Time Minimizing Transportation Problem Research Articles

An efficient algorithm for two-stage capacitated time minimization transportation problem with restricted flow

This paper discusses a two-stage capacitated time minimization transportation problem with the restricted flow (TSCTMTP-F) in which the transportation takes place in two stages and only a specified amount of commodity is transported in both stages. The total amount F_1 is transported during Stage-I and F_2 during Stage-II, and the objective is to minimize the sum of the transportation times for Stage-I and Stage-II.  In 2017, Kaur et al. (2017) studied this problem and developed a polynomially bounded iterative algorithm (Algorithm-A) to solve TSCTMTP-F. However, their proposed algorithm has some flaws and may not always yield an optimal solution to the problem TSCTMTP-F. An improved iterative algorithm  (Algorithm-C) is proposed in this paper that guarantees an optimal solution to the problem. Various theoretical results prove the convergence and efficiency of Algorithm-C to obtain an optimal solution to the problem TSCTMTP-F. Numerical problems are included in the support of theory along with a counter-example for which Algorithm-A fails to obtain its optimal solution. Computational experiments on a variety of test problems have been carried out to validate the convergence and efficiency of Algorithm-C.

A solution technique for capacitated two-level hierarchical time minimization transportation problem

As a generalization of two-level hierarchical time minimization transportation problem (2HTMTP), capacitated two-level hierarchical time minimization transportation problem (C2HTMTP) is a vital issue due to the route shipping capacity finiteness in reality, and no research report on C2HTMTP is available owing to the intractability. Only 4 approaches with defects are available in the literature to resolve 2HTMTP, and they cannot be applied directly to C2HTMTP although their generalized versions can. In this paper, by creating mathematical model along with auxiliary models and constructing network with lower and upper arc capacities, C2HTMTP is transformed into a collection of searching feasible flow in the constructed network, and consequently 4 iteration algorithms, with two respectively derived from other two by applying binary search, are developed as C2HTMTP’s solution method basing feasible flow algorithm. It is proved that the 4 iteration algorithms find C2HTMTP’s optimum solution in a polynomial computation time. Because of sufficiently exploiting C2HTMTP’s intrinsic network flow structure, the 4 iteration algorithms hold the merits such as no memory overflow, easier implementation on computer, higher computation efficiency, and easier extension to the capacitated multilevel hierarchical time minimization transportation problem with level exceeding two, and successfully overcome the deficiencies of extant solving approaches to 2HTMTP. Computational experiments validate that as compared to extant best solving method for 2HTMTP, the 4 iteration algorithms are robust and efficient C2HTMTP’s solution method able to serve as a powerful tool to solve other related complex optimization problems, where in terms of computation time, when applied to solve 2HTMTP, three algorithms vastly outperform (even the worst performing one rivals) extant best solving approach to 2HTMTP, especially when applied to solve C2HTMTP, two fast descent algorithms are rivals ranking first and second (one performs better for small and medium scale instances, another performs better for large scale instances), and significantly outperform the ranking third algorithm with rival as extant best 2HTMTP’s solving algorithm’s generalized version and great superiority to the ranking last algorithm.

Extension of Particle Swarm Optimization algorithm for solving two-level time minimization transportation problem

A time minimization transportation problem deals with the resource efficiency to minimize time taken by the transport systems to deliver the commodity from sources to destinations. In this paper, a two-level time minimization transportation problem has been considered that categorizes the source–destination links into Level-I and Level-II with respect to the higher and lower level priority. The optimal delivery schedule of Level-I is followed up with the same for the Level-II cells. The paper proposes a solution procedure consisting of new algorithms that have been hybridized within the Particle Swarm Optimization to solve the problem making efficient use of resources. The solution procedure provides a methodical approach to the transport enterprises.This procedure does away with the rigid constraints, such as the location and number of non-zero allocations, required to be met by the traditional techniques of solving the transportation problem. The procedure generates pairs of Level-I and Level-II times at each iteration and the best pair(s) amongst these is/are marked out as the optimal solution of the problem. The solution procedure is explained through a numerical illustration.

More-for-Less Paradox in Time Minimization Transportation Problem with Mixed Constraints

In emergency situations, such as at the time of the outbreak of infectious viruses (COVID-19, SARS, Ebola, MERS, etc.), strike of natural disasters (Earthquakes, tsunamis, cyclones, etc.), wars, terrorist attacks, etc., where distributing essential goods and services in minimum possible time is a major logistical challenge, the concept of more-for-less paradox could be helpful. In a minimization type transportation problem, this paradoxical situation occurs when the value of objective function falls below the optimum value by shipping a large number of total goods. In this article, a unified algorithm is developed to identify and resolve the existence of paradoxical situation in the time minimization transportation problem with mixed constraints using right-hand side parametric formulation. Using this prior approach, the paradoxical solution (if exists) can be found first, followed by an optimal solution. If the paradoxical part does not exist, it gets neglected. The conditions governing the existence of more transportation flow in less shipping time enable the decision-maker to extend the optimal solution in search of more-for-less opportunity at the time of emergency. The validity of an algorithm has been tested through numerical illustrations and by computational observations on matlab.

An iterative solution technique for capacitated two-stage time minimization transportation problem

An iterative solution technique for capacitated two-stage time minimization transportation problem

An alternate approach to solve two-level hierarchical time minimization transportation problem

This paper discusses a two-level hierarchical time minimization transportation problem, which is an important class of transportation problems arising in industries. This problem has been studied by various researchers (Sharma et al. in Eur J Oper Res 246:700–707, 2015; Sonia and Puri in TOP 12(2):301–330, 2004; Xie et al. in Comput Oper Res 86:124–139, 2017) and therefore, a number of polynomial time iterative algorithms are available to find its solution. All the existing algorithms, though efficient, have some shortcomings. The current study proposes an alternate solution algorithm for the problem that is more efficient in terms of computational time than the existing algorithms. The results justifying the underlying theory of the proposed algorithm are given. Further, a detailed comparison of the computational behaviour of all the algorithms for randomly generated instances of this problem, of different sizes validates the efficiency of the proposed algorithm.

Three-phase time minimization transportation problem

The present article discusses a special variant of the multiphase transportation problem dealing with the transportation of a commodity from various origins to terminals when the set of origin–terminal links is divided into three disjoint partitions, namely Phase-1 links, Phase-2 links and Phase-3 links. Transportation of the commodity via Phase-2 links starts only after the completion of transportation in Phase-1, and similarly transportation in Phase-3 commences only after transportation in Phase-2 has been completed. The aim is to find a solution that minimizes the total time of transportation of the commodity subject to the constraints pertaining to origin availabilities and terminal requirements. The overall availability of the commodity at various origins is equal to the total demand at various terminals. An iterative polynomially bounded algorithm is proposed for obtaining an optimal feasible solution of the three-phase problem. Theoretical justification and numerical illustrations are provided to support the algorithm.

A Shootout Method for Time Minimizing Transportation Problem with Mixed Constraints

In practical world applications, the time minimizing transportation problem provides a powerful framework for determining better ways for timely delivery of goods to consumers. In this article, we propose a shootout method to determine the optimum time of transportation for the time minimizing transportation problem with mixed constraints. The proposed problem is a variant of the time minimizing transportation problem in which constraints are of mixed nature. The cells of the transportation table with decreasing order of time are avoided for allocation one by one until feasibility is maintained. The method is structured in the form of an algorithm and computationally tested on MATLAB through problems of various sizes. It solves the time minimizing transportation problem with equality constraints also.

A modified Particle Swarm Optimization Algorithm to solve Time Minimization Transportation Problem

When the supply of items need urgent/earliest delivery to the destinations, the Time Minimization Transportation Problems (TMTPs) are indispensable. Traditionally these problems have been solved using the exact techniques, however, the (meta) heuristic techniques have provided a great breakthrough in search space exploration. Particle Swarm Optimization is one such meta-heuristic technique that has been applied on a wide variety of continuous optimization problems. For discrete problems, either the mathematical model of problem or the solution procedure has been changed. In this paper, the PSO has been modified to incorporate the discrete nature of variables and the non-linearity of the objective function. The proposed PSO is tested on the problems available in the literature and the optimal solutions are obtained efficiently. The exhaustive search capability of proposed PSO is established by obtaining alternate optimal solutions and the combinations of the allocated cells that are beyond (m  n  1) in number. This proposed solution technique, therefore, provides an effective alternate to the analytical techniques for decision making in the logistic systems.

A priority based time minimization transportation problem

This paper discusses a priority based time minimizing transporation problem in which destinations are prioritized so that the material is supplied, based upon the priorities of the destinations. All the destinations, which are at priority, are served first in Stage-I while the demands of the secondary destinations are met in Stage-II. It is assumed that secondary transportation can not take place until the primary transportation is done. The purpose is to transport in such a manner that the sum of the transportation time of primary and secondary destinations is minimum. To achieve this, two approaches are proposed. In the first approach, primary destinations are served optimally by giving weights while in the second approach, lexicographic optimization is used. From the generated pairs, the minimum sum of times corressponding to Stage-I and Stage-II times is picked up as the optimal solution. It is also shown, through Computational Details, that the lexicographic optimization converges to the optimal solution faster than the first approach as reported in Table 4.

A feasible flow-based iterative algorithm for the two-level hierarchical time minimization transportation problem

The THTMTP (two-level hierarchical time minimization transportation problem) is an important problem arising in industries. In literature, there are only two approaches with shortcomings to solve the problem. In this paper, the THTMTP is formulated as a mathematical model applicable to the case in which the total available supply at the sources is no less than the total demand at the destinations. A feasible flow-based iterative algorithm named THTMTP-A is proposed to solve the THTMTP by constructing network with lower and upper arc capacities. It is proved that the THTMTP-A algorithm can find the optimal solution to the THTMTP in a polynomial time. The proposed THTMTP-A algorithm has good performance in terms of computer implementation, computational time and required memory for computation, and hence overcomes successfully the shortcomings of the two existing approaches. Computational experiments validate that the THTMTP-A algorithm is an efficient and robust method to solve the THTMTP, and can serve as efficient tool to solve other related optimization problems.

Two stage intuitionistic fuzzy time minimizing transportation problem based on generalized Zadeh’s extension principle

In decision making of real life two stage time minimizing transportation problem (TSTMTP), we face here the state of uncertainty as well as hesitation due to some uncontrollable factors. Many authors have used fixed or fuzzy data in TSTMTP. But, TSTMTP with uncertainty and hesitation cannot be solved properly by taking fixed or fuzzy data. In present article, we have considered TSTMTP with uncertainty and hesitation of supply, demand and time. Here we have made decision of real life TSTMTP with uncertainty and hesitation by using triangular intuitionistic fuzzy numbers (TIFNs). We have generalized expected interval and value of triangular fuzzy numbers for TIFNs which converts each TIFN to a real number, also proposed and proved some related results. Further, a method is presented for the solution of real life TSTMTP and degrees of acceptance and rejection of optimal decision are computed by using generalized Zadeh’s extension principle. Finally, proposed method is implemented in an example.

Capacitated two-stage time minimization transportation problem with restricted flow

This paper discusses a capacitated time minimization transportation problem in which transportation operation takes place in two stages. In the first stage, due to some constraints, only a specified flow F 1 is transported from available sources to various destinations and in the second stage, a flow F 2 is transported depending upon the total demand of the destinations. The current problem is motivated by a production system of a steel industry where semi-finished jobs, initially located at various bins in its warehouse, are transported to various manufacturing facilities by transporters for the final processing and finishing. Due to some additional constraints, it is not possible to transport the number of semi-finished jobs equal to the exact number of final products to be manufactured, in one go. Therefore the transportation operation has to take place in two stages. Further, a capacity is associated with each bin-machine link which makes the current problem, a capacitated, two stage time minimization transportation problem with restricted flow. The objective is to minimize the sum of the transportation times for Stage-I and Stage-II. A polynomial time iterative algorithm is proposed that within each iteration, solves a restricted version of a related standard capacitated time minimization transportation problem and generates a pair of Stage-I and Stage-II times with Stage-II time strictly less than the Stage-II time of the previous iteration, whereas Stage-I time may increase. Out of these generated pairs, a pair with the minimum sum of transportation times of Stage-I and Stage-II is selected that gives the global optimal solution. Numerical illustration is included in the support of the theory.

An iterative algorithm for two level hierarchical time minimization transportation problem

This paper discusses a two level hierarchical time minimization transportation problem, in which the whole set of source–destination links consists of two disjoint partitions namely Level-I and Level-II links. Some quantity of a homogeneous product is first shipped from sources to destinations by Level-I decision makers using only Level-I links, and on its completion the Level-II decision maker transports the remaining quantity of the product in an optimal fashion using only Level-II links. The objective is to find that feasible solution for Level-I decision corresponding to which the optimal feasible solution for Level-II decision maker is such that the sum of shipment times in Level-I and Level-II is minimum. A polynomial time iterative algorithm is proposed to solve the two level hierarchical time minimization transportation problem. At each iteration a lexicographic optimal solution of a restricted version of a related standard time minimization transportation problem is examined to generate a pair of Level-I and Level-II shipment times and finally the global optimal solution is obtained by selecting the best out of these generated pairs. Numerical illustration is included in support of theory.

New Procedure of Finding an Initial Basic Feasible Solution of the Time Minimizing Transportation Problems

Minimization of transportation time is a great concern of the transportation problems like the cost minimizing transportation problems. In this writing, a transportation algorithm is developed and applied to obtain an Initial Basic Feasible Solution (IBFS) of transportation problems in minimizing transportation time. The developed method has also been illustrated numerically to test the efficiency of the method where it is observed that the proposed method yields a better result.

Linear Fractional Time Minimizing Transportation Problem with Impurities

In this study, the solution procedure of Linear Fractional Time Minimizing Transportation Problem with Impurities (LFTMTPI) in the commodity to generate total transportation solution schedules, is going to be presented. This LFTMTPI is related to a lexicographic linear fractional time minimizing transportation problem with impurities. The partial flows constituting a feasible transportation schedule may be partitioned according to the actual and standard transportation time involved. An LFTMTPI algorithm is also presented to solve such real life fractional decision priority problems. This algorithm takes into account the special structure of the problem due to impurities in the commodity and depends heavily on the optimality conditions. The optimality conditions reflects nothing else than dual feasibility.

Time minimizing transportation problem with fractional bottleneck objective function

This paper is aimed at studying the Time Minimizing Transportation Problem with Fractional Bottleneck Objective Function (TMTP-FBOF). TMTP-FBOF is related to a Lexicographic Fractional Time Minimizing Transportation Problem (LFTMTP), which will be solved by a lexicographic primal code. An algorithm is also developed to determine an initial efficient basic solution to this TMTP-FBOF. The developed TMTP-FBOF Algorithm is supported by a real life example of Military Transportation Problem of Indian Army.

CAPACITATED TWO-STAGE TIME MINIMIZATION TRANSPORTATION PROBLEM

A capacitated two-stage time minimization transportation problem is considered in this paper, in which the total availability of a homogeneous product at various sources is more than the total requirement of the same at destinations. Unlike the conventional imbalanced time minimization transportation problem, in the current problem, transportation takes place in two stages such that the minimum requirement of the destinations is satisfied in the first stage and the surplus amount is transported in the second stage. Each time the transportation from sources to destinations is done in parallel and the capacity on each route (source destination link) remains fixed, i.e., the total amount transported in both the stages cannot exceed its upper bound. In each stage, the objective is to minimize the shipment time and the overall goal is to find a solution that minimizes the sum of first and second stage times.

Bilevel time minimizing transportation problem

For a given time minimizing transportation problem comprising m sources and n destinations, the set of m sources is to be optimally partitioned into two mutually disjoint subsets L 1 and L 2 where, L 1 contains m 1 sources called Level-I sources and L 2 contains the remaining ( m − m 1 ) sources termed as Level-II sources. First, the Level-I decision maker sends the shipment from Level-I sources to partially meet the demand of destinations. Later, the Level-II decision maker sends the material from the Level-II sources to meet the left over demand of the destinations. A finite number of cost minimizing transportation problems are solved to judiciously generate a few of C m 1 m partitions of the set of m sources. The aim of this study is to find an optimal partition of the set of m sources such that the sum of times of transportation in the Level-I and Level-II shipments is the minimum. The proposed polynomial bound algorithm to find the global minimizer has been successfully coded in C++ and run on a variety of randomly generated test problems differing in input data.

Total time minimizing transportation problem

This paper shows the total transportation time problem regarding the time of the active transportation routes. If the multiple optimal solutions exist, it is possible to include other criteria as second level of criteria and find the corresponding solutions. Furthermore, if there is a multiple solution, again, the third objective can be optimized in lexicographic order. The methods of generation of the optimal solution in selected cases are developed. The numerical example is included. .