JASC: Journal of Applied Science and Computations ISSN NO: 1076-5131
A Graph Theory Approach for Optimizing 3D Bin Packing Problems with Weight Constraint
1 P. Sivasankar, 2 S K Rajesh Kanna, 3 N. Lingaraj, 4 G. Suresh 1Assistant Professor, 2 Professor, 3Assistant Professor, 4Professor 1 Department of Mechanical Engineering, 1 Rajalaksmi Institute of technology, India [email protected], [email protected], [email protected] ,[email protected]
Abstract : In the age of digital marking, the problem of storing and transporting arbitrary products all over the world in shortest time and with less transportation cost is an important research challenge in recent days needed both by the manufacturer and the logistic industries. This can be satisfied by packing the products into bins and the bins should be packed into the containers with a minimum storage space by changing its position and orientations and those problems are called as bin packing problems. This research mainly focuses on identifying optimal solution for bin packing problems using graph theory approach. Additionally, the developed module used to solve the benchmark instances and compared with their results. The obtained results are in par with the best known solutions.
IndexTerms - Bin Packing, Graph Theory, Optimization
I. INTRODUCTION The minimization of product transportation and logistics cost is an open research problem of recent days, as the sizes as well as the dimension of products ordered online are increasing day by day along with the increase in unpredictable volume. So, a need arises to identify an optimal solution in less amount of time which packs the bins in less container space and by considering the real time packing constraints. These type of problems arises in a wide variety of contexts and the study of one dimensional Bin Packing Problem (BPP)started in the early 1970 ‟s[Michael et al]. BPP states that there are an unlimited number of bins of
arbitrary sizes and shapes of volume V i>0, which to be packed into the container of capacity C, a set of items with their weights,
wi, Ʃwi of all the packed bins does not exceed the capacity of the container C and weight threshold W c. and the number of containers used for packing should be minimized. Thus the BPP became NP-Hard in strong sense [3]. The primary objective of this research is to minimize the number of containers required to pack all the available bins of prismatic shapes and arbitrary sizes. Thereby the transportation cost can be reduced and reduction in the logistic cost has been obtained by efficiently utilizing the available containers. Effective utilization of the container volume can be achieved by packing the bins without empty spaces, which the organization, seller and customers require. In this case, existing containers has been treated as rectangular prismatic shaped standard sized container, resource utilizations are packing of the bins of arbitrary volumes. Container capacity and maximum weight carrying capacity are the utilization threshold of the destination containers. The target is to identify the suitable bin packing pattern to minimize the destination containers and thereby maximizing the resource utilization, which in turn reduced the transportation cost of the product. Additionally, with more than one sized containers, the problem dimension increases. Besides this, there are many other considerations in the Bin Packing, but are not discussed in this paper. Not only Bin Packing Problems, but also the graph theory approach is also solving vast number of real world applications. Graph theory approach is an omnipotent mathematical tool which rephrases the real time applications into graphical form to provide unified solution to many classical problems in less time. In view of volume minimization problem, there exists various graph compression mechanisms. This research mainly focused on the solution of BPP in polynomial time, and to achieve the same, two offline graph theory based heuristics has been proposed. In the former approach, a vertex based weighted graph has been constructed from the available bins for packing, whereas each vertex in the graph represents the weight and volume of each bin to be packed in the container. Then, the heuristic chooses the real time packing constraints as the subset of the formulated vertices. The later heuristic approach is based on average weight criteria, which used to identify the minimal clique of the graph that to be packed in a container without exceeding the capacity of the container. The total number of clique partition is equal to the number of real time containers required. Volume VI, Issue I, January/2019 Page No:2945 JASC: Journal of Applied Science and Computations ISSN NO: 1076-5131 Also the developed approach can points towards a new era for solving multi-dimensional bin packing problem with conflicting constraints. The organization of this paper is as follows: The research work carried out by various researchers in the BPP is described in section II and the basic concepts of the related to the work are given in section III. Section IV elaborates the proposed algorithm and its implementations. Section V contains experimental results and in the final section VI concludes the research findings. II. LITERATURE REVIEW In this section, some of the major related works to solve the BPP are discussed in brief. Jnasen and Ohring [1997] consider time constrained scheduling problem where there was a set of jobs J with execution time t(j) to be machined in M machines and an undirected graph had constructed to identify the sequence of machining such a way the machines are utilized to the maximum [15]. Peter and Benny explained the representation of edge disjoint triangles in the graph having vertices n along with its compliments. Jansen and Ohring developed the heuristic approximation algorithm for solving the time constrained scheduling problems and in this research importance have not been given to the computational and packing time. Rajesh and Saravanan [2012] developed genetic algorithm for solving the ID bin packing problems without constraints. Epstein and Levin [2006] attempted to solve the BPP with conflicts using conflict graph approach by mainly improving the upper bounds on the graphs, interval graph and bipartite graphs along with the adaptation of weighting systems to analysis and identification of the best instances[16]. Rajesh kanna et al [2015] developed the ant colony based algorithm for solving the rectangular bin packing problem and proved that the algorithm is in par weith the standard heuristics. Mcclosekey and Shankar [2005] solved the BPP using the approach of Clique Graph and polynomial time approximation algorithm[17]. Rajesh et al [2012] developed a hybrid algorithm combines the genetic algorithm and tuning algorithm to solve the 3D container loading problem. Codenotti et al [2004] investigated the BPP by structurally arranging the available bins and packed them into the container from the lowest common ancestorhas low height[18]. Bujtas et al [2011] had used graph theory to solve BPP by constructing a graph G with lower and upper bound on its edges (bins to be packed into the container) and weights on its vertices (containers used to pack)[19]. Alexandr et al extended the Bollobás and Eldridge theorem on graph packing to hyper graphs packing and used two n vertex hyper graphs for experimentation. Rajesh and Malliga developed binary coded algorithm using evolutionary genetic approach to solve the bin packing problems with six major packing constraints. Richard et al formulated digraph approach for identifying the best packing paths and the algorithms used by the authors are Berge type augmenting configuration theorem, a min ‐max characterization, and a reduction to bipartite matching. Debajit and Samar developed a graph based algorithm for solving the 1D bin packing problems by considering the weight of the boxes. Raphael used regular tournament with n vertices having pair wise arc ‐disjoint directed triangles and constructed regular tournaments with a feedback arc set. Shachnai and Tami developed the heuristic algorithm for solving the online bin packing problems with constraints. Basu explained the analysis of algorithms in detail. Most of the above discussed literatures are hybridization of graph algorithms and exiting heuristics for solving BPP, as BPP categorized under NP hard problems. So in this research, attempt have been made to identify the minimum clique graph partition with weight constraint, which is the suitable bin packing sequence which utilizes less number of containers for multi container BPP. III. BIN PACKING PROBLEM USING GRAPH THEORY Initially the real time problems have to be converted in the graphical format, so that the graph theory approach has been used to identify the best packing sequence. The considerations made in this research to construct the graph are as follows. A Graph G is having set of vertices Vt(G) and set of edges E(G) connecting the vertices [West]. The vertices hold the weight values and the edge holds the relations associated between the end vertices. Vt(G) ϵ {bin1, bin2, …..bin n}, Whereas, bin represents the available items to be packed into the containers. Thus a vertex weighted graph has been constructed virtually. Vertex weighted graph is a unidirectional inter connected graph and the each vertex in the graph is assigned with a bin weight values. The values include the weight of the bin and its volume as positive values. Vt i(G) = {W i, V i} Whereas, W i and V i are the weight and volume of the bins to be packed into the container of capacity C and maximum weight threshold Wt(C). A Null Graph is a graph whose edge set isnull [Deo andNarsingh].This type of graph has been used to construct the dynamically varying bin packing problems and to hold the temporary bin data during execution of multi container loadingproblems. N(G) = { Null, Null} Volume VI, Issue I, January/2019 Page No:2946 JASC: Journal of Applied Science and Computations ISSN NO: 1076-5131 A Clique is a set of combination of bin sequence to be packed into a container represented in a graph G connected by adjacent vertices. Each clique in the graph represents the bins to be packed in the respective containers. For example 7 bins and 3 containers can be represented as follows. C(G) = {1-3-4} C1 {2-5-7} C2 {6} C3 Any vertex Vt(i) in the graph G having complete sub graph or clique in its adjacent Adj(i) is commonly called as simplicial vertex and can be used to arrange the vertices in an eliminating order. Chordal Graph is a simple graph inwhich every cycle of length four and greater has a cycle chord. Each vertex Vt iin the vertex weighted graph G, having weight W i and volume V i respectively and the sum of weight and volume of the clique graph vertices should be within the container capacity C j and Wt j. this problem is commonly called as Minimum Clique Partition withConstrained Weight (MCPCW) problem. The objective of the problem is identify the minimum number of cliques without exceeding the container capacity. Minimum Clique Partition withConstrained Weight (MCPCW) problem is NP-Hard and is defined as follows.Given theset S = {b 1, b 2, …, b n}of n number of bins bi, satisfying the container capacity threshold, for example, for 3 partition problem, container capacity threshold should satisfy C/4 < b i