Gouveia and Pires (European Journal of Operations Research 112(1999) 134-146) have proposed four extended formulations having precedence variables as extra variables and characterized the projections of three of the four formulations into the natural variable space. In Gouveia and Pires (Discrete Applied Mathematics 112 (2001)), they also have introduced some other extended formulations with the same extra variables and conjectured that the projection of one of the proposed formulations is equivalent to the one proposed by Dantzig, Fulkerson, and Johnson (Operations Research 2(1954) 393-410). In this paper, we provide a unifying framework based on which we give alternative proofs on the projections of three extended formulations and new proofs on those of two formulations appeared in Gouveia and Pires(1999, 2001).

This paper discusses a paradox in an Indefinite Quadratic transportation Problem. Here, the objective function is the product of two linear functions. A paradox arises when the transportation problem admits of a total cost which is lower than the optimum cost, by transporting larger quantities of goods over the same route. A sufficient condition for the existence of a paradox is established. Paradoxical Range of flow is obtained for any given flow in which the corresponding objective function value is less than the optimum value of the given transportation problem. It is illustrated with the help of a numerical example.

Regarding all methods of decision making as parts of the same subject, one is astounded to find that the field speaks to the practitioners about multicriteria decision making with a multi-forked tongue, different methods give different and conflicting outcomes even for very simple decisions. In this paper, five well-known decision theories are described and used to work out a simple decision to choose the best of three cars. The outcomes turn out not to be the same for all the methods, which is troubling because one would expect unique answers in decision making. Several "meta" criteria that are essential for making a decision theory to make a choice as to which decision approach is best.h is best.

In the present paper, we present method to solve a Fractional Bulk Transportation Problem(FBTP) in which the numerator is quadratic in nature and the denominator is linear. A related (FBTP) is formed whose feasible solutions are ranked to reach an optimal solution of the given problem. The method to find these feasible solutions makes use of parametric programming wherein a series of Ordinary Bulk Transportation Problems are solved by the usual methods.

This paper presents the reliability models for redundant systems composed of repairable components whose failure time and repair time distributions are phase-type. It is shown that the distribution of time to system failure is also phase-type. The dependency between components are considered and integrated into the model by the used of the rate adjustment factor. The phase-type representation is constructed for the system through algebraic operations on the parameters of components\` failure time and repair time distributions and the corresponding rate adjustment factors. Types of system structures considered are parallel, k-out-of-N system with load sharing scheme and standby system with operation priority.