CHAPTER TWO Survey of Literature
2.2. Supply Chain Network Design
2.4.2. Solution methods
LRP combines facility location and customer allocation to facilities with vehicle rout- ing. These problems are hard to solve since they merge two different problems in na- ture: facility location and vehicle routing. Mathematically, such problems are consid- ered as NP-hard problems (Karp 1972; Nagy and Salhi 2007; Marinakis and Marinaki 2008; Yu et al. 2010; Daskin et al. 2010; Perl and Daskin 1985; Hassanzadeh et al. 2009). Due to the computational complexity of such problems there is no unique solu- tion to them. Therefore, a solution space is defined by the use of a solution approach and an optimiser. Within the specified solution space the optimum solution(s) consider- ing the goal(s) of the problem and its constraints are found.
Hassanzadeh et al. (2009) categorises the solution methods into three groups, viz. exact, heuristic, and meta-heuristic. Brimberg and Hodgson (2011) argue that the classical location problems fall into a two-dimensional space and can be depicted in three ways, viz. continuous, discrete, network space. In network space, according to Brimberg and
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Hodgson (2011), ‘demands are expressed at vertices of networks, facility locations (in the case of medians) are selected from network vertices, and distances are calculated over the shortest paths in the network’. Nagy and Salhi (2007) categorise the methods used for solving LRPs in four main groups (Figure 2.7).
Figure 2.7 LRPs solution methods (adopted form Nagy and Salhi 2007)
Solution methods to facility location problem and its variants in the context of SC de- sign are depicted in Table 2.3:
Table 2.3 Solution methods for supply chain design problems
References Problem Solution Method
Cooper (1963-1964) Fixed charge FLP
Teitz and Bart (1968) Fixed charge FLP Exchange or ‘swap’ algorithm
Maranzana (1964) Uncapacitated fixed charge LP Neighbourhood search im-
provement algorithm
Geoffrion and Garves (1974) Fixed charge LP Lagrangian relaxation algorithm
Galovo et al. (2002) and Daskin (1995)
Uncapacitated fixed charge LP Lagrangian relaxation algorithm
Hansen and Mladenvic (1997) Fixed charge FLP Variable neighbourhood search
algorithm AL-Sultan and Al-Fawzan
(1999)
Uncapacitated fixed charge LP Tabu search Jayaraman and Pirkul (2001) Multi-Commodity, multi-plant,
capacitated facility FLP (supply chain design problem)
Heuristic approach base on Lagrangian relaxation
Jang et al. (2002) Design of SC network Lagrangian heuristics
Sayrif et al. (2002) Multi-source, single-product,
multi-stage SC network design problem
Spanning tree-based GA ap- proach
Shen et al. (2003) Basic model Column generation approach
Oszen et al. (2003) Model with routing cost estima-
tion
Lagrangian relaxation based solution algorithm
Jayaraman and Ross (2003) Designing of distribution net- work and management in supply chain environment
Heuristic approach based on simulated annealing
Shen and Daskin (2005) Model with service considera-
tion
Weighting method LRP Solution Methods,
based on the problme type
Exact Solution Methods for Deterministic Problems
Heuristic Solution Methods for Deterministic Problems
Stochastic & Dynamic Problems
Problems with Non-Standard Hierarchical Structure
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Shen and Daskin (2005) Model with service considera-
tion
Genetic algorithm
Shen (2000) Model with multiple commodi-
ties
Lagrangian relaxation embed- ded in a branch and bound algo- rithm
Yeh (2005) Multistage supply chain network
problem (MSCN)
Hybrid heuristic algorithm
Amiri (2006) Designing a distribution network
in a supply chain system with allow for multiple levels of capacities
Lagrangian based solution algo- rithm
Yeh (2006) Multistage supply chain network
problem (MSCN)
Memetic algorithm (MA)
Shen (2006) Profit maximising model with
demand choice flexibility
Branch-and-price algorithm
Altiparmak et al. (2006) Multi-objective SC network
design problem
New solution procedure based on genetic algorithm
Romeijn et al. (2007) Two-echelon SC design problem Column generation
Shen (2007) Model with unreliable supply Algorithm based on the bisec-
tion search and the outer ap- proximate algorithm
Synder et al. (2007) Model with parameter uncertain-
ty
Lagrangian relaxation based solution algorithm
Hinojosa et al. (2008) Dynamic supply chain design
with inventory
Lagrangian approach which relaxes the constraints connect- ing the distribution levels
Schütz et al. (2008) Two-stage stochastic supply
chain design problem
Sample average approximation in combination with dual de- composition
Altiparmak et al. (2009) Design of a single-source, multi- product, multi-stage SC network
Solution procedure based on steady-state genetic algorithm (SSGA)
Bischoff et al. (2009) Multi-dimensional mixed-
integer optimization problem (multi-facility location– allocation problem with polyhe- dral
barriers)
Heuristic methods (alternate location–allocation; alternate location-with-barriers allocation algorithm, & alternate location allocation-with-routes
algorithm)
Li et al. (2009) Capacitated plant location prob-
lem with multi-commodity flow
Lagrangian-based method in- cluding a Lagrangian relaxa- tion, a Lagrangian heuristic and a sub-gradient optimisation + Tabu search to further improve upper bounds provided by the Lagrangian procedure Liberatore et al. (2011) Stochastic R-interdiction median
problem with fortification (S- RIMF)
- Pre-processing techniques based on the computation of valid lower and upper bounds - Heuristic approach
Reddy et al. (2011) Single echelon supply chain two
stage distribution inventory optimisation
No solution offered!
Several optimisation algorithms are adopted by researchers in LRPs. A good number of these algorithms are heuristics/meta-heuristics. A detailed survey of the location-routing techniques is provided in (Madsen 1983; Min et al. 1998; Kenyon and Morton 2001;
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Nagy and Salhi, 2007). A synopsis of the optimisation techniques is illustrated in Table 2.4.
Table 2.4 A synopsis of the reported optimisation techniques for LRPs
Optimisation techniques used Publications
Self-organised optimisation using artificial neural network
Schwardt and Fischer (2009)
Honey bees mating optimisation Marinakis et al. (2008)
Ant colony optimisation Bell and McMullen (2004), Bin et al. (2009), Ting and Chen (2013)
Particle swarm optimisation Yang and Zi-Xia (2009); Liu et al. (2012)
Tabu search Gendreau et al. (1994), Melechovský et al. (2005), Albareda-Sambola et al. (2005), Caballero et al. (2007)
Simulated annealing Lin et al. (2002), Yu et al. (2010), Stenger et al. (2012).
Greedy randomised adaptive search optimisation
Prins et al. (2006), Duhamel et al. (2010), Nguyen et al. (2012)
Variable neighbourhood search op- timisation
Melechovský et al. (2005), Ghodsi and Amiri (2010), Derbel et al. (2011)
Genetic algorithms Zhou and Liu (2007), Marinakis and Marinaki (2008), Jin et al. (2010), Karaoglan and Altipa r- mak (2010)
Branch and cut optimisation Belenguer et al. (2011), Karaoglan et al. (2011) Mixed-integer programming; Integer
linear programming
Alumur and Kara (2007), Diabat and Simchi -Levi (2009); Laporte et al. (1989); Ambrosino and Scutella (2005)
NP-hard LRP is broken down into its components by heuristics/meta-heuristic methods. These components (Karp 1972) are solved and then the final solution to the problem is reached. These heuristic/meta-heuristic methods can follow location-allocation-routing algorithms or allocation-routing-location algorithms. Location-allocation-routing algo- rithms first locate the facilities, then allocated the customers to facilities and then define the connection routes. Instead, allocation-routing-location algorithms deal with alloca- tion of facilities and routing simultaneously. Usually allocation-routing-location algo- rithm defines a set of routes and assumes that all facilities are open, allocates customers to facilities and drops the unselected facilities from the system and updates the location and routing decision (Karp 1972; Perl 1983; Wu et al. 2002). Heuristics and meta- heuristics tend to search the solution space more electively when compared to conven- tional approaches.
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Considering the nature of the solution methods and optimisers, the next challenge in dealing with LRPs and any other form of SC network design model is finding a proper platform to solve the constructed model. Next section reviews the most used software packages and programs used as a solution platform for LRPs.