Duality and an algorithm for a class of continuous transportation problems. (English) Zbl 0538.90057

Anderson, E. J.; Philpott, A. B.

Math. Oper. Res. 9, 222-231 (1984).

The authors discuss duality theory for the following continuous transportation problem (CTP): \[ Minimize\quad \int_{X\times Y}c(x,y)d\rho(x,y) \] subject to \[ \int_{X\times Y}\hat f_ 1(x,y)d\rho(x,y)=\int_{X}f_ 1(x)d\mu_ 1(x), \] for all continuous functions \(f_ 1\) on X, \[ \int_{X\times Y}\hat f_ 2(x,y)d\rho(x,y)=\int_{Y}f_ 2(y)d\mu_ 2(y) \] for all continuous functions \(f_ 2\) on Y, \(\rho\geq 0\), where \(\rho\), \(\mu_ 1\) and \(\mu_ 2\) are nonnegative Radon measures and c is a continuous function, X and Y are compact spaces with \(\mu_ 1(X)=\mu_ 2(Y)\) and \(\hat f_ 1\) and \(\hat f_ 2\) are defined on \(X\times Y\) by \(\hat f_ 1(x,y)=f_ 1(x), \hat f_ 2(x,y)=f_ 2(y)\) for all \(x\in X\), \(y\in Y\). An algorithm to solve a simple version of CTP is given.

Reviewer: I.M.Stancu-Minasian

MSC:

90C08	Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)
90C48	Programming in abstract spaces
49N15	Duality theory (optimization)

Keywords:

infinite dimensional spaces; continuous transportation problem; nonnegative Radon measures