A Bonus Presentation on Location

A problem in location

Big Burger, a fast food chain, has 20 stores in the Southwest Ohio area located at the following coordinates where the units are in miles. Big Burger is considering constructing a receiving and distribution center for this region. Where to locate the distribution center has preoccupied top management for some time.

On the average, a delivery truck must stop at eachBig Burger every 1000 customers.

Facility customer x-coord y-coord1 501 0 02 226 100 303 92 80 1144 228 92 175 289 86 906 541 18 217 484 60 108 124 50 309 231 55 7010 431 10 211 208 117 4012 213 76 8413 154 30 6314 456 6 9015 88 95 10916 241 112 8417 377 50 6218 319 121 10119 233 119 9320 185 92 70

Customers per day/7-day week

Solve the following problems:

Criteria: Minimize

The weighted squared distance

Sum of the Euclidean distances

Weighted sum of the Euclidean distances

Maximum Euclidean distance

Sum of rectilinear distances

Weighted sum of the rectilinear distances

Maximum rectilinear distance

A Big Burger chef

The Solutions…      Euclidean   Rectilinear  

Criterion: Minimize x* y* DistanceWeighted Distance

Max distance Distance

Weighted Distance

Max distance

The weighted squared distance 57.50 50.00 1025.32 2068.18 81.44 1381.00 2801.76 114.50

Sum of the Euclidean distances 71.39 64.61 981.26 2153.77 96.29 1298.21 2883.26 136.00

Weighted sum of the Euclidean distances 55.90 53.31

1020.31 2065.24 80.70 1374.16 2798.53 112.79

Maximum Euclidean distance 60.06 51.03 1015.24 2071.14 78.81 1366.76 2801.33 111.09

Sum of rectilinear distances 78.37 69.94 988.39 2238.94 105.04 1289.00 2972.64 148.31

Weighted sum of the rectilinear distances

57.89 62.00 1000.22 2086.50 84.82 1331.45 2790.45 119.89

Maximum rectilinear distance 55.50 55.50 1016.32 2066.29 79.75 1367.00 2796.44 111.00

A problem in assignment…

The University must locate 5 new student activities. There is space in each of the following campus buildings for one new activity. Which activity should be located in which building?

New ActivitesHonors & Scholars Program

Computer Lab

Starbucks Franchise

Learning Center

Student Book Exchange

Available building space

Student Union

Kettering Labs

Jesse Philips Center

Ebeling Hall

Chaminade Hall

Sherman Hall

Some Data

Available building space

DormitoryStudent Union

Kettering Labs

Jesse Philips Center

Ebeling Hall

Chaminade Hall

Sherman Hall

number students

Founder's Hall 1 0.5 1 4 2 2.5 400Marycrest 5 6 6 8 4 3.5 860

Virginia Kettering 4 3 4 6 3.5 3 630Marianist Hall 3 2 2.5 3 5 5.5 770

Garden Apartments 3 4 4 3 6 6.5 530Campus South 4 5 3.5 4 6 6.5 160Stuart Complex 6 7 7 9 4 3.5 660

Distances in tenth's of a mile 4010

Observation: students normally leave their dorms to attend these activitiesand afterward return to their dorms.

Some More Data

Yet another observation: the fraction of students that make use of eachactivity each day has remained constant over time.

New Activites

Dormitory

Honors & Scholars Program

Computer Lab

Starbucks Franchise

Learning Center

Student Book Exchange

number students

Founder's Hall 40 240 280 120 160 400Marycrest 86 516 602 258 344 860

Virginia Kettering 63 378 441 189 252 630Marianist Hall 77 462 539 231 308 770

Garden Apartments 53 318 371 159 212 530Campus South 16 96 112 48 64 160Stuart Complex 66 396 462 198 264 660

Fraction of students 0.1 0.6 0.7 0.3 0.4 4010Students per Day

Let’s do some matrix multiplication…

Facility / Activity

Honors & Scholars Program

Computer Lab

Starbucks Franchise

Learning Center

Student Book Exchange

Student Union 1572 9432 11004 4716 6288Kettering Labs 1633 9798 11431 4899 6532

Jesse Philips Center 1730.5 10383 12113.5 5191.5 6922Ebeling Hall 2274 13644 15918 6822 9096

Chaminade Hall 1707.5 10245 11952.5 5122.5 6830Sherman Hall 1693 10158 11851 5079 6772

Student tenth’s of a mile per day

At B =

Let Solver do the work…

The Answer

Facility / Activity

Honors & Scholars Program

Computer Lab

Starbucks Franchise

Learning Center

Student Book Exchange

Student Union 1572 9432 11004 4716 6288Kettering Labs 1633 9798 11431 4899 6532Jesse Philips 1730.5 10383 12113.5 5191.5 6922Ebeling Hall 2274 13644 15918 6822 9096

Chaminade Hall 1707.5 10245 11952.5 5122.5 6830Sherman Hall 1693 10158 11851 5079 6772changing cells: sumStudent Union 0 0 1 0 0 1Kettering Labs 0 1 0 0 0 1Jesse Philips 1 0 0 0 0 1Ebeling Hall 0 0 0 0 0 0

Chaminade Hall 0 0 0 1 0 1Sherman Hall 0 0 0 0 1 1

sum 1 1 1 1 1 34427

A problem in campus layout

The University of Dayton, in its continuing efforts to make academic life as good as possible for the students, has decided to relocate its schools and departments and their respective classrooms. The objective is to minimize the total distances the students have to walk to get to their next class. By reducing travel time to the next class, classes can be extended from 50 to 55 minutes thus increasing by 10 percent the amount of learning that takes place.

The Data I

From the computerized registration system, a bright computer science major has compiled the following data based upon the last 10 semester’s of undergraduate student class schedules.

Students trips per day

DepartmentMath & Science

Social Sciences Humanities Engineering Business Arts

Math & Science 340 278 126 85 274Social Sciences 241 267 321 94 165

Humanities 113 214 253 276 322Engineering 278 301 278 24 133

Business 187 98 262 10 294Arts 56 87 387 247 264

The Data II

A bright civil engineering student, through direct measurements, obtained the following data:

Distances in tenth's of a mile

BuildingMariam

HallKettering

LabsJesse Philips

Center Ebeling HallChaminade

HallSherman

HallMariam Hall 3 2 4 3 3.5

Kettering Labs 3 1 6 4 4.5Jesse Philips Center 2 1 3 3.5 4

Ebeling Hall 4 6 3 5 5.5Chaminade Hall 3 4 3.5 5 0.5Sherman Hall 3.5 4.5 4 5.5 0.5

Which department/school should be placed in which building on campus?

Discrete Location

- A Meeting Place The Traveler Corporation has its home office in New

York and branch offices 15 other cities. The Company has monthly sales conferences in one

of these cities in which the entire sales force is brought together.

Currently, each branch office has taken turns in hosting these meetings.

However, the expense of these meetings has raised concern with top management.

The problem is to determine which city should be hosting these conferences in order to minimize the total travel expense to the company.

The Expenses

The primary expenses are the travel fares and the per diem that consists of an allowance for meals, lodging, and other incidental expenses (ML&IE). (GSA rates apply)

Unless the sales representative lives within 200 miles of the hosting city, the individual will fly at the cheapest commercial rate available. If they live within 200 miles, then it is assumed that the individual

drives and is reimbursed at the rate of 34 cents a mile. Flyers are assumed to take ¾ a day to reach their destination

and ¾ a day to return in which case they receive a corresponding partial per diem.

For per diem purposes, drivers are allowed ½ a day each for travel to and from the conference. Conferences normally last 4 days at the full per diem.

Individuals living in the host city receive no travel compensation.

Some DataDriving Miles Baltimore,

MDCinn-

cinnati, India-

napolis, IN Minnea-Philadel-phia, Pa

Air Fare polis, MNAtlanta,

GA 0 677 715 470 1405 800 535 805 2185 1135 665 865 774 2495 2785 639Baltimore,

MD 413 0 701 512 1673 1446 587 1072 2654 1110 1092 192 103 2819 2766 39Chicago,

IL 436 428 0 295 1000 1085 180 525 2020 410 1380 795 758 2135 2070 699Cinn-

cinnati, OH 287 312 180 0 1196 1064 114 595 2177 704 1131 631 567 2388 2360 508

Denver, CO 590 703 420 502 0 1120 1087 600 1025 915 2065 1780 1728 1270 1335 1671

Houston, TX 488 607 456 447 470 0 1021 795 1550 1230 1190 1635 1544 1930 2450 1408

India-napolis, IN 326 358 180 180 457 429 0 486 2068 590 1196 707 644 2277 2246 584

Kansas City, MO 491 450 320 363 366 485 296 0 1625 440 1470 1195 1129 1865 1900 1069

Los Angeles,

CA 765 929 707 762 431 651 724 683 0 1935 2740 2800 2711 385 1140 2651Minneapol

is, MN 477 466 250 429 558 517 360 268 813 0 1795 1200 1167 2010 2015 1107

Miami, FL 406 459 580 475 723 500 502 617 959 754 0 1280 1187 3115 3365 1052New York,

NY 528 180 485 385 748 687 431 502 980 504 538 0 93 3055 2860 228Philadel-phia, Pa 472 180 462 346 726 648 393 474 949 490 499 180 0 2886 2823 138

San Francisco,

CA 873 987 747 836 533 811 797 783 235 704 1090 1069 1010 0 810 2817Seattle,

WA 975 968 725 826 561 858 786 798 479 705 1178 1001 988 494 0 2763Washing-

ton, DC 390 180 426 310 702 591 356 449 928 465 442 180 180 986 967 0per diem

131 152 201 115 154 115 112 123 145 141 131 244 164 205 189 196

attendees 8 7 10 6 5 4 5 4 7 8 5 7 12 6 4 5

Los Angeles,

CA Miami, FLNew York,

NYWashing-

ton, DC

San Francis-

co, CASeattle,

WAKansas City, KSAtlanta, GA

Chicago, IL

Denver, CO

Houston, TX

The Alternatives Evaluated

Location costAtlanta, GA $117,560

Baltimore, MD $121,624Chicago, IL $143,991

Cinn-cinnati, OH $101,340Denver, CO $139,475

Houston, TX $119,475Indianapolis, IN $99,360Kansas City, MO $114,038Los Angeles, CA $149,434Minneapolis, MN $121,792

Miami, FL $132,257New York, NY $175,115

Philadelphia, Pa $125,555San Francisco, CA $188,010

Seattle, WA $184,588Washing-ton, DC $147,254

Global Airlines

Global Airlines operates a major hub for their North American operations in Hazard (pop 5400), Kentucky. They operate out of Terminal 2 that has 11 gates equally distant apart in a linear configuration as shown below.

TERMINAL BUILDING 2

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11

Each morning and each afternoon, there are 11 flights arriving at the terminal. Data has been obtained on the average number of passengers arriving on the morning flights that transfer between each pair of flights[1].

[1] This data was obtained from an extensive database containing over a years worth of passenger flight data using data mining techniques.

The DataFlight A B C D E F G H I J K

A 10 5 6 38 9 12 20 16 15 12

B 31 8 7 9 4 5 14 18 44

C 21 37 14 18 19 21 22 9

D 12 17 5 4 32 18 19

E 8 6 5 8 6 2

F 8 7 8 19 9

G 15 7 31 19

H 34 41 12

I 21 18

J 24

Assuming each gate is the same distant from its neighboring gate(s), assign each flight to a gate in order to minimize the total distance all transfer passengers must travel to arrive at their connecting flights.

A problem in permutations

Gate 1 2 3 4 5 6 7 8 9 10 11

flight K B I H G J C E D F A

An excellent problem for the student to solve. There are

only 11! = 39,916,800 alternatives to evaluate. It is suggested that the student

start evaluating at the earliest.

Total Distance = 3020 passenger units

Global Airlines Revisited

Global Air operates a major hub at its Kentucky facility. An objective of Global Air is to allow as many passengers that are

transferring to another flight to remain on the same aircraft. This minimizes baggage handling and passenger boarding while

maximizing passenger convenience. Thru-flight connections must involve flights requiring the same aircraft

type and have sufficient arrival and departure times to complete servicing.

A minimum of 30 minutes is required between the arrival of an aircraft and its departure.

Determine which arrival and departure flights should share the same aircraft.

The Data

The (historical) mean number of passengers that arrive on a particular flight and transfer to another departing flight.

FllightsGB40

47GB50

2GB20

2GB483 GB76

1GB123

1GB10

26GB12

7GB852 GB74

3GB58

0GB54

14GB53

54GB12

1arrival time

GB247 5.4 32.4     7.2   18 0     27 12.6 7.2   1:45 PM

GB561 5.7 0 5.7 6.84 7.98 2.28 10.2612.54 13.68 8.55 7.41 5.7 17.1 10.26 9:30 AM

GB120 24.8 7.44 12.4 17.36 0 0 34.7252.08 0 42.16 44.64 0 12.4 0 7:45 AM

GB404 30.2 0 0   0   70 5.25 43.75 0 0 35 12.25   10:00 AM

GB782 20.1 0 999   18.09   18.09 0     30.1524.1214.07   1:15 PM

GB123 14.64 9.15 16.47 1.83 3.66 0 0 18.3 27.45 32.94 27.4521.96 0 9.15 10:00 AM

GB201 11 0 13.2 0 9.9 6.6 0 0 24.2 12.1 5.5 0 11 16.5 8:30 PM

GB220 8.13 13.55 27.1 0 27.1 21.68 27.1 0 10.84 46.07 43.3618.97 8.13   9:40 PM

GB310 29.7 0 4.95 0 0 8.25 0 16.5 24.75 4.95 21.4524.75 0 29.7 8:50 AM

GB321 0 6.15 6.15 2.46 12.3 11.07 14.76 24.6 0 27.06 0 12.3 2.46 3.69 8:40 AM

GB300 25.6 19.2 22.4 0 0 3.2 0 32 67.2 54.4 80 0 16   9:20 AM

GB341 7.12 12.46 0 0 16.02 3.56 16.02 35.6 53.4 10.68 12.46 3.56 3.56 3.56 7:35 AM

GB353 10.15 4.06 7   16.24 5.4 12.18 0 0 56.84 56.84 0 0   10:00 AM

GB360 0 19.2 0 0 0 30.72 0 0 17.28 0 20.16 8.64 0 0 8:00 AM

GB372 5.4 11.1 15.1   15.84   0 0 10.08 3.84 3.36 2.4 1.92   10:45 PM

GB380 16.38 7.28 54.6 3.64 9.1 61.88 5.46 3.64 1.82 1.82 3.64 3.64 3.64 5.46 7:50 PM

dep time

3:29 PM

2:19 PM

1:42 PM

10:20 AM

3:11 PM

10:14 AM

2:38 PM

5:21 PM

11:24 AM

12:57 PM

2:00 PM

4:30 PM

2:51 PM

9:55 AM  

The Factory Floor

The Makeit Company assembles nine different products in the same plant. Each product is assembled from 13 different types of raw material and vendor supplied parts that are shipped to the company and stored within the plant.

There are 11 potential sites on the factory floor available for the assembly of each product.

A significant non-value added production cost is the material handling requirements for moving the raw material to the assembly sites.

All movement on the factory floor is by necessity rectilinear. The problem is to determine which products should be assembled at

which sites in order to minimize the total number of forklift feet required to support a single production shift.

The Data

Site 1 2 3 4 5 6 7 8 9 10 11X 6 10 15 18 20 30 40 55 62 70 86Y 3 5 6 10 12 15 25 34 40 55 71

raw material a b c d e f g h i j k l m

X 8 8 6 6 6 9 9 9 11 11 12 12 14Y 3 5 8 11 13 16 19 22 30 33 36 40 45

X,Y coordinates of each supplier bin

X,Y coordinates of assembly sites

One unit = one foot

900,0

75

Site 9

k

More Data

forklift trips per shift

  raw material 

product a b c d e f g h i j k l mz1 5 4 7 3 2 0 8 5 2 9 0 5 0z2 8 4 3 3 4 4 5 6 5 2 4 3 2z3 10 9 9 0 8 0 0 0 0 5 4 3 4z4 17 18 7 0 8 0 7 6 6 7 5 4 2z5 12 9 19 20 23 9 8 10 8 0 0 0 15z6 12 17 9 7 6 6 14 14 21 4 17 8 9z7 0 0 0 6 5 4 3 0 5 0 7 6 21z8 16 0 13 0 18 13 0 16 21 23 0 7 8z9 20 0 22 18 0 0 18 0 21 6 14 8 4

An Intermediate result

Distances in feetSite 1 2 3 4 5 6 7 8 9 10 11raw

materiala 2 4 10 17 21 34 54 78 91 114 146b 4 2 8 15 19 32 52 76 89 112 144c 5 7 11 14 18 31 51 75 88 111 143d 8 10 14 13 15 28 48 72 85 108 140e 10 12 16 15 15 26 46 70 83 106 138f 16 12 16 15 15 22 40 64 77 100 132g 19 15 19 18 18 25 37 61 74 97 129h 22 18 22 21 21 28 34 58 71 94 126i 32 26 28 27 27 34 34 48 61 84 116j 35 29 31 30 30 37 37 45 58 81 113k 39 33 33 32 32 39 39 45 54 77 109l 43 37 37 36 36 43 43 49 50 73 105

m 50 44 40 39 39 46 46 52 53 66 98

More Intermediate results

Site z1 z2 z3 z4 z5 z6 z7 z8 z91 961 1017 841 1390 2067 3192 1960 3015 26082 839 889 781 1222 1951 2736 1720 2639 23383 1015 1067 957 1562 2389 3148 1718 3031 27284 1065 1122 1093 1783 2500 3272 1661 3076 28455 1135 1188 1205 1951 2700 3438 1673 3192 30496 1607 1683 1769 2844 4083 4740 2128 4383 43267 2153 2291 2489 3964 6061 6120 2456 5653 57428 3099 3319 3459 5682 8903 8690 3162 8045 81129 3689 3932 4035 6721 10452 10290 3551 9620 9615

10 4839 5131 5191 8702 13361 13512 4652 12645 1258811 6439 6827 6855 11486 17617 18120 6476 16965 16780

product

Feet per shift = distance in feet x trips per shift

A Solver Solution

site z1 z2 z3 z4 z5 z6 z7 z8 z9 sum1 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 1 0 0 0 13 0 0 0 1 0 0 0 0 0 14 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 0 1 0 16 0 0 1 0 0 0 0 0 0 17 0 1 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 1 0 0 1

10 0 0 0 0 0 0 0 0 0 011 0 0 0 0 0 0 0 0 0 0

sum =1 =1 =1 =1 =1 =1 =1 =1 =1cost = 23112

product

Mall madness – an exercise in discrete location

House Depot, an extravagant home and garden center, desires to open several retail stores in the Miami Valley area.

Generally, they locate their stores near existing malls to take advantage of the additional number of people from outside the area that a mall will attract.

Zone City (potential sites ) Number of residential homes/apt

Median Household

Income

Site LocationPrimary zip code

1 Dayton 49257 $24,930 Arcade Square 454042 Kettering/Oakwood 19279 $43,926 Town & Country 454293 Vandalia/Huber Heights 15002 $48,520 North Park Mall 454244 Centerville/Wash. Township/Bellbrook 15488 $45,424 Cross Pointe 454585 Miamisburg/West Carrollton 9416 $40,125 Dayton Mall 453426 Fairborn/Beavercreek 21536 $41,083 Fairfield Commons 453247 Xenia/Yellow Spring 8758 $37,456 Westpark Square 453858 Springfield/New Carlisle 21096 $27,136 Upper Valley mall 454049 Lebanon/Franklin 7490 $40,534 Lebanon Mall 45036

10 Trotwood/Englewood 11716 $32,245 Salem Mall 45416

Mall madness

House Depot is a firm believer in the following retail-gravity model:

where Sij = annual sales in zone j made by residents of zone i

Pi = number of residential homes in zone i

ei = median household income

cij = cost of travel from zone i to zone j (usually proportional to the distance)

Wj = size (attractiveness) of the House Depot in zone j, measured in square feet

K = constant of proportionality determined empirically and found to be equal to 0.000000011 (1.1 x 10-8).

( )i i jij

ij

e P WS K

c

The Model

The gravity model is based upon several “reasonable” assumptions that include: (1) The flow of cash from zone i to j is proportional to the total amount of cash available for shopping in zone i; (2) The flow of cash from i to j is proportional to the “attractiveness” (i.e. size) of the store in zone j; and (3) The flow of cash from i to j is inversely proportional to the travel cost.

Reference: Wilson, A. G., Urban and Regional Models in Geography and Planning, Wiley, New York, 1974.

( )i i jij

ij

e P WS K

c

Data Collection begins…

zone 1 2 3 4 5 6 7 8 9 101 4 8.5 5 17.5 14.9 8.2 16.7 29.6 31.8 10.72 8.5 3 13 5.2 13.4 15.8 17.5 32.5 17.5 13.73 5 13 5 23.3 18.8 10.8 21.2 18.5 36.8 15.74 17.5 5.2 23.3 5 13 18.3 20 35 12.4 22.75 14.9 13.4 18.8 13 5 21.3 26.5 38.7 19.1 16.26 8.2 15.8 10.8 18.3 21.3 4.5 17.3 15.9 30.6 15.57 16.7 17.5 21.2 20 26.5 17.3 6 20.5 23.9 23.98 29.6 32.5 18.5 35 38.7 15.9 20.5 5.5 47.3 33.59 31.8 17.5 36.8 12.4 19.1 30.6 23.9 47.3 3 35.9

10 10.7 13.7 15.7 22.7 16.2 15.5 23.9 33.5 35.9 3

Road Distances in miles

Other considerations

Travel cost is estimated to be $ .40 per mile. Construction of a new store will incur a fixed cost (primary the cost

of the land) and a variable cost per square foot. For the Miami Valley region, the company has allocated a

construction budget of $5,000,000 and a total floor space capacity of 80,000 square feet.

By company policy, no more than 5 stores can be located in the same region and no two stores can be within ten miles of one another.

The problem: From the data, determine at which locations House Depot should open stores in the Miami Valley region and determine the size in square feet of each store in order to maximize total annual sales.

More data collection

Zone Site Location fixed cost

Variable cost per sq

ftsq feet

restriction1 Arcade Square $200,000 $63 11,0002 Town & Country $130,000 $58 10,0003 North Park Mall $110,000 $72 18,0004 Cross Pointe $95,000 $64 22,0005 Dayton Mall $105,000 $68 16,0006 Fairfield Commons $150,000 $55 14,0007 Westpark Square $92,000 $67 21,0008 Upper Valley mall $87,000 $54 17,0009 Lebanon Mall $86,000 $63 16,000

10 Salem Mall $125,000 $70 9,000

The Model

W1 = floor space of store in zone i

Yi = binary, 1 if store in zone i; 0 otherwise

MAX

22.26115W1+21.54661W2+18.34966W3+15.29732W4+11.60323W5+16.85215W6 +10.11839W7+9.079918W8+9.618649W9+13.73721W10

SUBJECT TO

W1+W2+W3+W4+W5+W6+W7+W8+W9+W10 <= 80,000 (total square feet)

W1-10000Y1 <= 0 W2-12000Y2 <= 0

W3-20000Y3 <= 0 W4-20000Y4 <= 0

W5-20000Y5 <= 0 W6-20000Y6 <= 0

W7-20000Y7 <= 0 W8-15000Y8 <= 0

W9-15000Y9 <= 0 W10-10000Y10 <=0

( )i i jij

ij

e P WS K

c

More Constraints

no more than 5 stores

Y1+Y2+Y3+Y4+Y5+Y6+Y7+Y8+Y9+Y10 <= 5

180000Y1+55W1+120000Y2+62W2+95000Y3+67W3+125000Y4+70W4+98000Y5+64W5+114000Y6+64W6+88000Y7+58W7+91000Y8+55W8+97000Y9+60W9+102000Y10+57W10 <=5,000,000 (construction budget)

Stores within 10 miles

Y1+Y2 <= 1

Y1+Y3 <= 1

Y2+Y4 <= 1

Y6+Y1 <= 1

The Solution

Objective function value = 1197326.

VARIABLE VALUE

Y1 0.000000

Y2 1.000000

Y3 1.000000

Y4 0.000000

Y5 1.000000

Y6 1.000000

Y7 0.000000

Y8 0.000000

Y9 0.000000

Y10 1.000000

W1 0.000000

W2 12000.0

W3 20000.0

W4 0.000000

W5 8390.625

W6 20000.0

W7 0.000000

W8 0.000000

W9 0.000000

W10 10000.0

Zone Site Location

1 Arcade Square

2 Town & Country

3 North Park Mall

4 Cross Pointe

5 Dayton Mall

6 Fairfield Commons

7 Westpark Square

8 Upper Valley mall

9 Lebanon Mall

10 Salem Mall

Gridlock County Gridlock is an agricultural county that has a population of 230,292 distributed throughout the

county as shown (Population distribution in hundreds. Each grid represents a square mile.)

1 0 1.2 1.8 1.6 2.3 3 2 2.1 1.1 1.8 0.9 0 0 0 0

2 2.1 51.6 28.7 0 0 4.3 3.3 1.2 2.8 31.8 44.3 78 64 57.4 0

3 3.4 36 45.2 90 78 3.4 2.1 1.9 3.4 3.3 31.2 42.8 110 86.2 0.7

4 5 7.1 8.1 83.9 77.2 6.4 5.4 0 0 2.3 51.0 65.6 73.8 43.2 3.2

5 5 6.1 9.4 9.2 8.6 7.1 6.3 4.2 3.2 1.8 1.9 2.1 27.9 31.7 3

6 3.2 5.5 11 12 7.6 6.5 3.4 3.5 6.2 2.1 3.4 4.5 37.1 23.4 1.2

7 1.2 1.7 1.9 2.1 2.4 2.7 3.2 1.8 1.5 0.8 0.7 0.5 0.1 0.07 0

8 0.08 0.07 1.2 1.8 2.6 2.8 3.1 3.2 2.7 1.8 0.1 0.4 0.05 0.06 0.02

9 0.07 0.09 0.16 0.34 0.45 3.2 11 12 18 11 9.3 0.1 0.2 0.3 0.1

10 1.2 0.9 0.78 0.9 10 9.1 12 14 13 9.8 6.5 1.6 2.3 1.2 0.9

11 2.3 1.2 1.2 1.9 7.6 6.7 8.9 10 9.8 7.6 5.7 3.4 4.2 3.4 1.1

12 3.2 4.3 6.5 2.4 8.7 5.6 4.5 2.3 4.5 8.6 3.4 5.4 3.2 3.2 2.3

13 0 0 5.6 2.3 4.5 7.8 9.1 5.6 3.4 4.7 2.1 2.2 1.8 1.9 2.1

14 0 1.2 4.5 3.3 5.6 9.8 5.4 2.3 2.1 3.4 1.9 1 0.9 0.5 0.7

15 0.1 0.9 3.4 4.5 7.6 11 6.8 3.4 1.2 4.3 2.3 0.9 0.6 0.4 0.3

16 1.2 2.3 4.3 6.7 9.2 9 5.8 1.2 0.9 0.8 0.8 0.75 0.34 0.12 0.08

17 4.5 6.8 8.9 9.5 9.8 11 8.7 5.6 1.2 0.45 0.34 0.65 0.76 0.23 0.12

18 6.7 9.3 11 14 15 12 10 8.6 7.8 2.3 1.1 0.08 0.07 0 0.09

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

The Problem

A countywide program to inoculate the entire population against the dreaded N1H1 virus is to begin shortly. The county health officials want to conduct the immunizations in the most convenient location within the county. Convenience is defined to be that location that will minimize the total miles traveled by all the residents of Gridlock County.

 (a) Approximating travel within the County as straight-line from one point to another, determine the most convenient location (grid) for conducting the immunizations.

 (b) Approximating travel within the County as rectilinear (horizontal and vertical travel only) from one point to another, determine the most convenient location (grid) for conducting the immunizations.

Assume the entire population within each grid is located at the center of the grid.

The solution

1 2 3 4 5 6 7 8 9 10 11 12 13 14 151 0 1.2 1.8 1.6 2.3 3 2 2.1 1.1 1.8 0.9 0 0 0 02 2.1 51.6 28.7 0 0 4.3 3.3 1.2 2.8 31.8 44.3 78 64 57.4 03 3.4 36 45.2 90 78 3.4 2.1 1.9 3.4 3.3 31.2 42.8 110 86.2 0.74 5 7.1 8.1 83.9 77.2 6.4 5.4 0 0 2.3 51 65.6 73.8 43.2 3.25 5 6.1 9.4 9.2 8.6 7.1 6.3 4.2 3.2 1.8 1.9 2.1 27.9 31.7 36 3.2 5.5 11 12 7.6 6.5 3.4 3.5 6.2 2.1 3.4 4.5 37.1 23.4 1.27 1.2 1.7 1.9 2.1 2.4 2.7 3.2 1.8 1.5 0.8 0.7 0.5 0.1 0.07 08 0.08 0.07 1.2 1.8 2.6 2.8 3.1 3.2 2.7 1.8 0.1 0.4 0.05 0.06 0.029 0.07 0.09 0.16 0.34 0.45 3.2 11 12 18 11 9.3 0.1 0.2 0.3 0.1

10 1.2 0.9 0.78 0.9 10 9.1 12 14 13 9.8 6.5 1.6 2.3 1.2 0.911 2.3 1.2 1.2 1.9 7.6 6.7 8.9 10 9.8 7.6 5.7 3.4 4.2 3.4 1.112 3.2 4.3 6.5 2.4 8.7 5.6 4.5 2.3 4.5 8.6 3.4 5.4 3.2 3.2 2.313 0 0 5.6 2.3 4.5 7.8 9.1 5.6 3.4 4.7 2.1 2.2 1.8 1.9 2.114 0 1.2 4.5 3.3 5.6 9.8 5.4 2.3 2.1 3.4 1.9 1 0.9 0.5 0.715 0.1 0.9 3.4 4.5 7.6 11 6.8 3.4 1.2 4.3 2.3 0.9 0.6 0.4 0.316 1.2 2.3 4.3 6.7 9.2 9 5.8 1.2 0.9 0.8 0.8 0.75 0.34 0.12 0.0817 4.5 6.8 8.9 9.5 9.8 11 8.7 5.6 1.2 0.45 0.34 0.65 0.76 0.23 0.1218 6.7 9.3 11 14 15 12 10 8.6 7.8 2.3 1.1 0.08 0.07 0 0.09

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Rectilinear Euclidean

Corner Store Location

Consider the location of a competitive store relative to an existing store in the NW corner of the following intersection:

 

   Hibshoosh, "A Principle for Corner Store Positioning under Intrinsic Modeling for Traffic Orientation", Decision Sciences, Vol 19, No.3 (summer 1988).

EastWest

South

North

The Rules of the RoadAssume perfectly symmetrical traffic flows from all four directions, and

upon arrival at the intersection, a vehicle performs one of the following traffic movements: (1) continues forward, (2) turns right, (3) turns left, or (4) makes a U-turn (assume legal).

 Let Pi = probability of approaching intersection from direction i, i = N, S, E, AND W. Then P1 = P2 = P3 = P4 = 1/4.

Pj = probability of executing traffic movement j (j = 1, 2, 3, or 4), then assume P1 > P2 > P3 > P4 represents the drivers preference for turning movements.

Further. assume a fixed proportion, c, of drivers who will pass a store will stop and make a purchase at that store. Let N represents the number of vehicles entering the intersection per hour, and K represents the average profit per buyer at the store. Then S = cNK is the average profit per vehicle per time period.

EastWest

South

North

cNk(1 + P1 + P3 + P4 )

No Competition P1 > P2 > P3 > P4

cNk(1 + P1 + P3 + P4 )cNk(1 + P1 + P3 + P4 )

cNk(1 + P1 + P3 + P4 )

EastWest

South

North

N (1 + P1 + P3 )

N (1 + P3 + P4 )P1 > P2 > P3 > P4

NE Corner

Competition

EastWest

South

North

N (1 + P1 + P4)

N (1 + P1 + P4 )P1 > P2 > P3 > P4

SE Corner

EastWest

South

North

N (1 + P3 + P4 )

N (1 + P1 + P3 )P1 > P2 > P3 > P4

SW Corner

Towards a Science of Management ScienceA Law of Corner Store Location

All other factors being equal, a conveniencecorner store should be positioned clockwiseadjacent to the already established rival store.

Bonus Problem - Locating on a Network - Discrete vs continuous

5

12

11

107

68

9

42

3

In which city should the county dump be locatedto minimize the total number of truck miles requiredto pick trash and bring it to the dump? Arc numbersare distances in miles. Travel from city i to city j is always via the shortest path.

trips city per day1 102 123 84 65 216 187 58 149 2010 811 612 4

3

2

5

9

47

4

38

5 5

3 2

2

3

1?

50