## 2 Literature Review
### Lagrangian Relaxation  
During the last decade, various publications appeared using heuristics and meta-heuristics to solve the MCLP. Lagrangian relaxation (LR) is one of these heuristics which has been used by many researchers to find bounds for different problems. For example, [PERSON] and [PERSON] [33] developed a Lagrangian heuristic for their modular capacitated location problem. There have been many other papers that used the LR-based algorithms to solve their models. Table 1 reviews the most important models in the area. There are various trends in using Lagrangian heuristics: applying LR just for producing lower bound, proposing heuristics for computing of both lower and upper bounds, using a fixed upper bound and try to propose lower bound heuristic and decomposing the main problem into sub-problems and then applying the mentioned procedures. Table 1 shows the most important papers related to the procedure used in this paper. From Table 1, it can be concluded that most researchers have proposed heuristics for both lower and upper bounds. [PERSON] et al. [34] has used an adaptive LR method to solve the problem of water supply and wastewater collection network design as a minimization problem to address the drought problem of Urmia lake in Iran. The adaptive LR solves the relaxed problem and its solution is accepted as the lower bound. Afterward, to obtain the feasible solution as the upper bound the solutions for continuous variables of the relaxed problem are fixed and used as parameters and the binary variables are modified in a way that the feasibility of the solution is approved. The Lagrangian multiplier is then updated and the procedure is repeated until the last defined iteration. Similarly, [PERSON] and [PERSON] [35] used LR method to solve the multi-objective model for a stochastic blood supply chain that was reformulated as a single objective and robust model before utilizing LR. The LR method developed by [PERSON] and [PERSON] [35] relaxed one of the constraints and the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Authors** & **Problem** & **Approach** & **Decom.** & **Just LB** & **Both LB and UB** \\ \hline [PERSON] et al. [39] & Resource-constrained modular scheduling & LR & & ✓ & \\ \hline [PERSON] et al. [37] & Hybrid flowshop scheduling problem & LR with cuts & ✓ & ✓ & \\ \hline [PERSON] et al. [38] & Automated guided vehicles routing problem & LR with Petri nets & ✓ & ✓ & \\ \hline [PERSON], [PERSON] and [PERSON] [40] & Large scale new variant of capacitated clustering problem & LR approach with two phases of dual optimization & ✓ & ✓ & \\ \hline [PERSON] and [PERSON] [41] & Two-stage capacitated facility location problem & Lagrangian heuristic producing feasible solutions & ✓ & \\ \hline [PERSON] et al. [42] & Multi-echelon joint inventory-location problem & Improved LR-based heuristics & ✓ & ✓ & ✓ \\ \hline [PERSON] et al. [43] & Two-level uncapacitated facility location problem with single-assignment constraints & Lagrangian-Based Branch-and-Bound Algorithm & ✓ & & ✓ \\ \hline [PERSON] et al. [23] & Multi-period stochastic covering location problems & LR based algorithm for high quality feasible solutions & ✓ & & ✓ \\ \hline [PERSON] et al. [44] & Three-echelon supply chain integrating inventory-location-routing problem under uncertainty & LR & & & ✓ \\ \hline [PERSON] and [PERSON] [35] & Multi-objective and stochastic blood supply chain & Lagrangian heuristic & & ✓ \\ \hline [PERSON] et al. [34] & Coordinated water supply and wastewater collection network design problem & Adaptive Lagrangian relaxation-based algorithm & & & ✓ \\ \hline [PERSON] et al. [36] & Train rescheduling and track emergency maintenance & Lagrangian-based decomposition & ✓ & & \\ \hline Current research & Dynamic modular capacitated maximal covering location problem & Lagrangian decomposition heuristic & ✓ & & ✓ \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Investigated studies using different approaches of Lagrangian relaxation method.