A RISK-PARAMETRIC DEFUZZIFICATION FRAMEWORK FOR SYMMETRIC
FUZZY TRANSPORTATION PROBLEMS: FROM POSSIBILISTIC QUANTILES
TO COHERENT TAIL RISK MEASURES

Kamal Gupta1, Dr. Vandana Bansal2
1Assistant Professor, Department of Mathematics, SPSK Khalsa College, Begowal
2Associate Professor, Head of Mathematics Deptt. Ramgarhia College, Phagwara

Abstract


Abstract. A class of symmetric Real-world transportation problems involves imprecise parameters arising from volatile fuel prices, labour costs and demand fluctuations. Traditional defuzzification strategies such as the Centroid of Gravity (COG) compute a crisp representative without accommodating the decision-maker’s risk attitude toward cost uncertainty. This paper proposes a unified risk-parametric framework grounded in asymmetric loss minimization. By minimizing a pinball loss function with parameter λ∈[0,1], we obtain a family of crisp equivalents that spans the spectrum from optimistic (λ→0) to pessimistic (λ→1) cost estimates. For smooth quadratic membership functions constructed via Newton’s Forward Interpolation, the λ-quantile (possibilistic VaR) admits an elegant closed-form trigonometric solution. We then derive a closed-form Possibilistic Conditional Value at Risk (PCVaR) — to our knowledge, the first such explicit formula for parabolic fuzzy numbers —that averages the worst (1-λ)-fraction of the possibility distribution and prove that it satisfies fundamental consistency axioms—monotonicity, translation invariance, positive homogeneity and exact linearity for symmetric parabolic fuzzy numbers—making it suitable for scalable transportation planning. Numerical experiments on two fuzzy transportation problems (3×4 and 4×4) demonstrate that the parameter λ not only shifts total cost but can fundamentally alter the optimal allocation pattern when route cost uncertainties exhibit heterogeneous spreads, while PCVaR provides a conservative, coherent alternative to simple quantile thresholding.

Received: 25 Nov 2023

Key Words and Phrases: Fuzzy transportation problem, defuzzification, asymmetric loss, possibilistic quantile, conditional value at risk, coherent risk measure, Newton forward interpolation.

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How to cite this paper?
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2023
Volume: 36
Issue: 6
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