<Para ID="Par1">A strongly polynomial-time algorithm is proposed for the strict homogeneous linear-inequality feasibility problem in the positive orthant, that is, to obtain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$x\in \mathbb {R}^n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math> </EquationSource> </InlineEquation>, such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$Ax 0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>A</mi> <mi>x</mi> <mo></mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$x 0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>x</mi> <mo></mo> <mn>0</mn> </mrow> </math> </EquationSource> </InlineEquation>, for an <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$m\times n$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </math> </EquationSource>...</equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></equationsource></equationsource></inlineequation></para>

We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582–599, <CitationRef CitationID="CR8">2002</CitationRef>). Existing global optimization methods to solve this...</citationref>