ASSIGNING UMPIRE CREWS The American Baseball League consists of 14 professional baseball teams organized into 3 divisions: the West Division with Seattle, Oakland, California, and Texas, the Central Division with Kansas City, Minnesota, Chicago, Milwaukee and Cleveland; and the East Division with Detroit, Toronto, Baltimore, New York, and Boston. In addition to the schedules for each team, the American League must determine the best way to assign the umpire crews to the various games played throughout the league. Umpire crews are assigned to specific home-team cities for the two-, three-, or four-game series in that city, but they are not assigned on an individual-game basis. Since there are 14 American League teams, as many as seven games can be played at the same time (doubleheaders count as one game in assigning crews); hence, seven umpire crews must be assigned. Several considerations are important in making the umpire crew assignments. Because of the amount of travel required, airline costs can be substantial. Thus, from a cost point of view, umpire crew assignments with minimum travel distances are desirable. However, a second consideration in the assignment of the umpire crews is that there should be a balance such that each crew works approximately the same number of games with each team and in each city. The considerations of minimizing travel distances and at the same time, balancing the crew assignments among the teams and cities are in conflict. In addition to these considerations, a number of requirements must be satisfied. The most important of these are 1. A crew cannot travel from city A to city B if the last game in city A is a night game and the first game in city B is an afternoon game on the next day. 2. A crew cannot travel from a West Coast location (Seattle, Oakland, or California) to Chicago, Milwaukee, Cleveland, or any East Division city without a day off. 3. Because of flight scheduling difficulties, a crew traveling into or out of Toronto must have a day off unless coming from or going to New York, Boston, Detroit, or Cleveland. 4. Any crew traveling from a night game in Seattle, Oakland, or California cannot be assigned to Kansas City or Texas for a game on the next day. 5. No crew should be assigned to the same team for more than two series in a row. The umpire crews have already been scheduled for the first four series of the five-series schedule shown in Table 5.8. Table 5.9 summarizes the crew assignments for the first four series in the schedule and shows the pairings for the fifth series. The superscript next to each team identification indicates the umpire crew assigned for that pairing. For example, for the fourth series, crew I is assigned to the Boston-Toronto games, crew 2 is assigned to the Detroit- California games, and so on. SEGMENT OF THE AMERICAN LEAGUE SCHEDULE SHOWING FIVE SERIES NY BOS BOS CLE TORBAL CHI KC CHI KC CHI CHI TOR TOR" CHI TOR- CHI TOR MIN CLE MIN CLE MIN CLE CHI MIN MIN MIN MIN Home Team Series Date SEA OAK CAL TEX KC MIN CHI MKE DET Mon CAL SEA TOR. NY 1 Tues CAL BOS SEA TORNY Wed CAL BOS SEA TOR NY Thuri. DET MKE KC 2 Fri DET NY MKE KC Sat DET NY MKE KC Sun DET NY(2) MKE KC Mon MKE NY Tues MKE DET NY CHI BOS Wed. MKE DET NY CHI BOS Thurs MKE DET NY CHI Fri NY MKE DET BAI" CLE" KC 1 Sat NY" MKE DET BAL CLE KC Sun NY MKE DET BAL CLE KC Mon. TEX CLE 5 Toes TEX" BOSCLE CAL SEA Wed TEX BOS CLEY CAL SEA Thurs TEX BOS CAL SEA BOS BOS BOS BOS BOS BAL BAL BAL OAK "Denote night game or early evening start. (2)Denots doubleheader (Two games in one day UMPIRE CREW ASSIGNMENTS FOR THE FIRST FOUR SERIES ARE SHOWN AS SUPER SCRIPTS BOS SEA Home Team Series SEA OAK CALTEX KC MIN CHI MKE DET CLE TOR BAL NY CAL BOS DET NY MKE KC TOR NY KO CHF MKE DET NY CHI BOS CH CLE! MIN NY MKE DETBAL CLE KC BOS TEX BOSCLE CAL SEA BAL OAK For the fourth series, umpire crew l is assigned to the Boston at Toronto series TOR MIN Table 5.10 shows the distances from the cities where the fourth series is being played to the cities where the fifth series is being played. League management would like to consider some other issues in assigning crews to the next series. Over the past nine series, crew 4 has umpired three series with Kansas City and three series with Milwaukee. Also, crew 5 has not been assigned to any games with New York, Toronto, or Detroit over the past month. Managerial Report Prepare a written recommendation to league management concerning the assignment of umpire crews to the fifth series that will minimize the distance traveled. DISTANCE COST MATRIX FOR UMPIRE CREW ASSIGNMENTS (SERIES 5) Crew From KC MIN CHI MKE DET TORNY SEA(7) 1825 1399 2007 1694 1939 2124 2421 OAK(3) 1498 1589 2125 1845 2079 2286 2586 CAL(2) 1363 1536 2035 1756 1979 2175 2475 TEX(6) 506 853 798 843 982 1186 1383 MIN(5) 394 0 334 297 528 780 1028 CHI(4) 403 334 0 74 235 430 740 TOR(1) 968 897 497 583 206 0 366 Note: The numbers in parentheses reference umpire crews. ASSIGNING UMPIRE CREWS The American Baseball League consists of 14 professional baseball teams organized into 3 divisions: the West Division with Seattle, Oakland, California, and Texas, the Central Division with Kansas City, Minnesota, Chicago, Milwaukee and Cleveland; and the East Division with Detroit, Toronto, Baltimore, New York, and Boston. In addition to the schedules for each team, the American League must determine the best way to assign the umpire crews to the various games played throughout the league. Umpire crews are assigned to specific home-team cities for the two-, three-, or four-game series in that city, but they are not assigned on an individual-game basis. Since there are 14 American League teams, as many as seven games can be played at the same time (doubleheaders count as one game in assigning crews); hence, seven umpire crews must be assigned. Several considerations are important in making the umpire crew assignments. Because of the amount of travel required, airline costs can be substantial. Thus, from a cost point of view, umpire crew assignments with minimum travel distances are desirable. However, a second consideration in the assignment of the umpire crews is that there should be a balance such that each crew works approximately the same number of games with each team and in each city. The considerations of minimizing travel distances and at the same time, balancing the crew assignments among the teams and cities are in conflict. In addition to these considerations, a number of requirements must be satisfied. The most important of these are 1. A crew cannot travel from city A to city B if the last game in city A is a night game and the first game in city B is an afternoon game on the next day. 2. A crew cannot travel from a West Coast location (Seattle, Oakland, or California) to Chicago, Milwaukee, Cleveland, or any East Division city without a day off. 3. Because of flight scheduling difficulties, a crew traveling into or out of Toronto must have a day off unless coming from or going to New York, Boston, Detroit, or Cleveland. 4. Any crew traveling from a night game in Seattle, Oakland, or California cannot be assigned to Kansas City or Texas for a game on the next day. 5. No crew should be assigned to the same team for more than two series in a row. The umpire crews have already been scheduled for the first four series of the five-series schedule shown in Table 5.8. Table 5.9 summarizes the crew assignments for the first four series in the schedule and shows the pairings for the fifth series. The superscript next to each team identification indicates the umpire crew assigned for that pairing. For example, for the fourth series, crew I is assigned to the Boston-Toronto games, crew 2 is assigned to the Detroit- California games, and so on. SEGMENT OF THE AMERICAN LEAGUE SCHEDULE SHOWING FIVE SERIES NY BOS BOS CLE TORBAL CHI KC CHI KC CHI CHI TOR TOR" CHI TOR- CHI TOR MIN CLE MIN CLE MIN CLE CHI MIN MIN MIN MIN Home Team Series Date SEA OAK CAL TEX KC MIN CHI MKE DET Mon CAL SEA TOR. NY 1 Tues CAL BOS SEA TORNY Wed CAL BOS SEA TOR NY Thuri. DET MKE KC 2 Fri DET NY MKE KC Sat DET NY MKE KC Sun DET NY(2) MKE KC Mon MKE NY Tues MKE DET NY CHI BOS Wed. MKE DET NY CHI BOS Thurs MKE DET NY CHI Fri NY MKE DET BAI" CLE" KC 1 Sat NY" MKE DET BAL CLE KC Sun NY MKE DET BAL CLE KC Mon. TEX CLE 5 Toes TEX" BOSCLE CAL SEA Wed TEX BOS CLEY CAL SEA Thurs TEX BOS CAL SEA BOS BOS BOS BOS BOS BAL BAL BAL OAK "Denote night game or early evening start. (2)Denots doubleheader (Two games in one day UMPIRE CREW ASSIGNMENTS FOR THE FIRST FOUR SERIES ARE SHOWN AS SUPER SCRIPTS BOS SEA Home Team Series SEA OAK CALTEX KC MIN CHI MKE DET CLE TOR BAL NY CAL BOS DET NY MKE KC TOR NY KO CHF MKE DET NY CHI BOS CH CLE! MIN NY MKE DETBAL CLE KC BOS TEX BOSCLE CAL SEA BAL OAK For the fourth series, umpire crew l is assigned to the Boston at Toronto series TOR MIN Table 5.10 shows the distances from the cities where the fourth series is being played to the cities where the fifth series is being played. League management would like to consider some other issues in assigning crews to the next series. Over the past nine series, crew 4 has umpired three series with Kansas City and three series with Milwaukee. Also, crew 5 has not been assigned to any games with New York, Toronto, or Detroit over the past month. Managerial Report Prepare a written recommendation to league management concerning the assignment of umpire crews to the fifth series that will minimize the distance traveled. DISTANCE COST MATRIX FOR UMPIRE CREW ASSIGNMENTS (SERIES 5) Crew From KC MIN CHI MKE DET TORNY SEA(7) 1825 1399 2007 1694 1939 2124 2421 OAK(3) 1498 1589 2125 1845 2079 2286 2586 CAL(2) 1363 1536 2035 1756 1979 2175 2475 TEX(6) 506 853 798 843 982 1186 1383 MIN(5) 394 0 334 297 528 780 1028 CHI(4) 403 334 0 74 235 430 740 TOR(1) 968 897 497 583 206 0 366 Note: The numbers in parentheses reference umpire crews