From: ypay@leland.Stanford.EDU (daniel cory)
Subject: National Mathematics Contest in Modeling
Summary: applied math contest (long)
Organization: DSG, Stanford University, CA 94305, USA
Date: Tue, 7 Apr 92 20:06:24 GMT

Every year, the Consortium for Mathematics and its Applications runs the
National Mathematics Contest in Modeling. Teams from colleges around the world
work on an applied math problem for a four day weekend.

I've mentioned the contest on this group before and thought I would post this
year's questions. The judging has already taken place so I think it's okay to
post the questions. If you want to see the official winners and more details of
the contest, see the UMAP journal (I think the winter issues)

I've entered this contest 5 times and now help organize the Stanford team.

Here are the questions from this year. If you want last year's questions,
please email me or see the journal mentioned above. If you are interested in
entering the contest, write to the address at the bottom of the post.

-Dan Cory

Problem A: Power Problem for an Air Traffic Control Radar

You are to determine the power to be radiated by an air traffic control 
radar at a major metropolitan airport. The airport authority wants to 
minimize the power of the radar consistent with safety and cost.

The authority is constrained to operate with its existing antennae and 
receiver circuitry. The only option that they are considering is upgrading 
the transmitter circuits to make the radar more powerful.

The question that you are to answer is what power (in watts) must be 
released by the radar to ensure detection of standard passenger aircraft at 
a distance of 100 kilometers.

Technical specifications:
a. The radar antenna is a section of a paraboloid of revolution with focal 
length of 1 meter. Its projection onto a plane tangent to its vertex is an 
ellipse with a major axis of 6 meters and a minor axis of 2 meters. The main 
lobe energy beam pattern, located at the focus, is an elliptical cone that 
has a major axis of one radian and a minor axis of 50 milliradians. The 
antenna and beam are sketched in the figures provided [Sorry, I don't know a
good way to send these pictures].

b. The nominal class of aircraft is one that has an effective radar 
reflection cross-section of 75 square meters. For the purposes of this 
problem, this means that in your initial model, the aircraft is equivalent 
to a 100% reflective circular disc of 75 square meters, which is centered on 
the axis of the antennae and is perpendicular to it. You many want to 
consider alternatives or refinements to this initial model.

c. The receiver circuits are sufficiently sensitive to process a return 
signal of 10 microwatts at the feed horn of the radar (which is located at 
the focus of the radar antenna).

Problem B: an Emergency Power Restoration System

Power companies serving coastal regions must have emergency response systems 
for power outages due to storms. Such systems require the input of data that 
allow the time and cost required for restoration to be estimated and the 
'value' of the outage judged by objective criteria. In the past, 
Hypothetical Electric Company (HECO) has been criticized in the media for 
its lack of a prioritization scheme.

You are a consultant to HECO power company. HECO possesses a computerized 
database with real time access to service calls that currently require the 
following information:

Time of report;
Type of requestor;
Estimated number of people affected;
Location (x,y).

Crew sites are located at coordinates (0,0) and (40,40), where x and y are 
in miles. The regions serviced by HECO is within -65<x<65 and -50<y<50. The 
region is largely metropolitan with an excellent road network. Crews must 
return to dispatch site only at beginning and end of shift. Company policy 
requires that no work be initiated until the storm leaves the area, unless 
the facility is a commuter railroad or hospital, which may be processes 
immediately if crews are available.

HECO has hired you to develop the objective criteria and schedule the work 
for the storm restoration requirements listed in Table 1 using the work 
force described in Table 2. Note that the first call was received at 4:20 
a.m. and that the storm left the area at 6:00a.m. Also note that many 
outages were not reported until much later in the day.

HECO has asked for a technical report for their purposes and an 'executive 
summary' in laymen's terms that can be presented to the media. Further, they 
would like recommendations for the future. To determine your prioritized 
scheduling system, you have to make additional assumptions. Detail those 
assumptions. In the future, you may desire additional data. If so, detail 
the information desired.

Table 1. Storm Restoration Requirements
                                                             Estimated
                                                             Repair Time
Time                                                         (Hours for
(a.m) Location  Type                             # Affected  Crew)
------------------------------------------------------------------------
4:20  (-10,30)  Business (cable TV)              ?             6
5:30  (3,3)     Residential                      20            7
5:35  (20,5)    Business (hospital)              240           8
5:55  (-10,5)   Business (railroad)              **            5
------------------------------------------------------------------------
6:00  All clear given; storm leaves area; crews can be dispatched
------------------------------------------------------------------------
6:05  (3,30)    Residential                      45            2
6:06  (5,20)    Area *                           2000          7
6:08  (60,45)   Residential                      ?             9
6:09  (1,10)    Government (city hall)           ?             7
6:15  (5,20)    Business (shopping mall)         200 workers   5
6:20  (5,-25)   Gov't (fire department)          15 workers    3
6:20  (12,18)   Residential                      350           6
6:22  (7,10)    Area *                           400           12
6:25  (-1,19)   Industry (newspaper)             190           10
6:40  (-20,-19) Industry (factory)               395           7
6:55  (-1,30)   Area *                           ?             6
7:00  (-20,30)  Gov't (high school)              1200 students 3
7:00  (40,20)   Gov't (elementary school)        1700          ?
7:00  (7,-20)   Business (restaurant)            25            12
7:00  (8,-23)   Gov't (police station & jail)    125           7
7:05  (25,15)   Gov't (elementary school)        1900          5
7:10  (-10,-10) Residential                      ?             9
7:10  (-1,2)    Gov't (college)                  3000          8
7:10  (8,-25)   Industry (computer manufacturer) 450 workers   5
7:10  (18,55)   Residential                      350           10
7:20  (7,35)    Area *                           400           9
7:45  (20,0)    Residential                      800           5
7:50  (-6,30)   Business (hospital)              300           5
8:15  (10,40)   Business (several stores)        50            6
8:20  (15,-25)  Gov't (traffic lights)           ?             3
8:35  (-20,-35) Business (bank)                  20            5
8:50  (47,30)   Residential                      40            ?
9:50  (55,50)   Residential                      ?             12
10:30 (-18,-35) Residential                      10            10
10:30 (-1,50)   Business (civic center)          150           5
10:35 (-7,-8)   Business (airport)               350 workers   4
10:50 (5,-25)   Gov't (fire department)          15            5
11:30 (8,20)    Area *                           300           12
* Area signifies a combination of two or more of the other
  classfication types.
** 25 workers; 75000 commuters

Table 2. Crew Descriptions

* Dispatch locations at (0,0) and (40,40).
* Crews consist of three trained workpersons.
* Crews report to the dispatch location only at the beginning and end of 
their shifts.
* One crew is scheduled for duty at al times on jobs assigned to each 
dispatch location. These crews would normally be performing routine 
assignments. Until the 'storm leaves the area,' they can be dispatched for 
'emergencies' only.
* Crews work eight (8) hour shifts.
* There are six crew teams available at each location.
* Crews can work only one overtime shift in a work day and receive time-and-
a-half for overtime.

Rules:
Each school may enter up to two teams of three undergraduates.
Any inanimate reference sources may be used.
You may start at 12:01AM Friday morning and must mail the solution in on Monday

For more information about the National Mathematics Contest in Modeling, write:
COMAP, Inc.
The Consortium for Mathematics and its Applications
60 Lowell Street
Arlington, MA 02174

Newsgroups: rec.puzzles
Subject: National Mathematics Contest in Modeling
Summary: applied math contest (long)
Followup-To: 
Distribution: world
Organization: DSG, Stanford University, CA 94305, USA
Keywords: 

Every year, the Consortium for Mathematics and its Applications runs the
National Mathematics Contest in Modeling. Teams from colleges around the world
work on an applied math problem for a four day weekend.

I've mentioned the contest on this group before and thought I would post this
year's questions. The judging has already taken place so I think it's okay to
post the questions. If you want to see the official winners and more details of
the contest, see the UMAP journal (I think the winter issues)

I've entered this contest 5 times and now help organize the Stanford team.

Here are the questions from this year. If you want last year's questions,
please email me or see the journal mentioned above. If you are interested in
entering the contest, write to the address at the bottom of the post.

-Dan Cory

Problem A: Power Problem for an Air Traffic Control Radar

You are to determine the power to be radiated by an air traffic control 
radar at a major metropolitan airport. The airport authority wants to 
minimize the power of the radar consistent with safety and cost.

The authority is constrained to operate with its existing antennae and 
receiver circuitry. The only option that they are considering is upgrading 
the transmitter circuits to make the radar more powerful.

The question that you are to answer is what power (in watts) must be 
released by the radar to ensure detection of standard passenger aircraft at 
a distance of 100 kilometers.

Technical specifications:
a. The radar antenna is a section of a paraboloid of revolution with focal 
length of 1 meter. Its projection onto a plane tangent to its vertex is an 
ellipse with a major axis of 6 meters and a minor axis of 2 meters. The main 
lobe energy beam pattern, located at the focus, is an elliptical cone that 
has a major axis of one radian and a minor axis of 50 milliradians. The 
antenna and beam are sketched in the figures provided [Sorry, I don't know a
good way to send these pictures].

b. The nominal class of aircraft is one that has an effective radar 
reflection cross-section of 75 square meters. For the purposes of this 
problem, this means that in your initial model, the aircraft is equivalent 
to a 100% reflective circular disc of 75 square meters, which is centered on 
the axis of the antennae and is perpendicular to it. You many want to 
consider alternatives or refinements to this initial model.

c. The receiver circuits are sufficiently sensitive to process a return 
signal of 10 microwatts at the feed horn of the radar (which is located at 
the focus of the radar antenna).

Problem B: an Emergency Power Restoration System

Power companies serving coastal regions must have emergency response systems 
for power outages due to storms. Such systems require the input of data that 
allow the time and cost required for restoration to be estimated and the 
'value' of the outage judged by objective criteria. In the past, 
Hypothetical Electric Company (HECO) has been criticized in the media for 
its lack of a prioritization scheme.

You are a consultant to HECO power company. HECO possesses a computerized 
database with real time access to service calls that currently require the 
following information:

Time of report;
Type of requestor;
Estimated number of people affected;
Location (x,y).

Crew sites are located at coordinates (0,0) and (40,40), where x and y are 
in miles. The regions serviced by HECO is within -65<x<65 and -50<y<50. The 
region is largely metropolitan with an excellent road network. Crews must 
return to dispatch site only at beginning and end of shift. Company policy 
requires that no work be initiated until the storm leaves the area, unless 
the facility is a commuter railroad or hospital, which may be processes 
immediately if crews are available.

HECO has hired you to develop the objective criteria and schedule the work 
for the storm restoration requirements listed in Table 1 using the work 
force described in Table 2. Note that the first call was received at 4:20 
a.m. and that the storm left the area at 6:00a.m. Also note that many 
outages were not reported until much later in the day.

HECO has asked for a technical report for their purposes and an 'executive 
summary' in laymen's terms that can be presented to the media. Further, they 
would like recommendations for the future. To determine your prioritized 
scheduling system, you have to make additional assumptions. Detail those 
assumptions. In the future, you may desire additional data. If so, detail 
the information desired.

Table 1. Storm Restoration Requirements
                                                             Estimated
                                                             Repair Time
Time                                                         (Hours for
(a.m) Location  Type                             # Affected  Crew)
------------------------------------------------------------------------
4:20  (-10,30)  Business (cable TV)              ?             6
5:30  (3,3)     Residential                      20            7
5:35  (20,5)    Business (hospital)              240           8
5:55  (-10,5)   Business (railroad)              **            5
------------------------------------------------------------------------
6:00  All clear given; storm leaves area; crews can be dispatched
------------------------------------------------------------------------
6:05  (3,30)    Residential                      45            2
6:06  (5,20)    Area *                           2000          7
6:08  (60,45)   Residential                      ?             9
6:09  (1,10)    Government (city hall)           ?             7
6:15  (5,20)    Business (shopping mall)         200 workers   5
6:20  (5,-25)   Gov't (fire department)          15 workers    3
6:20  (12,18)   Residential                      350           6
6:22  (7,10)    Area *                           400           12
6:25  (-1,19)   Industry (newspaper)             190           10
6:40  (-20,-19) Industry (factory)               395           7
6:55  (-1,30)   Area *                           ?             6
7:00  (-20,30)  Gov't (high school)              1200 students 3
7:00  (40,20)   Gov't (elementary school)        1700          ?
7:00  (7,-20)   Business (restaurant)            25            12
7:00  (8,-23)   Gov't (police station & jail)    125           7
7:05  (25,15)   Gov't (elementary school)        1900          5
7:10  (-10,-10) Residential                      ?             9
7:10  (-1,2)    Gov't (college)                  3000          8
7:10  (8,-25)   Industry (computer manufacturer) 450 workers   5
7:10  (18,55)   Residential                      350           10
7:20  (7,35)    Area *                           400           9
7:45  (20,0)    Residential                      800           5
7:50  (-6,30)   Business (hospital)              300           5
8:15  (10,40)   Business (several stores)        50            6
8:20  (15,-25)  Gov't (traffic lights)           ?             3
8:35  (-20,-35) Business (bank)                  20            5
8:50  (47,30)   Residential                      40            ?
9:50  (55,50)   Residential                      ?             12
10:30 (-18,-35) Residential                      10            10
10:30 (-1,50)   Business (civic center)          150           5
10:35 (-7,-8)   Business (airport)               350 workers   4
10:50 (5,-25)   Gov't (fire department)          15            5
11:30 (8,20)    Area *                           300           12
* Area signifies a combination of two or more of the other
  classfication types.
** 25 workers; 75000 commuters

Table 2. Crew Descriptions

* Dispatch locations at (0,0) and (40,40).
* Crews consist of three trained workpersons.
* Crews report to the dispatch location only at the beginning and end of 
their shifts.
* One crew is scheduled for duty at al times on jobs assigned to each 
dispatch location. These crews would normally be performing routine 
assignments. Until the 'storm leaves the area,' they can be dispatched for 
'emergencies' only.
* Crews work eight (8) hour shifts.
* There are six crew teams available at each location.
* Crews can work only one overtime shift in a work day and receive time-and-
a-half for overtime.

Rules:
Each school may enter up to two teams of three undergraduates.
Any inanimate reference sources may be used.
You may start at 12:01AM Friday morning and must mail the solution in on Monday

For more information about the National Mathematics Contest in Modeling, write:
COMAP, Inc.
The Consortium for Mathematics and its Applications
60 Lowell Street
Arlington, MA 02174




-- 
Dan Cory
If you look up through a eucalyptus tree, you see very few leaves.
Newsgroups: rec.puzzles,soc.culture.indian