Transportation Network Optimization A. Ogunbanwo, Brunel University, United KingdomA. Williamson, Brunel University, United KingdomM. Veluscek, Brunel University, United KingdomR. Izsak, Brunel University, United KingdomT. Kalganova, Brunel University, United KingdomP. Broomhead, Brunel University, United Kingdom

INTRODUCTION

The longevity of transportation networks dates back to the age of ancient civilization, since that time the main objective has in essence remained the same: to facilitate the transportation of goods from one location to another using the most cost effective mean available. To rephrase that in modern vernacular, the management and optimization of transportation networks in meeting business objectives. While the forces driving the interpretation of most cost effective means available have changed overtime with the introduction of new technologies, global trade links and governmental policies; the general expectation of a transportation network has remained constant. These driving forces have impacted on the perception of optimality; cost is not necessarily the sole objective these days. Optimization objectives such as energy costs and their variability, time, environmental impact but to name a few have growing in importance of late. Many of the existing approaches to supply chain management take a multi-objective optimization approach, combining several (possibly competing) objectives and optimize the network. As such there is a growing trend to perform balanced optimization across a number of objectives. Transportation networks are rapidly expanding due to the globalization of business and supply chains; as such the size and complexity of transportation networks has increased considerably in the last decade. Transportation network optimization is known to be a difficult and complex problem to solve, a deterministic solution is often not applicable or indeed available for such problems and the problems themselves are therefore categorized as NP-hard problems. In response to the failure of deterministic algorithms to solve NP-hard problems, other optimization techniques have been developed and applied.

The purpose of the paper is to review the current state of art in transportation network optimization. The paper is organized in five sections. In the Background section, we introduce the transportation problem, consider its theoretical aspects and implications, and perform a detailed analysis of the main contributions made in the field. In the Main Focus section, we present a critical analysis of the algorithms used, the objectives optimized and the complexity of the networks analyzed in the literature, and discuss the main problems that as yet remain to be addressed. In the Solutions and Recommendations section, we propose ideas, and possible solutions to these outstanding problems. Finally, in the Future Research Directions section, we discuss future and emerging trends.

BACKGROUND

A transportation or distribution network is a dynamic, stochastic and complex system that can be modeled as a graph where the nodes (vertices) represent entities that can in general can be categorized as representing producers, distribution centers and end customers (Ding, Benyoucef, & Xie, 2009) or in the more specialist case of manufacturing enterprises as manufacturing and distribution sites that procure raw material, process them into finished goods, and distribute the finish goods to customers (Ganeshan, 1999) (Figure 1 shows an example of a transportation network).

Figure 1 – Example of Transportation Network, where Si are sources, Dj destinations/dealers, and SPk and DPl are possible intermediate ports

The optimization of transportation networks is a specialization of the minimum-cost flow problem, a well-known optimization model, where the goal is to find a feasible flow of minimum cost in a network with capacity constraints and edge costs (Goldberg & Tarjan, 1987). As a specialization of a linear programming problem, it may be solved by applying the common algorithms from linear programming theory, e.g. the simplex method, branch and bound/cut, etc. Such methods are exact in the sense that they always terminate with a feasible solution, a solution that is also optimal. While it is desirable to have a solution that is theoretically guarantee to be the best, such exact methods are not always applicable. When the problem difficulty is high or the model is too complex (i.e. NP-Hard problems cannot always be solved exactly), or perhaps time or indeed the resources available are limited, then using an exact method is not always possible or a feasible solution strategy. In scenarios where it is not possible to apply exact methods, then finding an approximate solution could well be an acceptable tradeoff, particularly if the solution found is close to the optimal and the time/resource employed are reasonable.

In the case of many optimization problems it is often possible to design specific heuristics, heuristics that take advantage of specific problem properties and/or employ historical knowledge acquired from past experience. Clearly the qualities of such heuristic are heavily dependent on the level of domain knowledge and experience available in the design of the algorithm. In recent years there has been a growing interest in approaches based on general heuristics. This approach is applicable across a range of problem domains and often results in better performance than that achieved using specific heuristics, particularly in terms of solution quality and execution time. In the literature it is possible to detect a trend towards the use of meta-heuristic approaches as the solution basis for solving transportation networks problems. The most common approaches include (Multi Objective) Genetic Algorithm, (Fuzzy-) Ant Colony Optimization, and Swarm Particle Optimization.

One example of transportation network optimization is the trans-shipment problem of cross-docking networks where the goods are transferred from suppliers to retailers through cross-docking facilities, without storing them in distribution centers. Here the objective function is to minimize the transportation costs in the network by loading trucks in the supplier locations and routing them to customers directly or indirectly using cross-docking facilities where loads are consolidated (Musa, Arnaout; & Jung, 2010). The most common objective in transportation network optimization is in finding the shortest distant distribution on a given network i.e. to determine an optimal set of routes

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between suppliers and customers (Han & Ji, 2010). However there is growing interest in using optimization factors such as profit, energy, service level or resilience etc., possibly competing factors that result in an optimal solution that itself is possibly a tradeoff amongst these parameters.

An example of supply chain optimization based on multi-objective criteria can be found in (Ding, et al., 2009). H. Ding et al. addressed the design of production-distribution networks that includes both supply chain configuration and related operational decisions such as order splitting, transportation allocation and inventory control (Ding, et al., 2009). They developed a simulation framework based on Multi-Objective Genetic Algorithm (MOGA). Evolutionary algorithms have also been shown to be flexible. In (Chang, 2010), the author proposes a combination of the co-evolutionary mode with constraint-satisfaction mode to narrow down the possible solutions, as a means of reducing the exploration space. The co-evolutionary mode can adjust evaluation constraints dynamically to match a complex reality (Chang, 2010). See Figure 2 for an overview of Evolutionary Strategy variants used in supply chain optimization.

Another interesting approach is the ant colony optimization algorithm, which mimics the optimal way in which ants find their food. Ants deposit evaporating pheromone along their search paths that attracts other ants and influences the way in which they choose pathways. The ‘fastest’ routs tend to be those with the highest pheromone density, more ants that pass down a particular pathway then greater is the quantity of pheromone deposited, the strength of this marker increases the probability that other ants will travel along the same pathways. The deposited pheromone evaporates as a function of time; the degradation rate is faster on longer paths than on shorter ones. Consequently, after a number of tours to and from the nest, there is a higher concentration of pheromone on shorter paths than on the longer ones. In ACO, artificial ants work as simple computer agents within a network to probabilistically build solutions. While the ant colony algorithm in its original form has been successfully applied to transportation optimization problems (Musa, et al.,2010), some subsequent variants have improved the convergence behaviors (Han & Ji, 2010). See Figure 3 for details of Ant Colony Optimization variants.

Particle Swarm Optimization (PSO) has its roots in two main component methodologies. Perhaps the more obvious is its ties to artificial life (A-life) and in general to bird flocking, or fish schooling, and swarming theory in particular. However it is also related to evolutionary computation, and has ties to both genetic algorithms and evolutionary programming (Kennedy & Eberhart, 1995).PSO-based strategies can and have been applied to the solution of multi-objective transportation network optimization problems. M. Huang et al. (Huang, Li, & Wang, 2011) designed a Fourth-Party Logistics (4PL) network optimization model based on resilience and used a PSO method to solve the problem. Che Z. (Che, 2012) implemented a similar strategy in developing a decision methodology for the production and distribution planning of a multi-echelon unbalanced supply chain. In (Zhao &Dou, 2011), Zhao X. et al. proposed an improvement PSO approach. The authors embedded into the PSO algorithm a reduced variable neighborhood local search, in order to reduce the complexity of the solution space and enhance explorability.

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Figure 2 - Transportation Network Optimization approaches and Evolutionary Algorithms variants

Figure 3 - ACO Variants

MAIN FOCUS

Increasing Transportation Network Complexity

The field of transportation network optimization has been studied for a number of years, yet it remains a problem that is of great interest to both the industrial sector and the research community. The early transportation network optimization problems involved models based on a single objective function, and had small set of constraints. These networks also had a tendency towards lower complexity. These days the data and networks for transportation optimization problems usually comes from the industrial sector and are often multi-objective with a large and complex network topology.

Table 1 - Objectives investigated and algorithms used in existing approaches to Transportation Network Optimisation

Author, year AlgorithmsObjective(s)

Distance Cost Resilience Time Service level

Product quality

Environmental issues

(Xiang, Li, Huang, & Li, 2012) ACO (Bevilacqua, Costantino, & Dotoli, 2012) MOGA

(Che, 2012) PSO

(Sadjady & Davoudpour, 2012) MILP

(Zhao & Dou, 2011) PSO

(Boudahri, Sari, & Maliki, 2011) ILP

(Huang, et al., 2011) PSO

(Utama, Djatna, Hambali, Marimin, & Kusdiana, 2011)

ACO

(Anghinolfi, Paolucci, Sacone, & Siri, 2011) IP and ACO

(Yeh & Chuang, 2011) MOGA

(Georgiadis, Tsiakis, Longinidis, & Sofioglou, 2011) MILP and B&B

(Zhao, Kumar, Harrison, & Yen, 2011) PSO

(Musa, et al., 2010) ACO

(Han & Ji, 2010) ACO

(Ying-Hua, 2010) GA

(Ghoseiri & Nadjari, 2010) ACO

(Che & Chiang, 2010) MOGA

(Jiang, Zhao, & Sun, 2009) MOGA

(Ding, et al., 2009) MOGA

(Lin, Gen, & Wang, 2009) GA

(Chan & Kumar, 2009) ACO

(Lau, Chan, Tsui, & Ho, 2009) GA

In graph theory, several means exist to measure the complexity of a graph or network. Such measures are usually based on either the length of the whole network, the length of the shortest path, the number of cycles in the graph, or its level of connectivity.

Here we choose to measure the complexity of analyzed networks based on their beta index and the number of cycles measures. The beta index measurement take into account the level of connectivity of a graph, and requires prior knowledge of the number of graph vertices (nodes) and edges. Let e be the number of edge in a graph, and let v be the number of vertex in the same graph,

the beta index is define as: β= ev . The number of cycles measure is based on the maximum number of

independent cycles in a graph, and again requires prior knowledge of the number of vertices, edges,

and sub-graphs. Let e be the number of edge in a graph, let v be the number of vertices, and let p be the number of sub-graph. The number of cycles measure is defined as: u=e−v+ p. Table 2 reports the beta index and the number of cycle’s complexity measures for a number of networks as extracted from the literature

Table 2 - Network Complexity Analysis

Author, Year AlgorithmsNo

NodesNo

EdgesNo Sub-graph

Beta-index

No of Cycles

(Syarif, Yun, & Gen, 2002) GA 124 188 1 1.52 65(Yu, 2005) ACO 2300 3200 61 1.40 961

(Chen, Subprasom, & Ji, 2006) ES 4 10 1 2.5 7(Lin, et al., 2009) ES 35 210 1 6 176

(Jiang, et al., 2009) MOGA 20 64 1 3.2 45(Ghoseiri & Nadjari, 2010) ACO 4000 61783 1 15.46 57784

(Ying-Hua, 2010) GA 80 120000 1 1500 119921(Zhao & Dou, 2011) PSO 1000 1815 1 1.815 816(Huang, et al., 2011) B&B 14 40 1 2.86 27

(Sadjady & Davoudpour,2012) M-ILP 150 3600 1 24 3451

(Bevilacqua, et al., 2012) MOGA 9 18 1 2 10

We can deduce from this analysis, that the current trend is towards problems that are based on multiple-objective functions, larger data sets and higher complexity networks. From an industrial standpoint there is increased interest in including variability and uncertainty into the optimization models, so that they are more representative of real-world applications.

6%

52%

8%

20%

5%

6%5%

Distance CostResilience TimeService level Product qualityEnvironmental issues

Figure 4 - Frequency of appearance % of each objective

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30%

33%

12%

7%

5%

2%2%

2%5%

2%Ant Colony OptimisationGenetic AlgorithmParticle Swarm OptimsationBranch & BoundBenders’ decompositionBeam SearchLagrangian relaxationWagner-Whitin AlgorithmVogel's Approximation MethodMixed Integer Linear Programming

Figure 5 - Frequency of appearance % of each algorithm

Approaches from the domain of linear programming theory are not always well suited to solve large problems that involve multiple-objective functions and uncertainty. As such growing interest exists in the application of meta-heuristic methods. Although a preference by some researchers in the adoption of a (mixed) integer/linear programming technique, such as dual method, simplex method, or branch and bound/cut can still be observed, there is great preference and interest in adopting approximated methods (see Table 1 and Figure 5). The most frequently adopted methods are evolutionary strategy, genetic algorithm, ant colony system, and particle swarm optimization.

In general the literature analyzed discusses the solution to optimization problems that in the main relate to networks which are constrained to or operate in confined geographic locations/areas, which typically are no larger than a region or state. There is a growing interest from the logistics field to address problems that operate at a global level. In such cases the supply chains are complex, and their optimization is critical. Possible solutions to such problems are considerably less: it becomes difficult to build a distribution plan; at a global level more factors affect the solution. To address such problems, the underpinning mathematical models must include a greater number of constraints. The complexity of the problem increases such that for certain scenarios even approximated methods are difficult to be applied.

Many of the developed algorithms from the literature require a large amount of time to converge to an optimal or near optimal solution. Many authors identified the converging speed as a drawback of their algorithms. There is general agreement that future work should focus on reducing the convergence time required to achieve a good solution.

A recurring problem relates to how the general quality of the proposed algorithm/method is assessed. A more comprehensive supply chain simulation benchmarking framework should be established to apply proposed algorithms to a set of comparable test cases. Another important research direction is to account for risk related issues in the supply chain design. The approaches presented in the literature cannot be easily extended due to prohibitive computation time. Future research could be focused at establishing robust models that can accommodate changes to the parameters of the business environment for the life-time of the logistics network. Addressing demand uncertainty is a promising research avenue with significant practical relevance to the industrial sector.

Solutions and Recommendations

With the inevitable increases in problem complexity, the efficiency and scalability of the proposed solutions are key aspects. Meta-heuristic methods have already been proved to be scalable and effective when faced with large problems, but this may not always be sufficient. There is possibly some value in attempting to add specific domain knowledge from the field of transportation network optimization into these general methods. The availability of more specific information could very well help in trimming the solution/search space and hence allow better/faster convergence of the solution.

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The idea of specializing meta-heuristic methods could be implemented by applying hyper-heuristics. Hyper-heuristics are general heuristic methods that have the capability to automatically specialize to the problem at hand by the inclusion of artificial intelligence and machine learning techniques in the search process. Research in the field of hyper-heuristic is at an early stage, and, at the time of writing, there are very few studies that relate to the application of hyper-heuristic as a solution strategy for the transportation optimization problem.

Several approaches can be used to speed-up the convergence and reduce the computation time of metaheuristic algorithms. In (Tseng, Tsai, Chiang, & Yang, 2010), S. Tseng et al. modified ACO for the travel salesman problem to track the past iterations for the purpose of reducing redundant computations. In (Zhou & Deng, 2009), P. Zhou et al. developed a hybrid ACO with a taboo search to quickly reduce the solution search space.

From a technical point of view, there is undoubtedly value in introducing parallelization in meta-heuristic algorithms to allow the exploitation of current trends in multi-core hardware, and concurrent and distributed architectures. In (Randall & Lewis, 2002), M. Randall et al. developed a simple parallel version of the ACO algorithm for the travel salesman problem. Their parallelization strategy is based on the well-known master/slave approach. A master ant is used to coordinate the activities of the colony. The results showed an acceptable speed-up. However, one of the main disadvantages of this architecture is the large amount of communication required to maintain the pheromone matrix. Future work should concentrate on minimizing (both absolutely and relatively) the amount and frequency of this communication. Distributed agent based concurrency with its optimized message system is worthy of consideration.

When considering changes to the parameters of the business environment during the life-time of the logistics network, a promising first step would be the addition of stochastic simulation. A Monte Carlo simulation for instance could be used to update the parameters describing the transportation network with their sampled estimates. Instead of testing a proposed solution on a snapshot of the business, the simulations would cover a set of possible scenarios. These results would be of broader interest.

FUTURE RESEARCH

The solution to transportation network optimization will become key component in the field of logistics management. With the embracement of global supply chains it is unlikely that the business model of medium to large-size companies will be geographically constrained to localized areas. Developments in communication and transportation technologies have more than exceeded the limits that were common place just few years ago. The size and complexity of transportation problems will continue to increase, and the planning of resources distribution, production facilities and goods sourcing will inevitably become a more critical factor in the success of any business. Faster, accurate and more scalable solutions will undoubtedly be required. All the studies so far considered have in the main focused on the optimization of one or a few static features. To the knowledge of the authors, there has not been any proposal reported in the literature that relate to methods capable of including the dynamic behavior of external parameters, such as energy cost, transportation cost, and inventory cost fluctuation directly in the optimization model. The introduction of dynamic factors is important in bridging the gap that currently between research studies and the needs of real-world applications. Real-world problems are complex in nature, dynamic, and often involve stochastic parameters. Even the application of the best known methods requires certain simplification to be made in order to ensure they successfully converge to a solution. The problem is usually reduced in complexity to a much simpler one by ignoring certain factors or making assumptions about the operating environment. Consequently, the results produced by such models are not always of great interest to the actual stakeholders. What undoubtedly would be of interest is for future research directions that develop models and techniques capable of accommodating dynamic and stochastic factors.

In the near future, there is little doubt that logistic operations will be increasingly influenced by environmental factors. Environmental factors that currently affect the logistics sector are the increasing costs and taxes associated with non-green transportation technologies. Existing models may accommodate such factors by simply adding variables, and constraints. However, the resultant

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profit will inevitably be lower. Models should be changed to consider the possibility of investment in environmentally friendly technologies. This is not a straight forward strategy, since such investments could potentially lead to negative profits in the short term. Adopting such technologies is usually expensive and requires time before a positive return on any investment is made. Future research should focus on allowing higher tolerance and increasing the ‘line-of-sight’ of the models, such that the advantages resulting from the adoption of these technologies may be adequately taken into account. More options may ultimately be considered and eventually the simulation of such models could potentially lead to more desirable long term solutions.

CONCLUSION

Because of the increasing importance and complexity of supply chain management and logistic activities, this paper has presented the state of the art in studies relating to the field of transportation network optimization. Initially we presented the mathematical formalization of the problem, and discussed its reduction to the well-known problem of minimum cost flow optimization. We then investigated the most common approaches used to reach either an exact or an approximated solution. These methods included Mixed-Integer Linear Programming, Branch and Bound, (Multi Objective) Genetic Algorithm, (Fuzzy-) Ant Colony Optimization, and Swarm Particle Optimization.

In the remainder of the paper, we discussed research trends and the limitations of the currently proposed solutions. We commented on the trend in the need to have more complex models that truly represent the global transportation situation. On the failure of current algorithmic approaches in terms of their scalability, we suggested the exploration of more scalable methods, such as hyper-heuristics, and the exploration of parallelization techniques.

REFERENCES

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Bevilacqua, V., Costantino, N. & Dotoli, M. (2012). Strategic design and multi-objective optimisation of distribution networks based on genetic algorithms. International Journal of Computer Integrated Manufacturing, February 2013, 37-41.

Boudahri, F., Sari, Z. & Maliki, F. (2011). Design and Optimization of the Supply Chain of Agri-Foods: Application Distribution Network of Chicken Meat. CCCA 2011 : International Conference on Communication, Computing and Control Applications, 1-6.

Chan, F. T. S. & Kumar, N. (2009). Effective allocation of customers to distribution centres: A multiple ant colony optimization approach. Robotics and Computer-Integrated Manufacturing, 25(1), 1-12.

Chang, Y.-H. (2010). Adopting co-evolution and constraint-satisfaction concept on genetic algorithms to solve supply chain network design problems. Expert Systems with Applications, Oct, 37(10), 6919-6930.

Chen, A., Subprasom, K. & Ji, Z. (2006). A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem. Optimization and Engineering, Sep, 7(3), 225-247.

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Che, Z. (2012). A particle swarm optimization algorithm for solving unbalanced supply chain planning problems. Applied Soft Computing, Apr, 12(4), 1279-1287.

Che, Z. & Chiang, C. (2010). A modified Pareto genetic algorithm for multi-objective build-to-order supply chain planning with product assembly. Advances in Engineering Software, Jul, 41(7-8), 1011-1022.

Ding, H., Benyoucef, L. & Xie, X. (2009). Stochastic multi-objective production-distribution network design using simulation-based optimization. International Journal of Production Research, Jan, 47(2), 479-505.

Ganeshan, R. (1999). Managing supply chain inventories: A multiple retailer, one warehouse, multiple supplier model. International Journal of Production Economics, Mar, 59(1-3), 341-354.

Georgiadis, M. C., Tsiakis, P., Longinidis, P. & Sofioglou, M. K. (2011). Optimal design of supply chain networks under uncertain transient demand variations. Omega, Jun, 39(3), 254-272.

Ghoseiri, K. & Nadjari, B. (2010). An ant colony optimization algorithm for the bi-objective shortest path problem. Applied Soft Computing, Sep, 10(4), 1237-1246.

Goldberg, A. & Tarjan, R. (1987). Solving minimum-cost flow problems by successive approximation. Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing.

Han, N. & Ji, X. (2010). Optimization of logistics distribution routing problem based on improved ant colony algorithm. Machine Learning and Cybernetics (ICMLC), 2010 International Conference, July, 11-14.

Huang, M., Li, R. & Wang, X. (2011). Network Construction for Fourth-Party Logistics based on Resilience with Using Particle Swarm Optimization. Chinese Control and Decision Conference (CCDC), 3924-3929.

Jiang, Y., Zhao, L. & Sun, S. (2009). A Resilient Strategy for Meat-food Supply Chain Network Design. IEEE International Conference on Industrial Engineering and Engineering Management, 1479-1483.

Kennedy, J. & Eberhart, R. (1995). Particle swarm optimization. Proceedings of ICNN - International Conference on Neural Networks, 4, 1942-1948.

Lau, H. C. W., Chan, T. M., Tsui, W. T. & Ho, G. T. S. (2009). Cost Optimization of the Supply Chain Network Using Genetic Algorithms. IEEE Transactions on Knowledge and Data Engineering, 99, 1-36.

Lin, L., Gen, M. & Wang, X. (2009). Integrated multistage logistics network design by using hybrid evolutionary algorithm. Computers & Industrial Engineering, Apr, 56(3), 854-873.

Musa, R., Arnaout, J.-P. & Jung, H. (2010). Ant colony optimization algorithm to solve for the transportation problem of cross-docking network. Computers & Industrial Engineering, Aug, 59(1), 85-92.

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Randall, M. & Lewis, A. (2002). A parallel implementation of ant colony optimization. Journal of Parallel and Distributed Computing.

Sadjady, H. & Davoudpour, H. (2012). Two-echelon, multi-commodity supply chain network design with mode selection, lead-times and inventory costs. Computers & Operations Research, Jul, 39(7), 1345-1354.

Syarif, A., Yun, Y. & Gen, M. (2002). Study on multi-stage logistic chain network: a spanning tree-based genetic algorithm approach. Computers & Industrial Engineering, Jul, 43(1-2), 299-314.

Tseng, S.-P., Tsai, C.-W., Chiang, M.-C. & Yang, C.-S. (2010). A fast Ant Colony Optimization for Traveling Salesman Problem. IEEE Congress on Evolutionary Computation.

Utama, D. N. et al. (2011). Multi Objectives Fuzzy Ant Colony Optimization of Palm Oil Based Bioenergy Supply Path Searching. Advanced Computer Science and Information System (ICACSIS), 177-182.

Xiang, Q., Li, H., Huang, B. & Li, R. (2012). Improved Ant Colony Optimization for Multi-objective Route Planning of Dangerous Goods. 8th International Conference on Natural Computation (ICNC 2012), 772-776.

Yeh, W.-C. & Chuang, M.-C. (2011). Using multi-objective genetic algorithm for partner selection in green supply chain problems. Expert Systems with Applications, Apr, 38(4), 4244-4253.

Ying-Hua, C. (2010). Adopting co-evolution and constraint-satisfaction concept on genetic algorithms solve supply chain network design problems. Expert Systems with Applications, 37, 6919-6930.

Yu, B. (2005). Optimizing Bus Transit Netwrok with Parallel Ant Colony Algorithm. Proceedings of the Eastern Asia Society for Transportation Studies, 5, 374-389.

Zhao, K., Kumar, A., Harrison, T. P. & Yen, J. (2011). Analyzing the Resilience of Complex Supply Network Topologies Against Random and Targeted Disruptions. IEEE Systems Journal, 5(1), 28-39.

Zhao, X. & Dou, J. (2011). A Hybrid Particle Swarm Optimization Approach for Design of Agri-food Supply Chain Network. IEEE International Conference on Service Operations and Logistics and Informatics (SOLI), 162-167.

Zhou, P. & Deng, Q. (2009). Hybridizing fast taboo search with ant colony optimization algorithm for solving large scale permutation flow shop scheduling problem. 2009 IEEE International Conference on Granular Computing.

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ADDITIONAL READING SECTION

Additional reading about linear programming:

Chen, J., Lu, J., & Qi, S. (2010). Transportation network optimization of import crude oil in China based on minimum logistics cost. Emergency Management and Management Sciences (ICEMMS), 2010 IEEE International Conference, 335-338.

Bidhandi, H. M., Yusuff, R. M., Megat Ahmad, M. M. H., & Abu Bakar, M. R. (2009). Development of a new approach for deterministic supply chain network design. European Journal of Operational Research, 198(1), 121-128.

Creazza, A., Dallari, F., & Rossi, T. (2013). Applying an integrated logistics network design and optimisation model : the Pirelli Tyre case. International Journal of Production Research, 37-41.

Taylor, C., & Weck, O. d. (2006). Integrated transportation network design optimization. 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, May, 1-16.

Additional readings about evolutionary strategies:

Lianshuan, S., & Zengyan, L. (2009). An improved pareto genetic algorithm for multi-objective TSP. Fifth International Conference on Natural Computation, 585-588.

Dong, Y., Gu, J., & Li, N. (2007). Combination of genetic algorithm and ant colony algorithm for distribution network planning. Proceedings of the Sixth International Conference on Machine Learning and Cybernetics, Hong Kong, August, 19-22.

Wang, L., Kowk, S. K., & Ip, W. H. (2011). Design of an improved quantum-inspired evolutionary algorithm for a transportation problem in logistics systems. Journal of Intelligent Manufacturing, 23(6), 2227-2236.

Syarif, A., Yun, Y., & Gen, M. (2002). Study on multi-stage logistic chain network: a spanning tree-based genetic algorithm approach. Computers & Industrial Engineering, 43(1-2), 299-314.

Oliveira, F., Hamacher, S., & Almeida, M. R. (2009). Process industry scheduling optimization using genetic algorithm and mathematical programming. Journal of Intelligent Manufacturing, 22(5), 801-813.

Liu, Q. (2008). Study on multi-object optimization of logistics network based on genetic algorithm. International Conference on Computational Intelligence and Security, 210-214.

Ko, H. J., & Evans, G. W. (2007). A genetic algorithm-based heuristic for the dynamic integrated forward/reverse logistics network for 3PLs. Computers & Operations Research, 34(2), 346-366.

Hosseinzadeh, M., & Branch, A. (2012). An Optimization model for reverse logistics network under stochastic environment using genetic algorithm. International Journal of Business and Social Science, 3(12), 249-264.

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Chen, A., Subprasom, K., & Ji, Z. (2006). A simulation-based multi-objective genetic algorithm (SMOGA) procedure for BOT network design problem. Optimization and Engineering, 7(3), 225-247.

Additional reading about ant colony optimization strategies:

Yan, Y., Zhao, X., Xu, J., & Xiao, Z. (2011). A mixed heuristic algorithm for traveling salesman problem. Third International Conference on Multimedia Information Networking and Security, 229-232.

Stutzle, T., & Hoos, H. (1997). MAX-MIN Ant System and local search for the traveling salesman problem. Proceedings of 1997 IEEE International Conference on Evolutionary Computation (ICEC '97), 309-314.

Dorigo, M., Maniezzo, V., & Colorni, A. (1996). Ant system: optimization by a colony of cooperating agents. IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics: a Publication of the IEEE Systems, Man, and Cybernetics Society, 26(1), 29-41.

Blum, C. (2005). Ant colony optimization: Introduction and recent trends. Physics of Life Reviews, 2(4), 353-373.

Korukoglu, S., & Ball, S. (2011). An improved Vogel's approximation method for the transportation problem. Mathematical and Computational Applications, 16(2), 370-381.

Yu, B. (2005). Optimizing bus transit network with parallel ant colony algorithm. Proceedings of the Eastern Asia Society for Transportation Studies, 5, 374-389.

Additional reading about particle swarm optimization:

Huang, Y., Qiu, Z., & Liu, Q. (2010). Supply chain network design based on fuzzy neural network and PSO. Machine Learning and Cybernetics (ICMLC), 2010 International Conference, September, 2189-2193.

Additional reading about memetic algorithms:

Pishvaee, M. S., Farahani, R. Z., & Dullaert, W. (2010). A memetic algorithm for bi-objective integrated forward/reverse logistics network design. Computers & Operations Research, 37(6), 1100-1112.

Monte Carlo hyper-heuristic:

Ayob, M., & Kendall, G. (2003). A Monte Carlo hyper-heuristic to optimise component placement sequencing for multi head placement machine. Proceedings of the International Conference on Intelligent Technologies, InTech.

KEY TERMS & DEFINITIONS

Optimization Problem: The problem of finding the best value for a given max/min functions according to a set of constraints on the function variables.

Transportation Network: A given set of connections between producers and dealers/customers which may be used by the producers to transport finished goods to the dealers.

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Linear Programming: A mathematical technique used to solve optimization problem. Linear programming require the problem to be define as a mathematical model consisting of an objective function relative to a set of variables, and a set of constraints over those variables. Linear programming may only be applied if all the relationships in the model are linear.

Multi-Objectives: An objective function of an optimization model is defined as multi-objectives if it models more than one entity is to be optimized; the optimization problem has to be solved in more than one dimension. For instance, an objective function that includes variables of profit and time is multi-objectives.

Evolutionary Strategy: A strategy that mimics evolutionary models defined by Darwin’s laws of evolution. Evolutionary strategies require the definition a population, a crossover operator and fitness metric. The strategy used is to combine elements of the population, evolve them, and keep only the fittest/best ones. After a number of iterations, evolutionary strategy should produce a stronger population that represents a set of approximated solutions.

Ant Colony Systems: A system that mimics the behavior of ants in their search of food. Some experiments and observations of ant colonies have shown that ants very often are able to find the shortest path between the colony and their food source. It is the characterization of this behavior in a stochastic algorithm, and its application in finding the shortest path between a source and a destination or as in the case of transportation networks, between the producer and the dealer that defines the method.

Particle Swarm Optimization: A search strategy that starts from an initial set of candidate solutions (the particles) and try to improve them by looking at their neighbors in the solution spaces. A solution, or particle is moved according to a local criteria (i.e. the particle moves to the local best), in combination with a criteria based on the situation of all other particles (i.e. the particle moves to the best known position of the other particles).

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