Heuristics for the Worst Optimal Value of Interval Transportation Problems

TL;DR

The paper tackles the NP-hard problem of finding the worst finite optimal value $\overline{f}_{fin}$ in interval transportation problems by leveraging a strengthened quasi-extreme reduction that restricts attention to balanced quasi-extreme scenarios. It develops a local search method, a genetic algorithm, and a memetic algorithm that combines evolutionary search with local learning to approximate $\overline{f}_{fin}$ efficiently. The approaches are evaluated on standard benchmark instances, showing competitive performance and, in some cases, new best solutions within short runtimes, outperforming several existing methods. The results demonstrate that incorporating scenario encoding and balanced reductions yields practical, scalable heuristics for estimating worst-case values in uncertain transportation settings, with data and code openly available for reproducibility.

Abstract

An interval transportation problem represents a model for a transportation problem in which the values of supply, demand, and transportation costs are affected by uncertainty and can vary independently within given interval ranges. One of the main tasks of solving interval programming models is computing the best and worst optimal value over all possible choices of the interval data. Although the best optimal value of an interval transportation problem can be computed in polynomial time, computing the worst (finite) optimal value was proved to be NP-hard. In this paper, we strengthen a previous result showing a quasi-extreme decomposition for finding the worst optimal value, and building on the result, we design heuristics for efficiently approximating the value. Using a simplified encoding of the scenarios, we first derive a local search method and a genetic algorithm for approximating the worst optimal value. Then, we integrate the two methods into a memetic algorithm, which combines the evolutionary improvement of a genetic algorithm with individual learning implemented via local search. Moreover, we include numerical experiments for a practical comparison of the three different approaches. We also show that the proposed memetic algorithm is competitive with the available state-of-the-art methods for approximating the worst optimal value of interval transportation problems, this is demonstrated by finding the new best solutions for several instances, among others.

Figures (4)

• Figure 1: An interval plot of the maximal, minimal and average objective values attained by the four algorithms presented in this paper over $5$ runs for instances $1$-$5$ (above) and $6$-$10$ (below). The labels indicate the number of the algorithm used: local search (\ref{['alg:localSearch:first']} and \ref{['alg:localSearch:best']}), the genetic algorithm (\ref{['alg:geneticAlg']}) and the memetic algorithm (\ref{['alg:memeticAlg']}).
• Figure 2: An interval plot of the maximal, minimal and average running time of the four algorithms presented in this paper over $5$ runs for instances $1$-$5$ (above) and $6$-$10$ (below). The labels indicate the number of the algorithm used: local search (\ref{['alg:localSearch:first']} and \ref{['alg:localSearch:best']}), the genetic algorithm (\ref{['alg:geneticAlg']}) and the memetic algorithm (\ref{['alg:memeticAlg']}).
• Figure 3: The objective value and the number of generations of the genetic algorithm (left) and the memetic algorithm (right) on two instances for population size $N_{\text{pop}} \in \{10,30,50,100\}$. Different settings of parameters are used: fitness proportionate selection with strategy $\pi_S^3$ and $t_{\text{GA}} = 5$ (\ref{['plot:2:linshift:5']}), fitness proportionate selection with strategy $\pi_S^3$ and $t_{\text{GA}} = 20$ (\ref{['plot:2:linshift:20']}), tournament selection with $t_{\text{GA}} = 5$ (\ref{['plot:2:tournament:5']}) and tournament selection with $t_{\text{GA}} = 20$ (\ref{['plot:2:tournament:20']}).
• Figure 4: Instance 2. \ref{['plot:progress:genetics:10:linear:20']} and \ref{['plot:progress:memetics:10:linear:20']}: genetic and memetic with $N_\text{pop} = 10$, strategy $\pi^3_S$, $t_\text{GA} = 20$. \ref{['plot:progress:genetics:50:tournament:5']} and \ref{['plot:progress:memetics:50:tournament:5']}: genetic and memetic with $N_\text{pop} = 50$, tournament selection strategy, $t_\text{GA} = 5$. All algorithms found the same solution ($9200$). Numbers of generations: \ref{['plot:progress:genetics:10:linear:20']}: 80, but the cheapest iterations; \ref{['plot:progress:memetics:10:linear:20']} 26, but more complex generations (due to local search) makes it only 2 sec faster than genetic; \ref{['plot:progress:genetics:50:tournament:5']}: 50, the bigger population makes the generations more demanding; \ref{['plot:progress:memetics:50:tournament:5']}: 6, the local search is demanding here, this is even slower than the magenta algorithm with 4 times more generations. Note that \ref{['plot:progress:memetics:10:linear:20']} and \ref{['plot:progress:memetics:50:tournament:5']} spent most time on checking $t_\text{GA}$ generations with the same value.

Theorems & Definitions (5)

• Theorem 1: Garajova:QuasiextremeReductionInterval:2024, Carrabs:ImprovedHeuristicApproach:2021a
• Theorem 2: Garajova:QuasiextremeReductionInterval:2024
• Theorem 3
• proof
• Corollary 4