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SCM447: Logistics and Transportation
Management

Fall 2021
Homework 2

Due: November 14, 2021, 11:59pm

Part a: (2.5 points)

Logistics is important not only for providers of goods, but also for providers of services.
Transportation is often the primary concern for companies that provide on-site or in-person
services. One such example is a traveling circus like Ringling Brothers and Barnum &
Bailey, “The Greatest Show On Earth”.

The Ringling Brothers and Barnum & Bailey circus “cheats” by actually having two touring
circuses, the “red” circus and the “blue” circus, which tour the country simultaneously and
thus are able to cover more cities.

In this homework, we will be looking at the travels of a fictional circus with just a single
touring unit. This Atlanta-based circus plans to visit 24 cities, listed in the table below.
After visiting all of the cities, the circus will return to Atlanta.

As one might imagine, the travel costs associated with a circus are quite high. Special vehicles,
equipment, foods, etc. are needed to transport the tigers, elephants and other circus animals.
Other equipment, such as high-wire apparatus and set construction materials, are also
expensive to transport. Therefore, the circus would like to minimize its transportation costs
during its national tour. Of course, a benefit of minimizing travel time is that the circus
will have more time to perform, and thus generate more revenue. Because the special circus
vehicles can only travel on major highways, we can assume a constant speed for every
highway. Therefore, minimizing travel time is equivalent to minimizing travel distance in this
case.

The goal of the circus is to find a good tour of all the cities; that is, to find a path that visits
each city and then returns to its starting point. Of course, some tours are shorter than others,
and the optimal tour is the one which covers the least total distance.

Atlanta, GA Detroit, MI Lincoln, NE Phoenix, AZ
Boston, MA Indianapolis, IN Minneapolis, MN Pittsburgh, PA
Buffalo, NY Jacksonville, FL Nashville, TN Portland, OR
Charlotte, NC Kansas City, MO New Orleans, LA Salt Lake City, UT
Chicago, IL Las Vegas, NV Oakland, CA San Antonio, TX
Cleveland, OH Lexington, KY Oklahoma City, OK San Diego, CA

The raw city-to-city distance data (HW2data.xls) can be found in an Excel spreadsheet on
Canvas. All distances are symmetric; the distance from city A to city B is the same as
the distance from city B to city A.

1. Use your own intuition to get the shortest-distance circus tour you can find. Do

not use any of the algorithms from class – don’t worry, you’ll have plenty of
opportunity for that in the rest of the homework, and your grades will not depend
on the quality of the solution you find in this part. For now, just try using your
intuition and logic. Draw your best answer clearly on a copy of the map
(HW1map.pdf), and give the total distance.

2. Using the distances provided on the website (do not just go by eye, as the
graphical plot is not to scale), use the traditional Nearest Neighbor heuristic from
class to find the best route for the circus to take. Draw your answer clearly on
a copy of the map, provide a list of the edges in the order that you choose them,
and give the total distance.

Part B: (4.5 points)

3. A truck stationed at the depot (location 0) is to serve the demand of sales points
1 through 7, depicted in Figure 1, using a single tour. The distances between the depot
and the sales points are given in Table 1. The network is symmetric. (3 points)

0 1 2 3 4 5 6 7

0 – 27.9 54.6 42.0 56.5 37.0 30.9 34.1
1 – – 67.2 25.6 28.8 48.4 57.4 21.6
2 – – – 60.5 95.8 18.8 60.4 52.1
3 – – – – 39.4 43.1 70.2 12.2
4 – – – – – 77.2 84.5 44.4
5 – – – – – – 51.6 34.0
6 – – – – – – – 59.3
7 – – – – – – – –

Table 1: Distances between depot and sales points

Figure 1: Geographic Locations of Depot and Sales Points

2

3
5

1
0

4
6

7

2

4a) The first step of Christofides heuristic is to find the minimum spanning tree on the
network. Use Kruskal’s algorithm to identify the MST on the network, and draw your
solution on the copy of the network provided. Also, provide a list of the edges that you
added to the MST, in the order that you added them.

Figure 2: Minimum spanning tree

List of edges added:

4b) The second step of Christofides heuristic is to identify the set of nodes S with odd
degree with respect to the minimum spanning tree, and then find the minimum cost
perfect node matching, W∗, of the nodes in S. In the minimum spanning tree you
identified in (a), there should be 6 nodes with odd degree with respect to the minimum
spanning tree: S = {1, 2, 3, 4, 6, 7}. If this is not the case, go back and rework your
solution before proceeding. One of the pairs in the minimum cost perfect matching W∗
is (1,4). Determine the two additional pairs included in W∗, and show your work.

2

3
5

1
0

4
6

7

3

4c) Draw the minimum spanning tree again on the network below, but this time, also
include the arcs in W∗ as dotted lines. Then, specify a tour T C of the nodes that uses
each arc in your diagram exactly once. Note that in order to use each arc exactly once,
you will visit some nodes more than once in the tour T C .

Figure 3: Minimum spanning tree with perfect node matching

Tour using each arc in this diagram:

4d) Complete Christofides heuristic by introducing shortcuts to avoid revisiting nodes
in T C , such that a feasible solution to this instance of TSP results. Draw your TSP
tour on the diagram provided. Indicate the cost of the TSP tour.

Figure 4: TSP tour

Cost of TSP tour:

2

3
5

1
0

4
6

2

3
5

1
0

4
6

7

7

4

5

4. An online book retailer packs books into boxes, and the retailer has a contract with

a shipper specifying that any box weighing less than or equal to 20 pounds can be
shipped for $8 per box. Because the boxes are relatively large, and books relatively
small, it is safe to assume that the volume of the boxes is not a constraint for any
order. Instead, assume that the 20- pound weight is the relevant constraint on each
box. Obviously, one book cannot be split across separate boxes. An order is received
from an avid reader in Fayetteville, AR, for the books specified in Table 3. (1.5 points)

Table 2: Book weights (lbs)

5a) What is lower bound on the shipment cost for this order?

5b) Use the best fit bin packing heuristic to pack the books into boxes, and process
the books in the order they are indexed in Table 3 above. Specify the cost of the
associated shipment.

6

6

5c) Theoretically, offline bin packing heuristics have better performance than online
bin packing heuristics. In this case, we do have access to all item sizes when
starting a heuristic, so it should be advantageous to use an offline heuristic. Use the
first fit decreasing bin packing heuristic to pack the books into boxes, and specify
the cost of the associated shipment.

hw8rawdata

Por Oak SD Phx SLC Lin SAn NOr Nsh OkC KCy LVg Min Chi Det Ind Cle Lex Box Buf Pit Cha Atl Jax
Portland OR 0 534 931 993 626 1331 1728 2061 1959 1475 1489 744 1421 1750 1948 1878 2043 2012 2529 2147 2155 2279 2163 2444
Oakland CA 534 0 452 635 587 1367 1486 1917 1946 1368 1498 395 1574 1844 2062 1935 2148 2044 2682 2287 2249 2282 2127 2368
San Diego CA 931 452 0 298 631 1254 1133 1612 1741 1132 1346 267 1535 1734 1954 1788 2028 1870 2581 2189 2116 2074 1895 2096
Phoenix AZ 993 635 298 0 500 977 861 1324 1445 835 1058 256 1278 1452 1670 1498 1741 1576 2295 1907 1826 1776 1598 1802
Salt Lake Ci UT 626 587 631 500 0 788 1105 1442 1392 859 929 365 988 1259 1475 1357 1563 1474 2095 1699 1666 1724 1585 1849
Lincoln NE 1331 1367 1254 977 788 0 809 845 630 374 163 1045 343 484 704 569 785 689 1332 938 883 949 837 1130
San Antonio TX 1728 1486 1133 861 1105 809 0 513 834 435 737 1091 1130 1068 1235 1012 1262 1009 1774 1448 1303 1104 898 1014
New Orleans LA 2061 1917 1612 1324 1442 845 513 0 472 586 697 1525 1057 838 927 713 919 641 1356 1093 921 639 436 502
Nashville TN 1959 1946 1741 1445 1392 630 834 472 0 611 471 1584 695 396 456 248 452 178 941 632 472 336 212 517
Oklahoma Cit OK 1475 1368 1132 835 859 374 435 586 611 0 321 989 701 701 900 697 952 751 1500 1132 1021 942 768 1001
Kansas City MO 1489 1498 1346 1058 929 163 737 697 471 321 0 1156 397 400 614 444 682 547 1237 851 770 797 676 967
Las Vegas NV 744 395 267 256 365 1045 1091 1525 1584 989 1156 0 1302 1529 1749 1600 1829 1697 2377 1982 1924 1920 1756 1985
Minneapolis MN 1421 1574 1535 1278 988 343 1130 1057 695 701 397 1302 0 353 529 512 626 663 1119 729 740 939 900 1208
Chicago IL 1750 1844 1734 1452 1259 484 1068 838 396 701 400 1529 353 0 220 167 303 317 848 455 409 589 579 881
Detroit MI 1948 2062 1954 1670 1475 704 1235 927 456 900 614 1749 529 220 0 225 98 299 628 237 213 506 575 844
Indianapolis IN 1878 1935 1788 1498 1357 569 1012 713 248 697 444 1600 512 167 225 0 257 151 805 440 329 427 415 715
Cleveland OH 2043 2148 2028 1741 1563 785 1262 919 452 952 682 1829 626 303 98 257 0 278 554 185 115 437 536 784
Lexington KY 2012 2044 1870 1576 1474 689 1009 641 178 751 547 1697 663 317 299 151 278 0 768 455 295 277 280 570
Boston MA 2529 2682 2581 2295 2095 1332 1774 1356 941 1500 1237 2377 1119 848 628 805 554 768 0 396 480 728 921 1029
Buffalo NY 2147 2287 2189 1907 1699 938 1448 1093 632 1132 851 1982 729 455 237 440 185 455 396 0 183 553 690 904
Pittsburgh PA 2155 2249 2116 1826 1666 883 1303 921 472 1021 770 1924 740 409 213 329 115 295 480 183 0 372 509 723
Charlotte NC 2279 2282 2074 1776 1724 949 1104 639 336 942 797 1920 939 589 506 427 437 277 728 553 372 0 206 351
Atlanta GA 2163 2127 1895 1598 1585 837 898 436 212 768 676 1756 900 579 575 415 536 280 921 690 509 206 0 308
Jacksonville FL 2444 2368 2096 1802 1849 1130 1014 502 517 1001 967 1985 1208 881 844 715 784 570 1029 904 723 351 308 0