cost malmquist productivity index in supply chain

the index is excellent by the malmquist index as extended to productivity measurement. the index developed here is defined in terms of input cost rather than input quantity distance functions in supply chain. therefore, we propose productivity change is decomposed into overall efficiency and cost technical change. these decompositions provide a clearer situation of the root sources of supply chain productivity change, so that illustrated here in a sample of supply chain; so that results are computed using non-parametric mathematical programming.