Intexpress is a firm whose core business is express freight delivery all over North America. The services provided to customers are (a) delivery within 24 hours (next day service); (b) delivery within 48 hours (second day service); (c) delivery within 3–5 days (deferred service). In order to provide quick deliveries, long-haul transportation is made by plane.
326 LINKING THEORY TO PRACTICE
L H
Figure 8.17 Roll packing in the Waterworld example.
The Intexpress logistics system comprises a set ofshipment centres(SCs), of a single hub, of a fleet of airplanes and of a fleet of trucks. Loads are consolidated both in the SCs and in the hub. In particular, goods originating from the same SC are transported as a single load to the hub while all the goods assigned to the same SC are sent jointly from the hub. Freight is first transported to an originating SC, where it is consolidated; it is then transported to a final SC by air, by truck (ground service), or by using a combination of the two modes. Finally, freight is moved from the final SC to the destination by truck. Of course, an SC to SC transfer by truck is feasible only if distance does not exceed a given threshold. Air transportation is performed by a company-owned fleet (dedicated air service) or by commercial airlines (commercial air service). The outgoing freight is collected in the evening and delivered the morning after. Every day a company-owned aircraft leaves the hub, makes a set of deliveries, then travels empty from the last delivery point to the first pick-up point, where it makes a set of pick-ups and finally goes back to the hub. Each SC is characterized by an ‘earliest pick-up time’ and by a ‘latest delivery time’. Moreover, all arrivals in the hub must take place before a pre-established arrival ‘latest delivery time’ (cut-off time, COT) in such a way that incoming airplanes can be unloaded, sorted by destination, and quickly reloaded on to outgoing aircraft.
Since it is not economically desirable nor technically feasible that airplanes visit all SCs, a subset of SCs must be selected as aircraft loading and unloading points (air- stops, ASs). An SC that is not an AS is connected to an AS by truck (ground feeder route). Figure 8.18 depicts a possible freight route between an origin–destination pair.
Commercial air services are less reliable than dedicated air services and their costs are charged depending on freight weight. These are used when either the origin and its closest SC are so far apart that no quick truck service is possible, or when the overall transportation demand exceeds the capacity of company-owned aircraft.
Planning the Intexpress service network consists of determining
• the set of ASs served by each aircraft of the firm;
LINKING THEORY TO PRACTICE 327
Consolidate land shift Local pick-up/delivery Shift with a dedicated aircraft
Origin SC
SC
AS AS
AS
AS
AS Hub
Destination
Figure 8.18 A possible freight route between an origin–destination pair at Intexpress.
• truck routes linking SCs (which are not ASs) to ASs;
• the transportation tasks performed by commercial airlines.
The objective pursued is the minimization of the operational cost subject to ‘earliest pick-up time’ and ‘latest delivery time’ constraints at SCs, to the COT restriction, etc.
The solution methodology used by Intexpress is made up of two stages: in the first stage, the size of the problem is reduced (preprocessing phase) on the basis of a qualitative analysis; in the second stage, the reduced problem is modelled and solved as an IP program. In the first stage,
• origin–destination pairs that can be serviced by truck (in such a way that all operational constraints are satisfied) are allocated to this mode and are not considered afterwards;
• origin–destination pairs that cannot be served feasibly by dedicated aircraft or by truck are assigned to the commercial flights;
• low-priority services (deliveries within 48 hours or within 3–5 days) are made by truck or by using the residual capacity of a company-owned aircraft;
• the demands of origin/destination sites are concentrated in the associated SC.
Arouteis a partial solution characterized by
• a sequence of stops of a company-owned aircraft ending or beginning in the hub (depending on whether it is a collection or a delivery route, respectively).
• a set of SCs (not ASs) allocated to each aircraft AS.
Therefore, two routes visiting the same AS in the same order can differ because of the set of SCs (which are not ASs), or because of the allocation of these SCs to the ASs. If the demand of a route exceeds the capacity of the allocated aircraft,
328 LINKING THEORY TO PRACTICE the exceeding demand is transported by commercial flights. The cost of a route is therefore the sum of costs associated with air transportation, land transportation, and possibly a commercial flight if demand exceeds dedicated aircraft capacity.
The IP model solved in the second stage is defined as follows. LetKbe the set of available airplane types;Uk,k∈K, the set of the pick-up routes that can be assigned to an airplane of typek;Vk,k∈K, the set of the delivery routes that can be assigned to an airplane of typek;Rthe set of all routes (R =!
k∈K(Uk∪Vk));nk,k∈K, the number of company-owned airplanes of typek;Sthe set of SCs,oi, i ∈S, the demand originating at theith SC;di, i∈S, the demand whose destination is theith SC;cr,r ∈ R, the cost of router;qi, i ∈S, the cost paid if the whole demandoi is transported by a commercial flight;si, i ∈S, the cost paid if the whole demand di is transported by a commercial flight;αir, i ∈ S,r ∈R, a binary constant equal to 1 if routerincludes picking up traffic at theith SC, and 0 otherwise;δir, i∈S, r∈R, a binary constant equal to 1 if routerincludes delivering traffic to theith SC, and 0 otherwise;γir, i∈S,r∈ R, a binary constant equal to 1 if the first (last) AS of pick-up (delivery) routeris theith SC, and 0 otherwise. The decision variables of binary type arevi, i∈S, equal to 1 if demandoiis transported by commercial flight, and 0 otherwise;wi, i ∈ S, equal to 1 if demanddi is transported by commercial flight, and 0 otherwise;xr,r∈R, equal to 1 if (pick-up or delivery) routeris selected, and 0 otherwise.
The integer program is as follows.
Minimize
k∈K
r∈Uk∪Vk
crxr +
i∈S
(qivi+siwi) (8.28) subject to
k∈K
r∈Uk
xrαri +vi =1, i∈S, (8.29)
k∈K
r∈Vk
xrδir+wi =1, i∈S, (8.30)
r∈Uk
xrγir−
r∈Vk
xrγir =0, i∈S, k∈K, (8.31)
r∈Uk
xr ⩽nk, k∈K, (8.32)
xr ∈F, r∈R, (8.33)
xr ∈ {0,1}, r∈R, (8.34) vi ∈ {0,1}, i∈S, (8.35)
wi ∈ {0,1}, i∈S. (8.36)
The objective function (8.28) is the total transportation and handling cost. Con- straints (8.29) and (8.30) state that each SC is served by a dedicated route or by a
LINKING THEORY TO PRACTICE 329 commercial route; constraints (8.31) guarantee that if a delivery route of typek∈K ends in SCi∈S, then there is a pick-up route of the same kind beginning ini. Con- straints (8.32) set upper bounds on the number of routes which can be selected for each dedicated aircraft type. Finally, constraints (8.33) express the following further restrictions. The arrivals of the airplanes at the hub must be staggered in the period before the COT because of the available personnel and of the runway capacity. Simi- larly, departures from the hub must be scheduled in order to avoid congestion on the runways. Letnabe the number of time intervals in which the arrivals should be allo- cated;npthe number of time intervals in which the departures should be allocated;
fr,r ∈ R, the demand along router;at the maximum demand which can arrive to the hub in intervalt, . . . , na;At the set of routes with arrival time fromton;Pt the set of routes with departure time beforet;pt the maximum number of airplanes able to leave beforet. Hence, constraints (8.33) are
k∈K
r∈Uk∩At
frxr ⩽at, t =1, . . . , na, (8.37)
k∈K
r∈Vk∩Pt
xr ⩽pt, t =1, . . . , np. (8.38) Constraints (8.37) ensure that the total demand arriving at the hub is less than or equal to the capacity of the hub in each time intervalt =1, . . . , na, while constraints (8.38) impose that the total number of airplanes leaving the hub is less than or equal to the maximum number allowed by runaway capacity in each time intervalt = 1, . . . , np.
Other constraints may be imposed. For instance, if goods are stored in containers one must ensure that once a container becomes empty it is brought back to the orig- inating SC. To this end, it is necessary that the aircraft arriving at and leaving from each SC be compatible. In the Intexpress problem, there are four types of airplanes, indicated by 1, 2, 3 and 4. Aircraft of type 1 are compatible with type 1 or 2 planes, while aircraft of type 2 are compatible with those of type 1, 2 and 3. Therefore, the following constraints hold:
−
r∈U1
xrαir+
r∈V1∪V2
xrδir ⩾0, i∈S, (8.39)
r∈U1∪U2
xrαir−
r∈V1
xrδir ⩾0, i∈S, (8.40)
−
r∈U2
xrαir+
r∈V1∪V2∪V3
xrδir ⩾0, i∈S, (8.41)
r∈U1∪U2∪U3
xrαir−
r∈V2
xrδir ⩾0, i∈S. (8.42) Moreover, some airplanes cannot land in certain SCs because of noise restrictions or insufficient runaway length. In such cases, the previous model can easily be adapted by removing the routesr∈Rincluding a stop at an incompatible SC.
330 LINKING THEORY TO PRACTICE The variables in the model (8.28)–(8.32), (8.34)–(8.42) are numerous even if the problem is of small size. For example, in the case of four ASs (a,b,candd), there are 24 pick-up routes (abcd,acbd,adbc, etc.) each of which has a different cost and arrival time at the hub. If, in addition, two SCs (eandf) are connected by truck to one of the ASsa,b,candd, then the number of possible routes becomes 16×24=384 (as a matter of fact, for each AS sequence, each of the two SCseandf can be connected independently by land toa,b,cord). Finally, for each delivery route making its last stop at an ASd, one must consider the route making its last stop in a different SC g∈ S\ {d}. Of course, some of the routes can be infeasible and are not considered in the model (in the case under consideration the number of feasible routes is about 800 000).
The solution methodology is a classical branch-and-bound algorithm in which at each branching node a continuous relaxation of (8.28)–(8.32), (8.34)–(8.42) is solved.
The main disadvantage of this approach is the large number of variables. Since the number of constraints is much less than the number of variables, only a few variables take a nonzero value in the optimal basic solution of the continuous relaxation. For this reason, the following modification of the method is introduced. At each iteration, in place of the continuous relaxation of (8.28)–(8.32), (8.34)–(8.42), a reduced LP problem is solved (in which there are just 45 000 ‘good’ variables, chosen by means of a heuristic criterion); then, using the dual solution of the problem built in this manner, the procedure determines some or all of the variables with negative reduced costs, introducing the corresponding columns in the reduced problem (pricing out columns).
Various additional devices are also used to quicken the execution of the algorithm.
For example, in the preliminary stages only routes with an utilization factor between 30% and 185% are considered. This criterion rests on two observations: (a) because of the reduced number of company aircraft, it is unlikely that an optimal solution will contain a route with a used capacity less than 30%; (b) the cost structure of the air transportation makes it unlikely that along a route more than 85% of the traffic is transported by commercial airlines.
The above method was first used to generate the optimal service network using the current dedicated air fleet. The cost reduction obtained was more than 7%, cor- responding to a yearly saving of several million dollars. Afterwards, the procedure was used to define the optimal composition of the company’s fleet (fleet planning).
For this purpose, in formulation (8.28)–(8.32), (8.34)–(8.42), it was assumed thatnk was infinite for eachk∈K. The associated solution shows that five aircraft of type 1, three of type 2 and five of type 3 should be used. This solution yielded a 35% saving (about 10 million dollars) with respect to the current solution.