1.
Assume you have been hired by the Tweed Co. to design a new shopping mall. The mall will have a maximum of 300,000
square feet of useable space for shops of four different sizes, (1) small specialty shops, (2) intermediate, (3) medium, and (4) large department
stores. The following represents the
square footage requirements for each of the four stores, 1500, 4000, 7500,
20000. Tweed estimates that the monthly
lease revenues for each of the four types of stores will yield approximately 350,
1,500, 4,000, 15,000 dollars. The
developer wants to have at least two, but not more than four large department
stores. In addition, the developer would
like to allocate at least one-fifth of the total
available space available to stores of intermediate size. The square footage for small specialty shops
should be equal to at least one-fifth of the amount of space allocated to
medium-size stores. Finally for local
tax incentives, the lease revenue from small specialty and medium-sized stores
should be at least twice as much as the lease revenues from large department
stores. Formulate the LP model to determine how many stores of each size should
be included in the mall in order to maximize annul lease revenue. (Assume
fractional values for decision variables are acceptable)

1a.
What is the maximum monthly lease revenue? __________________

1b.
How many of each of the type of stores should be constructed.

1c.
How much space will not be used
(left over)? ___________

1d.
What is the total revenue generated from the lease of specialty shops?

___________

2 Frugal Rent-A-Car has five store lots in the
Greater St. Louis Metropolitan area. At
the beginning of each day, they would like to have a predetermined number of
cars available at each lot. However,
since customers renting a car may return the car to any of the five lots, the
number of cars available at the end of the day does not always equal the
designated number of cars needed at the beginning of the day. Frugal would like to redistribute the cars in
the lots to meet the minimum demand (desired) and minimize the time needed to
move the cars.

Table
I below, summarizes the results at the end of one particular day.

Table
IIbelow summarizes the time
required to travel between the lots.

Solve
the problem in order to determine how many cars should be transported from one
lot to the next.

Cars 1 2 3
4 5

Available 45 20 14 26 40

Desired 30 25 20 40 30

From 1 2 3
4 5

1 — 12 17 18 10

2 14 — 10 19 16

3 14 10 — 12
8

4 8 16 14 — 12

5 11 21 16 18 —

How
many cars will be sent from and to each destination?

From
Lot# To Lot# Number of Cars

________ ________ _______

________ ________ _______

________ ________ _______

________ ________ _______

________ ________ _______

Total
minutes required to transport all cars? ________

3.
Inferior Tile produces square vinyl
floor tile is three sizes, small (8 X 8), medium (12X12) and large (16 X
16). The tile produced on three machines
that vary in terms of the width of the tile produced. Machine 1 produces tile that is 12 inches
wide, machine 2 produces tile that is 16 inches wide, and machine 3 produces
tile that is 24 inches wide. The small
tile can be produced on any one of the three machines, however there will be
waste when Machine 1 is used. That is,
the width of the tile will be 8 inches, leaving 4 inches of waste. There will be no waste on the other to
machines since 2 smalls (8 X 8) produced on machine 2 will 16 inches wide and
three tiles produced on machine 3 will be 24 inches wide. The same logic applies to medium and large
tiles. The forecasted demand for the
following production period is to produce 48,000 small, 84000 medium and 54000
large tile squares. The machines have a
limited capacity and the time required in minutes to cut each tile varies on
the machine as shown in the table below.
A machine can cut more than one tile at the same time if the size is the
same for all tiles cut. For example,
Machine 2 can cut 2 small tiles in .4 minutes.
However Machine 3 cannot be set to cut a 16 inch tile and 8 inch tile at
the same time.

Time
in minutes per one square tile.

Machine 1
Machine 2 Machine 3

Small .2 .4 .9

Medium
.3 .3 .6

Large N/A .4 .5

Time
available 200 hrs 350 hrs 400 hrs

Formulate
the LP model to determine how many tile should be cut on each machine and meet
the forecasted demand and minimize waste.

.

3a.
How many small tiles should be produced on each of the three machines?

3b.
What would be the total waste if 85,000 medium sized tile were required?

3c.
What would be the total waste if machine 1 was available 220 hours per week?

4. The SoHo Museum director must
decide how many guards should be employed to control a new wing containing 25
rooms (Rooms A-Z – no V – as shown in the diagram below). Previously, a guard
was stationed in each room. Budget cuts
have forced the director to station guards in a doorway, guarding two rooms at
once (doorways are indicated by gaps in the lines separating the rooms).
Formulate the integer LP model to determine which doorways the guards should be
stationed and minimize the number of guards required. {I counted 26 decision
variables}

4a.
How many guards will be needed?
______________

4b.
Is there evidence of more than one optimal solution? Explain

4c.
How many rooms will be monitored by more than one guard? _________

5. Mr. Smothers, a rich
aristocrat, passed away leaving the legacy (the 13 items listed below) to bedivided between his only two sons, Tom and Dick. Formulate the integer LP model to determine
which son gets each item so that the difference between the assessed value of
the inherited items left to the two sons is minimized. (That is minimized the difference in the
assessed value of items of left to Tom (recommend scaled in thousands of
dollars) and the assessed value of items left to Dick)

Legacy
(Items left): Assessed Value

•A Caillebotte
picture: $5,000

•A bust of
Diocletian: $4,500

•A Yuan dynasty
Chinese vase: $20,000

•A 911 Porsche:
$40,000

•A Louis XV sofa:
$3,000

•
A 1966 signed John Lennon guitar $10,000

•A sculpture
dated 200 A.D.: $11,000

•A sailing boat:
$15,000

•A Harley Davidson motorbike: $7,250

•
A piece of furniture that once belonged to Cavour: $12,500

•Threediamonds: $9,000 each (items may be
divided up)

5a. What is the total assessed value of items
left to Dick? ____________

5b.
How much difference will there be between assessed value of items left to Tom
and Dick? __

5c. Which items will be left to Tom?

5d.
Is this the only division of items that will result in the optimal solution?