1. Locating Locomotive Refueling Stations Problem Description Kamalesh Somani, on behalf of RAS 2010 Problem Solving Competition Committee November 06, 2010

2. Operational Definitions Fuel trucks – Trucks which are used to fuel locomotives at a given railroad yard Train schedule - A train schedule defines — the sequence of yards in which a train stops on its route from origin to destination — the departure and arrival time and day — days of operation Train-start – Same train operated on different days of operation. If a train operates 3 days per week, say Monday, Wednesday and Friday, then the train-starts for that train at its origination yard are Monday, Wednesday and Friday

3. Background Every day, locomotives are assigned to trains, and along the way they are fueled at one or more of many fueling locations Fuel expenses are a significant part of any railroad's operating costs Fuel delivery costs differ from location to location because of the differences in distribution, marketing costs and other factors A railroad faces the problem of identifying a cost effective plan to fuel the locomotives that power its trains

4. Assumptions: Fuel Location All the fueling needs to be at a railroad yard. A yard may not have any fueling facility at all. The fueling locations can have one or more fuel trucks Individual fuel truck only serve one yard and available on 24/7 schedule The trucks have capacity constraints, A truck can fuel 25000 gallons of fuel per day The price of fuel per gallon is given for each railroad yard Fueling is instantaneous Truck weekly contracting cost is $4000

5. Assumptions: Trains The train schedule is fixed and known – it is repeated every week The locomotive requirement on each train is fixed and known (on locomotive id to train id level) and route is given as a cycle We consider a single locomotive type. The locomotive fuel tank capacity is 4500 gallons Fuel consumption rate is 3.5 miles/gallon Fixed cost of waiting for fueling and refueling (includes origin of the train, fueling is not allowed at destination) per stop is $250. Maximum Number of fueling and refueling during a train-start (not including origin) is 2. The planning horizon is 2 weeks

6. Problem Details Objective — Minimize the total fueling cost (contract cost of truck, fuel cost per gallon) Decision variables — Number of fuel trucks (0 or more) to position at each railroad yard — Number of gallons of fuel to add to individual locomotive at a railroad yard on its route Constraints — Fuel truck capacity (max 25000 gallons of fuel per day) — No locomotive can run out of fuel on line-of-road — Each train-start can stop at most 2 times for refueling (it does not include origin and fueling is not allowed at destination)

7. Problem Instance Partial train network taken from a real railroad 214 trains 214 locomotives 73 railroad yards