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ISDS-514Linear programming applications & Integer Linear Programming 3

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ISDS-514Linear programming applications & Integer Linear Programming

3.1 28. RESTAURANT CREW ASSIGNMENT. Burger Boy Restaurant is open from 8:00 a.m. to 10:00 P.M. daily. In addition to the hours of business, a crew of workers must arrive one hour early to help set up the restaurant for the day’s operations, and another crew of workers must stay one hour after 10:00 P.M. to clean up after closing. Burger Boy operates with nine different shifts:

Shift Type Daily Salary

|. 7AM-9AM Part-ume $15

2, 7AM-11AM Part-time $25

3. 7AM-3PM Full-ame $52

4. 1LAM-3PM Part-time $22

Shift Type Daily Salary

5. JIAM-7PM Full-ome $54

6. 3PM-7PM Part-ame $24

7. 3PM-11PM Full-time $55

8. 7PM-11PM Part-ume $23

9. OPM-11PM Part-time $16

A needs assessment study has been completed, which divided the workday at Burger Boy into eight 2-hour blocks. The number of employees needed for each block is as follows:

Time Block Employees Needed

7AM-9AM 8

9AM-1TAM 10

11AM-tPM 22

IPM-3PM 15

3PM-5PM 10)

SPM-7PM 20

7PM-9PM 16

9PM-11PM 8

Burger Boy wants at least 40% of all employees at the peak time periods of 11:00 4.M. to 1:00 P.M. and 5:00 p.m. to 7:00 P.M. to be full-time employees. At least two full-time employees must be on duty when the restaurant opens at 7:00 a.M. and when it closes at 11:00 p.m. ¢ Formulate and solve a model Burger Boy can use to determine how many employees it should hire for each of its nine shifts to minimize its overall daily employee costs.
3.2 LAW ENFORCEMENT. The police department of the city of Flint, Michigan, has divided the city into 15 patrol sectors, such that the response time of a patrol unit (squad car) will be less than three minutes between any two points within the sector.

 Unul recently, 15 units, one Jocated in each sector, patrolled the streets of Flint from 7:00 P.M. to 3:00

A.M. However, severe budget cuts have forced the city to eliminate some patrols. The chief of police has mandated that each sector be covered by at least one unit located either within the sector or in an adjacent sector.

 

The accompanying figure depicts the 15 patrol sectors of Flint, Michigan. Formulate and solve a binary model that will determine the minimum number of units required to implemenc the chief’s policy.

 

1 . & p

" | » 2 ]

! (ay

Police Patrol Sectors Flint, Michigan
 

3.3 34. SOFTWARE DEVELOPMENT. The Korvex Corporation is a company concerned with developing

CD-ROM software applications that it sells to major computer manufacturers to include as “packaged items” when consumers purchase systems with CD-ROM drives. The company is currently evaluating the feasibility of developing six new applications. Specific information concerning cach of these applications 1s summarized in the following table.

 

Projected Projected

Development Programmers — Present Worth

Applicanon Cost Required Net Profit

l $ 400,000 6 $2,000,000

2 $1,100,000 Ik $3,600,000

3 $ 940,000 20 $4,000,000

4 $ 760,000 16 $3,000,000

§ $1,260,000 28 $4,400,000

6 $1,800,000 34 $6,200,000

Korvex has a staff of 60 programmers and has allocated $3.5 million for development of new applications.

a. Formulate and solve a binary integer linear programming model for the situation faced by the Korvex Corporation.

 

3.4 Aire-Co produces home dehumidifiers at two different plants in Atlanta and Phoenix. The per unit cost of production in Atlanta and Phoenix is $400 and $360, respectively. Each plant can produce a maximum of 300 units per month. Inventory holding costs are assessed at $30 per unit in beginning inventory each month. Aire-Co estimates the demand for its product to be 300, 400, and 500 units, respectively, over the next three months. Aire-Co wants to be able to meet this demand at minimum cost.

a. Formulate an LP model for this problem.

 

b. Implement your model in a spreadsheet and solve it.

 

c. What is the optimal solution?

 

d. How does the solution change if each plant is required to produce at least 50 units per month?

 

e. How does the solution change if each plant is required to produce at least 100 units per month?

3.5 Carter Enterprises is involved in the soybean business in South Carolina, Alabama, and Georgia. The president of the company, Earl Carter, goes to a commodity sale once a month where he buys and sells soybeans in bulk. Carter uses a local ware- house for storing his soybean inventory. This warehouse charges $10 per average ton of soybeans stored per month (based on the average of the beginning and ending inventory each month). The warehouse guarantees Carter the capacity to store up to

400 tons of soybeans at the end of each month. Carter has estimated what he believes the price per ton of soybeans will be during each of the next six months. These prices are summarized in the following table.

 

Month 1 2 3 4 5 6

 

Price perfon $135 $110 $150 $175 $130 $145

Assume Carter currently has 70 tons of soybeans stored in the warehouse. How many tons of soybeans should Carter buy and sell during each of the next six months to maximize his profit trading soybeans?

a. Formulate an LP model for this problem.

b. Create a spreadsheet model for this problem, and solve it using Solver.

c. What is the optimal solution?

 

3.6 Hack Potts recently won $1,000,000 in Las Vegas and is trying to determine how to invest his winnings. He has narrowed his decision down to five investments, which are summarized in the following table.

 

Summary of Cash Inflows and Outflows

___lat beginning of years)

1 2 3 4

 

A —1 0.50 0.80

 

8 —] <> 1.25

 

¢C —4J <—»> <> 1.35

 

D -1 1.13

 

E —1 <> 1.27

If Jack invests $1 in investment A at the beginning of year 1, he will receive $0.50 at the beginning of year 2 and another $0.80 at the beginning of year 3. Alternatively, he can invest $1 in investment B at the beginning of year 2 and receive $1.25 at the beginning of year 4. Entries of “<—»” in the table indicate times when no cash in- flows or outflows can occur. At the beginning of any year, Jack can place money in a money market account that is expected to yield 8% per year. He wants to keep at least $50,000 in the money market account at all times and doesn’t want to place any more than $500,000 in any single investment. How would you advise Jack to invest his winnings if he wants to maximize the amount of money he'll have at the begin- ning of year 4?

a. Formulate an LP model for this problem.

b. Create a spreadsheet model for this problem, and solve it using Solver.

c. What is the optimal solution?