UV1318
JCG GLOBAL AIR SERVICES
Teaching Note
Synopsis
A pilot/manager in a large corporate aviation department must formulate a fueling plan
for an upcoming four-leg trip. Considerations for the fueling plan include different fuel prices at
each airport (fuel is much cheaper if bought at home) and ramp fees levied at destination airports
if fuel purchases fail to meet specified minimums. Other considerations include aircraft weight
limits for takeoffs and landings (with fuel being a large component of aircraft weight), fuel tank
capacity, and a company policy specifying a minimum amount of fuel required at each landing.
Objectives
To illustrate the use of mixed linear/integer programming to solve an exciting, easy-to-
understand business problem.
To develop skill in building Solver models.
Assignment Questions
1. Formulate a fuel plan for the upcoming trip.
2. Advise Sam Bursk on how to improve the fuel planning process.
Analysis
The case illustrates a situation for which mixed linear/integer programming is very well
suited. There is a clear objective—minimizing total fuel and ramp fee costs. Procedures adopted
by the company (e.g., aircraft must arrive with more than 2,400 pounds of fuel) and limitations
specified by the aircraft manufacturer (e.g., the aircraft must not weigh more than 36,400 pounds
, -2- UV1318
after fueling) lead to clearly defined constraints. The trickiest part of the model is the handling of
the ramp fees. Instead of using =IF() statements to account for ramp fees, it is better to use binary
decision variables. With binary decision variables for the ramp-fee payment decisions, a
calculation of minimum fuel purchase quantities (0 if ramp fee paid, the specified minimum if
not), and a constraint requiring that the amount of fuel purchased be greater than the calculated
minimums (after accounting for the ramp fee decision), the model is mixed integer/liner.
Exhibit TN1 shows a model and the optimal solution. The optimal plan is an interesting
one. The low fuel price at Moline, Illinois (KMLI), means we want to fill up the 13,000-pound
tank by purchasing (uploading) 6,000 pounds of fuel at KMLI. With only two passengers on the
first leg, the ramp weight of the aircraft is below the specified limit even with the full tank of
fuel. The 4,800 pounds of fuel burned on the first leg means the aircraft will land at KBOS 1,000
pounds lighter than the 31,800 pound landing limit. We avoid the expensive fuel at KBOS and
pay the $800 ramp fee. At KTEB, we take on just enough fuel to avoid the $450 ramp fee. At
KDAL, we upload enough fuel to get us back to KMLI with the minimum 2,400 pounds of fuel.
The 687 gallons uploaded at KDAL is enough to avoid the ramp fee.
For comparison purposes, Exhibit TN1 evaluates a “no tankering” plan. The algorithm
for determining this plan is to buy the amount of fuel at each stop such that you arrive at the next
with exactly 2,400 pounds. Included in the total cost and ramp-fee figure of $14,827 is the cost
required to fuel the aircraft back to 7,000 pounds in KMIL. The optimal plan saves the firm
almost $3,181 over the “no tankering” plan.
Clearly, the company procedure requiring 2,400 pounds of fuel upon arrival is a
convenient input to our model. Given uncertainty in fuel burn rates, there is no guarantee we will
actually arrive with 2,400 pounds. The 2,400 pounds is a planning figure. When weather
conditions are bad or unpredictable, the company raises the 2,400 requirement.
The CE750 limitations are also convenient assumptions for the model. I expect that the
risk associated with landing does not increase dramatically at exactly 31,800 pounds. But given
that this is the way the company operates, our model finds the fuel plan that minimizes total cost.
One limitation of the model is that it ignores the fact that fuel burn rates depend on the
weight of the plane. The company uses software to optimize the route flown on any leg in order
to minimize fuel use. The software accounts for current weather (wind speeds) as well as the
weight of the plane upon departure when it formulates the optimal route and reports the expected
amount of fuel burned. In practice, pilots revise the fuel burn numbers in column G if the fuel
burn numbers from the route optimizer are noticeably different from those in column G. Of
course, the new fuel burn numbers will result in a modified fuel plan, which will lead to
modified departure weights. If the new departure weights are markedly different, the pilots will
rerun the route optimization software. Note that there is the potential for small changes in fuel
burn numbers to effect big changes in departure weights. This can happen, for example, if the
change in burn numbers causes the model to shift to a dramatically different optimal solution.
JCG GLOBAL AIR SERVICES
Teaching Note
Synopsis
A pilot/manager in a large corporate aviation department must formulate a fueling plan
for an upcoming four-leg trip. Considerations for the fueling plan include different fuel prices at
each airport (fuel is much cheaper if bought at home) and ramp fees levied at destination airports
if fuel purchases fail to meet specified minimums. Other considerations include aircraft weight
limits for takeoffs and landings (with fuel being a large component of aircraft weight), fuel tank
capacity, and a company policy specifying a minimum amount of fuel required at each landing.
Objectives
To illustrate the use of mixed linear/integer programming to solve an exciting, easy-to-
understand business problem.
To develop skill in building Solver models.
Assignment Questions
1. Formulate a fuel plan for the upcoming trip.
2. Advise Sam Bursk on how to improve the fuel planning process.
Analysis
The case illustrates a situation for which mixed linear/integer programming is very well
suited. There is a clear objective—minimizing total fuel and ramp fee costs. Procedures adopted
by the company (e.g., aircraft must arrive with more than 2,400 pounds of fuel) and limitations
specified by the aircraft manufacturer (e.g., the aircraft must not weigh more than 36,400 pounds
, -2- UV1318
after fueling) lead to clearly defined constraints. The trickiest part of the model is the handling of
the ramp fees. Instead of using =IF() statements to account for ramp fees, it is better to use binary
decision variables. With binary decision variables for the ramp-fee payment decisions, a
calculation of minimum fuel purchase quantities (0 if ramp fee paid, the specified minimum if
not), and a constraint requiring that the amount of fuel purchased be greater than the calculated
minimums (after accounting for the ramp fee decision), the model is mixed integer/liner.
Exhibit TN1 shows a model and the optimal solution. The optimal plan is an interesting
one. The low fuel price at Moline, Illinois (KMLI), means we want to fill up the 13,000-pound
tank by purchasing (uploading) 6,000 pounds of fuel at KMLI. With only two passengers on the
first leg, the ramp weight of the aircraft is below the specified limit even with the full tank of
fuel. The 4,800 pounds of fuel burned on the first leg means the aircraft will land at KBOS 1,000
pounds lighter than the 31,800 pound landing limit. We avoid the expensive fuel at KBOS and
pay the $800 ramp fee. At KTEB, we take on just enough fuel to avoid the $450 ramp fee. At
KDAL, we upload enough fuel to get us back to KMLI with the minimum 2,400 pounds of fuel.
The 687 gallons uploaded at KDAL is enough to avoid the ramp fee.
For comparison purposes, Exhibit TN1 evaluates a “no tankering” plan. The algorithm
for determining this plan is to buy the amount of fuel at each stop such that you arrive at the next
with exactly 2,400 pounds. Included in the total cost and ramp-fee figure of $14,827 is the cost
required to fuel the aircraft back to 7,000 pounds in KMIL. The optimal plan saves the firm
almost $3,181 over the “no tankering” plan.
Clearly, the company procedure requiring 2,400 pounds of fuel upon arrival is a
convenient input to our model. Given uncertainty in fuel burn rates, there is no guarantee we will
actually arrive with 2,400 pounds. The 2,400 pounds is a planning figure. When weather
conditions are bad or unpredictable, the company raises the 2,400 requirement.
The CE750 limitations are also convenient assumptions for the model. I expect that the
risk associated with landing does not increase dramatically at exactly 31,800 pounds. But given
that this is the way the company operates, our model finds the fuel plan that minimizes total cost.
One limitation of the model is that it ignores the fact that fuel burn rates depend on the
weight of the plane. The company uses software to optimize the route flown on any leg in order
to minimize fuel use. The software accounts for current weather (wind speeds) as well as the
weight of the plane upon departure when it formulates the optimal route and reports the expected
amount of fuel burned. In practice, pilots revise the fuel burn numbers in column G if the fuel
burn numbers from the route optimizer are noticeably different from those in column G. Of
course, the new fuel burn numbers will result in a modified fuel plan, which will lead to
modified departure weights. If the new departure weights are markedly different, the pilots will
rerun the route optimization software. Note that there is the potential for small changes in fuel
burn numbers to effect big changes in departure weights. This can happen, for example, if the
change in burn numbers causes the model to shift to a dramatically different optimal solution.
