Types Of Linear Programming Problems Research Articles

Advanced method for solving the transportation problem

Transportation problem is one of the predominant areas of operations research. It is a type of linear programming problem designed to minimize the cost of distributing a single product from m sources to n destinations. Transportation problem is widely used as a decision making tool in different fields such as engineering, business management and many others. These types of problems can be solved by general network methods. In this paper The Advanced Method for Solving the Transportation Problem (AMSTP) is proposed to find initial basic feasible solution and optimal solution of transportation problem and compared its results with other existing methods.

A Short Literature on Linear Programming Problem

Researchers and scientists have developed various approaches and methodologies over time to model and analyze different types of linear programming problems, such as assignment problems and parametric programming problems. This paper provides a critical review and classification of existing modelling approaches and solution methods related to linear programming problems. Moreover, the simplex method is discussed in detail through a comprehensive literature review. The paper concludes by presenting an integrated research framework that is directly applicable to the present context, along with suggestions for future research directions.

Applying Particle Swarm Optimization Variations to Solve the Transportation Problem Effectively

The Transportation Problem (TP) is a special type of linear programming problem, where the objective is to minimize the cost of distributing a product from a number of sources to a number of destinations. Many methods for solving the TP have been studied over time. However, exact methods do not always succeed in finding the optimal solution or a solution that effectively approximates the optimal one. This paper introduces two new variations of the well-established Particle Swarm Optimization (PSO) algorithm named the Trigonometric Acceleration Coefficients-PSO (TrigAc-PSO) and the Four Sectors Varying Acceleration Coefficients PSO (FSVAC-PSO) and applies them to solve the TP. The performances of the proposed variations are examined and validated by carrying out extensive experimental tests. In order to demonstrate the efficiency of the proposed PSO variations, thirty two problems with different sizes have been solved to evaluate and demonstrate their performance. Moreover, the proposed PSO variations were compared with exact methods such as Vogel’s Approximation Method (VAM), the Total Differences Method 1 (TDM1), the Total Opportunity Cost Matrix-Minimal Total (TOCM-MT), the Juman and Hoque Method (JHM) and the Bilqis Chastine Erma method (BCE). Last but not least, the proposed variations were also compared with other PSO variations that are well known for their completeness and efficiency, such as Decreasing Weight Particle Swarm Optimization (DWPSO) and Time Varying Acceleration Coefficients (TVAC). Experimental results show that the proposed variations achieve very satisfactory results in terms of their efficiency and effectiveness compared to existing either exact or heuristic methods.

Extended DEA method for solving multi-objective transportation problem with Fermatean fuzzy sets

<abstract><p>Data envelopment analysis (DEA) is a linear programming approach used to determine the relative efficiencies of multiple decision-making units (DMUs). A transportation problem (TP) is a special type of linear programming problem (LPP) which is used to minimize the total transportation cost or maximize the total transportation profit of transporting a product from multiple sources to multiple destinations. Because of the connection between the multi-objective TP (MOTP) and DEA, DEA-based techniques are more often used to handle practical TPs. The objective of this work is to investigate the TP with Fermatean fuzzy costs in the presence of numerous conflicting objectives. In particular, a Fermatean fuzzy DEA (FFDEA) method is proposed to solve the Fermatean fuzzy MOTP (FFMOTP). In this regard, every arc in FFMOTP is considered a DMU. Additionally, those objective functions that should be maximized will be used to define the outputs of DMUs, while those that should be minimized will be used to define the inputs of DMUs. As a consequence, two different Fermatean fuzzy effciency scores (FFESs) will be obtained for every arc by solving the FFDEA models. Therefore, unique FFESs will be obtained for every arc by finding the mean of these FFESs. Finally, the FFMOTP will be transformed into a single objective Fermatean fuzzy TP (FFTP) that can be solved by applying standard algorithms. A numerical example is illustrated to support the proposed method, and the results obtained by using the proposed method are compared to those of existing techniques. Moreover, the advantages of the proposed method are also discussed.</p></abstract>

A two-stage data envelopment analysis approach to solve extended transportation problem with non-homogenous costs

<p style='text-indent:20px;'>The transportation problem is a particular type of linear programming problem in which the main objective is to minimize the cost. In marked contrast to the classical real-world transportation model, shipping supplies from one source to a destination cause several costs and benefits, each of which is incomparable to another. The extended transportation problem was first introduced in a study conducted by Amirteimoori [<xref ref-type="bibr" rid="b1">1</xref>]. In contrast, many important questions regarding the production possibility set, the place of costs, the benefits, and the essence of these costs were not fully addressed yet. Therefore, this paper focuses on transportation models that do not provide explicit output. This method is helpful because it is designed for a specific purpose: to send goods and supply-demand at the lowest cost and decision-maker; does not suffer from the confusion of costs and the various consequences of placing them costs and outputs. Furthermore, this model improves the contradiction between the essence of the problem and the input/output-oriented data envelopment analysis. In this paper, previous models that can not incorporate all the sources of inefficiency have been solved. We apply the slack-based measure(SBM) to calculate all identified inefficiency sources. A numerical example is considered to show the approach's applicability, as mentioned above, to actual life situations. As a result, the optimal costs achieved via the proposed method are more realistic and accurate by obtaining a more representative efficiency assessment. This example proved our proposed approach's efficiency, providing a more efficient solution by corporate all sources inefficiency and presenting efficient costs for each path.</p>

New Approach to Solve Assignment Problems with Branch and Bound Algorithm

This paper presents a new branch and bound algorithm for solving the assignment problems that uses a lower bound based with branching procedure. This is special type of linear programming problem dealing with the allocation of various resources to the various activities on one to one basis. Most of the business organization operates in traditional and old fashioned manner. Decision making is based on judgment and is quite subjective. Consequently, the entire process of decision making may lead to uncertain and unwanted results. Industries need more scientific and data driven strategies to achieve higher goals and to avoid risks of failure. The assignment problem deals with the maximum profit assigning of jobs or objects to agents such that each job is assigned to precisely one person subject to capacity restrictions on the agents. A new algorithm for the generalized assignment problem is presented that employs both column and row generation and also branch and bound to obtain optimal integer solutions to a set partitioning formulation of the problem. First, a generalized assignment issue that is solved by an existing Hungarian approach that only employs column generation is solved in this study by employing both row and column generation, and the optimal solution is obtained, which is similarly similar. The study is classified according to the objective, solution methodologies and related considerations. This paper hopes to give the reader an idea about the best job assigning that is modeled in this paper, discuss the formulation of assignment problem, its efficacy, enumerate the benefits gained and identify areas for further improvement.

A study of Transportation Problem for essential commodities as an OR model

    The purpose of this paper is to define the transportation problem as a type of linear programming problem. This paper considers a case study of Mizoram a state in India where we try to minimize the cost of distribution of essential commodities to cities that are remote and disconnected.

Vehicle routing problem with simulated annealing using python programming

Vehicle Routing Problems (VRP) are a type of linear programming problem. This paper tries to explain the completion of VRP using Python Programming with the Simulated Annealing algorithm. Python Programming is used as a tool by utilizing the wealth of packages in python. Python is a popular programming language. Python is easy to learn for beginners. However, there is still no example and guide for solving VRP using simulated annealing in python programming with a process paradigm. This design can be a reference and guide for beginners in dealing with similar problems or the development of existing problems.

Goal programming approach for solving heptagonal fuzzy transportation problem under budgetry constraint

Transportation problem (TP) is a special type of linear programming problem (LPP) where the objective is to minimize the cost of distributing a product from several sources (or origins) to some destinations. This paper addresses a transportation problem in which the costs, supplies, and demands are represented as heptagonal fuzzy numbers. After converting the problem into the corresponding crisp TP using the ranking method, a goal programming (GP) approach is applied for obtaining the optimal solution. The advantage of GP for the decision-maker is easy to explain and implement in real life transportation. The stability set of the first kind corresponding to the optimal solution is determined. A numerical example is given to highlight the solution approach.

Solving Linear Programming Problem in ℤn by Graphical Method

The aim of this paper is to find a new type of linear programming problems depending on the set ℤn (the set of integers modulo n) by using the graphical method. We gave a new definitions that is characterized by the set ℤn. We found that in this set many basic concepts in classic linear programming changed.

Optimisation by dual simplex approach in neutrosophic environment

In the present study, the coefficients of the objective function, technical coefficients, the right hand side coefficients and the decision variables in a crisp linear programming problem are considered as single valued neutrosophic numbers. Such type of linear programming problem is hereafter called a neutrosophic linear programming problem. The basic goal of this paper is to solve a neutrosophic linear programming problem by applying dual simplex method following the sense in crisp environment. In order to that, some necessary results are brought first and then an algorithm is developed. Finally, a classical problem has been solved to measure the efficiency of the proposed algorithm.

A new neural network model for solving the interval maximum flow problem

In this paper, a new nonlinear neural network for solving the interval maximum flow problem is presented. The our nonlinear neural network is able to generate optimal solution to the interval maximum flow problem. The interval maximum flow problem in network is formulated as a special type of linear programming problem and it is solved by appropriately defined neural networks. The performance of the our neural network is demonstrated by means of illustrative example.

Mehar Methods for Fuzzy Optimal Solution and Sensitivity Analysis of Fuzzy Linear Programming with Symmetric Trapezoidal Fuzzy Numbers

The drawbacks of the existing methods to obtain the fuzzy optimal solution of such linear programming problems, in which coefficients of the constraints are represented by real numbers and all the other parameters as well as variables are represented by symmetric trapezoidal fuzzy numbers, are pointed out, and to resolve these drawbacks, a new method (named as Mehar method) is proposed for the same linear programming problems. Also, with the help of proposed Mehar method, a new method, much easy as compared to the existing methods, is proposed to deal with the sensitivity analysis of the same type of linear programming problems.

Dynamic distributed unicast routing: optimal incremental paths

Developing new mathematical frameworks such as distributed dynamic routing algorithms for constructing optimal incremental paths from a node to another node is an important challenge in data communication networks. These new algorithms can model network resources optimally and increase network performances. A bundle of single routes in a current communication path, which starts from a source node and ends to a destination node, can consist of several successive nodes and links. The Incremental term emphasizes that the number of routes (links and nodes) in a current path can change so that achieving more data rate and optimal efficiency in the network. In this paper, our problem is to add/omit some routes consisting of some nodes and links to/from the current unicast path dynamically and optimally. We call this problem the Optimal Dynamic Distributed Unicast Routing (ODDUR) problem and it is a NP-complete problem. This problem can be formulated as a new type of Linear Programming Problem (LPP) for finding a minimum cost multichannel unicast path, which this path will minimize end-to-end delay and bandwidth consumption along the routes from the source node to the destination node. In this paper, at first a new mathematical framework will be constructed and then this framework will propose the new optimal dynamic distributed unicast routing algorithm for solving our LPP problem. This algorithm will compute an optimal solution for our LPP based on the simplex method and postoptimality computations and will reduce computations and consumed time. Simulation results will show that our new algorithm is more efficient than other available algorithms in terms of utilization of bandwidths and data rate.

Robust linear optimization under matrix completion

Linear programming models have been widely used in input-output analysis for analyzing the interdependence of industries in economics and in environmental science. In these applications, some of the entries of the coefficient matrix cannot be measured physically or there exists sampling errors. However, the coefficient matrix can often be low-rank. We characterize the robust counterpart of these types of linear programming problems with uncertainty set described by the nuclear norm. Simulations for the input-output analysis show that the new paradigm can be helpful.

Revised Distribution Method of finding Optimal Solution for Transportation Problems

Transportation problem is a special type of linear programming problem. This paper introduce a new approach to transportation problem namely, Revised Distribution method (RDI), for solving a wide range of such problems. The new method is based on allocating units to the cells in the transportation matrix starting with minimum demand or supply to the cell with minimum cost in the transportation matrix and then try to find an optimum solution to the given transportation problem. The proposed method is a systematic procedure, easy to apply and can be utilized for all types of Transportation problem with maximize or minimize objective functions. At the end, this method is illustrated with some numerical example.

A partitioning and bounded variable algorithm for linear programming

An interesting new partitioning and bounded variable algorithm (PBVA) is proposed for solving linear programming problems. The PBVA is a variant of the simplex algorithm which uses a modified form of the simplex method followed by the dual simplex method for bounded variables. In contrast to the two-phase method and the big M method, the PBVA does not introduce artificial variables. In the PBVA, a reduced linear program is formed by eliminating as many variables as there are equality constraints. A subproblem containing one ‘less than or equal to’ constraint is solved by executing the simplex method modified such that an upper bound is placed on an unbounded entering variable. The remaining constraints of the reduced problem are added to the optimal tableau of the subproblem to form an augmented tableau, which is solved by applying the dual simplex method for bounded variables. Lastly, the variables that were eliminated are restored by substitution. Differences between the PBVA and two other variants of the simplex method are identified. The PBVA is applied to solve an example problem with five decision variables, two equality constraints, and two inequality constraints. In addition, three other types of linear programming problems are solved to justify the advantages of the PBVA.

Wide-sense nonblocking Clos networks under packing strategy

In this paper, we study wide-sense nonblocking conditions under packing strategy for the three-stage Clos network, or /spl upsi/(m, n, r) network. Wide-sense nonblocking networks are generally believed to have lower network cost than strictly nonblocking networks. However, the analysis for the wide-sense nonblocking conditions is usually more difficult. Moore proved that a /spl upsi/(m, n, 2) network is nonblocking under packing strategy if the number of middle stage switches m/spl ges/[3/2n]. This result has been widely cited in the literature, and is even considered as the wide-sense nonblocking condition under packing strategy for the general /spl upsi/(m, n, r) networks in some papers. In fact, it is still not known that whether the condition m/spl ges/[3/2n] holds for /spl upsi/(m, n, r) networks when r/spl ges/3. In this paper, we introduce a systematic approach to the analysis of wide-sense nonblocking conditions for general /spl upsi/(m, n, r) networks with any r value. We first translate the problem of finding the nonblocking condition under packing strategy for a /spl upsi/(m, n, r) network to a set of linear programming problems. We then solve this special type of linear programming problems and obtain a closed form optimum solution. We prove that the necessary condition for a /spl upsi/(m, n, r) network to be nonblocking under packing strategy is m/spl ges/[(2-1/F/sub 2r-1/)n] where F/sub 2r-1/ is the Fibonacci number. In the case of n/spl les/F/sub 2r-1/, this condition is also a sufficient nonblocking condition for packing strategy. We believe that the systematic approach developed in this paper can be used for analyzing other wide-sense nonblocking control strategies as well.

A problem identification taxonomy for classification and automated formulation of multiple objective linear programming models

This paper presents a knowledge-based Decision Support System (DSS) for classification, formulation and solution of multiple objective linear programming problems. The authors propose a generic taxonomy which is used to classify commonly encountered types of linear programming problems. Classification of the problem, vis-a-vis the proposed taxonomy, is based on the interaction with the user to determine the attributes which best describe the context or setting of the problem. A total of twenty-four problem types are included in the taxonomy. Following classification, a problem type-specific rule base is invoked to assist the user in constraint formulation. A product blending linear programming problem is used to demonstrate this component of the system since these types of problems typically include more varied constraints, including ratio as well as additive types. A second rule base is invoked for formulation of the multiple criteria objective function, model solution and sensitivity analysis. An initial goal prioritization scheme is obtained by use of the Analytical Hierarchy Process [17]; the optimal goal ordering is obtained by interaction with the user in the form of pairwise attribute value tradeoffs. The system developed is intended for use by the OR-naive user who is more familiar with the content of a problem than he/she is with the mathematical tools needed to formulate and solve such models. The system is a model management tool designed to interpret user inputs and translate those inputs into a solvable multiple objective LP. This interface alleviates the technical burdens of content specification and solution. The approach expands previous formulation tools, such as those based on natural language processing [8], to a broader range of problem types in a multiple criteria environment. The system was implemented on a personal computer using VP-Expert, BASIC and LINDO, and is demonstrated on a multiple objective blending problem. The ability of the approach to accurately classify LP problems was tested on thirty-six subjects. Results suggest that correct classification of problems was more likely to occur when the system was used.

A goal programming network for linear programming

Tank and Hopfield have shown that networks of analog neurons can be used to solve linear programming (LP) problems. We have re-examined their approach and found that their network model frequently computes solutions that are only suboptimal or that violate the LP problem's constraints. As their approach has proven unreliable, we have developed a new network model: the goal programming network. To this end, a network model was first developed for goal programming problems, a particular type of LP problems. From the manner the network operates on such problems, it was concluded that overconstrainedness, which is possibly present in an LP formulation, should be removed, and we have provided a simple procedure to accomplish this.