Traveling Engineer Problem Andrew Burns, Victor Godoy-Cort´es, Kelsey Newman, Cody Pederson, and Westin Smith Team Astana, Engineering 102, University of Arizona, Tucson, AZ This problem asked us imagine we were an engineer selected by our Austin-based firm to travel to seven cities and demonstrate a breakthrough medical device to major hospitals. We were to develop two itineraries for this trip: one itinerary with the lowest airfare cost and another with shortest travel time; both itineraries were to be created using flight data provided in an Excel spreadsheet. To tackle this problem, we used a combination of conventional and computer-based heuristic alogorithms. Our heuristics produced one itinerary with a cost of $654 and another itinerary with a total travel time of 80 hours 20 minutes, for performance indices of 0.0122 and 2.390, respectively.
Nomenclature N Number of unique cities visited, not including connections p Ticket price P I Performance index Tf Time of final arrival at Austin International Airport Ti Time of initial departure from Austin International Airport Subscript A w.r.t. itinerary A B w.r.t. itinerary B
I.
Introduction
One major systems engineering problem is the vehicle routing problem. Each day millions of packages are delivered by the U.S. Postal Service and private parcel delivery companies like UPS and FedEx. Both the U.S. Postal Service and the private companies have fleets of vehicles to deliver the packages to the addressees. The problem is finding the best route to deliver all the packages with the fewest vehicles and as fast as possible. Likewise, airlines attempt to “deliver” passengers to various locations across the country and around the world as cheaply as possible. Like the delivery and airline companies, we used a combination of heuristic algorithms to quickly find both the cheapest flight itinerary and the fastest travel itinerary. The traveling engineer problem provided challenges similar to those real-world complications faced by both delivery and airline companies: time windows, flight schedules, routes, etc. and, moreover, it was an exercise in time management problem—time management with regard to the flight schedules, demonstration times, and return times of the hypothetical engineer, and time management with regard to formatting, examining, and interpreting data, creating algorithms, designing and delivering a presentation, and writing this report. Students should finish this problem with a greater understanding of the rigors of the design process. Our team in particular, worked even more than before on improving our time management and communication, two skills critical for employment in engineering today.
II.
Design Requirements
According to the problem, we [the engineer] were to depart from Austin International Airport (AUS) and travel to seven different cities, including: Philadelphia, PA; Columbus, OH; Chicago, IL; Boston, MA; Atlanta, GA; Washington, DC; and Denver, CO, to visit hospitals, demonstrate our [the engineer’s] firm’s new device for detecting and preventing type II Diabetes, and receive feedback from doctors on how to improve the 1 of 8 American Institute of Aeronautics and Astronautics
device for mass production. Before leaving, however; we were required to produce two itineraries, itineraries A and B, maximizing for two separate variables: airfare cost and elapsed time between departure and return to Austin, respectively. In developing these itineraries, we were mindful of both the itinerary performance index and the various explict and implicit constraints included with or inherent to the problem. A.
Performance Index (PI)
The quality of an itinerary was measured against one of two performance indices. These performance indices were used to make comparisons between teams and assess group performance. The first performance index, P IA , used for itinerary A, was equal to the quotient of the number of unique cities successfully visited and the sum of all airfare: N P IA = P
p
(1)
The second performance index, P IB , used for itinerary B, was equal to the quotient of the number of unique cities successfully visited and difference between the time of initial departure from Austin and the time of final return to Austin, in hours: P IB = B.
N Tf − Ti
(2)
Constraints
In addition to providing measures of performance, the problem contained both explicit and implicit constraints on design. The constraints explicitly listed included: 1. Only flight data provided in the Excel spreadsheet may be used in creating itineraries. 2. The earliest a demonstration may be given is 9 a.m. local time, and the latest is 8 p.m. local time. 3. Any flight taken must arrive before 11 p.m. local time. 4. No flight may be taken before 8 a.m. Sunday 25 January 2010. 5. The final return flight must arrive at AUS before 5 p.m. Saturday 31 January 2010. 6. Each demonstration takes a minimum of two hours. 7. It takes 30 minutes to travel to a hospital from an airport and vice versa. 8. Passengers must arrive one hour before a scheduled flight for check-in and security screening. Other constraints, though not listed, were, nevertheless, inherent to the problem: 1. No overlapping flights may be taken. 2. Only flights departing from the current city may be taken. 3. Performance indices may not exceed the performance indices for the optimal flight itineraries. C.
Design Description
1.
Itinerary A: Lowest Cost
Itinerary A (1) took 5 days, 4 hours, and 39 minutes to complete, from first departure from Austin International Airport (AUS) to Philadelphia International Airport (PHL) until final arrival at AUS from Denver International Airport (DEN). Total airfare—this itinerary’s minimized variable—amounted to just $654, or $81.75 per plane ticket. The most common carrier was Delta Air Lines.
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Table 1. Itinerary A
Route 9:10 a.m. Depart Austin (AUS) Arrive Philidelphia (PHL) 3:15 p.m. 6:00 a.m. Depart Philadelphia (PHL) Arrive Columbus (CMH) 10:01 a.m. 3:15 p.m. Depart Columbus (CMH) Arrive Washington, D.C. (DCA) 6:50 p.m. 1:00 p.m. Depart Washington, D.C. (DCA) Arrive Atlanta (ATL) 2:59 p.m. 7:25 a.m. Depart Atlanta (ATL) Arrive Boston (BOS) 10:00 a.m. 2:15 p.m. Depart Boston (BOS) Arrive Chicago (ORD) 4:25 p.m. 6:00 a.m. Depart Chicago (ORD) Arrive Denver (DEN) 7:34 a.m. 10:45 a.m. Depart Denver (DEN) Arrive Austin (AUS) 1:49 a.m. Total Time:
Date
Carrier
Cost
25-Jan
American Airlines
$96
26-Jan
Delta Air Lines
$59
26-Jan
Delta Air Lines
$79
27-Jan
Delta Air Lines
$84
28-Jan
Delta Air Lines
$89
28-Jan
American Airlines
$79
29-Jan
United
$94
30-Jan
Frontier Airlines
$74
5 d 4 hr 39 min
Total Cost:
$654
Table 2. Itinerary B
Route 8:35 a.m. Depart Austin (AUS) Arrive Denver (DEN) 9:41 p.m. 1:45 p.m. Depart Denver (DEN) Arrive Chicago (ORD) 5:20 p.m. 6:00 a.m. Depart Chicago (ORD) Arrive Washington, D.C. (DCA) 8:45 a.m. 1:30 p.m. Depart Washington, D.C. (DCA) Arrive Boston (BOS) 2:50 p.m. 7:10 p.m. Depart Boston (BOS) Arrive Atlanta (ATL) 10:07 p.m. 12:30 p.m. Depart Atlanta (ATL) Arrive Philadelphia (PHL) 2:34 p.m. 6:51 a.m. Depart Philadelphia (PHL) Arrive Atlanta (ATL) 9:19 p.m. 9:20 p.m. Depart Atlanta (ATL) Arrive Columbus (CMH) 10:55 p.m. 12:35 p.m. Depart Columbus (CMH) Arrive Austin (AUS) 4:55 p.m. Total Time:
Date
Carrier
Cost
25-Jan
United Airways
$84
25-Jan
United
$94
26-Jan
United
$97
26-Jan
United
$341
26-Jan
Delta Air Lines
$89
27-Jan
Delta Air Lines
$84
27-Jan
AirTran Airways
$84
27-Jan
Delta Air Lines
$79
28-Jan
American Airlines
$104
3 d 8 hr 20 min
Total Cost:
$1,056
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2.
Itinerary B: Shortest Time
Itinerary B (2) took 3 days, 8 hours, and 20 minutes to complete, from first departure from Austin International Airport (AUS) to Denver International Airport (DEN) until final arrival at AUS from Port Columbus International Airport (CMH). Total airfare amounted to $1,056, or $117.33 per plane ticket. The most common carrier was a tie between United and American Airlines. 3.
General
Both itineraries nicely maximized their key variable; however, in real life, one would be dull and the other impracticable. Two examples: (1) In Itinerary A, the flight on January 29 starts at 6:00 a.m. and ends at 7:34 a.m. If we assume that the drive to the demonstration takes 30 minutes and the demonstration itself lasts one hour, we [the engineer] are stuck waiting all day in Denver when we could have just as easily spent a few more dollars and left for home. (2) In Itinerary B, our [the engineer’s] flight arrives at Jackson-Hartsfield (ATL) at 9:19 p.m. and our [the engineer’s] next flight departs at 9:20 p.m. Even assuming that the planes arrived on time, it might be physically impossible to remove ourselves [him or herself] from the first plane in time to get to the next plane.
III.
Design Justification
Before we began working on the problem, the group decided that, because of the nature of the task, it would be most efficiently accomplished by dividing the work on Itinerary A and Itinerary B between group members—half were assigned Itinerary A and half Itinerary B. Itinerary A lent itself best to a computerbased algorithm because of the simplicity with which such an algorithm could be fashioned. Cody, along with Kelsey and Victor, decided to pursue the programming-based Itinerary A. Itinerary B, which was more intimately involved with time windows, a concept not easily dealt with using an Excel algorithm, and which the literature describes as extremely complicated, was given to Andrew and Westin to solve by hand, using basic heuristic principles.1 A.
Itinerary A
Itinerary A was approached with the assumption that any flight desired could be taken each day, for the lowest price.a In keeping with this assumption, a grid of costs was created and the lowest price between any two cities was placed where those two cities intersected. From here, several possibilities were suggested, and two were explored: a greedy algorithm and a minimum-cost insertion algorithm, both of which provided a reasonably low cost, but were ultimately off by between $20 and $100 from our final calculation. To explore the matter further, an Excel spreadsheet was created to apply an exchange heuristic to the flight data. (See Appendix for spreadsheet.) The comparisons this application provided were invaluable, because they allowed us explore minor route variations. We knew it to be very likely, statistically, that the global optimum would be surrounded by other routes with very similar costs and only minor variations in route order. This application sifted through each of these minor, localized variations of the current “cheapest” itinerary until it found the optimal itinerary. For example, we entered the itinerary shown in Table (3). After processing this itinerary through the exchange heurisitc algorithm, however; the itinerary was identical to Table (1). Based on the effectiveness of this automated, computer-based approach—an approach that took only a couple of hours to create—we find it very reasonable to assume, though it is by no means certain, that Itinerary A, herein provided, is the cheapest itinerary possible given the available flight options. B.
Itinerary B
Itinerary B was approached much more cautiously than Itinerary A for one major reason: time windows. Time windows are windows of time provided in a routing problem that may shift back and forth depending on certain variables, like for instance, the flight. Depending on which flight was chosen, another flight might not be available when the engineer arrived at the airport. Furthermore, even if a flight were available, it might be a nasty flight with numerous connections that took twelve hours to complete. Some flights departed at odd times and others arrived at prohibited times, etc. Consequently, these problems, seemed too complicated to a This
is a flawed assumption, but, for the purposes of this problem, it held.
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Table 3. Itinerary C
Route 8:05 a.m. Depart Austin (AUS) Arrive Philidelphia (PHL) 2:33 p.m. 6:00 a.m. Depart Philadelphia (PHL) Arrive Columbus (CMH) 9:50 a.m. 2:55 p.m. Depart Columbus (CMH) Arrive Chicago (ORD) 3:20 p.m. 6:00 a.m. Depart Chicago (ORD) Arrive Washington, D.C. (DCA) 8:45 a.m. 1:00 p.m. Depart Washington, D.C. (DCA) Arrive Atlanta (ATL) 2:59 p.m. 7:25 a.m. Depart Atlanta (ATL) Arrive Boston (BOS) 10:00 a.m. 2:30 p.m. Depart Boston (BOS) Arrive Denver (DEN) 5:20 p.m. 6:00 a.m. Depart Denver (DEN) Arrive Austin (AUS) 11:00 a.m. Total Time:
Date
Carrier
Cost
25-Jan
Continental Airlines
$96
26-Jan
Delta Air Lines
$59
26-Jan
American Airlines
$71
27-Jan
United
$97
27-Jan
Delta Air Lines
$84
28-Jan
Delta Air Lines
$89
28-Jan
United
$109
29-Jan
Frontier Airlines
$74
4 d 2 hr 55 min
Total Cost:
$679
automate for we were not entirely sure how to approach such a program, even with a programming language like C++ or Java. So, instead, we decided to do the problem by hand and compare our models with one and other, exchanging advice on how to minimize the time, until we could no longer find any faster route combinations. By the time we made our presentation, our Itinerary B looked like Table (4) Yet, after seeing another group give their presentation with a lower time than ours, Andrew compiled a chart of all of the flight data and color coded the times to indicate good times (i.e. flights that took the least amount of time) and bad times (i.e. excessively long flights). Using this chart provided in the Appendix, he calculated the new minimum shown in Table (2). Though we are not entirely certain that Itinerary B represents the absolute minimum time, we believe that we have exercised extreme rigor in composing Itinerary B, and that this itinerary represents the most reasonable of itineraries with the shortest time, even though it requires teleportation at Jackson-Hartfield.
IV.
Design Theory and System Model
As each itinerary was created using a different design theory and system model, it seems only fair to, again, give them individual treatment. A.
Itinerary A
The Excel spreadsheet used to examine the cost-minimized itinerary determined variations on a given itinerary and found the cheapest one. It did this by first assigning every city a number, thus giving every itinerary a corresponding code, i.e. 123456781. From an inputed seed itinerary, several possibile variants were examined: (1) switching two values in the code, (2) removing and reinsterting two consecutive values, (3) removing and reinserting one value, and (4) reversing the code. The cheapest itinerary code from each variation was examined in the next set of variations, achieving a “cascading effect� that reduced the input required to explore a seed itinerary. Although this method frequently discovered the best flight itinerary, it often discovered local minimum itineraries, or itineraries for which no single variation would reduce the cost but several subsequent itineraries might. For this reason, it is not proven that the provided itinerary (1) represents the global minimum; however, we estimate that the number of seed itineraries which produce the global maximum is rather large and so the number of seeds examined was sufficient. For some evidence of 5 of 8 American Institute of Aeronautics and Astronautics
Table 4. Itinerary B-Preliminary
Route 8:35 a.m. Depart Austin (AUS) Arrive Denver (DEN) 9:41 p.m. 1:45 a.m. Depart Denver (DEN) Arrive Chicago (ORD) 5:20 p.m. 6:35 a.m. Depart Chicago (ORD) Arrive Boston (BOS) 9:45 a.m. 2:50 p.m. Depart Boston (BOS) Arrive Washington, D.C. (DCA) 4:35 p.m. 9:00 p.m. Depart Washington, D.C. (DCA) Arrive Atlanta (ATL) 11:00 p.m. 1:55 p.m. Depart Atlanta (ATL) Arrive Philadelphia (PHL) 4:06 p.m. 6:00 a.m. Depart Philadelphia (PHL) Arrive Columbus (CMH) 9:50 a.m. 6:05 p.m. Depart Columbus (CMH) Arrive Austin (AUS) 10:08 p.m. Total Time:
Date
Carrier
Cost
25-Jan
US Airways
$84
25-Jan
United
$94
26-Jan
American Airlines
$71
27-Jan
American Airlines
$341
27-Jan
Delta Air Lines
$124
28-Jan
Delta Air Lines
$84
29-Jan
Delta Air Lines
$59
29-Jan
American Airlines
$143
3 d 13 hr 33 min
Total Cost:
$1,000
this and an explanation of a possible method for proving the global optimum, see “Route Transformation Theory” in the appendix. Another advantage of the Excel spreadsheet was that it enabled us to perform statistical analyses of the different itinerary possibilites and identify itinerary maximizing arbitrary characteristics, such as the median ticket price. From this, several interesting trends were observed. First, that the standard deviation and the total airfare cost are positively correlated. Interestingly, the size of this value compared to the size of the mean ticket price shrinks as ticket prices rise, meaning this expansion of median ticket price does not keep up with the growth of the average. We assume this is because the mean and median ticket prices do not line up, with the average ticket price lying well above the median ticket price for low prices but converging to the same number for higher ticket prices. This creates skewness. From this plot, it may be observed that most of the datapoints fit between curves represented by y=-0.5 and a distorted bell-shaped curve on top. This indicates most of the itineraries have a fairly balanced or positive skew. It can also be observed that the typical skew starts near 0 at the lowest mean, rises quickly until about 90, then recedes. For a comparison of these attributes with a more neutral and random sample, see “Statistics Comparison” in the Appendix. These statistics are useful for describing tendencies of the costs of flights within an itinerary. Except for more expensive itineraries, the mean ticket price lies above the median ticket price. As ticket prices fall, so does the spread or variance in the ticket prices. Further, in the cheapest itinerary, the prices are fairly centered around the mean, while in some, but not all, more expensive routes ($87-$110 mean) several ticket prices cluster at a value less than the mean. This tendency to cluster away from the mean is reduced as the mean approaches larger values. All this analysis is pertinent to Itinerary A because it provides valuable clues, and even alternative measures, when looking for the cheapest route. The cheapest itinerary should (a) have a small standard deviation, (b) have a large standard deviation compared to the mean, (c) be close to itineraries with medians well above the mean ticket price, or itineraries with large values of skewness, or (d) have a small value of skewness. It should be observed that Itinerary A, herein published, meets all of these qualititative criteria. B.
Itinerary B
As was explained in the previous section, in great detail, the computer-based algorithm provided great insight into the numerous factors affecting ticket pricing and provided enough substantive data to allow us
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to ponder several peculiar patterns. Itinerary B, the product of human trial-and-error, provided no such extensive insight. Routing optimization represents a very significant intersection between pure mathematics and practical engineering, but, as with most things that interesect disciplines, it is extremely complex, too complex for this group to provide any meaningful pronouncements.1 What design theory we did examine dealt mainly with our decision to eschew map-based approaches to solving this problem. Neither maps nor distance data was ever considered when approaching either Itinerary A or B, as, unlike many other teams, this team felt, with good evidence (see Appendix), that any correlations between distances, costs, and time were not statistically significant enough to play a significant role in the flight times. Given that most airplane companies operate off of a hub-and-spoke system of flight routing, with frequent connections and few direct flights, short distances were often swallowed up in three and four connections at dead-end airports with large lay-overs, etc. When approaching the flight data, this team stuck purely to the data provided. Judging by our results, we are satisfied that we took this approach, as other groups, lured by the tenuous connection between travel time and distance, ultimately produced lower PIs and may possibly receive lower grades.
V.
Test Procedures
Only one test of our models—the application of them to the data provided—could be done, so we feel it sufficient enough to describe in this section our method of handling each itinerary. We believe the extensive discussion in the previous section is sufficient to vouch for the rigors through which our computer-based algorithm was subjected, complete with numerical and statistical analysis. As to Itinerary B, when Andrew and Westin were assigned Itinerary B, they each produced two versions—an initial version, and a refined version. The two refined versions were compared, route by route, to one another to identify any time saving alterations. Kelsey, who found that she had extra time, provided additional help by providing a third refined version of an itinerary. When all three itineraries were collected, routes that saved time were swapped into a final copy, herein provided, of Itinerary B. Itinerary B represents the compiled work of all three team members.
VI.
Test Results
The results of our diligent work are reflected in the two itineraries. Extensive analysis and countless hours went into producing them and refining them to the best of our collective abilities. While, as mentioned earlier, we are not definitive about how near our final results were to the optimal itineraries, we are confident that they are as near as we are currently capable of identifying. Error was not as significant an issue with this problem as with the robot problem because we were dealing with a theoretical model rather than a real-world model. Most error for this problem was either human error, which is often called sloppiness, or software error.
VII.
Design Critique and Summary
In retrospect, we cannot see what else we could have possibly done to perfect our two itineraries. Perhaps, after reviewing other presentations and listening to other explanations, some regrets may arise; but, heretofore, no team member has voiced any concerns with the itineraries themselves. The giant regret—one giant sin—every member participated in was procrastination, some more than others. As future engineers, it is very necessary that we squeeze out these urges to put off for tomorrow what can be done today. While such behavior may only produce sleepless nights now, serious destruction and death may ensue if we engage in such vices later in our careers.
Appendix The Appendix has been moved to the back for convenience.
References 1 Potvin, Jean-Yves, and Jean-Marc Rousseau. “An Exchange Heuristic for Routeing Problems with Time Windows.” Journal of the Operational Research Society 46 (1995): 1433-446. Print.
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2 Dobson, Gregory, and Philip J. Lederer. “Airline Scheduling and Routing in a Hub-and-Spoke System.� Transportation Science 27.3 (1993): 281-97. Print.
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