Example 5.6
Non-logistics Network Models
Background Information

Braneast Airlines must staff the daily flights between
New York and Chicago shown in this table.
Flight Data for Braneast Problem
Flight
1
2
3
4
5
6
7
Leave
Chicago
6 A.M.
9 A.M.
NOON
3 P.M.
5 P.M.
7 P.M.
8 P.M.
Arrive
N.Y.
10 A.M.
1 P.M.
4 P.M.
7 P.M.
9 P.M.
11 P.M.
Midnight
Flight
1
2
3
4
5
6
7
Leave
N.Y.
7 A.M.
8 A.M.
10 A.M.
NOON
2 P.M.
4 P.M.
7 P.M.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Arrive
Chicago
9 A.M.
10 A.M.
NOON
2 P.M.
4 P.M.
6 P.M.
8 P.M.
Background Information –
continued

Each of Braneast’s crews lives in either New York or
Chicago.

Each day a crew must fly one New York-Chicago
flight and one Chicago-New York flight with at least
one-hour of downtime between flights.

For example, a Chicago-based crew could fly the 913 Chicago-New York flight and return on the 14-16
New York-Chicago flight.

This would incur a downtime of one hour. Braneast
wants to schedule crews so as to cover all flight and
minimize total downtime.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Solution

The trick is to set this up as an assignment model.
Because there are seven flights in each direction, we
need seven crews.

Each of the seven crews will be based in either
Chicago or New York.

We do not know at this point how many of each there
will be, but this information will be an output of the
model.

There are two sets of nodes and arcs, one for crews
based in Chicago and one for the crews based in
New York.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Solution – continued

Each such crew will be assigned two flights, the first
stating in Chicago, and the second starting in New
York.

Furthermore, such an assignment is possible only if
the flight from New York leaves at least one hour after
the flight from Chicago arrives in New York.

With this in mind, we can imagine an origin node
corresponding to the flight from Chicago, a
destination node corresponding to the flight from New
York, and an arc between them from origin to
destination.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Solution – continued

If the flow on this node is 1, a Chicago-based crew is
assigned to this pair of flights.

Of course, similar statements can be made for crews
based in New York.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
CREWSCHED.XLS

Once you understand the conceptual idea, the
implementation in Excel is fairly straightforward.

The completed model is shown on the next slide.

This file can be used to create the model.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Developing the Model

The model can be formed with the following steps:
– Enter inputs. Enter the given flight information in the ranges
B6:C12 and F6:G12. Because we plan to use this
information with lookup functions later on, we have named
the ranges A6:C12 and E6:G12 as C_Ltable and NY_Ltable.
The labels in columns A and E serve only to identify the
various flights.
– Find feasible assignments. To fill in the Chicago-based
crews section, find each early flight leaving from Chicago
that can be paired with a later flight leaving from New York
such that there is at least one hour of downtime in between.
Then enter the flight codes of all such pairs of flights in
columns A and B. Then do the same for the pairs that could
be handled by New York-based crews. Note that all this
information is entered in manually – there are no formulas
involved.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Developing the Model –
continued
– Downtimes for feasible assignments. Calculate the downtime
for each feasible pair of flights by using lookup functions to extract
the information form the flight schedules. Specifically, enter the
formula
=VLOOKUP(B17,NY_Ltable,2)-VLOOKUP(A17,C_Ltable,3)
in cell C17 and copy it down for other flight pairs starting in
Chicago. This simply subtracts the beginning time of the second
flight in the pair from the ending time of the first flight in the pair.
Similarly, enter the formula
=VLOOKUP(B28,C_Ltable,2)-VLOOKUP(A28,NY_Ltable,3)
in cell C28 and copy it down for other flight pairs in New York.
– Flows. Enter any flows in the Flows1 and Flows2 ranges in
column D. Remember that these will eventually be set at 0’s and
1’s, indicating that an assignments is either made or it isn’t.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Developing the Model –
continued
– Flow balance constraints. There is a node in the network
for each flight and a flow balance constraint for each node –
hence 14 flow balance constraints. However, things get a bit
tricky because each flight could be either the first or second
flight in a given flight pair. For example, consider flight 3C.
The network representation involving this flight appears
here.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Developing the Model –
continued
– Flight 3C is the later flight for the top two arrows, and it is the
earlier flight for the bottom arrow. Now comes the key
observation for this particular model. We require that flight
3C be flown exactly once, so exactly one of these arrows
must have flow 1 and the others must have flow 0.
Therefore, we add this node’s total inflow to its total outflow
and constrain this sum to be 1. To implement this in the
spreadsheet, enter the formulas
=SUMIF(Origins1,F17,Flows1) and
=SUMIF(Dests2,F17,Flows2) in cells G17 and H17, and
copy them to the range G18:H23 to take care of the flights
leaving from Chicago. Then enter the formulas
=SUMIF(Origins2,F24,Flows2) and
=SUMIF(Dests1,F24,Flows1) in cells G24 and H24, and
copy them to the range G25:H30 to take care of the flights
leaving form New York.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Developing the Model –
continued
– Finally, add these inflows and outflows in column I. As the
spreadsheet model indicates, we will eventually constrain
these sums to be 1.
– Total downtime. Calculate the total downtime in the
TotDowntime cell with the formula
=SUMPRODUCT(Downtimes1,Flows1)+SUMPRODUCT(D
owntimes2,Flows2).

Using Solver – The Solver dialog box should appear
as shown on the next slide. Note that the only
constraints are that the total flow into and out of each
node must be 1.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Developing the Model –
continued

This, plus the fact that network models with integer
inputs automatically have integer solutions, implies
that the flows on all arcs will be 0 or 1.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Solution – continued

The optimal solution that was shown indicates that
there should be two Chicago-based crews and five
New York-based crews.

This is because there are two 1’s in the Flows1 range
and five 1’s in the Flows2 range. These 1’s indicate
the crew assignments.

For example, one Chicago-based crew flies the 1C
and 4NY flights, and other flies the 2C and 7NY
flights. The total downtime for all seven crews in 26
hours.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Sensitivity Analysis

The only inputs to Braneast’s model are the flight
times, so we consider one possible sensitivity
analysis involving these flight times.

Suppose the 1C flight from Chicago is delayed by
one or more hours. How will this affect the optimal
solution?

We need to vary the input sections, as shown on the
next slide.

The original flight times are entered in cells L6 and
M6, a flight delay is entered in cell L9, and formulas
are entered in cells B6 and C8.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Sensitivity Analysis – continued

After running SolverTable, allowing the delay to vary
from 0 to 3 in increments of 1 and keeping track of
the total downtime and the numbers of crews based
in each city, we obtain the results in the table below.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Sensitivity Analysis – continued

But there is a subtle problem we forgot to take care
of.

The problem is that when we delay flight 1C, one or
more flight pairings that had at least one hour of
downtime originally might no longer have this minimal
required downtime.

In fact, Solver solution when the delay is 3 hours
schedules a crew to the pairing 1C-4NY. But this is
infeasible – the 1C flight gets into New York at times
13, and the 4NY flight leaves New York at time 12.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a
Sensitivity Analysis – continued

So the SolverTable solution reported for delays of 3
correspond to infeasible schedules.

Unfortunately there is no easy fix for running this
sensitivity analysis. Recall that we manually entered
the pairings.

To run this sensitivity analysis with SolverTable
correctly, we would need to modify the original model
so that Solver gets to choose from all possible
pairings, with the constraint that a pairing can be
chosen only if its downtime is at least 1.

The model would grow larger and somewhat more
complex, but it could be done.
5.1 | 5.2 | 5.3 | 5.4 | 5.5 | 5.7 | 5.8 | 5.9 | 5.10 | 5.10a