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41. One rather unrealistic assumption in the flight- scheduling model is that a given plane can fly two con- secutive flights

B D E Figure 5.36 Flight Arcs Flight capacity Flight net revenue 25 30 39 24 24 35 29 29 Flight flow 1 1 1 0 1 1 1 1 1 1 1 1

problem 41 of chapter 5 in practical management science 6th edition.
please show me how to apply the constraint on downtime.

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Answer #1

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With the network structure as it is, if a plane arrives, say, at WashDC at time 14 and there is a flight out of WashDC at time 14, then there is nothing from preventing the same plane flying both flights. To ensure a 1-hour downtime for any flight, the easiest fix is to increase the arrival times for all flights by an hour. I didn't change these arrival times in the data area to the left, but I changed the flight destinations the range B37:B59. No other changes are required. The optimal solution changes considerably. Now only 11 of the 23 flights are flown.

Flight scheduling model
Input data
Planes owned 4
Fixed cost per plane 15
Overnight cost 5
Flight information
Flight Number Origin Destination Departs Arrives Net Revenue
1357 Boston NY 8 9 25
8322 Boston NY 9.5 10.5 30
5903 Boston WashDC 12.5 14 39
1207 Boston WashDC 13.5 15 24
1671 Boston NY 13.5 14.5 24
5671 Boston WashDC 16 17.5 35
7133 Boston NY 16.5 17.5 29
4166 NY Boston 9.5 11 28
3842 NY Boston 12 13.5 13
1537 NY WashDC 13 14.5 18
9320 NY Boston 14 16 22
3042 NY Boston 16.5 18 28
3752 NY Boston 18 19.5 34
9677 NY WashDC 18 20 39
6212 NY Boston 18.5 20 15
6811 WashDC NY 9 11 12
9195 WashDC NY 12.5 14 28
8350 WashDC NY 13 15 13
9480 WashDC Boston 13.5 15.5 18
7555 WashDC NY 14 15.5 33
9041 WashDC Boston 14 15.5 28
7539 WashDC Boston 14.5 16.5 19
2710 WashDC Boston 16 17.5 15
Network formulation Flow balance constraints
Flight arcs
Flight origin Flight destination Flight net revenue Flight flow Flight capacity Node Net outflow Required
Boston8 NY10 25 1 <= 1 Boston8 0 = 0
Boston9.5 NY11.5 30 1 <= 1 Boston8.5 0 = 0
Boston12.5 WashDC15 39 0 <= 1 Boston9 0 = 0
Boston13.5 WashDC16 24 0 <= 1 Boston9.5 0 = 0
Boston13.5 NY15.5 24 1 <= 1 Boston10 0 = 0
Boston16 WashDC18.5 35 1 <= 1 Boston10.5 0 = 0
Boston16.5 NY18.5 29 1 <= 1 Boston11 0 = 0
NY9.5 Boston12 28 1 <= 1 Boston11.5 0 = 0
NY12 Boston14.5 13 1 <= 1 Boston12 0 = 0
NY13 WashDC15.5 18 1 <= 1 Boston12.5 0 = 0
NY14 Boston17 22 0 <= 1 Boston13 0 = 0
NY16.5 Boston19 28 1 <= 1 Boston13.5 0 = 0
NY18 Boston20.5 34 0 <= 1 Boston14 0 = 0
NY18 WashDC21 39 0 <= 1 Boston14.5 0 = 0
NY18.5 Boston21 15 0 <= 1 Boston15 0 = 0
WashDC9 NY12 12 0 <= 1 Boston15.5 0 = 0
WashDC12.5 NY15 28 0 <= 1 Boston16 0 = 0
WashDC13 NY16 13 0 <= 1 Boston16.5 0 = 0
WashDC13.5 Boston16.5 18 0 <= 1 Boston17 0 = 0
WashDC14 NY16.5 33 0 <= 1 Boston17.5 0 = 0
WashDC14 Boston16.5 28 1 <= 1 Boston18 0 = 0
WashDC14.5 Boston17.5 19 0 <= 1 Boston18.5 0 = 0
WashDC16 Boston18.5 15 1 <= 1 Boston19 0 = 0
Boston19.5 0 = 0
Ground arcs Boston20 0 = 0
Ground origin Ground destination Ground cost Ground flow NY8 0 = 0
Boston8 Boston8.5 0 1 NY8.5 0 = 0
Boston8.5 Boston9 0 1 NY9 0 = 0
Boston9 Boston9.5 0 1 NY9.5 0 = 0
Boston9.5 Boston10 0 0 NY10 0 = 0
Boston10 Boston10.5 0 0 NY10.5 0 = 0
Boston10.5 Boston11 0 0 NY11 0 = 0
Boston11 Boston11.5 0 0 NY11.5 0 = 0
Boston11.5 Boston12 0 0 NY12 0 = 0
Boston12 Boston12.5 0 1 NY12.5 0 = 0
Boston12.5 Boston13 0 1 NY13 0 = 0
Boston13 Boston13.5 0 1 NY13.5 0 = 0
Boston13.5 Boston14 0 0 NY14 0 = 0
Boston14 Boston14.5 0 0 NY14.5 0 = 0
Boston14.5 Boston15 0 1 NY15 0 = 0
Boston15 Boston15.5 0 1 NY15.5 0 = 0
Boston15.5 Boston16 0 1 NY16 0 = 0
Boston16 Boston16.5 0 0 NY16.5 0 = 0
Boston16.5 Boston17 0 0 NY17 0 = 0
Boston17 Boston17.5 0 0 NY17.5 0 = 0
Boston17.5 Boston18 0 0 NY18 0 = 0
Boston18 Boston18.5 0 0 NY18.5 0 = 0
Boston18.5 Boston19 0 1 NY19 0 = 0
Boston19 Boston19.5 0 2 NY19.5 0 = 0
Boston19.5 Boston20 0 2 NY20 0 = 0
NY8 NY8.5 0 1 WashDC8 0 = 0
NY8.5 NY9 0 1 WashDC8.5 0 = 0
NY9 NY9.5 0 1 WashDC9 0 = 0
NY9.5 NY10 0 0 WashDC9.5 0 = 0
NY10 NY10.5 0 1 WashDC10 0 = 0
NY10.5 NY11 0 1 WashDC10.5 0 = 0
NY11 NY11.5 0 1 WashDC11 0 = 0
NY11.5 NY12 0 2 WashDC11.5 0 = 0
NY12 NY12.5 0 1 WashDC12 0 = 0
NY12.5 NY13 0 1 WashDC12.5 0 = 0
NY13 NY13.5 0 0 WashDC13 0 = 0
NY13.5 NY14 0 0 WashDC13.5 0 = 0
NY14 NY14.5 0 0 WashDC14 0 = 0
NY14.5 NY15 0 0 WashDC14.5 0 = 0
NY15 NY15.5 0 0 WashDC15 0 = 0
NY15.5 NY16 0 1 WashDC15.5 0 = 0
NY16 NY16.5 0 1 WashDC16 0 = 0
NY16.5 NY17 0 0 WashDC16.5 0 = 0
NY17 NY17.5 0 0 WashDC17 0 = 0
NY17.5 NY18 0 0 WashDC17.5 0 = 0
NY18 NY18.5 0 0 WashDC18 0 = 0
NY18.5 NY19 0 1 WashDC18.5 0 = 0
NY19 NY19.5 0 1 WashDC19 0 = 0
NY19.5 NY20 0 1 WashDC19.5 0 = 0
WashDC8 WashDC8.5 0 1 WashDC20 0 = 0
WashDC8.5 WashDC9 0 1
WashDC9 WashDC9.5 0 1
WashDC9.5 WashDC10 0 1
WashDC10 WashDC10.5 0 1
WashDC10.5 WashDC11 0 1
WashDC11 WashDC11.5 0 1
WashDC11.5 WashDC12 0 1
WashDC12 WashDC12.5 0 1
WashDC12.5 WashDC13 0 1
WashDC13 WashDC13.5 0 1
WashDC13.5 WashDC14 0 1
WashDC14 WashDC14.5 0 0
WashDC14.5 WashDC15 0 0
WashDC15 WashDC15.5 0 0
WashDC15.5 WashDC16 0 1
WashDC16 WashDC16.5 0 0
WashDC16.5 WashDC17 0 0
WashDC17 WashDC17.5 0 0
WashDC17.5 WashDC18 0 0
WashDC18 WashDC18.5 0 0
WashDC18.5 WashDC19 0 1
WashDC19 WashDC19.5 0 1
WashDC19.5 WashDC20 0 1
Overnight arcs
Overnight origin Overnight destination Fixed cost Overnight cost Overnight flow
Boston20 Boston8 15 0 2
Boston20 NY8 15 5 0
Boston20 WashDC8 15 5 0
NY20 Boston8 15 5 0
NY20 NY8 15 0 1
NY20 WashDC8 15 5 0
WashDC20 Boston8 15 5 0
WashDC20 NY8 15 5 0
WashDC20 WashDC8 15 0 1
Constraint on planes
Number used Number owned
4 <= 4
Monetary values
Net revenues 273
Fixed costs 60
Overnight costs 0
Net profit 213
Flights flown 11
0 1
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