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This paper presents four different hybrid genetic algorithms for network design problem in closed loop supply chain. They are compared using a complete factorial experiment with two factors, viz. problem size and algorithm. Based on the significance of the factor “algorithm”, the best algorithm is identified using Duncan’s multiple range test. Then it is compared with a mathematical model in terms of total cost. It is found that the best hybrid genetic algorithm identified gives results on par with the mathematical model in statistical terms. So, the best algorithm out of four algorithm proposed in this paper is proved to be superior to all other algorithms for all sizes of problems and its performance is equal to that of the mathematical model for small size and medium size problems.

Supply chain management system has been being practiced as one of the main applications and research areas in the Industrial Engineering and Management discipline for the past two decades. A supply chain system is a complex network of business distribution channel which includes suppliers, manufacturers, distributors or wholesalers, warehouses, retailers, service providers and customers involved in the upstream and downstream flows of goods or services, information and finances. The effective and efficient management of the flows among these channel partners ensures the successive implementation of the entire supply chain. This can be achieved by adopting a coherent supply chain strategy with an appropriate network design, planning and implementation of the network for the execution of the flows in the supply chain system. The integration of entire supply chain flows into a closed loop network is the need of the hour now to ensure a business to be economically and environmentally sustainable with the changing trends in business and social environments, growing environmental consciousness in the society and government legislations to protect the environment as well as the business.

The integrated supply chain network design problems involve the decisions on the configuration of the network like, locating the right facilities, allocating the right capacity to the facilities, selection of the right mode of transportation and the route or path and optimizing the entire shipment of goods among the facilities or nodes in the entire distribution channel. These network design problems are often solved by using large optimization models. As these CLSC network design problems are combinatorial and complex in nature, handling of these problems with conventional optimization techniques like exact algorithms is quite difficult. Meta-heuristics algorithms like genetic algorithms and their hybrids are more suitable for these problems. In this context, this research work deals with a multi-echelon closed loop supply chain network design with forward and reverse logistics chains. In this research, an attempt has been made to develop a set of hybrid genetic algorithms and then select the best using through a complete factorial experiment.

The structure of this research paper is organized as follows. The next section gives the overview of the GAs and the third section deals with the literature review of the research works carried out in the past and recent periods and summarizes with the research gaps. The fourth section presents the statement of the problem identified and the fifth section presents the appropriate research methods adopted to solve the problem. The sixth section discusses the development and implementation of the mathematical model and the seventh section discusses the design, development and implementation of the hybrid genetic algorithms and its source codes. The eighth section discusses the comparison of the variants of hybrid genetic algorithms and determines the best HGA among them. The ninth section compares the best HGA with the mathematical model in terms of their performance measure for the optimization of the network. The last section concludes this research work with the summary of the main research outcome, scope and future research directions.

Genetic Algorithm (GA) is inspired by the theory of natural evolution and its principles. It is one of the directed random search techniques, employed to find a near-optimal solution for many larger problems in complex multi-dimensional search spaces. These algorithms encode a potential solution for complex problems on a simple chromosome like data structure and using techniques of natural evolution like inheritance, selection, mutation and crossover. After coding the solution in an appropriate way, GA iteratively works and evolves to get the global optimum without getting restricted at a local optimum. The individuals or chromosomes in a population are manipulated by the genetic operators to improve their fitness values while searching for global optimum solutions.

The steps involved in the formulation of Genetic Algorithms are mentioned as follows.

Chromosome Representation or Encoding: The individuals or the chromosomes are encoded or represented with a coding system to enable the computational processing of GA.

Initiation: Choose the initial population of the individuals.

Evaluation: Evaluate the fitness of each individual in population.

Breeding: Breeding is the process of reproduction with the following sub-steps which gets repeated until the termination condition gets satisfied.

Selection: Process of selecting the individuals with greater fitness for reproduction.

Crossover: Perform the crossover of selected individuals with a crossover probability for breeding new individuals which are offsprings.

Mutation: Perform the mutation process with a mutation probability on new individuals or offsprings for getting better offsprings.

Reevaluation: Evaluation the fitness of the each offspring created to find out the best or optimum fitness.

Termination: Terminate the process.

Genetic algorithms are theoretically and empirically proven to provide robust search and global solutions in complex scenarios for all sizes of problems. The Genetic Algorithms are capable of giving and efficient search in the problem domain and are finding wider range of applications in business, science, engineering and technology. These algorithms are more powerful in their search for improvement which has enabled the researchers to form different approaches of genetic algorithm. The genetic algorithms are classified as simple GA, parallel GA, hybrid GA, distributed GA, messy GA and adaptive GA etc., as given in

The literature is surveyed for the past 25 years and the varied research papers like concept papers, review papers, case studies, model papers and algorithmic research papers related to the principles, concepts of genetic algorithms, their application in the various closed loop and reverse supply chain network design problems are reviewed and presented in this section.

Goldberg and Deb [

problem, in the proposed algorithm.

Altiparmak et al. [

Altiparmak et al. [

Min and Ko [

Yun et al. [^{st} and 2^{nd} stages combined a new crossover operator called as weight mapping crossover (WMX). They applied a heuristic in the 3^{rd} stage to transportation of parts from processing center to manufacturer.

Gen et al. [

Costa et al. [

Zarei et al. [

Kaya et al. [

Hosseinzadeh and Roghanian [

Mehdizadeha and Afrabandpeia [

Rafsanjani and Eskandari [

Mahmoudi et al. [

Devika et al. [

Sarrafhaa et al. [

The literature is reviewed for the design of reverse supply chain systems as well as the closed loop supply chain systems i.e. the reverse supply chain networks in conjunction with forward supply chain networks under the following nine categories, viz. models, branch and bound algorithms, heuristics, genetic algorithms, simulated annealing algorithms, Petri net algorithms, simulation approaches and general approaches. The review outcome shows that the Meta heuristics particularly the Genetic Algorithm and its hybrids were applied majorly by the researchers to deal with the supply chain network design problems. The models developed were majorly based on MILP and a few were also on MINLP, depends on the complexity of the problems. The research gaps with respect to modeling and algorithmic approaches are identified and furnished as follows.

Most of the researches carried out in the prior periods deal with the CLSCND model very specific to a supply chain problem of a particular industry or a product category or a type of distribution network. There is a less or lack of research on the generic integrated closed loop network design model which can be applicable to a majority of the Industries or to the vast range of product categories or all kinds of distribution network. So, there is a need for the generic integrated model proposed in this research. This integrated model can be applicable to majority of the industries or product category or any kind of distribution network just by making minor addition or deletion in the stages or levels or echelons involved in the integrated model. Thus, this model can be applicable to various industrial sectors like Automobiles, Leather products, Textile products, Footwear, Consumer Electronics, Home appliances and Industrial equipment or machines, etc.

The past researches in the supply chain network design deal with the development of heuristics / meta heuristics mostly for forward chain network designs. A fair amount of research has been carried out to develop heuristics for reverse supply chain network designs. A very less research has been done to develop heuristics /Meta heuristics for the closed loop supply chains which are gaining importance in recent times. Also most of genetic algorithms developed in the past were designed with a specific set of GA parameters. A very less research has been done to study the performance of the GA with respect to changing GA parameters like selection methods, cross over methods, parent replacement methods etc., for the same problem. This gap is addressed in this research by developing new variants of hybrid genetic algorithm with different combination of GA parameters like using Elitism & Rank Selection methods, Chromosome Cross over techniques and Chromosome Replacement techniques.

The closed loop supply chain network model considered in this research work is a multi-echelon and multi stage network which includes manufacturers, wholesalers, retailers and first customers in the forward chain and service/repair centers, collector/dismantler/re-furbisher (CDR), remanufacturers, recyclers, disposal centers/land- fillers, resellers and second customers in the reverse chain. The network deals with the two types of product returns i.e. product returns due to repair and product returns due to end of use or life, in its reverse chain. The repair products are sent directly to the repair/service centre by the first customer for getting them repaired and the end of life products are returned either directly or through the retailer to the collector/dismantler/re-furbisher by the first customer for re-processing. The returned products thus collected in the CDR locations are sorted; parts are dismantled and segregated into recoverable and non-recoverable or waste items. The recoverable items are again segregated into re-furbishable, remanufacturable, recyclable items and shipped to the respective facilities or locations for recovery process. The non-recoverable or the waste items are shipped to the disposal centers/ Land-fillers and disposed through land filling or incineration. The recovered products through the process of refurbishing and remanufacturing are shipped to the second customers via resellers. The items recovered through recycling process are shipped to the raw material suppliers market by the recycler. Based on these discussions, it is clear that the returned products due to EOL are shipped from the first customers to the recovery facilities through a push mechanism and recovered products are shipped from the recovery facilities to the second customers through a pull logistics mechanism. In this context, the objective of this research work is to minimize the total cost which is the sum of the costs of both forward and reverse supply chains there by increasing the total profit of the closed loop network. After the text edit has been completed, the paper is ready for the template. Duplicate the template file by using the Save As command, and use the naming convention prescribed by your journal for the name of your paper. In this newly created file, highlight all of the contents and import your prepared text file. You are now ready to style your paper.

The research methods adopted for solving the research problem stated in the previous section are Modeling and Algorithmic research methods which are briefed as follows.

Modeling Research: A model is an abstraction of reality applied to the real life problems of business situations. Among the three types of modeling research methods, viz. symbolic model, mathematical model and simulation model, mathematical modeling research method is applied to design and develop the appropriate model for the specific CLSCND problem. The model is based on the Multi-Integer Non-Linear Progrmme (MINLP) as the problem is combinatorial and NP Hard in nature.

Algorithmic Research: An algorithm is a well-defined sequence of steps to solve a problem of interest in industry, business, etc. Exact algorithms, heuristics/meta-heuristics are applied to solve these problems based on whether the problem is polynomial or combinatorial in nature. In this research work, since the problem is combinatorial, complex and NP hard in nature, a Meta-heuristic i.e. Genetic Algorithm is applied to design and develop the appropriate algorithm for the specific CLSCND model. A Genetic Algorithm with hybrid architecture is planned due to its:

・ Global search nature leading to more accurate global optimal solutions rather than local optimal solutions by exact algorithms/heuristics;

・ Random pick up of chromosomes from the wider population and longer generations;

・ Faster in solving the problems and efficient in dealing the combinatorial NP hard problems.

Aravendan and panneerslvam [

In the closed loop network design model, in the forward supply chain, the manufacturers are responsible for manufacturing new or virgin products and supplying them to the wholesalers for distribution. The wholesalers are responsible for the distribution of new products to the retailers in their region. The retailers are responsible for selling the new products to the first customers as per their demands and also responsible for facilitating the after sales service. The customers’ nodes represent one or more customers or a group of customers. The first customers are responsible to return the products supplied to them as per the demand, either during the usage or after the usage of the products as either repair product returns or end of life product returns (EOL) respectively. In the reverse supply chain, the first customers return the repair products to the service/repair centre for getting them repaired and to reuse. The repair/service centers are responsible for providing the quality service to the customers and to ensure the prompt delivery of the repaired products to the first customers for reuse. The end of life products are returned to the collector/dismantler/re-furbisher (CDR) either directly or via retailers by the first customers. These CDR locations are responsible for collecting, dismantling and sorting the returned products and dismantled parts for refurbishing, remanufacturing, recycling and disposing via landfill or incineration. They are also responsible for supplying the remanufacturable to the manufacturers, recyclable to the recyclers, disposables to the land-fillers/incinerators. The CDR locations recondition the refurbishable products and distributing them directly to the resellers. The resellers also receive the remanufactured products from the remanufacturers and they sell both remanufactured and refurbished products to the second customers as per their demands. The recyclers are responsible for recycling the recyclable items received from CDR locations. The disposal centers/land-fillers are responsible for the safe disposal of the unusable wastes received from CDR locations either by landfilling or incinerations.

The assumptions and limitations of the proposed model are considered as follows: 1) The model is for a single product and single period network design, 2) The locations of the first customers and second customers are known and are with certain demands, 3) The quantities of products returned are certain and all the products supplied are returned as EOL products and repair products, 4) 60% of the products supplied are returned as EOL products and 40% are returned as repair product, 5) 50% of the EOL products are returned via retailers and remaining 50% of the EOL products are returned directly, to the Collectors/Dismantlers/Re-furbishers (CDR), 6) Out of the total returned EOL products, 30% are re-furbishable items, 45% are re-manufacturable items, 20% are recyclable items and 5% are non-recoverable and disposed by land-filler, 7) The quality of the remanufactured, refurbished and repaired products is different from that of the new product, 8) The potential locations of manufacturers, wholesalers, retailers, collectors/dismantlers/re-furbishers, repair/service centers, recyclers, land fillers and resellers are assumed, 9) The capacity of each location is known, 10) The costs parameters considered (viz., opening costs, operating costs, un-utilized capacity costs and transportation costs) are known for all the facilities and node, 11) The measure of quantity of products transported per trip is defined in the form of number of units per trip and 12) There is no shipment happening between the nodes in the same stage.

The various costs incurred at different nodes in the multi-echelons of the closed loop supply chain network are opening costs, operation costs and transportation costs for manufacturers, wholesalers and retailers in the forward chain, and the same costs are applied for repair/service centers, collectors/dismantlers/re-furbishers, land fillers, recyclers, remanufacturers and resellers in the reverse chain. An un-utilized capacity cost is also considered and added only to the hybrid centers manufacturers/remanufacturers. There is no cost considered for first customers and second customers as the products are picked up by them at the retail points in this particular model.

The proposed model has the objective of minimizing the total cost i.e. the cost of the shipment flows in both forward and reverse supply chains as given in

Subject to

Demand Constraints (DC):

Capacity Constraints (CC):

Balance Constraints (BC):

where,

Constraints (1) and (2) correspond to the demands of the first customers and second customers respectively. Constraint (3) makes sure that the sum of the outflows from each manufacturer to all the wholesalers does not exceed the capacity of the manufacturers. Constraint (4) makes sure that the sum of the outflows from each manufacturer to all the wholesalers plus sum of the outflows from the each manufacturer to all the resellers minus sum of the outflows from each CDR to all the manufacturers for remanufacturing does not exceed the capacity of the manufacturers. Constraint (5) makes sure that the sum of the outflows from each wholesaler to all the retailers does not exceed the capacity of the wholesalers. Constraint (6) makes sure that the sum of the outflows from each retailer to all the first customers and CDRs does not exceed the capacity of the retailers. Constraint (7) makes sure that the sum of the outflows from each repair center to all the first customers does not exceed the capacity of the repair centers. Constraint (8) makes sure that the sum of the outflows from each CDR to all the remanufacturers, resellers, recyclers and land fillers does not exceed the capacity of the CDRs. Constraint (9) makes sure that the sum of the inflows to land fillers from all CDRs does not exceed the capacity of the land fillers. Constraint (10) makes sure that the sum of the inflows to recyclers from all CDRs does not exceed the capacity of the recyclers. Constraint (11) makes sure that the sum of the outflows from each re-furbisher under CDRs to all the resellers does not exceed the refurbishing capacity of CDRs. Constraint (12) makes sure that the sum of the outflows from each remanufacturer to all the resellers does not exceed the capacity of the remanufacturers. Constraint (13) makes sure that the sum of the inflows to resellers from all the CDRs (refurbished items) and from all the remanufacturers does not exceed the capacity of the resellers. Constraint (14) makes sure that the sum of the outflows from each reseller to all the second customers does not exceed the capacity of the resellers. Constraint (15) makes sure that the sum of the quantities produced by the manufacturers is equal to the sum of the demands of the first customers. Constraint (16) makes sure that the sum of the quantities produced by the manufacturers is equal to the sum of the outflows from the manufacturers. Constraint (17) makes sure that the sum of the inflows to the wholesalers from each manufacturer is equal to the sum of outflows from the wholesalers to the retailers. Constraint (18) makes sure that the sum of the inflows to the retailers from each wholesaler is equal to the sum of the outflows from the retailers to first customers. Constraint (19) makes sure that the sum of the inflows to each first customer from retailer is greater than or equal to the sum of the outflows from each first customers to repair centers and CDRs. Constraint (20) makes sure that the sum of the outflows from each first customer to the repair center is less than or equal to the demand fraction for repair returns of the sum of the demands of the first customers. Constraint (21) makes sure that the sum of the inflows to each repair center from all the first customers is equal to the sum of the outflows from each repair center to all the first customers. Constraint (22) makes sure that the sum of the outflows from each first customer as EOL via retailer or directly to the CDRs is less than or equal to the demand fraction for EOL returns of the sum of the demands of the first customers. Constraint (23) makes sure that the sum of the outflows from each first customer as EOL via retailer to the CDRs is less than or equal to its return fraction of the sum of the EOL returns. Constraint (24) makes sure that the sum of the outflows from each first customer as EOL directly to the CDRs is less than or equal to its return fraction of the sum of the EOL returns. Constraint (25) makes sure that the sum of the EOL returns inflows to each CDR from first customer via retailer and directly is equal to the sum of the flow of CDR fractions exiting from each CDR to all the land fillers, recyclers, resellers and remanufacturers. Constraint (26) makes sure that the sum of the inflows to all the remanufacturers from each CDR is less than or equal to the sum of the outflows from the remanufacturers to resellers. Constraint (27) makes sure that the sum of the inflows to all the resellers from each CDR and each remanufacturer is equal to the sum of the outflows from the resellers to the second customers. Constraint (28) makes sure that the sum of the outflows from each reseller to all the second customers is equal to the sum of the demands of all the second customers.

Thus the mathematical model for the CLSC network design problem formulated as Multi Integer Non-Linear Programming model has been solved by Aravendan and Panneerselvam (2014) [

A hybrid genetic algorithm is the fusion or combination of two algorithmic structures i.e. the structure of the main genetic algorithm and a special algorithm specifically incorporated according to the nature of the problem. In this problem, the computation of total cost also includes the transshipment cost of the network in which the main genetic algorithm is fused with the transshipment algorithm which is the combination of VAM and U-V method Algorithms. This hybrid genetic algorithm is also developed with a combination of two selection methods i.e. Elitism and Rank selection. These combinations lead to the formation of variants of HGA developed.

This section presents the preliminaries of genetic algorithm adopted to develop the variants of .hybrid genetic algorithms.

Chromosome Encoding is a process of representing the individual genes in the chromosomes. The representation can be performed using bits, numbers, trees, arrays, lists or any other objects. In this particular problem, binary encoding (using bits) is performed as given as follows.

Binary Encoding

The most common way of encoding is a binary string in which each chromosome is encoded with a binary (bit) string. Each bit in the string represents a gene i.e. node or facility in the CLSC network design. So, every bit string is a solution but not necessarily the best solution. The binary encoding applied in this hybrid genetic algorithm for the example problem is illustrated in

A combination of elitism selection method and rank selection method are applied in all the HGA variants developed. The methods are briefly explained below.

1) Elitism Selection Method

In this method, the first best chromosome or the few best chromosomes are copied to the new population. The rest is done in a classical way. Such individuals can be lost if they are not selected to reproduce or if crossover or mutation destroys them. This method significantly improves the GA’s performance.

2) Rank Selection Method

Rank selection method ranks the population and every chromosome receives fitness from the ranking. The worst fitness has the last rank and the best fitness has the first rank. It results with slow convergence but prevents too quick convergence. It also keeps up selection process when the fitness variance is low. It preserves diversity and hence leads to a successful search.

An advantage of having more crossover points is that the problem space may be searched more thoroughly. The crossover methods applied for developing the variants of hybrid genetic algorithms are: 1) Two-point crossover method and 2) Uniform crossover method which are discussed as given in

The HGA variants are also developed by varying the chromosome replacement methods applied after crossover and mutation processes. The two methods, viz. both parent replacement method and weak parent replacement method, adopted in this research are briefed as follows.

1) Both Parent Replacement Method

In this method, the fitness values of the off-spring are replaced with that of the parent chromosomes irrespective of the nature of the fitness values. The fitness values of the parent chromosomes are not checked whether they are weaker or stronger than their corresponding offspring.

2) Weak Parent Replacement Method

In this method, the fitness values of the offspring are replaced only if they are stronger than the fitness values of the parent chromosomes. So, the subsequent generations will have chromosomes with better fitness values.

Facilities-Nodes/ Chromosomes | M | W | RT | FC | RP | CDR | LF | RC | RM | RS | SC | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 3 | 1 | 2 | 1 | 2 | 3 | |

Chromosome 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 |

Chromosome 2 | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |

M―Manufacturer, W―Wholesaler, RT―Retailer, FC―First Customer, RP―Repair Center, CDR―Collector/Dismantler/Re-furbisher, LF―Land Filler, RC―Re-Cycler, RM―Re-Manufacturer, RS―Re-Seller, and SC―Second Customer.

The various steps implemented sequentially to develop the hybrid genetic algorithms are furnished as follows.

Step 1: Input the following:

Number of stages = 11 (Manufacturers, Wholesalers, Retailers, First Customers, Repair Centers, Collector/ Dismantler/Re-furbisher, Recycler, Land filler, Remanufacturers, Resellers and Second Customers);

Maximum number of units/nodes in each stage from first to last stage. Ex. 3, 3, 3, 6, 2, 2, 2, 2, 3, 2, 3 (Medium size);

Maximum number of successive populations to be generated (N) = 50;

Maximum number of Chromosomes in Each Population-Population size (L) = 100;

The Generation Count GC = 1.

Step 2: Apply the binary encoding method for the chromosomes to decide on the status (open or close) of the genes (units) in the Chromosomes.

Step 3: Generate a random initial population of n chromosomes (suitable solutions for the problem). Let it be the larger population L = 100.

Step 4: Evaluate the Fitness function f(x) of each chromosome x in the population L. The fitness function is as used in the mathematical proposed by Aravendan and Panneerselvam [

Fitness Function Value FFV = OPC + OPRC + UCC + TC, Where OPC = Opening Cost, OPRC = Operation Cost, UCC = Un-utilized Capacity Cost and TC = Transportation cost. (The Transshipment Algorithm, a combination of VAM and U-V algorithms is incorporated to evaluate the transshipment cost in the Fitness function)

Step 5: Sort the population L by the objective function (fitness function) value in the ascending order for minimization problem.

Step 6: Apply Elite Count (say top 2) and Rank selection; Select a given percentage (say 30%) from the top of the larger population L leaving the elite count, to form a sub-population S.

Step 7: Randomly pick up any two unselected parent chromosomes from the sub-population S. Let them be the parents xand y.

Step 8: Perform the crossover of the parents x and y for a crossover probability P_{c} (say P_{c} = 0.3) to form their two new offspring x_{1} and y_{1}. If no crossover is performed, then assume the offspring x_{1} and y_{1} as the exact copies of the parent chromosomes x and y respectively. (Two point crossover and Uniform crossover methods are applied to form hybrids)

Step 9: Perform the mutation of each of the offspring x_{1} and y_{1} for a mutation probability P_{m} (say P_{m} = 0.06).

Step 10: Evaluate the fitness function value of each of the offspring x_{1} and y_{1}.

Step 11: Replace the FFV of parent chromosomes x and y in the larger population L, with the FFV of their respective offspring x_{1} and y_{1}. (Both parent replacement and Weak parent replacement methods are applied to form hybrids)

Step 12: Repeat Step 7 to Step 11 until all the chromosomes in the sub-population S are selected to create offspring.

Step 13: Increase the Generation Count (GC) by 1, i.e. GC = GC + 1.

Step 14: If GC ≤ N, then go to Step 5, else go to Step 15.

Step 15: Identify the chromosome in the larger population L, which has the best fitness function valueand printthe corresponding results.

Step 16: Stop.

The variants of hybrid genetic algorithm are developed by varying the GA parameters like selection method, crossover method and replacement method. Four hybrid genetic algorithms are thus constructed, viz. HGA1, HGA2, HGA3 and HGA4 with the following combination.

HGA1 has the combination of 2 point crossover method and both parent replacement method.

HGA2 has the combination of 2 point crossover method and weak parent replacement method.

HGA3 has the combination of uniform crossover method and both parent replacement method.

HGA4 has the combination of uniform crossover method and weak parent replacement method.

The configurations of these four hybrid genetic algorithms are illustrated in

The programme source codes for the four HGAs are developed using C#. Input data are fed by developing and implementing the store procedures in Microsoft SQL server. The whole programme is implemented in the Visual Basic. NET ver. 4.5 platform. This strategy has remarkably reduced the running time of the hybrid genetic algorithm for solving the problem.

The four hybrid genetic algorithms presented in this paper are compared using a complete factorial experiment with two factors, viz. “Problem Size” and “Algorithms”. The number of levels for the problem size is three, viz. small, medium and large. In all the three size problems, the number of nodes at manufacturer, wholesaler, retailer, first customers and second customers are varied according to the size of the problems. The number of levels for the algorithm is 4, viz. HGA1, HGA2, HGA3 and HGA4. In all these four algorithms, the selection method applied is common i.e. the combination of Elitism and Rank methods, but the crossover methods and chromosome replacement methods are different as illustrated in

Expt. No./ Problem Size | No. of Repns./ Expt. | No. of Nodes at the Various Stages in the Closed Loop Supply Chain Network Design | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

M | W | RT | FC | RP | CDR | LF | RC | RM | RS | SC | ||

1 Small | 6 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |

2 Medium | 6 | 3 | 3 | 3 | 6 | 2 | 2 | 2 | 2 | 3 | 2 | 3 |

3 Large | 6 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 2 | 4 |

M―Manufacturer, W―Wholesaler, RT―Retailer, FC―First Customer, RP―Repair Center, CDR―Collector/Dismantler/Re-furbisher, LF―Land Filler, RC―Re-Cycler, RM―Re-Manufacturer, RS―Re-Seller, and SC―Second Customer.

Expt. No./ Problem Size | Selection Method | Crossover Method | Crossover Probability | Mutation Probability | Parent Chromosome Replacement | ||
---|---|---|---|---|---|---|---|

1 Small | Rank | 2 pt. | Uniform | 0.3 | 0.06 | Both Parent | Weak Parent |

2 Medium | Rank | 2 pt. | Uniform | 0.3 | 0.06 | Both Parent | Weak Parent |

3 Large | Rank | 2 pt. | Uniform | 0.3 | 0.06 | Both Parent | Weak Parent |

The replications of the experiment were carried out in the above combinations for all the three problem sizes vis-à-vis the four HGAs and the results are plotted in

The different hypotheses proposed are as listed below.

・ Whether there are significant differences among the problem sizes in terms of the total cost;

・ Whether there are significance differences among the algorithms in terms of the total cost;

・ Whether there are significance differences among the interaction terms of problem size and algorithm in terms of the cost.

The results of ANOVA for the data given in

Hence, the best algorithm is obtained using Duncan’s multiple range tests. The result of Duncan’s multiple range tests are shown in

Expt. No./ Problem Size | Replications- Transportation Costs | Hybrid Genetic Algorithms (HGA)-Variants | |||
---|---|---|---|---|---|

HGA1 | HGA2 | HGA3 | HGA4 | ||

1 Small | Repln.1: TC = 1 | 118209336 | 118208336 | 118209336 | 115020093 |

Repln.2: TC = 3 | 116394408 | 116394408 | 119150408 | 115057808 | |

Repln.3: TC = 5 | 122613480 | 121389416 | 122613480 | 121253416 | |

Repln.4: TC = 6 | 122681016 | 121479480 | 122681016 | 121264802 | |

Repln.5: TC = 8 | 122816088 | 121916088 | 122806088 | 121716288 | |

Repln.6: TC = 10 | 122951160 | 122951160 | 122951150 | 121932160 | |

2 Medium | Repln.1: TC = 1 | 165602183 | 162519783 | 166398115 | 155925683 |

Repln.2: TC = 3 | 166398115 | 162788515 | 168316549 | 158710580 | |

Repln.3: TC = 5 | 168316549 | 168316549 | 168669898 | 159978749 | |

Repln.4: TC = 6 | 168669898 | 168669898 | 169105000 | 160969678 | |

Repln.5: TC = 8 | 166278430 | 169105000 | 169699983 | 162614783 | |

Repln.6: TC = 10 | 173616784 | 173616784 | 182700784 | 164954088 | |

3 Large | Repln.1: TC = 1 | 325416926 | 324416826 | 325416926 | 325406921 |

Repln.2: TC = 3 | 325766378 | 325756378 | 325776378 | 325696378 | |

Repln.3: TC = 5 | 326125830 | 326015830 | 326115930 | 326015830 | |

Repln.4: TC = 6 | 326773600 | 326582800 | 327133800 | 326637800 | |

Repln.5: TC = 8 | 327025008 | 326925008 | 327025008 | 326926008 | |

Repln.6: TC = 10 | 327230260 | 326989460 | 327230260 | 327230260 |

Source | Type III Sum of Squares | Degree of Freedom df | Mean Square | F | Sig. | Remarks |
---|---|---|---|---|---|---|

Algorithm | 1.677E+14 | 3 | 5.591E+13 | 6.417 | 0.001 | Significant |

Problem size | 5.596E+17 | 2 | 2.798E+17 | 32,114.712 | 0.000 | Significant |

Algorithm^{*} Problem Size | 1.931E+14 | 6 | 3.218E+13 | 3.694 | 0.003 | Significant |

Error | 5.227E+14 | 60 | 8.712E+12 | |||

Total | 5.605E+17 | 71 |

from other algorithms and its total cost is the least among the total costs all the algorithms. Hence, the hybrid genetic algorithm HGA4 is proved the best among all the four proposed hybrid genetic algorithms.

In the previous section, it is proved that the hybrid genetic algorithm 4 (HGA4) is proved to be the best among all the four proposed algorithms. Hence, in this section, it is benchmarked against the results of the mathematical model presented in section 6 using a complete factorial experiment in which the problem size is considered as one factor and the method of solving a problem is considered as another factor. The number of levels for the problem size is 3, viz. small, medium and large and the number of levels for the method is 2, viz. HGA4 and Model. The number of replications carried out under each experimental combination is 6. The results as per this complete factorial experiment are shown in

The different hypotheses of this comparison are listed below.

・ Whether there are significant differences among the problem sizes in terms of the total cost;

・ Whether there are significant differences among the two methods in terms of the total cost;

・ Whether there are significant differences among the interaction terms of problem size and method in terms of the cost.

The results of ANOVA executed for the experimental results shown in

Expt. No./ Problem Size | Replications- Transportation Costs | HGA4 | Mathematical Model |
---|---|---|---|

1 Small | Repln.1: TC = 1 | 115020093 | 114810000 |

Repln.2: TC = 3 | 115057808 | 114946700 | |

Repln.3: TC = 5 | 121253416 | 121066100 | |

Repln.4: TC = 6 | 121264802 | 121156700 | |

Repln.5: TC = 8 | 121716288 | 121302200 | |

Repln.6: TC = 10 | 121932160 | 121459500 | |

2 Medium | Repln.1: TC = 1 | 155925683 | 155849900 |

Repln.2: TC = 3 | 158710580 | 157710500 | |

Repln.3: TC = 5 | 159978749 | 159578800 | |

Repln.4: TC = 6 | 160969678 | 160503100 | |

Repln.5: TC = 8 | 162614783 | 162364900 | |

Repln.6: TC = 10 | 164954088 | 164234900 | |

3 Large | Repln.1: TC = 1 | 325406921 | No Solution |

Repln.2: TC = 3 | 325696378 | No Solution | |

Repln.3: TC = 5 | 326015830 | No Solution | |

Repln.4: TC = 6 | 326637800 | No Solution | |

Repln.5: TC = 8 | 326926008 | No Solution | |

Repln.6: TC = 10 | 327230260 | No Solution |

Source | Type III Sum of Squares | Degree of Freedom df | Mean Square | F | Sig. | Remarks |
---|---|---|---|---|---|---|

Algorithm | 8.121E+11 | 1 | 8.121E+11 | 0.079 | 0.782 | Not Significant |

Problem size | 1.010E+16 | 1 | 1.010E+16 | 979.781 | 0.000 | Significant |

Algorithm^{*} Problem Size | 8.261E+10 | 1 | 8.261E+10 | 0.008 | 0.930 | Not Significant |

Error | 2.0627E+14 | 20 | 1.031E+13 | |||

Total | 1.031E+16 | 23 |

In this paper, four different hybrid genetic algorithms have been proposed and they are compared through a complete factorial experiment with two factors, viz. problem size (three problem sizes) and Algorithm (four algorithms). It is found that there are significant differences among the four algorithms, viz. HGA1, HGA2, HGA3 and HGA4. Hence, using Duncan’s Multiple Range Test, it is found that the algorithm HGA4 is the best. In the next stage, HGA4 is compared with model using a complete factorial experiment with two factors, viz. Problem Size (Two sizes) and Method (HGA4 and Model). It is found that there is no significant difference between HGA4 and Model. So, HGA4 is proved to be superior in terms of giving very near optimal solution for small and medium size problems and provides the best solution when compared to all other hybrid genetic algorithms, viz. HGA1, HGA2 and HGA3.

Scope and Future Research DirectionsThe model and the hybrid genetic algorithm developed have good scope in the industrial application as they can be applied to a wide range of industries like automobiles, consumer electronics, textile apparels, fashion leather garments, luxury goods, footwear, home appliances, industrial equipment and machines, etc. As the incorporation of hybrid centers/facilities at the same location in the forward and reverse chains eliminates the establishment and maintenance costs of the separate facilities at different locations, the overall cost of the closed loop supply chain is minimized remarkably. The model and algorithm are proposed for the single product and single period system. Further research on the network design for multi-products and multi-periods CLSC system would be beneficial to the business sector. The future direction of research could also be an extension of this research work towards developing CLSC network design with incapacitated facilities to deal with the stochastic/uncertain demands of the customers.

The authors profusely thank M/S. Lindo Systems Inc., Chicago, USA for their technical support by providing the license of the extended version of LINGO 15 optimization software package. The authors also thank the anonymous referees for their constructive comments and suggestions, which have been incorporated in this research paper.

MuthusamyAravendan,RamasamyPanneerselvam, (2015) Development and Comparison of Hybrid Genetic Algorithms for Network Design Problem in Closed Loop Supply Chain. Intelligent Information Management,07,313-338. doi: 10.4236/iim.2015.76025