Most durable products have two distinct types of customers: first-time buyers and customers who already own the product, but are willing to replace it with a new one or purchase a second one. Firms usually adopt a price-discrimination policy by offering a trade-in rebate only to the replacement customers to hasten their purchase decisions. Any return flow of products induced by trade-in rebates has the potential to generate revenues through remanufacturing operations. In this paper, we study the optimal pricing/trade-in strategies for such durable, remanufacturable products. We focus on the scenario where the replacement customers are only interested in trade-ins. In this setting, we study three pricing schemes: (i) uniform price for all customers, (ii) age-independent price differentiation between new and replacement customers (i.e., constant rebate for replacement customers), and (iii) age-dependent price differentiation between new and replacement customers (i.e., age-dependent rebates for replacement customers). We characterize the roles that the durability of the product, the extent of return revenues, the age profile of existing products in the market, and the relative size of the two customer segments play in shaping the optimal prices and the amount of trade-in rebates offered. Throughout the paper we highlight the operational decisions that might influence the above factors, and we support our findings with real-life practices. In an extensive numerical study, we compare the profit potential of different pricing schemes and quantify the reward (penalty) associated with taking into account (ignoring) customer segmentation, the price-discrimination option, return revenues, and the age profile of existing products. On the basis of these results, we are able to identify the most favorable pricing strategy for the firm when faced with a particular market condition and discuss implications on the life-cycle pricing of durable, remanufacturable products.trade-in rebates, pricing, remanufacturing, durable products, product-age profile

Ivanov pointed out substantial analytical difficulties associated with self-gravitating, static, isotropic fluid spheres when pressure explicitly depends on matter density. Simplification achieved with the introduction of electric charge were noticed as well. We deal with self-gravitating, charged, anisotropic fluids and get even more flexibility in solving the Einstein-Maxwell equations. In order to discuss analytical solutions we extend Krori and Barua's method to include pressure anisotropy and linear or non-linear equations of state. The field equations are reduced to a system of three algebraic equations for the anisotropic pressures as well as matter and electrostatic energy densities. Attention is paid to compact sources characterized by positive matter density and positive radial pressure. Arising solutions satisfy the energy conditions of general relativity. Spheres with vanishing net charge contain fluid elements with unbounded proper charge density located at the fluid-vacuum interface. Notably the electric force acting on these fluid elements is finite, although the acting electric field is zero. Net charges can be huge ($10^{19}\,C$) and maximum electric field intensities are very large ($10^{23}-10^{24}\,statvolt/cm$) even in the case of zero net charge. Inward-directed fluid forces caused by pressure anisotropy may allow equilibrium configurations with larger net charges and electric field intensities than those found in studies of charged isotropic fluids. Links of these results with charged strange quark stars as well as models of dark matter including massive charged particles are highlighted. The van der Waals equation of state leading to matter densities constrained by cubic polynomial equations is briefly considered. The fundamental question of stability is left open.Comment: 22 Latex pages, 17 figures, Inclusion of new paragraph at the end of Conclusion & some of the old captions of the Figures are replaced with new caption

We propose a new model of a {\it gravastar} admitting conformal motion. While retaining the framework of the Mazur-Mottola model, the gravastar is assumed to be internally charged, with an exterior defined by a Reissner-Nordstr{\"o}m rather than a Schwarzschild line element. The solutions obtained involve (i) the interior region, (ii) the shell, and (iii) the exterior region of the sphere. Of these three cases the first case is of primary interest since the total gravitational mass vanishes for vanishing charge and turns the total gravitational mass into an {\it electromagnetic mass} under certain conditions. This suggests that the interior de Sitter vacuum of a charged gravastar is essentially an electromagnetic mass model that must generate the gravitational mass. We have also analyzed various other aspects such as the stress energy tensor in the thin shell and the entropy of the system.Comment: Minor addition, Accepted in Phys. Lett.

The singularity space-time metric obtained by Krori and Barua[1] satisfies the physical requirements of a realistic star. Consequently, we explore the possibility of applying the Krori and Barua model to describe ultra-compact objects like strange stars. For it to become a viable model for strange stars, bounds on the model parameters have been obtained. Consequences of a mathematical description to model strange stars have been analyzed.

We propose a unique stellar model under the f (R, T ) gravity by using the conjecture of MazurMottola [P. Mazur and E. Mottola, Report number: LA-UR-01-5067., P. Mazur and E. Mottola, Proc. Natl. Acad. Sci. USA 101, 9545 (2004).] which is known as gravastar and a viable alternative to the black hole as available in literature. This gravastar is described by the three different regions, viz., (I) Interior core region, (II) Intermediate thin shell, and (III) Exterior spherical region. The pressure within the interior region is equal to the constant negative matter density which provides a repulsive force over the thin spherical shell. This thin shell is assumed to be formed by a fluid of ultrarelativistic plasma and the pressure, which is directly proportional to the matter-energy density according to Zel'dovich's conjecture of stiff fluid [Y.B. Zel'dovich, Mon. Not. R. Astron. Soc. 160, 1 (1972).], does counterbalance the repulsive force exerted by the interior core region. The exterior spherical region is completely vacuum and assumed to be de Sitter spacetime which can be described by the Schwarzschild solution. Under this specification we find out a set of exact and singularity-free solution of the gravastar which presents several other physically valid features within the framework of alternative gravity.