# Examination _1 by chenmeixiu

VIEWS: 0 PAGES: 7

• pg 1
```									Problem 1

S&S Toy Company produces two dolls that are popular with young girls, the male Jack
doll and the female Jill doll. Both dolls are made from plastic and come with a variety of
outfits in standard packages.

   The Jack doll requires 5 ounces of plastic as compared to 3 ounces for the Jill
doll.
   The Jill doll uses 2 linear feet of cloth as compared to 1 linear foot of cloth of
clothes packaged with the Jack doll.
   There are 8000 linear feet of cloth for clothes available each week
   There are 9600 ounces of plastic each week available each week
   They have a total production time to make at most 3000 dolls
   The Jill doll is priced to net a unit profit of \$7 on each doll while each Jack doll
nets a profit of \$5.
   The goal is to maximize total profit

How many Jack Dolls should they manufacture per week? 0 JACK DOLL

How many Jill Dolls should they manufacture per week? 3000 JILL DOLL

What is the expected weekly profit? \$21,000

1 of 7
Problem 2

Howard Manufacturing produces five major appliances – stove, washers, electric dryers,
gas dryer, and refrigerators. All products go through three processes – molding/pressing,
assembly, and packaging.

   Each week there are 5600 minutes available for molding/pressing, 3000 available
for packaging, 1200 for stove assembly, 1200 for refrigerator assembly, and 2800
that can be used for assembling washers and dryers.

   The table below gives the gross unit net profit that includes the material and labor
as well as the molding/pressing, assembly and packaging times (in minutes)
required for the production of each type of appliance:

Molding/                                         Unit
Pressing        Assembly      Packaging         Profit
Stove                     5.5              4.5            4             \$110
Washer                    5.2              4.5           3.5             \$90
Electric Dryer            5.5               4            2.5             \$85
Gas Dryer                 5.1               3             2              \$80
Refrigerators             7.5               9             4             \$130

   Howard will like the number of washers to be equal the combined number of
dryers.
   The number of electric dryers should not exceed the number of gas dryers by
more than 100 per week

2 of 7
A Linear Programming model is formulated as follows:

DEFINE VARIABES

X1 = Number of stoves produced weekly
X2 = Number of washers produced weekly
X3 = Number of electric dryers produced weekly
X4 = Number of gas dryers produced weekly
X5 = Number of refrigerators produced weekly

OBJECTIVE FUNCTION
MAX 110X1 + 90X2 + 85X3 + 80X4 + 130X5

CONSTRAINTS
5.5X1 + 5.2X2 + 5.5X3 + 5.1X4 + 7.5X5         5600 (Molding/pressing)
4.5X1                                         1200 (Stove assembly)
4.5X2 + 4.0X3 + 3.0X4                 2800 (Washer/dryer assembly)
9.0X5         1200 (Refrigerator assembly)
4.0X1 + 3.5X2 + 2.5X3 + 2.0X4 + 4.0X5         3000 (Packaging)
X2 -    X3 -    X4                = 0 (Washers = Dryers)
X3 -    X4                 100 (E. Dryers  G. Dryers + 100)

NON-NEGATIVITY
All X's  0

3 of 7
A. Will you present the production schedule to the management even if the optimal
solution has fractional value? (for example number of stove = 123.45) 3 POINTS

Yes, the fractional number of appliance in optimal solution represents work in
progress.
B. Suppose the net profit of electric dryer is \$110, will you recommend produce
electric dryer? Why? 5 POINTS

Yes, when the profit of dryer is changed to \$110, it is out of optimality range,
resolving the problem reveals an optimal solution containing 221.88 e-dryers

C. Suppose the assembly time for washer and dryer costs you \$10 per minute now,
how much should Howard be willing to pay for an extra hour of washer and
dryer’s assembly time? Discuss for how many minutes your statement above is
valid. 6 POINTS

1. Howard should be willing to pay up to \$750 per hour (\$12.5 X 60)
2. For 510.5 minute (total of 3,310.5 minutes the statement should be true)

4 of 7
D. Suppose the molding/pressing time costs you \$12 per minute now, how much
should Howard be willing to pay for 100 extra minutes of molding/pressing time?
4 POINTS

No, Howard is not willing to pay for 100 extra minutes of molding/pressing time,
because shadow price = 0 and allowable increase = infinity

E. Would the optimal schedule changes if Howard underestimates the profit of stove
by \$10? Why or why not? 3 POINTS

No, optimal solution will not change if the profit of stove is underestimated by
\$10, because the allowable increase of objective function coefficient for stove is
\$13.63

5 of 7
Problem 3
Clancy’s Casino, in Muledeer, Nevada, is open 24 hours a day, seven days a week.
Along with all the other attractions and diversion, Clancy’s operates a variety of gaming
tables. Dealers at these tables are interchangeable. The casino has the following daily
requirements for dealers:

Time                               Minimum number
Block         From - To            of dealers needed
A          01:00 - 05:00                  7
B          05:00 - 09:00                  4
C          09:00 - 13:00                  9
D          13:00 - 17:00                 12
E          17:00 - 21:00                 15
F          21:00 - 01:00                 17

A dealer may start work at the beginning of any one of the six shifts and, having begun,
works eight consecutive hours.

Shift        Work Hours
1          01:00 - 09:00
2          05:00 - 13:00
3          09:00 - 17:00
4          13:00 - 21:00
5          17:00 - 01:00
6          21:00 - 05:00

Find the employee schedule that minimizes the total number of dealers required, meeting
the minimum level of each time block’s requirements.

6 of 7
LP Model

Variable Definition
Xi = the number of dealers beginning work at the start of shift i, where i = 1 through 6.

Objective Function

MIN     X1 + X 2 + X 3 + X4 + X5 + X 6

Constraints
X1 +                                X6        7    (Time block A)
X1 + X 2                                       4 (Time block B)
X2 + X 3                                9 (Time block C)
X3 + X 4                       12 (Time block D)
X4 + X 5                  15 (Time block E)
X5 + X 6         17 (Time block F)

X1, X2, X3, X4, X5, X6  0 and integer

Optimal Solution
Shift                       1         2          3            4     5          6       Total

Number of Dealers           4         0         9             3    12         5          33

7 of 7

```
To top