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The objective of this paper is to deal with a kind of fuzzy linear programming problem involving symmetric trapezoidal fuzzy numbers. Some important and interesting results are obtained which in turn lead to a solution of fuzzy linear programming problems without converting them to crisp linear programming problems.Keywords Fuzzy numbers . Ranking . Fuzzy linear programming AMS subject classification: 90C05 . 90C70 Bellman and Zadeh (1970) proposed the concept of decision making in fuzzy environments. After the pioneering work on fuzzy linear programming by Tanaka et al. (1974Tanaka et al. ( , 1984 and Zimmermann (1974), several kinds of fuzzy linear programming problems have appeared in the literature and different methods have been proposed to solve such problems. Numerous methods for comparison of fuzzy numbers have been suggested in the literature. Maleki, Mashinchi (1996, 2000) used the Rouben's method of comparison of fuzzy numbers and obtained an optimal solution. In this paper, we introduce a new type of fuzzy arithmetic for symmetric trapezoidal fuzzy numbers and propose a method for solving fuzzy linear programming problems without converting them to crisp linear programming problems. This paper is organized as follows: In section 1, we give the definitions of fuzzy linear programming, symmetric trapezoidal fuzzy numbers and some related results of fuzzy arithmetic on symmetric trapezoidal fuzzy numbers. In section 2, we prove fuzzy analogues of some important theorems of linear programming. A numerical example involving symmetric trapezoidal fuzzy numbers is also given to illustrate the theory developed in this paper.

High-pressure Raman scattering studies at ambient temperature are performed on n-heptane. We observe a liquid-solid transition around 1.5 GPa from the changes in the Raman spectra. This has been reported in earlier works. With increasing pressure, we observe large changes in the Raman modes and the spectra show a distinct change around 7.5 GPa. This marks the solid-solid transition at 7.5 GPa observed in n-heptane for the first time. As predicted in theoretical work, we observe dampening of methyl rotation in n-heptane below 7.5 GPa. With increase in pressure above 7.5 GPa we observe a definitive conversion of gauche to trans conformation in the solid phase. Upon release of pressure we do not observe any hysteresis, which suggests that the solid-solid transition takes place with no volume change or is a second-order transition. In this paper we propose this transition to be an orientational order-disorder transition driven by the dampening of the rotation of the methyl group.

Pressure-induced Raman spectroscopy studies on n-pentane have been carried out up to 17 GPa at ambient temperature. n-Pentane undergoes a liquid-solid transition around 3.0 GPa and a solid-solid transition around 12.3 GPa. The intensity ratio of the Raman modes related to all-trans conformation (1130 cm-1 and 2850 cm-1) to that of gauche conformation (1090 cm-1 and 2922 cm-1) suggests an increase in the gauche population conformers above 12.3 GPa. This is accompanied with broadening of Raman modes above 12.3 GPa. The high-pressure phase of n-pentane above 12.3 GPa is a disordered phase where the carbon chains are kinked. The pressure-induced order-disorder phase transition is different from the behavior of higher hydrocarbon like n-heptane.

We report high-pressure Raman studies on n-hexane up to 16 GPa. n-Hexane undergoes solid-solid transition around 9.1 GPa along with an already reported liquid-solid transition around 1.4 GPa. The intensity ratio of the Raman modes relating the all-trans conformation (1147 and 2872 cm-1) to that of the gauche conformation (1074 and 2923 cm-1) shows a sudden change across 9.1 GPa, suggesting an increase in the all-trans population conformers above 9.1 GPa. The disappearance of the torsional modes suggests a steric hindrance to the methyl end group, similar to the n-heptane case, suggesting that the high-pressure phase (above 9.1 GPa) is an orientationally disordered phase. In general, the transition pressure for the solid-solid transition is inversely proportional to the length of the carbon backbone in the medium chain length n-alkanes.

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A modified and convenient route using microemulsions (avoiding Ba-alkoxide) was evolved for the synthesis of uniform and monodisperse nanoparticles of BaTiO3 at low temperature (800 °C). X-ray line broadening and transmission electron microscopy studies show that the particle size varies in the range of 20–25 nm. Evidence for tetragonal distortion was found in these nano-sized (20–25 nm) particles of barium titanate from careful x-ray diffraction studies as well as from Raman spectroscopy. Our study showed that the critical size of the cubic to tetragonal transition in barium titanate may be much lower than suggested theoretically. The grain size showed an increase on sintering of 35 nm at 900 °C to 120 nm at 1100 °C, which was much lower than the grain size obtained at this temperature by the normal solid state route. The dielectric constant depends on sintering temperature and was found to increase from 210 (900 °C sintering) to 520 (1100 °C sintering) at 100 kHz. The dielectric constant was highly stable with temperature as well as frequency.

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