Hardcastle is a North European leader in intermodal transportation. In 2001 the com- pany operated nearly 240 000 containers, with an annual transportation cost of about 50 million euros.
Like other intermodal transportation companies, Hardcastle manages both full and empty containers. When a customer places an order for freight transportation, Hard- castle sends one or several empty containers of the appropriate type in terms of size, refrigeration, etc., to the pick-up point (see Figure 8.12). The containers are then loaded and sent to destination using a combination of modes (e.g. railway and sea transportation). At the destination, the containers are emptied and sent back to the company unless there is an outgoing load requiring the same kind of container (corre- sponding tocompensationbetween the demand and the supply of empty containers).
Unless compensation is possible, empty containers are then moved to a new pick- up point. Relocating empty containers is a resource-consuming activity whose cost should be kept at minimum. Unfortunately,compensationbetween the demand and the supply of empty containers is seldom possible for three main reasons:
• the origin–destination demand matrix is strongly asymmetrical (some loca-
318 LINKING THEORY TO PRACTICE
Consolidated load transfer Collection
Distribution Independent transfer
Rio de Janeiro
Brasilia San Paolo
(Depot) Genoa (Depot)
Vercelli Novara
Milan
Buenos Aires
Figure 8.12 Freight transportation at Hardcastle.
tions are mainly sources of materials while some others are mainly points of consumption);
• at a given location, the demand and supply for empty containers do not usually occur at the same time;
• containers may have a large number of sizes and features; as a result, it is unlikely that the containers incoming at a customer facility are suitable for outgoing goods.
For these reasons, the compensation between demand and supply is neglected in the following.
Because of the economies of scale in transportation, it is not convenient to move containers directly from supply to demand points. Instead, containers are sent to a nearby warehouse. Then, on a weekly basis, convoys of empty and full containers are moved between warehouses (see Figure 8.13). Warehouses are often public so that their location can easily be changed if necessary. Prior to its redesign, the logistics system contained 87 depots (64 close to a railway station and 23 close to a sea terminal). Moreover, empty container movements accounted for nearly 40% of the total freight traffic.
The management of the empty containers is a complex decision process made up of two stages (see Figure 8.14):
• at a tactical level, one has to determine, on the basis of forecasted origin–
destination transportation demands, the number and locations of warehouses, as well as the expected container flows among warehouses;
• at an operational level, shipments are scheduled and vehicles are dispatched on the basis of the orders collected and of short-term forecasts.
LINKING THEORY TO PRACTICE 319
Full containers Empty containers
Boston (Customer)
New Haven (Depot) New York
(Destination port)
Genoa (Origin port)
Pavia (Producer)
Milan (Depot)
Figure 8.13 Empty container transportation at Hardcastle.
Depot location Customer assignment
to depots
Empty container allocation Full and empty container routing
Orders Short-term
forecasts Medium-term
forecasts
Figure 8.14 Main decisions when managing containers at Hardcastle.
In order to redesign its logistics system, Hardcastle aggregated its customers into 300 demand points. Further, containers types were grouped into 12 types. LetCbe the set of customers,D the set of potential depots,P the set of different types of containers,fj,j ∈ D, the fixed cost of depotj,aijp, i ∈ C,j ∈ D,p ∈ P, the transportation cost of a container of typepfrom customerito depotj;bijp, i∈C, j ∈ D,p ∈ P, the transportation cost of a container of type p from depotj to customeri;cj kp,j ∈D,k∈D,p∈P, the transportation cost of an empty container of typepfrom depotj to depotk;dip, i ∈C,p∈P, the number of containers of typeprequested by the customeri;oip, i ∈C,p∈ P, the supply of containers of typep from customeri. Furthermore, letyj,j ∈ D, be a binary decision variable
320 LINKING THEORY TO PRACTICE equal to 1 if the depotj is selected, and 0 otherwise;xijp, i∈C,j ∈D,p∈P, the flow of empty containers of typepfrom customerito depotj;sijp, i∈C,j ∈D, p ∈ P, the flow of empty containers of typepfrom depotj to customeri;wj kp, j ∈D,k∈D,p∈P, the flow of empty containers of typepfrom depotj to depot k. The problem was formulated as follows.
Minimize
j∈D
fjyj+
p∈P
i∈C
j∈D
(aijpxijp+bijpsijp)+
j∈D
k∈D
cj kpwj kp
(8.9) subject to
j∈D
xijp=oip, i∈C, p∈P , (8.10)
j∈D
sijp=dip, i∈C, p∈P , (8.11)
i∈C
xijp+
k∈D
wkjp−
i∈C
sijp−
k∈D
wj kp=0, j ∈D, p∈P , (8.12)
p∈P
i∈C
(xijp+sijp)+
p∈P
k∈D
(wj kp+wkjp)
⩽yj
p∈P
i∈C
(oip+dip+2M), j ∈D, (8.13)
xijp⩾0, i∈C, j ∈D, p∈P , (8.14) sijp⩾0, i∈C, j ∈D, p∈P , (8.15) wj kp⩾0, j ∈D, k∈D, p∈P , (8.16) yj ∈ {0,1}, j ∈D, (8.17) whereMis an upper bound on thewj kpflows,j ∈D,k ∈ D,p ∈P. The objec- tive function (8.9) is the sum of warehouse fixed costs and empty container vari- able transportation costs (between customers and warehouses, and between pairs of warehouses). Constraints (8.10)–(8.12) impose empty container flow conservation.
Constraints (8.13) state that ifyj =0,j ∈D, then the incoming and outgoing flows from sitej are equal to 0. Otherwise, constraints (8.13) are not binding since
xijp ⩽oip, i∈C, j ∈D, p∈P , sijp ⩽dip, i∈C, j ∈D, p∈P , wj kp ⩽M, j ∈D, k∈D, p∈P .
The implementation of the optimal solution of model (8.9)–(8.17) yielded a reduction in the number of warehouses to 48 and a 47% reduction in transportation cost.
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