Deterministic multi-level problems with exact solution methods

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The multi-level FLPs are deterministic if all their input data is known by certainty. In some literature, the deterministic models are also known as static models [71]. The exact solution method is the solution procedure that guarantees the optimal solution [71]. There are several deterministic studies as discussed below.

In 1974, Geoffrion and Graves [25] presented a multi-level facility location problem as a distribution system design multi-commodity problem that is solved optimally. It is a two- level FLP which optimizes the location of DCs. The problem is formulated as a four-indexed

mixed integer linear program with multi-commodity flow from plants to customers through DCs [25]. It is a capacitated constrained problem on both plants and DCs. The problem studied by these authors is single-sourcing and single-period model. The model solution method is developed based on Benders decomposition techniques [25]. They solved a real life problem of a major food company using their model and algorithm. The optimal solution to the real problem is found with up to a composition of 17 commodities, 14 plants, 45 possible DC sites, and 121 customer points [25].

K¨oksalan et al. [45] in 1995 studied an application problem of a brewery company based in Turkey. It is a two-level location-distribution problem formulated as a mixed integer programming model. The malt factories are the higher level facilities and breweries are the lower level facilities that supplies to customer zones. They evaluated the existing transportation costs for shipping malt from the two malt factories to the three breweries, and shipping beer from the breweries to 300 different customer zones [45]. The model is solved optimally using interacting mathematical programming software (FORTRAN and LINDO). The company’s plan is to explore the best sites for opening new breweries. After the results of the study, two new breweries were then opened [45].

A production-distribution system design problem studied by Elhedhli and Goffin [22], is a two-level FLP. It is a supply chain based problem that is multi-product plant, single- sourcing DC with capacitated constraints. The optimal solution to the problem is based on Lagrangian relaxation, interior-point methods, and branch and bound.

A study by Hindi and Basta [29] considered a similar problem as the Geoffrion and Graves [25], but with three indexed formulation. The other difference is the absence of single-sourcing of DCs-to-customers’ service. The multi-products were transported from capacitated plants to capacitated DCs before the final destination to the customer points

with the known demand. The problem considered locations of DCs and their associated fixed costs. In the model the shipping costs from a plant to a possible DC and thereafter to the customer points were also considered. The problem is a mixed-integer programming model solved by a branch and bound algorithm. Again in 1998, Hindi et al. [28] presented a similar study that considered single-sourcing DC-to-customers’ service. In the model, they used a four indexed formulation where it was possible to trace the plant origin of each product quantity delivered to the customers. Generally, the objective was to choose the locations for opening DCs such that the total cost in the distribution system was minimized. An other similar study by Jiang et al. [38], used both heuristic and exact solution techniques.

A study by Tragantalerngsak et al. [81] is focused on a two-echelon, single-source, and capacitated facility location problem. The problem is formulated as a mixed integer linear program, with capacitated constraints. The model is solved optimally by a Lagrangian relaxation-based branch and bound algorithm. In their problem, the deliveries of products are made from the first-echelon facilities (they call them depots) to customers through the second echelon facilities (called facilities in the model). The main goal is to determine simultaneously, the number and location of facilities in each echelon, the flow of products between the facilities in different echelons, and finally the assigning of the customers to open facilities in the second echelon [81]. This problem has the following identified features:

• Two-echelon and single-source: In this case, each customer must be served by only

one facility from second echelon facilities. On the other hand, each facility in second echelon, will also receive products (deliveries) only from one depot in the first echelon depots. So, single-sourcing is applied to both layers of the distribution system.

• Capacitated and uncapacitated facilities: The second echelon facilities have specific

capacities that must not be violated, but the first echelon facilities (named as depots) are uncapacitated.

location decision for opening of facilities (second echelon); the second is the allocation of customers to open facilities, and the third is the allocation decision of allocating open facilities to depots (first echelon). All the decisions are made simultaneously.

The main applications mentioned in the study are in the telecommunication, distribution and transportation industries [81].

Ambrosino and Scutella [5] considered a complex distribution network design problem with capacitated facility location, warehousing, capacitated transportation and inventory levels. It is a network made up of four layers, namely; plants, central depots, regional depots and customers. The three types of routes are plant to central depots, central depots to regional depots, and finally, the routes from regional depots to customers. The major tasks on facility location, allocation, transportation (routing) and inventory were carried out optimally for some small instances using CPLEX software. The authors pointed out that, for solving larger problems and real instances, the only helpful methods have been heuristics [5].

The review paper by Klose [42] presents different problems of locating facilities and allocating

customers that covers the core topics of distribution system design. He pointed out

that, “model formulations and algorithms which address the issue vary widely in terms of fundamental assumptions, mathematical complexity and computational performance” [42]. In the paper, multi-level models are well discussed with the concerned variations. Other review papers that discuss this class of problems are by Melo [58], Narula [64], and Sahin and S¨ural [69].