6.6 Service Network Design Problems
6.6.2 The linear fixed-charge network design model
The linear fixed-charge network design (LFCND) problem is a particular FCND problem in which the transportation costs per flow unitckij are constant (hence the objective function (6.27) is linear).
More formally, the LFCND model can be formulated as follows.
Minimize
k∈K
(i,j )∈A
cijkxijk +
(i,j )∈A
fijyij subject to
{j∈V:(i,j )∈A}
xijk −
{j∈V:(j,i)∈A}
xj ik =
oik, ifi∈O(k),
−dik, ifi∈D(k), 0, ifi∈T (k),
i∈V , k∈K, xijk ⩽ukij, (i, j )∈A, k∈K,
k∈K
xijk ⩽uijyij, (i, j )∈A, xijk ⩾0, (i, j )∈A, k∈K,
yij ∈ {0,1}, (i, j )∈A.
The LFCND problem is NP-hard and branch-and-bound algorithms can hardly solve instances with a few hundreds of arcs and tens of commodities. Since instances arising in applications are much larger, heuristics are often used. To evaluate the quality of the solutions provided by heuristics, it is useful, as already observed in Chapter 3, to compute lower bounds on the optimal objective function valuez∗LFCND. In the following, two distinct continuous relaxations and a simple heuristic are illustrated.
The weak continuous relaxation
The weak continuous relaxation is obtained by relaxing the integrality requirement on the design variables.
LONG-HAUL FREIGHT TRANSPORTATION 227 Minimize
k∈K
(i,j )∈A
cijkxijk +
(i,j )∈A
fijyij (6.31)
subject to
{j∈V:(i,j )∈A}
xijk −
{j∈V:(j,i)∈A}
xj ik =
oik, ifi∈O(k),
−dik, ifi∈D(k), 0, ifi∈T (k),
i∈V , k∈K, (6.32) xijk ⩽ukij, (i, j )∈A, k∈K, (6.33)
k∈K
xijk ⩽uijyij, (i, j )∈A, (6.34) xijk ⩾0, (i, j )∈A, k∈K, (6.35)
0⩽yij ⩽1, (i, j )∈A. (6.36)
It is easy to verify that every optimal solution of such a relaxation satisfies each constraint (6.34) as an equality since fixed costsfij,(i, j ) ∈ A, are nonnegative.
Therefore, design variablesyij,(i, j ) ∈ A, can be expressed as a function of flow variablesxijk,(i, j )∈A,k∈K:
yij =
k∈Kxijk
uij , (i, j )∈A.
Hence, the constraints (6.36) can be replaced by the following conditions:
k∈K
xijk ⩽uij, (i, j )∈A.
The relaxed problem (6.31)–(6.36) can be therefore equivalently formulated as follows.
Minimize
k∈K
(i,j )∈A
ckij+fij uij
xijk (6.37)
subject to
{j∈V:(i,j )∈A}
xijk −
{j∈V:(j,i)∈A}
xj ik =
oik, ifi∈O(k),
−dik, ifi∈D(k), 0, ifi∈T (k),
i∈V , k∈K, (6.38)
228 LONG-HAUL FREIGHT TRANSPORTATION xijk ⩽ukij, (i, j )∈A, k∈K, (6.39)
k∈K
xkij ⩽uij, (i, j )∈A, (6.40) xijk ⩾0, (i, j )∈A, k∈K. (6.41) The model (6.37)–(6.41) is a minimum-cost flow problem with|K|commodities.
Let LB∗wbe the lower bound onz∗LFCNDgiven by the optimal objective function value of the above relaxation.
The strong continuous relaxation
The strong continuous relaxation is obtained by adding the followingvalidinequalities xijk ⩽ukijyij, (i, j )∈A, k∈K, (6.42) to the LFCND model and removing the integrality constraints on the design variables yij,(i, j )∈A. Taking into account the fact that constraints (6.6.2) are dominated by constraints (6.42), and can therefore eliminated, the relaxed problem is as follows.
Minimize
k∈K
(i,j )∈A
cijkxijk +
(i,j )∈A
fijyij (6.43)
subject to
{j∈V:(i,j )∈A}
xijk −
{j∈V:(j,i)∈A}
xj ik =
oik, ifi∈O(k),
−dik, ifi∈D(k), 0, ifi∈T (k),
i∈V , k∈K, (6.44) xijk ⩽ukijyij, (i, j )∈A, k∈K, (6.45)
k∈K
xijk ⩽uijyij, (i, j )∈A, (6.46) xijk ⩾0, (i, j )∈A, k∈K, (6.47)
0⩽yij ⩽1, (i, j )∈A. (6.48)
Let LB∗s be the lower bound onz∗LFCNDgiven by the optimal objective function value of the relaxation (6.43)–(6.48). Such problem has no special structure and can therefore be solved by using any general purpose LP algorithm. By comparing the two continuous relaxations, it is clear that LB∗s is always better than, or at least equal to, LB∗w, i.e.
LB∗s ⩾LB∗w.
This observation leads us to label the former relaxation asweak, and the latter as strong. Computational experiments have shown that LB∗w can be as much as 40%
lower than LB∗s.
LONG-HAUL FREIGHT TRANSPORTATION 229 Table 6.7 Forecasted transportation demand of refrigerated goods
(pallets per day) in the FHL problem.
Bologna Genoa Milan Padua
Bologna — 3 8 2
Genoa 0 — 1 2
Milan 4 2 — 1
Padua 3 1 1 —
Table 6.8 Forecasted transportation demand of goods at room temperature (pallets per day) in the FHL problem.
Bologna Genoa Milan Padua
Bologna — 3 4 2
Genoa 1 — 1 0
Milan 6 2 — 2
Padua 1 1 1 —
FHL is an Austrian fast carrier located in Lienz, whose core business is the trans- portation of small-sized and high-valued refrigerated goods (such as chemical reagents used by hospitals and laboratories). Goods are picked up from manufacturers’ ware- houses by small vans and carried to the nearest transportation terminal operated by the carrier. These goods are packed onto pallets and transported to destination termi- nals by means of large trucks. Then, the merchandise is unloaded and delivered to customers by small vans (usually the same vans employed for pick-up). In order to make capital investment in equipment as low as possible, FHL makes use of one-way rentals of trucks. Recently, the company has decided to enter the Italian fast parcel transportation market by opening four terminals in the cities of Bologna, Genoa, Padua and Milan. This choice made necessary a complete revision of the service net- work. The decision was complicated by the need to transport the refrigerated goods by special vehicles equipped with refrigerators, while parcels can be transported by any vehicle. The forecasted daily average demand of the two kinds of products in the next semester is reported in Tables 6.7 and 6.8.
Between each pair of terminals, the company can operate one or more lines (see Figure 6.15). Vehicles are of two types:
• trucks with refrigerated compartments, having a capacity of 12 pallets and a cost (inclusive of all charges) of€0.4 per kilometre;
• trucks with room-temperature compartments, having a capacity of 18 pallets and a cost (inclusive of all charges) of€0.5 per kilometre.
230 LONG-HAUL FREIGHT TRANSPORTATION In addition, the company considers the possibility of transporting goods at room temperature through another carrier, by paying€0.1 per kilometre for each pallet. A directed graph representation of the problem is given in Figure 6.15.
Distances between terminals are reported in Table 6.9. The least-cost service net- work can be obtained as the solution of an LFCND model with|K| =22 commodities (one for each combination of an origin–destination pair with positive demand and a kind of product). LetA1andA2be the set of lines operated by means of trucks having capacity equal to 12 pallets and 18 pallets, respectively, and letA3be the set of lines operated by an external carrier. Arc parameters are
ckij =0, (i, j )∈A1, k∈K, fij =0.4 dij, (i, j )∈A1, uij =12, (i, j )∈A1, ckij =0, (i, j )∈A2, k∈K, fij =0.5, dij, (i, j )∈A2, uij =18, (i, j )∈A2,
ckij =0.1, dij, (i, j )∈A3, k∈K, fij =0, (i, j )∈A3,
uij = ∞, (i, j )∈A3, wheredijis the distance between terminalsiandj.
The LFCND formulation is as follows.
Minimize
k∈K
(i,j )∈A1∪A2∪A3
cijkxijk +
(i,j )∈A1∪A2∪A3
fijyij subject to
{j∈V:(i,j )∈A1∪A2∪A3}
xijk −
{j∈V:(j,i)∈A1∪A2∪A3}
xj ik =
oki, ifi∈O(k),
−dik, ifi∈D(k), 0, ifi∈T (k),
i∈V , k∈K,
k∈K
xkij, (i, j )∈A1∪A2∪A3, xijk ⩾0, (i, j )∈A1∪A2∪A3, k∈K
yij ∈ {0,1}, (i, j )∈A1∪A2∪A3.
LONG-HAUL FREIGHT TRANSPORTATION 231
3
2 1
4
Milan Padua
Bologna Genoa
Figure 6.15 Graph representation of the FHL service network design problem (in order to make the picture clean, a single edge is drawn for each pair of opposite arcs).
Table 6.9 Distances (in kilometres) between terminals in the FHL problem.
Bologna Genoa Milan Padua
Bologna 0 225 115 292
Genoa 225 0 226 166
Milan 115 226 0 362
Padua 292 166 362 0
The strong continuous relaxation gives a lower bound LB∗s =€534.60 per day. A branch-and-bound algorithm based on the strong continuous relaxation generates 696 nodes. The least-cost solution is reported in Figure 6.16 and has a cost of€886.70 per day. In the optimal solution, 7 lines are operated by 12-pallet trucks while a single line is operated by an 18-pallet truck (travelling from Bologna to Milan with 5 pallets of parcels). The number of pallets transported between each pair of terminals by means of 12-pallet trucks and by the external carrier is reported in Tables 6.10 and 6.11.
Add–drop heuristics
Add–drop heuristics are simple constructive procedures in which at each step one decides whether a new arc has to be used (add procedure) or an arc previously used has to be left out (drop procedure). Several criteria can be employed to choose which arc has to be added or dropped. In the following, a very simple drop procedure is illustrated. In order to describe such an algorithm, it is worth noting that a candidate optimal solution is characterized by the set A ⊆ Aof selected arcs. A solution
232 LONG-HAUL FREIGHT TRANSPORTATION Table 6.10 Number of pallets of parcels/number of pallets of refrigerated goods
transported by means of 12-pallet trucks in the FHL problem.
Bologna Genoa Milan Padua
Bologna — 4/8 — 4/8
Genoa 0/7 — 2/10 —
Milan — 5/7 — —
Padua 8/3 4/5 — —
Table 6.11 Number of pallets of parcels/number of pallets of refrigerated goods transported by an external carrier in the FHL problem.
Bologna Genoa Milan Padua
Bologna — 1/0 — —
Genoa — — 4/0 9/0
Milan — — — —
Padua — — — —
3
2 1
4
Milan Padua
Bologna Genoa
Figure 6.16 Transportation lines used in the optimal solution of the FHL problem.
is feasible if the LMMCF problem on the directed graphG(V , A)induced byAis feasible. If so, the solution cost is made up of the sum of the fixed costsfij,(i, j )∈A, plus the optimal solution cost of the LMMCF problem. Moreover, it is worth noting that the LFCND solution associated withA =A, if feasible, is characterized by a large fixed cost and by a low transportation cost. On the other hand, a feasible solution associated with a setAwith a few arcs is expected to be characterized by a low fixed cost and by a high variable cost. Consequently, an improved LFCND solution can be
LONG-HAUL FREIGHT TRANSPORTATION 233
3
2 1
4
Milan Padua
Bologna Genoa
Figure 6.17 Solution provided by the drop heuristic in the FHL problem.
obtained by iteratively removing arcs from the setA=A, while the current solution is still feasible and the total cost decreases. The drop procedure is as follows.
Step 1. Seth = 0 andA(h)=A. Let xijk,(h),(i, j ) ∈ A, k ∈ K, be the optimal solution (if any exists) of the LMMCF problem on the directed graphG(V , A(h)) and let z(h)LFCND be the cost of the associated LFCND problem. If the LMMCF problem is infeasible,STOP, the LFCND problem is also infeasible.
Step 2. For each arc(i, j )∈A(h), setA(h)=A(h)\ {(i, j )}and solve the LMMCF problem on the directed graph(G(V , A(h)). If all the LMMCF problems are infea- sible,STOP, the set of the arcsA(h)and the flow patternxijk,(h),(i, j )∈A(h),k∈K, are associated with the best feasible solution found; otherwise, let(v, w)be the arc whose removal fromA(h)allows us to attain the least-cost LFCND feasible solution.
Step 3. SetA(h+1) =A(h)\(v, w),h=h+1 and go back to Step 2.
The number of iterations of the algorithm is no more than the number of arcs and, at each iteration, Step 2 requires the solution ofO(|A|)LMMCF problems.
By applying the drop heuristic to the FHL problem, a solution having a cost equal to€899 per day is obtained (see Figure 6.17).