IE 232 Operations Research Homework_3 1. The Tanglewood Chair by wuxiangyu


									                                   IE 232 Operations Research
1. The Tanglewood Chair Manufacturing Co. Has four plants located around the
country. The fabrication and assembly cost per chair and the minimum and maximum
monthly production for each plant are shown below.
          Plant      Cost/Chair    Max.Prod.    Min.Prod.
            1           $5            500           0
            2            7            750          400
            3            3           1000          500
            4            4            250          250

The company obtains the twenty pounds of wood required to make each chair from two
suppliers who have agreed to supply any amount ordered. In return, the company
guarantees purchase of at least eight tons of wood per month from each supplier. The cost
of wood is $0.10/lb from Supplier 1 and $0.075/lb from Supplier 2. The shipping cost in
dollars per pound of wood from each supplier to each plant is shown below.
                        Plant 1    Plant 2     Plant 3   Plant 4
         Supplier 1      0.01       0.02        0.04      0.04
         Supplier 2      0.04       0.03        0.02      0.02

The chairs are sold in New York, Houston, San Francisco, and Chicago. Transportation
costs in dollars per chair between the plants and the cities are:
                      N.Y.        Hou.   San.Fran.       Chi.
         Plant 1       1           1         2            0
         Plant 2       3           6         7            3
         Plant 3       3           1         5            3
         Plant 4       8           2         1            4

The following table shows the minimum demand that must be satisfied, the maximum
demand, and the selling price for chairs in each city.
                        Selling Price    Max.Demand         Min.Demand
           N.Y.              $20            2000                500
           Hou.               15             400                100
         San.Fran.            20            1500                500
           Chi.               18            1500                500

Construct an LP that minimizes the total cost while satisfying all of the restrictions cited

                              IE 232 Operations Research
2. Date-Match, Inc., has decided to expand their capabilities to offer not only a
computerized dating service but also a similar service for persons interest in more
permanent attachments.
Using slightly modified personal data forms, the data processing department has come up
with the following “pairing indices” for a four-person group and a seven-person group
who have indicated an interest in this service. In order to avoid possible biases,
knowledge of the gender of either group has been withheld. All that we know for certain
is that one group consists wholly of men and the other group wholly of women.
                                                   Group 1
    Group 2         1         2          3            4       5         6        7
       1           35        95         87            0      72        63       10
       2           53        90         85           15      65        69       75
       3           91        93         73           44      75        73       68
       4           15        20         22           45      30        42       12

The pairing indices are bounded between the limits of 0 and 100 and are a weighted
measure of probable compatibility in married couples. Formulate an LP that maximizes
the compatibility of married couples.
3. The Toys-R-4-U Company has developed two new toys for possible inclusion in its
product line for the upcoming Christmas season. Setting up the production facilities to
begin production would cost $50000 for toy 1 and $80000 for toy 2. Once these costs are
covered, the toys would generate a unit profit of $10 for toy 1 and $15 for toy 2.
The company has two factories that are capable of producing these toys. However, to
avoid doubling the start-up costs, just one factory would be used, where the choice would
be based on maximizing profit. For administrative reasons, the same factory would be
used for both new toys if both are produced.
Toy 1 can be produced at the rate of 50 per hour in factory 1 and 40 per hour in factory 2.
Toy 2 can be produced at the rate of 40 per hour in factory 1 and 25 per hour in factory 2.
Factories 1 and 2, respectively, have 500 hours and 700 hours of production time
available before Christmas that could be used to produce these toys.
It is not known whether these two toys would be continued after Christmas. Therefore,
the problem is to determine how many units (if any) of each new toy should be produced
before Christmas to maximize the total profit.

                               IE 232 Operations Research

4. You are assigned to arrange the songs on the cassette version of Madonna’s latest
album. A cassette tape has two sides (1 and 2). The songs on each side of the cassette
must total between 14 and 16 minutes in length. The length and type of each song is
given below in the table. Define your decision variable(s) and write down one constraint
to satisfy the condition given in each part (each part is independent).
       Song   Type                Length(min.)
         1    Ballad                   4
         2    Hit                      5
         3    Ballad                   3
         4    Hit                      2
         5    Ballad                   4
         6    Hit                      5
         7                             5
         8    Ballad and Hit           4

   a. Each side must have exactly two ballads
   b. Side 1 must have at least three hit songs.
   c. Either song 5 or song 6 must be side 1 but not both.
   d. If songs 2 and 4 are on side 1, then song 5 must be on side 2.
   e. If song 1 is on side 1, then song 8 must be on side 1 too (converse is not true).

5. When an accounting firm audits a corporation, the first phase of the audit involves
obtaining “knowledge of the business.” This phase of the audit requires the activities in
Table below. Draw the project network and determine the critical path for the network.
Also set up the LP that can be used to find the project’s critical path.
       Activity   Immediate Predecessors            Duration (days)
          A                     -                           3
          B                    A                            6
          C                    A                           14
          D                    C                            8
          E                    D                            4
          F                   B, E                          8
          G                     F                           9


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