Selective Breeding Model for Optimizing Multi Container Loading Problems with Practical Constraints

Multi container loading problem have been considered in this research for optimization of its packing pattern to yield maximum utilization of the container volume by satisfying the practical constraints such as boundary crossing constraint, weight constraint, stability constraint and placement constraint using selective breading algorithm. To be useful in real time packing, developed model also checks the feasibility of packing pattern and also uses best fit tuning algorithm for forbid possible empty spaces inside the container with the available bins and thereby avoid cargo displacements at the travel. The boundary crossing constraint conforms that the pallets are completely packed inside the containers without any overlap between themselves and with container boundary. Weight constraints are to check the total weights of the bins to be packed are within the threshold limit. Stability constraint is to satisfy the centre of gravity of the cargo is in line with the container and also the load bearing capacity of the bins. Placement constraint is to build pallets by considering the ease in loading and unloading respectively. In order to validate the developed model, the computational study had performed with large number of instances from ORLIB and the obtained solutions were satisfactory in most cases.


1.
Introduction The primary objective of Multi Container Loading Optimization Problems (MCLOP) is to pack the maximum number of available bins or boxes into the minimum number of containers by satisfying the practical constraints, if exist. Among the many variants of the MCLOP, in this research, the problem of categorizing the available bins into pallets and packing those pallets into the containers to reduce the freight rate and to serve the customers by delivering to the desired locations in time have been considered. Achieving an optimal packing solution for the container loading problems are having many economic and environmental pros such as reduced freight rate, ease in loading and unloading at multiple delivery points, maximum utilization of available container volume, avoiding the bin damages and maintain dynamic stability of bins and the containers.
In order to reduce the computational complexity and for easy in execution diagnosis, the entire research have been divided into five module, palletizing module, selective breeding module, pallet 3rd International Conference on Advances in Mechanical Engineering (ICAME 2020) IOP Conf. Series: Materials Science and Engineering 912 (2020) 032010 IOP Publishing doi: 10.1088/1757-899X/912/3/032010 2 optimization module, Tuning module and post processing module. In the first palletizing module, the user defined boxes are categorized based on the shape and size in the ascending order. Items of the similar product are categorized and arranged together to form a pallet. Similarly 'm' numbers of pallets have been formed with the available 'n' number of bins. The remaining bins are left unpalletized and will be used by the tuning algorithm in the tuning module later. Again the pallets are categorized with respect to the height to form the pallet layers. The packing pattern of this pallet layers will be optimized in this research using selective breading methodology. The dimensions of the pallet layers of each product type have been identified and will be given with the unique identifiers. Thus the computational complexity reduces by reducing the bin packing problem into the pallet loading problem. The second module is the selective breeding module, in which the algorithm have been developed for optimization and the parameters of the developed algorithm have been fine-tuned through sensitivity analysis to yield best result for the application.
The fitness function developed for this research is the combination of volume maximization and constraint satisfaction penalty function. The third module is the optimization of the pallet packing. The consideration taken into account for packing are that the company uses identical rectangular front open typecontainers, containers are large enough to fill all the orders and are of standard sizes, number of containers are more than the required, pallets are loaded in layer by layer arrangements and so on. In logistics transportation, it is severely monitored for the safety reasons which include the safety for both the containers and the products to be transported. This safety transportation introduces specific constraints such as the total weight of packed bins should be less than the container threshold weight carrying capacity and packed bins should be stable without empty spaces to prevent the bins from damages during transportation to various locations. Moreover, the stability of cargo has been obtained by making the center of gravity of the packed bins as close as possible to the center of gravity of the truck/container. The fourth module is the tuning module, in which the left out empty spaces have been filled with the remaining unpacked bins using best fit algorithm. Thereby the best packing pattern have been obtained in container loading. To be useful in practice, the final module is the post processing module, in which the graphical format of the identified packing pattern have been generated in the Cartesian coordinate system to make the packers or layman for ease in understanding the loading and unloading pattern.
Solutions to the MCLOP have beenfeasible and stable, because the realistic constraints such as dynamic stability, equilibrium of the items, risks in the loading and unloading processes, displacement of items during the journey, safety, etc. had considered in the fitness function of the optimization problem itself. Thus the developed modules had tested with the data sets available in the ORLIB and the results obtained are in satisfactory level. The major researches carried out in the considered research area are given in the following section.
In recent years, it is observed that there is drastic increase in the number of studies in the area of container loading problems with the practical constraints.Alonso et al (2019) developed the mathematical models which can generate the packing pattern for the multi container loading problems with practical constraints, but the author had some restriction in applying the model for universal problems. Antonio et al (2018) introduced the new load balance methodology for optimizing the container loading problem based on the weight in road transportation applications. Bischoff and Ratcliff (1995) explored various constraints involved in the development of optimization methodologies for the containerloading problems and used the BR instances, which has been used in this research for experimenting the developed selective breeding algorithm. Bortfeldt and Gehring (2001) enhanced the performance of the genetic algorithm by hybridization and proved that the hybridization yields better result compared to the raw algorithm for the container loading optimization problems. So in this research, the selective breeding algorithm has been hybridized with the best fit algorithm to yield best result.
Brand and Pedroso (2016) developed a generalized arc-flow formulation with graph compression methodology for solving the bin packing and the related problems and proved that the methodology produces better performance. Correcher

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Problem definition Based on the order placed by the customers, the distribution center started serving the needs by sending the product to the logistic centers. The lists of ordered products 'n' by the customer have to be completely transported to the customer location can be through the containers. Each and every product received by the logistic industries are having a predetermined pattern, with dimensions length, width and height represented by li,wi, and hi respectively, weight wti and constraint ci. In general, the logistic companies are using pallet packing by categorizing the available boxes either based on dimension or on delivery locations. The dimension of the pallets are represented by its length, width and height represented by lpwp, and hp respectively, weight wtp and constraint cp. Once the bins are categorized, similar bins have been piled up on pallet bases. These pallets are then loaded into trucks for delivering them to the required locations. The logistic companies are having set of 'P' identical containers of standard sizes and are off rectangular prismatic shapes of standard dimensions (L, W, H) and maximum threshold weight (WT). This weight includes the weight of the container and the packed bins, which will be assumed to be distributed equally. The mathematical equations for identifying the possible positions of the segregated pallets into the container are given in the Equation 1. (1) The Equation 1 is used to identify the number of permissible positions to pack the pallets inside the container along the length and width. Heights of the pallets have not been considered, in order to reduce the computational complexity and layer by layer packing doesn't have much influence over the Once the unfeasible points have been eliminated, then the best placement point needs to be identified. The best placement point is the point which is having the higher placement ratio , in the final stage, if the ratios are less than 0.5, then the pallets have to be placed in the row wise placements without exceeding the container boundary. Thus the pallets have to be loaded into the container.
The benchmark data set used in this research for experimenting the packing has been taken from the BR model. In the data set, the number of productsvaries from 1 to 142 and with the objective of identifying the minimum number of containers required for packing all the products. For the considered instances in the dataset, the constraints considered are the boundary crossing, weight, stability and the placement constraint. The boundary crossing constraint ensures that all the products are within the container boundary and the boundary includes the pallets and the spacers. The weight constraint ensures that the weight of all the packed products should be less than the threshold weight of the container i.e.
. The stability constraint checks the overhanging of the pallets in each and every layer and ensures that the base of the pallet has to be placed completely over the top of the bottom layer pallets. Also stability constraint checks the weight carrying capacity of the bottom bins. The placement constraint ensures that the products to be packed based on the delivery locations and in this research, it has been given with less penalty value, because this tendency has not been uniform compared to the other constraints and completely conflicts with other constraints and packing pattern.
In this research, in order to compute the packing pattern by satisfying the constraints in less computation time, selective breeding algorithm have been used. The adapted selective breeding algorithm consider all the constraints described and always returns a feasible solution for the corresponding configuration in every iterations, the advantage of the adopted algorithm is to identify the best from the same breed i.e. feasible solution again and again to yield the better optimal packing pattern.

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Proposed selective breeding algorithm Selective breeding algorithm (SBA) mimics the behaviour of the chicken or animal breeding methodology, i.e. identifying the best chromosome property from the existing and producing the 'n' number of breeds from the identified chromosomes. Thus the generated breeds having the desired properties. Again the same procedure has to be repeated till the termination condition achieved. The finally obtained breeds are having the user desired properties. In this research, each bin/pallet is assumed to be a gene in the chromosome. The length of the chromosome is equal to the number of instances in the set or the number of bins in the set to be packed into the container. This chromosome is allowed to breed its child to minimize the empty space inside the container. The consideration made in this research is that the size of the bins includes the bin size along with the size of spacer/wooden layer used for lifting the bins in loading and unloading. SBA algorithm consists of fivesteps. The first step is the initial population generation stage. In this stage, the chromosome size is set as that of the instances size. For example, a need arises to pack 1000 bins, then the chromosome size is set to 1000. These sizes will be dynamically varied or reduced if all the iterations are exceeds the container volume in the step of 10 %. Each gene or string in the chromosome represents the bin in the packing order. The sample initially generated 10 number of chromosome are shown in the Figure 2. The decimal numbers represents the bins types and the repetition of the numbers denotes the bins in the sequence of arrangement. The second stage of the SBA is the calculation of the breeding factor (Bf) value for the randomly generated parents and is given in the Equation 3. Then the generated parents are sorted based on the breeding factor value in ascending order. (3)

Whereas,
The third stage is the segregation of the dominant and recessive set. The entire population has been segregated into two sets based on the fitness value. The higher fitness set is called as dominant set having the best chromosomes parents and remaining set is called as recessive set having worst chromosomes parents. Then one dominant and one recessive parent from each set have to be allowed to breed its children's. In the above sample, 5 dominant and 5 recessive parents were clubbed together for breeding i.e. (1) D1r1 (2) D2r2 (3) D3r3 (4) D4r4 (5) D5r5.The fourth stage is the breeding stage, in which all the possible combinations of the chromosomes have to be breed from the available parents. The breed combinations are D1r1 x D2r2; D2r2 x D3r3; D3r3 x D4r4; D4r4 x D5r5; D1r1 x D3r3; D2r2 x D4r4; D3r3 x D5r5; D1r1 x D4r4; D2r2 x D5r5; D1r1 x D5r5. As the total, 80 breed can be generated for the above combinations. The fusion points can be the 50% of the length of the chromosome i.e. 50 fusion points and has to be generated at random for each and every breed. In the fusion points, the genes have to be swapped between the dominant and the recessive chromosomes. Thus as the result of breeding, 80 new chromosomes have been obtained along with the 10 initially generated parents. Because of the fusion in breeding, the solution may struck with the local optima and is called as in-breeding depression. To avoid this, in this work, 10% of chromosomes have been generated at random and added to the population. The fifth stage is the identification of the best chromosomes based on the breeding factor values. The best 10 chromosomes have been selected for the next iteration and the same procedure has to be followed till the termination conditions achieved. In this experimentation, the termination condition considered are the breeding values had to reach 1 or the number of iterations reached 100. The finally obtained set of chromosomes is the optimal packing pattern which can yield maximum container volume. Further to enhance the container volume utilization, the left out bins are given as the input to the best fit algorithm and the algorithm fits the bins in the available spaces inside the container and is the tuning module.

4.
Results and discussion In this research, experiments were run on 15 instance sets BR1, BR2, ..., BR15 [4]. Each set consists of 100 instances and have been categorized in to three classes based on the bin heterogeneity. Class 1 of the bins are of one types i.e. strongly homogeneous, whose instances are in BR0. Class 2 of the bins are of few different types i.e. weakly heterogeneous, whose instances from BR1-7. Class 3 of the bins are of different types i.e. strongly heterogeneous, whose instances are BR8-BR15. In order to validate the developed hybrid SBA module, the output obtained from the developed module has been compared with the previous research data in the Table 1. From the experimental output is clear that the developed SBA module without any constraints, produce comparatively better results, because it analysis all the possible combinations of bins, also breed the best from the better combinations. On the other hand, by including the constraints, the results have not up to the level expected, because the constraints added negative value to the breeding factor and thereby eliminates the best combinations which yield higher volume utilization. Thus the 3rd International Conference on Advances in Mechanical Engineering (ICAME 2020) IOP Conf. Series: Materials Science and Engineering 912 (2020) 032010 IOP Publishing doi:10.1088/1757-899X/912/3/032010 7 breeds from the best parent have eliminated and breed from the better parent only iterated. So by including the constraints, the results are not much satisfactory in using the SBA.
Stability constraint has been considered as one of the most essential issues in identifying the optimal solution, because it includes placement bin base size, allowable maximum height that the subset of layers can pile up and weight bearing capacity of the bottom bins. Similarly the placement constraint restricts some of the bin placements which can yield better packing pattern. For safety reasons, it is essential that the centre of gravity of the container should be located near to its geometric centre and in this research, layer by layer packing had been used, which automatically divides the load equally all over the container base. Thus the developed module can pack all the available bins in minimum number of containers and identifying the better results in less computational time. In addition to the solution, in this research, heuristic post processing module also developed i.e. the graphical models have also been generated for ease of packing and in layman understandable format, which is shown in the Figure 3. The layer by layer arrangement shown on the right of the Figure 3 is filled by the decimal numbers, which is filled with the bin numbers stored in the database. So that the packing layman can know the position and orientation of each and every bin. Also the developed heuristic module fills the decimal or bin numbers with the units equal to the size.

5.
Conclusion Revolution in the digital marketing increases the need of generating the high-quality solutions for multi objective loading problems by satisfying the constraints and involving multiple containers had been addressed in this research using hybrid selective breeding algorithm. In this paper, for experimenting, 15BR instances have chosen and the results obtained are in the satisfactory level and the optimal or quasi-optimal results are produced in reasonable times. Among the BR models, the experimental results show that the optimal solutions were obtained in most cases and on the other hand, for few instances for which optimality was not proven, but a feasible solution were found. Compared to the mathematical models, the SBA model generates results from the random initial solution and is independent of the parameters and the applications. In order to get the feasible solution, the negative penalty constraint function has been included in the objective function itself, so the invalid or unfeasible solutions breeds were eliminated in the initial stages itself. The boundary crossing constraints, weight constraint, stability constraint and placement constraint have also been considered. Also, the developed hybrid module generates the graphical loading plan to be useful in practice for the packers.In future, some interesting extensions have also been considered, such as top open truck, wall building approach, splitting demands over a time horizon, packing based on expected delivery dates, freight rate optimization, etc.