Applications of the operator theory of parametric programming for the transportation and generalized transportation problems

Abstract

Many operations research problems can be modelled as the transportation problem (TP) or generalized transportation problem (GTP) of linear programming together with parametric programming of the rim conditions (warehouse availabilities and market requirements) and/or the unit costs (and/or the weight coefficients in the case of the GTP). The authors have developed an operator theory for simultaneously performing such parametric programming calculation. The present paper surveys the application of this methodology to several classes of problems: (a) optimization models involving a TP or GTP plus the consideration of an additional important factor (e.g., production smoothing, cash management); (b) bicriterion TP and GTP (e.g., multimodal TP involving cost/time trade-offs, TP with total cost/bottleneck time trade-offs); (c) multi-period growth models (e.g., capacity expansion problems); (d) extensions of TP and GTP (e.g., stochastic TP and GTP, convex cost TP) (e) branch and bound problems involving the TP or GTP as subproblems (e.g., traveling salesman problem, TP with quantity discounts); and (f) algorithms for solving the TP and GTP (e.g., a polynomially bounded primal basic algorithm for the TP, a weight operator algorithm for the GTP). The managerial and economic significance of the operators is discussed.Key wordsTransporation NetworksParametric ProgrammingGeneralized Transportation ProblemsOperator TheoryNetwork Application