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.

Table 1                            Lot

Cars              1       2          3         4          5

Available        45        20     14       26       40                   

Desired         30        25      20       40      30       

Table 2                      To (in minutes)        

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?

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 be divided 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

• Three diamonds: $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?