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method

Heuristic is the term used in the field of optimization to characterize a certain kind of mathematical problem-solving procedures. As presented by Silver [73], “the term heuristic means a method which, on the basis of experience or judgement, seems likely to yield a reasonable solution to a problem, but which cannot be guaranteed to produce the

mathematically optimal solution”. Generally, due to the complexity of a great number

and variety of difficult mathematical problems, the heuristic solution method is needed in practice. The problems needed to be solved efficiently, and this has led to the development of efficient procedures in an attempt to find good or reasonable solutions, even if they are not optimal [71, 73]. In these methods, the process speed is an important measure in relation to the quality of the solution obtained. Heuristics are also known as approximate algorithms. They are mostly concerned with obtaining applicable solutions to the well defined mathematical representations (models) of real-world problem situations [73].

Klose [44] formulated a mixed integer programming model of a two-level capacitated facility location problem (TSCFLP). The model considered a single-product and single-source constraints. It is a linear programming based heuristic with three tasks. The first task is to find the optimal locations of depots from a set of possible depot sites in order to serve customers with a given demand; the second is the optimal assignments of customers to depots, and third, the optimal flow of product from plants to depots [44]. The model is solved by a heuristic approach based on the Lagrangian relaxation of the demand constraints. The procedure was tested on some problems with up to 10 plants, 50 possible depot sites and 500 customer points. “The computational results show that this method is able to compute near-optimal solutions and useful lower bounds for the TSCFLP in short computation times, even in the case of larger problem instances” [44]. In 2000, Klose [43] solved another similar problem but considering a Lagrangian heuristic based on the relaxation of the capacity

constraints. It is named as Lagrangian relax-and-cut approach. The resulting Lagrangian sub-problem is then solved efficiently by branch-and-bound methods. Next, the results are computed by means of a weighted Dantzig-Wolfe decomposition approach [43].

The studies by Amiri [7] and Lashine [50] are two-level FLPs addressed as the problem of designing a distribution network in a supply chain system. Lashine [50], also includes in his study, the routing decision. The studies by Amiri [7], Litvinchev et al. [52], and Landete & Marin [46], determined simultaneously the best location of both plants and warehouses, and the best strategy for distributing the products from the plants to the customers through warehouses. Amiri’s study allows the multiple levels of capacities available to the warehouses and plants [7]. In this case, it is possible to have several capacity values in plants and depots/warehouses. The study considers different values of DCs’ capacity during optimization of the model. Amiri [7] implemented an efficient heuristic solution procedure based on Lagrangian relaxation of the problem. The tested problems are up to 500 customers, 30 potential warehouses, and 20 potential plants. The two-level problem studied by Landete & Marin [46] is uncapacitated FLP where its solution was obtained by a heuristic approach that involves cuts . The study by Litvinchev et al. [52] considered a two-level CFLP with a single-product also uses a heuristic Lagrangian relaxation. The distribution network design problem studied by Jayaraman [37] uses simulated annealing (SA) to obtain nearly optimal distribution system design.

Hinojosa et al. [30] studied a multi-period and multi-commodity two-echelon capacitated facility location problem. This study considered multi-period planning horizon which has not yet been observed in the previous surveyed literature. They assumed that the capacities of plants and warehouses change over time (T) periods. This is also applied to demands and transportation costs. Seasonal known demands grouped in four periods are considered to influence the capacities and other parameters and/or variables determination for each period

[30]. Both plants and warehouses location were determined in each period. The authors have not provided any real life application, but the model presented applies to the situations where intermediate distribution and seasonal demand exist [30]. It is a cost minimization mixed integer programming problem where results were obtained by a Lagrangian relaxation with heuristic procedures.

A study by Ozsen et al. [65] was a two-echelon single product logistics system involving a single plant and several of potential warehouse sites. Only the location of warehouses is determined. It is a supply chain network design problem that is nonlinear integer- programming and solved by heuristic Lagrangian relaxation. This model considered multi- sourcing as customers (retailers) are sourced by more than one warehouse [65].

A two-level transportation problem studied by Gen et al. [24] is modelled in the supply chain system. The study aims to determine the distribution network that involves transportation problem and facility location to satisfy customers’ demands at minimum cost. The major constraints are the capacities of plants and DCs, and the minimum number of DCs to be selected. The constraints regarding the number of DCs to be selected is one of the component which distinguishes this study from the other models. It is very important when a manager has limited available capital [24]. In the three layers, the nature of transport is direct shipping without multiple stops. The model is solved by a heuristic method such as priority-based Genetic Algorithm (pb-GA) [24].

Generally, the deterministic problems found in the literature are the total cost minimization based on location and transportation strategies. They are also categorized as mini-sum problems [42, 69]. The target is satisfaction of customer demands with high quality service provision. Regardless of the number of commodities involved, and other attributes, the objective of the multi-level FLP is to design a distribution network for efficient transfer of

goods from supply to demand points. The structure of networks such as the number of facilities at different layers, their locations and capacities need to be determined optimally.

All of the previously discussed problems are deterministic that do not take into account the uncertainties or risks in the modelling or planning process. The next subsection considers studies that involve randomness (uncertainties or risks) concept.