A New Algorithm For The Initial Feasible Solutions Of Fixed Charge Transportation Problem

Maastricht University is offering a MBA program for people that have a bachelor degree and at least 5 years of working experience. Within the MBA program, students work in groups of 5 during a two year cycle. This thesis is about the formation of the student groups. The MBA program contains 60 students. Every year, two intake moments take place that usually allow 15 new students to enter. All 60 students follow the same course at the same time, implying that the order in which a student follows the courses depends only on the moment at which he/she starts the program. Every two periods, The university creates new student groups according to a set of hard and soft constraints, such that well-diversified groups are formed. Therefore, the student-with-student history, gender, nationality, and level of expertise of each student is taken into account. Hence a mapping from a set of students to groups is created that takes into account the corresponding constraints. The university chooses a group leader for each group. Two general solution methods are applied to the MBA sectioning problem. The first method uses the simplex algorithm to solve the problem. Therefore an integer linear program formulation of the problem was needed, and used as an input for an efficient ILP solver. The second approach starts with an initial feasible solution and improves upon this feasible solution using different improvement algorithms. The quality of each feasible solution depends on the calculated objective function value that measures the level of satisfaction of the different constraints. Different initial solution and improvement algorithms are discussed that help to obtain a feasible solution with an objective function value that is as low as possible. The implemented improvement algorithms are the Descent Improvement algorithm, Tabu Search, Simulated Annealing, and the Bipartite Weighted Matching Improvement algorithm. The first three algorithms make individual students swap between existing group formations. The Bipartite Weighted Matching Improvement algorithm iteratively selects a student from each group, and finds local optimal solutions for a bipartite matching problem in order to improve the overall objective value of the whole problem. In order to test the algorithms, one has to make sure that the instance on which the algorithms are tested mimics a real life example. Therefore, a simulation program is established that mimics the two year cycle and produces such an instance. Empirical results show that the best improvement algorithm considered is the Bipartite Weighted Matching Improvement algorithm. This algorithm, combined with an initial solution algorithm, is now being implemented into the current computer system of Maastricht University.

A Supply Selection Method for better Feasible Solution of balanced transportation problem

This paper studied the balanced transportation problem. Many algorithms were developed to solve optimally this problem. The most commonly used method is the modified distribution method (MODI). The MODI algorithm starts by generating an initial basic feasible solution using the northwest corner method (NCM). In general, the cost of the initial basic feasible solution generated by NCM is too high and far away from the optimal cost. For this reason, we propose in this paper a new method to generate an initial basic feasible solution to the balanced transportation problem to solve this issue. This method, named Global Minimum Method (GMM), was tested and compared with the NCM, the minimum cost method (MCM), the Vogel's approximation method (VAM), and the optimal cost obtained by MODI. The computational results proved that the new heuristic is promising for small size instances and outperforms all other methods for big size problems.

In this research, a new and efficient approach of finding an initial basic feasible solution to transportation problems is proposed. The proposed approach is named “Inverse Coefficient of Variation Method (ICVM)”, and the method is illustrated with seven numerical examples. Six existing methods; North West Corner Method (NWCM), Column Minimum Method (CMM), Least Cost Method (LCM), Row Minimum Method (RMM),Vogel’s Approximation Method (VAM), and Allocation Table Method (ATM) were compared with the proposed approach. It can be said conclusively that the proposed Inverse Coefficient of Variation Method (ICVM) provides an improved Initial Basic Feasible Solution to all the transportation problems used in the experiment. Further, the new method leads to the optimal solution to many of the problems considered.

Distribution is an important logistical problem in tape commodities because it has a short expiration time that requires speed and accuracy in shipping. This research collaborates implementation two methods. These are North West corner (NWC) and stepping stone. These methods make distribution scheduling model from many sources to destinations with minimum cost. This case has many steps. The first step is initial feasible solution with NWC. NWC is used this case because simple and fastest method for solving problem. NWC does not observe the cost of each shipping line in procedure. The second step revised with try and error the plan using stepping stone for optimization. In this research, collaboration these methods can reduce total cost of distribution equal to Rp.266.050. In fact, total distribution cost is Rp.317.500. After calculation, we can save cost Rp.51.450. Determination of the first source and destination in the North-west corner cell is greatly affect. If the first source and the destination have a small cost so the result is better. Stepping stone find optimization in second iteration because all of closed lanes have positive value. A positive value in repair index means that when given an allocation it will causing an increase in transportation costs. The allocation of product does not need to be changed because it will increase the cost of transportation so that the stepping stone work is completed. Percentage of differences is a formula to derive efficiency comparison between initial solution and optimal solution is 0.45%.

This work extends the theory of the fixed charge transportation problem (FCTP), currently based mostly on a forty-year-old publication by Hirsch and Danzig. This paper presents novel properties that need to be considered by those using existing, or those developing new methods for optimizing FCTP. It also defines the problem in an easier way, making it understandable to a wider spectrum of readers. While the analysis is limited to FCTP only, elements of it can easily be extended to the general fixed charge problem. Finally a novel, snap method for finding global minima for FCTPs with large fixed charge coefficients is introduced.

In this paper, we develop a new algorithm for the initial fuzzy basic feasible solution to a transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers or trapezoidal fuzzy numbers or real numbers. By applying the efficient ranking function and arithmetic operations the solution is obtained without changing it into its crisp equivalent form. Using the parametric form of fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using a proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.

Transportation of products from sources to destinations with minimal total cost plays an important role in logistics and supply chain management. The Uncapacitated Transportation Problem (UTP) is a special case of network flow optimization problem. The prime objective of this UTP is to minimize the total cost of transporting products from origins to destinations subject to the respective supply and demand requirements. The UTP consists of special network structure. Due to the special structure of this problem, the transportation algorithm is preferred to solve it. The transportation algorithm consists of two major steps: 1) Finding an Initial Feasible Solution (IFS) to TP and 2) Examining the optimality of this IFS. A better IFS generates a lesser number of iterations to obtain a Minimal Total Cost Solution (MTCS). Recently, Juman and Nawarathne (2019)’s Method was introduced to find an IFS to UTP. In this paper, the Juman and Nawarathne (2019)’s Method is improved to get a better IFS to a UTP. A comparative study on a set of benchmark instances illustrates that the new improved method provides better primal solutions compared to the Juman and Nawarathne (2019)’s Method. The proposed method is found to yield the minimal total cost solutions to all the benchmark instances.

In this paper we develop a new algorithm for the initial fuzzy basic feasible solution to a solid transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers. Using parametric form of triangular fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem without changing to its crisp equivalent form. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.

The transportation problem here is allocating goods from a place (source) to the destination (demand) optimally by considering minimal costs, effective allocation, and scheduling. To get this initial feasible solution using the Total Opportunity Cost Matrix-Modified Extremum Difference Method (TOCM-MEDM) approach. This method is a new method, which begins by making a case into the initial transportation problem table and then converting it into a TOCM (Total Opportunity Cost Matrix) matrix by calculating the difference between rows and columns. Then the entries turn into a TOCM matrix, the next step is to determine the row pointers and column pointers by subtracting the largest value from the smallest value in either row or column. This method is an approach from MEDM (Modified Extremum Difference Method). From these results, it can be continued to determine the maximum pointer cost by looking at the largest pointer. Then just calculate the minimum allocation of each row and column until supply and demand are met. The results of the analysis show that the results of solving the transportation problem to determine the initial feasible solution using the TOCM-MEDM method get the minimum yield costs of 410.

The fixed-charge transportation problem (FCTP) is still a challenging problem in the field of mathematical programming. In this paper, we consider fixed-charge transportation problem with logarithmic objective function. In the absence of any suitable algorithm to obtain the solution of this type of nonlinear transportation problem, we discuss the advantage of polynomial approximation. There exists a major difference between the two problems that the variables in the polynomial transportation problem have no upper bound but in the logarithmic transportation problem they are bounded. Using the expansion of logarithm we show the resemblance between the structural behaviour of linear and fixed-charge transportation problems. We illustrate a numerical example in support of the developed method.

There are two important approaches based on linear ranking functions for solving linear programming problems with cost coefficients as an auxiliary problem to obtain a fuzzy solution of fuzzy variable linear programming problem. The first approach uses the primal simplex method that assumes an initial primal feasible basic solution is at hand. The second approach is based on dual simplex method that begins with a basic dual feasible basic solution and proceeds by pivoting through a series of dual basic solutions until the associated complementary primal basic fuzzy solution is feasible. In this paper, we propose a new method called the primal-dual algorithm, which is similar to the dual simplex method and begins with dual feasibility and proceeds to obtain primal feasibility while maintaining complementary slackness. An important difference between the dual simplex method and the primal-dual method is that the primal-dual algorithm does not require a dual feasible solution to be basic. This algorithm is useful specially for solving minimum fuzzy cost flow problem in which finding an initial dual feasible solution turns out to be a trivial task.

This note presents an algorithm which is a completion of Russel's method for generating an initial feasible solution of the transportation problem when ties occur in selecting the basic variables. Interestingly, as a test problem shows, the initial feasible solution obtained by this complete Russel’s method turns out to be the optimal solution.

Solving fixed charge transportation problem with interval parameters

A common strategic and tactical decision in open pit mining is to determine the sequence of extraction for notional three-dimensional production blocks so as to maximize the net present value of the extracted orebody while adhering to precedence and resource constraints. This problem is commonly formulated as an integer program with a binary variable representing if and when a block is extracted. In practical applications, the number of blocks can be large and the time horizon can be long, and therefore, instances of this NP-hard precedence-constrained knapsack problem can be difficult to solve using an exact approach. The problem is even more challenging to solve when it includes explicit minimum resource constraints. We employ three methodologies to reduce solution times: (i) we eliminate variables which must assume a value of 0 in the optimal solution; (ii) we use heuristics to generate an initial integer feasible solution for use by the branch-and-bound algorithm; and (iii) we employ Lagrangian relaxation, using information obtained while generating the initial solution to select a dualization scheme for the resource constraints. The combination of these techniques allows us to determine near-optimal solutions more quickly than solving the monolith, i.e., the original problem. We demonstrate our techniques to solve instances containing 25,000 blocks and 10 time periods, and 10,000 blocks and 15 time periods, to near-optimality.