EL ESTUDIO: DISEÑO, PROCEDIMIENTO E INSTRUMENTO DE INDAGACIÓN

Constrained by solution technique or understanding to the problem, earliest researches focused on one single problem, either location, allocation or routing problem, in the network under strict assumptions of other influ- encing factors. For example, some studies focused on VRPs in tributary network with the assumption that hubs are fully interconnected337_{, while others planed the backbone network with a predetermined set of hubs}338_. Later, researchers proposed compound problems by realizing that these problems are actually interrelated. On accounts of the complexity of compound location problems, it is a common way to divide the whole prob- lem into several sub-problems (or stages), which are easier to solve separately than the former complete one. Different sub-problems of HLP/ FLP take up different positions in the solution process of the complete prob- lem with the evolution of understanding to the involving problems. In the following, we make literature re- view on this issue. As researches on HLRPs began much later than those on LRPs339_{, most of the literatures} we cite here are LRPs.

Sequential method

Sequential methods were first introduced. For example, Jacobson/ Madsen and Nambier et al340 _{first solved} the location problem by minimizing the sum of hub-to-customer distances and then solved the resulting route- planning problem based on the hub location decision. A more sophisticated method is to estimate beforehand the route length connecting hub and customers with certain formulation for location problem341_{. However,} since there is indeed no feedback or information exchange between location problem and routing problem, suboptimal solution for the problem is inevitable342_.

Cluster method

336 _{See Voss et al (1999), p. i.}

337 _{For example, Gavish and Balakrishnan et al studied algorithms for tributary network design. See Gavish (1991), pp.17-71; Balakrishnan et al (1991),}

pp.237-284; Balakrishnan et al (1995), pp.58-76.

338 _{See e.g. Agarwal (1989), pp.64-76; Altinkemer/Yu (1992), pp. 365-381; Balakrishnan/ Altinkemer (1992), pp.192-205; Baybars/ Edahl (1988), pp.}

503-528; Chang /Gavish (1993), pp.99-131; Gavish /Altinkemer (1990), pp. 236-245.

339 _{The earliest research on HLRP we can find is from Nagy and Salhi. See Nagy /Salhi (1998), pp. 261–275. However, research on LRP began as}

early as 1970s. See e.g. Waston-Gandy/ Dohrn (1973), pp. 321-329; Or/ Pierskalla (1979), pp. 86-94; Bednar/ Strohmeier (1979), pp.89-104.

340 _{See Jacobson/Madsen (1980), pp. 378–387; Nambier et al. (1989), pp.14–26.} 341 _{See Daganzo (2005)}

In contrast to up-down approach as sequential method, cluster method takes up down-up process by grouping nodes into clusters with some statistical techniques. It first partitions the customer set into clusters, then lo- cates a facility /hub in each cluster and finally solves VRP for each cluster343_{. Just as sequential method, no} feedback takes place.

Iterative method

Realizing this pitfall, Bookbinder& Reece and Perl& Daskin344 _{introduced iterative method. This method itera-} tively solve location, allocation, backbone routing and tributary routing problems by feeding information from one phase to another until some stopping criteria are met. A typical iterative framework is as follows345_. Step 1: Choose hub nodes, perhaps based on available information from an incumbent solution

Step 2: Assign non-hub nodes to tributary networks Step 3: Design the tributary networks

Step 4: Design the backbone network Step 5: Evaluate the solution

Step 6: If solution is acceptable, stop. Otherwise, feed information of current solution back to step 1, and repeat the process.

This method has been demonstrated to improve the solution quality compared with that of a sequential meth- od346_{. It is widely adopted by many studies with variations and omission}347_.

Hierarchical method

Nagy and Salhiargued that hierarchical method arose because the FLP (here also HLP) is essentially a location problem that takes the routing decision into consideration as well, and this leads to a hierarchy between locat- ing and routing348_{. The idea conforms to the view of management that location problem is the crux that has} much longer planning horizon than allocation and routing problems. Based on the concept of “nested method”, they proposed a hierarchical heuristic solution framework, in which the master algorithm is devoted to solving the location problem and refers in each step to a subordinate heuristic that solves the routing problem. They

343 _{See e.g. Kleinrock/ Kamoun (1980), pp. 221-248; Klincewicz (1991), pp.25-37; Saha/ Mukherjee (1995), pp.378-383. Recent researches include e.g.}

Barreto et al (2007), pp. 968-977; Billionnet et al (2005), pp.968-977.

344 _{See Bookbinder/Reece (1988), pp. 204–213; Perl/Daskin (1985), pp. 381–396.} 345 _{See Klincewicz (1998), p.314.}

346 _{See e.g. Salhi/ Nagy (2009), pp. 287-296.}

347 _{See e.g. Boorstyn/ Frank (1977), pp.29-47; Gerla/ Kleinrock (1977), pp. 48-60; Gavish (1982), pp.355-377; Monma /Sheng (1986), pp.946-965; Lin/}

Rath (1987), pp. 18-25; Gavish (1992), pp.167-191.

reported a 6% improvement from hierarchical over the sequential approach but with longer computation time349_.

Both iterative and hierarchical methods give feedback between sub-problems. However, sub-problems in itera- tive method are independent and parallel and are treated with equal importance, while location problem in hierarchical method is usually taken as the master problem which is incorporated with other sub-problems. In our opinion, the fundamental difference between hierarchical method and iterative method is the division ap- proach for the sub-problems. In iterative method one sub-problem may be contrary in certain aspect to anoth- er one, while in hierarchical method the sub-problem at lower stage is coherent with those at higher stage. Moreover, the result of the problem at higher stage depends on the outcome from lower stage. Sub-problems in other methods mentioned above can be solved individually, while the sub-problem at higher stage in hierar- chical method cannot be solved until the problems at lower stage are solved. We would like to explain this with an example that one HLP consists of two decisions: (1) where to locate the hubs so as to minimize the transportation costs; (2) how to construct a routing system for tributary network so as to satisfy the given service level. The iterative two-stage solution process proposed in the study repeatedly updates the transpor- tation cost rate in the location problem with the result from routing problem. In other words, the location sub- problem with direct spoke assumption is solved repeatedly by modifying the estimation of feeder link cost rate from the multi-stop routing problem. However, when this problem is solved with hierarchical method, the location problem must include routing decision rather than take direct feeder link as assumption. So the loca- tion problem cannot be solved until the feeder routes are determined.

Actually before Nagy and Salhi gave it the well-acknowledged name “hierarchical approach”, this method had already been adopted in former studies. For example, Nambiar et al.350 _{presented a method that uses the result} of their simple depot clustering heuristic as the starting point. Then, they consider in turn p =1, 2,…m depots being open. For each value of p, they reformulate the LRP as a p -median problem with tour lengths as varia- ble costs and solve it with an exact method. Routing is then solved with a savings method. If the cost of the LRP with p depots is more than that with p-1, the procedure is stopped. This can be viewed as a hierarchical method, since the routing costs are explicitly included in the location model. Nagy and Salhi’s351 _“nested method” consists of a location algorithm with LS that refers to a routing method when evaluating neighboring solutions. The location algorithm is based on TS and an add/drop/shift neighborhood. After each move, the routing solution is fully evaluated using a multi-depot VRP algorithm. Lin et al.352 _{first determine the mini-} mum number of facilities. Then the VRP solution is completely evaluated for all combinations of facilities. Vehicles are allocated to trips by completely evaluating all allocations. If the best routing cost found is more than the setup cost for an additional depot, the algorithm moves on to evaluating all sets of facilities that con- tain one more depot. In method by Albareda-Sambola et al353 _{an initial solution is found via the linear pro-} gramming relaxation of the model. The master algorithm for location decision follows the TS framework with

349 _{See Nagy/Salhi (1996a), pp.1166–1174.} 350 _{See Nambiar et al. (1981), pp. 183-189.} 351 _{See Nagy/Salhi (1996a), pp.1166–1174.} 352 _{See Lin et al (2002), pp. 5-25.}

LS moves of add, drop and shift. However, infeasible routing solutions are allowed and a penalty is included in the locational objective function for the violation.