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.


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 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.

Figure 2: Inventory in a Load Lane

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.

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.

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 cross-docks, 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,
  • 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.

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}

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.


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 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.


Table 2: Example Problems with Two Transportation Modes


Figure 3: Multi-level Cross-docking