Distribution and Inventory at Ford
H. Donald Ratliff
John Vande Vate
Mei Zhang
1. AUTOMOBILE DELIVERY
Most new automobiles manufactured in the US are transported by rail from
manufacturing plants to special railroad centers called ramps and then by truck to local
dealers. This is typically a load-driven system. Newly assembled automobiles are parked
in load lanes at the plant according to their destination ramp. Whenever a sufficient
number of vehicles destined for a single ramp accumulates, the vehicles are loaded on a
railcar, which is dispatched into the “loose car network”. Typically, the railcars used to
transport automobiles to the ramps are tri-levels capable of carrying 15 sedans, 5 on each
deck.
The “loose car network” is itself a load-driven cross-docking system for railcars.
The cross-docks in this system are switching yards where railcars headed in the same
direction are sorted into trains. In crossing the country, a railcar may pass through half a
dozen switching yards before reaching its final destination.
At the ramps, vehicles are off-loaded from the railcars and parked to await
delivery to their designated dealerships. When a sufficient number of vehicles destined
for dealerships in a given area accumulates, the vehicles are loaded on a rig and
delivered. Car hauling rigs typically carry between 8 and 12 sedans.
Recently, Ford ran a pilot study to examine the potential for special cross-docking
centers (called mixing centers) in the rail network. The pilot included 5 plants in the
eastern US and 15 rail ramps in the west and mid-west (see Figure 1). Prior to the pilot,
each plant dispatched railcars to each ramp via the loose car network. In an effort to
reduce the 12-day average delivery time from the plants to the dealerships, Ford
introduced a mixing center in Kansas City, Missouri and routed all vehicles from the 5
plants to the 15 ramps through this mixing center.
1
Orillia
Laurel
Portland
Omaha
Denver
Salt Lake
Louisville
Kansas City
Benicia
St.Louis
Norfolk
Belen
Mira Loma
Oklahoma City
Amarillo
El Mirage
Reisor
Alliance
Atlanta
Houston
Plant
Ramp
Figure 1: Ford Pilot Study
1.4 Plant Operations
In a system operating without any mixing centers, newly produced automobiles
are parked in load lanes according to their destination ramps. When a full railcar load of c
automobiles accumulates in a load lane, these vehicles are loaded onto a railcar, which is
dispatched into the loose car network. Consequently, if the production rate of vehicles
destined for the ramp is relatively constant, the average inventory of vehicles in the load
lane at the plant is (c-1)/2. It is perhaps surprising to observe that this number is
independent of the volume of vehicles shipped to the ramp.
To better understand why the average number of vehicles in a load lane is
independent of the volume of vehicles shipped to the corresponding destination ramp,
consider the following example. Suppose a given plant produces p1 vehicles per day for
ramp 1, p2 vehicles per day for ramp 2 and p3 vehicles per day for ramp 3. Figure 2
illustrates the inventory of vehicles at the plant in the load lanes for these three ramps
over time. Assuming a railcar carries c automobiles, the number of automobiles waiting
in a load lane never exceeds c-1, for once it reaches c, the vehicles are loaded into a
railcar. Consequently, the inventory at the plant is a function of the railcar capacity c and
2
the number of load lanes. Consolidating shipments through mixing centers reduces this
inventory by reducing the number of load lanes.
With a small supply rate (e.g., p1) it takes a longer time to build up a load. In
particular, vehicles destined for ramp i (with supply rate pi) will wait (c-1)/2pi on average
for a load to accumulate. Thus, while the average number of vehicles in each of the three
load lanes will be the same, (c-1)/2, the average times vehicles in these load lanes spend
waiting for a load to accumulate will be quite different. The total delay incurred waiting
in a load lane, however, is the product of the average delay per vehicle and the number of
vehicles incurring that delay. Thus, we have the following key observation
Observation: In a load-driven cross-docking system, the total delay incurred waiting for
transportation in a lane depends only on the capacity of the transportation units used on
the lane.
In our example, the total time delay incurred waiting for transportation is the same in all
three load lanes. It is simply (c-1)/2.
# autos
p1
c-1
1/2
Time
p2
c-1
1/2
Time
p3
c-1
1/2
Time
Figure 2: Inventory in a Load Lane
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2.1 Mixing Center Operations
At the mixing center, automobiles are unloaded from arriving railcars into load
lanes according to their destination ramps. When sufficient vehicles accumulate for a
given destination ramp, they are loaded onto an empty railcar and sent on. Any
remaining vehicles wait at the mixing center.
Thus, a mixing center serves as a load-driven cross-dock. Like all cross-docks, it
introduces additional handling of the product in order to reduce overall transportation
costs and, more importantly in this case, transportation time.
Routing shipments through a mixing center can reduce transportation time and
cost in two ways:
Faster Mode: Consolidating shipments out of each plant destined for a number of ramps
to a single mixing center can generate sufficient volume on the channel to warrant using
faster unit trains. Unit trains, consisting of 20 or more railcars with a common
destination, move directly from the plant to the destination ramp bypassing the switching
yards. A mixing center serving several plants can have a similar effect on shipments to
the ramps. Consolidating the different plants’ shipments to a ramp can facilitate the use
of unit trains.
Reduced Wait: Because the daily supply rates to some ramps (e.g., Laurel, Montana) are
much smaller than the capacity of a railcar, automobiles destined for these ramps may
wait several days for a full load to accumulate. Consolidating shipments through a
mixing center can eliminate these delays. Further, the number of vehicles waiting at the
plant influences both the average delivery time and the size of the lot at the plant.
Typically, automobile manufacturing plants are surrounded by suppliers’ facilities and, as
a result, land is scarce and expensive. Reducing the number of vehicles waiting at the
plant frees up valuable land for more productive uses.
In the remainder of this section, we discuss the effects of different operating policies at
the mixing center.
2.2 Equipment Balance Strategy
The equipment balance strategy is not a true load-driven strategy. It is designed to
ensure a predictable workload and to simplify the handling of railcars at the mixing
center. We say that a mixing center is fully balanced if each arriving railcar that is
unloaded can be reloaded with vehicles all destined for a common ramp. In this way, no
empty railcars are brought into or taken out of the mixing center (i.e., railcars come in
full and depart full). Note that if a mixing center is operated in a fully balanced manner it
will maintain a constant inventory level. The following lemma characterizes the
inventory of vehicles at the mixing center required to ensure the center can be fully
balanced.
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Lemma: A mixing center with (c-1)(r-1) vehicles, where c is the capacity of a railcar and
r is the number of ramps the center serves, can be operated in a fully balanced manner.
Proof. Consider a mixing center with (c-1)(r-1) vehicles in inventory. After a railcar is
unloaded, the number of vehicles at the center increases to (c-1)r+1. Since there are only
r destination ramps, there must be at least one to which c or more vehicles are headed.
Once c vehicles destined to this ramp are loaded on the railcar, the inventory at the
mixing center returns to (c-1)(r-1). □
Starting with no vehicles in inventory at the mixing center, one natural method for
building up the inventory required to support fully balanced operations is to take a railcar
away empty if there is no destination ramp with at least c waiting vehicles. It may take a
long time to build up an inventory of (c-1)(r-1) vehicles, but in the meantime, we only
need to deal with empty railcars leaving the mixing center. We will never have to worry
about delivering empty railcars to the center.
When operating in a fully balanced manner, every railcar on each arriving train
will be unloaded and then reloaded. To predict the workload at the mixing center, we
only need to know the number of railcars arriving. Thus, operating in a fully balanced
manner ensures a predictable workload at the mixing center without requiring detailed
knowledge of the vehicles on arriving trains. Note however, that operating in a fully
balanced manner, we may actually have a number of destinations with more than a full
railcar load of vehicles waiting at the mixing center. This will influence both the average
delivery time of vehicles and the size of the lot at the mixing center.
The average delay per vehicle incurred waiting for transportation at the mixing
center will be (c-1)(r-1)/P, where P is the total rate at which vehicles arrive at the center.
The total delay incurred waiting for transportation at the mixing center will simply be
(c-1)(r-1).
1.3 Minimum Inventory Strategy
The minimum inventory strategy is a true load-driven strategy. It attempts to
minimize the inventory of vehicles at the mixing center by bringing in empty railcars
whenever necessary to handle all available full loads. The maximum inventory at the
mixing center under this strategy is clearly (c-1)r. Typically, the inventory level at a
mixing center operating under the minimum inventory strategy will be close to half this
maximum level or (c-1)r/2. From our Observation, we know that the total delay incurred
waiting for transportation to each ramp served by the mixing center will be (c-1)/2.
Note that although the average inventory level of the minimum inventory strategy
will be significantly smaller than that of the equipment balance strategy, the center must
be large enough to handle (c-1)r vehicles. Further, under the minimum inventory strategy,
neither the inventory at the mixing center nor the workload will be easily predictable.
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The center will also need to maintain an inventory of empty railcars. We will add
to this inventory when there are insufficient loads at the mixing center to fill all the newly
arrived railcars and draw from it whenever extra empty railcars are required to handle the
loads.
3. MIXED INTEGER PROGRAMMING MODEL
There are two basic sets of decisions in designing a “load-driven” cross-docking
network: the location decisions, which deal with the number and positioning of crossdocks, and the routing decisions, which deal with how flow should be routed through the
selected cross-docks. Our objective is to minimize the average delay between the time a
vehicle is produced and the time it reaches its destination ramp. The two components of
this delay are the transportation delay (i.e., the time spent travelling) and the loading
delay (i.e., the time spent on waiting to be loaded on transportation units).
The total transportation delay incurred on a lane is simply the product of the
transportation time on that lane and the number of vehicles incurring that time.
As we have seen, with a constant supply rate, the total loading delay in each load
lane at the plant is (c-1)/2. The total loading delay in each load lane at the mixing center,
on the other hand, depends on the operating strategy. Under a minimum inventory
strategy, the average is (c-1)/2. Under the equipment balance strategy, however, we can
only describe the total over all the load lanes: It will be (c-1)(r-1). If we approximate the
total delay on each load lane to be (c-1)(r-1)/r, we will accurately capture the total delay
incurred at the mixing center under this strategy.
In this section, we model the problem of designing a load-driven cross-docking
network in which the mixing centers operate under the minimum inventory strategy as a
variant of the fixed charge design model (Magnanti, 1984). Our model assumes that
vehicles are either routed directly from the plant to the ramp or are routed from the plant
to a mixing center and from there to the ramp. Later, in Section 5, we generalize the
model to multi-tiered distribution systems in which vehicles may pass through several
mixing centers.
We let tij denote the travel time and fij the total loading delay incurred by vehicles
moving from point i to point j. Further, we assume the average supply rate from plant p
to ramp r, denoted by spr, is known. This assumption is consistent with North American
automobile distribution in so far as typically all or nearly all production of a given model
occurs at a single plant. In other settings, this assumption is justified when demand
through each distribution center is assigned to a single plant.
The variables in our model are:

xpr, the number of vehicles moving directly from plant p to ramp r per unit time,
6

ypcr, the number of vehicles moving from plant p to ramp r via mixing center c per
unit time and

zij, indicating whether or not any vehicles move directly from point i to point j.
The objective of our model is to minimize the average delay between the time a
vehicle is produced and the time it reaches its destination ramp. The first constraint
ensures that all deliveries are made. Constraints (2)-(4) enforce lane selection: no
vehicles can be shipped on a lane unless the delay on that lane is incurred.
min Σ{fpr zpr + tpr xpr: all plants p and ramps r} +
Σ{fpc zpc : all plants p and centers c} + Σ{fcr zcr : all centers c and ramps r} +
Σ{(tpc + tcr)ypcr : all plants p, centers c and ramps r}
s. t.
Σ {ypcr : all centers c} + xpr = spr for each plant p and ramp r
(1)
ypcr ≤ spr zpc for each plant p, center c and ramp r
(2)
ypcr ≤ spr zcr for each plant p, center c and ramp r
(3)
xpr ≤ spr zpr for each plant p and ramp r
(4)
xpr, ypcr ≥ 0 for each plant p, center c and ramp r
(5)
zij {0, 1} for each lane ij
(6)
Given potential sites for cross-docking centers, the model determines which of the
centers should be open and routes the flows from each plant to each rail ramp to
minimize the overall average delay.
4. COMPUTATIONAL RESULTS
To test the model, sample problems approximating the Ford new car network
were randomly generated. In these examples, the transportation delay was assumed to be
proportional to Euclidean distance. Random flow rates were generated for each
supply/demand pair again trying to be consistent with Ford production rates. Table 1
shows the computational results for a representative set of the problems tested. Of the 5
problems presented here, one problem required 3 branch-and-bound nodes to reach
optimality. For the rest, the LP relaxation provided an integral optimal solution.
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Number Number
Number
of
of
Number of
of Plants Centers Ramps Variables
1
2
3
4
5
10
10
25
25
30
15
15
10
10
15
30
30
40
40
60
Number of
Constraints
4800+900
4800+900
11000+1650
11000+1650
28800+3150
9600
9600
22000
22000
59600
Number Branch CPU
IP
of Open & Bound Time
Solution Centers Nodes (Sec)
236786.7
131784.2
556606.4
725904.7
1316414
5
3
6
6
10
3
0
0
0
0
38
54
297
168
1365
Table 1: Example Problems in Single Transportation Mode Network
Problems 1 and 2 have the same number of plants, centers and the ramps but they
differ in the actual locations of the nodes. The same is true for Problem 3 and Problem 4.
The fifth column in Table 1, indicates the number of continuous variables and the number
of binary variables, e.g., Problem 1 has 4,800 continuous variables and 900 binary
variables. The seventh column is the MIP optimal obtained using CPLEX 5.0. The next
column is the number of open centers in an optimal solution followed by the number of
branch-and-nodes required to find and prove the optimality of that solution. The last
column is the CPU time on an IBM RS6000 (Series 590) workstation. Problems 3 and 4
most closely approximate Ford’s new car distribution system in 1996. Branch-and-bound
requires so so few nodes to solve these problems because the binary variables are
strongly interrelated.
5. EXTENSIONS
As we mentioned earlier, consolidating flow on a channel in the automobile
delivery network can facilitate the use of faster unit trains. This faster transportation
mode involves more inventory (albeit on loaded railcars) and a longer loading delay as
the train cannot depart until a sufficient number of railcars are loaded. The average
inventory of waiting automobiles is approximately one-half the capacity of the train. The
model above extends in a straight forward way by simply allowing additional arcs for
each link where building a unit train is possible.
The following is the result of some randomly generated problems for the
automobile distribution system with optional unit trains between every pair of ship
points. Unit trains typically take 25-50 railcars and can travel up to 5 times faster than
individual railcars in the loose car network. For a unit train with 25 railcars, the fixed
delay for a link using a unit train is (25*15-1)/2 = 187. The problems in Table 2 are the
same as Table 1 but with both unit trains and loose cars allowed between every pair of
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ship points in the network. The objective values of all problems are reduced because in
some part of the network unit trains are used. Because the multi-mode networks are
much bigger than that of the single mode problems, the problems all required more CPU
time, especially the two larger problems.
Number Number Number
of
of
of
Plants Centers Ramps
1
2
3
4
5
10
10
25
25
30
15
15
10
10
15
30
30
40
40
60
Number of
Variables
Number of
IP
Constraints Solution
18600+1800
18600+1800
42000+2650
42000+2650
111600+6300
23400
23400
53000
53000
140400
Number of Branch CPU
Open
& Bound Time
Centers
Nodes (Sec)
167014.1
118019.5
333167.9
385276.3
632073.6
3
3
1
1
1
0
0
0
0
0
1442
975
5631
5281
36904
Table 2: Example Problems with Two Transportation Modes
Level 2
CD centers
Level 1
CD center
demand
nodes
supply
nodes
Figure 3: Multi-level Cross-docking
Our model extends in the natural way to distribution systems with more than one
tier of cross-docks as illustrated in Figure 4. To accommodate the possibility of passing
through more than one cross-dock in moving from a plant to a ramp, we extend the
variables ypcr to include all paths from the plants to the ramps including those that use
more than one ramp. For larger problems, these variables can be generated via standard
column generation techniques. Table 3 contains computational results for this extension
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of our model to networks with up to two tiers of cross-docks. In all of these problems the
LP relaxation provided an optimal integral solution. The optimal solutions to Problem 1
and Problem 2 include paths using both levels of cross-docking as well as paths using
only a single level of cross-docking. The last column of Table 3 is the CPU time on the
same IBM RS6000 (Series 590) workstation.
Number
of Plants
1
2
3
10
10
15
Number of Number of Number
Level 1
Level 2
of
Centers
Centers
Ramps
10
10
15
10
10
15
30
40
40
Number of
Variables
Number of
Constraints
36300+1200
48400+1500
153600+2475
42600
56800
172200
Branch & CPU
Bound
Time
Nodes (Secs)
0
0
0
964
3153
20877
Table 3: Example Problems with Two Levels of Cross-docking
Acknowledgements: The authors would like to thank Biff Wilson and the people at
Allied Systems for bringing the Ford pilot study to our attention and for providing advice
and data supporting the development and testing of our model.
REFERENCES
1. Donaldson, Harvey, Ellis L. Johnson, H. Donald Ratliff, and Mei Zhang (1998),
“Network Design for Schedule-Driven Cross-Docking Systems, Georgia Tech TLI
Report.
2. Magnanti, T. L. (1984) “Network Design and Transportation Planning: Models and
Algorithms,” Transportation Science 18(1), 1-55.
3. Mirchandani, Pitu B., and Richard L. Francis (1990) “The Uncapacitated Facility
Location Problem,” Discrete Location Theory, John Wiley & Sons, Inc.
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