some

flexibility may be available to BFI because of the unique capabilities of the new machines.

Authorized BFI dealers can purchase machines for 70% of the suggested retail price.

BFI’s president believes that the unique capabilities of the BodyPlus 200 can help position

BFI as one of the leaders in high-end exercise equipment. Consequently, he has stated that the

number of units of the BodyPlus 200 produced must be at least 25% of the total production.

Managerial Report

(a) What is the recommended number of BodyPlus 100 and BodyPlus 200 machines to

produce?

(b) How does the requirement that the number of units of the BodyPlus 200 produced be

at least 25% of the total production affect profits?

(c) Where should efforts be expended in order to increase profits?

2. Cloud Services Capacity Planning. Galaxy Cloud Services operates several data centers

across the United States containing servers that store and process the data on the Internet.

Suppose that Galaxy Cloud Services currently has five outdated data centers: one each in

Michigan, Ohio, and California and two in New York. Management is considering increasing

the capacity of these data centers to keep up with increasing demand. Each data center

contains servers that are dedicated to Secure data and to Super Secure data. The cost to

update each data center and the resulting increase in server capacity for each type of server

are as follows:

Data center Cost($millions) Secure Servers Super Secure Servers

Michigan 2.5 50 30 New York 1 3.5 80 40 New York 2 3.5 40 80

Ohio 4 90 60 California 2 20 30

The projected needs are for a total increase in capacity of 90 Secure servers and 90 Super

Secure servers. Management wants to determine which data centers to update to meet pro-

jected needs and, at the same time, minimize the total cost of the added capacity.

In particular you have been asked to

(a) Formulate a binary integer programming model that could be used to determine the

optimal solution to the capacity increase question facing management.

(b) Solve the model formulated in part (a) to provide a recommendation for management.

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3. Hub Location at Western Airlines Western Airlines has decided that it wants to design

a hub system in the United States. Each hub is used for connecting flights to and from cities

within 1000 miles of the hub. Western run flights among the following cities: Atlanta, Boston,

Chicago, Denver, Houston, Los Angeles, New Orleans, New York, Pittsburgh, Salt Lake City,

San Francisco, and Seattle. The company wants to determine the smallest number of hubs it

will need to cover all of these cities, where a city is ”covered” if it is within 1000 miles of at

least one hub. The following table lists the cities that are within 1000 miles of other cities.

Cities Cities within 1000 Miles)

Atlanta(AT) AT,CH,HO,NO,NY,PI Boston(BO) BO, NY, PI Chicago(CH) AT,CH,NY,NO,PI Denver(DE) DE, SL Houston(HO) AT, HO, NO

Los Angeles(LA) LA,SL,SF New Orleans(NO) AT,CH,HO,NO New York(NY) AT, BO, CH, NY, PI Pittsburgh(PI) AT, BO, CH, NY, PI

Salt Lake City (SL) DE LA, SL, SF, SE San Francisco (SF) LA, SL, SF, SE

Seattle(SE) SL, SF, SE

Formulate and solve a linear binary model to find the minimum number of hub locations that

can cover all cities. Please name those cities location which will cover all cities.

Note Since all the cities are binary, binary is the requirement for this problem. So you are

supposed to use @BIN(AT); @BIN(BO); ......@BIN(SE);

4. Distribution Plan to Minimize Cost. Consider the following network representation of a

transportation problem:

A network representation of a transportation problem. The diagram shows two origin nodes:

Jefferson City and Omaha with supplies 30 and 20; and three destination nodes: Des Moines,

Kansas City, and St. Louis with demands 25, 15, and 10. The origin node, Jefferson City

connects to the three destination nodes through distribution routes (arcs) that have trans-

portation costs per unit of 14, 16, and 7, respectively. The origin node, Omaha connects to

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the three destination nodes through distribution routes (arcs) that have transportation costs

per unit of 8, 10, and 5, respectively.

The supplies, demands, and transportation costs per unit are shown on the network. What

is the optimal (cost minimizing) distribution plan?

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