Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.03671

Bi-objective Two Phase Transportation Problem

Prabhjot Kaur·Kalpana Dahiya·Vanita Verma
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper discusses a bi-objective two-phase transportation problem (BTPTP) in which transportation takes place in two phases. The whole set of source-destination links is partitioned into two disjoint sets namely, Phase-I and Phase-II. The transportation among the source-destination links of Phase-II set is done only after the transportation among the source-destination links of Phase-I set is complete. The transportation among the source-destination links in each phase is done in parallel. The problem discussed in this paper concentrates on minimizing two objectives viz., sum of transportation times and sum of transportation costs of both the phases, simultaneously. Due to conflicting nature of the objectives, a polynomial time iterative algorithm named as BTPTP-Algorithm is developed that finds all of its non-dominated solutions. At each iteration, the BTPTP-Algorithm solves some specific restricted cost minimizing transportation problems and finds corresponding optimal feasible solutions. The restrictions imposed on the source-destination links of Phase-I and Phase-II sets, in each of these problems, depend upon their corresponding Phase-I and Phase-II transportation times obtained at the previous step. The algorithm is developed in such a way that it expels all the dominated solutions and records all the non-dominated solutions of the problem, systematically. A proof of the BTPTP-Algorithm's capability of recording all the non-dominated points is provided. Further, a numerical illustration is given in support of the theory and computational behavior of the BTPTP-Algorithm for various BTPTP instances of different sizes, is provided in terms of CPU time.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Prabhjot Kaur, Kalpana Dahiya, Vanita Verma. 2026-09-03. Bi-objective Two Phase Transportation Problem. https://arxiv.org/abs/2609.03671

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Dynamic Programming-Compatible Uncertainty Sets in Robust Markov Decision Processes

In this paper, we investigate the compatibility of robust Markov Decision Processes (RMDPs) with dynamic programming under various assumptions on the uncertainty set, i.e., we investigate when one can solve an RMDP by solving a fixed point equation. We show that in all generality, s-rectangular and sa-rectangular uncertainty sets are the only models of uncertainty that are compatible with dynamic programming. Our analysis shows that existing non-rectangular models, including r-rectangularity, are only weakly compatible with dynamic programming, as they require the assumption that rewards do not depend on the next state. In this case, our results imply that one can always construct a rectangular uncertainty set that is equivalent, for both policy evaluation and optimization, to the dynamic programming-compatible non-rectangular model. This highlights a key limitation: dynamic-programming-compatible non-rectangular uncertainty sets, although practically relevant for uncertainty quantification, do not provide a genuinely distinct assessment of policy performance. Interestingly, our proof techniques rely on identifying a novel simultaneous solvability property, which we show is central to several important properties of RMDPs, including the existence of stationary optimal policies and dynamic programming-based formulations. The simultaneous solvability property enables a unified approach to studying all existing models of uncertainty, rectangular and non-rectangular alike.

math.OC↗
arXiv2026

A simple and practical adaptive trust-region method

We present an adaptive trust-region method for unconstrained optimization that allows inexact solutions to the trust-region subproblems. Our method is a simple variant of the classical trust-region method of Ssorensen. The method achieves the best possible convergence bound up to an additive logarithmic term for finding an $ε$-approximate stationary point, i.e., $O( Δ_f L^{1/2} ε^{-3/2}) + \tilde{O}(1)$ iterations, where $L$ is the Lipschitz constant of the Hessian, $Δ_f$ is the optimality gap, and $ε$ is the termination tolerance for the gradient norm. This improves over existing trust-region methods whose worst-case bound is at least a factor of $L$ worse. We compare our performance with state-of-the-art trust-region (TRU) and cubic regularization (ARC) methods from the GALAHAD library on the CUTEst benchmark problems with at least 100 variables. We also compare with the recently developed Universal trust-region (UTR) method, using the same subproblem solver. In terms of shifted geometric mean of wall-clock times our method's is between $1.2\times$ and $2\times$ faster. We report similar improvements for number of function evaluations, factorizations, gradient evaluations and Hessian evaluations. Compared to the conference version of this paper, our revised method includes several practical enhancements. These modifications dramatically improved performance, including almost an order of magnitude reduction in the shifted geometric mean of wall-clock times. We also show that it suffices for the function to be continuously twice-differentiable to guarantee that either the minimum gradient norm converges to zero or the objective value tends towards negative infinity, even when the iterates diverge.

math.OC↗
arXiv2026

Rough Stochastic Pontryagin Maximum Principle and an Indirect Shooting Method

We derive first-order Pontryagin optimality conditions for stochastic optimal control with deterministic controls for systems modeled by rough differential equations (RDE) driven by Gaussian rough paths. This Pontryagin Maximum Principle (PMP) applies to systems following stochastic differential equations (SDE) driven by Brownian motion, yet it does not rely on forward-backward SDEs and involves the same Hamiltonian as the deterministic PMP. The proof consists of first deriving various integrable error bounds for solutions to nonlinear and linear RDEs by leveraging recent results on Gaussian rough paths. The PMP then follows using standard techniques based on needle-like variations. As an application, we propose the first indirect shooting method for nonlinear stochastic optimal control and show that it converges 10x faster than a direct method on a stabilization task.

math.OC↗