Activity 6.6 Module Problems - 6-1, 6-2, 6-3, 6-4, 6-5 and 6-6

Activity 6.6 Module Problems

Complete the following problems and submit the results in either a Microsoft Word document or
a Microsoft Excel spreadsheet. If you choose to use an Excel spreadsheet, place each problem on
a separate sheet and label the tab with problem number. Save your document with a descriptive
file name, including the assignment and your name.
6-1
The famous Y. S. Chang Restaurant is open 24 hours a day. Waiters and busboys report
for duty at 3am, 7am, 11am, 3pm, 7pm, or 11pm, and each works an 8-hour shift. The following
table shows the minimum number of workers needed during the six periods into which the day is
divided. Changs scheduling problem is to determine how many waiters and busboys should
report for work at the start of each time period to minimize the total staff required for one days
operation. (Hint: Let Xi equal the number of waiters and busboys beginning work in time period
i, where i = 1, 2, 3, 4, 5, 6.) If Changs were able to reduce the number of required waiters and
busboys by 1 during some period, during which period should they make the reduction?
Perio
d
1
2
3
4
5
6

Time

Number of Waiters
and Busboys Required

3am-7am
7am-11am
11am-3pm
3pm-7pm
7pm-11pm
11pm-3am

3
12
17
11
12
4

6-2
The Arden County, , superintendent of education is responsible for assigning
students to the three high schools in his county. He recognizes the need to bus a certain number
of students, for several sectors of the county are beyond walking distance to a school. The
superintendent partitions the county into five geographic sectors as he attempts to establish a
plan that will minimize the total number of student miles traveled by bus. He also recognizes
that if a student happens to live in a certain sector and is assigned to the high school in that
sector, there is no need to bus that student because he or she can walk to school. The three
schools are located in sectors B, C, and E.
The following table reflects the number of high school-age students living in each sector and the
busing distance in miles from each sector to each school:


Sector
A
B
C
D
E

School in
Sector B
5
0
4
7
12

Distance to School
School in
School in
Sector C
Sector E
9
6
5
12
0
7
3
5
8
0

Number of
Students
700
500
100
800
400
2,500

Each high school has a capacity of 900 students, but must have at least 700. Set up the objective
function and constraints of this problem using LP so that the total number of student miles
traveled by bus is minimized. Then solve the problem.
6-3
Coast-to-Coast Airlines is investigating the possibility of reducing the cost of fuel
purchases by taking advantage of lower fuel costs in certain cities. Since fuel purchases
represent a substantial portion of operating expenses for an airline, it is important that these costs
be carefully monitored. However, fuel adds weight to an airplane, and consequently, excess fuel
raises the cost of getting from one city to another. In evaluating one particular flight rotation, a
plane begins in Atlanta, flies from Atlanta to Los Angeles, from Los Angeles to Houston, from
Houston to New Orleans, and from New Orleans to Atlanta. When the plane arrives in Atlanta,
the flight rotation is said to have been completed, and then it starts again. Thus, the fuel on
board when the flight arrived in Atlanta must be taken into consideration when the flight begins.

Leg
Atlanta-Los Angeles
Los Angeles-Houston
Houston-New Orleans
New Orleans-Atlanta

Minimum Fuel
Required
(1,000 gal.)
25
16
8
12

Maximum Fuel
Allowed
(1,000 gal.)
36
23
18
20

Regular Fuel
Consumption
(1,000 gal.)
13
7
3
5

Fuel Price per
Gallon
4.25
4.55
4.15
4.22

Along each leg of this route, there is a minimum and a maximum amount of fuel that may be
carried. This and additional information is provided in the table above.
The regular fuel consumption is based on the plane carrying the minimum amount of fuel. If
more than this is carried, the amount of fuel consumed is higher. Specifically, for each 1,000
gallons of fuel above the minimum, 6% (or 60 gallons per 1,000 gallons of extra fuel) is lost due
to excess fuel consumption. For example, if 26,000 gallons of fuel were on board when the
plane takes off from Atlanta, the fuel consumed on this route would be 13 + 0.06 = 13.06
thousand gallons. If 27 thousand gallons were on board, the fuel consumed would be increased
by another 0.06 thousand, for a total of 13.12 thousand gallons.

Formulate this as a LP problem to minimize the cost.
a. How many gallons should be purchased in each city?
b. What is the total cost of his?
6-4
An air conditioning manufacturer produces room air conditioners at plants in Houston,
Phoenix, and Memphis. These are sent to regional distributors in Dallas, Atlanta, and Denver.
The shipping costs vary, and the company would like to find the least-cost way to meet the
demands at each of the distribution centers. Dallas needs to receive 800 air conditioners per
month, Atlanta needs 700, and Denver needs 250. Houston has 850 air conditioners available
each month, Phoenix has 800, and Memphis has 350. The shipping cost per unit from Houston
to Dallas is $8, to Atlanta is $12, and to Denver is $13. The cost per unit from Phoenix to Dallas
is $10, to Atlanta is $14, and to Denver is $8. The cost per unit from Memphis to Dallas is $11,
to Atlanta is $8, and to Denver is $14. How many units should be shipped from each plant to
each regional distribution center? What is the total cost for this?
6-5
Finnish Furniture manufactures tables in facilities located in three cities--Reno, Denver,
and Pittsburgh. The tables are then shipped to three retail stores located in Phoenix, Cleveland,
and Chicago. Management wishes to develop a distribution schedule that will meet the demands
at the lowest possible cost. The shipping cost per unit from each of the sources to each of the
destinations is shown in the following table:
From
Reno
Denver
Pittsburgh

Phoenix
10
12
18

To
Cleveland
16
14
8

Chicago
19
13
12

The available supplies are 130 units from Reno, 200 from Denver, and 160 from Pittsburgh.
Phoenix has a demand of 140 units, Cleveland has a demand of 160 units, and Chicago has a
demand of 200
units.
a. How many units should be shipped from each manufacturing facility to each of the retail
stores if cost is to be minimized?
b. What is the total cost?
6-6
The Gleaming Company has just developed a new dishwashing liquid and is preparing
for a national television promotional campaign. The firm has decided to schedule a series of
1-minute commercials during the peak homemaker audience viewing hours of 1pm to 5pm. To
reach the widest possible audience, Gleaming wants to schedule one commercial on each of four
networks and to have one commercial appear during each of the four 1-hour time blocks. The
exposure ratings for each hour, which represent the number of viewers per $1,000 spent, are
presented in the following table. Which network should be scheduled each hour to provide the
maximum audience exposure?

Viewing Hours
1-2pm
2-3pm
3-4pm
4-5pm

A
21.
2
18.
9
19.
2
11.5

B
18.
1
15.5

Network
C
11.3

Independent
9.5

17.1

12.3

18.5

10.9

9.7

19.4

16.8

12.8